Figure 1 shows the topology and control block diagram of a grid-forming PV system based on DC-voltage controller (hereinafter referred to as the DC-GFM system). Here, P _in is the output power of the PV array, and P _dc is the power injected into the converter. C _dc denotes the DC-side capacitor. L _f and R _f represent the filter inductor and equivalent resistance, respectively; C _f is the filter capacitor; L _g and R _g correspond to the grid-side inductance and equivalent resistance. I _f denotes the current at the converter port, while I _o and V _o represent the output current and voltage of the converter, respectively. The converter employs a DC-GFM control strategy, which mainly includes a DC voltage control loop, active power control loop, reactive power control loop, and a dual-loop control for voltage and current. V _dcref denotes the reference voltage on the DC side; P ₀ and Q ₀ are the reference active and reactive power outputs of the converter; V _ref and ω ₀ represent the rated output voltage and frequency, respectively. P _e and Q _e represent the active and reactive power outputs of the converter, respectively, while ω and θ denote the angular frequency and voltage phase angle at the converter output.

Based on the power balance on the DC side, the dynamic of the DC voltage can be derived as: V d c C d c d V d c d t = P i n − P d c (1)

Assuming the PV operates under maximum power point tracking and irradiance and temperature remain constant, the output power of the PV array P _in is steady. By expressing the variables in Equation 1 as the sum of their steady-state and perturbation components, the following expression can be obtained: C d c V d c 0 + Δ V d c d Δ V d c d t = P i n − P d c 0 + Δ P d c (2)

By eliminating the steady-state and higher-order perturbation components, the resulting relationship between the DC voltage and the DC output power is: Δ V d c = − Δ P d c s C d c V d c 0 (3)

By neglecting the power losses in the converter, P _dc in Equation 3 can be substituted with the output power of the converter P _e . The voltage and current dynamics across the LCL filter are given by: d I f d q d t = 1 L f V c d q − V o d q − R f I f d q ± ω 0 I f q d (4) d V o d q d t = 1 C f I f d q − I o d q ± ω 0 V o q d (5) d I o d q d t = 1 L g V o d q − V g d q − R g I o d q ± ω 0 I o q d (6)

Therefore, based on Equations 4–6, the output power of the converter can be derived using the instantaneous power theory as: P e = 1.5 * I o d V o d + I o q V o q Q e = 1.5 * I o d V o q − V o d I o q (7)

In DC-GFM, the DC voltage control loop maintains a constant DC voltage via a PI controller. Its control equation is given by: Δ P d c r e f = k p + k i s Δ V d c − Δ V d c r e f (8) Here, P _dcref is the output of the DC voltage control loop and serves as the reference input for the active power control loop. The active power control loop employs a proportional controller to regulate the output active power of converter and establish its frequency. The control equation is given by: ω = k p f Δ P d c r e f + Δ P 0 − Δ P e + ω 0 (9) Here, k _pf is the proportional gain of the active power control loop. The reactive power control loop adjusts the output reactive power of the converter and establishes the AC voltage. Its control equation is: V o = K q Q 0 − Q e + V r e f (10)

Subsequently, the output of the reactive power control loop is used as the input to the voltage and current control loop for rapid tracking of the grid-side voltage and current variations. Since the time scale of the voltage and current control loop is typically much faster than that of the power control loop, the dynamics of the voltage and current control loop can be neglected when studying the power-frequency dominated LFO issue (Qu et al., 2021). Therefore, the state space formed by Equations 1–10 can provide a detailed description of the system dynamics when low-frequency disturbances occur in the grid, namely,: X ˙ s y s = A X s y s + B U Y s y s = C X s y s + D U (11) Where X _sys is the state variable matrix of the state space, U and Y _sys represent the input and output matrices, respectively. A, B, C and D are the corresponding state-space matrices. Figure 2 presents the waveforms obtained from the theoretical model based on Equation 11 and the hardware-in-the-loop experiment, under a 0.4 MW load disturbance on the grid side occurring at 8 s. The close agreement between the two curves confirms the reliability of the developed state-space model.

To investigate how control parameters affect LFOs in the DC-GFM system, root locus diagrams corresponding to parameter variations are plotted based on Equation 11. Figures 3–5 respectively show how the root locus of the LFO mode evolves with increasing proportional gain k _p and integral gain k _i of the DC voltage control loop, and the proportional gain k _pf of the active power control loop. As shown in Figure 3, as k _p increases, the imaginary part of the eigenvalue corresponding to the LFO mode increases steadily, while the real part first increases and then decreases. The magnitude of change in the imaginary part is significantly greater than that of the real part, indicating that increasing k _p has limited impact on the duration of LFOs under the same disturbance, but it does result in a higher oscillation frequency.

As shown in Figure 4, when k _i is gradually increased, the imaginary part of the eigenvalue corresponding to the LFO mode remains nearly unchanged, while the real part decreases significantly. This indicates that for a given disturbance, increasing k _i has little effect on the oscillation frequency but noticeably extends the duration of oscillations. Moreover, if k _i becomes too large, the eigenvalue may cross the imaginary axis into the right-half plane, potentially causing system instability.

In contrast, as shown in Figure 5, with the gradual increase of k _pf , both the real and imaginary parts of the eigenvalue corresponding to the LFO mode increase, leading to an enhanced damping ratio. This indicates that under a disturbance, increasing k _pf raises the oscillation frequency while significantly reducing the oscillation duration.

As shown in the analysis of Section 2, an excessively large k _i or a too-small k _pf can compromise system stability under DC-GFM control. Although tuning these parameters can improve the damping of LFO mode, they often affect other aspects of system performance—for example, a small k _i may result in slow DC voltage regulation. An effective approach is to introduce an additional control loop to the existing control structure, providing the system with extra degrees of freedom. However, although the root locus-based parameters analysis can comprehensively reveal the evolution of system poles, it heavily relies on model parameters and falls short of uncovering the underlying mechanism of negative damping, thus making the design of supplementary control loops more challenging.

The damping torque method, based on classical control theory and the decomposition of the torque acting on the motion of generator rotor, quantifies the damping capability of synchronous generators. It is widely used in LFO risk assessment in traditional power systems. It provides clear physical insight into the occurrence of weak or negative damping modes, thereby guiding control design for oscillation suppression. Based on this, the damping torque method is employed in this paper to analyze the sources of negative damping in DC-GFM systems. In high-voltage grids, the grid impedance is primarily inductive, resulting in a decoupling of active and reactive power. Under this condition, the active power can be expressed by the power flow equation as follows: P e = V o V g X g sin ⁡ δ (12)

Here, δ represents the phase angle difference between the converter output voltage and the grid voltage. By linearizing Equation 12, we can obtain: Δ P e = V o V g X g Δ δ = K s Δ δ (13)

Here, K _s denotes the power synchronization coefficient of the converter, which reflects the power transfer limit of the DC-GFM system. Accordingly, by combining Equations 3, 8, 9 and 13, the power–frequency transfer function block diagram of the system is shown in Figure 6.

When the reference values remain unchanged, the power angle dynamics of the DC-GFM system can be derived from Figure 6 as follows: s Δ δ + k p f k p K s s C d c V d c 0 + k p f k i K s s 2 C d c V d c 0 Δ δ + k p f K s Δ δ = 0 (14)

Multiplying Equation 14 on both sides by the Laplace operator s leads to: s 2 Δ δ + k p f K s + k p f k i K s s 2 C d c V d c 0 ⏟ T D s Δ δ + k p f k p K s C d c V d c 0 ⏟ T S Δ δ = 0 (15)

According to the Phillips model, the power angle dynamics of a synchronous generator can be described as follows (Du and Wang, 2015): s 2 Δ δ + D M s Δ δ + ω 0 K 1 M Δ δ = 0 (16)

Here, the term D/M, which is associated with frequency, represents the damping torque that affects the ability of system to dampen the LFO. Its physical meaning is clear: it is a force (torque) proportional to velocity (angular velocity), acting as a resistive force (torque) against motion (angular displacement), thereby providing damping. The term ω _0K1/M, which relates to the power angle, represents the synchronizing torque that determines the ability of units to maintain synchronism. It primarily influences the oscillation frequency of the output power [19]. Therefore, by comparing Equations 15, 16, it can be observed that T _ᴅ is the key factor determining the damping capability of LFOs in DC-GFM systems. When a LFO mode with frequency ω _r exists in the system, setting s = jω _r yields T _ᴅ as: T D = K s k p f T D 1 − K s k p f k i ω r 2 C d c V d c 0 ⏟ T D 2 = K s k p f 1 − k i ω r 2 C d c V d c 0 (17)

Figure 7 shows the torque diagram of the DC-GFM system, where T _DVC represents the equivalent torque generated by the DC voltage control loop. It can be seen that T _D1, formed by component k _pf K _s , provides positive damping torque to the system, while T _D2, involving the k _i parameter, introduces negative damping torque. When the system damping is positive, an increase in the k _i parameter and a decrease in k _pf will gradually reduce the damping of the LFO mode, making the oscillations more pronounced. Therefore, as discussed in Section 2, increasing the k _i parameter leads to a decrease in the real part of the LFO mode, while increasing k _pf results in an increase in the real part. Meanwhile, in Equation 15, T _s primarily influences the oscillation frequency of the DC-GFM system. As k _p and k _pf increase, T _s also increases, leading to a higher oscillation frequency. Therefore, in Section 2, as the k _p and k _pf parameters increase, the imaginary part of the LFO mode gradually increases.

As shown in Figure 8, n DC-GFM units operate in parallel, where the output power of GFM_i (i = 1, 2, …, n) is denoted by P _i . The power-frequency transfer relationship at the point of common coupling (PCC) for each GFM_i is given by: G i s = − Δ P i Δ ω g (18)

For simplicity, the analysis begins with the case of n = 2. When a power disturbance P _L occurs on the grid side, the power balance at the PCC can be expressed as Equation 19: Δ P L = Δ P 1 + Δ P 2 = − G 1 s Δ ω g − G 2 s Δ ω g (19)

Therefore, the power–frequency relationship at the PCC can be derived as Equation 23: Δ ω g − Δ P L = 1 G 1 s + G 2 s = G 2 s − 1 1 + G 1 s / G 2 s (20)

Equation 20 can be interpreted as the closed-loop transfer function of a negative feedback system, as shown in Figure 9, where the feedforward gain is G ₂(s)⁻¹ and the feedback gain is G ₁(s). Therefore, G ₁(s)/G ₂(s) represents the open-loop transfer function of the system. According to the Nyquist stability criterion, the system shown in Figure 9 remains stable as long as the condition in Equation 21 is satisfied, ensuring a positive phase margin. ∀ ω r ∈ 0 , + ∞ , arg G 1 j ω r G 2 j ω r ∈ − π , π (21)

A sufficient condition for Equation 21 to hold is: ∀ ω r ∈ 0 , + ∞ , Re G i j ω r > 0 (22)

Based on Figure 6 and Equation 18, the expression for G _i (jω _r ) can be derived as follows: G i j ω r = j ω r K s k p f k p K s C d c V d c 0 − ω r 2 + k p f K s − k p f k i K s ω r 2 C d c V d c 0 j ω r (23)

Therefore, when Equation 24 is satisfied, Equation 22 holds, indicating that the system shown in Figure 9 is stable. k p f K s − k p f k i K s ω r 2 C d c V d c 0 > 0 (24)

By comparing Equations 17, 24, it can be concluded that the parallel system remains stable when both GFM converters exhibit positive damping torque. This analysis is then extended to the case of multiple DC-GFM units operating in parallel. Consider the jth inverter as a separate subsystem, while treating the remaining inverters as another subsystem. Based on the power balance relationship at the PCC shown in Figure 8, we can derive Equation 25: Δ P L = ∑ i = 1 n Δ P i = − G j s + ∑ i ≠ j n G i s Δ ω g (25)

Similarly, the power–frequency relationship at the PCC can be expressed as: Δ ω g − Δ P L = G j s − 1 1 + ∑ i ≠ j n G i s / G j s (26)

Equation 26 can also be viewed as the closed-loop transfer function of a negative feedback system. According to our earlier findings, Equation 22 is satisfied when each GFM converter contributes a positive damping torque. Hence, by applying the rules of complex addition, we obtain: ∀ ω r ∈ 0 , + ∞ , Re ∑ i ≠ j n G i j ω r > 0 (27)

Since the real part of G _j (jω _r ) is also positive, it follows that: ∀ ω r ∈ 0 , + ∞ , arg ∑ i ≠ j n G i j ω r / G j j ω r ∈ − π , π (28)

Equation 28 indicates that the equivalent negative feedback system described by Equation 26 maintains a positive phase margin at all times. Therefore, when n DC-GFM units operate in parallel, the system remains stable as long as each GFM converter provides positive damping torque. It is worth noting that Equations 27, 28 hold without imposing any constraints on the parameters of the DC-GFM converters. In other words, a multi-converter DC-GFM system with non-identical parameters remains stable as long as each DC-GFM provides positive damping torque.

As demonstrated in Sections 3.1 and 3.2, whether considering a single DC-GFM system or a system composed of multiple DC-GFMs, system stability fundamentally depends on ensuring that the damping torque of each DC-GFM remains positive. However, the integral gain k _i of the DC voltage control loop introduces negative damping torque into the system, which may lead to instability. Therefore, an additional damping control strategy is required to enhance system stability. As shown in Figure 10, the introduction of negative damping torque by the k _i parameter arises from the −180° phase lag introduced by two successive integration stages in the power-to-frequency feedback path. Therefore, this paper proposes the use of a lead-lag compensator to inject additional damping torque through power feedback, thereby counteracting the negative damping caused by the DC voltage control loop, as illustrated in Figure 10. The transfer function of the lead-lag compensator, G _p , is given by Equation 29: G p = s + T 2 s + T 1 (29) where, T ₁ = 10, T ₂ = 1, and T ₁ and T ₂ denote the lead and lag time constants of the lead-lag compensator, respectively. When T ₁ > T ₂, the phase of G _p is positive. As a result, the power feedback introduces an additional torque into the system, which acts as a positive damping torque that compensates for the negative damping torque caused by the k _i parameter in the DC voltage control loop.

As shown in Figure 11, the power waveforms under different proportional gains k _p of the DC voltage control loop are presented for a 0.4 MW load disturbance occurring at 8 s. It can be observed that the oscillation frequency increases with larger k _p values. However, the DC-GFM system reaches steady state at approximately 8.4 s across all k _p settings, indicating that k _p has little impact on the oscillation duration. This phenomenon has been explained in Sections 2 and 3. It is primarily due to the fact that k _p affects the synchronizing torque of the DC-GFM system, which is reflected in the root locus as a change mainly in the imaginary part of the LFO mode.

As shown in Figure 12, when a 0.4 MW load disturbance occurs at 8 s, the power waveforms under different integral gains k _i of the DC voltage control loop are presented. It can be observed that increasing k _i does not significantly affect the oscillation frequency of the DC-GFM system, but it does lead to a noticeable increase in oscillation amplitude, thereby requiring a longer time to reach steady state. Similarly, the influence of the k _i parameter can be explained based on the analysis presented in Sections 2 and 3. As k _i increases, it introduces negative damping torque into the system, causing the real part of the LFO mode to gradually decrease. This leads to a reduction in system damping and results in more pronounced oscillations.

Figure 13 shows the power profiles under a 0.4 MW load disturbance occurring at 8 s, with different proportional gains k _pf in the power control loop. It can be observed that as k _pf increases, the oscillation frequency of the DC-GFM converter rises while the amplitude decreases, indicating that a higher k _pf helps the converter stabilize more quickly. As indicated by the analysis of Equation 15, this behavior arises because the k _pf parameter influences both the damping torque and the synchronizing torque of the system. When the total system damping is positive, increasing k _pf enhances both torque components. As a result, the real and imaginary parts of the LFO mode increase, leading to a higher oscillation frequency and faster decay of oscillations.

To validate the effectiveness of the proposed damping enhancement strategy, compare the dynamic performance of the DC-GFM converter under identical disturbances using three approaches: conventional DC-GFM control, the damping method based on equivalent DC resistance (DMEDR) from reference (Ai et al., 2024), and the strategy proposed in this paper. Figures 14, 15 presents the power response waveforms with different control methods during a 0.4 MW load disturbance at 8 s. In this case, the parameter k _i is set to 30 p. u.

As shown in Figure 14, without any additional damping control, the DC-GFM converter diverges following the disturbance. In contrast, when applying the DMEDR from reference (Ai et al., 2024) and the strategy proposed in this paper, the converter returns to steady state after a period of oscillation. When the DMEDR from reference (Ai et al., 2024) is applied, the DC-GFM converter requires a relatively long period of oscillation to reach steady state. In contrast, the strategy proposed in this paper enables the system to return to steady state within just two or three oscillation cycles. This demonstrates that the proposed method provides stronger low-frequency damping for single DC-GFM converter systems.

Figure 15 shows the power waveforms during parallel operation of two DC-GFM units. GFM_1 has a k _i value of 40 p. u., while GFM_2 has a k _i value of 10 p. u. As illustrated in the Figure 12, the system remains stable when only GFM_2 is connected to the grid. However, as shown in Figure 15a, when the proposed damping enhancement strategy is not applied, the power outputs of both units begin to oscillate and gradually diverge after GFM_1 and GFM_2 are connected in parallel. The instability arises because the excessive negative damping torque generated by the high k _i value in GFM_1 drives the total system damping below zero. Therefore, even though GFM_2 itself provides positive damping torque, it still becomes unstable due to the gradual divergence of the electrical quantities of system. As shown in Figure 15b, after applying the DMEDR from reference (Ai et al., 2024), the equivalent resistance on the DC side provides some damping, which slows down the power divergence of GFM_1 and GFM_2. However, the system still gradually becomes unstable. In contrast, as shown in Figure 15c, when the proposed damping enhancement strategy is applied to both units, the output power of each unit reaches steady state around 9 s after experiencing a short oscillation. This demonstrates that the proposed strategy is also effective in suppressing oscillations in multi-unit systems.