The basic structure of the SVG in a renewable energy collection station is illustrated in Figure 1, where L _ac and R _ac represent the equivalent inductance and resistance of the filter, which include converter losses and the filter resistance, C _dc denotes the DC-side capacitance. For a cascaded H-bridge type SVG, a two-level structure can also be adopted after equivalence in the main circuit, ensuring that the external characteristics of the main circuit remain essentially consistent.

As shown in Figure 1, the expression for the ac side dynamics of SVG can be expressed as Equation 1: R ac + s L ac − ω L ac ω L ac R ac + s L ac i sd i sq = u sd u sq − u cd u cq (1) where u _sd and u _sq represent the d-axis and q-axis components of the ac grid voltage, respectively; u _cd and u _cq denote the d-axis and q-axis components of the SVG bridge arm output voltage; Z _ac = R _ac + sL _ac, and Z _wl = ωL _ac.

The expression for the dc side dynamics, considering power balance, can be written as Equation 2. C dc d u dc d t = 3 2 u cd i sd + u cq i sq u dc (2)

According to the requirements of automatic voltage control (AVC) for renewable energy collection stations, typical control modes of the SVG include constant reactive power control, constant ac voltage control, constant reactive current control, and high/low voltage ride-through control.

To achieve synchronous grid connection and precise power regulation, SVG is usually configured with the phase-locked loop (PLL) to accurately track the phase and frequency of the grid voltage. The tracking dynamics of the PLL cause a misalignment between the output phase and the theoretical phase reference, resulting in a mismatch between the system dq-frame and the control dq-frame. Let the symbol Δ and the subscript ‘0’ denote the small-signal variation and steady-state value of a state variable, respectively; the superscript ‘s’ indicates the corresponding quantity in the system coordinate frame oriented by the actual grid voltage, while the superscript ‘c’ represents the corresponding quantity in the control coordinate frame.

For the synchronous reference frame (SRF) PLL control structure shown in Figure 2, the expression for the small-signal quantity of the output phase θ _pll related to Δuc sq is: Δ θ pll = k ppll + k ipll s Δ u sq c s = G PLL s Δ u sq c s (3) where k_ppll and k_ipll represent the proportional and integral parameters of the PI controller used in the PLL.

The expression for the Δ u sq c and Δ u sd c can be expressed as: Δ u sd c Δ u sq c = cos ⁡ θ pll 0 sin ⁡ θ pll 0 − sin ⁡ θ pll 0 cos ⁡ θ pll 0 Δ u sd s Δ u sq s + u sq 0 c − u sd 0 c Δ θ PLL (4)

By substituting Equation 4 into 3 to eliminate Δ u sq c , Δ θ pll can be expressed as: Δ θ PLL = − sin ⁡ θ pll 0 G PLL s s + u sd 0 c G PLL s Δ u sd s + cos ⁡ θ pll 0 G PLL s s + u sd 0 c G PLL s Δ u sq s = G PLLd s Δ u sd s + G PLLq s Δ u sq s (5)

Denote Δ u sdq c = Δ u sd c , Δ u sq c T , Δ u cdq c = Δ u cd c , Δ u cq c T . By combining Equations 4, 5, the Δ u sdq c and Δ u cdq c can be represented by: Δ u sdq c = G θ s + G PLL us s Δ u sdq s Δ u cdq s = G θ − 1 s Δ u cdq c + G PLL uc s Δ u sdq s (6) where G θ s , G PLL us s and G PLL uc s can be expressed as Equations 7–9. G θ s = cos ⁡ θ pll 0 sin ⁡ θ pll 0 − sin ⁡ θ pll 0 cos ⁡ θ pll 0 (7) G PLL us s = u sq 0 c G PLLd s u sq 0 c G PLLq s − u sd 0 c G PLLd s − u sd 0 c G PLLq s (8) G PLL uc s = − u cq 0 s G PLLd s − u cq 0 s G PLLq s u cd 0 s G PLLd s u cd 0 s G PLLq s (9)

Similarly, denote Δ i sdq c = Δ i sd c , Δ i sq c T and the corresponding expression can be given by: Δ i sdq c = G θ s Δ i sdq s + G PLL i s Δ u sdq s (10) where G PLL i s is written as Equation 11. G PLL i s = i sq 0 c G PLLd s i sq 0 c G PLLq s − i sd 0 c G PLLd s − i sd 0 c G PLLq s (11)

Consequently, based on the power stage model, Δu _dc can be expressed as: Δ u dc = G udcd i s G udcq i s T Δ i sdq s + G dcd u s G dcq u s T Δ u cdq s = G d 1 s Δ i sdq s + G d 2 s Δ u cdq s (12) where G dcd u s , G dcq u s , G udcd i s and G udcq i s are shown expressed in Equation 13. G dcd i s = 3 u cd 0 s u dc 0 2 C dc u dc 0 2 s + 3 u cd 0 s i sd 0 s + 3 u cq 0 s i sq 0 s G dcq i s = 3 u cq 0 s u dc 0 2 C dc u dc 0 2 s + 3 u cd 0 s i sd 0 s + 3 u cq 0 s i sq 0 s G dcd u s = 3 i sd 0 s u dc 0 2 C dc u dc 0 2 s + 3 u cd 0 s i sd 0 s + 3 u cq 0 s i sq 0 s G dcq u s = 3 i sq 0 s u dc 0 2 C dc u dc 0 2 s + 3 u cd 0 s i sd 0 s + 3 u cq 0 s i sq 0 s G udcd i s = G dcd i s − G dcd u s Z ac − G dcq u s Z wl G udcq i s = G dcq i s + G dcd u s Z wl − G dcq u s Z ac (13)

Denote the PI regulator for the dc voltage loop as G _V(s), and the PI regulator for reactive power control as G _Q(s). Similarly, the PI regulators for ac voltage control and current loop can be denoted as G _U(s), and G _I(s), respectively. Thus, the control loops can be represented as: Δ i sd c * = G V s Δ u dc * − Δ u dc (14) Δ i sq c * = G Q s Δ Q * − Δ Q (15) Δ i sq c * = G U s Δ U * − Δ U (16) where ΔU denotes the small-signal quantity of voltage amplitude, and ΔQ represents the small-signal quantity of reactive power compensation.

Given the consistency of results between the system and control coordinate frame, this paper employs state variables in the system coordinate frame to calculate ΔQ and ΔU, thereby simplifying the computational process, and we can have: Δ Q = 1.5 i sq 0 s Δ u sd s − i sd 0 s Δ u sq s − u sq 0 s Δ i sd s + u sd 0 s Δ i sq s (17) Δ U = u sd 0 s U 0 u sq 0 s U 0 Δ u sdq s = G Udq s Δ u sdq s (18) where U ₀ denotes the steady-state voltage amplitude.

As shown in Figure 1, the small-signal equations of the inner current loop can be obtained as: Δ u cd c * = Δ u sd c + ω L ac Δ i sq c − G I s Δ i sd c * − Δ i sd c Δ u cq c * = Δ u sq c − ω L ac Δ i sd c − G I s Δ i sq c * − Δ i sq c (19)

For the modulation process and control delay, a first-order inertial link can be adopted for approximation, yielding: Δ u cdq c = 1 s T δ + 1 0 0 1 s T δ + 1 Δ u cdq c * + u cd 0 c * u dc 0 s T δ + 1 u cq 0 c * u dc 0 s T δ + 1 Δ u dc = G T 1 s Δ u cdq c * + G T 2 s Δ u dc (20) where T _δ denotes the control delay, and uc* cd0 and uc* cq0 represent the steady-state values of the d-axis and q-axis components of the modulation wave reference, which can be obtained by dividing uc cd0 and uc cq0 by the modulation gain.

Remark: The modulation process and control delay terms can also be approximated using higher-order link s, with the expression of these links derived through a Taylor expansion to achieve higher model accuracy. The first-order inertia link used in this study is sufficient to describe the impedance model characteristics.

To derive the impedance model for multi-condition grid-connected stabilization design of the SVG in a renewable energy collection station, the constant ac voltage control mode is taken as an example. Combining Equations 12, 14, 16, 18, the outer-loop small-signal equations are derived as: Δ i sdq c * = − G V s 0 0 G U s G dcd u s G dcq u s u sd 0 s U 0 u sq 0 s U 0 Δ u sdq s − G V s 0 0 G U s G udcd i s G udcq i s 0 0 Δ i sdq s = − K U s G p 1 s Δ u sdq s − K U s G p 2 s Δ i sdq s (21)

By combining Equations 6, 10, 16, the small-signal equations of inner-loop output can be expressed as: Δ u cdq c * = G θ s + G PLL us s + G wl s − G ci s G PLL i s Δ u sdq s + G wl s G θ s − G ci s G θ s Δ i sdq s + G ci s Δ i sdq c * (22) where G _ci(s) = diag {- G _I(s), - G _I(s)} and G _wl(s) can be expressed as Equation 23. G wl s = 0 ω L ac − ω L ac 0 (23)

By substituting Equation 12 into 20, the small-signal equation incorporating the dc-side dynamic characteristics can be given by: Δ u cdq c = G T 1 s Δ u cdq c * + G T 2 s Δ u dc = G T 1 s Δ u cdq c * + M 1 s Δ i sdq s + M 2 s Δ u sdq s (24) where M ₁(s) = G _T2(s) G _d1(s) and M ₂(s) = G _T2(s) G _d2(s).

By combining Equations 21, 22 and 24, a comprehensive small-signal block diagram of the SVG can be obtained, as illustrated in Figure 3. The dq-frame impedance expression of the SVG under constant ac voltage control is given by Equation 25. The derivation process for the reactive power control mode follows a similar approach, by combing Equations 12, 14, 15, 17, 19, 20, the specific expression is provided in Equation 26: Z U = Y ac E − G PLL uc − G θ − 1 M 2 + G T 1 G θ + G PLL us + G wl G PLL i − G ci G PLL i − G T 1 G ci K U G p 1 − 1 · E + Y ac G θ − 1 M 1 − G T 1 G ci K U G p 2 + G T 1 G wl G θ − G ci G θ (25) Z Q = Y ac E − G PLL uc − G θ − 1 M 2 + G T 1 G θ + G PLL us + G wl G PLL i − G ci G PLL i − G T 1 G ci K Q G p 3 − 1 · E + Y ac G θ − 1 M 1 − G T 1 G ci K Q G p 4 + G T 1 G wl G θ − G ci G θ (26) where K _Q(s), G _p3(s) and G _p4(s) are written as Equations 27–29. K Q s = diag G V s , G Q s (27) G p 3 s = G dcd u s 1.5 i sq 0 s G dcq u s − 1.5 i sd 0 s (28) G p 4 s = G udcd i s − 1.5 u sq 0 s G udcq i s 1.5 u sd 0 s (29)

When the system topology is complex, impedance collection is required for impedance analysis based on a single-section division, as shown in Figure 4.

In Figure 4, Z _o represents the aggregated load-side impedance of the system network, while Z _g represents the source-side impedance. Both the source-side and load-side impedances are represented as 2 × 2 matrices. Among them, the impendence renewable energy sides can be obtained by actual measurements, which is a grey/black box system with the unknown circuit and control parameters. The system stability is evaluated using the generalized Nyquist criterion. Specifically, L s = Z g s Y o s (30) where Y _o(s) represents the admittance form of the load-side impedance.

When L (s) shown in Equation 30 does not have right-half-plane poles, the system’s stability is determined by calculating the pair of eigenvalues {λ₁, λ₂} of L (s). The system is stable if and only if the eigenvalue trajectory does not encircle the point (−1, j0); otherwise, the system is unstable. When L (s) has right-half-plane poles, the necessary and sufficient condition for system stability is that the number of counterclockwise encirclements of the point (−1, j0) by the eigenvalues matches the number of right-half-plane poles. If this condition is satisfied, the system is stable; otherwise, the system is unstable.

When the system is complicated, especially involves network and multi-unit, impedance (or admittance) aggregation is required and cascading equivalent sections will influence stability analysis. In addition, several factors will complicate the stability determination, such as the difficulty in counting the encirclement of the GNC curve, the large disparity in the orders of the numerator and denominator of the impedance ratio, which leads to an incomplete curve.

In contrast, the admittance matrix is derived from the network topology, thus eliminating the need for aggregation processing and section selection. System stability can be determined directly by the distribution of the determinant roots, according to the consistency between the zeros of the admittance matrix determinant and the system’s eigenvalues. Although the calculation of roots is complex, the above approach utilizes the advantage of impedance method, thereby avoiding the complex integration process. This method is applicable to various system structures, and the analysis is no longer confined to a single section, enabling the observation of the stability of multiple nodes.

The roots of the determinant of the admittance matrix form a subset of the eigenvalues of the state matrix, and this subset contains all the dominant eigenvalues that characterize the interaction stability of the system. Therefore, when using the s-domain nodal admittance matrix Y (s) for the stability design of the SVG, the damping ratio and its sensitivity can still be calculated using Y (s), which can then be used for the coordinated stabilization design of the SVG.

Based on the damping ratio calculation results in Section 4.1, d dominant oscillation modes are selected according to their damping ratio values as key observation objects, and their damping ratios are denoted as ξ₁ to ξ_d. Using the damping ratios of dominant oscillation modes as stability design criteria, the lower bound for the damping ratio is set as ξ_ref, and the m parameters are assumed to participate in the tuning process. Let the required adjustment amounts of each parameter for stability improvement be denoted as |Δαᵢ| (where i ∈ [1, m]). The goal is to minimize the overall relative adjustment amount of the parameters, and the optimization objective is formulated as follows. min   Δ α sum = ∑ i = 1 m Δ α i α i 2 (36)

One of the constraints in the optimization model for parameters tuning is that the damping ratios meet the adjustment target, and its expression can be given by: ξ k + ∑ i = 1 m ∂ ξ k ∂ α i Δ α i ≥ ξ ref , k ∈ 1 , d (37)

The other constraint in the optimization model for parameters tuning is that the relative adjustment amount of each individual parameter does not exceed its limits, and its expression can be derived as: Δ α i ≤ X % α i , i ∈ 1 , m (38) where X% can be typically set to 20%–40%.

Equations 36–38 form a set of constraint conditions. Parameters tuning is performed under the aforementioned constraints, and the adjustment amounts of each participating parameter can thus be obtained. Figure 5 illustrates the parameter tuning process. After completing the parameter tuning, the system response characteristics at this point are still required verification by simulation or experiment. If indicators such as response speed and overshoot meet the operating requirements, the parameters tuning is completed, otherwise, the parameters tuning process is repeated with different setting of margins and adjustment range until the dynamic performance of SVGs are satisfied.

To validate the accuracy of the established impedance model, a comprehensive SVG simulation model has been developed in MATLAB/Simulink. The SVG circuit parameters and the control parameters for different modes are detailed in Table 1.

The measured frequency-scanning results and the theoretical analysis for the SVG under constant ac voltage control and reactive power control modes are presented in Figures 6, 7, respectively. The strong agreement observed in these figures confirms the accuracy of the proposed dq-impedance model.

As observed from Figures 6, 7, the impedance of the SVG under constant ac voltage control has a lower magnitude around the fundamental frequency, leading to a greater impact on the equivalent impedance during parallel integration. Moreover, this control mode introduces stronger negative damping in the vicinity of the fundamental frequency, thereby posing a threat to system stability.

Based on the grid impedance and SVG control parameters under different cases listed in Table 2, and using the unspecified parameters from the SVG electrical and control parameters in Table 1, a single-machine grid-connected model of the SVG with constant ac voltage control was established for stability analysis.

Figures 8, 9 present the theoretical results obtained from the eigenvalue analysis and the GNC method, respectively. For clarity, the dominant eigenvalues plot omits points located far from the imaginary axis. Figure 8a shows no points in the RHP, and the curve in Figure 9a does not encircle the critical point (−1, j0). Consequently, case 1 is determined to be stable. In contrast, Figure 8b contains RHP points, and the curve in Figure 9b encircles (−1, j0), leading to the conclusion that case 2 is unstable. These theoretical assessments are consistent with the simulation results presented in Figure 10. The FFT analysis in Figure 10 show the main oscillation frequency in abc coordinate system is 150 Hz, which is in good agreement with the frequency of 99.06 Hz in dq coordinate system corresponding to the imaginary part of the RHP dominant oscillation eigenvalues under case 2. The results above demonstrate the equivalence between the two stability assessment methods. Consequently, the determinant of the admittance matrix can be directly employed for stability judgment in subsequent studies of more complex systems.

The system architecture of the renewable energy collection station is depicted in Figure 11. In the diagram, impedance values marked in red are presented in per-unit values, while those in blue are in nominal values, with the per-unit system referenced to the local bus voltage and the individual equipment’s rated capacity. The wind turbine parameters are listed in Table 3. Combined with the constant ac voltage control SVG parameters from the earlier Table 1, the renewable energy collection station system with SVG integration was established. It is noted that the electrical and control parameters of the two SVGs in the system are identical.

A stability analysis was conducted on the established system. Figure 12 shows the terminal voltage waveforms of the two SVGs. At 2 s, the constant ac voltage loop control parameters of the SVGs were switched from Parameter Configuration 1 (as specified in Table 1) to Parameter Configuration 2 (as specified in Figure 12), while all other parameters remained unchanged. Approximately 1.5 s after the parameter switching, system oscillations occurred, indicating instability.

Figure 13 illustrates the eigenvalue distributions for the stability analysis of the collection station system under the two Parameter Configurations. In Figure 13a, the absence of eigenvalues in the RHP indicates that the system is small-signal stable under Parameter Configuration 1. Conversely, the presence of RHP eigenvalues in the subplot (b) demonstrates that the system becomes unstable under Parameter Configuration 2. These theoretical findings are consistent with the time-domain simulation results shown in Figure 12.

For the unstable system under Parameter Configuration 2, coordinated tuning of multi-equipment and multi-parameter is performed using the sensitivity calculation method in Section 4.1 and the parameter tuning method based on damping ratio sensitivity in Section 4.2.

The damping ratios of the dominant oscillation modes, ξ ₁=−0.0067 and ξ ₂=0.0275, are selected as the stability design criteria. The lower limit of the damping ratio adjustment is set to 0.02, and a total of 4 parameters (including the constant ac voltage loops and current loops of two SVGs) are chosen for coordinated optimization and tuning under the specified constraints. Table 4 presents the sensitivity of the damping ratio to each parameter, where the subscripts 1 and 2 denote SVG indices, and the superscripts U and I represent the constant ac voltage loop and current loop, respectively.

The Quadprog quadratic programming solver is utilized to search for the optimal solution with the objective of minimizing the overall relative adjustment amount of the parameters, subject to the constraints of the damping ratio adjustment target and the adjustment amount of each individual parameter. Finally, the constraint-satisfying control parameters, named Parameter Configuration 3, are acquired and presented in Table 5.

Figure 14 shows the simulated waveform of the SVG port voltage. At 2.5 s, the SVG is switched from Parameter Configuration 2 to Parameter Configuration 3, and the system gradually transitions from instability to stability. This is consistent with the theoretical stability assessment result in Figure 16a, which effectively verifies the effectiveness of the proposed multi-parameter coordinated optimization and tuning method.

The above verification indicates that after multi-parameter coordinated optimization, ξ ₁ and ξ ₂ are adjusted to meet the target value, thereby achieving system stability; however, the stability margin remains relatively small. To obtain a larger stability margin, the PI controllers for the dc voltage loop and PLL are added to the tuning process. With X% set to 50% and reference damping ratios ξ _1ref = 0.03 and ξ _2ref = 0.02, the damping ratio sensitivities to the newly added parameters are listed in Table 6, where the superscripts ‘V’ and ‘θ’ denote the dc voltage loop and PLL, respectively. Finally, the control parameters satisfying the constraints—designated as Parameter Configuration 4—are obtained, as shown in Table 7.

As can be seen from the simulated waveform in Figure 15, when the control parameters are switched from Parameter Configuration 2 to Parameter Configuration 4 at 2.5 s, the system achieves stability more rapidly, which is consistent with the theoretical stability assessment result in Figure 16b and realizes an improvement in the stability margin.