Resonance-Suppression Strategy for High-Penetration Renewable Energy Power Systems Based on Active Amplitude and Phase Corrector

Abstract

Due to the negative resistance effect of power electronic devices, power systems with a high proportion of renewable energy face a significant resonance risk. To address this, this paper proposes a resonance-suppression strategy for high-penetration renewable energy systems based on an active amplitude and phase corrector (APC). Firstly, by considering its internal dynamics and complete control loops, the impedance model of the APC is derived. Next, the similarities and differences between resonance stability and harmonic resonance are analyzed using the s-domain and frequency-domain admittance matrices, concluding that resonance suppression for low-damping s-domain modes can be handled in the frequency domain. Then, a supplementary APC control strategy in the abc-frame is proposed, which improves impedance magnitude at specific frequencies while keeping the phase almost unchanged. Finally, the proposed strategy is validated through case studies on an offshore wind power system in Zhejiang Province.

1. Introduction

To meet the “dual carbon” targets, renewable energy development has experienced rapid growth in recent years [1]. Renewable generation units, such as wind turbines (WTs) and photovoltaic systems, are typically interfaced with the grid via power electronic converters. Unlike traditional synchronous generators, these converters can exhibit negative resistance characteristics within certain frequency ranges due to the inherent dynamics introduced by their sophisticated control schemes [2]. In power systems with high renewable penetration, the collective negative resistance of multiple converters may induce wideband resonances, highlighting the need for effective resonance suppression strategies to mitigate associated risks.

Resonance stability refers to the damping characteristics of free oscillation components at natural resonance frequencies during electromagnetic transient processes [3]. In recent years, the impedance-based analysis has become the predominant approach for resonance stability analysis [4,5,6]. This method employs modular impedance models to represent components with complex dynamic behaviors, allowing stability to be assessed by examining impedance interactions according to certain criteria. Its practical advantage of direct measurability has facilitated widespread adoption in both research and engineering practice. Based on the impedance analysis framework, system impedance can be actively reshaped by adding amplitude and phase correctors (APCs) to enhance resonance stability. Due to their high adaptability, this paper focuses on cascaded active APCs based on multilevel conversion technology.

Impedance models are typically established in either the dq-domain or the sequence domain [7]. The dq-domain approach converts periodic time-varying signals into DC quantities, enabling small-signal modeling around DC operating points. Since converter control is also implemented in the dq-domain, this method naturally aligns with controller equations and has been widely adopted [8]. However, it faces challenges for converters with multi-harmonic characteristics, such as modular multilevel converters (MMCs), because harmonic components beyond the fundamental remain time-varying after dq-transformation, requiring multiple rotating reference frames and substantially increasing complexity. In contrast, sequence-domain modeling, initiated with the harmonic linearization method in [9] for two-level voltage source converters, represents time-varying signals through constant Fourier coefficients Sequence impedance models have since been developed for a variety of converters, including photovoltaic inverters, WT systems, and high-voltage DC transmission converters [10]. To handle converters with multi-harmonic characteristics, advanced impedance modeling techniques based on multi-harmonic linearization and harmonic state-space (HSS) methods have been developed, effectively capturing high-order harmonic interactions MMC impedance models using these approaches were reported in [11,12,13], with [13] further incorporating sequence extraction and frequency-shift matrices for a more compact formulation. Despite these advances, impedance modeling of cascaded active APCs employing multilevel conversion technology remains underexplored, with only limited studies available [14], highlighting the need for further investigation.

Active APCs typically achieve impedance reshaping through highly flexible supplementary control strategies, which can be categorized into phase compensation and virtual impedance control. Phase compensation control aims to enhance the phase of the equipment output impedance to mitigate negative resistance. For example, Ref. [15] proposed a current-loop-based phase compensation approach to improves the phase margin, while Ref. [16] selectively compensated the impedance phase within specific frequency ranges to provide directional damping. Ref. [17] incorporated an adaptive resonant integral filter at the phase-locked loop (PLL) front-end to reshape the PLL impedance, extending the feasible phase range toward lower frequency bands. Ref. [18] augmented the feedforward path with compensation control to suppress high-frequency negative resistance, and Ref. [19] analyzed control loop delays in the Luxi project, introducing phase compensation to enhance system dynamics Ref. [20] further achieved active damping of LCL resonance under wide grid impedance variations by employing capacitor current feedforward, extending the passive stability region up to the Nyquist frequency. Virtual impedance control, also known as active damping control, provides damping compensation through virtual impedance with clearer physical interpretation. This approach maps physical impedance circuits into the control loop to approximate equivalent damping branches. For instance, in inductor–capacitor–inductor (LCL)-filtered grid-connected inverters, impedance reshaping can be realized via capacitor current feedforward, capacitor voltage feedforward, or inductor current feedforward [21,22]. To mitigate grid influence, Ref. [23] proposed full-voltage feedforward compensation, effectively reducing inverter output impedance. Refs. [24,25] designed virtual impedances directly based on the target converter output characteristics, enabling effective damping with improved robustness. Ref. [26] investigated a converter connected to a weak AC grid via a series-compensated line, examining the combined effects of PLL, active damping, and virtual conductance on voltage stability, and provided design guidelines to enhance stability and suppress oscillations. In summary, these supplementary control strategies are predominantly implemented in the dq-frame, requiring coordinate transformations for signal transmission, whereas research on inherently more reliable abc-frame implementations remains limited.

To address the above issues, this paper investigates the modeling of APC and its supplementary control, with the specific contributions summarized as follows:

(1)

A comprehensive APC impedance model is developed in the HSS framework, capturing both the internal dynamics and the full control loops, with the power stage and control stage linearized separately and then integrated.

(2)

The mechanisms of resonance instability and harmonic amplification are clarified, and the relationship between s-domain and frequency-domain behaviors is established to enable practical frequency-domain analysis.

(3)

An abc-frame supplementary control is proposed, in which the output current is fed forward through a proportional gain and band-pass filter to the valve-side reference, avoiding coordinate transformation and frequency shifts while allowing reliable tuning of the characteristic frequency.

The rest of this paper includes four parts. In Section 2, the impedance model of the APC is derived. In Section 3, the similarities and differences between resonance stability and harmonic resonance are analyzed using the s-domain and frequency-domain admittance matrices, leading to the proposal of a supplementary APC control strategy in the abc-frame. In Section 4, the case studies of an offshore wind power system (OWPS) in Zhejiang Province are presented to demonstrate the effectiveness of the proposed strategy. The conclusions are summarized in Section 5.

2. APC Impedance Modeling

2.1. Linearized HSS Model of the APC Main Circuit

The main circuit topology of the chain-cascaded APC is shown in Figure 1.

According to Figure 1, for any phase j (j = A, B, C) of the APC, the dynamic characteristics of its main circuit can be described by the following set of equations (to simplify the expression, the subscript j is omitted below):

$$\left\{\begin{array}{l}{u}_{\mathrm{pcc}}=u-R_{0}i-L_0{\displaystyle \frac{di}{dt}}+u_{NO}\hfill \\ u=Nm{u}_{\mathrm{c}}\hfill \\ -mi=C{\displaystyle \frac{d{u}_{\mathrm{c}}}{dt}}\hfill \end{array}\right.$$

(1)

where u_pcc is the voltage at the point of common coupling (PCC); i is the bridge arm current; u_NO is the neutral point potential difference; u is the output voltage of the cascaded submodules; u_c is the average capacitor voltage of the submodules; m is the average switching function of the submodules; N is the number of submodules per bridge arm; R₀, L₀, and C are the equivalent resistance of the bridge arm, equivalent inductance of the bridge arm, and submodule capacitance, respectively. According to (1), during normal operation, the three phases of the APC are coupled through u_NO. To reduce the number and complexity of equations during the modeling process, it is necessary to eliminate u_NO. The role of u_NO is to cancel out the zero-sequence components in u_pcc and u, ensuring that the output current of the APC contains no zero-sequence components. Therefore, when the zero-sequence components of u_pcc and u are removed from the first equation of (1), u_NO can be eliminated while maintaining the balance of the equation, i.e.,:

$$\left\{\begin{array}{l}{L}_0{\displaystyle \frac{di}{dt}}+R_{0}i={(u-u_{\mathrm{pcc}})}^{\pm }\hfill \\ u=Nm{u}_{\mathrm{c}}\hfill \\ -mi=C{\displaystyle \frac{d{u}_{\mathrm{c}}}{dt}}\hfill \end{array}\right.$$

(2)

where the superscripts ± denote the positive and negative sequence components of the corresponding physical quantities. Substituting the second equation of (2) into the first equation and linearizing it yields

$$\left\{\begin{array}{l}{L}_0{\displaystyle \frac{d\mathsf{\Delta }i}{dt}}+R_0\mathsf{\Delta }i={(Nm\mathsf{\Delta }{u}_{\mathrm{c}}+N\mathsf{\Delta }m{u}_{\mathrm{c}}-\mathsf{\Delta }{u}_{\mathrm{pcc}})}^{\pm }\hfill \\ -m\mathsf{\Delta }i-m\mathsf{\Delta }i=C{\displaystyle \frac{d\mathsf{\Delta }{u}_{\mathrm{c}}}{dt}}\hfill \end{array}\right.$$

(3)

Further expressed in the HSS form

$$\left\{\begin{array}{l}{L}_{0}S\mathsf{\Delta }i+R_0\mathsf{\Delta }i={(Nm\mathsf{\Delta }{u}_{\mathrm{c}}+N{u}_{\mathrm{c}}\mathsf{\Delta }m-\mathsf{\Delta }{u}_{\mathrm{pcc}})}^{\pm }\hfill \\ -m\mathsf{\Delta }i-i\mathsf{\Delta }m=CS\mathsf{\Delta }{u}_{\mathrm{c}}\hfill \end{array}\right.$$

(4)

where S is the matrix representing the differential operation in the HSS formulation, as shown in (5).

$$S=\mathrm{diag}(\dots ,-j2{\omega }_{1}I,-j{\omega }_{1}I,O,j{\omega }_{1}I,j2{\omega }_{1}I,\dots )$$

(5)

Due to the multi-frequency coupling within the APC, when a disturbance component with frequency f_p appears at the APC port, it couples with the APC’s internal frequency components, giving rise to multiple disturbance components. To facilitate analysis and analytical modeling, the frequency–phase sequence relationships inside the APC under disturbance are summarized in

Table 1. Frequency–phase sequence relationships inside the APC under disturbance.

Phase Sequence	Phase Sequence
Steady State
Negative Sequence	−7f1	−4f1	−f1	2f1	5f1
Zero Sequence	−6f1	−3f1	0	3f1	6f1
Positive Sequence	−5f1	−2f1	f1	4f1	7f1
Positive-sequence Disturbance Injected at the AC Port
Zero Sequence	fp − 7f1	fp − 4f1	fp − f1	2f1	fp + 5f1
Positive Sequence	fp − 6f1	fp − 3f1	fp	fp + 3f1	fp + 6f1
Negative Sequence	fp − 5f1	fp − 2f1	fp + f1	fp + 4f1	fp + 7f1
Negative-sequence Disturbance Injected at the AC Port
Positive Sequence	fp − 7f1	fp − 4f1	fp − f1	fp + 2f1	fp + 5f1
Negative Sequence	fp − 6f1	fp − 3f1	fp	fp + 3f1	fp + 6f1
Zero Sequence	fp − 5f1	fp − 2f1	fp + f1	fp + 4f1	fp + 7f1

.

Therefore, when a positive-sequence disturbance is injected at the APC port, the following set of sequence component extraction matrices can be considered to obtain the sequence components of a physical quantity in the HSS:

$$\left\{\begin{array}{l}{E}_{-}^{+}=\mathrm{diag}(\dots ,0,1,0,0,1,0,0,\dots )\hfill \\ E_0^{+}=\mathrm{diag}(\dots ,0,0,1,0,0,1,0,\dots )\hfill \\ E_{+}^{+}=\mathrm{diag}(\dots ,1,0,0,1,0,0,1,\dots )\hfill \end{array}\right.$$

(6)

where the superscript + indicates that the injected disturbance is of positive sequence, while the subscripts +, −, and 0 denote the sequence components to be extracted (positive, negative, and zero sequence, respectively). Similarly, when a negative-sequence disturbance is injected at the APC port, the following set of sequence component extraction matrices can be considered to obtain the sequence components of a physical quantity in the HSS:

$$\left\{\begin{array}{l}{E}_{-}^{-}=\mathrm{diag}(\dots ,1,0,0,1,0,0,1,\dots )\hfill \\ E_0^{-}=\mathrm{diag}(\dots ,0,1,0,0,1,0,0,\dots )\hfill \\ E_{+}^{-}=\mathrm{diag}(\dots ,0,0,1,0,0,1,0,\dots )\hfill \end{array}\right.$$

(7)

where the superscript − indicates that the injected disturbance is of negative sequence, while the subscripts +, −, and 0 denote the sequence components to be extracted (positive, negative, and zero sequence, respectively).

Therefore, based on the internal disturbance-phase sequence relationship of the APC and utilizing the sequence component extraction matrices, (4) can be expressed in the following form:

$$\left\{\begin{array}{l}{E}_{\pm }\mathsf{\Delta }{u}_{\mathrm{pcc}}=E_{\pm }(Nm\mathsf{\Delta }{u}_{\mathrm{c}}+N{u}_{\mathrm{c}}\mathsf{\Delta }m)-L_{0}S\mathsf{\Delta }i+R_0\mathsf{\Delta }i\hfill \\ \mathsf{\Delta }{u}_{\mathrm{c}}={\displaystyle \frac{1}{C}}{S}^{-1}(-m\mathsf{\Delta }i-i\mathsf{\Delta }m)\hfill \end{array}\right.$$

(8)

where E_± represents the sum of the positive-sequence component extraction matrix and the negative-sequence component extraction matrix under positive-sequence or negative-sequence disturbance at the APC port.

Further rearrangement yields

$$E_{\pm }\mathsf{\Delta }{u}_{\mathrm{pcc}}=G_1\mathsf{\Delta }i+G_2\mathsf{\Delta }m$$

(9)

$$\mathsf{\Delta }{u}_{\mathrm{c}}=G_3\mathsf{\Delta }i+G_4\mathsf{\Delta }m$$

(10)

where the coefficient matrices G_j (j = 1, 2, 3, 4) are as follows:

$$\left\{\begin{array}{l}{G}_1=N{E}_{\pm }m{\displaystyle \frac{1}{C}}{S}^{-1}m-L_{0}S+R_{0}I\hfill \\ G_2=N{E}_{\pm }(m{\displaystyle \frac{1}{C}}{S}^{-1}i+u_{\mathrm{c}})\hfill \\ G_3=-{\displaystyle \frac{1}{C}}{S}^{-1}m\hfill \\ G_4=-{\displaystyle \frac{1}{C}}{S}^{-1}i\hfill \end{array}\right.$$

(11)

At this point, the linearized HSS model of the APC main circuit has been derived. The disturbance components of the port voltage and the submodule capacitor voltage can both be expressed in terms of the disturbance components of the port current and the modulation wave. Once the linearized HSS model of the control system is determined, combining it with the linearized HSS model of the main circuit allows for the elimination of the disturbance components of the modulation wave and the submodule capacitor voltage. This ultimately establishes the relationship between the disturbance components of the APC port voltage and the port current, enabling the solution of the APC impedance.

2.2. Linearized HSS Model of the APC Control System

To achieve independent control of active and reactive power in the APC, a classical dual-loop control strategy is applied to the control system design of the chain-based APC. The control block diagram is shown in Figure 2, which specifically includes submodule capacitor voltage control, reactive power control, inner-loop current control, and a PLL.

(1)

Coordinate Transformation Module

The coordinate transformations used in the APC include the Park transformation and its corresponding inverse transformation, as well as the Clark transformation. Under small disturbances, the transformation angle for the Park transformation and its inverse is θ = θ₀ + Δθ. The process of transforming a variable x = x₀ + Δx from the ABC coordinate system to the dq coordinate system can be expressed as (12), while the process of transforming a variable x_dq = x_0,dq + Δx_dq from the dq coordinate system to the ABC coordinate system can be expressed as (13)

$$T_{dq+}({\theta }_0+\mathsf{\Delta }\theta )(x_0+\mathsf{\Delta }x)\approx T_{dq+}({\theta }_0)x_0+T_{dq+}({\theta }_0)\mathsf{\Delta }x+T_{dq+}({\theta }_0+90°)x_0\mathsf{\Delta }\theta$$

(12)

$$T_{dq-}({\theta }_0+\mathsf{\Delta }\theta )(x_{0, dq}+\mathsf{\Delta }{x}_{dq})\approx T_{dq-}({\theta }_0)x_{0, dq}+T_{dq-}({\theta }_0)\mathsf{\Delta }{x}_{dq}+T_{dq-}({\theta }_0+90°)x_{0, dq}\mathsf{\Delta }\theta$$

(13)

The derivation process for (12) is as follows. Under small disturbances, when the transformation angle is θ = θ₀ + Δθ, the Park transformation matrix can be expressed as

$$\begin{array}{cc}\hfill T_{\mathrm{dq}+}({\theta }_0+\mathsf{\Delta }\theta )& \approx {\displaystyle \frac{2}{3}}\left[\begin{array}{c} \cos ({\theta }_0) \cos ({\theta }_0-120°) \cos ({\theta }_0+120°)\\ -\sin ({\theta }_0) -\sin ({\theta }_0-120°) -\sin ({\theta }_0+120°)\end{array}\right]\hfill \\ & -{\displaystyle \frac{2}{3}}\left[\begin{array}{c}\sin ({\theta }_0) \sin ({\theta }_0-120°) \sin ({\theta }_0+120°)\\ \cos ({\theta }_0) \cos ({\theta }_0-120°) \cos ({\theta }_0+120°)\end{array}\right]\mathsf{\Delta }\theta \hfill \\ & =T_{\mathrm{dq}+}({\theta }_0)\mathsf{\Delta }x+T_{\mathrm{dq}+}({\theta }_0+90°)\mathsf{\Delta }\theta \hfill \end{array}$$

(14)

The derivation process for (13) is as follows. Under small disturbances, when the transformation angle is θ = θ₀ + Δθ, the inverse Park transformation matrix can be expressed as

$$\begin{array}{cc}{T}_{\mathrm{dq}-}({\theta }_0+\mathsf{\Delta }\theta )& \approx {\displaystyle \frac{2}{3}}\left[\begin{array}{cc}\cos ({\theta }_0)& -\sin ({\theta }_0)\\ \cos ({\theta }_0-120°)& -\sin ({\theta }_0-120°)\\ \cos ({\theta }_0+120°)& -\sin ({\theta }_0+120°)\end{array}\right]\hfill \\ & -{\displaystyle \frac{2}{3}}\left[\begin{array}{cc}\sin ({\theta }_0)& \cos ({\theta }_0)\\ \sin ({\theta }_0-120°)& \cos ({\theta }_0-120°)\\ \sin ({\theta }_0+120°)& \cos ({\theta }_0+120°)\end{array}\right]\mathsf{\Delta }\theta \hfill \\ & =T_{\mathrm{dq}-}({\theta }_0)\mathsf{\Delta }x+T_{\mathrm{dq}-}({\theta }_0+90°)\mathsf{\Delta }\theta \hfill \end{array}$$

(15)

The Clark transformation is independent of angle, and its transformation process under small disturbances is shown in (16).

$$T_{\alpha \beta +}(x_0+\mathsf{\Delta }x)\approx T_{\alpha \beta +}{x}_0+T_{\alpha \beta +}\mathsf{\Delta }x$$

(16)

According to (12) and (13), in the linearized HSS model, the dq-frame disturbance components obtained by applying the Park transformation to Δx are

$$\left\{\begin{array}{l}\mathsf{\Delta }{x}_d=T_{d+}\mathsf{\Delta }x+x{\prime }_{d+}\mathsf{\Delta }\theta \hfill \\ \mathsf{\Delta }{x}_q=T_{q+}\mathsf{\Delta }x+x{\prime }_{q+}\mathsf{\Delta }\theta \hfill \end{array}\right.$$

(17)

The disturbance component in the ABC coordinate system obtained by applying the inverse transformation to Δx_d and Δx_q is

$$\mathsf{\Delta }x=T_{d-}\mathsf{\Delta }{x}_d+T_{q-}\mathsf{\Delta }{x}_q+(x{\prime }_{d-}+x{\prime }_{q-})\mathsf{\Delta }\theta$$

(18)

The disturbance components in the αβ coordinate system obtained by applying the Clark transformation to Δx are

$$\left\{\begin{array}{l}\mathsf{\Delta }{x}_{\alpha }=T_{\alpha }\mathsf{\Delta }x\hfill \\ \mathsf{\Delta }{x}_{\beta }=T_{\beta }\mathsf{\Delta }x\hfill \end{array}\right.$$

(19)

In (17) to (19), T_d+ and T_q+ represent the transformation to the d-frame and q-frame in the Park transformation, respectively, while T_d− and T_q− represent the transformation of the d-frame and q-frame components in the inverse Park transformation, respectively. T_α and T_β represent the transformation to the α-frame and β-frame in the Clark transformation. The steady-state quantity x₀ is transformed by T_dq+ (θ₀ + 90°), and the d-frame and q-frame components are extracted and expressed in Toeplitz matrix form as x’_d+ and x’_q+, respectively. The calculation of x’_d− and x’_q− follows the same principle.

The derivation of the coordinate transformation matrices in the HSS is presented below.

When Δx is a positive-sequence component (with amplitude ΔX and angular frequency ω_p), its Park transformation process can be written as

$$T_{dq+}({\omega }_{1}t)\left[\begin{array}{l}\mathsf{\Delta }X\cos ({\omega }_{p}t)\hfill \\ \mathsf{\Delta }X\cos ({\omega }_{p}t-120°)\hfill \\ \mathsf{\Delta }X\cos ({\omega }_{p}t+120°)\hfill \end{array}\right]=\left[\begin{array}{l}\mathsf{\Delta }X\cos [({\omega }_p-{\omega }_1)t]\hfill \\ \mathsf{\Delta }X\cos [({\omega }_p-{\omega }_1)t-90°]\hfill \end{array}\right]$$

(20)

(20) indicates that after the Park transformation, the frequency of the positive-sequence signal decreases by ω₁, while the q-frame component lags the d-frame component by 90°.

When Δx is a negative-sequence component (with amplitude ΔX and angular frequency ω_p), its Park transformation process can be written as

$$T_{dq+}({\omega }_{1}t)\left[\begin{array}{l}\mathsf{\Delta }X\cos ({\omega }_{p}t)\hfill \\ \mathsf{\Delta }X\cos ({\omega }_{p}t+120°)\hfill \\ \mathsf{\Delta }X\cos ({\omega }_{p}t-120°)\hfill \end{array}\right]=\left[\begin{array}{l}\mathsf{\Delta }X\cos [({\omega }_p+{\omega }_1)t]\hfill \\ \mathsf{\Delta }X\cos [({\omega }_p+{\omega }_1)t+90°]\hfill \end{array}\right]$$

(21)

(21) indicates that after the Park transformation, the frequency of the negative-sequence signal increases by ω₁, while the q-frame component leads the d-frame component by 90°.

Expressing (20) and (21) in the frequency domain:

$$\left[\begin{array}{l}\mathsf{\Delta }{X}_d({\omega }_p)\hfill \\ \mathsf{\Delta }{X}_q({\omega }_p)\hfill \end{array}\right]=\left[\begin{array}{l}1\hfill \\ \mp j\hfill \end{array}\right]\mathsf{\Delta }X({\omega }_p\pm {\omega }_1)$$

(22)

(22) indicates that the Park transformation produces different frequency-shift effects for different sequence components. The frequency up-shift matrix T_up and frequency down-shift matrix T_dw are defined as follows:

$$T_{\mathrm{up}}=\left[\begin{array}{llll}\ddots \hfill & \hfill & \hfill & \hfill \\ 1\hfill & 0\hfill & \hfill & \hfill \\ \hfill & 1\hfill & 0\hfill & \hfill \\ \hfill & \hfill & 1\hfill & \ddots \hfill \end{array}\right]$$

(23)

$$T_{\mathrm{dw}}=\left[\begin{array}{llll}\ddots \hfill & 1\hfill & \hfill & \hfill \\ \hfill & 0\hfill & 1\hfill & \hfill \\ \hfill & \hfill & 0\hfill & 1\hfill \\ \hfill & \hfill & \hfill & \ddots \hfill \end{array}\right]$$

(24)

The frequency up-shift matrix T_up indicates that after the Park transformation, the frequency of all components in the signal increases by ω₁; the frequency down-shift matrix T_dw indicates that after the Park transformation, the frequency of all components in the signal decreases by ω₁. Therefore, by combining the frequency-shift matrices with the sequence extraction matrices obtained in Section 2.1, the Park transformation matrices in the HSS can be derived:

$$T_{d+}=T_{\mathrm{dw}}{E}_{+}+T_{\mathrm{up}}{E}_{-}$$

(25)

$$T_{q+}=-j{T}_{\mathrm{dw}}{E}_{+}+j{T}_{\mathrm{up}}{E}_{-}$$

(26)

For the inverse Park transformation, the following frequency-shift effects occur in the frequency domain:

$$\mathsf{\Delta }X({\omega }_p)={\displaystyle \frac{1}{2}}[\mathsf{\Delta }{X}_d({\omega }_p-{\omega }_1)+\mathsf{\Delta }{X}_d({\omega }_p+{\omega }_1)+j\mathsf{\Delta }{X}_q({\omega }_p-{\omega }_1)-j\mathsf{\Delta }{X}_q({\omega }_p+{\omega }_1)]$$

(27)

Therefore, the transformation matrices for the inverse Park transformation in the HSS are

$$T_{d-}={\displaystyle \frac{1}{2}}{T}_{\mathrm{dw}}+{\displaystyle \frac{1}{2}}{T}_{\mathrm{up}}$$

(28)

$$T_{q-}={\displaystyle \frac{1}{2j}}{T}_{\mathrm{dw}}-{\displaystyle \frac{1}{2j}}{T}_{\mathrm{up}}$$

(29)

For the Clark transformation, when Δx is a positive-sequence component, the transformation process can be expressed as

$$\left[\begin{array}{l}\mathsf{\Delta }{X}_{\alpha }({\omega }_p)\hfill \\ \mathsf{\Delta }{X}_{\alpha }({\omega }_p)\hfill \end{array}\right]=\left[\begin{array}{l}1\hfill \\ -j\hfill \end{array}\right]\mathsf{\Delta }X({\omega }_p)$$

(30)

When Δx is a negative-sequence component, the transformation process can be expressed as

$$\left[\begin{array}{l}\mathsf{\Delta }{X}_{\alpha }({\omega }_p)\hfill \\ \mathsf{\Delta }{X}_{\alpha }({\omega }_p)\hfill \end{array}\right]=\left[\begin{array}{l}1\hfill \\ j\hfill \end{array}\right]\mathsf{\Delta }X({\omega }_p)$$

(31)

Therefore, the transformation matrices for the Clark transformation in the HSS are

$$T_{\alpha +}=E_{+}+E_{-}$$

(32)

$$T_{\beta +}=-j{E}_{+}+j{E}_{-}$$

(33)

(25), (26), (28), (29), (32), and (33) constitute the complete set of transformation matrices representing the coordinate transformation process in the HSS.

(2)

PLL

The PLL calculates the PCC voltage phase based on the measured PCC voltage, providing a synchronous reference signal for the APC. The derivation of the transfer function model from the disturbed PCC voltage to the disturbed output reference phase of the PLL in the HSS form is presented below.

When a disturbance occurs in the PCC voltage, the reference phase disturbance generated by the PLL (in the HSS form) is

$$\mathsf{\Delta }{\theta }_{\mathrm{PLL}}=\mathrm{diag}(T_{\mathrm{PLL}}({\omega }_{p+k}))\mathsf{\Delta }{u}_{\mathrm{pcc}q}$$

(34)

In the equation, Δu_pccq represents the q-frame component of the PCC voltage disturbance, and T_PLL (ω_p+k) denotes the open-loop transfer function of the PLL, expressed as

$$T_{\mathrm{PLL}}({\omega }_{p+k})={\displaystyle \frac{1}{j({\omega }_p+k{\omega }_0)}}(k_{\mathrm{pPLL}}+{\displaystyle \frac{k_{\mathrm{iPLL}}}{j({\omega }_p+k{\omega }_0)}})$$

(35)

According to the second equation in (17), Δu_pccq can be expressed as

$$\mathsf{\Delta }{u}_{\mathrm{pcc}q}=T_{q+}\mathsf{\Delta }{u}_{\mathrm{pcc}}+u{\prime }_{\mathrm{pcc}q+}\mathsf{\Delta }{\theta }_{\mathrm{PLL}}$$

(36)

Under steady-state conditions, Δu_pccq contains only the fundamental frequency component, and u’_pccq+ is, therefore, a diagonal matrix with all main diagonal elements being U₀cos(ω₀t + θ_u0). The transfer function matrix in the HSS form from the PCC disturbance voltage Δu_pcc to the PLL output disturbance reference phase Δθ_PLL is obtained by substituting (36) into (34), i.e.,:

$$\mathsf{\Delta }{\theta }_{\mathrm{PLL}}=G_{\mathrm{PLL}}\mathsf{\Delta }{u}_{\mathrm{pcc}}$$

(37)

where G_PLL is

$$G_{\mathrm{PLL}}=\mathrm{diag}({\displaystyle \frac{T_{\mathrm{PLL}}({\omega }_{p+k})}{1+U_0\cos {\theta }_{\mathrm{u}0}{T}_{\mathrm{PLL}}({\omega }_{p+k})}})T_{q+}$$

(38)

(3)

Submodule Capacitor Voltage Control

The purpose of submodule capacitor voltage control is to maintain the average capacitor voltage of all submodules in the three-phase bridge arms of the APC near the rated value. This control is implemented using a PI controller, with its input being the deviation between the average submodule capacitor voltage and the commanded capacitor voltage value, and its output being the d-frame current reference value for the inner current control loop. Its linearized HSS model is

$$\mathsf{\Delta }{i}_{d\mathrm{ref}}=G_{u\mathrm{c}}(\mathsf{\Delta }{u}_{\mathrm{c}}-\mathsf{\Delta }{u}_{\mathrm{cref}})$$

(39)

where G_uc is a diagonal matrix representing the gain of the PI controller in the submodule capacitor voltage control under the HSS form. The elements on its main diagonal are composed of the transfer functions of the PI controller at different frequencies, i.e.,:

$$G_{u\mathrm{c}}=\mathrm{diag}(k_{\mathrm{p}u\mathrm{c}}+{\displaystyle \frac{k_{\mathrm{i}u\mathrm{c}}}{j({\omega }_p+k{\omega }_0)}})$$

(40)

where k_puc and k_iuc represent the proportional coefficient and integral coefficient of the PI controller in the submodule capacitor voltage control, respectively.

(4)

Reactive Power Control

The purpose of reactive power control is to regulate the reactive power output of the APC to a specified value. This control is implemented using a PI controller, with its input being the deviation between the actual reactive power value and the commanded reactive power value, and its output being the q-frame current reference value for the inner current control loop.

According to the instantaneous power theory:

$$Q=u_{\mathrm{pcc}\beta }{i}_{\mathrm{ac}\alpha }-u_{\mathrm{pcc}\alpha }{i}_{\mathrm{ac}\beta }$$

(41)

Linearizing this equation and expressing it in the HSS form:

$$\mathsf{\Delta }Q=u_{\mathrm{pcc}\beta }\mathsf{\Delta }{i}_{\mathrm{ac}\alpha }-u_{\mathrm{pcc}\alpha }\mathsf{\Delta }{i}_{\mathrm{ac}\beta }-i_{\mathrm{ac}\beta }\mathsf{\Delta }{u}_{\mathrm{pcc}\alpha }+i_{\mathrm{ac}\alpha }\mathsf{\Delta }{u}_{\mathrm{pcc}\beta }$$

(42)

According to the control block diagram of the reactive power controller:

$$\mathsf{\Delta }{i}_{q\mathrm{ref}}=-G_Q\mathsf{\Delta }Q$$

(43)

where G_Q is a diagonal matrix representing the gain of the PI controller in the reactive power control under the HSS form. The elements on its main diagonal are composed of the transfer functions of the PI controller at different frequencies, i.e.,:

$$G_Q=\mathrm{diag}(k_{\mathrm{p}Q}+{\displaystyle \frac{k_{\mathrm{i}Q}}{j({\omega }_p+k{\omega }_0)}})$$

(44)

where k_pQ and k_iQ represent the proportional coefficient and integral coefficient of the PI controller in the reactive power control, respectively.

(5)

Inner-loop Current Control

The purpose of the inner-loop current control is to generate the d-frame modulation wave signal and the q-frame modulation wave signal based on the d-frame current reference value output from the submodule capacitor voltage control and the q-frame current reference value output from the reactive power control, respectively. The d-frame and q-frame modulation wave signals are then transformed into three-phase modulation wave signals via the inverse Park transformation, which are fed into the carrier phase-shift modulation stage to ultimately generate the triggering signals for each submodule.

According to the control block diagram of the inner-loop current controller:

$$\left\{\begin{array}{l}\mathsf{\Delta }{m}_d=G_i(\mathsf{\Delta }{i}_{d\mathrm{ref}}-\mathsf{\Delta }{i}_{\mathrm{ac}d})-K_i\mathsf{\Delta }{i}_{\mathrm{ac}q}+\mathsf{\Delta }{u}_{\mathrm{pcc}d}\hfill \\ \mathsf{\Delta }{m}_q=G_i(\mathsf{\Delta }{i}_{q\mathrm{ref}}-\mathsf{\Delta }{i}_{\mathrm{ac}q})+K_i\mathsf{\Delta }{i}_{\mathrm{ac}d}+\mathsf{\Delta }{u}_{\mathrm{pcc}q}\hfill \end{array}\right.$$

(45)

where G_i is a diagonal matrix representing the gain of the PI controller in the inner-loop current control under the HSS form. The elements on its main diagonal are composed of the transfer functions of the PI controller at different frequencies, i.e.,:

$$G_i=\mathrm{diag}(k_{\mathrm{p}i}+{\displaystyle \frac{k_{\mathrm{i}i}}{j({\omega }_p+k{\omega }_0)}})$$

(46)

where k_pi and k_ii represent the proportional coefficient and integral coefficient of the PI controller in the inner-loop current control, respectively.

According to (18), the modulation wave disturbance quantity in the three-phase stationary coordinate system can be expressed as

$$\mathsf{\Delta }m=T_{d-}\mathsf{\Delta }{m}_d+T_{q-}\mathsf{\Delta }{m}_q+(m{\prime }_{d-}+m{\prime }_{q-})\mathsf{\Delta }\theta$$

(47)

In summary, by combining (34) to (47) and eliminating the intermediate variables, the linearized HSS model of the APC control system can be obtained as

$$\mathsf{\Delta }m=G_5\mathsf{\Delta }{u}_{\mathrm{c}}+G_6\mathsf{\Delta }i+G_7\mathsf{\Delta }u$$

(48)

2.3. APC Impedance

By combining the linearized HSS model of the APC main circuit ((9) and (10)) with the linearized HSS model of the APC control system ((48)), the disturbance components of the modulation wave and the submodule capacitor voltage can be eliminated. This ultimately establishes the relationship between the disturbance components of the APC port voltage and the port current, allowing the solution of the APC impedance.

Specifically, first substitute (10) into (48) to eliminate the submodule capacitor voltage disturbance component in the linearized HSS model of the APC control system:

$$\mathsf{\Delta }m={(I-G_5{G}_4)}^{-1}[(G_5{G}_3+G_6)\mathsf{\Delta }i+G_7\mathsf{\Delta }u]$$

(49)

Then substitute (49) into (9) to eliminate the modulation wave disturbance quantity:

$$\begin{array}{cc}\hfill \mathsf{\Delta }i& =[G_1+G_2{(I-G_5{G}_4)}^{-1}({G_5{G}_3+G_6)]}^{-1}(E_{\pm }-G_2{(I-G_5{G}_4)}^{-1}{G}_7)\mathsf{\Delta }u\hfill \\ & =Y\mathsf{\Delta }u\hfill \end{array}$$

(50)

(50) represents the relationship between the APC port voltage disturbance component Δu and the port current disturbance component Δi. The matrix Y is the transfer function matrix describing their relationship. Thus, the port impedance of the APC at the disturbance frequency f_p is

$$Z_{\mathrm{APC}}={\displaystyle \frac{1}{Y(h+1,h+1)}}$$

(51)

Verification is performed using the APC with parameters shown in

Table 2. Parameters of the APC.

Structure	Parameter	Value
Main circuit	Rated AC voltage/kV	35
	Rated capacity/MW	30
	Bridge arm inductance/H	0.08
	Bridge arm resistance/Ω	0.001
	Capacitor/μF	1000
Control system	PLL proportional gain	30
	PLL integral gain	1000
	d-axis outer loop proportional gain	5
	d-axis outer loop integral gain	15
	q-axis outer loop proportional gain	0.2
	q-axis outer loop integral gain	15
	Inner-loop proportional gain	1
	Inner-loop proportional gain	10

. The electromagnetic transient model is built in PSCAD/EMTDC 4.6.2, and the simulation results of the port-impedance are obtained using the test signal method. The comparison between the analytical and simulation results of the port impedance frequency characteristics is shown in Figure 3. The analytical result closely matches the simulation one, validating the accuracy of the above analytical impedance model.

3. Supplementary Control Strategy of APC

3.1. Resonance Stability and Harmonic Resonance

The s-domain node admittance matrix (s-NAM) method is a common impedance-based analysis technique. The mathematical model of the s-NAM method is the s-NAM. The key to calculating the resonance modes is solving for the zeros of the determinant of the s-NAM [27,28], i.e.,

$$\det \left[Y\left(s_{\mathrm{res}}\right)\right]=\det \left[Y\left(-{\sigma }_{\mathrm{res}}\pm \mathrm{j}{\omega }_{\mathrm{res},s}\right)\right]=0$$

(52)

where Y(s) is the s-NAM; s_res is the s-domain resonance mode; σ_res is the damping factor; ω_res,s is the s-domain resonance angular frequency. σ_res determines resonance stability: if σ_res > 0, the system is stable; if σ_res ≤ 0, it is unstable.

Harmonic resonance is a steady-state phenomenon where the system exhibits a large (voltage) response to a (current) excitation at a particular frequency. This process reflects the frequency characteristics of the system, which can be described using the system’s frequency-domain node admittance matrix (F-NAM) [29,30].

The frequency-domain node voltage equation of the system is

$$Y\left(f\right)V\left(f\right)=R\left(f\right)\Lambda \left(f\right)L\left(f\right)V\left(f\right)=I\left(f\right)$$

(53)

where Y(f) is the F-NAM; V(f) and I(f) are the frequency-domain node voltage vector and the frequency-domain node injected current vector, respectively; Λ(f), R(f), and L(f) are the diagonal eigenvalue matrix, right eigenvector matrix, and left eigenvector matrix of Y(f), with Λ(f) = diag(λ₁, λ₂, …, λₙ) and L(f) = R(f)⁻¹.

Transforming (53) gives

$$L\left(f\right)V\left(f\right)=\Lambda {\left(f\right)}^{-1}L\left(f\right)I\left(f\right)\Rightarrow V_{\mathrm{mode}}\left(f\right)=\Lambda {\left(f\right)}^{-1}{I}_{\mathrm{mode}}\left(f\right)$$

(54)

where V_mode(f) and I_mode(f) are the frequency-domain modal voltage vector and frequency-domain modal current vector, respectively; Λ(f)⁻¹= diag(${\lambda }_1^{-1}$, ${\lambda }_2^{-1}$, …, ${\lambda }_n^{-1}$). The reciprocal of the eigenvalue is called the modal impedance. If an eigenvalue λₖ is very small, the modal current I_{mode_k} will trigger a very high modal voltage U_{mode_k}.

Each eigenvalue corresponds to a frequency-domain resonance mode, and the mode with the smallest eigenvalue is called the critical mode. If the smallest eigenvalue is sufficiently small at a certain frequency, a severe resonance will occur, and this frequency is called the frequency-domain resonance frequency.

The differences and similarities between resonance stability and harmonic resonance are illustrated using the simple parallel RLC circuit ($R<2\sqrt{L/C}$), shown in Figure 4.

The s-NAM of the simple parallel RLC circuit is

$$Y\left(s\right)=\left[\begin{array}{cc}{\displaystyle \frac{1}{R}}+sC& -{\displaystyle \frac{1}{R}}\\ -{\displaystyle \frac{1}{R}}& {\displaystyle \frac{1}{R}}+{\displaystyle \frac{1}{sL}}\end{array}\right]$$

(55)

The zeros of the corresponding determinant are

$$s_{\mathrm{res}}=-{\sigma }_{\mathrm{res}}\pm \mathrm{j}{\omega }_{\mathrm{res},s}=-{\displaystyle \frac{R}{2L}}\pm \mathrm{j}{\displaystyle \frac{\sqrt{4LC-R^2{C}^2}}{2LC}}$$

(56)

The F-NAM of the simple parallel RLC circuit is

$$Y\left(\mathrm{j}\omega \right)=\left[\begin{array}{cc}{\displaystyle \frac{1}{R}}+\mathrm{j}\omega C& -{\displaystyle \frac{1}{R}}\\ -{\displaystyle \frac{1}{R}}& {\displaystyle \frac{1}{R}}+{\displaystyle \frac{1}{\mathrm{j}\omega L}}\end{array}\right]$$

(57)

The corresponding eigenvalues are

$${\lambda }_{1,2}={\displaystyle \frac{1}{2}}\left\{\left[{\displaystyle \frac{2}{R}}+\mathrm{j}\left(\omega C-{\displaystyle \frac{1}{\omega L}}\right)\right]\pm \sqrt{\left({\displaystyle \frac{4}{R^2}}-{\displaystyle \frac{2C}{L}}\right)-\left({\omega }^2{C}^2+{\displaystyle \frac{1}{{\omega }^2{L}^2}}\right)}\right\}$$

(58)

Harmonic resonance focuses on the smallest eigenvalue, which depends on the range of R. A detailed discussion is not provided here; instead, the frequency characteristics of the smallest eigenvalue for different values of R are shown in Figure 5 and Figure 6. It can be observed that for harmonic resonance to occur, $R<\sqrt{L/C}$ must be satisfied, with frequency-domain resonance angular frequency ${\omega }_{\mathrm{res},f}=1/\sqrt{LC}$.

The two key parameters for resonance stability are σ_res, which represents the decay of free components, and ω_res,s, which represents their oscillation. For harmonic resonance, the key parameters are the critical modal impedance magnitude 1/|λ|_min, indicating the resonance intensity, and ω_res,f, the resonance frequency. The following two relationships hold between them:

$${\omega }_{\mathrm{res},f}=\sqrt{{\sigma }_{\mathrm{res}}^2+{\omega }_{\mathrm{res},s}^2}$$

(59)

For ω_res,s and ω_res,f, the two are different frequencies. As R decreases, meaning the damping in the system decreases, the two frequencies become closer. For σᵣₑₛ and 1/|λ|_min, as R decreases, σᵣₑₛ becomes smaller, meaning the time for free components to decay increases, and 1/|λ|_min becomes larger, indicating a more intense harmonic amplification. The above relationships are shown in Figure 7.

During resonance stability analysis, for dominant modes (with very small or negative damping), the resonance frequency and harmonic-resonance frequency are nearly identical, allowing the s-domain problem to be solved in the frequency domain.

3.2. Supplementary Control Channel Design

Based on the s-domain dominant mode resonance frequency characteristics, a band-pass filter (BPF) extracts the resonant frequency components for APC impedance reshaping. The input is the APC output current, and the designed current feedforward control channel is shown in Figure 8.

In Figure 8, K_SC is the supplementary control gain, which is typically a negative value; G_BPF is the BPF transfer function, as shown in (60).

$$G_{\mathrm{BPF}}\left(s\right)={\displaystyle \frac{s}{\left(Q/{\omega }_{\mathrm{c}}\right)s^2+s+Q{\omega }_{\mathrm{c}}}}$$

(60)

where Q is the quality factor; ω_c is the characteristic angular frequency.

Based on the derivation in Section 2, the impedance of the APC with supplementary control can be simply expressed as

$$\begin{array}{c}{Z}_{\mathrm{APC}\_\mathrm{SC}\_\mathrm{abc}}={\displaystyle \frac{G_I+\mathsf{\Delta }{G}_I-\left(s{L}_0+R_0\right)}{G_U+\mathsf{\Delta }{G}_U-1}}\\ \left\{\begin{array}{l}\mathsf{\Delta }{G}_I=0.5U_{\mathrm{dc}}{G}_{\mathrm{pu}\_i}{K}_{\mathrm{SC}}{G}_{\mathrm{BPF}}\hfill \\ \mathsf{\Delta }{G}_U=0\hfill \end{array}\right.\end{array}$$

(61)

where G_I and G_U are the transfer functions from current and grid-side voltage to valve-side voltage, respectively; ΔG_I and ΔG_U are the transfer functions from current and grid-side voltage to valve-side voltage through supplementary control, respectively; G_{pu_i} is the per-unit coefficient of the current.

The comparison of the APC impedance frequency characteristics near the characteristic frequency after supplementary control is shown in Figure 9, where K_SC = −8, Q = 20, and the characteristic frequency f_c = 320 Hz, with other parameters the same as those in Table 2. Near f_c, the impedance magnitude of the APC with supplementary control increases sharply, then decreases slightly, accompanied by a phase lag. The characteristic frequency is slightly higher than the local maximum magnitude frequency (f₁) and lower than the local minimum phase frequency (f₂), which in turn is lower than the local minimum magnitude frequency (f₃). As the phase decreases to the local minimum, the magnitude decreases steadily. When the phase recovers, the magnitude first decreases and then increases.

The three key parameters affecting the effectiveness of the supplementary control are K_SC, f_c, and Q, and their impacts on the APC impedance frequency characteristics are illustrated in Figure 10. As the magnitude of K_SC increases, the local maximum magnitude increases and the local minimum phase decreases, while the extent of the phase reduction gradually diminishes. f_c does not alter the local shape of the impedance frequency characteristics; instead, the local peak is in magnitude and the local valley is in phase shift with changes in f_c. Q does not change the local magnitude gain or phase lag; instead, the impedance frequency characteristics scale around f_c, and the curve becomes steeper as Q increases.

When the same control strategy is applied in the dq-frame, as shown in Figure 11, the impedance of the APC with supplementary control can be simply expressed as

$$\begin{array}{c}{Z}_{\mathrm{APC}\_\mathrm{SC}\_\mathrm{dq}}={\displaystyle \frac{G_I+\mathsf{\Delta }{G}_I-\left(s{L}_0+R_0\right)}{G_U+\mathsf{\Delta }{G}_U-1}}\\ \left\{\begin{array}{l}\mathsf{\Delta }{G}_I=0.5K_{\mathrm{m}}{U}_{\mathrm{dc}}{G}_{\mathrm{pu}\_i}{K}_{\mathrm{SC}}{G}_{\mathrm{BPF}}\hfill \\ \mathsf{\Delta }{G}_U=-0.5\mathrm{j}{I}_{\mathrm{m}0}{e}^{\mathrm{j}{\phi }_{i0}}{G}_{\mathrm{pu}\_i}{K}_{\mathrm{SC}}{G}_{\mathrm{BPF}}{G}_{\mathrm{PLL}}\hfill \end{array}\right.\end{array}$$

(62)

where K_m is the modulation coefficient; I_m0 are the magnitudes of the grid-side; φ_i0 is the phase difference between the grid-side current and voltage; G_PLL is the transfer function from the grid-side voltage perturbation to the phase perturbation through the PLL.

Under the same control parameters, the impedance frequency characteristics of the APC with supplementary control in different reference frames are shown in Figure 12. The supplementary control in the dq-frame results in different levels of magnitude gain and phase lag, because the corresponding ΔI terms differ by a coefficient K_m, while the effect of ΔU is negligible. Moreover, the dq-frame supplementary control introduces a waveform shift caused by the coordinate transformation, leading to a local rightward shift of 50°. Therefore, the selection of f_c is more straightforward and natural for the supplementary control implemented in the abc-frame.

4. Case Studies

4.1. Case Introduction

The case studied in this section is adapted from an OWPS in Zhejiang Province, China (Figure 13), consisting of two offshore wind farms (WFs), both connected to the onshore grid using the high-voltage AC transmission scenario.

The WTs in the target system are all direct-drive type. WF-A includes 32 units of 4.5 MW WTs and 18 units of 4 MW WTs, transmitting power via a 12.5 km three-core submarine cable and 4.6 km single-core land cables. WF-B includes 45 units of 6.25 MW WTs, with power delivered through an 18.34 km three-core submarine cable and 4.6 km single-core land cables. The topologies of the power collection system in WF-A and WF-B are plotted in Figure 14. Two APCs with parameters listed in Table 2 are arranged on the onshore centralized control bus.

The WTs on the same collection chain have identical types, close positions, and similar wind energy capture. Therefore, to facilitate subsequent analysis without loss of generality, each collection chain is simplified by aggregating all WTs into one node, retaining the longer submarine cables at both ends (highlighted in Figure 14) and ignoring the shorter ones between WTs.

In the OWPS, the output of all WTs is set to P = 1.0 pu and Q = 0.0 pu, and the output of all APCs is set to Q = 0.7 pu, with the target frequency range from 1 to 2000 Hz. In terms of analytical calculations, the supplementary control parameters of the APC are designed, and the resonance stability analysis results of the system before and after APC supplementary control are compared to verify the effectiveness of the proposed strategy. In terms of simulation verification, the dominant mode is observed in time-domain simulations to validate the accuracy of the analytical calculations.

4.2. Analytical Calculation

To apply the s-NAM method, each component is depicted as an s-domain impedance model. With all the models converted to the basic voltage level and arranged based on the system topology, the target system can be represented as an s-domain impedance network, further described as an s-NAM [31].

As presented in

Table 3. Resonance modes of the target system.

Mode	APC Before Supplementary Control			APC After Supplementary Control
No.	σres/s−1	fres,s/Hz	ζ	σres/s−1	fres,s/Hz	ζ
1	40.54	81.03	0.0794	40.53	81.03	0.0794
2	16.07	336.80	0.0076	72.94	323.76	0.0358
3	267.98	1296.93	0.0329	71.68	342.73	0.0333
4	372.88	1544.80	0.0384	267.98	1296.93	0.0329
5	/	/	/	372.88	1544.80	0.0384

Note: f_res,s is s-domain resonance frequency; ζ is damping ratio.

, the target system has four stable modes, with Mode 2 having a relatively small damping factor, so the APC supplementary control is designed for this mode.

The impedance of the OWPS is calculated as follows:

$$Z_{\mathrm{OWPS}}={\displaystyle \frac{Z_{\mathrm{WF}}{Z}_{\mathrm{APC}\&\mathrm{T}}}{Z_{\mathrm{WF}}+Z_{\mathrm{APC}\&\mathrm{T}}}}\Rightarrow \left|Z_{\mathrm{OWPS}}\right|={\displaystyle \frac{\left|Z_{\mathrm{WF}}\right|\left|Z_{\mathrm{APC}\&\mathrm{T}}\right|}{\left|Z_{\mathrm{WF}}+Z_{\mathrm{APC}\&\mathrm{T}}\right|}}$$

(63)

where Z_OPWS is the impedance of the OWPS; Z_WF is the impedance of the WFs; Z_APC&T is the impedance of the APC branch. The magnitude of Z_OPWS exhibits a peak at the resonance frequency corresponding to Mode 2 (Figure 15a), which is consistent with the characteristics of the dominant mode analyzed in Section 3.1. The magnitude peak of Z_OPWS is caused by the cancelation between the capacitive reactance of the WFs and the inductive reactance of the APC branch (Figure 15b).

Near the local maximum magnitude frequency, Z_WF exhibits strong resistive and weak reactive characteristics, with significant variation (Figure 15b). If Z_APC&T shows a similar strongly resistive and weakly reactive behavior but with much smaller variation and magnitude than Z_WF, the magnitude of Z_OPWS will stay below that of Z_APC&T over a broad frequency range (Figure 15b), thereby avoiding a pronounced peak in Z_OPWS. This result serves as the objective for designing the supplementary control parameters.

The design principles of the three key supplementary control parameters, K_SC, f_c, and Q, are discussed below. Due to the inductive impedance introduced by the APC branch transformer, |K_SC| must be large enough for the APC capacitive reactance to compensate this inductive component; however, it should not be excessively increased, since it would affect the impedance characteristics in other frequency ranges and, beyond a certain value, the phase-lag effect weakens while the magnitude gain remains unchanged, since the local secondary resistive frequency of Z_APC&T is higher than f_c can be set below the local maximum magnitude frequency of Z_WF. Moreover, Q should be chosen so that Z_APC&T varies as smoothly as possible near the local magnitude frequency of Z_WF. Additionally, both f_c and Q should not be too low, as this would impact the impedance characteristics across other frequency ranges. In summary, the objective function and constraints are given in (64).

$$\begin{array}{l}\min \left|Z_{\mathrm{APC}\&\mathrm{T}}\left(f_{\mathrm{p}}\right)\right|\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}-10°\le \mathrm{Arg}\left[Z_{\mathrm{APC}\&\mathrm{T}}\left(f_{\mathrm{p}}\right)\right]\le 10°\hfill \\ -11.20\le K_{\mathrm{SC}}\le -5.25\hfill \\ 270 \mathrm{Hz}\le f_{\mathrm{c}}\le 330 \mathrm{Hz}\hfill \\ 8\le Q\le 13\hfill \end{array}\right.\end{array}$$

(64)

where f_p is local maximum magnitude frequency of Z_WF.

Finally, K_SC = −11.13, f_c = 277 Hz, and Q = 12 are selected. After applying the supplementary control, in the frequency domain, the local peak magnitude of Z_OPWS is significantly reduced (Figure 15a), and in the s-domain, Mode 2 is replaced by two modes with much larger damping factors (Table 3).

4.3. Simulation Verification

A simulation model of the studied system is built in PSCAD/EMTDC 4.6.2. At 1.45 s, a three-phase grounding fault with a 40 Ω resistance is applied at the grid connection point (node 1), lasting for 0.05s. The (phase) voltages at Node 5, Node 9, and Node 25 are observed shortly after the fault clearing, with their time-domain waveforms and frequency spectra plotted in Figure 16a. It reveals a significant resonance mode with a frequency of approximately 335 Hz, corresponding to Mode 2. After the supplementary control, the component near 335 Hz in the transient voltage significantly decreases (Figure 16b), confirming the accuracy of the analytical calculation.

5. Conclusions

In response to the high resonance risk in power systems with a high proportion of renewable energy, this paper proposes a resonance-suppression strategy based on active APC. The main conclusions are as follows:

• This paper develops the APC impedance model using HSS, incorporating its internal dynamics and full control loops. The power stage and control stage models are linearized separately in HSS, and their integration results in the complete impedance model of the APC.
• Resonance instability originates from the right-half-plane zeros of the s-NAM, while harmonic amplification is caused by the maximum value of the reciprocal of the eigenvalues in the F-NAM. Although distinct, these two concepts are related. For s-domain resonance modes with low damping, the s-domain and frequency-domain resonance frequencies are nearly the same, allowing the problem to be handled in the frequency domain.
• This paper proposes an abc-frame supplementary control in which the output current is fed forward to the valve-side reference voltage through a proportional gain and a BPF. Compared with dq-frame supplementary control, the proposed approach does not require coordinate transformation, thereby avoiding frequency shift effects and enabling a more natural and reliable selection of the characteristic frequency f_c.

Building on the above conclusions, future work will further investigate supplementary control in other frequency ranges and seek validation through field measurements in practical engineering applications, complementing the simulation-based results presented in this study.