Harmonic Analysis and Elimination of Transmission Scheme Based on DRU for Medium-Frequency Offshore Wind Farms

Abstract

Offshore wind farm integration schemes based on diode rectifier unit (DRU) enhance economic efficiency and operational reliability, yet simultaneously introduce considerable harmonic distortion. This paper first establishes an accurate equivalent model of the DRU using a time-domain segmentation analytical method, validated through the time-domain simulations. Subsequently, by integrating the DRU model with the offshore wind farm network, a harmonic calculation method for the point of common coupling (PCC) is proposed. Analysis demonstrates that increasing the fundamental frequency and the number of DRU pulses effectively suppresses harmonic propagation. Furthermore, a structured six-step design methodology for AC filtering and reactive power compensation is introduced. Optimal AC filter configurations are developed for six distinct combinations of fundamental frequencies and DRU structures, providing theoretical foundations and practical guidance for ensuring power quality and enhancing the security of electrical system operations.

1. Introduction

Offshore wind power represents a critical component of the clean energy revolution, offering environmental, economic, and strategic value [1]. With technological advancements and policy support, it is poised to become a central pillar of future energy systems, driving the global transition toward a sustainable future [2,3]. Among various offshore wind farm integration schemes, the medium-frequency collection and direct current (DC) transmission system based on diode rectifier unit (DRU), as shown in Figure 1, has garnered significant attention due to its advantages of low cost and high reliability [4]. However, since this scheme uses DRU as a rectifier, it introduces substantial harmonic issues during operation, posing challenges to power quality and system stability.

The establishment of an accurate harmonic equivalent model serves as the foundation for harmonic analysis, and research in this area has been relatively well-developed. In Reference [5], to address harmonic resonance issues in offshore power grids, the authors developed a harmonic model of wind turbine converters, deriving positive- and negative-sequence impedances for the analysis. By comparing it with an ideal current source model, they achieved higher accuracy. Meanwhile, Reference [6] focused on the impact of disturbances on the AC side of grid-connected converters. Through vector diagram analysis and mathematical derivation, the authors established an equivalent converter model incorporating phase locked loops (PLL) and analyzed resonance instability caused by negative resistance.

There are two fundamental approaches to mitigating harmonic pollution from power electronic devices and other harmonic sources [7]. The first involves installing harmonic compensation devices, which are universally applicable to various harmonic sources. The second entails modifying the harmonic source itself to minimize or eliminate harmonic generation [8].

For the first method, the harmonic compensation device is typically an AC filter, which includes both passive AC filters and active AC filters. The technology of the former is more mature, while the latter exhibits characteristics such as proactivity, flexibility, and high efficiency. Reference [9] proposes a modeling framework for the Digital Power Twin (DPT) and introduces a compensation approach utilizing multilevel Shunt Active Power Filters (SAPFs) along with their adaptive control strategies to address power quality issues such as harmonic distortion and voltage regulation. Since the research objective of this paper focuses on offshore wind farms, where the harmonic characteristics are stable and predominantly composed of low-order harmonics, passive AC filters are better suited to fulfill the compensation requirements. Moreover, due to the high total power capacity of offshore wind farms, the required capacity of AC filters is considerably large. Employing active AC filters in such a scenario would lead to a substantial increase in costs. Therefore, passive AC filters are selected in this study for AC filtering.

The second method can be achieved by altering the structure of DRU [10]. Currently, DRU predominantly uses multi-pulse rectifier circuits, such as 12-pulse and 24-pulse rectifier circuits, with 12-pulse rectifier circuits being the most common. In Reference [11], the authors reviewed multi-pulse rectifier systems, comparing the advantages and disadvantages of 12-pulse, 24-pulse, and 36-pulse rectifiers, with a focus on the DC side of harmonic mitigation techniques. Additionally, Reference [12] theoretically analyzed the reduction in harmonic content when increasing the number of DRU’s pulse from 12 to 24, though no practical cases were studied.

In the domain of AC filtering and reactive power compensation, ensuring power quality and minimizing economic costs are critical factors in determining the configuration of AC filters. Reference [13] proposed a methodology and procedure for calculating the performance and steady-state ratings of capacitor commutated converter (CCC) AC filters, offering a foundational approach to AC filter configuration. Reference [14] introduced a three-stage stochastic robust model applicable to the optimization of renewable energy-based integrated energy systems. The optimization framework presented therein can be adapted to the design of optimal AC filter configuration schemes.

From the perspective of harmonic analysis and elimination of HVDC transmission schemes based on DRU for medium-frequency offshore wind farms, this paper carries out the following systematic investigations:

1.

A time-domain segmentation analytical method is proposed for accurate calculation of harmonic currents on the AC side in DRU;

2.

An equivalent modeling approach is developed for the offshore AC network within offshore wind farm integration systems to facilitate harmonic analysis at node PCC;

3.

A coordinated design methodology for AC filtering and reactive power compensation is introduced, tailored to different fundamental frequencies and DRU configurations.

2. Harmonic Analysis of DRU

This paper introduces the topology and model of DRU configured as a 12-pulse rectifier circuit. The time-domain segmentation analytical method is then used to calculate both the harmonic currents on the AC side and the commutation overlap angle of rectifier circuits. Through calculation of currents on the DC side, an iterative method is applied to enhance the calculation accuracy of currents on the AC side. On this basis, the same analytical method is extended to DRU configured as a 24-pulse rectifier circuit, ultimately establishing a comprehensive equivalent model for DRU. The symbols used in this section and their definitions are summarized in

Table 1. Symbol and definition.

Symbol	Definition
Ld	smoothing reactor
a, b, and c	the three phases
ω	the fundamental angular frequency of the system
n	the harmonic order
h	the highest harmonic order considered
uj(t), (j = a, b, or c)	the three-phase AC voltage at the node PCC
U(n) and U(−n)	the positive- and negative-sequence AC voltage amplitudes at the node PCC
θj(n), (j = a, b, or c)	the phase angles of the three-phase positive-sequence AC voltage on the valve side
θj(−n), (j = a, b, or c)	the phase angles of the three-phase negative-sequence AC voltage on the valve side
ujY(t), (j = a, b, or c)	the three-phase AC voltages on the valve side of the Y-Y converter transformers
ujD(t), (j = a, b, or c)	the three-phase AC voltages on the valve side of the Y-Δ converter transformers
UY(n) and UY(−n)	the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Y converter transformer
UD(n) and UD(−n)	the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Δ converter transformer
αij, (i = 1, 2; j = 1, 2,…,6) or (i = 1, 2, 3, 4; j = 1, 2,…,6)	the phase angle of the j-th bridge arm in the i-th three-phase rectifier circuit
Vij, (i = 1, 2; j = 1, 2,…,6) or (i = 1, 2, 3, 4; j = 1, 2,…,6)	the diode of the j-th bridge arm in the i-th three-phase rectifier circuit
p	the phase entering commutation
r	the phase exiting commutation
q	the phase not participating in the commutation process
uj, (j = p, r, or q)	the voltage of the phase j
ij, (j = p, r, or q)	the current of the phase j
k, (k = 1, 2, 3, 4, 5, 6)	the k-th commutation process and non-commutation process
uk, (k = 1, 2, 3, 4, 5, 6)	the voltage of the phase k
ik, (k = 1, 2, 3, 4, 5, 6)	the current of the phase k
μk	the commutation overlap angle of the k-th commutation process
Ak, Bk	constant
${\dot{U}}_{d(n)}$	the open-circuit harmonic equivalent voltage on the DC side of the n-th harmonic
Zi(n)	the equivalent internal impedance on the DC side of the n-th harmonic
Zd(n)	the impedance on the DC side formed by the smoothing reactor of the n-th harmonic
${\dot{I}}_{d(n)}$	the harmonic current on the DC side of the n-th harmonic
udY	the voltage on the DC side of a three-phase bridge rectifier circuit with Y-Y connected converter transformers
udΔ	the voltage on the DC side for a three-phase bridge rectifier with Y-Δ connected converter transformers
ud_j, (j = 12, or 24)	the resultant voltage on the DC side for the 12-pulse rectifier circuit
Ud_j(0), (j = 12, or 24)	the DC component of the n-th harmonic voltage on the DC side
Ud_j(n), (j = 12, or 24)	the magnitude of the n-th harmonic voltage on the DC side
θud_j(n), (j = 12, or 24)	the phase angle of the n-th harmonic voltage on the DC side
Zi(n)_1	the n-th harmonic during commutation process
Zi(n)_2	the n-th harmonic during non-commutation process
Zi(n)Y	the equivalent internal impedances under the n-th harmonic for the three-phase bridge rectifier circuits with Y-Y connected converter transformers
Zi(n)D	the equivalent internal impedances under the n-th harmonic for the three-phase bridge rectifier circuits with Y-Δ connected converter transformers
Zi(n)	the equivalent internal impedance for the rectifier circuit under the n-th harmonic
Zj, (j = p, r, or q)	the equivalent impedances of the commutation impedances of the phase j under the n-th harmonic
Id_j(0), (j = 12, or 24)	the DC component of the n-th harmonic current on the DC side
Id_j(n), (j = 12, or 24)	the magnitude of the n-th harmonic current on the DC side
θid_j(n), (j = 12, or 24)	the phase angle of the n-th harmonic current on the DC side
id (t)	the current on the DC side
ujYi(t), (j = a, b, or c; i = 1 or 2)	the three-phase AC voltages on the valve side of the Y-Y converter transformers (1 represents the upper, 2 represents the lower)
ujDi(t), (j = a, b, or c; i = 1 or 2)	the three-phase AC voltages on the valve side of the Y-Δ converter transformers (1 represents the upper, 2 represents the lower)
UYi(n) and UYi(−n), (i = 1 or 2)	the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Y converter transformer (1 represents the upper, 2 represents the lower)
UDi(n) and UDi(−n), (i = 1 or 2)	the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Δ converter transformer (1 represents the upper, 2 represents the lower)

.

2.1. Topological Structure of DRU Converter

In the DRU converter, since a smoothing reactor Ld is typically installed at the outlet of the DC side, the output characteristics of DRU resemble those of line commuted converter (LCC) and can be equivalently represented as a current source [15]. This paper investigates two DRU topologies, the 12-pulse and 24-pulse rectifier circuits, as shown in Figure 2. Specifically, Figure 2a depicts the structure of DRU based on a 12-pulse rectifier circuit, where the upper and lower three-phase bridge rectifier circuits are connected to converter transformers with Y-Y and Y-Δ configurations, respectively, distinguished by the subscripts Y and D for electrical quantities. Figure 2b shows the structure of DRU based on a 24-pulse rectifier circuit, where the upper and lower 12-pulse rectifier circuits are symmetrically configured [16]. The phases of voltages and currents of the upper circuit lead those of the lower circuit by π/12 (i.e., 15°), with the electrical quantities distinguished by the subscripts 1 and 2 to distinguish the upper and lower circuit.

The AC side of the DRU rectifier is connected to the node PCC, as shown in Figure 1. In practical power systems, perfectly symmetrical three-phase operation is often unattainable, with the most common asymmetry being background harmonics in the equivalent AC source at the node PCC, such as negative-sequence fundamental harmonics and other harmonics. Therefore, the three-phase AC voltage at the node PCC is expressed as in Equation (1):

$$\left\{\begin{array}{l}{u}_{\mathrm{a}}(t)={\displaystyle \sum _{n=1}^h{U}_{(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}\right)\hfill \\ u_{\mathrm{b}}(t)={\displaystyle \sum _{n=1}^h{U}_{(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}\right)\hfill \\ u_{\mathrm{c}}(t)={\displaystyle \sum _{n=1}^h{U}_{(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}\right)\hfill \end{array}\right.$$

(1)

where a, b, and c represent the three phases; ω is the fundamental angular frequency of the system; n is the harmonic order; and h is the highest harmonic order considered. u_a(t), u_b(t), and u_c(t) represent the three-phase AC voltage at the node PCC, which also serves as the three-phase AC voltage on the grid side for both the Y-Y and Y-Δ converter transformers. U_(n) and U_(−n) represent the positive- and negative-sequence AC voltage amplitudes at the node PCC, respectively. The phase angles of the three-phase AC voltage on the valve side, θ_a(n), θ_b(n), θ_c(n), θ_a(−n), θ_b(−n), and θ_c(−n), satisfy the relationship given in Equation (2):

$$\left\{\begin{array}{l}{\theta }_{\mathrm{a}(n)}={\theta }_{\mathrm{b}(n)}+{\displaystyle \frac{2\pi }{3}}={\theta }_{\mathrm{c}(n)}-{\displaystyle \frac{2\pi }{3}}\hfill \\ {\theta }_{\mathrm{a}(-n)}={\theta }_{\mathrm{b}(-n)}-{\displaystyle \frac{2\pi }{3}}={\theta }_{\mathrm{c}(-n)}+{\displaystyle \frac{2\pi }{3}}\hfill \end{array}\right.$$

(2)

Before establishing the model, this paper proposes the following assumptions to facilitate a more effective analysis of harmonic currents [17]:

• The DC current is treated as a constant when calculating the commutation overlap angle;
• Due to the decoupling effect of the DC system on AC interactions, the electrical quantities on the rectifier side and inverter side are mutually decoupled;
• The reactance in the system is significantly larger than the resistance, allowing the system resistance to be neglected;
• The winding configuration of the converter transformer prevents the zero-sequence current component from being transmitted to the valve side of the converter transformer. Therefore, the AC voltage considers only the positive-sequence and negative-sequence components, with no zero-sequence component.

2.2. Calculation Method of Harmonic Current of DRU Based on 12-Pulse Rectifier Circuit

2.2.1. Calculation of Harmonic Current on the AC Side

Taking the voltage at the node PCC as the reference, the phase of the voltage on the valve side in the Y-Y converter transformer remains identical to that of the node PCC. In contrast, for the Y-Δ converter transformer, the positive-sequence voltage on the valve side lags the voltage of the node PCC by π/6, while the negative-sequence voltage leads the voltage of node PCC by π/6, as expressed in Equations (3) and (4), respectively:

$$\left\{\begin{array}{l}{u}_{\mathrm{aY}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}\right)\hfill \\ u_{\mathrm{bY}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}\right)\hfill \\ u_{\mathrm{cY}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}\right)\hfill \end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{u}_{\mathrm{aD}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}-{\displaystyle \frac{\pi }{6}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}+{\displaystyle \frac{\pi }{6}}\right)\hfill \\ u_{\mathrm{bD}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}-{\displaystyle \frac{\pi }{6}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}+{\displaystyle \frac{\pi }{6}}\right)\hfill \\ u_{\mathrm{cD}}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}-{\displaystyle \frac{\pi }{6}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}+{\displaystyle \frac{\pi }{6}}\right)\hfill \end{array}\right.$$

(4)

where, u_aY(t), u_bY(t), u_cY(t), u_aD(t), u_bD(t), and u_cD(t) represent the three-phase AC voltages on the valve side of the Y-Y and Y-Δ converter transformers, respectively; U_Y(n) and U_Y(−n) represent the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Y converter transformer; while U_D(n) and U_D(−n) correspond to the positive- and negative-sequence AC voltage amplitudes on the valve side of the Y-Δ converter transformer.

In the three-phase bridge rectifier circuit, the operational states can be categorized into commutation and non-commutation processes, determined by the triggering instants of the converter valves. Taking the three-phase bridge rectifier circuit connected to the Y-Y converter transformer as an example, one power-frequency cycle [α₁₁, α₁₁ + 2π] is divided into six commutation intervals and six non-commutation intervals, as detailed in

Table 2. Time-domain interval.

Time-Domain Interval	Converter Valve			Phase Represented by p, r, and q
	Trigger	Close	Conduction	p	r	q
[α11, α11 + μ11]	V11	V15	V16	A	C	B
[α11 + μ11, α12]	/	/	V16, V11	A	/	B
[α12, α12 + μ12]	V12	V16	V11	C	B	A
[α12 + μ12, α13]	/	/	V11, V11	C	/	A
[α13, α13 + μ13]	V13	V11	V12	B	A	C
[α13 + μ13, α14]	/	/	V12, V13	B	/	C
[α14, α14 + μ14]	V14	V12	V13	A	C	B
[α14 + μ14, α15]	/	/	V13, V14	A	/	B
[α15, α15 + μ15]	V15	V13	V14	C	B	A
[α15 + μ15, α16]	/	/	V14, V15	C	/	A
[α16, α16 + μ16]	V16	V14	V15	B	A	C
[α16 + μ16, α11 + 2π]	/	/	V15, V16	B	/	C

. The analysis for the three-phase bridge rectifier circuit connected to the Y-Δ converter transformer follows a similar approach, except that the initial phase angle is delayed by π/6.

During each commutation or non-commutation process, the equivalent circuit topology of the converter remains identical, allowing for unified representation using fixed symbols: the letter p represents the phase entering commutation, r represents the phase exiting commutation, and q represents the phase not participating in the commutation process. The phase assignments for p, r, and q in different time intervals are as shown in Table 2, where ‘/’ stands for not available.

Taking the three-phase bridge rectifier circuit connected to the Y-Y converter transformer in Figure 2a as an example, the equivalent circuits for the k-th commutation process and non-commutation process are shown in Figure 3a and 3b, respectively (k = 1, 2, 3, 4, 5, 6), where the DC side is modeled as an equivalent DC current source.

Based on Kirchhoff’s voltage and current laws, the equations of state and boundary conditions for the k-th commutation process are given by Equations (5) and (6), respectively:

$$\left\{\begin{array}{l}{u}_p(t)-u_r(t)=L_p{\displaystyle \frac{d{i}_p}{dt}}-L_r{\displaystyle \frac{d{i}_r}{dt}}\\ i_p(t)+i_r(t)={(-1)}^{k+1}{I}_d\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}\omega t={\alpha }_k,i_p=0,i_r={(-1)}^{k+1}{I}_d\\ \omega t={\alpha }_k+{\mu }_k,i_p={(-1)}^{k+1}{I}_d,i_r=0\end{array}\right.$$

(6)

Taking the current i_p(t) of phase p as the research object and setting i_k(t) = i_p(t), Equations (5) and (6) can be simplified into the first-order ordinary differential equation and boundary conditions shown in Equations (7) and (8):

$$u_p(t)-u_r(t)=(L_p+L_r){\displaystyle \frac{d{i}_k}{dt}}$$

(7)

$$\left\{\begin{array}{l}\omega t={\alpha }_k,i_k=0\\ \omega t={\alpha }_k+{\mu }_k,i_k={(-1)}^{k+1}{I}_d\end{array}\right.$$

(8)

The general solution to the differential Equation (7) is obtained as follows:

$$\begin{array}{l}{i}_k(t)=A_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n\omega t+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n\omega t+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n\omega t+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n\omega t+{\theta }_{r(-n)}\right)\right]\end{array}$$

(9)

where A_k is a constant.

The boundary conditions from Equation (8) can be substituted into the general solution Equation (9):

$$\begin{array}{l}{i}_k({\displaystyle \frac{{\alpha }_k}{\omega }})=A_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n{\alpha }_k+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n{\alpha }_k+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n{\alpha }_k+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n{\alpha }_k+{\theta }_{r(-n)}\right)\right]=0\end{array}$$

(10)

$$\begin{array}{l}{i}_k({\displaystyle \frac{{\alpha }_k+{\mu }_k}{\omega }})\\ =A_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{r(-n)}\right)\right]={(-1)}^{k+1}{I}_d\end{array}$$

(11)

The constant A_k can be determined from Equation (9), while the commutation overlap angle μ_k can be obtained from Equation (10).

2.2.2. Calculation of Harmonic Current on the DC Side

For precise harmonic current analysis, the influence of harmonic currents on the DC side must be considered. The three-phase bridge rectifier circuit can be equivalently represented as a Thevenin equivalent circuit from the DC side. As for the n-th harmonic, the open-circuit harmonic equivalent voltage on the DC side is ${\dot{U}}_{d(n)}$, with an equivalent internal impedance Z_i(n). The equivalent circuit is shown in Figure 4, where Z_d(n) represents the impedance on the DC side formed by the smoothing reactor, and ${\dot{I}}_{d(n)}$ represents the harmonic current on the DC side.

The voltage on the DC side is calculated by the time-domain segmentation analytical method, which divides the process into commutation and non-commutation. Based on the k-th commutation process shown in Figure 3a, the equations of state can be derived as follows:

$$\left\{\begin{array}{l}{u}_d=u_p(t)-u_{Lp}(t)-(u_q(t)-u_{Lq}(t))\\ u_d=u_r(t)-u_{Lr}(t)-(u_q(t)-u_{Lq}(t))\\ i_p(t)+i_r(t)={(-1)}^{k+1}{I}_d\\ i_q(t)={(-1)}^{k+1}{I}_d\end{array}\right.$$

(12)

After simplification, the expression for the voltage on the DC side u_d during the commutation process can be obtained:

$$u_d={(-1)}^{k+1}\left[{\displaystyle \frac{L_r{u}_p(t)+L_p{u}_r(t)}{L_p+L_r}}-u_q(t)\right],\theta \in \left({\alpha }_k,{\alpha }_k+{\mu }_k\right)$$

(13)

For the k-th non-commutation process shown in Figure 3b, the voltage on the DC side u_d during the non-commutation process is derived as follows:

$$u_d={(-1)}^{k+1}\left[u_p(t)-u_q(t)\right],\theta \in \left({\alpha }_k+{\mu }_k,{\alpha }_{k+1}\right)$$

(14)

In summary, the complete cycle expression for the voltage on the DC side of a three-phase bridge rectifier circuit with Y-Y connected converter transformers, u_dY, can be obtained as follows:

$$u_{d\mathrm{Y}}=\left\{\begin{array}{l}{(-1)}^{k+1}\left[{\displaystyle \frac{L_r{u}_p(t)+L_p{u}_r(t)}{L_p+L_r}}-u_q(t)\right],\theta \in \left({\alpha }_k,{\alpha }_k+{\mu }_k\right)\\ {(-1)}^{k+1}\left[u_p(t)-u_q(t)\right],\theta \in \left({\alpha }_k+{\mu }_k,{\alpha }_{k+1}\right)\end{array}\right.,k=1,2,3,4,5,6$$

(15)

Similarly, the voltage on the DC side for a three-phase bridge rectifier with Y-Δ connected converter transformers, u_dΔ, can be derived. By superimposing the voltages on the DC side u_dY and u_dΔ, the resultant voltage on the DC side for the 12-pulse rectifier circuit, u_{d_12}, is obtained. Through Fourier decomposition, the complete time-domain expression of u_{d_12} is given by Equation (16), which represents the equivalent harmonic voltage source on the DC side.

$$u_{d\_12}(t)=U_{d\_12(0)}+{\displaystyle \sum _{n=1}^h{U}_{d\_12(n)}}\sin \left(n\omega t+{\theta }_{ud\_12(n)}\right)$$

(16)

where U_{d_12(0)} represents the DC component, while U_{d_12(n)} and θ_{ud_12(n)} represent the magnitude and phase angle, respectively, of the n-th harmonic voltage on the DC side.

For the Thevenin equivalent circuit on the DC side shown in Figure 4, the equivalent impedance Z_i(n) is calculated using the method for three-phase bridge rectifier circuits described in Reference [18]. Based on Figure 3a,b, the equivalent internal impedances Z_{i(n)_1} and Z_{i(n)_2} under the n-th harmonic during commutation and non-commutation process can be calculated using Equation (17):

$$\left\{\begin{array}{l}{Z}_{i(n)\_1}={\displaystyle \frac{Z_p{Z}_r}{Z_p+Z_r}}+Z_q\\ Z_{i(n)\_2}=Z_p+Z_q\end{array}\right.$$

(17)

where Z_p, Z_r, and Z_q represent the equivalent impedances of the commutation impedances under the n-th harmonic, respectively. The equivalent internal impedance Z_i(n) for the n-th harmonic is obtained by calculating the time-weighted average of the impedances during each commutation and non-commutation process over one complete cycle.

$$Z_{i(n)}={\displaystyle \sum _{k=1}^6\frac{1}{2\pi }\left[{\displaystyle {\int }_{{\alpha }_k}^{{\alpha }_k+{\mu }_k}{Z}_{i(n)\_1}d\theta }+{\displaystyle {\int }_{{\alpha }_k+{\mu }_k}^{{\alpha }_{k+1}}{Z}_{i(n)\_2}d\theta }\right]}$$

(18)

Z_i(n)Y and Z_i(n)D represent the equivalent internal impedances under the n-th harmonic for the three-phase bridge rectifier circuits with Y-Y and Y-Δ connected converter transformers, respectively. Then, the equivalent internal impedance Z_i(n) for the 12-pulse rectifier circuit under the n-th harmonic is obtained as the sum of these two components, expressed as follows:

$$Z_{i(n)}=Z_{i(n)Y}+Z_{i(n)D}$$

(19)

According to Figure 4, the harmonic current on the DC side can be derived as follows:

$${\dot{I}}_{\mathrm{d}(n)}={\displaystyle \frac{{\dot{U}}_{\mathrm{d}(n)}/\sqrt{2}}{Z_{\mathrm{c}(n)}+Z_{\mathrm{d}(n)}}}={\displaystyle \frac{I_{\mathrm{d}(n)}}{\sqrt{2}}}\angle {\theta }_{\mathrm{id}(n)}$$

(20)

where I_{d_12(0)} represents the DC component, while I_{d_12(n)} and θ_{id_12(n)} represent the magnitude and phase angle, respectively, of the n-th harmonic current on the DC side.

Consequently, the current on the DC side can be mathematically expressed as follows:

$$i_d(t)=I_{d(0)}+{\displaystyle \sum _{n=2}^h{I}_{d(n)}\sin (n\omega t+{\theta }_{id(n)})}$$

(21)

2.2.3. Iterative Calculation of Harmonic Current on the AC Side

To account for the influence of harmonic currents on the DC side on harmonic currents on the AC side, this paper revisits the solution for the k-th commutation process in Figure 3a. By substituting Equation (21) into Equations (5) and (6), we obtain the following:

$$\left\{\begin{array}{l}{u}_p(t)-u_r(t)=L_p{\displaystyle \frac{d{i}_p}{dt}}-L_r{\displaystyle \frac{d{i}_r}{dt}}\\ i_p(t)+i_r(t)={(-1)}^{k+1}{i}_d(t)\end{array}\right.$$

(22)

$$\left\{\begin{array}{l}\omega t={\alpha }_k,i_p=0,i_r={(-1)}^{k+1}{i}_d(t)\\ \omega t={\alpha }_k+{\mu }_k,i_p={(-1)}^{k+1}{i}_d(t),i_r=0\end{array}\right.$$

(23)

Setting i_k(t) = i_p(t), the following expression is derived:

$$u_p(t)-u_r(t)+L_r{\displaystyle \frac{d{(-1)}^{k+1}{i}_d}{dt}}=(L_p+L_r){\displaystyle \frac{d{i}_k}{dt}}$$

(24)

$$\left\{\begin{array}{l}\omega t={\alpha }_k,i_k=0\\ \omega t={\alpha }_k+{\mu }_k,i_k={(-1)}^{k+1}{i}_d({\displaystyle \frac{{\alpha }_k+{\mu }_k}{\omega }})\end{array}\right.$$

(25)

To solve the differential Equation (24), its general solution can be obtained as follows:

$$\begin{array}{l}{i}_k(t)=B_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n\omega t+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n\omega t+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n\omega t+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n\omega t+{\theta }_{r(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+\underset{\mathrm{The} \mathrm{term} \mathrm{added} \mathrm{after} \mathrm{considering} \mathrm{the} \mathrm{DC} \mathrm{current}}{\underbrace{{\displaystyle \frac{L_r}{L_p+L_r}}\cdot {(-1)}^{k+1}[I_{d(0)}+{\displaystyle \sum _{n=2}^h{I}_{d(n)}\sin (n\omega t+{\theta }_{id(n)})}]}}\end{array}$$

(26)

where B_k is a constant.

By applying the boundary conditions specified in Equation (25) to Equation (26), the following solution can be derived:

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{i}_k({\displaystyle \frac{{\alpha }_k}{\omega }})\\ =B_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n{\alpha }_k+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n{\alpha }_k+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n{\alpha }_k+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n{\alpha }_k+{\theta }_{r(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{L_r}{L_p+L_r}}\cdot {(-1)}^{k+1}[I_{d(0)}+{\displaystyle \sum _{n=2}^h{I}_{d(n)}\sin (n{\alpha }_k+{\theta }_{id(n)})}]\\ =0\end{array}$$

(27)

$$\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{i}_k({\displaystyle \frac{{\alpha }_k+{\mu }_k}{\omega }})\\ =B_k-{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{p(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{p(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{1}{L_p+L_r}}{\displaystyle \sum _{n=1}^h\frac{1}{n\omega }}\left[U_{\mathrm{Y}(n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{r(n)}\right)+U_{\mathrm{Y}(-n)}\cos \left(n({\alpha }_k+{\mu }_k)+{\theta }_{r(-n)}\right)\right]\\ \hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{L_r}{L_p+L_r}}\cdot {(-1)}^{k+1}[I_{d(0)}+{\displaystyle \sum _{n=2}^h{I}_{d(n)}\sin (n({\alpha }_k+{\mu }_k)+{\theta }_{id(n)})}]\\ ={(-1)}^{k+1}[I_{d(0)}+{\displaystyle \sum _{n=2}^h{I}_{d(n)}\sin (n({\alpha }_k+{\mu }_k)+{\theta }_{id(n)})}]\end{array}$$

(28)

The constant B_k can be determined from Equation (27), while the commutation overlap angle μ_k can be obtained from Equation (28). The number of iterations can be set according to the required computational accuracy.

Based on the equivalent circuit of the commutation process shown in Figure 3a, the three-phase harmonic currents on the AC side can be expressed as follows:

$$\left\{\begin{array}{l}{i}_p(t)=i_k(t)\hfill \\ i_r(t)={(-1)}^{k+1}{i}_{\mathrm{d}}(t)-i_p(t)\hfill \\ i_q(t)=-{(-1)}^{k+1}{i}_{\mathrm{d}}(t)\hfill \end{array}\right.$$

(29)

Based on the equivalent circuit of the non-commutation process shown in Figure 3b, the three-phase harmonic currents on the AC side can be mathematically expressed as follows:

$$\left\{\begin{array}{l}{i}_p(t)={(-1)}^{k+1}{i}_{\mathrm{d}}(t)\hfill \\ i_r(t)=0\hfill \\ i_q(t)=-{(-1)}^{k+1}{i}_{\mathrm{d}}(t)\hfill \end{array}\right.$$

(30)

By utilizing the time-domain segmentation intervals defined in Table 2, the harmonic currents on the converter valve side can be precisely calculated. These currents can then be converted to the grid side through the transformation ratio of the converter transformer. Following Fourier decomposition, harmonic current components with identical orders from both Y-Y and Y-Δ converter configurations are vectorially superimposed, ultimately yielding the resultant harmonic current values on the AC side. Increasing the number of iterations enhances the accuracy of the calculated values of harmonic currents on the AC side. At this stage, the computational flow diagram for determining harmonic currents on the AC side of DRU is shown in Figure 5, where the number of iterations is denoted as m.

2.3. Calculation of DRU Harmonic Current Based on 24-Pulse Rectifier Circuit

The 24-pulse rectifier circuit is formed by connecting two 12-pulse rectifier circuits in parallel on the AC side and in series on the DC side. Therefore, the fundamental principles of the circuit are the same as those of the 12-pulse rectifier circuit. While the three-phase AC voltage at the node PCC still follows Equation (1), in the upper 12-pulse rectifier circuit, the positive-sequence voltage phase on the valve side of the Y-Y converter transformer leads the node PCC voltage by π/24, while the negative-sequence voltage phase lags the node PCC voltage by π/24. Conversely, the positive-sequence voltage phase on the valve side of the Y-Δ converter transformer lags the node PCC voltage by π/8, while the negative-sequence voltage phase leads the node PCC voltage by π/8. In the lower 12-pulse rectifier circuit, the positive-sequence voltage phase on the valve side of the Y-Y converter transformer lags the node PCC voltage by π/24, while the negative-sequence voltage phase leads the node PCC voltage by π/24. Meanwhile, the positive-sequence voltage phase on the valve side of the Y-Δ converter transformer lags the node PCC voltage by 5π/24, while the negative-sequence voltage phase leads the node PCC voltage by 5π/24. The valve-side voltages of each converter transformer are given by Equation (31) to (32), as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{aY}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}+{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}-{\displaystyle \frac{\pi }{24}}\right)\hfill \\ u_{\mathrm{bY}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}+{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}-{\displaystyle \frac{\pi }{24}}\right)\hfill \\ u_{\mathrm{cY}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}+{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}-{\displaystyle \frac{\pi }{24}}\right)\hfill \end{array}\right.$$

(31)

$$\left\{\begin{array}{l}{u}_{\mathrm{aD}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}-{\displaystyle \frac{\pi }{8}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}+{\displaystyle \frac{\pi }{8}}\right)\hfill \\ u_{\mathrm{bD}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}-{\displaystyle \frac{\pi }{8}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}+{\displaystyle \frac{\pi }{8}}\right)\hfill \\ u_{\mathrm{cD}1}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}-{\displaystyle \frac{\pi }{8}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}1(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}+{\displaystyle \frac{\pi }{8}}\right)\hfill \end{array}\right.$$

(32)

$$\left\{\begin{array}{l}{u}_{\mathrm{aY}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}-{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}+{\displaystyle \frac{\pi }{24}}\right)\hfill \\ u_{\mathrm{bY}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}-{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}+{\displaystyle \frac{\pi }{24}}\right)\hfill \\ u_{\mathrm{cY}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}-{\displaystyle \frac{\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{Y}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}+{\displaystyle \frac{\pi }{24}}\right)\hfill \end{array}\right.$$

(33)

$$\left\{\begin{array}{l}{u}_{\mathrm{aD}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(n)}-{\displaystyle \frac{5\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{a}(-n)}+{\displaystyle \frac{5\pi }{24}}\right)\hfill \\ u_{\mathrm{bD}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(n)}-{\displaystyle \frac{5\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{b}(-n)}+{\displaystyle \frac{5\pi }{24}}\right)\hfill \\ u_{\mathrm{cD}2}(t)={\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(n)}-{\displaystyle \frac{5\pi }{24}}\right)+{\displaystyle \sum _{n=1}^h{U}_{\mathrm{D}2(-n)}}\sin \left(n\omega t+{\theta }_{\mathrm{c}(-n)}+{\displaystyle \frac{5\pi }{24}}\right)\hfill \end{array}\right.$$

(34)

where u_aY1(t), u_bY1(t), u_cY1(t), u_aD1(t), u_bD1(t), and u_cD1(t) represent the valve-side, three-phase AC voltages of the Y-Y converter transformer and the Y-Δ converter transformer, respectively, in the upper 12-pulse rectifier circuit. U_Y1(n) and U_Y1(−n) represent the magnitudes of the positive- and negative-sequence AC voltages on the valve side of the Y-Y converter transformer in the upper 12-pulse rectifier circuit, while U_D1(n) and U_D1(−n) represent the corresponding magnitudes for the Y-Δ converter transformer in the same circuit. By changing the subscript “1” to “2” for the aforementioned electrical quantities, the corresponding quantities for the lower 12-pulse rectifier circuit can be obtained.

The conduction sequence of the three-phase bridge rectifier circuits is as follows: the three-phase bridge rectifier is connected to the Y-Y converter transformer in the upper circuit, followed by the three-phase bridge rectifier connected to the Y-Y converter transformer in the lower circuit; then, the three-phase bridge rectifier is connected to the Y-Δ converter transformer in the upper circuit, and finally, the three-phase bridge rectifier is connected to the Y-Δ converter transformer in the lower circuit.

In the 24-pulse rectifier circuit, the preliminary calculation of the commutation overlap angle is identical to that described in Section 2.2. For the equivalent model on the DC side, the equivalent circuit diagram is the same as Figure 4, with the key difference being that the final 24-pulse-based DC voltage u_{d_24} is the superposition of four voltages on the DC side: u_dY1, u_dD1, u_dY2, and u_dD2. By performing Fourier decomposition on the voltage u_{d_24}, the full time-domain expression of u_{d_24} is obtained as shown in Equation (35), which serves as the equivalent harmonic voltage source on the DC side.

$$u_{d\_24}(t)=U_{d\_24(0)}+{\displaystyle \sum _{n=1}^h{U}_{d\_24(n)}}\sin \left(n\omega t+{\theta }_{ud\_24(n)}\right)$$

(35)

where U_{d_24(0)} is the DC component, while U_{d_24(n)} and θ_{ud_24(n)} represent the magnitude and phase angle, respectively, of the n-th DC harmonic voltage.

Similarly, the equivalent internal impedance under the n-th harmonic for the 24-pulse rectifier circuit, Z_i(n), is the sum of the equivalent internal impedances of the four three-phase bridge rectifier circuits:

$$Z_{i(n)}=Z_{i(n)Y1}+Z_{i(n)D1}+Z_{i(n)Y2}+Z_{i(n)D2}$$

(36)

where Z_i(n)Y1, Z_i(n)D1, Z_i(n)Y2, and Z_i(n)D2 represent the equivalent internal impedances of the upper Y-Y, upper Y-Δ, lower Y-Y, and lower Y-Δ converter transformer-connected rectifier circuits, respectively, at the n-th harmonic frequency.

Consequently, the current on the DC side can be determined, and by further using the iterative calculation method for harmonic currents on the AC side and commutation overlap angles described in Section 2.2, more precise values of the harmonic currents on the AC side can be obtained.

2.4. Analysis of Examples

This paper compares the time-domain segmentation analytical method with the time-domain simulation method to validate the computational accuracy of the former. The harmonic calculations using the time-domain simulation method are implemented through the electromagnetic transient simulation software PSCAD/EMTDC V5.0.

This paper takes a DRU rectifier station transmitting 1000 MW of active power, with a DC voltage of 640 kV and an AC voltage of 66 kV at the node PCC, as the calculation example. The DRU employs 12-pulse and 24-pulse rectifier circuits, with a per-unit leakage reactance of 0.1 for the converter transformer. In the calculations, the values of harmonic currents are independent of the fundamental frequency.

The characteristic values of the AC current harmonics and the Total Harmonic Distortion (THD) obtained from the above two methods are compared. The calculation results of harmonic currents for DRU based on 12-pulse and 24-pulse rectifier circuits are presented in

Table 3. Computational results of harmonic currents for DRU based on 12-pulse and 24-pulse rectifier circuits.

Order of Harmonic	DRU Based on 12-Pulse Rectifier Circuit		DRU Based on 24-Pulse Rectifier Circuit
	Time-Domain Segmented Analytical Method	Time-Domain Simulation Method	Time-Domain Segmented Analytical Method	Time-Domain Simulation Method
11	795.90	797.66	/	/
13	566.80	567.74	/	/
23	250.32	250.83	258.40	259.40
25	168.08	168.31	171.64	172.10
35	104.16	104.56	/	/
37	79.60	78.93	/	/
47	56.52	56.08	58.96	58.96
49	46.08	46.35	47.12	47.30
THD	10.96%	10.98%	3.39%	3.40%

, where ‘/’ stands for not available.

As evidenced by Table 3, under ideal conditions, the computational results obtained from the time-domain segmentation analytical method proposed in this study demonstrate close segmentation with those derived from time-domain simulation method. Consequently, the time-domain segmentation analytical method proves to be a viable and effective approach for calculating current harmonics on the AC side.

3. Calculation and Elimination of Harmonic of Offshore Wind Farm Integration Schemes

3.1. An Equivalent Model of Offshore Wind Farms for Harmonic Analysis

The offshore wind power transmission system based on DRU employs high-voltage direct current (HVDC) for power transmission. The DC segment effectively isolates the offshore AC system from the onshore AC grid, consequently focusing harmonic analysis primarily on the offshore AC system. The offshore AC system principally comprises wind turbines, submarine collector cables, AC harmonic filters, and DRU. In this study, the offshore wind farm adopts a radial configuration consisting of multiple wind turbine strings. Each sub wind farm connects to the node PCC before being integrated with the DRU through converter transformers, as illustrated in the offshore wind farm section of Figure 6a.

For the wind turbines, since the DRU lacks active commutation capability and cannot provide voltage support, this study employs grid-forming wind turbines. The equivalent model for the grid-side converter (GSC) and its downstream components is illustrated in Figure 6b. The GSC can be represented as a Norton equivalent circuit comprising a harmonic current source I_h(n) in parallel with a harmonic impedance Z_w(n). The wind turbine transformer is modeled using a π-equivalent circuit, and its equivalent impedance Z_Tπ(n) and shunt admittances Y_Tπ1(n) and Y_Tπ2(n) are all frequency-dependent parameters expressed as functions of fundamental frequency and harmonic order.

For the submarine cables, this study selects three-core armored submarine cables and employs the methodology described in Reference [19] to calculate the per-unit-length series impedance and shunt admittance at specific frequencies. The cable line equivalent model adopts a π-type equivalent circuit with hyperbolic function correction, as shown in Figure 6b. The calculation formulas for the relevant parameters in Figure 6b are given by Equation (37).

$$\left\{\begin{array}{l}Z={\displaystyle \frac{\sinh \gamma l}{\gamma l}}\cdot (r+jx)l\\ Y={\displaystyle \frac{\tanh (\gamma l/2)}{\gamma l/2}}\cdot (g+jb)l\\ \gamma =\sqrt{(r+jx)(g+jb)}\end{array}\right.$$

(37)

where the parameters r, x, g, and b represent the per-unit-length resistance, reactance, conductance, and susceptance of the submarine cable, respectively, while l denotes the total cable length. The propagation coefficient γ (dimension: km⁻¹) characterizes the wave propagation characteristics along the transmission line.

The equivalent structural diagram of the offshore wind farm case study is shown in Figure 7. By sequentially numbering each wind turbine and transmission line node, the offshore wind farm is transformed into an AC power network comprising 256 nodes, where the node PCC is designated as Node 256.

The nodal admittance matrix exhibits frequency-dependent characteristics. For the n-th harmonic component, the admittance matrix is denoted as Y_(n). Taking the ground as the reference node, the n-dimensional nodal voltage equation can be expressed as follows:

$$Y_{(n)}{\dot{U}}_{(n)}={\dot{I}}_{(n)}$$

(38)

where ${\dot{U}}_{(n)}$ represents the column vector composed of nodal voltages, while ${\dot{I}}_{(n)}$ represents the column vector of nodal current injections. Based on the AC power network configuration shown in Figure 7, a 256 × 256 nodal admittance matrix can be constructed, which serves as the foundation for calculating the Thevenin harmonic equivalent model parameters of the offshore wind farm.

3.2. Claculation of Harmonics

The AC filter is connected at the grid side of the converter transformer in the DRU rectifier station to effectively mitigate harmonic distortion and provide reactive power compensation. The equivalent circuit diagram of this configuration is presented in Figure 8 [20].

In Figure 8, the dashed box represents the AC filter with an impedance of Z_F(n). The harmonic voltage ${\dot{U}}_{PCC(n)}$ and harmonic current ${\dot{I}}_{PCC(n)}$ at the filter’s connection point to the AC system result from the superposition of two components: the harmonic contribution from the DRU’s equivalent harmonic current source and the Thevenin equivalent model of the offshore wind farm.

The harmonic voltage ${\dot{U}}_{PCC(n)}$ and current ${\dot{I}}_{PCC(n)}$ generated by the DRU’s equivalent harmonic current source and the Thevenin equivalent model of the offshore wind farm can be calculated using Equations (39) and (40), respectively:

$$\left\{\begin{array}{l}{\dot{U}}_{PCC(n)\_1}=\sqrt{3}\cdot {\displaystyle \frac{Z_{S(n)}\cdot Z_{F(n)}}{Z_{S(n)}+Z_{F(n)}}}{\dot{I}}_{DRU(n)}\\ {\dot{I}}_{PCC(n)\_1}={\displaystyle \frac{Z_{S(n)}}{Z_{S(n)}+Z_{F(n)}}}{\dot{I}}_{DRU(n)}\end{array}\right.$$

(39)

$$\left\{\begin{array}{l}{\dot{U}}_{PCC(n)\_2}={\displaystyle \frac{Z_{F(n)}}{Z_{S(n)}+Z_{F(n)}}}{\dot{U}}_{S(n)}\\ {\dot{I}}_{PCC(n)\_2}={\displaystyle \frac{{\dot{U}}_{S(n)}/\sqrt{3}}{Z_{S(n)}+Z_{F(n)}}}\end{array}\right.$$

(40)

The harmonic voltages and currents induced by the DRU equivalent harmonic current source on the AC filter, along with those generated by the Thevenin equivalent model of the offshore wind farm, can be analytically synthesized through Equation (41):

$$\left\{\begin{array}{l}{U}_{PCC(n)}=\sqrt{U_{PCC(n)\_1}^2+U_{PCC(n)\_2}^2+k\cdot U_{PCC(n)\_1}\cdot U_{PCC(n)\_2}}\\ I_{PCC(n)}=\sqrt{I_{PCC(n)\_1}^2+I_{PCC(n)\_2}^2+k\cdot I_{PCC(n)\_1}\cdot I_{PCC(n)\_2}}\end{array}\right.$$

(41)

The coefficient k in Equation (41) is an empirical parameter, the values of which are determined based on operational experience from existing HVDC projects in China. As summarized in

Table 4. Value of k.

Order of Harmonics	3	5	7	Greater than or Equal to 9 and All Even Harmonics
Value of k	1.62	1.28	0.72	0

, the recommended values for k are as follows [21]:

For harmonic calculations at the node PCC when AC filters are not installed, the analysis can be simplified by removing the AC filter branch from the equivalent circuit (Figure 8) and applying the superposition theorem with Equation (41). The harmonic voltage ${\dot{U}}_{PCC(n)}$ and current ${\dot{I}}_{PCC(n)}$ generated by the DRU’s equivalent harmonic current source and the Thevenin equivalent model of the offshore wind farm can be calculated using Equations (42) and (43), respectively:

$$\left\{\begin{array}{l}{\dot{U}}_{PCC(n)\_1}=\sqrt{3}\cdot Z_{S(n)}{\dot{I}}_{DRU(n)}\\ {\dot{I}}_{PCC(n)\_1}={\dot{I}}_{DRU(n)}\end{array}\right.$$

(42)

$$\left\{\begin{array}{l}{\dot{U}}_{PCC(n)\_2}={\dot{U}}_{S(n)}\\ {\dot{I}}_{PCC(n)\_2}={\displaystyle \frac{{\dot{U}}_{S(n)}}{Z_{S(n)}}}\end{array}\right.$$

(43)

3.3. Design Methodology for AC Filtering and Reactive Power Compensation

This section presents a systematic approach for designing AC filters and reactive power compensation systems in offshore converter stations, comprising six key steps:

1.

Calculate the harmonic voltage without AC filters;

2.

Determine the most severe harmonic scenario for a given frequency and DRU configuration;

3.

Calculate the required reactive power compensation;

4.

Design the configurations of AC filters;

5.

Calculate the harmonic voltage with AC filters;

6.

Determine the optimal filter design by minimizing reactive power compensation requirements and selecting the most cost-effective AC filter arrangement.

Steps 1 and 5 have been comprehensively detailed in Section 3.2, and the remaining steps are elaborated in this section.

The basis for determining the most severe harmonic conditions under a certain frequency and DRU structure is the power quality and the harmonic interference index of the AC system. According to Reference [22], the power quality of the 66 kV power grid and the harmonic interference index of the AC system are generally set for the voltage, as shown in

Table 5. Power quality of the 66 kV power grid and the harmonic interference index of the AC system.

Index		Value
Single Harmonic Distortion Rate (SHD)	Odd	2.4%
	Even	1.2%
THD		3.0%
Telephone Harmonic Form Factor (THFF)		1.0%

.

The reactive power capacity of AC filters must first be determined based on the reactive power consumption of DRU. According to Reference [23], the maximum commutation overlap angle is selected to calculate the power factor angle (φ) on the AC side of the DRU, thereby determining the reactive power demand of the DRU under different fundamental frequencies in the actual case study. This value corresponds to the total reactive power compensation requirement of the AC filters.

As shown in Figure 9, the filtering system in this study comprises shunt capacitors, single-tuned filters, and double-tuned filters, where the double-tuned filter can be equivalently modeled as two single-tuned filters connected in parallel. The parameters of the shunt capacitor banks are determined based on their rated capacity and voltage level, while the design of both single-tuned and double-tuned filters must consider their targeted harmonic orders. Priority should be given to selecting harmonic frequencies corresponding to the converter’s pulse number while simultaneously addressing adjacent characteristic harmonics.

For instance: in a 12-pulse rectifier-based DRU system, the single-tuned filter is designed for the 12th harmonic, and the double-tuned filter is configured for both 12th and 24th harmonics; in a 24-pulse rectifier-based DRU system, the single-tuned filter targets the 24th harmonic, and the double-tuned filter addresses both 24th and 48th harmonics.

Based on this methodology, AC filter configurations are systematically developed by integrating these diverse filtering components to achieve optimal harmonic suppression and reactive power compensation.

In this paper, the calculated value Q_DRU is taken as the initial reactive power compensation amount, and the reactive power compensation amount Q is changed with ΔQ as the step size to calculate the harmonics after installing the AC filter respectively. Under each reactive power compensation quantity Q, calculate the installation results of each AC filter scheme, and select the scheme that meets the power quality standards from them. Statistically analyze all the above schemes, and give priority to the one with a smaller reactive power compensation amount. On this basis, further select the scheme with the lowest cost. The flow diagram of the AC filtering and reactive power compensation design for the offshore converter station is shown in Figure 10.

4. Analysis of Examples

This paper systematically computes and comparatively analyzes the impacts of different variables on harmonic characteristics in offshore wind power transmission systems. Three critical variables are investigated:

• DRU power level: evaluated from 10% to 100% of rated power with 10% increments;
• Fundamental frequency: 50 Hz, 100 Hz, and 150 Hz;
• DRU topology: 12-pulse and 24-pulse rectifier circuit.

4.1. Key Parameters of the Offshore Wind Power Transmission System

The key parameters of the DRU-based medium-frequency gathering DC transmission system for offshore wind power with a rated transmission power of 1000 MW are shown in

Table 6. The main parameters of the transmission scheme for offshore wind farms.

Equipment	Parameter		Value
Wind turbines	Active power of one wind turbine (MW)		12.5
	Number of wind turbines (Number of clusters × number of wind turbines in one cluster)		85 (17 × 5)
	Rated frequency (Hz)		150
	Rated wind speed (ms−1)		12
	Rated terminal voltage (kV)		66
	Equivalent inductance (mH)		0.0125
	Equivalent capacitance (F)		2.0052 × 10−3
Transformer of wind turbine	Rated capacity (MVA)		13.8
	Ratio (kV)		0.69/66
	Leakage reactance of positive sequence (p.u.)		0.15
Converter transformer on the rectifier side of DRU	12-pulse rectifier circuit	Rated capacity (MVA)	2 × 550
		Ratio (kV)	66/236.95
		Leakage reactance of positive sequence (p.u.)	0.15
	24-pulse rectifier circuit	Rated capacity (MVA)	4 × 275
		Ratio (kV)	66/118.48
		Leakage reactance of positive sequence (p.u.)	0.15
HVDC	Rated DC voltage (kV)		±320
	Rated DC current (A)		1562.5
MMC	Rated capacity (MVA)		1100
	Number of submodules of the bridge arm		314
	Capacitance of submodule (mF)		9.71
	Reactance of bridge arm (mH)		52.70
	Equivalent capacity discharging time constant (ms)		40
Converter transformer on the inverter side of MMC	Rated capacity (MVA)		2 × 550
	Ratio (kV)		236.95/220
	Leakage reactance of positive sequence (p.u.)		0.15
Onshore AC power grid	Rated voltage (kV)		220

.

4.2. Harmonic Calculation and Analysis

The node PCC is the connection point between the offshore wind farm and the DRU rectifier station, and it is the voltage central point of the offshore AC part of the offshore wind power transmission system. Therefore, this section conducts harmonic calculation and analysis for the node PCC.

Under the same fundamental frequency and DRU structure, the harmonic voltages of node PCC at different power levels are analyzed by taking the combination of the offshore wind power transmission system of “an offshore wind farm with a fundamental frequency of 100 Hz + DRU based on a 12-pulse rectifier circuit” as an example, as shown in Figure 11. The relevant indicators are presented in

Table 7. THD and THFF under different power levels.

Power Level	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%
THD (%)	0.49	0.96	1.53	2.22	3.07	4.11	5.35	6.79	7.93	9.33
THFF (%)	0.56	0.97	1.29	1.54	1.71	1.81	1.85	1.87	1.88	1.89

.

It can be seen from Figure 11 that the harmonic voltage is mainly concentrated in the characteristic harmonics and the low-order non-characteristic harmonics from the second to the eighth, and its effective value basically conforms to the law of being proportional to the power level. It can be seen from Table 7 that the greater the power level, the greater the harmonic voltage, THD, and THFF. Therefore, the harmonic condition at the rated power is the most serious.

At the rated power, the harmonic voltages and currents at the node PCC of the offshore wind power transmission system with different fundamental frequencies and DRU structures are shown in Figure 12 and Figure 13, respectively, and the relevant indicators are presented in

Table 8. THD and THFF under different fundamental frequencies and DRU structures without AC filters.

Index	DRU Based on 12-Pulse Rectifier Circuit			DRU Based on 24-Pulse Rectifier Circuit
	50 Hz	100 Hz	150 Hz	50 Hz	100 Hz	150 Hz
THD (%)	0.49	0.96	1.53	2.22	3.07	4.11
THFF (%)	0.56	0.97	1.29	1.54	1.71	1.81

.

It can be known from Figure 13 that the smaller the fundamental frequency is, the larger the effective value of non-characteristic harmonics is, and the peaks all occur at 400–450 Hz. The increase in the number of DRU pulses leads to a significant reduction in the content of characteristic harmonics. It can be known from Table 8 that the increase in the fundamental frequency and the increase in the number of DRU pulses have led to a significant decrease in both THD and THFF.

4.3. Design Methodology for AC Filtering and Reactive Power Compensation

According to the process shown in Figure 10, the AC filtering and reactive power compensation designs are carried out for six combinations of different fundamental frequencies and DRU structures at the rated power. The calculation results of the maximum commutation overlap angle and reactive power of DRU under different fundamental frequencies and structures are shown in

Table 9. Maximum commutation overlap angle and reactive power of DRU under different fundamental frequencies and structures.

Index	DRU Based on 12-Pulse Rectifier Circuit			DRU Based on 24-Pulse Rectifier Circuit
	50 Hz	100 Hz	150 Hz	50 Hz	100 Hz	150 Hz
Maximum of commutation overlap angle (°)	33.8950	32.5910	32.0021	32.1986	32.0043	31.8315
Reactive power (%)	413.75	396.36	388.57	391.17	388.60	386.32

.

Let ΔQ = 20 Mvar. A total of four groups of AC filters are configured, each with the same capacity. The AC filters include double-tuned filters, single-tuned filters, and parallel capacitors. From this, 15 AC filter configuration schemes with costs ranging from high to low can be obtained, as shown in

Table 10. Fifteen AC filter configuration schemes with costs ranging from high to low.

Number	Scheme	Number	Scheme	Number	Scheme
1	4 × DTF	6	2 × DTF + 2 × PFC	11	4 × STF
2	3 × DTF + 1 × STF	7	1 × DTF + 3 × STF	12	3 × STF + 1 × PFC
3	3 × DTF + 1 × PFC	8	1 × DTF + 2 × STF + 1 × PFC	13	2 × STF + 2 × PFC
4	2 × DTF + 2 × STF	9	1 × DTF + 1 × STF + 2 × PFC	14	1 × STF + 3 × PFC
5	2 × DTF + 1 × STF + 1 × PFC	10	1 × DTF + 3 × PFC	15	4 × PFC

.

The configuration schemes of AC filters for offshore wind farm integration schemes with different fundamental frequencies and DRU structures are shown in

Table 11. Configuration schemes of AC filters for offshore wind farm integration schemes with different fundamental frequencies and DRU structures.

Fundamental Frequency	DRU Based on 12-Pulse Rectifier Circuit		DRU Based on 24-Pulse Rectifier Circuit
	Reactive Power Compensation capacity	Scheme	Reactive Power Compensation Capacity	Scheme
50 Hz	400 Mvar	3 × STF + 1 × PFC	240 Mvar	4 × PFC
100 Hz	360 Mvar	1 × STF + 3 × PFC	180 Mvar	4 × PFC
150 Hz	200 Mvar	4 × STF	/	/

.

The comparison of harmonic voltages at node PCC before and after the installation of AC filters is shown in Figure 14, and the related indicators such as THD and THFF are shown in

Table 12. THD and THFF under different fundamental frequencies and DRU structures with AC filters.

Index	DRU Based on 12-Pulse Rectifier Circuit			DRU Based on 24-Pulse Rectifier Circuit
	50 Hz	100 Hz	150 Hz	50 Hz	100 Hz	150 Hz
THD (%)	2.17	2.93	1.29	2.42	2.01	/
THFF (%)	0.9	0.84	0.57	0.48	0.17	/

, where ‘/’ stands for not available. Among them, since the combination of “150 Hz + 24 pulses” has already met the power quality requirements without AC filters, there is no need to configure AC filters.

After installing the AC filter, the effective values of all odd and even harmonic voltages were less than their limits, with THD and THFF being less than 3% and 1%, respectively, which met the power quality standards. It has been verified that the configuration scheme of the AC filter can meet the power quality standards at other power levels. The results show that both the increase of the fundamental frequency and the increase of the DRU pulse number in offshore wind farms can reduce the demand for AC filters.

5. Conclusions

Aiming at the harmonic problem of offshore wind farm integration schemes based on DRU, this paper starts from the equivalent model of offshore wind farms and the equivalent model of DRU, establishes the offshore AC part model of the offshore wind farm integration schemes, conducts harmonic calculation and analysis on this basis, and finally completes the design of AC filtering and reactive power compensation.

• This paper proposes a time-domain segmentation analytical calculation method applicable to the AC side harmonic current of DRU. The AC side harmonic current and the commutation overlap angle are derived by mathematical methods. The DC current decomposed by Fourier is substituted into the time-domain piecewise analytical method for iterative solution, thereby obtaining a more accurate equivalent model of DRU. It is compared with the time-domain simulation method to verify its correctness.
• This paper proposes a modeling method for the equivalent model of the offshore AC part of the offshore wind farm integration schemes and presents a harmonic calculation and analysis method for the harmonic analysis problem of node PCC under the dual influence of offshore wind farms and DRU. It proves that the increase of fundamental frequency and DRU pulse number has a suppression effect on harmonics.
• Aiming at the harmonic control problem of offshore wind farm integration schemes with different fundamental frequencies and different DRU structures, this paper proposes an AC filtering and reactive power compensation design scheme. Firstly, this paper constructs a collaborative design framework for AC filtering and reactive power compensation of the offshore wind power transmission system, including six key links. By comparing and analyzing the combinations of six different fundamental frequencies and DRU structures, the optimal AC filter configuration scheme is obtained, providing reference for the actual offshore wind power project.

Although this study focuses on offshore wind farms, the proposed AC filtering and reactive power compensation method—when integrated with appropriate power system component models—can be extended to other complex grid scenarios such as unbalanced grid conditions and renewable energy integration. This approach offers valuable insights and references for practical engineering applications.