Modeling and Analysis on AC-DC Harmonic Coupling of the Three-Phase Voltage Source Converter under Asymmetric Condition

Abstract

A three-phase voltage source converter (VSC) is the most common power electronic device in power systems, but its inherent nonlinearity leads to complex AC-DC harmonic coupling phenomena. Existing studies focus on the harmonic coupling characteristics of three-phase VSCs under three-phase symmetrical conditions, but the problem under asymmetrical conditions is rarely investigated. This paper proposes a practical modeling method for harmonic state space (HSS) modeling of the VSC topology, together with a classic T/4-delay-based decoupled double synchronous reference frame (DDSRF) control system. The steady-state phase shift characteristics of the T/4 delay link for each frequency component are first analyzed, and the expansion method of the DDSRF controller in the HSS frame is derived. Compared with the conventional method, the proposed technique can provide an accurate description of the steady-state effect of the delay link without introducing additional state variables and, therefore, can reduce model complexity and computation burden. Then, the small-signal close-loop HSS model of the VSC is established, and a harmonic transfer matrix between the AC positive sequence/negative sequence and DC current of different frequencies is developed, which reveals the global harmonic coupling relationship between AC and DC grids across the VSC. Analysis shows that the T/4-delay-link significantly weakens the AC-DC inter-harmonic coupling near the fundamental frequency. Finally, the electromagnetic transient simulations are carried out based on MATLAB/Simulink, and the validity of the proposed modeling method and the model and the correctness for the analysis of harmonic coupling characteristics are verified.

1. Introduction

The three-phase voltage source converter (VSC) achieves flexible AC-DC power and has the advantages of low commutation loss, high power quality, controllable power factor and fast control response. The VSC has been widely used in renewable power units, battery energy storage systems, electric vehicle charging systems, DC power transmission and distribution systems and flexible AC power transmission systems [1,2,3,4], etc. However, the inherent nonlinear characteristics of a three-phase VSC also render it one of the important harmonic sources in power systems [5,6]. In addition, the large-scale application of three-phase VSC has also significantly changed the harmonic transfer mechanism of power systems. The background harmonics on the AC/DC side can be coupled to the opposite side through the three-phase VSC, resulting in problems such as power quality degradation, accelerated equipment aging, increased resonance risk and so on [7,8]. With the rapidly growing penetration level of renewable energies, harmonic analysis, monitoring and suppression will become increasingly prominent [9,10]. Therefore, it is necessary to study the AC-DC harmonic coupling mechanism and transfer characteristics of the three-phase VSC to set up the theoretical basis for harmonic monitoring and suppression of the new power system.

Model linearization is the mainstream method used to study the harmonic coupling characteristics of three-phase VSCs. The state space averaging method presents the advantages of simplicity, intuity and scalability [11,12]. However, there is still strong coupling between the periodic time-varying modulation signals and state variables (i.e., voltage and current); therefore, further linearization is required. The three-phase AC time variables are transformed into time-invariant dq variables on the rotating synchronous frame, which not only simplifies the steady-state operating point calculation but also provides good data cohesion with the three-phase VSC controller [13,14,15].

However, under this framework, the harmonic information of all frequencies is mixed in the dq variables, which fails to provide a straightforward overview of the coupling relationship between AC and DC harmonics of different frequencies. To this end, the literature [16,17,18] apply the dynamic phasor method to model the converter on the basis of Fourier transform theory. By selecting the keen frequency components, the Fourier coefficient of each frequency component is modeled to obtain the normalization of the models, which can better describe the harmonic coupling relationship between different frequency components. Nevertheless, the specific key frequency components must be identified prior to quantitative modeling, which, however, lacks a universal criterion and may omit some important dynamic characteristics if the corresponding components are not included. Therefore, the dynamic phasor method has limited scalability for high-order component analysis. To address this issue, the Harmonic State Space (HSS) modeling method has been introduced for the modeling of power electronic equipment in recent years [19,20]. The method is also based on Fourier transform theory, but its modular and matrix-like mathematical structure avoids the key frequency screening problem and, therefore, significantly improves scalability in higher-order problem analysis. The HSS model of multiple types of converters was established in [21,22,23,24,25], which demonstrates the correctness and technical advantages of this method.

In light of the harmonic transfer characteristics of the three-phase VSC, existing research shows that the modulation process of the converter is the dominant path of interaction between the AC and DC systems. Because the fundamental frequency is the major component of the modulation signal, the coupling between the AC and DC harmonics of adjacent frequencies is dominant [7,8,9,10,11,12,13,14,15,16,17,18,19]. However, they assumed that the AC power grid is operated in an ideal symmetrical condition. Under asymmetric conditions, the controller of a VSC is different from a balanced operation with an extra positive-negative sequence separation link and negative-sequence current loop, which may lead to different harmonic coupling features. In the field, the T/4-delay-based sequence separation method, together with the decoupled double synchronous reference frame (DDSRF) current controller, is the most popular solution. Literature [26,27] investigate the impedance modeling method of VSC under unbalanced grid conditions, with special attention to the influence induced by the positive-negative sequence separation and the double synchronous reference frame current controller, respectively. However, the models are SISO transfer functions that lack enough information on the cross-coupling relationship between various AC and DC harmonics. In contrast, the literature [28] established the HSS model that leads to a MIMO harmonic transfer function, but the negative current loop is not considered. Therefore, the HSS modeling problem of the VSC under unbalanced grid conditions is still unsolved, and the inter-grid harmonic coupling feature is unclear.

To complement the existing research, an HSS modeling method for the VSC and the T/4-delay-based DDSRF current controller under unbalanced grid conditions is proposed. The bottleneck of this problem lies in the HSS linearization of the T/4 delay link. Conventional methods, such as Pade approximation, will introduce a series of state variables that considerably increase the model scale and computation burden and are not suitable for online applications where the computation capability is limited.

To this end, a practical HSS modeling method is proposed for steady-state harmonic coupling analysis by considering only the steady-state effect of the delay link. The steady-state phase shift characteristics of the T/4 delay link for each frequency component are first analyzed, and the expansion method of the DDSRF controller in the HSS frame is derived. Compared with conventional methods, the proposed method can maintain the salient steady-state feature while introducing no for the delay link, which is crucial for model simplicity and is of practical interest for online applications. Then, the harmonic transfer matrix between the AC positive sequence/negative sequence and the DC current of different frequencies is developed, and the influences of the T/4 delay link on the harmonic coupling characteristics are analyzed. In addition, the global harmonic coupling relationship between AC and DC systems is also revealed. Finally, the correctness of the proposed modeling method and model is verified by MATLAB/Simulink simulations.

The chapters of this paper are organized as follows: Section 1 introduces the basic principle and mathematical form of the HSS method; Section 2 presents the mathematical model of the three-phase VSC topology under asymmetrical conditions; Section 3 establishes the mathematical model of T/4-delay-based DDSRF control system; Section 4 develops the three-phase VSC small-signal model with a closed-loop control system, and deduces the AC and DC harmonic transfer matrix; Section 5 verifies the correctness of the modeling method and the model based on MATLAB/Simulink simulations; Section 6 concludes the paper.

2. Harmonic State Space Method

The typical expression for a linear time-period system is:

$$\left\{\begin{array}{l}\dot{x}(t)=A(t)x(t)+B(t)u(t)\\ y(t)=C(t)x(t)+D(t)u(t)\end{array}\right.$$

(1)

where x(t), y(t), and u(t) denote the state, input and output vectors, respectively; A(t), B(t), C(t) and D(t) denote the state, input, output and transfer matrices, respectively. Both the vectors and matrices are periodic time-varying with a period of $T_0$.

Mathematically, a periodic signal can be represented as a complex Fourier series as:

$$x(t)={\displaystyle \sum _{k=-\infty }^{\infty }{X}_k{e}^{jk{\omega }_{0}t}}$$

(2)

$$X_k=\frac{1}{T_0}{\displaystyle {\int }_{t-T_0}^{t}x(t)}{e}^{-jk{\omega }_{0}t}$$

(3)

where ${\omega }_0=2\pi /T_0$ denotes the fundamental angular frequency of the system; X_k is the kth Fourier coefficient of $x(t)$.

To describe the dynamic characteristics of the signal, the exponential periodic form of $x(t)$ is introduced on the basis of Equation (2):

$$x(t)={\displaystyle \sum _{k=-\infty }^{\infty }{X}_k{e}^{st}{e}^{jk{\omega }_{0}t}}$$

(4)

By substituting Equation (4) into Equation (1), the state space equation of the nth order harmonic component can be obtained via the harmonic balance principle:

$$\left\{\begin{array}{l}(s+jn{\omega }_0)x_n={\displaystyle \sum _{m\in Z}{A}_{n-m}}{x}_m+{\displaystyle \sum _{m\in Z}{B}_{n-m}}{u}_m\\ y_n={\displaystyle \sum _{m\in Z}{C}_{n-m}}{x}_m+{\displaystyle \sum _{m\in Z}{D}_{n-m}}{u}_m\end{array}\right.$$

(5)

where subscript n represents the nth-order Fourier coefficient of the corresponding variable and s denotes the Laplacian operator.

Based on Equation (5), the hth-order-truncated (i.e., considering all harmonic components between the −hth~hth order on the complex frequency domain) HSS model of the system can be expressed as:

$$\left\{\begin{array}{l}{\dot{x}}_{HSS}=(A_{HSS}-N_{HSS})·x_{HSS}+B_{HSS}·u_{HSS}\\ y_{HSS}=C_{HSS}·x_{HSS}+D_{HSS}·u_{HSS}\end{array}\right.$$

(6)

where x_HSS, u_HSS and y_HSS represent the harmonic coefficients of x(t), y(t) and u(t), respectively, which share the same mathematical structure. Taking x_HSS as an example:

$$x_{HSS}={\left[x_{-h}{}^T,\cdots ,x_{-1}{}^T,x_0{}^T,x_1{}^T,\cdots ,x_{-h}{}^T\right]}^T$$

(7)

A/B/C/D_HSS are Toeplitz matrices composed of the Fourier coefficients of A(t)/B(t)/C(t)/D(t), respectively, which also share the same mathematical formation. Taking A_HSS as an example, the general structure is expressed as:

$$A_{HSS}=\left[\begin{array}{ccccccc}{A}_0& A_{-1}& \cdots & A_{-h}& O& \cdots & O\\ A_1& \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ ⋮& \ddots & A_0& A_{-1}& \ddots & \ddots & O\\ A_h& \ddots & A_1& A_0& A_{-1}& \ddots & A_{-h}\\ O& \ddots & \ddots & A_1& A_0& \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & A_{-1}\\ O& \cdots & O& A_h& \cdots & A_1& A_0\end{array}\right]$$

(8)

where the submatrix A_n(n ∈ [−h, h]) is the nth-order Fourier coefficient of A(t); O denotes the zero matrix of the same size with A_n; N is a block diagonal matrix expressed as:

$$N=diag\left(-jh{\omega }_{0}I,\cdots ,-j{\omega }_{0}I,0·I,j{\omega }_{0}I,\cdots ,jh{\omega }_{0}I\right)$$

(9)

where I denotes the identity matrix with the same order as x(t).

3. HSS Modeling for the Three-Phase VSC Topology under Asymmetric Conditions

3.1. Average State Space Model of the Three-Phase VSC

The circuit topology of the three-phase VSC and the positive direction of the main electrical quantities are shown in Figure 1. In the figure, e_a, e_b and e_c are the three-phase voltages of the AC side; i_a, i_b and i_c denote the three-phase line-connected currents; v_C denotes the DC link capacitor voltage; i_in denotes the DC side input current; L and R denote the grid-interfaced filter inductance and equivalent resistance of the AC side, respectively. The KVL equation of a three-phase AC circuit can be expressed as:

$$L\frac{d}{dt}{i}_x=e_x-v_x-R·i_x,\hspace{1em}x\in \left\{a,b,c\right\}$$

(10)

where v_a, v_b and v_c denote the three-phase output voltages on the AC valve side of the VSC. Ignoring the high-frequency components caused by semiconductor switching and PWM delay, the average valve side voltage can be expressed as:

$$v_x=\frac{1}{2}{m}_x{v}_C,\hspace{1em}x\in \left\{a,b,c\right\}$$

(11)

where m_x denotes the modulation signal of phase x.

During steady-state operation, v_C is maintained around its rated value, whose dynamic equation can be deduced as follows:

$$C\frac{d{v}_C}{dt}=\frac{1}{2}{\displaystyle \sum _{x=a,b,c}{m}_x{i}_x}+i_{in}$$

(12)

where C denotes the DC link capacitance.

Based on Equations (10)–(12), the state space model of the three-phase VSC is established by utilizing v_C, i_a, i_b and i_c as the state variables, and i_in, e_a, e_b and e_c as the input variables, as Equations (13)–(17):

$${\dot{x}}_{cqt}^{org}\left(t\right)=A_{cqt}^{org}\left(t\right)x_{cqt}^{org}\left(t\right)+B_{cqt}^{org}\left(t\right)u_{cqt}^{org}\left(t\right)$$

(13)

$$x_{cqt}^{org}={\left[\begin{array}{cccc}{v}_C\left(t\right)& i_a\left(t\right)& i_b\left(t\right)& i_c\left(t\right)\end{array}\right]}^T$$

(14)

$$u_{cqt}^{org}\left(t\right)=\left[\begin{array}{cccc}{i}_{in}\left(t\right)& e_a\left(t\right)& e_b\left(t\right)& e_c\left(t\right)\end{array}\right]$$

(15)

$$A_{cqt}^{org}=\left[\begin{array}{cccc}0& m_a\left(t\right)/2C& m_b\left(t\right)/2C& m_c\left(t\right)/2C\\ -m_a\left(t\right)/2L& -R/L& 0& 0\\ -m_b\left(t\right)/2L& 0& -R/L& 0\\ -m_c\left(t\right)/2L& 0& 0& -R/L\end{array}\right]$$

(16)

$$B_{cqt}^{org}=diag\left(\begin{array}{cccc}1/C& 1/L& 1/L& 1/L\end{array}\right)$$

(17)

where the subscript cqt represents the circuit model and the superscript org represents the original mathematical model before HSS expansion.

3.2. HSS Modeling of the Three-Phase VSC

Based on the HSS principles introduced in Section 1, Equation (13) can be expanded into HSS form as follows:

$$x_{cqt}^{}=A_{cqt}{x}_{cqt}^{}+B_{cqt}{u}_{cqt}^{}$$

(18)

where A_cqt and B_cqt are of the formation shown in Equation (8), and the nth-order submatrices A_cqt,n and B_cqt,n are shown in Equations (19)–(21), respectively:

$$\begin{array}{l}{A}_{cqt,n}=\left[\begin{array}{cccc}0& m_{a,n}/2C& m_{a,n}/2C& m_{a,n}/2C\\ -m_{a,n}/2L& 0& 0& 0\\ -m_{a,n}/2L& 0& 0& 0\\ -m_{a,n}/2L& 0& 0& 0\end{array}\right]\\ \hfill \left(n\ne 0\right)\end{array}$$

(19)

$$A_{cqt,0}=\left[\begin{array}{cccc}0& m_{a,0}/2C& m_{a,0}/2C& m_{a,0}/2C\\ -m_{a,0}/2L& -R/L& 0& 0\\ -m_{a,0}/2L& 0& -R/L& 0\\ -m_{a,0}/2L& 0& 0& -R/L\end{array}\right]$$

(20)

$$B_{cqt,n}=\left\{\begin{array}{l}diag\left(\begin{array}{cccc}1/C& 1/L& 1/L& 1/L\end{array}\right),n=0\hfill \\ O_{4\times 4},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} n=others\hfill \end{array}\right.$$

(21)

where O_{4 × 4} represents a 4 × 4 zero matrix.

The mathematical forms of x_cqt and u_cqt are the same as in Equation (7). Since the VSC is generally connected to the main network through a Y-Δ transformer, only the positive and negative sequence components are considered under asymmetric conditions, while the zero-sequence component is ignored. By ignoring grid voltage harmonics, the nth-order components of e_a, e_b and e_c in u_cqt can be determined as Equations (22)–(24):

$$e_{a,n}=\left\{\begin{array}{ll}\left({\widehat{e}}_1+{\widehat{e}}_2\right)/2,\hfill & n=1\hfill \\ \left({\widehat{e}}_1^{*}+{\widehat{e}}_2^{*}\right)/2,\hfill & n=-1\hfill \\ 0,\hfill & n=others\hfill \end{array}\right.$$

(22)

$$e_{b,n}=\left\{\begin{array}{ll}\left({\widehat{e}}_1·e^{-j2\pi /3}+{\widehat{e}}_2·e^{j2\pi /3}\right)/2,\hfill & n=1\\ \left({\widehat{e}}_1^{*}·e^{j2\pi /3}+{\widehat{e}}_2^{*}·e^{-j2\pi /3}\right)/2,\hfill & n=-1\\ 0,\hfill & n=others\end{array}\right.$$

(23)

$$e_{c,n}=\left\{\begin{array}{ll}\left({\widehat{e}}_1·e^{j2\pi /3}+{\widehat{e}}_2·e^{-j2\pi /3}\right)/2,\hfill & n=1\\ \left({\widehat{e}}_1^{*}·e^{-j2\pi /3}+{\widehat{e}}_2^{*}·e^{j2\pi /3}\right)/2,\hfill & n=-1\\ 0,\hfill & n=others\end{array}\right.$$

(24)

where ${\widehat{e}}_1$ and ${\widehat{e}}_2$ are the positive and negative sequence grid voltage phasors, respectively; and the superscript ‘*’ represents the conjugation of corresponding complex variables.

Take the single-phase grounding fault of phase a as an example. If the remaining voltage of phase a is kE (0 < k < 1, E denotes the nominal voltage amplitude) on the primary side of the transformer, the positive- and negative-sequence voltage phasors can be written as:

$$\left\{\begin{array}{c}{\widehat{e}}_1=\frac{k+2}{3}\widehat{e}\\ {\widehat{e}}_2=\frac{k-1}{3}\widehat{e}\end{array}\right.$$

(25)

where $\widehat{e}$ denotes the grid voltage phasor under normal operating conditions. Subsequently, u_cqt in this case can be obtained by substituting Equation (25) into Equations (22)–(24). In addition, for other asymmetric scenarios, the positive- and negative-sequence phasors can be calculated and modified according to the specific grid-side conditions, and then the u_cqt for the target scenario can be obtained.

4. HSS Modeling of the VSC Control System under Asymmetric Conditions

The three-phase VSC commonly adopts the hierarchy double closed-loop control structure shown in Figure 2. The outer loop is responsible for controlling the energy balance control of the DC link, as well as the AC-side power factor or reactive power, while the inner loop is responsible for controlling the grid-tied current to track the commands issued by the outer loop. For asymmetric operations, the T/4-delay-based DDSRF current controller is widely utilized to achieve decoupled control of positive and negative sequence currents. The measured three-phase currents are first directed through a T/4-delay-based decoupling block to separate the positive- and negative-sequence components, and the separated components are subsequently regulated by the corresponding controller under the respective synchronous frames. This chapter will present the HSS modeling method of the T/4 delay link, inner loop, and outer loop sequentially.

4.1. T/4 Delay Link

In the T/4-delayed separation link, the instantaneously measured and T/4-delay three-phase currents i_a, i_b and i_c are first put through the Clarke transformation:

$${\left[\begin{array}{cc}{i}_{\alpha }\left(t\right)& i_{\beta }\left(t\right)\end{array}\right]}^T=T_{3s-2s}{\left[\begin{array}{ccc}{i}_a\left(t\right)& i_b\left(t\right)& i_c\left(t\right)\end{array}\right]}^T$$

(26)

$$\left[\begin{array}{c}{i}_{\alpha }\left(t-T/4\right)\\ i_{\beta }\left(t-T/4\right)\end{array}\right]=T_{3s-2s}\left[\begin{array}{c}{i}_a\left(t-T/4\right)\\ i_b\left(t-T/4\right)\\ i_c\left(t-T/4\right)\end{array}\right]$$

(27)

where T_3s–2s denotes the Clarke transformation matrix. In this paper, the constant-amplitude transformation is adopted:

$$T_{3s-2s}=\frac{2}{3}\left[\begin{array}{l}1\hspace{1em}-1/2\hspace{1em}-1/2\hfill \\ 0\hspace{1em}\sqrt{3}/2\hspace{1em}-\sqrt{3}/2\hfill \end{array}\right]$$

(28)

Then, the positive and negative sequence components can be calculated as follows:

$$\left[\begin{array}{c}{i}_{\alpha Pctrl}\left(t\right)\\ i_{\beta Pctrl}\left(t\right)\\ i_{\alpha Nctrl}\left(t\right)\\ i_{\beta Nctrl}\left(t\right)\end{array}\right]=\frac{1}{2}\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 1& 0\\ 1& 0& 0& 1\\ 0& 1& -1& 0\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\left(t\right)\\ i_{\beta }\left(t\right)\\ i_{\alpha }\left(t-T/4\right)\\ i_{\beta }\left(t-T/4\right)\end{array}\right]$$

(29)

where the subscripts Pctrl and Nctrl represent the positive- and negative-sequence components output by the T/4 delay link, respectively, which are also the measured values received by the positive- and negative-sequence current loops, respectively.

The T/4-delay link can be described by the following transfer function G_delay(s) that is expressed as an exponential function:

$$G_{delay}\left(s\right)=e^{-sT/4}$$

(30)

Pade approximation is a classic method for the linearization of exponential functions. However, for a delay time of 5 ms (T/4), the required Pade approximation order should be sufficiently high to ensure a satisfying accuracy, which will significantly increase the VSC model order and the computation burden and therefore hinder the online application of the model.

In this regard, a simplified modeling method for the study of steady-state characteristics is proposed. That is, only the steady-state effect of the delay link is considered since harmonic coupling is a pure steady-state problem. For the nth-order three-phase current, the steady-state influence of the T/4-delay can be obtained by letting s = jnω in Equation (30):

$${\left[\begin{array}{cc}{i}_{\alpha ,n}^{delay}& i_{\beta ,n}^{delay}\end{array}\right]}^T=e^{-jn\pi /2}{\left[\begin{array}{cc}{i}_{\alpha ,n}& i_{\beta ,n}\end{array}\right]}^T$$

(31)

where the superscript delay represents the variable after the delay.

Additionally, the T/4-delay algorithm realizes the positive-and negative-sequence separation of the fundamental frequency components (i.e., n = ±1). For the current components of other orders, {i_αPctrl, i_βPctrl} and {i_αNctrl, i_βNctrl} will still contain both positive and negative sequence components. To facilitate modeling of the DDSRF current loop, the nth-order components received by the positive- and negative-sequence current loops can be derived as follows:

$$\left[\begin{array}{c}{i}_{PPctrl,n}\\ i_{NPctrl,n}\\ i_{PNctrl,n}\\ i_{NNctrl,n}\end{array}\right]=\left[\begin{array}{cccc}1/2& j/2& 0& 0\\ 1/2& -j/2& 0& 0\\ 0& 0& 1/2& j/2\\ 0& 0& 1/2& -j/2\end{array}\right]\left[\begin{array}{c}{i}_{\alpha Pctrl,n}\\ i_{\beta Pctrl,n}\\ i_{\alpha Nctrl,n}\\ i_{\beta Nctrl,n}\end{array}\right]$$

(32)

where i_PPctrl,n and i_NPctrl,n are the nth-order positive- and negative-sequence components received by the positive-sequence current loop, respectively; and i_PNctrl,n and i_NNctrl,n are the nth-order positive- and negative-sequence components received by the negative-sequence current loop, respectively.

Based on Equations (29)–(32), the following equation can be obtained:

$$\left[\begin{array}{c}{i}_{PPctrl,n}\\ i_{NPctrl,n}\\ i_{PNctrl,n}\\ i_{NNctrl,n}\end{array}\right]=\frac{1}{4}\left[\begin{array}{cccc}1& j& j& -1\\ 1& -j& -j& -1\\ 1& j& -j& 1\\ 1& -j& j& 1\end{array}\right]\left[\begin{array}{c}{T}_{3s-2s}\\ e^{-jn\pi /2}{T}_{3s-2s}\end{array}\right]\left[\begin{array}{c}{i}_{a,n}\\ i_{b,n}\\ i_{c,n}\end{array}\right]$$

(33)

According to Equation (33), the proposed modeling method does not bring additional state variables and, therefore, will not increase the model complexity and computation burden. Consequently, the proposed method is more suitable for steady-state harmonic analysis.

4.2. DDSRF Current Loop

Due to the similarity of the positive- and negative-sequence current loops, only the HSS modeling of the positive-sequence loop is presented in detail. The structure of the positive sequence current controller is shown in Figure 3, whose control equation is expressed as:

$$m_{Pctrl}\left(t\right)=e_{Pctrl}\left(t\right)-j\omega L{i}_{Pctrl}\left(t\right)-\left(K_{P\_i}+\frac{K_{I\_i}}{s}\right)\left[i_{Pctrl}{}^{\mathrm{ref}}\left(t\right)-i_{Pctrl}\left(t\right)\right]$$

(34)

where m_Pctrl, e_Pctrl and i_Pctrl represent the output modulation signal, the voltage feedforward term and the current feedback term of the positive-sequence current loop expressed in complex form, respectively; the superscript ‘ref’ represents the reference value of the corresponding variable; K_{P_i} and K_{I_i} denote the current Proportional gain and integral gain of the proportional-integral (PI) control link.

m_Pctrl, e_Pctrl and i_Pctrl can all be decomposed into the complex Fourier series form, taking i_dqP for example:

$$i_{dqP}\left(t\right)={\displaystyle \sum _{n>0}{\widehat{i}}_{PPctrl,n}{e}^{j\left(n-1\right)\omega t}}+{\displaystyle \sum _{n>0}{\widehat{i}}_{NPctrl,n}{e}^{j\left(-n-1\right)\omega t}}+{\displaystyle \sum _{n<0}{\widehat{i}}_{PPctrl,n}{e}^{j\left(n+1\right)\omega t}}+{\displaystyle \sum _{n<0}{\widehat{i}}_{NPctrl,n}{e}^{j\left(-n+1\right)\omega t}}$$

(35)

By substituting Equation (35) into Equation (34), the model for each frequency component can be established. The block diagram of the nth-order component model is shown in Figure 4. In this section, the outputs of the integral channel in the PI controllers are selected as the state variables, i.e., x_PPctrl,n and x_NPctrl,n, and the modulation signals m_PPctrl,n and m_NPctrl,n are selected as the output variables. Subsequently, the nth-order (n > 0) model can be established as follows:

$$\left[\begin{array}{c}{\dot{x}}_{PPctrl,n}\\ {\dot{x}}_{NPctrl,n}\end{array}\right]=j\omega \left[\begin{array}{cc}-n+1& 0\\ 0& -n-1\end{array}\right]\left[\begin{array}{c}{x}_{PPctrl,n}\\ x_{NPctrl,n}\end{array}\right]+K_{I\_i}\left(\left[\begin{array}{c}{i}_{PPctrl,n}{}^{\mathrm{ref}}-i_{PPctrl,n}\\ i_{NPctrl,n}{}^{\mathrm{ref}}-i_{NPctrl,n}\end{array}\right]\right)$$

(36)

$$\begin{array}{r}\left[\begin{array}{c}{m}_{PPctrl,n}\\ m_{NPctrl,n}\end{array}\right]=\frac{2}{v_C{}^{\mathrm{ref}}}\left[\begin{array}{c}{x}_{PPctrl,n}\\ x_{NPctrl,n}\end{array}\right]-K_{P\_i}\left(\left[\begin{array}{c}{i}_{PPctrl,n}{}^{\mathrm{ref}}-i_{PPctrl,n}\\ i_{NPctrl,n}{}^{\mathrm{ref}}-i_{NPctrl,n}\end{array}\right]\right)\hspace{1em}\\ +\left[\begin{array}{c}{e}_{PPctrl,n}\\ e_{NPctrl,n}\end{array}\right]-j\omega L\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{i}_{PPctrl,n}\\ i_{NPctrl,n}\end{array}\right]\end{array}$$

(37)

For the nth-order (n < 0) component, the model can be established based on Figure 4c,d following the same procedure. Alternatively, it can be derived by conjugating Equations (36) and (37) based on the characteristic that the ±n order components are constantly conjugated.

The HSS model of the negative-sequence current loop can be established similarly, and the specific expressions will not be repeated.

Furthermore, to maintain cohesion with the circuit model established in Section 2, the modulation signals output by the DDSRF current loop must be converted to the three-phase form. The required conversion can be derived as follows:

$$\left[\begin{array}{c}{m}_{a,n}\\ m_{b,n}\\ m_{c,n}\end{array}\right]=T_{ctrl-cqt,n}\left[\begin{array}{c}{m}_{PPctrl,n}\\ m_{NPctrl,n}\\ m_{PNctrl,n}\\ m_{NNctrl,n}\end{array}\right]$$

(38)

where T_ctrl-cqt,n is the conversion matrix from the DDSRF frame to the stationary abc frame for the nth-order component, which is expressed as:

$$T_{ctrl-cqt,n}=\left\{\begin{array}{cc}\left[\begin{array}{llll}1& 1& 1& 1\\ e^{j2\pi /3}& e^{-j2\pi /3}& e^{j2\pi /3}& e^{-j2\pi /3}\\ e^{-j2\pi /3}& e^{j2\pi /3}& e^{-j2\pi /3}& e^{j2\pi /3}\end{array}\right],& n<0\hfill \\ \left[\begin{array}{llll}1& 1& 1& 1\\ e^{-j2\pi /3}& e^{j2\pi /3}& e^{-j2\pi /3}& e^{j2\pi /3}\\ e^{j2\pi /3}& e^{-j2\pi /3}& e^{j2\pi /3}& e^{-j2\pi /3}\end{array}\right],& n>0\hfill \end{array}\right.$$

(39)

4.3. DC-Link Voltage/Reactive Power Loop

Under asymmetrical conditions, the positive-sequence current is utilized for average DC link voltage control and reactive power regulation, while the negative-sequence current is suppressed to zero. The corresponding control equation is expressed as follows:

$$\left\{\begin{array}{c}{i}_{dP}{}^{\mathrm{ref}}=\left(K_{P\_vC}+K_{I\_vC}/s\right)\left(v_C{}^{\mathrm{ref}}-\overline{v_C}\right)\\ \begin{array}{l}{i}_{qP}{}^{\mathrm{ref}}=-\left(K_{P\_Q}+K_{I\_Q}/s\right)\left(Q^{\mathrm{ref}}-\overline{Q}\right)\\ i_{dN}{}^{\mathrm{ref}}=i_{qN}{}^{\mathrm{ref}}=0\end{array}\end{array}\right.$$

(40)

where K_{P_vC} and K_{I_vC} are the PI gains of the DC-link voltage controller; K_{P_vC} and K_{I_vC} are the PI gains of the reactive power controller; Q is the reactive power output by the three-phase VSC, with Q_ref representing its reference; and the upper line represents the average value of the corresponding variable. Under asymmetrical conditions, both v_C and Q contain significant double-frequency components, which are filtered by a 100 Hz second-order notch filter:

$$\left[\begin{array}{c}\overline{v_C}\\ \overline{Q}\end{array}\right]=G_{Notch}\left[\begin{array}{c}{v}_C\\ Q\end{array}\right]=\frac{s^2+{\omega }_0{}^2}{s^2+2{\omega }_{c}s+{\omega }_0{}^2}\left[\begin{array}{c}{v}_C\\ Q\end{array}\right]$$

(41)

Combining Equations (40) and (41), the outer-loop transfer function is established:

$$i_{dqP}{}^{\mathrm{ref}}=i_{dP}{}^{\mathrm{ref}}+j·i_{qP}{}^{\mathrm{ref}}=G_{outer}\left(\left[\begin{array}{c}{v}_C{}^{\mathrm{ref}}\\ Q^{\mathrm{ref}}\end{array}\right]-G_{Notch}\left[\begin{array}{c}{v}_C\\ Q\end{array}\right]\right)$$

(42)

where G_outer is the outer loop transfer function matrix.

For HSS modeling of the outer loop, considering the frequency shift effect of the Park transform, the following equations can be obtained:

$$i_{PPctrl,n+1}{}^{ref}=G_{outer}\left(\left[\begin{array}{c}{v}_{C,n}{}^{\mathrm{ref}}\\ Q_n{}^{\mathrm{ref}}\end{array}\right]-G_{Notch}\left[\begin{array}{c}{v}_{C,n}\\ Q_n\end{array}\right]\right),n\ge 1$$

(43)

$$i_{PPctrl,n-1}{}^{ref}=G_{outer}\left(\left[\begin{array}{c}{v}_{C,n}{}^{\mathrm{ref}}\\ Q_n{}^{\mathrm{ref}}\end{array}\right]-G_{Notch}\left[\begin{array}{c}{v}_{C,n}\\ Q_n\end{array}\right]\right),n\le -1$$

(44)

$$i_{PPctrl,\pm 1}{}^{ref}=\frac{1}{2}{G}_{outer}\left(\left[\begin{array}{c}{v}_{C,0}{}^{\mathrm{ref}}\\ \pm Q_0{}^{\mathrm{ref}}\end{array}\right]-G_{Notch}\left[\begin{array}{c}{v}_{C,0}\\ \pm Q_0\end{array}\right]\right)$$

(45)

4.4. HSS Model of the VSC Controller

Based on Equations (33)–(45), the HSS model of the VSC controller can be established and organized into the following form:

$${\dot{x}}_{ctrl}=A_{ctrl}{x}_{ctrl}+\left[\begin{array}{ccc}{B}_{ctrl1}& B_{ctrl2}& B_{ctrl3}\end{array}\right]\left[\begin{array}{c}{u}_{ctrl1}\\ u_{ctrl2}\\ u_{ctrl3}\end{array}\right]$$

(46)

$$y_{ctrl}=C_{ctrl}{x}_{ctrl}+\left[\begin{array}{ccc}{D}_{ctrl1}& D_{ctrl2}& D_{ctrl3}\end{array}\right]\left[\begin{array}{c}{u}_{ctrl1}\\ u_{ctrl2}\\ u_{ctrl3}\end{array}\right]$$

(47)

where u_ctrl1 denotes the v_C and three-phase current in the HSS form; u_ctrl2 represents the three-phase grid voltage in the HSS form; u_ctrl3 denotes the commands of the outer loop:

$$u_{ctrl1}=x_{cqt,HSS}$$

(48)

$$u_{ctrl2}=u_{cqt,HSS}$$

(49)

$$u_{ctrl3}=\left[\begin{array}{cc}{v}_C{}^{\mathrm{ref}}& Q^{\mathrm{ref}}\end{array}\right]$$

(50)

and y_ctrl denotes the three-phase modulation signal in the HSS form:

$$y_{ctrl}={\left[\begin{array}{lllll}{y}_{ctrl,-h}{}^T\hfill & \hfill \cdots \hfill & y_{ctrl,0}{}^T\hfill & \hfill \cdots \hfill & y_{ctrl,h}{}^T\hfill \end{array}\right]}^T$$

(51)

$$y_{ctrl,n}={\left[\begin{array}{ccc}{m}_{a,n}& m_{b,n}& m_{c,n}\end{array}\right]}^T$$

(52)

Combining the controller model of Equations (46)–(52) and the circuit model derived in Section 2, the closed-loop three-phase VSC state space model can be established:

$$\left[\begin{array}{c}{x}_{cqt}\\ x_{ctrl}\end{array}\right]=\left[\begin{array}{cc}{A}_{cqt}-N& O\\ B_{ctrl1}& A_{ctrl}\end{array}\right]\left[\begin{array}{c}{x}_{cqt}\\ x_{ctrl}\end{array}\right]+\left[\begin{array}{cc}{B}_{cqt}& O\\ B_{ctrl2}& B_{ctrl3}\end{array}\right]\left[\begin{array}{c}{u}_{cqt}\\ u_{ctrl3}\end{array}\right]$$

(53)

5. HSS Small-Signal Modeling

In the HSS model of Equation (53), the state transition matrix A_cqt in the circuit model still contains time-varying modulation signals. The effect of the control system on the circuit topology is reflected in the time-varying elements of A_cqt, which makes the system nonlinear. To facilitate subsequent analysis, the model needs to be further linearized around the steady-state operation point.

The small-signal models of the VSC topology can be expressed as follows:

$$\Delta {\dot{x}}_{cqt}=\left(A_{cqt}^S-N\right)\Delta x_{cqt}+B_{cqt}^{}\Delta u_{cqt}+\Delta A_{cqt}^{}{x}_{cqt}^S$$

(54)

where the superscript S represents the steady-state value of the corresponding variable; Δ represents the small-signal disturbance near the corresponding steady-state value. To maintain cohesion with the control system model, the third term in Equation (54) is rewritten as the small-signal expression of y_ctrl:

$$\Delta A_{cqt}^{}{x}_{cqt}^S=W·\Delta y_{ctrl}$$

(55)

where W denotes the small signal input matrix of the modulated signal:

$$W=diag\left(\begin{array}{lllll}{W}_{-h}\hfill & \hfill \cdots \hfill & W_0\hfill & \hfill \cdots \hfill & W_h\hfill \end{array}\right)$$

(56)

Based on Equations (19) and (20), W_n, the expression of the nth submatrix of W in Equation (56), is expressed as:

$$W_n=\left[\begin{array}{lll}{i}_{a,n}^S/2C\hfill & i_{b,n}^S/2C\hfill & i_{c,n}^S/2C\hfill \\ -v_{C,n}^S/2L\hfill & 0\hfill & 0\hfill \\ 0\hfill & -v_{C,n}^S/2L\hfill & 0\hfill \\ 0\hfill & 0\hfill & -v_{C,n}^S/2L\hfill \end{array}\right]$$

(57)

Consequently, the closed-loop VSC small-signal model can be established as:

$$\begin{array}{r}\left[\begin{array}{c}\Delta {\dot{x}}_{cqt}\\ \Delta {\dot{x}}_{ctrl}\end{array}\right]=\left[\begin{array}{cc}{A}_{cqt}-N+W·D_{ctrl1}& W·C_{ctrl}\\ B_{ctrl1}& A_{ctrl}\end{array}\right]\left[\begin{array}{c}\Delta x_{cqt}\\ \Delta x_{ctrl}\end{array}\right]\\ +\left[\begin{array}{cc}{B}_{cqt}+W·D_{ctrl2}& W·D_{ctrl3}\\ B_{ctrl2}& B_{ctrl3}\end{array}\right]\left[\begin{array}{c}\Delta u_{cqt}\\ \Delta u_{ctrl3}\end{array}\right]\end{array}$$

(58)

Abbreviate the above formula as:

$$\Delta {\dot{x}}_{syn}=A_{syn}\Delta x_{syn}+B_{syn}\Delta u_{syn}$$

(59)

where the subscript syn represents the synthetic close-loop model of the three-phase VSC model.

To investigate the coupling relationship between the DC side harmonic current and the AC side positive/negative sequence harmonics in the steady state, the output equation of the model is established as follows:

$$\Delta y_{syn}=C_{syn}{\left[\begin{array}{cc}\Delta x_{cqt}& \Delta x_{ctrl}\end{array}\right]}^T$$

(60)

$$\Delta y_{syn}={\left[\begin{array}{lllll}\Delta y_{syn,-h}{}^T\hfill & \hfill \cdots \hfill & \Delta y_{syn,0}{}^T\hfill & \hfill \cdots \hfill & \Delta y_{syn,h}{}^T\hfill \end{array}\right]}^T$$

(61)

$$\Delta y_{syn,n}={\left[\begin{array}{cc}{i}_{P,n}& i_{N,n}\end{array}\right]}^T$$

(62)

$$C_{syn}=diag\left(\begin{array}{lllll}{C}_{syn,-h}\hfill & \hfill \cdots \hfill & C_{syn,0}\hfill & \hfill \cdots \hfill & C_{syn,h}\hfill \end{array}\right)$$

(63)

$$C_{syn,n}=\left[\begin{array}{l}0\hspace{1em}1/3\hspace{1em}{e}^{j2\pi /3}/3\hspace{1em}{e}^{-j2\pi /3}/3\hspace{1em}O\hfill \\ 0\hspace{1em}1/3\hspace{1em}{e}^{-j2\pi /3}/3\hspace{1em}{e}^{j2\pi /3}/3\hspace{1em}O\hfill \end{array}\right]$$

(64)

Based on Equations (56) and (60), the transfer function between the AC and DC current harmonics can be constructed as:

$$H_{syn}=C_{syn}{\left(sI-A_{syn}\right)}^{-1}{B}_{syn}$$

(65)

Let s = 0 in Equation (65), the steady-state coupling matrix between the positive/negative sequence harmonic current on the AC side and the DC side harmonic current can be obtained, where each element represents the coupling gain between the corresponding AC and DC components.

Based on the established 9th-order model, the harmonic transfer matrix is derived according to the parameters shown in

Table 1. Bechmark Parameters.

Parameter	Value [Unit]
AC side rated voltage	10 [kV]
AC side asymmetry condition	Current of Phase A dropped to 0.1 p.u.
AC side control strategy	Suppression of negative sequence current
DC side input power	10 [MW]
AC side output power command	10 [MMW], 1 [Mvar]
DC link rated voltage	20 [kV]
AC side filter inductor	10 [mH]
AC side equivalent resistance	0.1 [Ω]
DC link capacitor value	100 [μF]
PWM carrier frequency	10 [kHz]
PI gain of voltage outer loop	(0.1, 10)
PI gain of reactive outer loop	(0.01, 1)
PI gain of current inner loop	(5, 100)

. The harmonic transfer gains between positive-sequence and DC-side harmonics are presented in Figure 5a, while the gains between negative-sequence and DC-side harmonics are presented in Figure 5b. Although the results are based on specific example parameters, they can still reflect the general laws of AC and DC harmonic current coupling.

According to Figure 5, the following conclusions can be drawn:

(1)

There is strong coupling between the AC-side positive and negative sequences and the DC-side current harmonics. For the n_Δth-order harmonic current on the DC side, the coupled AC-side positive- and negative-sequence currents are both of (n_Δ ± k)th-order (k = 1, 3, 5, …).

(2)

In general, the non-zero elements of the harmonic coupling matrix decrease with the increase of n_Δ and k. That is, in the figure, the amplitudes of the elements on both adjacent positions along the main diagonal are the largest, while the amplitudes decay from left to right. In addition, the amplitudes of elements far from the main diagonal are lower;

(3)

In symmetric cases without the T/4-delay link [7], along the main diagonal direction, the matrix elements decrease monotonically. However, with the influence of the positive and negative sequence separation links based on T/4 delay, the AC-DC inter-harmonic coupling relationship near the fundamental frequency is significantly weakened. This conclusion is supported by both Figure 5a,b, where the amplitudes of elements (2, 1) and (1, 2) are smaller than those of adjacent elements;

(4)

When modeling other asymmetric working conditions and outer loop control strategies, it is only necessary to modify u_cqt according to the positive and negative sequence voltage conditions of the AC port in a specific scenario. In the meantime, it just needs to modify u_ctrl3 and the correlation coefficient matrix B_ctrl3 and D_ctrl3 according to the specific control strategy. Therefore, the modeling method proposed in this paper exhibits good generality and extensibility.

6. Simulation Verification and Discussion

To verify the correctness and effectiveness of the proposed method and model, this paper establishes the electromagnetic transient simulation model, average value model and HSS model based on the same set of benchmark parameters shown in Table 1. The output results of the three methods are first compared in Figure 6. It can be seen that the steady-state outputs of the model are highly consistent with the electromagnetic transient simulation outcome, and the accuracy increases with an increase in the model order, which proves the correctness of the proposed modeling method and the established mathematical model.

To further verify the quantitative harmonic transfer gains, two additional tests are carried out with different perturbation currents injected into the DC-side current, namely, (20 A, 150 Hz) and (50 A, 100 Hz). In each case, the dominant positive- and negative-sequence current harmonics are extracted from the simulation waveforms and compared with the value output by the established model. The comparison results are shown in Figure 7. When a 150 Hz DC perturbance current is injected, the positive- and negative-sequence harmonics share the same spectrum, and the 150 Hz ± 50 Hz components are dominant in both sequences, which is consistent with the results in Figure 5. Quantitively, according to Figure 7a the amplitude of each frequency component calculated by the HSS model is consistent with the simulation results, while the maximum relative error of the 150 Hz ± 50 Hz component is 4.51%. When a 200 Hz DC perturbance current is injected, similar conclusions can be drawn from Figure 7b, and the maximum relative error is 6.10%. The increased relative error is because the injected harmonic current in the second case has a smaller amplitude and a higher frequency. Both factors lead to decreased coupled harmonic amplitude, which further results in an increase in the relative error.

To conclude, the quantitative analysis and comparison show that the AC-DC harmonic coupling gains generated by the HSS model have good matching performance with the electromagnetic transient simulation model, which verifies the accuracy of the established model and the correctness of the proposed modeling method.

7. Conclusions

This paper proposes a simplified small-signal HSS modeling method for a three-phase VSC with T/4-delay-based DDSRF control system under asymmetric conditions. The harmonic transfer matrix between the AC positive/negative sequence and the DC current is established, and the global harmonic coupling relationship between AC and DC currents is revealed. The following conclusions are drawn from this study:

(1)

The proposed modeling method provides a practical solution to the steady-state modeling problem of the T/4-delay-based DDSRF control system, which highlights the accurate steady-state characteristics description without increasing the model complexity. In addition, the proposed modeling method has good versatility and ductility, which can be easily extended to different symmetrical working conditions.

(2)

Strong coupling exists between the positive- and negative-sequence current harmonics and the DC current harmonics. For the n_Δth-order harmonic current on the DC side, the coupled AC-side positive- and negative-sequence currents are both of (n_Δ ± k)th-order (k = 1, 3, 5, …).

(3)

The T/4-delay-based sequence-separation link has considerable influence on the coupling relationship, i.e., compared with existing research results that utilize other separation techniques, the AC-DC inter-harmonic coupling near the fundamental frequency is significantly attenuated.

(4)

The output of the established model shows good matching performance with the electromagnetic transient simulation, indicating that it can correctly reflect the steady-state harmonic coupling characteristics of the three-phase VSC.

This paper focuses on the AC-DC harmonic coupling mechanism and modeling research of a single VSC device, which lays the foundation for harmonic calculation at the system level. The main contribution lies in the proposal of a method that solves the linear modeling problem of the nonlinear delay link without increasing the model complexity and computation burden, which is crucial for online harmonic monitoring and estimation toward a complex power grid. In the future, based on the AC-DC harmonic coupling mechanism revealed in this work, research on the unified harmonic modeling and analysis method of an AC-DC hybrid power grid with multiple VSCs will be carried out, and thus provides theoretical support for harmonic monitoring and governance of a new power system under the background of “double carbon”.