Pole-to-Pole Fault Current Impact Factors Analysis Based on Equivalent Impedance for Modular Multilevel Converter High-Voltage Direct Current System

Abstract

The fault current in MMC HVDC systems suffers from a quick rising rate and no zero-cross problems, which makes it hard to cut off and has become the main problem when promoting MMC dc systems. Among the different types of faults, the pole-to-pole fault is the most serious one. Hence, to clarify the discharging mechanism during pole-to-pole faults and determine the influential factors, this paper proposes a simplified modeling method for MMC HVDC fault current impact factors analysis. The proposed method uses the simplified MMC circuit when faults happen, and the frequency domain model is then established for fault current analysis. Considering that the fault period concerned always is less than 10 ms, the high-frequency part impedance characteristics of the established model are carefully studied when device parameters are varied. The smoothing reactor has a significant influence on the high-frequency impedance, while the arm inductance and arm resistance have much less impact on high-frequency impedance. This demonstrates, theoretically, that the pole-to-pole fault current is basically influenced by the smoothing reactor rather than other impact factors. The calculation results based on the model also prove that fault current will change a lot if the smoothing reactor’s parameters are different. The analysis verifies the accuracy of the equivalent model and supplies a theoretical analysis tool for MMC HVDC fault current analysis.

1. Introduction

The Modular Multilevel Converter (MMC) is a novel converter [1,2,3] characterized by low losses, easy modular cascade expansion, and the capability to achieve multilevel voltage output. Leveraging these advantages, the MMC has led to the development of the Modular Multilevel Converter Multi-Terminal High-Voltage Direct Current (MMC-MTDC) system. Flexible DC transmission technology, which meets the requirements for large-capacity and high-stability power transmission while enabling stable grid integration of clean energy generation, has become a focal point of current research [4,5].

Currently, multi-terminal DC transmission projects are sprouting up both domestically and internationally. In 2002, the United States officially proposed the “Smart Grid” project, planning over 60 flexible DC transmission projects over the next two decades to meet the significant demands of renewable energy development and the construction of a clean, flexible, and environmentally friendly smart grid [6]. In 2008, Europe utilized DC grid technology to integrate wind power from the North Sea and the Baltic Sea, as well as photovoltaic power from North Africa and the Middle East. In 2010, the United States commissioned its first MMC-MTDC project, significantly enhancing the stability and large-capacity power supply of the San Francisco power system. In 2011, the US built a backbone grid based on advanced DC transmission technology to accommodate the rapid expansion of new energy sources. In 2014, China commissioned a five-terminal project in Zhoushan, featuring diverse operation modes and enabling the transmission of green electricity from large-scale wind power integration.

Looking ahead, China will develop large-scale and complex AC-DC hybrid grids, necessitating further research. However, practical experience with MMC-MTDC systems is limited globally; thus, many unresolved challenges remain. Unlike AC systems, MMC-MTDC systems exhibit low inertia, numerous controllers, and susceptibility to submodule breakdown. Consequently, DC line short-circuit faults pose a significant overcurrent risk. Failure to effectively mitigate these faults can lead to converter station shutdowns and even the collapse of the DC system. As DC grids expand, the frequency of fault events is increasing.

In the early stages, the mainstream method used for fault current analysis and calculation in VSC HVDC grids was electromagnetic transient simulation [7,8]. By establishing electromagnetic transient simulation models, the fault characteristics of various faults in VSC HVDC systems were analyzed, and the evolution features of DC faults were summarized. Reference [9] conducted simulation studies on the fault characteristics and evolution mechanisms of three types of faults in MMC-HVDC systems: DC line breaks, pole-to-ground, and pole-to-pole short circuits. Reference [10] analyzed the DC fault characteristics in multi-terminal DC systems, simulated their response characteristics, and further proposed corresponding control strategies and protection sequences, enhancing the fault ride-through capability. Although simulation methods can yield accurate results, they are limited to obtaining current and voltage information for the specific faults set. Additionally, the detailed modeling process of power electronic converters is complex and computationally time-consuming, making it difficult to further quantitatively analyze the influence of circuit parameters, grid structures, and operating modes on fault currents. Therefore, the application of this method is limited.

Scholars both domestically and internationally have conducted extensive research on DC fault calculation, continuously refining computational methods. For the calculation of DC fault currents in a single MMC, the second-order discharge characteristics of the fault circuit are primarily utilized to solve differential equations. Reference [11] established a transient analysis model for DC pole-to-ground faults, categorizing the capacitive discharge process into underdamped oscillation and overdamped non-oscillation scenarios based on the grounding impedance, and providing time-domain expressions for capacitive voltage and discharge current in each case. Reference [12] equated the fault circuit of a pole-to-pole short-circuit fault to an RLC oscillatory discharge circuit, deriving formulas for capacitive voltage and arm current calculations, and presenting the peak value of the short-circuit current along with the time at which it occurs.

Reference [13] proposed a method for calculating inter-pole fault currents based on original differential equations, discussing the key components that influence the fault current. However, the high-order differential equations used in fault current calculations are cumbersome, limiting their application in fault current analysis for large-scale DC grids. To address this computational challenge, a state-space model is introduced in reference [14] to calculate inter-pole fault currents in multi-terminal DC systems, providing highly accurate and efficient numerical solutions. Based on the state-space model of MMCs, fault current characteristics can be analyzed to a certain extent. Reference [15] establishes state-space equations to solve the time-domain variation curves of fault currents and voltages in a four-terminal system, achieving high-precision solutions but involving a 18th-order matrix, making the solution process cumbersome and lacking in an analytical expression. To derive an analytical expression for fault currents, reference [16] equivalently transforms a three-terminal ring network into a two-terminal open network, yielding a clear expression with an error within 10%, which is slightly high. Reference [17] divides the DC grid lines into inner and outer zones based on the number of smoothing reactors separating them from the fault point, directly treating the outer zone as open-circuit, providing a valuable approach for studying analytical expressions. Reference [14] establishes a two-terminal system network model, providing initial and modified matrix equations for the DC grid before a fault, enhancing computational efficiency and quantitatively analyzing the transient characteristics of short-circuit currents in flexible DC grids under various operating conditions. However, it fails to analyze the intrinsic relationships between fault currents and various parameters. Reference [13] considers the impact of submodule arm currents, AC-side feeding, and converter control, concluding that the AC side has almost no influence on the injection of DC fault currents.

Flexible DC grids, composed of numerous power electronic devices with low inertia coefficients and incorporating a multitude of controllers, exhibit fault mechanisms that are fundamentally different from those of AC systems. In terms of mechanism research, most literature primarily focuses on establishing equivalent models to explore fault paths. Reference [18] introduces an approximate analysis model for calculating pole-to-ground faults in MMC-HVDC systems, laying the groundwork for fault mechanism studies. Reference [19] proposes fault protection strategies for pole-to-ground faults in MMC-HVDC systems, accompanied by a detailed analysis of fault paths. Reference [20] delves into the evolution of fault currents and the mechanisms leading to system overvoltage following pole-to-ground and pole-to-pole faults in MMC-HVDC systems based on a system fault model; however, this study remains at the theoretical level without considering practical engineering scenarios.

Reference [21], grounded in the actual Zhangbei project, investigates fault evolution processes, feature detection principles, and fault-influencing factors, contributing to practical engineering significance. Nonetheless, this theory is still at the qualitative analysis stage, lacking a theoretical derivation method. Reference [22] demonstrates that short-circuit fault currents are supplied by both the AC system feed and submodule capacitor discharge. Through theoretical and simulation analyses, the equivalent capacitance value of the converter station’s arm is determined, clarifying the sources of fault currents and the total discharge of submodules. Reference [23] further refines the accuracy of these capacitance values by proposing that both the modulation ratio and short-circuit ratio affect the equivalent capacitance, thereby correcting the capacitance value accuracy presented in earlier literature.

In summary, while the existing literature has conducted detailed analyses and calculations of DC fault characteristics in flexible DC grids, taking into account factors such as control systems and AC infeed [24], and proposing accurate methods for fault current calculation, these studies have also significantly simplified the computational process by leveraging the distribution characteristics of fault currents while maintaining practical engineering precision. This has provided valuable insights for the in-depth investigation of DC fault mechanisms and equipment selection. However, among the current calculation methods, there is a notable lack of research that quantitatively analyzes the factors influencing fault currents. Basically, the above methods can be divided into two types, namely the numerical computation and direct calculation method, based on discharging circuits. The following table lists the typical existing models for DC fault current calculations and their advantages and disadvantages. It can be seen from

Table 1. The typical existing models for DC fault current calculations.

The Typical Existing Models for DC Fault Current Calculations	Advantages	Disadvantages
Numerical computation methods	Suitable for different DC grids	May take a long time
Direct calculation method based on RLC discharging circuits	Can obtain the explicit expression	The theoretical basis is weak and the control’s impact is not considered

that each method has disadvantages.

This paper derives an explicit expression for calculating fault currents in VSC-DC transmission systems directly based on the theory and conducts an in-depth analysis of various factors affecting the level of DC fault currents using changes in frequency-domain impedance characteristics.

The novelties of this paper can be concluded as follows:

(1) The simplified equivalent model for the DC side fault current is proposed in this paper. The model considers the control loop’s impact and has more accuracy than traditional models.

(2) Different parameters’ impacts on MMC are analyzed based on the proposed model. The most influential components influencing the fault currents are studied, based on the high-frequency impedance evaluations.

The structure of the paper is as follows: Section 2 introduces the MMC circuit and model with different control modes, Section 3 proposes the MMC equivalent impedance model for fault current analysis, Section 4 analyzes the fault current impact factor based on impedance calculation and makes simulation verifications, and Section 5 concludes the paper.

2. MMC Circuit and Model in Different Control Modes

2.1. Equivalent Circuit for Converter

The internal commutation process of the Modular Multilevel Converter (MMC) is intricate, making it challenging to describe with a state-switching function. Furthermore, the studies discussed herein focus on the DC grid system and do not delve into the specific commutation processes within the converter. This is because the time scope for faults current analysis taken into consideration in this paper is very short: 10 ms. In such transient periods, the losses, AC–side interactions, electromagnetic coupling effects, and harmonics will not have an obvious impact on the fault current, because these factors always take time to have an impact.

Therefore, from the perspective of the external characteristics of the DC grid, based on the principle of power balance, the MMC can be equivalently represented as a controlled current source. The DC side of the MMC is simplified as an RLC circuit incorporating this controlled current source, as illustrated in Figure 1. Neglecting the losses within the converter station, it is assumed that all the AC power injected into the station is converted into the injected power on the DC side.

In the figure, u_g represents the amplitude of the equivalent power supply voltage; R_g and L_g denote the equivalent resistance and inductance of the equivalent power supply, respectively; R_t and L_t represent the equivalent resistance and inductance of the converter transformer, respectively; R_m and L_m are the resistance and inductance of the bridge arm, respectively; K_t is the turns ratio of the converter transformer; u_s and i_s are the voltage and current at the Point of Common Coupling (PCC), respectively; u_c and u_ceq are the AC equivalent internal electromotive force of the Modular Multilevel Converter (MMC) and the total DC component of the capacitor voltages of the invested submodules, respectively; i_dc and u_dc are the output DC and port DC voltage of the MMC, respectively; and C_eq = 6 C/N is the equivalent DC capacitance, where C is the submodule capacitance and N is the number of submodules in the bridge arm.

The mathematical model at DC side can be expressed as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{dc}}=u_{\mathrm{ceq}}-R_{\mathrm{eq}}{i}_{\mathrm{dc}}-L_{\mathrm{eq}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{dc}}}{\mathrm{d}t}}\hfill \\ i_{\mathrm{dc}}={\displaystyle \frac{p_{\mathrm{c}}}{u_{\mathrm{ceq}}}}-C_{\mathrm{eq}}{\displaystyle \frac{\mathrm{d}{u}_{\mathrm{ceq}}}{\mathrm{d}t}}\hfill \end{array}\right.$$

(1)

By conducting a small-signal analysis of Equation (1) at the steady-state operating point, we obtain the simplified small-signal linearized model of the converter as follows:

$$\Delta u_{\mathrm{dc}}={\displaystyle \frac{U_{\mathrm{ceq}0}}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}}\Delta p_{\mathrm{c}}-\left({\displaystyle \frac{U_{\mathrm{ceq}0}^2}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}}+R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)\Delta i_{\mathrm{dc}}$$

(2)

In the equation, variables denoted by uppercase letters with a subscript of 0 represent the steady-state values of the corresponding physical quantities; Δ signifies a small disturbance signal for that physical quantity.

2.2. The Modeling of the AC System and the Inner Control Loop

From Figure 1, the electromagnetic transient equation for the AC side of the Modular Multilevel Converter (MMC) can be derived.

$$\left\{\begin{array}{c}{L}_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{sd}}}{\mathrm{d}t}}=u_{\mathrm{sd}}-u_{\mathrm{cd}}-R_{\mathrm{s}}{i}_{\mathrm{sd}}-\omega L_{\mathrm{s}}{i}_{\mathrm{sq}}\\ L_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{sq}}}{\mathrm{d}t}}=u_{\mathrm{sq}}-u_{\mathrm{cq}}-R_{\mathrm{s}}{i}_{\mathrm{sq}}+\omega L_{\mathrm{s}}{i}_{\mathrm{sd}}\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{L}_{\mathrm{ac}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{sd}}}{\mathrm{d}t}}=u_{\mathrm{gd}}-u_{\mathrm{cd}}-R_{\mathrm{ac}}{i}_{\mathrm{sd}}-\omega L_{\mathrm{ac}}{i}_{\mathrm{sq}}\hfill \\ L_{\mathrm{ac}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{sq}}}{\mathrm{d}t}}=u_{\mathrm{gq}}-u_{\mathrm{cq}}-R_{\mathrm{ac}}{i}_{\mathrm{sq}}+\omega L_{\mathrm{ac}}{i}_{\mathrm{sd}}\hfill \end{array}\right.$$

(4)

$$\left\{\begin{array}{l}{L}_{\mathrm{s}}=L_{\mathrm{t}}+{\displaystyle \frac{L_{\mathrm{m}}}{2}}{K}_{\mathrm{t}}^2,R_{\mathrm{s}}=R_{\mathrm{t}}+{\displaystyle \frac{R_{\mathrm{m}}}{2}}{K}_{\mathrm{t}}^2\hfill \\ L_{\mathrm{ac}}=L_{\mathrm{g}}+L_{\mathrm{s}},R_{\mathrm{ac}}=R_{\mathrm{g}}+R_{\mathrm{s}}\hfill \end{array}\right.$$

(5)

Based on the control principle of the MMC’s DC side, the expression for the reference voltage can be obtained.

$$\left\{\begin{array}{l}{u}_{\mathrm{cd}}^{*}=u_{\mathrm{sd}}-\omega L_{\mathrm{s}}{i}_{\mathrm{sq}}-G_{\mathrm{id}}(i_{\mathrm{sd}}^{*}-i_{\mathrm{sd}})\hfill \\ u_{\mathrm{cq}}^{*}=u_{\mathrm{sq}}+\omega L_{\mathrm{s}}{i}_{\mathrm{sd}}-G_{\mathrm{iq}}(i_{\mathrm{sq}}^{*}-i_{\mathrm{sq}})\hfill \end{array}\right.$$

(6)

In the equation, G_id and G_iq represent the transfer functions of the inner-loop Proportional–Integral (PI) controllers for the d-axis and q-axis, respectively. Neglecting any delays and assuming that the output voltage is equal to the reference voltage, by substituting Equation (6) into the small-signal model of Equation (3), we obtain the following equation:

$$\left\{\begin{array}{l}\Delta i_{\mathrm{sd}}={\displaystyle \frac{G_{\mathrm{id}}}{R_{\mathrm{s}}+s{L}_{\mathrm{s}}+G_{\mathrm{id}}}}\Delta i_{\mathrm{sd}}^{*}=A_{\mathrm{d}}\Delta i_{\mathrm{sd}}^{*}\hfill \\ \Delta i_{\mathrm{sq}}={\displaystyle \frac{G_{\mathrm{iq}}}{R_{\mathrm{s}}+s{L}_{\mathrm{s}}+G_{\mathrm{iq}}}}\Delta i_{\mathrm{sq}}^{*}=A_{\mathrm{q}}\Delta i_{\mathrm{sq}}^{*}\hfill \end{array}\right.$$

(7)

2.3. The Modeling of the Outer Control Loop in Different Modes

■

Constant power control mode

When the converter adopts constant power control, the reference value for the inner current loop is given as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{sd}}^{*}=G_{\mathrm{od}}\left(P^{*}-P\right)\hfill \\ i_{\mathrm{sq}}^{*}=G_{\mathrm{oq}}\left(Q^{*}-Q\right)\hfill \end{array}\right.$$

(8)

In the equation, G_od and G_oq represent the transfer functions of the outer-loop Proportional–Integral (PI) controllers for the d-axis and q-axis, respectively. P* and Q* denote the reference values for active power and reactive power, respectively. By linearizing Equation (8), we obtain

$$\left\{\begin{array}{l}\begin{array}{ll}\Delta i_{\mathrm{sd}}^{*}& =G_{\mathrm{od}}\left(\Delta P^{*}-\Delta P\right)=G_{\mathrm{od}}\left(\Delta P^{*}-\left(U_{\mathrm{sd}0}\Delta i_{\mathrm{sd}}+I_{\mathrm{sd}0}\Delta u_{\mathrm{sd}}+U_{\mathrm{sq}0}\Delta i_{\mathrm{sq}}+I_{\mathrm{sq}0}\Delta u_{\mathrm{sq}}\right)\right)\\ & =G_{\mathrm{od}}\left[\Delta P^{*}\underset{B_{\mathrm{d}1}}{\underbrace{-\left(U_{\mathrm{sd}0}-\left(R_{\mathrm{g}}+s{L}_{\mathrm{g}}\right)I_{\mathrm{sd}0}+\omega L_{\mathrm{g}}{I}_{\mathrm{sq}0}\right)}}\Delta i_{\mathrm{sd}}\underset{B_{\mathrm{d}2}}{\underbrace{-\left(U_{\mathrm{sq}0}-\left(R_{\mathrm{g}}+s{L}_{\mathrm{g}}\right)I_{\mathrm{sq}0}-\omega L_{\mathrm{g}}{I}_{\mathrm{sd}0}\right)}}\Delta i_{\mathrm{sq}}\right]\\ & =G_{\mathrm{od}}\left(\Delta P^{*}+B_{\mathrm{d}1}\Delta i_{\mathrm{sd}}+B_{\mathrm{d}2}\Delta i_{\mathrm{sq}}\right)\end{array}\hfill \\ \begin{array}{ll}\Delta i_{\mathrm{sq}}^{*}& =G_{\mathrm{oq}}\left(\Delta Q^{*}-\Delta Q\right)=G_{\mathrm{oq}}\left(\Delta Q^{*}-\left(U_{\mathrm{sd}0}\Delta i_{\mathrm{sq}}+I_{\mathrm{sq}0}\Delta u_{\mathrm{sd}}-U_{\mathrm{sq}0}\Delta i_{\mathrm{sd}}-I_{\mathrm{sd}0}\Delta u_{\mathrm{sq}}\right)\right)\\ & =G_{\mathrm{oq}}\left[\Delta Q^{*}\underset{B_{\mathrm{q}1}}{\underbrace{+\left(U_{\mathrm{sq}0}+\left(R_{\mathrm{g}}+s{L}_{\mathrm{g}}\right)I_{\mathrm{sq}0}+\omega L_{\mathrm{g}}{I}_{\mathrm{sd}0}\right)}}\Delta i_{\mathrm{sd}}\underset{B_{\mathrm{q}2}}{\underbrace{-\left(U_{\mathrm{sd}0}+\left(R_{\mathrm{g}}+s{L}_{\mathrm{g}}\right)I_{\mathrm{sd}0}-\omega L_{\mathrm{g}}{I}_{\mathrm{sq}0}\right)}}\Delta i_{\mathrm{sq}}\right]\\ & =G_{\mathrm{oq}}\left(\Delta Q^{*}+B_{\mathrm{q}1}\Delta i_{\mathrm{sd}}+B_{\mathrm{q}2}\Delta i_{\mathrm{sq}}\right)\end{array}\hfill \end{array}\right.$$

(9)

By substituting (9) into (8), we obtain:

$$\left\{\begin{array}{l}\begin{array}{ll}\Delta i_{\mathrm{sd}}& ={\displaystyle \frac{\left(A_{\mathrm{d}}{G}_{\mathrm{od}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\right)\Delta P^{*}+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\Delta Q^{*}}{1-A_{\mathrm{d}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}-A_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{oq}}+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}}}\\ & =C_{\mathrm{d}1}\Delta P^{*}+C_{\mathrm{d}2}\Delta Q^{*}\end{array}\hfill \\ \begin{array}{ll}\Delta i_{\mathrm{sq}}& ={\displaystyle \frac{A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\Delta P^{*}+\left(A_{\mathrm{q}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\right)\Delta Q^{*}}{1-A_{\mathrm{d}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}-A_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{oq}}+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}}}\\ & =C_{\mathrm{q}1}\Delta P^{*}+C_{\mathrm{q}2}\Delta Q^{*}\end{array}\hfill \end{array}\right.$$

(10)

■

Constant DC voltage control mode

When the converter adopts constant DC voltage control, the reference value for the inner current loop is given as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{sd}}^{*}=G_{\mathrm{od}}\left(U_{\mathrm{dc}}^{*}-u_{\mathrm{dc}}\right)\hfill \\ i_{\mathrm{sq}}^{*}=G_{\mathrm{oq}}\left(Q^{*}-Q\right)\hfill \end{array}\right.$$

(11)

By linearizing Equation (11), we obtain

$$\left\{\begin{array}{l}\Delta i_{\mathrm{sd}}^{*}=G_{\mathrm{od}}\left(\Delta U_{\mathrm{dc}}^{*}-\Delta u_{\mathrm{dc}}\right)\hfill \\ \Delta i_{\mathrm{sq}}^{*}=G_{\mathrm{oq}}\left(\Delta Q^{*}-\Delta Q\right)=G_{\mathrm{oq}}\left(\Delta Q^{*}+B_{\mathrm{q}1}\Delta i_{\mathrm{sd}}+B_{\mathrm{q}2}\Delta i_{\mathrm{sq}}\right)\hfill \end{array}\right.$$

(12)

By substituting (12) into (8), the equivalent model for constant DC voltage control can be written as follows:

$$\left\{\begin{array}{l}\begin{array}{ll}\Delta i_{\mathrm{sd}}& =A_{\mathrm{d}}{G}_{\mathrm{od}}\Delta U_{\mathrm{dc}}^{*}-A_{\mathrm{d}}{G}_{\mathrm{od}}\Delta u_{\mathrm{dc}}\\ & =D_{\mathrm{d}1}\Delta U_{\mathrm{dc}}^{*}+D_{\mathrm{d}2}\Delta u_{\mathrm{dc}}\end{array}\hfill \\ \begin{array}{ll}\Delta i_{\mathrm{sq}}& ={\displaystyle \frac{A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\Delta U_{\mathrm{dc}}^{*}+A_{\mathrm{q}}{G}_{\mathrm{oq}}\Delta Q^{*}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\Delta u_{\mathrm{dc}}}{1-A_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{oq}}}}\\ & =D_{\mathrm{q}1}\Delta U_{\mathrm{dc}}^{*}+D_{\mathrm{q}2}\Delta Q^{*}+D_{\mathrm{q}3}\Delta u_{\mathrm{dc}}\end{array}\hfill \end{array}\right.$$

(13)

■

Droop control

When the converter adopts droop control, the reference value for the inner current loop is given as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{sd}}^{*}=G_{\mathrm{od}}\left[K_{\mathrm{d}}\left(U_{\mathrm{dc}}^{*}-u_{\mathrm{dc}}\right)+\left(P^{*}-P\right)\right]\hfill \\ i_{\mathrm{sq}}^{*}=G_{\mathrm{oq}}\left(Q^{*}-Q\right)\hfill \end{array}\right.$$

(14)

Similarly, by linearizing Equation (14) and substituting (14) into (8), the equivalent model for droop control can be written as follows:

$$\left\{\begin{array}{l}\begin{array}{ll}\Delta i_{\mathrm{sd}}& ={\displaystyle \frac{\left(A_{\mathrm{d}}{G}_{\mathrm{od}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\right)\left(\Delta P^{*}+K_{\mathrm{d}}\Delta U_{\mathrm{dc}}^{*}-K_{\mathrm{d}}\Delta u_{\mathrm{dc}}\right)+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\Delta Q^{*}+}{1-A_{\mathrm{d}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}-A_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{oq}}+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}}}\\ & =C_{\mathrm{d}1}\Delta P^{*}+K_{\mathrm{d}}{C}_{\mathrm{d}1}\Delta U_{\mathrm{dc}}^{*}-K_{\mathrm{d}}{C}_{\mathrm{d}1}\Delta u_{\mathrm{dc}}+C_{\mathrm{d}2}\Delta Q^{*}\end{array}\hfill \\ \begin{array}{ll}\Delta i_{\mathrm{sq}}& ={\displaystyle \frac{A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\left(\Delta P^{*}+K_{\mathrm{d}}\Delta U_{\mathrm{dc}}^{*}-K_{\mathrm{d}}\Delta u_{\mathrm{dc}}\right)+\left(A_{\mathrm{q}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}\right)\Delta Q^{*}}{1-A_{\mathrm{d}}{B}_{\mathrm{d}1}{G}_{\mathrm{od}}-A_{\mathrm{q}}{B}_{\mathrm{q}2}{G}_{\mathrm{oq}}+A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}1}{B}_{\mathrm{q}2}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}-A_{\mathrm{d}}{A}_{\mathrm{q}}{B}_{\mathrm{d}2}{B}_{\mathrm{q}1}{G}_{\mathrm{od}}{G}_{\mathrm{oq}}}}\\ & =C_{\mathrm{q}1}\Delta P^{*}+K_{\mathrm{d}}{C}_{\mathrm{q}1}\Delta U_{\mathrm{dc}}^{*}-K_{\mathrm{d}}{C}_{\mathrm{q}1}\Delta u_{\mathrm{dc}}+C_{\mathrm{q}2}\Delta Q^{*}\end{array}\hfill \end{array}\right.$$

(15)

3. The MMC Equivalent Impedance Model for Fault Current Analysis

3.1. MMC Detailed Equivalent Impedance Model

The linearized mathematical model for the active power injected into the converter is represented as follows:

$$\begin{array}{ll}\Delta p_{\mathrm{c}}& =U_{\mathrm{cd}0}\Delta i_{\mathrm{sd}}+I_{\mathrm{sd}0}\Delta u_{\mathrm{cd}}+U_{\mathrm{cq}0}\Delta i_{\mathrm{sq}}+I_{\mathrm{sq}0}\Delta u_{\mathrm{cq}}\\ & =\left(U_{\mathrm{cd}0}-\left(R_{\mathrm{ac}}+s{L}_{\mathrm{ac}}\right)I_{\mathrm{sd}0}+\omega L_{\mathrm{ac}}{I}_{\mathrm{sq}0}\right)\Delta i_{\mathrm{sd}}\\ & +\left(U_{\mathrm{cq}0}-\left(R_{\mathrm{ac}}+s{L}_{\mathrm{ac}}\right)I_{\mathrm{sq}0}-\omega L_{\mathrm{ac}}{I}_{\mathrm{sd}0}\right)\Delta i_{\mathrm{sq}}\\ & =E_1\Delta i_{\mathrm{sd}}+E_2\Delta i_{\mathrm{sq}}\end{array}$$

(16)

By substituting Equations (10), (13) and (15) into Equation (16), respectively, and converting them into nominal values, we obtain the expressions for the injected power under the three control modes:

$$\Delta p_{\mathrm{c}}=\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)\Delta P^{*}+\left(E_1{C}_{\mathrm{d}2}+E_2{C}_{\mathrm{q}2}\right)\Delta Q^{*}$$

(17)

$$\begin{array}{ll}\Delta p_{\mathrm{c}}& =\left(E_1{D}_{\mathrm{d}1}+E_2{D}_{\mathrm{q}1}\right){\displaystyle \frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}\Delta U_{\mathrm{dc}}^{*}+E_2{D}_{\mathrm{q}2}\Delta Q^{*}\\ & +\left(E_1{D}_{\mathrm{d}2}+E_2{D}_{\mathrm{q}3}\right){\displaystyle \frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}\Delta u_{\mathrm{dc}}\end{array}$$

(18)

$$\begin{array}{ll}\Delta p_{\mathrm{c}}& =\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)\Delta P^{*}+\left(E_1{C}_{\mathrm{d}2}+E_2{C}_{\mathrm{q}2}\right)\Delta Q^{*}\\ & +K_{\mathrm{d}}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right){\displaystyle \frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}\Delta U_{\mathrm{dc}}^{*}-K_{\mathrm{d}}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right){\displaystyle \frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}\Delta u_{\mathrm{dc}}\end{array}$$

(19)

By substituting Equations (17)–(19) into Equation (2), respectively, we obtain the Thevenin equivalent model of the converter.

$$\begin{array}{ll}\Delta u_{\mathrm{dc}}& ={\displaystyle \frac{U_{\mathrm{ceq}0}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)\Delta P^{*}}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}}+{\displaystyle \frac{U_{\mathrm{ceq}0}\left(E_1{C}_{\mathrm{d}2}+E_2{C}_{\mathrm{q}2}\right)\Delta Q^{*}}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}}\\ & -\left({\displaystyle \frac{U_{\mathrm{ceq}0}^2}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}}+R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)\Delta i_{\mathrm{dc}}\\ & =F_{\mathrm{PQ}\_\mathrm{Pr}\mathrm{ef}}\Delta P^{*}+F_{\mathrm{PQ}\_\mathrm{Q}r\mathrm{ef}}\Delta Q^{*}-F_{\mathrm{PQ}\_\mathrm{idc}}\Delta i_{\mathrm{dc}}\end{array}$$

(20)

$$\begin{array}{ll}\Delta u_{\mathrm{dc}}& ={\displaystyle \frac{\frac{U_{\mathrm{ceq}0}\left(E_1{D}_{\mathrm{d}1}+E_2{D}_{\mathrm{q}1}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}\Delta U_{\mathrm{dc}}^{*}+\frac{U_{\mathrm{ceq}0}{E}_2{D}_{\mathrm{q}2}}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\Delta Q^{*}}{1-\frac{U_{\mathrm{ceq}0}\left(E_1{D}_{\mathrm{d}2}+E_2{D}_{\mathrm{q}3}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}}\\ & -{\displaystyle \frac{\left(\frac{U_{\mathrm{ceq}0}^2}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}+R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)\Delta i_{\mathrm{dc}}}{1-\frac{U_{\mathrm{ceq}0}\left(E_1{D}_{\mathrm{d}2}+E_2{D}_{\mathrm{q}3}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}}\\ & =F_{\mathrm{UdcQ}\_\mathrm{Udc}r\mathrm{ef}}\Delta U_{\mathrm{dc}}^{*}+F_{\mathrm{UdcQ}\_\mathrm{Q}r\mathrm{ef}}\Delta Q^{*}-F_{\mathrm{UdcQ}\_\mathrm{idc}}\Delta i_{\mathrm{dc}}\end{array}$$

(21)

$$\begin{array}{ll}\Delta u_{\mathrm{dc}}& ={\displaystyle \frac{\frac{U_{\mathrm{ceq}0}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\Delta P^{*}+\frac{U_{\mathrm{ceq}0}{K}_{\mathrm{d}}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}\Delta U_{\mathrm{dc}}^{*}}{1+\frac{U_{\mathrm{ceq}0}{K}_{\mathrm{d}}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}}\\ & +{\displaystyle \frac{\frac{U_{\mathrm{ceq}0}\left(E_1{C}_{\mathrm{d}2}+E_2{C}_{\mathrm{q}2}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\Delta Q^{*}-\left(\frac{U_{\mathrm{ceq}0}^2}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}+R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)\Delta i_{\mathrm{dc}}}{1+\frac{U_{\mathrm{ceq}0}{K}_{\mathrm{d}}\left(E_1{C}_{\mathrm{d}1}+E_2{C}_{\mathrm{q}1}\right)}{P_{\mathrm{c}0}+s{C}_{\mathrm{eq}}{U}_{\mathrm{ceq}0}^2}\frac{P_{\mathrm{B}}}{U_{\mathrm{dcB}}}}}\\ & =F_{\mathrm{droop}\_\mathrm{P}r\mathrm{ef}}\Delta P^{*}+F_{\mathrm{droop}\_\mathrm{Udc}r\mathrm{ef}}\Delta U_{\mathrm{dc}}^{*}+F_{\mathrm{droop}\_\mathrm{Q}r\mathrm{ef}}\Delta Q^{*}-F_{\mathrm{droop}\_\mathrm{idc}}\Delta i_{\mathrm{dc}}\end{array}$$

(22)

The DC impedances of converters employing constant power control, constant voltage control, and droop control are denoted as F_{PQ_idc}, F_{UdcQ_idc}, and F_{droop_idc}, respectively.

3.2. The Simplified MMC Equivalent Impedance Model for Fault Current Analysis

The MMC DC impedance expression established in the previous section has a high order, leading to slow computation and low efficiency when analyzing the fault damping characteristics of large-scale DC grids. Therefore, it is necessary to obtain a simplified impedance model for the MMC by eliminating variables that have minimal impact on the damping characteristics during the initial stage of a fault.

Based on the simplified circuit of the MMC shown in Figure 1, the relationship between power, voltage, and current on the DC side can be derived as follows:

$$\left\{\begin{array}{l}\Delta p_c=u_{\mathrm{ceq}0}\cdot \Delta i_{\mathrm{d}}+i_{\mathrm{d}0}\cdot \Delta u_{\mathrm{ceq}}\hfill \\ \Delta u_{\mathrm{ceq}}=\Delta u_{\mathrm{dc}}+\left(R_{\mathrm{ceq}}+s{L}_{\mathrm{ceq}}\right)\Delta i_{dc}\hfill \\ \Delta i_{\mathrm{d}}=s{C}_{\mathrm{ceq}}\Delta u_{\mathrm{dc}}+\left[1+s{C}_{\mathrm{ceq}}\left(R_{\mathrm{ceq}}+s{L}_{\mathrm{ceq}}\right)\right]\Delta i_{\mathrm{dc}}\hfill \end{array}\right.$$

(23)

In the equation, i_d represents the current injected into the converter from the valve side.

As analyzed previously, the DC impedance of the converter is closely related to its control model. For converters employing constant active power control, it can be assumed that the AC power injected into the converter station matches the reference value set by the power controller. Even in the event of a DC line fault, the injected power remains constant in the short term. Furthermore, the power conversion efficiency of the MMC (Modular Multilevel Converter) is extremely high, allowing us to neglect active power losses. Therefore, it can be concluded that:

$$\Delta p_c=0$$

(24)

By substituting Equation (24) into Equation (23) and simplifying, we obtain the simplified DC impedance of the MMC as follows:

$${\mathrm{Z}}_{\mathrm{dc}}={\displaystyle \frac{{\mathrm{U}}_{\mathrm{ceq}0}\left[1+\mathrm{s}{\mathrm{C}}_{\mathrm{ceq}}\left({\mathrm{R}}_{\mathrm{ceq}}+\mathrm{s}{\mathrm{L}}_{\mathrm{ceq}}\right)\right]+{\mathrm{I}}_{\mathrm{d}0}\left({\mathrm{R}}_{\mathrm{ceq}}+\mathrm{s}{\mathrm{L}}_{\mathrm{ceq}}\right)}{{\mathrm{I}}_{\mathrm{d}0}+\mathrm{s}{\mathrm{C}}_{\mathrm{ceq}}{\mathrm{U}}_{\mathrm{ceq}0}}}$$

(25)

For converters employing constant DC voltage control, the power injected into the converter from the AC side, denoted as p_c, can be expressed as follows:

$$p_c=1.5{\mathrm{U}}_{\mathrm{sd}0}{\mathrm{i}}_{\mathrm{sd}}-1.5i_{\mathrm{sd}}^2\left(R_{\mathrm{s}}+s{L}_{\mathrm{s}}\right)$$

(26)

Due to the extremely fast response speed of the inner current loop, for the sake of simplifying the analysis, it is equivalent to a delay element. The d-axis component of the AC current, denoted as i_sd, can be represented as follows:

$$i_{\mathrm{sd}}=\left(U_{\mathrm{dc}}^{*}-u_{\mathrm{dc}}\right)\left(k_{\mathrm{p}1}+{\displaystyle \frac{k_{i1}}{s}}\right)\left({\displaystyle \frac{1}{1+\tau s}}\right)$$

(27)

In the equation, τ represents the time constant of the delay element. Substituting Equation (27) into Equation (26), we obtain:

$$\Delta p_c=-\left({\displaystyle \frac{k_{\mathrm{p}1}s+k_{\mathrm{i}1}}{s\left(1+\tau s\right)}}\right)\left[{\displaystyle \frac{3}{2}}{U}_{\mathrm{sd}0}-3I_{\mathrm{sd}0}\left(R_s+s{L}_s\right)\right]\Delta u_{\mathrm{dc}}$$

(28)

By substituting Equation (28) into Equation (27) and performing simplification, we obtain the simplified DC impedance of the MMC as follows:

$${\mathrm{Z}}_{\mathrm{dc}}={\displaystyle \frac{{\mathrm{U}}_{\mathrm{ceq}0}\left[1+\mathrm{s}{\mathrm{C}}_{\mathrm{ceq}}\left({\mathrm{R}}_{\mathrm{ceq}}+\mathrm{s}{\mathrm{L}}_{\mathrm{ceq}}\right)\right]+{\mathrm{I}}_{\mathrm{d}0}\left({\mathrm{R}}_{\mathrm{ceq}}+\mathrm{s}{\mathrm{L}}_{\mathrm{ceq}}\right)}{\left(\frac{{\mathrm{k}}_{\mathrm{p}1}\mathrm{s}+{\mathrm{k}}_{\mathrm{i}1}}{\mathrm{s}\left(1+\mathsf{\tau }\mathrm{s}\right)}\right)\left[\frac{3}{2}{\mathrm{U}}_{\mathrm{sd}0}-3{\mathrm{I}}_{\mathrm{sd}0}\left({\mathrm{R}}_{\mathrm{s}}+\mathrm{s}{\mathrm{L}}_{\mathrm{s}}\right)\right]+{\mathrm{I}}_{\mathrm{d}0}+\mathrm{s}{\mathrm{C}}_{\mathrm{ceq}}{\mathrm{U}}_{\mathrm{ceq}0}}}$$

(29)

3.3. The Fault Current Calculation Method Based on MMC Equivalent Model

It should be noted that the harmonic state space is also usually used for dynamic stability analysis, namely oscillation analysis, and the time scope for oscillation analysis is usually large, lasting up to several seconds. But the DC fault current time scope narrows to several milliseconds. Thus, the harmonic state space cannot be used for fault current analysis. To analyze the fault current in frequency domain, a suitable equivalent impedance model should be investigated, as proposed in this paper.

When a fault occurs on the DC line, the voltage at the fault point can be viewed as the superposition of the pre-fault voltage U_f and an additional voltage source −U_f. According to the superposition principle, the entire system can be regarded as the superposition of a normal operating network without the fault additional source and a fault additional network where only the fault additional source acts. The analysis of the fault current can be conducted solely within the fault additional network. The obtained voltages and currents are all fault components, and by adding them to their respective normal operating components, the total quantities after the fault can be obtained. The fault network of a bipolar short-circuit fault on the DC line in a two-terminal MMC-HVDC system is shown in Figure 2.

Thus, the fault current expression of I_f1 and I_f2 can be written as follows:

$$I_{f1}=U_f(s)/\left(Z_{dc1}+Z_{L1}+R_f{\displaystyle \frac{Z_{dc1}+Z_{dc2}+Z_{L1}+Z_{L2}}{Z_{dc2}+Z_{L2}}}\right)$$

(30)

$$I_{f2}=U_f(s)/\left(Z_{dc2}+Z_{L2}+R_f{\displaystyle \frac{Z_{dc1}+Z_{dc2}+Z_{L1}+Z_{L2}}{Z_{dc1}+Z_{L1}}}\right)$$

(31)

In the equation, the fault additional source −U_f takes the form of a step function, and its complex frequency domain expression is −U_f/s.

Next, the aforementioned fault current calculation method will be verified. The impedances of the converter stations are calculated using both the proposed simplified impedance method and the traditional RLC equivalent impedance method to determine the inter-pole short-circuit fault current in a two-terminal MMC-HVDC system. The conventional RLC model is widely used and can be taken as the comparable standard model. The parameters used for the calculations are listed in

Table 2. The parameters used for fault current calculations.

Parameters	Values
Rated capacity	3000 MVA
Rated DC voltage	±500 kV
Converter transformer ratio	750/570
Converter transformer inductance	0.16 p.u.
Converter transformer resistance	0.0022 p.u.
Arm inductance	66 mH
Arm resistance	3.71 Ω
Submodule capacitance	16.3 mF
Number of submodules per arm	495
Smoothing reactor	200 mH
Proportional parameter of the outer voltage control loop	8
Integral parameter of the outer voltage control loop	27.27 s−1
Line resistance	1 Ω
Line inductance	0.0082 H

, and the comparison results are shown in Figure 3.

The simulation results indicate that, since the fault current calculation within the first 10 ms is primarily influenced by the high-frequency components of the impedance, the simplified impedance of the constant power station, which only modifies the steady-state characteristics in the low-frequency range compared to the RLC impedance, yields results that are not significantly different. In contrast, the simplified impedance of the constant voltage station not only accounts for the steady-state characteristics in the low-frequency range but also incorporates the outer PI control, thereby correcting the impedance in the mid-to-high-frequency range. As the fault duration increases, the accuracy of the simplified impedance calculation becomes increasingly evident compared to the RLC impedance.

Specifically, the RLC model does not consider the inner and outer control loop of MMC, such that the control impact of converter cannot be calculated. Although the model proposed by this paper results in the same curves as the traditional RLC from Figure 3a, from Figure 3b, it can be seen that the proposed method has more accuracy compared with traditional RLC. This is because the converter in constant active power basically has no impact on the fault current, while the constant DC voltage control can have a greater influence on the fault current. As for the calculation speed, it can be clearly seen from (30) and (31) that the main difference in the proposed model is the equivalent impedance, which can be easily calculated once the control parameters have been determined. Thus, the calculation speed is basically the same as that achieved using a traditional model.

4. The Fault Current Impact Factor Analysis Based on Impedance Calculation

4.1. The Device Parameters’ Impact on Equivalent Impedance

Using the DC impedance expression and the fault current calculation formula, we have identified the device parameters that influence the damping characteristics, which include (1) the arm inductance; (2) the arm resistance; (3) the submodule capacitance; and (4) the smoothing reactor.

Next, taking the two-terminal MMC-HVDC system shown in Figure 2 as an example, we will analyze the individual impacts of these six device parameters on the damping characteristics. Assuming that the fault occurs at the midpoint of the DC line, we will plot the frequency domain amplitude diagram of the fault impedance for different values of each device parameter. The fault impedance includes the DC impedance of the converter station, the smoothing reactor, and the line inductance.

■

The arm inductance’s impact on equivalent impedance

With the arm inductance values set at 20 mH, 40 mH, 60 mH, 80 mH, and 100 mH, respectively, the frequency domain amplitude plots for Z_f1 and Z_f2 are illustrated in Figure 4, where f denotes the frequency domain, and its unit is Hz. The expressions of Z_f1 and Z_f1 are as shown in Equation (32).

$$\left\{\begin{array}{l}{Z}_{f1}={\mathrm{Z}}_{dc1}+R_{L1}+s\left(L_{dc}+L_{L1}\right)\hfill \\ Z_{f1}={\mathrm{Z}}_{dc2}+R_{L2}+s\left(L_{dc}+L_{L2}\right)\hfill \end{array}\right.$$

(32)

The left converter station employs constant active power control, resulting in consistent transmission power in the time domain and a constant impedance magnitude at low frequencies in the frequency domain. In contrast, the right converter station, utilizing constant DC voltage control, experiences variations in impedance with changes in transmission power, leading to a non-constant impedance magnitude at low frequencies. Alterations in the arm inductance have minimal impact on the impedance at low frequencies. However, at mid-to-high frequencies, as the arm inductance increases, the fault impedance magnitude also increases, exhibiting a uniform growth trend. According to the formula for calculating the fault current rise rate, the current state at the moment of fault is determined by the inductance in the circuit; a larger inductance value results in a slower fault rise rate, which corresponds to a higher impedance at high frequencies in the frequency domain. The fault impedance characteristics shown in Figure 4 align with this conclusion.

■

The arm resistance’s impact on equivalent impedance.

With the arm resistance values set at 1 Ω, 3 Ω, 5 Ω, 7 Ω, and 9 Ω, respectively, the frequency domain amplitude plots for Z_f1 and Z_f2 are illustrated in Figure 5.

The frequency domain amplitude plots of the fault impedance reveal that, regardless of the control mode employed, changes in the arm resistance do not affect the impedance at low and high frequencies. A closer inspection of the fault impedance details indicates that as the arm resistance increases, the impedance magnitude at mid-frequencies also increases.

■

The smoothing reactor’s impact on equivalent impedance.

With the smoothing reactor values set at 50 mH, 100 mH, 150 mH, 200 mH and 250 mH, respectively, the frequency domain amplitude plots for Z_f1 and Z_f2 are illustrated in Figure 6.

Similarly to the arm inductance, the smoothing reactor’s influence on fault impedance is primarily observed in the mid-to-high-frequency range, with a more pronounced effect at high frequencies. This is due to the fact that the inductance value of the smoothing reactor is significantly larger than that of the arm inductance. Therefore, in addition to its role in smoothing the DC, the smoothing reactor can also be utilized to limit DC faults.

■

Power control’s impact on equivalent impedance.

The power reference values for the constant active power converter station are set at 200 MW, 400 MW, 600 MW, 800 MW, and 1000 MW, respectively, and the frequency domain magnitude plots is drawn as shown in Figure 7.

As can be seen from Figure 7, the power level of the constant power control converter station primarily affects the low-frequency range of the DC impedance of the converter station. As the transmitted power increases, the impedance magnitude in the low-frequency range decreases, which is consistent with the impedance behavior of a steady-state circuit under constant voltage conditions.

4.2. The Device Parameters’ Impact on Fault Current

■

The arm inductance’s impact on fault current.

When the arm inductance values are set at 20 mH, 40 mH, 60 mH, 80 mH, and 100 mH, respectively, the fault currents plot for I_f1 and I_f2 are illustrated in Figure 8.

The arm inductance primarily affects the mid-to-high-frequency range of fault damping, with the amplitude in the high-frequency range increasing proportionally as the inductance value increases. This is reflected in the fault current in the time domain as a reduction in the rate of increase in the fault current. As shown in Figure 8, a larger arm inductance results in a smaller rate of increase in the fault current, validating the conclusions of the damping analysis.

■

The arm resistance’s impact on fault current

With the arm resistance values set at 1 Ω, 3 Ω, 5 Ω, 7 Ω, and 9 Ω, respectively, the fault current plots for I_f1 and I_f2 are illustrated in Figure 9.

The arm resistance primarily influences the mid-to-low-frequency range of fault damping. Consequently, the initial rate of current rise during a fault is unaffected by changes in resistance. As the fault progresses, the damping characteristics in the mid-to-low frequency range gradually become apparent, and the fault current decreases as the arm resistance increases. From the frequency domain plot in Figure 9, it can be observed that the impact of arm resistance is more pronounced in constant power stations compared to constant voltage stations. This is reflected in the time domain as a more significant difference in fault currents.

■

The smoothing reactor’s impact on fault current.

With the smoothing reactor values set at 50 mH, 100 mH, 150 mH, 200 mH, and 250 mH, respectively, the frequency domain amplitude plots for Z_f1 and Z_f2 are illustrated in Figure 10.

The smoothing reactor primarily affects the mid-to-high-frequency range of fault damping, and the smoothing reactor’s inductance value is usually larger than that of the arm inductance. Therefore, it has a particularly notable impact on the initial rate of increase in the fault current. Increasing the inductance of the smoothing reactor is beneficial for suppressing the rise in the fault current and substantially reducing its amplitude.

■

Power control’s impact on equivalent impedance.

The power reference values for the constant active power converter station are set at 200 MW, 400 MW, 600 MW, 800 MW, and 1000 MW, respectively. The line fault current curves on the side of the constant power control station are plotted in Figure 11.

The line resistance primarily affects the mid-frequency range of fault damping, exhibiting a similar influence pattern to the arm resistance. As the resistance value increases, the amplitude of the fault current decreases.

5. Conclusions

In this paper, the key factors influencing DC faults are summarized, and the impact of these factors on the amplitude of DC impedance and fault current is analyzed. The conclusions are as follows:

(1)

The low-frequency component of the equivalent impedance has no significant impact on the rate of increase and the peak value of the fault current during the initial stage (typically referring to the first 10 ms after the fault occurs), such as the DC power level. The mid-frequency component of the equivalent impedance does not affect the initial rate of increase in the fault current, but it does influence the current level several milliseconds after the fault occurs. Factors such as arm resistance and submodule capacitance, when their values change, will lead to variations in the fault current.

(2)

The high-frequency component of fault damping primarily influences the initial state of the fault current, specifically the initial rate of increase. Components such as the smoothing reactor, when their values change, can significantly alter the initial rate of increase in the fault current, thereby affecting the subsequent current levels. The damping characteristics in the high-frequency range have the most pronounced impact on the fault current.

Although these parameters are inherently fixed when converters are constructed, the fault current can also be limited through the DC grid topology optimization, and the fault current limiters can also be equipped with certain DC lines with high fault currents.