Analysis of the Impact of Short Circuit Faults in Converter Valve Submodules on Valve Power Transmission

Abstract

Faults of a Modular Multilevel Converter (MMC)-type converter valve significantly impact the reliability of flexible DC transmission systems. This paper analyzed the impact of ongoing short circuit faults in submodules on the power transmission of the MMC-type converter valve of which redundant submodules had been depleted. First, MMC’s working principle and its submodules’ possible operational states were investigated. Then, fault mechanisms for intra-submodule Insulated-Gate Bipolar transistor (IGBT) short circuits and inter-submodule short circuits were modeled to infer changes in power transmission during submodule faults. To quantify the impact of submodule faults on the energy transfer efficiency of the converter valve, an energy transfer efficiency index was proposed to obtain analytical expressions for energy transfer efficiency in the case of intra-submodule and inter-submodule short-circuit faults. Finally, the effectiveness of the proposed analytical model was verified through Simulink simulations. Simulation results indicate that ongoing intra-submodule and inter-submodule short circuits increase the input power of the converter valve, reducing energy transfer efficiency. Moreover, the energy transfer efficiency continues to decline with an increase in faulty submodules.

1. Introduction

The Modular Multilevel Converter (MMC) has been widely used in the field of flexible DC transmission projects due to its advantages such as low harmonic components in output voltage, low current distortion rate, ease of expansion, low switching loss, and strong fault ride-through capability [1,2]. However, unlike traditional transformer equipment, MMC-type converter valves are constructed by a large number of submodules (Submodule, SM) in cascade [3,4,5], so that the number and operating states of the submodules determine their transmission performance. Short circuit faults occurring in a submodule directly affect the power transmission of the converter valve, potentially jeopardizing the stable operation of the flexible DC transmission system [6,7]. Therefore, it is crucial to study the impact of short circuit faults in submodules on the power transmission of the converter valve.

The rapid escalation of fault current under DC short-circuit conditions imposes stringent requirements on detection speed and accuracy. To address this challenge, Liu proposed a deep neural network-based fault location method utilizing statistical features, which achieves superior classification accuracy and detection speed over conventional neural networks by extracting time–domain characteristics of fault currents [8]. Nevertheless, the fault complexity in MMC systems extends beyond DC-side short circuits, with submodule-level faults posing significant operational risks. Li developed a maximum information coefficient-based open-circuit fault localization strategy for submodules, enabling rapid identification of multiple concurrent faults within 20 ms through comparative analysis of capacitance voltage deviations between faulty and healthy submodules [9]. Furthermore, the existing detection methods commonly face limitations in threshold determination and insufficient noise immunity. Mao’s transient energy detection method overcomes these issues by constructing detection criteria using transient energy characteristics of submodule capacitors and DC inductors, demonstrating immunity to lightning-induced disturbances [10].

Rapid fault current suppression and resilient system recovery constitute critical challenges for ensuring the reliable operation of flexible DC grids. Zhang et al. introduced a dual adaptive controller for MMC systems, which dynamically adjusts DC voltage references and submodule capacitance voltages. This approach simultaneously curbs fault current rise rates and accelerates system recovery, providing novel insights into coordinated “current-limiting and recovery” optimization [11]. However, existing fault ride-through solutions often encounter trade-offs between economic viability and response speed. The Li research team addressed this dilemma in medium-voltage DC systems through topology optimization, enhancing bidirectional voltage withstand capability of submodules to achieve fault current clearance and system recovery within 20 ms, with significantly lower additional costs and operational losses compared to conventional solutions [12]. For more complex fault scenarios in multi-terminal DC networks, Kontos et al. proposed a post-fault coordinated control framework integrated with H-bridge MMC active discharge mechanisms. This innovation reduces pole-to-ground fault recovery time to 158 ms and pole-to-pole fault recovery time to 34 ms, eliminating dependence on costly DC circuit breakers [13].

The above research does not study the fault of the converter valve submodule itself. Each submodule in the MMC consists of two Insulated-Gate Bipolar transistors (IGBTs) and a capacitor. Because the submodule runs for a long time, the IGBT may occur an internal short circuit, which leads to the failure of the submodule. In addition, it is also found that the problem of submodules being bypassed due to short circuits between adjacent submodules also occurs. As a result, the problem of MMC stopping operation due to the failure of a submodule often occurs. Many researchers have employed fault detection methods based on AI algorithms to monitor the operating states of submodules and promptly isolate faulty submodules. Reference [14] proposed an AI fault diagnosis framework for submodules. References [15,16] performed adaptive SM fault diagnosis for submodules using time convolution and LSTM neural networks. Additionally, references [17,18] suggested that SM fault detection in MMC can be achieved by voltage variations in bridge arms observed by detectors or commutation.

To prevent MMC shutdown caused by submodule faults, it has been proposed that a certain number of redundant submodules and bypass switches be configured in addition to the specified number of submodules in the MMC [19]. When some submodules fail, the faulty submodules can be quickly isolated via the bypass switches while incorporating the redundant submodules to achieve the online isolation of the faulty submodules [20]. However, the use of excessive redundant submodules leads to significantly higher hardware costs and low utilization of submodules and makes the system more complex [21,22]. Therefore, it is not practical to configure many redundant submodules in actual applications. When consecutive submodule faults occur in actual engineering, the redundant submodules may be quickly depleted. Under the current operating rules of flexible DC systems, the MMC must be immediately shut down once the redundant submodules are depleted. At the peak of power transmission, this may result in a prolonged shutdown of the DC system, reducing economic efficiency, especially for offshore wind power transmission via flexible DC systems, where maintenance and repair of offshore converter stations are more challenging, resulting in more significant power loss caused by system shutdown. However, if the system is not shut down, unexpected catastrophic events may occur, leading to more severe safety and economic issues [23]. Therefore, from the perspectives of economic and safety considerations, MMC design, production, and operation personnel must understand the operational characteristics of the MMC where ongoing submodule faults occur after the redundant submodules are depleted. However, there is a lack of research regarding how submodule faults affect the operating states of the converter valve after redundant submodules are depleted.

This paper investigated the impact of ongoing short circuit faults in submodules on the power transmission of the converter valve after the redundant submodules are depleted. First, the working principle of the MMC valve was briefly described. On this basis, fault models for intra-submodule IGBT short circuits and inter-submodule short circuits were established based on the mathematical models of the MMC’s internal current and voltage. Next, to quantitatively analyze the impact of short circuit faults in submodules on the power transmission of the converter valve, we innovatively put forward the energy transfer ratio (ETR) of converter valve from the perspective of energy transfer efficiency and derived the analytical formula under different faults. The theoretical expressions for the energy transfer efficiency index under different fault conditions were derived. Finally, the effectiveness of the proposed analytical model was verified through Simulink simulations. Simulation results indicate that ongoing intra-submodule and inter-submodule short circuits increase the input power of the converter valve, reducing energy transfer efficiency. Moreover, the energy transfer efficiency continues to decline with an increase in faulty submodules.

2. Working Principle of MMC

The MMC-type converter valve is shown in Figure 1. The converter valve consists of six bridge arms, each composed of an arm inductor L₀, an equivalent resistor R₀, and N submodules connected in series [23]. The left side of the MMC is connected to the DC system, where i_dc represents the DC current and U_dc is the voltage between the positive and negative poles of the DC system. The right side of the MMC is connected to the AC system, where u_vj (j = a, b, c) denotes the three-phase voltage on the AC system side, u_j and i_j (j = a, b, c) represent the output three-phase AC voltage and current, respectively, and u_pj and u_nj (j = a, b, c) denote the upper and lower arm voltages of phase j, respectively. Similarly, i_pj and i_nj (j = a, b, c) represent the upper and lower arm currents of phase j, respectively.

The detailed structural composition of the submodule (SM) in the converter valve is illustrated in Figure 2.

In the submodule, T₁ and T₂ are two Insulated-Gate Bipolar transistors (IGBTs), D₁ and D₂ are two diodes connected in reverse parallel with the corresponding IGBTs, C₀ is a submodule capacitor, T₀ is a bypass thyristor at the input, and B is a bypass switch at the input. U_c is the voltage across the submodule capacitor, U_sm (U₀) is submodule outlet voltage, and i_arm is the current at the input.

At different times, the states of the two IGBTs, T₁ and T₂, inside the submodule are controlled by two opposite control signals ${\mathrm{g}}_1$ and ${\mathrm{g}}_2$, thereby achieving control over the submodule states. When ${\mathrm{g}}_1$ and ${\mathrm{g}}_2$ are set to 1, it indicates that the corresponding IGBT is conducting. Similarly, when ${\mathrm{g}}_1$ and ${\mathrm{g}}_2$ are set to 0, it indicates that the corresponding IGBT is turned off. The switching state of the submodule can be represented by Formula (1) [23]:

$$\mathrm{S}=\left\{\begin{array}{cc}1,\hfill & {\mathrm{g}}_1=1\hspace{1em},\hspace{1em}{\mathrm{g}}_2=0\hfill \\ 0,\hfill & {\mathrm{g}}_1=0\hspace{1em},\hspace{1em}{\mathrm{g}}_2=1\hfill \end{array}\right.$$

(1)

According to Figure 2, it is not difficult to observe that when S is set to 1, the submodule is activated, and when S is set to 0, the submodule is deactivated. For sub modules in different states, the charging and discharging of their internal capacitors are also different. Now, we will analyze the different situations:

①

When T₁ and D₁ are conductive, current charges the capacitor through D₁.

②

When T₁ is conducting, the current discharges the capacitor through T₁.

③

When T₂ is conducting, the current passes through T₂ to bypass the capacitor.

④

When T₂ and D₂ are conducting, the current passes through D₂ to bypass the capacitor.

⑤

When D₁ is conducting, current charges the capacitor through D₁.

⑥

When D₂ is conducting, the current passes through D₂ to bypass the capacitor.

The states are shown in

Table 1. Operating states of the submodule.

Type	State	T1	T2	D1	D2	Direction	Usm
①	Activated	1	0	1	0	+	Uc
②	Activated	1	0	0	0	−	Uc
③	Bypassed	0	1	0	0	+	0
④	Bypassed	0	1	0	1	−	0
⑤	Locked	0	0	1	0	+	Uc
⑥	Locked	0	0	0	1	−	0

.

As shown in the table above, the submodule has six operating states. However, the submodule is only “activated” in states ① and ②. In all other states, it is meaningless to study the impact of the submodule on the converter valve as it is bypassed. Therefore, the following analysis will only focus on the impact of short circuit faults on the converter valve when the submodule is activated.

3. Modeling of Short Circuit Faults in the Submodule

3.1. Internal IGBT Short Circuit in the Submodule

(1) When T1 is short-circuited, the state of the submodule is as shown in Figure 3:

The operation of the submodule in states ① and ② are as follows:

When T₁ is short-circuited, as shown in Figure 4a,b, if the on-state voltage drop of the IGBTs is neglected, the current path of the short-circuited T₁ is the same as the original current path. Therefore, the short circuit has little impact on the operation of the converter valve.

(2) When T₂ is short-circuited, the state of the submodule is as shown in Figure 5:

The operation of the submodule in states ① and ② are as follows:

Similarly, as shown in Figure 6a,b, short-circuited T₂ connects with T₁ to form an on-state circuit, leading to a rapid decrease in the submodule voltage and an increase in loss due to quick formation of a capacitor discharge circuit.

Based on the above analysis, it can be concluded that in the normal operating states of the submodule, if an IGBT is short-circuited, two scenarios may occur. One is that the operation remains the same as the original state, and the other is that the capacitor will discharge rapidly, causing a decrease in the capacitor voltage and an increase in losses, ultimately resulting in the voltage of the faulty submodule being deducted from the arm voltage.

To further simplify the discharge circuit in Figure 6b, the two IGBTs are equivalent to two resistors and then connected in series to form an equivalent resistor. The simplified circuit is shown in Figure 7. U_c is the capacitor voltage, i_c is the current in the discharge circuit.

Based on Figure 7, a differential equation for the capacitor discharge circuit is established as follows:

$$i_c=-C_0{\displaystyle \frac{d{U}_c}{dt}}$$

(2)

$$u_c+R{C}_0{\displaystyle \frac{d{U}_c}{dt}}=0$$

(3)

Taking the initial state U_c(0) = U₀, solving the above differential equation yields:

$$U_c\left(t\right)=U_0{e}^{-{\displaystyle \frac{t}{R{C}_0}}}$$

(4)

where U₀ represents the output voltage of each submodule under normal conditions, $U_{\mathrm{0}}=U_{dc}/(2\ast N)$.

Therefore, the DC voltage ${U_{dc}}^{\prime }$ of faulty MMC can be calculated by the pre-fault voltage $U_{dc}$ and the number of faulty submodules:

$${U_{dc}}^{\prime }=U_{dc}-n\ast U_0+n\ast U_0{e}^{-{\displaystyle \frac{t}{R{C}_0}}}$$

(5)

3.2. Inter-Submodule Short Circuit

Inter-submodule short circuits are less likely to occur than IGBT faults, but they more severely impact the converter valve. There have been several instances of adjacent submodules experiencing short-circuit faults in actual projects, so conducting a characteristic analysis on the inter-submodule short circuits is necessary.

Additionally, unlike internal IGBT short circuits, the inter-submodule short circuits occur at the submodule output, as shown in Figure 8 below.

Therefore, when an inter-submodule short circuit occurs, the output voltage of the submodule will immediately drop to zero, namely

$$U_{sm}=U_c\left(t\right)=0$$

(6)

Therefore, the DC voltage ${U_{dc}}^{\prime }$ of faulty MMC is calculated as follows:

$${U_{dc}}^{″}=U_{dc}-n\ast U_0$$

(7)

4. Transmission Performance Calculation Under Fault Conditions

4.1. Energy Efficiency Index

The ratio of AC output energy to DC input energy over time from T to T + △T is taken as an efficiency index for the converter valve’s energy transfer rate to define the energy transfer efficiency. The details are as follows:

$$\eta \%={\displaystyle \frac{3{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}{u}_j{i}_j\cos \phi dt}}{{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}{U}_{dc}{I}_{dc}dt}}}\times 100\%$$

(8)

4.2. Intra-Submodule IGBT Short Circuits

According to the topology of the MMC circuit in Figure 1, Kirchhoff’s voltage law (KVL) is written for the upper and lower bridge arms [24]:

$$R_0{i}_{pj}+L_0{\displaystyle \frac{d{i}_{pj}}{dt}}={\displaystyle \frac{U_{dc}}{2}}-u_{pj}-u_j$$

(9)

$$R_0{i}_{nj}+L_0{\displaystyle \frac{d{i}_{nj}}{dt}}={\displaystyle \frac{U_{dc}}{2}}-u_{nj}+u_j$$

(10)

KCL for the upper and lower bridge arms:

$$i_j=i_{pj}-i_{nj}$$

(11)

$$i_{\mathrm{diff}j}=(i_{pj}+i_{nj})/2$$

(12)

Since the AC output voltage u_j of the inverter side in the MMC is generally regarded as a set value under double closed-loop control, i.e., u_j′ = u_j, the arm voltages can also be considered consistent with those in the normal conditions, i.e., u_pj′ = u_pj and u_nj′ = u_nj.

By substituting ${U_{dc}}^{\prime }$ into Equations (9) and (10) using Equation (5), the arm currents under fault conditions can be derived ${i_{pj}}^{\prime }$${i_{nj}}^{\prime }$ as follows:

$${i_{pj}}^{\prime }=({\displaystyle \frac{U_{dc}-2\left(N-1\right)U_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}}-{\displaystyle \frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{L_0-C_{0}R{R}_0}})+C_p{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(13)

$${i_{nj}}^{\prime }=({\displaystyle \frac{U_{dc}-2N{U}_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}})+C_n{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(14)

where U is the voltage amplitude at the AC side.

By substituting Equations (13) and (14) into Equations (11) and (12), the post-fault ${i_j}^{\prime }$ and circulating current ${i_{diffj}}^{\prime }$ can be derived as follows:

$${i_j}^{\prime }=\left({\displaystyle \frac{U_0}{R_0}}-{\displaystyle \frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{L_0-C_{0}R{R}_0}}\right)+C_1{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(15)

$${i_{diffj}}^{\prime }=({\displaystyle \frac{U_{dc}-\left(2N-1\right)U_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}}-{\displaystyle \frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{2\left(L_0-C_{0}R{R}_0\right)}})+C_{diff}{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(16)

Additionally, according to the circulating current expression [25], the post-fault ${i_{dc}}^{\prime }$ is as follows:

$$\begin{array}{l}{i_{dc}}^{\prime }=3({i_{diffj}}^{\prime }-i_{cir})\\ \hspace{1em}\hspace{0.17em}=3({\displaystyle \frac{U_{dc}-\left(2N-1\right)U_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}}-{\displaystyle \frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{2\left(L_0-C_{0}R{R}_0\right)}}-i_{cir})+C_{dc}{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}\end{array}$$

(17)

where $i_{cir}$ can be obtained from system parameters and the converter manual.

Finally, according to Equation (8), the energy transfer efficiency ${\eta }^{\prime }$ under fault conditions is calculated, as shown in Equation (18):

$$\eta \%={\displaystyle \frac{3{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}U\left[\left(\frac{U_0}{R_0}-\frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{L_0-C_{0}R{R}_0}\right)+C_1{e}^{-\frac{R_{0}t}{L_0}}\right]}\sin t\cos \phi dt}{{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}{U}_{dc}\left[3(\frac{U_{dc}-\left(2N-1\right)U_0}{2R_0}+\frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}-\frac{C_{0}R{U}_0{e}^{-\frac{t}{C_{0}R}}}{2\left(L_0-C_{0}R{R}_0\right)}-i_{cir})+C_{dc}{e}^{-\frac{R_{0}t}{L_0}}\right]}dt}}\times 100\%$$

(18)

4.3. Inter-Submodule Short Circuits

Similarly to intra-submodule short circuits, ${U_{dc}}^{″}$${i_{pj}}^{″}$${i_{nj}}^{″}$. For inter-submodule short circuits, by substituting into Equations (9) and (10) via Equation (7), the bridge arm current under fault conditions can be obtained as follows:

$${i_{pj}}^{″}=({\displaystyle \frac{U_{dc}-2\left(N-n\right)U_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}})+C_p{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(19)

$${i_{nj}}^{″}=({\displaystyle \frac{U_{dc}-2N{U}_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}})+C_n{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(20)

Additionally, similar to intra-submodule faults, by substituting Equations (19) and (20) into Equations (11) and (12), inter-submodule post-fault ${i_j}^{″}$, ${i_{diffj}}^{″}$ can be calculated as follows:

$${i_j}^{″}={\displaystyle \frac{n{U}_0}{R_0}}+C_1{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(21)

According to the circulating current equation, the inter-submodule post-fault DC current ${i_{dc}}^{″}$ is as follows:

$${i_{dc}}^{″}=3({\displaystyle \frac{U_{dc}-\left(2N-n\right)U_0}{2R_0}}+{\displaystyle \frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}}-i_{cir})+C_{dc}{e}^{-{\displaystyle \frac{R_{0}t}{L_0}}}$$

(22)

Finally, according to Equation (8), the inter-submodule energy transfer efficiency ${\eta }^{″}$ under fault conditions is calculated, as shown in the following Equation (23):

$${\eta }^{″}\%={\displaystyle \frac{3{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}U\left(\frac{n{U}_0}{R_0}+C_1{e}^{-\frac{R_{0}t}{L_0}}\right)}\sin t\cos \phi dt}{{\displaystyle {\int }_T^{T+\mathrm{\Delta }T}{U}_{dc}\left[3(\frac{U_{dc}-\left(2N-n\right)U_0}{2R_0}+\frac{L_{0}U\cos t-R_{0}U\sin t}{L_0^2+R_0^2}-i_{cir})+C_{dc}{e}^{-\frac{R_{0}t}{L_0}}\right]}dt}}\times 100\%$$

(23)

5. Simulation Analysis

The simulation was performed on a PC with Windows 11 Pro Intel(R) Core(TM) Ultra 5 125H 1.20 GHz memory of 32.0 GB, The manufacturer is Lenovo, made in China. A VSC-HVDC DC simulation system was built based on SIMULINK in MATLAB R2023b to simulate the converter station’s actual operation and verify its effectiveness by comparing it with theoretical calculation results. In addition, the DC transmission system consists of two MMCs on the rectifier and inverter sides, respectively. Since the analytical expressions derived in this paper were for the inverter-side MMC in the DC-AC conversion, the validation and analysis were conducted using the inverter-side MMC. The VSC-HVDC DC simulation system diagram and parameters are shown in Figure 9 and

Table 2. Model parameters.

Parameter	Numerical Value	Unit
AC voltage	10	kV
DC voltage	21	kV
number of SM (a single bridge arm)	10	-
resistance of bridge arm	0.01	Ω
inductance of bridge arm	0.004	H
capacitance of SM	0.003	F
equivalent resistance of IGBT	0.001	Ω

, respectively. The traditional double closed-loop control was adopted. The specific simulation program is available at https://github.com/JJMalibo/VSC-HVDC, accessed on 12 January 2025.

5.1. Simulation Examples in Case of Intra-Submodule IGBT Short Circuits

Based on actual engineering scenarios, this section performs simulation calculations when the transmission power is set to 7.6 MW. Since the VSC-HVDC simulation system based on Simulink was in a transient state at 1 s before startup, the occurrence of the intra-submodule short circuit was set as 1 s. The DC power, AC power, and energy transfer efficiency during operation are shown in Figure 10a, Figure 10b, and Figure 10c, respectively.

As shown in Figure 10a–c, after the system stabilized at 1 s, the means of the solid and dashed lines in the same color were consistent, regardless of the number of submodules experiencing intra-submodule short circuit faults. Therefore, the computer-derived results of the theoretical model in this paper were consistent with the Simulink-based simulation results, indicating that the theoretical calculations in this paper are correct and can reflect the impact of short circuit faults in submodules on MMC power transmission when the redundant modules are depleted.

Secondly, from a numerical perspective, it could be observed that the DC side power continuously increased as the number of internally short-circuited submodules rose from 0 to 3. The reason for this phenomenon is as follows: In a flexible DC transmission system under double closed-loop control, the AC side of the MMC will output a constant set power. Figure 10b shows that even when intra-submodule short circuit faults occur in the MMC, the mean output power remains unchanged. Therefore, as the power loss of the MMC increases with the number of short-circuited submodules, the DC input power will increase to compensate for the MMC’s power loss.

Finally, the energy transfer efficiency calculated when faults occurred at 1 s is shown in Figure 10c. From Figure 10c, it can also be observed that the transfer efficiency decreased with an increase in the number of short-circuited submodules. Numerically, as 1 short circuit (10%), 2 short circuits (20%), and 3 short circuits (30%) occurred, the energy transfer efficiency decreased from 99.61% to 94.84%, 91.44%, and 88.50%, respectively, corresponding to reductions of approximately 4.77%, 8.17%, and 11.11%.

In summary, when the redundant modules in the MMC are depleted, the theoretical calculations in this paper can reflect the impact of short circuit faults in submodules on MMC power transmission. Additionally, when intra-submodule short circuit faults occur, the energy transfer efficiency of the MMC decreases with an increase in the number of short-circuited submodules.

5.2. Simulation Examples in Case of Inter-Submodule Short Circuits

Like intra-submodule short circuits, the VSC-HVDC simulation system based on Simulink was transient at 1 s before startup. Therefore, the occurrence of inter-submodule short circuit faults was set at 1 s. The DC power, AC power, and energy transfer efficiency during the operation are shown in Figure 11a, Figure 11b, and Figure 11c, respectively.

As shown in Figure 11a–c, similar to intra-submodule short circuits, after the system stabilized at 1 s, the means of the solid and dashed lines in the same color were consistent, regardless of the number of submodules experiencing inter-submodule short circuit faults. Therefore, the computer-derived results of the theoretical model in this paper were consistent with the Simulink-based simulation results, indicating that the theoretical calculations in this paper are correct and can reflect the impact of short circuit faults between submodules on MMC power transmission when the redundant modules are depleted.

Secondly, from a numerical perspective, it could be observed that the DC side power continuously increased as the number of internally short-circuited submodules rose from 0 to 3. This phenomenon occurs because the output voltage of submodules with short circuits drops to zero, causing the bridge arm to lose the power transmitted by one module. Therefore, the DC input power may increase under the double closed-loop control to ensure a constant set power output on the AC side of the flexible DC transmission system (as shown in Figure 11b).

Finally, the energy transfer efficiency calculated when faults occurred at 1 s is shown in Figure 11c. From Figure 11c, it can also be observed that the transfer efficiency decreased with an increase in the number of short-circuited submodules. Numerically, as 1 short circuit (10%), 2 short circuits (20%), and 3 short circuits (30%) occurred, the energy transfer efficiency decreased from 99.61% to 94.61%, 91.20%, and 88.31%, respectively, corresponding to reductions of approximately 5%, 8.41%, and 11.3%, similar to those in the intra-submodule short circuits. The reason for this phenomenon is as follows: the output voltage of the submodules with internal short circuits was consistent with that in inter-submodule short circuits as the capacitor voltage gradually dropped to zero. Therefore, when the redundant modules in the MMC are depleted, the theoretical calculations in this paper can effectively reflect the impact of inter-submodule short circuit faults on MMC power transmission.

In summary, when submodule short circuits occur in the MMC valve, the theoretical model in this paper can effectively reflect the impact of such faults on MMC power transmission. The impact pattern is that when submodule short circuit faults occur in the MMC, the DC input power increases on the inverter side. In contrast, the AC output power remains unchanged, and the energy transfer efficiency decreases.

5.3. Parameter Variation

In order to test the accuracy of the proposed theoretical model under different transmission power, in this section we increase the predetermined transmission power by 25% from 7.6 MW, i.e., 9.5 MW. When other parameters remain unchanged, the results of intra-submodule short circuits and inter-submodule short circuits are shown in Figure 12 and Figure 13, respectively:

From the blue curves of Figure 12a,b and Figure 13a,b, it can be seen that as the preset transmission power increases, the transmission power of MMC under the control algorithm increases to the preset value. Then, by comparing the dotted and solid lines in Figure 12 and Figure 13, it can be found that the results of the theoretical model are still consistent with the simulation results when the power is increased. Therefore, the theoretical calculation model is still accurate when the power changes. To sum up, the model proposed in this paper is generally valid.

6. Conclusions

This paper analyzed the impact of ongoing short circuit faults in submodules on power transmission of the MMC-type converter valve of which redundant submodules had been depleted. The contributions of this paper are as follows:

(1)

An energy transfer efficiency index was proposed to infer analytical expressions for energy transfer efficiency in the case of intra-submodule and inter-submodule short circuit faults. At the same time, the proposed analytical model was verified to be correct through the Simulink simulation program.

(2)

When the redundant submodules are depleted, the impact of short circuits in submodules on the valve power is as follows: both intra-submodule and inter-submodule short circuits increase the input power of the converter valve, reducing the energy transfer efficiency. Furthermore, the efficiency continuously decreases with an increase in the number of short-circuited submodules.

This research can help researchers to understand the operating limits of MMC and provide theoretical guidance for the design, optimization, and operation of MMC. For example, when submodule exhaustion occurs, this research can help operation and maintenance personnel to determine the risk of system continued operation when submodule failure continues. It also helps manufacturers to study how to further improve the energy transfer efficiency of MMC in normal and fault conditions by optimizing the topology.

In addition, it is very important to study the potential impact on the entire system and the occurrence of open circuit failures. This paper mainly studies the influence of short circuit faults on the converter valve. Subsequent work will be carried out to investigate the impact of the open-circuit failure and further potential cascading effects on the system.