# Reliability evaluation of modular multilevel converter based on Markov model

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## Abstract

The modular multilevel converter (MMC) is now the most attractive topology for medium and high voltage power conversion applications with several advantages over the traditional voltage source converter (VSC). However, due to a large number of sub-modules (SMs) in the MMC, system reliability is a big challenge in its practical application, where each SM may be considered as a potential point of failure. In this paper, a reliability evaluation based on the Markov model is proposed for the MMC. The failure rates of the power electronic devices and SMs are firstly analyzed. Then, the Markov model and the state transition equation of the system are built in detail. A general reliability evaluation function is established, in which the mean time to failure and reliability evaluation of the MMC with redundant SMs are also discussed. Finally, a practical direct current (DC) distribution example for reliability evaluation is analyzed, and the results verify that the reliability evaluation based on the Markov model could provide a useful reference for project design.

## Keywords

Modular multilevel converters Reliability evaluation Markov model## 1 Introduction

The offshore wind power has been widely developed in Europe, China, the United States and Australia [1]. Voltage source converter (VSC) based high voltage direct current (HVDC) transmission systems, including multi-terminal HVDC systems, are widely applied to transmit the power [2, 3], in which the modular multilevel converter (MMC) topology attracts the most attention of researchers [4]. The MMC topology is also used in DC distribution and power quality control, such as a unified power flow controller (UPFC) and static synchronous compensator (STATCOM) [5, 6]. Most research has focused on the operation and control strategy of MMCs [7, 8, 9, 10, 11], while their failure prediction has not received much discussion, although it is an important issue in power systems [12, 13]. Hence, there is a big motivation to find a solution for reliability evaluation and sub-module (SM) redundancy configuration of the MMC.

Some tentative work has been done in previous studies. For example, redundancy and reliability indexes of the MMC were defined in [14], but the general reliability function was not built. Also, using the k-out-of-n: G model and Gamma distribution, a reliability function of MMC was established [15], wherein the failure rates of SM components were based on the experience-hypothesis. Explicit models of components were not given. Some research has focused on the optimal number of SMs, without considering reliability evaluation [16, 17].

In this paper, the Markov model is proposed to analyze the failure rates of the MMC and its SMs, based on a graphical representation of system states. A general function for reliability evaluation of the MMC is given. The mean time to failure and reliability evaluation of the MMC with a redundant SM are both discussed.

The paper is organized as follows. In Section 2, the basic operation principles of the MMC are presented. In Section 3, the cause and impact of SM failure are discussed. Reliability analysis of MMC is given in Section 4, and with a redundant SM in Sections 5. Finally, an example analysis is provided in Section 6.

## 2 Basic operation principles of MMC

*n*SMs in each arm [18]. \(L_{0}\) is the arm inductor; \(U_{dc}\) is the DC bus voltage; \(u_{jm}\)(

*j=*a, b, c) is the output voltage of phase

*j*and \(u_{c}\) is the capacitor rated voltage.

Working mode of SM

State | \(S_{1}\) | \(S_{2}\) | \(i_{SM}\) | \(u_{SM}\) |
---|---|---|---|---|

ON | ON | OFF | \(u_{c}\) | |

OFF | OFF | ON | 0 | |

BLOCK | OFF | OFF | Positive | \(u_{c}\) |

Negative | 0 |

## 3 Cause and impact of SM fault

Each SM may be considered as a potential point of failure [19]. Hence, it is important to know the causes and impact of a SM fault.

### 3.1 Causes of SM fault

There are three main causes to trigger the failure of a SM.

- 1)
Damage to a power electronic component

The overload capability of power electronic components, such as insulated gate bipolar transistors (IGBTs) and diodes, is limited, so overvoltage and overcurrent might break them. Therefore, the damage to a power electronic component is one of the most common reasons to cause the SM fault.

- 2)
Damage to a passive component

The DC capacitor is a key passive component in a SM, and damage to it would also cause a SM fault. Fortunately, its failure rate is lower than that of power electronic devices.

- 3)
A faulty trigger pulse

If the trigger pulse signal experiences interference, it could not operate the SM correctly, and this might lead the SM to a fault in either short circuit or open circuit mode [20].

### 3.2 Impacts of SM fault

Once one SM is damaged, it would be bypassed and then replaced by the redundant SM, which can cause distortion of the output voltage and current and reduced performance of the converter. A fault of one SM can also cause the other components in the corresponding arm to fail and lead to total system collapse. When the quantity of broken SMs is more than the redundant ones, the MMC would work with asymmetrical operation. In the worst case, this might result in system failure.

## 4 Reliability analysis of MMC

To evaluate the MMC’s reliability, an analysis of the reliability of semiconductor devices (IGBTs and diodes) and capacitors is first required for use in the Markov model.

### 4.1 Component failure rate

- 1)
The average power dissipated by the diode is 53.47 W.

- 2)
The average power dissipated by IGBT is 346 W.

- 3)
The junction-ambient thermal resistance is 0.1618 °C/W.

Parameters for failure rates of components

Parameter | IGBT | Capacitor | Diode |
---|---|---|---|

\(\lambda_{0}\) | 2 | 0.7 | |

\(\pi_{U}\) | 1 | ||

\(\pi_{S}\) | 0.48 | ||

\(\lambda_{B}\) | 10 | 10 | |

\(\pi_{I}\) | 1 | 1 | |

\(\lambda_{EOS}\) | 40 | 40 | |

\(\tau_{off}\) | 0.942 | 0.942 | 0.942 |

\(\tau_{on}\) | 0.058 | 0.058 | 0.058 |

Failure rate | 224.535 | 18.33 | 194.32 |

### 4.2 Summary of Markov reliability model

The Markov model can be used to estimate a variety of reliability indexes, such as the failure rate, the mean time to failure (MTTF), the reliability and the availability [24, 25].

*t*, the probability of the system in the \(i^{\text{th}}\) state can be expressed as:

*t*is:

*μ*

_{10}is the returning rate; State 0 stands for the system works well; State 1 stands for the system has a fault, but could still work; State 2 stands for the system could no longer work.

### 4.3 Markov model of MMC reliability evaluation without redundant SMs

To reduce the order of the state equation and simplify the model, the same operating states and transition processes are assumed to apply to the SM. The reliability of the system could be considered as a summary of the reliability of multiple SMs. When the SM exits a fault state, it is considered as a permanent fault. Hence, the rate of return is equal to 0.

where State 0 is the normal working state, and State1 is the abnormal state, which means that there is one SM fault in any arm. The failure rate of SMs can be obtained by calculating the failure rates of the IGBTs, capacitors, and diodes in it.

*6n*SMs in the MMC, \(\lambda_{01}\) can be obtained as:

*n*is the number of SMs in any arm. Obviously, with an increasing number of SMs, the MTTF reduces proportionally.

*n*) are shown respectively in Fig. 4 and Fig. 5.

One can see that with fewer than 50 modules, the reliability does not drop rapidly with time, and before 1000 hours of operation, the reliability does not fall quickly with the number of SMs.

## 5 Reliability evaluation of MMC with a redundant SM by Markov model

As above, a general function for the reliability evaluation of the basic MMC is obtained.

- 1)
The redundant SMs do not participate in normal operation of the MMC. When a SM fails, it is removed, and the redundant SM is put into operation.

- 2)
The redundant SMs participate in normal operation. When a SM fails, it is removed, and the corresponding normal SM in the other phase is bypassed so that the system operates symmetrically.

- 3)
The redundant SMs participate in normal operation. When a SM fails, only that SM is bypassed.

Mode 3 leads to asymmetric operation. Mode 2 is not economical because the corresponding SM in the other phase is removed and the redundant SMs are aged by operation. Therefore, mode 1 is chosen for discussion.

*n*and one, respectively.

There are eight states in the model. State 0: All devices work well; State 1: One SM in any arm fails; State 2: Any two arms both have one SM failing; State 3, 4, 5: Respectively any three, four, five arms all have one SM failing; State 6: Six bridge arms all have one SM failing; State 7: The system can no longer work.

If another SM fails in one of the other five arms, the MMC can still run. The state transfers from state 1 to state 2. If another SM fails in the same arm, the MMC cannot run. The state transfers from state 1 to state 7.

The initial conditions are \(P_{0} \left( 0 \right) = 1\); \(P_{i} \left( 0 \right) = 0\; (i = 1\,\sim\, 7 )\).

*n*) is shown in Fig. 8.

The work above could provide a theoretical basis for a practical application project. By using Markov model, a general function and a mathematical calculation is firstly presented for the reliability of MMC, although it is always difficult to give a precise reliability evaluation for MMC in a practical power system project, with a large amount of power electronic components.

## 6 A reliability evaluation for a practical distribution network project

Some parameter values of the MMC used in a practical project in Zhejiang

Parameter | Value |
---|---|

Rated DC voltage of SM | 900 V |

Rated current of SM | 289 A |

Switching frequency | 1100 Hz |

Modulation ratio | 0.95 |

Power factor | 0.99 |

Comparison of two MTTFs

Case | MTTF (days) |
---|---|

Without redundant SM | < 400 |

One redundant SM | > 1200 |

The evaluation method proposed has a distinct effect to give a theoretical analysis and reliability evaluation for a practical project. For the DC distribution network project in Zhejiang, it could be found that the MTTF of the system with a redundant SM is more than 3 times that of the system without a redundant SM. Hence, it provides a theoretical proof that there should be at least one redundant SM in each arm to obtain a high reliability.

## 7 Conclusion

In a practical power system project, it is always difficult to provide a precise reliability evaluation for MMC because there is a large amount of power electronic components. In this paper, the Markov model is proposed to evaluate the reliability of the MMC. Based on the model, the mathematical calculation and a general function are firstly presented considering the failure of the electronic equipment in MMC. Based on that, the mean time to failure and the influence of redundant SMs are both discussed. Finally, they are applied to reliability evaluation for a practical DC distribution system. The system with a single redundant SM provides a factor of more than 3 improvement in reliability. The results give a further validation that the reliability evaluation based on Markov model could provide a useful reference for practical project design.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51607084), the National Key Research and Development Program of China (No. 2017YFB0903504), Jiangsu Electric Power Company Project (No. J2018076), State Grid Technology Project (No. 5210EF17002B) and the State Key Laboratory of Smart Grid Protection and Control.

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