## 2 Redundant Systems of Offshore Wind Turbines  
An offshore wind turbine is taken as an example to show the effectiveness and feasibility of the proposed methods. In the wind-power industry, redundancy designs play an important role in improving the system reliability of wind turbines. The redundancy of a component or system can improve the ability to maintain or restore its function when a failure of a member or connection has occurred. Redundancy can be achieved for instance by strengthening or introducing alternative load paths [20]. This paper focuses on a 5 MW doubly-fed induction generator with a four-point suspension produced by CSIC (Chongqing) Haizhuang Windpower Equipment Co., Ltd. from Chongqing, China. The system schematic of the offshore wind turbine (OWT) is presented in Figure 1. The OWT is a typical three-bladed, upwind, variable-speed, variable blade-pitch-to-feather-controlled turbine. The rated wind speed and the rated rotor speed are 12.6 m/s and 11.34 rpm, respectively. The design life of the OWT is 20 years. In Figure 1, the entire system of the OWT is a closed-loop control system, and every subsystem is closely related to the others. From the wind turbine structure and the design aspects, it can be seen that redundant systems of OWTs contain multiple types of functional redundancy including parallel redundancy (active redundancy), load-sharing redundancy (partially loaded redundancy), standby redundancy (passive redundancy), and \(k\)-out-of-\(n\) redundancy (change over redundancy) and redundancy designs exist in eleven subsystems: the pitch system (E\({}_{1}\)), the safety chain (E\({}_{2}\)), the communication system (E\({}_{3}\)), the speed measuring system (E\({}_{4}\)), the hydraulic system (E\({}_{5}\)), the cooling system of the nacelle (E\({}_{6}\)), the cooling system of the gearbox (E\({}_{7}\)), the power generation system (E\({}_{8}\)), the wind measuring system (E\({}_{9}\)), the temperature measuring system of the nacelle (E\({}_{10}\)), and the cooling system of the generator (E\({}_{11}\)), which are transformed into the fault tree (FT) of redundant systems shown in Figures 2 and 3. As it is shown in Figures 2 and 3, there are five types of gates in the FT of redundant subsystems: AND gate, OR gate, Voting OR gate, Standby gate, and Load Sharing gate. The AND and OR gates are the most basic types of gates in classical fault tree analysis. The basic symbolic descriptions are shown in Table 1. The logic symbol of \(k\)-out-of-\(n\) redundancy in a fault tree is represented as Voting OR gate in which the output event occurs if \(k\) or more of the input events occur (\(k\leq n\)). For standby redundancies, alike E\({}_{11}\) and E\({}_{12}\), when the active component fails, the \"standby\" backup component will take over the role of the failed component by the switch and the original system continues to function. Standby redundancy is one of the most commonly used redundancy methods in WTs. To illustrate the failure mechanism of load-sharing redundancy, the concept of load sharing in the field of WTs is proposed [21]. It is should be noted that components of the load-sharing redundancy will share a full load if both of them function, and the surviving component will suffer the full load if one component fails. In the OWT, the hydraulic system (E\({}_{5}\)), the cooling system of the nacelle (E\({}_{6}\)), the cooling system of the gearbox (E\({}_{7}\)) and the cooling system of the generator (E\({}_{11}\)) belong to load-sharing subsystems. The specific information of basic events is depicted in Table 2 including components' name, type of redundancy, mean time between failure (MTBF) and distribution type of failure times. The data of components' MTBF is real maintenance records provided by CSIC (Chongqing) Haizhuang Windpower Equipment Co., Ltd. Because reliability data of components can not be used directly, it needs to transform them into distribution parameters.