Fault-tolerant control of an open-winding brushless doubly-fed wind power generator system with dual three-level converter

Abstract

To improve the fault redundancy capability for the high reliability requirement of a brushless doubly-fed generation system applied to large offshore wind farms, the control winding of a brushless doubly-fed reluctance generator is designed as an open-winding structure. Consequently, the two ends of the control winding are connected via dual three-phase converters for the emerging open-winding structure. Therefore, a novel fault-tolerant control strategy based on the direct power control scheme is brought to focus in this paper. Based on the direct power control (DPC) strategy, the post-fault voltage vector selection method is explained in detail according to the fault types of the dual converters. The fault-tolerant control strategy proposed enables the open-winding brushless doubly-fed reluctance generator (BDFRG) system to operate normally in one, two, or three switches fault of the converter, simultaneously achieving power tracking control. The presented results verify the feasibility and validity of the scheme proposed.

Graphical abstract

Keywords

open-winding / brushless doubly-fed reluctance generator (BDFRG) / direct power control / fault-tolerant control / multi-level converter / wind power

1 Introduction

The brushless doubly-fed generator (BDFG) has the advantages of brushless structure, easiness in variable-speed constant-frequency operation, flexibly controllable active and reactive powers, and low converter capacity and cost, thus, suitable for wind power generation, especially the large offshore wind turbines/farms [1–6]. In a wind power generation system, the converter may fail due to the high energy density, complex conditions, and the relatively low reliability of power switches, which raises the concern about the reliability of BDFG [7–9]. The problem of improving system reliability can be tackled by improving the performance of the standard topologies and the early/post diagnostics of power devices [10,11] or developing new fault-tolerant topologies of power drives. Various converter structures for post-fault operation have been studied. A comparative performance study of the split-capacitor (SC) topology, the extra-leg split-capacitor (ELSC) topology and the extra-leg extra-switch (ELES) topology for voltage model (VM)-based direct torque control (DTC) brushless alternating current (BLAC) drives has been presented in Ref. [12]. In Ref. [13], many fault tolerant three-phase AC motor drive topologies are compared, including the switch-redundant topology, the double switch-redundant topology, the phase-redundant topology, the cascaded inverter topology, and the four-leg inverter topology. In a back-to-back converter, the phases of both sides (grid/machine) can be connected to each other through triacs [14]. In Ref. [14], a fault tolerant back-to-back converter for doubly-fed induction generator (DFIG)-based wind energy conversion application is studied. After the fault occurrence in one of the switches, the converter will continue its operation with the remaining five healthy legs. Proper gate signals are calculated and generated based on the actual five-leg structure and the vector control strategy. The multilevel converter is also an effective solution for fault tolerance because of more additional switching states. A novel active fault-tolerant space vector pulse width modulation (SVPWM) strategy for single-phase faults in three-phase multilevel converters is proposed in Ref. [15]. The multilevel converter is treated as a two-level converter (2LC) by introducing an offset vector. Its SVPWM strategy is designed according to the post-fault topology of the hybrid input switched seven-level converter. As a special multi-level converter structure, the dual converter topology connected to an open-winding machine has been discussed and applied in fault tolerant control. In Ref. [16], a method that allows the operation with direct torque control (DTC) in incipient faults of the power switching devices in single dc-link dual inverters is proposed. The control algorithm makes use of additional states to maintain control of the induction machine. In Ref. [17], a fault tolerant control strategy when faults occur in paralleled inverters fed permanent magnet synchronous motor (PMSM) drives is studied. It fully uses all the healthy phase legs including those in faulty inverters. The control scheme is based on rotor flux-oriented vector control and paralleled inverter structure.

Overall, the use of multilevel converters and dual converter structures has become an effective solution for post-fault operation of motor/generator systems for the following reasons. Compared to the traditional 2LC, the multilevel converter has the advantages of smaller output harmonic content, flexible control, small voltage stress on the switches, and lower loss. The open-winding topology has the advantages of non-neutral voltage fluctuation, smaller converter capacity, and stronger fault-tolerant capability [18,19]. Generally, the above topology can be applied to various systems, but the control strategies for post-fault operation are not universal. The post-fault operation strategies are usually obtained by modifying the original strategies (such as flux/voltage-oriented control, DTC) based on the converter topology after fault. Therefore, various systems with corresponding original control strategies have different strategies for post-fault operation. In this paper, the open-winding structure with dual three-level converter (3LC) is applied to the BDFRG system for wind power generation. All six terminals of control winding (CW) are drawn out to connect with the dual 3LC while power winding (PW) is grid connected. Besides, a post-fault operation strategy based on direct power control (DPC) is discussed. Due to the more switching states produced by dual 3LC, the vector selection method in the control strategy is more diversified, and the possibility of post-fault operation is significantly increased. Moreover, synthesized space voltage vector distributions of the dual 3LC are analyzed in detail when one switch open-/short-circuit fault occurs and two/three switches fault occur. The faulty switches are put under trigger suppression (no longer divided into open-circuit or short-circuit fault) to regain operation when more than one switch fault occur, which can simplify the analysis and system design. Based on the DPC strategy, the post-fault voltage vector selection method is explained in detail according to the fault types of the dual 3LC. A power deviation comparison module is designed to more appropriately select the voltage vector. The control strategy proposed can enable the open-winding brushless doubly fed reluctance generator (OW-BDFRG) system to operate without interruption when one, two, or three switches fault occurs in the dual 3LC, and simultaneously implement the power tracking control. The feasibility and validity of the fault-tolerant control strategy proposed based on DPC for the OW-BDFRG are analyzed and verified.

2 Mathematical model of OW-BDFRG

The space vector equations of voltages and fluxes for the OW-BDFRG in rotating reference frames are expressed as [1]

(1){\mathit{u}}_{\mathrm{p}}=R_{\mathrm{p}}{\mathit{i}}_{\mathrm{p}}+{\displaystyle \frac{\mathrm{d}{\mathit{\lambda }}_{\mathrm{p}}}{\mathrm{d}t}}+\mathrm{j}\omega {\mathit{\lambda }}_{\mathrm{p}},

(2){\mathit{u}}_{\mathrm{c}}=R_{\mathrm{c}}{\mathit{i}}_{\mathrm{c}}+{\displaystyle \frac{\mathrm{d}{\mathit{\lambda }}_{\mathrm{c}}}{\mathrm{d}t}}+\mathrm{j}({\omega }_{\mathrm{r}}-\omega ){\mathit{\lambda }}_{\mathrm{c}},

(3){\mathit{\lambda }}_{\mathrm{p}}=L_{\mathrm{p}}{\mathit{i}}_{\mathrm{p}}+L_{\mathrm{pc}}{\mathit{i}}_{\mathrm{c}}^{\mathrm{*}},

(4){\mathit{\lambda }}_{\mathrm{c}}=L_{\mathrm{c}}{\mathit{i}}_{\mathrm{c}}+L_{\mathrm{pc}}{\mathit{i}}_{\mathrm{p}}^{*},

where R_p and R_c are the resistances of PW and CW; L_p and L_c are the self-inductances of PW and CW; L_pc is the mutual inductance between PW and CW; u_p, i_p, and λ_p is the respective space voltage, current, and flux vectors of PW in the PW rotating reference frame; u_c, i_c, and λ_c is the respective space voltage, current and flux vectors of the CW in the CW rotating reference frame; and ω_r is the rotor electrical angular velocity. It should be noted that the space vector equations of the power and control windings are expressed in two different reference frames, the PW d_p–q_p reference frame rotating in arbitrary angular velocity ω and the CW d_c–q_c reference frame rotating at angular velocity ({\omega }_{\mathrm{r}}-\omega ). The CW flux linkage can be expressed in Eq. (5) by substituting Eq. (3) into Eq. (4).

(5){\mathit{\lambda }}_{\mathrm{c}}=\sigma L_{\mathrm{c}}{\mathit{i}}_{\mathrm{c}}+{\displaystyle \frac{L_{\mathrm{pc}}}{L_{\mathrm{p}}}}{\mathit{\lambda }}_{\mathrm{p}}^{*},

where \sigma =\mathrm{1}-L_{\mathrm{pc}}^{\mathrm{2}}/\left(L_{\mathrm{p}}{L}_{\mathrm{c}}\right).

The electromagnetic torque of the OW-BDFRG is given as

(6)T_{\mathrm{e}}={\displaystyle \frac{\mathrm{3}{p}_{\mathrm{r}}}{\mathrm{2}\sigma L_{\mathrm{c}}}}\left|{\mathit{\lambda }}_{\mathrm{c}}\times {\mathit{\lambda }}_{\mathrm{pc}}\right|,

where λ_pc is the PW flux linking CW, and \left|{\mathit{\lambda }}_{\mathrm{pc}}\right|={\displaystyle \frac{L_{\mathrm{pc}}}{L_{\mathrm{p}}}}\left|{\mathit{\lambda }}_{\mathrm{p}}\right|, but its rotational angular velocity is the same as CW flux. The mechanical power of the OW-BDFRG can be expressed as [1,2]

(7)P_{\mathrm{m}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}\mathrm{Re}\left\{\mathrm{j}\omega {\mathit{i}}_{\mathrm{p}}{\mathit{\lambda }}_{\mathrm{p}}+\mathrm{j}({\omega }_{\mathrm{r}}-\omega ){\mathit{i}}_{\mathrm{c}}^{\mathrm{*}}{\mathit{\lambda }}_{\mathrm{c}}\right\}=T_{\mathrm{e}}{\omega }_{\mathrm{rm}}=P_{\mathrm{pm}}+P_{\mathrm{cm}}={\displaystyle \frac{{\omega }_{\mathrm{p}}}{p_{\mathrm{r}}}}{T}_{\mathrm{e}}+{\displaystyle \frac{{\omega }_{\mathrm{c}}}{p_{\mathrm{r}}}}{T}_{\mathrm{e}},

where ω_p and ω_c are the applied frequencies to PW and CW respectively, and p_r is the rotor pole number. Neglecting the copper losses and the changes in field energy, the instantaneous electrical power becomes approximately the same as mechanical i.e., P_p≈ P_pm. Consequently, a method similar to torque control can be used to control power [20,21]. The rotor speed can be written as

(8)n_{\mathrm{r}}={\displaystyle \frac{\mathrm{60}(f_{\mathrm{p}}+f_{\mathrm{c}})}{p_{\mathrm{r}}}}.

The synchronous speed is defined, as expressed in Eq. (9) (control winding frequency is equal to 0).

(9)n_{\mathrm{r}\mathrm{0}}={\displaystyle \frac{\mathrm{60}{f}_{\mathrm{p}}}{p_{\mathrm{r}}}}.

3 Topology of OW-BDFRG fault-tolerant control

The structural diagram of the OW-BDFRG system is shown in Fig. 1. The machine side converter 1 (MSC1) and machine side converter 2 (MSC2) are both neutral point clamped (NPC) type 3LC and C₁, C₂, C₃, C₄ are equivalent capacitors. The OW-BDFRG stator comprises of two sets of windings with different pole numbers, the PW for generation and CW for excitation. The coupling relationship between the two sets is realized by the special rotor. The CW of OW-BDFRG is designed as the open-winding structure. The CW is opened and its two ends are connected with dual three-level bidirectional converter with the isolated DC bus connection mode, which avoids the flow of zero-sequence current.

According to Fig. 1, the phase voltage of the CW can be expressed as

(10)\{\begin{array}{l}{u}_{a\mathrm{1,\hspace{0.17em}}a\mathrm{2}}=u_{a\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{a\mathrm{2,\hspace{0.17em}}n\mathrm{2}}+u_{n\mathrm{1,\hspace{0.17em}}n\mathrm{2}},\hfill \\ u_{b\mathrm{1,\hspace{0.17em}}b\mathrm{2}}=u_{b\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{b\mathrm{2,\hspace{0.17em}}n\mathrm{2}}+u_{n\mathrm{1,\hspace{0.17em}}n\mathrm{2}},\hfill \\ u_{c\mathrm{1,\hspace{0.17em}}c\mathrm{2}}=u_{c\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{c\mathrm{2,\hspace{0.17em}}n\mathrm{2}}+u_{n\mathrm{1,\hspace{0.17em}}n\mathrm{2}}.\hfill \end{array}

To ensure normal operation of the generator, the sum of the three-phase voltages of the CW should be zero [22], thus, the CW phase voltage can be expressed as

(11)\left[\begin{array}{c}\hfill u_{a\mathrm{1,\hspace{0.17em}}a\mathrm{2}}\hfill \\ \hfill u_{b\mathrm{1,\hspace{0.17em}}b\mathrm{2}}\hfill \\ \hfill u_{c\mathrm{1,\hspace{0.17em}}c\mathrm{2}}\hfill \end{array}\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\left[\begin{array}{ccc}\hfill \mathrm{2}\hfill & \hfill -\mathrm{1}\hfill & \hfill -\mathrm{1}\hfill \\ \hfill -\mathrm{1}\hfill & \hfill \mathrm{2}\hfill & \hfill -\mathrm{1}\hfill \\ \hfill -\mathrm{1}\hfill & \hfill -\mathrm{1}\hfill & \hfill \mathrm{2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill u_{a\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{a\mathrm{2,\hspace{0.17em}}n\mathrm{2}}\hfill \\ \hfill u_{b\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{b\mathrm{2,\hspace{0.17em}}n\mathrm{2}}\hfill \\ \hfill u_{c\mathrm{1,\hspace{0.17em}}n\mathrm{1}}-u_{c\mathrm{2,\hspace{0.17em}}n\mathrm{2}}\hfill \end{array}\right].

In DPC, the active power (P_p) and reactive power (Q_p) of PW is controlled by adjusting the rotation and amplitude of λ_c which is regulated by u_c. In normal operation, the DPC principle of a common BDFRG system can be summarized as [20]: If λ_c lies in the kth sector, then vectors U_k+1 or U_k+2 would increase P_p, while U_k−1 or U_k−2 would decrease it (Fig. 2(a)). In the same sector, U_k−1, U_k, or U_k+1 would decrease Q_p, while U_k+2, U_k+3, or U_k−2 would increase it (Fig. 2(a)). Compared with 2LC, 3LC provides more available voltage vectors (Fig. 2(b)). P, O and N represent the three different output states of single-phase bridge arm. Taking the a-phase of MSC1 as an example, the output state of the a-phase in MSC1 is set as P when S_a1 and S_a2 are opened. The output state is set as O when S_a2 and S_a3 are opened. When S_a3 and S_a4 are opened, the output state is set as N. When CW is connected to dual 3LC, u_c is determined by three kinds of switching states in each phase of dual 3LC, and it can be synthesized by the voltage vectors output by MSC1 and MSC2 respectively. Nine switching states are present in each phase of dual 3LCs. Therefore, the amount of switching state combination is 9³ = 729, corresponding to 61 different space voltage vectors ultimately. In the above case, the effective voltage vectors for DPC are no longer limited to six (Fig. 2(a)), and the division of sectors can also be more detailed, as described in Section 4.

4 Fault analysis and voltage vector selection

Block diagram of the system proposed is demonstrated in Fig. 3. Switch fault diagnosis has been extensively studied, which is not the focus of this paper. This paper contemplates and focuses on the post-fault operation of the system, considering one, two, and three switches fault.

In Fig. 3, P_p and Q_p are expressed as

(12)\{\begin{array}{l}{P}_{\mathrm{p}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}(u_{\mathrm{p}\alpha }{i}_{p\alpha }+u_{\mathrm{p}\beta }{i}_{\mathrm{p}\beta }),\hfill \\ Q_{\mathrm{p}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}(u_{\mathrm{p}\beta }{i}_{\mathrm{p}\alpha }-u_{\mathrm{p}\alpha }{i}_{\mathrm{p}\beta }).\hfill \end{array}

The αβ-axis component of λ_c and its angle can be determined respectively as

(13)\{\begin{array}{l}{\mathit{\lambda }}_{\mathrm{c}\alpha }={\displaystyle \int (u_{\mathrm{c}\alpha }-R_{\mathrm{c}}{i}_{\mathrm{c}\alpha })\mathrm{d}t,}\hfill \\ {\mathit{\lambda }}_{\mathrm{c}\beta }={\displaystyle \int (u_{\mathrm{c}\beta }-R_{\mathrm{c}}{i}_{\mathrm{c}\beta })\mathrm{d}t,}\hfill \end{array}

(14)\theta ={\tan }^{-\mathrm{1}}{\displaystyle \frac{{\mathit{\lambda }}_{\mathrm{c}\beta }}{{\mathit{\lambda }}_{\mathrm{c}\alpha }}}.

Since θ ranges from –π/2 to π/2, θ is not the true position angle of CW, thus the CW flux angle needs to be corrected and revised as listed in Table 1.

4.1 Fault of one switch

Take the switch fault on the a-phase of MSC1 as the case of one switch fault. There are 8 types of one switch faults on a-phase arm of MSC1, which consist of S_a1 open-/short-circuit fault, S_a2 open-/short-circuit fault, S_a3 open-/short-circuit fault, and S_a4 open-/short-circuit fault.

4.1.1 S_a2 in short-circuit fault/S_a4 in open-circuit fault

If the output state of the a-phase arm is N, then a current loop is formed by S_a2–S_a3–S_a4–C₂–VD_a1 when the short-circuit fault of S_a2 occurs. Thus, to avoid secondary failures, the output state N is forbidden under this circumstance. When the open-circuit fault of S_a4 occurs, the output state N of the a-phase arm cannot be produced, hence it is lost. In both cases, the synthesized space voltage vector distribution of MSC1 and MSC2 is depicted in Fig. 4, where the voltage vectors used for DPC are indicated in blue. U_a-b represents the synthesized space voltage vector of the voltage vector U_a produced by MSC1 and U_b produced by MSC2. The red dots represent the voltage vectors that cannot be synthesized. The diversification of voltage vector selection makes the adjustment of P_p and Q_p more flexible and effective. Thus, to precisely utilize more voltage vectors, the sector plane is divided into 18 sectors as depicted in Fig. 4. Compared with the DPC scheme on the traditional 2LC, the CW voltage vector selection method is more diversified at this time. Hence, when CW flux linkage is located in sector II, the CW voltage vector located in region 1 can increase P_p and reduce Q_p. The CW voltage vector located in region 2 increases both P_p and Q_p. On the other hand, the CW voltage vector located in region 3 can reduce P_p and increase Q_p, while the CW voltage vector in region 4 can reduce P_p and Q_p. As illustrated above, when the CW flux linkage is located in sector II, U_14-8, U_14-9, and U_15-9 can increase P_p and reduce Q_p, but their effects are not consistent. U_15-9 has a stronger regulation effect on P_p than U_14-8 and U_14-9, on the contrary, U_15-9 has a weaker regulation effect on Q_p than U_14-8 and U_14-9. When CW is in other sectors, the influence of voltage vector on power can be analogous.

This paper focuses on adjusting P_p and Q_p by designing the power deviation comparison module in Fig. 5. According to DPC principles, the switching voltage vector selection table is reconstructed as tabulated in Table 1. S₁ is set as 1 when DP_p≥threshold, and S₁ is set as 0 when DP_p≤−threshold. Similar rules apply to S₂, while S₃ is employed to represent the numerical relationship between the errors of the active and reactive powers. S₃ is set as 1 when the error of the active power is more than that of the reactive power, and the selected voltage vector accelerates the change in the active power more effectively. Otherwise, S₃ is set at 0, and the selected voltage vector accelerates the change in the reactive power more effectively.

4.1.2 S_a3 in short-circuit fault/S_a1 in open-circuit fault

If the output state of the a-phase arm is P, a current loop is formed by S_a1–S_a2–S_a3–VD_a2–C₁ when the short-circuit fault of S_a3 occurs. To avoid secondary failures, the output state P is forbidden under this circumstance. When the open-circuit fault occurs in S_a1, the output state P of the a-phase arm cannot be produced, thus, it is lost. In both cases, the synthesized space voltage vector distribution of MSC1 and MSC2 is depicted in Fig. 6 where the voltage vectors used for control are indicated in blue, and the red points represent the voltage vectors that cannot be synthesized. The voltage vector selected are the same as that in Fig. 4, hence the switching voltage vector selection is the same as that in Table 2.

4.1.3 S_a2 in open-circuit fault

When the open-circuit fault of S_a2 occurs, the output state P and O of the a-phase arm cannot be produced. The synthesized space voltage vector distribution of MSC1 and MSC2 is depicted in Fig. 7, under this circumstance. The synthesized space voltage vectors used for DPC are indicated in blue, while the voltage vectors that cannot be synthesized are presented in red dots (Fig. 7). The entire space plane is evenly subdivided into 12 sectors in Fig. 7, while the switching voltage vector selection table is given in Table 3.

4.1.4 S_a3 in open-circuit fault

When the open-circuit fault of S_a3 occurs, the output state N and O of the a-phase arm cannot be produced. The synthesized space voltage vector distribution of MSC1 and MSC2 are depicted in Fig. 8. The voltage vectors used for DPC are indicated in blue, while the voltage vectors that cannot be synthesized are represented in red dots (Fig. 8). The voltage vector selection is the same as that in Fig. 7, thus the switching voltage vector selection table is the same as that in Table 3.

4.1.5 S_a1 or S_a4 in short-circuit fault

When the short-circuit fault of S_a1 or S_a4 occurs, the output state O is forbidden in order to prevent capacitor C₁ or C₂ from short-circuit. The synthesized space voltage vector distribution of MSC1 and MSC2 are illustrated in Fig. 9, where the synthesized space voltage vectors employed for the DPC are indicated in blue. It is important to mention that there is no non-synthetic vector in both cases. However, to simplify the design of the fault-tolerant strategy, the same voltage vector selection method is employed as shown in Figs. 4 and 6. Although this method reduces the bus voltage utilization compared to normal operation, the overall voltage vector selection table design of the system is more concise as compensation.

4.2 Fault of two switches

Once the fault of two or more switches is detected (whether open-circuit or short-circuit fault), the fault switches are put under trigger suppression to regain operation, which reduces the complexity of the analysis and system design. In this paper, the fault conditions of the following two switches are analyzed separately: two switches on one phase of one converter (S_a1 and S_a2, S_a1 and S_a3, S_a1 and S_a4, S_a2 and S_a3, S_a2 and S_a4, S_a3 and S_a4), two switches on two phases of one converter (S_a1 and S_b1, S_a1 and S_b2, S_a1 and S_b3, S_a1 and S_b4, S_a2 and S_b1, S_a2 and S_b2, S_a2 and S_b3, S_a2 and S_b4), and two switches on two phases of two converters (S_a1 and S_d1, S_a1 and S_d2, S_a1 and S_d3, S_a1 and S_d4, S_a2 and S_d1, S_a2 and S_d2, S_a2 and S_d3, S_a2 and S_d4 fault). The synthesized space voltage vector distributions of MSC1 and MSC2 in the above cases are shown in Fig. 10, in which the vectors that cannot be synthesized are represented in red dots, and the vectors that can be used for control are indicated in blue.

It is worth noting that the a-phase arm of MSC1 cannot output the P, N, and O state when the following faults occur: S_a1 and S_a3 fault, S_a2 and S_a3 fault, S_a2 and S_a4 fault. This means that MSC1 cannot output the effective voltage vector and the synthesized voltage vectors cannot be produced. MSC1 and MSC2 cannot produce the suitable composite vector when the following faults occur: S_a2 and S_b3 fault, S_a2 and S_d3 fault. Thus, the system cannot operate normally by adopting the strategy proposed when the above faults occur.

The voltage vector that can be used for control is the same as that in Figs. 4, 6, and 9 when the following faults occur: S_a1 and S_b1, S_a1 and S_d1, S_a1 and S_a4. The voltage vector that can be used for control is the same as that in Figs. 7 and 8 when the following faults occur: S_a1 and S_b2, S_a1 and S_b4, S_a1 and S_d2, S_a1 and S_d4, S_a2 and S_b1, S_a2 and S_b2, S_a2 and S_d1, S_a2 and S_d2, S_a1 and S_a2, S_a3 and S_a4. For simplicity and readability, the voltage vector selection table with the same available vectors as that in Section 4.1 is no longer repeated. This section only gives the voltage vector selection tables (Tables 4–7) in the following faults: S_a1 and S_b3, S_a1 and S_d3, S_a2 and S_b4, S_a2 and S_d4. The available voltage vectors in these cases are different from those described in Section 4.1. The system can still operate stably under the action of these voltage vectors.

4.3 Fault of three switches

The conditions of fault of three switches are analyzed separately: fault of three switches on one phase of one converter (S_a1, S_a2 and S_a3; S_a1, S_a2 and S_a4), fault of three switches on two phases of one converter (S_a1, S_a2 and S_b1; S_a1, S_a2 and S_b2; S_a1, S_a2 and S_b3; S_a1, S_a2 and S_b4; S_a2, S_a3 and S_b1; S_a2, S_a3 and S_b2; S_a2, S_a3 and S_b3; S_a2, S_a3 and S_b4), fault of three switches on two phases of two converters (S_a1, S_a2 and S_d1; S_a1, S_a2 and S_d2; S_a1, S_a2 and S_d3; S_a1, S_a2 and S_d4, S_a2, S_a3 and S_d1; S_a2, S_a3 and S_d2; S_a2, S_a3 and S_d3; S_a2, S_a3 and S_d4), fault of three switches on three phases of one converter (S_a1, S_b1 and S_c1; S_a1, S_b1 and S_c2; S_a1, S_b1 and S_c3; S_a1, S_b1 and S_c4; S_a1, S_b2 and S_c1; S_a1, S_b2 and S_c2; S_a1, S_b2 and S_c3; S_a1, S_b2 and S_c4; S_a2, S_b2 and S_c1; S_a2, S_b2 and S_c2; S_a2, S_b2 and S_c3; S_a2, S_b2 and S_c4), and fault of three switches on three phases of two converters (S_a1, S_b1 and S_d1; S_a1, S_b1 and S_d2; S_a1, S_b1 and S_d3; S_a1, S_b1 and S_d4; S_a1, S_b2 and S_d1; S_a1, S_b2 and S_d2; S_a1, S_b2 and S_d3; S_a1, S_b2 and S_d4; S_a2, S_b2 and S_d1; S_a2, S_b2 and S_d2; S_a2, S_b2 and S_d3; S_a2, S_b2 and S_d4).

The synthesized space voltage vector distributions of MSC1 and MSC2 in the above cases are shown in Fig. 11. It is worth noting that the a-phase arm of MSC1 cannot output the P, N and O state when the following faults occur: S_a1, S_a2 and S_a3; S_a1, S_a2 and S_a4; S_a2, S_a3 and S_b1; S_a2, S_a3 and S_b2; S_a2, S_a3 and S_b3; S_a2, S_a3 and S_b4; S_a2, S_a3 and S_d1; S_a2, S_a3 and S_d2; S_a2, S_a3 and S_d3; S_a2, S_a3 and S_d4. This means that MSC1 cannot output the effective voltage vector and the synthesized space voltage vectors cannot be produced. MSC1 and MSC2 cannot produce the suitable composite vector when the following faults occur: S_a1, S_a2 and S_b3; S_a1, S_a2 and S_d3; S_a1, S_b2 and S_c3; S_a2, S_b2 and S_c3; S_a1, S_b2 and S_d2; S_a2, S_b2 and S_d2; S_a2, S_b2 and S_d3. Thus, the system cannot operate normally by adopting the strategy proposed when the above faults occur.

The available voltage vectors in other cases are the same as those described in Sections 4.1 and 4.2. There is no doubt that the system can operate reliably under the control of these vectors. The simulation results of this part are omitted for simplicity.

5 Simulation results

The control strategy proposed (Fig. 3) for the OW-BDFRG is emulated and implemented using Matlab/Simulink. The generator parameters are: nominal power 23 kW, the number of pole-pairs are PW-4 and CW-2 respectively. The resistances of PW and CW are 0.3871 W and 0.3773 W respectively, self-inductances of PW and CW are 69.46 mH and 59.25 mH respectively, while the mutual-inductance between the two-stator windings is 56.52 mH. During simulation, the threshold of the active and the reactive power hysteresis comparator is 0.1 kW and 0.1 kvar, respectively. The PW is grid connected as evidenced in Fig. 1 (i.e., voltage amplitude is \mathrm{220}\sqrt{\mathrm{2}} V and frequency is 50 Hz). The DC-link voltage is set at 60 V (U_dc1 = U_dc2 = 60 V). The switch state is set to 0 to imitate open-circuit fault and 1 to imitate short-circuit fault, respectively. C₁ = C₂ = C₃ = C₄ = 2500 mF.

5.1 Operation without fault-tolerant control

The failure in power switches will affect the performance of the system and completely paralyze the system in severe cases. Taking the fault of S_a2 open-circuit as an example, the transitional response from healthy mode to fault mode is simulated as illustrated in Figs. 12 and 13. The fault occurred during the sub-synchronous speed operation is shown in Fig. 12. The system runs normally in the regular DPC scheme prior to 0.7 s and 2.5 s respectively. When the fault occurs, the PW and CW current are distorted irregularly, thus P_p and Q_p cannot track the set point and have a huge deviation.

Figure 13 illustrates the fault occurred during the super-synchronous speed operation. In healthy mode, when the OW-BDFRG is in sub-synchronous operation, the maximum active power deviation is less than 0.2 kW, and the maximum reactive power deviation is less than 0.1 kvar. However, when the system is operating super-synchronously, the maximum active power deviation is less than 0.5 kW, and the maximum reactive power deviation is less than 0.1 kvar.

5.2 Fault-tolerant operation

To verify the effectiveness of the fault-tolerant control proposed, the control method proposed is scrutinized in different fault types as discussed in Section 4. During simulation, the reference speed (n_rref), active power reference (P_pref) and reactive power reference (Q_pref) change stepwise at different times as shown in Fig. 14. The reference value of the active power is wind speed dependent; thus, the reference value of the reactive power needs to meet the power factor requirement of the grid. The power here is only given in the form of simulation to illustrate the effectiveness of the method (i.e., the negative sign - refers to generating active power and reactive power). The results presented include PW/CW currents, active/reactive power, and sector calculation.

Figures 15–26 show the steady and dynamic results for post-fault operation corresponding to different faults. The OW-BDFRG operates at sub-synchronous speed, synchronous speed, and super-synchronous speed respectively in different time periods to verify the variable-speed constant-frequency operation characteristics of the system. P_pref and Q_pref generate step changes at different times to observe power tracking. It is illustrated from the PW current that the PW current frequency is always 50 Hz regardless of whether the OW-BDFRG is running in sub-synchronous, synchronous, or super-synchronous operation. Moreover, it is shown that the PW current is sinusoidal. The CW current frequency varies according to n_rref. The CW current frequency is 5 Hz in sub-synchronous and super-synchronous operation, and it is 0 Hz in synchronous operation, which is consistent with Eqs. (8) and (9).

The PW current amplitude varies with P_p and Q_p. The CW current amplitude is related to the active and reactive power of PW and the speed. P_p changes to –10 kW and –15 kW at 1.3 s and 2 s respectively, which results in the increase of PW current amplitude at the corresponding time. Q_p affects the distribution of PW current and CW current to the excitation current. All excitation current is provided by CW current when Q_p = 0 kvar (0–0.9 s and 1.5–3.5 s), and the excitation current is provided by both CW current and PW current when Q_p = 5 kvar (0.9–1.5 s). At 0.9 s, Q_p changes from 0 kvar to 5 kvar, and PW current provides part of the excitation current. Therefore, the amplitude of PW current increases and the amplitude of CW current decreases at 0.9 s. At 1.5 s, Q_p changes from 5 kvar to 0 kvar, and PW current no longer provides the excitation current. All excitation current is provided by CW current, resulting in an increase in the amplitude of CW current and a decrease in the amplitude of PW current.

After the converter switch fails, the fault-tolerant control strategy proposed can make P_p and Q_p follow the set value to change rapidly (the transition time occurs within 0.01 s). The PW active power deviation can be kept within ±0.2 kW when the OW-BDFRG is in sub-synchronous stable operation and it can be kept within ±0.5 kW in super-synchronous stable operation. When the OW-BDFRG is in synchronous stable operation state, the active power deviation of PW can be controlled within ±0.1 kW. During this time, the power on the PW side is provided by the mechanical power input from the rotating shaft and the CW side in the DC excitation state. However, the synchronous speed is only a specific speed point and the OW-BDFRG usually operates in a doubly fed state.

The reactive power deviation is stably controlled within ±0.1 kvar throughout the entire operation process when one switch fails, and within ±0.4 kvar when two switches fails. The synthesized voltage vector available is significantly reduced, and the number of sectors is only 6, resulting in a relatively rough control when the following faults occur: S_a1 and S_b3, S_a1 and S_d3, S_a2 and S_b4, S_a2 and S_d4. But the system is still in stable operation. Due to the existence of the hysteresis comparator, the switching frequency is not fixed, thus the maximum switching frequency is 2.5 kHz. Figures 27–29 show the identification results of the sector where the CW flux vector is located. The sector identification result is shown in Fig. 27 when Table 2 is used. The entire plane is evenly subdivided into 18 sectors in this case. The sector value shows a decreasing step state when the generator is in sub-synchronous speed operation, indicating that the CW flux vector rotates clockwise. The sector value shows an increasing step state when the generator is in super-synchronous speed operation, indicating that the CW flux vector rotates counter-clockwise. Figures 28 and 29 show the situation when different switch tables are used. The explanation of Figs. 28 and 29 is similar to the discussion above. The system has a similar performance between post-fault operation and normal operation. The captured results proved that the strategy proposed makes the active and reactive powers track the changes of given values swiftly. Moreover, it makes the system operate stably when power switch fault occurs. The results have illustrated that the fault-tolerant control strategy can simultaneously implement variable-speed constant-frequency operation.

6 Conclusions

For the OW-BDFRG with dual 3LC system, an original fault-tolerant control strategy based on DPC scheme is proposed and investigated, which is the main contribution of this paper. The control strategy has demonstrated a high performance, simple structure, versatility, strong robustness, and good real-time which can improve the fault redundancy capability of the generation system. In addition, the control strategy comprises of a flexible control mode and reduces the converter capacity which is another very important feature of the control strategy proposed. The switching voltage vector selection tables in DPC are constructed under different fault conditions of the dual 3LCs and the system simulation has been verified. The results show that the fault-tolerant control strategy proposed makes the active and reactive powers of the OW-BDFRG track the power given values swiftly and stably. Such occurrence performs well for the fault of one, two, or three power switches and can simultaneously implement variable-speed constant-frequency operation.

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