A wind turbine consists of several components capable of converting kinetic energy into electrical energy. The blades on the wind turbine can convert wind energy into mechanical energy, which is then transmitted through the drive system to the generator, where it is further transformed into electrical energy. Therefore, the wind turbine is a primary and critical component of a WPG system, directly impacting the efficiency of wind power generation.

According to aerodynamic principles, it is possible to express the airflow power as Equation 1. P w = 1 2 ρ A v w 3 (1) where ρ represents air density, under normal conditions, ρ = 1.2 k g / m 3 ; A denotes swept area; v w is wind speed.

The blades capture wind power can be expressed as Equation 2. P m = 1 2 ρ A v w 3 C p (2) where C p denotes wind energy utilization coefficient. According to the Betz limit, the maximum theoretical value of this coefficient is 0.59. The sweep area A of the wind turbine is only related to the physical size of the wind motor, air density is equal to 0.2 generally. Under the wind speed is given, the wind energy utilization coefficient C p determines the power obtained by wind turbine.

When the blade rotates, the ratio of the tip speed to the input wind speed is defined as the tip speed ratio λ . λ can be indicated as Equation 3. λ = ω m R v w (3) where ω m represents the angular velocity of the blade, R represents blade radius.

For variable pitch wind turbine, C p can be expressed as Equation 4. C p λ , β = 0.5176 116 λ i − 0.4 β − 5 e − 21 λ i + 0.0068 λ (4) where β represents pitch angle, λ i is determined by Equation 5. 1 λ i = 1 λ + 0.08 β − 0.035 β 3 + 1 (5)

Through computing P m , according to Equation 6, the output torque of wind turbine is obtained and inputs into the PMSG. T m = P m ω m (6)

PMSG uses permanent magnet material to replace the excitation winding, and the permanent magnet generates rotor excitation, which is a brushless motor. Since there is no rotor winding, its size and weight are greatly reduced, and there is no rotor winding loss.

Figure 2 shows the equivalent model of PMSG. The time domain model of PMSG in the stationary coordinate system can be represented by voltage equation, flux equation and rotor motion equation. The three-phase stator winding voltage equation can be described as Equation 7. u s a = R s i s a + p ψ s a u s b = R s i s b + p ψ s b u s c = R s i s c + p ψ s c (7) where u s a , u s b , u s c represent three-phase winding phase voltage; i s a , i s b , i s c represent three-phase winding phase current; ψ s a , ψ s b , ψ s c denote three-phase winding flux linkage; p = d d t . Three phase winding flux equation can be written as Equation 8. ψ s a ψ s b ψ s c = L a a L a b L a c L b a L b b L b c L c a L c b L c c i s a i s b i s c + ψ f a ψ f b ψ f c (8) where L a a , L b b , L c c are three-phase winding inductance; M a b = M b a , M a c = M c a , M b c = M c b denote mutual inductance between three phase windings; ψ f a , ψ f b , ψ f c denote the flux linkage between the rotor and the stator, which can be described as Equation 9. ψ f a ψ f b ψ f c = ψ f cos ⁡ θ cos θ − 2 π 3 cos θ + 2 π 3 (9)

According to the theory of permanent magnet motor, the motor motion equation can be written as Equation 10. J d ω m d t = T e − T m − B m ω m (10) where T m represents input mechanical torque, which can be acquired in (6), T e denotes mechanical torque of PMSG; B m = 0 is coefficient of rotational viscosity.

In the static coordinate system, the uneven air gap leads to the asymmetry of the fixed rotor magnetic field structure, and the projection of the rotor flux on the three-phase stator winding is related to the rotor position Angle. The mathematical model of the synchronous motor is a set of nonlinear time-varying equations related to the instantaneous position of the rotor, which is difficult to analyze and control. After Park transformation, the stator winding is equivalent to the d and q axis winding and the rotor winding are relatively stationary, so that the inductance parameters of the d and q axes become fixed, and the stator voltage, current and flux vector are all constant direct flow that is relatively stationary with the rotor. In PSCAD, the modeling of WPG system can be realized by applying coordinate transformation.

The three-phase full-bridge inverter’s topology is described in Figure 3, and all of the symbols utilized in the subsequent LBFC design process are defined in Table 1.

Obviously, the states of S j P and S j N are reversed, let S j P serves as the working state of the switches on j − p h a s e and rewrite it as S j b , S j b ∈ { 0,1 } . Then the three-phase inverter modelling in this paper can be written as follows Equation 11. L filter p i L j = V d c S j b + u O N − R filter i L j − v j C filter p u C j = i L j − i j u O N = − V d c 3 S a b + S b b + S c b (11) where u O N denotes the voltage between point O and N , u C j represents the filter capacitor voltage. The output current of the inverter would need to be controlled in this paper. As a result, the differential Equation 11 has to be stated in general linear single-input, single-output (SISO) form (Liberzon and Trenn, 2013). Then i L j is both the system’s output y j and one of the state variables, which have been set as X j = [ i L j , u C j ] . The control variable of the system is defined as u j = S j b . Thus the j − p h a s e system in the inverter can be given by Equations 12, 13. p X j t = F X j + G X j u j t y j t = h j t (12) where F x j = − 1 L filter R filter x 1 + V d c 3 S k b + V d c 3 S l b + v j 1 C filter x 1 − i j G x j = 2 V d c 3 L filter 0 h j t = x 1 t (13) where k and l indicate the two phases aside from j − p h a s e , their operation states S k b and S l b are treated as constant variables during discussing j − p h a s e to allow the independence of logic switching control for each phase.

The order of the LBFC varies depending on the relative degree r of the system. Specifically, the relative degree of control objective h j ( t ) with respect to system’s input u j ( t ) is to differentiate output h j ( t ) until input u j ( t ) explicitly appears in h j r ( t ) , namely, as Equation 14.. h j r t = L F r h j t + L G L F r − 1 h j t u j t (14) when L G L F r − 1 h j ( t ) ≠ 0 holds. Thus, the approach for determining the relative degree of system (12) is shown in Equation 15. h j 1 t = L F h j t + L G h j t u j t = − R filter L filter x 1 t − V d c S k b + V d c S l b + 3 v j 3 L filter + 2 V d c 3 L filter u j t (15) Therefore, it has r = 1 and the first-order LBFC would be adopted here. The switching logic of the first-order LBFC can be simply defined as Equation 16. q t = G e t , φ 0 + − ε 0 + , φ 0 − + ε 0 − , q t − = G e t , e ̄ , e ̲ , q t − = e ≥ e ̄ ∨ e > e ̲ ∧ q t − q 0 − ∈ t r u e , f a l s e (16) where q ( t ) ∈ { t r u e , f a l s e } ,having q ( t − ) ≔ l i m ε → 0 q ( t − ε ) , is the switching logic’s output deduced by the tracking error e ( t ) = i Lji . The chosen funnel boundaries are shown by φ 0 ± , ε 0 ± represent the safety distances, and the upper trigger e ̄ and lower trigger e ̲ are made up of these two. The BBFC enables to limit the fault current in the funnel through logical switching control. The funnel boundaries can be obtained by several simulation trials, combining with the limit value of current. In most instances, the value of ε 0 ± is set as 0. The scheme of LBFC in PSCAD can be described by Figure 4.

On the basis of L g h j i ( t ) = 2 V dci 3 L filter > 0 , the control law of the LBFC designed to suppress the inverter outlet currents is given as Equation 17. S j b t = 0 , if q ( t )   =  true 1 ,  if q ( t )   =  false (17)

The state-dependent strategy T , depicted in Figure 1, is the foundation upon which the switching control scheme created for the overcurrent suppression of the PMSM power system operates. It is explained in Figure 5.

Assume that following a short-circuit malfunction at the system, the absolute value of any j − p h a s e current at the inverter’s outlet is | e j ( t ) | . Then the switching strategy is stated as that the inverter bridge arm switching control switches from the GSC Control to LBFC if T 1 is satisfied and switches from the LBFC to the GSC Control on condition that T 2 is satisfied, where T 1 and T 2 are illustrated as follows (Liu et al., 2016a):

T 1 : { | e j ( t ) | ≥ ϖ } , T 2 : {The switching frequency of the control signal generated by ad LBFC reaches its maximum} ∨ {Signal for fault clearance, it can be V ̄ t j > τ }where ϖ , τ are design parameters of the switching strategy of the system, V ̄ t j is the three phase voltage of the short-circuit point.

On bus 1 at t = 1.5 s , a three-phase current fault occurred. Figures 8, 9 display the dynamics of WPG system. Where the WPG system in Figure 9 is equipped with LBFC during current fault, and system in Figure 8 is only under the vector control. Due to the current fault of three-phase, the magnitudes of bus voltages dropped, and the three-phase voltages measured on bus 1 are shown in Figures 8A, 9A. Fault was cleared after 90 ms. During the current fault time, the short-circuit currents of the system without LBFC reached 3 times the nominal current value, as shown in Figure 8B.

In contrast, the switching control of vector control and LBFC had an optimizing effect on WPG system during fault time. When the short-circuit current was up to the switching criterion ϖ , the inverter bridge arm switching control was switched from the vector control to the LBFC, which in turn controlled the current the set boundary values φ 0 − to φ 0 + , as seen in Figure 9B, which represented that BBFC was able to control the fault current. Furthermore, with the magnitudes of bus voltages declined, the active power both in Figures 8C, 9C decreased. As presented in Figures 8D, 9D, WPG system’s reactive power output fluctuated following the voltages’ volatility when system converted from LBFC to vector control.

Similarly, three-phase-to-ground fault occurred on bus 2 at 1.5 s . The dynamics of WPG system were displayed in Figures 10, 11. When a three-phase current fault occurred at 1.5s, the three-phase voltages decreased and were close to zero during the fault time, as seen in Figures 10A, 11A. WPG system in Figure 10 utilized vector control only, while system in Figure 11 employed the switching control of vector control and LBFC. The results were the same as above content, LBFC could operate effectively, it could control the fault current of WPG system, as displayed in Figures 10B, 11B. The fault was cleared after 90 ms, when the system switched from LBFC to vector control, and voltage and current oscillations occurred during this process, which also caused active and reactive power fluctuations, as seen in Figures 10C, D and Figures 11C, D.