The three-phase converter considered here has a constant power input P in , which is modeled by connecting a controlled current source in parallel to the DC-side capacitor, with the output current being P in / V dc , as shown in Figure 1. R f and L f are the filter resistance and filter impedance, respectively. R g and L g are the gird resistance and gird impedance, respectively. V ̄ d 1 , V ̄ q 1 , i ̄ d 1 , and i ̄ q 1 are the per unit value output current and voltage of the converter in the dq-frame. The grid-following control is realized in the dq reference frame. The d-axis control loop is used to regulate the capacitor voltage V d c , and the q-axis control loop is used to regulate the reactive power output Q of the converter. A PLL is used to capture the phase angle and angular frequency of the voltage vector v ⃗ of the filter capacitors C f . The phase angle output θ pll of the PLL is used as the phase angle input of the Park transformation to generate the dq components of three-phase voltages and currents, and it is also used as the phase angle reference to generate the three-phase reference voltages for the converter, that is, e ̄ * a , e ̄ * b , and e ̄ * c .

Inspired by the control block in the study by Huang and Vanfretti (2023), the structure of the FLF for the GFL controller is also presented in Figure 1. The FLF is realized by subtracting the three-phase flux linkages Ψ ̄ a , Ψ ̄ b , and Ψ ̄ c of the filter capacitor node from the three-phase reference voltages generated by the current loop. The obtained three-phase signals are used as the inputs of the pulse width modulation (PWM) block, using which triggering pulses s a , s b , and s c are generated for converter bridges. If overmodulation does not occur, the following are the three-phase output voltages of the converter: e ̄ a = e ̄ a * − Ψ ̄ a , e ̄ b = e ̄ b * − Ψ ̄ b , e ̄ c = e ̄ c * − Ψ ̄ c . (1)

The three-phase flux linkages of the filter capacitor node are calculated by Equation 2. Ψ ̄ a = Ψ ̄ a ′ − M e a n Ψ ̄ a ′ Ψ ̄ b = Ψ ̄ b ′ − M e a n Ψ ̄ b ′ Ψ ̄ c = Ψ ̄ c ′ − M e a n Ψ ̄ c ′ (2) where p Ψ ̄ a ′ = 2 π f n v a / v n , p Ψ ̄ b ′ = 2 π f n v b / v n , p Ψ ̄ c ′ = 2 π f n v c / v n , (3) where p = d / d t in Equation 3 represents the differential operator, M e a n ( ⋅ ) represents the mean value of a variable over a period of 1 / f n , the M e a n block is to eliminate the asymmetric component of single-phase flux linkages, f n is the rated frequency of the system in Hz, v n denotes the nominal peak phase-to-ground voltage in V, and v a , v b , and v c are the three-phase voltages of the filter capacitor node.

Applying Clark transformation on the two sides of Equations 3, 4, we obtain p Ψ ̄ x ′ Ψ ̄ y ′ = 2 π f n v ̄ x v ̄ y , (4) where the zero sequence components are neglected as the symmetrical three-phase circuit is considered here. The vector of node flux linkage can be written as Ψ ̄ ′ ⃗ = Ψ ̄ x ′ + j Ψ ̄ y ′ , and the vector of node voltage can be denoted as v ̄ ⃗ = v ̄ x + j v ̄ y . The vector form of Equation 4 can be written as p Ψ ̄ ′ ⃗ = 2 π f n v ̄ ⃗ . By writing the vectors in the polar coordinate form, we have Ψ ̄ ′ ⃗ = Ψ ̄ ′ e j θ Ψ ̄ , v ̄ ⃗ = v ̄ e j θ v ̄ , and Equation 5 can be obtained as p Ψ ̄ ′ + j Ψ ̄ ′ p θ Ψ ̄ e j θ Ψ ̄ = 2 π f n v ̄ e j θ v ̄ . (5)

In the steady state, the magnitude and angular frequency of sinusoidal signals remain constant, that is, p Ψ ̄ ′ = 0 and p θ Ψ ̄ = ω Ψ ̄ . Therefore, the steady-state relationship between the voltage and flux linkage vectors can be described by Equation 6. Ψ ̄ ′ = v ̄ , θ Ψ ̄ + π 2 = θ v ̄ , v ̄ ⃗ = j Ψ ̄ ⃗ ′ , (6) where ω ̄ Ψ ̄ = ω Ψ ̄ / ( 2 π f n ) = 1 . Magnitudes of the per unit flux linkage and voltage vectors are the same, and the flux linkage vector leads the node voltage vector by 90 ° in the steady state.

According to Ma et al. (2022), parameters of the PLL have a significant impact on the synchronous stability of GFL converters. Taking advantage of the data-driven technique, a DFF neural network is employed to study the nonlinear expression of the overshoot and the stabilization time of system frequency in terms of PLL parameters, and the GA is used to search for the optimal solution of this expression to obtain the optimal parameters for the PLL. The DFF-GA-based parameter optimization for the PLL is constructed as shown in Figure 2. f s and t s denote the overshoot and settling time of the system frequency, respectively, where C o s t train = 2.4 f s + 1.2 t s . K pop and K iop are the optimal parameters with DFf-GA. A total of 10,000 sets of data are randomly divided into 8,000 sets of the training set and 2,000 sets of the test set, which are obtained by the Simulink model. The input data are the proportional–integral (PI) parameters of the PLL shown in Figure 5. Each input layer of the DFF is expressed by Equation 7. o i l n = x i n , l = 1 f P j l − 1 n + b l − 1 n , l = 2,3,4 (7) Here, x i ( n ) indicates the input of the DFF; x 1 = K p and x 2 = K i , n = 1 , … , 8000 , and f ( ⋅ ) represent the transfer function of the hidden layer, for example, f ( x ) = m a x ( 0 , x ) ; and l denotes the layer number of the DFF. The output of each layer is defined as Equation 8. P j l n = ∑ i = 1 q W i , j l n o i l n l = 1,2,3,4 . (8)

The output of DFF is expressed as Equation 9.. C o s t DFN = g P j l n + b n l , l = 4 , (9) where g ( x ) = x , and the values of i , j , q can be obtained in Table 1.

The performance index function is expressed by the quadratic error function as Equation 10. L n = 1 2 C o s t train n − C o s t DFN n 2 . (10)

The weight W i , j ( l ) ( n ) and bias b ( l ) ( n ) update functions are performed using the gradient descent method as Equation 11. W i , j l n + 1 = W i , j l n − η ∂ L n ∂ W i , j l n b l n + 1 = b l n − η ∂ L n ∂ b l n , (11) where η denotes the network learning rate, and η = 0.01 . GA is equipped to solve the global minimum value of the obtained nonlinear function. With the advantages of excellent global search capabilities, outstanding adaptability to nonlinear objective functions, and universality, the GA is utilized to search the optimal solution, which includes the process of selection, crossover, and mutation, and its process is shown in Figure 3.

In order to validate the theoretical observations above, the exact ROA and estimated ROA are investigated. For acquiring the exact ROA, time-domain simulation is implemented to determine whether the collected operating point is a stable equilibrium point. For acquiring the estimated ROA, the Lyapunov function and its level set have to be searched by adding the coordinate transformation of the equality constraints into Equation 15 and Equation 18, and then solving the following SOS programming problem which are divided into Equations 19, 20, 22, 23. The coordinate transformation can be described as z ( 2 i − 1 ) = s i n θ pll , z ( 2 i ) = 1 − c o s θ pll . S O S P 0 a : search V , s 1 ∈ Σ M z , V 0 = 0 , λ 1 , λ 2 ∈ P M z V s . t . V − λ 1 ⊤ G − q 1 ∈ Σ M z − s 1 β − p − V ̇ − λ 2 ⊤ G − q 2 ∈ Σ M z , (19) where q 1 ( z ) = ϵ 1 z 2 2 and q 2 ( z ) = ϵ 2 z 2 2 , and ϵ 1,2 indicate small positive real numbers. G denotes the vector established by equality constraints. After a feasible V ( z ) is found and rewritten as V 0 a ( z ) , it is necessary to search for a maximum level set satisfying V ̇ 0 a ( z ) < 0 to hold ⋁ z ∈ V 0 a γ ( z ) / 0 , which can be achieved by solving the following SOS programming problem. S O S P 0 b : max s 2 , s 3 ∈ Σ M z , λ 2 ∈ P M z γ s . t . − s 2 γ − V 0 a − s 3 V ̇ 0 a − λ 2 ⊤ G − q 1 ∈ Σ M z . (20)

As the − s 2 γ in SOSP0b is a bilinear term, it should be solved by a bisection search on γ to gain the optimal solution γ 0 b . Then, a new function V 0b ( z ) = V 0 a ( z ) / γ 0 b can also be regarded as an initial Lyapunov function for Equation 15 and Equation 18. In order to obtain the largest estimated ROA of the state equation, an iterative searching for V ( z ) is implemented by the following SOS programming program. S O S P 1 : search V 1 i + 1 , s 4 , s 5 , s 6 ∈ Σ M z , V 1 i + 1 0 = 0 , λ 3 , λ 4 , λ 5 ∈ P M z V 1 i + 1 s . t . − s 4 1 − V 1 i + 1 − V 1 i + 1 − λ 3 ⊤ G ∈ Σ M z V 1 i + 1 − λ 4 ⊤ G − q 1 ∈ Σ M z − s 5 1 − V 1 i + 1 − s 6 V ̇ 1 i + 1 − λ 5 ⊤ G − q 2 ∈ Σ M z , (21) where i = 0,1,2   … , and the initial Lyapunov function is set as V 1 0 ( z ) = V 0 b ( z ) . s 1 , s 2 , … , s 6 denote the SOS polynomials. λ 1 , λ 2 , … , λ 5 denote polynomials. Moreover, the bilinear terms in SOSP1 are s 5 V 1 i + 1 and s 6 V ̇ 1 ( i + 1 ) , and a coordinate-wise iterative method is employed here. Then, SOSP1 is transformed into a linear semi-definite programming problem at each iteration. Thus, SOSP1 can be realized by solving the two SOS programming problems given below. S O S P 1 a : search s 5 ∈ Σ M z , λ 5 ∈ P M z s 5 , s 6 s . t . − s 5 1 − V 1 i − s 6 V ̇ 1 i − λ 5 ⊤ G − q 2 ∈ Σ M z . (22)

If the satisfied s 5 and s 6 are found and rewritten as s 5 ( 1 a ) and s 6 ( 1 a ) , the second problem is written as S O S P 1 b : search V 1 i + 1 ∈ Σ M z , V 1 i + 1 0 = 0 , λ 3 , λ 4 , λ 5 ∈ P M z V i + 1 s . t . V 1 i + 1 − V 1 i − λ 3 ⊤ G ∈ Σ M z V 1 i + 1 − λ 4 ⊤ G − q 1 ∈ Σ M z − s 5 1 a i 1 − V 1 i + 1 − s 6 1 a V ̇ 1 i + 1 − λ 5 ⊤ G − q 2 ∈ Σ M z . (23)

Through SOSP1a to SOSP1b, an expanding interior algorithm is realized, and the final optimal Lyapunov function for the state is obtained, Equation 15 and Equation 18. Thus, the exact ROA and the estimated ROA of the system with and without the FLF scheme are shown in Figure 6.

In Figure 6, the exact ROA and estimated ROA of the system with the FLF scheme are larger than those of the system without the FLF scheme, which verifies the observations found in Section 3.

To validate the effectiveness of the FLF control and parameter optimization, simulation studies were carried out for the converter system shown in Figure 7 using MATLAB/Simulink. The control structure in Figure 7 is shown in Figure 1. The test system adopts grid-following with the FLF–DFF scheme. Two cascades of dc-linked voltage control and current control are included in the control strategy in Figure 7. The system parameters of the test system are chosen as shown in Table 2. The simulation step length was set to be 20 μ s. The parameter optimization results by the DFF-GA method are shown in Figure 8. A three-phase-to-ground fault was applied on the PCCB at t = 1s, and the simulation results under different cases are shown in Figure 9.

Parameter optimization is carried out based on the system with FLF control. As shown in Figure 8b, the MSE of the output of DFF drops to 1 0 − 4 , which indicates that the error between the predicted value by DFF and the real value is acceptable. The cost value of the DFF with respect to K p and K i is shown in Figure 8c; with the K p close to 100 and K i dropping to 1,500, the cost value is lower, indicating that the PLL system shows better transient response. Figure 8d shows the convergence curve of GA. When the iteration reaches to 130, the GA algorithm achieves convergence. The system with FLF after DFF-GA optimization shows the best performance. These results agree with the theoretical analysis and manifest the improved synchronous stability attributed to the FLF and DFF-GA optimization. As shown in Figure 8a, the original parameters K pori and K iori are 40 and 1,600, respectively. Through the DFF-GA optimization of the PLL parameters, the oscillation damping performance of the GFL converter has been enhanced greatly, which shows the smaller overshoot and faster transient response, where the optimal PI coefficients are obtained as K pop = 88.7933 and K iop = 1411.3992 .

Figure 9 presents the results of the system with conventional PLL, conventional PLL with FLF, and conventional PLL with FLF–DFF-GA control strategies. According to the simulation results, the maximum fault clearing time of the system without FLF was 0.249 s, whereas that of the system with FLF was 0.329 s. The converter system will become unstable if the fault is not cleared within the maximum clearing time. Figure 9 shows the results of the two systems, in which the length of fault was set as 0.25 s for the one without FLF and the length of fault was set as 0.329 s for the one with FLF. According to Figure 9, the capacitor voltage, filter inductor current, and the active power output of the converter system with FLF stabilize to the pre-fault steady-state value after the 0.329 s′ fault is cleared. In contrast, the system without FLF becomes unstable after the 0.25 s′ fault is cleared. Moreover, after parameter optimization by DFF-GA, the transient stability of the system with FLF was further improved. As shown in Figure 9c, the active power of the system with FLF after DFF-GA optimization shows smaller overshoot and faster transient response than the system without FLF.

To compare the proposed method with the PLL damping enhancement-based method (Huang et al., 2022), the hardware-in-loop (HIL) studies are carried out. The experiments are carried out with a 2 M W three-phase grid-connected converter, whose system parameters are also shown in Table 2. The topology of the experiment system is presented in Figure 10, in which the grid-connected system is fed by RTDS. The control strategies are configured in the dSpace real-time simulator for simulation. Communication is conducted between RTDS and dSpace via optical fibers, and the simulation step size is also 20 μ s. A three-phase-to-ground-fault was applied on PCCB at t = 2 s, and the single-phase voltage and active power output results are shown in Figure 12. The strategy method scheme (Wu et al., 2022) is shown in Figure 11.

According to the HIL results, when restrained by the control parameters of the PLL, the GFL system with PLL damping enhancement control becomes unstable when the fault clearing time (CCT) is set at 0.16 s, whereas the system with the same PLL control parameters and FLF control is larger than 0.229 s. Comparing the results in Figures 12a, b, the output active power of the system with the FLF scheme stabilizes to pre-fault steady-state values through a 0.15-s transient process after the fault is cleared. This is because of the damping coefficient introduced by the FLF strategy, which enhances the system equivalent damping. In contrast, the excess power generated by fault cannot be dissipated by the PLL damping enhancement method; this is because the equivalent damping is restricted, and then the system becomes unstable after 0.16 s′ fault is cleared. The experimental results of the system with different control schemes under three-phase-to-ground fault are presented in Table 3, which shows that the system with the hybrid scheme of FLF–DFF shows the best performance.