The output power of the grid-following converter is constant (Du et al., 2021), and it is seen as a PQ node. Its typical configuration is shown in Figure 1A. When small-signal perturbations are added to the grid voltage, the controller d-q frame is no longer aligned with the system d-q frame. There exists a difference between two d-q frames. The relationship between vectors in two frames is [ Δ v d c Δ v q c ] = [ Δ v d s + Δ θ V q s Δ v q s − Δ θ V d s ] , (1) where c represents symbols in the controller d-q frame, s represents symbols in the system d-q frame, v _d and v _q represent voltage perturbation in d frame and q frame, and V _d and V _q represent voltage stable values in the d frame and q frame, respectively. For clarity, uppercase letters (for example, V _d ) in this article are used to denote voltage and current stable values, and lowercase letters (for example, v _d ) represent the small-signal perturbation.

Δθ is related to PLL, which is Δ θ = G P L L Δ v q s , (2) where G _PLL is the PLL closed-loop transfer function, which is as follows (Wen et al., 2016): G P L L = s k p P L L + k i P L L s 2 + s V d s k p P L L + V d s k i P L L . (3)

By adding (2) to (1) [ Δ v d c Δ v q c ] = [ 1 V q s G P L L 0 1 − V d s G P L L ] [ Δ v d s Δ v q s ] . (4)

Then, G PLL v is defined as follows: G PLL v = [ 1 V q s G P L L 0 1 − V d s G P L L ] . (5)

For current and converter output voltage, the similar equations can be derived as follows: [ Δ i d c Δ i q c ] = [ 0 G P L L I q s 0 − G P L L I d s ] [ Δ v d s Δ v q s ] + [ Δ i d s Δ i q s ] , (6) [ Δ v r d s Δ v r q s ] = [ Δ v r d c Δ v r q c ] − [ 0 G P L L V r q s 0 − G P L L V r d s ] [ Δ v d s Δ v q s ] . (7)

Then, G PLL i and G PLL d are defined as follows: G PLL i = [ 0 G P L L I q s 0 − G P L L I d s ] , (8) G PLL d = [ 0 G P L L V r q s 0 − G P L L V r d s ] . (9)

According to (4), (6), and (7), the small-signal model of the PQ node is developed in Figure 1B. Solving the equations represented by Figure 1B, the output impedance of the PQ node is as follows: Z = ( Y L + Y L G PLL d + Y L G del G PI G PLL i ) − 1 ( E + Y L G del G PI ) , (10) where Y_L represents the filter transfer function matrix, G_del represents time delay due to digital control and PWM, G_PI represents the PI controller in the current loop, and E represents the second-order identity matrix. The detailed expressions of the abovementioned transfer matrices are shown in the Supplementary Appendix.

The output active power and voltage amplitude are constant in the grid-forming converter, and it is seen as a PV node. The configuration of the PV node is shown in Figure 2A. Different from the PQ node, the transformation phase angle is provided by the power loop, and Δθ is related to active power. Δ θ = − ω f s ω g ( D p + J s ) ( s + ω f ) Δ P . (11)

However, Δ P = 3 2 ( V c d c Δ i d c + V c q c Δ i q c ) + 3 2 ( I d c Δ v c d c + I q c Δ v c q c ) . (12)

Combining (11) and (12), similar small-signal equations can be determined for voltage and current in the PV node, which yields [ Δ v c d c Δ v c q c ] = G vv [ Δ v c d s Δ v c q s ] + G iv [ Δ i d s Δ i q s ] , (13) [ Δ i d c Δ i q c ] = G vi [ Δ v c d s Δ v c q s ] + G ii [ Δ i d s Δ i q s ] , (14) [ Δ i L d c Δ i L q c ] = [ Δ i L d s Δ i L q s ] + G vl [ Δ v c d s Δ v c q s ] + G ii [ Δ i d s Δ i q s ] , (15) [ Δ v r d c Δ v r q c ] = [ Δ v r d s Δ v r q s ] + G vr [ Δ v c d s Δ v c q s ] + G ir [ Δ i d s Δ i q s ] . (16)

G_i (i = vv,iv … … ,ir) models the small-signal perturbation path from system vectors to controller vectors. Figure 2B shows the small-signal model of the PV node. G^v _Q and Gⁱ _Q model the small-signal perturbation path from the voltage and current to reactive power. G_LPF represents the low-pass filter transfer function matrix. G_kq represents the reactive power drop coefficient transfer function matrix. G^v _PI and Gⁱ _PI represent the PI controller in the voltage loop and current loop, respectively. Y_C represents the capacitor transfer function. The detailed expressions of the abovementioned transfer functions are shown in the Supplementary Appendix.

The output impedance can be derived from Figure 2B, which is Z = { [ Y L G d e l G P I i G P I v ( G k q G L P F G Q v G v v + G k q G L P F G Q i G v i − G v v ) − ( Y L G d e l G P I i G v l + Y L G v r + Y L ) ] − ( E + Y L G d e l G P I i ) Y c } − 1 × { ( E + Y L G d e l G P I i ) − [ ( Y L G d e l G P I i G P I v ) ( Y L G d e l G P I i G P I v ( G k q G L P F G Q v G i v + G k q G L P F G Q i G i i - G i v ) ) − ( Y L G d e l G P I i G i l + Y L G i r ) ] } . (17)

Due to the existence of large capacitors in PET, the middle-voltage side and low-voltage side can achieve decoupling. Thus, the middle-voltage side can be equivalent to constant DC voltage. The simplified PET configuration is shown in Figure 3A. Due to the constant output voltage amplitude and frequency, PET is seen as a slack bus. Different from PQ and PV nodes, the transformation phase angle in a slack bus is constant, and two d-q frames are aligned together, which means Δθ is zero. According to Figure 3A, the small-signal model of a slack bus is developed in Figure 3B. The output impedance of the slack bus is derived from Figure 3B, which is Z = ( Y L G del G PI i G PI v + Y L + Y C + Y L G del G PI i Y C ) -1 ( E + Y L G del G PI i ) . (18)

The system parameters are given in Table 1. When PET connects to the PQ node, the root loci with the change of the line length are shown in Figure 5. As shown in Figure 5A, when the R/X ratio is smaller than 1, the root locus enters the right-half plane, and the system becomes unstable by an increase in the line length. As shown in Figure 5B, when the R/X ratio is larger than 1, the phenomenon is the same as that when the R/X ratio is smaller than 1. This is because the grid-following converter is suitable for the strong grid. By an increase in the line length, the grid is weaker than before, and the grid-following converter becomes unstable.

In addition to the influence of the line length, the R/X ratio also affects system stability. In order to investigate the influence of the R/X ratio, the stability margin is introduced and defined as the distance between the closed-loop dominant pole and imaginary axis. The stability margin is high if the closed-loop dominant pole is far away from the imaginary axis. The relationship between the R/X ratio and stability margin is shown in Figure 6. Comparing Figure 6A and Figure 6B, the R/X ratio larger than 1 is more beneficial for system stability in most cases which means this system is suitable for the resistive line.

The low-voltage side converter in PET will cause coupling problems with the connected converters, and its parameters also affect the system stability. Figure 7 shows the influence of the voltage loop proportional gain k _pv in PET. By increase of k _pv , the system becomes unstable. This indicates that the lower value of k _pv is beneficial for the system stability.

When PET connects to the PV node, the root loci with the change of the line length are shown in Figure 8. As shown in Figure 8A, when the R/X ratio is smaller than 1, the root locus enters the left-half plane, and the system becomes stable by an increase in the line length. As shown in Figure 8B, when the R/X ratio is larger than 1, the phenomenon is the same as that when the R/X ratio is smaller than 1. Under different R/X ratios, by an increase in the line length, the root locus enters the left-half plane, and the system becomes stable. This is because the grid-forming converter is suitable for the weak grid. By an increase in the line length, the connected grid is weaker, and the grid-forming converter is more stable.

The relationship between the R/X ratio and stability margin is shown in Figure 9. The R/X ratio smaller than 0 shows that the system is unstable under this circumstance. Comparing Figure 9A and Figure 9B, the R/X ratio smaller than 1 is more beneficial for system stability in most cases, which means that this system is suitable for the inductive line.

Figure 10 shows the influence of k _pv . By increase of k _pv , the system becomes unstable. It indicates that the larger k _pv is not beneficial for system stability.

In this section, a parameter sensitivity analysis is performed, which investigates the effects of parameter variations on system stability and the coupling effect among parameters. Among all the control parameters in the system, k _pv and line length are investigated. The parameter sensitivity is defined as dσ/dx, where σ is the real part of the closed-loop dominant pole, and x is k _pv and line length. The parameter sensitivity smaller than 0 represents this parameter is not conductive to the system stability. The parameter sensitivity larger than 0 represents this parameter improves the system stability.

When PET and PQ nodes are connected, the parameter sensitivity of k _pv and line length are shown in Figure 11A and Figure 11B, respectively. As shown in Figure 11A, a small range of k _pv will result in a large change in the system stability, and its influence on stability varies with the line length. This indicates that k _pv plays a major impact on system stability, and the line length will influence the effect of k _pv on system stability. Comparing Figure 11A and Figure 11B, k _pv sensitivity is larger than that of the line length, which means it is more important to set a proper value of k _pv in designing the system.

When PET and PV nodes are connected, the parameter sensitivity of k _pv and line length are shown in Figure 11C and Figure 11D, respectively. As shown in Figure 11D, the parameter sensitivity is larger than 0. This corresponds with the previous analysis. The increase in the line length improves system stability. Comparing Figure 11C and Figure 11D, k _pv sensitivity is larger than that of the line length.

Overall, the aforementioned conclusions are shown in Table 1.