As a special type of grid-forming control, the block diagram of the V/f-controlled MMC station is illustrated in Supplementary Figure S4, where the connected AC system is labeled as G _grid (s). Moreover, the voltage controller of the grid-forming MMC adopts the widely used default u d * − i d * and u q * − i q * pairing scheme (Scheme 1) shown in Supplementary Figure S4. In the most common passive load supply scenario, there are two typical cases: (1) purely resistive load supply and (2) purely inductive load supply. Passive loads with other power factors are situated between these two cases. The following analysis aims to qualitatively explain the small-disturbance stability of Scheme 1.

Scheme 1 is stable with a small disturbance. Suppose that this system is already operating in a steady state. When there is a small increase in u d * , the d-axis current reference i d * increases due to the voltage controller, leading to increase in i _d . For a resistive load, the increase in i _d means increase in the load current, causing an increase in the d-axis component u _d of the PCC voltage.

For an inductive load, the increase in i _d causes an increase in u _q because of the inductive load characteristics. Subsequently, under regulation by the voltage controller, the q-axis current reference i q * decreases, causing a decrease in i _q . Consequently, there is an increase in the d-axis component u _d of the PCC voltage by the inductive load characteristics.

The aforementioned two processes are plotted in Supplementary Figure S5A,B, where the blue and red arrows respectively indicate the changes in the reference voltage and response characteristics. However, when there is a synchronous machine in the islanded network, the physical meaning of Scheme 1 is not intuitive. Considering the example of Supplementary Figure S2, the output active and reactive powers of the MMC station can be calculated as in Eq. (4): P s = U 2 Z s cos ⁡ α − U E Z s cos δ − α Q s = U 2 Z s sin ⁡ α + U E Z s sin ⁡ α δ − α , (4) where α is the impedance angle of Z _s . It is noted that α is usually close to π/2, so it can be concluded that the active power is closely related to the voltage phase angle difference between the PCC and Thevenin equivalent voltages and that the reactive power is closely related to the PCC voltage magnitude. Considering that the steady-state phase angle of the PCC voltage is 0, the voltage phase angle difference and PCC voltage magnitude are dominated by the q- and d-axis components of the PCC voltage, respectively.

The output active and reactive powers of the MMC station can be also calculated using the d- and q-axis components of the PCC voltage and current as in Eq. (5): P s = u d i d + u q i q Q s = − u d i q + u q i d . (5)

Since u _d and u _q are approximately 1.0 p.u. and 0 p.u., respectively, the output active and reactive powers are dominated by the d- and q-axis components of the MMC output current. The above analysis indicates that the u d * − i q * pairing scheme would be more reasonable than the default u d * − i d * pairing scheme since the MMC station output reactive power is dominated by u _d and i _q . The effect of i _d in the reactive power control is not as significant as that of i _q . Similarly, the u q * − i d * pairing scheme would be more reasonable than the default u q * − i q * scheme. For simplicity, the u d * − i q * and u q * − i d * pairing scheme for the voltage controller is denoted as Scheme 2.

Based on Supplementary Figure S4, the block diagram of variable pairing for the voltage controller is outlined in Supplementary Figure S6, where the connected AC system, voltage controller, and MMC station plus current controller are labeled as G _grid (s), G _v (s), and G _MMC (s), respectively. It is noted that the aim of the voltage controller is to make u _d (u _q ) track its reference u d * ( u q * ) and that the variable pairing for the voltage controller G _v (s) essentially helps determine the pairing scheme between the input (u _d , u _q ) and output ( i d * , i q * ).

This work aims to determine the variable pairing scheme based on concept of the relative gain array (RGA) (Bristol, 1966). The RGA relies on calculating the coupling degree between the input and output of the system’s remaining part, except for the controller whose variables are to be paired. Here, the system’s remaining part includes the connected AC system and MMC station plus current controller, which is denoted as G(s) in Supplementary Figure S6. G(s) is a typical two-input two-output system, and the relative gain between the jth input and ith output (element in row i and column j of the RGA Λ) of G(s) is defined as Eq. (6): λ i j ≜ g i j g ^ i j = ∂ y o u t i ∂ u i n j ∆ u i n k = 0 k ≠ j ∂ y o u t i ∂ u i n j ∆ y o u t i = 0 i ≠ j , (6) where u _{in_j} and y _{out_i} are the jth input and ith output of system G(s); the numerator is the open-loop gain when all other loops are open, and the denominator is the open-loop gain when all other loops are closed with perfect control.

The variable paring can be selected on the basis of the RGA according to the following criteria: (1) when λ _ij is closer to 1, u _{in_j} and y _{out_i} are paired better; (2) when λ _ij is between 0.8 and 1.2, then u _{in_j} and y _{out_i} form a suitable pair, but λ _ij beyond this range does not necessarily mean an infeasible pair. However, the RGA does not provide any information on the stability of the system. The RGA Λ can be calculated as Λ = G 0 . ∗ G 0 − 1 T , (7) where G(0) is defined as G(s)|s = 0; the operation “.*” indicates the Hadamard product; G(0)⁻¹ is the inverse matrix of G(0); and the superscript “T” is the matrix transpose.

For the scheme shown in Supplementary Figure S4, the mathematical model of the MMC station plus current controller is described by Eqs. (11, 12): u d i f f d s = u d s − ω L i q s + i d * s − i d s k p 2 + k i 2 s u d i f f q s = u q s + ω L i d s + i q * s − i q s k p 2 + k i 2 s , (11) i d s = u d i f f d s − u d s + ω L i q s / s L i q s = u d i f f q s − u q s − ω L i d s / s L , (12) where k _p2 and k _i2 are the proportion and integral gains of the current controller, respectively.

After substituting Eq. (11) into Eq. (12), G _MMC(s) can be derived as in Eq. (13), and G _MMC(0) is a 2 × 2 identity matrix. Therefore, the steady-state sensitivity matrix between [ i d * , i q * ]^T and [u _d , u _q ]^T becomes the steady-state sensitivity matrix between [i _d , i _q ]^T and [u _d , u _q ]^T. G M M C s = s k p 2 + k i 2 s 2 L + s k p 2 + k i 2 0 0 s k p 2 + k i 2 s 2 L + s k p 2 + k i 2 . (13)

Three types of load models are considered in this work: constant impedance, constant current, and constant power loads. Suppose that the voltage and current phasors of the load node are U ˙ l and I ˙ l (flowing out of the load), respectively; the steady-state voltage–current characteristics of the constant impedance load can be generalized as in Eq. (14): I ˙ l = − Y l U ˙ l , (14) where Y _l (Y _l = G _l + jB _l ) is the admittance of the load. Thus, the steady-state voltage–current small-disturbance characteristics of the constant impedance load is as follows: Δ I l d Δ I l q = − G l − B l B l G l Δ U l d Δ U l q , (15) where ΔI _ld and ΔI _lq are the real and imaginary parts of vector Δ I ˙ l , respectively; ΔU _ld and ΔU _lq are the real and imaginary parts of vector Δ U ˙ l , respectively.

For the constant current load, the steady-state voltage–current small-disturbance characteristics are described by Eq. (16): Δ I l d Δ I l q = 0 0 . (16)

For the constant power load, the steady-state active and reactive powers can be calculated using Eq. (17), and the derived steady-state voltage–current small-disturbance characteristics are given by Eq. (18). − P l = U l d I l d + U l q I l q − Q l = − U l d I l q + U l q I l d , (17) Δ I l d Δ I l q = − U l d U l q U l q − U l d − 1 I l d I l q − I l q I l d Δ U l d Δ U l q . (18)

Given the existence of the V/f-controlled MMC station (slack bus), a synchronous machine can be treated as either the PQ or PV bus in the steady-state analysis. Therefore, both the steady-state active power and steady-state reactive power (or terminal voltage) should be taken into account. The synchronous machine’s steady-state electrical power P _g is approximately equal to its mechanical power P _m provided the equivalent damping of the synchronous machine is small enough. For synchronous generators, P _m has a linear relationship with the rotor angular frequency as given in Eq. (19), where P _m0, ω ₀, ω _g , and R _d are the reference mechanical power, rated rotor speed, steady-state rotor speed, and rotor speed-governor speed droop coefficient, respectively. For synchronous condensers, P _g is approximately equal to zero. P g = P m = P m 0 + ω 0 − ω g / R d . (19)

Considering the frequency support from the V/f-controlled MMC station, ω _g is always equal to ω ₀ in the steady state, and the synchronous machine’s steady-state electrical power remains constant. Therefore, the small disturbance of the synchronous machine’s steady-state active power is 0. Δ P g = Δ P m 0 = 0 . (20)

Based on the above analysis, the steady-state voltage–current small-disturbance characteristics in accordance with the active power are derived as in Eq. (21), where the subscripts d and q respectively mean the real and imaginary parts of the phasor in the V/f-controlled MMC’s dq reference frame (also the d- and q-axis components). U g d U g q Δ I g d Δ I g q = − I g d I g q Δ U g d Δ U g q . (21)

The next step involves modeling the steady-state voltage–current small-disturbance characteristics based on the reactive power (or terminal voltage). The first mode is the constant reactive power control mode, where the synchronous machine’s steady-state reactive power Q _g is constant. The steady-state voltage–current small-disturbance characteristics are as shown in Eq. (22): U g q − U g d Δ I g d Δ I g q = I g q − I g d Δ U g d Δ U g q . (22)

The second mode is the constant terminal voltage control mode, where the steady-state terminal voltage U _g is constant. Equation (23) gives the steady-state voltage–current small-disturbance characteristics in this mode. U g d U g q Δ U g d Δ U g q = 0 . (23)

The third mode is the constant reactive power control mode, where the steady-state reactive power Q _g has a linear relationship with the terminal voltage U _g , as shown in Eq. (24). The steady-state voltage–current small-disturbance characteristics are described by Eq. (25), where Q _g0 and K _q represent the output reactive power reference and terminal voltage output reactive power droop coefficient. Q g = K q U g 0 − U g + Q g 0 , (24) U g q − U g d Δ I g d Δ I g q = I g q − K q U g d U g − K q U g q U g − I g d Δ U g d Δ U g q . (25)

The power grid network acts as a bridge to connect the individual components and establish the relationship between i _dq and u _dq . The steady-state analysis is based on the nodal voltage equations of Eq. (26), where the power grid network is represented by the nodal admittance matrix Y (= G + j B). By expanding Eq. (26) into the real and imaginary parts, Eq. (27) can be derived. I + Δ I = Y U + Δ I I = YU , (26) Δ I d = G Δ U d − B Δ U q Δ I q = B Δ U d + G Δ U q . (27)

In Eqs. (26, 27), I(=[ I ˙ 1 , I ˙ 2 ,…, I ˙ n ]^T) and U(=[ U ˙ 1 , U ˙ 2 ,…, U ˙ n ]^T) are the injecting current phasor vector and node voltage vector of the power grid network, ΔI and ΔU are small disturbances in vectors I and U, ΔI _d and ΔI _q are the real and imaginary parts of the vector ΔI, and ΔU _d and ΔU _q are the real and imaginary parts of the vector ΔU, respectively.

With the exception of the constant current load nodes and V/f-controlled MMC station, suppose that there are k load nodes and r synchronous machine nodes in the islanded network. By rearranging Eq. (27) in the order of the V/f-controlled MMC station, load, and synchronous machine nodes, Eq. (28) can be derived. For compactness, the subscript “dq” indicates vectors containing both the d-axis and q-axis components in the V/f-controlled MMC station’s dq reference frame. Δ I 1 d q Δ I l 1 d q ⋮ Δ I l k d q Δ I g 1 d q ⋮ Δ I g r d q = Y 11 Y 12 Y 13 Y 21 Y 22 Y 23 Y 31 Y 32 Y 33 Δ U 1 d q Δ U l 1 d q ⋮ Δ U l k d q Δ U g 1 d q ⋮ Δ U g r d q . (28)

For brevity, the load current and voltage small-disturbance vectors are denoted by ΔI _Ldq and ΔU _Ldq , and the synchronous machine current and voltage small-disturbance vectors are denoted by ΔI _Gdq and ΔU _Gdq , respectively. Then, Eq. (28) can be expanded into Eqs. (29–31). Δ I 1 d q = Y 11 Δ U 1 d q + Y 12 Δ U L d q + Y 13 Δ U G d q , (29) Δ I L d q = Y 21 Δ U 1 d q + Y 22 Δ U L d q + Y 23 Δ U G d q , (30) Δ I G d q = Y 31 Δ U 1 d q + Y 32 Δ U L d q + Y 33 Δ U G d q . (31)

As noted in Section 4.2 and Section 4.3, the relationships between ΔI _Ldq and ΔU _Ldq as well as ΔI _Gdq and ΔU _Gdq can be generalized in compact form as Eqs. (32, 33), respectively. Δ I L d q = E Δ U L d q , (32) F Δ I G d q = H Δ U G d q + J Δ I 1 d q + K Δ U 1 d q . (33)

By substituting Eqs (32, 33) into Eqs (30, 31), ΔI _Gdq and ΔI _Ldq can be eliminated, and the vectors ΔU _Gdq and ΔU _Ldq can be solved as linear combinations of ΔU _1dq and ΔI _ldq , as shown in Eq. (34): Δ U L d q = N Δ I 1 d q + W Δ U 1 d q Δ U G d q = M − 1 J Δ I 1 d q + M − 1 L Δ U 1 d q , (34) where L = K − F Y 31 + F Y 32 Y 22 − E − 1 Y 21 M = F Y 33 − H − F Y 32 Y 22 − E − 1 Y 23 N = E − Y 22 − 1 Y 23 M − 1 J W = E − Y 22 − 1 Y 21 + Y 23 M − 1 L . (35)

Substituting Eq. (34) into Eq. (29), the steady-state characteristics between ΔI _ldq and ΔU _1dq can be derived as Eq. (36), where I is the identity matrix. Thus, the matrix G(0) (also the steady-state sensitivity matrix) can be calculated using Eq. (37). I − Y 12 N − Y 13 M − 1 J Δ I 1 d q = Y 11 + Y 12 W + Y 13 M − 1 L Δ U 1 d q , (36) G 0 = Y 11 + Y 12 W + Y 13 M − 1 L − 1 I − Y 12 N − Y 13 M − 1 J . (37)

The test case is shown in Supplementary Figure S8, where the power load in the islanded network is supplied by an MMC station and an AC tie line. When the AC tie line L₁ is tripped, there is a need for control-mode switching (PQ to V/f mode) in the MMC station. G_C in Supplementary Figure S8 represents a synchronous condenser. The parameters of this system are listed in Table 1. In this case, the constant impedance load model is adopted, and the synchronous condenser regulates the terminal voltage (20 kV side) at 1.0 p.u. The PCC voltage and frequency that are regulated by the MMC station in the V/f mode are set as 1.0 p.u. and 50 Hz, respectively. Before disconnection of AC tie line L₁, both the active and reactive powers of the MMC station are both set to 0.

First, the RGA Λ is calculated for different operating conditions with the load power restricted within 100 MVA, and the calculated elements of Λ are plotted in Supplementary Figure S9. It is observed that the default u d * − i d * and u q * − i q * pairing of Scheme 1 has certain areas where the elements of Λ can be equal to 1, but these areas are very small. Thus, this pairing is only effective for very limited operating conditions. In contrast, the u d * − i q * and u q * − i d * pairing of Scheme 2 can achieve values of the elements of Λ equal to 1 in most areas, meaning that it is effective for most operating conditions and is therefore more reasonable.

The voltage controller for Scheme 2 is depicted in Supplementary Figure S10. To demonstrate the advantages of Scheme 2, simulation comparisons are performed between the two pairing schemes under two typical operating scenarios wherein the load is set to 80 MW + j50 MVar. Accordingly, an electromagnetic transient model of the system in Supplementary Figure S8 was built in the time-domain simulation software PSCAD/EMTDC.

Before 1.0 s, the island is connected to the external system, and the MMC station operates in the PQ mode in which both the active and reactive powers are 0. At 1.0 s, the island is disconnected from the external system by tripping the AC tie line L₁, and the MMC station switches to the V/f control mode at 1.05 s. The system response characteristics for the two input–output pairing schemes for the outer loop are plotted in Supplementary Figures S11, S12.

As seen from Supplementary Figures S11, S12, Scheme 2 is stable during control-mode switching while Scheme 1 loses stability. After the AC tie line is tripped, the rotor speed of G_C decreases because the active power absorbed by the load is supplied by the rotor kinetic energy; at the same time, the PCC voltage decreases. When the MMC station switches to the V/f control mode, the system shows different dynamics for the two pairing schemes.

From the perspective of active power and frequency, since the steady-state frequency is set to the rated frequency, the active power of the MMC station should increase to accelerate the rotor and supply the load. From the perspective of reactive power and voltage, the MMC station should increase the output reactive power immediately after the 1.05 s mark since the steady-state PCC voltage is larger than the voltage during the 1.0–1.05 s period.

For Scheme 2, it is seen in Supplementary Figure S11 that the active and reactive powers of the MMC station generally increase during the 1.05–1.5 s period, which coincides with the active and reactive power demand mentioned above. For Scheme 1 shown in Supplementary Figure S12, as the d-axis and q-axis voltages are smaller than their reference values shortly after 1.05 s, both the q-axis and d-axis currents increase as shown in Supplementary Figure S12B. The increase in the d-axis current means increase in active power, which meets the MMC active power increase demand. However, the increase in q-axis current decreases the MMC station reactive power and conflicts with its increased demand. Thus, the reactive power is insufficient in this system, and the voltage collapse occurs as shown in Supplementary Figure S12C. The system is therefore unstable with Scheme 1. The advantage of the u d * − i q * and u q * − i d * pairing of Scheme 2 in the control-mode switching scenario is thus proved.

In this scenario, the AC tie line is assumed to be switched off, and the MMC station operates in the V/f control mode. Before 1.0 s, the islanded system already enters steady state. At 1.0 s, an equivalent 35 MVar of reactive power load is tripped, and the system response characteristics for the two input–output pairing schemes for the outer loop are shown in Supplementary Figures S13, S14.

According to Supplementary Figures S13, S14, since the equivalent reactive power load is tripped, the reactive power output from the MMC station should also decrease to achieve balance. For Scheme 2, the system quickly reaches a steady state without any large oscillations in the process. However, the system is unstable with Scheme 1. Therefore, the simulation results prove the advantage of the u d * − i q * and u q * − i d * pairing of Scheme 2 for the load step-change scenario.

In this case, the AC tie line is assumed to be switched off, and the MMC station operates in the V/f control mode. Before 1.0 s, the islanded system already enters steady state. At 1.0 s, a solid three-phase-to-ground fault occurs at the PCC bus and is cleared 100 ms later. The system response characteristics of the two input–output pairing schemes for the outer loop are shown in Supplementary Figures S15, S16. During PCC AC fault, the PCC voltage decreases, and the PCC voltage and rotor speed decrease at the same time. When the fault is cleared, the active power of the MMC station increases to accelerate the rotor speed, and the reactive power of the MMC station should also increase to supply the load.

For Scheme 2, it is seen in Supplementary Figure S15 that the active and reactive powers of the MMC station increase when the fault is cleared, coinciding with the active and reactive power demand mentioned above. However, the system is unstable with Scheme 1. In Supplementary Figure S16, the reactive power continues decreasing to negative when the fault is cleared (during 1.1–1.4 s) regardless of the fact that the q-axis voltage keeps decreasing. The positive feedback between the reactive power and q-axis voltage deteriorates the voltage stability, and the voltage collapse at approximately 1.4 s is seen in Supplementary Figure S16. Therefore, the simulation results prove the advantage of the u d * − i q * and u q * − i d * pairing of Scheme 2 in the PCC AC fault scenario.