As shown in the figure is a single-machine, single-load system, E is the Davening-equivalent potential of the AC large system, Z _s is the system-equivalent impedance, and P _L + jQ _L is the load power. After the power system is subjected to small perturbations, no spontaneous oscillations or non-periodic out-of-steps occur, and it can automatically recover to the initial operating state, and the power system is statically stable. When this system reaches the static critical stability, the maximum load power is shown in Equation 1. P E m a x = E U L Z s (1)

When the load absorbed power exceeds this limit, then the system will be destabilised. Therefore, the static stable transmission power limit of the system should be enhanced as much as possible to improve the static stability margin and reduce the risk of static instability of the system as much as possible.

In this paper, the theory related to static stability of high percentage new energy systems is studied from the impedance perspective analysis, and the equivalent impedance of the load when the single-unit, single-load system reaches the static stability limit will be analysed in the following.

For a specific system, its parameters are certain, and when the load equivalent impedance is changed, its absorbed power will be changed. Let the load impedance be expressed as Z L ∠ α L , the system impedance be expressed as Z S ∠ α S , then the system output complex power can be expressed as S W = U L I L * = Z L cos ⁡ α L + j ⁡ sin ⁡ α L E 2 Z L 2 + 2 Z L Z s ⁡ cos α L − α s + Z s 2 (2) where, U _L and I _L denote the load voltage and load current. Equation 2 reflects the complex power absorbed by the load from the system, it is not only related to the load impedance magnitude, but also affected by the impedance phase.

The active power absorbed by the load can be expressed as in Equation 3. P L = E 2 Z L ⁡ cos ⁡ α L Z L 2 + 2 Z L Z s ⁡ cos α L − α s + Z s 2 (3)

Taking the partial derivatives for the magnitude and phase angle respectively, we have ∂ P L ∂ Z = cos ⁡ α L Δ 2 Z s 2 − Z L 2 ∂ P L ∂ α L = − Z s 2 + Z L 2 sin ⁡ α L + 2 Z L Z s ⁡ sin ⁡ α s Δ 2 Δ = Z s 2 + Z L 2 + 2 Z L Z s ⁡ cos α L − α s (4)

It can be shown in Equation 4 that the load power achieves its maximum value when and only when Z _L = Z _s and α _L = −α _s, i.e., when the load impedance is equal in magnitude and opposite in phase to the system. The corresponding limit power is P L m a x = E 2 4 Z s ⁡ cos ⁡ α s (5)

From Equation 5, reducing the system impedance and phase angle can improve the system limit output power.

SIMULINK is used to build a single machine and single load system as shown in Figure 1, the system potential E = 1, R _s + jX _s = 0.4e^j80°, the load complex impedance is Z _L e ^jα , and the three-dimensional diagram of the absorbed active power with impedance magnitude and phase is shown below, which shows that when the load impedance is equal to the system impedance and the phase is opposite, the absorbed active power reaches the maximum value.

The new energy single-machine single-feed system is shown in Figure 2, where U _w, P _w, Q _w are the new energy station voltage, active output and reactive output, and the meaning of other symbols is the same as that in Figure 1.

Comparing Figures 3, 3, although the power transmission direction of the stand-alone single-load system is opposite to that of the new energy’s stand-alone single-feeder system, they have similar structures, and therefore, they have similar properties.

In the stand-alone single-load system, with the increase of the load power, a voltage dive will occur and lead to voltage instability, while in the stand-alone single-feeder system, a similar phenomenon will also occur. For example, the single machine single-feed system shown in Figure 3 is built in Simulink. Set R S + j X S = 0 + 0.15 j and E = 1 in Figure 3. Increase the active output of the new energy station W continuously, and the voltage and power of the new energy station is as follows:

As can be seen in Figure 4, with the increase of the new energy output power, the voltage of the new energy field station gradually decreases to reach the maximum output power of the new energy, and then continue to increase the active power, the voltage of the new energy field station will plummet, and there is a voltage destabilisation, which is similar to that of the stand-alone, single-fed-in system.

Further, considering the new energy single-feeder system, whether there is a similar single-feeder system, that is, the new energy unit equivalent impedance and the AC system amplitude is equal to the system is in a critical stable state, and can be based on the magnitude of the new energy unit equivalent impedance to determine the voltage stability of the system.

After the above analysis, the new energy grid-connected system has a similar conclusion with the load, when its equivalent impedance is equal to the system impedance, the new energy output power reaches the limit. When the equivalent impedance is greater than the system impedance, the power is less than the limit and the system is stable. When the equivalent impedance is smaller than the system impedance, the output power is greater than the stability limit, and the system is unstable. In this paper, the system impedance is called the critical impedance, based on which the static voltage stability margin of the new energy station is defined. Z Δ = Z w − Z c r (6) where Z _∆ is the static voltage stability margin expressed as an impedance vector, Z _w is the equivalent impedance of the new energy field station, and Z _cr is the system impedance or critical impedance.

Based on this index, static voltage stability determination criteria can be described as Equation 7. Z Δ > 0 , stable Z Δ = 0 , critical stable Z Δ < 0 , unstable (7)

Figure 5 gives the curve of the impedance determination index defined in this paper versus the static voltage stability of the system.

Figure 6 shows a typical grid-forming converter topology and control method. Where: V _d is the DC side voltage of the converter, which can be identified as a constant value; e _a, e _b and e _c are the internal potentials of the converter; i _a, i _b and i _c are the output currents of the VSG; i _ga, i _gb and i _gc are the grid-connecting currents of the converter; v _a, v _b and v _c are the output terminals of the converter; and L _f, C _f and R _f are the filtering inductances, filtering capacitances and damping resistances of the converter, respectively; L _g and R _g are the equivalent line inductance and resistance of the grid, respectively; v _ga, v _gb and v _gc are the equivalent AC system voltages.

The active control strategy of the grid-forming converter draws on the inertia and primary frequency regulation characteristics of the synchronous generator; the reactive control strategy simulates the primary voltage regulation characteristics of the synchronous generator. For the voltage-controlled converter, the following expressions are available: T s + ω N − ω v D p − T e = J s ω v T e = P e ω v ≈ P e ω N T e = P s ω v ≈ P s ω N − ω v = s θ Q s + 2 D q V N − V − Q e = 2 K Q s E v (8) where: J is the virtual rotational inertia; ω _v and ω _N are the output angular frequency of the converter and the rated angular frequency of the grid, respectively; T _s and T _e are the given torque and the electromagnetic torque, respectively; D _p and D _q are the active damping coefficient and the reactive damping coefficient, respectively; P _s and Q _s are the active and reactive power given, respectively; P _e and Q _e are the instantaneous output active and reactive powers, respectively are the instantaneous output active power and reactive power; θ is the phase of the inverter’s internal potential; K _Q is the reactive inertia coefficient; V _N and V are the rated voltage RMS and output voltage RMS, respectively; and E _v is the rms value of the inverter’s internal potential.

The output active power P _e and reactive power Q _e of the converter can be calculated according to the instantaneous power theory, which is given by Equation 9. P e = 1.5 ν α i α + ν β i β Q e = 1.5 ν β i α − ν α i β (9) where: i _α , i _β are the output current of the converter in the αβ coordinate system; v _α , v _β are the output voltage of the converter in the αβ coordinate system.

The modulating waveform of VSG is jointly determined by the outputs of active and reactive controllers with the Equation 10. e ma = 2 E v ⁡ cos ⁡ θ e mb = 2 E v ⁡ cos θ − 2 π 3 e mc = 2 E v ⁡ cos θ + 2 π 3 (10)

The converter is controlled in a stationary coordinate system and therefore cannot be modelled in a synchronous rotating coordinate system for small signal linearisation. In this study, the harmonic linearisation method is used to derive the positive sequence output impedance model of the converter. The f _p frequency positive sequence voltage perturbation is injected at the point of common coupling (PCC), and according to the 2-fold mirror frequency coupling effect, the main circuit can be decomposed into f _p frequency and f _p -2f ₁ frequency equivalent circuits, and its small-signal model is s L f + R f i a f = e a f − s 2 L f C f + s R f C f + l v a f f = ± f p s L e + R e i a f = v a f − v ap f (11) s L f + R f i a f = e a f − s 2 L f C f + s R f C f + l v a f f = ± f p − 2 f 1 s L e + R e i a f = v a f (12) where the expressions for the components of the filter capacitor voltage v _a[f], output current i _a[f], and inductor current i _La[f] at different frequencies are shown in Equations 13–15. v a f = V 1 , f = ± f 1 V p , f = ± f p V p 2 , f = ± f p − 2 f 1 (13) i a f = I 1 , f = ± f 1 I p , f = ± f p I p 2 , f = ± f p − 2 f 1 (14) i L a f = I L 1 , f = ± f 1 I L p , f = ± f p I L p 2 , f = ± f p − 2 f 1 (15) where: V 1 = V 1 / 2 ; V p = V p / 2 e ± j ϕ v p ; V p 2 = V p 2 / 2 e ± j ϕ v p 2 ; I 1 = I 1 / 2 e ± j ϕ i 1 ; I p = I p / 2 e ± j ϕ i p ; I p 2 = I p 2 / 2 e ± j ϕ i p 2 ; I L 1 = I L 1 / 2 e ± j ϕ i L 1 ; I Lp = I Lp / 2 e ± j ϕ i Lp ; I Lp 2 = I Lp 2 / 2 e ± j ϕ i Lp 2 . X 1 , X p , and X p 2 are the amplitudes or initial phase angles of the fundamental frequency, the positive-order perturbation component, and the negative-order coupling component of the associated steady state value of X. The amplitudes or initial phase angles of the positive-order perturbation component and the negative-order coupling component of the associated steady-state value of X are the amplitudes or initial phase angles.

The positive sequence voltage perturbation passes through the power control loop causing a phase angle perturbation quantity Δφ such that φ = φ ₁ + Δφ, where φ ₁ = ω _1t is the fundamental voltage phase angle.

v _a[f], i _a[f] and i _La[f] are park transformed to obtain the voltage-current expression in the dq coordinate system as shown in Equations 16–21. v d f = V d , f = 0 Δ v d 0 + V q Δ φ , f = ± f p − f 1 (16) v q f = V q f = 0 Δ v q 0 − V d Δ φ f = ± f p − f 1 (17) i d f = I d   f = 0 Δ i d 0 + I q Δ φ   f = ± f p − f 1 (18) i q f = I q   f = 0 Δ i q 0 − I d Δ φ   f = ± f p − f 1 (19) i L d f = I L d , f = 0 Δ i L d 0 + I L q Δ φ , f = ± f p − f 1 (20) i L q f = I L q , f = 0 Δ Δ i L q 0 − I L d Δ φ , f = ± f p − f 1 (21) where: V d = V 1 , V d = 0 , Δ v d 0 = V p + V p 2 , Δ v q 0 = ∓ j V p ± j V p 2 , I d = I 1 ⁡ cos ⁡ ϕ i 1 , I q = I 1 ⁡ sin ⁡ ϕ i 1 , Δ i d 0 = I p + I p 2 , Δ i q 0 = ∓ j I p ± j I p 2 , I Ld = I L 1 ⁡ cos ⁡ ϕ i 1 , I Lq = I L 1 ⁡ sin ⁡ ϕ i 1 , Δ i Ld 0 = I Lp + I Lp 2 , Δ i Lq 0 = ∓ j I Lp ± j I Lp 2 .

After neglecting the high frequency components, the frequency domain expressions for active and reactive power of the converter are obtained by Equations 22, 23 using the convolution theorem in the frequency domain: P e f = 1.5 V d I d + I q V q   f = 0 1.5 ( I d Δ v d 0 + V d Δ i d 0 + I q Δ v q 0 + V q Δ i q 0 f = ± f p − f 1 (22) Q e f = 1.5 V q I d − I q V d   f = 0 1.5 I d Δ v q 0 + V q Δ i d 0 − I q Δ v d 0 − V d Δ i q 0 f = ± f p − f 1 (23)

According to the mathematical model of the power control loop in Equation 8, ignoring the small signal quantity of the quadratic term, we obtain the frequency domain expressions shown in Equations 24, 25 for φ and E _v φ f = φ 1 , f = 0 G p p s I d Δ v d 0 + V d Δ i d 0 + I q Δ v q 0 + V q Δ i q 0 f = ± f p − f 1 (24) E v f = E v 0 , f = 0 G p q s I d Δ v q 0 + V q Δ i d 0 − I q Δ v d 0 − V d Δ i q 0 + H p s Δ v d 0 f = ± f p − f 1 (25) where: G pq s = − 3 / 2 / K s ;   H p s = − D q / K s ;   G pp s = − 3 / 2 / J s + D p ω n s .

The voltage-current control dual-loop voltage loop reference is generated by the power control loop, where v dref f = E v f and v qref f = 0 , and is inverted by Park’s coordinate inverse transformation to obtain the a-phase bridge-arm voltage response e _a[f] as shown in Equation 26 e a f = V 1 + s L f + R f I 1 f = ± f 1 M p s ∓ j ω 1 V p + N p s ∓ j ω 1 I p + − G pi s ∓ j ω 1 + j ω n L f I Lp + L p s ∓ j ω 1 I p 2 f = ± f p M p 2 s ± j ω 1 V p + N p 2 s ± j ω 1 I p + − G pi s ± j ω 1 + j ω n L f I Lp 2 + L p 2 s ± j ω 1 I p 2 f = ± f p − 2 f 1 (26) where M p s ∓ j ω 1 , N p s ∓ j ω 1 , L p s ∓ j ω 1 , M p 2 s ± j ω 1 , N p 2 s ± j ω 1 , L p 2 s ± j ω 1 are the coefficients of voltage-current related components.

To eliminate the f p − 2 f 1 frequency coupling current, (Equation 26), is substituted into (Equation 12) to obtain I _p2 with respect to the filter capacitor voltage and output current expressions: I p 2 = M p 2 s ∓ j ω 1 V p + N p 2 s ∓ j ω 1 I p D p 2 s ∓ j ω 1   f = ± f p (27) where D_{p 2}(s) can be described as Equation 28. D p 2 s = s ∓ j ω 1 L e + R e − L p 2 s + { s ∓ j ω L f + R f — G p i s − j ω n L f ) × s ∓ j ω 1 L e + R e s ∓ j ω 1 C f + 1 (28)

Bringing Equation 27 into Equation 26 and combining Equations 11, 12, the positive sequence impedance of the converter is obtained by solving as shown in Equation 29. Z p s = N p s − j ω 1 + L p s − j ω 1 N p 2 s − j ω 1 D p 2 s − j ω 1 + j ω n L f − G p i s − j ω 1 − s L f + R f M p s − j ω 1 + L p s − j ω 1 M p 2 s − j ω 1 D p 2 s − j ω 1 − 1 + j ω n L f − G p i s − j ω 1 − s L f + R f s C f + s L e + R e (29)

By analyzing the impact of each control parameter in the impedance expression on the amplitude and phase of the converter impedance, it is evident that only the reactive voltage sag coefficient exerts significant influence. Therefore, this study selects the reactive voltage sag coefficient as the variable for regulating converter impedance. The influence of other relevant parameters on grid-forming converter impedance characteristics is minimal and can be disregarded. It should be noted that when compared with (Cespedes and Jian, 2014), under identical parameter conditions, the equivalent impedance of a grid-forming converter is lower than that of a grid-following converter, thereby reducing its static voltage stability margin to some extent after connection to the system. Consequently, urgent research into methods for improving static voltage stability in grid-forming converter grid-connection scenarios at this stage is warranted.

To verify the correctness of the conclusions of this paper based on the impedance characteristics of the converter, a simulation model of a single machine single-feed system as shown in Figure 3 is constructed. It is proved that the reactive voltage sag coefficient has a large influence on the magnitude and phase of the impedance. The Figure 7 shows the influence of the reactive voltage sag coefficient on the impedance magnitude and phase of the field station.

It can be seen from Figure 7 that the reactive voltage sag coefficient has little effect on the phase of the power station impedance and a larger effect on the amplitude. Therefore, the influence of the control parameter on the impedance phase is no longer considered in this paper. The following section further proposes the static voltage stability limit enhancement strategy based on the matching of impedance parameters by changing the control parameter to affect the impedance replication of the converter.

Firstly, the applicability of the strategy proposed in this paper is analysed in a simple single-fed system as shown in Figure 9, and the main parameters of the grid-forming converter are shown in Table 1. The AC system impedance is 2j, and the system voltage is 1. All the simulations in this paper are carried out under the reference value of 100MVA, and the voltage level is set to 110V.

In the Table 1, V _d denotes the inverter DC voltage, L _f, R _f and C _f denote the filter inductance, resistance and capacitance on the AC side, respectively. L _e and R _e denote the line inductance and resistance. f _s is the switching frequency of the inverter. k _ip and k _ii denote current inner loop proportional integral coefficients. k _vp and k _vi denote voltage outer loop proportional integral coefficient. The meaning of the remaining variables has been described in the previous content.

In the system based on the initial parameters listed in Table 2, the outgoing power of the new energy field station is continuously increased by increasing the impedance amplitude and the corresponding impedance amplitude is recorded until the system becomes unstable and the static stabilization limit power is recorded. The equivalent impedance and limit power of the new energy field station at the original converter control parameters and after changing the parameters are given in Table 3. According to the results, the equivalent steady-state impedance of the new energy field station can be improved to a certain extent by changing the reactive voltage sag factor, i.e., the static voltage stability margin is improved. Although the improvement of the steady state margin at steady state is small, the improvement for the ultimate power is still considerable.

Figure 10A gives the curve of the PCC voltage with the output power of the new energy field station during the whole process under the original control parameters, and the power decreases rapidly after reaching the maximum value, indicating that the phenomenon of static voltage destabilisation occurs. At this time, the corresponding voltage is the critical stabilisation voltage, and the output impedance of the new energy field station is 2.04 (equivalent to the high-voltage side of the system, the same below), which is equal to the size of the AC system impedance, which verifies the static voltage stabilisation method of critical stabilisation impedance proposed in this paper.

Now, the control parameters of the grid-forming converter are adjusted to verify whether the control strategy proposed in this paper can improve the new energy static voltage stability margin. The reactive voltage sag coefficients are changed to 15 and 22, at which time the voltage and impedance in the process of delivering power increase are recorded, which are given in Figures 10B, C, respectively. The power-voltage curves under three different parameter conditions are given in Figure 11. According to the results, the limiting output power of the new energy field station is increased after changing the control parameters, and the impedance amplitude of the field station is larger at the same power, and the static voltage stability margin characterised by impedance is higher. The converter impedances of the three curves at the critical point are 2.01 p.u. and 1.98 p.u., which are close to the critical impedance of 2 p.u., respectively.

Before validating the effectiveness of the methodology of this paper in multi-feed-in systems, the changes in the limit power and the total power limit of the section are briefly analysed for each field station after applying the proposed strategy.

If the static stability margin of the current field station 2 is low, its static stability limit needs to be improved. When the control parameters are changed so that X ₂ is increased, for the new energy field station 1, the equivalent system impedance is calculated by Equation 30. X eq 1 = X 2 X g X 2 + X g (30)

The static stability margin of the new energy field station 1 is obtained by Equation 31. Z Δ = X 1 − X eq (31)

The increase of the grid-connected impedance of station 2 will lead to the increase of the equivalent impedance X _eq, which will lead to the decrease of the static stability margin of station 1, thus increasing the risk of static voltage instability of the system.

Therefore, it is set in this paper that in a multi-site grid-connected system, the site with smaller static voltage stability margin should be adjusted with priority; Secondly, after adjusting the parameters, it should be verified whether the margin of each site meets the requirements.

It is noted from Equation 32 that although increasing the impedance of a new energy station individually reduces the static voltage stability margins of other grid-connected stations to a certain extent, the overall static stability margins are improved from the point of view of the whole aggregation station, which can withstand a higher power delivery limit. Z Δ 1 = X 1 − X g X 2 X g + X 2 Z Δ 2 = X 2 − X g X 1 X g + X 1 Z Δ g = X 1 X 2 X 1 + X 2 − X g (32)

The application of the method proposed in this paper in the multi-field station grid-connected system is now described. The new energy multi-feed-in system shown in Figure 12 is constructed, the converters are all network-controlled, and the initial control parameters of each station are the same (the same as the single-machine single-feed-in system). The power reference value of the new energy multi-feeder system is 100MVA, the voltage level is 350V, the impedance of the system is 1j, and the relevant parameters of each station under the initial working condition are shown in the following table.

The power-voltage curves of the two stations and the pooling bus are shown in Figures 13–15, respectively, when the control parameters are changed so that the equivalent impedance of Station 1 is increased. After changing the parameters, the static stability margin of the new energy station 1 increases due to the increase of its own impedance, while the margin of the station 2 decreases slightly, but the limit power sent from the pooling bus increases compared with the previous one. The power curves of the pooling bus before and after the parameter change are shown in Figure 16.

As can be seen from Figures 13–15, changing the control parameters of a station can appropriately increase the static stability limit margin of that station, but it will cause the static voltage stability margin of other stations to slightly decrease. Overall, the static phone stability margin of the whole grid-connected system is improved greatly, which also proves the feasibility of the method proposed in this paper. The power sequence before and after in Figure 16 shows that the static stability margin of the system is improved after the converter control parameters are changed.