The schematic diagram of the equivalent model of the two-area AC system is shown in Figure 1.

In the simplified model, E 1 ∠ δ 1 , E 2 ∠ δ 2 denote the internal potential and rotor angle, x 1 and x 2 are the equivalent line reactance, x 12 is the tie line reactance between area 1 and 2, P G 1 and P G 2 are the mechanical power of the equivalent generators, U 1 ∠ θ 1 and U 2 ∠ θ 2 denote the respective bus voltage and phase angle, P L 1 and P L 2 are the load powers. P A 12 is the AC transmission power between area 1 and area 2.

For the convenience of analysis, the following assumptions are made for the system:

1) Ignoring the mechanical power change of equivalent generators G1 and G2, it is assumed that the values of PG1 and PG2 are constant

Under this assumption, and the rotor motion of the equivalent generator in area 1 and area 2 of the hybrid power system is: 2 H 1 ω 0 ⋅ d 2 δ 1 d t 2 = P G 1 − P A 12 − P L 1 (1) 2 H 2 ω 0 ⋅ d 2 δ 2 d t 2 = P G 2 − P A 12 − P L 2 (2) Where, H 1 and H 2 are the inertia of the equivalent generators in area 1 and area 2, respectively; ω 0 is the synchronous angular frequency.

The Eq. 1, 2 is further simplified, and the results are as follows: d 2 δ d t 2 = P m − P e (3) In the formula, the equivalent rotor angle is δ = δ 1 − δ 2 ; the equivalent mechanical power of the two-area AC system is P m = ( P G 1 M 1 − P G 2 M 2 ) − ( P L 1 M 1 − P L 2 M 2 ) , the equivalent electromagnetic power is P e = ( 1 M 1 + 1 M 2 ) U 1 U 2 x 12 sin ⁡ δ M 1 = 2 H 1 ω 0 , M 2 = 2 H 2 ω 0 .

According to Eq. 3, the power characteristic curve of AC system is shown in Figure 4A.

The two-area network model for AC/DC parallel transmission model was shown in Figure 2, and its schematic diagram of the equivalent model is shown in Figure 3.

In the simplified model, the sending and the receiving power grids are interconnected synchronously through the AC/DC transmission channels. P D 12 is the DC transmission power between area 1 and area 2. G 1 is the equivalent machine of 13 generator sets at the sending end, and G 2 is the equivalent machine of 10 generator sets at the receiving end. The meanings of other parameters are the same as those in Figure 1.

Under this same assumption in previous section, the rotor motion equation of the equivalent generator in area 1 and area 2 of the hybrid power system is: { 2 H 1 ω 0 ⋅ d 2 δ 1 d t 2 = P G 1 − P A 12 − P DC − P L 1 2 H 2 ω 0 ⋅ d 2 δ 2 d t 2 = P G 2 − P L 2 + P A 12 + P DC (4) the two-machine system shown in Eq. 4 is further simplified, and the results are as follows: d 2 δ ′ d t 2 = P m ′ − P e ′ (5)

In the formula, the equivalent rotor angle is; the equivalent mechanical power of the two-area AC/DC hybrid system is P ′ m = ( P G 1 M 1 − P G 2 M 2 ) − ( P L 1 M 1 − P L 2 M 2 ) , P ′ m = P D C α , α is the proportion of the DC transmitted power; the equivalent electromagnetic power is P ′ e = M P D C + M U 1 U 2 x 12 sin ⁡ δ ′ , M = 1 M 1 + 1 M 2 , M 1 = 2 H 1 ω 0 , M 2 = 2 H 2 ω 0 .

From the perspective of energy, the problem of the power system transient stability is whether the impact energy of the system can be essentially absorbed by itself (Bhui and Senroy, 2017). For the traditional two area AC system, when the three-phase grounding short-circuit fault occurs in the receiving AC system, the impact energy accumulated in the transient process of the system after the fault can be approximately expressed by the acceleration area S δ 0 a b δ c on the power angle characteristic curve, as shown in Figure 5A, the braking energy of the system can be approximately expressed by the deceleration area S b c d . The system maintains stable operation when S b c d > S δ 0 a c δ c , after the fault is removed. When S b c d < S δ 0 a b δ c after the fault is removed, and the system loses rotor angle stability. When, S b c d = S δ 0 a b δ c after the fault is removed, the system can maintain a critical stable state.

It can be concluded that the criterion of power system transient power angle stability is the difference η between the maximum deceleration area and the acceleration area of the system after a fault occurs. When η ≥ 0 , the system can maintain transient stability after the fault is removed, otherwise the system is unstable.

Similarly, when the acceleration and deceleration areas of AC/DC hybrid system under different working conditions are determined, the transient rotor angle stability criterion of hybrid system under different working conditions can be obtained.

The power characteristic curve of constant AC operating point operation state is shown in Figure 5B. At δ 0 , CF occurs, and the system’s stable operation point moves from A to the stable critical point D along the corresponding power curve, and δ c is the limiting excision angle. The system stability margin index η 1 = S B C D − S δ 0 A B δ c , which is shown in Eq. 6. η 1 = α ⋅ M ⋅ P m ′ ⋅ δ 0 − α ⋅ M ⋅ P m ′ ⋅ δ c + P m ′ ⋅ δ 0 + M U 1 U 2 x 12 cos δ c − M U 1 U 2 x 12 cos δ h − P m ′ ⋅ δ h (6) The power characteristic curve of constant total transmission power operation state is shown in Figure 5C. The system stability margin index η 2 = S F G H − S δ ′ o ′ E F δ ″ c ′ , which is shown in Eq. 7. η 2 = − αM P m ′ + αM P ′ δ h ′ + P m ′ δ 0 + M U 1 U 2 x 12 cos δ c ′ − M U 1 U 2 x 12 cos ⁡ δ h ′ + P m ′ δ h ′ (7)

According to Eqs 6, 7, the main factors affecting the stability of AC/DC hybrid system are DC power transmission ratio α and fault duration t . The longer the fault duration, the worse the transient stability of the hybrid system after the fault is removed, and the relationship between DC transmission power and transient stability needs to be further explored.

For Eqs 6, 7, the sensitivity of DC transmission power ratio α is analyzed: d η 1 dα = M P m ′ δ 0 − M P m ′ δ c < 0 (8) d η 2 dα = − M P m ′ δ c ′ + M P m ′ δ h ′ > 0 (9)

From the Eq. (8), when the AC operating point does not change, the system stability margin index η 1 is a decreasing function of α , i.e., when the DC transmission power increases, the system stability margin is decreased, and the stability of the hybrid system is decreased.

From the Eq. (9), when the total transmitted power remains unchanged, the system stability margin index η 2 is an increasing function of α , i.e., when the DC transmission power increases, the stability margin of the system increases, and the stability of the hybrid system also increases.

In the basic control strategy of the AC/DC hybrid system, the PI regulator on the rectifier side corresponds to the constant current control mode, and when its output reaches the lower limit (5°), it switches to the minimum firing angle control mode. Two different PI regulators on the inverter side correspond to two control modes of constant current and constant extinction angle respectively, and take the minimum output of the two regulators as the firing angle command on the inverter side (Faruque et al., 2005). The basic control strategy diagram of the inverter side is shown in Figure 6.

In order to improve the commutation failure immunity performance of the inverter side, the commutation failure prevention control (CFPREV) is added into the control strategy of the hybrid system, and is shown in Figure 7.

The upper part is the asymmetric fault detection, when the zero-sequence component of AC voltage exceeds the threshold V diff during the fault, a certain trigger offset angle will be output. The lower part is the three-phase fault detection, when the AC voltage drop exceeds the threshold V abz , a certain trigger offset angle will also be output. The two trigger offset angles will take the maximum value as the output of CFPREV.

In order to avoid more serious damage to the system, the blocking strategy is usually adopted after three commutation failures in the DC system.

In Eqs 6, 7, set η 1 = 0 and η 2 = 0 , respectively, the limit cut-off angle and the maximum fault duration t ′ of the AC/DC hybrid system corresponding to different DC ratios can be determined. When the duration of fault exceeds the maximum allowed duration, DC blocking strategy is adopted.

The power characteristic curves of AC/DC hybrid system after two working conditions when the DC blocking strategy is adopted, are shown in Figure 8.

During the HVDC single-pole blocking, the transmission power of the DC line is reduced to half of the original transmission power. When the AC operating point of the system remains unchanged, equivalent electromagnetic power and the curve of the equivalent mechanical power move down. When the total transmission power of the system does not change, only the equivalent electromagnetic power curve moves down.

To maintain the stability of the system, when the HVDC single-pole blocking strategy is only used the acceleration energy increases, the deceleration energy accumulated during two working conditions of the transient process decreases, the stability margin index η and the system stability reduces. Therefore, when the system cannot recover its stability by its inertia, then other emergency control measures should be adopted after DC blocking.

After the HVDC blocking, the sending side system has a large surplus of active power, and the difference in rotor angle between two-area equivalent units is further widened, it is necessary to combine the conventional stability control measures of the power grid to maintain the stability of the system rotor angle. Machine-cutting and load-cutting measures can reduce the transient accumulated energy during the process of system failure. In two area equivalent units, the increase in the difference between the generator’s speed and the rotor angle difference, is restrained, which is helpful to keep the rotor angle stable.

From the above analysis, when the system is unstable, only using DC blocking as an emergency control measure will worsen the rotor angle stability of the system. To maintain transient stability of AC/DC hybrid system, it is necessary to adopt coordinated control measures for DC blocking and for the generator and load cutting. The flow chart for coordinated control strategy of AC/DC hybrid system is shown in Figure 9.

After the coordinated control strategy is adopted for the AC/DC hybrid system, the equivalent power characteristic curve for two operating condition is shown in Figure 10.

When using coordinated control, to minimize the impact on the stable operation of the AC/DC hybrid system, to maximize the reliability of the power supply, re-draw the stability margin index η ′ 1 and η ′ 2 for the two operating conditions, and to determine the minimum control cost.

According to Figure 10, after the coordinated control strategy is adopted, the stability margin index of two operation condition is η ′ 1 = S e q i u + S k i h − S δ ′ 0 A Q δ ′ c _, η ′ 2 = S f m s v + S o s z − S δ ′ 0 ′ E f δ ′ c ′ , which are shown in Eqs 10, 11 _. η 1 ′ = β P m ′ δ m − β P m ′ δ h ′ − P m ′ δ 0 − M P m ′ ⋅ α ⋅ δ m + 3 2 αM P m ′ δ c ′ − M U 1 U 2 x 12 cos ⁡ δ c ′ − 1 2 αM P m ′ δ h ′ + P m ′ δ h ′ + M U 1 U 2 x 12 cos ⁡ δ h ′ (10) η 2 ′ = − P m ′ δ 0 ′ + αM P m ′ δ c ′ − M U 1 U 2 x 12 cos ⁡ δ c ′ + β ′ P m ′ δ m ″ − αM P m ′ δ h ′ − β ′ P m ′ δ h ′ + M U 1 U 2 x 12 cos ⁡ δ h ′ (11) Let η ′ 1 = 0 , η ′ 2 = 0 for determining the relationship between the minimum amount of machine cut-off and load cut-off β corresponding to the different DC ratios under the two working conditions with coordinated control.