Due to the unique randomness and intermittent nature of wind power generation, wind turbines must be connected to the grid through converters. Usually, the grid-connected converter (GFL) needs additional frequency devices for virtual inertia support, which does not behave directly in the frequency characteristics, so it can be equivalent to a negative constant load (Sajadi, Kenyon and Hodge, 2022) instability problems of large power grids.

Direct-drive wind turbines are usually controlled by constant power. When the grid is disturbed, the wind power will enter the low-voltage ride-through mode to provide reactive power to the grid (Alexandrova, Semken, and Pyrhönen, 2014). When the grid runs normally, the active power output of direct-drive wind generators remains unchanged and is positive, providing active power to the grid, while the reactive power remains zero, so the direct-drive wind generators under no fault conditions can be equivalent to a negative resistance. Suppose that the voltage of the wind power connecting point is U and the fault current is I during the fault period, then the power provided by the direct-drive wind generators to the grid can be expressed as follows: S = P + j Q = − U ˙ I ˙ * , (1) I ˙ * = U Z , (2) Z = r + j x , (3) P + j Q = − U 2 r w r w 2 + x w 2 − j U 2 x w r w 2 + x w 2 , (4) where P and Q, respectively, represent the active and reactive power output by the direct-drive wind generators to the system during the fault period; r _w and x _w represent the equivalent resistance and reactance of the direct-drive wind generators, respectively. During the occurrence of a voltage drop, the active and reactive power output by the direct-drive wind generators is positive. When studying the stability situation at the global level of the large power grid, the details that have no essential influence can be ignored, and the direct-drive wind generators during fault are equivalent to parallel negative resistance and negative reactance. Figure 1 shows the simplified model of systems before and during a fault, where SG1 represents the sum of sending-end grid units; SG2 represents the sum of receiving-end thermal power units; and PMSG represents the sum of receiving-end wind turbines.

According to the complex network theory, the topology conceals the physical information on the network and abstracts the relationship between nodes for study. When the object to be analyzed is a network structure with a large number of nodes and lines, the influence of the network topology structure is usually focused, while the internal details of the node elements are ignored. Therefore, when the research target is a large-scale power grid, the specific internal structure and operation mode of the turbine can be ignored.

Figure 1A shows a simplified system model before the fault. Figure 1B shows a simplified system model during the fault.

It has been proven that the doubly fed wind turbine can be simply equivalent to a pure impedance model when analyzing the power angle stability of large power grids (Liu, Shi, and Pang, 2020). In order to prove that the direct-drive wind generator can also be similarly equivalent, this paper analyzes the synchronous stability under disturbance (three-phase short circuit) when the doubly fed wind generator and the direct-drive wind generator are, respectively, connected to three power systems with different scales. The results are shown in Figure 2. It can be seen that with the increase in the scale of the power grid, the synchronous control capabilities provided by the direct-drive wind turbine and the doubly fed wind turbine tend to be the same. Combining the conclusions given in the previous literature studies, when studying the synchronous stability of large-scale power grids in this paper, the direct-drive wind generator is also similarly equivalent to impedance.

Figure 2A shows the IEEE 9-node power grid (maximum difference: 6.92°). Figure 2B shows the IEEE 39-node power grid (maximum difference: 5.15°). Figure 2C shows the Guangdong power system with 21,596 nodes in China (maximum difference: 0.1268°).

In order to study the influence of wind turbine replacement on the power angle stability of the system under fault conditions, a short-circuit fault in the power grid at the receiving end is taken as an example. First, the system situation is analyzed before the fault occurs.

Figure 3A shows a simplified circuit without a wind turbine connected to the grid, where S is the equivalent node of the power grid at the sending end, R is the equivalent node of the power grid at the receiving end, and W is the position connected to the wind turbine; X _dT1 is the equivalent reactance of generators and transformers of the sending-end grid, X _L represents the equivalent reactance of the power transmission channels between the sending-end power grid and the receiving-end power grid, and X _dT2 is the equivalent reactance of the receiving-end power grid units, transformers, and loads. Figure 3B is a corresponding simplified model of the system after the replacement in the receiving-end grid before the fault. Since the wind turbine only outputs active power, the equivalent impedance has only the real part. Let the capacity of a single wind turbine be the reference value of the per-unit system; hence, N represents the per-unit value of the total capacity of synchronous units and wind turbines in the receiving-end power grid, and k represents the proportion of thermal units replaced by equal-capacity wind turbines in the receiving-end power grid.

Figure 3A shows a simplified model of a power grid without a wind turbine. Figure 3B shows the simplified model of a power grid partially replaced by wind turbines before the fault. Figure 3C shows a simplified model of a power grid without a wind turbine under the fault. Figure 3D shows the simplified model of a power grid partially replaced by wind turbines under the fault.

Suppose that m = 1 / X dT 1 + X L , n = N / X dT 2 , for Figure 3A, the system transfer impedance without a wind turbine is as follows: Z SR = − j m + n m n = − j 1 − k m + n 1 − k m n . (5)

For Figure 3B, the system transfer impedance with a wind turbine is as follows: Z SR ′ = − j m + n − k n + k N r w 1 − k m n . (6)

One can construct function F ₁ as follows: F 1 k = 1 − k m n 2 ⋅ Z S R ′ 2 − Z SR 2 = N 2 r w 2 − m 2 − 2 m n k 2 + 2 m n + 2 m 2 k = a 1 k 2 + b 1 k , (7) where a ₁>0, b ₁ > 0, F ₁ (0) = 0, so function F ₁ is a concave function, and the extremum point is located in the left half of the coordinate system, and Z S R ′ < Z SR before the fault occurs; the equivalent electrical distance of the system with wind turbines is greater than that without those turbines.

Figure 3C is a simplified model of the system without wind turbine access during a fault, and Figure 3D is a corresponding simplified model of the system with wind turbines during a fault, where X _Δ is the additional reactance during a short-circuit fault.

For Figure 3C, the transfer impedance between the system sending-end power grid S and receiving-end power grid R is as follows: Z SR = − j m + n + 1 X Δ m n = − j 1 − k m + n + 1 X Δ 1 − k m n . (8)

For Figure 3D, the transfer impedance of system with wind turbines is as follows: Z SR ′ = − j m + n − k n + 1 X Δ + k N x w + k N r w 1 − k m n . (9)

The function F ₂ similar to Equation 4 is constructed as follows: F 2 k = 1 − k m n 2 ⋅ Z S R ′ 2 − Z SR 2 = d − k n + k N x w 2 + k 2 N 2 r w 2 − 1 − k 2 d 2 = N 2 x w 2 + N 2 r w 2 − 2 N n x w − n 2 − d 2 k 2 + 2 d N x w − 2 d n + 2 d 2 k = a 2 k 2 + b 2 k , (10) where d = m + n+1/X _Δ. d = m + n + 1 X Δ = 1 X d T 1 + X L + N X d T 2 + 1 X Δ . (11)

For large networks, when N is large enough, d could be approximated as d = N / X dT 2 = n . For function F ₂, a ₁>0, b ₁ < 0, function F ₂ is a concave function, and the extremum point is located in the right half of the coordinate system. When k = 0, F ₂ (0) = 0; when k = 1, F 2 1 = d − n + N x w 2 + N 2 r w 2 > 0 . The suggestive curve of function F ₂ can be roughly represented in Figure 4.

When 0 < k<s, Z S R ′ < Z SR , and the equivalent electrical distance between the sending- and receiving-end power grids with wind turbines decreases along with the growth of k.

When s<k<l, Z S R ′ < Z SR , and the equivalent electrical distance between the sending- and receiving-end power grids with wind turbines is still shorter than that between grids without wind turbines but increases along with the growth of k.

When k>l, Z S R ′ < Z SR , and the equivalent electrical distance between the sending- and receiving-end power grids with wind turbines is longer than that between grids without wind turbines.

The power characteristic equation of a generator during a fault is shown in Equation 9. P S = E q 2 Z 11 sin ⁡ α 11 + E q U Z 12 sin δ − α 12 , (12) where Z 11 = Z S S ′ = − j d − k n + k N X w + k N r w − m n − m X Δ + k m n − k m N X w − j k N r w . (13)

Suppose that the numerator of Z ₁₁ is (c + ej) when the denominator is removed. c = k N r w − 1 − m d − k n + k N X w . (14) e = − j d − k n + k N X w m 2 − m d − k n + k N X w + j k 2 N 2 r w 2 . (15)

As previously mentioned, for large networks, N is large, so the value of the imaginary part e containing N ² is much larger than that of the real part c. It can be approximated that c ≈ 0, sinθ ₁₁ ˜ 1, θ ₁₁ ˜ 90°, α ₁₁ ˜ 0°, and sinα ₁₁ ˜ 0. Now, Equation 9 can be expressed as follows: P S = E q U Z 12 sin δ − α 12 = E q U Z S R ′ sin δ − α S R ′ . (16)

For curve F in Figure 5A, when 0 < k ₁<k ₂<s, the acceleration and deceleration areas under different wind turbine ratios are shown in Figures 5B–D, where I, II, and III are, respectively, characteristic curves before a fault, during a fault, and after the clearance of the fault without a wind turbine; IV, V, and VI are, respectively, the characteristic curves before a fault, during a fault, and after the clearance of the fault when the wind turbine ratio is k ₁; VII, VIII, and IX are, respectively, the characteristic curves before a fault, during a fault, and after the clearance of the fault when the wind turbine ratio is k ₂. Z S R ′ < Z SR before a fault, so curves IV and VII are below curve I, and the peak value of the curve decreases with the increase in k. For the same reason, during a fault, curves V and VIII are over curve II, and the peak value of the curve increases with the increase in k. At the same time, the curves shift α ₁ (α ₂) to the right, and α ₁<α ₂. At the early stage of the fault, the wind generator outputs reactive power to the system, and the generator is equivalent to a resistance and a reactance with a negative value in parallel. However, in the later stage of the fault, reactive power is not output, and the reactance is 0. There is only negative resistance, so the curve after the fault of the wind generator replacement system consists of two parts, and the curve VI is lower than IX. Replacement of the thermal power unit by wind power leads to a reduction in system input power; so when the replacement ratio is k ₁ or k ₂, the input power curves PT ₁ and PT ₂ are lower than the input power curve PT ₀ in the system without wind turbines. Suppose that the fault is cleared when the power angle is δ′, with no wind turbine accessed, the operating point of the generator before the fault is a ₀, and the acceleration and deceleration areas are, respectively, S a 0 b 0 d c and S i b 0 k j . When k = k ₁, the operating point of the generator before the fault is a ₁, the acceleration area is S a 1 b 1 f e , and the deceleration area is S l b 1 n m . When k = k ₂, the operating point of the generator before fault is a ₂, the acceleration and deceleration areas are, respectively, S a 2 b 2 h g and S o b 2 q p . To meet the requirements of the equal area rule, when the power angle moves to δ ₀, S a 0 b 0 d c = S i b 0 k j ; when the power angle moves to δ ₁, S a 1 b 1 f e = S l b 1 n m ; and when the power angle moves to δ ₂, S a 2 b 2 h g = S o b 2 q p . Analysis shows that δ ₂<δ ₁<δ ₀, and the larger the wind turbine replacement ratio, the faster the system can meet the requirements of the equal area rule, so with the increase in the wind power replacement ratio, the power angle stability of the system is gradually enhanced for small k, and the stability is stronger than that of the system without wind turbines.

Figure 5A shows acceleration and deceleration areas under different turbine replacement ratios with wind power access when 0 < k<s. Figure 5B shows the acceleration and deceleration areas under different turbine replacement ratios without wind power access. Figure 5C shows the acceleration and deceleration areas when k = k ₁. Figure 5D shows the acceleration and deceleration areas when k = k ₂.

When s<k ₃<k ₄<l, the acceleration areas and the deceleration areas under different wind turbine replacement ratios are shown in Figure 6. When the proportion of wind power increases gradually, in order to ensure the safe and stable operation of the power grid and consider the law of the development of the real power grid, it is usually necessary to replace thermal units with wind turbines and ensure the output of the thermal power unit at the same time. Therefore, it is assumed that before the fault, the power curve and the input power curve of the prime mover remain unchanged, where I is the power characteristic curve before the fault. II and III are, respectively, the power characteristic curves during the fault and after clearance of the fault when the wind turbine replacement ratio is k ₃. IV and V are, respectively, the power characteristic curves during the fault and after clearance of the fault when the wind turbine replacement ratio is k ₄. Suppose that the fault is cleared when the power angle is δ'. When k = k ₃, the acceleration area is S a b f e , and the deceleration area is S l b g h . When k = k ₄, the acceleration area is S a b d c and the deceleration area is S i b j k . To meet the requirements of the equal area rule, S a b f e = S l b g h when δ = δ ₃, and S a b d c = S i b j k when δ = δ ₄. Analysis shows that δ ₃<δ ₄ and that systems with a larger wind turbine replacement ratio meet the requirements of the equal area rule more slowly, so with the increase in the wind turbine replacement ratio, the power angle stability of the system is weakened.

When k>l, with the increase in the replacement proportion, the power angle curve and the prime mover input power curve during the fault gradually decrease, and compared with the condition that wind turbines cannot be accessed, the change cannot be ignored, so it will be more difficult to accurately find the critical value of the power angle with the equal area rule. In the actual operation of the power grid, when the proportion of wind power reaches a considerable value, it will cause instability of the grid voltage, frequency, and other problems. Furthermore, the operation of the actual power grid does not allow the elimination of excessive thermal power units, so this paper does not analyze the power angle stability of the scenario when k>l.

As an important coastal province, Guangdong has abundant offshore resources for the development of offshore wind power. At the same time, in order to ensure the high-load operation of the Guangdong Power Grid, it needs to rely on the power support of Yunnan, Guizhou, and Guangxi, which cause the Guangdong grid to become a typical receiving-end power grid in the Southern Power Grid.

The 2022 summer high-load operation mode of the China Southern Power Grid is taken as an example, in which the Guangdong Power Grid is the receiving-end network (R), as can be seen in Figure 7A. By replacing thermal power units in coastal cities with direct-drive wind turbines to improve the replacement ratio of the Guangdong Power Grid, six simulation scenarios are obtained, where k ₁ = 2.5%, k ₂ = 12.5%, k ₃ = 25%, k ₄ = 39.8%, k ₅ = 51.1%, and k ₆ = 60.28%.

Figure 7A shows the global situation of power transmission from west to east in the China Southern Power Grid. Figure 7B shows the position of disturbed lines. Figure 7C shows the position of the turbine access.

At 0.2 s, two three-phase short-circuit disturbances are applied separately. The first fault happens on the transmission line from node Yuexuanhai21 near the integration point of the Guangdong Power Grid Xuanwu Offshore Wind Plant to node Yuexuanwu21, and the second fault happens on the transmission line from node Yuexianling51 to node Yuehuadu51, far away from the wind power integration point. Both of the two faults were removed at 0.32 s. Taking the power angle change of unit YueyuhaiG1 as an example, the approximate position of the disturbed line is shown in Figure 7B. Figures 8A,B respectively, show the power angle curves of YueyuhaiG1 in the wind power ratio of k ₁, k ₂, and k ₃ under the two faults.

Figure 8A shows the Yuexuanhai21–-Yuexuanwu21 line (a three-phase short circuit occurs). Figure 8B shows the Yuexianling51–-Yuehuadu51 line three-phase short circuit.

As can be seen in Figure 8, when the proportion of wind power is relatively small, with the increase in k, the swing amplitude of the system power angle decreases and the stability of the power angle increases, which is consistent with the results of the analysis in Figure 8.

Thermal power is continuously replaced with wind power, and the same disturbances are set. Figure 9 shows the power angle curve of YueyuhaiG1 when the proportion of wind power is k ₄, k ₅, and k ₆.

As can be seen in Figure 9, when the proportion of wind power increases further, the swing amplitude of the system power angle will increase and the stability of the power angle will become weaker, which is consistent with the results of the analysis in Figure 6.

Figure 9A shows the Yuexuanhai21–-Yuexuanwu21 line (a three-phase short circuit occurs). Figure 9B shows the Yuexianling51–-Yuehuadu51 line three-phase short circuit.

According to the simulation results, compared to the fault far away from the wind power access point, the condition of the system power angle is more stable when the fault occurs near the wind power access point, and the whole swing of the system power angle is more consistent with the theoretical derivation results given in Section 2.1.