The converter station of the DC transmission line generally adopts constant active power or constant DC voltage control, which does not have a power frequency characteristic, so it cannot participate in frequency regulation. By equipping the FLC to the power outer loop of the constant active power control, the DC transmission lines can adjust the power and realize the frequency modulation. At the same time, in order to ensure the safe and stable operation of the DC line and avoid frequent fluctuation, the DC FLCs generally add a dead zone link. To improve the sensitivity of the response and sufficient power transmission, the FLC adopts the PI control of isochronous control and suppresses the large-scale fluctuation of the system by power reset. The control logic of the DC FLC is divided into two types: reverse frequency difference regression model and integral negative feedback regression model.

The control block diagram of the reverse frequency difference regression model is shown in Figure 1, and the integral negative feedback regression model is shown in Figure 2.

In Figure 1, Δf represents the system frequency deviation detected by the DC FLC, which is the difference between the current frequency f and the reference frequency f _ref of the system; f _H and f _L represent the upper and lower critical values of the frequency deviation dead zone, respectively; S represents the Laplace operator; K _P and K _I represent the proportional coefficient and integral coefficient of the PI controller, respectively; x _max and x _min represent the upper and lower limit of the integral state quantity, respectively; symbol x represents the integral state quantity of the PI controller; ΔP _max and ΔP _min represent the upper and lower limit of DC power modulation, respectively; ΔP represents DC power modulation; P _ref represents the reference value of the DC transmission power; P _L ^D represents the set value of the DC transmission power; and Δf ₀ represents the frequency deviation through the dead zone.

If the system frequency deviation Δf is in the dead zone, the frequency state quantity Δf ₀ is 0, and the DC power modulation amount is also 0, that is, the FLC does not act on the frequency deviation within this range, and the DC line power is transmitted according to the rated value. If the frequency deviation Δf exceeds the upper limit of the dead zone f _H, and Δf ₀ is greater than 0, it is superimposed into the DC modulation amount ΔP after the proportional link and the integral link, and the DC FLC transmission power setting value is added to the power reference value. After the DC transmission power increases, the frequency slowly recovers until the frequency deviation reaches the dead zone. This stage is the FLC regression link. At this moment, the state quantity Δf ₀ gradually becomes smaller, and the modulation quantity still exists after passing the proportional integral module. When the frequency deviation reaches the dead zone, the integral state quantity x is reduced to 0, and the DC modulation power is also reduced to 0 so that the DC transmission power setting value is consistent with the reference value. The response and regression speed of the FLC depend on the proportional and the integral coefficient. When the frequency deviation Δf is lower than the lower limit f _L, the frequency adjustment process and the regression process are similar to the above analysis.

In Figure 2, x ₁, x ₂, x ₃, and x ₄ represent the intermediate state variables of the control process. K represents the regression coefficient; t represents the integral time constant; other symbols are the same as those shown in Figure 1. The function of control logic 1 is that when |Δf ₀|≥0.005 Hz, a delay of 100 ms makes the output state x ₁ to Δf ₀, and the FLC is started; when |Δf ₀|<0.005 Hz, the forced output state x ₁ is 0. The function of control logic 2 is that when x ₁ ≠ 0, the output signal x ₃ selects x ₂; when x ₁ = 0, the output signal is selected as x ₄ after a 60-s delay.

According to the above logic and control, we can see that when the frequency deviation is greater than the upper limit of the dead zone, the DC FLC will delay the action by 100 ms, control logic 1 outputs the frequency state quantity Δf ₀ after passing through the dead zone, control logic 2 selects the output x ₂, and the negative feedback regression part stops. At this time, the DC modulation amount ΔP increases continuously so that the DC FLC transmission power setting value will also increase, thereby reducing the system frequency deviation. This process is consistent with the modulation process of the inverse frequency difference regression model. When the frequency is restored to the dead zone of the FLC, the output of control logic 1 is 0, the output of control logic 2 is x4, the FLC enters the regression state, the output state quantity x of the integral negative feedback link gradually decreases to 0, and the transmission power set value P _L ^D is consistent with the reference value P _ref. When the frequency deviation is lower than the lower limit of the dead zone, the frequency adjustment process is similar to the above analysis.

The control logic of the integral negative feedback regression model is similar to the reverse frequency difference regression model. The structural difference is mainly reflected in the integral negative feedback link and two logic control parts. The functional difference is mainly reflected in the regression process during frequency recovery.

The SFR model is a simplified model in ASF. It is assumed that the whole power system frequency is uniform, and the frequency modulation control of all units is equivalent to a reheat link speed control system, and the system is converted into a single machine model. The model is shown in Figure 3.

In Figure 3, H represents the rotor inertia time constant; D represents the damping coefficient of the load; T represents the equivalent inertia time constant of the turbine; a represents the characteristic coefficient of the turbine, where a = −2 for the water turbine and 0<a<1 for the steam turbine; R represents the adjustment coefficient of the governor; ΔE represents the angular frequency adjustment; ΔP _m represents the output power of the generator (mechanical power); ΔP _L represents the disturbance power (load power); ΔP _e represents the active power difference (electromagnetic power); ω _ref denotes the reference angular frequency; and Δω _r represents the angular frequency deviation of the generator. In the AC synchronous system, the spatial distribution difference of frequency is very small, and the output frequency of the generator can be considered as the frequency of the whole system (Anderson and Mirheydar, 1990).

For power systems with renewable energy integration, it is necessary to introduce parameters that characterize the permeability of renewable energy based on the SFR model (Dai et al., 2022). The renewable energy integration method of the synchronous power grid is mainly realized by replacing synchronous units with renewable energy units. However, renewable energy generators generally do not have frequency modulation capabilities (Ye et al., 2016). Therefore, the synchronous machine per unit coefficient K _G participating in frequency modulation is introduced in the turbine part and generator part. The renewable energy permeability λ is expressed as 1-K _G.

According to the analysis in Section 2, the frequency regulation process of the DC FLC is mainly based on the PI controller of isochronous control, and some limiting links are ignored. The simplified model of the bilateral DC FLC is obtained, as shown in Figure 4.

In the figure, ω _rec represents the frequency of the rectifier side, ω _inv represents the frequency of the inverter side, and ΔP represents the power modulation amount. When the DC FLC is configured on one side of the rectifier side, the FLC only monitors the frequency deviation of the rectifier side. The DC power modulation amount will be accessed through the additional control of the power outer loop of the converter station. The power adjustment amount only responds to the frequency deviation of the rectifier side, adjusts the DC transmission power, and affects the power distribution at both ends of the transmission and reception.

For multi-area asynchronous interconnected power grids, the power regulation of DC lines will be used as the main frequency regulation measure, and the power regulation effect of the FLC is equivalent to reducing power disturbance (Liu et al., 2023). Therefore, the DC modulation power can be connected to the disturbance power to offset the frequency deviation. According to the characteristics of power generation and load type, each synchronous power grid can be equivalent to an SFR model, which comprises a single turbine, governor, and generator. The interconnection of the three-area asynchronous interconnected power grid can be equivalent to three DC lines and the FLC. By adjusting the signal of FLC input control, one-sided configuration and two-sided configuration of the DC FLC can be selected.

The frequency analysis model of the three-area asynchronous interconnected power grid is shown in Figure 5. In the figure, R ₁, a ₁, T ₁, H ₁, and D ₁ represent the equivalent adjustment coefficient, equivalent turbine characteristic coefficient, equivalent turbine inertia time constant, equivalent rotor inertia time constant, and equivalent load damping coefficient of synchronous power grid 1, respectively. K _P1 and K _I1 represent the proportion coefficient and integral coefficient of the FLC between synchronous power grids 1 and 2. K _G1 represents the unit coefficient of the synchronous machine participating in the frequency modulation of synchronous power grid 2. Other parameters are similar.

In order to verify the rationality and stability of the proposed model, the theoretical analysis is carried out. Ignoring the dead zone and amplitude limiter module, ω _ref is set to 0, and only the transfer function relationship between ΔP _L1, ΔP _L2, and ΔP _L3 and the frequency of the three regions Δω ₁, Δω ₂, and Δω ₃ are considered.

Considering only ΔP _L1, the whole transfer model shown in Figure 5 is split and simplified to obtain the simplified transfer function model shown in Figure 6.

In the Figure 6, G₁(s) represents the forward path of the transfer function in synchronous grid 1, H₁(s) represents the negative feedback path, and T₁(s) represents the DC line transfer function between grid 1 and grid 2. Other symbols are similar.

For the convenience of calculation, the transfer function is expressed by symbols. Taking synchronous grid 1 as an example, G₁(s), H₁(s), and T₁(s) are expressed as follows: G 1 s = 1 2 H 1 K G 1 s + D 1 , (1) H 1 s = K G 1 a 1 T 1 s + 1 R 1 T 1 s + 1 , (2) T 1 s = K P 1 + K I 1 s . (3)

According to the simplified rules of the transfer function structure diagram, the negative feedback module is simplified as follows: G 1 s 1 + G 1 s H 1 s = R 1 T 1 s + R 1 2 H 1 R 1 T 1 s 2 + R 1 D 1 T 1 + 2 H 1 R 1 + a 1 T 1 s + 1 . (4)

G₂(s), H₂(s), T₂(s), G₃(s), H₃(s), and T₃(s) are similar. The transfer function equations of synchronous power grids 1, 2, and 3 are listed as follows: a b c Δ P L 1 − Δ ω 1 s − Δ ω 3 s T 3 s − Δ ω 1 s − Δ ω 2 s T 1 s G 1 s 1 + G 1 s H 1 s = Δ ω 1 s Δ ω 1 s − Δ ω 2 s T 1 s + Δ ω 3 s − Δ ω 2 s T 2 s G 2 s 1 + G 2 s H 2 s = Δ ω 2 s Δ ω 1 s − Δ ω 3 s T 3 s − Δ ω 3 s − Δ ω 2 s T 2 s G 3 s 1 + G 3 s H 3 s = Δ ω 3 s . (5)

The transfer function relations between ΔP _L1 and Δω ₁(s), Δω ₂(s), and Δω ₃(s) are obtained by solving Eqs 1–5. Similarly, the transfer function relations between ΔP _L2, ΔP _L3, and Δω ₁(s), Δω ₂(s), and Δω ₃(s) are calculated. The transfer function matrix of the frequency deviation of each region under different disturbance inputs, as shown in Eq 6, is established by the calculation results. The detailed calculation process is shown in Supplementary Material. Δ ω 1 s Δ ω 2 s Δ ω 3 s = A 11 s A 12 s A 13 s A 21 s A 22 s A 23 s A 31 s A 32 s A 33 s Δ P L 1 Δ P L 2 Δ P L 3 . (6)

For the above matrix expression, the poles of the first-column parameters of the matrix are calculated under typical parameters, as shown in Figure 7. All poles appear in the negative half-plane. According to the necessary and sufficient conditions for the stability of the linear system (Xiao et al., 2023), the model has good stability and can meet the frequency analysis requirements of multi-area asynchronous interconnected power grids.

According to the frequency response model of the multi-area asynchronous interconnected power grid proposed in this section, the following will take a three-area asynchronous power grid as an example: grid I is rich in hydropower resources and is a sending-end power grid. Grid II is a receiving-end power grid, with a small scale of power generation and load, low grid strength, and poor frequency anti-disturbance ability. Grid III is the main receiving-end area, its power generation and load scale are large, the power grid strength is high, and the frequency stability is good. The following section studies two topics by combining the permeability changes of renewable energy. On one hand, the influence of renewable energy penetration on the lowest frequency decrease is studied, and on the other hand, the relationship between the FLC configuration scheme and the maximum acceptance ratio of renewable energy is studied.

Due to the influence of the single-sided and double-sided configuration of the DC FLC on the frequency support capability of each region, the fault type is analyzed by taking the blocking loss of DC lines between synchronous grids I and II as 5,000 MW. At this moment, the two ends belong to complementary power disturbance, only the DC lines and FLC between synchronous grids I and II are considered, and other power grids and lines are not considered.

First, we analyze the maximum frequency deviation of the sending and receiving power grids when the penetration rate of grid II changes. The penetration rate is simulated to increase from 0% to 60% at a step size of 10%. The asynchronous grid connection line is set on the grid-I side and the bilateral FLC. The maximum frequency deviation and quasi-steady state frequency of the two regions are extracted, and the simulation and theoretical analysis are carried out.

The simulation results of one-sided FLC configuration are shown in Figure 8. The frequency decrease of power grid II has reached the set value of low-frequency load shedding, which seriously affects the stable operation of the grid system. With the increase in renewable energy permeability, the frequency of the power grid continues to decrease, and the downward trend of the lowest point becomes larger. The quasi-steady state frequency deviation decreases slowly, but it also continues to increase. The frequency response index of grid I and the FLC transmission power remain.

The increase in grid II permeability leads to a decrease in frequency modulation capacity, and the frequency deviation becomes larger. Since only the one-sided FLC is configured on the grid-I side, the FLC transmission power is determined by the frequency deviation of the one-sided power grid. The permeability of the grid-I side remains, and the frequency response curve is unchanged, so the FLC transmission power remains unchanged. The deterioration in the frequency characteristics of the grid-II side is mainly due to the reduction in frequency modulation resources.

The simulation results of the two-sided FLC configuration are shown in Figure 9. With the increase in the permeability of grid II, the frequency of grid II gradually decreases, the downward trend of the lowest point gradually increases, and the quasi-steady state deviation increases. The frequency deviation of grid I becomes smaller, the trend of the deviation becomes approximately linear, and the quasi-steady state deviation is basically stable. The FLC transmission power increases gradually, and the peak power increases accordingly.

Because the permeability of grid II becomes larger, its frequency modulation resources decrease, and the frequency deviation becomes larger. The power support of grid I obtained through the bidirectional FLC increases, and the effect equivalent to the disturbance power of grid I decreases, so the frequency deviation becomes smaller.

Combined with the above analysis, we can see that the bilateral FLC can adjust the transmission power of the DC line according to the frequency decrease difference between the two sides, which has more advantages in alleviating two-end complementary power disturbances, and can restore the frequency deviation to the range of non-triggering under-frequency load shedding. However, the single-sided FLC can only respond to the permeability change on the FLC configuration side, and the weakening of the frequency modulation resources on the other side cannot be adjusted.

In order to study the influence of different configuration schemes of the DC FLC on the penetration rate of renewable energy, taking a three-area grid as an example, the fault type of blocking 5,000 MW of DC lines between power grids I and II is also set up, and the simulation is carried out considering different FLC configurations. The influence of the permeability change of synchronous grid III on the frequency response index of each grid is analyzed, and the conclusion of DC FLC configuration under the condition of renewable energy integration is summarized.

When the FLC is configured only on the grid-I side, the simulation results are shown in Figure 10. After the fault occurs, grids I and III have a positive frequency deviation, and grid II has a negative frequency deviation. With the increase in the renewable energy penetration rate of grid III, the maximum frequency deviation of grid III increases, the quasi-steady state frequency change is consistent with the maximum frequency deviation trend, and the frequency characteristics of grids I and II do not change. The adjustment of the FLC of each DC line remains unchanged.

At this time, non-complementary power disturbances occur in grids I and III. Since only the grid-I side is equipped with the FLC, the DC transmission line only responds to the frequency deviation of the grid-I side, and the change in the renewable energy penetration rate in grid III will not affect the transmission power between grids I, II, and III. The blocking fault of the transmission line leads to an active power surplus of grid I, and the system frequency has a positive deviation so that the power is transmitted to grids II and III through the FLC to adjust the power and reduce the frequency deviation. The active power shortage of grid II causes a negative frequency deviation. The transmission power from grids I to II can reduce the frequency deviation of grid II, but it is not enough to compensate for it completely. Grid III receives the power adjustment transmitted by grid I, resulting in a power surplus and a positive frequency deviation. At the same time, the increase in permeability leads to the decrease in the frequency modulation capacity of power grid III, the weakening of active power consumption capacity, and the increase in the frequency deviation of grid III.

When grid I–II DC lines are configured with the one-sided FLC and grid I–III lines are configured with the bilateral FLC, the simulation results are shown in Figure 11. After the fault occurs, grids I and III have a positive frequency deviation, and grid II has a negative frequency deviation. With the increase in the renewable energy penetration rate of grid III, the frequency positive deviation of grids I and III becomes larger, but the frequency variation of grid I is smaller, and the frequency negative deviation of grid II becomes smaller.

Grid I will transmit the adjustment power to grids II and III through the FLC. Due to the active power surplus caused by the acceptance power of grid III, the frequency has a positive deviation. The increase in the renewable energy penetration rate of grid III leads to a decrease in frequency modulation capacity, the positive frequency deviation becomes larger, and the frequency difference between grids I and III decreases. According to the control characteristics of the bidirectional FLC, the transmission power of the grid I–III DC FLC is reduced; grid I gains more active surplus, and the positive frequency deviation becomes larger. According to the characteristics of the one-sided FLC, the transmission adjustment power to grid II increases. More active power compensates the power shortage of grid II, and the reverse frequency deviation of grid II decreases.

When only grid I and III sides are configured with the FLC, the results given in Figure 11 are completely consistent. When configuring the FLC on the grid-III side of the grid II–III lines, the frequency deviation range of grid III is in the dead zone of the FLC due to the large frequency modulation capacity. The FLC configured on the line does not respond, and the FLC adjustment power is still zero, that is, the frequency response index of each grid is consistent that given in Figure 11.

When only the I–II lines of the grid are configured with the one-sided FLC, and the other lines are configured with the bilateral FLC, the simulation results are shown in Figure 12. After the fault occurs, grid I has a positive frequency deviation, and grids II and III have a negative frequency deviation. With the increase in the penetration rate of renewable energy in grid III, the negative deviation of the frequency of grids II and III becomes larger, and the positive deviation of grid I frequency decreases.

Because only one-sided FLC is configured in the grid I–II lines, the other lines are configured on both sides. After the fault occurs, grid I surpluses a large amount of power and transmits the power to power grids II and III, and grid II loses a large amount of power and absorbs power from grid III through the line. Because the frequency modulation capacity of grid II is smaller than that of grid I, the frequency deviation is larger than that of grid I. Through the bilateral FLC characteristics, power grid III mainly undertakes the function of transmitting power to power grid II, resulting in frequency reverse deviation. The increase in the permeability of grid III leads to an increase in the negative frequency deviation, and the frequency difference between grids I and II becomes larger. Therefore, the transmission power of grid I to grid III becomes larger, and the power of grid III to absorb power from grid II through the FLC becomes larger. The power disturbance of grid I decreases, and the frequency deviation becomes smaller.

When all DC lines are configured with the bilateral FLC, the simulation results are shown in Figure 13. There is a positive frequency deviation in grid I and a negative frequency deviation in grids II and III. With the increase in the permeability of grid III, the frequency deviation of grids II and III becomes larger, the frequency deviation of grid III becomes more obvious, and the frequency deviation of grid I decreases slightly. After the renewable energy penetration rate increases, the transmission power of grid II–III lines decreases, and the transmission power of grid I–II and I–III lines becomes larger.

Due to the configuration of the bilateral FLC on all lines, after the fault occurs, grid I surpluses a large amount of power and transmits the power to the power grids II and III through the FLC. Grid II absorbs power from grids I and III through the line FLC due to the loss of a large amount of power. Similarly, because the frequency modulation capacity of grid II is smaller than that of grid I, grid III mainly undertakes the function of transmitting power to grid II, resulting in a negative frequency deviation. The frequency modulation capacity of grid III is reduced, the power support provided to grid I becomes larger, and the power support provided to grid II becomes smaller, so the positive frequency deviation of grid I becomes smaller, and the negative frequency deviation of grid II becomes larger. Because the frequency modulation capacity of grid II is small, the frequency deviation of grid II decreases more, so the frequency difference between the grid I and II is larger. According to the characteristics of the bilateral FLC, the FLC adjustment of the line between grids I and II is slightly larger.

Combined with the above analysis and comparison, it can be seen that the bilateral FLC has limitations in alleviating the non-complementary power disturbance at both ends. The fault side cannot make full use of the contralateral frequency modulation resources, and the power regulation effect of the DC FLC is not good. At the same time, when the frequency modulation capacity at one end is small, the configuration of the bilateral FLC will lead to excessive power support on the opposite side. After the integration of renewable energy, the frequency deviation on the support side increases in reverse.