The rotor motion equation of a synchronous generator describes the variation in rotor speed under unbalanced torque when fluctuations occur in the mechanical power output by the prime mover or the electromagnetic power output by the synchronous machine. It can be expressed as: Δ P m − Δ P e = 2 H s y s d Δ f d t (1) where ΔP _m is the mechanical power output of the prime mover; ΔP _e is the electromagnetic power output of the synchronous machine; Δf is the frequency variation (since the frequency f is directly proportional to the rotor angular velocity ω, for a more intuitive representation of the power-frequency relationship, the frequency deviation Δf will be used as a substitute for the angular velocity variation Δω; H _sys represents the equivalent inertia time constant of the generator set: H s y s = ∑ i = 1 n H i S i ∑ i = 1 n S i = ∑ i = 1 n H i S i S B S G (2) where n represents the number of synchronous units in the system, S _i and H _i are their respective rated capacities and inertia time constants, while S _B(SG) denotes the total rated capacity of conventional SG. The system load’s response to frequency deviations is primarily encapsulated in the load damping constant D. When a frequency deviation occurs, the variation in load power is given by: Δ P L = D Δ f (3) where ΔP _L represents the power variation of frequency-sensitive loads, and D is the load damping constant. Applying Laplace transforms to Equations 1 and 3 yields the equivalent model of the generator and load, as illustrated in Figure 1A. Figure 1B depicts the simplified version of this model.

Traditional thermal power generation units employ steam turbines as their prime movers, and the mechanical power output of the prime mover can be controlled by adjusting the valve opening of the steam turbine. The process of loading and unloading the steam chamber and inlet pipe takes a certain amount of time. Therefore, the process of load-frequency control through regulating the steam flow passing through the steam turbine using control valves can be represented by an inertial element with a time constant T _t , as shown in Figure 2. The value of the time constant T _t typically ranges from 0.2 to 0.3 s.

To distribute loads among multiple generator set reasonably, the governor system should be capable of reducing rotational speed when load power increases. This regulating characteristic can be achieved using an integral element with steady-state feedback, as illustrated in Figure 3.

In the figure, R represents the equivalent droop coefficient of the generator set, which physically signifies the ratio between the frequency variation and the change in generator output power. By simplifying Figure 3 and combining it with Figure 2, we obtain the equivalent model of the prime mover and governor as shown in Figure 4, where T _s = 1/(K _m R) represents the inertia time constant of the governor.

From Equations 1–3, it is evident that during active power disturbances in a power system containing synchronous machines, the system frequency undergoes an abrupt change. The generator sets, due to their rotational inertia, can provide energy proportional to the rate of frequency change, offering transient support to the system frequency. Larger inertia time constants of the generator set result in the release of more rotational kinetic energy. However, renewable energy systems such as wind and solar power generation are typically characterized by low inertia, and their output power is decoupled from the grid frequency, rendering them incapable of responding to frequency variations. With a high penetration of these renewable sources into the power system, the overall system inertia inevitably decreases, reducing the rotational inertia available to counter frequency changes and leading to degradation of frequency dynamic performance.

The VSM technology enables renewable energy interfaces, such as converters, to mimic the virtual inertia characteristics of synchronous machines by appending control loops. The basic working principle involves adjusting the output power of renewable energy units in response to frequency deviations during system disturbances, thereby equipping them with the capability to respond to frequency variations. Analogous to the rotational inertia effect of synchronous machines, the variation in output power of renewables through virtual inertia control in response to frequency changes is given by: Δ P e 1 = K d d Δ f d t (4) where ΔP _e1 is variation in output power of renewable energy units under virtual inertia control, where k _d represents the virtual inertia coefficient. Applying the Laplace transform to Equation 4 yields the SFR of renewables equipped with virtual inertia characteristics, as depicted in Figure 5.

To achieve a reasonable load distribution among multiple generator sets, traditional energy sources, under the influence of droop-equipped governors, exhibit a characteristic where output power increases with load increase. In contrast to rotational inertia, which provides transient support to the system frequency, the droop characteristic of generator sets reduces the steady-state error in system frequency after disturbances. The droop rate of generator sets can be expressed by Equation 5: R % = ω N L − ω F L ω 0 (5)

The magnitude of PFR capability in generator sets is intimately tied to their droop rates. When integrating low-inertia sources such as large wind farms or photovoltaic power stations into the grid, they are typically required to possess a certain level of PFR capability. By leveraging VSM technology to control the inverters of renewable energy interfaces, RES can exhibit a similar droop characteristic to traditional energy sources during frequency variations. In contrast to virtual inertia control, which can instantly respond to frequency changes, inverters executing droop control for frequency regulation require a certain time delay. Compared to the governor dynamics of traditional energy sources, the SFR of RES with droop characteristics is illustrated in Figure 6.

Where K _R is the droop control coefficient of RES, also known as the PFR gain, is denoted as T _R , which represents the droop control time constant. ΔP _e2 signifies the variation in output power of renewable energy units under the influence of droop control.

After renewable energy units acquire virtual inertia and droop characteristics similar to synchronous machines through VSM technology, these converter-based power sources can be aggregated with synchronous machines into a single-machine equivalent model. The SFR that considers the frequency regulation capability of renewables is depicted in Figure 7.

Where K _RES and K _m represent the proportions of actual output power contributed by renewable energy units and thermal power units, respectively, to the total system generation.

Without considering the virtual inertia provided by non-synchronous power sources, after replacing synchronous units with non-synchronous power sources of equal capacity, the system equivalent inertia time constant based on the total rated capacity of the system can be expressed by Equation 6: H S Y S ′ = ∑ i = 1 n − m H i S i ∑ i = 1 n − m S i + S B N S G = ∑ i = 1 n H i S i − ∑ i = m + 1 n H i S i ∑ i = 1 n − m S i = H S Y S − Δ H (6) where ΔH represents the equivalent inertia time constant of the synchronous units that have been replaced by non-synchronous power sources.

From an energy perspective, replacing synchronous units with non-synchronous power sources of equal capacity directly reduces the number of conventional synchronous units in operation. Consequently, the total rotational kinetic energy of the system decreases as the number of synchronous units diminishes, leading directly to a reduction in the system’s equivalent inertia level.

Under the premise of not considering the participation of asynchronous power sources in PFR, after replacing synchronous units with asynchronous power sources of equal capacity, the mechanical power gain coefficient of steam turbines, K _m can be expressed by Equation 7: K m s ′ = S B s ′ S B S G ′ + S B N S G = S B s − Δ S B s S B S G = K m s − Δ K m s (7) where ΔS _B(s) refers to the capacity of hydraulic turbines and steam turbines that have been replaced by non-synchronized power sources. ΔK _ms represents the variation in the mechanical power gain coefficient for hydraulic turbines and steam turbines.

Accordingly, the equivalent droop coefficient of the system, R′ can be expressed by Equation 8: 1 R ′ = 1 R s y s − 1 R s Δ K m s = 1 R s y s − 1 Δ R (8) where ΔR represents the variation in the equivalent droop coefficient of the system after it has been partially replaced by non-synchronous power sources.

In summary, since the total system capacity remains unchanged, replacing synchronous units with non-synchronous power sources of equal capacity directly alters the number of conventional synchronous units in operation. Consequently, the number of prime movers and governors of generating units decreases accordingly, leading to a gradual weakening of the frequency regulation capability of the governor system of generating units. Therefore, when synchronous units are replaced by non-synchronous power sources, the equivalent droop coefficient R′ of the system gradually increases, which is equivalent to reducing the system’s PFR capability.

The SFR shown in Figure 7 can be further simplified, and by letting H _SYS = H _sys + K _d , the rotational inertia of conventional units and the virtual inertia of renewable energy units can be aggregated into the overall system inertia. Consequently, the SFR can be expressed as:

Where T _sys represents the equivalent response time constant of the system, and K denotes the participation factor of renewable energy in frequency regulation, as shown in Equation 9. K = 1 − k SG + k RE (9) where k _SG is the replaced portion of synchronous machines is denoted as the proportion of substitution, and k _RE represents the proportion of power electronic sources that provide both inertia and frequency regulation (if power electronic sources do not provide frequency regulation, then k _RE = 0). At this point, the equivalent inertia of the system is H _SYS , and the equivalent droop coefficient is K/R. For simplicity, this equivalent model is used in subsequent analysis to investigate the impact of PFR parameters on the system.

Based on the SFR derived in the previous section from Figure 8, the transfer function from load disturbance to frequency variation can be expressed by Equation 10: Δ f = ω n 2 K D + R − 1 1 + T S Y S s P d s s 2 + 2 ζ ω n + ω n 2 (10)

Where ω _n and ζ represent the undamped natural frequency and damping ratio, respectively, and their expressions are shown in Equation 11: f n 2 = D R + K 2 H S Y S R T S Y S ζ = 2 H S Y S R + D R + K T S Y S 2 D R + K f n (11)

Assuming the magnitude of the disturbance is ΔP, applying the Laplace inverse transform to Equation 10 yields the expression of the system frequency response in the time domain, enabling the further derivation of evaluation parameters for the dynamic characteristics of system frequency, as shown in Equation 12. Δ f t = R Δ P D R + K 1 + α e − ζ ω n t ⁡ sin ω r t + ϕ ω r = ω n 1 − ζ 2 α = 1 − 2 T S Y S ζ ω n + T S Y S 2 ω n 2 1 − ζ 2 ϕ = arctan ω r T S Y S 1 − ζ f n T S Y S − arctan 1 − ζ 2 − ζ (12)

At t = 0, the maximum rate of change of frequency (RoCoF) can be obtained. Based on the above analysis, it can be concluded that: d Δ f d t t = 0 = Δ P 2 H S Y S + K (13)

At t = ∞, the quasi-steady-state frequency deviation of the system can be obtained. Δ f set = Δ f ∞ = Δ P D + K / R (14)

Using typical parameters from the reference (Banarkar et al., 2006) as the model’s parameters, and under a power disturbance of 10% of the synchronous generator capacity, the system frequency response curves are compared after wind turbines replace SG at different proportions, as shown in Figure 9.

From Figure 9A, it can be observed that as the proportion of wind power replacing SG increases (K decreases), the maximum frequency deviation, the maximum rate of change of frequency, and the quasi-steady-state frequency deviation all exhibit an increasing trend. Thus, all three evaluation indicators of system frequency response deteriorate, undoubtedly indicating that wind power integration has a negative impact on system frequency response characteristics, and this negative impact becomes more severe with an increasing proportion of wind power integration.

Figure 9B demonstrates that variations in H _SYS affect the magnitude of the system frequency rate of change post-disturbance. A smaller H _SYS results in a larger frequency rate of change throughout the PFR process, including an increased maximum value at the initial moment of the disturbance, consistent with the formula for calculating the maximum frequency rate of change given in Equation 13. Additionally, H _SYS significantly influences the system’s frequency nadir, with a smaller H _SYS leading to an earlier occurrence of the nadir and a larger maximum frequency deviation at this point.

According to Figure 9C, changes in D primarily impact the system maximum frequency deviation. A smaller D results in a larger maximum frequency deviation. While Equation 14 indicates that D also influences the quasi-steady-state frequency deviation, this effect is generally limited due to the typically small damping coefficients in power systems.

Finally, Figure 9D shows that variations in R affect the quasi-steady-state frequency deviation. A larger R delays the occurrence of the frequency nadir and leads to a larger maximum deviation at this point, as well as a larger quasi-steady-state frequency deviation.

According to above section, the K and R significantly influence the Δf _set. Thus, the relationship between K, R, and Δf _set needs to be further establish to control the value of Δf _set under different conditions of K by adjusting R.

According to Equation 14, the R can be calculated by: Equation 12: R = K Δ f set Δ P − Δ f set D (15)

It can be seen that the governing coefficient is directly proportional to K, while it is inversely prwxoportional to ΔP and D. Assuming a permissible quasi-steady-state frequency deviation of Δf _set = 0.2 Hz, the required droop coefficient R of the system is dependent on K, the integration of power electronic sources, and the active power disturbance ΔP. When the frequency regulation contribution of power electronic sources in the system decreases, an increase in the droop coefficient is necessary. Conversely, a larger active power disturbance requires a reduction in the equivalent droop coefficient to stabilize the system frequency.

Finally, a flowchart is given by Figure 10 to obtain the equivalent droop coefficient R based on the proposed SFR in this paper.

In order to verify the accuracy of the proposed SFR model, the active power disturbances of the load increase are respectively set as 8 MW, 12 MW, 16 MW (accounting for 10%, 15%, 20% of the system capacity separately) in PSCAD simulation model and SFR model. The frequency response curves and the errors between the PSCAD and SFR are respectively shown in Figure 12 and Table 1.

From the above figure and table, it can be observed that the output of the SFR model closely matches the results of the time-domain simulation under various load increment scenarios. Even in the case of 20% load increment, which caused a frequency drop of 0.46Hz, the error in the minimum frequency value was only 2.377%. These simulation results convincingly demonstrate the effectiveness of the proposed SFR model.

When the system suffers a 20% active power disturbance (18 MW), to maintain the quasi-steady-state deviation Δf _set = 0.23 Hz, substituting parameters into Equation 15 yields a equivalent droop coefficient of 1.4%. As shown in Figure 13, which compares the frequency response curves before and after the adjustment, by appropriately modifying the droop coefficient, the quasi-steady-state deviation Δf _set can be maintained at 0.23 Hz even under larger active power disturbances.

Under the scenario that the proportion of new energy sources participating in frequency regulation decreases, the overall frequency regulation capability of the system declines, leading to a shift in the demand for the droop coefficient. Keeping the power disturbance scenario in IV.A unchanged, the proportion of wind turbines participating in frequency regulation K is set to 0.5 and the simulation results are shown in Figure 14.

Compared to the 10%-power-disturbance original waveform in IV.A, the decrease in K results in a deterioration of the system frequency stability, manifested by enlarged frequency fluctuations and an increase in Δf _set. To keep the Δf _set to its initial value of 0.23 Hz, substituting the relevant parameters into Equation 14, yields a required R of 1.4%. As shown in Figure 14, upon adjusting R using the proposed method, Δf _set is successfully reinstated to meet the original specification in IV.A, despite the reduction in K. However, due to the diminished participation of wind turbines in frequency regulation, the frequency fluctuation profile exhibits greater deviations compared to the original waveform.

To verify the impact of D on frequency regulation effectiveness, keeping the 10%-power-disturbance scenario outlined in Section 4.1 unchanged, the equivalent damping constant D is set to 0.5 and the simulation results are shown in Figure 15. According to Equation 15, the Δf _set D is much smaller than ΔP, so the impact of Δf _set D can be ignored, the droop regulation coefficient calculated by Equation 15 has not changed significantly. Thus, the droop coefficient doesn’t need to be adjusted with the change of D.