The total operational cost of the system includes power generation, reserve, and penalty costs for the curtailment of renewable energy generation. The specific expressions are introduced in detail in Section 4.

The energy utilization efficiency of WTs and PVs can be quantitatively evaluated using the real-time generation penetration rate (Chen et al., 2020) and other indices. In this paper, the system’s real-time penetration rate c w , t , c p v , t (Equation 5) and consumption η w , t , η p v , t (Equation 6) of WTs and PVs are selected as evaluation indices to analyze the applicability of the proposed reserve capacity allocation model. c w , t = ∑ w ∈ N w P w , t max − P w , t , c p v , t = ∑ p v ∈ N p v P p v , t max − P p v , t , (5) η w , t = ∑ w ∈ N w P w , t max − P w , t P w , t max , η p v , t = ∑ p v ∈ N p v P p v , t max − P p v , t P p v , t max , (6) where P w , t max and P p v , t max are the predicted power of WT and PV at time t, respectively. P w , t and P p v , t are the actual power output of WT and PV at time t, respectively.

The inertia level of the system at period t is expressed by the equivalent inertia constant H _sys,t of the system (Liu et al., 2021): H s y s , t = E s y s , t S B , s y s , t , (7) E s y s , t = ∑ i ∈ N i P i max ⋅ H i ⋅ u i , t + ∑ w ∈ N w P w , t max ⋅ H w + ∑ m ∈ N m P D m , t max ⋅ H m , (8) S B , s y s , t = ∑ i ∈ N i P i max ⋅ u i , t + ∑ w ∈ N w P w , t max + ∑ p v ∈ N p v P p v , t max , (9) where E s y s , t (Equation 8) is the total equivalent kinetic energy of the system, and S B , s y s , t (Equation 9) is the total startup capacity of the system, both of which are related to the capacity of the SGs participating in frequency regulation and the startup/shutdown status u _i,t of the SGs. H _i , H _w , and H _m are the inertia constants of the SGs, WTs, and PV battery storage, respectively.

The dynamic frequency response process of a system with a disturbance is shown in Figure 2. The frequency dynamics can be expressed by Equation 10 using the rotor motion equation of the generator: 2 H s y s f 0 d Δ f d t = Δ P m − Δ P e S B , s y s , (10) where Δ P m and Δ P e are the mechanical and electromagnetic power of the generator, respectively, and Δ f is the system frequency deviation.

Assuming that the mechanical power input by the generator remains unchanged for a relatively short period after a disturbance, the initial unbalanced power of the system is the disturbance power Δ P . The initial RoCoF can be calculated by Equation 11: R o C o F 0 = − Δ P ⋅ f 0 2 H s y s ⋅ S B , s y s . (11)

Considering that the power response of the generator governor responds to a power imbalance, the rotor motion equation of the generator can be modified as 2 H s y s f 0 d Δ f d t = v i P F R ⋅ t − Δ P S B , s y s , (12) where v i P F R denotes the primary frequency response rate of the i-th generator. By integrating both ends of Equation 12 simultaneously, the system frequency deviation in the time domain can be obtained: f t = f 0 ⋅ v i P F R 4 H s y s S B , s y s t 2 − Δ P ⋅ f 0 2 H s y s S B , s y s t + f 0 − Δ f d b , (13) where Δ f d b is the frequency dead band of generator governors.

Using S B , s y s as the base power, the frequency regulation parameters of a system comprising multiple units, including renewable energy units, are aggregated. The equivalent inertia parameter for inertia response is presented in Equation 7, and the aggregation of the primary frequency regulation rate parameter in the system frequency deviation model is shown in Equation 14. v P F R = ∑ i ∈ N i v i P F R u i , t + ∑ w ∈ N w v w P F R + ∑ p v ∈ N p v v p v P F R / S B , s y s . (14)

Assuming that the primary frequency regulation rate of the generator is constant, it can be seen from Equation 13 that the postfault system frequency deviation f(t) is a quadratic function of time. The frequency nadir will appear at t _nadir when d f / d t = 0 . By taking the derivative of both sides of Equation 13 with respect to t, t _nadir can be obtained: t n a d i r = Δ P v P F R + − Δ f d b R o C o F 0 . (15)

By substituting t _nadir into Equation 13, the frequency nadir of the system is f n a d i r = f 0 − Δ f d b − f 0 ⋅ Δ P 2 4 H s y s S B , s y s v P F R . (16)

Combined with Equations 15, 16, the relationship between f _nadir and t _nadir can be derived as Equation 17: f n a d i r = f 0 − Δ f d b − f 0 ⋅ Δ P 2 4 H s y s S B , s y s Δ P t n a d i r + Δ f d b R o C o F 0 . (17)

The frequency nadir directly determines the actions of the protection relays. Therefore, it is necessary to restrict the system frequency nadir, f _nadir , under credible contingencies: f n a d i r ≤ f U F L S ⇒ t n a d i r max ≤ 4 H s y s S B , s y s f 0 − Δ f d b − f U F L S f 0 ⋅ Δ P + 2 H s y s S B , s y s Δ f d b f 0 ⋅ Δ P , (18) where f _UFLS is the threshold of under-frequency load-shedding protection devices. The frequency dynamics are integrated into the reserve model. The constraint of f _nadir is transferred to the limitation of t _nadir in Equation 18, which means that the frequency regulation of the system should have a sufficient response speed.

Combined with Equation 16, the constraints on the reserve of units participating in primary frequency regulation can be derived as y i u p ≤ v i P F R ⋅ t n a d i r max = v i P F R ⋅ 4 H s y s S B , s y s f 0 − Δ f d b − f U F L S f 0 ⋅ Δ P + 2 H s y s S B , s y s Δ f d b f 0 ⋅ Δ P , (19) where y i u p represents the upward reserve primary frequency regulation of the i-th unit.

The frequency security limits of the power grid considered in this study are summarized as follows:

1. Maximum RoCoF limit

The RoCoF describes the speed of the system frequency change following a disturbance. To avoid exceeding the threshold R o C o F max , the following constraint is enforced: R o C o F 0 ≤ R o C o F max . (20)

2. Frequency nadir limit

To avoid triggering the action of frequency protection relays with power loss/increase disturbance, the frequency nadir of the system should be limited using the upward reserve constraint derived in Equation 19.

3. Quasi-steady frequency limit

The quasi-steady-state frequency deviation Δ f s s is related to the system power disturbance, primary frequency regulation rate, and overall primary frequency reserve capacity R P F R and is restricted by Δ f s s = Δ P − R P F R D ⋅ P l o a d ≤ Δ f s s max , (21) where D is the load damping constant of the system, and P l o a d is the total load demand.

The constraints of the generation scheduling layer include the output of various types of generations, the unit commitment of SGs, and system power balance limitations. The output limits of the generation units are expressed in Equation 26: u i , t ⋅ P i min ≤ P i , t 0 ≤ u i , t ⋅ P i max 0 ≤ P w , t 0 ≤ P w , t max 0 ≤ P p v , t 0 ≤ P p v , t max , (26) where P t 0 is the output power of a unit in period t, and the subscripts i, w, and pv represent the SGs, WT, and PV indices, respectively.

The WT must provide some power to participate in the system frequency adjustment. The reserve capacity provided by the w-th WT in period t is denoted by Equation 27: R U w , t = P w , t max − P w , t 0 . (27)

Equations 28, 29 determine ramp limits and startup and shutdown times: R G i d o w n ≤ P i , t 0 − P i , t − 1 0 ≤ R G i u p , (28) ∑ m = 0 T i o n − 1 u i , t + m ≥ T i o n ⋅ u i , t − u i , t − 1 , ∑ m = 0 T i o f f − 1 1 − u i , t + m ≥ T i o f f ⋅ u i , t − 1 − u i , t , (29) where R G i d o w n and R G i u p are the ramp-up/ramp-down limits of the i-th SG, and T i o n , T i o f f are the minimum continuous startup/shutdown time of i-th SG. ∑ i ∈ N i P i , t 0 + ∑ w ∈ N w P w , t 0 + ∑ p v ∈ N p v P p v , t 0 = ∑ l ∈ N l P l , t + ∑ d c ∈ N d c P d c , t . (30)

Equation 30 limits the power balance of the system. N l , N d c are, respectively, the number of load and HVDC links, and P l , t , P d c , t are the load demand and HVDC power at period t, respectively.

The second-level model is discretized with a time scale of τ ₁, and the time interval is set to Δn. System frequency security is verified for each time interval.

Table 1 presents the economics of the different scenarios from multiple perspectives, including the generation cost, reserve cost, wind curtailment, and total operating cost. The curtailment provided in the table is the average wind curtailment (MW). The total operating and generation costs of the system in Scenario 2 are the lowest of the three scenarios because Scenario 2 does not consider the WT to provide frequency support, and system frequency security constraints are not enforced. For Scenario 3, because only SGs ensure system frequency security under predefined disturbances, more WT output is curtailed to maintain a higher SG online capacity; therefore, the curtailment penalty cost is significantly higher than that of Scenario 1.

Owing to the utilization of the MPPT operation strategy and participation in frequency regulation through battery storage of PVs, there was no PV curtailment in the three scenarios. The wind curtailment in the power grids of Scenarios 1 and 3 during the dispatch period is shown in Figure 7. In Scenario 1, the wind curtailment rate of the system was maintained below 5%, and the average wind curtailment rate during the entire dispatch period was 4.57%. In Scenario 3, the wind curtailment rate of the system at hour 1 was 27.2%, and the wind curtailment amount was 4615 MW. During the entire dispatch period, the average wind curtailment rate was 13.45%, and more wind-power generation was curtailed.

Figure 8 compares the inertia levels in the three scenarios. It can be seen that the proposed optimization model considers the virtual inertia support of renewable energy. Hence, the fluctuation in the system equivalent inertia level in Scenario 1 was smaller and could be maintained at a high level (approximately 5.23 s). Owing to the single source (only SGs) of inertia in Scenarios 2 and 3, the system inertia level was not significantly improved even when the wind curtailment rate reached 13.45%, and the low-inertia risk was noticeable.

Figure 9 compares the postfault initial RoCoF during the scheduling period of the three scenarios. Scenario 1 had a smaller RoCoF than the traditional reserve optimization methods (Scenarios 2 and 3). Because frequency security is not considered in Scenario 2, the frequency change rate exceeds the threshold, and system frequency security cannot be guaranteed. A comparison of the frequency nadir for the three scenarios is shown in Figure 10. Similar to the frequency change rate index, the proposed reserve optimization method (Scenario 1) had a smaller frequency nadir with the same disturbance as the other traditional methods (Scenarios 2 and 3). The frequency nadir in Scenario 2 exceeded the threshold.

Figure 11 shows the detailed allocation optimization results of the primary reserve capacity in Scenario 1. The primary frequency regulation of the system is dominated by SGs, supplemented by WT and PV battery storage. According to the primary reserve Equation 39, the primary frequency regulation reserve capacity of the system is related to the disturbance. Therefore, the primary frequency regulation reserve in Scenario 1 is nearly equal to the initial system disturbance.

Figure 12 shows the configuration results of the secondary reserve capacity for the three scenarios. The upward adjustment space for the secondary reserve capacity of the system fluctuated significantly in all three scenarios. Because of the significant wind curtailment in Scenario 3, the upward adjustment space of the SGs was compressed, and the secondary reserve capacity in Scenario 3 was relatively small. The tertiary reserve capacity of the system during the dispatch period is shown in Figure 13. In Scenario 1, because of the upward reserve reservation for the WT, the SGs support more load demand than in the traditional scheme, and the total tertiary reserve capacity of the system in Scenario 1 is smaller than that in Scenario 2 without wind curtailment measures. Similar to the secondary reserve, the tertiary reserve adjustment space for the SGs in Scenario 3, where there is a large amount of wind curtailment, is also compressed. Thus, the total tertiary reserve of Scenario 3 was significantly lower than that of Scenario 2, which had no wind curtailment, and lower than that of Scenario 1, which had only a small amount of wind curtailment.