The thermal power DPR compensation case is related to the output reduction behavior at dispatch time. Taking the compensated peaking service case in Hubei Province, China, as an example, the reference standard was set at 50% of the unit’s maximum output. Compensation is based on the generation of thermal power below the minimum technical output. The compensation case was divided into five levels, as listed in Table 1 (National Energy Administration and Central China Regulatory Bureau, 2022). where B i , t , peak G is the peak regulation compensation cost for the thermal power unit i; p j , peak G is the peak regulation compensation price for the j level of thermal power unit; P i , j , t , peak G is the compensated peak regulation power for the j level of the thermal power unit i; and Δ t is the calculation time step.

Thus, the total peak regulation compensation expenditure at time t is B t , peak = ∑ i = 1 N G ∑ j = 1 5 p j , peak G P i , j , t , peak G Δ t (2) where N G is number of thermal power units.

Wind, photovoltaic, and thermal power share DPR compensation costs. To further improve each entity’s enthusiasm for participating in peak regulations, a different approach was adopted for the capacity-sharing mechanism. Unlike the electricity allocation mechanism (Zhao et al., 2022), the capacity allocation mechanism was determined based on the proportion of each unit’s maximum output, whereas the allocation of thermal power units was determined based on the proportion of their non-DPR capacity. W l , t WT = P l , t , max WT B t , peak ∑ i = 1 N G P i , basic G + ∑ l = 1 N WT P l , t , max WT + ∑ m = 1 N PV P m , t , max PV (3) W m , t PV = P m , t , max PV B t , peak ∑ i = 1 N G P i , basic G + ∑ l = 1 N WT P l , t , max WT + ∑ m = 1 N PV P m , t , max PV (4) W i , t , basic G = P i , basic G B t , peak ∑ i = 1 N G P i , basic G + ∑ l = 1 N WT P l , t , max WT + ∑ m = 1 N PV P m , t , max PV (5) P i , basic G = P i , max G − P i , min G (6) where W l , t WT , W m , t PV , and W i , t , basic G represent the peak regulation cost allocation for the wind farm, photovoltaic station, and thermal power unit at time t; P l , t , max WT and P m , t , max PV represent the maximum output of the wind farm and photovoltaic station at time t; N WT is the number of wind farms; N PV is the number of photovoltaic plants; P i , basic G is the non-DPR capacity of the thermal power unit i; and P i , max G and P i , min G represent the maximum and minimum output of thermal power unit i, respectively.

The proposed peak regulation initiative was quantified by the extra profits obtained from each entity participating in the service. The wind power profit and photovoltaic peak regulation are composed of the profit from electricity sales, the allocation cost, and the penalty for abandoning wind and light. The thermal power peak regulation profit is composed of compensation, allocation, and DPR costs. These are shown in Eqs. 7–9. The thermal power unit DPR transfers peaking power generation rights to renewable energy. However, renewable energy compensates for thermal power units. When the profit is positive, each entity is willing to participate in the peak regulation auxiliary service to ensure fairness of the compensation and cost allocation mechanism.

(1) Wind power peak regulation profit:

The stimulated demand response (IDR) mainly includes two types: shiftable and sheddable loads.

(1) Shiftable Load

A shiftable load generally refers to a load, that is, not interrupted but can shift the entire electricity consumption period, such as in air conditioning and factories. The subsidizing shiftable load’s cost C TL in the system is denoted by C TL = ∑ i = 1 N TL ∑ t = 1 T c TL P i , t , out TL Δ t (13) where c TL represents the unified subsidy unit price for the shiftable load in the system, P i , t , out TL represents type i shiftable load’s power transfer-out at time t, N TL is the number of shiftable load types, and T is the total number of scheduling time periods.

(2) Sheddable load Sheddable load refers to loads such as lighting and computers that cannot be shifted but can be reduced or interrupted for electricity consumption. The load schedule after adopting the sheddable load control is

The system’s subsidy cost for sheddable load is represented as C RL = ∑ t = 1 T c RL , t P RL , t Δ t (15) where c RL , t is the system’s unified subsidy unit price for RL.

The PDR changes the way users use electricity by adjusting the price signal based on the principle of consumer psychology. The electricity price elastic matrix is used to model the PDR, as shown in Eqs. 16–18. E = ε 11 ε 12 ⋯ ε 1 T ε 21 ε 22 ⋯ ε 2 T ⋮ ⋮ ⋱ ⋮ ε T 1 ε T 2 ⋯ ε T T (16) ε = Δ D D p Δ p (17) Δ D 1 D 1 ⋮ Δ D T D T = E Δ p 1 p 1 ⋮ Δ p T p T (18) where E represents the electricity quantity and price’s elasticity matrix; ε is the demand price elasticity coefficient; and Δ D and Δ p represent the electricity quantity D and electricity price p’s increments, respectively.

Thermal power units bear the residual net load deducting other peaking resources (Li et al., 2020). In order to reduce the peak regulation pressure of thermal power units and maximize the use of resources such as wind, solar, water storage and DR to participate in peak regulation auxiliary services, the minimum fluctuation of thermal power output is taken as the objective function, as shown in Eq. 19. min f 1 = 1 T ∑ t = 1 T P t G − P av G (19) P t G = ∑ i = 1 N G P i , t G (20) P av G = 1 T ∑ t = 1 T P t G (21) where P t G is the output of all thermal power units at time t and P av G is the average thermal power output in time period T.

The comprehensive electricity cost of the system includes the comprehensive operating costs of the thermal power units, ES, and load, as shown in Eq. 22. min f 2 = C ES + C MT − C LD (22)

(1) Comprehensive operating cost of thermal power units

a) Fuel and start-stop unit cost C basic = C coal + C ss (23) C i , t , coal = a i P i , t G 2 + b i P i , t G + c i U i , t G (24) C coal = ∑ t = 1 T ∑ i = 1 N G C i , t , coal (25) C ss = ∑ t = 2 T ∑ i = 1 N G S i , start U i , t G 1 − U i , t − 1 G + S i , stop U i , t − 1 G 1 − U i , t G ] (26) where C basic is the basic operation cost of thermal power; C coal and C ss represent the cost of coal consumption and the thermal power unit’s start-stop operations, respectively; a i , b i , and c i represent the secondary, primary, and constant coefficients of unit i’s consumption, respectively; S i , start is unit i’s start-up cost; and S i , stop is unit i’s shutdown cost.

The method used in Li et al. (2022) can reduce the difficulty of solving and increase the rate, which linearizes the quadratic function of unit i’s coal consumption cost.

According to Eq. 25, the abscissa of the cost function represents the power output of the thermal power units and the ordinate represents the cost. Assuming that the abscissa is divided into m segments, with each segment having a length l, the expression is as follows: L = P i , max G − P i , min G m (27) P i , t , l G is introduced to represent unit i’s actual output value in period t and segment l; thus, Eq. 24 can be rewritten as C i , t , coal = ∑ l = 1 m k l P i , t , l G + U i , t G P i , 0 G (28) where k l represents each segment’s slope, calculated by Eq. 29. P i , 0 represents the unit cost at the lowest output, calculated by Eq. 30. k l = a i P i , t , l + 1 G 2 + b i P i , t , l + 1 G + c i − a i P i , t , l G 2 + b i P i , t , l G + c i P i , t , l + 1 G − P i , t , l G (29) P i , 0 = A i P i , min 2 + B i P i , min + C i (30)

b) Unit DPR loss cost

The additional unit losses caused by the unit DPR are approximately calculated with reference to the widely used Manson-Coffin formula (Lin and Tian, 2017), as shown in Eq. 31. C i , t , unit = β S unit , i / 2 N f P i , t G (31) where β is the influence unit operation’s coefficient, S unit , i represents unit i’s purchase cost; and N f P i , t G represents the fatigue cycle until rotor fracture occurs.

Linear fitting of the actual unit loss data was performed to obtain the linear formula of the unit loss under different outputs: C i , t , unit = − 0.000045 P i , t G P N + 0.000033 S unit , i T (32) where P N is the unit’s rated capacity, and T represents the test impact frequency of 96 times/day.

c) Unit DPR-assisted fuel combustion cost C i , t , oil = Q i , t , oil p oil (33) where Q i , t , oil is unit i’s DPR stage fuel injection quantity at time t, and p oil is the oil price.

Based on the different operating conditions of the thermal power units, the comprehensive operating cost is expressed as C M T = C basic , P i , min G < P i , t ≤ P i , max G C basic + ∑ i = 1 N G ∑ t = 1 T C i , t , unit , P i , a < P i , t ≤ P i , min G C basic + ∑ i = 1 N G ∑ t = 1 T C i , t , unit + C i , t , oil , P i , b < P i , t ≤ P i , a (34) where P i , b is the limit of the DPR-assisted combustion power for the thermal power units; P i , a is the limit of the non-DPR-assisted combustion power for thermal power units.

(2) ES comprehensive operating costs

a) ES operating cost model

The ES operating cost model considers charging and discharging. C S = ∑ t = 1 T c S P t dis + P t char (36) where c S is the charge and discharge power cost coefficient of ES.

b) ES operating profit model

The ES operating profit depends mainly on its electricity sales and environmental profits, as shown in Eq. 37. B S = ∑ t = 1 T p t grid P t dis η d − p t grid P t char η c + ∑ t = 1 T ∑ k M η d p k poll P t dis ξ k poll (37) where p t grid is the on-grid electricity price; η c and η d represent the charging and discharging efficiencies of the ES, respectively; P t char and P t dis represent the charging and discharging power at time t of the ES, respectively, respectively; p k poll is the penalty coefficient for pollutant k; ξ k poll is the environmental benefit coefficient of k pollutant’s reducing emissions by replacing power generation from the higher-level power grid with ES; and M is the pollutant type number.

(3) Comprehensive load electricity cost

The profit from electricity sales to internal users can be expressed as follows C pro = ∑ t = 1 T p t PDR P t PDR (39) where p t PDR is the time-of-use electricity price for internal users in the system after PDR and P t PDR is the internal user load in the system after PDR.

The minimum total abandoned consumption of wind power, photovoltaic power, and hydropower in T periods is taken as the objective function, as shown in Eq. 40: min f 3 = ∑ t = 1 N T ∑ l = 1 N WT P l , t , cur WT Δ t + ∑ t = 1 N T ∑ m = 1 N PV P m , t , cur PV Δ t + ∑ t = 1 N T ∑ h = 1 N H P h , t , cur H Δ t (40) where P h , t , cur H is hydropower station h’s spilled water power in time period t, and N H is the number of hydropower plants.

(1) Power balance constraints

(4) ES constraints S t S = S t − 1 S 1 − δ + P t char η c Δ t E S − P t dis Δ t η d E S S min ≤ S t S ≤ S max (48) S T S = S 0 S (49) where S t S is the state of charge (SoC) of the ES at time t; δ represents the self-discharge rate of the ES; E S is the ES’s capacity; S max and S min represent upper and lower limits of the charging state of the ES, respectively; and S 0 S and S T S represent ES’s charge state at the beginning and end of the time, respectively.

(5) Power grid transmission capacity constraint

(6) IDR constraint ∑ t = 1 T Q t , out TL = ∑ t = 1 T Q t , in TL (51) 0 ≤ Q t , out TL ≤ Q max TL (52) 0 ≤ Q t , in TL ≤ Q max TL (53) 0 ≤ P t RL ≤ P t , max RL (54) where Q t , out TL and Q t , in TL represent the transferred and received loads at time t, respectively; Q max TL is the maximum transferred and received loads at each time; and P t , max RL are the shiftable loads at time t.

(7) PDR constraint F t FR ≤ F t , max FR (55) F t FR = P t PDR − P t − 1 PDR P t PDR (56) P min L ≤ P t PDR ≤ P max L (57) ∑ t = 1 T P t PDR = ∑ t = 1 T P t L (58) 1 − ∑ t = 1 T P t PDR − P t L ∑ t = 1 T P t L ≥ m s (59) 1 − ∑ t = 1 T p t PDR P t PDR − p t L P t L ∑ t = 1 T p t L P t L ≥ m p (60) where F t FR is the load fluctuation rate at time t; F t , max FR is the maximum load fluctuation rate at time t; P min L and P max L represent the original load’s minimum and maximum values, respectively; P t L is the original load’s forecasted value at time t; m s and m p represent user satisfaction with electricity usage and satisfaction lower limit with electricity expenses, respectively; and p t L is the original electricity purchase price for users at time t.

The modified IEEE 30-bus system was used for the simulation analysis to verify the effectiveness and applicability of the proposed method, as shown in Figure 2 (Cao et al., 2023). The case study included one wind farm, one photovoltaic power station, one hydroelectric power station, five thermal power units, and one energy-storage system. The parameters of each unit are listed in Table 2 and Table 3 (Li et al., 2019).

The on-grid electricity price for wind power is 570 yuan/(MWh), and that for photovoltaic power is 921 yuan/(MWh). The penalty coefficient for wind and photovoltaic abandonment was 536 yuan/(MWh). The wind power, photovoltaic power, and load curves are shown in Figure 3 (Li et al., 2023). The on-grid electricity price of the thermal power was 375 yuan/(MWh), and the thermal power unit purchase cost was 4,394 yuan/kW. The fuel consumption during the DPR stage of the unit was 4.8 t/h, and the fuel price was 6,130 yuan/t. The profits from electricity purchases and ES sales are based on the peak and grid valley electricity prices, as shown in Figure 4. The charge and discharge cost was 50 yuan/(MWh), and the ES environmental benefit was converted to 314 yuan/(MWh) (Yang et al., 2022). In PDR, a three-level load price is used to represent the peak, valley, and flat power consumption characteristics, as shown in Figure 4. Assuming that 5% of the users participate in the PDR every hour, the elastic matrix parameters of the electricity price are listed in Table 4 (Cui et al., 2021b). In the IDR, the user compensation fee for the shiftable and sheddable load was 140 yuan/(MWh), and the parameter settings are shown in Table 5 and Table 6 (Hou et al., 2022b).

To develop an efficient and economical experimental design model, we used an L 4 2 3 orthogonal array in the orthogonal experimental design (Hou et al., 2022c). Considering the dependency relationship between the depth peak regulation and the peak regulation initiative, four cases were designed, as listed in Table 7.

The above four cases were simulated and analyzed, and the power output curves and optimization results for each entity are presented in Figure 5 and Table 8.

By comparing Cases 2 and 4, it can be seen that the renewable energy consumption significantly improved after the implementation of the demand response. The load curve was optimized by guiding users to change their power consumption behavior through demand response, and the wind power abandonment rate was reduced from 2.2% to 1.2%. In addition, the demand response can effectively reduce the peak–valley difference in the system net load, peak load pressure, and energy storage of the thermal power units. By comparing the output of the thermal power units in Figure 5, we can see that in Case 4, the thermal power unit output fluctuation is smaller and the operating cost is lower.

By comparing Cases 2 and 3, we found that Case 3 gives up the DPR because it considers the peak regulation initiative. This is because of the following reasons: although DPR improves the system’s flexibility, with an increase in the peak regulation depth, thermal power units face greater economic pressure owing to the unit loss cost and fuel injection cost and the reduced power generation. The peak regulation compensation obtained by thermal power units cannot fully compensate for the increased cost due to the DPR; therefore, they choose to withdraw from the peaking cooperation voluntarily, which proves the fairness of the cost compensation and capacity-sharing mechanism.

Case 3 can be viewed as a case in which DPR is not considered. Compared with Case 3, Case 2 considers DPR and ES access to the thermal power units. Despite the increased loss and fuel injection costs of the thermal power units, the DPR reduces the wind abandonment rate by 4.4% and increases the profit by 285,000 yuan. This is achieved while ensuring the lowest total operating cost of the system. In summary, the thermal power unit DPR has a significant effect on the renewable energy consumption and system operating costs.

To explore the impact of different renewable energy penetration rates on the system peak regulation, cases 1 and 2 were compared. The results show that, in Case 1, the operating cost of the thermal power units increases significantly with a decrease in renewable energy penetration. This is because thermal power units operate in the conventional peak regulation stage, instead of renewable energy to provide more electricity, need to face more coal consumption costs and start-stop costs. Despite the increased cost of the thermal power units, the system can still operate well in high-permeability wind and low-permeability renewable energy systems, thereby demonstrating good adaptability.

To further explore the effectiveness of the proposed model, this section analyzes the output of each thermal power unit, as shown in Figure 6.

In case 2, thermal power unit 1 is in the DPR state from 10:00 to 15:00, whereas units 4 and 5 perform the start-stop peak regulation. From the perspective of the system peak regulation effect, the DPR can improve the flexibility of the system. However, in Case 3, the DPR-related cost of the thermal power units cannot be adequately compensated for. Unit 1 includes the DPR and enters a normal peak load-balancing state after the active peak load-balancing constraint is introduced. This is because of the lack of compensation in the peak regulation service or the design of the compensation mechanism. Considering the peak load balancing initiative, it is essential to explore appropriate incentive mechanisms and compensation strategies. These efforts aim to attract thermal power units to participate actively in deep-peak regulations, which can significantly improve the system economy.

Similar to Case 3, the thermal power units in Case 1 did not participate in the DPR for different reasons. In Case 3, the thermal power units exit the DPR because the system fails to provide adequate compensation for the increased peaking cost. In Case 1, although there is a large peak and valley difference in the net load curve, the start–stop cost of small-capacity units is relatively low because of the large number of thermal power units; therefore, they can replace DPR by start-stop peaking. The system can achieve better economy and scheduling flexibility through a flexible start-stop peak regulation strategy. This means that thermal power units can not only combine depth peaking and start-stop peaking strategies to provide a stable power supply to the system, but can also reduce the peak-valley difference of the system. Thus, it is an economical and efficient choice for system operations.