The prediction error of uncertain resources is the main factor that affects the accurate execution of the dispatching plan. To ensure reliable power supply to the system. Analysis of renewable energy prediction error characteristics is needed to study the impact of prediction error on the development of dispatching plans. Using the ratio of the difference between the ultra-short-term forecast data and short-term forecast data of renewable energy output to the installed capacity of the unit to express its forecast error, it can be expressed as ξ r , t = ( P r , t r o l l − P r , t d a ) / P r s (1) Where r represents the wind farm or PV plant. P r , t r o l l and P r , t d a are the ultra-short-term predicted and short-term predicted power of the wind farm or PV plant. P r s is the wind farm or PV plant size. ξ r , t is the error size of wind power or PV in both time scales.

The information of the output forecast data of a wind farm and PV power station from 01.01 to 03.01 days is counted with a resolution of 15 min. The error values between the intraday ultra-short-term forecast data and the day-ahead short-term forecast data for wind and PV output are given in Figure 1. It can be seen from Figure 1 that the forecast errors of wind and PV fluctuate within the range of ±30% and ±25%. This error is sufficient to cause the system to abandon wind and lose load due to the mismatch of unit response rate (Wang et al., 2021). It occurs frequently especially in the systems with established operation plan units and those containing heat-determined units. This also indicates the need for intraday revisions. To cope with the uncertain demand variation of the system. This paper will extend the probabilistic method to generate sufficient probability scenarios to portray the stochasticity of the forecast error; and use the following units with high flexibility for recalibration during actual operation.

In this paper, the nonparametric kernel density estimation method is used to obtain the probability distribution of the respective predictions based on the statistical historical data of the scenarios. Compared with the parametric method, the error distribution function does not have to be assumed in advance, which reduces the influence of uncertain factors on the probability model (Li et al., 2019). If p r = { p r 1 , p r 2 , ⋯ , p r n } is the sample space of the renewable energy forecast error data, n is the sample size. Then the probability density function of the renewable energy forecast error can be expressed as f ( p r ) = 1 n h ∑ i = 1 n K ( p r − p r i h ) = 1 n ∑ i = 1 n K ( p r − p r i ) (2) K ( p ) = 1 2 π exp ( − 1 2 p 2 ) (3) Where p r i is the forecast error sample i of wind power or PV output. h is the bandwidth. K ( p ) is the Gaussian kernel function.

A high or low bandwidth h will directly affect the probability density distribution of the prediction error. In order to minimize the prediction error, the optimal bandwidth model is described in Lang et al. (2020). The Gaussian kernel function is settled and the normal distribution N ( μ , σ ) is used as the reference distribution of the probability density function f ( p r ) . The optimal bandwidth can be obtained as h A M I S E = 1.06 σ n − 1 / 5 (4) Where σ is the sample standard deviation.

After using the above method to determine the probability distribution function of wind power output forecast error, and based on the idea of stratified sampling, latin hypercube sampling is used to generate the initial scenarios of large-scale error. In order to effectively simulate the error uncertainty and ensure the model calculation efficiency. The synchronous back-substitution reduction method is used to remove a large number of similar scenarios, retain some representative scenarios, and obtain the corresponding probability of each scenario through calculation. As a result, the uncertain problem is transformed into a deterministic problem. The detailed steps are as follows:

1) Suppose x 1 , x 2 , ⋯ , x T are T independent random variables, and their cumulative probability distribution function (wind or PV) is as follows

Consider the existence of random variables and the inconsistent response speed of decision variables. The slow-response decision variable (starting and stopping of the unit) needs to be determined before random variables start to appear. Decision variables with faster response speed (unit ramping, etc.) are not limited, and can be determined after more accurate random variables (such as the regularly updated ultra-short-term forecast output of renewable energy sources). Therefore, this paper introduces a two-stage stochastic programming model (George, 1955), the form is as follows { min z . c T z + E s { Q ( z , s ) } s . t .       A z ≤ b       z ≥ 0       Q ( x , s ) = { min y s q s T y s s . t .   W s y s ≤ h s − T s z y s ≥ 0 ,       ∀ s ∈ Ω (8) Where z and y s are the decision variables of the first and second stages. The variables in both stages are greater than zero. c T , q s T , b , h s , A , T s and W s are known vectors and coefficient matrices. E s is the expected value under scenario s in the second-stage planning, which can be used to evaluate the volatility of the random variable in the objective function c T z + Q ( z , s ) . Ω is the set of all possible scenarios. Since the equation contains random variables, it cannot be solved directly. Random variables can be replaced by calculating expected values. Convert uncertain problems to deterministic ones.

This paper establishes a source-load-storage rolling schedule based on a two-stage stochastic programming algorithm. The first stage is to determine the start-stop combination of the following units. Based on the determined start-stop decision of the unit, after the random variable arrives, the output of the unit is adjusted in the second stage to meet the changing demand of the renewable energy output. Intraday dispatch mainly utilizes the feature that the forecast accuracy of renewable energy is negatively correlated with forecast horizon. Combined with regularly updated ultra-short-term forecast data of wind, PV and load, periodic adjustment and revision of the system’s day-ahead plan aims to achieve the effect of global optimization of the system’s output plan. The intraday scheduling of this paper takes 15 min as an interval and 4 h as a cycle. The system automatically updates and obtains ultra-short-term forecast information of wind, PV and load for the next 4 h every 15 min. The rolling timing is shown in Figure 2.

Considering the operating cost of thermal power units in the objective function as a nonlinear quadratic function, the quadratic function in the objective function can be linearized by its linearization through segment linearization (Carrion and Arroyo, 2006). The core idea of the linearization process is to divide the quadratic function into m segmented functions. Then, the slope of each segmented function is found and converted into a primary function with respect to the horizontal coordinate, and thus the linearization of the quadratic function is realized. The linearization principle diagram of thermal power unit operating cost segments is shown in Figure 3. The operating interval of thermal power unit i is divided into m ( m = 3 ) segments, and the value of m should be chosen appropriately. the smaller m is, the less linearization accuracy will be achieved. the larger m is, the higher linearization accuracy will be achieved, and at the same time, it will increase the size of the decision variables, which makes it difficult to solve the calculation later. Z t h , i 0 denotes the operating cost of thermal power unit i at the lowest level of output. Z t h , i s e g is the segmentation point of segment s e g ( s e g = { 1,2,3 } ). Δ P t h , i s e g is the length of each segment. length of each segment interval. K t h , i s e g is the slope of segment s e g . Z t h , i s e g , Δ P t h , i s e g and K t h , i s e g should satisfy the following constraints { Δ P t h , i s e g = P t h , i max − P t h , i min m K t h , i s e g = C t h , i , t ( Z t h , i s e g ) − C t h , i , t ( Z t h , i s e g − 1 ) Z t h , i s e g − Z t h , i s − 1 Z t h , i s e g = Z t h , i 0 + s e g ⋅ Δ P t h , i s e g Z t h , i 0 = P t h , i min (36)

After linearization, the operating cost of thermal power unit i is converted to the following model: { C t h , i , t ( P t h , i , t ) = A i , 0 + ∑ m s e g = 1 ( K t h , i s e g ⋅ Δ P t h , i s e g ) P t h , i , t = P t h , i min + ∑ m s e g = 1 ( Δ P t h , i s e g ) 0 ≤ Δ P t h , i s e g ≤ Z t h , i s e g − Z t h , i s e g − 1 A i , 0 = α t h , i ( P t h , i , t ) 2 + β t h , i P t h , i , t + γ t h , i (37)

Referring to the linearization process of thermal power units, the operating cost of gas turbines can be linearized similarly.

The start-up cost of pumped storage in the objective function is a bilinear nonlinear programming problem that can be linearized using McCormick’s inequality (Castro and Pedro, 2015). Taking the pumped start-up cost I i , t p I i , t - 1 p as an example, by introducing a new binary integer variable I i , t y and making I i , t y = I i , t p I i , t - 1 p , the equation can be equated to the following linear constraint { 0 ≤ I i , t y ≤ I i , t p I i , t y ≤ I i , t - 1 p I i , t p + I i , t - 1 p − 1 ≤ I i , t y ,             I i , t y = I i , t p I i , t - 1 p (38)

Referring to the processing method of I i , t p I i , t - 1 p , I i , t g I i , t − 1 g and I j , t I j , t − 1 can be similarly linearized.

After the linearization process, the model built in this paper belongs to the mixed integer linear programming problem. By writing a program in the YALMIP environment of MATLAB and calling the solver GUROBI to solve the model, the optimal output combination of each unit is obtained.

In the case study, the power system in this paper includes five thermal power units, two gas turbines, a pumped-storage power station, a wind farm and a PV power station. The installed capacity of wind farm and PV power station in the system is 400 and 250 MW respectively. There are five thermal power units and two gas turbines, and the specific parameters are shown in Table 1. A pumped storage power plant with an installed capacity of 60 MW, the upper reservoir storage limit and the initial reservoir capacity of this pumped storage power plant are 600 MWh and 300 MWh. Figure 4 shows the rolling updated ultra-short-term power forecasting curves for wind, PV and load. Figure 5 gives the day-ahead operation plan curves for thermal units, where the day-ahead plan identifies three thermal units to be put into operation.

Figure 6 shows three representative sequences of wind power errors obtained after generating 200 initial scenarios using the method described in Section 2, with probabilities of 0.46, 0.305 and 0.235, respectively. The PV error curve can also be obtained this manner. The following three scenarios are set to verify the effectiveness of the proposed dispatching model for making dispatching plans in high-penetration renewable energy power systems.

Case 1: Day-ahead scheduling without the participation of quick start and stop groups.

The intraday output curves of thermal units and gas turbines for case 2 and 3 are shown in Figures 7, 8. Since case 2 does not consider the start-stop combination of fast units during the day, the gas turbines are on during the optimized hours. The downward adjustment space of the system in the night abandonment interval is reduced, which leads to the limitation of wind power feed-in power. The gas turbines in case 3 need to be called up only during the nighttime peak of the load (19:30-20:15) according to the regulation demand. This reduces the operating cost of the gas turbine and provides more adjustable space for the system in case 3 during the optimization period. The output curve of the thermal unit shows that the thermal unit in case 3 has a smoother climb.

Figures 9, 10 show the change curves of pumped storage unit output and reservoir capacity under case 2 and case 3. The model built in this paper relaxes the capacity of the end of the reservoir and considers the correction cost of inconsistency between the beginning and end of the reservoir. It can effectively avoid the phenomenon that the pumped-storage unit has been pumping water and generating electricity when the program is running, and the effect of the unit call is not obvious. As can be seen from the figure below, pumped storage is mainly used to store energy during the nighttime wind power generation, and is used as the power generation side to provide increased capacity for the system during the peak load period. The analysis shows that the number of power generation starts of the pumped storage unit in the two cases remains the same. However, the number of pumping starts in case 3 is reduced by two on the basis of case 2, which increases the flexibility, safety and economy of the system.

Figure 11 represents the wind power abandoned by the system for each case. Among them, case 1 has a larger amount of wind abandonment. cases 2 and 3 consider fast start-up and shutdown of units during the day, and their abandoned wind power intervals are shortened from the period of 0:15-5:30 at night to 2:15-4:00 and 2:30-3:15, respectively. the total amount of abandoned wind power is reduced from 565.478 MW-h to 25.9581 MWh and 9.1021 MW. Although the abandoned power in case 2 is significantly reduced, the wind power in case 3 is almost fully online. It is able to further increase the dispatchability of renewable energy and the acceptance rate into the grid.

Table 2 shows the total intraday operating costs of the system under each case. Among them, the set system power abandonment penalty fee is 50 yuan/MW. The operating cost of pumped storage includes the start-up and shutdown of pumped storage and the cost of storage capacity correction. Compared with case 1, although case 2 increases the flexibility of unit participation in regulation, the cost of wind curtailment is reduced by 107,903.972 yuan. However, without taking into account the start-stop combination of rapid units under the daily scale, the thermal power operating cost has increased by 51,091.975 yuan on the basis of the previous plan. After considering the running cost of the fast unit, the total cost of the system has increased by 52,984.09 yuan on the basis of the day-ahead plan. In case 3, a two-stage stochastic programming model is adopted and the decision of starting and stopping of rapid units is considered, and the cost of wind curtailment of the system is further reduced by 4,277.0007 yuan. At the same time, the gas turbine and pumped storage unit are called on demand, and the final total system operating cost is reduced by 67,115.3984 yuan on the basis of the previous plan. This shows that the economics of the method in this paper and the absorption effect of wind power are better.

Table 3 compares the cost optimization rate of the intraday scheduling method proposed in this paper with other intraday scheduling methods. Compared with the researches (Cui, et al., 2021.; Jin, et al., 2020) that configures fast units to participate in regulation but does not consider the start-stop combination of fast units, the intraday cost optimization rate is increased by 4.9775 and 2.1219%, respectively. Compared with the research (Ran et al., 2016) based on the rolling scheduling strategy of short-term unit combination, this paper combines the two-stage stochastic programming theory and considers the fast unit participation in the intraday cost optimization rate, which can improve the cost optimization rate by 4.9838%.