4.1.1. Effectiveness Verification
The modified IEEE118 node system includes 16 thermal power plants, three cascade hydropower plants, three wind power plants with a capacity of 200 MW, and two solar power plants with a capacity of 150 MW. The specific parameters of wind and solar power plants are shown in
Table 1. To verify the effectiveness of DR-DOPF in reducing the comprehensive power generation cost and water spillage, the scenario displayed in
Figure 1 is selected from the sample. Under this scenario, the real output of the sum of wind and solar power considerably exceeds the forecasted output in periods 5–9 and 15–19.
For convenience of expression, the traditional dynamic optimal power flow without considering water spillage is abbreviated as TN-DOPF. The TN-DOPF model does not consider the uncertainty of wind and solar output, and the objective function does not include the cost of water spillage. Regarding the DR-DOPF model, the confidence level of the Wasserstein ambiguity set is set to 0.95, and the risk factor is set to .
The result calculated is the day-ahead generation scheduling. To obtain the operation curve and water spillage of hydropower plants in real-time dispatching, the two models are treated as follows.
Regarding the DR-DOPF model, the solution of the real-time output value is given by Equation (6). Specifically, at the real-time dispatching stage, the real output of wind and solar power and the forecasting errors are already known, and the participation factors
,
and the unit planned output
,
are calculated. Hence, the real output of each plant can be directly determined according to Equation (6). Actually, this method dynamically distributes the unbalanced power to each unit according to the participation factor. The flowchart of the DR-DOPF model is shown in
Figure 2.
Regarding TN-DOPF, because this model does not involve a participation factor, the actual output cannot be directly determined, but this issue can be addressed according to the actual dispatching situation. To embody the principle by which the dispatcher prioritizes hydropower adjustment in practical operation, it is assumed that 90% of the wind and solar power forecast error is balanced by the hydropower plants and the remaining 10% by the thermal plants, and the power imbalance is assumed to be evenly distributed between the three hydropower plants.
Figure 3 shows the optimal output of the day-ahead scheduling and the real-time output of the power plant under the DR-DOPF model when the sample size is 100. The real-time output of hydropower and thermal power is lower than that based on the generation plan, because the real output of wind and solar power exceeds the predicted value and the output curve of thermal power is relatively smooth in the previous generation plan but fluctuates in the real-time output curve, especially during periods 5–10 and 15–19. During these two periods, the actual output of wind and solar power is significantly higher than expected, which indicates that the thermal power unit under the DR-DOPF model greatly participates in system power regulation. Although this suggests that the thermal units will produce higher regulation cost, compared to the regulation mode mainly relying on hydropower, the joint optimal regulation mode of hydropower and thermal power can produce higher economic benefits. This point is discussed in detail later.
Figure 4 displays the day-ahead and real-time output of hydropower plant 1 under the DR-DOPF and TN-DOPF models; the other two hydropower plants are similar and are not described here.
Figure 4a,c reveal that neither model produces water spillage in the day-ahead scheduling. However, at the real-time stage, both models produce water spillage (
Figure 4b,d), which is considerably smaller for DR-DOPF than TN-DOPF. The reason is that when dealing with forecasting errors of wind and solar power output, the hydropower plants under these two models must bear a certain amount of the power imbalance. Because the real value of the wind power output during periods 5–9 and 15–19 greatly exceeds the predicted value, under real-time dispatching, to maintain the system power balance, hydropower plant 1 must greatly reduce the power generation based on the previous power generation plan, resulting in water spillage. In addition,
Figure 5 shows that the adjusting power of hydropower plant 1 is lower under the DR-DOPF model than under the TN-DOPF model, especially during periods 5–9 and 15–19; consequently, the water spillage amount under the DR-DOPF model is much smaller.
Figure 6 displays the total unbalanced power and adjusting power of thermal power and hydropower under the two models. It can be seen that the adjusting thermal power under DR-DOPF is larger than that of TN-DOPF because it is determined by the participation factors calculated by the day-ahead scheduling. The day-ahead plan considers the uncertainty of wind and solar output, and the objective function contains the water spillage cost. To minimize the water spillage cost, this inevitably requires that the decisions made according to the participation factors can optimally assign the unbalanced power and fully utilize the regulation capacity of the thermal power units. Hence, the regulation burden of the hydropower units can be reduced, and consequently, the water spillage can be decreased. The NDOPF model is modeled on the actual dispatching situation, in which dispatchers tend to prioritize the hydropower plant-adjusting power, and thermal power units participate in regulation to a lesser degree. In this case, power regulation mainly depends on hydropower. Due to the high regulation burden of the hydropower units and the unreasonable distribution of regulation, when the real value of wind and solar power output greatly exceeds the expectation, water spillage easily occurs.
Under the TN-DOPF model, the proportion of thermal power participating in regulation is low, which can reduce the regulation cost of thermal power units; however, this phenomenon results in large water spillage. Although the water spillage is not reflected in the cost function of TN-DOPF, it cannot be ignored. To fairly compare the comprehensive cost of TN-DOPF and DR-DOPF, the water spillage of TN-DOPF is converted into the water spillage cost according to Equation (17) and added to its objective function value to obtain the comprehensive cost.
Table 2 presents the comprehensive generation cost and total water spillage according to the two methods. Compared with the TN-DOPF model, the total water spillage of DR-DOPF decreases considerably, with a decrease ratio of more than 86%, and the comprehensive cost decreases by more than 12%, which indicates that this model can effectively reduce the total water spillage and comprehensive generation cost. This analysis confirms the previous conclusion that the joint optimal regulation of hydropower and thermal power can produce higher comprehensive economic benefits compared to regulation mainly relying on hydropower.
Table 3 summarizes the test results of DR-DOPF for various sample sizes. The Wasserstein radius exhibits a negative correlation with sample size, because when more sample data are used, more sufficient information can be obtained about the real probability distribution. Therefore, impossible distributions can be excluded from the Wasserstein ball, resulting in a narrower radius range, less conservativeness of the ambiguity set, and a lower comprehensive cost and water spillage. It can be seen that when the sample size increases from 20 to 5000, the comprehensive power generation cost of the system reduces from 2.613 × 10
5 USD to 2.409 × 10
5 USD, a reduction of 7.8%; and total water spillage reduces from 9.458 × 10
5 m
3 to 9.201 × 10
5 m
3, a reduction of 2.7%. This result verifies that with the increase in the sample size, the conservatism of the model lessens and the economy of the model improves. In addition, it should be noted that the number of constraints and variables of DR-DOPF do not increase with the sample size, and the operation time basically remains stable from 28–30 s, which indicates that the DR-DOPF model achieves good computational performance.