*5.1. Illustrate the E*ff*ectiveness of the NSGA-II*

The parameters of the genetic algorithm, multi-objective particle swarm optimization (MOPSO), NSGA-II, and optimized NSGA-II are set in Table 2. Verifying the validity and stability of NSGA-II based on constrained processing is a crucial part of optimizing the feasible search space. Since NSGA-II belongs to a posteriori algorithm, the effectiveness of the multi-objective reservoir scheduling is verified by selecting the 1981 data, and the comparison results are shown in Table 3.

**Table 2.** Parameter setting of the genetic algorithm (GA) and the improved NSGA-II.




*Water* **2020** , *12*, 915

In order to determine the accuracy and stability of the GA, MOPSO, NSGA-II, and the improved NSGA-II algorithm, take the result of running five times as an example, in which the different indexes of each algorithm are obtained. The upper and lower limits of the optimal value of power generation and water shortage, the mean value, the standard deviation, and the average running time are shown in Table 3. It can be seen that:

(1) The upper limit of the power generation of optimized NSGA-II is 10.315 billion kW·h. Compared with NSGA-II and MOPSO, however, both of which improved by 1 million kW·h, there is almost no change relative to the GA. The lower limit of the power generation of optimized NSGA-II is the same as that of NSGA-II and MOPSO.

(2) From the view of ecological water shortage, the lower limit of ecological deficiency is consistent. The upper limit of ecological water shortage for optimized NSGA-II is 506 million m3, which is higher than NSGA-II but lower than MOPSO.

(3) From the stability of results, the maximum and minimum standard deviation of ICS optimization results is 0.03, which is lower than the other three algorithms. In addition, the first results of MOPSO significantly premature convergence, the lower limit of the amount of water is non-zero. Meanwhile, the second result of NSGA-II has local convergence, and the power generation does not reach the maximum. Therefore, the results of optimized NSGA-II are relatively stable.

(4) From the optimization time, NSGA-II is longer than GA, and different optimization strategies will make the algorithm optimization time longer. While optimized NSGA-II is shorter than NSGA-II, which reflects that the improved strategy proposed in this paper accelerates the population convergence under the premise of ensuring a certain accuracy. Overall, the optimized NSGA-II is superior to GA, NSGA-II, and MOPSO, indicating that the optimization strategy is effective.

#### *5.2. Analysis of the Impact of Ecological Goals on Dispatching Results*

For the single-objective model with the maximum power generation as the target, a strong ecological constraint is added to form Model 2. The maximum degree of influence of ecological goals on the scheduling results can be found by comparing Model 1 with Model 2. Among the long series of results, the total power generation was 8.759 billion kW · h, which was 0.87% lower than the 8.836 billion kW · h of Model 1. However, the ecological guarantee rate of the Huayuankou section increased by 18.75%, which met the planned 90% guarantee rate. It shows that using ecological goals as a constraint can significantly increase the guarantee rate of ecological water demand, but the power generation benefits are slightly lost.

Model 2 is a single-objective model with maximum power generation after adding strong ecological constraints, and the impact of ecological objectives on the dispatching results can be compared between Model 1 and Model 2. Select 1981–1982, 1960–1961, and 1972–1973 as wet, normal, dry. Among these dates, July–October is the flood season, November–March is the dry season, and April–June is the water-supply period. The process of reservoir water level, reservoir discharge, and the power generation of Model 1 and Model 2 are compared and analyzed, as shown in Figure 6, Figure 7, and Figure 8.

**Figure 6.** Water level change of the Xiaolangdi reservoir in each typical year.

**Figure 7.** Variation of discharge from the Xiaolangdi reservoir in each typical year.

**Figure 8.** Variation of the monthly generating capacity of Xiaolangdi in each typical year.

(1) The water level of each month in Model 2 is basically lower than Model 1, especially in the non-flood period of the wet year, the normal, and the dry year, and the water level of the reservoir is obviously maintained at the low water level. This is because, in order to meet the downstream ecological water storage, the reservoir needs to increase the discharge, which causes the water level to drop. Therefore, considering the ecological constraints, the water level during the water supply period of the Xiaolangdi dam during the dispatching period has been changed.

(2) Restricted by ecological constraints, the monthly outflow of the Xiaolangdi dam in Model 2 basically meets the ecological requirements. In Model 1, the discharge flow of individual months is lower than the ecological water demand and cannot meet the ecological requirements. This is because, in order to maximize the power generation, Model 1 needs to discharge as little and as much water as possible during the storage period and store it at the normal storage level as soon as possible. As a result, the amount of water discharged during the storage period cannot meet the ecological water demand. The destruction of the month occurred in the dry season. The number of damaging months is 2, 4, and 4, respectively, and the degree of damage for the wet year, normal year, and dry year are as follows.

(3) The power generation of Xiaolangdi Hydropower Station shows the same variation in all typical years: the flood season is large, the dry season is small, and the water supply period is large. In Model 2, the power generation in the dry season is obviously larger than Model 1. The main reasons are as follows: In Model 1, the destruction of ecological water demand are all in the dry season. In order to ensure the ecological flow, Model 2 runs at the low water level during the dry year, which increases the discharge and power generation.

Table 4 shows the annual electricity generation and the number of eco-friendly months of different models in each typical year. As can be seen from Table 4, the monthly number of ecological guarantees in each typical year of Model 2 was increased by 2, 4, and 4 months, while the power generation decreased by 0.33, 0.86 and 1.27 billion kW·h. The ecological target has a great impact on power generation during the year and has the greatest impact on power generation in the dry year.

Therefore, adding Model 2 with strong ecological constraints, there is a clear change in the operation process and the magnitude of the reservoir during the dispatch cycle, and Model 2 sacrifices more power generation to complete the ecological target.


**Table 4.** Analysis of power generation and ecological guarantee in each typical year.

#### *5.3. Result Analysis of the Multi-Objective Model*

In this paper, we use the optimized NSGA-II to solve the multi-objective Model 3. The results of the multi-objective optimization scheduling model are plotted as a Pareto-front curve, as shown in Figure 9.

**Figure 9.** Multi-objective Pareto-front curves in each typical year.

In the Pareto-front curve of multi-objective optimal scheduling, the annual variation in the annual water power generation and the comprehensive water shortage in the wet year are 9.564–9.597 billion kW·<sup>h</sup> and 0–0.530 billion m3, respectively; the annual variation in the annual water power generation and the comprehensive water shortage in normal year are 6.707–6.795 billion kW·h and 1.547–3.246 billion m3, respectively; and the annual variation in the annual water power generation and the comprehensive water shortage in dry year are 5.875–6.002 billion kW·h and 0–2.835 billion m3, respectively.

It can be seen that with the decrease in typical annual runoff that the decrease in hydropower generation and the increase in ecological comprehensive water shortage make the contradiction between ecological target and power generation target aggravating. The total water shortage in normal year is larger than that of the dry year, which is mainly caused by the annual runoff process and the integrated water demand process.

Due to the different water requirement processes in typical years, Model 1 and Model 2 are different in Pareto-front curves:

(1) The annual generation capacity of the model 1 is 9.597 billion kW·h, and the comprehensive water shortage is 0.530 billion m3, which is located at the right extreme value of the Pareto-front curve; the annual generation capacity of Model 2 is 9.564 billion kW·h, and the comprehensive water shortage is 0 billion m3, which is located at the left extreme value of the Pareto-front curve.

(2) The annual generation capacity of Model 1 and Model 2 in the normal water year is 6.795 billion kW·h and 6.707 billion kW·h, respectively, and the comprehensive water shortage is 3.246 billion m<sup>3</sup> and 1. 547 million m3, respectively.

(3) The annual generation capacity of Model 1 and Model 2 in the dry water year is 6.002 billion kW·h and 5.875 billion kW·h, respectively, and the comprehensive water shortage is 2.835 billion m<sup>3</sup> and 0 million m3, respectively.

(4) With the increase in the total amount of incoming water, the power generation in the dry year, the normal year, and the wet year increase sequentially, and the water shortage also increases in turn. There is a lot of water in the normal year, but it is restricted by the constraints of poor distribution

in the water year, and the ecological water demand is greater than in the dry year, which causes the maximum water shortage to increase.

It can be seen that the single-objective calculation results fall on the Pareto-front curve of the global equilibrium solution. With the decrease in incoming water, the power generation decreased drastically, the water deficit increased significantly, and the contradiction between the target of ecological comprehensive water demand and the power generation goal was aggravated.

On the Pareto-front curve of the multi-objective global equilibrium solution, we select the power generation index, and generate Schemes 1 to 5 shown from small to large (the position is shown in Figure 9). The outflow of Xiaolangdi in each typical year is shown in Figures 10–12. The scheduling results of each scheme are shown in Table 5.

**Figure 10.** Comparative analysis of the outflow of Xiaolangdi in the wet year.

**Figure 11.** Comparative analysis of the outflow of Xiaolangdi in the normal year.

**Figure 12.** Comparative analysis of the outflow of Xiaolangdi in the dry year.


**Table 5.** Comparative analysis of various schemes in typical years.

Table 5 shows that in wet year, the multi-objective scheme can meet the ecological requirements under the condition of minimum power generation. From Schemes 1–5, the power generation increased from 9.565 billion kW·h to 9.595 billion kW·h, and the outflow decreased during the dry year. From the ecological destruction duration, the monthly number of guarantees is reduced from 12 months to 9 months; from the depth of ecological destruction, the ecological water shortage increased from 0 to 0.462 billion m3, and Schemes 2–5 failed to meet the ecological requirements.

Compared with the wet year, the power generation Schemes 1–5 consist of 6.709–5 kW·h increased to 6.792 billion kW·h, from the diachronic perspective of ecological destruction, which dropped from 11 months to 8 months. From the depth of ecological damage, the ecological water shortage of 1.601 billion m<sup>3</sup> increased to 2.957 billion m3.

Like the normal year, the power generation of each scheme is reduced to a minimum value of 5.875 billion kW·h, which can basically guarantee the minimum ecological runoff process in the dry year. The ecological flow of Scheme 2 to Scheme 5 has been destroyed in varying degrees. Under the condition of a continuous decrease in incoming water, the contradiction between power generation target and ecological target is aggravated day by day.

From the above analysis, the following conclusions are drawn:

(1) Due to the difference in incoming water and ecological water demand, the Pareto-front optimal solution set shows the solution spatial distribution and the dimension change of each typical single-multi-objective model. The results of the single-objective Model 1 or Model 2 are basically on the curve, and the results of Model 1 are close to the maximum target of power generation, and Model 2- s results are close to the minimum target of water shortage.

(2) The location of the single-objective result in the curve shows that it has some limitations. The multi-objective model considers the power generation and ecological targets and gives a multi-objective global equilibrium solution set. The multi-objective model has superiority that can fully meet the ecological flow requirements. It provides a multi-objective dispatching solution for decision-makers in reservoir operation and watershed management.

(3) In the multi-objective model, except for Scheme 1, other schemes cannot meet the downstream ecological objective. The safety of ecological flow in the lower reaches of the Yellow River is serious, and as the inflow decreases, the amount of power generation will be correspondingly reduced, the number of ecological assurances and the amount of ecological water deficit will increase, and the ecological targets will be harder to meet. The contradiction between ecological and power generation targets is increasingly aggravated—it is bound to sacrifice the efficiency of power generation in exchange for ecological benefits.

### *5.4. Sensitivity Analysis*

In order to reveal the law of mutual transformation between power generation and ecology, this paper analyzes the sensitivity of various dispatching schemes in each typical year. Using the dimensionless method, three new indicators are creatively proposed: the ratio of power generation to the installed capacity is defined as the coefficient of elasticity (*f* 1); the ratio of the difference in the total water requirement and water deficit to the total water demand is defined as the ecological elastic coefficient (*f* 2), as shown in Table 5.

*Water* **2020**, *12*, 915

The increase in power generation benefit when reducing the 1% ecological benefit is the ratio of ecological- and power-loss benefits (*k*), just like the Formula (4). The greater the *k*, the smaller the impact of ecological benefits on power generation benefits, and the greater the overall benefits of the scheme.

$$k(i) = \left\{f\_1(i) - f\_1(i-1)\right\} / \left\{f\_2(i) - f\_2(i-1)\right\} \tag{10}$$

Based on the Scheme 1, the k of each typical year can be obtained as shown in Figures 12 and 13.

**Figure 13.** Correlation between power generation and ecological elasticity in each typical year.

It can be seen from Figure 14 that the linear fitting relationship between *f* <sup>1</sup> and *f* <sup>2</sup> in each typical year is better, and the linear slopes of the typical years are 15.03, 14.70, and 22.45, respectively, showing a consistent and significant increasing trend. In response to the sensitivity of power generation and ecology, the normal year is the smallest, followed by the wet year and the dry year. That is, with the decrease in inflow, the amount of power generation, eco-elastic coefficient, and the number of ecological guarantee months are all reduced, the ecological water deficit is greatly increased, the restrictive relationship between power generation and ecological targets is strengthened, and the sensitivity between the two is enhanced. The decrease in the ecological elasticity coefficient will make the restoration of ecological environment more severe; the reservoir should take the ecological as the main goal in the dry year and take the phased measures to alleviate the ecological deterioration situation.

**Figure 14.** *k* value of different scheme intervals in each typical year.

It can be seen from Figure 13 that from Scheme 1 to Scheme 5, the k value in each typical year shows a continuous decreasing trend, and the maximum value of *k* in each typical year is 0.082, 0.109, and 0.071, respectively. The corresponding optimal mediation scheme all are Schemes 1–2 intervals. The recommended best coordination solution can maximize the comprehensive benefit.
