**1. Introduction**

Rivers are the most important life support system for mankind—they are the key to maintaining the material and energy cycle of a region or river basin. On the other hand, the hydrological situation of rivers affects all aspects of natural ecosystems. Good hydrological conditions are important for maintaining the health of rivers and maintaining aquatic life. Ecosystem integrity plays a very important role. As the mother river of China, the Yellow River is the largest water supply source

in Northwest China and North China. It undertakes water supply tasks for 15% of the country's cultivated land diversion irrigation, 12% of the population, and more than 50 large and medium-sized cities. As we all know, the Yellow River is the river with the highest sand content in the world. Its largest sediment is located in the middle reaches and flows through the Loess Plateau. The large amount of sediment carried in the Yellow River caused great damage to the national economy and ecological environment. Thus, there is an urgent need to mitigate this problem by means of reservoir management. The construction of cascade reservoirs in the Yellow River basin has made great economic benefits, resulting in serious ecological problems at the same time. The ecological problems in the middle and lower reaches of the Yellow River are mainly reflected in the following aspects: -<sup>1</sup> The reservoir operation has a great effect on the morphological changes of the river. The water and sediment regulation of the Xiaolangdi dam is beneficial to river erosion, but the erosion amplitudes are becoming smaller [1–3]. It is predicted that the river downstream of Xiaolangdi will quickly be desilted after 2020, and the amount of desilted river will reach the pre-reservoir level by around 2028. By then, the Zhongshui River Channel will be difficult to maintain, and the ecological flow in some sections will be destroyed [4,5]. -<sup>2</sup> The seasonal basic ecological flow in the river channel cannot be guaranteed, and the species diversity in the river channel is severely damaged [6,7]. -<sup>3</sup> The water quality of the downstream river water deteriorates [8], the oxygen content of the water body decreases, and BOD, COD, and other indicators exceed the normal threshold, resulting in the self-healing and dirt-holding capacity being significantly reduced.

The multi-objective optimal dispatching of reservoirs has become a hot spot in the research of reservoir operation in China. In the past, the study on the optimal operation of reservoirs has been carried out by power generation [9], flood control [10,11], irrigation [12], and other single target to maximize the reservoir economic benefits. With the rapid development of social economy, the reservoir urgently needs to improve the utilization of water and play its comprehensive benefit. The operation models and the optimization algorithms from single target [13,14] to multi-objective [15–26] have become the future overall trend of reservoir optimization. The operation model was based on a single-objective model of power generation originally. Robin W. [27] established a single-objective model of power generation and constructed a reservoir [28] that established an optimal power generation operation model, which is simple and easy to implement and provides an effective method for the optimal operation of cascade hydropower stations. In today's world, the reservoir has to undertake more and more tasks, so the reservoir began to use the multi-objective model to guide the actual operation. Authors such as Huang Cao [29] established a power generation, flood control, and ecological multi-objective joint operation model—a model that can better reflect the interrelationships and changes between these objectives. Yang Na [30] established a multi-objective reservoir operation model that takes into account the profiting requirements of a reservoir and the ecological requirements of a river. The optimization results show that the scheduling method of preference for eco-environmental targets is more favorable for the implementation of the water diversion project of the South-to-North Water Transfer Project. Jin Xin [31] constructed a multi-objective ecological operation model for water supply reservoirs, which can provide theoretical support for the ecological operation decision of the northern reservoir for water supply. In 1962, domestic scholars began to apply a genetic algorithm as a classical single-objective solution algorithm [32] to reservoir operation. At the beginning of this century, Chang Jianxia [33] coded an algorithm in decimal mode that is now widely used in reservoir operation. The genetic algorithm can also transform the multi-objective into a single-objective solution by constraint processing [34]. However, with the emergence of dimensionality, the genetic algorithm (GA) also derives the evolutionary algorithm for solving multi-objective models, which is the non-dominated sorting genetic algorithm (NSGA-II) [35–39]. NSGA-II is an improvement on the basis of GA. At the beginning of the twenty-first century, a multi-objective evolutionary algorithm is applied to the field of reservoir operation [40], and Yuan Ruan [41] use NSGA-II to solve multi-objective optimization scheduling model. This example analysis shows that the algorithm has a good accuracy in solving the multi-objective problem. Zhu Jie [42] uses improved NSGA-II to obtain

a better dispatching scheme for the Zhanghe reservoir. NSGA-II has shown its superiority in the application of multi-objective reservoir scheduling.

However, the existing NSGA-II stochastic search algorithm still has some shortcomings, such as the instability of results and a long computation time. At the same time, in the current research on the ecological regulation of reservoirs, the ecological demand was mostly a fixed process [43–49], or even a fixed value, and there was no distinction between different incoming water years, which is not a comprehensive consideration of the ecological needs of the basin. In view of this, this paper set different ecological needs in each typical year and conducted both a single- and multi-objective optimization scheduling of the Xiaolangdi reservoir, and studied the impact of different ecological objectives on the operation mode of the reservoir. The used genetic algorithm and improved NSGA-II algorithm based on constraint processing and the spatial optimization technology-multi-objective model to obtain the single-objective optimal solution and its distribution characteristics in the multi-objective Pareto-front optimal solution set, revealing the multi-objective scheduling mutual conversion rules between targets. This paper chooses index factors to quantify the ecological and mutual feedback conversion relationship between power generation targets and finally recommends the use of a global optimal balance solution in a multi-target solution set to minimize the impact of the reservoir on the surrounding environment adverse effects [50]. The research results provide a decision-making basis for the dispatchers of the Xiaolangdi reservoir and the Xixiayuan reservoir. This paper has important practical significance and application value for improving the ecological environment of the lower Yellow River and improving the comprehensive utilization of reservoirs.

#### **2. Research Area and Data**

The Yellow River originated from Maqu in the Yoguzonglie basin at the northern foot of the Bayankala Mountains on the Qinghai-Tibet Plateau, and finally flowed into the Bohai Sea in Shandong Province. The basin area is 795,000 km2, and the total length of the main stream is 5464 km, which is the second longest river in China. The Xiaolangdi Reservoir has great regulation and storage capacity, and plays an important role in ecological protection of the Yellow River and sediment control in the river. This paper selects Xiaolangdi as the research object, whose location is on the main stream of the Yellow River, north of Luoyang. The location overview shown is in Figure 1. The main parameters of the Xiaolangdi reservoir are shown in Table 1.

**Figure 1.** Location overview of Xiaolangdi.


**Table 1.** Main parameters of Xiaolangdi.

Long series runoff collected during the historical period from 1961 to 2009 (hydrological year) of the Xiaolangdi Reservoir was used as input data, and the data was input according to the reservoir optimization model. The monthly average natural runoff of the Xiaolangdi Long-series is shown in Figure 2.

**Figure 2.** Monthly average runoff process of the Xiaolangdi reservoir.

This article integrates the collected industrial, agricultural, domestic, and ecological water demand processes downstream of the Xiaolangdi reservoir to obtain the comprehensive water demand process downstream of the Xiaolangdi reservoir, and uses it as a constraint for optimal scheduling research. The comprehensive water demand process is shown in Figure 3.

**Figure 3.** Integrated water requirement process in the lower reaches of Xiaolangdi.

In Figure 3, the minimum ecological water requirement is the flow process needed to maintain the downstream ecosystem without degradation during the minimum water requirement. The suitable ecological water requirement during the process of water demand is the flow process to maintain the suitable habitat of the lower reaches and ensure the normal survival and reproduction of the downstream species; the maximum ecological water requirement is the flow process to maintain the balance of river scouring and deposition and to restore the capacity of river pollution.

#### **3. Modeling**

Model 1: Maximize power generation

$$\text{MaxE} = \sum\_{t=1}^{T} kQ\_{\mathbb{S}}(t)H(t) \times \Delta t \tag{1}$$

where *E* is the total power generation of hydropower stations during dispatching period, and *t* and *T* are the time serial numbers and the total number of periods, respectively. In addition, *k* is the comprehensive output coefficient for power station; *Qo*(*t*) and *H*(*t*) are the power discharge and water head in the *t*-th period of the power station, respectively.

Model 2: Maximize power generation while meeting the ecological flow requirement

The objective function and constraints are the same as Model 1. In addition, the ecological flow requirement needs to be considered:

$$Q\_d(t) \le Q\_o(t) \tag{2}$$

where *Qd*(*t*) is the ecological flow requirement of the downstream channel during the *t*-th period. Model 3: Multi-objective optimal operation considering both ecology and power generation

$$\text{Cov}\,b\_1\text{:}\qquad\text{Max}\,\text{E}=\sum\_{t=1}^{T}k\mathbb{Q}\_o(t)H(t)\times\Delta t\tag{3}$$

$$Obj\_2 \colon \qquad MinW = \sum\_{t=1}^{T} Q\_s(t) \times \Delta t \tag{4}$$

where *W* and *Qs*(*t*) are the total water shortage in dispatching periods and water shortage flow in the *t*-th period. The constraints are the same as Model 1.

The constraints for models one to three are as follows:

(1) water balance constraints

$$V(i, t) = V(i, t - 1) + [Ql(i, t - 1) - QO(i, t - 1)] \times \Delta t \tag{5}$$

where *V*(*i*, *t*) and *V*(*i*, *t* − 1) are the initial storages of the *i*th reservoir at times *t* and *t* − 1, respectively. *QI*(*i*, *t* − 1) and *QO*(*i*, *t* − 1) are the inflow and outflow of the *i*th reservoir at time *t* − 1, respectively. *t* is the time interval.

(2) Outflow constraints

$$
\mathcal{Q}O\_{\text{min}}(i, t) \le \mathcal{Q}O(i, t) \le \mathcal{Q}O\_{\text{min}}(i, t) \tag{6}
$$

$$QO(i, t) = QI(i, t) - \left[V(i, t+1) - V(i, t)\right] / \Delta t \tag{7}$$

where *QO*min(*i*,*t*) and *QO*min(*i*,*t*) are the minimum and maximum allowable outflows of the *i*th reservoir at time *t*, respectively.*QO*(*i*,*t*) and *QI*(*i*,*t*) are the outflow and inflow of the *i*th reservoir at time *t*, respectively. *V*(*i*, *t* + 1) and *V*(*i*,*t*) are the initial and final storages of the *i*th reservoir at times *t* + 1 and *t*, respectively.

(3) water level constraints

$$Z\_{\min}(i, t) \le Z(i, t) \le Z\_{\max}(i, t) \tag{8}$$

where *Z*min and *Z*max are the minimum and the maximum water levels of the *i*th reservoir at time *t*, respectively.

(4) Hydropower outputs constraints

$$N\_{\min}(i, t) \le N(i, t) \le N\_{\max}(i, t) \tag{9}$$

where *N*min(*i*, *t*) and *N*max(*i*, *t*) are the minimum and maximum hydropower outputs of the *i*th reservoir at time *t*, respectively. In general, *N*min is the guaranteed output and *N*max is the installed capacity.

#### **4. Methodology**

#### *4.1. Single-Objective Solution*

Genetic algorithm (GA) is a common method for solving the single-objective optimal dispatching model [51]. This article intends to use the basic genetic algorithm, such as references, to solve the single-objective Model 1 and Model 2.

#### *4.2. Multi-Objective Solution*

There are many methods to solve the multi-objective optimization model of reservoir, such as the constraint method, the weight coefficient method, the multi-objective evolutionary algorithm, and so on. The constraint method is to transform the multi objective into a single objective, and the weight coefficient method is to set the weight of each target and transform it into a single objective. Indeed, the two objective problems are transformed into a single-objective solution, while it is impossible to reveal the transformation rules among the targets from the global equilibrium solution to clarify the sensitive relationship among the targets. In this multi-objective optimal dispatching model, which considers both ecology and power generation, the key of each dispatching objective is the process of outflow. Specifically, the expected discharge process of power generation target has strong pulse in non-flood season, reducing the outflow volume, to obtain high water level and maximum power generation; the expected outflow process in Model 3 is similar to ecological water demand, and the pulse is weakened. In the non-flood season, the steady outflow of the reservoir can be maintained to meet the gentle ecological base flow of the lower reaches, so that the difference between them is reduced, and the minimum amount of water shortage is obtained. Hence, one can see that the ecology and power generation objectives are contradictory and counter-productive contradictions. Thus, this paper intends to use the non-dominated ranking genetic algorithm (NSGA-II), which is based on constraint processing to optimize the feasible search space, and treats each dispatching objective equally to make it survive the competition in the process of optimization so as to obtain the global equilibrium solution.

In order to improve the accuracy and efficiency of the NSGA-II, this paper optimizes the NSGA-II by optimizing the feasible search space. The reservoir multi-objective optimization scheduling model has a highly nonlinear characteristic, and different treatment methods should be adopted for different constraints. In this paper, the constraints are divided into the convertible constraints and the non-convertible constraints, in which the convertible constraints are used to optimize the initial search space. The convertible constraints are water level constraints, discharge constraints, ecological water demand constraints. The non-convertible constraints include flow balance constraints, water balance constraints, and power output constraints.

The specific feasible search space optimization steps are as follows:

*Step*1: The minimum and maximum water level of the reservoir form the initial search space, as shown in Figure 1;

*Step*2: In the convertible constraints, the constraint of reservoir capacity and the flow constraint of reservoir outflow are converted to the upper and lower limits of the water level, and it is intersected with the water level constraints of the reservoir in order to eliminate the search space that does not satisfy the convertible constraint conditions.

*Step*3: The initial population of each target is generated in the feasible search space, and the optimization is finished by the constraint processing and the search space optimization.

The optimized feasible search space is shown in Figure 4. Among them, Z represents the water level, the unit is meter, T represents time in seconds. Z(T) represents the water level of the reservoir during the T period, Z<sup>0</sup> represents the initial water level of the reservoir, Zmax represents the highest water level, and Zmin represents the lowest water level.

**Figure 4.** Optimization of the feasible search space.

The optimization of feasible search space: the infeasible solution space is removed from the initial search space, which, in a sense, not only does not reduce the diversity of the population, but also improves the quality of initial population, which makes easier to converge.

When the initial population is optimally calculated, the stage solutions of all the generations of each group can meet the transformable constraints, only to determine the non-transitive constraints. When the non-transformable constraint is not satisfied, the penalty function will be adopted to reduce the fitness function value, thereby reducing the chance that the individual will inherit to the next generation and ensure the optimal path for the optimization. The optimized NSGA-II, on the one hand, optimizes the initial search space to the feasible search space, eliminates the infeasible search space, reduces the search range of the global equilibrium solution, and improves the optimization speed; on the other hand, all the solutions of all generations in the optimal search satisfy the convertible constraints, and only need to determine whether the non-convertible constraints are satisfied or not, thus avoiding the cumbersome and redundant judgment of all the constraints. After the NSGA-II is optimized, it not only avoids falling into the local optimal solution, but also accelerates the convergence rate and improves the search efficiency and the precision of the algorithm. The operation flow chart of the NSGA-II in which the feasible search space was optimized is shown as Figure 5.

**Figure 5.** The operation flow chart of the non-dominated genetic algorithm (NSGA-II).
