*Article* **Copula-Based Research on the Multi-Objective Competition Mechanism in Cascade Reservoirs Optimal Operation**

**Menglong Zhao 1, Shengzhi Huang 1,\*, Qiang Huang 1, Hao Wang 2, Guoyong Leng 3, Siyuan Liu <sup>1</sup> and Lu Wang <sup>1</sup>**


Received: 10 April 2019; Accepted: 9 May 2019; Published: 12 May 2019

**Abstract:** Water resources systems are often characterized by multiple objectives. Typically, there is no single optimal solution which can simultaneously satisfy all the objectives but rather a set of technologically efficient non-inferior or Pareto optimal solutions exists. Another point regarding multi-objective optimization is that interdependence and contradictions are common among one or more objectives. Therefore, understanding the competition mechanism of the multiple objectives plays a significant role in achieving an optimal solution. This study examines cascade reservoirs in the Heihe River Basin of China, with a focus on exploring the multi-objective competition mechanism among irrigation water shortage, ecological water shortage and the power generation of cascade hydropower stations. Our results can be summarized as follows: (1) the three-dimensional and two-dimensional spatial distributions of a Pareto set reveal that these three objectives, that is, irrigation water shortage, ecological water shortage and power generation of cascade hydropower stations cannot reach the theoretical optimal solution at the same time, implying the existence of mutual restrictions; (2) to avoid subjectivity in choosing limited representative solutions from the Pareto set, the long series of non-inferior solutions are adopted to study the competition mechanism. The premise of sufficient optimization suggests a macro-rule of 'one falls and another rises,' that is, when one objective value is inferior, the other two objectives show stronger and superior correlation; (3) the joint copula function of two variables is firstly employed to explore the multi-objective competition mechanism in this study. It is found that the competition between power generation and the other objectives is minimal. Furthermore, the recommended annual average water shortage are 1492 <sup>×</sup> <sup>10</sup><sup>4</sup> m3 for irrigation and <sup>4951</sup> <sup>×</sup> 104 <sup>m</sup><sup>3</sup> for ecological, respectively. This study is expected to provide a foundation for selective preference of a Pareto set and insights for other multi-objective research.

**Keywords:** multi-objective competition mechanism; cascade reservoirs operation; copula function; Pareto set

#### **1. Introduction**

Real-world systems always refer to multiple objective optimization in their operations. Multi-objective optimization problems (MOPs) usually require the simultaneous optimization of some incommensurable and competitive objectives. A reservoir system, for example, serves various purposes and involves multi-objective decision-making in the implementation process. With continuous development and utilization of water resources and hydropower resources and the expansion of

management and the protection objectives of natural resources, watershed development has shifted from a traditional single task such as power generation and flood control to comprehensive utilization taking account of the ecological environment. While on the basis of the total amount of water resources availability, there are often contradictions among the development goals in the basin due to unreasonable utilization [1] such as deterioration of the ecological environment caused by the increase of irrigation water usage and large power loss as a result of large discharge in the flooding season and insufficient water after the flooding season.

The proper operation of cascade reservoirs is a multi-stage, nonlinear, high-dimensional and strictly controlled optimization process pertinent to the planning and management of water resources [2–5]. From the optimization direction, the optimization methods could be categorized as gradient search, direct search and meta-heuristic search [6]. From the development process, algorithms utilized to achieve optimal scheme of reservoir operation could be divided into classic methods and intelligence optimization algorithms. Traditionally, many researchers have adopted classical techniques such as linear programming, nonlinear programming and dynamic programming to deal with the MOPs, either a weighted or a constrained approach, without considering all objectives simultaneously [7–9]. The classic methods are relatively simple but are limited by the possibilities such as not attaining global optima, convergence to local optima and being hampered by dimensionality. Natural phenomena drive the development of the latter intelligence optimization methods. One of the strengths of intelligence optimization algorithms is their convergence to be virtually global optimal for any well-defined optimization problem [10–12]. In recent years, many intelligent optimization algorithms and improved algorithms, such as Evolutionary Algorithms (EA), Artificial Neutral Network (ANN), Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), Particle Swarm Optimization (PSO) and Cuckoo Search (CS), have been proved to be useful in the multi-reservoir systems optimization for maximizing benefits of multi-objectives [13–18]. This study employs improved NSGA-II to solve the MOPs of reservoir optimal dispatching.

As the MOPs usually contain multiple incommensurable and competing objectives, it is impossible to find a single optimal solution to satisfy all the targets. Instead, the solution exists in the form of alternative trade-offs, known as the Pareto optimal solutions [19]. There is a contradiction between objectives of every Pareto optimal solution, identified as universality and particularity. To better understand the evolution process of Pareto optimality and choose the optimum, it is necessary to explore the competition mechanism among multiple objectives and the interactions between two or three objectives.

Currently, most of studies on the competition mechanism between multiple objectives always focus on the objectives of several typical schemes which are unavoidably subjective, while few studies address the competition rules among multi-objectives (≥3) of long series to reflect the statistical law of each objective [20–22]. Copulas [23] are known as an effective means for describing the dependence between random variables, thus expected to be suitable for studying competition or dependence among multi-objectives. Recently, different copulas have been employed for the multivariate analysis of spatiotemporal change in probabilistic forecasting of seasonal droughts [24], multivariate real-time droughts assessment [25], joint return periods of precipitation and temperature extremes [26], flood frequency analysis [27,28], risk analysis [29], energy environmental optimization [30–32], stochastic hydrological simulation [33] and so on, while its application in the research field of multi-objective competition relationship has not yet emerged. A key feature of copula is to characterize the dependency structure of two or more variables, either cross-correlation or auto-correlation [34], making it a promising method for the multi-objective questions. In this study, copula function is chosen to construct joint sequence values of two targets. This is the first time that copula is applied to explore multi-objective competition mechanism whereby the overall impact of one objective on the other two objectives can be evaluated.

The Heihe River Basin, as the second largest inland river basin of China, is the study area. The primary goals of this study are: (1) to analyze the three-dimensional and two-dimensional spatial

distributions of the Pareto set obtained on account of reservoir dispatching obtained in the preliminary work; (2) to establish a formula for quantitatively describing the relationship between two objectives and to inform the law of water use; and (3) to explore the multi-objective competition mechanism for the overall impact of one objective on the other two objectives with the copula function constructing the joint sequence of two targets. The goal of this study is to reveal the multi-objective competitive mechanism and alleviate the competitive nature of water usage. It also provides guidance for the unified allocation of water resources and provides insights for other multi-objective problems.

#### **2. Study Area and Data**

As the second largest inland river basin of China, Heihe River Basin is situated within 98◦ E~102◦ E and 37.5◦ N~42.4◦ N, with an area of about 134,000 km2. With an average annual precipitation of 400 mm and an average annual potential evaporation of 1600 mm, the basin lays in the interior of the Eurasian continent and characterized by arid hydrological features. The predominant land use types are desert and grass lands, accounting for roughly 60% and 25% of the total area, respectively [35]. Under water and ecological stresses, the Heihe River Basin suffers water-table decline, terminal lakes dryness, grassland degeneration as well as widespread desertification largely owing to the combined impacts of climate change and human activities [36,37]. Given its key role in water resources planning and management in northwestern China, the Heihe River Basin has long been a focus of studies on inland rivers in arid regions and it is therefore selected for this case study.

The main stream of Heihe River is about 928 km long from the birthplace of the Qilian Mountains to the tail of Juyan Lake. It is divided into upstream, middle and downstream areas by the Yingluo Gorge and the Zhengyi Gorge. The upstream is the main water production area. The middle is the main irrigation water area. The downstream is the main ecological water consumption area (as shown in Figure 1). The first control project in the Heihe river basin is Huangzangsi hydro-junction, covering an area of 7648 km<sup>2</sup> and controlling almost 80% incoming water in the upper reaches of the Heihe River (Figure 1). Huangzangsi is a within-year reservoir which has 4.06 <sup>×</sup> 108 m3 total storage and 3.34 <sup>×</sup> 108 m3 regulating storage, with 6.02 MW guaranteed output and 49 MW installed capacity. Huangzangsi Reservoir and seven run-off hydropower stations, which are Baopinghe, Sandaowan, Erlongshan, Dagushan, Xiaogushan, Longshou-II and Longshou-I, combining with the Zhengyixia Reservoir at the end of the middle reaches, to constitute the structure of cascade reservoirs that have '2 reservoirs 7 stations.'

In the previous work, based on the long series of monthly run-off data from July 1957 to June 2014, the model of cascade reservoirs dispatching in the Heihe River Basin has been established and solved, which will be the basis of this study.

**Figure 1.** Location and sections distribution of the Heihe River Basin.

#### **3. Research Methodology**

#### *3.1. Description of Multi-Objective Model of the Heihe Cascade Reservoirs Operation*

#### Objective Function

The comprehensive utilization of water resources in the main stream of Heihe River is mainly embodied by three aspects: power generation in the upper reaches, irrigation in the middle reaches and ecological water use in the lower reaches. Consequently, this configuration has three objectives, namely minimum irrigation water shortage, minimum ecological water shortage and maximum generation capacity of cascade hydropower stations. The allocation of water resources in the Heihe River Basin requires that the 'electricity regulation' complies with the 'water regulation,' that is, the irrigation water and the ecological water need to be given priority while taking into account power generation requirements. With the decision variables of the final water level of the time interval of Huangzangsi Reservoir and Zhengyixia Reservoir, the objective functions are as follows:

Objective 1: Minimum Irrigation Water Shortage

$$\min f\_1 = \sum\_{t=1}^{T} \sum\_{i=1}^{l} \alpha(t, i) \left[ Q\_{irr}^d(t, i) - Q\_{irr}^s(t, i) \right] \Delta t \tag{1}$$

$$\alpha(t,i) = \begin{cases} \ 0 \ Q\_{irr}^d(t,i) \le \mathcal{Q}\_{irr}^s(t,i) \\\ 1 \ \mathcal{Q}\_{irr}^d(t,i) > \mathcal{Q}\_{irr}^s(t,i) \end{cases} \tag{2}$$

Objective 2: Minimum Ecological Water Shortage

$$\min f\_2 = \sum\_{t=1}^{T} \beta(t) \left[ Q\_{\text{acc}}^d(t) - Q\_{\text{acc}}^s(t) \right] \Delta t \tag{3}$$

$$\beta(t) = \begin{cases} \ 0 \ Q\_{\rm coo}^d(t) \le Q\_{\rm coo}^\epsilon(t) \\\ 1 \ Q\_{\rm coo}^d(t) > Q\_{\rm coo}^\epsilon(t) \end{cases} \tag{4}$$

Objective 3: Maximum generation capacity

$$\max f\_3 = \sum\_{t=1}^{T} \sum\_{j=1}^{I} K\_j Q\_{\mathbb{S}^c}(t, j) H(t, j) \Delta t \tag{5}$$

Subject to the following constraints:

$$V(t,j) = V(t-1,j) + [Q\_{\text{in}}(t,j) - Q\_{\text{out}}(t,j)]\Delta t \tag{6}$$

$$Z\_l(j) \le Z(t, j) \le Z\_{\text{h}}(j) \tag{7}$$

$$Q\_{\min}(j) \le Q\_{\text{out}}(t, j) \le Q\_{\max}(j) \tag{8}$$

$$0.6 \times N\_{\mathbb{S}}(j) \le N(t, j) \le N\_{\mathbb{S}}(j) \tag{9}$$

All variables need to satisfy non-negative constraints where, *f*<sup>1</sup> (104 m3) is irrigation water shortage; *<sup>f</sup>*<sup>2</sup> (10<sup>4</sup> <sup>m</sup>3) denotes ecological water shortage; *<sup>f</sup>*<sup>3</sup> (10<sup>8</sup> kW·h) indicates generation capacity of cascade hydropower stations; *t* is time interval number and *T* is total number of time intervals, *T* = 684; *i* is river number and *I* is total river reach number, *I* = 5; *Qd irr*(*t*, *<sup>i</sup>*) and *Qs irr*(*t*, *i*) indicates irrigation water demand and water supply flow of time *t* and river *i* respectively, m3/s; α(*t*, *i*) denotes the irrigation coefficient of time *t* and river *I*; <sup>Δ</sup>*t* is unit time, <sup>Δ</sup>*t* = 2.63 <sup>×</sup> 10<sup>6</sup> s; *Qd eco*(*t*) and *Q<sup>s</sup> eco*(*t*) represents the ecological water demand and water supply flow of time *t* respectively; β(*t*) is the ecological coefficient in time *t*; *j is* hydropower station number and *J* is total number of hydropower stations, *J* = 9; *Kj* is comprehensive output coefficient of station *j*; *Qge*(*t*, *j*) indicates generation flow in time *t* of station *j*; *H*(*t*, *j*) (m) denotes the net water head in time *t* of station *j*; *V*(*t*, *j*) indicates the terminal storage capacity of time *t* and station *j*; *Q*in(*t*, *j*) and *Q*out(*t*, *j*) denote the input and outflow. *Zl*(*j*) and *Zh*(*j*) describe the lowest water level and the highest water level for power generation in Reservoir *j*; *Z*(*t*, *j*) is the final reservoir level of time *t* and station *j*; *Q*min(*j*) and *Q*max(*j*) indicates the minimum discharge and maximum safe discharge of station *j*; *Ng*(*j*) and *Nc*(*j*) are the guarantee output and installed capacity respectively of station *j*; *N*(*t*, *j*) is the output of time *t* and station *j*.

The constraints include: reservoir water balance constraint; reservoir water level constraint; reservoir discharge; hydropower station output; boundary initial conditions; and non-negative variables.

#### *3.2. ICGC-NSGA-II Algorithm*

NSGA-II is recognized as one of the most effective multi-objective optimization algorithms. By non-dominated sorting of multi-objective problems, NSGA-II replaces the traditional multi-objective methods in transforming objective functions (such as weighted or constrained transformation). Its inherent parallelism makes it possible to search multi-objective non-inferior solution sets simultaneously and to deal with non-inferior frontier irregular optimization problems [7,38,39]. Moreover, it can better maintain the diversity of the population and has a strong robustness. However, multi-objective reservoir dispatching is a multi-stage decision-making problem. The value of decision variables in the previous stage usually affects the scope of the feasible region in the next stage. When evolutionary algorithm is adopted to solve the problem, the feasible region takes a small proportion in the search space and the efficiency of the evolutionary algorithm is very low. Reducing the proportion

of infeasible areas in search area is thus one of the most effective methods for improving the efficiency of operation. In this study, ICGC-NSGA-II is used to solve the reservoir dispatching model and the specific procedures can be found in Reference [21].
