Coupling ecological water requirement to optimize water resources under changing climatic conditions


 Meeting ecological water requirements (EWR) is important for guaranteeing watershed system stability in many arid and semi-arid lands. Rainfall–runoff relationships under the effects of climate change may induce many adverse changes for EWR. Inherent uncertainties in water resources management and potential variations in EWR should both be considered to obtain desired water allocation strategies under changing climatic conditions. In this research, an integrated approach was proposed through incorporation of copula functions and a Markov Chain Monte Carlo (MCMC) simulation into a general chance-constrained programming (CCP) modeling framework. The proposed method was found effective for water resource management with respect to the following: (a) tackling correlated features in watershed rainfall and inflow under climate change based on copula–MCMC simulations, (b) obtaining runoff distributions using the copula sampling method under multiple climate change scenarios, (c) analyzing fluctuations of EWR based on varied monthly flows with consideration of diverse runoff distributions, and (d) obtaining desired water allocation strategies through the developed CCP model with consideration of EWR and water shortage risks. Application of the developed method to water resources management in the city of Dalian (China) indicated that the EWR in the watersheds of Dalian would suffer large variations under changing climatic conditions. Moreover, in comparison with the supply in 2025, the increase of water supply from transferring water from the Dahuofang Reservoir (Hun River) would be 6942–33,772, 6942–25,472, and 2849–14,259 Mt with risk tolerance levels of 20%, 50%, and 80%, respectively.


Introduction
Water resources management is regarded as a complicated, adaptive, and integrated decision-making unit for maintaining ecological, economic, and social functions of water systems (Mayer et al., 2014;RazaviToosi and Samani, 2016). Satisfaction of ecological water requirement (EWR) is thus an important way to maintain watershed structure and the corresponding ecosystems in many arid and semi-arid lands (Ling et al., 2016).
However, EWR could be influenced by adverse spatiotemporal variations in water availabilities under combined changes of climate change conditions (Bracken et al., 2018;Giuliani and Castelletti, 2016;Prasad et al., 2015). Many countries are facing increasing challenges in balancing water resources utilization with ecological protections (Benda et al., 2016;Gao et al., 2014). Thus, decision-making approaches to long-term water resources management highlight the importance of nonstationary and uncertain perspectives of EWR and water allocation strategies.
Ecological flow assessment has been applied widely to describe the amount of water flow required to sustain watershed ecosystems and human livelihoods (Acreman, 2016). Zeng et al. (2017) focused on tradeoffs between ecological and irrigative water usages. Zhang et al. (2016a) analyzed multiyear trends of the annual and seasonal changes in ecological flow. Pastor et al. (2014) proved that two methods were effective in analyzing ecological flow (i.e., the variable monthly flow (VMF) and Tessmann methods) for many large-scale  . For example, Veettil and Mishra (2016) incorporated EWR within a framework of water security assessment under climate change impacts. Incorporating Bayesian theory into the framework for EWR analysis, O'Brien et al. (2018) proposed a robust regional-scale approach in order to achieve the desired strategies for ecological protection. It is necessary to tackle the nonstationary frequency of water availability for EWR, especially when influenced by correlated features in rainfall and inflow of a watershed, in conjunction with the subjective decisions of stakeholders in water resources management (Bracken et al., 2018;Chen et al., 2019;Haghighi et al., 2018;Nasr-Azadani et al., 2017;Xue et al., 2016). Previously, statistical methods, such as Monte Carlo simulation, Latin hypercube sampling, and Copula functions, were incorporated to support nonstationary analysis in water resources management (Pal and Talukdar, 2020). For example, Trindade et al. (2019) analyzed variations in long-term water infrastructure investments based on Monte Carlo sampling.
Traditional methods of EWR analysis should also be improved to reflect variations in both water supply and demand, especially under the climate-change background (Gibbs et al., 2018). A water management system (WMS) is generally proposed to provide sustainable strategies regarding adapting to water availability, resolving human vulnerability, and fulfilling ecosystem and human needs with balanced water supply and demand (Cook and Bakker, 2012;Xue et al., 2016). Inherent uncertainties of a WMS and potential interactions with EWR could influence the effectiveness of water allocation strategies (Brookfield and Gnau, 2016;Wang and Huang, 2014). Specifically, given changing 5 environmental, economic, and social conditions, it is difficult to predict definitively the characteristics of future water demand and supply (Brookfield and Gnau, 2016).
Concurrently, integration of future technological innovation could partly offset any increase in water demand (Psomas et al., 2017). It is likely that the additional time and cost requirements of adaptive alternatives would be met to mitigate the uncertain conditions (Brookfield and Gnau, 2016). Although previous research has considered the inherent uncertainties in a WMS, uncertainties of EWR under the background of climate change have rarely been considered in terms of water resources allocation or reservoir operation (He et al., 2018).
The following methods have been proposed previously for analysis of WMS uncertainties: 1) uncertainties of water availability, especially under climate-change effects, were considered based on multiple series of representative scenarios (e.g., Representative Concentration Pathway (RCP) 4.5 and RCP 8.5) (Xie et al., 2017). For example, previous climate research provided multiple scenarios of the relative effects of uncertain climate change on water availability (Gaivoronski et al., 2012;Whateley and Brown, 2016).
Moreover, scenarios of future conditions have been applied widely in uncertainty analysis of socioeconomic and climate change indicators (Huskova et al., 2016;Safavi et al., 2015); 2) Sampling methods (e.g., Monte Carlo, Latin hypercube, and copula sampling) were used to address the randomness of available water resources . Zhang et al. (2017) proposed historical and future operating methods for deriving adaptive operating rules considering both historical information and future projections of a WMS. The varied availability of surface water could be obtained by Monte Carlo 6 sampling in a numerical model ; and 3) Correlated uncertainties could also be determined using statistical methods (e.g., relevance analysis and copula functions) . As flexible statistical approaches, Bayesian and copula methods can capture the joint distributions of correlated parameters at different time scales (Ren et al., 2020;Xu and Valocchi, 2015;Yang et al., 2018). O'Brien et al. (2018) proposed a hybrid approach based on ecological flow and Bayesian theory to support decision-making strategies for adaptation practices. Copula theory has been applied broadly in multivariate frequency analysis to evaluate the characteristics of nonstationarity (Bracken et al., 2018). Nguyen-Huy et al. (2017) analyzed the joint influence of the El Niño-Southern Oscillation and the Interdecadal Pacific Oscillation Tripole Index on seasonal precipitation based on vine copulas. Based on copula theory, Miao et al. (2016) presented comprehensive analysis of the joint probabilistic characteristics of precipitation and temperature on the Loess Plateau of China. Many other joint distributions of drought characteristics in watersheds have been constructed using copula functions (Zhang et al., 2015). Previous, single-or multi-objective optimizing models were applied in water resource management, considering multiple water resources requirements in socio-economic activities (Tarebari et al., 2018). Almazan-Gomez et al. (2018) introduced a water management model to simulate scarcity scenarios and to measure the associated default rates of ecological flow. Hydrological simulations were also integrated into optimizing management models to obtain desired water allocation plans in consideration of watershed characteristics (Lobanova et al., 2017). In order to tackle the issues of 7 stochastic processes in stream flow of WMS, uncertainty analysis methods can also be incorporated within a general framework of optimizing techniques (Wang and Huang, 2014;Zhang et al., 2020). For example, Lv et al. (2018) proposed an interval chance-constrained programming (CCP) model based on a Monte Carlo simulation for regional ecosystem planning under uncertainty. Tan et al. (2016) proposed a robust interval fuzzy programming approach for identifying sustainable agricultural and industrial production strategies at the watershed scale in a highly uncertain environment. Yu et al. (2018) proposed a copula-based stochastic programming method for allocating regional resources that considered random and uncertain parameters in objective function and constraint conditions. Lei et al. (2018) proposed a stochastic optimization model for hydropower generation reservoirs, in which a transition probability matrix was calculated based on copula functions. Furthermore, owing to random variations of parameters, a WMS is challenged by failure of occurrence in water allocation to multiple users (e.g., water shortage risk). Thus, a risk-based balance inexact optimization model is required to reflect the risky conditions in water resources management (Xie et al., 2016).

Optimizing water resources considering ecological water requirements under climate change
In long-term water resources management, decision makers should pay attention to nonstationary and uncertain characteristics of ecological flow and water allocation strategies. The nonstationary features of ecological flow arise from fluctuations in watershed runoff. Previously, watershed runoff, which is correlated with rainfall, has generally been analyzed using rainfall-runoff models (Watson et al., 2019). Traditionally, 9 climate change effects are described into multiple downscaled climate-change scenarios related to future rainfall and temperature. Therefore, watershed runoff analysis must be improved through incorporation of the relationship between runoff and rainfall under various climate change scenarios. Also, methods of EWR analysis should incorporate runoff variations with consideration of the relationship between rainfall and runoff under the effects of climate change. Moreover, the risk attitude of stakeholders to water allocation strategies influences the effectiveness of water allocation strategies. The risk tolerance level (RTL) of stakeholders to water shortages should be considered to obtain

Inflow analysis
As precipitation and runoff are two correlated random variables, copula functions are introduced to indicate the fluctuations of runoff and rainfall influenced by climate change effects (Guo et al., 2017). According to Sklar's theorem (Sklar, 1959;Zhang et al., 2016b), copula functions can be used to describe the joint distributions of correlated random variables. The joint probability density function (PDF) of the two random XY f x y ] can be expressed as Equation 1 (Vergni et al., 2015): where X and Y are two correlated random variables (i.e., precipitation and runoff), FX(x) and FY(y) are their marginal cumulative distribution functions (CDFs), respectively, c() is the fitted copula function, and fX(x) and fY(y) represent the probability density functions (PDFs) of X and Y, respectively. Multiple copula functions have been applied widely in watershed inflow analysis (e.g., Archimedean copulas, and Gumbel-Hougaard copula) (Kong et al., 2015). For example, one-parameter Archimedean copulas have been used widely in hydrologic frequency analysis (Kong et al., 2015). To select the best-fitted copula functions, two indicators (i.e., root mean square error (RMSE) and Nash-Sutcliffe efficiency (NSE)) were used to examine the performance of the copula functions (Equations 2 and 3) (Sadegh et al., 2017).
where the indicators of indicate the extents of variations in observed and predicted values, f % represents the joint probability in observed variables of precipitation and runoff, and f is their joint probability predicted by copula functions.

Ecological water requirement under uncertain conditions
The VMF method can be used to analyze EWR (Pastor et al., 2014;Steffen et al., 2015).
where EW is the minimum amount of ecological water requirement, m indicates MMF, and a is mean annual flow.

Water resources management based on chance-constraint programming
The CCP method is effective in reflecting the reliability of water shortage risk under uncertainty. The CCP method does not require that all constraints be totally satisfied; instead, they can be satisfied in a proportion of cases within given probabilities. A general stochastic linear programming problem can be formulated as shown in equation 5 (Wu et al., 2015): where X is a vector of decision variables, and A(t), B(t), and C(t) are sets with random elements defined on probability space T, where tT  . To solve model (5) When ai are deterministic and bi are random, constraint (5b) can be converted into Model 5 is transformed into the following CCP model (Equation 8): Consider a practical problem of water allocation to multiple water users during a dry season. The manager could present the problem by minimizing allocation costs and fulfilling water demand under random changes of watershed flow. Uncertain information 14 regarding water supply and ecological runoff would occur under the background of climate change. Thus, the CCP model could be adopted as follows (Equation 9). where f is cost of water allocation management, ijk c is per unit cost of water supplied by the i th river for the k th water user of the j th district,

Correlations between runoff and precipitation under climate change
Statistical data related to precipitation and runoff in Dalian were obtained from hydrology yearbooks. Association parameters of precipitation and runoff were estimated based on the copula-MCMC simulation in the framework of the Multivariate Copula Analysis Toolbox (Sadegh et al., 2017). Suitability of the association parameters in the copula functions was analyzed based on the methods of RMSE and NSE. The results of the association parameters of precipitation and runoff are presented in Table 1. As indicated by the indicators of RMSE and NSE, the best-fitted association parameters could be described by the Frank copula function.

Variations in runoff and precipitation under climate change
Incorporating the association parameters between precipitation and runoff, data analysis was conducted based on the copula sampling method. From the correlated sampling data   (Table S3)

Ecological water requirement
The amount of EWR was influenced greatly by variation in runoff velocity under the various climate change scenarios. Fluctuations of EWR are illustrated in Figure 3. It can be seen that the Biliu and Yingna rivers would require more ecological water in comparison with the other rivers in Dalian. As indicated by the CVs in ecological water requirement (Table S4)

Water resources optimization under changing climatic conditions (1) Water resources management model
A water resources optimization model was applied to water allocation for multiple water users (i.e., agricultural sector, industrial sectors, service sector, residents, and other users) in the eight districts of Dalian (Table S5). Dalian's water demands and supplies are listed in Table S6. In consideration of the water allocation project from the Hun River to Dalian, considerable energy would be supplied to the water resources management system in Dalian (Table S7). Thus, energy consumption was chosen as the objective of the water resources optimization model. The model could be presented to find desired water allocation strategies by minimizing allocation energy consumption, guaranteeing ecological runoff, and fulfilling water demands under the background of climate change.
Therefore, model (9) was changed into model (10) Therefore, the current supply capacity of the Songshu and Liuda reservoirs should be increased by 0.99-2.21 and 0.67-2.18 times, respectively, by 2025.

(2) Energy consumption
Under the desired allocation strategies of water resources, the amount of energy 24 consumption in the Dalian WMS is presented in   To obtain efficient strategies regarding water allocation in Dalian, it is crucial to consider the influence of EWR and water shortage risk in water resources optimization.

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