Journal of Hydrology 578 (2019) 124134
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Defining the robust operating rule for multi-purpose water reservoirs under deep uncertainties
Kang Rena, Shengzhi Huanga, , Qiang Huanga, Hao Wangb, Guoyong Lengc, Yunchen Wua ⁎
State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an 710048, China State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China c State Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China b
This manuscript was handled by Geoff Syme, Editor-in-Chief
Depending on the concept of optimality, the decision makers responsible for the planning and management of water resources seek to maximize the optimization performance. However, the optimized solutions are vulnerable to failure because decision making in water resources management usually involves factors with deep uncertainties (e.g. runoff conditions, water demand growth, climatic forces, etc.). In this study, we further contribute to the Many-Objective Robust Decision Making framework for defining the robust operating rules of a water supply system of interest. The multi-objective optimization and uncertainty analysis tools were used to reveal the trade-offs between the competing objectives and discover the sensitive factors for the plausible states of the system, respectively. The robust operating rules are demonstrated for the Han to Wei inter-basin water transfer project, which is the most important water diversion project in Shaanxi Province, China. Results show that, although the current operating rules are optimized for the water supply system over the long term, these operating rules cannot deal with the problem of performance degradation under deep uncertainties associated with runoff conditions and water demand growth. The two uncertain factors are the sensitive factors responsible for the failure or success of the system, and robustness of the system may be achieved by reducing the effect of key uncertainties (e.g. reduced water demand growth). Furthermore, there are obvious differences in the robust operating rules across different months, with the key or sensitive months providing the uncertain ranges to likely sustain the success of the system. Finally, the successful frequency of the system derived from the most robust operating rules is 18% higher than that obtained from the current operating rules for alternative states of the world. The robust operating rules offer critical insights into the challenges posed by deep uncertainties and provide a management template for decision making on climate change and complex human activities.
Keywords: Operating rule curve Uncertainty analysis Sensitivity analysis Water supply
1. Introduction Over the past century, a large number of reservoirs and dams have been built for providing water resources, flood control, and power generation. These hydraulic projects establish the water resource base for the development and progress of human society. In developed countries, the social and ecological costs of dam construction are no longer acceptable, since the best reservoirs and dams have already been developed, and the degradation in river ecosystems caused by dams has received increasing attention (Vörösmarty et al., 2010; Ansar et al., 2014; Moran et al., 2018; Zhao et al., 2019). As a result, the construction of large dams in such countries is largely absent, and has been replaced by the question “how can the existing reservoirs and dams be managed more effectively for balancing the competing objectives and mitigating the risks associated ⁎
with uncertainties? (Kasprzyk et al., 2012; Borgomeo et al., 2016; Wild et al., 2018)” In developing countries, especially those in South America, South Asia, and Africa, the development and construction of reservoirs and dams for the future is still at its peak, and managing the existing reservoirs and dams while providing a management strategy for future reservoirs and dams is a key concern for the decision makers (Winemiller et al., 2016; Sabo et al., 2017; Latrubesse et al., 2017). In water resources system analysis, the uncertainties derive from the inability of decision makers to adequately describe the possible states of the world for quantifying the possible risks in the system (Kasprzyk et al., 2012; Huang et al., 2016). Furthermore, water resources management, which is affected by the increase in the uncertainty as a result of climate change and human activities, has become a risk-based decision-making process (Morgan and Mellon, 2011; Brown et al., 2015; Guo et al., 2019). Such
Corresponding author at: State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an 710048, China. E-mail address: [email protected]
https://doi.org/10.1016/j.jhydrol.2019.124134 Received 8 July 2019; Received in revised form 22 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0022-1694/ © 2019 Elsevier B.V. All rights reserved.
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management has become necessary for sustainable development of human society. Optimal operation of reservoirs provides a feasible way to reduce the cost and risk associated with reservoir management and balance the beneficial relationship between the competing objectives (Fayaed, et al., 2013; Giuliani et al., 2014). Despite the progress in the use of system engineering theory and complex non-linear optimization algorithms in reservoir operation, most of these methods remain theoretical and are rarely used in practical application (Maier et al., 2014). The gap between theoretical research and practical application of reservoir optimization operation is obvious, which is why it is difficult for decision makers to use the results of optimization as a guide in practical operation (Labadie, 2004; Brown et al., 2015). On the one hand, because the complexity of the optimization operation model makes it hard for decision makers to understand deeply its principle and usage conditions, it is difficult to devise a reasonable management strategy in practice which is directly employable. On the other hand, the root cause of the gap is that most of the optimal operation processes are modelled with deterministic runoff sequences, which indicate that all the information (observed/ historical information) is known beforehand during the operation period. However, in practical reservoir operation, the future runoff information is completely unknown and cannot be accurately predicted (Herman et al., 2015; Borgomeo et al., 2018). As a result, an operation strategy based on historical information is not available owing to the uncertainty in runoff and other factors (e.g. water demand growth and precipitation variability). Furthermore, in the context of climate change, the stationarity of runoff series has been destroyed, which has led to more severe challenges in devising operation strategies based on historical information (Milly et al., 2015). At present, most reservoirs are operated according to the operation rule curves as part of actual management (You and Cai, 2008). The traditional operation rule curves can be identified according to the basic storage relationship of a reservoir. In recent years, the parametersimulation-optimization approach has offered the possibility of better guidance of reservoir operation (Dariane and Momtahen, 2009). The most representative of the parameter-simulation-optimization methods is the evolutionary multi-objective direct policy search (EMODPS), which combines direct policy search, nonlinear approximating network, and multi-objective evolutionary algorithm to define Pareto-approximate operating rules for multi-purpose reservoir operation (Giuliani et al., 2015a,b). EMODPS is used to improve water reservoir operation through direct use of the hydro-meteorological data (e.g. snow water equivalent, cumulative inflow, and ENSO state) (Denaro et al., 2017; Libisch-Lehner et al., 2019). To mitigate the gap between the theoretical research and practical application of reservoir optimization operation, the parameter-simulation-optimization approach will become the focus of future research. Although the methods based on “making the most of data” can better determine the reservoir operation strategy, they are still affected by the uncertainties in decision-making process (Giuliani et al., 2015b). In the context of uncertainty, decision makers are aiming at realizing multiple performance objectives in water resources management, along with ensuring that the differences between the expected and actual performances are minimized in the possible future states (Kasprzyk et al., 2012; Bhave et al., 2018). Based on these goals, robustness is most widely defined as “the insensitivity of the system design to errors, random or otherwise, in the estimates of those parameters affecting design choice” (Matalas and Fiering, 1977). Although the specific approach is different when applying this concept, decision makers seek alternative strategies for water resources systems to minimize the possibility of an undesirable outcome across future states. These alternatives have been advocated by frameworks recently developed for decision making under uncertainties, including Decision Scaling (Brown et al., 2012), Robust Decision Making (Lempert, 2002), Information-Gap (Ben-Haim, 2004), and Many-Objective Robust Decision Making (MOROD) (Kasprzyk et al., 2013). These frameworks should be applied to reservoir operation to
assess whether the existing operation rules are suitable for uncertain scenarios and to clarify whether they are robust in the expected future states of the world (Herman et al., 2015). Furthermore, existing water projects often involve multiple tasks (e.g. flood control, water supply, power generation, etc.), and the performances objectives of multistakeholder in reservoir management are different according to their preference (Labadie, 2004; Reed et al., 2013; Jiang et al., 2019). To use the existing water-related infrastructures more efficiently and equitably, multi-objective decision making should be employed to coordinate the conflicting relationships between multiple stakeholders. This coordination is reflected in the achievement of a Pareto (trade-off) solution across multiple water utilities. Therefore, this study focuses on the gap between the reservoir optimal operation and practical application, and analyzes the multi-objective characteristic of reservoir operation to mitigate the performance conflicts. Based on above mentioned considerations, we further contribute to the MOROD framework and define the robust operation rule for multi-purpose water reservoir under deep uncertainties. For achieving these objectives, this study evaluated a water supply system and explored the ability of the robust operating rule to deal with deep uncertainties of the water resources system for future planning (2020–2030). The Han to Wei inter-basin water transfer (IBWT) Project, which is a water supply system located in Shaanxi Province of China, was selected for a case study. To reveal the drawback of the current operating rule under deep uncertainties, the pre-optimized operating rules derived by Ming et al. (2017) were evaluated through inputting the uncertainty factors into the simulation model. A multi-objective reservoir optimization model based on the MORDM approach was established to identify the robust operating rule of the water reservoir of interest. Moreover, the scenario and the sensitivity of the solutions of the operation model were analyzed by Patient Rule Induction Method (PRIM). Finally, the robust operating rule of the water supply reservoir was defined by combined analysis of the tradeoffs and sensitivity in multi-objective space. This study establishes the robust operating rule for reservoir operation in practical application, which can better address the performance degradation of water resources system under deep uncertainties. 2. Study area and pre-optimized operating rules 2.1. The Han to Wei IBWT project The Han River, the largest tributary of the Yangtze River, is located in the subtropical monsoon region of southern China. The Han River basin covers an area of 159,000 km2 and receives an average annual precipitation of 800–1200 mm. The abundant rainfall in this basin produces an average annual natural runoff of 56 billion m3, most of which occurs during the flood season from May to October. The runoff also varies greatly from year to year, with Cv values of 0.3–0.5, and the maximum annual runoff is 4 times larger than the minimum annual runoff (Fig. 1c). From the Han River basin to the north through the Qinling Mountains (the North-South boundary of China), the study area reaches the Wei River Basin which is located in the transitional region between a semi-humid region and an arid region. The Wei River basin covers an area of 134,766 km2 and receives an average annual precipitation of 500–800 mm. The rainfall of Wei River is unevenly distributed across the year, with the maximum occurring in July to October. The rainfall of this basin produces an average annual natural runoff of 10.4 billion m3, which is the basis of the water resources of 76 major cities with a total population of 22 million in the Guanzhong Plain which are used for domestic living, industry, and irrigation. With the increasing impact of climate change and human activities (Huang et al., 2014, 2017), the water resources of the Wei River basin are not sufficient to support sustainable socio-economic development of the Guanzhong Plain. Furthermore, the increased water stress of the Wei River basin reflects a broader difference between the North and South of China (Liu et al., 2018; Huang et al., 2019; Fang et al., 2019). Although the Han River has been selected as the water source (i.e. the 2
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Fig. 1. (a) Map of the Wei River basin and the Han River basin; (b) the components of the Han to Wei IBWT project; (c) the annual inflow of the Huangjinxia reservoir and the Sanhekou reservoir. The Danjiangkou reservoir is the main water sources of the middle line of the South-to-North water transfer project.
Danjaingkou reservoir) for the South-to-North water transfer project, a new project (the Han to Wei IBWT project) for water transfer from the southern to the northern region is planned which is expected to solve the increasingly serious problem of water shortage in the Guanzhong Plain (Fig. 1a). The water transfer system of the IBWT project is located in the upper reaches of the Han River basin (Fig. 1b), and contains two reservoirs (the Hunagjinxia reservoir and the Sanhekou reservoir) and related water supply facilities (hydropower stations, pumping stations, and pipelines). These two reservoirs are the core components of the water transfer project, and are mainly used to store water for reducing the temporal deviation in the water supply. The Huangjinxia reservoir, a run-of-river reservoir, is located in the upper reaches of the Han River; it raises the water level so that the pumping station can supply water more easily. The Sanhekou reservoir is a multi-year regulation reservoir which is located in the tributary (the Ziwu River) of the Han River. If there is excess water in the water supply process, this water will be stored in the reservoir for supply during the dry period. The electrical energy consumed by the pumping stations of this project during the water supply process is mainly provided by the Huangjinxia and Sanhekou hydropower stations. The more the water diverted, the more is the electricity consumed. Therefore, the main competing objectives of the project are the reliability of water supply and the net electricity generation of the hydropower stations. The project aims to reduce water shortages in the Guanzhong Plain, and it is planned to diverted 1000 million m3 to the Wei River basin in 2025, increasing to 1500 million m3 in 2030. The diverted water resources are mainly used to supply 22 users, including meeting domestic and industrial demands (Ren et al., 2019). Furthermore, these volumes of diverted water are not defined arbitrarily, and are based on water supply and demand analysis of the benefited areas (in the Wei River basin) over the planning horizon. This analysis takes into account the growth of industrial, agricultural, domestic, ecological, and reused or discharged water to pre-determine the thresholds of water supply.
hydropower generation. For rational utilization of a reservoir, reservoir operating rules are used to determine when and how to operate the reservoir for storage and release over various runoff conditions. At the planning stage of a reservoir, the standard operating rule curves are designed to guide the relationship between storage and release. However, these rule curves may not be applicable in practice, especially when drought occurs, and should be modified or refined (Herman et al., 2014). Considering the shortcomings of the standard operating rule curves for the IBWT project, optimized operating rule curves, which are based on a multi-objective optimization model, were developed by Ming et al. (2017) (Fig. 2). For different annual water demands, the optimized operating rule curves inherit the characteristic curves of the standard operating policy and contain one water transfer rule curve and two hedging rule curves. As shown in Fig. 2, the active storage of the Sanhekou reservoir is divided into different operating zones by the different rule curves. In Zone Ⅰ, when the water level of the Sanhekou reservoir is above the water transfer rule curve, the water demand is fully satisfied by the Sanhekou reservoir, and the pumping station of the Hunagjinxia reservoir does not operate. In Zone Ⅱ, the Huangjinxia reservoir is preferred for meeting the water demand, and the available water is transported to the node (Fig. 1b) through the Hunagjinxia pumping station. If the water demand is greater than the amount of diverted water, the deficit water will be compensated for by the Sanhekou reservoir. Furthermore, if the demand is less than the amount of diverted water, the Sanhekou pumping station will operate to pump the excess water into the Sanhekou reservoir for storage. The water supply for industry and domestic living is not hedged in this Zone. In Zones III and Ⅳ, the basic relationship between the water supply and demand is similar to that in Zone Ⅱ. However, the water supply for the industry is hedged in Zone III, whereas that for both industry and domestic living is hedged in Zone Ⅳ. The rationing factor of the water supply for industry and domestic living is 0.9. These pre-optimized operating rules are based on the long-term historical runoff data obtained from the Changjiang Water Resources Commission for the period 1954 to 2010, and the water demand of the recipient basin is fixed at two standards of 1000 and 1500 million m3 per year. However, the planning horizon of water supply for this project is 10 years (2020–2030), therefore, focusing on the runoff conditions over the 10-year window period is more important to the operation of
2.2. Pre-optimized operating rules of the IBWT project Reservoir operation usually involves the realization of multiple objectives, including flood control, drought management, water supply, and 3
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Fig. 2. Optimized operating rules of the water supply system under different water demand, (a) water supply for 1000 million m3 per year, (b) water supply for 1500 million m3 per year. Table 1 Uncertainty factors and sampling thresholds for robust decision-making. Uncertainty factor*
Climate variability Water demand
Inflow sampled threshold Demand growth multiplier
* The uncertainty factors were considered based on rigorous uncertainty analysis of a previous work which considered 13 uncertainty factors (Herman et al., 2014).
In this current study, runoff conditions and future water demand are considered as the typical uncertain exogenous factors. In order to correlate the optimization model output with the planned horizon of the project, the operation period is set to 10 years (2020–2030). As shown in Fig. 1c, the runoff conditions over the 10-year windows show significant differences, which indicate whether samples were taken from a specific window or from the full sequence, which may not fully represent the possible runoff conditions in the future. Therefore, the runoff uncertainty ensemble was sampled from three 10-year runoff windows, including high, middle, and low windows, through the LHS method (Fig. 3a, b). The sampled runoff sequences increased the frequency and magnitude of the extreme values. However, the sampling is based on a mathematical algorithm with the historical distributions of the runoffs. The magnitude of the extreme value of some sampled runoff sequences may be unreasonable, and can be much larger than that of the historical runoffs. Therefore, multipliers (i.e. the upper bound of 1.2 and the lower bound of 0.8 in Table 1) are used to limit the range of extreme values and further select the sampled runoff sequences. Another uncertainty ensemble which needs to be sampled is the future water demand, which is determined by the capacity of the IBWT project. This uncertainty ensemble was sampled by setting the upper and lower scaling factors of the uncertain dimension. As shown in Fig. 3c, the blue line represents the current baseline demand projection, whereas the projected uncertain range of demand growth is colored according to the values of the scaling factors.
the project. As shown in Fig. 1c, the three 10-year runoffs display low, middle, and high conditions, and the differences between these runoff conditions may be more pronounced in the future owing to the destruction of the runoff stationary as a result of climate change and human activities. Furthermore, the regional water demand is unlikely to remain fixed in the near future. Therefore, this optimization falls far short of the goal of practical application when decision makers deal with the deep uncertainties in the input information of the system. In this study, the uncertainty factors listed in Table1 were considered based on rigorous uncertainty analysis of a previous work which considered 13 uncertainty factors (Herman et al., 2014). 2.3. Sampling uncertainty and testing the pre-optimized operating rules In this section, the possible runoff conditions were sampled with historical runoff data by the Latin Hypercube Sampling (LHS) method, and the uncertain set of water demand growth was established through the scaling factors. These uncertainties include two components (the uncertainties of runoff and water demand) making up the plausible future states of the world, which are used to test the pre-optimized operating rules. 2.3.1. Sampling uncertainty The deep uncertainties mainly result from the incorrect (or inaccurate) projection of the future states of the world. Therefore, decision making based on the status quo or expected future states of the world may fail to account for the risks associated with these uncertainties. Although accurate predictions of future states of the world cannot be made, alternative decision-making options can be provided by tracing the probability distributions of the input parameters to the model output. These input parameters of the model are the deeply uncertain exogenous factors (e.g. future runoff conditions, water demand, population growth, etc.), which can be sampled to create an uncertainty ensemble. Each ensemble member represents a set of exogenous factors for a state of the world in which the future condition has been determined.
2.3.2. Testing of the pre-optimized operating rules Although the pre-optimized operating rules were proposed from an optimization model, the inputs of this model were the deterministic historical runoff data, and the water demand was fixed to 1000 and 1500 million m3 per year over the operation years, respectively. Therefore, when both runoff conditions and water demand are uncertain ensembles, the pre-optimized operating rules are tested for whether they meet the performance requirements of the stokeholds. As 4
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Fig. 3. Sampling deeply uncertain inflow and water demand factors.
mentioned in Section 3.2, four performance metrics were considered, along with the corresponding threshold values. The uncertain ensembles are inputted to the simulation model, and based on the preoptimized operating rules, these performance metrics can be calculated. If all the metrics satisfy the requirements of the stokeholds, the water supply system will be in successful states, otherwise, it is in failed states. Fig. 4 shows the operational consequences of the pre-optimized operating rules for 1000 (Fig. 4a) and 1500 million m3 per year (Fig. 4b). The green points represent the solutions which successfully meet the requirements in the alternative states of the world, whereas the magenta points represent the states of the water supply system in which the solutions fail in this regard. The plotted inflow and water demand factors show a clear separation between the alternative states of the world where the consequences of the solutions are successful and failed. This result indicates that both the uncertain factors are sensitive factors, and the increased water demand and decreased runoffs would prompt the water supply system towards the failed states, and vice versa. Furthermore, Herman et al. (2014) considered a 13-dimensional uncertainty space of an urban water supply case study and revealed that the factors of inflows and demand growth are the most sensitive factors and that the other uncertain factors (e.g. evaporation, water supply allocation, consumer reductions, etc.) do not provide additional information on whether the system succeeds or fails. As shown in Fig. 4a, the successful states of the system are 40% of all the states of the world. By contrast, the proportion of the successful states of the system reported in Fig. 4b is large than 70%. This result indicates that choosing different operating rules has a significant impact on the state of the system under deep uncertainties,
which will confuse decision makers in their decision making. If the inflow conditions and water demand are specific (e.g. inflow condition of a dry year and a water demand of 800 million m3 per year), there may not be a reasonable operating rule which can be selected. Therefore, for defining the robust operating rules, a framework for robust decision making is presented in following sections. 3. Methodology In this section, the methods for defining the robust operating rule are presented by formulating the multi-objective optimization model and application. Furthermore, the robustness measures of solutions and PRIM for scenario discovery and sensitivity analysis are also introduced in detail by specifying them in this study. 3.1. Problem formulation 3.1.1. Optimization operation model As mentioned above, the competing objectives of the IBWT project are the reliability of water supply and the energy generation of the two reservoirs. Therefore, this study established a multi-objective optimization operation model which aimed to minimize the water shortage index of water supply and maximize the net revenue of the hydropower generation. The ecological/environmental objective is considered as a constraint (a fixed eco-flow constraint) in the optimization model. The two objectives are formulated as follow: Objective one: minimize the water shortage index
min WSI =
max(WtD WtS, 0) × 100% WtD
where WSI is the water shortage index (unitless); T is the total number of operation periods (tth month); WtDis the total water demand of the users at the node (m3); WtS is the actual water supply to the users (m3). Objective two: maximize the net revenue
max NR =
t t i Qi Hi
gqit hit i
E1 Nit E2 Pit 10 8
(2) (3) (4)
where NR is the net revenue (yuan); i = 1 denotes the Hunagjinxia reservoir, i = 2 denotes the Sanhekou reservoir; E1is the feed-in tariff, which is set as 0.3 (yuan/kWh), E2 is the electricity price, which is set as 0.5 (yuan/kWh); Nit is the power generation of the hydropower plants
Fig. 4. The solutions derived from the pre-optimized operating rules fail or succeed to meet the performance requirements of stakeholder, (a) pre-optimized operating rule under annual water demand for 1000 million m3, (b) preoptimized operating rule under annual water demand for 1500 million m3. 5
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(kW); Pit is the electrical consumption of the pumping stations (kW); t is the duration of the time step (h); i is the coefficient of the hydropower stations (unitless); Qit is the discharge for power generation (m3s 1); Hit is the hydraulic head for the power generation (m); g is the gravitational acceleration (9.81Nkg 1); qit is the rate of water supply (m3s 1); hit is the delivery head for water supply (m); i is the coefficient of the pumping stations (unitless). In this study, the hydraulic head of the Huangjinxia hydropower station is considered as a constant over the long-term operation period because the Huangjinxia reservoir is a run-of-river reservoir. The constraints of the optimization operation model are the physical and operational constraints, which include mass balance, water diversion capacity, operating rule curves, and power output capacity. The details of the constraints of the optimization model can be found in the paper by Ming et al. (2017).
The regret-based measure R quantifies the performance deviation Di of an objective between the current state of the world and the baseline state of the world. This measure is maximized as follows:
R = max[Di,90 : P (Di i
Di, j =
|F (x )i, j
Di,90) = 0.90]
F (x )i | (6)
F (x )i
where F (x )i denotes the value of objective i in the baseline state of the world; F (x )i, j denotes the value of objective i determined in the current state of the world j; Di, j is the performance deviation of a solution between the current state of the world and the baseline state of the world; Di,90 is 90th percentile performance deviation. The first satisficing-based measure S1 is defined as the fraction of N states of the world in which the performance requirements of decision makers are satisfied by a solution in one or more objectives.
3.1.2. Parameterization-simulation-optimization approach The decision variable of the traditional reservoir optimization model is the storage capacity of the reservoir, which can be iteratively optimized through optimization algorithms (e.g. evolutionary algorithms) (Clarkin et al., 2018). In this study, we consider direct optimization of the operating rule curves, which should be parameterized first. The parameters (i.e. the decision variables) used to represent the operating rule curves are the water levels of the reservoir in different months. There are three operating rule curves which should be optimized and each curve includes 12 months, covering the annual operation period. In order to reduce the variability of the rule curves and the time taken for optimization, the rule curves are classified into four operation periods for one year according to the hydrological conditions. These operation periods are the pre-flood season (from January to March), main flood season (from April to June), post-flood season (from July to October), and dry season (November and December). Each suboperation period is represented using the same parameter so that each operation rule curve can be optimized with four parameters. Therefore, although there are three curves (36 parameters) which need to be optimized, the optimization parameters are reduced to 12 through this classification. In the initial step, these parameters can be randomly generated along with the search range, and then, decision makers operate the water supply system according to the pre-parametric rule curves to simulate the results of water supply and power generation. Finally, the optimization algorithm is used to iteratively improve the results by evaluating the two objective functions and fitness value. In this study, the cuckoo search (CS) algorithm was used to optimize these parameters. The CS algorithm, which is an evolutionary algorithm inspired from the obligate brood parasitic behavior of some birds, was established by the Yang and Deb (2010). The CS algorithm further improves the search efficiency by using the Lévy flight process, which has been demonstrated by previous studies to solve the non-linear and highly complexity optimization problems (Yang and Deb, 2013; Ming et al, 2017, 2018).
N s, j
where, if solution s meets the requirements in the state of the world j, s, j =1, and s, j = 0 otherwise. The most important feature of this measure is that it incorporates the performance requirements of multiple stakeholders. In this case study, although the objectives of the optimization pertain to the water shortage index of the water supply and the net benefits of the hydropower generation, the reliability of the water supply ( Hashimoto et al., 1982) and minimum water supply index are also considered to strictly influence the selection of the solutions. Therefore, the requirements of water shortage index less than 0.5%, net benefits above 0, reliability above 95%, and minimum water supply index above 70% are considered in this study. The second satisficing-based measure S2 is defined as the uncertainty horizon over which the system can withstand failures. This measure is derived from the Info-Gap approach and can be formulated as:
= max[ : min F (x )j j U( )
where is the maximum uncertainty horizon which can be accepted without performance falling below r . In this study, the reliability of the water supply was used to calculate S2, and the threshold value r is 0.95. 3.3. Scenario discovery and sensitivity analysis with PRIM The main purpose of using scenario discovery in water resource system analysis is to identify the regular patterns in the system, which can be used to interpret and predict its characteristics (Bryant and Lempert, 2010; Kasprzyk et al., 2012). Generally, a water resources system involves a lot of input parameters to control its output, and there are differences in the contributions of different parameters to the output. These differences can be identified through sensitivity analysis. Scenario discovery and sensitivity analysis are important for decision making in analytic applications involving water resource systems under deep uncertainties because the deterministic patterns or information can be obtained from the uncertainties. This information or patterns will be useful to decision makers for reducing the complexity in decision making. These two methods are based on statistical or data-mining algorithms, and, fortunately, PRIM can be used to accomplish both tasks at the same time. The following section begins with an introduction of the several core concepts applied in this method. In previous work (Bryant and Lempert, 2010), the regions of input parameters space are characterized as multi-dimensional “boxes” through a specific algorithm. An individual box which is constrained by partial input parameters is called a scenario, and a set of boxes is interpreted as a set of scenarios. The scenario discovery applied here is to seek a box or a set of boxes which exhibits the maximum explanatory power for input-relevant cases.
3.2. Robustness measures of the solutions Using the multi-objective evolutionary algorithm, many sets of Paretoapproximate (non-dominated) solutions can be obtained under many plausible states of the world. The robust solution, which is a solution that provides improved performance across as many states of the world as possible, will be more easily accepted by decision makers. Robustness measures of solutions can be used to quantify and rank which one is the most robust solution of these sets of Pareto-approximate solutions. However, a previous study has shown that selecting different robustness measures will significantly affect the selection of the robust solution (Giuliani and Castelletti, 2016). Therefore, for selecting the robust solution reasonably, one regret-based measure and two satisficing-based measures were considered in this research; they were identified by Lempert and Collins (2007) and further discussed by Herman et al. (2015). 6
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According to previous studies (Bryant and Lempert, 2010; Kasprzyk et al., 2012; Herman et al., 2014), there are three reasonable measures of quality for the purpose of scenario discovery, that is, density, coverage, and interpretability. In order to provide useful information in decision-making, a selected box should capture as many proportions as possible of the total number of input-relevant cases (high coverage), capture primarily input-relevant cases (high density), and display high interpretability for decision making. Coverage is the ratio of the total number of input-relevant cases xi in scenario B to the total number of input-relevant cases X , which can be formulated as follows:
year lag, and the total of 57 years of runoff could be divided into 47 runoff durations. These established runoff conditions and the uncertain ensemble of water demand growth identified in section 2.3.1 are used to define the robust operating rules. Fig. 7 shows the cumulative distributions of the four performance metrics across the alternative states of the world and those calculated by the parameterization-simulation-optimization approach. Each line represents a solution across the alternative states of the world. The Paretoapproximate solutions calculated through multi-objective optimization are shown as dark green lines. The robust solutions, which are selected through the robustness measures, are indicated in blue, red, and green lines. The performance requirements of the stakeholders are represented as black vertical lines. Several salient features can be observed in Fig. 7. First, although the set of Pareto-approximate solutions provides options for decision makers to improve the quality of alternative decisions, these solutions do not reduce the risk of performance loss when the system is across the alternative states of the world (i.e. the deep uncertainties). In the space of the deep uncertainties, the proportions of the performance metrics for the system in the successful states are 50%, 50%, 42%, and 30%, respectively. This result indicates that multiple uncertain factors exhibit a significant impact on the performance of the system and that robustness of the system may be achieved by reducing the effect of the key uncertainties (Herman et al., 2015, see the solid circle in Fig. 4). Second, as demonstrated by Herman et al. (2015), different robustness measures reveal significantly different ranking for the solutions. In this study, although the robust solutions identified based on the robustness metrics show differences in the rankings, these solutions are concentrated in the performance frontiers. This result facilitates the selection of the most robust solutions. Fig. 8 shows the parallel axis plots of the most robust solutions obtained according to the robustness measures. The values of these performance metrics are normalized between their minimum and maximum values so that these metrics can be plotted on a common axis. Each line represents a solution at the corresponding performance values for a specific state of the world. The ideal solutions are represented as red lines, which correspond to the minimum water demand growth and positive runoff conditions. Fig. 8 shows clear conflicts (or trade-offs), especially when seeking to minimize the water shortage index or maximize the net revenue. The attainment of a low water shortage for the water supply in the IBWT project is in conflict with the sustenance of a positive net revenue. The net revenue is more sensitive across the alternative states of the world than over the Pareto-approximate sets (Fig. 7), indicating that water demand and supply is the key factors determining the performance of the IBWT project. Finally, the most robust solutions according to the robustness measures show similar patterns of the performances. For brevity, we chose the robust solution derived from regret-based measure R for further analysis. Fig. 9 shows the box plots of the robust operating rule according to the robustness measure R. For clarity, the water transfer rule curves and the hedging rule curves for industry and domestic living are plotted in subgraphs, which are represented as high, middle, and low curves, respectively. The high curves are distributed between the upper and lower limits of the reservoir's storage capacity, especially the operation rule of the wet season (e.g. July and August), which is distributed over a wider range compared to that of another season. The middle and low curves are distributed over a narrow range, however, the rule curves of the dry season (e.g. November and December) are distributed between the upper and lower limits. This result indicates that there are key months for the operating rules, and, the reasonability of the operating rules may determine the performance of the system over the entire operation period.
y xi B i y xi X i
where yi = 1 if xi ∈ B and yi = 0 otherwise. Density is the ratio of the total number of input-relevant cases xi in a scenario B to the total number of cases x in the scenario:
y xi B i x B
The measure of interpretability is highly subjective because it requires decision makers to select a specific scenario which is based on the characteristics of the problem. Generally, these measures are conflicting, therefore, the ideal set of scenarios (high coverage, high density, and high interpretability) is not observed. For a given dataset, these three measures generally define a multi-dimensional efficiency frontier. Therefore, PRIM is used to generate alternative scenarios at different points of the frontier, and then, users choose the most reasonable scenario based on the preferences of the decision makers. The PRIM is a bump-hunting algorithm which was proposed by Friedman and Fisher (1999). The algorithm presents decision makers with a visualization of the “peeling trajectory” by plotting the density of each box against its coverage. Scenario discovery can be easily achieved using this method through the R toolkit which is available at http://cran.r-project.org/ web/packages/sdtoolkit/index.html. For more details on this method, the reader can refer to the report of Bryant and Lempert (2010). 4. Results and discussion Our framework for defining the robust operating rule, which depends on the MORDM approach, is shown in Fig. 5. Sections 4.1–4.3 detail the results and demonstrate this framework by using the Han to Wei IBWT project test case. 4.1. Selecting the robust solution across the alternative states of the world Each solution across the alternative states of the world can be calculated using the parameterization-simulation-optimization approach described in Section 3.1. Although the runoff uncertain ensemble has been sampled in Section 2.3.1, this ensemble is only used to test the reasonability of the operating rules. Since the water supply system considered in this study has two water sources (namely, the Hunagjinxia reservoir and the Sanhekou reservoir), the positive correlation between their monthly inflows is significant (Fig. 6). Reasonable runoff conditions of these two water sources should consider their synchronous and asynchronous encounter situations, however, this problem is beyond the consideration of this study. The sampled runoff uncertain ensemble regenerated the relationships of magnitude and frequency between the two water sources and ignores this problem. Therefore, this ensemble may only be used to test the operating rules, rather than to identify the robust operating rules. Keeping these points in mind, only the historical runoff combination of the two sources is reasonable. For establishing the alternative runoff conditions, the 57year (1954–2010) runoff time series used to define the pre-optimized operating rules was split into shorter 10-year durations for each of the two water sources. Two adjacent runoff durations maintained a one-
4.2. Scenario discovery and sensitivity analysis There are differences in the response sensitivity of the output of a system to the input parameters of the system, and the purpose of identifying these differences is to find out the key factors affecting the 7
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Fig. 5. The main steps for defining the robust operating rule.
Fig. 6. The correlation between the Huangjinxia reservoir monthly inflow and the Sanhekou reservoir monthly inflow.
performance of the system. As shown in Fig. 4, both the uncertainties of inflow conditions and water demand growth are responsible for the vulnerabilities of the system. However, this result derived from the preoptimized operating rules was evaluated by a qualitative and visual sensitivity analysis. Based on the results of multi-objective optimization, sensitivity analysis can be strengthened by using a formal and quantitative method (PRIM). Therefore, this section first discovered the scenarios and analyzed the most sensitive uncertain factors, and then, explored the relationships between the optimized operating rule curves and the system performance. First, the inputs of the PRIM method require that each runoff condition should be represented by a value, and that the magnitude of the runoff conditions should be equal to the magnitude of each performance metric. However, each runoff condition is a long-term time series, which is difficult to represent by a fixed value. Therefore, for
Fig. 8. Most robust solutions optimized according to the robustness measures across the alternative states of the world.
rationality, we chose a series of characteristic values (e.g. maximum and minimum values, mean and median values, standard deviation, skew coefficient, variation coefficient, interquartile range, etc.) which represent the overall conditions of runoff as the input variables of this method. The water demand increases strictly, no matter which characteristic value can represent its overall state. Fig. 10 shows the ranges of the sensitive factors identified by PRIM which are likely to produce a particular outcome. The x-axis is normalized with the corresponding maximum and minimum values of the sensitive factors. For water demand shortage, the inflow of the Huangjinxia reservoir and the water
Fig. 7. Cumulative distribution of Pareto-approximate solutions across the alternative states of the world. 8
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Fig. 11. The overall measures of selected scenarios with the PRIM approach, (a) high water transfer curve, (b) middle hedging curve, and (c) low hedging curve.
and July of the high rule curves, the April, July, and November of the middle rule curves, and the January of the low rule curves. Furthermore, for the net revenue of hydropower generation, the sensitive operation months are the July and November of the high rule curves, the November of the middle rule curves, and the July and November of the low rule curves. These results indicate that, when the water levels of the reservoir are above the threshold values of the high rule curves, the available water resources of the Sanhekou reservoir are sufficient to meet the water demand without the need for supply from the Huangjinxia reservoir. The high rule curves are in the higher position, and the moments when water supply is hedged occur earlier. This strategy ensures minimum water shortage for long-term water supply. The sensitive months of the middle rule curves are in the wet season, and the middle rule curves are distributed over wide ranges, therefore, the inflows of the two reservoirs are sufficient for water supply. Furthermore, when the water levels are below the low rule curves, both the water supply for industry and domestic living are hedged, which suggest that the water demand cannot be meet indefinitely. Finally, for the net revenue of hydropower generation, the lower position of the high rule curves, the smaller is the power consumed in pumping water. This strategy ensures that positive net revenue can be realized. The sensitive months of the middle and low rule curves are in the wet season, and the available water resources are sufficient. Therefore, their reasonable operating curves should depend on the specific conditions of runoff and water demand.
Fig. 9. Most robust operating rule according to the robustness measure R across the alternative states of the world, where the water transfer rule curves and the hedging rule curves for industry and domestic life are plotted in subgraphs, which are represented with high, middle, and low curves, respectively.
demand growth are identified by the method as the sensitive factors, which suggests that a maximum value of the inflow greater than 1051 m3s 1 and an average water demand less than 111 million m3 per year will most likely ensure success of the system. Furthermore, for the net revenue of hydropower generation, the inflows of the Huangjinxia and Sanhekou reservoirs and the water demand growth are identified by the method as the sensitive factors. Maximum values of the inflows of the two reservoirs greater than 1529 and 214 m3s 1, respectively, and an average water demand less than 67 million m3 per year will most likely ensure the success of the system. Second, Figs. 11 and 12 provide the detailed results of the discovered scenarios, from the optimized operating rules to the system performance. Fig. 11 shows the overall metrics of quality of the selected scenarios. As observed, most of the values of density and coverage are greater than 0.9, therefore, they provide high confidence levels for the selected scenarios. The competitive relationships between them are well balanced, especially for the high rule curves, which are stronger than the middle or low rule curves. However, in Fig. 11b and c, some lower values of density are observed, which indicate that the discovered scenario may not accurately describe the input-output relations of the system in the middle and low hedging curves. Furthermore, scenario discovery of the middle and low operating rule curves is more difficult than that of the high operating rule curves. Fig. 12 shows the most sensitive operation months based on the robust operating rules for the selected scenarios of Fig. 11. For the success of the system, the key water levels of the sensitive months which need to be controlled are displayed above on green bars, and the corresponding values of density and coverage are shown in the right tables. For water demand shortage, the sensitive operation months are the January, April,
4.3. Defining the robust operating rule In this section, the key question raised in the Introduction section is answered, that is, how can the robust operating rule be defined for a water supply reservoir under deep uncertainties? First, the uncertain ensembles sampled in Section 2.3 were used to test the optimized operating rules, which were selected according to the robustness measures of solutions (Section 4.1). Second, based on the performance requirements of the stakeholders, the performance calculated from the optimized operating rules was compared with that derived from the preoptimized operating rules to choose the best-performing solutions. Fig. 13 presents the results of the best-performing solutions and shows the performance space of the maximum success frequency. As shown in Fig. 13a, the positions of the operating rule curves are ranked according to the values of the successful frequency. This result is consistent with the findings reported in Section 4.2, that is, the high rule curves are in the higher position, the moments when the water supply is hedged occur earlier, and the water demand is easier to meet. The green line shows the water transfer rule curve of pre-optimization, which is located in higher positions in the dry seasons and in lower positions in the wet seasons. This rule curve hedges the water supply later in the dry seasons and earlier in the wet seasons, compared to the robust rule curves, which indicate that the water supply during the year is not properly allocated when the system experiences the deep uncertainties. Fig. 13b and c show the middle and low rule curves, which are
Fig. 10. The most sensitive input factors with the sensitivity analysis approach. 9
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Fig. 12. The most sensitive operation months to the robust operating rules according to the selected scenarios of Fig. 11, where the “D” and “C” denote the “Density” and “Coverage”, respectively.
compared with the pre-optimized operating rule curves. Although the positional ranks of these rule curves are not related to the successful frequency, the rule curves of the maximum successful frequency are located in lower positions. Furthermore, these results are consistent with that shown in Fig. 9. There are more outlier values in the middle and low curves, which result to the scenarios cannot be accurately discovered in these curves. Compared with the pre-optimized operating rule curves, the robust operating rule curves delay the moments of restricted water supply and expand the operational range of the water level, which are significant for ensuring the performance of the system.
As shown in Fig. 13d, the successful proportion of the performance space is significantly larger than that derived from the pre-optimized operating rules (Fig. 4b). The maximum frequency of success is about 0.88, which is more than the original successful frequency of roughly 0.18. Finally, when the runoff conditions are in the high range, the system is always in the successful states, indicating that this strategy is a better way of achieving the desired performance of the system than the original operating rules under deep uncertainties. The robust operating rule curves highlight the important role of robustness analysis in identifying the key sources of uncertainties, while
Fig. 13. Comparison of the selected most robust operating rules and the pre-optimized operating rule. 10
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providing the available operating rules for practical applications (Giuliani et al., 2014). Although precise prediction of future water demand growth and runoff conditions is lacking, we demonstrate that the robust operating rule can mitigate the gap between optimization operation and actual reservoir operation. This mitigation is based on the operating rule used in practical reservoir operation, which is optimized by combining the optimization algorithm (e.g. multi-objective evolutionary algorithms) with parameterization-simulation-optimization approach. Furthermore, the framework we proposed for identifying the robust operating rule balances the competing objectives and integrates the modern analysis tools for water resource systems under deep uncertainties.
making the most of data can be integrated into the parameterizationsimulation-optimization framework to better define the robust operating rules, and improve the ability to integrate the concept of deep uncertainty into optimization.
We sincerely appreciate the editor and four anonymous reviewers for their helpful and constructive comments. This research is jointly funded by the National Natural Science Foundation of China (51879213 and 51709221), the National Key Research and Development Program of China (2017YFC0405900), the Planning Project of Science and Technology of Water Resources of Shaanxi (2017slkj-16 and 2017slkj-19), the Key laboratory research projects of the education department of Shaanxi province (17JS104), the Open Research Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin (China Institute of Water Resources and Hydropower Research) (IWHR-SKL-KF201803), and the Belt and Road Special Foundation of the State Key Laboratory of HydrologyWater Resources and Hydraulic Engineering (2018490711).
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments
The deep uncertainties derived from climate variability and human activities will challenge water resources management in the coming decades. This requires advancing the optimization model for complex water resource systems in order to obtain the robust operating rules for an uncertain world. This study contributes to the parameterization-simulation-optimization framework for defining the robust operating rules of water resource systems for the alternative states of the world based on the MORDM framework and multi-objective optimization for better capturing the uncertain factors and their interactions with the water supply system in order to evaluate the performance of the system. The operating rule is parameterized according to the parameters of the operation rule curves, which were simulated for the actual operation process and optimized using the evolutionary algorithms. The application to the case study of the Han to Wei IBWT project allows evaluating the different uncertain factors to identify the robust operating rules for improving the current operating rules. The current operating rule for a water resource system may not address the challenges of the deep uncertainties. The hedging rule curves derived from the parameterization-simulation-optimization framework provide a feasible way to reduce the gap between actual operation and optimizing operation. Results show that the hedging rules are more robust and effective than the pre-optimized operating rules. The successful frequency of the case system based on the hedging rules is 18% higher than that of the pre-optimized operating rules for the plausible future states of the world. The hedging rules used in the actual operation process only require the input information of the current operation period, therefore, they are more operable for decision makers. Moreover, according to sensitivity analysis and scenario discovery approach, the inflow conditions and water demand growth are the sensitive factors governing the success or failure of the system. Robustness of a system may be achieved by reducing the effect of key uncertainties (e.g. reduced the water demand growth). For optimized rule curves, there are differences in the sensitive months of the water transfer rule curves and the hedging rule curves. The several ranges of the sensitive factors which are likely to produce a particular outcome are identified. Finally, compared with the pre-optimized operating rules, the robust operating rules provide a range of operation spaces for better addressing the challenges posed by the deep uncertainty. The framework proposed in this study can provide alternative management strategies for decision makers. When the risk sources are broad or the performance requirements of stakeholders are different, this framework is more robust in resolving risk escalation and performance degradation than the traditional operation strategies. With the increasing impact of climate change and human activities on water resource systems, the management of water resources will involve paying more attention to robust solutions instead of “optimality” under deep uncertainties (Herman et al., 2014). The root cause of uncertainty is the inability to accurately predict the future state of the world. Therefore, the range of alternative states of the world should be appropriately constrained for improving the credibility of decision making and reducing the computational expense. The approach of
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