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WARSYP: A robust modeling approach for water resources system planning under uncertainty
Abstract
In this work we present a solution procedure for multiperiod water resources system planning, where the aim is to obtain the optimal policy for water resources utilization under uncertainty. The target levels to be achieved are related to the following parameters: reservoir capacity, hydropower demand and other demand uses for urban, industrial, irrigation, ecological and other purposes. The approach allows for the conjunctive use of surface systems together with groundwater. The hydrological exogenous inflow and demand of different kinds are considered via a scenario analysis scheme due to the uncertainty of the parameters. So, a multistage scenario tree is generated and, through the use of full recourse techniques, an implementable solution is obtained for each scenario group at each stage along the planning horizon. A novel scheme is presented for modeling the constraints to preserve the reserved stored water in (directly and non-directly) upstream reservoirs to satisfy potential future needs in demand centers at given time periods. An algorithmic framework based on augmented Lagrangian decomposition is presented. Computational experience is reported for the deterministic case.
... This circumstance allows to use Stochastic Integer Programming (SIP) for solving multistage mixed 0-1 programs under uncertainty. It has a broad application field, mainly, in production planning [3,4,6], energy generation planning [17,19], finance [14,15], vehicle routing [22] and water resources management [13], among others. ...
... It is represented by a set of scenarios. The methodology can be adapted to the integrated surface-underground system presented in [13]. By considering a splitting variable mathematical representation of the Deterministic Equivalent Model (DEM) and exploiting the structure of the problem, the so-called Branch-and-Fix Coordination (BFC) algorithmic framework can be used. ...
... The water resource system management is a sector where the benefits of being able to solve large scale stochastic problems have a major impact on society at large. It can include a surface subsystem and a groundwater subsystem, see [13]. This paper is devoted to model the minimization of the risk of water usage failure in the surface subsystem. ...
We present a framework for solving the strategic problem of assigning inter- boundary water resources to demand centers under uncertainty in the water ex- ogenous inflow in the reservoirs and other segments of the basin system along the time horizon. The functional to maximize is the probability of satisfying different targets on the stored water and different demands over a set of scenarios. A scenario tree based scheme is used to represent the Deterministic Equivalent Model of the stochastic mixed 0-1 program with complete recourse. The constraints are modelled by a splitting variable representation via scenarios and, so, a Stochastic Integer Pro- gramming scheme can be used to exploit the excess probability functional structure
... This problem is coordinated with the long-term problem [2,3] by means of a future cost function, which represents the cost associated with the use of water resources in the future (long-term planning horizon). Additionally, the MTOP is strongly influenced by the water inflow uncertainty [4], and thus it is a stochastic problem [5]. ...
... Progressive Hedging (PH) is an AL-based method that has shown good performance for very difficult or large-scale problems [9,10]. Although there are applications of this algorithm in some areas (financial applications [10,11] and network flow problems [12,13]), in power system operation, to the best knowledge of the authors, there is only one reference [4]. Given the pioneering characteristic of this reference, room for improvements exists. ...
... Although not often discussed in the literature, studying the algorithm properties and the problem structure to assess the best ways to decompose the original problem can provide more efficient alternatives (regarding the CPU time) for a particular problem. 4 Furthermore, the PH algorithm requires special attention regarding the penalty parameter setting, because it is an AL-based method, which usually requires the use of heuristics 2 [14]. ...
This paper addresses the solution of the Medium-Term Operation Planning (MTOP) problem. This operation scheduling problem aims to define the output of each power plant to minimize the expected production cost over a medium-term planning horizon. In hydrothermal systems, the MTOP is strongly influenced by the amount of water inflow to the reservoirs of hydroelectric plants, which is uncertain. Thus, the System Operator (SO) must consider these uncertainties in the problem resolution, which can be solved by means of Stochastic Programming (SP) techniques, such as the Progressive Hedging (PH) proposed in this paper. This paper presents suitable decomposition schemes to reduce the CPU time, such that it is possible to use a detailed model for the problem. Additionally, a parallel computational approach based on the PH algorithm is also implemented. To demonstrate the efficiency of the proposed schemes, a large hydrothermal system is investigated in the case studies.
... Richardson (2013) provides a review of early work in this area. A key result from these studies is that the net effect of PEVs on power system costs and Edirisinghe et al. (2000), Escudero (2000), Nowak and R € omisch (2000), Philpott et al. (2000), Cai et al. (2001), Pritchard and Zakeri (2003), Lino et al. (2003), Castro and Gonz alez (2004) How does the uncertainty introduced by climate change affect hydropower operations? ...
The electric power industry is undergoing dramatic change driven by rapid technological transformation, concern over climate change, and evolving market structures. As a result, there is a need for new models to help the industry better utilize resources in a time of increasing uncertainty and to help government policy makers better understand the impacts of their regulatory decisions. We provide a structured review of the operations research and management science literatures to describe the current operational and policy issues in the electric power industry, with a particular focus on issues surrounding electricity market design, renewable integration, effects of climate policy on electric power infrastructure, rise of electric powered vehicles, energy storage, and the growing interdependence between natural gas and electric power sectors. We identify the current research frontier and classify the existing research into clusters with respect to managerial issues addressed. We offer a forecast for where the electric power industry is going and describe some important public policy issues. Finally, we highlight research opportunities and discuss how the management science community can contribute by considering the interaction between operational considerations and electric power policy. This article is protected by copyright. All rights reserved.
... Given the many uncertainties in water resources problems, Stochastic Optimization (SO) models have long been used [Hall and Howell, 1970;Escudero, 2000;Higgins et al., 2008;Rosenberg and Lund, 2009]. While there are many SO models and methods, e.g. ...
This paper investigates the application of techniques from distributionally robust optimization (DRO) to water allocation under future uncertainty. Specifically, we look at a rapidly-developing area of Tucson, Arizona. Tucson, like many arid and semi-arid regions around the world, faces considerable uncertainty in its ability to provide water for its citizens in the future. The main sources of uncertainty in the Tucson region include (1) the unpredictable future population growth, (2) the availability of water from the Colorado River in light of competing claims from other states and municipalities, and (3) the effects of climate variability and how this relates to water consumption. This paper presents a new data-driven approach for integrating forecasts for all these sources of uncertainty in a single optimization model for robust and sustainable water allocation. We use this model to analyze the value of constructing additional treatment facilities to reduce future water shortages. The results indicate that DRO can provide water resource managers important insights to minimize their risks and, in revealing critical uncertainties in their systems, plan for the future.
... Although the application of robust optimization is considerable in water resource problems, the probabilistic and non-probabilistic robust optimization should be distinguished. During the past decades, application of probabilistic (scenario-based) robust optimization in water resource management has been extensive (Watkins and McKinney 1997;Escudero 2000;Pallottino et al. 2005;Jia and Culver 2006;Kang and Lansey 2012;Chen et al. 2013). However, its complexity cannot be ignored (Watkins and McKinney 1997). ...
In this study, a new methodology is proposed to balance environmental and economic issues in water allocation under uncertainty. Two objective functions, including maximizing economic income (EI) and minimizing environmental pollution (EP), were considered as two groups of players to construct a deterministic multi-objective bargaining methodology (DMOBM). In the next step, it is enhanced to a robust multi-objective bargaining methodology (RMOBM), which is capable of incorporating the main uncertainties exist in the problem. A large-scale inter-basin water transfer case study was utilized to investigate the applicability of the developed model. The outputs of the models showed that Nash equilibrium provide a rather narrow range of solutions. According to the results, the required rounds to reach Nash equilibrium raised as the uncertainty level increased. In addition, higher levels of uncertainty lead to higher reduction in water allocating of receiving basin. Sensitivity analysis showed that economic income values are less sensitive to changes of uncertain parameters than the environmental objective function. The developed methodology could provide a framework to incorporate the behavior of different stakeholders. Furthermore, the proposed method can be reliable under the condition of facing water allocation uncertainties.
... Therefore, operating the water resource facilities in an efficient manner during drought conditions through the coordinated operation of existing dams is required to minimize drought damage. Some optimizations minimize flow deficits, rather than setting hard constraints [19,20]. Goor et al. [21] revealed that the operation of the coordinated reservoirs enable an average annual saving of at least 2.5 billion m 3 , and would have significant positive impacts on hydropower generation and irrigation. ...
Multifunctional weirs can be used to maintain water supply during dry seasons and to improve downstream water quality during drought conditions through discharge based on retained flux. Sixteen multifunctional weirs were recently constructed in four river systems as part of the Four Rivers Restoration Project. In this study, three multifunctional weirs in the Geum River Basin were investigated to analyze the environmental effects of multifunctional weir operation on downstream flow. To determine seasonal vulnerability to drought, the basin was evaluated using the Palmer Drought Severity Index (PDSI). Furthermore, the downstream flow regime and the effect on water quality improvement of a coordinated dam–multifunctional weir operation controlled by: (a) a rainfall–runoff model; (b) a reservoir optimization model; and (c) a water quality model, were examined. A runoff estimate at each major location in the Geum River Basin was performed using the water quality model, and examined variation in downstream water quality depending on the operational scenario of each irrigation facility such as dams and weirs. Although the water quality was improved by the coordinated operation of the dams and weirs, when the discharged water quality is poor, the downstream water quality is not improved. Therefore, it is necessary to first improve the discharged water quality on the lower Geum River. Improvement of the water quality of main stream in the Geum River is important, but water quality from tributaries should also be improved. By applying the estimated runoff data to the reservoir optimization model, these scenarios will be utilized as basic parameters for assessing the optimal operation of the river.
... Finally, in the presence of sufficient and reliable historical data, stochastic programming approaches are employed to handle uncertainty (e.g. Kapelan et al., 2005;Babayan et al., 2005;Kang and Lansey, 2012;Housh et al., 2013;Pallattino et al., 2005;Escudero, 2000;Huang and Loucks, 2000). Meanwhile, in the cases where various types of data constitute a hybrid uncertain environment, approaches like fuzzy stochastic programming (see Wang and Huang, 2011;Li et al., 2009;Maqsood et al., 2005) and interval stochastic programming (see Luo et al., 2003;Li et al., 2006;Li and Huang, 2007) turn out to be appropriate. ...
In this paper, a mixed scenario-based and probabilistic two-stage stochastic programming model is proposed for the design of integrated water supply and wastewater collection systems. None of the existing models simultaneously takes into account both business-as-usual and hazard uncertainties. To explore suitable solutions in a reasonable time, a solving procedure comprised of the (1) sample average approximation method, (2) Bezdek fuzzy clustering method and (3) Benders decomposition algorithm is developed. In order to expedite the convergence of the applied Benders decomposition algorithm, different acceleration techniques especially the local branching method are utilized. The performance of the proposed mathematical model and solution procedure is analyzed computationally through a real case study which the results show the usefulness of the developed stochastic programming model as well as the efficiency of the solution approach.
... Multistage Stochastic Programming is the proper problem formulation when decisions are interdependent and some of them can be taken after uncertainty is resolved [5]. Multistage Stochastic Programming has been the subject of many studies [5,13], even specific to MPC and control of dynamic systems [27], or to water management [14,29,50,51]. Yet, its major practical drawback is its intractability. ...
... For this reason, the controller has to calculate control inputs that are valid for all the scenarios in the branch. Once the bifurcation point has been reached, the uncertainty is solved and the controller can calculate specific control inputs for the scenarios in each of the new branches [39]. Hence, the outcome of TB-MPC is not a single sequence of control inputs, it is a tree with the same structure of that of the disturbances. ...
Control of drinking water networks is an arduous task, given their size and the presence of uncertainty in water demand. It is necessary to impose different constraints for ensuring a reliable water supply in the most economic and safe ways. To cope with uncertainty in system disturbances due to the stochastic water demand/consumption and optimize operational costs, this paper proposes three stochastic model predictive control (MPC) approaches, namely, chance-constrained MPC, tree-based MPC, and multiple-scenario MPC. A comparative assessment of these approaches is performed when they are applied to real case studies, specifically, a sector and an aggregate version of the Barcelona drinking water network in Spain. Copyright
... Applications of this type of RO in water resources range from water distribution system design (Cunha and Sousa 2010) and wastewater treatment design (Afonso and Cunha 2007) to the design of large-scale water systems (Escudero 2000), as well as the design of groundwater pump and treatment systems (Ricciardi, Pinder, and Karatzas 2009). Nearly all previous water resources applications involve only feasibility robustness, and do not consider solution robustness. ...
“Confronting Climate Uncertainty in Water Resources Planning and Project Design describes an approach to facing two fundamental and unavoidable issues brought about by climate change uncertainty in water resources planning and project design. The first is a risk assessment problem. The second relates to risk management. This book provides background on the risks relevant in water systems planning, the different approaches to scenario definition in water system planning, and an introduction to the decision-scaling methodology upon which the decision tree is based.
The decision tree is described as a scientifically defensible, repeatable, direct and clear method for demonstrating the robustness of a project to climate change. While applicable to all water resources projects, it allocates effort to projects in a way that is consistent with their potential sensitivity to climate risk.
The process was designed to be hierarchical, with different stages or phases of analysis triggered based on the findings of the previous phase. An application example is provided followed by a descriptions of some of the tools available for decision making under uncertainty and methods available for climate risk management. The tool was designed for the World Bank but can be applicable in other scenarios where similar challenges arise.”
... The operation of electric power systems 1 covers a broad spectrum of activities or studies, among which stands out the operation planning/scheduling [1,2]. In general, this problem [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] is divided into steps (Long-term, Medium-term and Short-term), which have different planning horizons and, consequently, prioritize in each step distinct details of the problem modeling. Briefly, the global problem involves the analysis of the important operational aspects of the system to define the optimal level of energy production to supply the demand in an economical and reliable manner. ...
The operation planning of hydrothermal systems is, in general, divided into coordinate steps which focus on distinct modeling details of the system for different planning horizons. The Medium-Term Operation Planning (MTOP) problem, one of the operation planning steps and the focus of this chapter, aims to define a weekly generation for each power plant with the minimum expected operational cost over a specific planning horizon, regarding especially the uncertainties related to reservoir inflows. Consequently, it is modeled as a stochastic problem and solving it requires the use of multistage stochastic programming algorithms. In this sense, the objective of this chapter is to discuss the problem features, its particularities and its importance to the operational planning. Additionally, the stochastic methods usually used to solve the problem and some applications are presented.
... The scenario generating approach considers a set of statistically independent scenarios, and exploits the inner structure of their temporal evolution in order to obtain a robust decision [13]. Another examples of scenario approach in WR management can found in [12] and [14]. ...
... Applications of this type of RO in EWR range from water distribution system design (Cunha and Sousa 2010) and wastewater-treatment design (Afonso and Cunha 2007) to the design of large-scale water systems (Escudero 2000), as well as the design of groundwater pump and treatment systems (Ricciardi et al. 2009). Nearly all previous EWR applications involve only feasibility robustness, minimize ...
Much progress has been made in the standardization of uncertainty analysis techniques for simulation modeling but less progress has been made in optimization modeling. Among the various techniques used for optimization modeling under uncertainty, robust optimization (RO) uniquely allows for evaluation and control of the various risks of poor system performance resulting from input parameter uncertainties in water-resources problems. A model formulation was developed that addresses an inadequacy in a previous RO formulation. The importance of evaluating, through postprocessing, RO model results with respect to a range of performance metrics, has been demonstrated rather than a single metric, as has been common in previous studies. An analysis of the tradeoffs between solution robustness (nearness to optimality across all scenarios) and feasibility robustness (nearness to feasibility across all scenarios) illustrates the importance of including these terms in multiobjective water resources decision models.
... We should differentiate between water resource planning (usually long-term) and water distribution scheduling (usually daily basis). In the case of planning, we distinguish between the deterministic environment as proposed by Andreu et al. [2] and the stochastic environment which consider uncertainty among its main parameters such as water inflow and needs as presented by Escudero [3]. We focus on the latter case. ...
Most of the easier structural solutions for greater utilization of water resources
have already been implemented. The principal objective of this paper is to
develop a methodology for helping to produce irrigation scheduling policies for
the deployment of water resources for agricultural use. We have, therefore,
developed WISCHE (Water Irrigation Scheduling), a decision support system for
irrigation scheduling in an arborescent network. Computational experience has
been implemented with satisfactory results in \“La Comunidad de Regantes,
Riegos de Levante, Canal 2nd”, eastern Spain. Its irrigation area covers 2188
Hectares, with 2831 nodes distributed in 20 sectors. Irrigation is needed on a
daily basis in five time periods. Network topology, periodical demands and
historical consumptions are the inputs and scheduling the hydrants and turns are
the output. Assignments must satisfy two types of constraints: the first one based
on physical network limitations and the other one based on previous
consumptions and demands. In order to deal with the first limitations, a mixed
0−1 separable quadratic program is proposed. Statistical information about
previous consumptions on turns demanded and non-attended demands is used to
assign a priority to hydrants. Thus, the system favours hydrants which used what
they applied for and hydrants which could not be well attended previously.
Keywords: water resource scheduling, agriculture irrigation, mixed 0–1
separable quadratic programs.
... Recently, Bender and Simonovic (2000) proposed a fuzzy compromise approach for water resource systems planning. A robust modeling approach for water resources system planning under uncertainty was presented by Escudero (2000). Huang and Loucks (2000) introduced inexact two-stage stochastic programming for water resources management. ...
Planning of water resources systems is often associated with many uncertain parameters and their interrelationships are complicated. Stochastic planning of water resources systems is vital under changing climate and increasing water scarcity. This study proposes an interval-parameter two-stage optimization model (ITOM) for water resources planning in an agricultural system under uncertainty. Compared with other optimization techniques, the proposed modeling approach offers two advantages: first, it provides a linkage to pre-defined water policies, and; second, it reflects uncertainties expressed as probability distributions and discrete intervals. The ITOM is applied to a case study of irrigation planning. Reasonable solutions are obtained, and a variety of decision alternatives are generated under different combinations of water shortages. It provides desired water-allocation patterns with respect to maximum system benefits and highest feasibility. Moreover, the modeling results indicate that an optimistic water policy corresponding to higher agricultural income may be subject to a higher risk of system-failure penalties; while, a too conservative policy may lead to wastage of irrigation supplies.
... MSP has been the subject of many studies [37,36], even specific to MPC and control of dynamic systems, [38], or to water management [39,40,41,42], yet its major practical drawback is its intractability. In MSP in fact, every time some uncertainty is resolved, the system dimension is multiplied by the number of possible outcomes of the uncertain variable. ...
Short-term water system operation can be realised using Model Predictive Control (MPC). MPC is a method for operational management of complex dynamic systems. Applied to open water systems, MPC provides integrated, optimal, and proactive management, when forecasts are available. Notwithstanding these properties, if forecast uncertainty is not properly taken into account, the system performance can critically deteriorate.
Ensemble forecast is a way to represent short-term forecast uncertainty. An ensemble forecast is a set of possible future trajectories of a meteorological or hydrological system. The growing ensemble forecasts’ availability and accuracy raises the question on how to use them for operational management.
The theoretical innovation presented here is the use of ensemble forecasts for optimal operation. Specifically, we introduce a tree based approach. We called the new method Tree-Based Model Predictive Control (TB-MPC). In TB-MPC, a tree is used to set up a Multistage Stochastic Programming, which finds a different optimal strategy for each branch and enhances the adaptivity to forecast uncertainty. Adaptivity reduces the sensitivity to wrong forecasts and improves the operational performance.
TB-MPC is applied to the operational management of Salto Grande reservoir, located at the border between Argentina and Uruguay, and compared to other methods.
... This model [1] has being improved continuously, aiming to produce a satisfactory response, which opens room for contributions, such as the proposal of this paper. This problem is particularly complex owing to some characteristics specially related to randomness of water inflows to the reservoirs [2]. Thus, solutions obtained by models that do not recognize this uncertainty produce unsatisfactory results. ...
The Medium-Term Operation Planning (MTOP) of hydrothermal systems aims to define the generation for each power plant, minimizing the expected operating cost over the planning horizon. Mathematically, this task can be characterized as a linear, stochastic, large-scale problem which requires the application of suitable optimization tools. To solve this problem, this paper proposes to use the Nested Decomposition, frequently used to solve similar problems (as in Brazilian case), and Progressive Hedging, an alternative method, which has interesting features that make it promising to address this problem. To make a comparative analysis between these two methods with respect to the quality of the solution and the computational burden, a benchmark is established, which is obtained by solving a single Linear Programming problem (the Deterministic Equivalent Problem). An application considering a hydrothermal system is carried out. Mathematical subject classification: Primary: 06B10; Secondary: 06D05.
... Over the past years, a number of optimization methods have been proposed for dealing with uncertainties in water resources management [6][7][8][9][10][11]. For example, Chung et al. [12] applied a robust optimization approach in a water supply system to minimize the total system cost; this approach could address parameter uncertainty without excessively affecting the system. ...
... The scenario analysis approach considers a set of statistically independent scenarios, and exploits the inner structure of their temporal evolution in order to obtain a "robust" decision policy, in the sense that the risk of wrong decisions is minimised. Some examples are given in Escudero [2000], Pallottino et al. [2005] for water resources management, in Mulvey and Vladimirou [1989] for investment and production planning, in Glockner [1996] for air traffic management and in Hoyland and Wallace [2001] for insurance policy and production planning. In this paper he authors improve the approach already presented in Pallottino et al. [2005] and propose a reoptimization procedure that follows the scenario analysis. ...
In water resources management problems, uncertainty is mainly associated with the value of hydrological exogenous inflows and demand patterns. Deterministic models are inadequate to represent these problems and traditional stochastic optimization models cannot be used if there is insufficient statistical information to support the model. In this paper the uncertainty is modelled by a scenario approach in a multistage environment which includes different possible system configurations in a wide time horizon. A robust chance optimization model is used in order to obtain a so-called barycentric value with respect to decision variables. The successive reoptimization step, based on this barycentric solution, allows reducing the consequences deriving from a wrong decision. The improved version of WARGI DSS performs scenario analysis by identifying trends and essential features on which to base a robust decision policy. The current version of WARGI can be linked to commercial solvers as well as to some free solvers such as IdrScen. IdrScen is a new package for large dimension problems based on open source philosophy, that exploits the speed of network simplex methods in order to obtain very efficient solutions to the scenario problems. Moreover, the application to a real water resource system in Sardinia, Italy, shows the usefulness of the scenario analysis in water resources problems affected by a high level of uncertainty in data input. It appears that IdrScen can be a promising alternative tool to commercial codes for large size optimization problems coming for complex real resource systems.
... Multistage Stochastic Programming is the proper problem formulation when decisions are interdependent and some of them can be taken after uncertainty is resolved [5]. Multistage Stochastic Programming has been the subject of many studies [5,13], even specific to MPC and control of dynamic systems [27], or to water management [14,29,50,51]. Yet, its major practical drawback is its intractability. ...
... Multistage Stochastic Programming is the proper problem formulation when decisions are interdependent and some of them can be taken after uncertainty is resolved [5]. Multistage Stochastic Programming has been the subject of many studies [5,13], even specific to MPC and control of dynamic systems [27], or to water management [14,29,50,51]. Yet, its major practical drawback is its intractability. ...
Model Predictive Control (MPC) is a control algorithm that shows promising advantages for the operational water management of hydraulic structures. In this paper, MPC is applied for the real-time control of the Salto Grande hydropower plant on River Uruguay at the Argentinean/Uruguayan bolder. Objectives of the control are the maximization of hydro
power benefits as well as flood mitigation upstream and downstream of the dam.
The project is ongoing. The results available at this moment are obtained feeding MPC
with historical inflow series, equivalent to having exact inflow forecasts and representing
an upper boundary of the potential result. For the operational use, real forecasts are required and these forecasts are uncertain. Deterministic forecast neglects uncertainty and can lead to lower control performances. On the other hand, ensemble forecasts, made of a representative set of possible future trajectories, consider forecast uncertainty explicitly. Tree-Based MPC uses ensemble data in a multistage stochastic programming version of the MPC problem, which means that the resolution of uncertainty in time is explicitly taken into account. This paper presents available results obtained using MPC with exact inflow and describes the Tree-Based MPC method concisely, which will be applied to this test case as a follow up of the present work.
... We should differentiate between water resources planning, which is usually long-term planning, and water distribution scheduling, which is usually performed on a daily basis. In the case of planning, see Andreu et al. (1996) for the deterministic environment and Pereira and Pinto (1991) and Escudero (2000) for the stochastic case. This is done by considering the uncertainty in the main parameters (i.e., water inflow and needs). ...
We present a mixed zero–one separable non-linear approach to the optimization of the management of water resources used for agricultural irrigation purposes. It provides dynamic planning of the daily irrigation scheduling for a given land area by considering the irrigation network topography, water flow conditions and logistical operation constraints in order to optimize the use of water stored in a reservoir. We present a mixed zero–one separable non-linear program and a solution procedure that iteratively solves a mixed zero–one linear approximation of the model. We are not aware of any previous attempt to solve large-scale mixed zero–one separable non-linear programs of this kind. Some computational experiences on a large-scale real-life problem are reported.
... As one would expect, the term "robust optimization" is quite popular, and the descriptor "robust" is so commonly applied that its specific meaning, and its implications for model results, has become elusive. For example there are a number of papers that cite Mulvey et al. (1995) in the derivation of their optimization methods, but do not actually use Mulvey et al.'s methods (Escudero 2000;Jia and Culver 2006;Kuhn and Madanat 2006;Rosenberg and Lund 2009;Suh and Lee 2002). There are also a number of papers in water resources, mostly water distribution system design, that use techniques very similar to the RO techniques presented by Mulvey et al. (1995), but do not reference Mulvey et al. (1995). ...
The term "robust optimization" is quite popular, and the descriptor "robust" so commonly applied in the water resources literature that its specific meaning, and its implications for model results, have become elusive. We here provide a brief summary of the applications of robust optimization to water resources planning and management problems in the past 15 years. We describe and demonstrate concerns regarding Mulvey et al.'s (1995) multi-objective approach to robust optimization. We evaluate the idea of minimization of the expected costs as the primary design criterion by contrasting solutions to a simple robust optimization problem in terms of common performance measures, including reliability, vulnerability, and sustainability.
... Studies that focused on economic issues used monetary objectives and dynamic programming methods (Edwards et al., 1999;Hodge, 2001). Some optimizations minimized flow deficits, rather than setting a hard constraint (Escudero, 2000;Bessler et al., 2003). For these, violations of minimum flow regulations were possible, particularly if minimum flows constraints were not assigned a high priority relative to other, non-environmental constraints. ...
Hydroelectric power provides a cheap source of electricity with few carbon emissions. Yet, reservoirs are not operated sustainably, which we define as meeting societal needs for water and power while protecting long-term health of the river ecosystem. Reservoirs that generate hydropower are typically operated with the goal of maximizing energy revenue, while meeting other legal water requirements. Reservoir optimization schemes used in practice do not seek flow regimes that maximize aquatic ecosystem health. Here, we review optimization studies that considered environmental goals in one of three approaches. The first approach seeks flow regimes that maximize hydropower generation, while satisfying legal requirements, including environmental (or minimum) flows. Solutions from this approach are often used in practice to operate hydropower projects. In the second approach, flow releases from a dam are timed to meet water quality constraints on dissolved oxygen (DO), temperature and nutrients. In the third approach, flow releases are timed to improve the health of fish populations. We conclude by suggesting three steps for bringing multi-objective reservoir operation closer to the goal of ecological sustainability: (1) conduct research to identify which features of flow variation are essential for river health and to quantify these relationships, (2) develop valuation methods to assess the total value of river health and (3) develop optimal control softwares that combine water balance modelling with models that predict ecosystem responses to flow. Copyright © 2007 John Wiley & Sons, Ltd.
... We should differentiate between water resources planning, which is usually long-term planning, and water distribution scheduling, which is usually performed on a daily basis. In the case of planning, see Andreu et al. (1996) for the deterministic environment and Pereira and Pinto (1991) and Escudero (2000) for the stochastic case. This is done by considering the uncertainty in the main parameters (i.e., water inflow and needs). ...
In this paper we present the models and the algorithms which are being used in a decision support system (DSS) to determine water irrigation scheduling. The DSS provides dynamic scheduling of the daily irrigation for a given land area by taking into account the irrigation network topology, the water volume technical conditions and the logistical operations. The system has been validated by the Agriculture Community of Elche (Spain) and annexed to their Supervisory Control and Data Acquisition system (SCADA). We present two heuristic approaches to solve the mixed 0–1 separable nonlinear program for irrigation scheduling implemented with free software.
... To overcome the above difficulties, in this paper we analyse the scenario approach for WR offering some general rules for organising a predefined set of scenarios into the scenario-tree and for identifying a complete set of decision variables relative to all the scenarios under investigation. The scenario-tree is obtained by aggregate common portions of scenarios; the aggregation condition guarantees that the solution (that is, the decisions) in any given period is independent of the information not yet available, as detailed in Section 3. Scenario analysis approach for WR was proposed in (Escudero, 2000); and in Wam-Me EU project, , and tested on some real physical systems. Our proposal is to embed an evolution of the above WR scenario analysis tool into a DSS that allows in depth investigation of the robustness of the solution and, if necessary, refinement of decisions. ...
In this paper we present a scenario analysis approach for water system planning and management under conditions of climatic and hydrological uncertainty. The scenario analysis approach examines a set of statistically independent hydrological scenarios, and exploits the inner structure of their temporal evolution in order to obtain a “robust” decision policy, so that the risk of wrong decisions is minimised. In this approach uncertainty is modelled by a scenario-tree in a multistage environment, which includes different possible configurations of inflows in a wide time-horizon. In this paper we propose a Decision Support System (DSS) that performs scenario analysis by identifying trends and essential features on which to base a robust decision policy. The DSS prevents obsolescence of optimiser codes, exploiting standard data format, and a graphical interface provides easy data-input and results analysis for the user. Results show that scenario analysis could be an alternative approach to stochastic optimisation when no probabilistic rules can be adopted and deterministic models are inadequate to represent uncertainty. Moreover, experimentation for a real water resources system in Sardinia, Italy, shows that practitioners and end-users can adopt the DSS with ease.
This study explores optimal control in the case when certain input uncertainties cannot be well quantified. In these scenarios the decision maker prefers to have the most flexibility, which is defined to be the largest range of options for decision variables, and still achieve the objectives of the operation while satisfying all constraints. Each decision variable is generalized to be a range of potential actions. These ranges are modeled with random variables, thus uncertainty quantification techniques are employed to compute expected values of objectives and probabilities of chance constraints. The proposed framework determines optimal probability densities for the decision variables by treating the amount of flexibility as an additional objective. There is a clear trade-off between the amount of flexibility that can be allowed and the resulting expected values of the other objectives. A dimension reduction technique is employed to ensure a reasonable dimension for the search space.
This paper focuses on the capacity uncertainty in water supply chains that occurs when facilities face disruption. A combination of scenario-based two-stage stochastic programming with the min-max robust optimization approach is proposed to optimize the water supply chain network design problem. In the first stage, the decisions are made on locations and capacities of reservoirs and water-treatment plants while recourse decisions including amount of water extraction, amount of water refinement, and consequently amount of water held in reservoirs are made at the second stage. The proposed robust two-stage stochastic programming model can help decision makers consider the impacts of uncertainties and analyze trade-offs between system cost and stability. The literature reveals that most exact methods are not able to tackle the computational complexity of mixed integer non-linear two-stage stochastic problems at large scale. Another contribution of this study is to propose two metaheuristics - a particle swarm optimization (PSO) and a bat algorithm (BA) - to solve the proposed model in large-scale networks efficiently in a reasonable time. The developed model is applied to several hypothetical cases of water resources management systems to evaluate the effectiveness of the model formulation and solution algorithms. Sensitivity analyses are also carried out to analyze the behavior of the model and the robustness approach under parameters variations.
《黑河流域模型集成》系统地介绍了黑河流域生态-水文-经济集成建模的总体思路、建模方法，以及在上游寒区生态水文模型集成、中游干旱区地表地下水耦合建模及生态水文模型集成、整个流域经济模型、流域水资源管理决策支持系统等方面取得的研究成果。我们希望藉此书，向同行全面介绍黑河模型集成的阶段性成果，并在此基础上，拓清继续努力的方向。
流域科学研究中的模型集成可概括为“水－土－气－生－人”集成模型的发展。它一般应包括地表水、地下水、水质、能量平衡、植被动态/作物生长、碳氮等生物地球化学循环、土壤侵蚀等模块，有些模型还实现了和大气模型的单向（大气模型作为驱动）或双向耦合，并进一步耦合土地利用、水资源及社会经济模型。
《黑河流域模型集成》（程国栋等著. 北京：科学出版社, 2019.11）一书系统地反映了2004～2018 年，中国科学院寒区旱区环境与工程研究所（简称“寒旱所”）流域模型集成研究团队在流域集成模型方面的进展。其中，2004～2010年主要是准备阶段，寒旱所启动了“黑河流域交叉集成研究的模型开发和模拟环境建设”，初步建立了刻画黑河流域上游冰冻圈水文过程和中下游地表-地下水相互作用的水文模型，开发了建模环境，大大提升了流域模型集成研究的能力，锻造了一支从事模型集成研究的队伍。2010 年，国家自然科学基金委员会启动了“黑河流域生态-水文过程集成研究”重大研究计划（简称“黑河计划”）。“黑河计划”是将我国流域科学研究推进到国际先进行列的重大举措，也是一次陆地表层系统科学研究方法的全面实践。随着“黑河计划”的启动，黑河流域模型集成进入快车道，国内精英力量汇聚一堂，系统地开展了黑河流域生态-水文-经济集成建模研究。期间，寒旱所的建模人员作为骨干，或主持、或参加了有关项目，特别是对研发山区冰冻圈生态水文模型、中下游地表-地下水耦合模型、建模环境、决策支持系统作出较大贡献。这些研究人员也都是本书的主要作者。
黑河流域是我国第二大内陆河流域，面积共计约14.3×10^4 km²，从流域的上游到下游，以水为纽带形成了“冰雪／冻土－森林－草原－河流－湖泊－绿洲－沙漠－戈壁”的多元自然景观，流域内寒区和干旱区并存，山区冰冻圈和极端干旱的河流尾闾地区形成了鲜明对比。同时，黑河流域开发历史悠久，人类活动显著地影响了流域的水文环境，2000 多年来，这一地区的农业开发，屯田垦殖，多种文化的碰撞交流、此消彼长，无不与生态、水文深刻地联系在一起。自然和人文过程交汇在一起，使黑河流域成为开展流域综合集成研究的一个十分理想的试验流域。
与我国西北内陆河流域其他地区一样，黑河上游的山区有较多的降水和冰雪融水，是流域水资源的形成区；中下游的盆地内则降水稀少，是水资源的耗散区。水出山后流入盆地，“有水是绿洲，无水成荒漠，水多则盐碱化”，水是控制黑河流域生态状况的决定因素。过去几十年，西北干旱区不同的内陆河流域都在讲述着同样的故事：随着上中游人口持续增加，经济不断增长，耗水日益增多，越来越多地挤占下游的生态用水，最终导致下游尾闾湖干涸，沙尘暴迭起，胡杨林成批死亡，酿成严重的生态灾难。这些都毫无例外地涉及水、生态和经济。问题的实质在于如何用有限的水资源，既保证经济发展，又维系生态系统的健康。
20 世纪末，黑河下游生态明显恶化，国家实施生态调水，拯救下游生态取得了成功。并由此引发了对黑河流域水、土、气、生、人等要素的大量研究。这些研究对内陆河流域的可持续发展具有重要的指导意义。黑河的生态研究已经走出了就生态论生态的小圈子，正在走向探索“以流域为单元，以水为主线”协调水-生态-经济系统中的各种关系的新阶段。近30 年来，黑河流域已成为我国内陆河研究的基地，具有较为完善的观测网络和各种科学研究与实验积累下来的大量资料。同时，它也是近年来开展内陆河综合治理的典型案例，是建设节水型社会的基地。
从以上两方面看，黑河流域既是一个陆地表层系统科学集成研究（integrated study）的试验流域，也是实践流域水资源综合管理（integrated management）的基地。集成研究必须打好观测、模型、数据基础，必须建立一个3M［观测（monitoring）、模型（modeling）、数据分析处理（manipulating）］一体化平台。为此，我们自2000年始，以数据、观测、模型为三个阶段性重点，开展了建立3M 平台的不懈努力，已完成了数据平台（数字黑河）和观测平台（黑河遥感试验及黑河流域观测系统）的建设，也在集成模型发展——特别是在水文、生态、经济模型两两耦合方面取得了长足进步。
The optimal spatial allocation of irrigation water under uncertainty has become a serious concern because of irrigation water shortage and uncertain factors that affect irrigation water allocation. In this study, an optimal multi-objective model for irrigation water allocation under uncertainty is developed to maximise the economic benefit of crops and minimise the operation cost and water deficit of crop irrigation. The original and optimal plantation structure, irrigation mode and soil water content are acquired through geospatial technology. A bilayer nested optimisation (BLNO) algorithm is designed to produce multiple individuals of design vectors using an ant colony neural network algorithm for an outer optimisation. Meanwhile, a continuous adaptive ant colony (CAAC) algorithm is used for inner optimisation to calculate the interval values of the uncertain model. The crop distribution and irrigation mode are obtained to parameterise the planting area and the water demand of each crop and each block in the multi-objective model. This model is then solved and analysed. Compared to the optimal schemes obtained from an inexact two-stage fuzzy-stochastic programming and the CAAC, respectively, BLNO can effectively and efficiently solve the optimal spatial allocation of irrigation water under uncertainty. This method can spatially maximise the economic benefit of crops and minimise the operation cost and water deficit of crop irrigation using lower and upper bound maps whilst visually obtaining the exact crop type, reasonable irrigation method and precise water demand for each block and for the entire irrigated area.
The planning of operations of hydrothermal systems is, in general, divided into coordinated steps which focus on distinct modeling details of the system for different planning horizons. The medium-term operation planning (MTOP) problem, one of the operation planning steps and the focus of this chapter, aims at defining weekly generation for each power plant with theminimum expected operational cost over a specific planning horizon, with regard especially to the uncertainties related to reservoir inflows. Consequently, it is modeled as a stochastic problem and solving it requires the use of multistage stochastic optimization algorithms. In this sense, the objective of this chapter is to discuss the problem features, its particularities, and its importance in the overall operational planning. The stochastic methods usually used to solve this problem and some applications are also presented.
In this paper, we propose a scenario-based robust optimization model for the design of a water supply system considering the risk of facility failure, which is represented as an uncertainty set generated by a finite set of scenarios. New facilities are planned to be built to hedge against the possible failure of existing system facilities that would potentially damage the capacity of the system to meet given user demands. The goal is to build facilities that are both cost-effective and make the system robust. The system robustness is defined as the ability to satisfy user demands for every data realization in the uncertainty set. The proposed model is shown to be equivalent to a large-scale mixed-integer linear program that is solved by a Benders decomposition algorithm. Computational results demonstrate the efficiency of the proposed algorithm, and show that substantial improvement in system robustness can be achieved with minimal increase in system cost.
Optimal reservoir operation under uncertainty is a challenging engineering problem. Application of classic stochastic optimization methods to large-scale problems is limited due to computational difficulty. Moreover, classic stochastic methods assume that the estimated distribution function or the sample inflow data accurately represents the true probability distribution, which may be invalid and the performance of the algorithms may be undermined. In this study, we introduce a Robust Optimization (RO) approach, Iterative Linear Decision Rule (ILDR), so as to provide a tractable approximation for a multi-period hydropower generation problem. The proposed approach extends the existing LDR method by accommodating nonlinear objective functions. It also provides users with the flexibility of choosing the accuracy of ILDR approximations by assigning a desired number of piecewise linear segments to each uncertainty. The performance of the ILDR is compared with benchmark policies including the Sampling Stochastic Dynamic Programming (SSDP) policy derived from historical data. The ILDR solves both the single and multi-reservoir systems efficiently. The single reservoir case study results show that the RO method is as good as SSDP when implemented on the original historical inflows and it outperforms SSDP policy when tested on generated inflows with the same mean and covariance matrix as those in history. For the multi-reservoir case study, which considers water supply in addition to power generation, numerical results show that the proposed approach performs as well as in the single reservoir case study in terms of optimal value and distributional robustness. This article is protected by copyright. All rights reserved.
Decision theory refers to the analysis, formalization, and prediction, through mathematical models, of optimal and real decision-making; it involves the process of selection of perceived solutions, actions, and outcomes to a given problem from a set of possible alternatives. Depending on the extent of possible quantification, presence of uncertainty, number of decision makers, and number of objectives, decision theory is classified as quantitative or qualitative, deterministic or stochastic, single or multiple party, and single or multiobjective. The management of water resources systems involves unavoidable natural and social conditions of risk and uncertainty and multiple competing or conflicting parties and objectives, which introduce the risks of high economic and social costs due to wrong decisions, necessitating the formulation of models that adequately represent a given situation by incorporating all factors affecting it. Hence, the adequate modeling of such systems should incorporate risk and uncertainty; decision theory under risk and uncertainty is called decision analysis or risk management. Risk management approaches may be broadly categorized as nonprobability and probability based techniques. Non-probability based techniques include sensitivity analysis, decision criteria, analytical hierarchy, and game theory; probability based techniques include scenario analysis, moments, decision trees, expected value, stochastic optimization, Bayesian and fuzzy analysis, and downside risk measures such as Value-at-Risk and Conditional Value-at-Risk metrics. Many of these methods, though very useful, suffer from critical shortcomings: sensitivity and scenario analysis only provide some intuition of risk, expected value fails to highlight extreme-event consequences, decision trees and hierarchical approaches fail to generate robust and efficient solutions in highly uncertain environments, central moments do not account for fat-tailed distributions and penalize positive and negative deviations from the mean equally, recourse does not provide means to control risk, and Value-at-Risk does not provide information about the extent and distribution of the losses that exceed it and is not coherent. Game theory, used in situations of multiple party decision making, suffers from several systematic violations, such as the common consequence, preference reversal, and framing effects. To account for some of these shortcomings, several extensions and alternatives have been suggested such as the conditional-Value-at-Risk and behavioral game theory.
This paper presents the development and the first application of a superiority–inferiority-based inexact fuzzy-stochastic quadratic programming (SI-IFSQP) approach for sustainable water supply under multiple uncertainties. SI-IFSQP improves conventional nonlinear programming by tackling multiple uncertainties within an individual parameter; SI-IFSQP is also superior to existing inexact methods due to its reflection of economies of scale and reduction of computational requirements. An interactive solution algorithm with high computational efficiency was also proposed. The application of SI-IFSQP to long-term planning of a multi-source multi-sector water supply system demonstrated its applicability. The close reflection of system complexities, such as multiple uncertainties, scale economies and dynamic parameters, could enhance the robustness of the optimization process as well as the acceptability of obtained results. Corresponding to varied system conditions and decision priorities, the interval solutions from SI-IFSQP could help generate a series of long-term water supply strategies under a number of economic, environmental, ecological, and water-security targets.
In this paper, the robust counterpart (RC) approach (Ben-Tal et al., 2009) is applied to optimize management of a water supply system (WSS) fed from aquifers and desalination plants. The water is conveyed through a network to meet desired consumptions, where the aquifers recharges are uncertain. The objective is to minimize the net present value cost of multiyear operation, satisfying operational and physical constraints. The RC is a min-max guided approach, which converts the original problem into a deterministic equivalent problem, requiring only that the uncertain parameters resides within a user-defined uncertainty set. The robust policy obtained by the RC approach is compared with polices obtained by other decision-making approaches including stochastic approaches.
Keywords
Problem Description
Stochastic Approach
Nonanticipative Water Resources Policies
Conclusions
See also
References
Real-time control is becoming an accepted engineering solution for reducing the impact of undesired situations occurring in water systems. Model Predictive Control (MPC), which is generally used in a deterministic environment, can be used effectively to exploit best the information contained in an forecast. In case of Ensemble Forecast, the problem of optimal control in MPC can be reshaped in a probabilistic formulation, but such a formulation is very complex and requires calculation time that is often not suitable for a use in real time. Scenario reduction techniques can be used to reduce the calculation time until an acceptable value, while minimizing the decrease in performance due to such simplification. Uncertainty of the prediction increases further in time and the ensembles spread out. We can reduce the number of scenarios by pruning the ensemble at different stages. The way in which the pruning takes place changes with time. Tree-shaped scenario reduction refers to a corresponding decision tree to take into account that the decision at the present moment is influenced by the knowledge that new and more accurate information will be available in the future. Tree-shaped scenario reduction has been applied in a Model Predictive Controller to a drainage canal system in Delfland (The Netherlands) and compared with results obtained using deterministic MPC. To conclude, we summarize the advantages and limits of this novel method, and possible future solutions to overcome these limits.
This paper presents a new methodology to generate a tree from an ensemble. The reason to generate a tree is to use the ensemble in multistage stochastic programming. A correct tree structure is of critical importance because it strongly affects the performance of the optimization. A tree, in contrast to an ensemble, specifies when its trajectories diverge from each other. A tree can be generated from the ensemble data by aggregating trajectories over time until the difference between them becomes such that they can no longer be assumed to be similar, at such a point, the tree branches. The proposed method models the information flow: it takes into account which observations will become available, at which moment, and their level of uncertainty, i.e. their probability distributions (pdf). No conditions are imposed on those distributions. The method is well suited to trajectories that are close to each other at the beginning of the forecasting horizon and spread out going on in time, as ensemble forecasts typically are.
Open water systems are one of the most externally influenced systems due to their size and continuous exposure to uncertain meteorological forces. In this paper we use a stochastic programming approach to control a drainage system in which the weather forecast is modeled as a disturbance tree. A model predictive controller is used to optimize the expected value of the system variables taking into account the disturbance tree. This technique, tree-based model predictive control (TBMPC), is solved in a parallel fashion by means of dual decomposition. In addition, different possibilities are explored to reduce the communicational burden of the parallel algorithm. Finally, the performance of this technique is compared with others such as minmax or multiple model predictive control.
This paper presents a new methodology to generate a tree from an ensemble. The reason to generate a tree is to use the ensemble in multistage stochastic programming. A correct tree structure is of critical importance because it strongly affects the performance of the optimization. A tree, in contrast to an ensemble, specifies when its trajectories diverge from each other.
A tree can be generated from the ensemble data by aggregating trajectories over time until the difference between them becomes such that they can no longer be assumed to be similar, at such a point, the tree branches.
The proposed method models the information flow: it takes into account which observations will become available, at which moment, and their level of uncertainty, i.e. their probability distributions (pdf). No conditions are imposed on those distributions.
The method is well suited to trajectories that are close to each other at the beginning of the forecasting horizon and spread out going on in time, as ensemble forecasts typically are. Copyright © 2012 John Wiley & Sons, Ltd.
This paper provides an overview of the key contributions within the planning and scheduling communities with specific emphasis on uncertainty analysis. As opposed to focusing in one particular industry, several independent sectors have been reviewed in order to find commonalities and potential avenues for future interdisciplinary collaborations. The objectives and physical constraints present within the planning and scheduling problems may vary greatly from one sector to another; however, all problems share the common attribute of needing to model parameter uncertainty in an explicit manner. It will be demonstrated through the literature review that two-stage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, conditional value-at-risk, and other risk mitigation procedures have found widespread application within all of the analyzed sectors. This review is the first work which attempts to provide a comprehensive description of the advances and future directions for planning and scheduling under uncertainty within a variety of sectors.
This paper presents the development and the first application of an inexact quadratic programming (IQP) approach for sustainable
water supply under multiple uncertainties. The developed IQP improves conventional nonlinear programming by tackling multiple
uncertainties within an individual parameter; IQP is also superior to existing inexact methods due to its reflection of economies
of scale and reduction of computational requirements. An interactive solution algorithm with high computational efficiency
was also proposed. The application of IQP to long-term planning of a multi-source multi-sector water supply system demonstrated
its applicability. The close reflection of system complexities, such as multiple uncertainties, scale economies and dynamic
parameters, could enhance the robustness of the optimization process as well as the acceptability of obtained results. Corresponding
to varied system conditions and decision priorities, the interval solutions from IQP could help generate a series of long-term
water supply strategies under a number of economic, environmental, ecological, and water-security targets.
KeywordsWater supply–Sustainable development–Decision making–Uncertainty–Inexact quadratic programming
A novel parallel decomposition algorithm is developed for large, multistage stochastic optimization problems. The method decomposes the problem into subproblems that correspond to scenarios. The subproblems are modified by separable quadratic terms to coordinate the scenario solutions. Convergence of the coordination procedure is proven for linear programs. Subproblems are solved using a nonlinear interior point algorithm. The approach adjusts the degree of decomposition to fit the available hardware environment. Initial testing on a distributed network of workstations shows that an optimal number of computers depends upon the work per subproblem and its relation to the communication capacities. The algorithm has promise for solving stochastic programs that lie outside current capabilities.
Dealing with significantly large design problems for water-resource systems a mixed optimization procedure based on network linear programming and the subgradient method will be described. Using a linear problem formulation, the procedure uses network linear programming as a subproblem that assumes the knowledge of design variables. Since inside its domain, the global objective functions is a convex piecewise linear function, a subgradient method is used to obtain the direction of the improvement of design variables at each iteration using the solutions of the network subproblem. The mixed technique permits an efficient evaluation of the design variables in order to reach a good approximation of the global objective function optimum. The solution technique performs well in the purely linear case and, moreover, allows some kinds of nonlinearities in the cost functions of design variables.
A computational scheme that is able to determined at each stage the most economical generation decision for each plant of a hydrothermal system is presented. The algorithm is based on the stochastic extension of Benders decomposition and can be implemented from already existing operation models. The results can be used in the weekly or monthly generation scheduling activities in real-time operation. A case study with 37 reservoirs of the Brazilian system is presented and discussed.
This paper surveys recent work on dynamic stochastic programming problems and their applications. New results are included on the measurability and interpretation-in terms of the expected value of perfect information (EVPI)-of the dual multiplier processes corresponding to these problems. A final section reports preliminary computational experiments with algorithms for 2-stage problems
Multistage stochastic linear programming has many practical applications for problemswhose current decisions have to be made under future uncertainty. There are a variety ofmethods for solving the deterministic equivalent forms of these dynamic problems, includingthe simplex and interior-point methods and nested Benders decomposition, which decomposesthe original problem into a set of smaller linear programming problems and hasrecently been shown to be superior to the alternatives for large problems. The Benderssubproblems can be visualised as being attached to the nodes of a tree which is formed fromthe realisations of the random data process determining the uncertainty in the problem. Thispaper describes a parallel implementation of the nested Benders algorithm which employsa farming technique to parallelize nodal subproblem solutions. Differing structures of thetest problems cause differing levels of speed-up on a variety of multicomputing platforms:problems with few variables and constraints per node do not gain from this parallelisation.We therefore employ stage aggregation to such problems to improve their parallel solutionefficiency by increasing the size of the nodes and therefore the time spent calculating relativeto the time spent communicating between processors. A parallel version of a sequentialimportance sampling solution algorithm based on local expected value of perfect information(EVPI) is developed which is applicable to extremely large multistage stochastic linearprogrammes which either have too many data paths to solve directly or a continuous distributionof possible realisations. It utilises the parallel nested Benders algorithm and a parallelversion of an algorithm designed to calculate the local EVPI values for the nodes of the treeand achieves near linear speed-up.
Mathematical programming models with noisy, erroneous, or incomplete data are common in operations research applications. Difficulties with such data are typically dealt with reactively—through sensitivity analysis—or proactively—through stochastic programming formulations. In this paper, we characterize the desirable properties of a solution to models, when the problem data are described by a set of scenarios for their value, instead of using point estimates. A solution to an optimization model is defined as: solution robust if it remains “close” to optimal for all scenarios of the input data, and model robust if it remains “almost” feasible for all data scenarios. We then develop a general model formulation, called robust optimization (RO), that explicitly incorporates the conflicting objectives of solution and model robustness. Robust optimization is compared with the traditional approaches of sensitivity analysis and stochastic linear programming. The classical diet problem illustrates the issues. Robust optimization models are then developed for several real-world applications: power capacity expansion; matrix balancing and image reconstruction; air-force airline scheduling; scenario immunization for financial planning; and minimum weight structural design. We also comment on the suitability of parallel and distributed computer architectures for the solution of robust optimization models.
For nonconvex problems, the saddle point equivalence of the Lagrangian approach need not hold. The nonexistence of a saddle point causes the generation of a dual gap at the solution point, and the Lagrangian approach then fails to give the solution to the original problem. Unfortunately, dual gaps are a fairly common phenomenon for engineering system design problems.
Methods which are available to resolve the dual gaps destroy the separability of separable systems. The present work employs the method of multipliers by Hestenes to resolve the dual gaps of engineering system design problems; it then develops an algorithmic procedure which preserves the separability characteristics of the system. The theoretical foundations of the proposed algorithm are developed, and examples are provided to clarify the approach taken.
Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing.
La utilización racional del agua implica una gestión eficiente, integral y sostenible del recurso. Los sistemas de recursos hídricos y las soluciones para aliviar sus problemas son cada vez mas complejos. Por consiguiente, a efectos de analizar los sistemas de forma integrada y de abordar incertidumbres clásicas relacionadas con los usos, demandas o recursos, así como nuevos temas tales como impactos de posibles cambios climáticos, será necesario utilizar herramientas tecnológicamente avanzadas. El desfase existente actualmente entre el estado del arte del análisis de sistemas de recursos hidráulicos y la práctica cotidiana de la toma de decisiones en el mundo real puede y debe reducirse. La factibilidad y realidad de este proceso es demostrada en este artículo, en el que se describe un sistema soporte de decisión para recursos hídricos, que está siendo utilizado en la actualidad en la Confederación Hidrográfica del Segura para responder a cuestiones enfocadas al uso racional del agua. Después de una breve descripción de la problemática de la cuenca, se llega a la conclusión de que, en sistemas tan complejos, el uso de herramientas como la que se describe es, más que algo conveniente, la única posibilidad de analizar con éxito la planificación y las políticas de gestión del sistema, obteniendo una idea del rendimiento y de la fiabilidad del mismo. El sistema soporte incluye dos módulos principales: un módulo para la optimización de esquemas de sistemas de recursos, y un módulo para la simulación de la gestión de dichos sistemas incluyendo el uso conjunto de aguas superficiales y subterráneas. La creación de esquemas y la introducción de datos, así como el análisis de resultados, se ven facilitados por interfases y postprocesadores gráficos interactivos. La aplicación de estas herramientas en una cuenca tan compleja como la del Segura tiene a su vez un efecto de promoción de su uso en otras cuencas, confirmando que, bajo ciertas condiciones, herramientas tecnológicamente avanzadas pueden ser usadas y aceptadas en la práctica cotidiana de la planificación y gestión hidrológicas.
Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/44253/1/10479_2005_Article_BF02031741.pdf
Multistage stochastic linear programming has many practical applications for problems whose current decisions have to be made under future uncertainty. There are a variety of methods for solving the deterministic equivalent forms of these dynamic problems, including the simplex and interior point methods and nested Benders decomposition # which decomposes the original problem into a set of smaller linear programming problems and has recently been shown to be superior to the alternatives for large problems. The Benders subproblems can be visualised as being attached to the nodes of a tree which is formed from the realisations of the random data vectors determining the uncertainty in the problem. Parallel versions of the nested Benders algorithm involvetwo obvious techniques for parallelising the associated tree structure for multiprocessors or multicomputers # subtree parallelisation or a nodal parallelisation # both of which utilise a farming approach. The nodal parallelisation techni...
The River Segura basin, in southeastern Spain, is a semiarid Mediterranean region with a highly developed agriculture. The demand of water far exceeds the renewable surface and ground water resources. Mining of aquifers has been a practice for many years and still continues. There is also an external source of water: the Tagus-Segura interbasin transfer. Proper management is crucial in order to avoid or minimize shortages and conflicts. Systems analysis techniques are being used and a hierarchy of models has been developed to be used as tools for the planning and management of the basin. These include models for operational hydrology, as well as optimization and simulation models. This paper introduces the OPTIRED optimization model and the SGC simulation model. The models have been designed as general models so they can be used for other water resources systems.
Drought problems must be analyzed and solved at a regional level. Drought studies must focus not only on the hydrological characterization of the phenomena but also formulate the results in terms of engineering risk. In this paper concepts and methods to characterize droughts as regional processes are presented. The concepts of risk, reliability, resiliency and vulnerability are applied to characterize regional droughts and provide an engineering framework to deal with drought problems. A Portuguese river basin (rio Ave) is used to illustrate the concepts and techniques presented in the paper. A multivariate stochastic model is used to generate a synthetic series of monthly precipitation at six distinct regions defined in the river basin. Thirty-three years of monthly data are used to calibrate the parameters of this model. The generated series is used to perform the study of regional droughts from an engineering risk point of view.
We present a general modeling framework for robust optimization of linear programs with uncertainty in the values of the objective function coefficients and the values of the right-hand-side. The methodology presented here is straightforward applicable to the uncertainty in the constraints matrix coefficients as well. In contrast to traditional mathematical programming approaches, we model uncertainty by using scenarios to characterize the objective function and right-hand-side coefficients. Solutions are obtained for each scenario and, then, these individual scenario solutions are aggregated to yield a non-anticipative or implementable policy that minimizes the regret of wrong decisions. Such approach makes it possible to consider a variety of recourse decision types. A given solution is termed robust if it minimizes the expected difference over the set of scenarios between the objective function value of the solution and the objective function value of the optimal solution for each scenario, while satisfying certain non- anticipativity constraints. This approach results in a huge model with a submodel per scenario group at each period. Different augmented- Lagrangian strategies are proposed. Our approach allows the parallel solution of the decomposed submodels. Some ideas for problem solving are explored.
This chapter discusses the method of multipliers for equality constrained problems. By solving an approximate problem, an approximate solution of the original problem can be obtained. However, if a sequence of approximate problems can be constructed that converges in a well-defined sense to the original problem, then the corresponding sequence of approximate solutions would yield in the limit a solution of the original problem. The basic idea in penalty methods is to eliminate some or all of the constraints and add to the objective function a penalty term that prescribes a high cost to infeasible points. A parameter that determines the severity of the penalty and as a consequence the extent to which the resulting unconstrained problem approximates the original constrained problem is associated with the penalty methods.
This paper describes an approach for modeling two-stage stochastic programs that yields a form suitable for interior point algorithms. A staircase constraint structure is created by replacing first stage variables with sparse “split variables” in conjunction with side- constraints. Dense columns are thereby eliminated. The resulting model is larger than traditional stochastic programs, but computational savings are substantial — over a tenfold improvement for the problems tested. A series of experiments with stochastic networks drawn from financial planning demonstrates the attained efficiencies. Comparisons with MINOS and the dual block angular stochastic programming model are provided as benchmarks. The split variable approach is applicable to general two- stage stochastic programs and other dual block angular models.
This paper presents a methodology for the solution of multistage stochastic optimization problems, based on the approximation of the expected-cost-to-go functions of stochastic dynamic programming by piecewise linear functions. No state discretization is necessary, and the combinatorial “explosion” with the number of states (the well known “curse of dimensionality” of dynamic programming) is avoided. The piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to Benders cuts in a stochastic, multistage decomposition framework. A case study of optimal stochastic scheduling for a 39-reservoir system is presented and discussed.
Simulation and optimization are among the most commonly used elements in the OR toolkit. Often times, some of the data elements used to define an optimization problem are best described by random variables, yielding a stochastic program. If the distributions of the random variables cannot be specified precisely, one may have to resort to simulation to obtain observations of these random variables. In this paper, we present conditional stochastic decomposition (CSD), a method that may be construed as providing an algorithmic interface between simulation and optimization for the solution of stochastic linear programs with resource. Derived from the concept of the stochastic decomposition of such problems, CSD uses randomly generated observations with a Benders decomposition of the problem. In this paper, our method is analytically verified and graphically illustrated. In addition, CSD is used to solve several test problems that have appeared in the literature. Our computational experience suggests that CSD may be particularly well suited for situations in which randomly generated observations are difficult to obtain.
Consideration is given to a multistage stochastic program with recourse, with discrete distribution, quadratic objective function and linear inequality constraints. It is shown that under reasonable assumptions, solving such a program is equivalent to solving a nested sequence of piecewise quadratic programs and the author extends the algorithm presented in an earlier report to the multistage situation. The application of the method to an energy investment problem and report on the results of numerical experiments are presented.
This paper presents a parallel computation approach for the efficient solution of verylarge multistage linear and nonlinear network problems with random parameters. Theseproblems result from particular instances of models for the robust optimization of networkproblems with uncertainty in the values of the right-hand side and the objective functioncoefficients. The methodology considered here models the uncertainty using scenarios tocharacterize the random parameters. A scenario tree is generated and, through the use offull-recourse techniques, an implementable solution is obtained for each group of scenariosat each stage along the planning horizon.
This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.
A common,approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really strong enough to support a stochastic programming model, has been to set up and analyze a number of scenarios. The aim then is to identify trends and essential features on which a robust decision policy can be based. This paper develops for the rst time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios. The scenarios are bundled at various levels to reect the availability of information, and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight. Keywords: optimization under uncertainty, scenario analysis, progressive hedging, infor-
Planning is an integral part of water resources development and management. Whether or not particular plans or programs are eventually implemented, the planning process itself forces us to think about what we are or should be doing to address a particular set of problems or needs. The planning process should lead to a better understanding of what will happen if we do or do not act and, if we decide to do something, which of many possible actions is likely to be the best. Such planning requires information. Models are an increasingly important source of information, but such information is never complete, is rarely if ever certain, and hence is never a substitute for the judgment of experienced planners and managers. This paper attempts to reflect on the broader context in which models are used in the practice of planning, the inherent limitations of models in this broader context, and hence the challenges modelers have when addressing the information needs of planners and managers. Some criteria for judging the degree to which models have met these needs are reviewed. The paper concludes with some thoughts on the current state of planning models and supporting technology.
For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages.
Large-scale stochastic linear programs can be efficiently solved by using a blending of classical Benders decomposition and a relatively new technique called importance sampling. The paper demonstrates how such an approach can be effectively implemented on a parallel (Hypercube) multicomputer. Numerical results are presented.
We present a variant of Karmarkar's algorithm for block-angular structured linear programs, such as stochastic linear programs. By computing the projection efficiently, we give a worst-case bound on the order of the running time that can be an order of magnitude better than that of Karmarkar's standard algorithm. Further implications for approximations and very large-scale problems are given.
Nested decomposition is extended to the case of arborescent nonlinear programs. Duals of extensive forms of nonlinear multistage
stochastic programs constitute a particular class of those problems; the method is tested on a set of problems of that type.
A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.
Uncertainty in the parameters of a mathematical program may present a modeller with considerable difficulties. Most approaches in the stochastic programming literature place an apparent heavy data and computational burden on the user and as such are often intractable. Moreover, the models themselves are difficult to understand. This probably explains why one seldom sees a fundamentally stochastic model being solved using stochastic programming techniques. Instead, it is common practice to solve a deterministic model with different assumed scenarios for the random coefficients. In this paper we present a simple approach to solving a stochastic model, based on a particular method for combining such scenario solutions into a single, feasible policy. The approach is computationally simple and easy to understand. Because of its generality, it can handle multiple competing objectives, complex stochastic constraints and may be applied in contexts other than optimization. To illustrate our model, we consider two distinct, important applications: the optimal management of a hydro-thermal generating system and an application taken from portfolio optimization.
We describe and compare stochastic network optimization models for investment planning under uncertainty. Emphasis is placed on multiperiod a sset allocation and active portfolio management problems. Myopic as well as multiple period models are considered. In the case of multiperiod models, the uncertainty in asset returns filters into the constraint coefficient matrix, yielding a multi-scenario program formulation. Different scenario generation procedures are examined. The use of utility functions to reflect risk bearing attitudes results in nonlinear stochastic network models. We adopt a newly proposed decomposition procedure for solving these multiperiod stochastic programs. The performance of the models in simulations based on historical data is discussed.
Portfolio managers in the new fixed-income securities have to cope with various forms of uncertainty, in addition to the usual interest rate changes. Uncertainy in the timing and amount of cashflows, changes in the default and other risk premia and so on, complicate the portfolio manager's problem. We develop here a multi-period, dynamic, portfolio optimization model to address this problem. The model specifies a sequence of investment decisions over time that maximize the expected utility of return at the end of the planning horizon. The model is a two-stage stochastic program with recourse. The dynamics of interest rates, cashflow uncertainty, and liquidity, default and other risk premia, are explicitly modeled through postulated scenarios. Simulation procedures are developed to generate these scenarios. The optimization models are then integrated with the simulation procedures. Extensive validation experiments are carried out to establish the effectiveness of the model in dealing with uncertainty. In particular the model is compared against the popular portfolio immunization strategy, and against a portfolio based on mean-absolute deviation optimization.
This paper describes an algorithm for calculating optimal operating strategies in a multi-reservoir hydroelectric system, which can take into account inflow stochasticity and does not require discretization of the state space. The solution approach, called stochastic dual dynamic programming (SDDP), is based on the approximation of the expected-cost-to-go functions of stochastic dynamic programming at each stage by piecewise linear functions. These approximate functions are obtained from the dual solutions of the scheduling problem at each stage and may be interpreted as Benders' cuts in a stochastic, multistage decomposition algorithm. Case studies with the Brazilian systems are presented and discussed.
This paper describes a generic decision-support system (DSS) which was originally designed for the planning stage of dicision-making associated with complex river basins. Subsequently, it was expanded to incorporate modules relating to the operational stage of decision-making. Computer-assisted design modules allow any complex water-resource system to be represented in graphical form, giving access to geographically referenced databases and knowledge bases. The modelling capability includes basin simulation and optimization modules, an aquifer flow modelling module and two modules for risk assessment. The Segura and Tagus river basins have been used as case studies in the development and validation phases. The value of this DSS is demonstrated by the fact that both River Basin Agencies currently use a version for the efficient management of their water resources.
An augmented Lagrangian method is proposed for handling the common rows in large scale linear programming problems with block-diagonal structure and linking constraints. Using a diagonal quadratic approximation of the augmented Lagrangian one obtains subproblems that can be readily solved in parallel by a nonlinear primal-dual barrier method for convex separable programs. The combined augmented Lagrangian/barrier method applies in a natural way to stochastic programming and multicommodity networks.
In this study we present the weekly optimization of a power system consisting of hydro and thermal units. The mathematical model, a mixed integer (linear) programming problem, is discussed and the solution of the unit commitment and economic dispatch problem is determined over a period of a week on an hourly basis. The system contains 840 integer variables and 1344 continuous variables, with sparsity of 0.11%. This work allows both the water in the hydro unit and in the pump-storage unit to be costed.
This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.
The present stage of developments in stochastic programming gives already a good base for real-life applications. The possibility of using alternative models is studied on a small-size but meaningful example connected with water management of a real-life water resource system in Eastern Czechoslovakia. Both of the considered conceptually different stochastic programming models take into account intercorrelations within a group of random parameters and provide comparable optimal decisions. At the same time, these models are used for comparison of existing numerical procedures for stochastic programming, namely, approximation schemes that result in large-size linear programs, stochastic quasigradient methods and special techniques for handling joint chance constraints.
We present a modeling framework for a robust simulation of
multiperiod hydrothermal power management and energy trading under
uncertainty in generators availability, fuel procurement, transport and
stock costs, exogenous water inflow at river basins and energy demand
per subperiod at each time period of a given planning horizon. A
deterministic treatment of the problem provides unsatisfactory results
for medium term (1-2 years) planning horizon. We use a 2-stage scenario
analysis based on a partial recourse approach, where the generation
decision policy can be implemented for a given set of initial periods
and the solution for the other periods does not need to be anticipated
and, then, it depends on the scenario to occur. We have used an
augmented Lagrangean decomposition scheme by dualizing the coupling
constraints splitting control variables (fuel stock and stored water)
for the last implementable period. We present computational results
including different simulations of a Spanish generation subsystem
composed of 87 thermal generators, 57 hydro plants and reservoirs, 7
fuel types and 5 time periods. 160 scenarios are simultaneously
considered
The authors present a modeling framework for the robust solution
of hydroelectric power generation management problems with uncertainty
in the values of the water inflows and outflows. A deterministic
treatment of the problem provides unsatisfactory results, except for
very short time horizons. They describe a model based on scenario
analysis that allows a satisfactory treatment of uncertainty in the
model data for medium and long-term planning problems. Their approach
results in a huge model with a network submodel per scenario plus
coupling constraints. The size of the problem and the structure of the
constraints are adequate for the use of decomposition techniques and
parallel computation tools. They present computational results for both
sequential and parallel implementation versions of the codes, running on
a cluster of workstations. The codes have been tested on data obtained
from the reservoir network of Iberdrola, an electric utility owning 50%
of the total installed hydroelectric capacity of Spain, and generating
40% of the total energy demand
The aim is to show an application of stochastic dual dynamic
programming to seasonal planning in a part of the Norwegian
hydro-dominated power system. The subsystem under study has 35
reservoirs on 28 watercourses. It is found that for the study system the
new procedure is entirely feasible and gives good results. Two
implementation details are studied more closely: use of relaxation in
the solution of the subproblems, and a starting technique, called
pre-segment, to save iterations in the overall problem. Both are found
to have a significant effect on computer time. The test showed that dual
dynamic programming has reasonable computing times and can be a useful
tool in stochastic scheduling in a hydro-dominated system
Large-scale linear programming techniques in stochastic programming, in: Numerical Techniques for Stochastic Optimization
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Some considerations about the Influence of the Spanish Reservoirs' Operation on the Floods Felt Dounstream in Portugal
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A. Podda, G.M. Sechi, P. Susnik and P. Zuddas, Network algorithms for water resource system optimization, Hydrosoft'94, Port Carras, Greece (1994).
Generation expansion planning under uncertainty in demand, economic environment, generation availability and book life
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Algorithms for large scale water resource structured models with network flow kernels
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Drought contingency model for water control at Corps reservoirs
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A large-scale water resources network optimization algorithm, ICOTA Conference
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