**3.1 Optimization and simulation**

Reservoir system analysis models have traditionally been categorized as simulation, optimization, and hybrid combinations of both. Development and application of decisionsupport tools within the water resources development agencies in the United States have focused on simulation models. The published literature on modeling reservoir systems is dominated by optimization techniques.

The term *optimization* is used synonymously with *mathematical programming* to refer to a mathematical algorithm that computes a set of decision variable values which minimize or

Some reservoir/river system management models simulate water quality constituents along with water quantities. However, generalized water quality models, not covered in this chapter, are designed specifically for particular types of river and/or reservoir system water quality analyses. The typically relatively simple water quality features of the models explored in this chapter are secondary to their primary function of detailed modeling of

Modeling applications often involve a system of several models, utility software products, and databases used in combination. A reservoir/river system management model is itself a modeling system, which often serves as a component of a larger modeling system that may include watershed hydrology and river hydraulics models, water quality models, economic evaluation tools, statistical analysis methods, databases and various software tools for

The models discussed here are used for various purposes in a variety of settings. Planning studies may involve proposed construction projects or reallocations of storage capacity or other operational modifications at existing projects. Reservoir operating policies may be reevaluated periodically to assure responsiveness to current conditions and objectives. Studies may be motivated by drought conditions, major floods, water quality problems, or environmental losses. Operating plans for the next year or next season may be updated routinely based on a modeling system. Models support the administration of treaties, agreements, water right systems, and other water allocation mechanisms. Real-time modeling applications may involve decision-support for water management and use curtailment actions during droughts. Likewise, real-time flood control operations represent

**3. Models for analyzing development and operation of reservoir systems** 

Pioneering efforts in computer simulation of reservoir systems include U.S. Army Corps of Engineers studies of six reservoirs on the Missouri River initiated in 1953, International Boundary and Water Commission simulations of the Rio Grande in 1954, and a simulation study of the Nile River Basin in 1955 (Maass et al., 1966). Several books on modeling and analysis of reservoir operations are available (Votruba and Broza, 1989; Wurbs, 1996; ReVelle, 1999; Nagy et al., 2002). Labadie (2004) summarizes the extensive and complex research literature on reservoir system optimization models. Wurbs (1993, 2005a) presents state-of-the-art reviews of reservoir system analysis from a practical applications

Reservoir system analysis models have traditionally been categorized as simulation, optimization, and hybrid combinations of both. Development and application of decisionsupport tools within the water resources development agencies in the United States have focused on simulation models. The published literature on modeling reservoir systems is

The term *optimization* is used synonymously with *mathematical programming* to refer to a mathematical algorithm that computes a set of decision variable values which minimize or

water development, regulation, allocation, and management.

managing time series, spatial, and other types of data.

another type of application.

**3.1 Optimization and simulation** 

dominated by optimization techniques.

perspective.

maximize an objective function subject to constraints. Optimization is covered by water resources systems books (Karamouz et al., 2003; Jain & Singh, 2003; Simonovic, 2009) as well as numerous operations research and mathematics books. Thousands of journal and conference papers have been published since the 1960's on applying variations of linear programming, dynamic programming, gradient search algorithms, evolutionary search methods such as genetic algorithms, and other optimization techniques to reservoir system analysis problems. Various probabilistic methods for incorporating the stochastic nature of stream flows and other variables in the optimization models have been proposed (Labadie2004).

This chapter focuses on generalized simulation models. A simulation model is a representation of a system used to predict its behavior under a given set of conditions. Alternative executions of a simulation model are made to analyze the performance of the system under varying conditions, such as for alternative operating plans. Although optimization and simulation are two alternative modeling approaches with different characteristics, the distinction is obscured by the fact that models often contain elements of both. An optimization procedure may involve automated iterative executions of a simulation model. Optimization algorithms may be embedded within simulation models either to perform certain periphery computations or to provide the fundamental computational framework for the simulation model.

#### **3.2 Network flow linear programming**

Of the many mathematical programming methods available, linear programming (LP), particularly network flow LP, has been the method most often adopted in practical modeling applications in support of actual water management activities. The general LP formulation described in many mathematics and systems engineering textbooks is as follows.

$$\text{Minimize or Maximumize } Z = \sum\_{j=1}^{n} \mathbf{c}\_{j} \mathbf{x}\_{j} \tag{1}$$

$$\text{subject to } \Sigma \mathbf{a}\_{\vec{\mathbf{n}}} \mathbf{x}\_{\vec{\mathbf{j}}} \le \mathbf{b}\_{\mathbf{i}} \text{ for } \mathbf{i} = \mathbf{1}, \dots, \mathbf{m} \text{ and } \mathbf{j} = \mathbf{1}, \dots, \mathbf{n} \tag{2}$$

$$\mathbf{x}\_{\mathbf{j}} \ge \mathbf{0} \text{ for } \mathbf{j} = \mathbf{1}, \dots, \mathbf{n} \tag{3}$$

A LP solution algorithm finds values for the n decision variables xj that optimize an objective function subject to m constraints. The cj in the objective function equation and aij and bi in the constraint inequalities are constants.

A number of generalized reservoir system simulation models including several discussed later in this chapter are based on network flow programming, which is a computationally efficient form of LP. Network flow programming is applied to problems that can be formulated in a specified format representing a system as a network of nodes and arcs having certain characteristics. The general form of the formulation is as follows.

$$\text{Minimize or Maximize } \sum \mathbf{c}\_{\vec{\text{ij}}} \mathbf{q}\_{\vec{\text{ij}}} \text{ for all arcs } \tag{4}$$

Generalized Models of River System Development and Management 9

*oriented* presented at the beginning of this chapter. Many models are developed for a specific reservoir system rather than being generalized. Most of the numerous reservoir system optimization models reported in the literature were developed in university research studies

Under the sponsorship of the U.S. Army Corps of Engineers (USACE) Institute for Water Resources, Wurbs (1994, 1995) inventoried generalized water management models in the categories of demand forecasting, water distribution systems, ground-water, watershed runoff, stream hydraulics, river and reservoir water quality, and reservoir/river system operations. Wurbs (2005a) later reviewed generalized reservoir/river system operations models in greater detail for the USACE. Most of the models cited in these inventories were developed by government agencies in the United States and are in the public domain,

Public domain generalized modeling systems play important roles in many aspects of water management in the United States (Wurbs, 1998). Of the many water-related models used in the U.S., the Hydrologic Modeling System (HMS) and River Analysis System (RAS) are probably applied most extensively. These and other models developed by the Hydrologic Engineering Center (HEC) of the USACE are available at the website shown in Table 1. HEC-HMS watershed precipitation-runoff and HEC-RAS river hydraulics modeling systems are combined with HEC-ResSim in the integrated Corps Water Management System for modeling reservoir system operations described later. However, most applications of HEC-HMS and HEC-RAS by government agencies and consulting firms are for urban floodplain delineation or design of urban stormwater management facilities. The number of agencies and individuals that model operations of major multiple-purpose reservoir systems is much smaller than the number of users of HEC-HMS, HEC-RAS, and various other generalized models used for other purposes. However, generalized reservoir system models are

A Hydrologic Modeling Inventory (HMI) is maintained at Texas A&M University at the web site http://hydrologicmodels.tamu.edu/ in collaboration with the U.S. Bureau of Reclamation. The HMI is updated periodically, including an update during 2010. Models are organized in various categories with summary descriptions provided for each model. The HMI includes the MIKE BASIN, CALSIM, MODSIM, RiverWare, and WRAP models cited later in this chapter. In addition to developing and maintaining the HMI, Singh and Frevert (2006) edited a book inventorying models focused primarily on watershed hydrology but also including several river/reservoir system management models including RiverWare (Zagona et al., 2006), MODSIM (Labadie, 2006), and WRAP (Wurbs, 2006).

The following review focuses on several of the generalized reservoir/river management modeling systems that have been extensively applied by water management agencies and/or their consultants to support actual planning and/or operations decisions. The models cited below along with other similar models are discussed in more detail by Wurbs

This presentation focuses on modeling systems developed in the United States largely because the author's professional experience has been limited primarily to the United States. The U. S. is somewhat unique compared to most other countries in that generalized models are available in the public domain free-of-charge. Most, though not all, water management

and have not been applied by model-users other than the original model developers.

meaning they are available to interested model-users without charge.

significantly contributing to effective river basin management.

(2005a).

subject to ∑ q q0 ij ji −∑ = for all nodes (5)

$$1\_{\vec{\mathbf{u}}} \le \mathbf{q}\_{\vec{\mathbf{u}}\mathbf{j}} \le \mathbf{u}\_{\vec{\mathbf{u}}} \text{ for all arcs} \tag{6}$$

where qij is the flow rate in the arc connecting node i to node j cij is a penalty or weighting factor for qij lij is a lower bound on qij uij, is a upper bound on qij

The system is represented as a collection of nodes and arcs. For a reservoir/river system, the nodes are sites of reservoirs, diversions, stream tributary confluences, and other pertinent system features as illustrated by the control points of Figure 1. Nodes are connected by arcs or links representing the way flow is conveyed. Flow may represent a discharge rate, such as instream flows and diversions, or a change in storage per unit of time.

A solution algorithm determines the values of the flows qij in each arc which optimize an objective function subject to constraints including maintaining a mass balance at each node and not violating user-specified upper and lower bounds on the flows. The weighting factors cij in the objective function are defined in various ways such as unit costs in dollars or penalty or utility terms that provide mechanisms for expressing relative priorities. Each arc has three parameters: a weighting, penalty, or unit cost factor cij associated with qij; lower bound lij on qij; and an upper bound uij on qij. Network flow programming problems can be solved using conventional LP algorithms. However, the network flow format facilitates the use of much more computationally efficient algorithms that allow analysis of large problems with thousands of variables and constraints.

### **3.3 Caution in applying simplified representations of the real world**

Models are necessarily simplified representations of real world systems. Many references discuss shortcomings of the mathematical representations used to model systems of rivers and reservoirs. Rogers and Fiering (1986) outlined reasons that water management practitioners were reluctant to apply mathematical optimization algorithms proposed by researchers that included deficiencies in databases, modeling inadequacies, institutional resistance to change, and the fundamental insensitivity of many actual systems to wide variations in design choices. Iich (2009) explores limitations of network flow programming. McMahon (2009) highlights the various complexities of applying computer models and concludes that models can be quite useful despite their imperfections when considered in the context of data uncertainties, real-world operator experience, social priorities for water management, and externally imposed constraints on actual operational practice.

Powerful generalized software packages are playing increasingly important roles in water management. Computer models greatly contribute to effective water management. However, models must be applied carefully with professional judgment and good common sense. Model-users must have a thorough understanding of the computations performed by the model and the capabilities and limitations of the model in representing the real-world.

### **4. Generalized user-oriented river/reservoir system models**

Many hundreds of reservoir/river system models are described in the published literature. However, only a small number of these models fit the definitions of *generalized* and *user-*

subject to ∑ q q0 ij ji −∑ = for all nodes (5)

The system is represented as a collection of nodes and arcs. For a reservoir/river system, the nodes are sites of reservoirs, diversions, stream tributary confluences, and other pertinent system features as illustrated by the control points of Figure 1. Nodes are connected by arcs or links representing the way flow is conveyed. Flow may represent a discharge rate, such

A solution algorithm determines the values of the flows qij in each arc which optimize an objective function subject to constraints including maintaining a mass balance at each node and not violating user-specified upper and lower bounds on the flows. The weighting factors cij in the objective function are defined in various ways such as unit costs in dollars or penalty or utility terms that provide mechanisms for expressing relative priorities. Each arc has three parameters: a weighting, penalty, or unit cost factor cij associated with qij; lower bound lij on qij; and an upper bound uij on qij. Network flow programming problems can be solved using conventional LP algorithms. However, the network flow format facilitates the use of much more computationally efficient algorithms that allow analysis of

Models are necessarily simplified representations of real world systems. Many references discuss shortcomings of the mathematical representations used to model systems of rivers and reservoirs. Rogers and Fiering (1986) outlined reasons that water management practitioners were reluctant to apply mathematical optimization algorithms proposed by researchers that included deficiencies in databases, modeling inadequacies, institutional resistance to change, and the fundamental insensitivity of many actual systems to wide variations in design choices. Iich (2009) explores limitations of network flow programming. McMahon (2009) highlights the various complexities of applying computer models and concludes that models can be quite useful despite their imperfections when considered in the context of data uncertainties, real-world operator experience, social priorities for water

Powerful generalized software packages are playing increasingly important roles in water management. Computer models greatly contribute to effective water management. However, models must be applied carefully with professional judgment and good common sense. Model-users must have a thorough understanding of the computations performed by the model and the capabilities and limitations of the model in representing the real-world.

Many hundreds of reservoir/river system models are described in the published literature. However, only a small number of these models fit the definitions of *generalized* and *user-*

where qij is the flow rate in the arc connecting node i to node j

as instream flows and diversions, or a change in storage per unit of time.

large problems with thousands of variables and constraints.

**3.3 Caution in applying simplified representations of the real world** 

management, and externally imposed constraints on actual operational practice.

**4. Generalized user-oriented river/reservoir system models** 

cij is a penalty or weighting factor for qij

uij, is a upper bound on qij

lij is a lower bound on qij

ij ij ij lqu ≤ ≤ for all arcs (6)

*oriented* presented at the beginning of this chapter. Many models are developed for a specific reservoir system rather than being generalized. Most of the numerous reservoir system optimization models reported in the literature were developed in university research studies and have not been applied by model-users other than the original model developers.

Under the sponsorship of the U.S. Army Corps of Engineers (USACE) Institute for Water Resources, Wurbs (1994, 1995) inventoried generalized water management models in the categories of demand forecasting, water distribution systems, ground-water, watershed runoff, stream hydraulics, river and reservoir water quality, and reservoir/river system operations. Wurbs (2005a) later reviewed generalized reservoir/river system operations models in greater detail for the USACE. Most of the models cited in these inventories were developed by government agencies in the United States and are in the public domain, meaning they are available to interested model-users without charge.

Public domain generalized modeling systems play important roles in many aspects of water management in the United States (Wurbs, 1998). Of the many water-related models used in the U.S., the Hydrologic Modeling System (HMS) and River Analysis System (RAS) are probably applied most extensively. These and other models developed by the Hydrologic Engineering Center (HEC) of the USACE are available at the website shown in Table 1. HEC-HMS watershed precipitation-runoff and HEC-RAS river hydraulics modeling systems are combined with HEC-ResSim in the integrated Corps Water Management System for modeling reservoir system operations described later. However, most applications of HEC-HMS and HEC-RAS by government agencies and consulting firms are for urban floodplain delineation or design of urban stormwater management facilities. The number of agencies and individuals that model operations of major multiple-purpose reservoir systems is much smaller than the number of users of HEC-HMS, HEC-RAS, and various other generalized models used for other purposes. However, generalized reservoir system models are significantly contributing to effective river basin management.

A Hydrologic Modeling Inventory (HMI) is maintained at Texas A&M University at the web site http://hydrologicmodels.tamu.edu/ in collaboration with the U.S. Bureau of Reclamation. The HMI is updated periodically, including an update during 2010. Models are organized in various categories with summary descriptions provided for each model. The HMI includes the MIKE BASIN, CALSIM, MODSIM, RiverWare, and WRAP models cited later in this chapter. In addition to developing and maintaining the HMI, Singh and Frevert (2006) edited a book inventorying models focused primarily on watershed hydrology but also including several river/reservoir system management models including RiverWare (Zagona et al., 2006), MODSIM (Labadie, 2006), and WRAP (Wurbs, 2006).

The following review focuses on several of the generalized reservoir/river management modeling systems that have been extensively applied by water management agencies and/or their consultants to support actual planning and/or operations decisions. The models cited below along with other similar models are discussed in more detail by Wurbs (2005a).

This presentation focuses on modeling systems developed in the United States largely because the author's professional experience has been limited primarily to the United States. The U. S. is somewhat unique compared to most other countries in that generalized models are available in the public domain free-of-charge. Most, though not all, water management

Generalized Models of River System Development and Management 11

Water Resources Engineering Simulation Language (WRESL) was developed for the model to allow the user to express reservoir/river system operating requirements and constraints. The user-supplied statements written in the WRESL language are used by the model to define the LP formulation. Time series data are stored, manipulated, and plotted using the Hydrologic Engineering Center (1995, 2009) Data Storage System (HEC-DSS), which is also used with WRAP, discussed later, as well as with HEC-ResSim and other HEC simulation

The Texas Water Development Board (TWDB) Surface Water Resources Allocation Model and Multiple-Reservoir Simulation and Optimization Model simulate and optimize the operation of an interconnected system of reservoirs, hydroelectric power plants, pump canals, pipelines, and river reaches using a monthly computational time step. The daily time step MONITOR also simulates complex surface water storage and conveyance systems operated for hydroelectric power, water supply, and low flow augmentation (Martin, 1983, 1987). The TWDB has adopted the WRAP modeling system, described later, for statewide and regional planning studies conducted in recent years, replacing these early TWDB

The early TWDB models, original California Department of Water Resources model, and the original versions of HEC-PRM and MODSIM discussed later are all based on the same network flow programming solution algorithm. An early version of WRAP was also developed using the same algorithm, but another simulation approach was actually adopted for WRAP. The original solution algorithms in HEC-PRM and MODSIM were later replaced

Most of the large federal reservoirs in the U.S. were constructed and are operated by the U.S. Army Corps of Engineers (USACE) or U.S. Bureau of Reclamation (USBR). The USACE has over 500 reservoirs in operation across the nation as well as many navigation locks, hydropower plants, and flood control structures. The USACE operates essentially all of the reservoir projects that it has constructed. The USBR has transferred operation of many of its projects to non-federal sponsors upon completion of construction but continues to operate about 130 reservoirs and appurtenant structures in the 17 western states. The USACE plays a dominant role in the U.S. in operating large reservoir systems for navigation and flood control. The USBR water resources development program was originally founded upon constructing irrigation projects to support development of the western U.S. The responsibilities of the two agencies evolved over time to emphasize comprehensive

The USACE and USBR developed many models for specific reservoir systems during the 1950's-1970's (Wurbs, 1996, 2005a). Many of these system-specific models have since been replaced with generalized models. The USBR currently uses RiverWare and MODSIM, which are described later in this chapter, and several remaining system-specific models. The USACE Hydrologic Engineering Center (HEC) maintains a suite of generalized simulation models that are widely applied by water agencies, consulting firms, and universities throughout the U.S. and the world. This chapter later focuses on HEC-ResSim but several

with much more computationally efficient network flow programming algorithms.

**4.3 Models developed by federal agencies in the United States** 

multiple-purpose water resources management.

other HEC products are also noted below.

models.

models.

software products developed with government funding in the U. S. are made accessible to the professional water management community without charging a fee for the software.
