The textbook water resources planning problem is usually set up as a deterministic optimization problem, which does not explicitly consider uncertainty in hydrologic or economic parameters. The form of the objective function that is maximized (or minimized) varies. The most appropriate economic criterion might be to maximize total economic net benefits derived from the infrastructure in question, but other common formulations are to minimize investment costs subject to constraints on water supply reliability or firm hydropower generation capacity, or to maximize total benefits subject to cost or other constraints (ReVelle and McGarity, 1997). A variety of studies using different objective functions can be found in the literature (Loucks et al., 1981); a detailed summary is not presented here. Some of today’s water resources modeling packages – for example Riverware (Zagona et al., 2001) or WEAP (Yates et al., 2005) – can be parameterized to allow economic optimization.
Water resources planning textbooks acknowledge that optimization with economic objectives is insufficient to thoroughly evaluate the reliability of infrastructure projects (Maass et al., 1962; Loucks et al., 1981; Loucks et al., 2005). The key shortcoming associated with optimization is the
assumption of perfect foresight; water resources managers are allowed to know river flows with certainty, and to operate the system based on that knowledge. To address this limitation, more complete project planning (Figure 4) usually relies on studies that use operating rules developed using optimization to then conduct repeated simulations that test the sensitivity of results to
uncertainty about model parameters and system flows, or modification of the individual parameters or flow sequences used in the constraint structure of the optimization problem. As shown, these
planning studies usually assume that historical climate conditions will be maintained.
Figure 4. The stylized “traditional” framework for economic appraisal of water resources investments
The constraint structure of such optimization and simulation models is largely provided by system continuity equations. These continuity equations are written for both storage and intermediate flow nodes in the system, although the interpretation of these two node types varies slightly. For storage nodes in the system (i.e. lakes and reservoirs), the continuity equations take the form of
Historical hydrology Future Demand Projection(s)
Water Resources System
Optimization
Modeling Improved modeling
approach includes stochastic simulation Interpretation of results New Project(s) Historical Climate Sensitivity Analysis: Simulation Physical System Impacts
Hydropower, demands met, flood flows, etc. Converted to Economic Impacts
(via monetization) Costs: Capital, land, O&M, etc.
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equation 1 below (adapted from Loucks et al. (1981)). These equations are written for a time step of length t; all the variables in equation 1 such that all quantities correspond to the same length of time t.
Ss,t+1 = Ss,t + Qs,t - Rs,t - Es,t - Ls,t - Ds,t, (1) where Ss,t = storage in a reservoir at node s and time t;
Qs,t = inflow to node s and time t;
Rs,t = outflow from node s at time t;
Es,t = net evaporation losses from node s at time t;
Ls,t = seepage (storage in groundwater) losses at node s at time t;
Ds,t = water withdrawal (for consumptive use, or alternatively some fraction of which is
consumed, with the balance returned to the system further downstream) from node s at time t.
Equation 1 forms the basis for the reservoir models used in this research. For flow at modeled river nodes without storage infrastructures, there is no storage (Ss,t and Ss,t-1 are therefore
zero) and the evaporation and seepage terms Es,t and Ls,t are usually set equal to zero, such that the
constraint in equation 1 simplifies to:
Qs,t = Rs,t + Ds,t, 6
(2)
which simply says that inflows at node s are equal to outflows net of water withdrawals, at time t. Integrated water resources system models rely on a collection of such reservoir and flow nodes, which correspond to the configuration of the system in question and to the availability of flow gauge and water demand data. The flows of water into and out of those nodes are calculated based on the general relationships presented in equations 1 and 2. An illustrative representation of these equations around a central reservoir node is shown in Figure 5.
6
Note that this assumes that such loss terms are incorporated into the inflow quantity Qs,t; an alternative approach could include a loss term that combines evaporation and seepage and applies to the reach between nodes s-1 and s)
Figure 5. Illustrative representation of a reservoir node connected to ordinary flow nodes as modeled in a typical water resources system using equations 1-4.
In this model, inflows Qs,t can be represented as the combination of outflows from the
upstream node s-1 and the local increment to natural streamflow Ls,t between nodes s-1 and s:
Qs,t = Rs-1,t + Ls,t = Rs-1,t + Fs,t - Fs-1,t , (3) where Fs,t = historical flow measured at gauging station at node s and time t (Cohon, 1978).
Several aspects of equation 3 should be noted. The flow increment Ls,t is not simply the local
runoff into the river between nodes s-1 and s. Unless losses are otherwise modeled, it includes evaporation and seepage losses between these nodes, and may include unmeasured consumptive use by people or industries located along the river. As a result, it can sometimes be difficult to predict or measure the various components that make up Ls,t. In the absence of data and/or more
complicated hydraulic studies, modelers have often relied on two types of approximations for
specifying Ls,t. In the first approach, when the contribution of local runoff (from tributaries that join the
river system between two nodes in the model) is important, it is customary to back-calculate Ls,t from
flow records at gauging stations in the system, and to assume that a) losses from seepage, evaporation and unmonitored consumption of water are small and/or constant in time or not
necessary to include explicitly and b) the lag time which governs the movement of water through the system is implicitly accounted for in the incremental flow series Ls,t for all t. This approach can lead to
systematic bias in the measurement of inflows, especially if these factors vary over time.
Reservoir node (s) with dS/dt = Ss,t+1 - Ss,t Ordinary river node (s +1) Ordinary river node (s-1) Demand Ds,t Evaporation Es,t Outflow Rs,t Inflow Qs,t = Rs-1,t + Ls,t Seepage Ls,t Outflow Rs-1,t Local inflow Ls,t Demand Ds+1,t Outflow Rs+1,t Downstream system Upstream system Demand Ds-1,t Inflow Qs-1,t Inflow Qs+1,t Local inflow Ls+1,t
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On the other hand, when there are no important sources of local runoff (for example in rivers that pass through arid or semi-arid reaches or when all tributaries are fully gauged, flow calibration can be conducted using various techniques applied to the historical gauge records at nodes s-1 and s, such as regression or neural network modeling. These approaches typically express the observed flow in the historical record Fs,t at node s as some combination of n lagged flow terms at nodes s and
s-1 plus a constant term, for example:
Fs,t = Fs-1,t +…+ Fs-1,t-n + Fs-1,t-1 +…+ Fs-1,t-n + ks. (4)
Such models are subject to the same types of biases as those suggested using the first approach described above.
In the absence of more sophisticated hydraulic flow models (in many data-constrained systems), the planning problem thus relies on analysis using models composed of nodes for which inflows and outflows are determined using equations 1-4. In the simplest application of these models, the planner relies solely on historical flow sequences and explores system impacts (model outputs such as hydropower generated, water demands met, etc., as shown in Figure 4) that result from adding new infrastructures or withdrawals to the system, and/or shifting water allocations or operating strategies within it. The use of optimization for this analysis thus assumes perfect foresight of water flows, and enables allocation of water to its highest value uses (Harou et al., 2009). The analyst determines the storage, inflow, outflow and demand variable values that maximize net economic benefits. In many cases, planners may have additional objectives to the net benefits criterion, which can be considered by incorporating additional constraints, or modifying the objective function.
In sensitivity analysis, the planner may relax the assumption of perfect foresight, and conduct simulation using the historical record combined with realistic reservoir operating rules (often
developed using optimization procedures, and modified in iterative fashion to respond to results in the sensitivity analysis). At best, the planner will supplement these simulations based on the historical record with ones that use stochastic simulation, to determine the risks posed by natural flow variability and associated with the various favored system designs and/or operating rules. The literature is rich
with successful application of such tools, which can be found in textbooks and model reviews (Loucks et al., 1981; Yeh, 1985; Harou et al., 2009). In addition, stochastic linear programming or dynamic programming techniques can be used to help guide design of operating rules (Yeh, 1985). The objective function of these models is however generally limited to physical objectives, such as maximization of water supply reliability or hydropower generation, rather than economic criteria.
3.2 Challenges to using the basic planning framework
Historical context
In the past, it has been argued that the textbook planning model depicted in Figure 4 is very infrequently used in real-world systems design (Rogers and Fiering, 1986) and that reservoir operating rules have only rarely been developed based on results obtained from systems analysis techniques (Yeh, 1985). There is little reason to believe that this reality has changed considerably in the past twenty years (Harou et al., 2009). More precisely, it is clear that hydrological routing models and general water resources modeling tools are widely used by engineers and planners around the world, but that systems optimization applications – particularly economic optimization – remain rare outside the academic world. This is somewhat surprising considering the number of such applications in the water resources literature. A noteworthy exception to the lack of use of systems techniques is the application of the CALVIN model for water resources planning applications in California.
These model reviews offer a number of explanations for why use of such models has not been more widespread. Rogers and Fiering emphasize strongly institutional resistance to the use of systems optimization techniques, whether in the US or developing countries. Also in the US, these planning techniques developed too late to influence the construction of large infrastructures, such that the payoff of using them is now seen as small. Infrastructures were built piecemeal and river basin plans were rarely centrally planned. For developing countries, which have a chance to benefit from this model, key constraints include insufficient high-quality data to inspire confidence in results, and a lack of validation tools (because few models exist already which can be used to test the robustness of the optimal solutions). Lack of confidence in modeling results is also a consequence of the fact that
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models are frequently developed by outsiders who may not spend sufficient time training stakeholders to use the models effectively.
Rogers and Fiering also highlight technical and interpretation problems with the use of optimization models. Three specific issues appear to be most important: a) the challenge of how to deal with general planning uncertainty, b) the insensitivity of many systems to wide variations in design alternatives, and c) the existence of multiple near optimal alternatives. These three issues are of course interrelated, since uncertainty in optimization model parameters could lead one to different ‘optimal’ solutions depending on their assumed values. Multiple designs can often be justified, and optimization models, which are costly to develop, typically have little to say about the relative
strengths of different development options. Beyond this issue, Harou et al. (2009) point to the inability to easily represent social, political and/or environmental objectives and risk aversion preferences in the mathematical expressions of optimization models.
My own view is that optimization tools can in fact provide valuable information to the planning process for new water resources infrastructures. These models are particularly useful in narrowing the choice set of project alternatives, to allow a focus on specific regions or types of infrastructures that are likely to generate significant economic benefits. Optimization tools have in fact been used in this way in the Eastern Nile to direct attention towards storage projects in the Blue Nile canyon, by demonstrating the large hydropower benefits of such projects (Whittington et al., 2005; Wu and Whittington, 2006). Like other water resources modeling tools, however, they should not be used in isolation. Such models are unable to evaluate comprehensively system performance in the presence of large uncertainties. Since water resources projects tend to be capital intensive and deliver benefits over a long time horizon about which relatively little is known, uncertainties will always loom large.
Types of planning uncertainties
The focus of this research is on the use of planning models within the context of future uncertainty about the behavior and performance of water resources systems, so we now turn to this issue. Use of the basic textbook framework and equations 1-4 present a number of challenges to hydrologists and planners, even if it is assumed that historical conditions will be roughly maintained. First, the specification of Ds,t (water demand) is problematic; it is very difficult to forecast how
withdrawals for consumptive use will vary over the length of the planning horizon. Second, Ls,t (the
seepage term) is difficult to estimate without detailed hydraulic study, and is typically ignored or included using statistical models or other types of approximations. Third, the data used to simulate the continuity equations tend to be approximate; it is common to rely on average net evaporation rates from storage structures and water demand quantities, where the averages pertain to the
model’s time step, and may not vary from year-to-year. Finally, some relevant costs and benefits may be difficult to monetize, and so may be omitted entirely from the net benefits equation.
If the assumption of sustained historical flow conditions cannot be assumed, the problems with the basic approach shown in Figure 4 become more significant. Embedded in the appraisal framework are a series of assumptions related to climate, flow variability, future water demands, and the types of economic outputs derived from the system in question. Planning agencies and
consultants generally struggle to deal with the larger shortcomings of the standard assessment approach. This may be one of the reasons why ex post reviews of large water projects have often been so negative, contradicting the predictions of ex ante analyses. For example, the World Commission on Dams (2000) found that hydropower production from many large dams was often much lower than predicted in planning studies, suggesting that the standard methods used to set reservoir rules on reservoir releases do not yield realistic operational regimes.
It is instructive to think generally about the scope of these problems, first by unpacking the various components of the system water balance, and then by considering how the traditional planning framework is used to determine economic benefits. Here we focus most closely on the threats posed by climate change. Assume that the planner uses the standard analytical framework as best as she can with existing tools, relying on a combination of well-calibrated optimization and simulation models, and using stochastic methods to assess the probabilities associated with the outcomes of interest. The planner also incorporates the latest knowledge of climate change to inform her prediction of runoff into the system under climate change conditions.
Consider the components of the water balance equations that make up the water resources system (Figure 6). The variables of interest are Ss,t, Qs,t, Rs,t, Es,t, Ls,t, Ds,t, Ls,t. Precisely what
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Figure 6. Illustrative representation of nodes modeled in a water resources system, with description of typical assumptions
To begin, consider the local inflow contributions Ls,t that occur throughout the system and
contribute to the inflow terms Qs,t in equation 3. These terms are uncertain due to a variety of factors.
First, there could be technical problems related to gauge stability and the accuracy of gauge
measurements; in short, problems with the determination of Fs,t and Fs-1,t. These could be addressed
by improving data collection and gauging procedures. More importantly, there may be concerns over the stability of the “black box” calibrated model relationships developed based on equation 4. There is no guarantee that these relationships will continue to hold if the magnitude of flows within the system changes dramatically (as physical changes such as groundwater-surface water interactions and river channel flow may be permanently altered in important ways), or if there are important changes in non- measured consumptive water use along the river.
Third, questions could be raised about whether the historical record adequately represents the natural variability of runoff, especially if that record is short. There is ample evidence from periods prior to the instrumental record suggesting that natural variability in some rivers exceeds that
observed in flow records, even when these extend for more than 50 years.7 Fourth, there may be systematic changes in the quantity, variability, and timing of runoff due to climate or land use
changes. While state-of-the-art climate change analyses use sophisticated downscaling and modeling
7
A notable example is the Nile Basin (see Hassan, 1981; Shahin, 1985; Davies and Walsh, 1997; Nicholson, 2001; Nicholson and Yin, 2001; Marchant and Hooghiemstra, 2004).
Outflow Rs-1,t from upstream
Assumptions: Upstream operating rules, use patterns, precipitation, reservoir balances,
other upstream effects Reservoir node (s)
with dS/dt* = Ss,t+1 - Ss,t
Demand Ds,t
Assumptions: Growth predictions, crop-water use, cropping patterns
Evaporation Es,t
Assumptions: Precipitation, temperature
Inflow Qs,t = Rs-1,t + ∆Fs,t
Seepage Ls,t