Multiobjective optimization of urban water resources: Moving toward more practical solutions

Authors


Abstract

[1] The issue of drought security is of paramount importance for cities located in regions subject to severe prolonged droughts. The prospect of “running out of water” for an extended period would threaten the very existence of the city. Managing drought security for an urban water supply is a complex task involving trade-offs between conflicting objectives. In this paper a multiobjective optimization approach for urban water resource planning and operation is developed to overcome practically significant shortcomings identified in previous work. A case study based on the headworks system for Sydney (Australia) demonstrates the approach and highlights the potentially serious shortcomings of Pareto optimal solutions conditioned on short climate records, incomplete decision spaces, and constraints to which system response is sensitive. Where high levels of drought security are required, optimal solutions conditioned on short climate records are flawed. Our approach addresses drought security explicitly by identifying approximate optimal solutions in which the system does not “run dry” in severe droughts with expected return periods up to a nominated (typically large) value. In addition, it is shown that failure to optimize the full mix of interacting operational and infrastructure decisions and to explore the trade-offs associated with sensitive constraints can lead to significantly more costly solutions.

1. Introduction

[2] Recent Australian experience with arguably the severest drought on record and a potentially shifting climate has highlighted the vulnerability of urban water supplies to “running out of water.” As storages dwindled in the major urban centers of Sydney, southeast Queensland, Perth, Melbourne, and Adelaide, agencies responsible for urban water supply triggered drought contingency plans which started with the imposition of restrictions and, in most cases, led to the development of climate-independent sources of water such as desalination and wastewater reclamation. To secure Australian cities against drought, investments totaling tens of billions of dollars have been committed.

[3] In an Australian industry position paper describing a framework for urban water resource planning, Erlanger and Neal [2005, p. 6] state that “A safe and reliable water supply system is of utmost importance to the community. It is expected and understood that water utilities manage their water resources so that communities never run out of water.” Erlanger and Neal recognize that failure to supply minimum water needs for an extended period would most likely result in disastrous social and economic losses that could conceivably threaten the very existence of the urban community.

[4] Managing drought security in urban water supply is a complex and costly task, typically tackled using a two-pronged risk management approach, implementing short- and long-term options. The risk of exposure to severe drought is managed by application of long-term options such as policies that affect water use efficiency and provision of long-lead time infrastructure. Specifically, these long-term options control the probability of triggering short-term options or drought contingency plans, which may involve restrictions or rationing and short lead time (and usually very expensive) source augmentation or substitution.

[5] This paper considers the question, what is the best mix of long- and short-term options in an urban headworks system? Here “headworks” is used to refer to that part of the urban water supply infrastructure that harvests, stores and distributes water to major consumption zones. In seeking an answer to this question, several practical considerations deserve particular attention:

[6] 1. The maximization of drought security conflicts with the objectives of minimizing cost and environmental impacts. Recognizing the difficulty of quantifying environmental and social impacts solely in economic terms, multiobjective optimization [Deb, 2001] is needed to identify the trade-offs between conflicting objectives.

[7] 2. The consequences of an urban area running out of water are so severe that most systems are designed to have very high levels of security. This means that the probabilities of triggering drought contingency plans, particularly during extreme drought, are likely to be very small, while the probability of running out of water should be even lower. Because drought security criteria are often expressed in terms of probabilities of trigger events [Erlanger and Neal, 2005], it is vital that such probabilities be accurately estimated.

[8] 3. The performance of an urban headworks system is jointly dependent on the mix of short- and long-term options. Therefore, in a search for the best solution, it is essential that both short- and long-term options be evaluated jointly.

[9] Our review of the water resource optimization literature in section 2 concludes that no previous work has adequately addressed all these practical considerations. The principal contribution of this paper is twofold. First, the problem of optimizing the planning and management of urban water resources is formulated in a manner that addresses all these considerations. Specifically, the formulation addresses the practical challenges of identifying approximate Pareto optimal solutions involving the full mix of short- and long-term options, while realistically accounting for drought risk and the trade-offs between economic, security and environmental factors. Second, a case study demonstrates the practical importance of addressing these challenges. It shows that failure to address these challenges can result in solutions that are significantly inferior and of limited practical value to headworks managers.

[10] This paper is organized as follows: Following a review of the literature, the shortcomings of existing methods are identified and motivate a new approach that more fully deals with the requirements of practical multiobjective urban water resource planning. An extensive hypothetical case study based on the headworks system for Sydney (Australia) demonstrates the practical importance of adopting this new approach and illustrates the challenges and insights identifying the approximate Pareto optimal solutions that trade-off economic costs, environmental and drought-related social impacts.

2. Review of Urban Water Resources Optimization Works

[11] In the quest for securing urban water supplies against drought, water utilities use a mix of short- and long-term options to manage supply and demand. The short-term response to drought is embodied in the drought contingency plan (DCP). It is common practice to develop a staged DCP that progressively imposes severer restrictions on consumption while accessing emergency sources of water. The fundamental proposition is that the DCP reduces (and nowadays with the availability of climate–independent sources of water such as desalination, potentially eliminates) the risk of the system running out of water. A number of optimization studies have explored the benefit of imposing restrictions on demand to mitigate drought. For instance, Shih and ReVelle [1994, 1995] developed hedging rules for a single reservoir to reduce demand during drought. Tu et al. [2003, 2008] developed a mixed integer linear programming model that jointly considers reservoir release and hedging rules to minimize the shortages in current and future water supply. A limitation of these studies is that the social and economic cost of imposing restrictions was not addressed. Although imposing restrictions on demand reduces the risk of running out of water, frequent restrictions are not socially acceptable in major Australian cities [Erlanger and Neal, 2005].

[12] In response to reducing the frequency of restrictions yet maintaining security, water utilities consider a range of long-term options to reduce demand and increase supply. However, each option imposes a cost on the community and environment. A number of studies have developed models to find the least cost combination of short- and long-term options. Lund [1987] evaluated the integration of water conservation measures with capacity expansion options showing that costs could be minimized by applying conservation measures to delay water treatment plant expansion. Rubinstein and Ortolano [1984] demonstrated the application of demand management in long-term water supply planning. In a similar vein, Dziegielewski et al. [1992] developed a framework to balance long-term water supply alternatives with short-term drought responses in order to identify the most cost-effective investments offering long-term drought protection. Subsequently, Wilchfort and Lund [1997] minimized the expected cost of a combination of long-term and short-term options. Jenkins and Lund [2000] integrated shortage management and yield models to identify operating rules that minimize operating and shortage costs. However, as Dziegielewski et al. [1992] emphasized, the usefulness of these approaches depends on the accuracy and validity of costs associated with short-term demand reduction measures.

[13] Because of difficulties in estimating costs associated with restrictions or shortages, a number of studies [Randall et al., 1990; Ko et al., 1992; Liang et al., 1996; Kim et al., 2006; Reddy and Kumar, 2006; Chen et al., 2007; Yang et al., 2007; Kim et al., 2008; Chang and Chang, 2009; Kasprzyk et al., 2009] have adopted a multiobjective optimization approach. All of these studies except those by Yang et al. [2007] and Kasprzyk et al. [2009] have focused on short-term decisions associated with reservoir releases and restriction rules. However, there is an interaction between short- and long-term options as demonstrated by Lund [1987]. Yang et al. [2007] investigated the interaction between reservoir operating rules and reservoir capacity but did not incorporate any DCPs. Kasprzyk et al. [2009] focused on water marketing and portfolio-based management strategies in the context of a single reservoir system; they did not optimize infrastructure options nor DCPs.

[14] The rationale for multiobjective optimization is strengthened when environmental impacts are considered. Rivers downstream of dams typically experience a hydrologic regime change which can adversely impact on the health of riverine ecosystems [Shiau and Wu, 2007]. In recent years, in an effort to support sustainable ecosystems, releasing sufficient water to meet in-stream flow requirements (environmental flows) has received considerable attention from the water resources management community [Richter et al., 2006].

[15] In many past studies, environmental flows have been considered as a constraint [Tu et al., 2003, 2008]. However, this hides the trade-offs between cost, supply security and environmental impact. Suen and Eheart [2006] circumvented this shortcoming using multiobjective optimization to demonstrate the trade-off between human and ecosystem needs where the ecosystem objective was to maximize similarity between natural and flow released from the reservoir. Likewise, Shiau and Wu [2007] applied multiobjective optimization to optimize weir operation to balance ecosystem and human needs. However, these studies ignored the cost dimension and only focused on operational rules. By explicitly presenting the trade-offs between cost, drought security and environmental impact, Erlanger and Neal [2005] suggest communities may be prepared to pay more in return for less environmental damage.

[16] To evaluate the performance of an urban headworks system, a simulation model is typically constructed to model the behavior of the system in response to a time series of hydroclimatic and demand inputs; see Labadie [2004] for a review. The length of the time series used as input is critical. Given that urban systems typically operate with high levels of reliability, the time series must be long enough to enable a meaningful assessment of drought risks. The significance of this issue is best illustrated by an example. The annual probability of triggering a DCP, math formula, can be estimated by counting the number of years the DCP is triggered in a simulation and dividing by the number of simulation years N. Assuming annual independence, the standard error (SE) of the estimate based on binomial probability model considerations is

display math

[17] Suppose in a 100 year simulation, the DCP was triggered once. Then math formula and the standard error is 0.010. This large uncertainty can be presented more intuitively using return periods; it can be shown that the 95% confidence limits on the return period for the DCP trigger are 23 and 1580 years. This uncertainty arises solely because of the insufficient length of the simulation.

[18] This example suggests that evaluating drought risks and associated drought security criteria using simulation models with insufficiently long input time series borders on being meaningless with the results being sensitive to the choice of the input time series. Indeed, Ajami et al. [2008] suggest that use of historical data can lead to development of inefficient water management rules.

[19] One way to reduce this sampling uncertainty is to increase the length of the input time series. This can be done by generating long stochastic input time series by sampling from probability models fitted to historical data [Salas, 1992]. All but three of the reviewed multiobjective optimization applications to urban water resource systems used historical data. Though Kim et al. [2008] and Shiau [2009] used 100 and 40 years of synthetic data, respectively, such record lengths are considered completely inadequate for use with high-security urban systems. In their study of many-objective portfolio planning Kasprzyk et al. [2009] evaluated the performance of each proposed portfolio with 5000 10 year Monte Carlo samples. However, their Monte Carlo strategy involved resampling 10 year samples from a 33 year historical record, which is statistically unlikely to include severe drought; that said, Kasprzyk et al. did consider solution robustness by investigating sensitivity to initial conditions and extreme drought-demand scenarios. As a result, all of the reviewed studies suffer from the potentially serious limitation that the Pareto solutions are not robust in the sense of the solutions being sensitive to the choice of input data used in the simulation.

3. A More Practical Multiobjective Optimization Methodology for Urban Water Supply

[20] Section 3 formulates a multiobjective optimization methodology for an urban headworks system which addresses all the shortcomings identified in previous work on this subject. In section 4 a case study is used demonstrate the practical significance of addressing these shortcomings.

[21] Generally, the urban headworks multiobjective optimization problem can be formulated as follows:

display math

where x is a vector of decision variables that are to be optimized.

[22] The function math formula represents the headworks simulation model which takes as input math formula, a matrix of streamflow and climate values at multiple sites for an N year period, and math formula, a matrix of unrestricted demand at multiple sites for the same N year period, to produce simulation outputs math formula. There are many simulation models in the literature [Labadie, 2004] capable of simulating urban headworks systems. All that matters is that the model satisfactorily simulates the actual operation of the headworks system using information that would be available to the operators. The simulation outputs are used to evaluate math formula, the vector of criterion (or objective function) values. The function math formula is a vector of constraints.

[23] The constraint math formula is essential to the urban headworks optimization problem. It requires that no unplanned demand shortfalls, denoted by math formula, occur during the simulation. Unplanned shortfalls occur when the demand, permitted by the DCP, cannot be supplied: such shortfalls typically would occur when reservoirs run dry or when limitations in transfer capacity result in demand zones being supplied less than the minimum permitted by the DCP.

[24] The optimization problem (2) is largely intractable using classical optimization approaches which typically impose severe constraints on the form of the simulation model math formula and constraints math formula and therefore restrict the inclusion of variables in the decision space. However, the advent of evolutionary optimization algorithms (see Deb [2001]) has made solution of (2) significantly more tractable. In the water resources field, many researchers have recognized and exploited this opportunity; see the recent review by Labadie [2004] and Nicklow et al. [2010]. Of particular importance to this study is the greater freedom in specifying the decision vector; this enables optimization of the full mix of decision variables affecting the short and long term.

[25] Formulation (2) differs from previous formulations in the way it deals with drought security. The specification of drought security in the sense used by Erlanger and Neal [2005], namely, urban “communities never run out of water,” is problematic. Unless climate-independent sources of water (such as desalination) can guarantee a minimum supply, there will always be a finite probability that the system will run out of water. This is unavoidable. The best that one can do is manage the risk of running out of water.

[26] The optimal solutions in (2) are conditioned on the input QN. A more useful interpretation is that the Pareto optimal solutions (2) secure the system against droughts with return periods up to an expected value of N years. Seen this way, the expected return period N defines the drought security risk level for the system. As will be demonstrated, the explicit recognition of this risk level is vital to practical optimization outcomes.

4. Case Study

[27] Section 4 presents a case study to illustrate the application of the multiobjective optimization formulation (2) and to identify important insights arising from its application. It is motivated by the headworks system that supplies Sydney, Australia's largest city, serving a current population of 4.5 million.

4.1. Optimization Implementation Issues

[28] We briefly describe our implementation of (2). Similar to Cai et al. [2001], Cui and Kuczera [2005], and Yang et al. [2007] we adopted a two-level optimization approach. For the simulation model math formula, we adopted the WATHNET model [Kuczera, 1992; Kuczera et al., 2009], which uses network linear programming with side constraints to allocate water within the system. A scripting language within WATHNET enables the user to specify quite complex runtime functions to assign arc capacities and costs and side constraints to the network linear program. The decision vector x is accessible to all scripts and, therefore, can fully control the specification of the network linear program. While Cai et al. and Cui and Kuczera used a single-objective genetic algorithm, we had to select a multiobjective evolutionary algorithm. We adopted the epsilon dominance multiobjective evolutionary algorithm (εMOEA) of Laumanns et al. [2002] with modifications by Jefferson et al. [2005]. We selected εMOEA because we found in our experiments that it converged faster than the well-known NSGA-II (non-dominated sorting genetic algorithm). The scripts within WATHNET process the decision variable values exported by εMOEA to WATHNET and evaluate the criterion values exported by WATHNET to εMOEA.

[29] A major implementation issue was the computational time to solve the optimization problem (2). The total computational time is proportional to N, the number of years of simulation. One would expect water utilities to be risk averse to running out of water and, therefore, N would be much larger than historical record lengths and likely to range from 500 years upward. In this case study, a 10,000 year simulation using monthly time steps takes approximately 60 s on an Intel T7700 CPU running at 2.40 GHz. If the multiobjective optimization algorithm trials 20,000 different decision vectors, the total run time will be about 14 days. To reduce the turnaround time, parallel computing described by Cui and Kuczera [2005] was used to take advantage of the inherent parallelization of εMOEA.

4.2. Description of Sydney Headworks System

[30] The case study considers a simplified representation of the Sydney headworks system which, nonetheless, accounts for many of the interesting dynamics of the Sydney system. It considers several scenarios involving a hypothetical mix of short- and long-term options that cater for a future population of 7 million.

[31] Figure 1 presents a WATHNET schematic of the Sydney headworks system with the nodes labeled R representing reservoirs, S representing stream nodes, D representing demand zones, and W representing waste and sink nodes. A network of reservoirs, pumping stations and water treatment plants supplies water to two demand zones labeled Sydney and South in Figure 1. The existing system has a total storage capacity of 3343,487 ML (megaliters). Warragamba reservoir is the largest reservoir in the system with a capacity of 2031,000 ML. The Sydney demand zone, which serves approximately 90% of the population, is supplied by Warragamba reservoir together with a number of smaller reservoirs, Avon, Woronora, Cataract, Nepean and Cordeaux. In contrast, the South demand zone, which serves the remaining 10% of the population, is only supplied by Nepean and Avon reservoirs. An interbasin transfer scheme augments the natural inflows into Warragamba and Nepean-Avon reservoirs. The transfer scheme is located on the Shoalhaven River and involves a small pondage at Lake Yarrunga from which water is lifted over 500 m using two pumping stations to transfer water to Wingecarribee reservoir from where it can be transferred to Warragamba or Nepean reservoirs. The pump stations have a monthly transfer capacity of 46,600 ML.

Figure 1.

WATHNET schematic of Sydney water supply headworks system. The nodes labeled R represent reservoirs, S represents stream nodes, D represents demand zones, and W represents waste and sink nodes.

[32] For the purposes of this case study, environmental flow considerations are restricted to the Wollondilly River between Wingecarribe and Warragamba reservoirs. The primary issue is limiting high flows when pump transfers from the Shoalhaven are in progress, to avoid adverse impacts on riverine ecosystem function. Scott and Grant [1997] investigated the impacts of high flows on the riverine ecosystem and recommended maximum monthly regulated flows to avoid ecological impacts.

[33] In this case study, two options for augmenting the supply are available. The first is a new dam at Welcome Reef on the Shoalhaven River, upstream of Lake Yarrunga. The second is a desalination plant serving the Sydney demand zone. This latter option is strategically different to Welcome Reef in that it provides a climate-independent supply of water.

[34] The supply zones, Sydney and South, are each disaggregated into three demand nodes representing domestic indoor, outdoor watering and commercial/industrial consumption. In this case study, the DCP only restricts the outdoor watering usage; it is recognized that rationing during severe drought would be extended to the other usage categories.

4.3. Streamflow and Demand Data

[35] The Sydney system experiences high natural climate variability; for instance, the annual coefficient of variation for inflows to Warragamba reservoir is about 1.1. In view of this variability and the multiyear persistence of droughts, the reservoirs in the Sydney system have significant over-year carryover capacity. Therefore, when generating stochastic hydroclimate data for this system, it is important that the stochastic model produce sequences that are consistent with the multiyear observed statistics such as cumulative overlapping n year sums (with n ranging from 1 to 5 years). The following two-step algorithm was used to generate stochastic streamflow and climate data: (1) annual values were generated using the Matalas [1967] lag 1 multisite model calibrated to noncontiguous historical streamflow and climate records up to 84 years long using the missing-data EM algorithm [Kuczera, 1987], and (2) monthly values were obtained by disaggregating the annual flows using the method of fragments. Extensive testing has revealed this model produces multiyear statistics consistent with the observed data. Indeed, Thyer et al. [2006] argue that more complex stochastic models describing decadal to multidecadal-scale variability are not identifiable using historical records of the length available in the case study.

[36] To explore the sensitivity of the approximate optimal solutions to the choice of drought security return period, two sets of stochastic data were used: one 500 years long and the other 10,000 years long. It is noted that the 500 year series corresponds to the first 500 years of the 10,000 year series.

[37] Demand data for the 7 million population scenario was disaggregated into indoor domestic, outdoor domestic, and commercial/industrial types following the procedure described by Cui [2003]. Because outdoor domestic demand is correlated with rainfall, it was stochastically generated using the stochastically generated rainfall as input. This ensures that the higher outdoor water usage during droughts is preserved in the stochastic data.

4.4. Decision Variables

[38] A large number of options is available to ensure a secure water supply for Sydney's 7 million population scenario. In this case study, 11 decision variables, listed in Table 1, were identified as being potentially important. They are classified as either infrastructure (which corresponds to a physical asset) or operational (which affects the way the system is operated).

Table 1. List of Decision Variables
Decision VariableDescriptionLower LimitUpper LimitCategory
1Pump mark Warragamba0.31Operational
2Pump mark Avon0.31Operational
3Level 1 restriction trigger0.050.95Operational
4Trigger increment0.050.25Operational
5Desalination plant capacity (ML d−1)01000Infrastructure
6Desalination plant trigger0.050.95Operational
7Welcome Reef capacity (ML)01000000Infrastructure
8Warragamba base gain800012,000Operational
9Warragamba incremental gain10200Operational
10Maximum Wollondilly flow during Sept to Mar (ML month−1)12,200100,000Operational
11Maximum Wollondilly flow at other times (ML month−1)18,300100,000Operational

[39] Decisions x1 and x2 control the pump transfer of water from the Shoalhaven basin. x1 is a pump mark that defines the Warragamba storage fraction which triggers transfer of water from Shoalhaven to Warragamba; if the storage fraction in Warragamba is below the pump mark x1 at the start of a month, the maximum pump transfer capacity is activated. A separate pump mark x2 is applied to Avon on account of it being the main supply to the South demand zone.

[40] Decisions x3 and x4 define the first stage of the DCP. When the total storage fraction falls below the trigger x3, the first level of restrictions is imposed on outdoor domestic water use with a target reduction of 33%. If the total storage fraction falls below (x3 – x4), then the second level of restrictions is imposed with outdoor domestic water use reduced by 67%. If the total storage fraction falls below (x3 – 2x4), then the third level of restrictions is imposed with outdoor domestic water use totally banned.

[41] Decisions x5 and x6 define the second stage of the DCP. When the total storage fraction falls below the trigger x6, the already-constructed desalination plant with daily capacity of x5 ML d−1 is activated. Decision x7 defines the capacity of welcome reef reservoir.

[42] Decisions x8 and x9 define the priority for storing water in Warragamba. All the reservoirs in the WATHNET network linear program were assigned 20 carryover arcs which “store” water for the next time step. Each carryover arc has a capacity equal to 1/20 of the reservoir capacity and a gain (i.e., negative cost) defined by

display math

where BG is the base gain and IG is the incremental gain. Because of Warragamba's dominant storage, all reservoirs except Warragamba were assigned a base gain of 10,000 and an incremental gain of 100; this implements the so-called space rule which seeks to keep each reservoir with the same storage fraction. Decisions x8 and x9 define the base and incremental gain for Warragamba, respectively. Depending on the values assigned to x8 and x9, water may be preferentially stored in Warragamba or in the rest of the system.

[43] Finally, decisions x10 and x11 define the maximum monthly Wollondilly transfer capacity during September to March and at other times, respectively. The lower limit on these decisions corresponds to that recommended by Scott and Grant [1997]. These two decisions are active in the three-objective scenario and fixed in the other scenarios.

4.5. Objectives and Constraints

[44] Three objectives were judged to be relevant to the case study.

[45] 1. The first objective is to minimize frequency of restrictions (%). Erlanger and Neal [2005] state that the supply system should be capable of maintaining an adequate level of supply most of the time. Accordingly, the frequency of restrictions describes the fraction of the time consumers will not have an adequate level of supply. Cui and Kuczera [2005] used willingness-to-pay concepts to estimate the economic cost of restrictions from which they estimated the economically optimal frequency of restrictions. Here, the restriction frequency is made an explicit criterion in recognition of the difficulty of accurately estimating the economic cost and the political/social sensitivity that is associated with imposition of restrictions.

[46] 2. The second is to minimize the present worth cost ($). The present worth cost is the sum of capital and discounted expected operating costs and the costs of unplanned shortfalls. The capital cost represents the cost of building new infrastructure, which in this case study, is the Welcome Reef dam and/or the desalination plant. Table 2 summarizes the capital costs for Welcome Reef and the desalination plant. The capital cost model uses a binary function: if the asset is selected by the optimization, then the total cost is the sum of a fixed setup cost and a cost proportional to the size of the asset; however, if the asset is not selected, the capital cost is zero. The operating cost includes the costs for pumping transfers from the Shoalhaven and operation of the desalination plant. A 5% discount rate was used.

Table 2. Cost Summary for Infrastructure Decision Variables
Decision VariableFixed and Unit Costs
Desalination plant capacity (ML d−1)$1250,000,000 + $4000,000 ML d−1
Welcome Reef capacity (ML)$100,000,000 + $1000 ML−1 storage

[47] To ensure the DCP adequately copes with all droughts during the simulation period, solutions are constrained to avoid unplanned demand shortfalls. In this case study, an unplanned shortfall occurs when the system is unable to supply domestic indoor and commercial/industrial demand. This would occur when the highest restriction level, that bans all outdoor water use, is in force and the reservoirs become empty.

[48] The constraint on unplanned shortfalls is imposed using a penalty function approach. Here, a penalty of $100,000 per ML unplanned shortfall is added to the present worth cost. This penalty was selected to steer the optimization search away from solutions which allow reservoirs to “run dry” with consequent failure to supply minimum water needs.

[49] 3. The third objective is to minimize environmental stress on the Wollondilly River. In this case study, the Wollondilly River between Wingecarribee and Warragamba reservoirs has been identified as ecologically important. There is a vast literature that examines the ecological impacts of altered flow regimes. For example, Tharme [2003] documented over 200 individual environmental flow methodologies which have been utilized in 44 countries. Arthington et al. [2004] outlined the characteristics, strengths and limitations of the category of techniques termed holistic methodologies. In another study, Petts [2009] reviewed the advances in environmental flow science over the past 30 years. In more specific view, Dewson et al. [2007] reviewed literature on the consequences of natural low flows and artificially reduced flows on habitant conditions and on invertebrate community structure, behavior and biotic interactions. These studies underscore the difficulty in characterizing ecological response. As the purpose of this case study is illustrative, a notional response function is developed based on the field studies by Scott and Grant [1997] who identified potentially adverse impacts of altered flow regimes on platypus and water bird populations in the Wollondilly River. To avoid these impacts, they recommended that the maximum monthly regulated flow be limited to 18,300 ML during the winter months from April to August, and to 12,200 ML during the summer months. The ecological impact of exceeding these recommended maxima is not well understood [Grant and Temple-Smith, 2003]. Nonetheless, it is known that during the summer months, high flows have the highest impacts on the breeding of platypus and water bird populations, while the impacts of high flows are significantly less severe during the winter months. Accordingly, the following environmental stress metric was adopted to penalize the adoption of maximum regulated flow limits, x10 and x11, in excess of those recommended by Scott and Grant.

display math

where math formula is the actual regulated release in the Wollondilly in month m and math formula is the penalty for exceeding the recommended flow limits in month m. The environmental stress criterion is the sum of the monthly stresses over the simulation. Unlike the first two criteria, the environmental stress criterion is based on limited field data and relies on subjective judgments such as the impact in summer months is 5 times that of winter months and that the impact is cumulative. Consequently, the trade-offs between environmental stress and the other criteria need to be interpreted with the understanding that there is considerable epistemic uncertainty about the environmental impacts.

[50] Apart from the constraint on unplanned shortfalls, which was implemented using a penalty function approach, the only other constraints were the limits on the decision variables summarized in Table 1.

4.6. Case Study Scenarios

[51] Seven case study scenarios are used to illustrate the importance of using an optimization formulation that deals with the shortcomings identified in the literature review. The first two scenarios demonstrate the importance of jointly optimizing the full mix of decisions, particularly when there are interactions between short- and long-term and/or operational and infrastructure decision variables. Then, three different scenarios are used to demonstrate the influence of environmental constraints on system behavior and the beneficial aspects of treating an environmental constraint as an objective. Finally, a comparison of two sets of scenarios with different input data length highlights the practically serious shortcoming arising from use historical or short-length synthetic data when there are expectations of high levels of drought security. A summary of these scenarios is presented in Table 3 as a reference.

Table 3. Summary of Case Study Scenarios
ScenarioDecisionsOptimization CriteriaRecord Length (Years)Purpose
11–9; 10, 11 set to upper limitPresent worth cost, restriction frequency500Study consequence of fixing operational decisions
25, 6, 7; 10, 11 set to upper limitPresent worth cost, restriction frequency500Study consequence of fixing operational decisions
31–9; 10, 11 set to lower limitPresent worth cost, restriction frequency500Contrast use of environmental constraints against environmental trade-offs
41–9; 10, 11 set to upper limitPresent worth cost, restriction frequency500Contrast use of environmental constraints against environmental trade-offs
51–11Present worth cost, restriction frequency, environmental stress500Contrast use of environmental constraints against environmental trade-offs
61–9; 10, 11 set to upper limitPresent worth cost, restriction frequency500Contrast solutions with different levels of drought security
71–9; 10, 11 set to upper limitPresent worth cost, restriction frequency10,000Contrast solutions with different levels of drought security

[52] For all seven scenarios, the εMOEA algorithm was run with 10 different initial random number seed. Recognizing that an evolutionary algorithm cannot guarantee convergence to the Pareto optimal solutions, the approximate Pareto optimal solutions are taken here to be the nondominated solutions from the 10 runs. The first four scenarios were run for 30,000 generations, while scenario 5 was run for 100,000 generations and scenarios 6 and 7 for 10,000 generations; εMOEA terminated its search if the maximum number of generations was reached or if the Pareto solutions did not change after 1000 generations. The εMOEA parameters were tuned to ensure good coverage and diversity along the Pareto front; the tuned parameters were mutation rate of 0.01, crossover rate of 1.00, population size of 100, and the hyperbox epsilons for the restriction criterion of 0.005, for the present worth cost criterion $1000 and for environmental stress criterion 0.001.

4.7. Joint Optimization of Operating and Infrastructure Decision Variables: Scenarios 1 and 2

[53] Section 4.7 compares the approximate Pareto optimal solutions for two scenarios which differ in the mix of decisions to be optimized. In scenario 1, all operational and infrastructure decisions were optimized except for decisions 10 and 11 which were fixed at their upper limit. In contrast, in scenario 2, the optimization problem is akin to asking what is the best capacity expansion option with the rest of the system operated as normal. Accordingly, only two infrastructure decisions, the desalination plant and Welcome Reef reservoir capacities, and the desalination operational decision, the desalination plant trigger, were optimized; the remaining operational decisions were set to the following values guided by the desire to minimize operating costs and the frequency of restrictions: x1 = 0.3; x2 = 0.3; x3 = 0.5; x4 = 0.05; x8 = 10,000; x9 = 100. In each scenario, two objectives were considered, namely, minimizing the present worth cost and restriction frequency.

[54] Figure 2 shows the approximate Pareto optimal fronts for the two scenarios. There is a considerable gap between the two Pareto fronts with the scenario 1 Pareto front dominating the scenario 2 front. By optimizing all the operational decisions, considerably lower present worth costs can be achieved for the same restriction frequency. Clearly fixing some of the operational decisions severely limited the ability of the optimization to take full advantage of the desalination plant and Welcome Reef. In scenario 2, because the restriction trigger x3 was set to 0.50, it was impossible to produce outcomes with a restriction frequency greater than 20%. Likewise, the restriction frequency could not fall below 2.5%, because the desalination plant and Welcome Reef capacities were at their upper bounds.

Figure 2.

Approximate Pareto optimal fronts for scenario 1 (all decisions optimized) and scenario 2 (two infrastructure decisions and one operational decision optimized).

[55] While the gap between the Pareto fronts is dependent on the choice of values assigned to the decisions not optimized in scenario 2, the important conclusion to be drawn is that when operational and infrastructure decisions interact, the failure to optimize all decisions can lead to inferior outcomes. Importantly, the ability to solve the optimization problem (2) makes it practically feasible to explore the whole decision space.

4.8. Moving From Environmental Constraints to Trade-Offs: Scenarios 3–5

[56] Section 4.8 investigates the insights and benefits that arise from considering environmental trade-offs rather than imposing environmental constraints. Three scenarios are considered. The first two, scenarios 3 and 4, establish the sensitivity of the system to decisions x10 and x11, which determine maximum regulated flows in the Wollondilly River.

4.8.1. Sensitivity to Environmental Flow Constraints: Scenarios 3 and 4

[57] The sensitivity of the system to the specification of environmental flow constraints is explored using scenarios 3 and 4. In scenario 3, x10 and x11 are fixed at the values recommended by Scott and Grant [1997], while in scenario 4, x10 and x11 are fixed at an arbitrarily large value that would not impose constraint on transfers from the Shoalhaven. Scenario 3 imposes nominally no environmental stress, while scenario 4 would allow imposition of maximal environmental stress.

[58] Figure 3 presents the approximate Pareto optimal solutions for scenarios 3 and 4. The imposition of the environmental flow constraint on the Wollondilly River substantially shifts the Pareto front outward. For example, for a 10% restriction frequency, the imposition of the Wollondilly environmental flow constraint increases the present worth cost by ∼$1,600 million. The reason for this sensitivity will be explained subsequently. Here the point to be made is that the imposition of environmental flow constraints can hide important trade-offs [Suen and Eheart, 2006] and consequently it may be more helpful to treat environmental needs as a criterion, albeit poorly defined, to better understand the trade-offs with other criteria.

Figure 3.

Approximate Pareto optimal front for scenario 3 (with environmental flow constraints) and scenario 4 (without environmental flow constraints). The solid symbols represent solutions that include the desalination plant.

[59] The solutions presented by the solid symbols in Figure 3 are the only solutions in which the desalination plant has been selected; the steepening of the Pareto front just before a desalination plant is included in the solution set is attributed to the high fixed cost of constructing the desalination plant. What is particularly striking about the filled solutions is the sensitivity of the desalination plant to the Wollondilly environmental flow constraint. When no constraint is imposed (scenario 4), the desalination plant is only selected if solutions produce restriction frequencies of less than 5%. In contrast, if the constraint is imposed (scenario 3), a desalination plant is selected for all solutions with restriction frequencies less than 17%.

[60] Each solution on the Pareto front in Figure 3 corresponds to a particular decision vector. To gain a better understanding of the sensitivity of the solutions to the Wollondilly constraint, the relationships between subsets of the approximate Pareto optimal decisions for scenarios 3 and 4 are analyzed. Figure 4a shows the relationship between decision x5, the desalination plant capacity, and x6, the desalination plant trigger, for the solutions that adopt desalination. Regardless of the desalination plant capacity, the trigger level lies between 0.5 and 0.75 for both scenarios. However, when there is no constraint on Wollondilly releases (scenario 4), the desalination capacity lies in the range 200 to 300 ML d−1. In contrast, for scenario 3, the capacity ranges from 100 to 500 ML d−1 suggesting interaction with other variables. Figure 4b shows the relationship between decision x8, the base gain for Warragamba, and x9, its incremental gain. When the Wollondilly flow constraint is imposed (scenario 3), all base gains except one, are greater than or very close to 10,000 and most of the incremental gains are greater than 100. This means the WATHNET simulation model assigns the highest preference to keeping water in Warragamba and thus will seek to supply the Sydney zone from other sources before accessing Warragamba. In contrast, when no constraint is imposed (scenario 4), the situation is more complex, with a negative linear relationship between base and incremental gain: increasing the base gain by 100 is offset by a reduction in incremental gain of about 50. This suggests there are complex interactions between the Warragamba gains and other decisions and, therefore, no simple interpretation can be made. Figure 4c displays the relationship between the capacity of Welcome Reef and restriction frequency. When the Wollondilly flow constraint is imposed, Welcome Reef has a consistently smaller capacity reflecting the fact that the Wollondilly constraint limits the utility of storage on the Shoalhaven River. Figure 4d shows the relationship between the decision x4, the level 1 restriction trigger, and restriction frequency. There is little difference between scenarios 3 and 4, with a lower trigger associated with lower restriction frequencies. Furthermore, in virtually all cases, decision x5 was at its lower limit of 0.05. This suggests that for the adopted criteria, the optimal strategy is to impose the severest restrictions as soon as possible; that said, such a strategy would unlikely to be socially acceptable.

Figure 4.

Comparison of approximate Pareto optimal decisions for scenario 3 (with environmental flow constraints) and scenario 4 (without environmental flow constraints): (a) desalination plant capacity (ML d−1) versus desalination plant trigger level, (b) Warragamba base and incremental gain, (c) Welcome Reef capacity as a function of restriction frequency, and (d) restriction frequency versus level 1 restriction trigger.

[61] Figure 5 displays the relationship between the two pump marks, x1 and x2, and the level 1 restriction trigger x4 against the restriction frequency for each scenario. For the scenario with no environmental constraint (scenario 4), the Warragamba pump marks associated with the lowest restriction frequency are low because the presence of the desalination plant reduces the dependence of the system on transfers from the Shoalhaven. Without the desalination plant, however, the Warragamba pump mark jumps close to 1 and then declines to about 0.3 as the restriction frequency increases. With the exception of some interaction with Warragamba pump marks for restriction frequencies between 10% and 20%, the Avon pump mark largely lies in the range 0.3 to 0.4. In contrast, scenario 3 (environmental constraint imposed) reveals a very different behavior for the Warragamba pump mark which is mainly in excess of 0.7. This suggests the Wollondilly flow constraint forces transfers to start much earlier in the Warragamba drawdown. As a result, there is a higher chance that Warragamba will spill resulting in a wasted transfer and an overall higher pump cost than would occur with a lower pump mark.

Figure 5.

Plot of Warragamba and Avon pump marks and level 1 restriction trigger against restriction frequency for scenario 3 (with environmental flow constraints) and scenario 4 (without environmental flow constraints).

[62] The comparison of the scenario 3 and 4 solutions highlights the complexity of the relationships between decisions. While Figure 3 displays a substantial cost trade-off between scenarios 3 and 4, the analysis of Figures 4 and 5 suggests that it is not straightforward to interpret the difference in solutions. The interactions between decisions appear to involve, in many cases, more than two variables. This underscores the importance of conducting optimization using the full decision space.

4.8.2. Three-Objective Case Study: Scenario 5

[63] Scenarios 3 and 4 represent the extremes in terms of environmental stress on the Wollondilly River and suggest there is a significant trade-off between the environmental stress and other objectives. For this reason, it is worth exploring the trade-offs more fully by undertaking an optimization using all three objectives, namely, minimize restriction frequency, minimize present worth cost and minimize environmental stress on the Wollondilly River; this represents scenario 5.

[64] Figure 6 presents all the approximate Pareto optimal solutions plotted against restriction frequency and present worth cost with a color-coded scale for environmental stress. For a given restriction frequency, reducing the environmental stress increases the present worth cost. However, what is of greater interest and practical significance is the variability in trade-offs between present worth cost and environmental stress as restriction frequency changes. For restriction frequencies less than 7%, the difference in present worth cost between the best and worst environmental outcomes ranges between $600 to $700 million. However, between 7% and 18%, the present worth cost difference increases by about a factor of 2; this coincides with the transition to desalination. Beyond restrictions frequencies of 18%, no solution uses desalination and the cost gap rapidly closes.

Figure 6.

Approximate Pareto optimal solutions for scenario 5 plotted against restriction frequency and present worth cost. The color code describes the environmental stress.

4.9. The Drought Security Cost Trade-Off: Scenarios 6 and 7

[65] So far all the scenarios were based on an expected drought security return period of 500 years; that is, the approximate Pareto optimal solutions ensure the system can cope with droughts having an expected return period up to 500 years without running out of water. In Section 4.9, the sensitivity of the approximate Pareto optimal solutions to the drought security return period is examined. We consider two scenarios, 6 and 7, which use 500 and 10,000 year hydroclimate time series, respectively. For both scenarios, restriction frequency and present worth cost are minimized with no environmental constraint on releases in the Wollondilly River.

[66] As a prelude, one of the approximate Pareto optimal solutions for the 500 year scenario was simulated using the 10,000 year input series: this solution had no desalination and a relatively high restriction frequency of 22%. Figure 7a shows the plot of total storage for the most severe drought in the first 500 years. It is observed that during this drought, the system ran dry but just avoided unplanned shortfalls. The fact that the optimized decisions just avoided unplanned shortfalls in the 500 year scenario would suggest the system becomes vulnerable when exposed to severer droughts. This is confirmed in Figure 7b, which shows a plot of unplanned shortfall expressed as a percentage of total demand for the 10,000 year scenario. The limitations of the 500 year return period solution are abundantly clear. Unplanned shortfalls of up to 95% of demand, sustained for periods up to 6 months, would most likely lead to catastrophic outcomes. This vulnerability is unavoidable in systems totally reliant on climate-dependent sources of water.

Figure 7.

Pareto solution from scenario 6: (a) Time series of total storage during the most severe drought in the first 500 years and (b) time series of unplanned shortfalls, expressed as a percentage of total demand for 10,000 years.

[67] Figure 8 presents the approximate Pareto optimal fronts for the present worth cost and restriction frequency criteria for the 500 and 10,000 year scenarios. The shift in the Pareto front is striking. For a 10% restriction frequency, the present worth cost increases from $2,600 million to $8,300 million. This large jump in cost arises from the need to avoid unplanned shortfalls in droughts considerably more severe than experienced in the 500 year scenario. To better understand the impact of using the 10,000 year scenario, Table 4 presents three pairs of solutions on the Pareto fronts selected so that each pair had a similar restriction frequency. There are four key differences between the 500 and 10,000 year scenarios:

Figure 8.

Comparison of approximate Pareto frontier for scenario 6 (500 year record) and scenario 7 (10,000 year record).

Table 4. Summary of Labeled Solutions on Pareto Fronts in Figure 8
 Solution
 6-17-16-27-26-37-3
Objectives      
 Drought return period (years)50010,00050010,00050010,000
 Restriction frequency (%)131322224545
 Present worth cost ($m)240081401860791015707450
Decisions      
 Pump mark Warragamba0.400.650.300.500.300.53
 Pump mark Avon0.650.310.300.310.300.32
 Level 1 restriction trigger0.550.560.630.610.820.80
 Trigger increment0.050.050.050.050.050.05
 Desalination plant capacity (ML d−1)092309800922
 Desalination plant trigger0.520.500.54
 Welcome Reef capacity (ML)994,342970,629981,469890,867873,783894,823
 Warragamba base gain10,06410,26710,073998410,07310,141
 Warragamba incremental gain104111072410213

[68] 1. For the 500 year scenario no desalination plant was selected, while in the 10,000 year scenario, all solutions had the desalination plant capacity set close to the upper limit of 1000 ML d−1.

[69] 2. The Warragamba pump mark jumps from 30% in the 500 year scenario to 68% in the 10,000 year scenario to commence transfers from the Shoalhaven much earlier in any drought.

[70] 3. For the 10,000 year scenario, the Warragamba base and incremental gains ensure that Warragamba is preferentially drawn down. This strategy triggers transfers from the Shoalhaven earlier than if all reservoirs were balanced according to the space rule.

[71] 4. All three solutions for the 500 year scenario opt for Welcome Reef close to its maximum capacity of 1,000,000 ML. In contrast, for the 10,000 year scenario, the size of Welcome Reef decreases with increasing restriction frequency because the desalination plant capacity remains essentially unchanged close its maximum capacity.

4.10. Discussion

[72] The seven case study scenarios have demonstrated the value of an optimization methodology that addresses the three shortcomings identified in the previous literature. The overarching conclusion from the case study is that, in the case of urban headworks systems, failure to optimize the full mix of operational and infrastructure decisions, failure to allow for droughts with high-return periods and failure to explore trade-offs implicit in “soft constraints” can produce demonstrably inferior solutions.

[73] The issue of drought security is of paramount importance for cities located in regions subject to severe prolonged droughts. The prospect of running out of water for an extended period would threaten the very existence of the city and its social and economic fabric. The case study highlighted the potentially serious shortcomings of solutions based on short historical or synthetic streamflow records. Very different approximately optimal solutions were found when securing against an expected 500 and a 10,000 year drought. For a 10% restriction frequency, the optimal solution for the 10,000 year record incurred a present worth cost over three times that for the 500 year record. While the optimal solution for the 500 year scenario just avoided unplanned shortfalls in the worst drought of the 500 year record, the more severe droughts in the 10,000 year record resulted in extended and unsustainable periods of unplanned shortfalls. It is therefore critically important that simulation record lengths, over which system performance is evaluated, are sufficiently long to match drought security expectations. It is not an uncommon industry practice to design for the worst historical drought and then add a reserve [e.g., Cloke and Samra, 2009]. Such an approach does not communicate the risk of running out of water, which in the case of a large urban system could be potentially catastrophic. By rerunning the optimization problem (2) for different record lengths (as done in scenarios 6 and 7), the trade-off between drought security and other criteria can be explicitly explored to enable an informed decision.

[74] The issue of confidence in the drought return period deserves comment. The Pareto optimal solutions given by (2) secure the system against droughts with return periods up to an expected value of N years; the actual return period may differ from the expected value. If one needs more confidence in the return period, the following preconditioning algorithm can be used to reduce uncertainty in the return period for the N year record: Generate M replicates of length N years; rank the replicates using a suitable low-flow statistic such as the minimum k year sum; select the replicate corresponding to the median rank.

[75] Even if drought security is adequately accounted for, failure to optimize the full mix of infrastructure and operational decisions and explore trade-offs implicit in “soft” constraints can result in solutions that involve far greater economic cost than is necessary. The issue of “soft” constraints can be particularly challenging. To transform an environmental constraint into an objective requires the formulation of an environmental response function that maps decisions into a meaningful metric of environmental response. It is widely accepted that this is a difficult task constrained by limited data and difficulties in identifying causal mechanisms. We acknowledge that the environmental stress function given by (4) is subjective and most likely incomplete. Accordingly, the main insight is not quantitative but an awareness that there are very significant trade-offs between environmental response and cost and that these trade-offs are a nonlinear function of restriction frequency. In view of this, a strong case could be made to invest in studies to better inform the specification of the environmental response function and so better inform the trade-off process. Seen from this perspective, the optimization methodology advanced in this study is part of an iterative process involving progressive refinement of information and objectives.

5. Conclusions

[76] This paper has formulated and demonstrated a multiobjective optimization methodology for urban water supply headworks planning and management that produces solutions with demonstrably greater practical value. Its principal contribution is the identification of three practically significant shortcomings in the literature and a methodology to resolve these shortcomings. The case study motivated by the headworks system for Sydney, Australia, demonstrated the significant manner in which these shortcomings can compromise the practical value of so-called Pareto optimal solutions.

[77] Urban headworks systems are typically planned and operated in a manner that ensures a very low risk of running out of water or catastrophic water shortages. The case study demonstrates the very considerable sensitivity of Pareto optimal solutions to the return period of the worst drought. While this may seem self-evident, the literature has largely ignored this issue and repeatedly published optimal solutions conditioned on historical records or short stochastic records. Where high levels of drought security are required, such solutions are flawed and methodologies that produce such solutions should be avoided. Our approach addresses drought security explicitly. It identifies near-optimal solutions that are constrained so that the system does not “run dry” in severe droughts with expected return periods up to a specified value.

[78] In many cases, the operating rules that control the operation of the headworks system are conditioned on the system infrastructure. It is therefore vital in optimization studies, in which new system infrastructure is to be added or existing infrastructure modified, that key operating rules are optimized jointly with the infrastructure options. While this may substantially increase the dimension of the decision space, it is not worth the risk of obtaining significantly inferior solutions. In a similar vein, the imposition of “soft” constraints, such as the environmental flow constraints in the case study, runs the risk of missing potentially good solutions. In the case of constraints to which system performance is sensitive, their reformulation as objectives within a multiobjective optimization framework can enable a more thorough and computationally efficient assessment of trade-offs, an outcome that would be difficult to achieve using conventional sensitivity analysis.

[79] With the advent of evolutionary multiobjective optimization algorithms and readily available parallel computing technology, it is becoming practically feasible to optimize urban headworks systems in a manner that produces results of practical relevance to managers. We have shown that not properly dealing with important characteristics of urban headworks systems can produce so-called Pareto optimal solutions that are severely compromised. We strongly advocate that optimization studies involving urban headworks systems adequately deal with the shortcomings we have identified.

[80] It is important that “good” solutions be found for the “right” problem. This study has made a significant contribution toward this goal by addressing identifiable shortcomings. However, in practice, planners have to deal with scenario uncertainty in which assumptions have to be made about model structure and exogenous factors such as system forcing and political and social constraints. There can be a considerable and difficult-to-quantify uncertainty about these scenarios. In such cases, one can argue that “good” solutions need to produce good outcomes across the range of plausible scenarios; in other words, “good” solutions need to be robust [Matalas and Fiering, 1977]. There is growing literature on robust optimization (see, e.g., Deb and Gupta [2006]) whose concepts can be applied to the urban headworks problem.

Acknowledgments

[81] This work was financially supported by the eWater Cooperative Research Centre. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not reflect the views of the eWater CRC or any other organization.

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