Economic Valuation of Inter-Annual

Reservoir Storage in Water Resources

Systems:

Theory, Development, and Applications

A thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Science and Engineering

1/8/2019

Majed Khadem

School of Mechanical, Aerospace and Civil Engineering

2

Contents

Abstract ................................................................................................................................ 11

Declaration ........................................................................................................................... 13

Copyright Statement ............................................................................................................ 14

Acknowledgements .............................................................................................................. 15

The Author ........................................................................................................................... 16

1 Chapter one: Introduction ............................................................................................ 17

1.1 Background ........................................................................................................... 18

1.1.1 Reservoir operation ........................................................................................ 18

1.1.2 Water as an economic good ........................................................................... 20

1.1.3 Hydro-economic modelling ........................................................................... 22

1.1.4 Hydrological foresight ................................................................................... 24

1.1.5 Deterministic vs stochastic models ................................................................ 25

1.2 Research questions ................................................................................................ 27

1.3 Aims and objectives .............................................................................................. 28

1.4 Literature review ................................................................................................... 28

1.4.1 Stochastic Dual Dynamic Programming ........................................................ 28

1.4.2 Model Predictive Control ............................................................................... 29

1.4.3 Carryover storage value ................................................................................. 31

1.4.4 Hydro-economic vs non-hydro-economic models ......................................... 33

1.4.5 Simulation vs optimisation ............................................................................. 35

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1.4.6 Optimisation methods .................................................................................... 37

1.5 Case study: California Central Valley ................................................................... 38

1.6 Summary of the methodology ............................................................................... 42

1.7 Thesis structure ...................................................................................................... 46

1.8 Contributions to research ....................................................................................... 47

References ........................................................................................................................ 48

2 Chapter two: How do hydro-economic model formulations impact their

recommendations?................................................................................................................ 61

Abstract ............................................................................................................................ 62

2.1 Introduction ........................................................................................................... 63

2.2 Methodology ......................................................................................................... 64

2.2.1 Annual objective function .............................................................................. 65

2.2.2 Extension 1: Carry-over Storage Value Functions (Models B and D)........... 66

2.2.3 Extension 2: Dynamic Groundwater Pumping Costs (Models C and D) ...... 66

2.2.4 Data validation and further refinements ......................................................... 67

2.2.5 Implementation .............................................................................................. 68

2.3 Application to California ....................................................................................... 68

2.4 Results ................................................................................................................... 71

2.4.1 Part I ............................................................................................................... 71

2.4.2 Part II .............................................................................................................. 74

2.4.3 Part III ............................................................................................................ 78

2.5 Discussion ............................................................................................................. 79

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2.6 Conclusions ........................................................................................................... 80

References ........................................................................................................................ 82

3 Chapter three: Estimating the economic value of inter-annual reservoir storage in

water resource systems ......................................................................................................... 84

Abstract ............................................................................................................................ 85

3.1 Introduction ........................................................................................................... 85

3.2 Methodology ......................................................................................................... 89

3.2.1 Carry-over storage value functions ................................................................ 89

3.2.2 Solution strategy ............................................................................................ 91

3.3 Application ............................................................................................................ 92

3.3.1 Annual optimization model ............................................................................ 96

3.3.2 Multi-objective problem and resolution ......................................................... 98

3.4 Results ................................................................................................................... 99

3.4.1 Marginal water values .................................................................................... 99

3.4.2 Basin-wide inter-annual operation ............................................................... 102

3.4.3 Sensitivity analysis ....................................................................................... 105

3.5 Discussion ........................................................................................................... 108

3.6 Conclusion ........................................................................................................... 110

Acknowledgments .......................................................................................................... 111

References ...................................................................................................................... 112

4 Chapter four: Investigating historical valuation of reservoirs – a California Central

Valley case study................................................................................................................ 119

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Abstract .......................................................................................................................... 120

4.1 Introduction ......................................................................................................... 120

4.2 Material and Methods .......................................................................................... 122

4.2.1 Proposed methodology ................................................................................. 122

4.2.2 Model implementation ................................................................................. 125

4.3 Illustrative example ............................................................................................. 126

4.3.1 The Central Valley of California.................................................................. 126

4.3.2 Historical approximation .............................................................................. 128

4.4 Results ................................................................................................................. 129

4.4.1 Reservoir storage valuation .......................................................................... 129

4.4.2 Calibration .................................................................................................... 135

4.5 Discussion ........................................................................................................... 137

4.6 Conclusion ........................................................................................................... 140

References ...................................................................................................................... 141

5 Chapter five: Discussion and Conclusion .................................................................. 145

5.1 Discussion ........................................................................................................... 146

5.2 Future work ......................................................................................................... 148

5.3 Conclusion ........................................................................................................... 148

Appendix A. Model input data ........................................................................................... 150

Appendix B. Effect of evolutionary search configuration on model performance ............ 172

Appendix C. Difference between storage trajectories of Model A and CALVIN Optimised

............................................................................................................................................ 174

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Appendix D. Proposed model formulation ........................................................................ 176

Appendix E. Pareto solution analysis................................................................................. 178

Appendix F. Historical approximation by CALVIN ‘base case’ run................................. 181

Thesis word count: 29460 (excluding appendices, references, acknowledgement, copyright

statement, and declaration)

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List of figures

Figure 1. A Standard linear operating policy (adopted from Draper (2001)). ..................... 19

Figure 2. Deterministic and stochastic process (adopted from Hillier and Lieberman

(2004)). ................................................................................................................................. 26

Figure 3. Network schematic of the CALVIN model from. Northern parts (left side) form

the upstream of the water system. The area shown in the box is the extent of the CALVIN

model used in this thesis. ..................................................................................................... 41

Figure 4. Relation between demand curve and COSVF of a reservoir. ............................... 43

Figure 5. Network schematic of the Central Valley of California case-study application. . 70

Figure 6. Annual aggregated surface reservoir storage volumes of Model A compared to

CALVIN Optimised and historical approximation during: a) 1922-57; and b) 1958-93. ... 73

Figure 7. Annual aggregated groundwater sub-basin storage volumes compared to

CALVIN optimised and historical approximation. .............................................................. 74

Figure 8. Comparison of the four models’ annual aggregated surface reservoirs’ storage

volume during: a) 1922-57; and b) 1958-93. ....................................................................... 75

Figure 9. Comparison of the four models’ annual aggregated groundwater sub-basins

storage volume over the planning horizon. .......................................................................... 76

Figure 10. Comparison of annual mean unit pumping cost of groundwater sub-basins. ..... 76

Figure 11. Comparison of annual water scarcity volume as the percentage of target demand

from combined agricultural and urban sectors. .................................................................... 77

Figure 12. Results of the proposed model before and after modifications made by the

author (corresponding to Model D-pre correction and Model D respectively): a) Aggregate

irrigation deficit; and b) Aggregate groundwater storage trajectory. ................................... 78

Figure 13. Comparison of annual water scarcity in demand sites: Model D-GS vs Model D.

.............................................................................................................................................. 80

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Figure 14. Proposed model workflow. ................................................................................. 92

Figure 15. The California Central Valley storages and river system. .................................. 95

Figure 16. Non-dominated solution points showing the Pareto-optimal trade-off between

the two objective functions: economic benefits and mean water marginal values (arrows

show the direction of preference). ...................................................................................... 100

Figure 17. Distribution of average stored water marginal value in the Central Valley.

Values in parenthesis are average marginal value. ............................................................ 102

Figure 18. Annual aggregated surface reservoirs’ storage level comparison during : a)

1922-57; and b) 1958-93. ................................................................................................... 103

Figure 19. Annual aggregated groundwater storage level. ................................................ 104

Figure 20. Comparison of a) water scarcity as the percentage of target delivery and b) the

corresponding scarcity cost in demand sectors (combined agricultural and urban demands).

............................................................................................................................................ 105

Figure 21. Envelope showing the distribution of river inflows in the synthetic ensemble (in

grey) and the historical inflow data (black line) during: a) 1922-57; and b) 1958-93. ...... 106

Figure 22. Probability of exceedance of: a) aggregated 72-year shortage volumes; and b)

worst 3-year shortage volume. The reference scenario is the one obtained with historical

inflow, and with COSVF, i.e. the limted foresight model. ................................................ 107

Figure 23. Comparison water shortage and water availability during the worst 3-year

drought. .............................................................................................................................. 108

Figure 24. Relation between demand curve and COSVF of a reservoir. ........................... 123

Figure 25. Flowchart of the proposed model workflow. .................................................... 125

Figure 26. The Central Valley reservoir and river system. ................................................ 127

Figure 27. Comparison of historical approximation and observation of storage level of

Shasta. ................................................................................................................................ 129

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Figure 28. Pareto non-dominated solutions of the two fitness functions (arrows show the

direction of preference). ..................................................................................................... 130

Figure 29. Dispersion of historical water marginal value solutions from zone of

concentration at: a) dead storage, and b) full storage......................................................... 130

Figure 30. Distribution of the maximum COSVF from the solution points of the zone of

concentration. Red points show the maximum COSVF of the “optimised” model run. ... 131

Figure 31. Calibrated storage trajectories with average, minimum and maximum valuations

in: a) Don Pedro, b) New Melones, and c) Pine Flat. ........................................................ 132

Figure 32. (a) The three basins of the Central Valley (adopted from Jenkins et al. (2001)),

COSVF of reservoirs in (b) Sacramento Valley, (c) San Joaquin basin, and (d) Tulare

basin. .................................................................................................................................. 135

Figure 33. Comparison of the calibrated storage trajectories of major reservoirs to the

historical approximation..................................................................................................... 137

Figure 34. Comparison of the historical approximation and the optimised model for: a)

surface reservoirs over 1922-57; b) surface reservoirs over 1958-93; and c) groundwater

over 1922-93. ..................................................................................................................... 139

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List of tables

Table 1. Formulations of the model used in this paper. Model A is the ‘benchmark model’

and Model D is the ‘proposed model’. ................................................................................. 65

Table 2. Marginal economic value of stored surface water in September at major California

Central Valley reservoirs evaluated by Model D. Reservoirs are from north to south.

Maximum capacity varies per month due to flood control rules. Net inflow includes

deductions for evaporative and seepage losses. ................................................................. 100

Table 3. Historical marginal water values of end-of-year surface reservoirs’ storage in the

Central Valley, listed from north to south. ......................................................................... 132

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Abstract

This thesis develops and applies methods that blend physical science, engineering and

economics to the management of water resources. The core intuition of the work is that

multi-disciplinary models of water systems which integrate hydrological, engineering and

economic dimensions of the problem will be most effective at identifying best modes of

management at system scale. The problem area of focus in the thesis is the question of

inter-annual reservoir operation in large-scale multi-reservoir systems. Excessively

generous releases will threaten future supplies but exaggerated pre-emptive saving of water

supplies will create unnecessary economic hardship downstream. What’s the appropriate

amount of water to carry-over from one year to the next? How can effective and efficient

carry-over storage strategies be determined? This thesis proposes to address this question

with a generalised approach for large-scale water resource systems using end-of-year

carry-over storage value functions, i.e., curves that quantify the economic value of

maintaining various amounts of water storage for subsequent years. The proposed

approach uses hydro-economic optimisation models to simulate the economic allocation of

water over space and time within human managed water systems. The model breaks up the

simulated period of study into shorter periods and performs sequential runs of the

optimisation model which allocates water from source nodes to water demands or storage

nodes. The final state from the previous year provides the initial condition to each year-

long problem and COSVF acts as a terminal condition representing the value of stored

water for future use. These COSVFs have a concave shape to reflect the fact that the value

of stored water is high when water is scarce and low when abundant. COSVF parameters

that optimise performance can be determined using an external multi-objective

evolutionary algorithm (EA), thus enabling to estimate the storage valuation which brings

the highest overall regional economic benefits from water use. The scholarly contribution

of the thesis includes investigating how hydro-economic model features impact their

results, a new method for optimising water storage strategies in large complex non-convex

managed water systems, and an investigation of how historical reservoir release data can

be used to reveal the implicit economic value attributed to stored water. Existing

approaches for valuation of carryover storage either suffer from curse of dimensionality,

i.e. they fail as the size of the problem increases, or are unable to handle non-convexity

(nonlinearity) of the natural phenomena. Above contributions are applied to a large-scale

California Central Valley hydro-economic system where groundwater head-dependent

pumping costs make the problem non-convex. Initially, it is investigated how an improved

groundwater formulation that considers non-linear groundwater pumping costs leads to

reduced overdrafting of aquifers. Also, it is shown that use of shadow prices for water

marginal values, does not lead to an efficient management practice, especially in case of

conjunctive use with non-linear groundwater pumping representation. This finding

contradicts to what was being used in similar existing approaches for valuation of water

storage e.g. Stochastic Dual Dynamic Programming. The hydro-economic optimisation

model is then used to find COSVFs that lead to economically efficient management of the

water system. Results show improved scarcity management evidenced by a reduction of

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scarcity (80% in scarcity volume and 98% in scarcity costs) compared to historical

estimates. Finally, COSVF are calibrated to derive historical valuation of end-of-year

storage for the region. This application reveals the implicit over-year storage values for 30

reservoirs in California’s Central Valley; results are discussed. The economic valuation of

storage estimated in our case-studies can help inform water storage management decisions.

Conclusions and a discussion of the three contributions are included.

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Declaration

No portion of the work referred to in the thesis has been submitted in support of an

application for another degree or qualification of this or any other university or other

institute of learning.

14

Copyright Statement

1. The author of this thesis (including any appendices and/or schedules to this thesis) owns

certain copyright or related rights in it (the “Copyright”) and s/he has given The University

of Manchester certain rights to use such Copyright, including for administrative purposes.

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3. The ownership of certain Copyright, patents, designs, trademarks and other intellectual

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and/or Reproductions.

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commercialisation of this thesis, the Copyright and any Intellectual Property University IP

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The University’s policy on Presentation of Theses.

15

Acknowledgements

There are many people that have played an invaluable role in supporting me over the

course of this PhD project. I wish to express my sincere gratitude to all of them. First and

foremost, I would like to thank my supervisors, Prof Julien Harou and Prof Peter Stansby,

for giving me the opportunity to undertake this research. Their guidance and support has

been fantastic and invaluable over the course of this project.

My gratitude also goes to my current and past colleagues for their help, support and

encouragement. All members of the Prof Harou’s research team have provided

fundamental support in this study. However, special thanks should go to Dr Charles Rouge,

Dr Silvia Padula, Dr Steven Knox, and Dr Khaled Mohammed.

I would like to thank the UK Engineering and Physical Sciences Research Council

(EPSRC), University College London (UCL), and The University of Manchester for

providing the funding for this PhD. I would also like to thank the GAMS (Generalized

Algebraic Modelling System) Corporation for providing a cluster license to support this

research, and The University of Manchester for the use of Computational Shared Facility

for the high performance computing performed in this research.

Finally, I would like to say a special thank you to my family and friends who have

endlessly supported me throughout this process. In particular I would like to thank my

mum and dad who have always encouraged me to pursue a career that I enjoy and to do the

best that I can. Their financial support over the 4 years has also allowed me to focus on my

research without having to have any additional stresses. And my most sincere gratitude

here goes to my lovely wife, Afsoon, who was by my side during all the ups and downs of

my past 5 years, and remained patient and strong over the course of this PhD.

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The Author

The author of this thesis, Majed Khadem, was awarded a Bachelor of Science in Civil

Engineering (2:1) from Babol Noshirvani University of Technology in 2008. There, he

worked on his final-year project of “Hydraulic evaluation of open channels: case study of

Marzrood, Iran”. He was awarded a Master of Science in Civil Engineering-Water

Resources Management (1st) from Iran University of Science and Technology in 2011. His

dissertation was titled “Groundwater utilization optimization using decomposition

methods”.

In 2014, he was awarded an EPSRC Studentship and oversees student tuition scholarship

to complete this PhD at the University of Manchester. He was also awarded a CASE

Award from the Halcrow Group Limited (CH2M Hill). In 2016, he was nominated for the

“Best postgraduate that teaches” at the University of Manchester for the academic year

2015-2016. He is now expecting his son in October 2018.

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1 Chapter one: Introduction

18

1.1 Background

Planning and management of water resources covers a wide range of problems, as stated

by Loucks, Stedinger, and Haith (1981) “Water: too much, too little, too dirty. Throughout

the world, these are the conditions that prompt water resources planning”. A further

problem is that the overall water situation is likely to deteriorate as a result of global

changes (Loucks & van Beek, 2005). To tackle these problems, researchers from a wide

range of disciplines including water engineers, economists, political scientists, planners,

and conservationists have gained significant experience in designing, constructing,

operating structures, and implementing non-structural measures that will permit improved

management of natural water resources (Loucks et al., 1981). Reservoirs are among the

largest human interventions on earth with this respect. However, building a new piece of

infrastructure is not always a solution. Several studies reported on removing of existing

dams for ecological or socio-economic concerns (Lisius, Snyder, & Collins, 2018; Noda,

Hamada, Kimura, & Oki, 2018; Peters, Liermann, McHenry, Bakke, & Pess, 2017;

Tornblom, Angelstam, Degerman, & Tamario, 2017; Turner, Chase, & Bednarski, 2018).

As stated by J. J. Harou et al. (2009), management of water resources in the future will

move from building new water supply systems to better operating existing ones. Operating

existing infrastructures more efficiently in a rapidly changing world, rather than planning

new ones, is a crucial challenge to balance competing demands (Gleick & Palaniappan,

2010). Holistic approaches promoting efficient water allocation in water systems are still

needed (J. R. Lund, X. Cai, and G. W. Characklis (2006a) and X. Cai (2008)).

1.1.1 Reservoir operation

The number of dams is staggering and still increasing globally. Approximately half of the

dams create reservoirs for irrigation purposes, the other half have reservoirs for

hydropower generation, flood control and water supply, either as single-purpose or in

combination (Deltares, 2017). With electricity and freshwater demand still growing,

storage reservoirs are vital components in the sustainable development of many countries.

Storage reservoirs for hydropower and other purposes influence the amount, timing and

quality of the water available to different users in a river basin. Therefore, planning of

reservoirs requires assessment of current and future water use and availability at the river

basin scale. Operation of a single reservoir for a single function does not present many

analytical obstacles, but the same is not true when a reservoir fulfils a number of

19

potentially conflicting objectives or where several reservoirs are operated conjunctively. A

further complication arises when decision has to be made between releasing for immediate

demands and storing for later uses.

Reservoir operating rule curves are the most commonly used approach for guiding and

managing the reservoir operation (Chang, Chen, & Chang, 2005). They indicate limiting

rates of reservoir releases required or allowed during various seasons of the year to meet

all functional objectives. Operating rules specify reservoir releases as a function of

deviations from the ideal storage volume and other state variables such as hydrologic

conditions (Draper, 2001a).

The simplest form of the reservoir operation rule is the standard linear operating policy

(SLOP; Loucks et al. (1981)). These policies accept some present delivery deficit to reduce

the probability of greater water or energy shortage in the future (Bower, Hufschmidt, &

Reedy, 1962). According to SLOP, if in a particular period, the amount of water available

in storage is less than the target demand, whatever quantity is available would be released.

If the water available is more than the target but less than target demand plus available

storage capacity, then a release equal to the target demand is made and the excess water is

stored in the reservoir. An example of such rule is depicted in Figure 1. Hashimoto,

Stedinger, and Loucks (1982) demonstrated that where the loss function (on releases) is

linear, the SLOP is the best policy.

Figure 1. A Standard linear operating policy (adopted from Draper (2001)).

Yet, reservoir operation for water supply is not always rational to satisfy the full current

demand, because of the possibility of larger water scarcity in the future. Hedging rule

20

policies, as illustrated in Figure 1, are designed for rationing water supply in appropriate

preparation for potential low inflows in the near future (J. -Y You & X. Cai, 2008). There

are some requirements for hedging rules to be effective: (1) hedging rules are suitable for

reservoirs with low refill potentials subject to variable annual inflows and operated for

over-year storage (Draper, 2001); (2) for hedging to be optimal requires a convex, non-

linear loss function (Hashimoto et al., 1982); and (3) they require not only a convex and

non-linear loss function but also that the hydrology have substantial probability of

prolonged of dry periods (Klemes, 1977). A hydrology that, perhaps oddly, has very severe

droughts of one period followed by extremely wet conditions which always fill the

reservoir would never have hedging be optimal. Hedging was interpreted with stronger

implications for operations than for economic efficiency: ‘‘providing only portion of the

target release, when in fact all or at least more of the target volume could be provided’’

(Hashimoto et al., 1982). Following these studies, hedging has been explored to resolve

reservoir operation problems focusing on minimizing utility loss or water supply deficit

over drought periods (e.g., Shih and ReVelle (1994) and Shih and Revelle (1995)). J. -Y

You and X. Cai (2008) found that hedging is trivial when water demand is small relative to

water availability and/or reservoir capacity is small and evaporation loss reduces the role

of hedging rule in reservoir operation. Once prerequisites hold for optimal hedging rules,

there exist several methods in the literature for eliciting one. If economic measures are not

involved in the process of determining hedging rules, and once they fail to operate

reservoirs effectively, catastrophic consequences to water users may be imposed.

1.1.2 Water as an economic good

Economic valuation of uses of water resources has the potential to bring a more balanced

perspective to the allocation and management of water resources (Loomis, 2000). Human

access to clean water for basic needs and sufficient environmental and public use

allocation are compatible with and encouraged by an economic approach to water

management (R. Young, 2005). A key element in quantifying outcomes of planning and

managing water resources is to determine benefits of allocating water (or loss incurred) by

(not) meeting demands. These all imply that water must be considered as an economic

commodity in the planning and management process. As stated by Principle 4 of Dublin

statement (U.N., 1992) “Water has an economic value in all its competing uses and should

be recognized as an economic good”. Interpretation and applicability of this economic

21

principle continues to be debating among water professionals (Tilmant, Pinte, & Goor,

2008).

Perhaps the best example of using economic value of water for decision on water

allocation in a real-life case could be the one narrated by Loomis (2000): The case

included tributary water flows into Mono Lake in California versus diverting such flows

for urban and industrial uses in Los Angeles. These diversions were depleting tributary

streams and decreasing the lake level. In 1983, the California State Supreme Court

demanded a re-evaluation of the city of Los Angeles’s 1941 water rights and required a

balancing of the public trust uses of water. The Los Angeles Department of Water and

Power projected the re-evaluation debate in terms of ‘‘300 Fish versus 28,000 People?’’

The amount of willingness to pay to protect the Mono Lake ecosystem were contrasted to

hydropower and water supply benefits and costs in the economic analysis of the different

water allocation alternatives (Jones&StokesAssociates, 1993). Eventually, the state ordered

the flows into Mono Lake to be increased and Los Angeles’s water rights to be reduced by

nearly half (Loomis, 1995).

Yet, the need for appropriate water valuation is underscored by regulatory frameworks that

promote an economically efficient allocation of water, e.g. the Water Framework Directive

(EU-Commission, 2000, 2012) in the European Union, or the emergence of water markets

in various places, including the western United States (Hadjigeorgalis, 2009; Hansen,

Howitt, & Williams, 2014; S. Wheeler, Garrick, Loch, & Bjornlund, 2013; S. A. Wheeler,

Loch, Crase, Young, & Grafton, 2017), Australia (Garrick, Hernández-Mora, &

O’Donnell, 2018; Lewis & Zheng, 2018; Owens, 2016), or the UK (Erfani, Binions, &

Harou, 2015; Parker, 2007). However, in the latter case, since water markets are usually

absent or ineffective, the value of water cannot be directly derived from market activities

and allocation decisions can seldom rely on market prices (Tilmant et al., 2008). Instead,

they must rather be assessed through modelling and analytical approaches. Economists

have developed and implemented various nonmarket valuation techniques to water

resources management problems (Loomis, 2000), especially in the field of water quality

such as stream and wetland restoration (Collins, Rosenberger, & Fletcher, 2005;

Woodward & Wui, 2001). Most of the nonmarket valuation studies reported in the

literature focus on single water use, long-run policy problems, and assume that the water

availability is given (Tilmant et al., 2008).

22

When water is considered as a variable input in production, the value of water can be

measured by the value marginal product. This is defined as the value of an additional

quantity of product due to the use of an additional unit of water. Marginal water values can

be used in river basin management to signal water scarcity, to prioritise zones where soil

and water conservation measures must be implemented, to increase the productivity of

water by (re)allocating it to more productive uses. There are essentially two modelling

approaches in water resources systems analysis to derive marginal water values: simulation

versus mathematical programming (Tilmant et al., 2008). Both approaches require the

integration of a hydrological model and an economic model. With the simulation approach,

the marginal water values can only be derived after analysing the results of a large number

of runs associated with small perturbations in water availability. The main advantage is

that it can handle nonlinear relationship and can be fairly detailed. The disadvantages come

from it being computationally cumbersome as the number of simulations is quite large.

However, with today’s advances in computational infrastructures, the latter should not

impose hardship to adopt such approaches. In mathematical programming approaches, the

marginal water values correspond to the Lagrange multipliers associated with the mass

balance equations which are all available at the optimal solution. The Lagrange multipliers

give the change in the objective function due to a small perturbation in the constraint right-

hand side. These multipliers, also called shadow prices, would correspond to market prices

if water were being traded on a market. Mathematical programming is one of the deductive

methods for non-market valuation of water (R. Young, 2005). This group of approaches

take advantage of being computationally efficient and easily implemented. However, they

cannot be very detailed and are unable to handle non-convexity involved in many real-

world phenomena. Nonlinear hydropower generation and groundwater pumping cost are

only two instances where this prerequisite does not hold.

1.1.3 Hydro-economic modelling

Hydro-economic modelling is the act of representing regional scale hydrologic,

engineering, environmental and economic aspects of water resources systems within a

coherent framework. The key element is to operationalise economic concepts by

incorporating them at the heart of water resource management models (J. J. Harou et al.,

2009). In non-economic system models, water demands are commonly represented by

fixed water ‘‘requirements” or delivery targets. Economics aids water managers to move

23

from a static view of water demand (e.g. water rights, priorities and projections of

population growth and agricultural and industrial water requirements) to a view of demand

related to the economic concept of value. Water value changes with the quantity and type

of use (J. J. Harou et al., 2009). Monetising all water uses provides an even-handed

comparison among users. Identifying the value of resources with conflict in usage helps

diffuse contests by introducing clarity and revealing the often relatively modest sums

involved (Fisher et al., 2002). It worth noting that hydro-economic models are different

from related tools such as engineering models that minimise financial costs or economic

models such as dynamic optimisation of groundwater stocks, economy-wide general

equilibrium models, input-output analysis, cost-benefit analysis, agent-based models, etc.

In hydro-economic models, water allocation is incentivised or evaluated by the economic

values it generates.

Inclusion of economic criteria in hydro-economic models adds a layer of complexity

beyond traditional water planning models that may be difficult or controversial for water

managers to accept (J. J. Harou et al., 2009). Several barriers exist to directly use hydro-

economic model results. Simplification and aggregation of physical, economic and

regulatory processes and data is imminent for timely construction and resolution of

regional models. If physical aggregation (e.g. merging number of sites such as reservoirs

into one for simplicity) is coarser (more severe) than the existing simulation models,

managers may consider the hydro-economic model as too theoretical or inadequately

detailed to support local decision making. Models with simplified process equations are

accused of reduced parameter set to accurately represent the system. Simplification may

contribute to lack of robustness at the local scale; for instance a small perturbation in

conveyance cost on a link (e.g. canal) could results in flows to change route to totally a

different one. However, at the larger regional scale such local impacts tend to negligible,

leading to generally robust system-wide results in terms of system operation and responses

to different scenarios and policies. It is also difficult to make simplified regional models

well imitate observed data and calibrate these models to historical data (X. Cai & Wang,

2006; Draper, Jenkins, Kirby, Lund, & Howitt, 2003). Another bias in this context was

described by Tilmant et al. (2008): most of the integrated economic-hydrologic models

reported in the literature do not explicitly consider the hydrologic uncertainty and are

therefore likely to overestimate the performance of the water resources system. This can be

improved by incorporating an appropriate hydrological foresight into such models.

24

Most hydro-economic models share basic elements i.e. hydrologic flows, water

management infrastructure, economic water demands, operating costs, and operating rules

(J. J. Harou et al., 2009). Other components of hydro-economic models are as follows.

Water demands, consumption, and other features where water incurs a cost or benefit are

represented as “nodes” in the model, when following the “node-arc” representation of the

water system (Maass et al., 1962). Boundary conditions in the form of inflows or outflows

can occur anywhere in the water network. Short-term forecasts of inflows based on

weather predictions and current hydrologic conditions can be used for operating purposes.

External system inflow data can be obtained from historical flow gage records or synthetic

time series generated by stochastic hydrology models. Minimum and maximum capacities

and operating costs are specified for each component. In a hydro-economic model,

economic water demands are introduced by functions defining net economic benefits

generated during a particular model time-step (Bear, Levin, & Buras, 1964). If the

objective of the model is to minimise cost(s), water scarcity costs incurred by lower

deliveries can be represented by penalty functions (B. D. Newlin, Jenkins, Lund, & Howitt,

2002). Environmental water concerns may be alternatively represented with operating rules

or constraints, where an objective function valuation is unavailable. Operating costs

include pumping, treatment, artificial recharge and other costs to move water between

network nodes. Water quality costs can be introduced as operating costs, so they could be

evaluated and varied depending on the source of water delivered to each urban area, where

incoming water quality varied primarily with source (Draper et al., 2003).

1.1.4 Hydrological foresight

Hydro-economic models, depending on how they handle time dynamics and possess

knowledge of future hydro-climatic conditions, can have no foresight (myopia), limited

foresight, or perfect foresight (hyperopia). Models with myopia (from myopic or short-

sighted view for the future) (Madani & Dinar, 2012; Madani & Hipel, 2011; Pinches,

1982) have no information of what is happening beyond the current modelling time-step

(e.g. month) and do not apply a boundary condition at the end of each modelling time-step.

Hui, Lund, and Madani (2016) report that the myopia causes extra costs and inefficiencies

in river systems, happening historically when rational players made decisions. Traditional

rule curves also suffer from policy myopia as they fail to explore the full set of trade-offs

between evolving multisector objectives and preferences in river basins (Giuliani, Herman,

25

Castelletti, & Reed, 2014). On the other hand, perfect foresight implies that full knowledge

of hydrologic condition is available over the entire planning horizon. This is at odds with

actual knowledge and operation and can lead to decisions anticipating wet and dry years in

advance. The attribute of perfect foresight causes unrealistic storage operations: large

carryover storage prior to drought; and little storage prior to wet years (Draper, 2001). He

also stated that operating rules derived from perfect hydrological foresight may not be

optimal given the stochastic nature of inflows and the inherent uncertainty of reservoir

operating decisions.

In order to partially overcome these limitations, researchers have developed and used the

concept of limited foresight (Draper, 2001; Draper & Lund, 2004; J. -Y You & X. Cai,

2008; J. -Y. You & X. Cai, 2008). As described by Draper (2001), the limited foresight

eliminates any knowledge of the hydro-climatic conditions beyond the current water year

while assuming a perfect within-year foresight. The latter assumption is correct for

majority of cases globally. In many watersheds, early spring measurements of the

precipitation and water content of the snowpack provide reasonably accurate forecasts of

hydrological inflow to the end of the water year (Draper, 2001). Models with limited

foresight can hedges deliveries as an insurance against forthcoming drought. These hedges

are sometimes successful and sometimes not. A crucial part of implementing limited

foresight is to accurately assign inter-annual (also known as over-year or carry-over)

storage. This is the amount of water stored in reservoirs at the end of each year for future

uses which is the missing bit of the myopic models. The carry-over storage can either be

implemented as a target storage capacity, or following the concept of hydro-economic

modelling, can be enforced through economic valuation. Thus, the use of limited foresight

approach in hydro-economic models can bring an opportunity to directly estimate the

economic valuation of reservoirs’ carryover storage.

1.1.5 Deterministic vs stochastic models

Depending on how input data to a model is used and how its output is calculated two

classes of models emerge, namely deterministic and stochastic modes. Models in which

outcomes are determined through known relationships among states and events, without

any room for random variation are known as deterministic models. In such models, a given

input will always produce the same output. Modelling output is often determined by single

run of the model. In contrast, stochastic models use ranges of values for variables.

26

Stochastic models normally require several model runs in order to generate output. Hillier

and Lieberman (2004) define deterministic and stochastic models as follow: Deterministic

models simplify the problem of incomplete or erroneous data often faced in real word

systems. In the deterministic approach, input data are fixed and predictable quantities, and

uncertainty of future outcomes is not considered probabilistically, although it can be

accounted for with safety factors. In stochastic models some or all input data are random

variables and therefore the output is obtained with some range of uncertainty (Figure 2).

Figure 2. Deterministic and stochastic process (adopted from Hillier and Lieberman (2004)).

Both limited foresight and perfect foresight approaches described in the previous section

fall within the deterministic class. However, as per Draper (2001), they are implicitly

stochastic models regarding the way they capture the element of risk or uncertainty. This

class of deterministic models (limited and perfect foresight models) uses a long historic

flow record or a synthetic streamflow sequence to represent the range and frequency of

possible inflows which is a characteristic of stochastic models. However, because they are

not represented in a form of probabilities, such deterministic models are only called

implicit stochastic models (as opposed to explicit stochastic models which directly work

with probabilities). Difficulties in explicitly stochastic formulations have led modellers of

large integrated systems to rely on implicitly stochastic modelling techniques (Labadie,

1997). Some of these difficulties include: stochastic hydrology still poses many theoretical

difficulties (Jackson, 1975; Klemeš, 1974); for large systems it entails calculation of auto-

and cross-correlation coefficients (Draper, 2001) which quantifies how flow time series are

related from one year to another and how similar different flow time series are,

respectively; and inflows must be modelled using only one of few available statistical

models (Loucks et al., 1981); the required computational effort makes most of the explicit

27

stochastic approaches infeasible for complex systems (Draper, 2001). Thus, implicit

stochastic models are suitable candidates for tackling reservoir operation problems because

such problems are multistage dynamic stochastic control problem in nature (Marino &

Loaiciga, 1985).

1.2 Research questions

This section describes what research questions this dissertation is trying to answer. One of

the major challenges that reservoir operators/water resources managers face is that how

much water should be released for immediate downstream demands and how much should

be stored for later uses. This is also known as inter-annual reservoir operation. Excessive

release will threaten future supplies while unnecessary hedging creates economic hardship

downstream. What’s the appropriate amount of carry-over storage in reservoirs? What

socio-economic consequences will these management decisions have? How can economic

valuation of over-year stored water help in such circumstances? What are the limitations of

the existing approaches for valuating stored water? How can we bypass those limitations?

What are the immediate applications of economic valuation?

This dissertation aims to answer above questions. Principle 4 of Dublin statement (U.N.,

1992) clearly asks for any management decision to be made with consideration of

economic value of water. Loomis (2000) described a case where explicit valuation of water

use was applied as an indicator to solve a real-life allocation problem between two water

demand zones. Reservoirs can be considered as demand sites with the amount of active

storage expressed as their demand target. Then, a comparison similar to Loomis (2000) can

be true between a demand site and a reservoir to decide whether to store water (for future

uses) or to release for downstream demands. This is the basis of the work presented in this

dissertation. For this purpose, an approach capable of determining and comparing the

marginal value of water in demand sites as well as reservoirs is developed. The approach

must be able to quantify the economic consequences of any decision made. It should show

improvement to the existing approaches. Finally, the approach must be generalisable. By

definition, a generalisable approach is expressed in theories, principles, or statements of

relationships that can be generally applied to other experiences. In that sense, the proposed

approach is generalisable as it provides a framework that can be readily implemented to

solve any problem of the above nature (i.e. determining the economic value of reservoirs’

storage in large-scale multi-reservoir systems) without the need for redoing from scratch.

28

Several methods/approaches have been developed to address all or part of the above

research questions. Yet, some biases exist that hinders the applicability of the existing

methods. “Curse of dimensionality” (R.E. Bellman & Dreyfus, 1962) affects model by

exponentially growing the dimension of solution space when the dimension of the problem

(number of variables) increases. This can be further deteriorated by presence of multiple

reservoirs and demand sites which generally makes the analytical derivation and even the

numerical resolution of such problems intractable in practice with most methods (Labadie,

2004). All deterministic models using a perfect foresight are known to suffer from curse of

modelling. “Non-convexity” (Mas-Colell, 1987) prerequisite of some of the existing

approaches begs for concave behaviour of natural phenomena i.e. all components should

be represented with linear equations. The next section reviews some of the existing

approaches as well as fundamental modelling concept that are used in this context.

1.3 Aims and objectives

Following the research questions outlined in the previous section, there exists an

opportunity to develop an approach that determines inter-annual storage operation which

lead to economic boost of water allocation. This is the aim of this thesis. Few approaches

exist that govern over-year reservoirs’ release/storage. Yet, they are not assumption-free

which reduces their applicability. The objective of this thesis is to propose an approach to

obtain the economic value of end-of-year storage. The valuation method is used to

construct reservoirs’ inter-annual operation. The proposed approach is not hindered by

assumptions (linearity) or scale of the problem, contributing to a methodological advance.

1.4 Literature review

1.4.1 Stochastic Dual Dynamic Programming

First application of Stochastic Dynamic Programming (SDP; R.E. Bellman (1958)) in

optimal reservoir operation dates back to 80s (Stedinger, Sule, & Loucks, 1984; Trezos &

Yeh, 1987). SDP determines release decisions whilst maximising current benefits plus the

expected benefits from future operation. Despite its wide range of applications, SDP is

hindered by curse of dimensionality. Functional optimisation of SDP is particularly too

complex to be solved numerically. Hence, applications are limited to systems that include

29

few variables, and state transitions must be defined explicitly, which requires a stochastic

representation of the inflow process (Raso & Malaterre, 2017). To cure this plague, an

extension of SDP, Stochastic Dual Dynamic Programming (SDDP; Pereira (1989), Pereira

and Pinto (1991)) was initially developed for short and midterm large-scale hydropower

generation systems. SDDP can output a comprehensive set of hydro-economic indicators

under a wide range of hydro-climatic conditions. This means SDDP can explicitly consider

hydrologic uncertainty (Rougé & Tilmant, 2016).

Several studies has been carried out to extend SDDP and its applications, including

assessment of marginal water values at reservoirs (Tilmant et al., 2012; Tilmant et al.,

2008), which can approximate the carry-over storage value of water within those reservoirs

(Tilmant, Arjoon, & Marques, 2014), management of systems with both groundwater and

surface water reservoirs (Macian-Sorribes, Tilmant, & Pulido-Velazquez, 2017), long-term

hydropower scheduling (e.g., Gjelsvik, Mo, and Haugstad (2010); Homem-de-Mello, de

Matos, and Finardi (2011);and Bezerra, Veiga, Barroso, and Pereira (2012)), and obtaining

multiple near optimal solutions (Rougé & Tilmant, 2016).

To avoid the curse of dimensionality that hinders optimisation of large-scale systems,

algorithms such as SDDP rely on key approximations or assumptions. For instance, SDDP

assumes that the benefit-to-go (or future benefits) function is convex, whereas head-

dependent pumping costs (Davidsen, Liu, Mo, Rosbjerg, & Bauer-Gottwein, 2016) or

endogenous hydropower prices (Mo, Gjelsvik, and Grundt (2001); and T. Kristiansen

(2004)) are two instances with concave functions.

1.4.2 Model Predictive Control

Model predictive control (MPC; Richalet, Rault, Testud, and Papon (1978a); Morari and

Lee (1999); and Mayne, Rawlings, Rao, and Scokaert (2000)) uses a model to predict

future system dynamics and solves a control problem with optimisation, considering

system constrains over a finite prediction horizon (Camacho & Alba, 2013). The accuracy

and complexity of the prediction model significantly influence the control performance of

MPC in terms of control accuracy and computation time (Xu, 2017). Only recently, several

studies applied MPC for various types of problems. Delgoda, Malano, Saleem, and

Halgamuge (2016) proposed a theoretical framework based on MPC for irrigation control

to minimize both root zone soil moisture deficit and irrigation amount under a limited

30

water supply. Shahdany, Majd, Firoozfar, and Maestre (2016) introduced a solution

strategy using MPC to handle drastic inflow changes without constructional modification

in the main canal shape and the off-take structures. Sharafi and Safavi (2016) implemented

a multi-agent MPC for a real large scale system, Rhine Meuse Delta, to control the flow

through the system. Grosso, Velarde, Ocampo-Martinez, Maestre, and Puig (2017)

proposed three stochastic MPC approaches, namely chance-constrained MPC, tree-based

MPC, and multiple-scenario MPC, to cope with uncertainty in system disturbances due to

the stochasticity of water demand/consumption and in order to optimise operational costs.

Tian, Aydin, Negenborn, van de Giesen, and Maestre (2017) proposed an approach using

MPC to link the spill from canal dikes with the softened constraint. Tian, Negenborn, et al.

(2017) proposed and assessed the model performance of multi scenario MPC, considering

an ensemble streamflow forecast. They then developed the Adaptive Control Resolution

(ACR) approach as a computationally efficient scheme to practically reduce the number of

control variables. Wang, Ratnaweera, Holm, and Olsbu (2017) provided a MPC-based

approach to optimise full-scale wastewater treatment plants performance and reduce

operation cost in practice. Xu (2017) introduced a dynamic target trajectory approach to

calculate changes of control targets that are used by MPC. They applied the approach to

the Central Main Canal in Arizona. Uysal, Alvarado-Montero, Schwanenberg, and Şensoy

(2018) demonstrated a real-time flood control case in Turkey with consideration of

streamflow forecast uncertainty especially for limited storage multi-purpose reservoirs

using synthetic ensemble inflows and the mass-conservative tree-based MPC method.

Uysal, Schwanenberg, Alvarado-Montero, and Sensoy (2018) consider improving short

term operation strategies by addressing long term water supply and short term flood

control purposes, where the novelty of the approach lied within developing a joint

optimisation-simulation operating scheme using predictive control techniques for short

term optimisation of a multi-purpose reservoir.

Although no MPC application reported to be developed for valuating reservoir storage, it is

a widely-used approach for incorporate data (inflow) uncertainty without suffering from

curse of dimensionality. Yet, it is more suitable for short-term operation of reservoir

systems (Raso & Malaterre, 2017). Raso and Malaterre (2017) proposed an extension to

MPC in order to handle long-term reservoir operation problems, but applicability of their

method to handle large systems remained questionable.

31

1.4.3 Carryover storage value

Carryover storage, also known as over-year or inter-annual storage, is the amount of water

stored in reservoir at the end of each year for later uses. This concept was first used in least

cost capacity expansion problem (Bogle & Osullivan, 1980). They realised that the main

difficulty with capacity expansion problem is with the effect of carryover storage which

makes the cost of a decision dependant to the decision made in other periods. For instance,

it would be clearly the short term least cost solution to use all the local supply until it runs

out, but this strategy would be expensive in the long term. It was not until Draper (2001)

that direct valuation of carryover storage was used as a tool for reservoir operation

decisions. Draper (2001) proposed that an optimised terminal or carryover storage value

function can be used to manage reservoir systems with limited hydrological foresight. The

carryover storage value function (COSVF), once optimally determined, is implemented as

a boundary condition to yearly optimisation problems in order to prevent reservoirs from

depletion at the end of each year. Draper (2001) also showed that quadratic carryover

storage value functions can well fit a range of reservoir operations settings. Draper and

Lund (2004) analytically demonstrated that the optimal hedging policy for water supply

reservoir operations depends on a balance between beneficial immediate release and

carryover storage values. They illustrated that searching for the optimal COSVF and then

optimising the hedging rule could be easier, but not for large-scale problems as it will be

affected by curse of dimensionality. For clarity, the analytical approach of Draper and

Lund (2004) for determining the optimal COSVF, considering quadratic COSVF and

demand benefit function (as the most common form of such functions), is elaborated here.

As per Draper and Lund (2004), release and carryover storage decisions should be made to

maximise the sum of immediate uses and carryover storage benefits. This can be

mathematically represented as:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑍 =∑𝐵(𝐷) + 𝐶(𝑆)

𝐷,𝑆

(1)

subject to

𝑆 + 𝐷 = 𝐴 (2)

0 ≤ 𝑆 ≤ 𝑘 (3)

0 ≤ 𝐷 ≤ 𝑑𝑚 (4)

32

where Z is the net benefit objective function, B is the benefit function of demand node, D is

the allocation to demand node, C is the COSVF for surface reservoir, S is the storage in the

reservoir, k is the maximum storage capacity, and dm is the maximum (target) water

demand. Above formulation applies only when water availability (A) is less than maximum

demand plus storage capacity (A<dm+k). When A>dm+k, hedging is irrelevant because

plenty of water exists to supply all demands, fill the reservoir, and spill. The Lagrangian

for this problem, within the bounds of the inequality constraints where hedging is relevant,

is

𝐿 = 𝐶(𝑆) + 𝐵(𝐷) + (𝐴 − 𝑆 − 𝐷) (5)

The first-order conditions for solving this problem include

𝜕𝐿

𝜕𝑆= 0 =

𝜕𝐶(𝑆)

𝜕𝑆− (6)

𝜕𝐿

𝜕𝐷= 0 =

𝜕𝐵(𝐷)

𝜕𝐷− (7)

𝜕𝐿

𝜕= 0 = 𝐴 − 𝑆 − 𝐷 (8)

Equation (8) gives the constraint of (2) whereas equations (6) and (7) lead to

𝜕𝐶(𝑆)

𝜕𝑆=𝜕𝐵(𝐷)

𝜕𝐷 (9)

Equation (9) means that at optimality, the marginal benefits of storage must equal the

marginal benefits of allocation. Equations (8) and (9) can be used to obtain the optimal

hedging rules for a range of conditions. Assuming a quadratic form for both C(S) and

B(D), i.e. C(S)=as+bsS+csS2 and B(D)=ad+bdD+cdD

2, combining equations (8) and (9) to

give optimal release D* as a function of total water availability A leads to

𝑏𝑠 + 2𝑐𝑠𝑆∗ = 𝑏𝑑 + 2𝑐𝑑𝐷

∗ 𝑜𝑟

𝑏𝑠 + 2𝑐𝑠(𝐴 − 𝐷∗) = 𝑏𝑑 + 2𝑐𝑑𝐷

∗ (10)

𝐷∗ =𝑏𝑠 − 𝑏𝑑 + 2𝑐𝑠𝐴

2(𝑐𝑠 + 𝑐𝑑) (11)

This linear form of hedging would apply in the region where inequalities (3) and (4) do not

bind. In addition, as equation (11) suggests, this analytical approach is only applicable to

cases with single reservoir and one demand node. You and Cai (2006) followed the

analysis of Draper and Lund (2004) in attempt to extend the analysis with consideration of

33

uncertainty and imperfect information, which complicates the hedging rule analysis. They

applied the approach to a two-period (now and then) problem with single reservoir.

Approaches involving COSVF and hedging require not only that the loss function be

convex and non-linear but also that the hydrology have substantial probability of

persistence of dry periods. A hydrology that, perhaps oddly, has very severe droughts of

one period followed by extremely wet conditions which always fill the reservoir would

never have hedging be optimal. Consequently, implementing COSVF will not provide

strong enough incentive to preserve water for later uses. The persistence of annual

droughts is important in the determination of rules governing carryover storage. Another

group of studies (e.g. Ximing Cai, McKinney, and Lasdon (2002) and J.-T. Shiau (2011))

applied carryover storage as boundary condition for short-term (annual) optimisation

problems but without economic valuation of such storage volumes. Instead, these studies

use target carryover storage volume in order to prevent over-exploitation of water

resources. This approach, although introducing fewer variables into the problem, requires a

good understanding of the topology (physical and geographical attributes) and hydrology

of the problem, which makes them case-dependent and reduces the generalisability of the

approach.

1.4.4 Hydro-economic vs non-hydro-economic models

Using COSVF requires modelling and simulating the water resources system with

consideration of economic drivers. This is what hydro-economic models are developed for.

Such models simulate the water system using economically characterised demand values,

often represented in the form of utility (benefit) functions rather than traditional target

demand constraints. Moreover, hydro-economic models seek to maximise system-wide net

economic benefits from water allocation and reservoir operation. The first use of this

modelling approach roots back to early 60s, where several studies used economic water

demand curves as a mean to optimise and manage water resources systems including Bear

et al. (1964); Bear and Levin (1967), Rogers and Smith (1970), Gisser and Mercado

(1972), and Gisser and Mercado (1973). Since then researchers have used different names

to refer to applications and extensions of this hydrologic engineering – economic water

modelling approach including: hydrologic–economic (Gisser & Mercado, 1972),

hydroeconomic (Noel & Howitt, 1982), economic–hydrologic–agronomic (Lefkoff &

Gorelick, 1990), institutional (Booker & Young, 1994), integrated hydrologic–economic-

34

institutional (Booker, 1995), integrated river basin optimization (Ward & Lynch, 1996),

efficient allocation (Diaz & Brown, 1997), integrated economic–hydrologic (McKinney,

Cai, Rosegrant, Ringler, & Scott, 1999; Rosegrant et al., 2000), economic-engineering

(Draper et al., 2003; Lund et al., 2006a; B. D. Newlin et al., 2002), integrated hydrologic–

agronomic– economic (X. Cai, McKinney, & Lasdon, 2003), demand and supply (R. C.

Griffin, 2016), integrated hydrologic–economic (X. Cai et al., 2003; Pulido-Velazquez,

Andreu, & Sahuquillo, 2006; Ringler, von Braun, & Rosegrant, 2004), holistic water

resources–economic (X. Cai, 2008; X. Cai & Wang, 2006), integrated hydrodynamic–

economic (Jonkman, Bockarjova, Kok, & Bernardini, 2008), and integrated ecological–

economic (Volk et al., 2008).

Some of recent studies using hydro-economic modelling include Amin, Iqbal, Asghar, and

Ribbe (2018), Bekchanov, Ringler, Bhaduri, and Jeuland (2016), Davidsen et al. (2015),

Escriva-Bou, Pulido-Velazquez, and Pulido-Velazquez (2017), Foster, Brozovic, and Speir

(2017), Hassanzadeh, Elshorbagy, Wheater, and Gober (2016), Jalilov, Keskinen, Varis,

Amer, and Ward (2016), Kahil, Ward, Albiac, Eggleston, and Sanz (2016), Kim and

Kaluarachchi (2016), Lopez-Nicolas, Pulido-Velazquez, Rouge, Harou, and Escriva-Bou

(2018), Medellín-Azuara et al. (2015), Momblanch, Connor, Crossman, Paredes-Arquiola,

and Andreu (2016), Satti, Zaitchik, and Siddiqui (2015), Wan et al. (2016), and Zhu,

Marques, and Lund (2015). Despite vast application of hydro-economic modelling

approaches, no work has been carried out to date to consider uncertainty issues among

recent hydro-economic modelling (Momblanch et al., 2016). This can be addressed by

incorporating limited hydrological foresight (Draper, 2001) into hydro-economic models.

Implementation of limited foresight in hydro-economic models demands accurate

estimation of COSVFs. This can be done by linking an Evolutionary Algorithms (EAs) to

hydro-economic models-this has not yet been addressed in literature. A major gap in

developments of hydro-economic modelling has been the weak integration of physically-

based representations of different components of water sources and uses to inform complex

basin scale policy choices (Kahil et al., 2016). Another drawback of hydro-economic

models can be poor simulation of actual water markets since individual agent behaviour

and transaction costs cannot be represented easily (R.C. Griffin, 2006; R. A. Young, 1986).

35

1.4.5 Simulation vs optimisation

Use of hydro-economic models and the potential of linking them to EAs require defining

and distinguishing between simulation and optimisation models. Simulation is the

technique used for the evaluation of the consequences of decisions without being

implemented in the real system concerned. This is somewhat a too generic definition in the

field of water resource systems and applies to a broader meaning of simulation. Votruba

(1988) narrows down this definition for water resource problems: In water resource

system design, simulation is a modelling technique in which the operation of the water

resource system is represented by mathematical and logical relationships in a chosen time

step based on specific inputs (inflows of water into the system, demands for the supply of

water and for water-derived products and services), capital costs of hydroelectric power

plants, capacities of diversion tunnels, etc., and on some predetermined operational policy.

Following this definition, any model that fulfils the above requirements can fall into

simulation class such as hydro-economic models. Use of optimisation formulation in

hydro-economic models implies that simulation models can consist of more than

traditional set of policy rules.

There are four major groups of simulation models: rule-based, optimisation-based, agent-

based, and hybrid approaches. Rule-based water system simulation models (e.g. Aquator;

OxfordSicentificSoftware (2014), IRAS-2010; Matrosov, Harou, and Loucks (2011), and

LARaWaRM; Walsh et al. (2016)) offer a flexible framework to evaluate management

plans with a set of governing rules. They are used to analyse the impact of a finite number

of proposed planning strategies on the environmental system by answering the question

‘what if’ (Padula, 2015). Rule-based models applied to real-world problems can quickly

become inefficient and tedious (Draper et al., 2004) as rules and specifications regarding

how and when they should be applied can quickly become complex.

Optimisation-based models (e.g. WRIMS; Munévar and Chung (1999) and Draper et al.

(2004), MODSIM; Labadie (2005), OASIS; Randall, Cleland, Kuehne, Link, and Sheer

(1997), SISAGUA; Barros, Zambon, Delgado, Barbosa, and Yeh (2005), WATHNET;

Kuczera (1992), and WEAP; Yates, Sieber, Purkey, and Huber-Lee (2005)) identify the

solution to a problem, among an infinite number of possible alternatives, by answering the

question ‘what’s best’ (Padula, 2015). Optimisation-based models determine water

allocation and/or reservoir operation by taking them as decision variables, locating their

36

optimal values which maximise or minimise an objective function (or set of objective

functions) subject to budgets, environmental, or resource availability constraints (Castillo,

Conejo, Pedregal, García, & Alguacil, 2001). Optimisation-based models can quickly be

plagued by curse of dimensionality, especially in presence of non-linear constraints and

objectives which complicates the identification of a ‘global optimum’ solution (Edgar,

Himmelblau, & Lasdon, 2001). This, however, is the case when mathematical

programming methods are used. When it is not possible to solve for a global optimal

solution, heuristic methods are commonly used in place of classical mathematical

optimisation techniques (Padula, 2015). Heuristic methods cannot guarantee global optimal

solution, but they can locate a near-optimal or a locally optimal one.

Agent-based models (e.g. REPAST; North, Howe, Collier, and Vos (2005), and

ABSTRACT; van Oel, Krol, Hoekstra, and Taddei (2010)) are composed of autonomous

entities or agents which have only limited knowledge and information processing

capacities. The agent-based concept is a mind-set rather than a technology, where a system

is described from the perspective of its constituent parts (Bonabeau, 2002). Nonetheless,

from a pragmatic modelling point of view, there are several features that are common to

most agents (Wooldridge and Jennings (2009)- extended and explained further by Franklin

and Graesser (1997); Epstein (1999); Torrens (2004); Macal and North (2010)). Agent-

based models are comprised of multiple, interacting agents situated within a model or

simulation environment. A relationship between agents is specified, linking an agent to

other agents within a system. An agent-based model has to be built at the right level of

description for every phenomenon, judiciously using the right amount of detail for the

model to serve its purpose (Couclelis, 2002). Castle and Crooks (2006) point out agent-

based models are often not generalizable and are developed in a case-specific manner.

Finally, hybrid approaches (e.g. RiverWare; Zagona, Fulp, Shane, Magee, and Goranflo

(2001), and SOURCE; Welsh et al. (2013)) combine one or more of the aforementioned

techniques to simulate the water system. Depending on what techniques are included in

this class of models, they can inherit their drawbacks and merits. One common issue is

that, because of incorporating various modelling approaches and being complex, they are

often computationally intense and easily plagued by curse of dimensionality. A simulation

model of any aforementioned type can be linked to an external model in order to overcome

the curse of dimensionality. As initially detailed by Geoffrion (1972) and later expanded

37

by X. Cai, McKinney, and Lasdon (2001) and Ximing Cai et al. (2002), a set of

“complicating variables” are chosen from the original model, which make the problem

much easier to solve (e.g. linear) once they are known/fixed. The complicating variables

are sought for by an external EA (e.g. Genetic Algorithm (GA), Simulated Annealing

(SA), or Tabu Search (TS)) and a now-simpler model is solved to simulate the system for

each set of fixed/known values suggested by the EA. When an EA is used to vary the

complicating variables, there are no restrictions on how these variables can appear in the

model. Indeed, this generality is obtained at a price: convergence is not guaranteed and it is

often slow (X. Cai et al., 2001).

1.4.6 Optimisation methods

Optimisation techniques fall within two main categories: mathematical programming and

heuristic search algorithms. Mathematical programming (e.g. Linear Programming, Non-

Linear programming, Mixed Integer Linear programming, etc.) relies on calculation of

derivative of the objective function, the performance criterion to be minimised or

maximised. Hence, problems with discontinues objective function cannot efficiently be

solved using this class of methods. In addition, in non-linear problems, it becomes more

likely that the algorithm is trapped in the local optima as the size of the problem increases.

When it comes to multi-objective problems, implementation of mathematical programming

would be problematic as they rely on combining several objectives into one using weights.

This makes the method sensitive to the weights chosen and the results can vary

significantly with different weights. Yet, mathematical programming methods are often

fast and free from extensive parameterisation of the algorithm. Heuristic search algorithms,

especially evolutionary algorithms, explore the solution space and evolve to the optimal

solution. Such methods can never guarantee reaching an optimal solution in an absolute

sense. However, available approaches reported in literature help in ensuring a near-optimal

solution can be obtained. Most evolutionary algorithms require configuration of search

parameters and the solution can sometimes be sensitive to this parametrisation. They are

also time-consuming compared to mathematical programming as they often need

thousands of objective function evaluations (iterations) to converge to a near-optimal

solution.

Recent advances in high performance computation facilities made EAs an interesting

option for researchers. Perhaps this is the reason why EAs have continued improvement

38

and development in the past three decades. One of the most recent and novel EAs is BORG

MOEA (Hadka and Reed, 2013). BORG’s auto-adaptive nature controls algorithms

parameterisation, meaning that users need to configure very few parameters which will

have minimum effect on the quality of the obtained solution. Users are required to pre-

determine the number of initial population as BORG modifies this number throughout its

iterations to set the ‘best’ estimate of population. Additionally, maximum number of

function evaluations is input to the algorithm by users. In case of multi-objective

optimisation problems, search resolution (ɛ) is required by users. These features make

BORG one of favourite EAs, which can be found in many studies (Chilkoti et al, 2018;

Giuliani et al, 2018; Eckart et al, 2018; Yan et al, 2017; Zatarain Salazar et al, 2017; and

Al-Jawad and Tanyimboh, 2017). Niayifar and Perona (2017) compared the performance

of BORG and Non-dominated Sorting Genetic Algorithm-II (NSGA-II) on a water

allocation problem and reported that NSGA-II, with the need to tune search parameters,

took twice the time BORG required to solve the problem. For these reasons, BORG is used

in this thesis as optimisation algorithm.

1.5 Case study: California Central Valley

For illustration purposes, the large-scale water resources system of California’s Central

Valley is adopted from the CALVIN (CALifornia Value Integrated Network; Draper et al.

(2003)) model and is modified for this thesis. The CALVIN model explicitly integrates the

operation of water facilities, resources, and demands for California’s great inter-tied

system. It is the first model of California water system where surface water resources,

groundwater, and water demands are managed simultaneously state-wide. This model

covers 92% of California’s population and 88% of its irrigated land, with approximately

1200 spatial elements, including 51 surface reservoirs, 28 groundwater basins, 18 urban

economic demand areas, 24 agricultural economic demand areas, 39 environmental flow

locations, 113 surface and groundwater inflows, and numerous conveyance and other links

representing the vast majority of California’s water management infrastructure. Figure 3

shows the network schematic of CALVIN where nodes include surface reservoirs,

groundwater aquifers, urban and agricultural demand zones, junctions, outflow (sink) sites,

etc.; arcs (links) include man-made canals, pipelines, aqueducts, rivers, springs, etc.

39

The fundamental optimisation framework for CALVIN is network flow optimization with

gains and losses, also known as generalized network flow optimisation. The general

mathematical form is as follows (Jensen & Barnes, 1980):

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒 𝑍 =∑∑𝐶𝑖,𝑗𝑋𝑖,𝑗𝑗𝑖

(12)

subject to

∑𝑋𝑗,𝑖 =∑𝑎𝑖,𝑗𝑋𝑖,𝑗 + 𝑏𝑗 ∀ 𝑛𝑜𝑑𝑒 𝑗

𝑖𝑖

(13)

𝑋𝑖,𝑗 ≤ 𝑢𝑖,𝑗 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑙𝑖𝑛𝑘 𝑓𝑟𝑜𝑚 𝑛𝑜𝑑𝑒 𝑖 𝑡𝑜 𝑛𝑜𝑑𝑒 𝑗 (14)

𝑋𝑖,𝑗 ≥ 𝑙𝑖,𝑗 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑙𝑖𝑛𝑘 𝑓𝑟𝑜𝑚 𝑛𝑜𝑑𝑒 𝑖 𝑡𝑜 𝑛𝑜𝑑𝑒 𝑗 (15)

where Z is the total cost of flows throughout the network, Xi,j is flow leaving node i

towards node j, ci,j is the economic costs (agricultural or urban), bj is the external inflows to

node j, ai,j represents gains/losses on flows in arc i,j, ui,j is the upper bound on flow in arc

i,j, and li,j is the lower bound on flow in arc i,j. Some extensions to the above simple model

can be implemented. For instance, convex piece-wise linear cost functions on single arcs

can be represented by using several arcs to represent one physical arc, with each sub-arc

having an appropriate upper bound and unit cost. Also, the losses (ai,j in equation (13)) can

be used to encompass reservoir evaporation, conveyance losses, consumptive use, and

reuse.

The basic network flow optimization (without gains and losses) has long been used to

model water problems (Labadie, 1997). The addition of gains and losses enables a more

explicit representation of return flows, system losses, and differences between applied and

consumptive water use. While there is little novelty in this problem formulation, its speed

and simplicity allows for the solution of larger and more detailed problems than would

otherwise be possible (Jenkins et al., 2001).

The objective of the above optimisation is to maximise the year 2020 economic benefits

(or minimise costs) of water operations and delivery to agricultural and urban water users

throughout the state-wide inter-tied system over the range of hydrologic conditions

represented by the 1922-1993 historical hydrology (Jenkins et al., 2001). In CALVIN,

Water is valued according to the standard economic principle of willingness-to-pay, i.e.

water is worth what the users are willing to pay for it. Environmental requirements are

implemented by a series of minimum and fixed flow constraints at selected stream and

40

wetland locations. Where possible, these environmental constraints are made to vary

according to projected 2020 environmental regulations. The hydrologic representation in

the model includes surface water and groundwater inflows, and return flows to surface and

groundwater resulting from urban and agricultural water delivery. These are taken to

represent year 1922-1993 monthly hydrologies under year 2020 development conditions.

The proposed model of this thesis is inspired from the CALVIN mode. All modifications

are explained in details in chapter two of this thesis. Additionally, this thesis uses a slightly

modified water network than to the one shown in the box in Figure 3. In the CALVIN

model each agricultural demand node is represented by two nodes to account for return

flows to surface and groundwater resources. In this thesis they are aggregated and a link

for a flow of the sum of return flows (groundwater and surface water) represents the

percentage of delivered water transferred back to the water network. Similarly, each urban

demand is divided into residential and industrial demand in the CALVIN model, whereas

they are merged into one node in this thesis. Additionally, water treatment nodes are

excluded in this thesis. Appendix A demonstrates the input data used for the work of this

thesis.

Figure 3. Network schematic of the CALVIN model from. Northern parts (left side) form the upstream of the water system. The area shown in the box is the extent of the

CALVIN model used in this thesis.

1.6 Summary of the methodology

To address the research question, this dissertation contributes to a generalisable approach

of estimating the economic valuation of end-of-year storage in large-scale water resources

system, where existing approaches fail due to non-convexity and/or curse of

dimensionality often involved in real-life cases. The proposed approach provides a

generalizable and direct method of valuating stored water by implementing COSVF. This

to avoid alternative approaches such as SDDP which rely on the Lagrange multiplier

(shadow value) associated to mass balance constraint to extract water value. Such

approaches are applicable to systems with components having linear (convex) behaviour.

Use of COSVF implies that the proposed approach a) need to use a hydro-economic

modelling approach to decide on inter-annual reservoir operation question, and b) must

implement the limited foresight in order to be able to assign a COSVF at the end of each

water year. These two points result in a more complex model compared to traditional

deterministic non-hydro-economic models. This rings the alarm that the proposed model

can easily be affected by curse of dimensionality. To avoid this, and in the light of the

work of X. Cai et al. (2001), a hybrid approach is used that links an EA to a hydro-

economic model. Borg-MOEA (Hadka & Reed, 2013) was used as the EA. Borg’s self-

adaptive features increase its robustness and effectiveness while minimizing the search

parametrization by the user. The non-convex hydro-economic model used in this

dissertation was coded in GAMS and solved using the Minos solver version 5.5 (Murtagh

& Saunders, 1998). Minos applies the Generalised Reduced Gradient (GRG) method

which is suitable for nonlinear programming problems with linear constraints (Labadie,

2004).

Reservoirs’ COSVF can be used to derive reservoirs’ demand curve. By definition, a

reservoir demand curve is the derivative of its COSVF (Figure 4). Quadratic COSVF is a

common form for such utility function (Draper, 2001). This is because an additional unit of

reservoir’s water is positively valued (positive first derivative of COSVF) and any

additional unit of water has a declining marginal value (negative second derivative of

COSVF). Derivative of such quadratic function is a negatively sloped linear function-the

demand curve. When water is scarce (at dead storage) any additional unit of water in

reservoir is highly valued, when it is plenty (at full capacity) the marginal value of water is

very small (sometimes zero). Obtaining the demand curve of reservoirs in a water system

43

provides valuable information for regulators and water resources managers. This

information can be applied as a proxy for economic water scarcity (Pulido-Velazquez,

Alvarez-Mendiola, & Andreu, 2013) which suggests to decision-makers where to focus for

new policy decisions/regulations, investments, etc. and highlights zones eligible for further

development. End-of-year COSVF are quadratic functions of storage in each surface

reservoir, depending on two water marginal values (p1, p2) at maximum and minimum

storage (Smax and Smin) respectively defined by:

{

𝐶𝑂𝑆𝑉𝐹(𝑝1, 𝑝2, 𝑠𝑚𝑖𝑛) = 0𝑑𝐶𝑂𝑆𝑉𝐹

𝑑𝑆|𝑆=𝑠𝑚𝑖𝑛

= 𝑝1

𝑑𝐶𝑂𝑆𝑉𝐹

𝑑𝑆|𝑆=𝑠𝑚𝑎𝑥

= 𝑝2

(16)

Figure 4. Relation between demand curve and COSVF of a reservoir.

where S is the storage level. There are two requirements to effectively use COSVFs: (1)

this function is used only for the storage of the last month of any year, and (2) it must not

vary over years. The requirement (1) is because limited foresight includes a perfect within-

year foresight and no foresight beyond each year. When the model “sees” the hydro-

climatic condition within a given year (intra-annual; e.g. months 1 to 11) and optimises to

achieve maximum economic benefit from allocation, implementing COSVF for any month

X, 1 ≤ X ≤ 11, to incentivise reservoirs to keep water for future [X-12], will be ineffective.

This is the basic concept of the limited hydrological foresight. Since no foresight beyond

the current year is embedded into such models, a COSVF for end-of-year storage serves as

boundary condition that prevents reservoirs from depletion. The requirement (2) is in fact

44

an adherence of limited foresight approaches. Varying COVFs implies that information of

future events beyond the current year exists.

It should be noted that surface and groundwater storages have asymmetric roles in water

valuations. Without value functions for surface reservoirs, their use would be free in a

hydro-economic model, leading to their more aggressive depletion. Therefore, surface

water storage valuation is crucial to represent the uncertainty value of stored surface water.

Moreover, surface water storage is filled and depleted every year or every few years at

most. This short timescale compared with the study period makes interpretation of

COSVFs unambiguous. This is not the case for groundwater. It is proposed that (1) the cost

of using groundwater seen by the hydro-economic model is “pumping cost + carryover

storage value.” In most aquifers, pumping costs are large enough that the COSVF is near

zero; (2) as a consequence of (1), integration (or not) of COSVF has little effect on

management outcomes; and (3) large, multi-decadal variations in the aquifer storage make

their COSVF (when it exists) difficult to interpret.

In order to challenge the proposed methodology against curse of dimensionality, a large-

scale hydro-economic model inspired from CALVIN is used to simulate the water

resources system of California’s Central Valley while maximising the system-wide

economic benefit from water allocation. The hydro-economic model follows the node-arc

representation of the water resources system. Nodes include surface and groundwater

reservoirs, urban and agricultural demand points, junctions, etc., and arcs (links) include

canals, pipes, natural streams, etc. (Shamir, 1979). This network comprises over 300

nodes, including 30 surface reservoirs, 22 groundwater sub-basins, 21 agricultural demand

sites, 30 urban demand sites, 220 junction and 4 outflows nodes; and over 500 links (river

channels, pipelines, canals, diversions, and recharge and recycling facilities).

The hydro-economic model simulates the water system by annually maximising the net

economic benefit from water allocation. This model also applies a COSVF for each surface

reservoir at the end of each annual run in order to prevent over-exploitation of water in

reservoirs. Parameters that define each COSVF (p in equation (16)) must be optimally

determined to result in an economically efficient solution. Different components of the

hydro-economic model used in this thesis are described in detail in Chapter 2.

45

BORG is used to vary the valuation of end-of-year storage of surface reservoirs. Recall

that COSVF parameters can be translated into marginal values through derivation. Each

population member of BORG suggests a set of p (or marginal values at dead and full

storage) which then becomes fixed and known to the hydro-economic model. The hydro-

economic model then simulates the system to assess the resulting net economic gain over

the planning horizon. This will be the metric (fitness function) to evaluate the performance

of COSVF parameterisation. A set of COSVF parameters that lead to higher overall net

benefit will be considered as the “fitter” solution and survives to the next generation. This

process continues until a stopping criterion is met.

Use of single objective EA with the above fitness function would become problematic.

Since the valuation of end-of-year storage is a tool to prevent depletion of reservoirs, if a

valuation is enough to do so, any valuation above that “true” value will also lead to

preserving storage. For instance, if the true marginal water value of a reservoir is $10 per

cubic meter (this is the amount below which reservoir starts to drain out), a marginal value

of $100 per cubic meter or $1000 per cubic meter will also encourage the reservoir to keep

water in storage. In order to eliminate EA’s locating such unreasonably high values, a

second objective is introduced which minimises the arithmetic mean of stored water

marginal value at dead and full storage. In this manner, the optimal solution found by

BORG is the “true” marginal value of water, lowest possible water marginal value that

keeps water in storage at the end of each year. The introduction of the second objective

requires using a multi-objective EA. The auto-adaptive BORG MOEA can tailor the

various parameters that control the behaviour of the algorithm to the specific

characteristics of the problem (Reed, Hadka, Herman, Kasprzyk, & Kollat, 2013). Using a

MOEA offers the option of including other ecological, risk-related, and socioeconomic

metrics, if needed.

The MOEA evaluates solutions based on the generated economic profit. To calculate this

profit, the water system is simulated by the hydro-economic model using a historical time-

series of streamflow. An argument arises here that the performance of the proposed

approach can be sensitive on the input hydro-climatic data. In order to investigate the

sensitivity of the marginal water value solutions to the choice of streamflow and to

challenge the robustness of the proposed methodology, a sensitivity analysis is carried out.

An ensemble of 100 synthetic scenarios is built by bootstrapping over the historical

46

scenario. Then, the water system is re-simulated using the obtained optimal water valuation

solution and 100 different inflow scenarios. Results show that the proposed model is robust

and water marginal value solutions are insensitive against a wide range of hydro-climatic

scenarios.

Several applications can be derived from the proposed generalisable methodology. One

interesting applications is determination of historical valuation of a multi-reservoir system:

how does the historical operation of reservoirs translate to valuation of water? How

operators valuated water in reservoirs in the past? This valuation can be used (1) in

comparative studies, (2) to assess how different historical operation was from the

optimised (economy-efficient) one, and (3) to discover reservoirs that were eligible for

expansion or removal. In order to find the historical water valuation, the main fitness

function (net economic profit) of the proposed model has to be replaced by a calibration

function. The calibration function minimises the deviation between the resulting operation

(quantified by reservoir storage level) and the observed operation. In this manner, the

solution of the MOEA will be a set of reservoir valuation that leads to best mimicking

historical operation.

1.7 Thesis structure

The remainder of this thesis is structured as follows: Chapter ‎2 represents the first paper

entitled “An extended hydro-economic model – Application to California’s water resource

system”. This paper introduces the hydro-economic model used in this dissertation along

with the extensions/improvements made to this model. The second paper entitled

“Estimating the economic value of inter-annual reservoir storage in water resource

systems” forms Chapter ‎3. This chapter includes linking the hydro-economic model to the

EA, water value solutions of California’s Central Valley water resources system, and a

sensitivity analysis of the results. In Chapter ‎4, the third paper namely “Investigating

historical valuation of water resources – a California Central Valley case study” is

presented. This paper describes methodological steps taken to obtain the historical

valuation of water in the Central Valley, interpretation of historical valuation and

comparison to those of Chapter ‎3, and what these valuations mean. Finally, Chapter ‎5

brings discussion over the proposed methodology as well as concluding remarks. Appendix

A represents the set of input data used to run the model proposed in this thesis. Appendix B

investigates how different search parametrisation of the evolutionary algorithm used in this

47

thesis, may influence results. Appendix C elaborates the cause of differences between

Model A (of Chapter ‎2) results to those of CALVIN, on which Model A is built. Appendix

D describes model formulation used in this thesis. Appendix E explains a post-process

stage of how to interpret and extract results from the obtained Pareto trade-off after

implementing the MOEA. Appendix F describes how historical operation is approximated

and how well they match observed data. It should be emphasised that due to the journal

format of this thesis, one shall expect overlap in the material presented in this chapter and

in others.

1.8 Contributions to research

Contribution of this thesis to research will be highlighted in each of technical chapters (i.e.

Chapters ‎2-‎4). Instead, in this section, the contribution of the PhD candidate to each paper

is represented as a requirement for thesis submitted in the journal format.

In Chapter ‎2, the PhD candidate corrected the limited foresight implementation of the

hydro-economic model, added hydropower generation to the model along with gathering

the associated data, modified the benefit function of the agricultural demand sites which

were creating bias for the model, spotted mistakes in the input data and fixed them,

performed all the modelling after above modifications and drafted the paper. In Chapter ‎3,

linking the hydro-economic to BORG and running parallel jobs on University of

Manchester’s High Performance Computing (HPC) cluster (CSF), proposing inclusion of

the second objective (initial runs was with a single objective which lead to multiple near-

optimal solution), generating the synthetic ensemble by bootstrapping and performing a

sensitivity analysis, proposing the rationale for extracting water value solution from the

Pareto trade-off (Appendix E), producing figures, and drafting the paper was the

contribution of the PhD candidate. In Chapter ‎4, the PhD candidate contributed to the idea

of calibrating to get historical water values, carrying out technical and modelling tasks,

interpretation of results, and drafting the paper.

48

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ZATARAIN SALAZAR, J., REED, P. M., QUINN, J. D., GIULIANI, M. &

CASTELLETTI, A. 2017. Balancing exploration, uncertainty and computational demands

in many objective reservoir optimization. Advances in Water Resources, 109, 196-210.

ZHU, T., MARQUES, G. F. & LUND, J. R. 2015. Hydroeconomic optimization of

integrated water management and transfers under stochastic surface water supply. Water

Resources Research, 51, 3568-3587.

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2 Chapter two: How do hydro-economic model formulations

impact their recommendations?

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How do hydro-economic model formulations impact their

recommendations?

Abstract

The proposed model has limited (intra-year) hydrologic foresight and a dynamic

representation of groundwater pumping resulting in a nonlinear formulation. To understand

the impact of these features, we compare the proposed model’s results to those of other

closely related model formulations with perfect foresight and static groundwater costs. The

limited foresight feature is implemented by maximizing net benefits over consecutive

annual runs using historical data. These runs are optimized individually using a carry-over

storage benefit function to prevent emptying of reservoirs. Each annual optimization is

linked to the previous one by end-of-year storage. Dynamic groundwater pumping is

implemented using the storage coefficient method for independent groundwater sub-basins.

The piezometric head in each period is calculated as a function of piezometric head in the

previous one, groundwater withdrawals and a lumped storage coefficient for the

groundwater sub-basin. This head is then used to calculate the cost of pumping water out

of the groundwater sub-basin. This paper describes a proposed hydro-economic model and

applies it to the California water resource system (excluding isolated northern areas and

Southern California). The proposed model simulates an idealized version of California’s

water market using an optimization model where water allocations, groundwater pumping,

reservoir releases and storage levels are optimized for each year independently. The

proposed open-source model and its formulation variations offer practitioners to build

customized hydro-economic models. Such models suggest how to increase economic

efficiency of water systems by suggesting allocative, institutional and engineering

measures.

Keywords: Hydro-economic, Central Valley, CALVIN, Optimization, Limited foresight,

Dynamic pumping cost

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2.1 Introduction

Various tools combining economy and engineering have been applied to aid water

resources management (Pulido-Velazquez, Andreu, Sahuquillo, & Pulido-Velazquez,

2008). Hydro-economic models (Julien J. Harou et al., 2009) use economic principles to

support decision making on water allocation, alternative evaluation, and institutional

design (Braden, 2000; R. C. Griffin, 1998; J. R. Lund, X. M. Cai, & G. W. Characklis,

2006). Economic demand curves are used instead of water requirements to reflect the value

of water depending on its availability (J. M. Griffin & Steele, 1980). In hydro-economic

optimization models, these demand curves are used to direct the allocation of water, the

idea being that simulating such an idealized water market can provide insight into more

efficient water management (i.e., best value allocation between users and over time) and

system expansion.

In order to carry out this optimization, typically all the modelled time-periods are

combined into one large mathematical program. Optimising over a single time horizon

means the model can optimise the inter-temporal management (i.e. store or use now) but it

also means the models benefits from ‘perfect hydrological’ foresight (Draper, 2001),

meaning the allocations over time are efficient because the model considers future

conditions and can anticipate (e.g., allocate less before a dry period). To remove or limit

this foresight, for example by splitting the modelling into annual sub-models, however is

not easy as it would require to pre-select target storages which violates the purpose of

hydro-economic optimization analysis (finding the most efficient management).

Another issue with hydro-economic optimization modelling is that keeping the model

linear (or at least quadratic) makes them easier to solve. Because they tend to be large

(often for the reason in the prior paragraph), linearity is required to enable them to be

solvable at all (i.e. the model is so large, with hundreds of thousands of decision variables,

that a linear model is better to ensure the optimality of the solution).This encourages model

builders to make simplifications and remove non-linearities (e.g., frequent targets in water

systems are linearising hydropower production or groundwater pumping).

In this paper we propose two solutions to the deterministic hydro-economic optimization

modelling conundrum: introducing limited foresight via carry-over storage value functions,

64

and introducing non-linear groundwater pumping costs. In some sense, the first enables the

second as a 100-year monthly hydro-economic (solved in one go) can take days to solve,

whereas an annual one can take seconds or minutes (because the speed of solution of

mathematical programmes non-linearly expands with the number of decision variables).

The smaller annual model enables more complexity to be introduced; and we examine how

non-linear pumping costs impact model recommendations.

The different proposed model formulations are applied to California’s water management

system (excluding northern isolated basin and Southern California). Water management in

California, as the world’s sixth largest economy and fifth largest supplier of food, treats

water like an economic good i.e. it can be traded in markets. Competition among

agricultural, urban, and environmental sectors has intensified with population growth and

increasing environmental allocation. The California Value Integrated Network (CALVIN)

model has been used extensively for water management studies in California (Draper et al.,

2003; Jenkins et al., 2004). This is a deterministic hydro-economic optimization model

which assumes perfect foresight and fixed pumping costs. To deal with the large model

size, this model uses a network flow formulation to increase the speed at which it can be

solved. In this paper we slightly simplify this model and implement it within a generalized

optimization scripting software environment so that we can implement various

formulations which show the impact of different combinations of assumptions on foresight

and linearity. The rest of the paper is organised as follows, section 2 outlines methods,

section 3 describes the application to California, and section is 4 results, followed by

discussion and conclusion.

2.2 Methodology

The proposed model is a generalized hydro-economic model that uses optimization to

simulate California’s water market under different hydrological or other scenarios. Flows

and storage are determined via constrained maximization of the net benefits from water

allocation to urban and agricultural uses, constrained with environmental flows and

conveyance and storage capacities. There are 4 versions of the model, which are

summarized in Table 1.

65

Table 1. Formulations of the model used in this paper. Model A is the ‘benchmark model’ and Model D is

the ‘proposed model’.

Perfect foresight Limited foresight

Static pumping costs Model A Model B

Dynamic pumping costs Model C Model D

Versions B and D of the model implement limited foresight, i.e., each year is optimized

independently with the final storage at the end of year 𝑦 serves as initial storage at the

beginning of year 𝑦 + 1. An end-of-year carryover storage valuation is used to balance

within-year uses with future uses. Limited foresight here still implies a perfect within-year

forecast.

Versions C and D of the model implement dynamic pumping costs, i.e., costs that are

dependent to the piezometric head. This reflects the fact that the energy required to pump

groundwater is directly proportional to the height over which that water must be lifted. In

contrast, models A and B use a constant pumping cost per unit of water extracted.

2.2.1 Annual objective function

Given a deterministic inflow sequence, monthly operation decisions determine flows 𝑥𝑡

through each link, as well as piezometric head ℎ𝑡 in aquifers and storage 𝑠𝑡 in reservoirs.

Then, the model maximizes the objective function Z that equals the sum of net benefits

𝑓𝑡(𝑥𝑡, ℎ𝑡, 𝑠𝑡) over each month (t) of the year, and of a terminal value function 𝐶𝑂𝑆𝑉𝐹(𝑠𝑇)

with T=12 which represents the economic value of carryover storage:

𝑍 =∑𝑓𝑡(𝑥𝑡, ℎ𝑡, 𝑠𝑡)

𝑇

𝑡=1

+ 𝐶𝑂𝑆𝑉𝐹(𝑠𝑇) (17)

This maximization problem is constrained by water balance equations at all nodes in the

model and by environmental lower and physical upper limits to flows in the links, as well

as capacity constraints on storage in reservoirs and piezometric head in aquifers. Besides, a

66

major aspect of California’s hydrology is return flows from agricultural and urban

activities (M.W. Jenkins et al., 2001). Return flows of applied water from agricultural and

urban water use to surface and groundwater deep percolation are included in the proposed

model. They are expressed as a percentage of water used at each demand node. Detailed

description of the mathematical representation of the proposed model can be found in

Appendix D. Here, we explain the two proposed extensions only.

2.2.2 Extension 1: Carry-over Storage Value Functions (Models B and D)

Carry-over storage value functions prevent the model from depleting surface reservoirs at

the end of each water year (Draper, 2001). Carry-over storage value increases as the

storage level (S) in a reservoir sr rises. We use a linear COSVF in this study:

𝐶𝑂𝑆𝑉𝐹(𝑠𝑠𝑟,𝑇) =∑𝑚𝑠𝑟𝑠𝑠𝑟,𝑇𝑠𝑟

(18)

Here, T indicates the last month in each water year (September). This preliminary version

of the model uses only linear parameter for COSVFs, msr, that are derived from shadow

values of mass balance constraints of Model A.

2.2.3 Extension 2: Dynamic Groundwater Pumping Costs (Models C and D)

The CALVIN model determines pumping costs by multiplying the unit pumping cost of

$0.20 per af/ft lift by an estimate of the average pumping head in each groundwater sub-

basin (Hansen, 2007). This is a very simplified representation of the physical process of

groundwater pumping (J. J. Harou & J. Lund, 2008), therefore Models C and D add a

module whereby pumping costs vary with piezometric head. Groundwater pumping costs,

𝐺𝑊𝑡, are represented in the objective function as follows.

𝐺𝑊𝑡(ℎ𝑡, 𝑥𝑡) =∑C𝑡𝑔𝑗(ℎ𝑡

𝑔). 𝑥𝑡

𝑔𝑗

𝑔,𝑗

∀𝑡 (19)

where g and j are the two nodes at the extremities of the groundwater pumping link – g

being the aquifer. In models A and B, coefficients C𝑡𝑔𝑗

are constant regardless of the head.

67

However, in models C and D, the change in groundwater pumping costs as related to

piezometric head is reflected in the cost of pumping from groundwater sub-basin g to node

j as shown below.

C𝑡𝑔𝑗(ℎ𝑡

𝑔) = 𝑢𝑛𝑖𝑡𝑐 ∙ (𝑒𝑙𝑒𝑣𝑔 − ℎ𝑡

𝑔) ∀𝑔, 𝑗, 𝑡 (20)

where 𝑢𝑛𝑖𝑡𝑐 ($/ft/Kaf) is the cost of energy required to lift by a unit length a unit of water,

𝑒𝑙𝑒𝑣𝑔 (ft) is the mean ground elevation above aquifer g, so that the difference in equation

(20) is the height over which water must be lifted.

J. J. Harou and J. Lund (2008) suggest that the storage coefficient formulation is the most

economical method to model both lumped groundwater volume and head functions. The

storage coefficient relates the volume of water released or absorbed into or from storage

(net stress) per unit surface area of confined aquifer per unit change in piezometric head.

As per J. J. Harou and J. Lund (2008) the hydraulic head in each groundwater sub-basin is

calculated as follows:

ℎ𝑡𝑔= ℎ𝑡−1

𝑔+𝑖𝑛𝑓𝑔

𝑡 + ∑ 𝑥𝑡𝑗𝑔− ∑ 𝑥𝑡

𝑔𝑗𝑗𝑗

𝑠𝑐𝑔 ∙ 𝑎𝑟𝑒𝑎𝑔

(21)

where 𝑖𝑛𝑓𝑔𝑡 is recharge from precipitation; 𝑥𝑡

𝑗𝑔 is artificial recharge flow from node j into

aquifer g, e.g., through percolation of irrigation water; 𝑥𝑡𝑔𝑗

is pumping, the flow from

aquifer g to node j; 𝑠𝑐𝑔 is the mean storage coefficient of aquifer g; 𝑎𝑟𝑒𝑎𝑔 is the aquifer’s

area.

2.2.4 Data validation and further refinements

Extensions 1 and 2 described in the previous sections were the outcome of studies of

several students and researchers prior to the author of this thesis. The author of this thesis

made several other amendments and refinements after the first two extensions were

implemented, these include fine-tuning the implementation of extensions 1 and 2.

Following a scrutiny of the input data, the author of this thesis observed several mistakes

and mismatches, including aquifers’ storage coefficients, aquifers’ initial head, and storage

nodes’ initial storage. Furthermore, the limited (plus the perfect within year) foresight of

68

the model means that the boundary conditions, i.e. COSVFs, must be implemented at the

end of each annual optimisation, whereas the model initially had COSVFs employed at the

end of each month within each perfect-foresight optimisation runs. This is not sensible and

essentially a wrong implementation of COSVFs for a limited foresight model, although

this might not affect results. The author of this thesis fixed this issue. The author also

included hydropower generation in the model, as well as amending urban and agricultural

benefit functions. The model initially used benefit functions that showed zero marginal

value of allocating water to demand nodes at their target demand. This was problematic

especially for farms where unit pumping cost is contrasted to the value of per unit of water

to determine allocation. At full demand, there was initially zero benefit for farmers

according to the old benefit functions. This encouraged farmers not to pump any water at

vicinity of full demand because the unit groundwater pumping cost was much higher than

the marginal value of water for agricultural yields. This resulted in unnecessary drought in

demand nodes. The author of this thesis fixed this issue by replacing the old agricultural

benefit function with its piece-wise linear counterpart (data used to do so was obtained

from the CALVIN database) and the old benefit function for urban water uses was replaced

with a new quadratic function with a non-zero slope at full demand (marginal water value

greater than zero). The latter was done using data from M.W. Jenkins et al. (2001) with

water retail prices (Black&Veatch, 1995) to represent urban willingness-to-pay at target

demand.

2.2.5 Implementation

Equations (17) to (21) and (A 1) to (A 3) form a non-linear optimization formulation,

which are implemented in GAMS (General Algebraic Modelling System; Rosenthal

(2016)), a high-level modelling system for mathematical programming and optimization.

All four models are solved using MINOS (Murtagh & Saunders, 2013) as the solver.

MINOS applies the Generalised Reduced Gradient (GRG) method which is suitable for

nonlinear programming problems with linear constraints (Labadie, 2004).

2.3 Application to California

The benchmark model and data are built using CALVIN (Draper et al., 2003; Marion W.

Jenkins et al., 2004). The model formulations are an extension of various PhD theses

including Draper (2001), Hansen (2007), Medellín-Azuara (2006), J. J. Harou and J. Lund

69

(2008). The groundwater data are taken from the CVRASA groundwater model built by

the USGS (Faunt, 2009).

In the context of California, the perfect intra-annual foresight is reasonably consistent with

the observation that early spring measurements of the depth and water content of the

snowpack in many watersheds provide reasonably precise forecasts of reservoir inflow to

the end of the water year (Draper, 2001). The impact of perfect within-year forecast on

winter operations is limited because Central Valley inflows are dominated by springtime

snowpack melt.

70

Figure 5. Network schematic of the Central Valley of California case-study application.

Extent of the CALVIN model included in this application

71

2.4 Results

We present three sets of results below. In Part I we show how the benchmark model

(Model A) compares to CALVIN results (both the constrained which reproduces ‘historical

approximation and unconstrained model runs denoted as ‘CALVIN Optimised’). In the

second part, we compare results of the 4 formulations. Although the main contribution is

the improvement from Model A to Model D, we include all four models should one seek to

investigate the sole impact of implementing each of the proposed extensions. For models

with limited foresight, we used shadow values from the mass balance constraint (storage

nodes only) of the benchmark model version (Model A) to derive carry-over storage value

function (COSVF). Shadow values from the last month of each water year were extracted

and a mean value was used as a constant water value in storage. This value serves as the

slope of the linear COSVF. In the last part, we compare the results of the proposed model

pre- and post-corrections made by the author of this thesis, as outlined in section ‎2.2.4.

2.4.1 Part I

This section presents the results of the benchmark model compared to those of CALVIN.

The benchmark ‘Model A’ run, with perfect foresight and fixed groundwater pumping

costs, is an attempt to reproduce the results of the CALVIN optimised case after which

Model A is based. In CALVIN optimised, the model allows for an open water market

limited only by water availability, facility capacity and environmental and flood control

restrictions, and optimizes the water network to achieve maximum net economic gains.

Results are also compared with those of the historical simulation, which is a constrained

run of CALVIN in order to meet historical operation and allocation policies and is the

representative of historical trends.

Similar to CALVIN Optimised, the benchmark version optimises water allocation and total

net benefits over the entire 72-year historic period as opposed to annually as in the limited

foresight models. This run uses a slightly different code compared with the three other

runs. Initial values for flows, storages and deliveries are included in the code; these values

assist the solver in approaching an optimal solution. The inclusion of initial values is

particularly important for nonlinear models which do not have a single global optimum and

will generally terminate when a local optimal solution is found. The inclusion of initial

values attempts to ensure that the correct optimum solution is found. The initial values for

72

storage and delivery are the output from the CALVIN Optimised, the initial values for

flows in the south links are from the CALVIN Optimised and the initial values in the north

links are from the outputs of the Sacramento Valley Model (SVM) (Antoniou, 2011). It

should be noted that these initial values may hinder the comparability between CALVIN

Optimised and the Model A run because they guide the model to an optimal solution that is

the same or similar to that of CALVIN.

Figure 6 and Figure 7 illustrate the surface reservoir storage and the groundwater storage

of Model A as compared to those resulting from CALVIN Optimised and historical

approximation. As Model A is based on the CALVIN Optimised case it would be expected

that Model A results match those of CALVIN Optimised. The figures show that Model A

produces results that are similar to CALVIN Optimised for both surface reservoir storage

and groundwater sub-basin storage; however, they are not identical. That is because the

north (CALVIN region 1 and 2) and south (CALVIN region 3 and 4) portions of Model A

were built separately and combined afterwards; the north portion has a simplified network

representation, while the south portion has an almost identical network to CALVIN.

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Figure 6. Annual aggregated surface reservoir storage volumes of Model A compared to CALVIN

Optimised and historical approximation during: a) 1922-57; and b) 1958-93.

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Figure 7. Annual aggregated groundwater sub-basin storage volumes compared to CALVIN optimised and

historical approximation.

It is proposed that the differences between the CALVIN Optimised and Model A are due to

choices of initial flow values and network simplifications in the north portion of the region

- Appendix C elaborates this.

2.4.2 Part II

Model B and D benefit from hydrologic limited foresight. This implies that they should

valuate keeping water in reservoirs for later utilisation. We use linear COSVF for surface

reservoirs storages to embed such valuation. Groundwater sources are managed through

implementation of pumping cost schemes. Figure 8 compares surface reservoir operation

of the four model versions.

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Figure 8. Comparison of the four models’ annual aggregated surface reservoirs’ storage volume during: a)

1922-57; and b) 1958-93.

Models using limited hydrological foresight (Model B and D) obtained higher level of

water stored in surface reservoirs but very similar to each other. This is due to both

implementing a COSVF at the end of each water year to prevent reservoirs from being

drained out. Consequently, as a general rule, limited foresight leads to a more conservative

operation for surface reservoirs. This is vividly observed in the major drought of 1978. The

similarity in the trajectory obtained in Model B and Model D is because surface water in

both models used the same COSVF. As indicated in Figure 9, Limited foresight models

(Model B and D) which more conservatively utilize water in surface reservoirs, must rely

on groundwater sources to meet demands. However, Model B possessed even lower level

of groundwater storages. This lies within the fact that the Model D is imposing higher, yet

more realistic, pumping cost by calculating a head-dependent cost. Pumping cost in models

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76

with fixed rate (Model A and B) is generally underestimated; higher pumping cost means

less use of groundwater resources.

Figure 9. Comparison of the four models’ annual aggregated groundwater sub-basins storage volume over

the planning horizon.

Model A and Model C were able to keep more water in groundwater storages. This is a

consequence of more liberal use of surface water resources in models with perfect

foresight. This is attributed to lack of incentive to keep water at the end of each water year

for later uses (COSVF). The more realistic representation of groundwater pumping cost in

models using dynamic pumping cost scheme leads to preserving more water in aquifers.

This is illustrated in Figure 10.

Figure 10. Comparison of annual mean unit pumping cost of groundwater sub-basins.

Mean groundwater unit pumping cost of the 21 aquifers in models with dynamic pumping

cost was calculated much higher than those with fixed pumping cost. As the piezometric

head drops down by extracting aquifers, unit pumping cost increases. Increase in the

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77

pumping cost prevents further exploitation of groundwater resources, helping aquifers to

replenish. This is the reason for the decline in unit pumping cost after every rise happening

during droughts. Higher pumping cost in Model D compared to Model C is justified

through how surface water resources are used. No COSVF implemented for surface water

in the Model C leads to more use of surface reservoirs compared to Model D. More use of

surface reservoirs decrease the stress on groundwater aquifers. This, in turn, translates to

higher piezometric head and lower unit pumping cost.

Different allocation obtained by these four models creates different level of water shortage

for downstream users. We investigate this through observation of scarcity level that

demand sectors experienced (

Figure 11).

Figure 11. Comparison of annual water scarcity volume as the percentage of target demand from combined

agricultural and urban sectors.

The perfect foresight assumption in Model A and Model C, although being irrational, leads

to no water scarcity in demand sites. Because these models are aware of severe droughts in

advance they can share scarcity costs across time in an economically efficient way. Model

A encounters no scarcity as it is a replicate of CALVIN Optimised and uses its deliveries

as target demands. Model C has an annual scarcity of 0.016% in average. Limited foresight

models, on the other hand, were unprepared whenever a drought occurred. In the most

severe drought of 1976-77, these models poorly handled water scarcity (peaking at 13% in

Model B and 7% in Model D). This is due to the fact that Model B and Model D applied

limited foresight with an imprecise assumption for COSVF parameters. This is further

discussed in the next section. Underestimated pumping cost in Model B allowed for more

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78

freely extracting water from groundwater resources, leaving the system more vulnerable to

droughts.

2.4.3 Part III

Here we compare the results of the proposed model with and without amendments

proposed by the author of this thesis (see section ‎2.2.4). For this purpose, we illustrate the

groundwater storage trajectories, and agricultural shortages of the two versions. The two

versions of the model are labelled as Model D (post correction) and Model D-pre

correction. As it can be seen, the differences, although not very significant, are more

notable during major droughts.

Figure 12. Results of the proposed model before and after modifications made by the author (corresponding

to Model D-pre correction and Model D respectively): a) Aggregate irrigation deficit; and b) Aggregate

groundwater storage trajectory.

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79

Use of agricultural demand curves with zero slope (marginal water value from allocation)

at full demand in the ‘pre correction’ version of Model D means farmers prefer not to

pump anymore when water is allocated to a level near their target water demand. This is

simply because farmers would find it more sensible not to pump and incur costs. This lead

to higher water shortage in agriculture sector which, as seen in Figure 12.a, was avoidable

by replacing the old quadratic benefit curves with their piece-wise linear counterpart. In

addition, Figure 12.b shows the errors found in the input data of Model D-pre correction

contributed to over estimation of groundwater level in the Central Valley.

2.5 Discussion

This study proposes an extension to the current hydro-economic model for the

management of California’s Central Valley water resource system. First, we proposed a

dynamic pumping cost for groundwater sub-basins. This more realistically represent head-

dependant pumping cost as opposed to the previously used approaches that apply a fixed

rate pumping cost. This extension also provides the distribution of piezometric heads in the

region. Next, we proposed using a limited hydrologic foresight instead of a perfect

foresight. Two key assumptions are involved. The first assumption is a perfect intra-annual

foresight which includes full knowledge of inflows within each year. This is factual in the

context of the California because early spring measurements of the depth and water content

of the snowpack in many watersheds provide reasonably precise forecasts of reservoir

inflow to the end of the water year. For cases where such forecast is not available, the time

scale of the proposed limited foresight methodology can be downgraded into, for example,

monthly runs for which full knowledge of inflows is proved to be accurate.

The second assumption is employing a linear COSVF with the slope (water marginal

value) being extracted from shadow values of the mass balance constraint. Results revealed

that the use of shadow values caused economically inefficient water allocation. This

implies that water marginal value can be assigned so that the model outputs more

economic gains. A coarse grid search was conducted to find water marginal values that

lead to the highest amount of net economic gains over the planning horizon. A marginal

value of 10 $/af used for all surface reservoirs produced the highest revenue. Figure 13

shows how the resulting water scarcity compares to those of Model D (Model D result that

uses grid search values is labelled as Model D-GS).

80

Figure 13. Comparison of annual water scarcity in demand sites: Model D-GS vs Model D.

While model D has an average scarcity level of 0.84% per year, Model D-GS could reduce

it to 0.64%. This is 24% reduction in the average annual scarcity volume compared to

Model D. This implies that relying on shadow values (Lagrangian multipliers) will not

guarantee an efficient management outcome. Model D-GS, equipped with an improved

limited foresight, was able to more successfully hedge against severe drought of 1977 and

those of 1988-92. This is manifested by higher scarcity caused by Model D-GS in the

earlier years such as in 1961. This figure suggests that an effective optimization tool can be

linked to the proposed model for locating a set of water marginal values that boost the

economy of the region. Once such hybrid model is built, more complicated and accurate

form of COSVF (e.g. quadratic) can be sought.

2.6 Conclusions

This study introduces a hydro-economic model for the management of the California’s

Central valley water system. The model is built based on the available model for the region

– CALVIN. This model uses an optimization formulation to search for the optimal

allocation which maximizes the net benefit over the 72-year planning horizon. While

CALVIN uses a perfect hydrological foresight and a fixed pumping cost scheme, we

propose two extensions to more realistically simulate the water resources system. These

include limited foresight and dynamic pumping cost scheme. Implementation of limited

foresight requires introducing carry-over storage value function-a function used to assess

the value of stored water at the end of each water year. Applying this function helps the

models to operate more conservatively by keeping water in storage for future uses.

Dynamic pumping cost is calculated in accordance to changes in the piezometric head

following the storage coefficient method. Four model versions were created to investigate

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81

the impact of the proposed extensions. Results showed that the benchmark model was able

to imitate CALVIN Optimised to a satisfactory level. The remaining three models were

built on this model. The shadow values from mass balance constraint of the benchmark

model were used to derive COSVF in limited foresight models. The proposed limited

foresight model operated more conservatively and closer to historical records. High

scarcity level generated by limited foresight models shows that the assumption of linear

COSVF with shadow values as water marginal values does not lead to economically

efficient solutions. It is suggested that a search for accurate estimate of COSVF parameters

can improve model behaviour in terms of handling water scarcity and economic gains.

82

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Draper, A. J., Jenkins, M. W., Kirby, K. W., Lund, J. R., & Howitt, R. E. (2003).

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84

3 Chapter three: Estimating the economic value of inter-annual

reservoir storage in water resource systems

85

Estimating the economic value of inter-annual reservoir storage in water

resource systems

Abstract

Reservoir operators can face pressure to release water for immediate demands. If they

release too much water they threaten future supplies, not enough and they’ve created

economic hardship downstream. What’s the appropriate amount of carry-over storage?

This paper examines this question for complex large water resource systems by economic

valuation of end-of-year carry-over storage. The use of economic carry-over storage value

functions (COSVF) helps represent inter-annual inflow uncertainty within water resource

optimization models. The approach divides a perfect foresight optimization problem into

year-long (limited foresight) sub-problems that the within-year optimization engine solves

sequentially to find optimal short-term operations. The final state from the previous year

provides the initial condition to each year-long problem, and end-of-year COSVFs are the

final condition. Here, the COSVF parameters that maximize the inter-annual benefits from

river basin operations (release/extraction and allocation) are found by evolutionary

computation. This generalizable solution approach can handle non-convexity in large-scale

water resources systems. The approach is illustrated with a regional model of the

California Central Valley water system including 30 reservoirs, 22 aquifers, and 51 urban

and agricultural demand sites. Head-dependent pumping costs make the optimization

problem non-convex. Optimized inter-annual reservoir operation improves over extra-

cautious operation in the historical approximation, reducing the average annual scarcity

volume and costs by 80% and 98%, respectively. Obtained economic valuation of storage

can help inform water storage management decisions.

3.1 Introduction

This work is concerned with advancing algorithmic methods and analytical water valuation

approaches for large-scale water resources systems, even when simplifying assumptions

such as convexity do not apply. In particular, it proposes a generalizable approach to value

inter-year water storage no matter the size of the water system, nor the mathematical

characteristics of the associated economic optimization problem. Such approaches are

warranted at a time where the focus of water engineering – at least in industrialized

countries – shifts away from the planning and construction of new storage facilities and

86

towards the management of existing ones. This in a context where water is valuable for

competing uses, but its value varies across space and time (Harou et al., 2009). Holistic

approaches promoting efficient water allocation in water systems are needed (Lund et al.

(2006); Cai, 2008). The need for appropriate water valuation is underscored by regulatory

frameworks that promote an economically efficient allocation of water, e.g. the Water

Framework Directive (EU-Commission, 2000, 2012) in the European Union, or the

emergence of water markets in various places, including the western United States

(Hadjigeorgalis, 2009; Hansen et al., 2014; S. Wheeler et al., 2013; S. A. Wheeler et al.,

2017), Australia (Garrick et al., 2018; Lewis & Zheng, 2018; Owens, 2016), or the UK

(Erfani et al., 2015; Parker, 2007).

Most approaches for the efficient allocation of reservoir storage are limited by the so-

called “curse of dimensionality” which causes the necessary computational time and

memory to increase exponentially with the number of storage units (R.E. Bellman &

Dreyfus, 1966; Giuliani, Castelletti, Pianosi, Mason, & Reed, 2016). Examples include

dynamic programming (Banihabib, Zahraei, & Eslamian, 2017; Ji, Li, Wang, Liu, &

Wang, 2017; Mansouri, Pudeh, Yonesi, & Haghiabi, 2017), stochastic dynamic

programming (Scarcelli, Zambelli, Soares, & Carneiro, 2017; Soleimani, Bozorg-Haddad,

& Loáiciga, 2016; Zhou, Peng, Cheng, & Wang, 2017) and model predictive control

(MPC; Richalet, Rault, Testud, and Papon (1978b); Morari and Lee (1999); and Mayne et

al. (2000); Raso and Malaterre (2017)). Other studies (Ximing Cai et al., 2002; J. T. Shiau,

2011) used nonlinear optimization formulation with constrained carry-over storage

volumes. Such approaches require a good understanding of the topology and hydrology of

the problem, which makes them case-dependent and reduces the generalizability of the

approach. In fact, few solutions strategies are fit for the optimization of large-scale

systems. An example is stochastic dual dynamic programming (SDDP; Pereira and Pinto

(1991)), a method initially created for large-scale hydropower generation systems and that

has since then been extended to the diagnostics of large-scale transboundary issues

including hydropower and irrigation (Tilmant and Kelman, 2007) but also other uses

(Tilmant, Beevers, & Muyunda, 2010) . Still, SDDP relies on the key assumptions that the

benefit-to-go (or future benefits) function is convex. Non-convexities are found for

instance in head-dependent pumping costs (Davidsen et al., 2016) or endogenous

hydropower prices (Mo et al. (2001); and T. Kristiansen (2004)). While SDDP has been

87

extended to systems with both groundwater and surface water reservoirs (Macian-Sorribes

et al., 2017), these models omit head-dependent pumping costs.

These remarks extend to the analytical economic valuation of water. The analytical

valuation of carryover storage (e.g. Draper and Lund, 2004, You and Cai, 2008) is limited

to cases with a few reservoirs. SDDP is able to tackle water valuation in large-scale

systems in the presence of hydrological uncertainty (Tilmant et al., 2008), but only in the

absence of significant groundwater abstractions that introduce non-convexities if the head-

dependence of pumping costs is accounted for.

To avoid both non-convexity and curse of dimensionality, the current paper proposes a

generic hybrid approach using Evolutionary Algorithms (EAs). EAs have been used in

conjunction with mathematical programming to deal with the irregular topology of highly

constrained decision spaces in global-local hybrid search for complex multi-reservoir

systems (Nicklow et al., 2010), where one performs local optimizations that help the other

find global optima. With a few exceptions (e.g., Tospornsampan, Kita, Ishii, and Kitamura

(2005)), the EA has generally been selected as the global search tool, often paired with a

linear program (Afshar, Zahraei, & Marino, 2010; Ahn & Kang, 2014; X. Cai et al., 2001;

Reis, Bessler, Walters, & Savic, 2006) or other methods, e.g., stochastic dynamic

programming in mainly parallel multi-reservoir systems (Huang, Yuan, & Lee, 2002).

There remains an opportunity to build a generic hybrid approach that can handle 1)

complex multi-reservoir systems featuring serial and parallel reservoirs as well as 2) non-

convexity and 3) inter-annual uncertainty.

The proposed approach divides the multiyear horizon into year-long sub-horizons. These

year-long optimization problems are solved sequentially, using reservoirs’ end-of-year

carryover storage value functions (COSVFs; Draper, 2001; Draper and Lund, 2004) as

final (boundary) states that contain information on the expected value of water for use

during the following years. Contrary to previous analytical approaches fit for systems

comprising a limited number of reservoirs, here the COSVFs parameters are determined

through implementation of a genetic or evolutionary algorithm that finds the valuation of

end-of-year storage defined by COSVF parameters, leading to optimal multiyear

operations. Through this hybrid method, this work contributes to a generalizable approach

that offers explicit valuation of stored water. This is done without formulating the

88

convexity assumptions in conjunctive use (surface water and groundwater) systems, or

assumptions on the stochasticity of inflows.

To our knowledge, this is the first application of EAs to explicitly elicit the value of water

in a hydro-economic model (J.J. Harou et al., 2009). In contrast, some hybrid GA-LP

approaches use evolutionary computation to find end-of-year conditions such as storage

targets, to prevent reservoirs from being emptied by a within-year mathematical program.

Yet this end-of-year state either depends on hydrological conditions in the following year

(e.g., X. Cai et al. (2001)), in which case it implies year-ahead foresight, or on linear

weights (e.g. Reis et al. (2006)), which contradicts the economic intuition that the marginal

value of storage decreases as reservoirs fill, analytically demonstrated for a single reservoir

(e.g., Draper and Lund (2004); J. Y. You and X. Cai (2008)). A non-linear concave

COSVF, as the one used in this study, can maintain this economic intuition.

The proposed approach also enables a realistic determination of inter-annual reservoir

storage which is missing from the existing perfect foresight models, e.g., which assume a

perfect knowledge of hydro-climatic conditions over the period of interest, potentially

years in advance. Such models have enabled the integration of significant multi-sectoral

complexity in large-scale systems, but naturally, the perfect forecast assumption is at odds

with the uncertain information water managers have to deal with. It can lead to suboptimal

reservoir policies if their results are interpreted too prescriptively (Philbrick & Kitanidis,

1999). Yet, their ability to formulate and solve complex water resources problems means

perfect foresight has remained attractive (Bharati et al., 2008; Fowe, Nouiri, Ibrahim,

Karambiri, & Paturel, 2015; Mendes, de Barros, Zambon, & Yeh, 2015; Parehkar,

Mousavi, & Kim, 2016; Vieira et al., 2011; Yang & Yang, 2013; Zambon et al., 2012;

Zarghami, Safari, Szidarovszky, & Islam, 2015). This approach provides a convenient and

rigorous methodology for integrating inter-annual uncertainty into existing models without

having to reformulate them from scratch. One benefit of deterministic perfect foresight

models is that they are relatively easy to apply to large real-world systems, so we propose

that providing an approach that permits to reduce their hydrological foresight whist

allowing estimating the economic value of over-year storage is a valuable contribution.

A synthetic large-scale and multi-reservoir water system inspired from the California’s

Central Valley illustrates the approach. It is based on existing models of the region,

primarily CALVIN (CALifornia Value Integrated Network; (Draper et al., 2003)) a large-

89

scale hydro-economic optimization model with perfect foresight. In the remainder of this

work, section two describes the proposed methodology, section three presents the

California Central Valley application, results are shown in section four, followed by

discussion and conclusions in sections five and six respectively.

3.2 Methodology

3.2.1 Carry-over storage value functions

The objective of maximizing benefit (or minimizing cost) from operating infrastructure –

reservoirs, demand sites, etc. – in a river basin is classically formulated as a stochastic

multistage decision-making problem (R. E. Bellman, 1964):

𝑍 = 𝐸 [∑𝑓𝑡(𝑥𝑡, 𝑢𝑡, 𝑞𝑡)

𝑇

𝑡=1

+ 𝜈𝑇+1(𝑥𝑇+1, 𝑢𝑇+1)] (22)

where [1,T] is the time frame over which the optimization takes place, E[.] is the

expectation operator, 𝑓𝑡(. ) is the benefit function at stage t, 𝑢𝑡 are the decisions taken at t,

𝑥𝑡 is the state of the system, typically including reservoir storage, 𝑞𝑡 is the vector of

stochastic inflows, and 𝜈𝑇+1(. ) is a final value function. Note that this final value function

is incorporated to avoid the emptying of storage units at the end of the horizon. This

optimization is carried out under constraints such as the water balance, physical constraints

on flows and storages, and institutional and regulatory constraints.

Few strategies exist to tackle the “curse of dimensionality” that often makes optimization

computationally intractable in large-scale systems. This is especially true when objectives

are non-convex. A common strategy has been to eliminate uncertainty by solving instead

for a predetermined sequence of inflows 𝑄 = (𝑞𝑡)𝑡∈[1,𝑇], such as the historic sequence of

inflows. The maximization of objective 𝑍 is approximated by its perfect foresight

counterpart 𝑍𝑃𝐹:

𝑍𝑃𝐹(𝑄) =∑𝑓𝑡(𝑥𝑡, 𝑢𝑡, 𝑞𝑡)

𝑇

𝑡=1

+ 𝜈𝑇+1(𝑥𝑇+1, 𝑢𝑇+1) (23)

90

Perfect foresight (or deterministic) optimization assumes that all future inflows are known,

which can lead to decisions anticipating wet and dry years in advance. This work proposes

dividing the time frame [1,T] into K year-long time frames [𝑡𝑘 + 1, 𝑡𝑘+1]. For instance

with a monthly time step and K years, 𝑡𝑘 = (𝑘 − 1) × 12 so [𝑡1 + 1, 𝑡2] = [1,12] and

[𝑡𝐾 + 1, 𝑡𝐾+1] = [𝑇 − 11, 𝑇]. A maximization sub-problem can be proposed for each year,

with the following objective:

𝑍𝑘(𝑄, 𝑝) = ∑ 𝑓𝑡(𝑥𝑡, 𝑢𝑡 , 𝑞𝑡)

𝑡𝑘+1

𝑡=𝑡𝑘+1

+ 𝐶𝑂𝑆𝑉𝐹𝑘(𝑝; 𝑥𝑡𝑘+1 , 𝑢𝑡𝑘+1) (24)

where the final condition 𝐶𝑂𝑆𝑉𝐹(𝑝; 𝑥𝑡𝑘+1 , 𝑢𝑡𝑘+1) is the COSVF of reservoirs, which

describes the expected value of stored water for use beyond the end of the current water

year. Assuming a functional form, reservoirs’ COSVF can be described by the parameters

p of this function – e.g., in this work, two parameters for a quadratic COSVF with zero

value at dead storage (see equation (34)).

The K sub-problems described by equation (24) are solved sequentially. The initial

condition of sub-problem 𝑘 + 1 is given by the final state from sub-problem 𝑘. The

sequential optimization of objectives 𝑍1 to 𝑍𝐾 leads to maximizing a limited foresight

objective 𝑍𝐿𝐹:

𝑍𝐿𝐹(𝑄, 𝑝) = ∑(max𝑢𝑡{𝑍𝑘(𝑄, 𝑝)} − 𝐶𝑂𝑆𝑉𝐹𝑘(𝑝; 𝑥𝑡𝑘+1 , 𝑢𝑡𝑘+1))

𝐾

𝑘=1

(25)

where according to equation (24), the term between brackets corresponds to the sum of

operational benefits over year 𝑘. The limited foresight objective 𝑍𝐿𝐹 still assumes perfect

foresight in the short term, but is limited to the end of the sub-time frame. After that, future

inflows are uncertain. The benefits and associated river basin operations yielded by

maximizing 𝑍𝐿𝐹 depend on the parameters p describing the COSVF. 𝑍𝐿𝐹 computes the sum

of operational benefits. Contrary to Z in equation (22), the existence of the COSVF into

each Zk ensures that there will not be any unrealistic behaviour (emptying reservoirs) at

the end of the time horizon. Therefore, the final boundary condition of equation (22) does

91

not need to feature into equation (25), and maximization of the overall objective 𝑍 can be

approximated by finding the set of parameters p that maximizes 𝑍𝐿𝐹(𝑄, 𝑝).

3.2.2 Solution strategy

Finding max𝜋 𝑍𝐿𝐹(𝑄, 𝑝) is a double maximization problem, with (i) a series of within-year

deterministic optimizations, and (ii) an optimization in the parameter space of the COSVF

(Draper 2001). Maximization (i) is carried out for a given set of COSVF parameter values

p using deterministic optimization. Maximization (ii) is then implemented through

evolutionary computation, taking COSVF parameter space as the evolutionary algorithm’s

decision space. Carrying out maximization (ii) finds economically meaningful optimized

carry-over storage values.

Yet, there can be a problem of interpretation of the resulting COSVF coefficients in the

case where some reservoirs within the system fill every year. For these reservoirs, the

search for the highest performing economic valuation of storage becomes insensitive to

COSVF parameterisation and so a 2nd

objective must be added. The second objective is

there to force these reservoirs to adopt a meaningful valuation, by finding the lowest

valuation which ensures best overall economic performance. Maximization (ii) is carried

out as part of the resolution of the following multi-objective problem:

min𝜋(𝐹1, 𝐹2) (26)

where the first fitness function is that of finding parameter values that maximize benefits

from operations in the limited foresight operations:

𝐹1 = −max𝜋𝑍𝐿𝐹 (𝑄, 𝑝) (27)

The second fitness function aims to eliminate sets of parameters that lead to unreasonably

high marginal values of water, and therefore, unreasonably high values of carry-over

storage – recall that the marginal value of storage is a COSVF’s derivative. Therefore,

fitness function 𝐹2 accounts for the average marginal water value Asr of each reservoir 𝑠𝑟

with nsr being the number of reservoirs:

92

𝐹2 =1

𝑛𝑠𝑟∑𝐴𝑠𝑟𝑠𝑟

(28)

For a quadratic COSVF, Asr is the arithmetic mean of marginal water value at dead and full

storages. 𝐹2 weighs all reservoirs the same regardless of size to avoid undervaluing storage

in smaller reservoirs. Figure 14 shows the flowchart of the proposed approach.

Figure 14. Proposed model workflow.

3.3 Application

This approach is applied to a model inspired from CALVIN (Draper et al., 2003), an

existing optimization model specifically developed for the management of California water

93

resources system. CALVIN is a hydro-economic optimization model with perfect foresight

aimed at maximizing the economic gains from water allocation and management

throughout the system over the historical period. CALVIN represents many features of the

California water resources system, such as the integration of surface water and

groundwater supplies, the choice of optimization over rule-based simulation models, and

the use of economic drivers to allocate water rather than existing system of water rights

and contracts (Draper, 2001). Yet, it suffers from the limitations of perfect foresight. In the

model used here, inspired from CALVIN, hydrological uncertainty of future inflows is

introduced by dividing the monthly 72-year deterministic model into 72 shorter periods of

one year each. In the context of California, perfect intra-annual foresight is reasonably

consistent with the observation that early spring measurements of the depth and water

content of the snowpack enable predicting discharge months ahead with reasonable

accuracy and until the end of the water year (Draper, 2001). The impact of perfect within-

year forecast on winter operations is limited because the Central Valley inflows are

dominated by springtime snowpack melt. For cases where this condition does not hold, one

can apply the proposed approach with shorter time frames for which inflow forecasts are

sufficiently accurate.

California’s Central Valley (see map on Figure 15) covers 20,000 square miles and is one

of the world’s most productive agricultural regions (Faunt, 2009). This area serves over 30

million people and over 2.3 million ha of irrigated farmland (CDWR, 2009). More than

250 different crops are grown in the Central Valley with an estimated value of $17 billion

per year (GreatValleyCenter, 2005). About 75 percent of California’s irrigated land is in

the Central Valley, which relies heavily on surface water diversions and groundwater

pumping (Faunt, 2009). Another major demand is hydropower which is 9 to 30% of the

electricity used in the state, depending on hydro-climatic conditions (Group & Cubed,

2005). The study area is bound by the Cascade Mountain Range to the north, the Sierra

Nevada to the east, the Tehachapi Mountains to the south and the Coast Ranges and San

Francisco Bay to the west (Faunt, 2009). The northernmost reservoirs in the study area are

Shasta and Whiskey town, and the southernmost one is Isabella.

There is a significant imbalance in the spatial and temporal distributions of water supply

and demand in California. Nearly 75 percent of renewable water supply originates in the

northern third of the state in the wet winter and early spring. Nearly 80 percent of

94

agricultural and urban water use is in the southern two-thirds of the state in the dry late

spring and summer (CNRA, 2009). California’s Central Valley often suffers from

droughts. Historic dry periods include 1918-20, 1923-26, 1928-35, 1947-50, 1959-62,

1976-77, 1987-92, 2007-09, and 2012-16 (CDWR, 2015).

An arc-node representation of the water system is used. Nodes include surface and

groundwater reservoirs, urban and agricultural demand points, junctions, etc., and arcs

(links) include canals, pipes, natural streams, etc. (Shamir, 1979). This network comprises

over 300 nodes, including 30 surface reservoirs, 22 groundwater sub-basins, 21 agricultural

demand sites, 30 urban demand sites, 220 junction and 4 outflows nodes; and over 500

links (river channels, pipelines, canals, diversions, and recharge and recycling facilities).

95

Figure 15. The California Central Valley storages and river system.

Input data from the CALVIN model has been adapted to this distinct model. In particular,

hydrological data is from a 72-year historical inflow data covering 1922 to 1993 (Marion

W. Jenkins et al., 2004). Demand data adopted from the CALVIN model are projected at

2020 levels according to the California Department of Water Resources (DWR) data on

per capita urban water use by county and population by detailed analysis unit (DAU)

assembled for Bulletin 160-98. Please refer Appendix A for a full list of input data used in

this study.

96

3.3.1 Annual optimization model

For year 𝑘 ∈ [1; 72], benefits are computed over a monthly time step, and the benefit

maximization objective from equation (24) translates into:

𝑍𝑘(𝑄, 𝜋) = ∑ (∑𝑈𝐵𝑡𝑢𝑟

𝑢𝑟

+∑𝐴𝐵𝑡𝑎𝑔

𝑎𝑔

+∑𝐻𝐵𝑡ℎ𝑝

ℎ𝑝

𝑡=12𝑘

𝑡=12×(𝑘−1)+1

−∑𝑁𝐶𝑡𝑖,𝑗−∑𝑃𝐶𝑡

𝑔𝑤−∑𝐼𝐶𝑡

𝑖

𝑖𝑔𝑤𝑖,𝑗

) +∑𝐶𝑂𝑆𝑉𝐹𝑡=12𝑘𝑠𝑟

𝑠𝑟

(29)

Here, 𝜋 shows the set of choices of COSVF parameters (p1, p2). The sums of monthly

benefits (between brackets) are in order of: urban benefits summed over urban demand

sites ur, agricultural benefits summed over agricultural demand sites ag, hydropower

benefits summed over hydropower plants hp, network costs summed over all links between

any pair of nodes (i,j), pumping costs summed over all exploited aquifers gw, and

infeasibility penalties summed over all nodes i. The end-of-year COSVF condition over all

surface reservoirs sr is the same as in equation (28). For each month, the model is subject

to the water balance constraint; lower/upper bounds on flows and storage levels; and

hydropower generation capacity. In addition, a major aspect of California’s hydrology is

return flows from agricultural and urban activities (M.W. Jenkins et al., 2001). Return

flows of applied water from agricultural and urban water use to surface and groundwater

deep percolation are included in the proposed model. They are expressed as a percentage

of water used at each demand site.

Economic benefits come from water use by urban and agricultural demand sites, and from

hydropower generation. Benefit functions used convey the economic intuition that

allocating an additional unit of water increases benefits as long as demand is not fully met

(positive first derivative) but that marginal returns are decreasing (negative second

derivative). Piece-wise linear benefit functions (AB) for agricultural demand sites are

identical to those of CALVIN. Quadratic urban benefit functions (UB) use data from M.W.

Jenkins et al. (2001) with water retail prices (Black&Veatch, 1995) to represent urban

willingness-to-pay at target demand.

97

In California, it is assumed that the presence of “high-head” facilities where the effect of

reservoir storage on turbine head is small allows for a linear relationship between head and

hydropower generation (Madani & Lund, 2007; Vicuna et al., 2008):

𝐻𝐵𝑡ℎ𝑝 = 𝑅𝑡

ℎ𝑝𝑃𝐹ℎ𝑝𝑝𝑡 (30)

where R is the release of the reservoir for the power plant hp, PF is the power factor which

relates release to hydropower generation, and p is the monthly-varying hydropower unit

price. Costs in the objective function include network costs (NC) for conveyance, treatment

and conjunctive use operations; costs for infeasibilities (IC); and energy costs for

groundwater pumping (PC). Network costs are linear with respect to flows through a link,

i.e., a constant unit cost for each link. To guarantee algorithmic feasibility, artificial

inflows can be made available at each node, similar to Draper et al. (2003) for CALVIN.

These flows only exist to allow for constraints being met, so they are penalized by a

penalty (cost) several orders of magnitude above other costs. These are particularly

valuable for identifying and debugging infeasibilities.

The CALVIN model represents pumping costs by multiplying the unit pumping cost of

$49.42 per MCM/m lift ($0.20 per af/ft lift; MCM is a million 𝑚3) by a static estimate of

the average pumping head in each groundwater sub-basin (Hansen, 2007), the current

model has pumping costs that dynamically vary with head in the aquifer, following the

equations proposed by J. J. Harou and J. Lund (2008). System-wide groundwater pumping

costs are represented as follows:

𝑃𝐶𝑡𝑔𝑤

= 𝑢𝑐𝑡𝑔𝑤

∑ 𝑄𝑡𝑔𝑤,𝑗

𝑔𝑤,𝑗|𝑔𝑤,𝑗∈𝐶𝑂

(31)

𝑢𝑐𝑡𝑔𝑤

= 𝑐𝑔𝑤𝐿𝑡𝑔𝑤 (32)

In above equations, uc is pumping unit cost, CO is the connectivity matrix which defines

how nodes are linked, c is the unit cost per lift, and L is the height water being lifted to

reach the ground elevation. J. J. Harou and J. Lund (2008) suggest that the storage

coefficient formulation is a parsimonious method to model both lumped groundwater

98

volume and head functions. The storage coefficient relates the volume of water released or

absorbed into or from storage (net stress) per unit surface area of the confined aquifer per

unit change in piezometric head. Piezometric head in each groundwater sub-basin is

calculated as follows (lift is set equal to the difference between ground elevation and the

piezometric head level):

𝐿𝑡𝑔𝑤

= 𝐿𝑡−1𝑔𝑤

−𝑖𝑡𝑔𝑤+ ∑ 𝑙𝑖,𝑔𝑤𝑄𝑡

𝑖,𝑔𝑤𝑖,𝑔𝑤|𝑖,𝑔𝑤∈𝐶𝑂 − ∑ 𝑄𝑡

𝑔𝑤,𝑗𝑔𝑤,𝑗|𝑔𝑤,𝑗∈𝐶𝑂

𝑠𝑔𝑤𝑎𝑔𝑤

(33)

where i is the net recharge from precipitation, l is the loss coefficient in links (due to

evaporation and/or seepage), s is the mean storage coefficient, and a is the aquifer’s area.

Finally, end-of-year COSVF are quadratic functions of storage in each surface reservoir,

depending on two parameters (𝑝1𝑠𝑟 , 𝑝2

𝑠𝑟) defined by:

{

𝐶𝑂𝑆𝑉𝐹𝑠𝑟(𝑝1

𝑠𝑟 , 𝑝2𝑠𝑟; 𝑠𝑚𝑖𝑛

𝑠𝑟 ) = 0

𝑑𝐶𝑂𝑆𝑉𝐹𝑠𝑟

𝑑𝑠|𝑠=𝑠𝑚𝑖𝑛

𝑠𝑟= 𝑝1

𝑠𝑟

𝑑𝐶𝑂𝑆𝑉𝐹𝑠𝑟

𝑑𝑠|𝑠=𝑠𝑚𝑎𝑥

𝑠𝑟= 𝑝2

𝑠𝑟

(34)

The nonlinear model of the California system is coded in GAMS and solved using the

Minos solver version 5.5 (Murtagh & Saunders, 1998). Minos applies the Generalized

Reduced Gradient (GRG) method which is suitable for nonlinear programming problems

with linear constraints (Labadie, 2004).

3.3.2 Multi-objective problem and resolution

The multi-objective problem formulation is as described in the method section, equations

(26) to (28). Using the parametrization of end-of-year COSVFs, the fitness of the carryover

storage objective is given by:

𝐹2 =1

𝑛𝑠𝑟∑

𝑝1𝑠𝑟 + 𝑝2

𝑠𝑟

2𝑠𝑟

(35)

Borg-MOEA (Hadka & Reed, 2013) was used for multi-objective optimization because

Borg’s self-adaptive features increase its robustness and effectiveness while minimizing

99

the search parametrization by the user. There are 30 surface reservoirs, so there are 60

decision variables for solution by the evolutionary algorithm. Carryover storage value can

only have positive values and are bounded by the maximal value taken by any of the urban

and agricultural demand curves. For the case-study, an initial population size of 100,

100,000 maximum number of function evaluations as the stopping criterion, and epsilon

(search resolution) value of $1,000,000 and 8107 $/MCM (10 $/af) for the fitness functions

(equations (27) and (28), respectively) were used. There is no need to configure

evolutionary search parameters (e.g. mutation and cross over factors, selection probability,

etc.) in BORG, as BORG’s auto-adaptive feature determines an optimal value for these

parameters using its internal functions. Only maximum number of function evaluations

and search resolution has to be input by the user. This is discussed in details in Appendix

B. The case presented here was solved using 96 Intel processors working jointly on a Unix-

based computing cluster.

Results are presented as a set of ‘non-dominated’ solutions, known as the Pareto Front,

whereby any improvement with respect to one objective is at the expense of the other.

Evolutionary algorithms are heuristic search methods that approximate the Pareto curve

without ever reaching it in an absolute mathematical sense. Formally therefore, the trade-

offs are ‘Pareto-approximate’ although they are subsequently being referred to as ‘Pareto-

optimal’ to simplify the discussion (Hurford, Huskova, & Harou, 2014). Finally, in order to

assess the sensitivity of the proposed model to the input streamflow time-series, an

ensemble of synthetic scenarios was created by bootstrapping from the historical time-

series. 100 time-series were generated which leads to 7200 simulations (considering the

72-year time frame and annual model runs).

3.4 Results

3.4.1 Marginal water values

To capture the trade-off between the two fitness functions, a Random Seed (RS) analysis

with five seeds was performed. Figure 16 shows the Pareto optimal solution points. The

Pareto front quickly becomes nearly flat regarding the main (economic) fitness function 𝐹1,

suggesting that the economic optimization problem possesses multiple near-optimal

solutions. As detailed in Appendix E, the COSVF parameters leading to each of these near-

optimal solutions are very similar, with differences mainly for small reservoirs. The

100

remainder of this results section uses averages of the COSVF parameters across these

simulations, displayed in Table 2; this is also justified by Appendix E. It should be noted

that limited foresight model in this chapter is equivalent to Model D of Chapter two with

optimal COSVF parameters. Hence, for the sake of consistency in naming, results from the

limited foresight run with the optimised COSVF parameters are denoted as ‘Model D’ in

this chapter. The perfect foresight counterpart is the same as Model C of Chapter two;

‘Model C’ is used as the label for perfect foresight results in this chapter.

Figure 16. Non-dominated solution points showing the Pareto-optimal trade-off between the two objective

functions: economic benefits and mean water marginal values (arrows show the direction of preference).

Table 2. Marginal economic value of stored surface water in September at major California Central Valley

reservoirs evaluated by Model D. Reservoirs are from north to south. Maximum capacity varies per month

due to flood control rules. Net inflow includes deductions for evaporative and seepage losses.

Reservoir

End-of-year

active storage

(MCM)

Annual average

net inflow

(MCM)

Marginal

benefits from

hydropower

generation

($/MCM)

Marginal value

at dead storage

($/MCM)

Marginal

value at full

storage

($/MCM)

Shasta 3,344 6,816 7,475 51,659 7,493

Whiskeytown 138 1,144 9,258 70,557 9,288

Black Butte 122 488 0 785 0

Oroville 2,682 4,966 11,180 22,263 22,263

New Bullards Bar 560 1,496 21,719 55,829 21,720

Camp Far West 126 458 0 190 25

Indian Valley 731 529 0 21,636 13

Folsom 701 3,271 5,245 64,711 5,246

Berryessa 1,926 438 0 21,311 0

Pardee 235 840 0 26,668 0

New Hogan 263 184 0 30,807 25

New Melones 1,507 1,285 9,015 29,984 9,015

EBMUD aggregate 63 0 0 95 0

660

710

760

810

860

910

960

1010

1060

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000 60,000

Do

wn

stre

am n

et

be

nfi

t (B

$)

Mean water marginal value ($/MCM)

101

Los Vaqueros 41 0 0 16 0

Lloyd-Eleanor 333 542 27,953 27,953 27,953

Hetch Hetchy 399 936 0 1,403 1,402

Del Valle 23 0 0 552 0

Don Pedro 1,727 792 7,815 39,475 7,957

Turlock 69 0 0 363 0

McClure 907 1,128 5,221 35,396 5,231

SF aggregate 277 0 0 0 0

Eastman 99 82 0 490 37

Santa Clara 209 156 0 154 23

Hensley 79 101 0 61,784 0

San Luis 1,958 0 0 2 0

Millerton 495 2,082 0 74 0

Pine Flat 1,177 2,041 2,910 8,563 2,971

Kaweah 101 581 0 1,825 0

Success 81 170 0 10,773 0

Isabella 453 876 0 900 151

Table 2 shows that marginal water values are low for surface reservoirs with very low

annual net inflow (e.g. Los Vaqueros, Del Valle, Turlock, San Francisco (SF) aggregate,

and San Luis), suggesting that the Central Valley economy usually does not rely on them

(at the margin) for water supply. Surface reservoirs in the northern regions (upstream) and

those on the eastern range of the Central Valley have higher marginal values for stored

water (e.g. Shasta, Whiskeytown, Folsom, Oroville, New Bullards Bar, New Melones,

etc.). Reservoirs producing hydropower normally show higher marginal values. These

reservoirs are also on the eastern range (Figure 15). This is consistent with taller mountains

and higher volumes of inflow. Table 2 demonstrates how valuable water is at different

points in the basin, a proxy for economic water scarcity (Pulido-Velazquez et al., 2013).

This suggests to decision-makers where of focus for new policy decisions – regulations,

investments, etc. Figure 17 shows a map of surface reservoirs’ mean marginal water value

in California Central Valley. This figure depicts that geographical distribution of reservoirs

is the main reason for variation in the valuation.

102

Figure 17. Distribution of average stored water marginal value in the Central Valley. Values in parenthesis

are average marginal value.

3.4.2 Basin-wide inter-annual operation

Inter-annual reservoir operation results compare the approach proposed here (Model D)

with perfect foresight results for the same model (Model C), and with historical conditions

as estimated by the CALVIN model (Marion W. Jenkins et al., 2004) using a highly

constrained model calibrated to represent operation policies in 1998. All models use

103

identical starting storages. Model C also has a final boundary condition to avoid emptying

surface reservoir and groundwater aquifers in the final years of the record.

Figure 18. Annual aggregated surface reservoirs’ storage level comparison during : a) 1922-57; and b) 1958-

93.

Total surface storage time series is shown in Figure 18. Model C uses more of the available

storage because it hedges ideally against future droughts. COSVF in Model D encourages

saving water for subsequent potentially dry years, and thus this model leads to a more

cautious allocation strategy to hedge against droughts. Historical operations (as estimated

by CALVIN) were even more conservative than Model D. This is due to 1) historical

demand being less than the projected 2020 demand levels – use of 2020 demands was a

requirement set out by California Department of Water Resources Bulletin 160-98, 2)

greater groundwater use than is predicted by Model D (Figure 19), and 3) a more cautious

approach by real-world reservoir operators who lack perfect inter-annual foresight.

0

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(M

CM

)

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(a)

Model D Model C Historical approximationMinimum storage Maximum storage

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15,000

20,000

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(b)

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104

Regarding total groundwater storage, the main feature is the dynamic pumping cost

(equations (31)-(33)) which incentivizes conserving and replenishing groundwater (Figure

19) to reduce subsequent pumping costs. The more liberal use of surface reservoirs in

Model C avoids pumping costs by maintaining storage levels in groundwater sub-basins

close to full capacity (Figure 18). The conservative operation of surface reservoirs in the

historical case means the state relies more on groundwater sources as reflected with its

more intensive use in Figure 19. Model C hedges against future droughts using

groundwater resources. This is why aquifer storages reach near full capacity over the few

years prior to every drought.

Figure 19. Annual aggregated groundwater storage level.

560,000

570,000

580,000

590,000

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610,000

620,000

630,000

640,000

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(M

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Model D Model C Historical approximation

0.00.20.40.60.81.01.21.41.61.82.0

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(a)

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105

Figure 20. Comparison of a) water scarcity as the percentage of target delivery and b) the corresponding

scarcity cost in demand sectors (combined agricultural and urban demands).

Difference in the operation of storage nodes leads to different allocation results. The

simulated allocation is seen in water scarcity among demand sectors (Figure 20). Model C

anticipates droughts to store additional water and hedge lower value uses which leads to a

small but constant water scarcity – there is no shortage of urban demand and negligible

0.024% agricultural shortage. Scarcity costs are shared across time in an economically

efficient way. The proposed hybrid optimization approach is geared towards avoiding large

costs incurred by severe droughts, at the expense of recurrent shortage for the least

valuable water uses – in agriculture, with scarcity up to 1.4% in 1977. It still avoids almost

any water scarcity to cities, with peaks at 1.2% shortage in the severe 1977 drought. Yet,

average scarcity level remains quite small in the model (0.3% of target demands per year).

Cautious operation obtained by the run constrained to near-historical operations incurs

higher scarcity of deliveries (1.5% of target demands per year), perhaps reflecting some

real historical water scarcity and historical demand levels smaller than those modelled

here. The reduction in the average annual scarcity volume from the historical operation to

Model D was equal 80%. Comparison of annual scarcity costs indicates that the efficient

hedging in Model D decreased the average annual scarcity cost by 98% (Figure 20.b). The

reduction in the average annual scarcity volume and cost was respectively 95% and 100%

from the proposed Model D to Model C.

3.4.3 Sensitivity analysis

This section investigates the robustness of the COSVF coefficients found in the

optimization results to different streamflow conditions within the historical range, i.e.,

within climatic conditions similar to those of the 72-year time-series used for Model D run.

020406080

100120140160180

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106

We generated a set of 100 monthly time series of 72 year length by bootstrap resampling of

the historical streamflow time series (J. Harou et al. (2007); J. J. Harou et al. (2010);

Anghileri et al. (2016); Knight et al. (2018)). This was done by randomly re-ordering the

annual blocks of 72-year streamflow data. Below we show the shortage and drought

indicators generated by operating these systems while using the same COSVF coefficients

as in Table 2– and Sections 3.4.1 and 3.4.2. Figure 21 depicts the range of monthly inflows

in the synthetic ensemble and compares it to the historical trend.

Figure 21. Envelope showing the distribution of river inflows in the synthetic ensemble (in grey) and the

historical inflow data (black line) during: a) 1922-57; and b) 1958-93.

We used the aggregate 72-year water shortage volume and the volume of the worst 3-year

shortage (that is the duration of the worst drought in the historical event) as an indicator to

compare the performance of the synthetic ensemble to those of the proposed Model D and

the historical approximation. The performance is illustrated as an exceedance probability

chart (Figure 22). Each point from this chart shows the percentage of times that scenarios

produced a value equal to or greater than the one of that point.

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Figure 22. Probability of exceedance of: a) aggregated 72-year shortage volumes; and b) worst 3-year

shortage volume. The reference scenario is the one obtained with historical inflow, and with COSVF, i.e. the

limted foresight model.

The aggregate 72-year shortage volume of all synthetic scenarios was lower than that of

the historical operation. 91% of scenarios produced half or lower of the historical 72-year

shortage volume. This indicates that the valuation of surface water through the COSVF

using historical inflows can robustly improve the management of water resources under a

range of conditions.

37% of synthetic time series showed higher worst 3-year shortage compared to the

historical approximation, i.e., the 3-year period with the highest shortage volume. This is

generally due to 1) worse-than-historical 3-year droughts in the synthetically generated

ensemble, and 2) less favourable conditions entering into that drought, i.e., less surface

water storage to begin with, typically as a result of a drier-than-average year(s) prior to

that 3-year period. The combination of the two is expressed as available water – the total

runoff during the three-year period plus the initial surface water storage, which contrary to

groundwater is available without pumping costs – in Figure 23. 97 out of 100 of these

worst 3-year drought periods on the Figure feature less available water than in the

historical case. In fact on average, the amount of stored water available across the

ensemble prior to the worst three years of the average scenarios is just 55.8 % of the

5,000

10,000

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30,000

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108

historical time series. In spite of this, shortage is greater in the historical approximation

63% of the time.

Figure 23. Comparison water shortage and water availability during the worst 3-year drought.

3.5 Discussion

This paper proposes an approach to evaluate inter-annual reservoir storage in non-convex

and non-linear large-scale optimization models of water resources systems. It uses

optimized end-of-year COSVFs for surface water reservoirs to account for the expected

value of water beyond the current water year. These COSVF are quadratic to reflect the

fact that the value of water increases when it is scarce and reservoir levels are low. Multi-

year perfect foresight problems can be reformulated as a suite of multi-period

mathematical programing problems that are solved sequentially with a) storage calculated

at the end of each sub-problem serving as the initial storage condition for the next one, and

b) COSVFs representing the imperfect information that system operators have about future

inflows (in our application a water year). COSVFs are represented by quadratic functions

whose parameter values are found by evolutionary search methods. This is the first

instance of coupling between an evolutionary algorithm and a hydro-economic model to

provide an economic valuation of water in large-scale systems where the associated

optimization problem is non-convex.

Values obtained with respect to only the first objective function (F1) of the MOEA suggest

an upper threshold for COSVF parameters rather than a direct estimate. Introducing the

second objective (𝐹2) to optimize these parameter values helps find the lowest possible

marginal water values that keep reservoirs from being over-depleted at the end of each

year. Thanks to the use of MOEA, other management objectives could be integrated into

the valuation of carry-over storage; this is left to future work.

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109

An application of the proposed Model D to California’s Central Valley is compared to

results obtained by its perfect foresight counterpart, Model C, and another model run

representing operations constrained to resemble historical ones. These provided useful

information on the consequences of management with hyperopia and myopia, respectively.

Model C, taking advantage of the assumption of full knowledge of future hydrological

inflows, relies heavily on surface reservoirs rather than groundwater aquifers. Simulated

historical results which were used as a representative for real-life operation over the time

period of study, demonstrated that its myopic behaviour can lead to poor outcomes:

conservative use of surface water resources implies more intensive use of groundwater,

and greater overdrafting of groundwater (J. J. Harou & J. R. Lund, 2008; Nelson et al.,

2016). The historical operating policy case also shows substantial water scarcity at demand

sites incurring an average of 1.5% of target demands per year. The proposed Model D

showed that its operation is cautious enough to manage future droughts, even though

without information about the long-term future hydro-climatic conditions as in Model C.

The proposed Model D reported values of end-of-year storage. This can inform operators

and water managers about the economic value of keeping water in storage for subsequent

potential dry years and be used as a proxy to highlight zones eligible for further

development. Implementing non-convex head dependent pumping costs in the proposed

model provides piezometric head levels that would have manifested under economic

historical water management. This is not possible with methods that depend on convex

(linear) behaviour of the model such as SDDP. In addition, simulating the case considered

in this study reduced run time from nearly 30 hours for Model C to 5 minutes on the same

machine for Model D and enabling the link to a heuristic search algorithm. The search for

optimal COSVFs required 87 hours per random seed per core, using 96 CPU cores on a

Unix computing cluster.

Yet, some limitations exist for the work in this paper. High nonlinearity and long run-times

of the proposed approach linking the model to many-objective heuristic search restricts its

extendibility. For example, considering the common nonlinear relation for hydropower

generation for similar cases of the same scale could make the approach computationally

impracticable. This is due to increase in the number of variables (height of water in

reservoirs) and nonlinearity of the model. However, even with the current model, this issue

could potentially be addressed by choosing an efficient algorithm for the annual

110

optimization phase of the hydro-economic model. Also, in the current work on carry over

storage, only the value of surface water reservoirs is considered; dynamic pumping costs

are considered a proxy for groundwater value in order to make the problem more tractable

(less storage units to optimize COSVFs parameters for).

It should be noted that surface and groundwater storages have asymmetric roles with

respect to water valuations. Without value function for surface reservoirs, the use of these

resources will be free in a hydro-economic model. Then, this model tends to depletes

surface reservoirs first. This is why valuation of surface water storage is crucial. Besides,

surface water storage is filled and depleted every year, or every few years at most. This

short time-scale compared with the study period makes the interpretation of carry-over

storage value functions unambiguous. This is not the case for groundwater however. We

tested the incorporation of COSVF for groundwater and report the following: (1) the cost

of using groundwater “seen” by the hydro-economic model is “pumping cost + carry-over

storage value”. In most aquifers, pumping costs are large enough that the COSVF is zero;

(2) as a consequence of (1), integration (or not) of COSVF has very little effect on

management outcomes; and (3) large, multi-decadal variations in the aquifer storage make

their COSVF (when it exists) difficult to interpret.

3.6 Conclusion

Inter-annual reservoir operation in large water resource systems has long been a challenge.

Approaches using models with hyperopia (perfect foresight optimization) assume full

knowledge of future supply and demand which is unavailable to water managers. In

contrast, a model with myopia, such as the one used to approximate historical operating

policies, manages reservoirs overcautiously, imposing excessive economic scarcity during

major droughts or over-hedges in non-drought years. In this paper we present an approach

to address this modelling problem by limiting hydrological foresight (to represent the

annual forecasting afforded by California snow storage estimation), which requires

determining the economic value of end-of-year carry-over storage. The proposed approach

discretizes the full planning horizon to shorter periods (in our case 1 hydrological year) and

performs sequential runs. The carry-over storage value function acts as a boundary

condition representing the value of stored water for future use (beyond each optimized

period) and is optimally determined using an external many-objective search algorithm.

This approach enables determining the inter-annual release decisions, and it introduces a

111

method for valuation of carry-over storage in large-scale water resources systems with

non-convexity.

The method was applied to a large-scale water resources system: California’s Central

Valley. Borg, an auto-adaptive evolutionary algorithm was used to search for the optimized

economic values of storage in surface reservoirs through repeated use of an optimization-

driven hydro-economic simulation formulated as a series of non-linear mathematical

programs. Results showed an improvement in scarcity management evidenced by a

reduction of scarcity (80% in scarcity volume and 98% in scarcity costs) compared to a

historical approximation. Groundwater results show how considering non-linear

groundwater pumping costs in management models leads to reduced recommended

overdrafting of aquifers. A sensitivity analysis showed that the proposed approach is robust

and the obtained solution performs well against a wide range of hydro-climatic scenarios.

Using a many-objective search algorithm offers the flexibility to consider more objectives.

Acknowledgments

The work was supported by the UK Engineering and Physical Sciences Research Council

(ref. EP/G060460/1), University College London, and The University of Manchester. The

GAMS (Generalized Algebraic Modeling System) Corporation provided a cluster license

to support this research. The University of Manchester’s Computational Shared Facility

was used for the high performance computing.

112

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4 Chapter four: Investigating historical valuation of reservoirs – a

California Central Valley case study

120

Investigating historical valuation of reservoirs – a California Central

Valley case study

Abstract

Water is not only a valuable commodity, but is also valued in different ways depending on

social, ecological, and historical condition. Valuation of water helps in solving conflicts

where allocation decisions must be made based on economic profitability. Assessing the

historical economic value of water resources is even a further challenging task, especially

when market valuation is not available and existing non-market valuation methods fail due

to scale or complexity of the problem. This paper proposes a generalisable approach for

direct estimate of historical willingness-to-pay (WTP) of end-of-year storage in large-scale

reservoir systems. The approach uses quadratic carry-over storage value functions

(COSVFs) for end-of-year storage as terminal condition to prevent depletion of reservoirs.

Parameters of COSVFs, which can be translated to WTPs, are calibrated by an

Evolutionary Algorithm (EA) while minimising a fitness function of mean squared error

between the modelled storage levels and the historical values. Through this hybrid

approach we are able to capture the WTPs of reservoir over-year storage that leads to best

imitation of historical operation. The proposed hybrid approach is not plagued by non-

convexity and curse of dimensionality which often hinder application of the existing

valuation methods. This is illustrated via a large-scale regional model of the California

Central Valley water system including 30 reservoirs where head-dependent pumping cost

makes the problem non-convex. Results show that historical WTP was set higher for the

larger reservoirs than the smaller ones, hoping to preserve their storage as a water bank for

future drought. Further analysis reveals that such strategy was sub-optimal.

4.1 Introduction

Valuation of water resources can be vital for allocation decisions and policy making. As

exemplified by Loomis (2000), a hypothetical market method followed by a survey was

applied to solve the allocation case of “300 Fish versus 28,000 People?” in California.

Khan (2007) focuses on surface water valuation for irrigation purposes and pricing aspects

in Pakistan. More recently, an online survey was used to derive ecological versus social

WTP for restoration of Everglades of south Florida (Seeteram, Engel, & Mozumder, 2018).

No method has been developed for estimating the historical WTP.

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Existing valuation methods fall within two main categories: market and non-market

techniques. In market valuation techniques, determining the value of an asset is based on

the selling price or the price that consumers are willing to pay for a commodity in market.

Water markets are common in Western US (Hadjigeorgalis (2009); Hansen, Howitt, and

Williams (2014); and S. A. Wheeler, Loch, Crase, Young, and Grafton (2017)), Australia

(S. Wheeler, Garrick, Loch, and Bjornlund (2013); Garrick, Hernández-Mora, and

O’Donnell (2018); Lewis and Zheng (2018); and Owens (2016)), and the UK (Erfani,

Binions, and Harou (2015); and Parker (2007)), but they are not available globally and are

usually ineffective (Tilmant, Pinte, & Goor, 2008). A non-market valuation determines the

economic value of a commodity that cannot be traded directly in markets and there is no

market price for it to evaluate. Non-market approaches are survey-based (e.g. Contingent

Valuation; Thayer (1981)) or modelling-based (e.g. SDDP; Pereira and Pinto (1991)). In

order to assess the historical WTP of reservoir storage, only modelling-based non-market

approaches are applicable. Available examples of such group of approaches rely on the

Lagrangian multiplier associated to reservoirs’ mass balance equation (shadow value) to

capture water marginal value (Tilmant et al., 2008). Such approaches suffer from either

curse of dimensionality (e.g. SDP; Scarcelli, Zambelli, Soares, and Carneiro (2017);

Soleimani, Bozorg-Haddad, and Loáiciga (2016); and Zhou, Peng, Cheng, and Wang

(2017)) or non-convexity (e.g. SDDP; Tilmant et al. (2008); Tilmant, Beevers, and

Muyunda (2010); and Macian-Sorribes, Tilmant, and Pulido-Velazquez (2017)) which is

present in most real-life cases.

This paper contributes to proposing a hybrid approach for assessing the historical WTP of

over-year storage in water systems affected by non-convexity and curse of dimensionality.

The proposed approach couples an EA with the extended version of an existing hydro-

economic model. To prevent emptying reservoirs, COSVF is implemented as a boundary

(terminal) condition to each year-long simulation carried out by the hydro-economic

model. These COSVFs are quadratic to reflect the fact that the value of water increases

when it is scarce and declines when it becomes plenty. Reservoirs WTP can be obtained

owing to the fact that reservoirs’ demand curve featuring WTP at dead and full storage is

the derivative of the COSVF. Two parameters defining COSVFs (or WTPs defining the

corresponding demand curve) are calibrated by the EA while minimising a mean squared

error fitness function between the modelled storage level and its historical value. It is likely

for the EA to find unrealistically high WTP. That is because if a WTP is enough to keep

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reservoir full at the end of a given year, a value 100 times higher would also have the same

effect. To eliminate this, a second fitness function is added which forces the EA to find the

lowest possible WTP that closely reproduce historical condition. Once calibration is

carried out and historical WTPs are obtained, they can be used to: (1) be contrasted with

the current pricing, (2) inspect management decision made in the past and perhaps learn

lessons for the future, and (3) to compare with its market value if available. In this paper,

the second aspect of historical valuation is discussed. A California’s Central Valley is used

as the case study. Results show that the proposed approach is able to find WTP values that

lead to accurate imitation of historical operation. Further analysis demonstrates that

historical operation was far from being efficient, hence historical WTPs are sub-optimal.

4.2 Material and Methods

4.2.1 Proposed methodology

In the proposed methodology, valuation of end-of-year storage begins with assigning a

COSVF for the last month of each year. To do this, the hydrological foresight of the hydro-

economic model is amended from a perfect foresight to limited foresight, i.e. we assume a

perfect within-year foresight but no information of hydro-climatic condition beyond the

current year is available. This is done by dividing the N-year planning horizon by N annual

periods. The starting condition for each period would be the ending condition of the

previous one. The COSVF is implemented at the end of each annual period to avoid

emptying reservoirs. COSVFs are quadratic and concave functions that a) represent the

potential benefits of keeping water at storage for later release, and b) reflect the fact that

the value of water increases when it is scarce and decreases when it is surplus. By

definition, reservoirs’ demand curve is the derivative of their COSVF. Therefore, the two

parameters that define the COSVF can be translated to reservoirs WTPs at dead (Smin) and

full (Smax) storage, p1 and p2, as (S is reservoir storage capacity):

{

𝐶𝑂𝑆𝑉𝐹(𝑝1, 𝑝2, 𝑠𝑚𝑖𝑛) = 0𝑑𝐶𝑂𝑆𝑉𝐹

𝑑𝑆|𝑆=𝑠𝑚𝑖𝑛

= 𝑝1

𝑑𝐶𝑂𝑆𝑉𝐹

𝑑𝑆|𝑆=𝑠𝑚𝑎𝑥

= 𝑝2

(36)

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Figure 24. Relation between demand curve and COSVF of a reservoir.

The hydro-economic model simulates the water system in each year by maximising the

system-wide net economic benefit of water allocation, thanks to the following annual (yr)

objective function:

max 𝑍𝑦𝑟 (𝑄, 𝑆, 𝑝1, 𝑝2)

= ∑ (∑𝑈𝐵𝑚𝑛𝑢𝑟

𝑢𝑟

+∑𝐴𝐵𝑚𝑛𝑎𝑔

𝑎𝑔

+∑𝐻𝐵𝑚𝑛ℎ𝑝

ℎ𝑝

12

𝑚𝑛=1

−∑𝑁𝐶𝑚𝑛𝑖,𝑗−∑𝑃𝐶𝑚𝑛

𝑔𝑤−∑𝐼𝐶𝑚𝑛

𝑖

𝑖𝑔𝑤𝑖,𝑗

) +∑𝐶𝑂𝑆𝑉𝐹𝑚𝑛=12𝑠𝑟

𝑠𝑟

(37)

where Z is the objective function, Q is the flow in links, S is the storage capacity in storage

nodes, p1 and p2 are reservoirs’ WTPs at dead and full storage respectively, UB and AB are

quadratic (concave) monthly (mn) economic revenue from allocating water to urban (ur)

and agricultural (ag) nodes respectively, HB is the linear hydropower generation revenue at

hydropower plants (hp), NC is network cost incurred due to conveyance, treatment, etc. in

the link between nodes i and j, PC is the dynamic groundwater pumping cost in

groundwater nodes (gw) which varies as the pumping head changes, IC is the infeasibility

cost. In the Central Valley, it is assumed that the presence of “high-head” facilities allows

for a linear relationship between head and hydropower generation. This is because the

effect of reservoir storage on turbine head is small (Madani & Lund, 2007; Vicuna,

Leonardson, Hanemann, Dale, & Dracup, 2008). Hence, HB is a linear function in this

study. The set of constraints include water balance constraint; lower/upper bounds on

flows and storage levels; and hydropower generation capacity. Additionally, a major aspect

124

of California’s hydrology is return flows from agricultural and urban activities (Jenkins et

al., 2001). Return flows from agricultural and urban water use to surface and groundwater

deep percolation are included in the proposed model as a percentage of water used at each

demand site.

The next step is to link the hydro-economic model to an external search algorithm, the EA,

to calibrate the set of COSVF parameters (or WTP values) that lead to best mimic of the

historical operation. For this purpose, after all yearly simulations are performed, the

following fitness function is minimised in every function evaluation of the EA:

𝐹1 =1

𝑁𝑠𝑟𝑁𝑦𝑟𝑁𝑚𝑛∑∑∑(𝑆𝑦𝑟,𝑚𝑛

𝑠𝑟 − �̂�𝑦𝑟,𝑚𝑛𝑠𝑟 )

2

𝑠𝑟𝑚𝑛𝑦𝑟

(38)

The above function computes mean squared error between the modelled storage levels (S)

and the historical ones (�̂�) with yr and mn being the time index representing years and

months of the planning horizon respectively, and Nsr, Nyr, and Nmn being the number of

surface reservoirs, years, and months respectively. In the simplest form, where a single

reservoir releases for a single downstream demand node, various estimation of reservoir

WTP can be obtained following the above setting. For example, assuming a flat demand

curve for both the reservoir and the demand node, if the WTP of demand node is, say,

$1000 per MCM (million m3), the reservoir won’t keep water in storage at the end of each

year if its WTP is obtained below $1000 per MCM. But if the reservoir has a WTP of more

than $1000 per MCM, no matter how much above that value, it won’t release for the

downstream demand as it implies that keeping water for the future uses would be more

beneficial. Hence, it is very likely that the EA finds a different value every time it is run.

Some of these values could lead to optimum fitness function (F1) and some could be even

sub-optimal. This condition can quickly exacerbate in the presence of multiple storage

nodes and demand sites, similar to the case of this paper. It would be very difficult to

scrutinise this condition with the above single objective formulation. To cure this, we

introduce a second fitness function (F2) to now-a-multi-objective EA which minimises the

sum of average WTPs of reservoirs:

𝐹2 =1

𝑁𝑠𝑟∑

𝑝1𝑠𝑟 + 𝑝2

𝑠𝑟

2𝑠𝑟

(39)

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The second fitness function weighs all reservoirs the same regardless of their size to

prevent undervaluing storage in smaller reservoirs. Implementing F2 helps MOEA to find

the lowest possible WTPs that minimises F1 and to eliminate the chance of finding

unrealistically high solutions. Decision variables of the MOEA (WTPs; 𝑝1𝑠𝑟 , 𝑝2

𝑠𝑟) are

positive and bounded to the highest WTPs among all demand nodes.

The workflow of the proposed approach starts by the MOEA randomly generating set of

WTPs. Then F2 is calculated before passing WTPs to the hydro-economic model. The

hydro-economic model first converts WTPs to COSVF parameters following equation

(36). With known COSVF, the hydro-economic model annually simulates the water system

using the objective function of equation (37). At the end of annual simulations, the hydro-

economic model calculates F1 and reports it back to the MOEA. Next, MOEA performs its

evolutionary operators (e.g. mutation, crossover, selection, etc.) and proceeds to the next

generation until a stopping criterion is met. The stopping criterion used in this study is a

pre-defined number of function evaluations. The proposed workflow is illustrated as a flow

chart in Figure 25.

Figure 25. Flowchart of the proposed model workflow.

4.2.2 Model implementation

Borg (Hadka & Reed, 2013) was used as the MOEA in this study. The auto-adaptive

BORG MOEA can adjust different parameters that control the behaviour of the algorithm

to the specific characteristics of the problem (Reed, Hadka, Herman, Kasprzyk, & Kollat,

2013). This increases its robustness and effectiveness while minimising the search

parametrisation required by the user. Having 30 surface reservoirs in the Central Valley

case means BORG include 60 decision variables. An initial population size of 100,

maximum number of function evaluations of 100,000 as the stopping criterion, and epsilon

(search resolution) value of 65.73 MCM2 (1000 kaf

2) and 8107 $/MCM (10 $/af) for the

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fitness functions (equations (38) and (39), respectively) were adopted in this study. Please

refer Appendix B for further details. The case presented here was solved occupying 96

Intel processors, working jointly on a Unix-based high-performance computing (HPC)

cluster at the University of Manchester. For simulation purpose, the non-linear hydro-

economic model was coded in GAMS and solved using the Minos solver version 5.5

(Murtagh & Saunders, 1998). By applying the Generalized Reduced Gradient (GRG)

method, Minos is suitable for non-linear programming problems with linear constraints

(Labadie, 2004).

4.3 Illustrative example

4.3.1 The Central Valley of California

California’s Central Valley (Figure 26) is one of the world’s most productive agricultural

regions (Faunt, 2009) with over 2.3 million ha of irrigated farmland (CDWR, 2009). More

than 250 different crops are grown in the Central Valley with an estimated value of $17

billion per year (GreatValleyCenter, 2005). About 75 percent of California’s irrigated land

is in the Central Valley, which depends heavily on surface water diversions and

groundwater pumping (Faunt, 2009). Nearly 75 percent of renewable water supply

originates in the northern third of the state in the wet winter and early spring while almost

80 percent of agricultural and urban water use is in the southern two-thirds of the state in

the dry late spring and summer (CNRA, 2009). In the context of California, perfect within-

year foresight is consistent with early spring measurements of the depth and water content

of the snowpack which enable predicting discharge months ahead with reasonable

accuracy and until the end of the water year (Draper, 2001). The Central Valley often

suffers from droughts such as 1918-20, 1923-26, 1928-35, 1947-50, 1959-62, 1976-77,

1987-92, 2007-09, and 2012-16 (CDWR, 2015).

127

Figure 26. The Central Valley reservoir and river system.

The illustrative case of this paper is built upon CALifornia Value Integrated Network

(CALVIN; Draper, Jenkins, Kirby, Lund, and Howitt (2003)). CALVIN Optimised, a

hydro-economic model (Harou et al., 2009) with perfect foresight, is the ‘unconstrained’

run of CALVIN used to simulate the Central Valley water system by maximising the

system-wide net economic benefit from water allocation. CALVIN Optimised applies

economic drivers to allocate water rather than existing system of water rights and contracts

(Draper, 2001). Yet, the perfect hydrological foresight of CALVIN Optimised limits its

applicability. We use an extended version of CALVIN Optimised for calibration which

corrects the perfect foresight by dividing the planning horizon into year-long runs with

initial condition of each run being the ending condition of the previous one and an end-of-

128

year COSVF, representing the potential benefit of allocating water for future uses, set as

the terminal condition of each run. Another extension to CALVIN Optimised comes from

improving the groundwater pumping cost scheme. The CALVIN Optimised model

represents pumping costs by multiplying the unit pumping cost of $49.42 per MCM/m lift

($0.20 per af/ft lift; MCM is a million m3) by a static estimate of the average pumping head

in each aquifer (Hansen, 2007). The extended version of CALVIN Optimised includes

pumping costs that dynamically vary with head in the aquifer. This head-dependent

pumping cost introduces non-convexity into the problem.

The water system is represented as a network of nodes and arcs (Maass et al., 2013), where

nodes include surface and groundwater reservoirs, urban and agricultural demand points,

junctions, etc., and arcs (links) include canals, pipes, natural streams, etc. (Shamir, 1979).

The water network of the Central Valley comprises 30 surface reservoirs, 22 groundwater

sub-basins, 21 agricultural demand sites, 30 urban demand sites, 220 junction and 4

outflows nodes; and over 500 links (river channels, pipelines, canals, diversions, and

recharge and recycling facilities). The planning horizon is 72 years, 1922-93.

4.3.2 Historical approximation

In order to calibrate the model to produce historical WTPs, historical reservoirs’ storage

data needs to be available. Since observed data was not attainable for all reservoirs and for

the entire planning horizon, storage capacity time-series of an already calibrated model is

used. We take reservoir capacity time-series from ‘historical approximation’, a ‘base case’

or ‘constrained’ run of CALVIN which applies constraints to reproduce historical event.

Hereafter, we refer to such results as historical approximation.

CALVIN 'base case' run used projected 2020 demands as a requirement set out by CDWR'

Bulletin 160-98. CALVIN was initially built with the aim of informing policy-making in

the California context. Therefore, the model was forward-looking and had no intention in

carrying out a historical simulation purely. One should consider that CALVIN 'base case'

run is a benchmark which only extrapolates historical operations using historical hydrology

and projected demands. To the best of our knowledge, 'base case' results are the only

available data which closely approximates historical trend of water system operation of

California's Central Valley. Here, we verify the historical approximation by comparing the

129

storage trajectory of Shasta, the largest reservoir of the region, from historical

approximation and historical observation – see Figure 27.

Figure 27. Comparison of historical approximation and observation of storage level of Shasta.

The historical approximation shows a close match to the observed data for Shasta. Slight

difference in the approximation of historical approximation is negligible. This comparison

denotes that the use of historical approximation as an approximation for historical events is

correct.

4.4 Results

4.4.1 Reservoir storage valuation

In order to obtain the Pareto trade-off and to ensure convergence, a Random Seed (RS)

analysis with five seeds was used. This is to check that various starting points lead to the

same set of final optima. The Pareto front (Figure 28) consists of non-dominated solutions

with respect to the two fitness functions. No improvement can be made to the calibration

fitness function (F1) without deteriorating the other fitness function (F2).

0

1000

2000

3000

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rage

(M

CM

)

Water Years

Historical approximationObserved data

80,000

100,000

120,000

140,000

160,000

180,000

200,000

220,000

240,000

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000

Squ

are

d d

evi

atio

n (

MC

M2 )

Mean water marginal value ($/MCM)

Zone of concentration

130

Figure 28. Pareto non-dominated solutions of the two fitness functions (arrows show the direction of

preference).

A zone of concentration is identified within the set of Pareto solutions. The next immediate

solution point outside of this zone has significant difference in either of fitness function

values. Concentration of solution points in this zone suggest that the estimate for historical

water marginal values can be sought there. We investigate this by looking at historical

marginal water valuation at dead and full storage (Figure 29) from each solution point of

the zone of concentration.

Figure 29. Dispersion of historical water marginal value solutions from zone of concentration at: a) dead

storage, and b) full storage.

0

20,000

40,000

60,000

80,000

100,000

120,000

Shas

ta

Wh

iske

yto

wn

Bla

ck B

utt

e

Oro

ville

Ne

w B

ulla

rds

Bar

Cam

p F

ar W

est

Fols

om

Ind

ian

Val

ley

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rye

ssa

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de

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w H

oga

n

Los

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UD

agg

rega

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Llo

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Het

ch H

etch

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San

ta C

lara

SF a

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gate

Kaw

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Pin

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lat

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Mill

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n

McC

lure

Hen

sley

East

man

Do

n P

edro

Mar

gin

al w

ate

r va

lue

($

/MC

M)

Reservoirs

(a)

0

10,000

20,000

30,000

40,000

50,000

60,000

Shas

ta

Wh

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yto

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Bla

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utt

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ar W

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ian

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rye

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Vaq

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UD

agg

rega

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Llo

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lean

or

Het

ch H

etch

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San

ta C

lara

SF a

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gate

Kaw

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Succ

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Pin

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lat

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Lu

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Del

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Hen

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East

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Do

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edro

Mar

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al w

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r va

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($

/MC

M)

Reservoirs

(b)

131

Historical water marginal value solutions from the zone of concentration are quite diverse

for some reservoirs. This is the case with small reservoirs, whereas WTP values for large

reservoirs such as Shasta are consistent across solution points. Small storage size of these

reservoirs diminishes the variation in the maximum benefit (COSVF) each reservoir can

produce. This is examined and illustrated through a box plot showing the range of

maximum COSVF values for each reservoir (Figure 30).

Figure 30. Distribution of the maximum COSVF from the solution points of the zone of concentration. Red

points show the maximum COSVF of the “optimised” model run.

According to Figure 30, few reservoirs experience significantly different valuations of their

maximum end-of-year COSVF. The most notable are Pine Flat, New Melones, and Don

Pedro. Here, we investigate how this diversity in reservoir valuation manifests in the

operation of these reservoirs. The storage trajectory of these three reservoirs is simulated

using valuation solutions that created average, minimum, and maximum values in Figure

30. These three valuations of storage are used in three separate runs of the model, with all

other parameters unchanged (including COSVF from other reservoirs). This is depicted in

Figure 31. Resulting end-of-year storage levels of the above three reservoirs prove to be

identical regardless of which marginal value of water is chosen (Figure 31). Therefore, it is

the “average” valuation that is reported in Table 3.

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Reservoirs

Historical valuationBOX 1'Optimised model'

132

Figure 31. Calibrated storage trajectories with average, minimum and maximum valuations in: a) Don Pedro,

b) New Melones, and c) Pine Flat.

Table 3. Historical marginal water values of end-of-year surface reservoirs’ storage in the Central Valley,

listed from north to south.

Reservoir

End-of-year active

storage (MCM)

(1)

Annual average

net inflow (MCM)

(2)

Historical marginal value at dead storage

($/MCM) (3)

Historical marginal value at full storage

($/MCM) (4)

Historical average

marginal value ($/MCM)

(5)

‘Optimised’ average

marginal value ($/MCM)

(6)

400

900

1,400

1,900

2,400

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(MC

M)

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(b)

Calibrated with average values Calibrated with maximum values Calibrated with minimum values

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133

Shasta 3,344 6,816 52,666 38,549 45,608 29,576

Whiskeytown 138 1,144 12,484 9,542 11,013 39,923

Black Butte 122 488 480 189 334 393

Oroville 2,682 4,966 45,830 42,139 43,985 22,263

New Bullards Bar 560 1,496 32,382 28,429 30,406 38,775

Camp Far West 126 458 1,872 1,817 1,844 108

Indian Valley 731 529 15 13 14 10,825

Folsom 701 3,271 27,638 5,830 16,734 34,979

Berryessa 1,926 438 18,915 18,865 18,890 10,656

Pardee 235 840 107 14 60 13,334

New Hogan 263 184 347 120 234 15,416

New Melones 1,507 1,285 74,067 9,108 41,588 19,500

EBMUD 63 0 516 286 401 48

Los Vaqueros 41 0 145 27 86 8

Lloyd-Eleanor 333 542 28,437 28,129 28,283 27,953

Hetch Hetchy 399 936 7,911 7,104 7,507 1,403

Del Valle 23 0 1,273 1,181 1,227 276

Don Pedro 1,727 792 69,837 10,213 40,025 23,716

Turlock 69 0 369 94 232 182

McClure 907 1,128 42,123 5,940 24,032 20,314

SF aggregate 277 0 1,071 650 860 0

Eastman 99 82 959 10 485 264

Santa Clara 209 156 1,766 727 1,246 89

Hensley 79 101 329 135 232 30,892

San Luis 1,958 0 1,275 467 871 1

Millerton 495 2,082 35,190 747 17,969 37

Pine Flat 1,177 2,041 63,661 7,752 35,706 5,767

Kaweah 101 581 5,884 5,747 5,816 913

Success 81 170 281 8 144 5,387

Isabella 453 876 1,670 1,011 1,340 526

Column (1) of Table 3 contains active storage (full storage – dead storage) of surface

reservoirs in the last month of the water year, i.e. September. Note that maximum capacity

varies per month due to flood control requirements. The second column shows the annual

net inflow calculated as annual surface runoff minus any loss (e.g. seepage and

evaporation). Columns (3) and (4) include the obtained historical marginal water values by

averaging values depicted in Figure 29.a and Figure 29.b respectively. Values in column

(5) are the average of values in columns (3) and (4). This shows the flat water valuation

required to reach the same amount of maximum COSVF at full storage as in the linear

(Figure 24) reservoirs’ demand curve. Column (5) can be used as an economic proxy for

comparing reservoirs valuation, and to contrast historical valuation against economically

efficient valuation. The latter is demonstrated in column (6) and the model run is labelled

134

as ‘optimised’. To obtain these values, the model was run with a net economic benefit

fitness function computed for the entire planning horizon which replaces the current

calibration fitness function (equation (38)). Hence, the resulting valuation is not those that

imitate historical operation, but those leading to the most profitable operation of surface

reservoirs.

Similar to the ‘optimised’ model, the historical operation valued major reservoirs and those

located on the eastern range of the Central Valley (e.g. Oroville, New Bullards Bar,

Folsom, New Melones, Lloyd & Eleanor, Don Pedro, McClure, an Pine Flat) higher than

others. However, the historical model assigned a higher marginal water value than the

optimised run for large reservoirs - examples are Shasta, Oroville, Berryessa, New

Melones, Don Pedro, San Luis, and Pine Flat. This is also apparent in Figure 30, where the

maximum COSVF of the optimised model was lower than those of the historical operation

in most reservoirs. This more strict valuation has two consequences: (1) groundwater

resources are first to supply water; and (2) when it comes to surface reservoirs, smaller

reservoirs are prioritised for release. COSVF for each reservoir can be derived from values

in columns (3) and (4) of Table 3. Recall that these two marginal values form reservoirs’

linear demand curve and COSVF is the integral of the demand curve. We report COSVF of

the Central Valley reservoirs for each of its three main basins (Figure 32).

0

25

50

75

100

125

150

0 0.2 0.4 0.6 0.8 1

CO

SVF

(M$

)

Storage

(b)

Whiskeytown, Black Butte, Camp Far West, Indian Valley

Shasta

Oroville

New Bullards Bar

Folsom

Berryessa

(a)

135

Figure 32. (a) The three basins of the Central Valley (adopted from Jenkins et al. (2001)), COSVF of

reservoirs in (b) Sacramento Valley, (c) San Joaquin basin, and (d) Tulare basin.

Storage capacities in Figure 32 are normalised for the ease of comparison. This figure

shows that the valuation, hence the COSVF, of reservoirs is also dependant to how

upstream they are located. For instance, the maximum economic benefit generated from

keeping water in storage in the largest reservoir of Sacramento Valley basin (i.e. Shasta) is

higher than that of San Joaquin basin (i.e. Don Pedro) and Tulare basin’s (i.e. Pine Flat).

4.4.2 Calibration

In this section, we investigate how closely the calibrated reservoir operation matches

historical case represented by ‘historical approximation’. To obtain storage trajectories,

the Central Valley water network was simulated using water marginal values of Table 3,

columns (3) and (4). We compare the storage capacity time series of the largest reservoirs

possessing the highest WTPs (Figure 33).

0

50

0 0.2 0.4 0.6 0.8 1

CO

SVF

(M$

)

Storage

(c)

Pardee, Turlock, EBMUD, New Hogan, Los Vaqueros, Hensley,Santa Clara, Eastman, SF aggregate, San Luis, Del ValleLloyd&Eleanor

Hetch Hetchy

New Melones

Millerton

Mc Clure

Don Pedro

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1

CO

SVF

(M$

)

Storage

(d)

Kaweah, Isabella, Success

Pine Flat

2,000

2,500

3,000

3,500

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4,500

5,000

5,500

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Shasta

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)

Water Year

Don Pedro

137

Figure 33. Comparison of the calibrated storage trajectories of major reservoirs to the historical

approximation.

It can be seen that the calibrated model was able to closely reproduce historical trend. This

validates that values reported in Figure 32 and Table 3 are indeed the correct estimate of

the Central Valley reservoirs’ COSVF and end-of-year marginal values respectively.

4.5 Discussion

This paper is concerned with the estimation of historical economic value of reservoirs’

end-of-year storage. A hybrid calibration model is built using a hydro-economic model

coupled with an EA. End-of-year storage in the hydro-economic model is governed by the

COSVF, a quadratic function used to express the fact that the value of water increases

when it is scarce and decreases when water is plenty. Parameters of the quadratic COSVF

is optimally located by the EA while the hydro-economic model simulates the water

system. This approach can easily be used when market valuation is absent or inefficient

and when non-market methods are plagued with non-convexity and/or curse of

dimensionality. The proposed approach can be extended for estimation of historical

valuation in other sectors. For instance, how decision-makers valued fisheries in the past?

The proposed approach is illustrated through valuation of 30 surface reservoirs in

California’s Central Valley water system. The over-cautious historical operation of this

region was used to obtain how reservoir operators and regulators valued water in storage

during 1922-93. Results show that they a) prioritised groundwater resources over surface

reservoirs, and b) preferred to release from smaller reservoirs first and keep water in larger

ones as a water bank. The former is demonstrated via comparing columns (5) and (6) of

Table 3. The historical operation imposed more strict use of surface reservoirs than the

optimised model, meaning that they relied more on aquifer extraction. Additionally,

0

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1,200

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(M

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)

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Pine Flat

Calibrated Historical

138

contrasting the surface and groundwater resources of historical operation to the optimised

model reveals the reason behind the overdrafting of groundwater in the history of Central

Valley (Harou & Lund, 2008; Nelson et al., 2016). This is indicated in Figure 34. Columns

(1) and (5) of Table 3 along with Figure 32 show that the historical operation favoured

smaller reservoirs when it comes to surface reservoir releases. This can become

problematic when a small reservoir is the sole supplier to a demand site (e.g. New Hogan

supplying Stockton). This strategy seems to be not effective. With depleting groundwater

resources and hoping that large reservoirs can be trusted as a water bank, historical

operation reached the same level of storage as in the optimised model during severe

droughts such as 1976-78.

0

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(a)

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(b)

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139

Figure 34. Comparison of the historical approximation and the optimised model for: a) surface reservoirs

over 1922-57; b) surface reservoirs over 1958-93; and c) groundwater over 1922-93.

Yet, some limitations hinder the proposed approach. The comparison of historical

approximation vs optimised operation carried out in this chapter suggests way in which the

system's modelling could be further improved. In this chapter, a higher valuation of surface

storage in the historical case could also reflect the existence of operational constraints not

yet accounted for in the optimisation framework. For example, benefits for maintaining

stream temperature that is not seen in this study. Also, approaches including COSVF

require that the hydrology have substantial probability of persistence of dry periods. A

hydrology that, perhaps oddly, has very severe droughts of one period followed by

extremely wet conditions which always fill the reservoir would never have implementing

COSVF be optimal as COSVFs will not provide strong enough incentive to preserve water

for later uses. Moreover, such hybrid approaches demand high computational requirement

and run time. Employing the proposed hybrid model was only feasible with the help of

High Performance Computing (HPC) facilities. The search for COSVFs in this paper

required 93 hours per random seed per core, using 96 CPU cores.

It should be noted that the quality of fit and interpretation of results is debatable. Since the

optimisation problem (finding the set of COSVF parameters that best reproduce historical

operation) has high number of decision variables to optimally locate, the problem has high

degree of freedom. This means the chance of getting near-perfect calibration is high

because the EA can ‘play’ with different combinations of COSVF parameters until it finds

the best match. Another issue is that by calibrating to the full time series, it cannot be

guaranteed that obtained COSVF parameters reflect the implicit historical value placed on

carryover storage by operators. The ideal approach to address above remarks is to carry out

out-of-sample validation. In this way, a portion of the 72-year time series is used for

calibration purpose (yr in equation (38) would be, say, 1, 2… 60). The rest of the time

560,000

580,000

600,000

620,000

640,000

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(M

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)

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(c)

Historical approximation Optimised model

140

horizon is used to re-simulate the water resources system to investigate if the COSVF

parameters obtained from the first portion of the time horizon is still able to reproduce

historical operation in the remaining time segment (the last 12 years for instance). The

latter stage is called validation. Carrying out this last stage was not possible due to time

and computational constraints that exist for the author of this thesis. Such validation

exercise is left for future works.

4.6 Conclusion

This paper proposes a hybrid approach for estimating the historical valuation of surface

reservoir storage. This is done via implementing quadratic COSVFs for the last month of

annual runs of a hydro-economic model as the terminal condition to prevent depletion of

reservoirs. WTPs (or parameters of COSVFs) are calibrated by an EA while minimising a

mean squared error fitness function between the modelled storage levels and the historical

values. A second fitness function is introduced to guide the EA towards locating the lowest

possible WTPs that keep reservoirs at or close to historical storage level. Through this

hybrid approach we are able to capture the valuation of reservoirs’ over-year storage that

leads to best imitation of historical approximation. Use of the hybrid approach makes the

proposed model applicable to cases where non-convexity and curse of dimensionality

plagues most of the existing approaches. The large-scale water system of California’s

Central Valley is used for illustration purposes. Results show that the reservoir operators

and regulators highly valued surface reservoirs and relied mostly on groundwater.

Additionally, smaller reservoirs were first to release then the larger ones. While such

operation hoped to keep water in larger reservoir as a water bank, a comparison of surface

and groundwater trend between the historical approximation and an optimised model (the

same as the calibration model but maximising the system-wide net economic benefits)

reveals that historical operation was not economically efficient. They caused overdrafting

of groundwater resources while having the same volume of water in storage in surface

reservoirs during major drought of 1975-78.

141

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144

145

5 Chapter five: Discussion and Conclusion

146

5.1 Discussion

The aim of this thesis was to answer the question ‘what is the appropriate amount of inter-

annual (over-year) storage in large-scale multi-reservoir systems which leads to economic

boost from water allocation?’ This thesis addresses this question through development of a

generalisable approach that uses storage valuation method to determine end-of-year storage

- this was the objective of the thesis. Few approaches exist that govern over-year

reservoirs’ release/storage. Yet, they are not assumption-free which reduces their

applicability. The proposed approach is not hindered by such assumptions (requiring

linearity) or scale of the problem, contributing to a methodological advance. Following the

successful application of SDDP for storage valuation in convex problems only, this is the

first non-market valuation approach that is not suffering from non-convexity and curse of

dimensionality to the best of the author’s knowledge. It uses optimised end-of-year

COSVFs for surface water reservoirs to account for the expected value of water beyond the

current water year. To challenge the proposed approach the large-scale case of California’s

Central Valley is chosen, where head-dependant groundwater pumping cost is the source

of non-convexity. In order to simulate this water system with COSVFs implemented at the

end of each annual run, an existing hydro-economic model of the region, i.e. CALVIN, is

adopted and improved by introducing two extensions. This was detailed in Chapter two,

where static pumping cost scheme of CALVIN was replaced by a dynamic head-dependent

one and its perfect hydrological foresight was modified to a limited foresight. Chapter two

demonstrated that CALVIN’s unrealistic assumption of knowledge of future events, and its

undervalued groundwater abstraction cost can successfully be amended.

Following the development of the hydro-economic model of the Central Valley in Chapter

two, in Chapter three, it was linked to an external EA to optimally locate parameters of the

quadratic COSVF. BORG was used for this purpose. Results of optimal COSVF

parametrisation revealed that once reservoirs’ end-of-year storage is accurately evaluated,

they can lead to better management of water resources and reduce shortage in water-scarce

regions. This was evidenced by comparing such management with the historical one,

where optimised inter-annual reservoir operation improved over extra-cautious operation

in the historical approximation, reducing the average annual scarcity volume and costs by

80% and 98%, respectively.

147

Another application of the proposed methodology of this thesis was demonstrated in

Chapter four. In this Chapter, instead of determining COSVF parameters that maximise the

system-wide net economic benefit, they were calibrated to reproduce historical operation.

The resulting COSVFs represent end-of-year storage valuation that regulators and reservoir

operators implicitly set in the past. Further analysis showed that the management strategy

that they adopted was not effective. They prioritised groundwater resources over surface

water and smaller reservoirs over larger ones. In this manner, they hoped to have large

reservoirs as water banks for extreme droughts. However, this strategy lead to overdrafting

of aquifer sub-basins and large reservoirs became empty during major droughts. The over-

conservative historical operation was evidenced by comparing it to the operation of an

‘optimised’ model which evaluated over-year storage to maximise the economic benefit of

water allocation. While the aggregate surface water supply in the ‘optimised’ model of the

central Valley had a marginal value of $11,784 per MCM, the historical operation used

average marginal value of $12,572 per MCM in average for all reservoirs.

The proposed approach represented in this thesis is not limitation/assumption free. The

proposed approach assumed a perfect within year foresight. While this is reported to be

correct for the Central Valley (and many other regions globally), one can use a shorter time

frame with perfect foresight. For example, if only monthly weather forecast is accurate,

monthly runs (simulations) can be used instead of annual runs. Then, CSOVF has to be

assigned for end-of-month storage. This requires different COSVF for each month but

consistent across all years (i.e. January valuation, Februarys valuation, etc.). It should be

noted that this new setting introduces more variable to the EA but reduces the run time

required for simulation.

Approaches including COSVF require that the hydrology of the region have substantial

probability of persistence of dry periods. A hydrology that, perhaps oddly, has very severe

droughts of one period followed by extremely wet conditions which always fill the

reservoir would never have implementing COSVF be optimal because COSVFs will not

provide strong enough incentive to preserve water for later uses. Moreover, such hybrid

approaches demand high computational requirement and run time. Employing the

proposed hybrid model is not feasible without using parallel processors and HPC facilities.

148

5.2 Future work

The studies presented in this thesis can be taken forward in several ways. The work

presented in Chapter three can be extended to include groundwater COSVFs and to

investigate the effect of different pumping cost schemes on aquifers marginal value. How

the modelling stage should be carried out is still under development and discussion. This

requires further research and testing various modelling approaches. A multi-sectoral

application of the proposed approach is also under consideration. This involves assessing

storage marginal values in attempt to not only maximising off-stream benefits (from urban

and agricultural water consumption) but also taking into account in-stream objectives

(maximising an ecological performance metric such as the number fish habitat in the

natural streams). Because different valuation of end-of-year storage leads to different

reservoir operation (release pattern) estimation of COSVFs can influence water

temperature in reservoirs and rivers. Perturbation in stream temperature can change the

survival rate of natural habitat, e.g. Chinook salmon in the case of the Central Valley

streams. Therefore, a future work can include a new objective function in the MOEA

where survival rate of Chinook salmon is maximised and stream temperature is introduced

to the problem as a new variable. Thanks to the use of MOEA, other management

objectives such as resilience can be integrated into the valuation of carry-over storage; this

is also left to future work.

5.3 Conclusion

This thesis contributes to a novel modelling-based non-market valuation of surface water

resources. This approach is generalisable, meaning that it can be readily applied to

different cases with new set of inputs and without needing to redo the modelling stage.

Existing modelling-based non-market valuation approaches suffer from either curse of

dimensionality such as SDP, or non-convexity such as SDDP. The proposed approach,

thanks to the use of hybrid approach which couples an MOEA to a hydro-economic model,

is not plagued by the above issues.

Two application of the proposed approach was represented in this thesis and several others

were proposed for future studies. The first application was to find surface water valuation

that lead to economically efficient operation of California’s Central Valley water system. It

was shown that water scarcity could have been avoided if the accurate estimate of end-of-

149

year storage valuation were considered. In the second application estimate of historical

valuation of end-of-year storage was tackled. When previous contingent valuation is not

available, the only option for this purpose is to use modelling-based valuation techniques.

A Central Valley case comprising 30 reservoirs was used as the illustrative example.

Results depict that overcautious historical operation of Central Valley water system is

consistent with more strict valuation obtained for surface reservoirs.

150

Appendix A. Model input data

This section provides all the input data used for the proposed model of this thesis.

Table A 1. Minimum allowable storage in storage nodes. This is Smin in equations 1, 19, and 21 and in

minimum storage capacity constraints. Volumes are in Kilo acre foot (Kaf) with 1 Kaf = 1.23348 Million

Cubic Meter (MCM).

Node Node Node

GW1 1400 GW6 11277 SF aggregate 31

GW10 19597.5 GW7 7403 Black Butte 10

GW11 8629.23 GW8 12945 Camp Far West 1

GW12 8700 GW9 14224 Indian Valley 0

GW13 30761.36 GWSC 197.16 EBMUD 83

GW14 45262 New Melones 80 Hetch Hetchy 36

GW15 69548 San Luis 80 Berryessa 10.3

GW16 0 Del Valle 9.8 Isabella 0.184

GW17 6621 Millerton 120 Kaweah 0.57

GW18 33454 McClure 115 Lloyd-Eleanor 30.1

GW19 42491 Whiskeytown 10 Success 0.557

GW2 9000 Shasta 116 Los Vaqueros 72

GW20 22722 Hensley 4 New Bullards Bar 251

GW21 47351 Eastman 10 New Hogan 17.5

GW3 10500 Oroville 29.6 Pine Flat 45.38

GW4 7900 Folsom 83 Pardee 12.2

GW5 8912 Don Pedro 100 Santa Clara 37

Turlock 11

Note: GW=groundwater, SF=San Francisco, and EBMUD= East Bay Municipal Utility District

Table A 2. Maximum allowable storage in storage nodes. This is Smax used in maximum storage capacity

constraints. Volumes are in Kaf with 1 Kaf = 1.23348 MCM.

Month

Node 1 2 3 4 5 6 7 8 9 10 11 12

GW1 5448 5448 5448 5448 5448 5448 5448 5448 5448 5448 5448 5448

GW10 29250 29250 29250 29250 29250 29250 29250 29250 2925

0

2925

0

29250 29250

GW11 15543 15543 15543 15543 15543 15543 15543 15543 1554

3

1554

3

15543 15543

GW12 13919 13919 13919 13919 13919 13919 13919 13919 1391

9

1391

9

13919 13919

GW13 47484 47484 47484 47484 47484 47484 47484 47484 4748

4

4748

4

47484 47484

GW14 65235 65235 65235 65235 65235 65235 65235 65235 6523

5

6523

5

65235 65235

GW15 90978 90978 90978 90978 90978 90978 90978 90978 9097

8

9097

8

90978 90978

GW16 11650 11650 11650 11650 11650 11650 11650 11650 1165

0

1165

0

11650 11650

GW17 13492 13492 13492 13492 13492 13492 13492 13492 1349

2

1349

2

13492 13492

GW18 59544 59544 59544 59544 59544 59544 59544 59544 5954

4

5954

4

59544 59544

151

GW19 68266 68266 68266 68266 68266 68266 68266 68266 6826

6

6826

6

68266 68266

GW2 24162 24162 24162 24162 24162 24162 24162 24162 2416

2

2416

2

24162 24162

GW20 40814 40814 40814 40814 40814 40814 40814 40814 4081

4

4081

4

40814 40814

GW21 81622 81622 81622 81622 81622 81622 81622 81622 8162

2

8162

2

81622 81622

GW3 22127 22127 22127 22127 22127 22127 22127 22127 2212

7

2212

7

22127 22127

GW4 15362 15362 15362 15362 15362 15362 15362 15362 1536

2

1536

2

15362 15362

GW5 24162 24162 24162 24162 24162 24162 24162 24162 2416

2

2416

2

24162 24162

GW6 22864 22864 22864 22864 22864 22864 22864 22864 2286

4

2286

4

22864 22864

GW7 12270 12270 12270 12270 12270 12270 12270 12270 1227

0

1227

0

12270 12270

GW8 32842 32842 32842 32842 32842 32842 32842 32842 3284

2

3284

2

32842 32842

GW9 23395 23395 23395 23395 23395 23395 23395 23395 2339

5

2339

5

23395 23395

GWSC 655 655 655 655 655 655 655 655 655 655 655 655

New

Melones

1975 1950 1950 1950 1950 2020 2196 2400 2400 2300 2150 2015

San Luis 610.4 732.1 970.3 1350.

4

1605.

8

1508.

3

1224.

9

1015.

2

821 606.8 546.6 525

Del Valle 27.83 27.76 27.74 27.72 27.67 27.55 27.36 27.06 26.71 26.32 25.99 25.75

Millerton 327.87 351.5

7

424.0

7

435.5

4

435 521 521 521 521 521 426.6

9

307.6

2

McClure 676 676 676 676 676 737 851 969 1024 1024 982.3

8

851

Whiskeytow

n

220 240 240 240 240 240 240 240 240 240 240 235

Shasta 3400 3252 3368 3828 4552 4330 4552 4552 4552 4300 4000 3700

Hensley 22.54 20.87 28 34 42 51 73 64.48 39.59 42.37 39.63 39.1

Eastman 46.39 46.29 91.89 105 111 120 136 145.4 124 81.8 58.58 58

Oroville 3164 3164 3164 3164 3164 3164 3471 3539 3539 3539 3531 3352

Folsom 721 575 575 576 600 681 801 976 976 951 801 651

Don Pedro 1690 1690 1690 1690 1690 1690 1713 1990 2030 2030 2030 1773

SF aggregate 135.9 145.1 156.2 167.1 170.6 169.2 162.2 154.5 144.6 134.9 125.9 128

Black Butte 99 80 69 58 58 77 125 147 150 150 129 109

Camp Far

West

103 103 103 103 103 103 103 103 103 103 103 103

Indian Valley 601 590 486 486 550 567 590 606 614 602 592 593

EBMUD 133 133 136 144 149 152 153 151 147 142 137 134

Hetch

Hetchy

360 360 360 360 360 360 360 360 360 360 360 360

Berryessa 1571 1569 1602 1601 1601 1599 1598 1594 1591 1583 1573 1572

Isabella 241 185 184 183 331 364 426 485 534 460.9 330.1 297.2

Kaweah 43.49 11.2 10 9 9 11 67 116 136 141 141 77.1

Lloyd-

Eleanor

301 287 290.6 271.8 282.4 274.7 298 301 301 301 301 301

Success 31 11 10 9 13 22 54 78 82 83 59 55.2

Los

Vaqueros

105 105 105 105 105 105 105 105 105 105 105 105

New 660 645 645 600 600 685 825 930 890 830 755 705

152

Bullards Bar

New Hogan 225 191 152 165 188 206 250 266 258 245 240 231

Pine Flat 683.9 664.1 648.1 654 730 773 813 854 935 995 968.7

6

763.2

9

Pardee 198 193 188 183 188 193 198 203 210 210 210 203

Santa Clara 79.65 83.14 113.7 149.4 156.4 169 168.6 160.3 142.1 116.3 94.93 94

Turlock 66.4 66.7 66.9 66.9 66.8 66.6 66.4 66 65.8 65.6 65.7 66

Note: GW=groundwater, SF=San Francisco, and EBMUD= East Bay Municipal Utility District. Month 12 is

September.

Table A 3. Initial pumping lift in groundwater nodes, i.e. the distance water has to be lifted at the first time-

step to reach the surface. This is used in equation 18. Distances are in foot (ft) with 1 ft = 0.3048 m.

Node Node Node

GW1 159.4 GW16 171.96 GW3 146.81

GW10 69.972 GW17 153.44 GW4 177.5

GW11 121.38 GW18 186.84 GW5 195.68

GW12 146.54 GW19 270.77 GW6 98.258

GW13 165.69 GW2 214.59 GW7 116.26

GW14 393 GW20 364.66 GW8 159.5

GW15 168.17 GW21 408.11 GW9 100.69

GWSC 358.57

Note: GW=groundwater.

Table A 4. Aquifers’ surface area in acre with 1 acre = 4046.86 m2. This is area in equation 6 and a in

equation 18.

Node Node Node

GW1 328.12 GW16 395 GW3 679.71

GW10 1236 GW17 420 GW4 346.53

GW11 568 GW18 988 GW5 607.43

GW12 395 GW19 890 GW6 648.2

GW13 1137 GW2 688.77 GW7 344.8

GW14 593 GW20 519 GW8 884.62

GW15 1112 GW21 618 GW9 716.83

GWSC 10

Note: GW=groundwater.

Table A 5. Aquifers’ storage coefficient. This is sc in equation 6 and s in equation 18.

Node Node Node

GW1 0.0775 GW16 0.0867 GW3 0.1019

GW10 0.1803 GW17 0.0958 GW4 0.0918

GW11 0.0916 GW18 0.1187 GW5 0.0807

GW12 0.0784 GW19 0.1022 GW6 0.0969

GW13 0.1021 GW2 0.0919 GW7 0.0755

GW14 0.0658 GW20 0.1047 GW8 0.0914

GW15 0.1275 GW21 0.1142 GW9 0.0977

GWSC 0.08

Note: GW=groundwater.

153

Table A 6. Aquifers’ fixed pumping cost. This is equivalent to C in equations 4 and 5 and uc in equation 16

and 17 for models that use a static pumping cost scheme. The unit is $/Kaf, with 1 Kaf = 1.23348 MCM.

Node Node Node

GW1 30000 GW16 29800 GW3 23800

GW10 15600 GW17 31600 GW4 16000

GW11 20600 GW18 45200 GW5 18800

GW12 23600 GW19 68400 GW6 18200

GW13 30000 GW2 28200 GW7 28800

GW14 76400 GW20 67200 GW8 28600

GW15 46600 GW21 69600 GW9 20400

GWSC 20000

Note: GW=groundwater.

Table A 7. Linear coefficients of the quadratic benefit functions for demand nodes. The unit is $/Kaf, with 1

Kaf = 1.23348 MCM. Month

Nodes 1 2 3 4 5 6 7 8 9 10 11 12

AG1 85117.

74

0 0 0 0 70102.

39

95160.

7

96132.

05

98322.

05

98789.

16

10072

0.7

96189.

27

AG2 13477

1.7

0 0 0 12171

4.3

11930

7.9

12877

9.9

12226

8.1

12343

9.4

12357

7.3

12357

7.3

12614

3.3

AG3 97693.

42

0 0 0 12336

4.7

88630.

33

13468

5.2

12468

5.9

12684

7.6

12233

5.3

11694

0.3

89976.

99

AG4 47573.48

0 0 0 121871.4

52018.78

129936.3

119890.3

119826.7

115343.69

109747.98

73192.69

AG5 48579.

3

0 0 0 10461

5.4

38844.

9848

10938

0.9047

10176

8.6622

10629

5.395

10325

8.1834

99038.

53737

62685.

88602

AG6 41118.13187

0 0 0 106703.7975

81313.0793

114326.7991

101080.1271

104328.3089

106390.2244

98676.32578

82950.59718

AG7 61267.

08861

0 0 0 98000 48586.

2069

10438

7.8657

10079

0.6634

10252

0.5189

10085

7.3913

97966.

59243

77660.

29458

AG8 133301.5165

0 0 0 113333.3333

87152.74151

127091.8245

116175.662

104892.3111

102720.1928

109032.0963

122970.1719

AG9 21649

2.1569

0 0 0 14430

0

18856

2.2047

17813

8.2049

17806

3.8298

17326

6.9462

16867

3.0098

16462

7.5116

19722

0.1439

Bfield 3611000

3611000

3611000

3611000

3611000

3611000

1816714.25

1816714.25

1816714.25

1816714.25

1816714.25

1816714.25

CC 89546

67

89546

67

89546

67

89546

67

89546

67

89546

67

45051

43

45051

43

45051

43

45051

43

45051

43

45051

43

CVPM10

72100.59375

0 0 0 263679.125

151156.2656

141846.3438

133757.2344

149791.5781

156508.2813

149030.7344

119643.8984

CVPM

10UR

19320

00

19320

00

19320

00

19320

00

19320

00

19320

00

97200

0

97200

0

97200

0

97200

0

97200

0

97200

0

CVPM11

61949.90625

0 0 0 0 72269.80469

111442.8516

113018.9688

123711.5

124432.6563

113354

89694.58594

CVPM

11UR

44466

66.5

44466

66.5

44466

66.5

44466

66.5

44466

66.5

44466

66.5

22371

42.75

22371

42.75

22371

42.75

22371

42.75

22371

42.75

22371

42.75

CVPM12

183652.1875

0 0 0 100951.26

212993.3438

199541.3281

205104.5156

236187.0781

234728.0625

215662.5156

198682.7813

CVPM

12UR

16713

33.375

16713

33.375

16713

33.375

16713

33.375

16713

33.375

16713

33.375

84085

7.125

84085

7.125

84085

7.125

84085

7.125

84085

7.125

84085

7.125

CVPM13

129923.0781

0 0 0 184286.283

175049.4375

176914.4063

160085.7344

173610

175336.0469

173481.25

151986.875

CVPM

13UR

33043

33.25

33043

33.25

33043

33.25

33043

33.25

33043

33.25

33043

33.25

16624

28.625

16624

28.625

16624

28.625

16624

28.625

16624

28.625

16624

28.625

CVPM14

317542.1563

0 0 0 515095.5313

414172.1875

211476.3594

179930.7031

289838.375

324331.9688

363600

328873.6875

CVPM

14UR

25376

66.75

25376

66.75

25376

66.75

25376

66.75

25376

66.75

25376

66.75

12767

14.25

12767

14.25

12767

14.25

12767

14.25

12767

14.25

12767

14.25

CVPM15

222340.1719

0 0 0 213672.413

192240.0469

182764.3438

159416.4375

174160.2656

175377.7031

175610.5469

181166.1094

CVPM

15UR

25376

66.75

25376

66.75

25376

66.75

25376

66.75

25376

66.75

25376

66.75

12767

14.25

12767

14.25

12767

14.25

12767

14.25

12767

14.25

12767

14.25

154

CVPM

16

68762.

64063

0 0 0 0 13507

9.6406

17738

3.145

20852

1.5625

21459

6.4531

21928

9.6406

20508

6.9219

16103

6.75

CVPM17

272552.125

0 0 0 726948.375

312286.4063

324849

290358.625

298475.6875

298919

297720.9688

289629.2188

CVPM

17UR

39790

00

39790

00

39790

00

39790

00

39790

00

39790

00

20018

57.125

20018

57.125

20018

57.125

20018

57.125

20018

57.125

20018

57.125

CVPM18

244238.625

0 0 0 271524.9375

245463.9844

227670.7656

187839.9531

213139.8906

210861.8594

212715.2188

201026.7344

CVPM

18UR

28673

33.25

28673

33.25

28673

33.25

28673

33.25

28673

33.25

28673

33.25

14425

71.375

14425

71.375

14425

71.375

14425

71.375

14425

71.375

14425

71.375

CVPM19

119007.8125

0 0 0 95711.53125

163159.6563

150527.75

145528.2188

154734.9063

158396.8906

159123.5938

141680.6406

CVPM

19UR

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

15660

00

15660

00

15660

00

15660

00

15660

00

15660

00

CVPM20

335269.8438

0 0 0 352136.5938

372433.9688

308929.375

260270.5625

278588.0313

288832.8125

300831.125

294439.5625

CVPM

20UR

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

15660

00

15660

00

15660

00

15660

00

15660

00

15660

00

CVPM

21

17974

7.7344

0 0 0 12097

8.7813

26162

8.6094

22133

0.4531

20720

4.6563

22126

3.3281

22923

9.8125

23017

0.3438

23225

3.2656

CVPM

21UR

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

31126

66.75

15660

00

15660

00

15660

00

15660

00

15660

00

15660

00

EB 5405000

5405000

5405000

5405000

5405000

5405000

2719285.75

2719285.75

2719285.75

2719285.75

2719285.75

2719285.75

FRES

NO

23766

66.75

23766

66.75

23766

66.75

23766

66.75

23766

66.75

23766

66.75

11957

14.25

11957

14.25

11957

14.25

11957

14.25

11957

14.25

11957

14.25

NAPA 62866

66.5

62866

66.5

62866

66.5

62866

66.5

62866

66.5

62866

66.5

31628

57.25

31628

57.25

31628

57.25

31628

57.25

31628

57.25

31628

57.25

OakFla

t

56810

00

56810

00

56810

00

56810

00

56810

00

56810

00

28581

42.75

28581

42.75

28581

42.75

28581

42.75

28581

42.75

28581

42.75

Reddin

g

25606

67

25606

67

25606

67

25606

67

25606

67

25606

67

12882

86

12882

86

12882

86

12882

86

12882

86

12882

86

SAC 38870

00

38870

00

38870

00

38870

00

38870

00

38870

00

19555

71.375

19555

71.375

19555

71.375

19555

71.375

19555

71.375

19555

71.375

SFPU

C

46153

33.5

46153

33.5

46153

33.5

46153

33.5

46153

33.5

46153

33.5

23220

00

23220

00

23220

00

23220

00

23220

00

23220

00

STOC 32200

00

32200

00

32200

00

32200

00

32200

00

32200

00

16200

00

16200

00

16200

00

16200

00

16200

00

16200

00

StBarb

ara

93456

67

93456

67

93456

67

93456

67

93456

67

93456

67

47018

57

47018

57

47018

57

47018

57

47018

57

47018

57

UD2 25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

UD3 25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

UD4 25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

25606

66.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

12882

85.75

UD5 48070

00

48070

00

48070

00

48070

00

48070

00

48070

00

24184

28.5

24184

28.5

24184

28.5

24184

28.5

24184

28.5

24184

28.5

UD6 62866

66.5

62866

66.5

62866

66.5

62866

66.5

62866

66.5

62866

66.5

31628

57.25

31628

57.25

31628

57.25

31628

57.25

31628

57.25

31628

57.25

UD8 32200

00

32200

00

32200

00

32200

00

32200

00

32200

00

16200

00

16200

00

16200

00

16200

00

16200

00

16200

00

UD9 7179833.5

7179833.5

7179833.5

7179833.5

7179833.5

7179833.5

3612214.25

3612214.25

3612214.25

3612214.25

3612214.25

3612214.25

YUBA 48070

00

48070

00

48070

00

48070

00

48070

00

48070

00

24184

28.5

24184

28.5

24184

28.5

24184

28.5

24184

28.5

24184

28.5

Note: the last month is September.

Table A 8. Quadratic coefficients of benefit functions for demand nodes. The unit is $/Kaf 2, with 1 Kaf =

1.23348 MCM. 1 2 3 4 5 6 7 8 9 10 11 12

AG1 -

7057.8561

0 0 0 0 -

2540.32848

-

2540.32848

-

2004.11852

-

2004.11852

-

1556.71539

-

1796.65816

-

2775.2241

AG2 -

3178.57779

0 0 0 -

289795.918

-

896.796269

-

896.796269

-

485.981772

-

485.981772

-

397.539772

-

516.195808

-

868.75517

AG3 -

3213.59938

0 0 0 -

36283.737

-

240.33766

-

240.33766

-

178.608348

-

178.608348

-

167.30763

-

225.318439

-

834.665997

AG4 - 0 0 0 - - - - - - - -

155

1420.9

5211

43525.

5102

390.57

4351

390.57

4351

247.22

846

247.22

846

225.42

207

292.90

8573

816.15

3937

AG5 -735.15

8865

0 0 0 -10059

1.716

-195.71

4473

-195.71

4473

-145.23

213

-145.23

213

-139.90

1072

-177.52

0232

-348.87

5145

AG6 -470.67

4586

0 0 0 -13506.

8098

-368.60

5878

-368.60

5878

-286.96

3112

-286.96

3112

-266.40

1804

-347.01

1977

-562.91

1219

AG7 -

1938.83192

0 0 0 -

122500

-

565.481396

-

565.481396

-

452.389546

-

452.389546

-

438.510397

-

545.471005

-

1216.86453

AG8 -

2972.82597

0 0 0 -

62962.963

-

761.758718

-

761.758718

-

292.635619

-

292.635619

-

238.086855

-

341.750553

-

777.111804

AG9 -

10612.

3606

0 0 0 -

51535.

7143

-

707.45

9114

-

707.45

9114

-

302.70

2562

-

302.70

2562

-

267.48

0193

-

424.07

9113

-

1418.8

4996

CVPM

10

-

536.46

2769

0 0 0 -

55371.

5078

-

492.65

8875

-

492.65

8875

-

229.90

387

-

229.90

387

-

238.80

5405

-

262.06

4331

-

622.16

0217

CVPM11

-590.86

5723

0 0 0 0 -577.65

155

-577.65

155

-436.95

7825

-436.95

7825

-412.60

2478

-450.56

842

-551.18

6523

CVPM12

-3550.3

4375

0 0 0 -15640.

204

-1179.4

4775

-1179.4

4775

-843.22

4121

-843.22

4121

-777.50

2686

-964.76

0315

-1535.8

429

CVPM13

-975.92

6025

0 0 0 -21565.

554

-530.19

1834

-530.19

1834

-264.48

8098

-264.48

8098

-256.73

7122

-294.28

5431

-553.88

8062

CVPM

14

-

19129.0469

0 0 0 -

16404.3164

-

769.004944

-

769.004944

-

425.857147

-

425.857147

-

509.795624

-

742.040833

-

8654.57031

CVPM

15

-

4307.24854

0 0 0 -

3977.96502

-

677.607727

-

677.607727

-

224.653351

-

224.653351

-

207.326752

-

236.946518

-

1197.88745

CVPM

16

-

1311.8

1348

0 0 0 0 -

1864.3

365

-

1864.3

365

-

1431.8

6487

-

1431.8

6487

-

1393.8

3728

-

1548.7

843

-

1799.0

9229

CVPM

17

-

5658.8

3447

0 0 0 -

48755

7.594

-

2779.1

001

-

2779.1

001

-

1039.6

9519

-

1039.6

9519

-

980.38

3728

-

1174.7

1973

-

1959.6

0242

CVPM18

-4025.0

2686

0 0 0 -9953.2

5977

-507.55

9235

-507.55

9235

-232.84

3079

-232.84

3079

-222.27

8061

-290.20

6024

-973.58

9356

CVPM19

-2021.8

7927

0 0 0 -1845.5

752

-934.60

669

-934.60

669

-459.34

4849

-459.34

4849

-416.54

8889

-450.44

3268

-1522.4

6545

CVPM

20

-

10234.1221

0 0 0 -

27181.0996

-

2429.49219

-

2429.49219

-

1224.34753

-

1224.34753

-

1152.01343

-

1423.85046

-

3328.20435

CVPM

21

-

2906.6582

0 0 0 -

2070.13648

-

1085.80481

-

1085.80481

-

499.871979

-

499.871979

-

474.577301

-

553.773315

-

2106.79663

YUBA -

10609

13.69

-

14928

57.17

-

17061

22.42

-

17563

02.44

-

17272

72.67

-

15714

28.52

-

49080

2.34

-

33297

9.28

-

25962

7.33

-

23664

8.42

-

24175

8.25

-

31318

6.82

SAC -

57610.

36

-

79529.

41

-

98113.

21

-

10706

3.67

-

10174

5.93

-

96709.

58

-

31711.

28

-

23070.

1

-

17907.

92

-

16129.

29

-

16325.

61

-

19250.

12

NAPA -

57787

1.73

-

72214

8.81

-

80039

0.39

-

83333

3.34

-

81713

9.97

-

76031

5.21

-

26563

0.05

-

20788

4.38

-

18260

7.72

-

17151

2.23

-

17802

8.65

-

20551

3.77

STOC -37433

1.55

-50541

5.17

-58091

2.84

-61269

1.44

-58823

5.27

-54054

0.56

-17069

7.01

-12684

9.89

-10723

8.61

-95617.

53

-98928.

28

-11787

8.19

CC -70531

4.03

-89399

1.62

-10286

21.78

-10740

23.03

-10594

10.48

-98941

1.32

-31691

7.65

-25072

4.5

-21641

6.54

-20548

9.1

-21188

2.09

-24217

2.93

EB -

191290.19

-

229044.84

-

242018.53

-

243775.92

-

238457.64

-

229604.3

-

85532.3

-

73810.4

-

66521.98

-

62985.8

-

63824.01

-

70257.61

Reddin

g

-

35626

6.71

-

49481

4.89

-

54441

7.36

-

58137

5.18

-

56371

3.16

-

52392

1.64

-

17101

8.97

-

12458

0.4

-

95238.

11

-

80530.

45

-

83635.

92

-

10746

4.62

156

UD2 -

457221.11

-

634377.99

-

698014.64

-

747203.6

-

722943.76

-

674747.51

-

219881.5

-

160115.06

-

122344.32

-

103389.58

-

107464.61

-

138102.14

UD3 -

19195

40.35

-

27154

47.27

-

30925

92.57

-

31809

52.54

-

31361

50.43

-

28547

01.05

-

88359

7.88

-

60017

9.7

-

47009

1.51

-

42793

0.82

-

43774

5.74

-

56466

6.1

UD4 -

57094

02.1

-

79523

81.18

-

92777

78.29

-

96811

59.56

-

92777

78.29

-

85641

03.16

-

26507

93.63

-

18005

39.24

-

14033

61.38

-

12723

80.99

-

13072

40.71

-

17040

81.68

UD5 -94570

1.34

-13312

10.15

-15200

00

-15655

43.03

-15424

35.4

-14026

84.55

-43693

3.81

-29659

4.14

-23145

0.73

-21100

4.55

-21557

5.03

-27903

8.71

UD6 -10472

54.17

-14462

08.09

-17806

73.17

-19454

32.97

-18468

46.77

-17577

70.65

-57564

0.54

-41911

5.75

-32539

6.83

-29322

3.67

-29656

4.18

-34968

0.17

UD8 -

1327014.28

-

1794871.86

-

2058823.51

-

2170542.7

-

2089552.19

-

1917808.17

-

606060.6

-

449438.19

-

380952.37

-

338983.05

-

350877.19

-

418118.48

UD9 -

1026864.07

-

1387407.44

-

1592687.08

-

1682839.2

-

1613264.51

-

1482977.06

-

468601.46

-

348400.31

-

294358.01

-

262582.38

-

271646.12

-

323544.64

CVPM

10UR

-

50909

0.91

-

68852

4.59

-

78873

2.39

-

83168

3.17

-

80000

0

-

73362

4.45

-

23225

8.06

-

17266

1.87

-

14574

8.99

-

12996

3.9

-

13457

9.44

-

16035

6.35

CVPM

11UR

-

21152

4.43

-

28578

4.68

-

32823

9.95

-

34616

5.32

-

33275

9.61

-

30566

5.35

-

96513.

85

-

71706.

74

-

60634.

57

-

54049.

02

-55909 -66659

CVPM

12UR

-

16840

4.79

-

22743

8.71

-

26139

0.89

-

27577

4.83

-

26472

3.74

-

24343

9.42

-

76895.

94

-

57090.

48

-

48283.

5

-

43044.

72

-

44553.

44

-

53099.

5

CVPM13UR

-22447

9.17

-30309

4.23

-34828

2.83

-36743

3.93

-35298

9.35

-32430

3.99

-10244

8.3

-76108.

07

-64337.

96

-57355.

78

-59345.

96

-70731.

11

CVPM14UR

-73311

18.49

-24491

30.6

-10025

74.59

-27548

89.72

-51437

45.14

-21935

05.63

-78352

4.68

-62713

1.49

-37632

8.8

-32443

0.29

-40904

5.97

-18801

47.69

CVPM

15UR

-

451261.08

-

614670.38

-

758304.7

-

855297.16

-

766203.7

-

709539.12

-

200788.6

-

147767.86

-

120780.88

-

109584.51

-

116038.56

-

139075.63

CVPM

17UR

-

528244.27

-

719334.72

-

887179.49

-

997118.16

-

896373.06

-

827751.2

-

234629.29

-

172625.98

-

141224.49

-

128053.29

-

135668.54

-

162593.98

CVPM

18UR

-

21890

5.47

-

29824

5.61

-

36774

8.28

-

41417

4.97

-

37158

4.7

-

34343

4.34

-

97319.

8

-

71572.

1

-

58583.

96

-

53083.

53

-

56270.

22

-

67417.

76

CVPM

19UR

-

14710

14.49

-

20049

38.27

-

26025

64.1

-

28491

22.81

-

25777

77.78

-

21827

95.7

-

64804

4.69

-

48535

5.65

-

38538

2.06

-

35365

8.54

-

38032

7.87

-

46400

0

CVPM20UR

-61098

5.7

-83282

0.51

-10826

66.67

-11768

11.59

-10698

28.72

-90523

9.69

-26851

8.52

-20173

9.13

-15977

9.61

-14683

5.44

-15782

3.13

-19237

1.48

CVPM21UR

-11666

66.67

-15828

46

-20661

57.76

-22369

14.6

-20350

87.72

-17239

91.51

-51327

4.34

-38410

5.96

-30446

1.94

-27951

8.07

-30051

8.13

-36708

8.61

OakFlat -

91617.

21

-

11451

0.89

-

12797

9.27

-

13003

4.22

-

12443

3.25

-

11846

5.23

-

40651.

74

-

33680.

29

-

29376.

2

-

28172.

23

-

28828.

2

-

32258.

77

SFPUC -

202284.95

-

222591.98

-

232927.07

-

231449.44

-

223584.03

-

226358.34

-

86737.27

-

81865.78

-

77407.74

-

16861.11

-

17867.95

-

21409.58

Bfield -

153997.06

-

209612.82

-

273281.11

-

296226.42

-

269527.9

-

228529.84

-

67725.93

-

50839.22

-

40302.91

-

37000.67

-

39778.73

-

48511.69

StBarba

ra

-

68870

0.56

-

85905

5.67

-

96287

5.2

-

98505

0.51

-

91930

6.18

-

88913

2.02

-

30471

1.91

-

26668

1.25

-

23692

9.06

-

22734

0.54

-

23564

6.63

-

25856

4

Fresno -

69514.

52

-

94714.

33

-

11689

2.91

-

13155

1.03

-

11809

5.24

-

10911

6.51

-

30904.

2

-

22739.

78

-

18603.

53

-

16861.

11

-

17867.

95

-

21409.

58

Note: the last month is September.

Table A 9. Initial storage capacity of groundwater aquifers and surface reservoirs. This is used for the mass

balance constraint of storage nodes at the first time-step. Volumes are in Kaf with 1 Kaf = 1.23348 MCM.

157

Month

Nodes 1 2 3 4 5 6 7 8 9 10 11 12

GW1 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0 1902.0

GW10 22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

22213.0

GW11 10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

10948.

0

GW12 10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

10380.0

GW13 31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

31143.

0

GW14 51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

51075.0

GW15 70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

70494.

0

GW16 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0 6359.0

GW17 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0 7311.0

GW18 40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

40775.

0

GW19 43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

43085.

0

GW2 11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

11843.

0

GW20 22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

22630.

0

GW21 51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

51595.

0

GW3 13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

13345.

0

GW4 10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

10350.

0

GW5 15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

15552.

0

GW6 17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

17948.0

GW7 10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

10025.

0

GW8 22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

22366.0

GW9 17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

17744.

0

GWSC 425.0 425.0 425.0 425.0 425.0 425.0 425.0 425.0 425.0 425.0 425.0 425.0

New

Melones

1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0

San Luis 525.0 525.0 525.0 525.0 525.0 525.0 525.0 525.0 525.0 525.0 525.0 525.0

Millerton 176.0 176.0 176.0 176.0 176.0 176.0 176.0 176.0 176.0 176.0 176.0 176.0

McClure 229.0 229.0 229.0 229.0 229.0 229.0 229.0 229.0 229.0 229.0 229.0 229.0

Whiskeyto

wn

200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0

Shasta 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0 2496.0

Hensley 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0

Eastman 58.0 58.0 58.0 58.0 58.0 58.0 58.0 58.0 58.0 58.0 58.0 58.0

Oroville 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0 2555.0

Folsom 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0 549.0

Don Pedro 373.0 373.0 373.0 373.0 373.0 373.0 373.0 373.0 373.0 373.0 373.0 373.0

SF

aggregate

128.0 128.0 128.0 128.0 128.0 128.0 128.0 128.0 128.0 128.0 128.0 128.0

Black Butte 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2

Camp Far

West

35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0

Indian Valley

306.7 306.7 306.7 306.7 306.7 306.7 306.7 306.7 306.7 306.7 306.7 306.7

158

EBMUD 117.9 117.9 117.9 117.9 117.9 117.9 117.9 117.9 117.9 117.9 117.9 117.9

Hetch

Hetchy

330.6 330.6 330.6 330.6 330.6 330.6 330.6 330.6 330.6 330.6 330.6 330.6

Berryessa 806.3 806.3 806.3 806.3 806.3 806.3 806.3 806.3 806.3 806.3 806.3 806.3

Isabella 281.8 281.8 281.8 281.8 281.8 281.8 281.8 281.8 281.8 281.8 281.8 281.8

Kaweah 77.0 77.0 77.0 77.0 77.0 77.0 77.0 77.0 77.0 77.0 77.0 77.0

Lloyd-

Eleanor

216.6 216.6 216.6 216.6 216.6 216.6 216.6 216.6 216.6 216.6 216.6 216.6

Success 41.9 41.9 41.9 41.9 41.9 41.9 41.9 41.9 41.9 41.9 41.9 41.9

Los

Vaqueros

88.3 88.3 88.3 88.3 88.3 88.3 88.3 88.3 88.3 88.3 88.3 88.3

New

Bullards

Bar

600.0 600.0 600.0 600.0 600.0 600.0 600.0 600.0 600.0 600.0 600.0 600.0

New Hogan 159.0 159.0 159.0 159.0 159.0 159.0 159.0 159.0 159.0 159.0 159.0 159.0

Pine Flat 550.0 550.0 550.0 550.0 550.0 550.0 550.0 550.0 550.0 550.0 550.0 550.0

Pardee 195.0 195.0 195.0 195.0 195.0 195.0 195.0 195.0 195.0 195.0 195.0 195.0

Santa Clara 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0

Turlock 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0

Del Valle 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0

Note: GW=groundwater, SF=San Francisco, and EBMUD= East Bay Municipal Utility District. The last

month is September.

Table A 10. Hydropower plants characteristics.

Nodes Power factor Maximum hydropower generation

capacity (GWhr/month)

Month Unit price

New Melones 0.467 216000 1 26000

Folsom 0.278 143064 2 26000

Oroville 0.588 463860 3 26000

Shasta 0.395 452880 4 20000

Whiskeytown 0.519 129600 5 20000

Hetch Hetchy 0.96 87768 6 20000

Pine Flat 0.3 136800 7 18000

Don Pedro 0.41 146304 8 18000

McClure 0.28 68040 9 18000

New Bullards Bar 1.14 234000 10 20000

11 20000

12 20000

Table A 11. Flow multiplier in links. This is used in mass balance constraint to represent the proportion

(percentage) of flow that is not lost during conveyance. A flow multiplier of 1 denotes no loss of flow in the

link. Link Link Link Link

AG1-C5 1 C64-C63 0.89 D605-D606 1 D98-C36 0.92

AG1-GW1 1 C64-C689 1 D606-C46 0.86 D98-D517 1

AG2-C4 1 C65-C100 1 D606-C50 0.85 FRESNO-D605 1

AG2-GW2 1 C65-C63 0.89 D606-C609 1 GW10-C84 1

AG3-C305 1 C65-C66 1 D608-C90 0.84 GW10-CVPM10UR

1

AG3-GW3 1 C65-C97 1 D608-C91 0.99 GW11-C172 1

AG4-D61 1 C66-CVPM21 1 D608-D731 1 GW11-

CVPM11UR

1

159

AG4-GW4 1 C66-GW21 1 D612-D676 1 GW12-C45 1

AG5-C307 1 C67-C314 1 D615-D640 1 GW12-

CVPM12UR

1

AG5-GW5 1 C67-C34 0.93 D616-C10 1 GW13-C46 1

AG6-GW6 1 C67-C8 1 D616-C42 1 GW13-

CVPM13UR

1

AG7-C8 1 C688-C60 0.88 D619-D691 1 GW14-C91 1

AG7-GW7 1 C688-C62 1 D61-C301 1 GW15-C90 1

AG8-D517 1 C689-C66 0.95 D622-D624 1 GW15-CVPM15UR

1

AG8-GW8 1 C68-AG9 1 D624-C46 0.86 GW16-C50 1

AG9-D509 1 C68-D523 1 D624-C48 1 GW16-FRESNO 1

AG9-GW9 1 C68-GW9 1 D632-D634 1 GW17-C55 1

BANKS-D801 1 C69-C13 1 D634-C46 0.86 GW17-

CVPM17UR

1

Bfield-dum 1 C69-D76b 1 D634-C47 1 GW18-C60 1

C100-

CVPM19

1 C6-AG2 1 D640-D694 1 GW18-

CVPM18UR

1

C100-D850 1 C6-C1 1 D642-D643 1 GW19-C100 1

C100-GW19 1 C70-C71 1 D643-D645 1 GW19-

CVPM19UR

1

C103-outflow2 1 C71-CC 1 D645-C45 0.83 GW19-D851D852 1

C10-C84 0.9 C72-C46 0.86 D645-C46 0.86 GW1-C3 1

C11-C302 0.95 C73-C100 0.99 D645-D646 1 GW1-Redding 1

C11-C6 0.93 C73-D859 1 D646-D647 1 GW20-C63 1

C12-C13 1 C73-SRBLV 1 D647-D649 1 GW20-

CVPM20UR

1

C12-D76b 1 C74-C100 0.99 D649-C45 0.83 GW21-Bfield 1

C13-C302 1 C74-C63 0.89 D649-C46 0.86 GW21-C66 1

C14-AG4 1 C74-C66 0.95 D649-D695 1 GW21-C98 1

C14-D30 1 C74-D752 1 D653a-D653b 1 GW21-

CVPM21UR

1

C14-GW4 1 C75-C90 0.84 D653b-D672 1 GW2-dum4 1

C14-UD4 1 C75-D845 1 D662-C172 0.83 GW3-C303 1

C15-C301 1 C76-C55 0.81 D662-C45 0.83 GW3-UD3 1

C15-C313 1 C76-C688 1 D662-D663 1 GW4-C14 1

C172-

CVPM11

1 C78-C79 1 D663-D664 1 GW5-C26 1

C172-D689 1 C79-SRASF 1 D664-C172 0.83 GW5-UD5 1

C172-GW11 1 C80-C26 0.96 D664-C45 0.83 GW5-YUBA 1

C17-AG6 1 C84-CVPM10 1 D664-D683 1 GW6-C17 1

C17-C314 1 C84-D731 1 D66-D30 1 GW6-NAPA 1

C17-GW6 1 C84-GW10 1 D66-dum5 0.95 GW6-UD6 1

C18-D511 1 C85-C84 0.9 D670-SRTR 1 GW7-C34 1

C1-C4 1 C86-dum2 1 D672-C172 0.83 GW7-SAC 1

C1-C6 0.93 C87-D77 1 D672-D675 1 GW8-C36 1

C201-SREB 1 C88-C78 1 D675-D676 1 GW8-STOC 1

C26-AG5 1 C89-C56 1 D676-D616 1 GW8-UD8 1

C26-C311 1 C8-D503 1 D683-D687 1 GW9-C68 1

C26-GW5 1 C8-SAC 1 D687-D688 1 GW9-UD9 1

C2-D74 1 C90-C75 1 D688-D689 1 GWSC-OakFlat 1

C301-D43 1 C90-CVPM15 1 D689-C10 1 KernRef-SRTLB 0.001

160

C302-dum5 1 C90-GW15 1 D689-C172 0.83 LV-SRLV 1

C303-AG3 1 C91-C92 1 D689-D612 1 MS-C71 1

C303-C305 0.95 C91-CVPM14 1 D691-D694 1 Mendota-D615 0.25

C303-GW3 1 C91-GW14 1 D692-D693 1 OLDR-C310 1

C303-SWR 0.75 C92-C91 0.99 D693-D619 1 OakFlat-dummy1 1

C305-C15 1 C92-D745 1 D694-C46 0.86 PMPCC-C70 1

C305-C303 1 C95-D752 1 D694-D695 1 Pixley-SRTLB 0.001

C306-C18 1 C95-KernRef 0.87 D695-D697 1 Redding-GW1 1

C307-C311 1 C97-Bfield 1 D697-D698 1 SAC-C8 1

C307-D42 1 C97-C73 1 D698-D699 1 SER-D42 0.16

C309-D59 1 C98-C66 0.95 D699-C10 1 SR10-D670 1

C309-OLDR 1 C98-D855D857 1 D699-C45 0.83 SR12-D714 1

C30-C84 0.9 C98-GW21 1 D699-D683 1 SR12-D816 1

C310-C70 1 C9-C12 1 D701-C30 1 SR15-D892 1

C310-LV 1 C9-C6 0.93 D701-D703 1 SR18-C49 1

C311-SER 1 CVPM10UR-GW10

1 D703-D710 1 SR18-C72 1

C313-C306 1 CVPM10-D612 1 D710-D712 1 SR18-D605 1

C313-C314 1 CVPM10-GW10 1 D710-D814 1 SR20-D642 1

C314-C17 0.93 CVPM11UR-

GW11

1 D712-D722 1 SR3-D5 1

C315-C316 1 CVPM11-GW11 1 D714-C316 1 SR3-dum1 0.97

C315-OakFlat 1 CVPM12UR-

GW12

1 D714-OakFlat 1 SR4-D5 1

C316-GWSC 1 CVPM12-D697 1 D722-C30 1 SR52-D622 1

C318-D692 1 CVPM12-GW12 1 D722-D723 1 SR53-D632 1

C31-D37 1 CVPM13UR-

GW13

1 D723-D724 1 SR6-C31 1

C32-D42 1 CVPM13-D695 1 D724-D608 1 SR6-C80 0.96

C34-AG7 1 CVPM13-GW13 1 D731-C46 0.86 SR6-YUBA 1

C34-C67 1 CVPM14UR-GW14

1 D731-C84 0.9 SR81-D662 1

C34-GW7 1 CVPM14-GW14 1 D731-D732 1 SR8-C39 1

C36-AG8 1 CVPM15UR-

GW15

1 D732-D733 1 SR8-C8 1

C36-C37 1 CVPM15-C59 1 D732-Mendota 0.85 SR8-SAC 1

C36-GW8 1 CVPM15-GW15 1 D733-D619 1 SRASF-OakFlat 1

C37-C36 0.92 CVPM16-C59 1 D73-C2 1 SRASF-SFPUC 1

C37-D98 1 CVPM16-GW16 1 D742-C85 1 SRBBL-C9 1

C39-dum6 1 CVPM17UR-GW17

1 D742-D743 1 SRCFW-C26 1

C3-AG1 1 CVPM17-C59 1 D743-D744 1 SRCFW-C34 0.93

C3-D74 1 CVPM17-GW17 1 D744-C85 1 SRCFW-D37 1

C3-GW1 1 CVPM18UR-

GW18

1 D744-C92 1 SRCLI-C17 0.93

C42-D521 1 CVPM18-GW18 1 D745-D747 1 SREB-C310 1

C44-C88 1 CVPM19UR-

GW19

1 D747-D749 1 SREB-EB 1

C45-CVPM12 1 CVPM19-GW19 1 D749-D750 1 SRHHR-C44 1

C45-D699 1 CVPM20UR-GW20

1 D74-C3 0.97 SRHHR-SR81 1

C45-GW12 1 CVPM20-C73 1 D74-C5 1 SRLB-C18 1

161

C46-CVPM13 1 CVPM20-GW20 1 D750-

CVPM14UR

1 SRLB-NAPA 0.94

C46-D694 1 CVPM21UR-GW21

1 D750-D751 1 SRLI-C65 1

C46-GW13 1 CVPM21-GW21 1 D751-C75 1 SRLK-C89 1

C47-D692 1 D16-C172 0.83 D752-C74 1 SRLL-C44 1

C48-C47 1 D16-D653a 1 D752-D855D857 1 SRLL-SR81 1

C49-C50 0.85 D30-C14 1 D76b-D66 1 SRLS-C57 1

C49-C76 1 D30-D31 1 D77-C11 1 SRLV-C310 1

C4-C69 1 D31-C26 0.96 D77-C6 0.93 SRNBB-C26 0.96

C50-C53 1 D31-D61 1 D77-dum2 1 SRNBB-C31 1

C50-CVPM16 1 D37-C32 1 D801-D803 1 SRNBB-SRCFW 1

C50-GW16 1 D42-C34 0.93 D801-D891 1 SRNHL-C42 1

C51-C53 1 D42-D43 1 D803-C84 0.9 SRNHL-STOC 1

C52-C59 1 D43-C306 1 D803-D804 1 SRPF-C51 1

C52-C90 0.84 D43-C67 1 D804-D814 1 SRPR-C39 1

C53-C50 0.85 D503-D511 1 D814-D816 1 SRPR-D98 1

C53-C54 1 D507-C68 1 D814-SR12 1 SRSCV-C315 1

C53-C55 0.81 D507-D509 1 D816-D712 1 SRTR-D16 1

C54-C52 1 D509-DELTA 1 D816-D818 1 STOC-C42 1

C54-D608 1 D511-D507 1 D818-D820 1 SWR-C15 1

C55-C53 1 D511-D513 1 D820-D742 1 TRACY-D701 1

C55-CVPM17 1 D511-NAPA 1 D845-D847 1 UD2-GW2 1

C55-GW17 1 D513-D515 1 D845-D850 1 UD3-GW3 1

C56-C59 1 D513-D521 1 D847-C100 1 UD4-GW4 1

C56-C60 0.88 D515-C68 1 D847-D848 1 UD5-GW5 1

C56-C90 0.84 D515-D522 1 D848-D849 1 UD6-GW6 1

C57-C58 1 D517-D515 1 D849-StBarbara 1 UD8-GW8 1

C58-C59 1 D521-C68 1 D850-C100 0.99 UD9-GW9 1

C58-C60 0.88 D521-D522 1 D850-D851D852 1 WC-C201 1

C59-SRTLB 1 D522-D523 1 D851D852-D853 1 YUBA-D37 1

C5-C87 1 D523-C68 0 D853-C95 1 dum1-C3 0.97

C609-C48 1 D523-D525 1 D855D857-C98 1 dum1-D5 0.97

C609-D608 1 D525-D528 1 D855D857-D859 1 dum1-D73 1

C60-C688 1 D525-D550 1 D859-D860 1 dum2-C1 1

C60-CVPM18 1 D528-D509 1 D860-D863 1 dum4-C6 1

C60-GW18 1 D528-MS 1 D863-C103 1 dum4-GW2 1

C60-Pixley 0.5 D550-C309 1 D863-C98 1 dum4-UD2 1

C62-C100 0.99 D550-PMPCC 1 D891-D892 1 dum5-C303 1

C62-C64 1 D59-BANKS 1 D891-SR15 1 dum6-WC 1

C63-C65 1 D59-TRACY 1 D892-D896 1 dum-GW21 1

C63-CVPM20 1 D5-D73 1 D896-C316 1 dummy1-C316 1

C63-GW20 1 D5-Redding 1 D896-OakFlat 1 dummy1-OakFlat 1

Table A 12. Maximum allowable flow in links. This is used in maximum flow constraint. Volumes are in Kaf

with 1 Kaf = 1.23348 MCM. Link Link Link Link

162

AG1-C5 1000000

0

C64-C63 69.4 D605-D606 1000000

0

D98-C36 26.4

AG1-GW1 10000000

C64-C689 10000000

D606-C46 2.2 D98-D517 1000000

AG2-C4 1000000

0

C65-C100 84.7 D606-C50 1000000

0

FRESNO-D605 1000000

0

AG2-GW2 10000000

C65-C63 135.5 D606-C609 10000000

GW10-C84 197.88

AG3-C305 1000000

0

C65-C66 135.5 D608-C90 17 GW10-

CVPM10UR

1000000

0

AG3-GW3 10000000

C65-C97 10000000

D608-C91 4.9 GW11-C172 52.22

AG4-D61 1000000

0

C66-CVPM21 1000000

0

D608-D731 1000000

0

GW11-

CVPM11UR

1000000

0

AG4-GW4 10000000

C66-GW21 10000000

D612-D676 10000000

GW12-C45 80.56

AG5-C307 1000000

0

C67-C314 1000000

0

D615-D640 1000000

0

GW12-

CVPM12UR

1000000

0

AG5-GW5 1000000

0

C67-C34 49.1 D616-C10 1000000

0

GW13-C46 290.96

AG6-GW6 1000000

0

C67-C8 1000000

0

D616-C42 1000000

0

GW13-

CVPM13UR

1000000

0

AG7-C8 10000000

C688-C60 172.3 D619-D691 10000000

GW14-C91 332.85

AG7-GW7 1000000

0

C688-C62 1000000

0

D61-C301 1000000

0

GW15-C90 407.88

AG8-D517 1000000

0

C689-C66 28.8 D622-D624 1000000

0

GW15-

CVPM15UR

8.63

AG8-GW8 1000000

0

C68-AG9 1000000 D624-C46 57.2 GW16-C50 60.76

AG9-D509 1000000

0

C68-D523 73.5 D624-C48 1000000

0

GW16-FRESNO 1000000

0

AG9-GW9 1000000

0

C68-GW9 1000000

0

D632-D634 1000000

0

GW17-C55 152.39

BANKS-D801 1000000

0

C69-C13 1000000

0

D634-C46 42.9 GW17-

CVPM17UR

1000000

0

Bfield-dum 1000000

0

C69-D76b 1000000

0

D634-C47 1000000

0

GW18-C60 384.95

C100-

CVPM19

1000000

0

C6-AG2 1000000

0

D640-D694 1000000

0

GW18-

CVPM18UR

1000000

0

C100-D850 1000000

0

C6-C1 1000000 D642-D643 1000000

0

GW19-C100 171.1

C100-GW19 1000000

0

C70-C71 1000000

0

D643-D645 1000000

0

GW19-

CVPM19UR

1000000

0

C103-outflow2 1000000

0

C71-CC 1000000

0

D645-C45 5.4 GW19-D851D852 1000000

0

C10-C84 40.8 C72-C46 89.5 D645-C46 111.4 GW1-C3 20.76

C11-C302 10000000

C73-C100 10000000

D645-D646 10000000

GW1-Redding 10000000

C11-C6 1000000

0

C73-D859 1000000

0

D646-D647 1000000

0

GW20-C63 108.1

C12-C13 10000000

C73-SRBLV 10000000

D647-D649 10000000

GW20-CVPM20UR

10000000

C12-D76b 1000000

0

C74-C100 1000000

0

D649-C45 12.2 GW21-Bfield 33

C13-C302 1000000

0

C74-C63 1000000

0

D649-C46 4.3 GW21-C66 228.31

C14-AG4 1000000

0

C74-C66 30.6 D649-D695 1000000

0

GW21-C98 1000000

0

C14-D30 1000000 C74-D752 1000000

0

D653a-D653b 1000000

0

GW21-

CVPM21UR

1000000

0

C14-GW4 1000000

0

C75-C90 74.16 D653b-D672 1000000

0

GW2-dum4 153.23

C14-UD4 1000000

0

C75-D845 1000000

0

D662-C172 66 GW3-C303 170.98

C15-C301 1000000

0

C76-C55 12.9 D662-C45 107.1 GW3-UD3 1000000

0

C15-C313 1000000

0

C76-C688 1000000

0

D662-D663 1000000

0

GW4-C14 110.47

C172-

CVPM11

1000000

0

C78-C79 1000000

0

D663-D664 1000000

0

GW5-C26 225.65

C172-D689 1000000

0

C79-SRASF 1000000

0

D664-C172 2.5 GW5-UD5 1000000

163

C172-GW11 1000000

0

C80-C26 1000000

0

D664-C45 2 GW5-YUBA 1000000

0

C17-AG6 1000000 C84-CVPM10 10000000

D664-D683 10000000

GW6-C17 148.06

C17-C314 1000000

0

C84-D731 1000000

0

D66-D30 1000000

0

GW6-NAPA 1000000

C17-GW6 10000000

C84-GW10 10000000

D66-dum5 10000000

GW6-UD6 10000000

C18-D511 1000000

0

C85-C84 27.5 D670-SRTR 1000000

0

GW7-C34 96.02

C1-C4 10000000

C86-dum2 10000000

D672-C172 10 GW7-SAC 31.1

C1-C6 41.4 C87-D77 1000000

0

D672-D675 1000000

0

GW8-C36 208.38

C201-SREB 10000000

C88-C78 10000000

D675-D676 10000000

GW8-STOC 10

C26-AG5 1000000 C89-C56 1000000

0

D676-D616 1000000

0

GW8-UD8 1000000

C26-C311 1000000

0

C8-D503 1000000

0

D683-D687 1000000

0

GW9-C68 73.77

C26-GW5 1000000

0

C8-SAC 10.3 D687-D688 1000000

0

GW9-UD9 1000000

C2-D74 10000000

C90-C75 10000000

D688-D689 10000000

GWSC-OakFlat 30.5

C301-D43 1000000

0

C90-CVPM15 1000000

0

D689-C10 1000000

0

KernRef-SRTLB 1000000

0

C302-dum5 1000000

0

C90-GW15 1000000

0

D689-C172 3 LV-SRLV 1000000

0

C303-AG3 1000000

0

C91-C92 1000000

0

D689-D612 1000000

0

MS-C71 1000000

0

C303-C305 1000000 C91-CVPM14 391.7 D691-D694 1000000

0

Mendota-D615 1000000

0

C303-GW3 1000000

0

C91-GW14 1000000

0

D692-D693 1000000

0

OLDR-C310 1000000

0

C303-SWR 1000000

0

C92-C91 232.8 D693-D619 1000000

0

OakFlat-dummy1 1000000

0

C305-C15 1000000

0

C92-D745 1000000

0

D694-C46 0.5 PMPCC-C70 1000000

0

C305-C303 361.1 C95-D752 1000000

0

D694-D695 1000000

0

Pixley-SRTLB 1000000

0

C306-C18 1000000

0

C95-KernRef 1000000

0

D695-D697 1000000

0

Redding-GW1 1000000

0

C307-C311 1000000

0

C97-Bfield 36.37 D697-D698 1000000

0

SAC-C8 1000000

0

C307-D42 1000000

0

C97-C73 1000000

0

D698-D699 1000000

0

SER-D42 1000000

0

C309-D59 1000000

0

C98-C66 98.13 D699-C10 1000000

0

SR10-D670 1000000

0

C309-OLDR 1000000

0

C98-D855D857 1000000

0

D699-C45 4.5 SR12-D714 1000000

0

C30-C84 142.5 C98-GW21 1000000

0

D699-D683 1000000

0

SR12-D816 1000000

0

C310-C70 10000000

C9-C12 10000000

D701-C30 10000000

SR15-D892 10000000

C310-LV 1000000

0

C9-C6 1000000

0

D701-D703 1000000

0

SR18-C49 1000000

0

C311-SER 10000000

CVPM10UR-GW10

10000000

D703-D710 10000000

SR18-C72 81.4

C313-C306 1000000

0

CVPM10-D612 1000000

0

D710-D712 1000000

0

SR18-D605 1000000

0

C313-C314 10000000

CVPM10-GW10 10000000

D710-D814 10000000

SR20-D642 10000000

C314-C17 1000000

0

CVPM11UR-

GW11

1000000

0

D712-D722 1000000

0

SR3-D5 1000000

0

C315-C316 10000000

CVPM11-GW11 10000000

D714-C316 10000000

SR3-dum1 10000000

C315-OakFlat 75.15 CVPM12UR-

GW12

1000000

0

D714-OakFlat 75.15 SR4-D5 1000000

0

C316-GWSC 20 CVPM12-D697 10000000

D722-C30 10000000

SR52-D622 10000000

C318-D692 1000000

0

CVPM12-GW12 1000000

0

D722-D723 1000000

0

SR53-D632 1000000

0

C31-D37 1000 CVPM13UR- 1000000 D723-D724 1000000 SR6-C31 1000000

164

GW13 0 0 0

C32-D42 1000000

0

CVPM13-D695 1000000

0

D724-D608 1000000

0

SR6-C80 1000000

0

C34-AG7 10000000

CVPM13-GW13 10000000

D731-C46 10.3 SR6-YUBA 10000000

C34-C67 1000000 CVPM14UR-

GW14

1000000

0

D731-C84 118.1 SR81-D662 1000000

0

C34-GW7 10000000

CVPM14-GW14 10000000

D731-D732 10000000

SR8-C39 10000000

C36-AG8 1000000 CVPM15UR-

GW15

1000000

0

D732-D733 1000000

0

SR8-C8 1000000

0

C36-C37 1000000 CVPM15-C59 10000000

D732-Mendota 10000000

SR8-SAC 64.8

C36-GW8 1000000

0

CVPM15-GW15 1000000

0

D733-D619 1000000

0

SRASF-OakFlat 13.5

C37-C36 13.4 CVPM16-C59 10000000

D73-C2 10000000

SRASF-SFPUC 10000000

C37-D98 1000000

0

CVPM16-GW16 1000000

0

D742-C85 1000000

0

SRBBL-C9 1000000

0

C39-dum6 10000000

CVPM17UR-GW17

10000000

D742-D743 10000000

SRCFW-C26 10000000

C3-AG1 1000000

0

CVPM17-C59 1000000

0

D743-D744 1000000

0

SRCFW-C34 1000

C3-D74 1000000 CVPM17-GW17 10000000

D744-C85 10000000

SRCFW-D37 10000000

C3-GW1 1000000

0

CVPM18UR-

GW18

1000000

0

D744-C92 1000000

0

SRCLI-C17 1000000

0

C42-D521 10000000

CVPM18-GW18 10000000

D745-D747 10000000

SREB-C310 10000000

C44-C88 1000000

0

CVPM19UR-

GW19

1000000

0

D747-D749 1000000

0

SREB-EB 1000000

0

C45-CVPM12 10000000

CVPM19-GW19 10000000

D749-D750 10000000

SRHHR-C44 10000000

C45-D699 1000000

0

CVPM20UR-

GW20

1000000

0

D74-C3 32.6 SRHHR-SR81 1000000

0

C45-GW12 1000000

0

CVPM20-C73 1000000

0

D74-C5 1000000

0

SRLB-C18 1000000

0

C46-CVPM13 1000000

0

CVPM20-GW20 1000000

0

D750-

CVPM14UR

3.42 SRLB-NAPA 1000000

0

C46-D694 10000000

CVPM21UR-GW21

10000000

D750-D751 10000000

SRLI-C65 10000000

C46-GW13 1000000

0

CVPM21-GW21 1000000

0

D751-C75 1000000

0

SRLK-C89 1000000

0

C47-D692 10000000

D16-C172 111.3 D752-C74 30.62 SRLL-C44 10000000

C48-C47 1000000

0

D16-D653a 1000000

0

D752-D855D857 1000000

0

SRLL-SR81 1000000

0

C49-C50 6.8 D30-C14 10000000

D76b-D66 10000000

SRLS-C57 10000000

C49-C76 1000000

0

D30-D31 1000000

0

D77-C11 1000000

0

SRLV-C310 1000000

0

C4-C69 10000000

D31-C26 361.1 D77-C6 10000000

SRNBB-C26 10000000

C50-C53 1000000

0

D31-D61 1000000

0

D77-dum2 1000000

0

SRNBB-C31 1000000

0

C50-CVPM16 1000000

0

D37-C32 1000000

0

D801-D803 1000000

0

SRNBB-SRCFW 1000000

0

C50-GW16 1000000

0

D42-C34 33 D801-D891 1000000

0

SRNHL-C42 1000000

0

C51-C53 1000000

0

D42-D43 1000000

0

D803-C84 1.2 SRNHL-STOC 1000000

0

C52-C59 1000000

0

D43-C306 1000000

0

D803-D804 1000000

0

SRPF-C51 1000000

0

C52-C90 461 D43-C67 1000000

0

D804-D814 1000000

0

SRPR-C39 1000000

0

C53-C50 130.2 D503-D511 1000000

0

D814-D816 1000000

0

SRPR-D98 1000000

0

C53-C54 1000000

0

D507-C68 1000000

0

D814-SR12 1000000

0

SRSCV-C315 1000000

0

C53-C55 217.4 D507-D509 1000000

0

D816-D712 1000000

0

SRTR-D16 1000000

0

C54-C52 1000000

0

D509-DELTA 1000000

0

D816-D818 1000000

0

STOC-C42 1000000

0

165

C54-D608 1000000

0

D511-D507 1000000

0

D818-D820 1000000

0

SWR-C15 1000000

0

C55-C53 10000000

D511-D513 900 D820-D742 10000000

TRACY-D701 10000000

C55-CVPM17 1000000

0

D511-NAPA 1000000

0

D845-D847 1000000

0

UD2-GW2 1000000

0

C55-GW17 10000000

D513-D515 10000000

D845-D850 10000000

UD3-GW3 10000000

C56-C59 1000000

0

D513-D521 1000000

0

D847-C100 25.16 UD4-GW4 1000000

0

C56-C60 179.6 D515-C68 115.9 D847-D848 10000000

UD5-GW5 10000000

C56-C90 29.7 D515-D522 1000000

0

D848-D849 1000000

0

UD6-GW6 1000000

0

C57-C58 10000000

D517-D515 10000000

D849-StBarbara 10000000

UD8-GW8 10000000

C58-C59 1000000

0

D521-C68 33.4 D850-C100 237.42 UD9-GW9 1000000

0

C58-C60 23.1 D521-D522 1000000

0

D850-D851D852 1000000

0

WC-C201 1000000

0

C59-SRTLB 1000000

0

D522-D523 1000000

0

D851D852-D853 1000000

0

YUBA-D37 1000000

0

C5-C87 10000000

D523-C68 0 D853-C95 10000000

dum1-C3 10000000

C609-C48 1000000

0

D523-D525 1000000

0

D855D857-C98 88.6 dum1-D5 1000000

0

C609-D608 1000000

0

D525-D528 1000000

0

D855D857-D859 1000000

0

dum1-D73 1000000

0

C60-C688 1000000

0

D525-D550 1000000

0

D859-D860 1000000

0

dum2-C1 1000000

0

C60-CVPM18 1000000

0

D528-D509 1000000

0

D860-D863 1000000

0

dum4-C6 1000000

0

C60-GW18 1000000

0

D528-MS 1000000

0

D863-C103 1000000

0

dum4-GW2 1000000

0

C60-Pixley 1000000

0

D550-C309 1000000

0

D863-C98 14.6 dum4-UD2 1000000

0

C62-C100 3.5 D550-PMPCC 1000000

0

D891-D892 1000000

0

dum5-C303 1000000

0

C62-C64 1000000

0

D59-BANKS 1000000

0

D891-SR15 1000000

0

dum6-WC 1000000

0

C63-C65 1000000

0

D59-TRACY 1000000

0

D892-D896 1000000

0

dum-GW21 1000000

0

C63-CVPM20 1000000

0

D5-D73 1000000

0

D896-C316 1000000

0

dummy1-C316 6

C63-GW20 1000000

0

D5-Redding 1000000

0

D896-OakFlat 75.15 dummy1-OakFlat 1.33

Table A 13. Minimum allowable flow. This is used in minimum flow constraint to reflect the environmental

flow requirements in some links. Volumes are in Kaf with 1 Kaf = 1.23348 MCM. Link Link Link Link

AG1-C5 0 C64-C63 0 D605-D606 0 D98-C36 0

AG1-GW1 0 C64-C689 0 D606-C46 0 D98-D517 0

AG2-C4 0 C65-C100 0 D606-C50 0 FRESNO-D605 0

AG2-GW2 0 C65-C63 0 D606-C609 0 GW10-C84 0

AG3-C305 0 C65-C66 0 D608-C90 0 GW10-CVPM10UR 0

AG3-GW3 0 C65-C97 0 D608-C91 0 GW11-C172 0

AG4-D61 0 C66-CVPM21 0 D608-D731 0 GW11-CVPM11UR 0

AG4-GW4 0 C66-GW21 0 D612-D676 0 GW12-C45 0

AG5-C307 0 C67-C314 0 D615-D640 0 GW12-CVPM12UR 0

AG5-GW5 0 C67-C34 0 D616-C10 0 GW13-C46 0

AG6-GW6 0 C67-C8 0 D616-C42 0 GW13-CVPM13UR 0

AG7-C8 0 C688-C60 0 D619-D691 0 GW14-C91 0

AG7-GW7 0 C688-C62 0 D61-C301 241.5 GW15-C90 0

166

AG8-D517 0 C689-C66 0 D622-D624 0 GW15-CVPM15UR 0

AG8-GW8 0 C68-AG9 0 D624-C46 0 GW16-C50 0

AG9-D509 0 C68-D523 0 D624-C48 0 GW16-FRESNO 0

AG9-GW9 0 C68-GW9 0 D632-D634 0 GW17-C55 0

BANKS-D801 0 C69-C13 0 D634-C46 0 GW17-CVPM17UR 0

Bfield-dum 0 C69-D76b 0 D634-C47 0 GW18-C60 0

C100-CVPM19 0 C6-AG2 0 D640-D694 0 GW18-CVPM18UR 0

C100-D850 0 C6-C1 0 D642-D643 0 GW19-C100 0

C100-GW19 0 C70-C71 0 D643-D645 0 GW19-CVPM19UR 0

C103-outflow2 0 C71-CC 0 D645-C45 0 GW19-D851D852 0

C10-C84 0 C72-C46 0 D645-C46 0 GW1-C3 0

C11-C302 0 C73-C100 0 D645-D646 0 GW1-Redding 0

C11-C6 0 C73-D859 0 D646-D647 0 GW20-C63 0

C12-C13 0 C73-SRBLV 0 D647-D649 0 GW20-CVPM20UR 0

C12-D76b 0 C74-C100 0 D649-C45 0 GW21-Bfield 0

C13-C302 0 C74-C63 0 D649-C46 0 GW21-C66 0

C14-AG4 0 C74-C66 0 D649-D695 0.97 GW21-C98 0

C14-D30 0 C74-D752 0 D653a-D653b 3.925 GW21-CVPM21UR 0

C14-GW4 0 C75-C90 0 D653b-D672 0 GW2-dum4 0

C14-UD4 0 C75-D845 0 D662-C172 0 GW3-C303 0

C15-C301 0 C76-C55 0 D662-C45 0 GW3-UD3 0

C15-C313 0 C76-C688 0 D662-D663 0.604 GW4-C14 0

C172-CVPM11 0 C78-C79 0 D663-D664 0 GW5-C26 0

C172-D689 0 C79-SRASF 0 D664-C172 0 GW5-UD5 0

C172-GW11 0 C80-C26 0 D664-C45 0 GW5-YUBA 0

C17-AG6 0 C84-CVPM10 0 D664-D683 0 GW6-C17 0

C17-C314 0 C84-D731 0 D66-D30 0 GW6-NAPA 0

C17-GW6 0 C84-GW10 0 D66-dum5 0 GW6-UD6 0

C18-D511 0 C85-C84 0 D670-SRTR 0 GW7-C34 0

C1-C4 0 C86-dum2 0 D672-C172 0 GW7-SAC 0

C1-C6 0 C87-D77 0 D672-D675 0 GW8-C36 0

C201-SREB 0 C88-C78 0 D675-D676 0 GW8-STOC 0

C26-AG5 0 C89-C56 0 D676-D616 0 GW8-UD8 0

C26-C311 0 C8-D503 0 D683-D687 0 GW9-C68 0

C26-GW5 0 C8-SAC 0 D687-D688 0 GW9-UD9 0

C2-D74 0 C90-C75 0 D688-D689 0 GWSC-OakFlat 0

C301-D43 0 C90-CVPM15 0 D689-C10 0 KernRef-SRTLB 0

C302-dum5 0 C90-GW15 0 D689-C172 0 LV-SRLV 0

C303-AG3 0 C91-C92 0 D689-D612 0 MS-C71 0

C303-C305 0 C91-CVPM14 0 D691-D694 0 Mendota-D615 0

C303-GW3 0 C91-GW14 0 D692-D693 0 OLDR-C310 0

C303-SWR 0 C92-C91 0 D693-D619 0 OakFlat-dummy1 0

C305-C15 0 C92-D745 0 D694-C46 0 PMPCC-C70 0

C305-C303 0 C95-D752 0 D694-D695 0 Pixley-SRTLB 0

167

C306-C18 0 C95-KernRef 0 D695-D697 0 Redding-GW1 0

C307-C311 0 C97-Bfield 0 D697-D698 0 SAC-C8 0

C307-D42 0 C97-C73 0 D698-D699 0 SER-D42 0

C309-D59 0 C98-C66 0 D699-C10 0 SR10-D670 0

C309-OLDR 0 C98-D855D857 0 D699-C45 0 SR12-D714 0

C30-C84 0 C98-GW21 0 D699-D683 0 SR12-D816 0

C310-C70 0 C9-C12 0 D701-C30 0 SR15-D892 0

C310-LV 0 C9-C6 0 D701-D703 0 SR18-C49 0

C311-SER 0 CVPM10UR-GW10 0 D703-D710 0 SR18-C72 0

C313-C306 0 CVPM10-D612 0 D710-D712 0 SR18-D605 0

C313-C314 0 CVPM10-GW10 0 D710-D814 0 SR20-D642 0

C314-C17 0 CVPM11UR-GW11 0 D712-D722 0 SR3-D5 0

C315-C316 0 CVPM11-GW11 0 D714-C316 0 SR3-dum1 0

C315-OakFlat 0 CVPM12UR-GW12 0 D714-OakFlat 0 SR4-D5 0

C316-GWSC 0 CVPM12-D697 0 D722-C30 0 SR52-D622 0

C318-D692 0 CVPM12-GW12 0 D722-D723 0 SR53-D632 0

C31-D37 60.4 CVPM13UR-GW13 0 D723-D724 0 SR6-C31 0

C32-D42 0 CVPM13-D695 0 D724-D608 0 SR6-C80 0

C34-AG7 0 CVPM13-GW13 0 D731-C46 0 SR6-YUBA 0

C34-C67 0 CVPM14UR-GW14 0 D731-C84 0 SR81-D662 0

C34-GW7 0 CVPM14-GW14 0 D731-D732 0 SR8-C39 0

C36-AG8 0 CVPM15UR-GW15 0 D732-D733 0 SR8-C8 11.35266

C36-C37 0 CVPM15-C59 0 D732-Mendota 0 SR8-SAC 0

C36-GW8 0 CVPM15-GW15 0 D733-D619 0 SRASF-OakFlat 0

C37-C36 0 CVPM16-C59 0 D73-C2 0 SRASF-SFPUC 0

C37-D98 0 CVPM16-GW16 0 D742-C85 0 SRBBL-C9 0

C39-dum6 0 CVPM17UR-GW17 0 D742-D743 0 SRCFW-C26 0

C3-AG1 0 CVPM17-C59 0 D743-D744 0 SRCFW-C34 0

C3-D74 0 CVPM17-GW17 0 D744-C85 0 SRCFW-D37 0

C3-GW1 0 CVPM18UR-GW18 0 D744-C92 0 SRCLI-C17 0

C42-D521 0 CVPM18-GW18 0 D745-D747 0 SREB-C310 0

C44-C88 0 CVPM19UR-GW19 0 D747-D749 0 SREB-EB 0

C45-CVPM12 0 CVPM19-GW19 0 D749-D750 0 SRHHR-C44 0

C45-D699 0 CVPM20UR-GW20 0 D74-C3 0 SRHHR-SR81 0

C45-GW12 0 CVPM20-C73 0 D74-C5 0 SRLB-C18 0

C46-CVPM13 0 CVPM20-GW20 0 D750-CVPM14UR 0 SRLB-NAPA 0

C46-D694 0 CVPM21UR-GW21 0 D750-D751 0 SRLI-C65 0

C46-GW13 0 CVPM21-GW21 0 D751-C75 0 SRLK-C89 0

C47-D692 0 D16-C172 0 D752-C74 0 SRLL-C44 0

C48-C47 0 D16-D653a 0 D752-D855D857 0 SRLL-SR81 0

C49-C50 0 D30-C14 0 D76b-D66 0 SRLS-C57 0

C49-C76 0 D30-D31 0 D77-C11 0 SRLV-C310 0

C4-C69 0 D31-C26 0 D77-C6 0 SRNBB-C26 0

C50-C53 0 D31-D61 0 D77-dum2 0 SRNBB-C31 0

168

C50-CVPM16 0 D37-C32 0 D801-D803 0 SRNBB-SRCFW 0

C50-GW16 0 D42-C34 0 D801-D891 0 SRNHL-C42 0.121

C51-C53 0 D42-D43 60.3 D803-C84 0 SRNHL-STOC 0

C52-C59 0 D43-C306 0 D803-D804 0 SRPF-C51 0

C52-C90 0 D43-C67 0 D804-D814 0 SRPR-C39 0

C53-C50 0 D503-D511 181.1 D814-D816 0 SRPR-D98 0

C53-C54 0 D507-C68 0 D814-SR12 0 SRSCV-C315 0

C53-C55 0 D507-D509 0 D816-D712 0 SRTR-D16 0

C54-C52 0 D509-DELTA 0 D816-D818 0 STOC-C42 0

C54-D608 0 D511-D507 0 D818-D820 0 SWR-C15 0

C55-C53 0 D511-D513 0 D820-D742 0 TRACY-D701 0

C55-CVPM17 0 D511-NAPA 0 D845-D847 0 UD2-GW2 0

C55-GW17 0 D513-D515 0 D845-D850 0 UD3-GW3 0

C56-C59 0 D513-D521 0 D847-C100 0 UD4-GW4 0

C56-C60 0 D515-C68 0 D847-D848 0 UD5-GW5 0

C56-C90 0 D515-D522 0 D848-D849 0 UD6-GW6 0

C57-C58 0 D517-D515 0 D849-StBarbara 0 UD8-GW8 0

C58-C59 0 D521-C68 0 D850-C100 0 UD9-GW9 0

C58-C60 0 D521-D522 0 D850-D851D852 0 WC-C201 0

C59-SRTLB 0 D522-D523 0 D851D852-D853 0 YUBA-D37 0

C5-C87 0 D523-C68 0 D853-C95 0 dum1-C3 0

C609-C48 0 D523-D525 0 D855D857-C98 0 dum1-D5 0

C609-D608 0 D525-D528 0 D855D857-D859 0 dum1-D73 3.019

C60-C688 0 D525-D550 0 D859-D860 0 dum2-C1 0

C60-CVPM18 0 D528-D509 0 D860-D863 0 dum4-C6 0

C60-GW18 0 D528-MS 0 D863-C103 0 dum4-GW2 0

C60-Pixley 0 D550-C309 0 D863-C98 0 dum4-UD2 0

C62-C100 0 D550-PMPCC 0 D891-D892 0 dum5-C303 0

C62-C64 0 D59-BANKS 0 D891-SR15 0 dum6-WC 0

C63-C65 0 D59-TRACY 0 D892-D896 0 dum-GW21 0

C63-CVPM20 0 D5-D73 196.3 D896-C316 0 dummy1-C316 0

C63-GW20 0 D5-Redding 0 D896-OakFlat 0 dummy1-OakFlat 0

Table A 14. Return flow coefficient in links exiting demand nodes. This is used in the mass balance

constraint and shows the proportion (percentage) of flow that is not deducted due to returning to the network

(due to seepage for example). A return flow coefficient equal 1 indicates that no proportion is flow is

returned to the network.

Link Link Link Link

AG1-C5 0.560531 AG9-D509 0.3 CVPM14-

GW14

1 CVPM20-

GW20

0.99

AG1-GW1 0.439469 AG9-GW9 0.7 CVPM15UR-

GW15

1 CVPM21UR-

GW21

1

AG2-C4 0.230189 Bfield-dum 1 CVPM15-C59 0.6 CVPM21-

GW21

1

AG2-GW2 0.769811 CVPM10UR-

GW10

1 CVPM15-

GW15

0.4 FRESNO-

D605

1

AG3-C305 0.219737 CVPM10-

D612

0.74 CVPM16-C59 0.69 OakFlat-

dummy1

1

169

AG3-GW3 0.780263 CVPM10-

GW10

0.26 CVPM16-

GW16

0.31 Redding-GW1 1

AG4-D61 0.820191 CVPM11UR-

GW11

1 CVPM17UR-

GW17

1 SAC-C8 1

AG4-GW4 0.179809 CVPM11-

GW11

1 CVPM17-C59 0.39 STOC-C42 1

AG5-C307 0.25 CVPM12UR-

GW12

1 CVPM17-

GW17

0.61 UD2-GW2 1

AG5-GW5 0.75 CVPM12-

D697

0.62 CVPM18UR-

GW18

1 UD3-GW3 1

AG6-GW6 1 CVPM12-

GW12

0.38 CVPM18-

GW18

1 UD4-GW4 1

AG7-C8 0.45 CVPM13UR-

GW13

1 CVPM19UR-

GW19

1 UD5-GW5 1

AG7-GW7 0.55 CVPM13-

D695

0.66 CVPM19-

GW19

1 UD6-GW6 1

AG8-D517 0.78992 CVPM13-

GW13

0.34 CVPM20UR-

GW20

1 UD8-GW8 1

AG8-GW8 0.21008 CVPM14UR-

GW14

1 CVPM20-C73 0.01 UD9-GW9 1

YUBA-D37 1

Table A 15. Links with network cost due to pumping, treatment, or conveyance. This is the cost per unit of

flow incurred in links. The unit is $/Kaf with 1 Kaf = 1.23348 MCM. Link Link Link Link

C100-GW19 5000 C3-GW1 5000 C71-CC 50000 SR8-SAC 50000

C14-GW4 5000 C45-GW12 5000 C84-GW10 5000 SRASF-OakFlat 100000

C172-GW11 5000 C46-GW13 5000 C8-SAC 70000 SRASF-SFPUC 125000

C17-GW6 5000 C50-GW16 5000 C90-GW15 5000 SREB-EB 50000

C26-GW5 5000 C55-GW17 5000 C91-GW14 5000 SRLB-NAPA 65000

C303-GW3 5000 C60-GW18 5000 D511-NAPA 75000 SRNHL-STOC 40000

C315-OakFlat 115000 C63-GW20 5000 D714-OakFlat 349000 dum4-GW2 5000

C34-GW7 5000 C66-GW21 5000 D896-OakFlat 349000 dummy1-C316 33000

C36-GW8 5000 C68-GW9 5000 SR6-YUBA 50000 dummy1-OakFlat 350000

Table A 16. Target demand in demand nodes. Volumes are in Kaf with 1 Kaf = 1.23348 MCM. Node Month

1 2 3 4 5 6 7 8 9 10 11 12

YUBA 3.94 2.8 2.45 2.38 2.42 2.66 3.65 5.38 6.9 7.57 7.41 5.72

SAC 58.67 42.5 34.45 31.57 33.22 34.95 45.68 62.79 80.89 89.81 88.73 75.25

NAPA 9.46 7.57 6.83 6.56 6.69 7.19 8.82 11.27 12.83 13.66 13.16 11.4

STOC 7.48 5.54 4.82 4.57 4.76 5.18 7.03 9.46 11.19 12.55 12.13 10.18

CC 11.04 8.71 7.57 7.25 7.35 7.87 10.53 13.31 15.42 16.24 15.75 13.78

EB 24.57 20.52 19.42 19.28 19.71 20.47 23.55 27.29 30.28 31.98 31.56 28.67

Redding 6.25 4.5 4.09 3.83 3.95 4.25 5.58 7.66 10.02 11.85 11.41 8.88

UD2 4.87 3.51 3.19 2.98 3.08 3.3 4.34 5.96 7.8 9.23 8.88 6.91

UD3 1.16 0.82 0.72 0.7 0.71 0.78 1.08 1.59 2.03 2.23 2.18 1.69

UD4 0.39 0.28 0.24 0.23 0.24 0.26 0.36 0.53 0.68 0.75 0.73 0.56

UD5 4.42 3.14 2.75 2.67 2.71 2.98 4.1 6.04 7.74 8.49 8.31 6.42

UD6 5.22 3.78 3.07 2.81 2.96 3.11 4.07 5.59 7.2 7.99 7.9 6.7

170

UD8 2.11 1.56 1.36 1.29 1.34 1.46 1.98 2.67 3.15 3.54 3.42 2.87

UD9 6.08 4.5 3.92 3.71 3.87 4.21 5.71 7.68 9.09 10.19 9.85 8.27

CVPM10UR

3.3 2.44 2.13 2.02 2.1 2.29 3.1 4.17 4.94 5.54 5.35 4.49

CVPM11U

R

18.28 13.53 11.78 11.17 11.62 12.65 17.17 23.11 27.33 30.66 29.64 24.86

CVPM12U

R

8.63 6.39 5.56 5.27 5.49 5.97 8.1 10.91 12.9 14.47 13.98 11.73

CVPM13U

R

12.8 9.48 8.25 7.82 8.14 8.86 12.02 16.18 19.14 21.47 20.75 17.41

CVPM14U

R

0.301 0.901 2.201 0.801 0.429 1.006 1.207 1.508 2.513 2.915 2.312 0.503

CVPM15U

R

4.89 3.59 2.91 2.58 2.88 3.11 4.71 6.4 7.83 8.63 8.15 6.8

CVPM17U

R

6.55 4.81 3.9 3.47 3.86 4.18 6.32 8.59 10.5 11.58 10.93 9.12

CVPM18U

R

11.39 8.36 6.78 6.02 6.71 7.26 10.98 14.93 18.24 20.13 18.99 15.85

CVPM19U

R

1.84 1.35 1.04 0.95 1.05 1.24 1.79 2.39 3.01 3.28 3.05 2.5

CVPM20U

R

4.43 3.25 2.5 2.3 2.53 2.99 4.32 5.75 7.26 7.9 7.35 6.03

CVPM21U

R

2.32 1.71 1.31 1.21 1.33 1.57 2.26 3.02 3.81 4.15 3.86 3.16

OakFlat 53.92 43.14 38.6 37.99 39.7 41.7 52.08 62.86 72.07 75.15 73.44 65.63

SFPUC 19.84 18.03 17.23 17.34 17.95 17.73 19.83 21.01 22.22 22.79 22.45 21.61

Fresno 29.73 21.82 17.68 15.71 17.5 18.94 28.66 38.95 47.61 52.53 49.57 41.37

Bfield 20.39 14.98 11.49 10.6 11.65 13.74 19.87 26.47 33.39 36.37 33.83 27.74

StBarbara 11.8 9.46 8.44 8.25 8.84 9.14 11.43 13.06 14.7 15.32 14.78 13.47

AG1 6.03 0 0 0 0 2.93 18.73 23.93 24.53 31.73 28.03 17.33

AG2 21.2 0 0 0 0.21 8.9 71.8 120.1 127 155.3 119.7 72.6

AG3 15.2 0 0 0 1.7 24.4 280.2 273.2 355.1 365.6 259.5 53.9

AG4 16.74 0 0 0 1.4 17.04 166.34 165.9

4

242.3

4

255.8

4

187.3

4

44.84

AG5 33.04 0 0 0 0.52 19.74 279.44 300.34

365.95

369.04

278.95

89.84

AG6 43.68 0 0 0 3.95 77.68 155.08 169.9

8

181.7

8

199.6

8

142.1

8

73.68

AG7 15.8 0 0 0 0.4 11.6 92.3 95.11 113.31

115 89.8 31.91

AG8 22.42 0 0 0 0.9 15.32 83.42 138.2

2

179.2

2

215.7

2

159.5

2

79.12

AG9 10.2 0 0 0 1.4 12.7 125.9 169.2 286.2 315.3 194.1 69.5

CVPM10 67.2 0 0 0 2.381 157.49

143.96 212.63

325.77

327.69

284.34

96.152

CVPM11 52.42

3

0 0 0 0 52.42

3

96.462 132.3

3

141.5

6

150.7

9

125.7

9

81.36

5

CVPM12 25.864

0 0 0 3.2273

34.864

84.591 113.86

140.05

150.95

111.77

64.682

CVPM13 66.56

4

0 0 0 4.272

7

132.4

7

166.84 247.2

9

328.2 341.4

7

294.7

5

137.2

CVPM14 8.3 0 0 0 15.7 181.2 137.5 230.9 340.3 318.1 245 19

CVPM15 25.81 0 0 0 26.85

7

264.9

5

134.86 187.8

1

387.6

2

422.9

5

370.5

7

75.61

9

CVPM16 26.20

9

0 0 0 0 32.20

9

47.5727

3

75.57

3

74.93

6

78.66

4

66.20

9

44.75

5

CVPM17 24.08

2

0 0 0 0.745

5

25.80

9

58.445 140.5

4

143.5

4

152.4

5

126.7

2

73.9

CVPM18 30.34 0 0 0 13.64 173.5 224.28 319.9 457.6

9

474.3

2

366.4

9

103.2

4

CVPM19 29.43 0 0 0 25.93 149.1

3

80.53 89.83 168.4

3

190.1

3

176.6

3

46.53

CVPM20 16.38 0 0 0 6.477

6

56.19

6

63.579 100.6

8

113.7

7

125.3

6

105.6

4

44.23

4

CVPM21 30.92 0 0 0 29.22 133.1

2

101.92 141.0

2

221.3

2

241.5

2

207.8

2

55.12

171

172

Appendix B. Effect of evolutionary search configuration on model

performance

The choice of maximum number of function evaluations was bounded by the regulation of

HPC facilities at University of Manchester. Jobs (model runs) submitted to this HPC

cluster had the run time limit of 7 days. The maximum number of function evaluation was

chosen so that the run time doesn’t exceed the designated 7 days. If the run time goes

beyond 7 days, the job is killed and no output is generated (regardless of how close the run

was close to completion). However, this may influence effective convergence of the

algorithm. In such cases, random seed (RS) analysis is recommended in the literature. RS

carries out several identical runs of the evolutionary algorithm, each starting with different

and random initial solutions (seed). The choice of 5 seeds was informed by consulting

other members of the research group at University of Manchester, investigating similar

multi-objective optimisation problems of the similar size, and the observation that different

seeds only added solution points to the flat part of the Pareto front. The latter is illustrated

below for a sample run of the Central Valley case but with different initial storage and

groundwater characteristics (before implementing modification highlighted in

section ‎2.2.4).

Figure A 1. Non-dominated solutions of a sample run with 4 seeds (arrows show the direction of preference

for each fitness function). Note that seed 4 uses a different search resolution in the MOEA.

In Figure A 1, seed 4 was run with different search resolution i.e. $1,000,000 and 81070

$/MCM (100 $/af) for the fitness functions (equations (27) and (28), respectively).

600

650

700

750

800

850

900

950

1000

1050

35

00

85

00

13

50

0

18

50

0

23

50

0

28

50

0

33

50

0

38

50

0

43

50

0

Do

wn

stre

am n

et

be

nfi

t (

B$

)

Mean water marginal value ($/MCM))

Seed 1

Seed 2

Seed 3

Seed 4 (higher epsilon)

173

Generally, higher epsilon value means that more non-dominated solution can enter the

Pareto non-dominated set as epsilon is the dimension of non-dominancy box. But it

essentially implies that these solutions points might be far from being optimal. As it can be

seen in the above figure, the run using higher epsilon value found more non-dominated

solution points compared to other runs (with smaller epsilon) at the cost of the majority of

the solution points being ‘less optimal’ (dominated) in contrast to other runs. The choice of

epsilon for the runs in Chapters 3 and 4 was following few trial-and-error efforts and after

consulting with members of the research group at University of Manchester.

174

Appendix C. Difference between storage trajectories of Model A and

CALVIN Optimised

This appendix scrutinises the differences between aggregated storage trajectory of

CALVIN optimised to that of Model A. Because Model A is built based on CALVIN

Optimised, one expects they both show similar storage time-series over the planning

horizon. But this is not the case as shown in Figure 6. This is due to choices of initial flow

values and network simplifications in the north portion of the region. To prove this,

aggregate storage time series are split into ‘North’ vs ‘South’ portions (Figure A 2).

Figure A 2. Aggregated storage comparison between Model A and CALVIN Optimised for: a) North

portion, and b) South portion of the Central Valley.

The results from the north portion of Model A’s network are similar to the CALVIN

results but not exact; while the south portion gives almost exactly the same results as

CALVIN Optimised. This indicates that the variations in Model A from the CALVIN

results are due mainly to the north portion of the network.

4,000

6,000

8,000

10,000

12,000

14,000

16,000

19

22

19

24

19

26

19

28

19

30

19

32

19

34

19

36

19

38

19

40

19

42

19

44

19

46

19

48

19

50

19

52

19

54

19

56

19

58

19

60

19

62

19

64

19

66

19

68

19

70

19

72

19

74

19

76

19

78

19

80

19

82

19

84

19

86

19

88

19

90

19

92

Sto

rage

(M

CM

)

Water Year

(a)

Model A CALVIN Optimised

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

19

22

19

24

19

26

19

28

19

30

19

32

19

34

19

36

19

38

19

40

19

42

19

44

19

46

19

48

19

50

19

52

19

54

19

56

19

58

19

60

19

62

19

64

19

66

19

68

19

70

19

72

19

74

19

76

19

78

19

80

19

82

19

84

19

86

19

88

19

90

19

92

Sto

rage

(M

CM

)

Water Year

(b)

Model A CALVIN Optimised

175

176

Appendix D. Proposed model formulation

This section provides mathematical representation of the hydro-economic model used in

this study. The followings describe different terms composing the annual net benefit

function.

The net benefits 𝑓𝑡 at month 𝑡 are the sum of monthly urban, agricultural, and hydropower

benefits (𝑈𝑅𝑡(𝑥𝑡), 𝐴𝐺𝑡(𝑥𝑡) , and 𝐻𝑃𝑡(𝑥𝑡) respectively); monthly costs are network costs

𝑁𝑊𝑡(𝑥𝑡) incurred for conveyance, treatment and conjunctive use, groundwater pumping

costs 𝐺𝑊𝑡(ℎ𝑡, 𝑥𝑡), and penalties, 𝐼𝑁𝐹𝑡(𝑥𝑡, ℎ𝑡 , 𝑠𝑡), associated with constraint violations:

𝑓𝑡(𝑥𝑡, ℎ𝑡, 𝑠𝑡) = 𝑈𝑅𝑡(𝑥𝑡) + 𝐴𝐺𝑡(𝑥𝑡) + 𝐻𝑃𝑡(𝑥𝑡) − 𝑁𝑊𝑡(𝑥𝑡) − 𝐺𝑊𝑡(ℎ𝑡 , 𝑥𝑡) −

𝐼𝑁𝐹𝑡(𝑥𝑡, ℎ𝑡, 𝑠𝑡)

(A 1)

The CALVIN model derives urban benefit functions from observed water prices, quantities

consumed, and an assumption of constant elasticity (M.W. Jenkins et al., 2001). Urban

benefits 𝑈𝑅𝑡(𝑥𝑡) are derived following the method suggested by Hansen (2007), and using

raw data from the CALVIN model. This results in a quadratic function of water delivery

with negative quadratic coefficients – decreasing marginal benefits.

Likewise, agricultural demand curves are derived from CALVIN data using the

methodology proposed by Hansen (2007). However, we then used a piece-wise linear

translation of the quadratic agricultural benefit function. That is to avoid zero water

marginal value at target demands which could have tempted farmers to stop pumping in

order to incur less cost.

Network costs 𝑁𝑊𝑡(𝑥𝑡) refer to conveyance, treatment and conjunctive use costs. They are

assumed to be a linear function of flow in each link. It should be noted that less nodes are

considered in this model compared to CALVIN, as some nodes have been merged in order

to simplify the network.

Numerical infeasibilities may appear making the network problem infeasible. In order to

guarantee feasibility artificial inflows, titled infeasibility flows, are made available to the

model at each node (Draper, 2001). These flows are included in model’s conservation of

mass equations to ensure that such flows are accounted for. These artificial flows which

are in fact they are slack/surplus variables in a mathematical programming context, are not

177

desirable therefore in order to deter the model from introducing infeasibility flows they are

penalized by a high cost coefficient in the objective function. This can be seen in equation

(A 2). Infeasibility flows are useful to identify model infeasibilities and assist in the

detection of modelling errors or data inconsistencies (Antoniou, 2011).

𝐼𝑁𝐹𝑡(𝑥𝑡, ℎ𝑡, 𝑠𝑡) = ∑ 𝑖𝑛𝑓𝑒𝑎𝑠𝑖,𝑡 ∙ 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒

𝑖∈𝑛𝑜𝑑𝑒𝑠

∀𝑡 (A 2)

where infeas is the artificial flow introduced at each node to track physical infeasibilities

and multiple is a large enough cost used to penalize and avoid the occurrence of such

infeasibilities. In California, the presence of “high-head” facilities where the effect of

reservoir storage on turbine head is small allows for a linear relationship between head and

hydropower generation (Madani & Lund, 2007; Vicuna, Leonardson, Hanemann, Dale, &

Dracup, 2008):

𝐻𝑃𝑖,𝑡 = 𝑅𝑖,𝑡𝑃𝐹𝑖𝑝𝑡 (A 3)

where R is the release of the reservoir for the power plant i, PF is the power factor which

relates release to hydropower generation, and p is the monthly-varying hydropower unit

price.

178

Appendix E. Pareto solution analysis

The Pareto front shown in Figure 16 contains a flat part with respect to the main

(economic) fitness function. This means that different combinations of water marginal

values lead to a near-maximum net economic benefit. Part of the reason for this is that

when the end-of-year marginal value of storage at a given reservoir is enough to fill the

reservoir up almost completely every year, using marginal values that are greater by

several orders of magnitude yields similar results. This is why the search minimizes

objective F2, as this enables us to look for the minimal marginal water value that fills the

reservoir. Concentration of solution points around the beginning of the flat part suggest

that the estimate for water marginal value solution can be sought there. Solution points

from F1>1020 B$ and F2<15000 $/MCM are considered for this purpose. The dispersion of

marginal value solutions from this range is represented in Figure A1.

Figure A1. Distribution of reservoirs’: a) maximum and b) minimum water marginal value solutions (i.e. at

a) minimum and b) maximum storage). Colours represent different solution point from the flat part of the

Pareto front (contains 32 different solution points and colours).

010,00020,00030,00040,00050,00060,00070,00080,00090,000

100,000

Shas

ta

Wh

iske

yto

wn

Bla

ck B

utt

e

Oro

ville

Ne

w B

ulla

rds

Bar

Cam

p F

ar W

est

Fols

om

Ind

ian

Val

ley

Ber

rye

ssa

Par

de

e

Ne

w H

oga

n

Los

Vaq

uer

os

EBM

UD

Turl

ock

Llo

yd&

Elea

no

r

Het

ch H

etch

y

San

ta C

lara

SF a

ggre

gate

Kaw

eah

Succ

ess

Isab

ella

Pin

e F

lat

Ne

w M

elo

nes

San

Lu

is

Del

Val

le

Mill

erto

n

McC

lure

Hen

sley

East

man

Ne

w D

on

Ped

ro

Mar

gin

al w

ate

r va

lue

(

$/M

CM

)

Reservoirs

(a)

0

5,000

10,000

15,000

20,000

25,000

30,000

Shas

ta

Wh

iske

yto

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(

$/M

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Reservoirs

(b)

179

Variations in the value of the second fitness function F2 in the flat part of the Pareto front

are due to differences in the maximum marginal values. This dispersion occurs in small

reservoirs, whereas values for large, key reservoirs such as Shasta are consistent across

simulations. Small storage size of these reservoirs limits the variation in the maximum of

COSVF each reservoir can produce. This is illustrated in Figure A2.

Figure A2. Maximal total value of end-of-year carry-over storage (i.e. total value of carry-over storage if

reservoirs are full). Colours represent different solution point from the flat part of the Pareto front.

According to Figure A2, few reservoirs experience significantly different valuations of

their total end-of-year carry-over storage. In fact, this does not have much influence on

reservoir operations. This is illustrated by the reservoir with the most variations in total

carry-over storage, Folsom. A sensitivity analysis for this reservoir is carried out by

clustering maximum marginal water values (i.e. marginal water values at minimum

storage) into two groups – consistent with panel a) from Figure A1 Average values from

both groups determine a “low” and “high” valuation of storage, and the average of all

values determines an “average” valuation. These three valuations of storage are used in

three separate runs of the model, with all other parameters unchanged (including COSVF

from other reservoirs). Resulting end-of-year storage levels prove to be almost identical

regardless of which cluster of marginal value of water is chosen (Figure A3). Therefore, it

is the “average” valuation that is reported in Table 2; similar analyses have been conducted

for other reservoirs with diverging marginal values of storage.

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

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($)

Reservoirs

180

Figure A3. End-of-year storage level in Folsom using different sets of water marginal value.

0100200300400500600700800900

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181

Appendix F. Historical approximation by CALVIN ‘base case’ run

The CALifornia Value Integrated Network (CALVIN) is an economically-driven

engineering “optimization” model. The model can operates facilities and allocates water so

as to maximize state-wide agricultural and urban economic value from water use

(CALVIN Optimized; OP) or it can be further constrained to meet operating or allocation

policies (CALVIN base case; BC). The latter reproduces historical trend and is labelled as

‘historical approximation’ in this thesis. For both versions of the model, readers are

encouraged to refer B. Newlin, Lund, Kirby, and Jenkins (1999), Howitt et al. (1999),

M.W. Jenkins et al. (2001), and Marion W. Jenkins et al. (2004). The model referred to as

“perfect foresight” in Chapter 3, is similar to and inspired from CALVIN Optimised,

expect for the representation of groundwater pumping and hydropower generation,

exclusion of the Central Valley region 5, and network simplification in regions 1-4.

Although historical approximation results are not exactly the same as the observed data,

they show an acceptable match. We compared the historical observation of storage in

Shasta, the largest reservoir of the region, to the approximated storage derived by historical

approximation (Figure A4). It should be noted that the observed data prior to 1953 was not

available for Shasta. Hence, this comparison spans only from 10/1953 to 1/1993. Figure

A4 shows that historical approximation was able to accurately imitate storage trajectory of

Shasta. Slight difference in the approximation of historical approximation is negligible.

This comparison denotes that the use of historical approximation results as an

approximation for historical events is correct.

Figure A4. Comparison of the storage capacity of Shasta: historical approximation vs observed data.

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