Water Resources Planning and Management: an overview
1.1 How would you define ‘Integrated Water Resources Management” and what distinguishes it from “Sustainable Water Resources Management”?
1.2 Can you identify some common water management issues that are found in many parts of the world?
Governments often provide aid to regions affected by droughts or floods, but this approach lacks incentives for implementing sustainable land use management practices Without encouraging proactive measures, such as altering agricultural methods or improving water conservation strategies, communities may remain vulnerable to future disasters This reliance on aid can perpetuate a cycle of dependency, hindering long-term resilience and adaptation to climate change It is essential for policies to integrate support with incentives that promote responsible land management, ensuring that communities are better prepared for future environmental challenges.
1.4 What tools are available for integrated water resources planning and management?
1.5 What structural and non-structural measures can be taken to better manage water resources?
• % freshwater resources worldwide available for drinking:
• Number of people who die eac h year from diseases associated with unsafe drinking water:
• % freshwater resources in polar regions:
• U.S per capita annual withdrawal of cubic meters of freshwater:
• World per capita annual withdrawal of cubic meters of freshwater:
• Tons of pollutants entering U.S lakes and rivers daily:
• Average number of gallons of water consumed by humans in a lifetime:
• Average number of kilometers per day a woman in a developing country must walk to fetch fresh water:
1.7 Briefly describe the 6 greatest rivers in the world.
In my region, significant water resource management issues include over-extraction of groundwater, pollution from agricultural runoff, and inadequate infrastructure for water distribution To address these challenges, implementing sustainable water management practices such as rainwater harvesting, enhancing wastewater treatment, and promoting conservation efforts can be effective These alternatives not only help mitigate water scarcity and improve water quality but also offer economic benefits through job creation in green technologies and social advantages by ensuring equitable access to clean water for all communities.
Water resource systems are complex networks composed of interdependent components that include inputs such as precipitation, surface water, groundwater, and human activities, while outputs consist of water distribution, consumption, and wastewater The inclusion of specific elements in these systems is determined by their relevance to water availability and management, while the decision on spatial and temporal detail is guided by the scale of analysis and the dynamics of water flow and usage over time.
Sustainability in renewable resource management refers to the responsible use and conservation of resources, ensuring that water supplies meet current and future demands in a changing environment It is applicable over various spatial and temporal scales, allowing for the assessment of long-term viability of water management plans To determine the sustainability of a management policy, one must evaluate its ecological impact, resource availability, and adaptability to shifting conditions This concept closely aligns with adaptive management, which emphasizes flexibility and learning from outcomes to enhance resource stewardship and resilience.
1.11 Identify and discuss briefly some of the major issues and challenges facing water managers today.
Water Resource Systems Modelling: its role in planning and management
2.4 Why develop and use models?
2.5 What is a decision support system?
2.6 What is shared vision modeling and planning?
2.7 What characteristics of water resources planning or management problems make them suitable for analysis using quantitative systems analysis techniques?
Water resource systems planning often faces several specific problems, including water allocation, flood management, and water quality control For water allocation, the objective is to optimize the distribution of water among various users, with unknown decision variables such as the amount of water to allocate to each sector Constraints may include legal regulations, environmental impacts, and the availability of water supply In flood management, the goal is to minimize flood damage while ensuring public safety, with decision variables like the design of flood control structures Constraints can involve budget limitations and land-use regulations Lastly, for water quality control, the objective is to maintain acceptable water quality levels, with unknown decision variables related to pollution control measures Constraints include regulatory standards and the capacity of treatment facilities.
A review of recent issues of journals focused on water resources and related fields such as engineering, economics, planning, and operations research reveals several key publications that feature articles on water resources systems planning and analysis These journals discuss a variety of current topics and challenges, including sustainable water management, optimization of water distribution systems, economic impacts of water resource allocation, and innovative planning strategies to address water scarcity By examining these sources, one can gain valuable insights into the latest trends and research developments in water resources systems.
Water resource systems planning often faces challenges in quantifying certain critical factors, making it difficult to integrate them into mathematical models for evaluating alternative solutions Despite their limitations, these quantitative models can still provide valuable insights during the planning process, especially when considering non-quantifiable aspects such as community values, environmental impacts, and social equity Examples of planning problems with intangible objectives include balancing ecological preservation with water supply needs, addressing the cultural significance of water resources to local communities, and managing public perceptions and stakeholder engagement in water management decisions.
2.11 Define integrated water management and what that entails as distinct from just water management.
2.12 Water resource systems serve many purposes and can satisfy many objectives What is the difference between purposes and objectives?
2.13 How would you characterize the steps of a planning process aimed at solving a particular problem?
If you reside in a region where the sole affordable water source is a non-rechargeable aquifer, it is essential to create a sustainable water management plan This plan should include strategies for monitoring water levels, implementing conservation practices, and prioritizing water use for essential needs Additionally, consider investing in alternative water sources or technologies, such as rainwater harvesting or desalination, to reduce reliance on the aquifer Regular assessments and community engagement will ensure that the plan adapts to changing conditions and promotes long-term water sustainability.
Modelling methods for Evaluating Alternatives
3.1 Briefly outline why multiple disciplines are needed to efficiently and effectively manage water resources in major river basins, or even in local watersheds
3.2 Describe in a page or two what some of the issues are in the region where you live
3.3 Define adaptive management, shared vision modeling, and sustainability
3.4 Distinguish what a manager does from what an analyst (modeler) does
3.5 Identify some typical or common water resources planning or management problems that are suitable for analysis using quantitative systems analysis techniques.
3.6 Consider the following five alternatives for the production of energy (10 3 kwh/day) and irrigation supplies (10 6 m 3 /month):
Alternative Energy Production Irrigation Supply
In evaluating the best alternative for hydropower energy and irrigation supply, decision makers may choose alternative E despite the availability of options that offer greater benefits This choice could be influenced by factors such as cost-effectiveness, environmental impact, or long-term sustainability Additionally, alternative E might align better with regional development goals or community needs, making it a more appealing option despite its lower energy output Ultimately, the decision reflects a balance between immediate benefits and broader strategic considerations.
To minimize the total cost of storing a specified volume of water in a cylindrical tank, define a mathematical model analogous to Equations 3.1 to 3.3 The unknown decision variables in this model include the radius and height of the tank, while the model parameters consist of the volume of water to be stored and the costs associated with materials and construction An iterative approach can be developed to systematically adjust the dimensions of the tank, optimizing for cost efficiency while ensuring the required volume is maintained.
3.8 Briefly distinguish between simulation and optimization.
3.9 Consider a tank, a lake or reservoir or an aquifer having inflows and outflows as shown in the graph below
The analysis of water outflow reveals key metrics: the inflow reached its maximum and minimum values at specific times, while the outflow recorded its lowest point at a particular moment Additionally, the storage volume peaked and dipped at designated intervals To quantify these changes, a mass balance equation can be formulated for the time series of storage volumes, assuming that inflows and outflows remain constant throughout each period.
3.10 Describe, using words and a flow diagram, how you might simulate the operation of a storage reservoir over time To simulate a reservoir, what data do you need to have or know?
3.11 Identify and discuss a water resources planning situation that illustrates the need for a combined optimization-simulation study in order to identify the best alternative solutions and their impacts.
To analyze the storage volumes in a tank or reservoir with varying inflows and constant outflows, one must sketch a plot reflecting these changes over time The slope of the storage volume plot at any moment can be determined by calculating the difference between the inflow and outflow rates depicted in the accompanying graph This approach allows for a clear understanding of how the storage volume fluctuates in response to the inflow and outflow dynamics.
3.13 Write a flow chart/computer simulation program for computing the maximum yield of water that can be obtained given any value of active reservoir storage capacity, K, using
Find the values of the storage capacity K required for yields of 2, 3, 3.5, 4, 4.5 and 5.
To ensure at least a 95% probability that the best solution derived from simulations of a water resource system falls within the top 5% of all possible solutions, a significant number of simulations must be conducted This requires assumptions such as the distribution of potential solutions being known and that the simulations are representative of the entire solution space While the best solution from the simulations may closely approximate the optimal solution, it cannot be guaranteed to match it exactly, highlighting the inherent uncertainty in simulation-based approaches.
3.15 Assume in a particular river basin 20 development projects are being proposed Assume each project has a fixed capacity and operating policy and it is only a question of which of the
To maximize net benefits to the region, 20 projects must be evaluated, requiring 5 minutes of computer time for each combination Even with the assumption that 99% of alternative combinations can be eliminated through good judgment, the total time needed would still amount to 36 days of computer processing This highlights the significant computational demands of simulation in regional interdependent multiproject water resources planning, emphasizing the importance of efficient decision-making and resource allocation.
To determine the optimal allocation of water (X j) to three users (j) who derive benefits (R j(X j)), start by assessing the total available water (Q) Create a flow chart that outlines the steps to evaluate each user's benefit function, ensuring the combined benefits are maximized The flow chart should include identifying the constraints of water distribution, calculating individual benefits based on varying allocations, and iteratively adjusting the distribution until the highest total benefit is achieved This systematic approach will help in determining the most efficient allocation of water resources.
3.17 Consider the allocation problem illustrated below
The allocation priority in each simulation period t is:
First 10 units of streamflow at the gage remain in the stream.
Next 20 units go to User 3.
Next 60 units are equally shared by Users 1 and 2.
Next 10 units go to User 2.
In this study, we examine a gage site with the assumption of no incremental flow along the stream and no return flow from users We define the allocation policy at each site, presenting it as a graph that illustrates the allocation as a function of the flow at the allocation site To evaluate this allocation policy, we simulate it using river basin simulation models such as RIBASIM, WEAP, or Modsim, applying a specified inflow series that ranges from 0 to 130 units.
Optimization Methods
4.1 Consider two alternative water resource projects, A and B Project A will cost $2,533,000 and will return $1,000,000 at the end of 5 years and $4,000,000 at the end of 10 years Project
B will cost $4,000,000 and will return $2,000,000 at the end of 5 and 15 years, and another
$3,000,000 at the end of 10 years Project A has a life of 10 years, and B has a life of 15 years. Assuming an interest rate of 0.1 (10%) per year:
(a) What is the present value of each project?
(b) What is each project’s annual net benefit?
(c) Would the preferred project differ if the interest rates were 0.05?
By extending each series of projects to a common terminal year, such as 30 years, and assuming that each project is replaced with a similar one that maintains the same cost and return time stream, it can be demonstrated that the annual net benefits for each series will align with the findings from part (b) This approach ensures a consistent comparison of the projects over the extended period, highlighting the equivalence in their net benefits.
If compounding takes place at the end of m equal-length periods within a year, where the nominal interest rate is r, the effective annual interest rate (r') can be expressed as follows: r' = (1 + r/m)^(m) - 1 This formula illustrates the relationship between the nominal rate and the effective rate, highlighting how the frequency of compounding impacts the overall interest earned over the year Understanding this concept is crucial for investors and borrowers alike, as it affects the true cost of loans and the yield on investments.
When compounding is continuous, the compound interest factor that transforms a present value into a future value over a period of T years is represented by the formula e^(rT) This relationship is derived from the limit as the number of compounding periods approaches infinity, specifically lim(k→∞) (1 + 1/k)^(k) = e, which is the foundation of natural logarithms.
The term "capitalized cost" represents the present value (PV) of an infinite series of equal end-of-year payments, denoted as A By assuming an interest rate of r, it can be demonstrated that as the terminal period T approaches infinity, the capitalized cost reflects a consistent valuation of these perpetual payments.
The internal rate of return (IRR) for any project is defined as the interest rate that equates the present value of all income with the present value of all costs In Exercise 4.1, the IRR for Project A is approximately 8%, while Project B has an IRR of around 6% These interest rates, denoted as r, effectively satisfy the equation for each respective project.
In Exercise 4.1, the selection of plans was based on maximizing annual benefits as an economic criterion Another important criterion is the benefit-cost ratio, which is calculated by dividing annual benefits by annual costs; a ratio of at least one is essential for annual benefits to surpass annual costs For illustration, consider two projects, I and II, to evaluate their respective benefit-cost ratios.
What additional information is needed before one can determine which project is the most economical project?
Bonds are commonly issued to fund investments in water resource projects, representing a commitment to pay a predetermined interest amount, typically on a semiannual basis, along with the bond's face value at a future date The selling price of a bond can vary from its face value, as current market interest rates influence its purchase price, while the interest payments remain fixed.
A bond with a face value of $10,000 pays an annual coupon of $500 for a decade, resulting in a coupon interest rate of 5% When purchased at its face value, the actual interest rate received by the bondholder remains at 5%.
In a competitive market, a bond with a 5% coupon rate will struggle to sell if comparable bonds offer a 10% interest rate To attract buyers, the bond's purchase price must be adjusted so that the effective interest rate for the new owner aligns with the prevailing 10% This adjustment ensures that the bond remains competitive and appealing to investors seeking similar risk profiles.
Municipal bonds offer tax-exempt interest, potentially freeing investors from both state and federal income taxes For instance, a 5% municipal bond can be comparable to a 7% taxable bond for those in the 30% income tax bracket This tax advantage not only lowers local tax obligations for funding municipal bond interest but also presents appealing investment options for individuals in higher tax brackets.
4.8 What is the meaning of the Lagrange multiplier associated with the constraint of the following model?
Maximize Benefit(X) – Cost(X) Subject to: X ≤ 23
4.9 Assume water can be allocated to three users The allocation, x j, to each use j provides the following returns: R(x 1) = (12x 1 – x 1 2), R(x 2) = (8x 2 – x 2 2) and R(x 3) = (18x 3 – 3x 3 2)
To maximize the total return, F(X), from three allocations while ensuring that the total does not exceed 10 units, we must determine the preferred allocation for each user At the optimal solution, the marginal returns, represented as ∂(R(xj))/∂xj, will equal the shadow price or Lagrange multiplier (λ) linked to the water availability constraint If 15 units of water were available for allocation among the three users, the solution would adjust accordingly, and the value of the Lagrange multiplier would reflect the change in total return per additional unit of water allocated.
4.10 In Exercise 4.9, how would the Lagrange multiplier procedure differ if the objective function, F( X ), were to be minimized?
To minimize the sum of squared deviations between actual allocations \( x_j \) and target allocations \( T_j \), we must develop a planning model considering a water supply \( Q \) that is less than the total of all target allocations \( T_j \) This involves constructing a Lagrangian that incorporates the constraints of the water supply However, solving the partial differential equations derived from the Lagrangian may not yield a global minimum due to the nature of the constraints and the potential for non-convexities in the solution space.
Using Lagrange multipliers, it can be demonstrated that the optimal design for a cylindrical storage tank of any volume \( V > 0 \) allocates one-third of the total cost to its base and top, while two-thirds is attributed to its side This cost distribution remains consistent irrespective of the varying costs per unit area for the base or side Such principles are commonly included in engineering design handbooks.
An industrial firm produces two unique products, A and B, which rely heavily on water, their scarce resource, while other resources are abundant They have the flexibility to set the unit prices for these products; however, past experience indicates that higher prices lead to decreased sales The relationship between unit price and sales quantity is defined by specific demand functions.
(a) What are the amounts of A and B, and their unit prices, that maximize the total revenue obtained?
(b) Suppose the total amount of A and B could not exceed some amount T max
What are the amounts of A and B, and their unit prices, that maximize total revenue, if i) T max = 10 ii) T max = 5
Water is essential for the production of goods A and B, with the production functions defined as A = 0.5 X A for product A and B = 0.25 X B for product B Each unit of A requires a specific amount of water, denoted as X A, while each unit of B requires a different amount, represented as X B.
(c) Find the amounts of A and B and their unit prices that maximize total revenue assuming the total amount of water available is 10 units.
(d) What is the value of the dual variable, or shadow price, associated with the 10 units of available water?
4.14 Solve for the optimal integer allocations x 1, x 2, and x 3 for the problem defined by
Exercise 4.9 assuming the total available water is 3 and 4 Also solve for the optimal allocation policy if the total water available is 7 and each x j must not exceed 4
Quantity of product A Quantity of product B
Fuzzy Optimization
5.1 An upstream reservoir serves as a recreation site for swimmers, wind surfers and boaters.
The reservoir functions as a flood storage system and allows for the diversion of its releases to an irrigation area Downstream, a wetland receives both the unallocated portion of the reservoir release and the return flow from irrigation, which poses a risk to the ecosystem due to its salinity concentration To address this, there are specific targets for recreation lake levels, irrigation allocations, and wetland flow and salinity The primary challenge is to optimize reservoir releases and irrigation allocations to effectively meet these targets, known as the 'crisp' problem However, if these targets are considered fuzzy rather than precise, fuzzy membership functions can be developed to derive the most effective reservoir release and allocation strategy, thus transforming the issue into the 'fuzzy' problem.
During period 2 the flood storage capacity is 5 mcm.
Irrigation return flow fraction: 0.3 (i.e., 30% of that diverted for irrigation);
Salinity concentration of reservoir water: 1 ppt;
Salinity concentration of irrigation return flow: 20 ppt;
Reservoir average inflows for four seasons, respectively: 5, 50, 20, 10 mcm;
Target maximum salinity concentration in wetland: 3 ppt;
Target storage target for all seasons: 20 mcm;
The seasonal minimum flow targets for the wetland are set at 10 mcm in spring, 20 mcm in summer, 15 mcm in autumn, and 15 mcm in winter, while the maximum flow targets are 20 mcm, 30 mcm, 25 mcm, and 25 mcm for the same seasons Additionally, the target irrigation allocations are 0 mcm in spring, 20 mcm in summer, 15 mcm in autumn, and 5 mcm in winter To optimize reservoir releases that align with these flow and salinity targets, a crisp problem approach will first be employed Subsequently, fuzzy membership functions will be developed to replace the established targets, facilitating a more flexible solution to the problem.
Data-Based Models
To apply genetic algorithms for determining the parameters \( a_{ij} \) in a water quality prediction model, start by creating a flow chart that outlines the process This model predicts the concentration downstream of an upstream discharge site using observed values of mass inputs \( W_i \), concentrations \( C_j \), and flows \( Q_j \) at site \( j \) The flow chart should illustrate the steps involved in data collection, initialization of the genetic algorithm, evaluation of fitness based on observed data, selection of parent parameters, crossover and mutation processes, and iterative refinement until optimal parameters are identified.
The goal of fitness is to reduce the total differences between the observed values (Cj) and the calculated values (Cj) To transform this into a maximization objective, one might employ a suitable mathematical approach.
Utilize the genetic algorithm program GANLC to forecast the parameter values specified in problem 6.1, followed by employing the artificial neural network (ANN) to create a predictor for downstream water quality based on these parameter values.
ANN are contained on the attached CD You may use the model and data presented in
Section 5.2 of Chapter 4 if you wish.
Utilize a genetic algorithm program, such as GANLC found on the accompanying CD, to determine the optimal allocations Xi that maximize the overall benefits for the three water users along a stream, each with distinct individual benefits.
Assume the available stream flow is some known value (ranging from 0 to 20).
Determine the effect of different genetic algorithm parameter values on the ability to find the best solution.
6.4 Consider the wastewater treatment problem illustrated in the drawing below.
The initial stream concentration just upstream of site 1 is 32 The maximum concentration of the pollutant just upstream of site 2 is 20 mg/l (g/m 3 ), and at site 3 it is
25 mg/l Assume the marginal cost per fraction (or percentage) of the waste load removed at site 1 is no less than that cost at site 2, regardless of the amount removed.
Utilize the genetic algorithm program GANLC, available on the CD, or an alternative suitable genetic algorithm program, to determine the most cost-effective wastewater treatment solutions for sites 1 and 2 Ensure that these solutions meet the quality constraints required at sites 2 and 3, respectively.
Discuss the sensitivity of the GA parameter values in finding the best solution You can get the exact solution using LINGO as discussed in Section 5.3 in Chapter 4.
To develop an artificial neural network for flow routing, utilize the ANN tool provided in the accompanying CD Begin by training the network with one set of upstream and downstream flow data over a period of five intervals to determine the unknown weights and variables Subsequently, validate the calculated parameters, including weights and bias constants, using a separate set of flow data.
Develop the simplest artificial neural network you can that does an adequate job of prediction.
Time period Upstream flow Downstream flow
[These outflows come from the following model, assuming an initial storage in period 1 of
50, the detention storage that will remain in the reach even if the inflows go to 0 For each period t:
Outflow(t) = 1.5(-50 + initial storage(t) + inflow(t) ) 0.9 ] where the outflow is the downstream flow and inflow is the upstream flow.
Concepts in Probability, Statistics and Stochastic Modelling
An example of a water resources planning study is the analysis conducted for the Colorado River Basin, which focuses on sustainable water management amid increasing demand The basic information utilized in this study includes historical water usage data, climate patterns, population growth projections, and ecological impact assessments The methods employed to transform this information into actionable decisions involve statistical modeling, scenario analysis, stakeholder engagement, and cost-benefit assessments Ultimately, the study culminates in recommendations for water conservation strategies, infrastructure investments, and policy adjustments to ensure the long-term viability of water resources in the region.
(a) Indicate the major sources of uncertainty and possible error in the basic information and in the transformation of that information into decisions, recommendations, and conclusions.
(b) In systems studies, sources of error and uncertainty are sometimes grouped into three categories:
1 Uncertainty due to the natural variability of rainfall, temperature, and stream flows which affect a system’s operation.
2 Uncertainty due to errors made in the estimation of the models’ parameters with a limited amount of data.
3 Uncertainty or errors introduced into the analysis because conceptual and/or mathematical models do not reflect the true nature of the relationships being described.
Indicate, if applicable, into which category each of the sources of error or uncertainty you have identified falls
7.2 The following matrix displays the joint probabilities of different weather conditions and of different recreation benefit levels obtained from use of a reservoir in a state park:
(a) Compute the probabilities of recreation levels RB 1, RB 2, RB 3, and of dry and wet weather.
(b) Compute the conditional probabilities P(wetRB 1), P(RB 3dry), and P(RB 2wet).
In flood protection planning, the design flow is commonly based on the 100-year flood, representing the 0.99 quantile estimate This approach assumes that floods occurring in different years are independently distributed.
(a) Show that the probability of at least one 100-year flood in a 5-year period is
(b) What is the probability of at least one 100-year flood in a 100-year period?
(c) If floods at 1000 different sites occur independently, what is the probability of at least one 100-year flood at some site in any single year?
7.4 The price to be charged for water by an irrigation district has yet to be determined
There is currently a 60% likelihood that the price of water will reach $10 per unit, while there is a 40% chance it will be priced at $5 per unit This pricing scenario significantly impacts the demand for water.
POSSIBLE RECREATION BENEFITS uncertain The estimated probabilities of different demands given alternative prices are as follows:
(a) What is the most likely value of future revenue from water sales?
(b) What are the mean and variance of future water sales?
(c) What is the median value and interquartile range of future water sales?
(d) What price will maximize the revenue from the sale of water?
7.5 Plot the following data on possible recreation losses and irrigated agricultural yields
The use of expected storage levels or allocations tends to underestimate the expected value of convex functions that represent reservoir losses, while simultaneously overestimating the expected value of concave functions related to crop yields A concave function, denoted as f(x), satisfies the inequality f(x) ≤ f(x₀) + f’(x₀)(x – x₀) for any chosen x₀ This implies that employing f(E[X]) will consistently lead to an overestimation of the expected value of a concave function f(X), where X is treated as a random variable.
Prob of Quantity Demanded given Price
Uncertainty regarding the useful life of a project can complicate its economic evaluation For instance, when planning a reservoir, the exact duration until its useful life ends due to silting remains uncertain While a high discount rate may diminish the impact of this uncertainty for long-lived projects, it becomes more significant for shorter projects, such as wastewater treatment facilities In such cases, relying on expected life to assess present costs or benefits can lead to misleading conclusions.
This project is anticipated to yield $1,000 in net benefits annually for a duration ranging from 10 to 30 years, with equal probability for each year within the 11 to 30-year timeframe Utilizing a discount rate of 10%, the financial viability and long-term value of the project can be effectively assessed.
(a) Compute the present value of net benefits NBo, assuming a 20-year project life. (b) Compare this with the expected present net benefits E[NBo] taking account of uncertainty in the project lifetime.
(c) Compute the probability that the actual present net benefits is at least $1000 less than NBo, the benefit estimate based on a 20-year life.
(d) What is the chance of getting $1000 more than the original estimate NBo?
The beta distribution is a continuous random variable that effectively models the proportion of fish or other animals exhibiting distinctive features across various large samples Its probability density function is defined for parameters a and b (where a > 0 and b > 0) as cx^(α-1)(1-x)^(β-1) for values of x within the range of 0 to 1.
0 otherwise (a) Directly calculate the value of c and the mean and variance of X for α= β= 2.
In general, the constant c is defined as c = Γ(α + β) / (Γ(α)Γ(β)), where Γ(α) represents the gamma function, equivalent to (α – 1)! for integer values of α Utilizing this definition, one can derive the general expressions for the mean and variance of the random variable X It is important to ensure that the value of c is determined in such a way that the integral of the density function over the interval (0, 1) equals one for any values of α and β.
7.8 The joint probability density of rainfall at two places on rainy days could be described by
(a) F XY (x, y), the joint distribution function of X and Y.
(b) F Y (y), the marginal cumulative distribution function of Y, and f Y (y), the density function of Y.
(c) f Y X (y x), the conditional density function of Y given that X = x, and F Y X (y x), the conditional cumulative distribution function of Y given that X = x (the cumulative distribution function is obtained by integrating the density function)
F Y X (y x = 0) > F Y (y) for y > 0 Find a value of x o and y o for which
7.9 Let X and Y be two continuous independent random variables Prove that
E[g(X)h(Y)] = E[g(X)]E[h(Y)] for any two real-valued functions g and h Then show that Cov(X, Y) = 0 if X and Y are independent.
In many cases, observations of quantities such as flow (X) and concentration (Y) lead to the calculation of a derived quantity, g(X, Y), like mass flux To accurately estimate the standard deviation of g(X, Y), it is essential to consider the standard deviations of X and Y, as well as their correlation By employing a second-order Taylor series expansion, one can derive the mean of g(X, Y) based on its partial derivatives alongside the means, variances, and covariance of X and Y Furthermore, a first-order approximation of g(X, Y) allows for the estimation of its variances, which depend on the partial derivatives and the statistical moments of X and Y It is important to note that the covariance between X and Y plays a crucial role in these calculations.
A study was conducted to analyze the behavior of water waves interacting with a breakwater on a sloping beach, using a small tank for experimentation Researchers measured the height of the waves, from crest to trough, at various distances from the wave generator and along the beach The data collected included mean wave heights and their standard errors at multiple points near the breakwater.
At which points were the wave heights significantly different from the height near wave generator assuming that errors were independent?
Standard Error of Mean (cm)
Near wave generator 1.9 cm from breakwater 1.9 cm from breakwater 1.9 cm from breakwater
The experimenter focuses on the ratio of wave heights near the breakwater compared to initial wave heights in deep water By utilizing the results from Exercise 7.10, the standard error of this ratio can be estimated at three specific points, assuming that measurement errors at these locations and near the wave generator are independent The analysis reveals at which point the ratio significantly deviates from 1.00.
Using the results of Exercise 7.10, show that the ratio of the mean wave heights is probably a biased estimate of the actual ratio Does this bias appear to be important?
In this section, we derive Kirby's bound, as represented in Equation 7.45, by calculating the sample estimates of skewness for the most skewed sample that could be observed Additionally, we establish the upper limit of (n - 1)^(1/2) for the estimate of the population coefficient of variation, ν, under the condition that all observations are nonnegative This analysis provides crucial insights into the behavior of skewness and variation in statistical samples.
7.13 The errors in the predictions of water quality models are sometimes described by the double exponential distribution whose density is
What are the maximum likelihood estimates of α and β? Note that β x−β d d = -1 x > β
Is there always a unique solution for β?
To derive the equations for maximum likelihood estimates of the parameters α and β in the gamma distribution, one must recognize that an analytical expression for dΓ(α)/dα is not available, preventing a closed-form solution for the maximum likelihood estimate of α However, the maximum likelihood estimate of β can be expressed as a function of the maximum likelihood estimates of α This relationship is crucial for statistical analysis involving the gamma distribution, as it provides a pathway to estimate parameters effectively despite the complexities involved.
7.15 The log-Pearson Type-III distribution is often used to model flood flows If X has a log- Pearson Type-III distribution then
Y = ln(X) – m has a two parameter gamma distribution where e m is the lower bound of X if β > 0 and e m is the upper bound of X if β < 0 The density function of Y can be written
Calculate the mean and variance of X in terms of α, β and m Note that
E[X r ] = E[(exp(Y + m)) r ] = exp(rm) E[exp(rY)]
To evaluate the necessary integrals, it's crucial to recognize that the constant terms in the definition of f_Y(y) guarantee that the integral of this density function over the range of y equals one, provided that α > 0 and βy > 0 The mean of X fails to exist for specific values of r and β Additionally, the parameters m, α, and β significantly influence the shape and scale of the distribution.
When comparing empirical and fitted distributions of streamflows or other variables, it is essential to assign cumulative probabilities to each observation, known as plotting positions For the i-th largest observation, X_i, this assignment is crucial for accurate analysis.
Thus the Weibull plotting position i/(n + 1) is one logical choice Other commonly used plotting positions are the Hazen plotting position (i – 3 /8)/(n + ẳ) The plotting position (i –
3/8)/(n + ẳ) is a reasonable choice because its use provides a good approximation to the expected value of X i In particular for standard normal variables
Modelling Uncertainty
8.1 Can you modify the deterministic discrete DP reservoir-operating model to include the uncertainty, expressed as P ij t , of the inflows, as in Exercise 7.25?
The operating policy determines the release or final storage for each season based on both the initial storage levels and the inflow rates As inflow conditions fluctuate, the release and final storage volumes may also adjust, necessitating the discretization of both inflows and storage volumes In this model, inflow and storage are treated as state variables, with the ability to predict inflow with certainty at the beginning of each period Consequently, each node in the network reflects a specific initial storage and inflow value However, future inflows can only be estimated probabilistically, creating a dynamic network structure influenced by these variables.
To determine the optimal steady-state operating policy for a reservoir with discrete inflows, we consider two possible inflows, Q it, occurring in two periods each year, each with associated probabilities P it The goal is to minimize the expected sum of squared deviations from the storage target of 4 and the release target of 2, with storage volumes constrained to integer values between 3 and 5 By analyzing the initial reservoir volumes and the known inflows at the start of each period, we can derive an effective release strategy The resulting annual expected sum of squared deviations from both the storage and release targets will provide insight into the reservoir's operational efficiency.
This is an application of Exercise 7.26 except the flow probabilities are independent of the previous flow
To develop a linear model for determining the optimal joint probabilities of predefined discrete initial storage volumes, inflows, and final storage volumes in a reservoir for each period t, we define various indices: k for initial storage volumes (S kt), i for inflows (Q it), l for final storage volumes (S l,t+1), j for discrete inflows (Q j,t+1), and m for discrete final storage volumes (S m,t+2) in the subsequent period t+1 The unknown joint probability, PR kilt, represents the relationship between the initial storage, inflow, and final storage volume in period t, aiming to maximize expected net benefits, denoted as B kilt These net benefits are contingent upon the specific combinations of k, i, and l, along with the conditional inflow probabilities, P ij t = Pr{Q j,t+1|.
Q it}, are known Show how the optimal operating policy can be determined once the values of the joint probabilities, PR kilt, are known
The same policy can be found by DP Develop a DP model to find the optimal operating policy
In Exercise 8.3, instead of using final volume subscripts l and m for calculating joint probabilities PR kilt, we can denote different reservoir release volumes with subscripts d and e This adjustment would impact the linear programming model developed for period i = 1 and time t, necessitating a reevaluation of the model's structure and outcomes.
0.830.71 altered to include d and e in place of l and m? How would the dynamic programming recursion equation be altered?
8.5 Given joint probabilities PR kilt found from Exercise 8.3, how would one derive the probability distribution of reservoir releases and storage volumes in each period t?
Model Sensitivity and Uncertainty Analysis
9.1 Distinguish between sensitivity analysis and uncertainty analysis.
In the allocation model discussed in previous chapters, three water consumers receive allocations \( x_i \) from a total water supply \( Q \) The benefits derived from these allocations are represented by the equations: \( 6x_1 - x_1^2 \) for the first consumer, \( 7x_2 - 1.5x_2^2 \) for the second consumer, and \( 8x_3 - 0.5x_3^2 \) for the third consumer.
Discuss possible sources of uncertainty in model structure and model output, and identify and display parameter sensitivity
9.3 Discuss how model output uncertainty is impacted by both model input uncertainty as well as parameter sensitivity
In water resources studies, significant focus is often placed on uncertainties related to water supplies such as precipitation, streamflows, and evaporation, while less emphasis is given to uncertainties surrounding management objectives, infrastructure costs and benefits, and political support for various decisions To address this imbalance, it is essential to develop a straightforward water resources planning model that incorporates both water quantity and quality management This model can demonstrate that uncertainties in management objectives may have a more substantial impact on outcomes than the hydrologic uncertainties typically analyzed.
Conduct a deterministic sensitivity analysis for Consumer 1 from Exercise 9.2, focusing on three key parameters: Q, 6, and 1, with the latter two representing the benefit function The low, most likely, and high values for these parameters are set at 3, 3, 0.5; 6, 6, 1; and 12, 9, 1.5, respectively Present the findings through a Pareto chart, a tornado diagram, and a spider plot to effectively illustrate the sensitivity of the analysis.
In addressing the water allocation problem outlined in Exercise 9.2, we consider the uncertainty surrounding the available water amount Q, which follows a cumulative probability distribution given by q/(6+q) for q ≥ 0 With an expected value of Q set at 6, an uncertainty analysis can be conducted to approximate the distribution and its implications for effective water management strategies This analysis will help identify potential risks and inform decision-making processes regarding water resource allocation.
• Estimating the mean and standard deviation of the outputs
• Estimating the probability the performance measure will exceed a specific threshold
• Assigning a reliability level on a function of the outputs, e.g., the range of function values that is likely to occur with some probability
• Describing the likelihood of different potential outputs of the system
Show the application of Monte Carlo sampling and analysis, Latin hypercube sampling, generalized likelihood uncertainty estimation and factorial design methods.
Performance Criteria
10.1 Distinguish between multiple purposes and multiple objectives and give some examples of complementary and conflicting purposes and objectives of water resources projects.
Farmers' demand for water, represented as a linear function of price, can be expressed as q = a - bp, where a and b are positive constants To determine their willingness to pay for a specific quantity of water q, we must analyze the cost of delivering that quantity, denoted as cq, where c is a positive constant A public agency should supply water at a level that maximizes the difference between farmers' willingness to pay and total costs In contrast, if a private firm operates the local water district with the goal of maximizing profits, it will supply a different quantity of water, resulting in distinct profit levels Farmers' consumer surplus, defined as their willingness to pay minus the actual payment for water, will vary between these two scenarios A comparison reveals whether farmers incur greater losses than the profits gained by the private firm when transitioning from a socially optimal supply to a profit-maximizing point This relationship can be effectively illustrated with a graph depicting the demand curve and the unit cost of water, highlighting the areas that represent the firm's profits and the farmers' consumer surplus.
In the previous chapters, we examined the water allocation problem, where the returns, B i(X i), from distributing a specific amount of water, X i, among three different uses are analyzed Each use has its own optimal allocation strategy to maximize returns.
Consider this a multi-objective problem Instead of finding the best overall allocation that maximizes the total return assume the objectives are to maximize the returns from each user
The weighting, constraint, goal attainment, and goal programming methods are effective tools for analyzing trade-offs among three objectives within a limited total water supply, such as 6 units By employing these techniques, decision-makers can evaluate the relative importance of each objective, establish constraints based on available resources, and determine the optimal balance to achieve the desired goals This approach not only facilitates a clearer understanding of the interdependencies between objectives but also aids in making informed choices that maximize water utilization.
10.4 Under what circumstances will the weighting and constraint methods fail to identify efficient solutions?
A reservoir is being designed to support irrigation and enhance water quality through low flow augmentation, with an annual storage capacity of 6 million cubic meters The maximum demand for irrigation is set at 4 million cubic meters In this context, let X1 represent the water allocation for irrigation and X2 denote the allocation for downstream flow augmentation, highlighting the dual objectives of water management in this project.
(a) Write the multi-objective planning model using a weighing approach and a constraint approach.
(b) Define the efficient frontier This requires a plot of the feasible combinations of
(c) Assume that various values are assigned to a weight W 1 for Z1 whereas weight W 2 for Z2 is constant and equal to 1, verify the following solutions to the weighing model.
10.6 Show that the following benefit, loss, and cost functions can be included in a linear optimization problem for finding the active storage volume target T s , annual release target
T R and the actual storage releases R t in each within-year period t, and the reservoir capacity
The goal is to optimize the annual net benefits derived from the construction and operation of the reservoir, taking into account the known inflows for each of the 12 periods within a year It is important to note that the loss function related to reservoir recreation does not depend on the value of T s, in contrast to the loss function associated with reservoir releases To achieve this, a comprehensive linear programming model must be structured, clearly defining all variables that are not already specified Additionally, let δ t T R represent the known release target for each period t.
The river basin under consideration has potential reservoir locations at sites 1, 2, and 4, with the possibility of constructing a diversion canal between sites 1 and 2 The cost associated with each reservoir, denoted as C_i(K_i), depends on its active storage capacity K_i Additionally, the cost of the diversion canal is represented by C_i(Q_i), where Q_i signifies the flow capacity of the canal Furthermore, the expense incurred for diverting a flow Q_ijt from site i to site j is also taken into account.
To minimize costs while meeting specific target allocations (demands) T it for users at sites 3 and 5, a model is needed that accounts for the known return flow from use 3, which is 40% of its allocated amount Additionally, the natural streamflows Q i t at each site i during each period t are predetermined, providing essential data for optimizing resource distribution.
10.8 Suppose that there exist two polluters, A and B, who can provide additional treatment,
X A and X B, at a cost of C A(X A) and C B(X B), respectively Let W A and W B be the waste produced at sites A and B, and W A(1 – X A) and W B(1 – X B) be the resulting waste discharges at site A and
Discharges must adhere to the maximum effluent standards E A max and E B max The pollution concentration at various sites, represented by the equation a Aj (W A(1 – X A)) + a Bj (W B(1 – X B)) + q j, should not surpass the stream standards S j max Additionally, it is important to consider total costs and cost inequity, expressed as C A(X A).
C B(X B) and C A(X A) – C B(X B)] are management objectives to be determined
(a) Discuss how you would model this multi-objective problem using the weighting and constraint (or target) approaches.
(b) Discuss how you would use the model to identify efficient, non-inferior (Pareto- optimal) solutions.
Effluent standards at sites A and B, along with ambient stream standards at site J, may be substituted with alternative planning objectives, such as reducing the volume of waste released into the stream These objectives can be integrated into a multi-objective model by defining specific targets for waste minimization and developing strategies to achieve them while balancing environmental and operational considerations.
10.9 (a) What conditions must apply if the goal attainment method is to produce only non- inferior alternatives for each assumed target T k and weight w k ?
(b) Convert the goal programming objective deviation components w i (z i * −z i ( x ) ) to a form suitable for solution by linear programming.
Water quality objectives can be challenging to measure accurately, leading to various efforts to consolidate the diverse aspects of water quality into unified indices One notable index was introduced by Dinius, focusing on social accounting systems for the evaluation of water quality.
Resources, Water Resources Research, Vol 8, 1972 pp 1159-1177) Water quality, Q, measured in percent is given by n n n w w w
The use of a quality index in multi-objective water resources planning, represented by the equation Q_i, where Q_i denotes the i-th quality constituent such as dissolved oxygen or chlorides, and w_i reflects its relative importance, offers a structured approach to assess water quality However, this method can be critiqued for potentially oversimplifying complex water quality dynamics, as it may not adequately capture the interactions between different constituents Additionally, the subjective determination of weights (w_i) can introduce bias, leading to inconsistent evaluations Therefore, while the index serves as a useful tool, its limitations must be acknowledged to ensure comprehensive water resource management.
10.11 Let objective Z1(X) = 5X 1 – 2X 2 and objective Z2(X) = - X 1 + 4 X 2 Both are to be minimized Assume that the constraints on variables X1 and X2 are:
(a) Graph the Pareto-optimal or non-inferior solutions in decision space.
(b) Graph the efficient combination of Z1 and Z2 in objective space.
(c) Reformulate the problem to illustrate the weighting method for defining all efficient solutions of part (a) and illustrate this method in decision and objective space.
(d) Reformulate the problem to illustrate the constraint method of defining all efficient solutions of part (a) and illustrate this method in decision and objective space.
(e) Solve for the compromise set of solutions using compromise programming as defined by
Minimize [w 1(Z 1 *- Z 1) α + w 2(Z 2 *- Z 2) α ] 1/α where Z i * represents the best value of objective i with all weights w equal to 1 and α equal to 1, 2, and ∞
To select among three plans, each with three objectives, we utilize indifference analysis, where Z_ji denotes the value of objective i for plan j Given the values for each objective across the plans, we aim to maximize each objective An identical indifference function suggests a willingness to trade off twice the units of a higher objective for one unit of a lower objective For instance, if we compare two plans with objective values (30, 5, 10) and (20, 5, 15), we find them equally preferable By applying this method, we can effectively rank the three plans based on their respective objective values, establishing a clear order of preference.
River Basin Planning Models
To predict the natural unregulated flow at any site along the Han River in South Korea, utilize the drainage area and discharge data to develop a predictive equation By plotting average flow against catchment area, the relationship can be analyzed, revealing that the slope of this function represents the flow per unit area, which is critical for understanding river dynamics and managing water resources effectively.
In watersheds with substantial elevation changes, it is possible to create predictive equations that estimate average annual runoff per hectare based on elevation To estimate the natural average annual flow at any gage within such a watershed, one would apply these equations by inputting the specific elevation of the gage location This approach is particularly effective in areas where water loss from stream channels due to evaporation or seepage is minimal, allowing for accurate predictions of flow based on the elevation-driven runoff calculations.
11.3 Compute the storage yield function for a single reservoir system by the mass diagram and modified sequent peak methods given the following sequences of annual flows: (7, 3, 5,
The analysis reveals that each year consists of two distinct hydrologic seasons: a wet season (t = 1) and a dry season (t = 2) It is observed that 80% of the annual inflow occurs during the wet season, while 80% of the desired yield is required in the dry season By employing the modified sequent peak method, the study demonstrates the increased storage capacity needed to maintain the same annual yield due to the redistribution requirements within the year.
Each year is divided into a wet season (1) and a dry season (2), with 80% of inflow occurring during the wet season and 80% of the desired yield in the dry season By applying the modified sequent peak method, we can demonstrate the increased storage capacity needed to maintain the same annual yield due to the redistribution requirements within the year This approach highlights the critical balance between seasonal inflow and yield demands, emphasizing the need for enhanced storage solutions to optimize water resource management.
To estimate the maximum constant reservoir release or yield \( Y \) given a fixed reservoir capacity \( K \), a linear programming model can be formulated that maximizes \( Y \) while adhering to the constraints of historical flow data over \( T \) time periods Conversely, to determine the minimum reservoir capacity \( K \) required for a specified yield \( Y \), another linear programming model can be developed that minimizes \( K \) under the same flow constraints These models can be utilized to create a storage capacity-yield function, which illustrates the relationship between the available yield \( Y \) and the corresponding reservoir capacity \( K \).
To develop an optimization model for determining the least-cost combination of active storage capacities, K1 and K2, for two reservoirs on a single stream, we aim to ensure a reliable constant annual flow downstream Given that the cost functions Cs(Ks) for each reservoir are predefined and considering that there is no dead storage or evaporation, we will utilize 10 years of monthly unregulated flow data for analysis The model will maintain the cost functions in their original form without linearization, focusing on optimizing the storage capacities to minimize costs while meeting the required flow yield.
First bridge of the Han River
(b) Describe the two-reservoir operating policy that you would incorporate into a simulation model to check the solution obtained from the optimization model.
11.6 Given the information in the accompanying tables, compute the reservoir capacity that maximizes the net expected flood damage reduction benefits less the annual cost of reservoir capacity.
11.7 Develop a deterministic, static, within-year model for evaluating the development alternatives in the river basin shown in the accompanying figures Assume that there are t = 1,
The objective of the model is to maximize the total annual net benefits within a defined basin over multiple periods This involves determining the reservoir capacities, including both active and flood storage, as well as the annual allocation targets Additionally, it requires calculating the levee capacity necessary to protect site 4 from a T-year flood and the specific water allocations for sites 3 and 7 throughout the year All variables and functions must be clearly defined, and a solution method should be outlined to achieve the maximum net benefit.
FLOOD STAGE FOR FLOOD OF
10 a 25 30 40 70 a 10 is fixed cost if capacity > 0; otherwise, it is 0.
Levee Potential flood damage site Irrigation area
In this simplified model, we assume there are no evaporation losses or dead storage requirements We define key variables: T represents the target, K is the reservoir capacity, D indicates the deficit, E denotes the excess, and P stands for the power plant capacity The natural streamflows are represented by Q t, while R t indicates the reservoir releases, and S t reflects the initial reservoir storage volumes during period t Additionally, K f signifies the flood storage capacity at site 2, which is designed to accommodate a peak flow of QS, while QR represents the flood flow capacity of the downstream channel.
The function κ(QS) defines the relationship between QS and K f, while the unregulated design flood peak flow that necessitates protection is represented as QN Additionally, KWH denotes the kilowatt hours of energy, H signifies the net storage head, and ht refers to the hours within a specific period t Furthermore, the variable q represents the allocation of water supply.
Benefit functions will be B( ), L( ) will denote loss functions and C( ) will denote the cost functions
11.8 List the potential difficulties involved when attempting to structure models for defining:
(a) Water allocation policies for irrigation during the growing season.
(b) Energy production and capacity of hydroelectric plants.
(c) Dead storage volume requirements in reservoirs.
(d) Active storage volume requirements in reservoir.
(e) Flood storage capacities in reservoirs.
(f) Channel improvements for flood damage reduction.
(g) Evaporation and seepage losses from reservoirs.
(h) Water flow or storage targets using long-run benefits and short- run loss functions.
To address the anticipated growth in water supply demand, modeled as t(60 - t) over the next 30 years, it is essential to determine the minimum present value of construction costs for a selection of water supply options Given that the current water supply network lacks excess capacity, immediate project implementation is necessary The capacities of these projects are independent and can be aggregated Utilizing a discount factor of exp(-0.07t), careful analysis will yield the optimal solution for meeting future demand effectively.
Potential reservoir for water supply, Potential reservoir for water supply, flood control
Diversion to a use, 60% of allocation returned to river
Existing development, possible flood protection from levee
Potential reservoir for water supply, recreation
Hydropower; plant factor = 0.30 Potential diversion to an irrigation district
To create a simulation model for defining a storage-yield function of a single reservoir with known monthly inflows over n years, construct a flow diagram that outlines the process The model should incorporate evaporation rates and the relationship between storage volume and surface area To achieve a steady-state solution that is independent of the initial storage volume, implement a method that allows the system to stabilize over time, ensuring that the results reflect consistent reservoir behavior rather than the influence of arbitrary starting conditions.
To estimate the probability that a specific reservoir capacity, K, will meet a series of known downstream release demands, r_t, in the face of uncertain future inflows, i_t, a flow diagram for a simulation model can be utilized This model will outline the process of assessing various scenarios of inflow sequences and their impact on reservoir performance The simulation will focus on evaluating the relationship between inflow variability and the ability to satisfy release demands, ultimately providing insights into the reliability of reservoir capacity under changing conditions.
To create a cost-effective flood protection strategy, an optimization model is developed that balances the storage capacity of an upstream reservoir with necessary channel improvements at a downstream site The model aims to safeguard the downstream area from a specified design flood with a return period of T Key variables and functions are defined to quantify the relationship between storage capacity, channel enhancements, and flood risk mitigation, ensuring optimal resource allocation for effective flood management.
(b) How could this model be modified to consider a number of design floods T and the benefits from protecting the potential damage site from those design floods? Let
BF T be the annual expected flood protection benefits at the damage site for a flood having return periods of T.
(c) How could this model be further modified to include water supply requirements of
A t to be withdrawn from the reservoir in each month t? Assume known natural flows s
Q t at each site s in the basin in each month t.
To enhance the model by incorporating recreational benefits or losses at the reservoir site, let T_s represent the target storage volume Define D_t_s as the difference between the actual storage volume S_t_s and the target T_s when S_t_s exceeds T_s, while E_t_s indicates the difference when T_s surpasses S_t_s The annual recreational benefits B(T_s) are influenced by the target storage volume T_s, with losses L_D D(t_s) and L_E E(t_s) arising from the deficit D_t_s and excess E_t_s in storage volumes.
To optimize reservoir capacity, flood storage, and recreation volume, we must develop a linear programming model that maximizes the combined annual expected benefits from flood control and recreation, while accounting for losses due to deficits and excesses during the recreation season, as well as the costs associated with reservoir capacity The model requires a constant yield of 30 in each period, with the flood season spanning from period 3 to period 6.
B(T) = 9T where T is a particular unknown value of
11.13 The optimal operation of multiple reservoir systems for hydropower production presents a very nonlinear and often difficult problem
Use dynamic programming to determine the operating policy that maximizes the total annual hydropower production of a two-reservoir system, one downstream of the other The releases