CIVE 646 Problems, Fall 2011

Water Resource Systems Planning and Management Daniel P. Loucks & Eelco van Beek Exercises Content Chapter 1 Water Resources Planning and Management: an overview..........................1 Chapter 2 Water Resource Systems Modelling: its role in planning and management 2 Chapter 3 Modelling methods for Evaluating Alternatives ..........................................3 Chapter 4 Optimization Methods..................................................................................7 Chapter 5 Fuzzy Optimization....................................................................................23 Chapter 6 Data-Based Models ....................................................................................24 Chapter 7 Concepts in Probability, Statistics and Stochastic Modelling....................26 Chapter 8 Modelling Uncertainty ...............................................................................42 Chapter 9 Model Sensitivity and Uncertainty Analysis..............................................44 Chapter 10 Performance Criteria ................................................................................45 Chapter 11 River Basin Planning Models...................................................................49 Chapter 12 Water Quality Modelling and Prediction .................................................60 Chapter 13 Urban Water Systems...............................................................................65 Chapter 14 A Synopsis................................................................................................67 Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 1 Chapter 1 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? 1.3 Comment on the common practice of governments giving aid to those in drought or flood areas without any incentives to alter land use management practices in anticipation of the next flood or drought. 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? 1.6 Find the following statistics: · % freshwater resources worldwide available for drinking: · Number of people who die each 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. 1.8 Identify some of the major water resource management issues in the region where you live. What management alternatives might effectively reduce some of the problems or provide additional economic, environmental, or social benefits. 1.9 Describe some water resource systems consisting of various interdependent components. What are the inputs to the systems and what are their outputs? How did you decide what to include in the system and what not to include? How did you decide on the level of spatial and temporal detail to be included? 1.10 Sustainability is a concept applied to renewable resource management. In your words define what that means and how it can be used in a changing and uncertain environment both with respect to water supplies and demands. Over what space and time scales is it applicable, and how can one decide whether or not some plan or management policy will be sustainable? How does this concept relate to the adaptive management concept? 1.11 Identify and discuss briefly some of the major issues and challenges facing water managers today. Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 2 Chapter 2 Water Resource Systems Modelling: its role in planning and management 2.1 What is a system? 2.2 What is systems analysis? 2.3 What is a mathematical model? 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? 2.8 Identify some specific water resource systems planning problems and for each problem specify in words possible objectives, the unknown decision variables whose values need to be determined, and the constraints or that must be met by any solution of the problem. 2.9 From a review of the recent issues of various journals pertaining to water resources and the appropriate areas of engineering, economics, planning and operations research, identify those journals that contain articles on water resources systems planning and analysis, and the topics or problems currently being discussed. 2.10 Many water resource systems planning problems involve considerations that are very difficult if not impossible to quantify, and hence they cannot easily be incorporated into any mathematical model for defining and evaluating various alternative solutions. Briefly discuss what value these admittedly incomplete quantitative models may have in the planning process when non-quantifiable aspects are also important. Can you identify some planning problems that have such intangible objectives? 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? 2.14 Suppose you live in an area where the only source of water (at a reasonable cost) is from an aquifer that receives no recharge. Briefly discuss how you might develop a plan for its use over time. Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 3 Chapter 3 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 (103 kwh/day) and irrigation supplies (106 m3/month): Alternative Energy Production Irrigation Supply A 22 20 B 10 35 C 20 32 D 12 21 E 6 25 Which alternative would be the best in your opinion and why? Why might a decision maker select alternative E even realizing other alternatives exist that can give more hydropower energy and irrigation supply? 3.7 Define a model similar to Equations 3.1 to 3.3 for finding the dimensions of a cylindrical tank that minimizes the total cost of storing a specified volume of water. What are the unknown decision variables? What are the model parameters? Develop an iterative approach for solving this model. 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. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (days) Inflow Outflow Flows (m3/day) Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 4 a) When was the inflow its maximum and minimum values? b) When was the outflow its minimum value? c) When was the storage volume its maximum value? d) When was the storage volume its minimum value? e) Write a mass balance equation for the time series of storage volumes assuming constant inflows and outflows during each time 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. 3.12 Given the changing inflows and constant outflow from a tank or reservoir, as shown in the graph below, sketch a plot of the storage volumes over the same period of time. Show how to determine the value of the slope of the storage volume plot at any time from the inflow and outflow graph below. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (days) Flows (m3/day) 100 50 0 Inflow Outflow Change in Storage (m3) 300 150 0 -150 -300 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (days) Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 5 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. 3.14 How many different simulations of a water resource system would be required to ensure that there is at least a 95% chance that the best solution obtained is within the better 5% of all possible solutions that could be obtained? What assumptions must be made in order for your answer to be valid? Can any statement be made comparing the value of the best solution obtained from the all the simulations to the value of the truly optimal solution? 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 20 projects would maximize the net benefits to the region. Assuming 5 minutes of computer time is required to simulate and evaluate each combination of projects, show that it would require 36 days of computer time even if 99% of the alternative combinations could be discarded using “good judgment.” What does this suggest about the use of simulation for regional interdependent multiproject water resources planning? 3.16 Assume you wish to determine the allocation of water Xj to three different users j, who obtain benefits Rj(Xj). The total water available is Q. Write a flow chart showing how you can find the allocation to each user that results in the highest total benefits. 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. Remainder goes downstream. Year y Flow Qy Year y Flow Qy 1 5 9 3 2 7 10 6 3 8 11 8 4 4 12 9 5 3 13 3 6 3 14 4 7 2 15 9 8 1 User 1 User 2 User 3 Gage site Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 6 a) Assume no incremental flow along the stream and no return flow from users. Define the allocation policy at each site. This will be a graph of allocation as a function of the flow at the allocation site. b) Simulate this allocation policy using any river basin simulation model such as RIBASIM, WEAP, Modsim, or other selected model (see CD) for any specified inflow series ranging from 0 to 130 units. Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 7 Chapter 4 Optimization Methods Engineering economics: 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? (d) Assuming that each of these projects would be replaced with a similar project having the same time stream of costs and returns, show that by extending each series of projects to a common terminal year (e.g., 30 years), the annual net benefits of each series of projects would be will be same as found in part (b). 4.2 Show that 1 (1 ) T t t r - = + å T T r r r ) 1( 1 ) 1( + - + = . 4.3 a) Show that if compounding occurs at the end of m equal length periods within a year in which the nominal interest rate is r, then the effective annual interest rate, r’, is equal to 1 1 - ÷ ø ö ç è = æ + ¢ m m r r b) Show that when compounding is continuous (i.e., when the number of periods m® ¥), the compound interest factor required to convert a present value to a future value in year T is erT. [Hint: Use the fact that limk® ¥ (1 + 1/k)k = e, the base of natural logarithms.] 4.4 The term “capitalized cost” refers to the present value PV of an infinite series of end-of- year equal payments, A. Assuming an interest rate of r, show that as the terminal period T ® ¥, PV = A/r. 4.5 The internal rate of return of any project is or plan is the interest rate that equals the present value of all receipts or income with the present value of all costs. Show that the internal rate of return of projects A and B in Exercise 4.1 are approximately 8 and 6%, respectively. These are the interest rates r, for each project, that essentially satisfy the equation ( ) ) ( 0 1 0 = + - - =å t T t t t r C R 4.6 In Exercise 4.1, the maximum annual benefits were used as an economic criterion for plan selection. The maximum benefit-cost ratio, or annual benefits divided by annual costs, is another criterion. Benefit-cost ratios should be no less than one if the annual benefits are to exceed the annual costs. Consider two projects, I and II: Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 8 What additional information is needed before one can determine which project is the most economical project? 4.7 Bonds are often sold to raise money for water resources project investments. Each bond is a promise to pay a specified amount of interest, usually semiannually, and to pay the face value of the bond at some specified future date. The selling price of a bond may differ from its face value. Since the interest payments are specified in advance, the current market interest rates dictate the purchase price of the bond. Consider a bond having a face value of $10,000, paying $500 annually for 10 years. The bond or “coupon” interest rate based on its face value is 500/10,000, or 5%. If the bond is purchased for $10,000, the actual interest rate paid to the owner will equal the bond or “coupon” rate. But suppose that one can invest money in similar quality (equal risk) bonds or notes and receive 10% interest. As long as this is possible, the $10,000, 5% bond will not sell in a competitive market. In order to sell it, its purchase price has to be such that the actual interest rate paid to the owner will be 10%. In this case, show that the purchase price will be $6927. The interest paid by the some bonds, especially municipal bonds, may be exempt from state and federal income taxes. If an investor is in the 30% income tax bracket, for example, a 5% municipal tax-exempt bond is equivalent to about a 7 % taxable bond. This tax exemption helps reduce local taxes needed to pay the interest on municipal bonds, as well as providing attractive investment opportunities to individuals in high tax brackets. Lagrange Multipliers 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, xj, to each use j provides the following returns: R(x1) = (12x1 – x1 2), R(x2) = (8x2 – x2 2) and R(x3) = (18x3 – 3x3 2). Assume that the objective is to maximize the total return, F(X), from all three allocations and that the sum of all allocations cannot exceed 10. a) How much would each use like to have? b) Show that at the maximum total return solution the marginal values, ¶(R(xj))/ ¶xj, are each equal to the shadow price or Lagrange multiplier (dual variable) associated with the constraint on the amount of water available. c) Finally, without resolving a Lagrange multiplier problem, what would the solution be if 15 units of water were available to allocate to the three users and what would be the value of the Lagrange multiplier? 4.10 In Exercise 4.9, how would the Lagrange multiplier procedure differ if the objective function, F(X), were to be minimized? Project I II Annual benefits 20 2 Annual costs 18 1.5 Annual net benefits 2 0.5 Benefit-cost ratio 1.11 1.3 Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 9 4.11 Assume that the objective was to minimize the sum of squared deviations of the actual allocations xj from some desired or known target allocations Tj. Given a supply of water Q less than the sum of all target allocations Tj, structure a planning model and its corresponding Lagrangian. Will a global minimum be obtained from solving the partial differential equations derived from the Lagrangian? Why? 4.12 Using Lagrange multipliers, prove that the least-cost design of a cylindrical storage tank of any volume V > 0 has one-third of its cost in its base and top and two-thirds of its cost in its side, regardless of the cost per unit area of its base or side. (It is these types of rules that end up in handbooks in engineering design.) 4.13 An industrial firm makes two products, A and B. These products require water and other resources. Water is the scarce resource-they have plenty of other needed resources. The products they make are unique, and hence they can set the unit price of each product at any value they want to. However experience tells them that the higher the unit price for a product, the less amount of that product they will sell. The relationship between unit price and quantity that can be sold is given by the following two 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 Tmax. What are the amounts of A and B, and their unit prices, that maximize total revenue, if i) Tmax = 10 ii) Tmax = 5 Water is needed to make each unit of A and B. The production functions relating the amount of water XA needed to make A, and the amount of water XB needed to make B are A = 0.5 XA, and B = 0.25 XB, respectively. (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? Dynamic programming 4.14 Solve for the optimal integer allocations x1, x2, and x3 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 xj must not exceed 4. Quantity of product A Quantity of product B Po Unit price Po Unit Price 8 – A 6 – 1.5B Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 10 4.15 Consider a three-season reservoir operation problem. The inflows are 10, 50 and 20 in seasons 1, 2 and 3 respectively. Find the operating policy that minimizes the sum of total squared deviations from a constant storage target of 20 and a constant release target of 25 in each of the three seasons. Develop a discrete dynamic programming model that considers only 4 discrete storage values: 0, 10, 20 and 30. Assume the releases cannot be less than 10 or greater than 40. Show how the model’s recursive equations change depending on whether the decisions are the releases or the final storage volumes. Verify the optimal operating policy is the same regardless of whether the decision variables are the releases or the final storage volumes in each period. Which model do you think is easier to solve? How would each model change if more importance were given to the desired releases than to the desired storage volumes? 4.16 Show that the constraint limiting a reservoir release, rt, to be no greater than the initial storage volume, st, plus inflow, it, is redundant to the continuity equation st + it – rt = st+1. 4.17 Develop a general recursive equation for a forward-moving dynamic programming solution procedure for a single reservoir operating problem. Define all variables and functions used. Why is this not a very useful approach to finding a reservoir operating policy? 4.18 The following table provides estimates for the recent values of the costs of additional wastewater treatment plant capacity needed at the end of each 5-year period for the next 20 years. Find the capacity expansion schedule that minimizes the present values of the total future costs. If there is more than one least cost solution, indicate which one you think is better, and why? The cost in each period t must be paid at the beginning of the period. What was the discount factor used to convert the costs at the beginning of each period t to present value costs shown above. In other words how would a cost at the beginning of period t be discounted to the beginning of period 1, given an annual interest rate of r? (Only the algebraic expression of the discount factor is asked, not the numerical value of r.) 4.19 Consider a wastewater treatment plant in which it is possible to include five different treatment processes in series. These treatment processes must together remove at least 90% of the 100 units of influent waste. Assuming the Ri is the amount of waste removed by process i, the following conditions must hold: 20 £ R1 £ 30 0 £ R2 £ 30 0 £ R3 £ 10 0 £ R4 £ 20 0 £ R5 £ 30 DISCOUNTED COST OF ADDITIONAL CAPACITY: UNITS OF ADDITIONAL CAPACITY Period Years 2 4 6 8 10 1 1-5 12 15 18 23 26 2 2 6-10 8 11 13 15 6 3 11-15 6 8 8 4 16-20 4 10 Additional Required Capacity at End of Perioda aThis is the total required capacity that must exist at the end of period. Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 11 (a) Write the constrained optimization-planning model for finding the least-cost combination of the removals Ri that together will remove 90% of the influent waste. The cost of the various discrete sizes of each unit process i depend upon the waste entering the process i as well as the amount of waste removed, as indicated in the table below. (b) Draw the dynamic programming network and solve this problem by dynamic programming. Indicate on the network the calculations required to find the least- cost path from state 100 at stage 1 to state 10 at stage 6 using both forward- and backward-moving dynamic programming solution procedures. (c) Could the following conditions be included in the original dynamic programming model and still be solved without requiring R4 to be 0 in the first case and R3 to be 0 in the second case? (i) R4 = 0 if R3 = 0, or (ii) R3 = 0 if R2 £ 20. 4.20 The city of Eutro Falls is under a court order to reduce the amount of phosphorus that which it discharges in its sewage to Lake Algae. The city presently has three wastewater treatment plants. Each plant i currently discharges Pt kg/day of phosphorus into the lake. Some or all plants must reduce their discharges so that the total for the three plants does not exceed P kg/day. Let Xt be the percent of the phosphorus by additional treatment at plant i, and the Ci(Xi) the cost of such treatment ($/year) at each plant i. a) Structure a planning model to determine the least cost (i.e., a cost effective) treatment plant for the city. PROCESS i: Influent, Removal, Ii Ri 100 20 5 100 30 10 1 2 3 4 5 Annual Cost = Ci(Ii, Ri) 80 10 3 3 1 80 20 9 2 80 30 13 70 10 4 5 2 70 20 10 3 70 30 15 60 10 6 2 3 60 20 4 6 60 30 9 50 10 7 3 4 50 20 5 8 50 30 10 40 10 8 5 5 40 20 7 12 40 30 18 30 10 8 8 30 20 10 12 20 10 8 Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 12 b) Restructure the model for the solution by dynamic programming. Define the stages, states, decision variables, and the recursive equation for each stage. c) Now assume P1 = 20; P2 = 15; P3 = 25; and P = 20. Make up some cost data and check the model if it works. 4.21 Find (draw) a rule curve for operating a single reservoir that maximizes the sum of the benefits for flood control, recreation, water supply and hydropower. Assume the average inflows in four seasons of a year are 40, 80 60, 20, and the active reservoir capacity is 100. For an average storage S and for a release of R in a season, the hydropower benefits are 2 times the square root of the product of S and R, 2(S R)0.5 and the water supply benefits are 3R0.7 in each season. The recreation benefits are 40-(70-S)2 in the third season. The flood control benefits are 20 – (40 – S)2 in the second season. Specify the dynamic programming recursion equations you are using to solve the problem. 4.22 How would the model defined in Exercise 4.21 change if there were a water user upstream of this reservoir and you were to find the best water allocation policy for that user, assuming known benefits associated with these allocations that are to be included in the overall maximum benefits objective function? 4.23 Suppose there are four water users along a river who benefit from receiving water. Each has a water target, i.e., each expects and plans for a specified amount. These known water targets are W(1), W(2), W(3), and W(4) for the four users respectively. Show how dynamic programming can be used to find two allocation policies. One is to be based on minimizing the maximum deficit deviation from any target allocation. The other is to be based on minimizing the maximum percentage deficit from any target allocation. 4-24 An industrial firm makes two products, A and B. These products require water and other resources. Water is the scarce resource-they have plenty of other needed resources. The products they make are unique, and hence they can set the unit price of each product at any value they want to. However experience tells them that the higher the unit price for a product, the less amount of that product they will sell. The relationship between unit price and quantity that can be sold is given by the following two demand functions. (a) What are the amounts of A and B, and their unit prices, that maximize the total revenue that can be obtained? (You can use calculus to solve this problem if you wish.) (b) Suppose the total amount of A and B could not exceed some amount Tmax. What are the amounts of A and B, and their unit prices, that maximize total revenue, if iii) Tmax = 10 iv) Tmax = 5 Quantity of product A Quantity of product B Po Unit price Po Unit Price 8 – A 6 – 1.5B Water Resources Systems Planning and Management Exercises Daniel P. Loucks & Eelco van Beek 13 Water is needed to make each unit of A and B. The production functions relating the amount of water XA needed to make A, and the amount of water XB needed to make B are A = 0.5 XA, and B = 0.25 XB, respectively. (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. Use discrete dynamic programming, both forward- and backward-moving algorithms. You can assume integer values of each water allocation X for this exercise. Show your work on a network. For the backward moving algorithm, also show your work using tables showing the state, Si, the possible decision variables XA and XB and their values, the best decision, and the best value, Fi(Si), associated with the best decision. Gradient “Hill-climbing” methods 4.25 Solve Exercise 4.24(b) using hill-climbing techniques and assuming discrete integer values and Tmax = 5. For example, which product would you produce if you could make only 1 unit of either A or B? If you could make another unit of A or B, which would you make? Continue this process up to 5 units of products A and/or B. Linear and non-linear programming 4.26 Consider the industrial firm that makes two products A and B as described in Exercise 4.24(b). Using Lingo (or any other program you wish): (a) 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. (b) What is the value of the dual variable, or shadow price, associated with the 10 units of available water? (c) Suppose the demand functions are not really certain. How sensitive are the allocations of water to the parameter values in those functions? How sensitive are the allocations to the parameter values 0.5 and 0.25 in the production functions? 4.27 Assume that there are m industries or municipalities adjacent to a river, which discharge their wastes into the river. Denote the discharge sites by the subscript i and let Wi be the kg of waste discharged into the river each day at those sites i. To improve the quality downstream, wastewater treatment plants may be required at each site i. Let xi be the fraction of waste removed by treatment at each site i. Deve