FA23_CEE434_HW4

CEE 434 Environmental Systems I, Fall 2023 Homework #4 (GA, Reservoir Modeling) Due Nov. 19 th Problem 1 (60 points). Recall the water allocation example that we discussed in the class (GA Lecture). The problem is to determine the optimal values of diversions to the three firms located along the river to maximize the total net benefits obtained from the firms. The total amount of water available for the firms is limited to 50% of the river flow in the upstream (Q). i) (10 points) Formulate the optimization problem. ii) (20 points) Assume that ࠵? = 16 . Use the MATLAB function and script provided for that example and solve the optimization problem using the GA parameters in the table below (do not change other parameters in the options). For each case presented in the table below, run the GA three times and complete the solution tables (see next page). N (Population Size) G (generations) Crossover Fraction Case 1 4 5 0 Case 2 4 10 0 Case 3 4 10 0.20 Case 4 4 10 0.80 Case 5 10 50 0.20 Case 6 10 50 0.80 Case 7 20 100 0.20 Case 8 20 100 0.80 iii) (10 points) Discuss the impact of GA parameters (N, G, Crossover fraction) iv) (10 points) Based on your results, pick the best combination of GA parameters, and solve the optimization problem for ࠵? = 8 , and ࠵? = 32 . Discuss the impact of river flow constraint on each firm’s net benefit and total net benefit. v) (5 points) Assume that you are NOT able to insert the river flow constraint into GA formulation in MATLAB. Suggest another way that you can account for the constraint in GA optimization.

Problem I i NB M 6x_x 2 N13242 7 ㄨ 2 15 ㄨ 22 NB 了 ⼼ 8 ㄨ 3 05 ㄨ 了 2 objective function max È N Bi Xi constraints xtxvtx.CO5Q ii see below iii The optimization solution becomes more converged as N and G increase Excessive crossover fraction maydisrupt potential good solutions before they dominate the population while a too small crossover fraction can result in the loss of valuable solutions iv N 50 G 100 crossover fraction o 2 Q 8 254102 25.4105 05017 0.6035 28958 25.4102 25.4097 ⼼ 4106 o.mg 29685 v5.4094 25.4016 0.3964 00481 29564 以 4073 5.4073 a 2 49.1607 49.1607 3 以 8 49.1607 49.1607 3 2.33 8 49.1607 49.1607 233 8 49.1607 49.1607 Qj the influence of 1 unit of flow constraint on net benefit and total benefit l

vi) (5 points) You saw this example in Nonlinear Programming (NLP) and Dynamic Programming (DP) lectures as well. a. Discuss the differences between GA, NLP and DP in solving this problem b. Comment on the advantages and disadvantages of each optimization method Solution tables Case 1 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A Case 2 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A Case 3 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A

Case 4 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A Case 5 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A Case 6 Best Fitness Average Fitness x1 x2 x3 Run 1 Run 2 Run 3 Average N/A N/A N/A Case 7 Best Fitness Average Fitness x1 x2 x3 Run 1

Problem 3 (60 points). Use the inflow data provided in Problem 2 to formulate and solve an optimization model for three cases: 1 ) given reservoir capacity K = 15 cms-month, find the firm yield Y; 2 ) given the firm yield as Y=6.5 m 3 /s, find K. Use GAMS or other programming tool to solve the problem; 3 ) assuming the reservoir storage is K = 18 cms-month, optimize annual water supply assuming the annual demand is 80 cms-month and monthly water demand coefficient (monthly fractions) are 0.06, 0.06, 0.07, 0.08, 0.09, 0.10, 0.13, 0.11, 0.09, 0.08, 0.07, and 0.06 for January to December, respectively; the minimum reservoir storage is 1.5 cms-month, and the initial reservoir storage is 5 cms-month.