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Water Resources Planning and Management Daene C. McKinney Optimization

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Reservoirs Hoover Dam 158 m 35 km3 2,074 MW Grand Coulee Dam 100 m 11.8 km3 6,809 MW Toktogul Dam 140 m 19.5 km3 1,200 MW

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Dams Masonry dams –Arch dams Gravity dams Embankment dams rock-fill and earth-fill dams Spillways

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Reservoir QtQt RtRt StSt K RtRt K StSt QtQt

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Example Allocate reservoir release R t to 3 users and provide instream flow Q t Operating Policy Allocation Policy release R t inflow I t storage S t

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Optimization Benefit Decision variables Objective: Constraints: Optimization model

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Simulation Operating Policy Allocation Policy

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Simulation vs Optimization Simulation models: Predict response to given design Optimization models: Identify optimal designs or policies

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Modeling Process Problem identification –Important elements to be modeled –Relations and interactions between them –Degree of accuracy Conceptualization and development –Mathematical description –Type of model –Numerical method - computer code –Grid, boundary & initial conditions Calibration –Estimate model parameters –Model outputs compared with actual outputs –Parameters adjusted until the values agree Verification –Independent set of input data used –Results compared with measured outputs Problem identification and description Model verification & sensitivity analysis Model Documentation Model application Model calibration & parameter estimation Model conceptualization Model development Data Present results

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Example – Water Users Allocate release to users and provide instream flow Obtain benefits from allocation of x i, i = 1,2,3 B i (x i ) = benefit to user i from using amount of water x i

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Example Decision variables: Note: if sufficient water is available the allocations are independent and equal to How? Objective: Optimization model: Constraint:

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Optimization Problems Objective function Decision variables Constraint set

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Optimization Problems while satisfying constraints x f(x)f(x) x* minimum x* x f(x)f(x) X ab X={x: a<x< b} Feasible region Find the decision variables, x, that optimize (maximize or minimize) an objective function

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Example

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Existence of Solutions Weierstrass Theorem –Describes conditions on the objective function and the constraint set so that we are guaranteed that solutions will always exist Constraint set is compact (closed and bounded) Objective function is continuous on the constraint set x* x f(x)f(x) X ab X={x: a<x< b} Feasible region

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Convex Sets convex nonconvex x y x y If x and y are in the set, then z is also in the set, i.e., don’t leave the set to get from x to y

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Convex Functions Line segment joining points on a convex function does not lie below the function Linear functions are convex.

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Existence of Global Solutions Local-Global Theorem (maximization) –Describes conditions for a local solution to be global Constraint set is compact and convex Objective function is continuous on the constraint set and concave Then a local maximum is global x f(x)f(x) x* Global maximum Concave function X

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Solutions – Global or Local? Global Max Local Max

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Solutions Local - Global Theorem: 1.If X is convex and f(x) is a convex function, then a local minimum is a global minimum x f(x)f(x) x* Global minimum Convex function X x f(x)f(x) x* Global maximum Concave function X 2.If X is convex and f(x) is a concave function, then a local maximum is a global maximum

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Types of Optimization Problems Nonlinear Program Linear Program Classic Program

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No Constraints Single Decision Variable First-order conditions for a local optimum x f(x)f(x) x* Global minimum Convex function X Second-order conditions for a local optimum No constraints Tangent is horizontal Curvature is upward Scalar

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No Constraints Multiple Decision Variables First-order conditions for a local optimum n - simultaneous equations No constraints Vector

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Classical Program General Form Example All equality constraints

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Single Constraint Multiple Decision Variables One constraint Vector

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Single Constraint Multiple Decision Variables Lagrangian First-order conditions N+1 equations

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Example Lagrangean First – order conditions Notice the signs

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Example Decision variables: Note: if sufficient water is available the allocations are independent and equal to How? Objective: Optimization model: Constraint:

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Example Lagrangean First – order conditions Notice the signs

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Example Equal marginal benefits (slopes) for all users

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Release Allocation Rule Allocation rule tells you the amount of released water allocated to each use

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Classical Programming Vector Case – Multiple Constraints Lagrangian First-order conditions N+M equations M constraints Vector N equations M equations

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Example From Revelle, C. S., E. E. Whitlach, and J. R. Wright, Civil and Environmental Systems Engineering, Prentice Hall, Upper Saddle River, 1997

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Nonlinear Program General FormExample

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Reservoir with Power Plant Hoover Dam earliest known dam - Jawa, Jordan - 9 m high x1 m wide x 50 m long, 3000 BC

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Reservoir with Power Plant QtQt RtRt StSt K EtEt RtRt K StSt EtEt HtHt QtQt

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Reservoir with Power K QtQt RtRt StSt EtEt Q Inflows 3 Q t Inflows (L 3 /time period) S t Storage volume 3 S t Storage volume (L 3 ) K Capacity 3 K Capacity (L 3 ) R t Release ( 3 /period) R t Release (L 3 /period) E t Energy (kWh) H t Head (L) k Coefficient (efficiency, units) Maximize power production given capacity and inflows Nonlinear

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