# Water Resources Planning and Management Daene C. McKinney Optimization.

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Water Resources Planning and Management Daene C. McKinney Optimization

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

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

Reservoir QtQt RtRt StSt K RtRt K StSt QtQt

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

Optimization Benefit Decision variables Objective: Constraints: Optimization model

Simulation Operating Policy Allocation Policy

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

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

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

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

Optimization Problems Objective function Decision variables Constraint set

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

Example

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

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

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

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

Solutions – Global or Local? Global Max Local Max

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

Types of Optimization Problems Nonlinear Program Linear Program Classic Program

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

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

Classical Program General Form Example All equality constraints

Single Constraint Multiple Decision Variables One constraint Vector

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

Example Lagrangean First – order conditions Notice the signs

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

Example Lagrangean First – order conditions Notice the signs

Example Equal marginal benefits (slopes) for all users

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

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

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

Nonlinear Program General FormExample

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

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

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