Water Resources Systems Modeling for
Planning and Management
•••••••••
An Introduction to the Development and Application
of
Optimization and Simulation Models
for Aiding in
Water for Resources Planning, Management
and
for Addressing Operational Issues and Problems.
A River Basin System
Water for:
• Water Supply (M,I,A)
•
•
•
•
•
Recreation
Nature
Hydropower
Pollution Control
Navigation
A River Basin System
Infrastructure:
• Reservoirs, wells, pumps
•
•
•
•
Diversion canals, pipelines
Recreation facilities
Hydropower plants
Water & Wastewater
Treatment Plants
• Navigation locks
• Flood Control Res. & Levees
• Distribution/Collection Sys.
A River Basin System
Why Model:
• What to do or design?
• Where to do or design it?
• How much or how big
and how to operate?
• When to implement?
• Why? What are the
Hydrologic, Economic,
Ecosystem, and Social
Impacts?
A River Basin System
When to Model:
• There exists a problem or
opportunity.
• A decision is to be made.
• Many alternatives.
• Best alternative not obvious.
• Quantitative aspects.
A System – Interdependent Components
River Basin:
• Lakes, Reservoirs,
• Wetlands, River,
• Aquifers, wells,
• Pumps,
• Treatment Plants,
• Diversions,
• M, I, & A Users,
• Hydropower Plants
A System – Interdependent Components
Municipality:
• Water Treatment,
• Water distribution
network,
• Aquifers wells,
• Pumps,
• Storage tanks,
• Sewerage
collection network,
• Wastewater
treatment.
A System – Interdependent Components
Irrigation:
• Diversion canals,
• Drainage system,
• Crop areas,
• Equipment,
• Labor,
• Fertilizer,
• Pesticides, etc.
A System – Interdependent Components
THE SYSTEM
OUTPUTS
INPUTS
COMPONENTS
FOCUS: Performance of System
not necessarily of its individual components.
A System – Interdependent Components
THE SYSTEM
OUTPUTS
INPUTS
COMPONENTS
GOAL: Maximize System Performance.
Water Resources Systems
Water Resources Systems
Water Resources Systems Engineering
Topics:
• Modeling Approaches &Applications
• Shared Vision Modeling
• System Performance Criteria
• Integrating Hydrology and Aquatic
Ecosystems – a Case Study
Water Resources Systems Modeling
A Model:
A mathematical description of some system.
Model Components:
Variables, parameters, functions, inputs,
outputs.
A Model Solution Algorithm:
A mathematical / computational procedure for
performing operations on the model – for
getting outputs from inputs.
Water Resources Systems Modeling
Model Types:
• Descriptive
(Simulation)
• Prescriptive (Optimization)
• Deterministic
• Probabilistic or Stochastic
• Static
• Dynamic
• Mixed
Water Resources Systems Modeling
Algorithm Types:
• Descriptive
(Simulation)
• Prescriptive (Constrained Optimization)
• Mathematical Programming
•
•
•
•
Lagrange Multipliers
Linear Programming
Non-linear Programming
Dynamic Programming
• Evolutionary Search Procedures
•
Genetic Algorithms, Genetic Programming
Water Resources Systems Modeling
Simulation:
System
Inputs
System Design and
Operating Policy
WATER RESOURCE
SYSTEM
Optimization:
System
Inputs
System
Outputs
System Design and
Operating Policy
WATER RESOURCE
SYSTEM
System
Outputs
Water Resources Systems Modeling
Modeling Example
• Problem.
Need a water tank of capacity  V.
• Performance Criterion.
Cost minimization.
• Numerous alternatives.
Shape, dimensions, materials.
• Best design not obvious.
Water Resources Systems Modeling
Modeling Example Continued
Consider a cylindrical tank  V.
having radius R and height H.
Average costs per unit area:
Ctop
Cside
Cbase
R
H
Water Resources Systems Modeling
Modeling Example Continued
Model:
Minimize Total_cost (Objective)
subject to: (Constraints)
Volume = (R2H)  V.
Total_cost = $_Side+$_Base+$_Top
$_Side = Cside(2RH)
$_Base = Cbase(R2)
$_Top = Ctop(R2)
Water Resources Systems Modeling
Modeling Example Continued
Solution:
$_Side / Total_cost = 2/3
($_Base+$_Top) / Total_cost = 1/3
No matter what shape and unit costs.
Water Resources Systems Modeling
Modeling Example Continued
Solution: a tradeoff between cost and volume.
Total
Cost
Tank Volume
Water Resources Systems Modeling
Other Modeling Examples
Water Pollution Control
Water Allocations to Competing Uses
Tradeoffs!
Water Resources Systems Modeling
Other Modeling Examples
Water Quality – Aquatic Ecosystems
Silt
Acid
Mine
Drainage
Point-Source
Pollution
Fish Kill
Ecosystem
Enhancement
Stakeholder Participation:
Shared Vision Modeling
Shared Vision Modeling
A multi-purpose river basin planning example:
Shared Vision Modeling
A multi-purpose river basin planning example:
Gage
•
Recreation
Flood storage
Pumped storage
hydropower
Irrigation
Levee
protection
Urban area
Water Resource Systems Engineering
Planning & Management Objectives
Types of Objectives or Measures of Performance:
• Physical
• Statistical
• Economic
• Environmental – Ecological
• Social
• Combinations
• Multi-objective analyses.
Water Resource Systems Engineering
Planning & Management Objectives
Broad Goals  Aims  Objectives  Specific Strategies:
Why?
• National Security and Welfare.
How?
• Self Sufficiency.
• Regional Economic Development.
• Public and Environmental Health.
• Economic Efficiency and Equity.
• Environmental Quality.
• Ecosystem Biodiversity and Health.
• System Reliability, Resilience, Robustness.
• Water supply: quantity, quality, reliability, cost.
• Flood protection, flood plain zoning.
• Energy and food production.
• Recreation, navigation, wildlife habitat.
• Water and wastewater treatment.
Water Resource Systems Engineering
Planning & Management Objectives
Overall measures of system performance:
• Mean – average or expected value.
• Variance – average of squared deviations from
the mean value.
• Reliability – Prob(satisfactory state).
• Resilience – Prob(sat. state following unsat. state).
• Robustness – adaptability to other than design
input conditions.
• Vulnerability – expected magnitude or extent of
failure when unsatisfactory state occurs.
Water Resource Systems Engineering
Planning & Management Objectives
Time series of system performance values:
System
Performance
Measure
Mean
Failure threshold
Time
Water Resource Systems Engineering
Planning & Management Objectives
System
Performance
Measure
Mean
Failure threshold
Time
Same:
Different:
Mean and Variance
Reliability, Resilience and Vulnerability
Water Resource Systems Engineering
Planning & Management Objectives
System
Performance
Measure
System
Performance
Measure
Mean
Failure threshold
Time
Mean
Failure threshold
Compare Reliabilities, Resiliences, Vulnerabilities.
Water Resource Systems Engineering
Planning & Management Objectives
Objectives expressed as functions to be maximized or
minimized or as constraints that have to satisfied.
Economic objectives:
• Maximize benefits: improvement in income,
welfare, or willingness to pay.
• Minimize costs: benefits forgone, opportunity
costs, adverse externalities.
• Maximize net benefits: benefits less losses and
costs.
• Minimize inequity: differences in distributions
of net benefit among stakeholders.
Water Resource Systems Engineering
Planning & Management Objectives
Economic objectives:
Maximize Net Revenue (Private):
Marginal Revenue = Marginal cost
Maximize Net Social Benefits (Public):
Unit Price = Marginal cost
Po
Unit price = Po – bQ
2b b
P*pri.
Marginal revenue = Po – 2bQ
Marginal cost = c
Private:
Consumer’s surplus
Producer’s surplus
P*pub.
Q
Q*pri.
Q*pub.
Public: All consumer surplus.
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
OBJECTIVES
ALTERNATIVE
ALTERNATIVE PROJECTS
Relative impact.
Relative importance.
Alternative Codes:
1-10, ++ + 0 - --, A B C D, S F.
22
$3
57
Sat
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Other Multi-objective Methods:
• Satisficing
• Dominance
• Lexicography
• Indifference Analyses
• Obj. Weights or Obj. Constraints
• Goal Attainment and Programming
• Compromise Programming
• Interactive Methods
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Satisficing (setting improving targets for objectives
that are functions of decision variables in vector X.)
•
F C
•
D•
OBJ2(X)
•A
B
•
Second Iteration: C
First Iteration: C, D, F.
E•
OBJ1(X)
Alternatives Considered: A, B, C, D, E, F.
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Dominance (eliminating alternatives that are
inferior with respect to all objectives.)
•
F C
•
D•
OBJ2(X)
• A
B
•
A dominated by C and F
B dominated by C, D, F
D dominated by C
E•
OBJ1(X)
Alternatives Considered: A, B, C, D, E, F.
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Lexicography (rank objectives from most important
to least important. If a tie go to next most
important objective, etc.)
OBJ2(X)
•A
B•
•
F C
•
D•
If OBJ1 is most
important, pick E.
E•
If OBJ2 is most
important, pick F.
OBJ1(X)
Alternatives Considered: A, B, C, D, E, F.
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Objective Weights (identify Pareto efficiency frontier
by varying weights associated with each objective.)
Maximize {w1• OBJ1(X) + w2• OBJ2(X)}
Subject to model constraints gi(X)  bi i
OBJ2(X)
F
•
C
Changing weights in objective space
identifies dominant solutions on
efficiency frontier.
•
E•
OBJ1(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Objective Weights (identify Pareto efficiency frontier
by varying weights associated with each objective.)
Maximize {w1• OBJ1(X) + w2• OBJ2(X)}
Subject to model constraints gi(X)  bi i
OBJ2(X)
•
F
Changing weights in objective space
identifies dominant solutions on convex
efficiency frontier. It misses others.
C•
•
E•
OBJ1(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Objective Constraints (include all objectives but one
as constraints having bounds. Vary bound values to
identify Pareto efficiency frontier.)
Maximize OBJ1(X)
OBJ2(X)
Subject to:
gi(X)  bi
Discrete frontier
i
OBJ2(X)  L2
L2
F
•
C
•
E•
OBJ1(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Objective Constraints (include all objectives but one
as constraints having bounds. Vary bound values to
identify Pareto efficiency frontier.)
Continuous frontier
Maximize OBJ1(X)
OBJ2(X)
Subject to:
gi(X)  bi
i
OBJ2(X)  L2
L2
F
•
C
•
E•
OBJ1(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Goal Attainment (minimize maximum weighted
deviation from preselected targets for each objective.
Vary weight values to identify efficiency frontier.)
Minimize D
Subject to:
gi(X)  bi
OBJ2(X)
i
wk•{Tk – OBJk(X)}  D k
T2
F
•
C
•
E•
T1
OBJ1(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Goal Programming (minimize sum of weighted
deviations from preselected targets for each objective.
Vary weight values to identify efficiency frontier.)
Minimize Sk [wdk(Dk) + wek(Ek)]
Ek
Subject to:
gi(X)  bi
wdk(Dk)
i
OBJk(X) = Tk – Dk + Ek k
Dk
wek(Ek)
Tk
OBJk(X)
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Compromise Programming (minimize nth root of
weighted sum of deviations from best value for each
objective raised to the nth power. Vary weights and n
to identify portion of efficiency frontier.)
Minimize { Sk wkn[Zk - OBJk(X)]n}1/n
Subject to:
gi(X)  bi
OBJ2(X)
Z2
i
Zk = Max. feasible value of OBJk k
n=2
n=
OBJ1(X)
Z1
Water Resource Systems Engineering
Planning & Management Objectives
Decision Making with Multiple Objectives:
Multi-objective Methods:
• Interactive Methods (user(s) involved in defining
improvements in all objectives, as desired.)
OBJ2(X)
OBJ2(X)
3
•
4
•
•2 1
•
OBJ1(X)
Iterating along
efficiency frontier.
5 6••
3 ••4
•
• •2
1
7
OBJ1(X)
Iterating toward the
efficiency frontier.