water resources planning and management daene c. mckinney water quality
TRANSCRIPT
Water Resources Planning and Management
Daene C. McKinney
Water Quality
Water Quality Management• Critical component of overall water management in a basin• Water bodies serve many uses, including
– Transport and assimilation of wastes – Assimilative capacities of water bodies can be exceeded WRT intended
uses • Water quality management measures
– Standards• Minimum acceptable levels of ambient water quality
– Actions• Insure pollutant load does not exceed assimilative capacity while
maintaining quality standards– Treatment
Water Quality Management Process
• Identify – Problem– Indicators – Target Values
• Assess source(s)• Determine linkages
– Sources Targets
• Allocate permissible loads
• Monitor and evaluate• Implement
Physical Processes Controlling Flux
• Advection– Solutes carried along by flowing water
• Diffusion– Transport by molecular diffusion
• Dispersion– Transport by mechanical mixing
Models
• Advection + dispersion - major processes by which dissolved matter is distributed throughout a water body (e.g., river)
C = concentration (M/L3)V = Average velocity in reach (L/T)D = Longitudinal dispersion coefficient (L2/T)t = timex = longitudinal distance
Advection term
Dispersion term
Source term
Reaction term
Eq. 10.10 in text
Steady-state Model• Steady-state
– Where k = decay rate (1/T) kC
x
CV
x
CD
2
20
2
41
V
kDm
– Solution is
– W = loading (M/T) at x = 0
Eq 10.12 in text
Advection - Dispersion
Advection Dominated Flow
Water Quality Example
• W1, W2 = Pollutant loads (kg/day)• x1, x2 = Waste removal efficiencies (%)• P2
max, P3max = Water quality standards (mg/l)
• P2, P3 = Concentrations (mg/l)• Q1, Q2, Q3 = Flows (m3/sec)• a12, a13, a23, = Transfer coefficients
Water Quality Example
Water Quality ExampleParamet
erUnits Value
Q1 m3/s 10
Q2 m3/s 12
Q3 m3/s 13
W1 kg/day 250,000
W2 kg/day 80,000
P1 mg/l 32
P2max mg/l 20
P3max mg/l 20
a12 - 0.25
a13 - 0.15
a23 - 0.60
Water Quality Example
0.11 x79.128.1 21 xx
8.01 x
0.12 x
Water Quality Example
• Cost of treatment at 1 greater than cost at 2 (bigger waste load at 1)
• Marginal cost at 1 greater than marginal cost at 2, c1 > c2 for same level of treatment
Water Quality Example
2,10.10.0
79.128.1
8.0
toSubject
Minimize
21
1
2211
ix
xx
x
xcxc
i
Cost of treatment at 1 >= cost at 2marginal cost at 1, c1, >= marginal cost at 2, c1, for the sameamount of treatment.
Water Quality Example
Example
• Irrigation project– 1800 acre-feet of water per year
• Decision variables
– xA = acres of Crop A to plant?
– xB = acres of Crop B to plant?
1,800 acre feet = 2,220,267 m3
400 acre = 1,618,742 m2
Crop A Crop B
Water requirement (Acre feet/acre) 3 2
Profit ($/acre) 300 500
Max area (acres) 400 600
Example
2
4
6
8
10
2 4 6 8 10xA (hundreds acres)
x B (
hu
nd
red
s ac
res)
xB< 600
xA> 0 xA< 400
3xA +2 xB < 1800
xB > 0
Example
2
4
6
8
10
2 4 6 8 10xA (hundreds acres)
x B (
hu
nd
red
s ac
res)
xB< 600
xA> 0 xA< 400
xB > 0
Z=3600=300xA +500xB
Z=2000=300xA +500xB
Z=1000=300xA +500xB
(200, 600)
GAMS CodePOSITIVE VARIABLESxA, xB;
VARIABLESobj;
EQUATIONS objective, xAup, xBup, limit;
objective.. obj =E= 300*xA+500*xB;xAup.. xA =L= 400.;xBup.. xB =L= 600.;limit.. 3*xA+2*xB =L= 1800;
MODEL Calibrate / ALL /;SOLVE Calibrate USING LP MAXIMIZING obj;
Display xA.l;Display xB.l;
Marginal, Lagrange multiplier, shadow price, dual variable
GAMS Output LOWER LEVEL UPPER MARGINAL
---- EQU objective . . . 1.000---- EQU xAup -INF 200.000 400.000 .---- EQU xBup -INF 600.000 600.000 300.000---- EQU limit -INF 1800.000 1800.000 100.000
LOWER LEVEL UPPER MARGINAL
---- VAR xA . 200.000 +INF .---- VAR xB . 600.000 +INF .---- VAR obj -INF 3.6000E+5 +INF .
Marginal
Marginals
• Marginal for a constraint = Change in the objective per unit increase in RHS of that constraint.– i.e., change xB
– Objective = 360,000– Marginal for constraint = 300– Expect new objective value = 360,300
600Bx 601Bx
New Solution LOWER LEVEL UPPER MARGINAL
---- EQU objective . . . 1.000---- EQU xAup -INF 199.333 400.000 .---- EQU xBup -INF 601.000 601.000 300.000---- EQU limit -INF 1800.000 1800.000 100.000
LOWER LEVEL UPPER MARGINAL
---- VAR xA . 199.333 +INF .---- VAR xB . 601.000 +INF .---- VAR obj -INF 3.6030E+5 +INF .
180023 BA xxNote: Adding 1 unit to xB adds 300 to the objective, but constraint 3 says
and this constraint is “tight” (no slack) so it holds as an equality, therefore xA must decrease by 1/3 unit for xB to increase by a unit.
Unbounded Solution
00
400
toSubject
500300Maximize
BA
A
BA
xx
x
xxZ
2
4
6
8
10
2 4 6 8 10xA (hundreds acres)
x B (
hu
nd
red
s ac
res)
xA> 0 xA< 400
xB > 0
unbounded
Take out constraints3 and 4, objective can Increase without bound
Infeasibility
00
300023
600
400
toSubject
500300Maximize
BA
BA
B
A
BA
xx
xx
x
x
xxZ
2
4
6
8
10
2 4 6 8 10xA (hundreds acres)
x B (
hu
nd
red
s ac
res)
xB< 600
xA> 0 xA< 400
3xA +2 xB > 3000
xB > 0
Change constraint 4to >= 3000, then no intersection of constraints exists and no feasible solution can be found
Multiple Optima
BA xxZ 200300Objective
Change objective coefficient to 200, then objective has same slope as constraint and infinite solutions exist 2
4
6
8
10
2 4 6 8 10xA (hundreds acres)
x B (
hu
nd
red
s ac
res)
xB< 600
xA> 0 xA< 400
xB > 0
Z=1800=300xA +200xB
Infinite solutions on this edge
3xA +2 xB < 1800