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Supply enough clean drinking water. Henri P . Gavin Tess Kretschmann (Duke CEE 2005) CEE 201L. Uncertainty, Design, and Optimization Department of Civil & Environmental Engineering Duke Univ. Spring 2016 system schematic Schematic of a

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SLIDE 1

Supply enough clean drinking water.

Henri P . Gavin Tess Kretschmann (Duke CEE 2005)

CEE 201L. Uncertainty, Design, and Optimization Department of Civil & Environmental Engineering Duke Univ.

Spring 2016

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SLIDE 2

system schematic

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SLIDE 3

Schematic of a water treatment system — three systems in series

  • Cp

Vu, Cu Vt, Ct Vu_max Vt_max Vr_max Vr, Cr q1 q2 q3 Qr_min Qe Cu Qu,o Qr,o Qt,o Qr

watershed area transpiration

Qt

evaporation water treatment treated water untreated water reservoir river supply

T, P

  • verflow
  • verflow

river flow

Vg_max Vg

groundwater storage community water demand

Qs, Cs Qg Qi

input precipitation

Qp Qp Qu Qd

  • Water Shed

Reservoir Water treatment plant

◮ Flows Q (Mgal/d) and

contaminant concentrations C (ppm) through the system satisfy mass balance.

◮ Determining random environmental variables:

Precipitation, Temperature, Population

Supply clean drinking water. system schematic CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 1 / 29

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SLIDE 4

Schematic of a water treatment system - watershed

  • Cp

Vu, Cu Vt, Ct Vu_max Vt_max Vr_max Vr, Cr q1 q2 q3 Qr_min Qe Cu Qu,o Qr,o Qt,o Qr

watershed area transpiration

Qt

evaporation water treatment treated water untreated water reservoir river supply

T, P

  • verflow
  • verflow

river flow

Vg_max Vg

groundwater storage community water demand

Qs, Cs Qg Qi

input precipitation

Qp Qp Qu Qd

  • ◮ Qi : input precipitation - a compound random process

◮ Qt : transpiration ◮ Vg : volume of ground water in watershed ◮ Vg max : capacity of ground water in watershed ◮ Qs : stream flow into reservoir y) ◮ Cs : contaminant concentrations in stream ◮ Qg : ground water flow into reservoir

Supply clean drinking water. system schematic CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 2 / 29

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SLIDE 5

Schematic of a water treatment system - reservoir

  • Cp

Vu, Cu Vt, Ct Vu_max Vt_max Vr_max Vr, Cr q1 q2 q3 Qr_min Qe Cu Qu,o Qr,o Qt,o Qr

watershed area transpiration

Qt

evaporation water treatment treated water untreated water reservoir river supply

T, P

  • verflow
  • verflow

river flow

Vg_max Vg

groundwater storage community water demand

Qs, Cs Qg Qi

input precipitation

Qp Qp Qu Qd

  • ◮ Qe : evaporation from reservoir

◮ Qr : river flow ◮ Qr min : minimum allowable river flow from reservoir ◮ Qro : reservoir overflow ◮ Vr : volume of water in reservoir ◮ Vr max : capacity of reservoir ◮ Cr : contaminant concentrations in reservoir ◮ Qu : flow from reservoir into the untreated water tank

Supply clean drinking water. system schematic CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 3 / 29

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SLIDE 6

Schematic of a water treatment system - water treatment

  • Cp

Vu, Cu Vt, Ct Vu_max Vt_max Vr_max Vr, Cr q1 q2 q3 Qr_min Qe Cu Qu,o Qr,o Qt,o Qr

watershed area transpiration

Qt

evaporation water treatment treated water untreated water reservoir river supply

T, P

  • verflow
  • verflow

river flow

Vg_max Vg

groundwater storage community water demand

Qs, Cs Qg Qi

input precipitation

Qp Qp Qu Qd

  • ◮ Vu, Vu max : volume of water in untreated tank, and capacity

◮ Cu : contaminant concentrations in untreated tank ◮ Quo : over flow from untreated tank ◮ Qp, Qp max : flow processed through water treatment plant, and capacity ◮ q1, q2, q3 : treatment intensity of three treatment types ◮ Cp : contaminant concentrations of processed water ◮ Vt, Vt max : volume of water in treated tank, and capacity ◮ Qto : over flow from treated tank ◮ Ct : contaminant concentrations in treated tank ◮ Qd : community water demand

Supply clean drinking water. system schematic CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 4 / 29

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precipitation modeling

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Precipitation — data analysis

1 2 3 4 5 6 7 2000 2002 2004 2006 2008 2010 2012 2014 2016 precipitation, in. USGS 02087182 FALLS LAKE ABOVE DAM NEAR FALLS, NC - 1998-01-01 -- 2016-03-06

  • 20
  • 15
  • 10
  • 5

5 10 15 20 2000 2002 2004 2006 2008 2010 2012 2014 2016 1-yr SPI, in

  • 30
  • 20
  • 10

10 20 2000 2002 2004 2006 2008 2010 2012 2014 2016 2-yr SPI, in

Daily and cumulative rainfall measured at USGS 02087182 (Falls Lake above Dam, near Falls, NC)

Supply clean drinking water. precipitation modeling CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 5 / 29

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Precipitation — statistical analysis — PDF & CDF

10-2 10-1 100 101 0.5 1 1.5 2 2.5 3 PDF, fR(r) rainfall data fR(r) ~ gamma fR(r) ~ exponential 0.312 inches per rainfall avg 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 CDF, FR(r) measured precipitation per rainfall, r, in. rainfall data FR(r) ~ gamma FR(r) ~ exponential avg i.p.r. = 0.312 cov i.p.r. = 1.478

Distribution of rainfall measured at USGS 02087182 (Falls Lake above Dam, near Falls, NC)

Supply clean drinking water. precipitation modeling CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 6 / 29

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Precipitation — statistical modeling

◮ rainy-day return period, Tr

Tr = total number of days total number of days with rain = 6638 2014 = 3.30 days between rainfalls (1)

◮ Probability of a wet day, W ∈ (0, 1)

Prob[wet day] = Prob[W = 1] = 1/Tr = Prob[U ≤ 1/Tr] (2)

◮ rainfall amount, R

Prob[R ≤ r] = 1 − e−r/¯

r

exponential (3)

  • r

Prob[R ≤ r] = 1 Γ(1/c2

r ) · γ

1 c2

r

, r c2

r ¯

r

  • gamma

(4)

◮ mean and c.o.v. of rainfall amount per rainfall ¯

r, cr = σr/µr ¯ r = (0.095 inches per day) (3.30 days between rainfalls) = 0.312 inches per rainfall (5) cr = 1.48 (6)

Supply clean drinking water. precipitation modeling CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 7 / 29

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Precipitation, USGS 02087182, 1998-1-1 - 2016-3-6

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 2000 2002 2004 2006 2008 2010 2012 2014 2016 rainfall return period 1/2 yr avg 1 yr avg 2 yr avg 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 2000 2002 2004 2006 2008 2010 2012 2014 2016 inches per rainfall 1/2 yr avg 1 yr avg 2 yr avg 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2000 2002 2004 2006 2008 2010 2012 2014 2016 daily rainfall 1/2 yr avg 1 yr avg 2 yr avg

Supply clean drinking water. precipitation modeling CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 8 / 29

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Falls Lake elevations, USGS 02087182

240 245 250 255 260 265 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Falls Lake Elevation, ft Year USGS 02087182 FALLS LAKE ABOVE DAM NEAR FALLS, NC 1998-01-01 -- 2016-03-05

Supply clean drinking water. precipitation modeling CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 9 / 29

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modeling trends in precipitation and temperature

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System Simulation - climate change & future rainfall

◮ Climate Change Time Scale CCTS — ? — a random variable ◮ Less frequent rainfall

Tr(d) = 3.30 (1 + (d/(CCTS × 365)) (7)

◮ More intense rainfall — overall same annual rainfall

¯ r(d) = 0.095 Tr(d) (8)

◮ increasing temperatures

— but with periodic and random fluctuation T(d) = ¯ T − T1 cos(2π(d − 15)/365) + T7 sin(2πd/(7 · 365) +Tcd/(365 × CCTS) + δT (9)

Supply clean drinking water. climate change CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 10 / 29

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CCTS = 100 year

1 2 3 4 5 6 2000 2010 2020 2030 2040 2050 rainfall, in year

  • 20
  • 15
  • 10
  • 5

5 10 15 20 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 1-yr SPI, in

  • 30
  • 20
  • 10

10 20 30 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2-yr SPI, in 100 101 102 103 104 0.5 1 1.5 2 2.5 3 number of rain-falls rainfall amount, r, in simulated data gamma distribution 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 C.D.F., FR(r) rainfall amount, r, in simulated data gamma distribution 2 3 4 5 6 7 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 rainfall return period, days CCTS = 100 years 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 inches per rainfall 20 30 40 50 60 70 80 90 100 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050

  • temp. deg F

year CCTS = 100 years

Supply clean drinking water. climate change CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 11 / 29

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SLIDE 16

CCTS = 50 year

2 4 6 8 10 12 2000 2010 2020 2030 2040 2050 rainfall, in year

  • 20
  • 15
  • 10
  • 5

5 10 15 20 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 1-yr SPI, in

  • 30
  • 20
  • 10

10 20 30 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 2-yr SPI, in 100 101 102 103 104 0.5 1 1.5 2 2.5 3 number of rain-falls rainfall amount, r, in simulated data gamma distribution 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 C.D.F., FR(r) rainfall amount, r, in simulated data gamma distribution 2 3 4 5 6 7 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 rainfall return period, days CCTS = 50 years 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 inches per rainfall 20 30 40 50 60 70 80 90 100 2005 2010 2015 2020 2025 2030 2035 2040 2045 2050

  • temp. deg F

year CCTS = 50 years

Supply clean drinking water. climate change CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 12 / 29

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modeling trends in population

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Durham NC Population growth, P(d)

50 100 150 200 250 300 350 400 450 1970 1980 1990 2000 2010 2020 2030 2040 Durham County NC Population/1000 Year P(y) = Po + 1556(y-yo) + (48 +/- 6%)(y-yo)2 + δ P population growth model +99% prediction interval

  • 99% prediction interval

U.S. Census Bureau Data

P(d) = Po + P1(d/365) + P2(d/365)2 + δP (10)

Supply clean drinking water. population CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 13 / 29

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SLIDE 19

system dynamics

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Schematic of a water treatment system — three systems in series

  • Cp

Vu, Cu Vt, Ct Vu_max Vt_max Vr_max Vr, Cr q1 q2 q3 Qr_min Qe Cu Qu,o Qr,o Qt,o Qr

watershed area transpiration

Qt

evaporation water treatment treated water untreated water reservoir river supply

T, P

  • verflow
  • verflow

river flow

Vg_max Vg

groundwater storage community water demand

Qs, Cs Qg Qi

input precipitation

Qp Qp Qu Qd

  • Water Shed

Reservoir Water treatment plant

◮ Flows Q (Mgal/d) and

contaminant concentrations C (ppm) through the system satisfy mass balance.

◮ Determining random environmental variables:

Precipitation, Temperature, Population

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 14 / 29

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Mass (volume) balance, in general

◮ Volumes and Flows

volume tomorrow = volume today + inflow - outflow V(d + 1) = V(d) + Qi(d) − Qo(d) (11) . . . subject to limits on capacities. Overflow goes to the river.

◮ Concentrations

mass = (concentration) × (volume) mass tomorrow = mass today + inflow mass - outflow mass C(d + 1)V(d + 1) = C(d)V(d) + Ci(d)Qi(d) − C(d)Qo(d) (12)

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 15 / 29

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SLIDE 22

Watershed dynamics

◮ Input precipitation flow (Ar: rainfall area (Mgal/in))

Qi = Ar WR

◮ Ground water volume

Vg(d + 1) = Vg(d) + Qi(d) − Qt(d) − Qg(d) − Qs(d) (13)

◮ Transpiration flow

Qt(d) = (αt + βtT(d)) Vg(d)/Vg,max (14)

◮ Ground water flow to reservoir

Qg(d) = αgVg(d)/Vg,max . (15)

◮ Stream flow to reservoir

Qs(d) = αsVg(d)/Vg,max + 0.5Vg(d) − Vg,max (16) Cs(d) = ¯ Cs + cpP(d) + csQs(d) (17)

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 16 / 29

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Reservoir dynamics

◮ Volume of water in the reservoir

Vr(d + 1) = Vr(d) +Qs(d) + Qg(d) (18) −Qe(d) − Qu(d) − Qr,min − Vr(d) − Vr,max

◮ Evaporation flow

Qe(d) = (αe + βeT(d)) Vr(d)/Vr,max (19)

◮ Contaminant Concentrations

Cr(d + 1)Vr(d + 1) = Cr(d)Vr(d) +Cs(d)Qs(d) (20) −Cr(d)Qu(d) − Cr(d)Qr,min −Cr(d)Vr(d) − Vr,max,

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 17 / 29

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SLIDE 24

Water treatment dynamics

◮ Volume in the untreated tank

Vu(d + 1) = Vu(d) + Qu(d) − Qp(d) − Vu(d) − Vu,max. (21)

◮ Contaminant concentrations in the untreated tank

Cu(d + 1)Vu(d + 1) = Cu(d)Vu(d) + Cr(d)Qu(d) (22) −Cu(d)Qp(d) − Cu(d)Vu(d) − Vu,max

◮ Contaminant concentrations in processed water

Cp(d) = Cu(d) exp [− [R] q(d)/Qp(d)] . (23)

◮ Volume in the treated tank

Vt(d + 1) = Vt(d) + Qp(d) − Qd(d) − Vt(d) − Vt,max (24)

◮ Contaminant concentrations in the treated tank

Ct(d + 1)Vt(d + 1) = Ct(d)Vt(d) + Cp(d)Qp(d) (25) −Ct(d)Qd(d) − Ct(d)Vt(d) − Vt,max

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 18 / 29

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SLIDE 25

Effectiveness of Water Treatment Processes, R

Cp(d) = Cu(d) exp [− [R] q(d)/Qp(d)] .

◮ R(i, j) shows the effectiveness of treatment “j” for pollutant “i”

R =               chlorine filters activated carbon micro-organisms 1000 1000 250 suspended solids 50 2500 50 petro-chemical 100 100 1000              

◮ R is diagonally-dominant ...

Treatment “i” is most effective for pollutant “i”

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 19 / 29

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SLIDE 26

Community water demand water conservation regulations, and river flow

Qd(d) = P(d)

  • 100 + 0.4(T(d) − ¯

T)

  • ×

(26)

  • 1 − Cc

0.50Vr,max − Vr(d) 0.50Vr,max − Vr(d)

  • + δQd
  • Qr(d) = Qr,min + Vr(d) − Vr,max + Vu(d) − Vu,max + Vt(d) − Vt,max . (27)

Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 20 / 29

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SLIDE 27

Reservoir levels, water demand and conservation

240 242 244 246 248 250 252 254 256 258 260 262 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Falls Lake Elevation, ft Year USGS 02087182 FALLS LAKE ABOVE DAM NEAR FALLS, NC 2007-01-01 -- 2016-03-05 5 10 15 20 25 30 35 40 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 None D0 D1 D2 D3 D4 Water Demand, Mgpd Drought Level Year www.ncwater.org Supply clean drinking water. system dynamics CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 21 / 29

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SLIDE 28

the design objective

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Performance and Cost

◮ Total cost to build and operate the system for fifty years ◮ Construction Costs

reservoir . . . $0.01/gallon untreated water tank . . . $0.5M+$0.10/gal treated water tank . . . $0.5M+$0.20/gal water treatment plant . . . $0.10/gal/day

◮ Operating Costs

chlorine . . . $0.03/gal filters . . . $0.01/gal activated carbon . . . $0.05/gal of decontaminant.

◮ Fines

  • If Vt falls below 0.1Vt,max, then a penalty of $0.05 per gallon is charged for

the water consumed that day.

  • A penalty of $0.1M per day is charged for days in which the pollutant

concentrations in the treated water exceed their respective limits.

  • A penalty of $0.1M per day is charged for days with excessive down-stream

flooding.

Supply clean drinking water. the design objective CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 22 / 29

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SLIDE 30

two-input → two-output control system

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SLIDE 31

Two-Input, Two-Output (TITO) standard control problem

cost

Z

ctrls

y=[Vr, Vu, Vt, Cu, Ct]

msmnts environment

x=[Vg,Vr,Vu,Vt, Cr,Cu,Ct, Z]

CONTROLLER YOUR THE PLANT

w=[Qi, Cs, T, Qd] u=[Qu, q, Qp]

Discrete-time dynamics x(d + 1) = f( x(d), u(d), w(d) ) (28)

Supply clean drinking water. TITO control CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 23 / 29

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SLIDE 32

your job

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SLIDE 33

Your Job

◮ Design the system.

The design parameters are: [Vr,max, Vu,max, Vt,max, Qp,max]

◮ Control the system.

Determine a function F (the control rule) relating the measurements to the controls

  • n a day-to-day basis.

[Qu, q1, q2, q3, Qp] = F([Vr, Vu, Vt, Cu, Ct])

◮ In order to . . .

minimize the 84% quantile (mean + standard deviation) of the total cost at the end of the operation life.

◮ Note that all constraint violations (running out of water, delivering dirty

water) are montetized and added to the objective. With this formulation, there are no separate constraints to consider.

Supply clean drinking water. your job CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 24 / 29

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SLIDE 34

example plots

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SLIDE 35
  • ex. Precipitation and Transpiration

0.2 0.4 0.6 0.8 1 2010 2015 2020 2025 2030 2035 2040

statistical averages

  • avg. inch per rainfall
  • avg. percent days with rain

200 400 600 800 1000 2010 2015 2020 2025 2030 2035 2040

cumulative inches

cumulative precipitation cumulative transpiration

  • 10
  • 5

5 10 15 2010 2015 2020 2025 2030 2035 2040

1-yr precipitation index, in year Supply clean drinking water. example plots CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 25 / 29

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SLIDE 36
  • ex. Ground water, Stream, Evaporation, River, Tank Vol.

5 10 15 20 25 30 2010 2015 2020 2025 2030 2035 2040

Mgal / day

groundwater flow evaporation 0.2 0.4 0.6 0.8 1 1.2 2010 2015 2020 2025 2030 2035 2040

volumes / capacities

ground water reservoir untreated treated 1 10 100 1000 10000 2010 2015 2020 2025 2030 2035 2040

Mgal / day

stream flow river flow

Supply clean drinking water. example plots CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 26 / 29

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SLIDE 37
  • ex. Population, Consumption, Cost

40 60 80 100 120 140 160 180 200 220 2010 2015 2020 2025 2030 2035 2040

population/1000

60 70 80 90 100 110 120 130 2010 2015 2020 2025 2030 2035 2040

consumption, gpppd

100 150 200 250 300 2010 2015 2020 2025 2030 2035 2040

cost, M$ Supply clean drinking water. example plots CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 27 / 29

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SLIDE 38
  • ex. Water Quality

0.001 0.01 0.1 1 10 100 1000 2010 2015 2020 2025 2030 2035 2040

micro-organisms

Cs Cr Cu Ct 0.001 0.01 0.1 1 10 100 1000 2010 2015 2020 2025 2030 2035 2040

suspended solids

Cs Cr Cu Ct 0.001 0.01 0.1 1 10 100 1000 2010 2015 2020 2025 2030 2035 2040

petro-chemical

Cs Cr Cu Ct

Supply clean drinking water. example plots CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 28 / 29

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SLIDE 39

uncertainty propagation

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SLIDE 40

Uncertainty Propagation

◮ Primary random variables

mean c.o.v. climate change time scale CCTS 50 y 0.30 temperature rise Tc 4 deg/(365 × CCTS). 0.20 population growth P2 48 people/day2 0.05 conservation effectiveness Cc 0.15 0.30

◮ Secondary random variables

temperature δT σ = 5 deg. F normal rainfalls N Tr = 3.30(1 + (d/(CCTS × 365))) days rainfall amnt. R ¯ r = 0.095Tr, cr = 1.48 gamma rainfall area Ar med.= 0.4Aw, c.o.v.=0.90 lognormal population δP µ = 0, σ = 1500 people normal demand δQd µ = 0, σ = 5 gpppd normal

Supply clean drinking water. uncertainty propagation CEE 201L. Duke Univ. H.P .G, T.K. Spring 2016 29 / 29