Filling Pond Control Filling Pond Control Minimizing Water loss on Minimizing Water loss on Scipio RiverScipio River(Comparison between PI Controller and Fractional Order Controllers)

By Nicolas MONEGIER DU SORBIER

OutlineOutline

1.Introduction2.System Specifications3.System Modeling and Parameters

Calculation4.System Control & Simulations (PI, Piα,

PIIα)5.Comparison : PI vs Piα vs PIIα

Controllers6.Conclusion

1. Introduction1. IntroductionThe aim of this project was to control the

filling of each pond of the Scipio Open Channel in the right time, minimizing the water loss at the end of the canal.

The most difficult was that only a positive action was available on the system.

Many controllers were tested in this project to find the best one.

2. System Specification2. System Specification

Sensors and actuators can be used just one time each hour.

Gate height, Diversion water level and Pond water Level are available for each Pond.

Gate height, Water level and Outflow are available for the Reservoir.

3.System Modeling and 3.System Modeling and Parameters CalculationParameters Calculation

Diversion Flow CalculationDiversion Flow Calculation

For each diversion gate, the flow was calculated using the derivative of the pond volume variations.

Thus, the points related to the gate height, the diversion pond height and the flow were plotted.

Seeing these plots, so many points had very low flows for many values of gate heights and diversion pond heights. That is why a Look-up Table, based on 5 values of diversion pond heights and 5 values of gate heights, was made for the model.

So for each cell of the Look-up Table, the average was taken in the neighborhood of the point.

Reservoir Flow Calculation Reservoir Flow Calculation

-20-15

-10-5

0

0

0.5

1

0

0.5

1

1.5

Reservoir Height (feet)

Look-Up Table : Reservoir

Gate (feet)

Flo

w (

m3 /s

)

Height (feet)/Gate(feet)

-23 -7 -2 3

0 0 0 0 0

0.1 0.148 0.172 0.192 0.215

0.3 0.301 0.326 0.347 0.397

0.5 0.526 0.547 0.583 0.614

0.7 0.744 0.746 0.783 0.824

0.9 0.922 0.936 0.958 0.994

1.1 1.168 1.184 1.216 1.237

1.3 1.31 1.324 1.333 1.348

Johnnie’s Diversion Flow Johnnie’s Diversion Flow Calculation Calculation

Height (feet)/Gate(feet)

2 3 4 5 6

0 0 0 0 0 0

0.4 0.036 0.05 0.063 0.073 0.09

0.8 0.099 0.112 0.128 0.147 0.162

1.2 0.175 0.197 0.229 0.252 0.283

1.6 0.278 0.327 0.403 0.443 0.473

23

45

6

00.5

11.50

0.1

0.2

0.3

0.4

0.5

Diversion Pond Height (feet)

Look-Up Table : Johnnie's Diversion Pond

Gate (feet)

Flo

w (

m3 /s

)

Tom’s Diversion Flow Tom’s Diversion Flow CalculationCalculation

Height (feet)/Gate(feet)

-1 0.5 2 3.5 5

0 0 0 0 0 0

0.4 0.046 0.066 0.076 0.099 0.102

0.8 0.092 0.114 0.139 0.141 0.179

1.2 0.148 0.176 0.193 0.211 0.242

1.5 0.257 0.326 0.399 0.414 0.482

-10

12

34

5

0

0.5

1

1.50

0.1

0.2

0.3

0.4

0.5

Diversion Pond Height (feet)

Look-Up Table : Tom's Diversion Pond

Gate (feet)

Flo

w (

m3 /s

)

City’s Diversion Flow City’s Diversion Flow CalculationCalculation

Height (feet)/Gate(feet)

1 3 5 7 9

0 0 0 0 0 0

0.25 0.03 0.044 0.067 0.092 0.115

0.5 0.054 0.063 0.092 0.128 0.202

0.75 0.078 0.091 0.146 0.189 0.254

1 0.118 0.174 0.283 0.383 0.514

24

68

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

Diversion Pond Height (feet)

Look-Up Table : City's Diversion Pond

Gate (feet)

Flo

w (

m3 /s

)

Cemetary’s Diversion Flow Cemetary’s Diversion Flow CalculationCalculation

Height (feet)/Gate(feet)

-1 0.5 2 3.5 5

0 0 0 0 0 0

0.4 0.03 0.031 0.034 0.038 0.041

0.8 0.054 0.076 0.101 0.136 0.159

1.2 0.075 0.124 0.157 0.284 0.331

1.5 0.099 0.176 0.294 0.392 0.469

23

45

67

0

0.5

1

1.5

0

0.1

0.2

0.3

0.4

0.5

Diversion Pond Height (feet)

Look-Up Table : Cemetary's Diversion Pond

Gate (feel)

Flo

w (

m3 /s

)

Diversion Flow ModelingDiversion Flow Modeling

Time Delay CalculationTime Delay Calculation

To Calculate the Time Delay between the reservoir and each diversion, the Rootcanal software was used: http://www.neng.usu.edu/bie/faculty/merkley/BIE6300.htm

Through Google Earth the topographic datas of the Scipio channel were obtained to model it on this software (elevations, distances)

Thus, 3 Look-Up Tables were made to calculate the delay relative to the Reservoir outflow for each part of the canal.

So each part of the canal was represented by a delay : y (t)= u (t-τ)

Time Delay CalculationTime Delay Calculation

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2 2.5

R e se rv o ir fl o w (in m ^ 3/s)

Tim

e de

lay

(in m

inut

es)

J ohnnie

Tom

C ity /C em etary

Flow (in m^3/s)

Johnnie Diversion (in min)

Tom Diversion (in min)

City/Cemetary Diversion (in min)

0.045 465 592 621

0.095 304 398 419

0.15 246 322 341

0.21 210 279 295

0.25 195 260 274

0.3 182 242 255

0.4 164 220 230

0.5 150 200 212

0.6 141 190 200

0.7 133 180 191

0.8 130 174 184

0.9 123 168 178

1 120 162 178

1.5 106 145 154

2 99 135 144

Delay ModelingDelay ModelingFor each part of the canal, Delays were modeled like this :

But for parts between diversions and Ponds, Delays were modeled by constant delays.

Johnnie’s Pond Delay=0.2 hours, Tom’s Pond Delay=0.2 hoursCity’s Pond Delay=0.1 hours and Cemetary’s Pond Delay=0.5 hours

Pond Volume CalculationPond Volume Calculation

Where q (t) is the flow and C the initial Pond Volume

Model OverviewModel Overview

4. System Control & 4. System Control & Simulations (PI, PiSimulations (PI, Piαα,, PIIPIIαα))

Pond PI ControllerPond PI Controller

Reservoir PI ControllerReservoir PI Controller

Johnnie’s Pond (PI)Johnnie’s Pond (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

Johnnie's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gate Height

Gat

e (f

eet)

Tom’s Pond (PI)Tom’s Pond (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

Tom's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e (f

eet)

Gate Height

City’s Pond (PI)City’s Pond (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hour)

Fill

ing

City's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hour)

Gat

e (f

eet)

Gate Height

Cemetary’s Pond (PI)Cemetary’s Pond (PI)

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Fill

ing

Time (hours)

Cemetary's Pond

0 5 10 15 20 25 30-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (hours)

Gat

e (f

eet)

Gate Height

Reservoir (PI)Reservoir (PI)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gat

e (f

eet)

Reservoir Gate Height

For PI Global error=156830 m 3 and Water loss = 8158 m3

Johnnie’s Pond in Use (PI)Johnnie’s Pond in Use (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Johnnie's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gate Height in Use

Gat

e in

use

(fe

et)

Tom’s Pond in Use (PI)Tom’s Pond in Use (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Tom's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

City’s Pond in Use (PI)City’s Pond in Use (PI)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

City's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

Cemetary’s Pond in Use Cemetary’s Pond in Use (PI)(PI)

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (hours)

Fill

ing

in u

se

Cemetary's Pond in Use

0 5 10 15 20 25 30-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (hours)

Gate Height in Use

Gat

e in

use

(fe

et)

Reservoir in Use (PI)Reservoir in Use (PI)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gat

e in

use

(fe

et)

Reservoir Gate in Use

For PI Global error=168450 m 3 and Water loss =7868 m3

Pond PIPond PIαα Controller Controller

Reservoir PIReservoir PIαα Controller Controller

Johnnie’s Pond (PIJohnnie’s Pond (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (feet)

Fill

ing

Johnnie's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (feet)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

Tom’s Pond (PITom’s Pond (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

Tom's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

City’s Pond (PICity’s Pond (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (feet)

Fill

ing

City's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (feet)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.6

a=0.9

a=0.3

a=0.6

a=0.9

Cemetary’s Pond (PICemetary’s Pond (PIαα))

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time (hours)

Fill

ing

Cemetary's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.6a=0.8

a=0.3

a=0.6

a=0.8

Reservoir (PIReservoir (PIαα))

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gate

(fe

et)

Reservoir Gate Height

a=0.3

a=0.6

a=0.8

For α = 0.6 Global error=157320 m 3 and Water loss = 8026 m3

For α = 0.8 Global error=164330 m 3 and Water loss = 7447 m3

For α = 0.3 Global error=156950 m 3 and Water loss = 8066 m3

Johnnie’s Pond in Use (PIJohnnie’s Pond in Use (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time (hours)

Fill

ing

in U

se

Johnnie's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

Tom’s Pond in Use (PITom’s Pond in Use (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time (hours)

Fill

ing

in u

se

Tom's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

City’s Pond in Use (PICity’s Pond in Use (PIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

City's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

Cemetary’s Pond in Use Cemetary’s Pond in Use (PI(PIαα))

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (hours)

Fill

ing

in u

se

Cemetary's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.6

a=0.8

a=0.3

a=0.6

a=0.8

Reservoir in Use (PIReservoir in Use (PIαα))

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gate

in u

se (

feet)

Reservoir Gate Height in Use

a=0.3

a=0.6

a=0.8

For α = 0.6 Global error=163620 m 3 and Water loss = 7176 m3

For α = 0.8 Global error=168010 m 3 and Water loss = 7745 m3

For α = 0.3 Global error=161750 m 3 and Water loss = 7912 m3

Pond PIIPond PIIαα Controller Controller

Reservoir PIIReservoir PIIαα Controller Controller

Johnnie’s Pond (PIIJohnnie’s Pond (PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

Johnnie's Pond

0 5 10 15 20 25 30

0

0.5

1

1.5

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.7

a=0.3

a=0.7

Tom’s Pond (PIITom’s Pond (PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

Tom's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.7

a=0.3

a=0.7

City’s Pond (PIICity’s Pond (PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

City's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.7

a=0.3

a=0.7

Cemetary’s Pond (PIICemetary’s Pond (PIIαα) )

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time (hours)

Fill

ing

Cemetary's Pond

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e (f

eet)

Gate Height

a=0.3

a=0.7

a=0.3

a=0.7

Reservoir (PIIReservoir (PIIαα))

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gate

(fe

et)

Reservoir Gate Height

a=0.3

a=0.7

For α = 0.3 Global error=155680 m 3 and Water loss = 7369 m3

For α = 0.7 Global error=158170 m 3 and Water loss = 8118 m3

Johnnie’s Pond in Use Johnnie’s Pond in Use (PII(PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Johnnie's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.7

a=0.3

a=0.7

Tom’s Pond in Use (PIITom’s Pond in Use (PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Tom's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.7

0.3

0.7

City’s Pond in Use (PIICity’s Pond in Use (PIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

City's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.7

a=0.3

a=0.7

Cemetary’s Pond in Use Cemetary’s Pond in Use (PII(PIIαα))

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (hours)

Fill

ing

in u

se

Cemetary's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

a=0.3

a=0.7

a=0.3

a=0.7

Reservoir in Use (PIIReservoir in Use (PIIαα))

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gate

in u

se (

feet)

Reservoir Gate Height in Use

a=0.3

a=0.7

For α = 0.3 Global error=158320 m 3 and Water loss = 6979 m3

For α = 0.7 Global error=161820 m 3 and Water loss = 7891 m3

5. Comparison : PI vs Pi5. Comparison : PI vs Piα α vsvs PIIPIIαα

Johnnie’s Pond in Use (PI vs PiJohnnie’s Pond in Use (PI vs Piαα

vsvs PIIPIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Johnnie's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

PI

PIa\a=0.6

PIIa\a=0.3

PI

PIa\a=0.6

PIIa\a=0.3

Tom’s Pond in Use (PI vs PiTom’s Pond in Use (PI vs Piαα

vsvs PIIPIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

Tom's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

PI

PIa\a=0.6

PIIa\a=0.3

PI

PIa\a=0.6

PIIa\a=0.3

City’s Pond in Use (PI vs PiCity’s Pond in Use (PI vs Piαα

vsvs PIIPIIαα))

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (hours)

Fill

ing

in u

se

City's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

PI

PIa\a=0.6

PIIa\a=0.3

PI

PIa\a=0.6

PIIa\a=0.3

Cemetary’s Pond in Use (PI vs PiCemetary’s Pond in Use (PI vs Piαα

vsvs PIIPIIαα))

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (hours)

Fill

ing

in u

se

Cemetary's Pond in Use

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

Time (hours)

Gat

e in

use

(fe

et)

Gate Height in Use

PI

PIa\a=0.6

PIIa\a=0.3

PI

PIa\a=0.6

PIIa\a=0.3

Reservoir in Use (PI vs PiReservoir in Use (PI vs Piαα vsvs

PIIPIIαα))

For PIα α = 0.6 Global error=163620 m 3 and Water loss = 7176 m3

For PIIα α = 0.3 Global error=158320 m 3 and Water loss = 6979 m3

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

Time (hours)

Gate

in u

se (

feet)

Reservoir Gate in Use

PI

PIa\a=0.6

PIIa\a=0.3

For PI Global error=168450 m 3 and Water loss = 7868 m3

ComparisonComparison

Normal Condition Using Condition

Value (m^3) PI relative Value (m^3) PI relative

PI Global Error 156830 0.00% 168450 0,00%

PI Water Loss 8158 0.00% 7868 0,00%

PIα Global Error 157320 +0.31% 163620 -2.87%

Piα Water Loss 8026 -1.62% 7176 -8.79%

PIIα Global Error 155680 -0.73% 158320 -6.01%

PIIα Water Loss 7369 -9.67% 6979 -11.29%

6. Conclusion6. Conclusion

The efficiency, for this system, of the PIIα

controller compared to others controllers was shown in this study.

Also an other strategy based on cooperative control could be applied but this one is hard to control due to the system complexity.

ReferencesReferences

www.sevierriver.org

“Rootcanal software” by Dr. MERKLEY Biological & Irrigation Engineering Department (USU,Logan)

“Open-Channel flows” by Subhash C. JAIN

“Introduction to Hydrology” by Warren VIESSMAN and Gary L. LEWIS

ThanksThanks

Thanks to all for listening to my presentation

Particular thanks to Dr. Chen for making me come to Logan and to Christophe for his help during this project