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Filling Pond Control Minimizing Water loss on Scipio River (Comparison between PI Controller and Fractional Order Controllers) By Nicolas MONEGIER DU SORBIER

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Outline 1.Introduction 2.System Specifications 3.System Modeling and Parameters Calculation 4.System Control & Simulations (PI, Pi α, PII α ) 5.Comparison : PI vs Pi α vs PII α Controllers 6.Conclusion

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1. Introduction The 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.

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2. 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.

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3.System Modeling and Parameters Calculation

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Diversion 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.

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Reservoir Flow Calculation Height (feet) / Gate(feet) -23-7-23 00000 0.10.1480.1720.1920.215 0.30.3010.3260.3470.397 0.50.5260.5470.5830.614 0.70.7440.7460.7830.824 0.90.9220.9360.9580.994 1.11.1681.1841.2161.237 1.31.311.3241.3331.348

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Johnnie’s Diversion Flow Calculation Height (feet) / Gate(feet) 23456 000000 0.40.0360.050.0630.0730.09 0.80.0990.1120.1280.1470.162 1.20.1750.1970.2290.2520.283 1.60.2780.3270.4030.4430.473

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Tom’s Diversion Flow Calculation Height (feet) / Gate(feet) 0.523.55 000000 0.40.0460.0660.0760.0990.102 0.80.0920.1140.1390.1410.179 1.20.1480.1760.1930.2110.242 1.50.2570.3260.3990.4140.482

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City’s Diversion Flow Calculation Height (feet) / Gate(feet) 13579 000000 0.250.030.0440.0670.0920.115 0.50.0540.0630.0920.1280.202 0.750.0780.0910.1460.1890.254 10.1180.1740.2830.3830.514

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Cemetary’s Diversion Flow Calculation Height (feet) / Gate(feet) 0.523.55 000000 0.40.030.0310.0340.0380.041 0.80.0540.0760.1010.1360.159 1.20.0750.1240.1570.2840.331 1.50.0990.1760.2940.3920.469

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Diversion Flow Modeling

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Time 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 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 )

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Time Delay Calculation Flow (in m^3/s) Johnnie Diversion (in min) Tom Diversion (in min) City/Cemetary Diversion (in min) 0.045465592621 0.095304398419 0.15246322341 0.21210279295 0.25195260274 0.3182242255 0.4164220230 0.5150200212 0.6141190200 0.7133180191 0.8130174184 0.9123168178 1120162178 1.5106145154 299135144

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Delay Modeling For 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 hours City’s Pond Delay=0.1 hours and Cemetary’s Pond Delay=0.5 hours

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

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Model Overview

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4. System Control & Simulations (PI, Pi α, PII α )

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Pond PI Controller

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Reservoir PI Controller

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Johnnie’s Pond (PI)

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Tom’s Pond (PI)

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City’s Pond (PI)

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Cemetary’s Pond (PI)

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Reservoir (PI) For PI Global error=156830 m 3 and Water loss = 8158 m 3

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Johnnie’s Pond in Use (PI)

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Tom’s Pond in Use (PI)

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City’s Pond in Use (PI)

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Cemetary’s Pond in Use (PI)

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Reservoir in Use (PI) For PI Global error=168450 m 3 and Water loss =7868 m 3

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Pond PI α Controller

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Reservoir PI α Controller

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Johnnie’s Pond (PI α )

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Tom’s Pond (PI α )

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City’s Pond (PI α )

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Cemetary’s Pond (PI α )

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Reservoir (PI α ) For α = 0.6 Global error=157320 m 3 and Water loss = 8026 m 3 For α = 0.8 Global error=164330 m 3 and Water loss = 7447 m 3 For α = 0.3 Global error=156950 m 3 and Water loss = 8066 m 3

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Johnnie’s Pond in Use (PI α )

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Tom’s Pond in Use (PI α )

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City’s Pond in Use (PI α )

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Cemetary’s Pond in Use (PI α )

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Reservoir in Use (PI α ) For α = 0.6 Global error=163620 m 3 and Water loss = 7176 m 3 For α = 0.8 Global error=168010 m 3 and Water loss = 7745 m 3 For α = 0.3 Global error=161750 m 3 and Water loss = 7912 m 3

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Pond PII α Controller

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Reservoir PII α Controller

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Johnnie’s Pond (PII α )

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Tom’s Pond (PII α )

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City’s Pond (PII α )

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Cemetary’s Pond (PII α )

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Reservoir (PII α ) For α = 0.3 Global error=155680 m 3 and Water loss = 7369 m 3 For α = 0.7 Global error=158170 m 3 and Water loss = 8118 m 3

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Johnnie’s Pond in Use (PII α )

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Tom’s Pond in Use (PII α )

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City’s Pond in Use (PII α )

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Cemetary’s Pond in Use (PII α )

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Reservoir in Use (PII α ) For α = 0.3 Global error=158320 m 3 and Water loss = 6979 m 3 For α = 0.7 Global error=161820 m 3 and Water loss = 7891 m 3

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5. Comparison : PI vs Pi α vs PII α

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Johnnie’s Pond in Use (PI vs Pi α vs PII α )

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Tom’s Pond in Use (PI vs Pi α vs PII α )

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City’s Pond in Use (PI vs Pi α vs PII α )

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Cemetary’s Pond in Use (PI vs Pi α vs PII α )

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Reservoir in Use (PI vs Pi α vs PII α ) For PI α α = 0.6 Global error=163620 m 3 and Water loss = 7176 m 3 For PII α α = 0.3 Global error=158320 m 3 and Water loss = 6979 m 3 For PI Global error=168450 m 3 and Water loss = 7868 m 3

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Comparison NormalConditionUsingCondition Value (m^3)PI relativeValue (m^3)PI relative PI Global Error1568300.00%1684500,00% PI Water Loss81580.00%78680,00% PIα Global Error157320+0.31%163620-2.87% Piα Water Loss8026-1.62%7176-8.79% PIIα Global Error155680-0.73%158320-6.01% PIIα Water Loss7369-9.67%6979-11.29%

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6. Conclusion The efficiency, for this system, of the PII α 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.

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References 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

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Thanks 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

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