Optimal Control of GREENHOUSE CULTIVATION Optimal Control of GREENHOUSE CULTIVATION Gerrit van Straten • Gerard van Willigenburg Eldert van Henten • Rachel van Ooteghem CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5963-2 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com v Contents Preface .xi Acknowledgments xv Authors .xvii Notation Conventions xix 1Chapter Introduction and Problem Statement 1 1.1 Greenhouse-Crop Cultivation—Benets and Challenges . 1 1.2 Automatic Control . 2 1.3 Elementary Description of the Greenhouse-Crop System 2 1.4 Measurements and Instrumentation 6 1.5 Decomposition, Fluxes, and Information Flows . 7 1.6 General State–Space Representation . 10 1.7 Hierarchical Computerized Control 11 1.8 Current Status of Computerized Control . 13 1.9 How Is This Book Organized? 14 Reference 14 2Chapter Introduction to Optimal Control of Greenhouse Climate 15 2.1 Introduction and Motivation 15 2.2 A Simple Illustrative Example 16 2.3 General Formulation of Optimal Control Problems 17 2.4 Benets and Difculties Associated with Optimal Control 21 3Chapter Open-Loop Optimal Control 25 3.1 Introduction .25 3.2 Optimal Control Theory 25 3.3 Optimal Control Algorithms .30 3.3.1 Indirect Methods 31 3.3.2 Direct Methods and Control Parameterization 33 References 38 4Chapter Closed-Loop Optimal Control . 39 4.1 Introduction . 39 4.2 State Estimation .40 4.3 Linear Quadratic Feedback . 41 4.3.1 Feedback by Receding Horizon Control 42 4.3.1.1 The Problem of Widely Different Time Scales 42 4.3.1.2 Feedback Design for Optimal Greenhouse Climate Control 44 4.3.2 Conclusions 47 References 48 vi Contents 5Chapter Greenhouse Cultivation Control Paradigms 49 5.1 Introduction . 49 5.2 Optimal Control Revisited . 49 5.2.1 Generic Problem Statement 49 5.2.2 Open-Loop Solution of the Whole Problem .50 5.2.3 The Choice of the Weather . 51 5.2.4 Closed-Loop Solution of the Whole Problem 52 5.2.4.1 Online Solution by Repeated Optimization 52 5.2.4.2 Online Solution by Using Stationarity of the Hamiltonian 54 5.2.5 Time-Scale Decomposition 54 5.2.5.1 Ofine Solution of the Slow Subproblem . 55 5.2.5.2 Online Implementation . 55 5.2.5.3 Hierarchical Control, Setpoint Tracking 56 5.2.5.4 Receding Horizon Optimal Control with Slow Costates as Inputs . 57 5.2.5.5 Explaining the Difference: The Sailing Analogy . 58 5.3 Earlier Surveys of Greenhouse Climate Control Solutions .60 5.4 Classication of Proposed Greenhouse Climate Control Solutions 61 5.4.1 Focus on Feedback Control of Fast Greenhouse and Fast Crop Subsystems . 63 5.4.1.1 General Overview 63 5.4.1.2 Realizing a Given Greenhouse Climate .64 5.4.1.3 Control of Greenhouse Climate within Operational Bounds 67 5.4.1.4 Greenhouse Climate Control with Cost Minimization 68 5.4.1.5 Controlling Fast Crop Processes: The “Speaking Plant” 71 5.4.2 Focus on Strategies Driven by Slow Crop Processes . 72 5.4.2.1 Assessing Economics by Simulation or Local Optimization . 73 5.4.2.2 Optimal Strategies Using Dynamic Optimization . 74 5.4.3 Integration, Application, and Implementation . 78 5.4.3.1 Expert Systems . 79 5.4.3.2 Implementation of Optimal Control—Overview . 79 5.4.3.3 Direct Application of Computed Controls 79 5.4.3.4 Hierarchical Control with Settings .80 5.4.3.5 Implementations of Optimal Control Using Meta- Information 80 5.4.3.6 Tracking the Slow Variables—Crop Development 81 5.4.3.7 Integrated Optimal Control 81 5.5 Discussion and Conclusion 81 References 82 6Chapter A Seminal Case: Lettuce 89 6.1 Introduction .89 6.2 Models .90 6.3 The Optimal Control Problem .94 6.4 Optimal Control Case Studies .97 Contents vii 6.4.1 Analysis of the Optimal Control Problem 97 6.4.2 Comparison of Optimal Control with Climate Control Supervised by a Grower . 102 6.4.2.1 Materials and Methods . 102 6.4.2.2 Results . 103 6.4.2.3 Discussion . 107 6.4.2.4 Concluding Remarks 108 6.4.3 Sensitivity Analysis of the Optimal Control Problem 109 6.4.3.1 Materials and Methods . 109 6.4.3.2 Results and Discussion . 110 6.4.3.3 Concluding Remarks 113 6.4.4 Time-Scale Decomposition 114 6.4.4.1 Materials and Methods . 114 6.4.4.2 Results . 115 6.4.4.3 Concluding Remarks 120 6.5 Concluding Remarks . 120 References 121 7Chapter An Experimental Application: Tomato 123 7.1 Introduction . 123 7.2 Tomato Model 124 7.2.1 Working with Leaves Instead of Generative Parts . 126 7.2.2 Assimilate Pool 126 7.2.3 Leaf and Fruit Biomass 129 7.2.4 Losses . 129 7.2.5 Constitutive Relations . 130 7.2.5.1 Photosynthesis 130 7.2.5.2 Growth Demand . 130 7.2.5.3 Maintenance Respiration 131 7.2.5.4 Development State 131 7.2.5.5 Harvest Rate . 132 7.3 Greenhouse Climate Model . 133 7.3.1 Heat Balances . 135 7.3.1.1 Soil 138 7.3.1.2 Heating Pipe System . 138 7.3.2 Mass Balances 140 7.3.2.1 Water Vapor in the Greenhouse Air . 140 7.3.2.2 Carbon Dioxide in the Greenhouse Air 142 7.3.3 Comparison of Lumped Model with Control Input by Actuators or by Fluxes 142 7.4 State–Space Form of the Complete Greenhouse-Crop Model 143 7.5 Calibration and Model Results 145 7.5.1 Calibration of the Big Leaf–Big Fruit Model . 145 7.5.2 Calibration of the Heating Pipe and Greenhouse Climate Model . 148 7.5.3 Conclusions about the Models 150 7.6 Open-Loop Optimization 151 7.6.1 Problem to Be Solved . 151 7.6.2 Method . 152 7.6.3 Results 153 7.6.4 Recapitulation of the Open-Loop Step . 161 viii Contents 7.7 Two-Time-Scale Receding Horizon Optimal Controller (RHOC) . 163 7.7.1 Problem to Be Solved . 164 7.7.2 Method . 165 7.7.3 Results 165 7.8 Evaluation of Optimal Control 167 7.8.1 Sensitivity of RHOC to Modeling Errors . 167 7.8.2 Sensitivity of Slow Costates to the Nominal Weather . 169 7.8.3 Sensitivity of RHOC to Slow Costates . 169 7.8.4 Sensitivity of RHOC to Weather Forecast and Prediction Horizon . 169 7.9 Assessment of Economic Result as Compared with Conventional Control . 172 7.9.1 Simulated Comparison . 173 7.9.1.1 Initial Conditions 173 7.9.1.2 Matching the Humidity Constraint Violation . 173 7.9.1.3 Humidity Penalty and Heat Input . 174 7.9.1.4 Results . 174 7.10 Discussion and Conclusions 176 References 177 8Chapter An Advanced Application: The Solar Greenhouse 179 8.1 Introduction . 179 8.2 Description of the Solar Greenhouse Concept 180 8.3 System Description 181 8.3.1 Greenhouse Conguration . 181 8.3.2 Assumptions . 182 8.4 The Solar Greenhouse Model 183 8.4.1 Carbon Dioxide Model . 189 8.4.1.1 Carbon Dioxide Supply . 191 8.4.1.2 Photosynthesis and Respiration 191 8.4.1.3 Carbon Dioxide Transport due to Ventilation 192 8.4.1.4 Carbon Dioxide Transport past the Screen . 192 8.4.2 Water Vapor Model 192 8.4.2.1 Canopy Transpiration . 193 8.4.2.2 Condensation of Water 193 8.4.2.3 Water Vapor Transport due to Ventilation 194 8.4.2.4 Water Vapor Transport past the Screen 194 8.4.3 Thermal Model . 194 8.4.3.1 Convection 195 8.4.3.2 Longwave Radiation Absorption 197 8.4.3.3 Shortwave Radiation Absorption 197 8.4.3.4 Conduction 200 8.4.3.5 Latent Heat Exchange . 201 8.4.4 Modeling the Screen 201 8.4.4.1 Screen Closure 202 8.4.4.2 Volume Flow Air past the Screen . 203 8.4.4.3 Temperatures and Concentrations of CO 2 and H 2 O When the Screen Is Open .203 8.4.5 Modeling Ventilation 204 8.4.5.1 Volume Flow of Air through Windows and Leakage .204 Contents ix 8.4.6 Modeling the Heating and the Cooling System .206 8.4.6.1 Heating System Boiler and Condenser .206 8.4.6.2 The Aquifer .209 8.4.6.3 Heating System Heat Pump 209 8.4.6.4 Cooling System Heat Exchanger 212 8.5 Model of Crop Biophysics . 214 8.5.1 Evapotranspiration . 215 8.5.2 Crop Photosynthesis and Respiration . 218 8.5.2.1 Photosynthesis Model . 218 8.5.3 Temperature Integration . 225 8.6 Sensitivity Analysis, Calibration, and Validation 228 8.6.1 Conventional versus Solar Greenhouse Model 228 8.6.1.1 Control Inputs .228 8.6.1.2 External Inputs 228 8.6.1.3 States . 229 8.6.2 Sensitivity Analysis 229 8.6.3 Parameter Estimation . 230 8.7 Optimal Control . 232 8.7.1 Cost Function . 232 8.7.1.1 Derivation Bounds for Aquifer Energy Content . 235 8.7.2 Receding Horizon Optimal Control . 239 8.7.3 Control Inputs .240 8.7.3.1 Initial Guess Control Inputs 242 8.7.3.2 State-Dependent Control Input Bounds 243 8.7.3.3 Example Grid Search 244 8.7.4 External Inputs: The Weather Predictions .245 8.7.5 Initial Values States 246 8.7.6 Optimization Method: Gradient Search .246 8.7.7 Results RHOC with Gradient Search .248 8.7.7.1 A Priori versus A Posteriori Results .249 8.7.7.2 Inuence of the Separate Solar Greenhouse Elements . 257 8.7.8 Conclusions and Discussion . 258 References 259 Appendices . 261 A. Solar Radiation Parameters . 261 A.1 Solar Parameters . 261 A.2 Radiation Parameters 262 B. Humidity Parameters 264 B.1 Saturation Pressure and Concentration .264 B.2 Relative Humidity . 265 B.3 Dewpoint Temperature .266 9Chapter Developments, Open Issues, and Perspectives . 267 9.1 Introduction . 267 9.2 Developments in the Greenhouse Industry and Consequences for Control . 267 9.2.1 Recent Advances in the Greenhouse Industry . 267 9.2.2 Future Developments in the Greenhouse Industry . 268 9.2.2.1 Innovations Motivated by Sustainability: Energy and CO 2 .269