Model Predictive Control for Operational Water Management: A Case Study of the Dutch Water System

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M

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A CASE

STUDY OF THE

DUTCH

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SYSTEM

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 29 september 2015 om 10:00 uur

door

Xin TIAN

Bachelor of Applied Mathematics East China Normal University

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This dissertation has been approved by the promotor: Prof. dr. ir. N. C. van de Giesen

Composition of the doctoral committee: Rector Magnificus

chairman

Prof. dr. ir. N.C. van de Giesen CiTG, TU Delft, promotor Dr. J. M. Maestre Universidad de Sevilla Dr. ir. R.R. Negenborn 3mE, TU Delft

Independent members:

Prof. dr. ir. J. Hellendoorn 3mE, TU Delft

Prof. dr. ir. Q. He East China Normal University Prof. dr. ir. P. van der Zaag CiTG, TU Delft & UNESCO-IHE Prof. dr. ir. H.H.G. Savenije CiTG, TU Delft

Prof. dr. ir. G.S. Stelling CiTG, TU Delft, reserved

Keywords: Model Predictive Control, operational water management, the Dutch water system

Printed by: Ipskamp Drukkers

Copyright © 2015 by X.Tian All rights reserved. No parts of this publication may be repro-duced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission of the author.

ISBN 978-94-6259-863-8

An electronic version of this dissertation is available at

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In memory of

dr. ir. Peter-Jules van Overloop

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S

UMMARY

W

ater is needed everywhere to satisfy domestic, agricultural and industrial water demands, to maintain navigation systems, and to preserve healthy and sustain-able ecosystems. In order to protect us from floods and to reallocate water resources in a man-made environment, the ‘hardware’, water-related structures, is constructed in many water systems. However, the ‘software’, suitable control approaches to oper-ationally maintain and regulate these structures, is still a challenge. Model Predictive Control, standing out from various control technologies, has shown its powerful ability to address multiple objectives, large-scale systems, distributed systems and long-term prediction problems in the field of operational water resources management.

This thesis aims to answer the following main research question:

How can we apply generic Model Predictive Control and its extensions to sat-isfy operational objectives in water resources management?

As the first element of Model Predictive Control, an internal model needs to be for-mulated to calculate system states at present and in the future. The linearized Saint Venant equations can well describe hydrodynamics of open water systems, which fit a linear internal model. Second, a quadratic objective function needs to be formulated based on actual management goals, which usually require the water level or discharge to be kept around a pre-defined setpoint. Last, physical and operational limitations are considered as constraints in the optimization problem, such as the height of the river-bank or the capacity of the pump. By describing a water resources management problem in terms of one of the elements, Model Predictive Control is able to address problems of flood management, water supply, irrigation and so on.

Chapter 5 applies the standard formulation of Model Predictive Control to flood man-agement in the Dutch water system. Instead of continued reinforcing and heightening water-related structures, Model Predictive Control shows the potential of making the most of the existing infrastructure to reduce potential damages and manage the delta in an operational and holistic way.

Different from conventional control, the computational complexity of the optimiza-tion is an important issue to consider in Model Predictive Control. Chapter 6 proposes two additional schemes, the adaptive prediction accuracy scheme and the large time step scheme, to achieve computational efficiency. These two schemes are practical and advantageous, especially when implementing Model Predictive Control on large-scale problems.

In reality, a large number of water resources management problems includes multi-ple management objectives at the same time, which can be addressed either in a cen-tralized way or a distributed way. Chapter 7 proposes both the cencen-tralized and dis-tributed Model Predictive Control for multi-input-multi-output water resources man-agement. Centralized Model Predictive Control includes all the objectives into a single

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viii SUMMARY

objective function while distributed Model Predictive Control considers the objectives in each subsystem individually and allows adjacent subsystems to communicate.

The results of this thesis suggest that Model Predictive Control is a useful technique for different kinds of water resources management problems, especially with constraints and multiple inputs and multiple outputs. The proposed additional schemes can enable Model Predictive Control to fulfill more real-time requirements, such as a large-scale problem.

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S

AMENVAT TING

W

ater is nodig om te voorzien in de waterbehoefte van huishoudens, landbouw en industrie; om scheepvaart mogelijk te maken en om gezonde en duurzame eco-systemen te behouden. Om ons tegen water te beschermen en om water te verdelen is veel ‘hardware’ ontwikkeld: de kunstwerken in watersystemen. De ‘software’ om die kunstwerken te bedienen en te beheren is daarentegen nog steeds een uitdaging. ‘Model Predictive Control’ in het bijzonder heeft laten zien een krachtig instrument te zijn dat kan omgaan met verschillende doelstellingen in het waterbeheer, in grote systemen en systemen met decentrale aansturing, en als er sprake is van lange-termijnvoorspellingen op het gebied van operationeel waterbeheer.

Dit proefschrift beantwoordt de volgende hoofdvraag:

Hoe kunnen ‘Model Predictive Control’ (MPC) en de voorzettingen daarvan toege-past worden om operationele doelen in waterbeheer te behalen?

MPC vereist als eerste element het formuleren van een intern model om de huidige en toekomstige systeemtoestanden te berekenen. Hierbij kan gebruik gemaakt worden van de gelinealiseerde Saint Venant vergelijkingen die de hydrodynamica van open wa-tersystemen goed kunnen beschrijven. Vervolgens moet een kwadratische doelfunctie worden opgesteld die is gebaseerd op de werkelijke beheerdoelen, die meestal vereisen dat het waterniveau op een vooraf gedefinieerd peil gehouden wordt. Tenslotte moeten fysieke en operationele randvoorwaarden in acht genomen worden, zoals de hoogte van de oevers of de capaciteit van de pomp. Door een waterbeheerprobleem te beschrijven in termen van deze elementen, kan MPC gebruikt worden bij het beheer van overstro-mingen, watertoevoer, irrigatie en dergelijke.

Hoofdstuk 5 laat een toepassing zien van de standaardformulering van MPC voor overstromingsbeheer in het Nederlands watersysteem. Als alternatief voor het steeds versterken en verhogen van kunstwerken wordt hier het potentieel getoond van het op-timaliseren van het gebruik van de bestaande infrastructuur om mogelijke schade te ver-minderen en de delta integraal en systematisch te beheren.

Bij MPC, anders dan bij standaard beheersmethoden, is het belangrijk de complexi-teit van de optimaliseringberekeningen te beschouwen. In hoofdstuk 6 worden daarom twee methoden uitgewerkt om de berekeningen efficiënter te laten plaatsvinden: de ‘adaptive prediction accuracy’ methode en de ‘large time step’ methode. Deze twee me-thoden zijn praktisch en nuttig, vooral als MPC op problemen van een grote schaal wordt toegepast.

In werkelijkheid behelzen veel waterbeheerproblemen meerdere managementdoe-len tegelijkertijd, die met of zonder centrale sturing aangepakt kunnen worden. In hoofd-stuk 7 wordt zowel gecentraliseerde als gedecentraliseerde MPC beschreven voor situa-ties met meerdere inputs en outputs. In MPC bij systemen met centrale sturing worden alle doelen in een enkele kostenfunctie geïntegreerd, terwijl bij MPC met decentrale stu-ring de doelen in ieder subsysteem individueel worden beschouwd en aangrenzende

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x SAMENVATTING

subsystemen met elkaar afgestemd worden.

De resultaten van dit proefschrift demonstreren dat Model Predictive Control een nuttige techniek is voor verschillende soorten waterbeheerproblemen, in het bijzonder als er sprake is van opgelegde beperkingen en bij meerdere inputs en outputs. De ont-wikkelde aanvullende methoden kunnen MPC geschikter maken voor situaties waarbij een grote behoefte is aan real-time informatie voor het nemen van beheerbeslissingen, zoals bij problemen op een grote schaal.

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A

BBREVIATIONS AND

N

OTATIONS

Abbreviations

APA adaptive predictive accuracy

CMPC centralized Model Predictive Control DMPC distributed Model Predictive Control

DWS Dutch water system

LTS large time step

MIMO multi-input-multi-output

MPC Model Predictive Control

MSL mean sea level

NSC the North Sea Canal

PI proportional-integral

QP quadratic programming

SISO single-input-single-output

WRM water resources management

WRS water resources system

Notations

∆

= defined as

∵ because

∴ therefore

H Â 0 matrix H is positive definite

HT transposed H

∇ the gradient operator

⇔ necessary and sufficient conditions

⊗ Kronecker tensor product

dim(S) dimension of space S

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C

ONTENTS

Summary vii

Samenvatting ix

1 Introduction 1

1.1 Water Resources Management . . . 2

1.2 The Dutch Water System and Its Water Resources Management Problems . 4 1.3 Classic Control Approaches for Operational Water Management . . . 6

1.4 Model Predictive Control for Operational Water Management . . . 7

1.5 Problem statement . . . 8

1.6 Dissertation Outline . . . 9

2 Model Predictive Control 11 2.1 Basic Formulation of Model Predictive Control . . . 12

2.1.1 The Internal Model . . . 13

2.1.2 The Cost Function . . . 15

2.1.3 The Constraint . . . 16

2.2 Solving an Model Predictive Control Problem . . . 17

2.2.1 The Final Form as a Quadratic Programming Problem . . . 17

2.2.2 Solving an Unconstrained Model Predictive Control Problem . . . . 17

2.2.3 Solving a Constrained Model Predictive Control Problem . . . 18

2.3 Concluding Remarks . . . 18

3 Model Predictive Control for Water Resources Management 19 3.1 Modeling Water Systems . . . 20

3.1.1 Modeling Hydrodynamics of Rivers and Canals . . . 20

3.1.2 Modeling Hydraulic Structures. . . 21

3.2 Applying Model Predictive Control to Water Resources Management Prob-lems. . . 23

3.2.1 Flood Defense . . . 25

3.2.2 Water Shortage Management . . . 25

3.2.3 Navigation System Maintenance . . . 26

3.2.4 Water Supply for Various Purposes . . . 26

3.3 Applying Model Predictive Control to Water Resources Management in Time and Space . . . 26

3.3.1 Handling Predictions . . . 26

3.3.2 Time-variant References . . . 26

3.3.3 Managing Large-scale Water Systems . . . 27

3.3.4 From Centralized Management to Distributed Management. . . 27

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xiv CONTENTS

4 Modeling the Dutch Water System 29

4.1 Modeling the Rivers and Canals of the Dutch Water System . . . 30

4.2 Structure Settings . . . 34

5 Operational Flood Management in the Low-lying Dutch Water System 37 5.1 Floods in Low-lying Deltas . . . 38

5.2 Model Predictive Controller Design for Preventing Riverine and Coastal Floods. . . 38

5.3 Flood Prevention in the Dutch Water System . . . 39

5.3.1 Preventing Riverine Floods. . . 39

5.3.2 Preventing Coastal Floods . . . 48

6 Efficient Model Predictive Control for Managing Water Resources in Real-time 55 6.1 Problems of Applying Model Predictive Control to Water Resources Man-agement in Real-time . . . 56

6.2 The Adaptive Prediction Accuracy Scheme for Handling Long-term Pre-dictions in Real-time Control . . . 57

6.3 Simulations and Results of Applying the Adaptive Prediction Accuracy Scheme to Control Problems with Long-term Predictions . . . 59

6.3.1 Simulation Design . . . 59

6.3.2 Using Long-term Predictions for Flood Management . . . 62

6.3.3 The Reduction of the Computation Time . . . 62

6.3.4 Performance Assessment . . . 64

6.4 Model Prediction Control with the Large Time Step Scheme . . . 65

6.5 Simulations and Results of Applying the Large Time Step Scheme to Con-trol Problems with Long-term Predictions . . . 68

7 Multi-objective Water Resources Management 71 7.1 Water Resources Management Problems with Multiple Objectives . . . 72

7.2 The Centralized Model Predictive Controller Design . . . 72

7.3 Simulation of Using Centralized Model Predictive for Multi-objective Wa-ter Resources Management . . . 74

7.4 Results and Discussions of Using Centralized Model Predictive for Multi-objective Water Resources Management . . . 81

7.4.1 Scenario 1: No Setpoint Setting . . . 81

7.4.2 Scenario 2: Low Penalty on Water Supply . . . 81

7.4.3 Scenario 3: Low Penalty on Navigation. . . 82

7.5 The Distributed Model Predictive Controller Design . . . 85

7.6 Simulation and Results of Using Distributed Model Predictive for Multi-objective Water Resources Management . . . 88

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CONTENTS xv

8 Conclusions and Recommendations 93

8.1 Conclusions. . . 94

8.1.1 Conclusions on Model Predictive Control . . . 94

8.1.2 Conclusions on Application of Model Predictive Control for Water Resources Management . . . 94

8.2 Recommendations . . . 95

8.2.1 Model Predictive Control or Conventional Control? . . . 95

8.2.2 Linear or Non-linear? . . . 95 8.2.3 Centralized or Distributed? . . . 95 8.2.4 Outlook . . . 95 References 97 Curriculum Vitae 103 List of Publications 105

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1

I

NTRODUCTION

Water is the nature of all things. Thales (624B.C. ∼ 547B.C.)

The highest goodness is like water. Water benefits all things and does not compete.

上善如水水善利万物而不争

Lao Zi (Taoist, 571B.C. ∼ 471B.C.)

When we enjoy benefits of water, we also struggle with continuous threats and troubles. In order to reduce the damage induced by water, we never cease to explore ways to manage water resources. The ‘hardware’, artificial water-related structures, has been constructed in many water systems. The ‘software’, suitable control approaches to operationally main-tain and regulate these structures, is still a challenge. Model Predictive Control, standing out from various control technologies, has shown its powerful ability to address multiple objectives, large-scale systems, distributed systems and long-term prediction problems in the field of operational water resources management.

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2 1.INTRODUCTION

1.1. W

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ater is an everlasting attraction. Our ancestors established habitats along rivers due to the easy access to available water resources. Even today, people prefer to live near rivers and seas because of attractive views and water-related recreation.

Water is a precious resource, which is needed almost everywhere. Water is required to satisfy domestic, agricultural and industrial water demands, to maintain navigation systems, and to preserve healthy and sustainable ecosystems.

However, water is also a global problem. It is estimated that 780 million people lack access to clean water worldwide and approximately 3.4 million die from a water-related disease annually (www.water.org). Moreover, with the population booming and even ‘saturating’, for example, in India, China, Egypt and Algeria, water demands for agricul-ture are considerably increasing. These demands are already far beyond the maximum available amount. Furthermore, cities in estuaries, such as Rotterdam and Shanghai, are facing problems of the coastal flood and urban water-logging, induced by a potential storm surge. Besides, countries of a large area usually need to address unequal water distribution such as the USA, Australia, and China.

In order to protect and reallocate water resources in a man-made environment, dikes, dams and adjustable infrastructures, such as sluices, barriers, and pumps, have been constructed in many water systems. Water resources, together with the water-related infrastructure, constitute a water resources system (WRS), which is defined as follows [Koudstaal et al., 1989]:

A Water Resources Systems (WRS) is geographically defined and consists of: the natural subsystem of rivers, brooks, ditches and lakes and their embank-ments and bottoms; the infrastructural subsystem, such as canals, reservoirs, dams, weirs, sluices, pumping plants, and waste water treatment plants.

To plan and make decisions for water resources management (WRM), multiple de-mands for different purposes, often conflicting, have to be met. Realizing WRM is similar to running a computer: not only hardware is required but software is also a platform of great importance for full and smooth functioning. On the basis of existing structures (the ’hardware’ of a WRS), developing a good plan (the ’software’ of a WRS) to manipu-late them is as critical as a robust and efficient computer operating system.

Past events teach us that the lack of a reasonable plan to manipulate structures and manage water resources usually directly leads to an unsuccessful implementation. For instance, the Yellow River in China has been fully canalized to protect against floods, which gives rise to the even worse consequence that the waterway frequently dries up in the downstream catchment. Implementing the Snowy Mountains Scheme project, aiming to address unequal distribution in south-east Australia, led to flooded habitats of plants and animals. However, failure teaches success. Engineers can always accu-mulate experience and seek more effective solutions. This results in the emergence and development of every scientific subject and WRM is not an exception.

Water resources management is often implemented in a hierarchical structure, as shown in Figure 1.1. The administrative level at the top is primarily in charge of es-tablishing legislation in order to support a healthy and developing WRS. The general

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1.1.WATERRESOURCESMANAGEMENT

1

3 Implementing Level Operational Level Administrative Level

Figure 1.1: Hierarchical structure of Water resources management.

and overall goals, instead of detailed acts, are focused at the administrative level, for in-stance, a task of flood management may be allocated to a certain parliament or a water organization by the administrator at this level. The operational level in the middle fo-cuses on seeking technical measures to design a feasible plan in detail. The operational level designs rules, according to which structures can be manipulated. Once the rules for controlling the structures have been settled, the implementing level at the bottom level is in charge of ensuring the implementation of the designed plan.

The operational level plays a communicative role between the other two levels. Gen-erally, the operational level receives feedback information from the implementing level, such as in-situ measurements or structural failures, and then accordingly adjusts control rules. If no optimal plans can be made to realize the administrative target based on the existing hydrological conditions and structures, the administrative level is supposed to adjust the legislation and the established plan, for example, adding more structures or regulating river-ways. Next, the operational level can validate the feasibility of the new plan and update the control rule in technical terms so that either the further feedback can be given to administrative level or the goals can be fulfilled based on the updated rule.

When making water management plans at the operational level, a variety of targets usually needs to be taken into consideration. This thesis also focuses on this kind of WRM problem, the so-called integrated WRM problem, which is defined by the Global Water Partnership (http://www.gwp.org/) as follows:

Integrated water resources management is a process which promotes the coor-dinated development and management of water, land, and related resources, in order to maximize the resultant economic and social welfare in an

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equi-1

4 1.INTRODUCTION

table manner without compromising the sustainability of vital ecosystems. In other words, integrated WRM requires a holistic plan to include all local, regional and national management objectives. Integrated WRM is an important and practical topic since most WRSs do not have only one problem but chained and often conflicting prob-lems.

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YSTEM AND

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ROBLEMS

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he Dutch water system (DWS), also known as the Rhine-Meuse-Scheldt delta system, is the main case study in this thesis. The DWS is a low-lying joint area formed by the confluence of three Rivers Rhine, Meuse, and Scheldt, see Figure 1.2. The whole area of the DWS, including the territory and all the water bodies, amounts to 41,160 km2 [Netherlands Hydrological Society (NHV), 2011], over a quarter of which lies below mean sea level (MSL).

Generally, the DWS plays an important role with respect to water transportation. The excessive amount of water flowing through the DWS is diverted into the North Sea via barriers and sluices. On the other hand, a considerable amount of cargoes needs to be transported everyday via navigable ways, such as the New Water Way (‘Nieuwe water-weg’ in Dutch) and the North Sea Canal (‘Noordzeekanaal’ in Dutch).

The history of the DWS shows a continuous struggle between man and nature. The storm surge of 1916 gave rise to the closure of the inland Zuiderzee and the reclamation of large polder areas. Later, in 1953, another disastrous coastal flooding, also primarily caused by a storm surge at the North Sea, resulted in 1800 deaths and enormous eco-nomic losses [Gerritsen, 2005]. After this notorious event, the Delta Project was initiated in 1958 to close the southwestern part of the Netherlands. Later a series of dams, bar-riers, large gates and pumps were installed inland and along the coastline. This typical man-made environment enables the system to be managed at an operational level.

Even though the area is protected through this enormous man-made effort, the DWS is still facing different kinds of problems, especially when climate change is considered:

• Flood defense: Including preventing both riverine and coastal floods, flood de-fense is still the most important task in the DWS.

• Water supply: Water is required for various users, such as household, industry, agriculture, polder flushing, and canal salt water flushing.

• Navigation: As the DWS is the main entrance of waterways to central Europe, a certain depth of the navigable channel has to be guaranteed for smooth naviga-tion.

• Drought management: Though droughts do not happen very frequently in the DWS, they result in wider and more serious damages and economic losses. Water needs to be stored ahead of droughts and operationally allocated during droughts. As a water system with complex problems, the DWS is in need of operational measures to be managed, which inspired the use of the DWS as an illustrative case study.

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1.2.THEDUTCHWATERSYSTEM ANDITSWATERRESOURCESMANAGEMENTPROBLEMS

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ontrol techniques, including classic (or conventional), modern and robust control theories, are commonly used in almost all fields of engineering and science [Ogata, 2010]. Along with the development of control theory, operational WRM was realized by the classic control approaches initially in the 20th century, for example, feedback control [Liu et al., 1994; van Overloop, 2006a; Clemmens and Wahlin, 2004; Schuurmans, 1997] and feedforward control [Ahn, 2000; Clemmens and Wahlin, 2004; Bautista et al., 2003; Schuurmans et al., 1999].

Control inputs and outputs need to be defined beforehand in a control problem. In the thesis, the inputs and outputs of a WRS are defined either as the water level or the discharge [van Overloop, 2006b]. Other objects, such as the reservoir volume, can be converted into a water level. Depending on the number of inputs and outputs of the considered system, control problems can be categorized as:

• SISO: single-input-single-output (SISO) problems

• MIMO: multi-outputs (MIMO) problems, also including

multi-inputs-single-output (MISO) and single-input-multi-outputs (SIMO) problems.

Depending on objectives of the considered system, control problems can also be cate-gorized as local control problems, in which a group of controllers is regulated in order to fulfill local objectives, and global control problems, in which different groups of con-trollers, usually geographically distributed, are regulated in order to fulfill separate ob-jectives. A global control problem is a set of local control problems, which can be solved in the following manners:

• In a centralized manner: All objectives are merged in a single agent.

• In a decentralized manner: All objectives are solved individually by separate agents

and the communication between agents is not allowed.

• In a distributed manner: All objectives are solved individually by separate agents

and the communication between agents is allowed.

The following paragraphs in this subsection give an overview of the application of classic control approaches to WRSs.

The feedback control approach is also referred to as closed-loop control, which has successfully addressed a large range of WRM problems. Feedback controllers work in a straightforward way, relying on the principle that the deviation error between the actual value and the pre-defined reference value is corrected in every closed loop. Feedback controllers are robust and usually make a system resilient to external disturbances [Astr and Murray, 2012]. However, the repetitive correction of errors is always triggered after the present moment, resulting in significant time delays.

The feedforward control approach, also referred to as open-loop control, is an alter-native to overcome time delays. Since the future evolution of the system is considered by taking the prediction into account, the required amount of adjustment is calculated

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1.4.MODELPREDICTIVECONTROL FOROPERATIONALWATERMANAGEMENT

1

7

and implemented in advance. In this way, the system can respond early to a potential upcoming event. However, due to the prediction accuracy and system limitations, the adjustment ahead cannot guarantee the water level and discharge to reach the reference value.

Feedback controllers are commonly combined with feedforward controllers to com-pensate for the shortcomings of both. Feedback and feedforward controllers are easy to tune and efficient to obtain solutions. So they are fairly practical for a local SISO problem. Nonetheless, most WRM cases are primarily global and MIMO. Moreover, the constraints, such as the height of the dike or the capacity of the pump, is an important element to be considered in WRM, which cannot be handled in either feedback control, feedforward control or the combination. Hence, a more advanced control approach is required to overcome these weaknesses and better manage water resources.

1.4. M

ODEL

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ONTROL FOR

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ATER

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ANAGEMENT

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ince its appearance, Model Predictive Control (MPC) has had a widespread impact on almost every control industry [Darby and Nikolaou, 2012; Maciejowski, 2002], for example, chemical process control [Eaton and Rawlings, 1992; Qin and Badgwell, 2003], power system management [Kouro et al., 2009], and operational water management [van Overloop, 2006b]. The advantage of using MPC is that feedback control, feedforward control and explicit constraints on system states can be included. MPC uses an internal model to calculate the behavior of the system on account of system predictions over a finite prediction horizon (in open loop). Afterward, an objective function is built up to define the real-time control goals. The optimal solutions are obtained by solving the objective function. At the next time step, the optimal control problem is reformulated and resolved over the next prediction horizon, meanwhile taking account of updated measurements and predictions. The procedure is repeated for every prediction horizon in a receding fashion.

As a modern control approach, MPC preserves all the good features of conventional control approaches and can additionally address time delays and constraints. Further-more, extended schemes are flexible to be fused with the standard MPC structure to suit a wide variety of applications, such as robust MPC, tree-based MPC, distributed MPC. Owing to its ability and flexibility, MPC is especially powerful and is gaining popularity in multi-variable process control [Borrelli et al., 2014; Holkar and Waghmare, 2010], such as operational WRM [van Overloop, 2006b]. Successful applications of MPC in WRM can be found in: irrigation and drainage system control[Zafra-Cabeza et al., 2011; van Over-loop, 2006a; Negenborn et al., 2009; Hashemy Shahdany et al., 2015], flood defense [Tian et al., 2014; Breckpot et al., 2013; Blanco et al., 2010], integrated water quantity and qual-ity management [Xu et al., 2013], reservoir management [Galelli et al., 2014; Raso, 2013], and WRM with forecast uncertainties [Raso et al., 2014; Maestre et al., 2013].

Different from classic control approaches, the computation time of solving an opti-mization in MPC is a significant issue that has to be considered. When employing MPC on WRSs, there are two typical ways. One way is to solve an optimization off-line using a multi-parametric program solver, in order to generate a look-up table [Borrelli et al.,

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1

8 1.INTRODUCTION

2014]. However, off-line MPC cannot include model uncertainties [Kouvaritakis et al., 2002] and its computation time grows rapidly with the problem size (state dimension and prediction horizon) [Wang and Boyd, 2010; Buerger, 2013]. This means that the off-line MPC can only be applied to rather small-scale problems. The other way to employ MPC is to solve an optimization on-line, for example, over each prediction horizon. This on-line scheme takes realistic system measurements and predictions into account. On-line MPC can be applied to WRM of large water systems, especially with long predictions and uncertainties. The on-line scheme is therefore chosen in this thesis for large-scale problems.

The optimal control problem in MPC is commonly framed as a quadratic program-ming (QP) problem due to available efficient algorithms to solve a convex optimization problem in MPC. Nonetheless, solving a QP problem on-line over a long prediction hori-zon still results in substantial time and computational resources [Borrelli et al., 2014]. Moreover, when extended schemes are applied in MPC, a larger amount of computa-tion is generated accordingly. Thus, the challenge remains of how to obtain solucomputa-tions efficiently in order to implement MPC in real-time on large problems [Buerger, 2013].

Last but not least, a special structure used in MPC, hierarchical MPC, is worth men-tioning to complete this subsection, which is frequently applied in irrigation and drainage systems [Sadowska et al., 2014]. Hierarchical MPC addresses a global MIMO problem within two layers. The upper layer makes a holistic plan for the whole system and deter-mines a setpoint for every local subsystem while the local subsystem in the lower layer executes the demand delivered from the upper layer by running a simple classic con-troller or a centralized MPC concon-troller.

1.5. P

ROBLEM STATEMENT

T

his thesis aims to answer the main research question:

How can we apply generic MPC and its extensions to satisfy operational ob-jectives in water resources management?

In order to answer this question, a series of objectives are fulfilled step by step.

Firstly, the thesis studies the possibility of applying MPC to WRM at the operational level, linking a general WRM problem with one or a few principles of MPC. Based on the features of MPC, the thesis demonstrates how to include and meet different basic requirements of WRM problems.

Secondly, the thesis studies the potential of applying MPC to more complicated WRM problems by involving some specific extended schemes. Illustrative cases studied in this thesis include operational riverine and coastal flood management, multi-objective WRM, distributed WRM.

Thirdly, the thesis studies the efficiency of applying MPC for WRM in real-time. In practice, WRM of large-scale WRSs and WRM with long-term predictions usually lead to a control problem with a heavy computational burden. In order to reduce the com-putational complexity of the optimization in MPC, two practical schemes, namely the adaptive prediction accuracy scheme and the large time step scheme, are added to the standard MPC structure.

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1.6.DISSERTATIONOUTLINE

1

9

Last but not least, the thesis also studies the potential of utilizing MPC at the opera-tional level to improve the decision-making process at the administrative level and the plan - implementing process at the implementing level.

1.6. D

ISSERTATION

O

UTLINE

I

n Chapter 2, the underlying theory and the basic formulation of generic MPC is intro-duced, including the introduction of the key elements of MPC, i.e. the internal model, the constraint, the setpoint and the objective function.

Chapter 3 focuses on the way to apply MPC to any general WRM problem. A hy-drodynamic model is prerequisite to implement control, which is introduced in Section 3.1. A series of WRM examples are discussed in detail, including flood defense, drought management, and navigation system maintenance.

Chapter 4 lays the foundation for subsequent chapters. Since the DWS is used as the test case in this thesis, this chapter specifically deals with the hydrodynamic modeling and the water-related structure settings.

Chapters 5 - 7 study practical WRM problems in detail. Chapter 5 illustrates how to protect against riverine and coastal floods in a low-lying delta. Chapter 6 illustrates how to reduce the computation time of MPC for the sake of real-time control on large-scale systems or with long-term predictions. Chapter 7 illustrates how to include multiple objectives into MPC, in centralized and distributed manners, in order to make a holistic WRM plan.

Chapter 8 summarizes the contributions of this dissertation and elaborates on the recommendations.

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2

M

ODEL

P

REDICTIVE

C

ONTROL

Starting to weave fishing nets is far better than merely coveting fish in the river.

临渊羡鱼弗如退而结网

Liu An (Taoist, 179B.C. ∼ 122B.C.)

Chapter 1 has discussed several types of classic and modern control approaches that can be applied to operational WRM. Model Predictive Control is chosen and studied in this thesis due to its ability to deal with multiple variables, time delays, uncertainties and especially constraints, which are relevant to water resources management.

This chapter introduces the basic rationale of MPC. Section 2.1 describes the way to build up a standard formulation of MPC. First of all, an internal model needs to be built up to calculate system states at present and in the future. Then, an objective function is formu-lated to include all control targets. Meanwhile, constraints on system states and control variables have to be satisfied. The procedure is implemented in a receding horizon man-ner. In Section 2.2, the way to solve an MPC problem is briefly introduced for the purpose of discussions in sequential chapters.

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2

12 2.MODELPREDICTIVECONTROL

2.1. B

ASIC

F

ORMULATION OF

M

ODEL

P

REDICTIVE

C

ONTROL

M

odel Predictive Control (MPC), also known as model-based predictive control or receding horizon control [Camacho and Bordons, 1999; Maciejowski, 2002], stands for a family of control techniques. These names reflect two important features of MPC:

(a) MPC is a model-based control approach, in which a dynamic system model, also referred to as an internal model, describes the evolution of the system states and outputs as a function of the inputs and disturbances applied. Then a constrained optimization problem, typically a quadratic programming problem, is solved over a certain prediction horizon. Later, only the first control action, which is the opti-mal solution relating to the present moment, is implemented in the system. (b) At the next time step, the optimal control problem is reformulated and resolved

over the next prediction horizon, meanwhile taking account of updated measure-ments and predictions. The procedure is repeated at each time step, resulting in a receding horizon strategy.

These two features are easily understood through an example. Assume a pump is controlled hourly to keep the water level of a canal close to a predefined setpoint. At the beginning of every hour, the precipitation and water demand of the water system is predicted, for example, for the next two days. Later, the optimal control actions, which are the flow adjustment via the pump, for the next two days are obtained by solving an optimization problem. Next, only the control action for the next hour is implemented, because when the beginning of the next hour is reached, new control actions are ob-tained from a newly solved optimal control problem with updated measurements and predictions.

The way to formulate a standard MPC structure has been discussed thoroughly in lit-erature, for example, see [Maciejowski, 2002; Camacho and Bordons, 1999; Borrelli et al., 2014]. For the sake of comprehension and further discussion, the formulation is shown in the following paragraphs.

Consider a system that needs to be controlled to meet several targets. The follow-ing constrained optimization problem, startfollow-ing from any time step k over a prediction horizon Np, can be defined as follows:

min u(k),...,u(k+Np−1) J (k) = Np−1 X i =0 f (x(k + i + 1),u(k + i )) (2.1) subject to x(k + i + 1) = s(x(k + i ),u(k + i ),d(k + i )) i = 0,..., Np− 1 (2.2) g (x(k + i + 1),u(k + i )) ≤ b(k + i ) i = 0,..., Np− 1 (2.3)

where Np∈ N+is the length of the prediction horizon, x(i ) ∈ Rnis the state at time step i ,

u(i ) ∈ Rmis the control variable at time step i , d (i ) ∈ Rnis the disturbance, f :Rn+m_{→ R} is the stage cost function (linear or non-linear), s :Rn+m_{→ R}n is the function describ-ing system dynamics, g :R(n+m)×l→ Rl is the constraint function, and b(i ) ∈ Rl is the constraint vector. The feasible optimal solution (if it exists) is found by solving the opti-mization problem (2.1)-(2.3).

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2.1.BASICFORMULATION OFMODELPREDICTIVECONTROL

2

13

2.1.1. T

HE

I

NTERNAL

M

ODEL

The internal model s in (2.2) plays a role in describing the present and future behav-ior of the controlled system, which reveals the evolution of the system along the predic-tion trajectory. The internal model can be obtained by system identificapredic-tion, which aims to set up a relationship between inputs and outputs by performing a series of experi-ments [Maciejowski, 2002].

The internal model can also be obtained from known physical principles. For in-stance, hydrodynamics of open water systems can be described via the Saint-Venant equation set, based on the assumptions that the flow is one-dimensional, the streamline curvature is small, vertical accelerations are negligible, the slope of the average channel is small, and the variation of channel width is small as well [Litrico and Fromion, 2009].

Remark 1 Most physical equations describing principles of nature are non-linear and

dif-ferential. However, they are usually linearized in control problems because linear prob-lems are more efficient to solve and many experimental works have shown that linear models also have good performances. Besides, based on the linearity, the constraint (2.2) can be substituted into the objective function (2.1) using recursion. To obtain a linear form from a non-linear physical equation, the first-order Taylor’s expansion can be applied to the equation in a smooth interval.

Once the linear model of system dynamics has been obtained, Equation (2.2) can be further expressed in the following form:

x(k + i + 1) = A(k + i )x(k + i ) + Bu(k + i )u(k + i ) + Bd(k + i )d(k + i ) (2.4)

where A(k) ∈ Rn×n, Bu(k) ∈ Rn×mand Bd(k) ∈ Rn×lare coefficient matrices derived from

physical parameters of the system and the linearization of non-linear equations. Because of the linearity of the internal model (2.4), the states x’s in the time interval [k, k + t] can be represented by the initial state as follows:

x(k + t) = A(k + t − 1)x(k + t − 1) (2.5)

+Bu(k + t − 1)u(k + t − 1) + Bd(k + t − 1)d(k + t − 1)

= A(k + t − 1)A(k + t − 2)x(k + t − 2)

+A(k + t − 1)Bu(k + t − 2)u(k + t − 2) + Bu(k + t − 1)u(k + t − 1)

+A(k + t − 1)Bd(k + t − 2)d(k + t − 2) + Bd(k + t − 1)d(k + t − 1) ∀l ∈[2,t ]∩N = Ã l Y i =1 A(k + t − i ) ! x(k + t − l ) (2.6) + l X j =2 Ã_{j −1} Y i =1 A(k + t − i ) ! Bu(k + t − j )u(k + t − j ) + l X j =2 Ã_{j −1} Y i =1 A(k + t − i ) ! Bd(k + t − j )d(k + t − j ) +Bu(k + t − 1)u(k + t − 1) + Bd(k + t − 1)d(k + t − 1)

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2

l =t = Ã _t Y i =1 A(k + t − i ) ! x(k) (2.7) + t X j =2 Ã_{j −1} Y i =1 A(k + t − i ) ! Bu(k + t − j )u(k + t − j ) + t X j =2 Ã_{j −1} Y i =1 A(k + t − i ) ! Bd(k + t − j )d(k + t − j ) +Bu(k + t − 1)u(k + t − 1) + Bd(k + t − 1)d(k + t − 1)

The equation (2.7) can also be formulated in matrices: ˜ x(k, t ) = ˜A(k, t )x(k) + ˜Bu(k, t ) ˜u(k, t ) + ˜Bd(k, t ) ˜d (k, t ) (2.8) where ˜ x(k, t ) = [x(k + 1),··· , x(k + l ),··· , x(k + t)]T (2.9) ˜

u(k, t ) = [u(k),··· ,u(k + 1 − 1),··· ,u(k + t − 1)]T (2.10) ˜ d (k, t ) = [d(k),··· ,d(k + 1 − 1),··· ,d(d + t − 1)]T (2.11) ˜ A(k, t ) =              A(k) .. . l −1 Q i =0A(k + i ) .. . t −1 Q i =0A(k + i )              (2.12) ˜ Bu(k, t ) = (2.13)                   Bu(k) A(k + 1)· Bu(k) Bu(k + 1) 0 . . . . .. µ_{l −1} Q i =1A(k + i ) ¶ · Bu(k) · · · _BA(k + l − 1)· u(k + l − 2) Bu(k + l − 1) . . . . .. µ_{t −1} Q i =1A(k + i ) ¶ · Bu(k) · · · Bu(k + t − 1)                  

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2.1.BASICFORMULATION OFMODELPREDICTIVECONTROL

2

15 and ˜ Bd(k, t ) = (2.14)                   Bd(k) A(k + 1)· Bd(k) Bd(k + 1) 0 . . . . .. µ_{l −1} Q i =1A(k + i ) ¶ · Bd(k) · · · _BA(k + l − 1)· d(k + l − 2) Bd(k + l − 1) . . . . .. µ_{t −1} Q i =1A(k + i ) ¶ · Bd(k) · · · Bd(k + t − 1)                  

The equations (2.7) and (2.8) indicate the relationship between x(k + t) and x(k), for example, the state x(k + t), in fact, is a linear combination of the initial state, the control variables over the prediction, and the disturbances.

2.1.2. T

HE

C

OST

F

UNCTION

In practice, the cost function f presented in (2.1) is usually formalized according to the actual aim of the control problem, such as penalizing the deviation between the system state and a reference trajectory. Considering the positive and negative deviations, a quadratic form is commonly formulated as follows:

J (k) = Np−1 X i =0 f (x(k + i + 1),u(k + i )) (2.15) ∆ = Np−1 X i =0 · x(k + i + 1) − r (k + i + 1) u(k + i ) ¸T· Q 0 0 R ¸ · x(k + i + 1) − r (k + i + 1) u(k + i ) ¸ (2.16) = ( ˜x(k, t ) − ˜r(k, t))TQ ( ˜˜ x(k, t ) − ˜r(k, t)) + ˜u(k, t )TR ˜˜u(k, t ) (2.17) E q.2.8

= u(k, t )˜ TH ˜u(k, t ) + F ˜u(k, t ) +C (2.18)

where Q ∈ Rn×n, R ∈ Rm×mare quadratic penalties on states and control variables, r ∈ Rn is the reference vector with respect to the system states and

˜ Q= diag([Q, · · · , Q])∆ (2.19) ˜ R_{= diag([R, · · · , R])}∆ (2.20) H= ˜∆B_uT(k, t ) ˜Q ˜Bu(k, t ) + ˜R (2.21) F= 2∆ ¡ ˜x(k, t)T_{A(k, t )}_˜ T + ˜d (k, t )TB˜d(k, t )T− ˜r(k, t )T ¢ _˜ Q ˜Bu(k, t ) (2.22)

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2

C_{= ˜r(k, t )}∆ TQ ˜˜r (k, t ) + ˜x(k, t)TA(k, t )˜ TQ ˜˜A(k, t ) ˜x(k, t ) (2.23) + ˜d (k, t )TB˜d(k, t )TQ ˜˜Bd(k, t ) ˜d (k, t ) + ˜x(k, t)TA(k, t )˜ TQ ˜˜Bd(k, t ) ˜d (k, t )

−2 ˜r(k, t )TQ ˜˜A(k, t ) ˜x(k, t ) − 2 ˜r(k, t)TQ ˜˜Bd(k, t ) ˜d (k, t )

In many cases in the field of WRM, Q is semi-definite and R is definite, and both are diagonal, which guarantees the objective function (2.18) to be convex.

Remark 2 The interaction of states is usually considered in the internal model rather than

in the objective function and the controllers are independent of each other. In other words, Q and R are diagonal. Besides, if any state of the system x is not considered in the control problem, the related element on the diagonal of Q can be set as zero. However, u, as defined as the adjustment of the structure, cannot have a zero penalty. Otherwise, u would be any value within its constraints, which is not expected in reality. To conclude, Q is a diagonal matrix with non-negative elements on its diagonal and thereby semi-definite and R is a diagonal matrix with positive elements on its diagonal and thereby definite.

Remark 3 The term ( ˜x(k + 1) − ˜r(k + 1))TQ( ˜˜ x(k +1)− ˜r(k +1)) in (2.16) represents the goal that the state vector is regulated towards the reference vector over the prediction horizon while the term ˜u(k) ˜R ˜u(k) implies the goal that the operation of controllers should not be too frequent.

Remark 4 In (2.18), the first term and the second term stand for the quadratic and linear

forms respectively, which determines the solution of the optimization problem (the op-timum). The remainder C is a constant, which does not contribute to characteristics of optimums. The constant term can be left out when searching the optimum.

The values of penalties Q and R are determined by the relative importance of sub-objectives. These penalties are also referred to as ‘weights’. A relatively heavy weight on a system state x indicates that this state should evolve towards the reference trajectory r as fast as possible and a relatively heavy weight on a control variable u indicates that this variable should be relatively small, which means the control effort is penalized.

An MPC formulation with a linear internal model and a quadratic cost function can be well connected with the theory of linear systems and control [Maciejowski, 2002] and can fit a real-time control problem. In other words, because of a linear internal model and a quadratic cost function, the optimization problem usually can be solved by efficient algorithms, such as the interior-point method or the active-set method [Ma-ciejowski, 2002].

2.1.3. T

HE

C

ONSTRAINT

Constraints are the limitations on optimization solutions. They originate from phys-ical restrictions or operational requirements. Actually, constraints exist in a very wide range of problems, for example, a pump has a capacity or a water level has an upper or lower boundary. The most significant characteristic of MPC is its ability to take the constraint into account in an explicit manner.

Two types of constraints, hard constraints and soft constraints, are basically used in MPC [Zheng and Morari, 1995; Prasath and Jø rgensen, 2009; Bemporad et al., 2002;

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2.2.SOLVING ANMODELPREDICTIVECONTROLPROBLEM

2

17

Wang and Boyd, 2010]. Hard constraints are rigid limits on the system outputs or con-trol variables while soft constraints are less rigid and are allowed to be violated to some extent. For instance, it does not cause a serious problem if the water level in a river rises higher than the safety level for a very short period. This means the safety level can be somehow violated and treated as a soft constraint.

It is very practical to introduce soft constraints in control. On the one hand, the violation, in reality, does happen. The soft constraint can make reality be modeled in a more accurate way. On the other hand, the infeasibility problem can be avoided since the feasible area is expanded.

To implement a soft constraint, a coupled virtual state and virtual input signal need to be applied to the internal model [van Overloop et al., 2008] as follows:

· x(k + 1) ˆ x(k + 1) ¸ = · A(k) 0 A(k) 0 ¸ · x(k) ˆ x(k) ¸ + · Bu(k) 0 Bu(k) −1 ¸ · u(k) ˆ u(k) ¸ + · Bd(k) B_d(k) ¸ d (k) (2.24)

where ˆx and ˆu are coupled. In particular, ˆx, weighted with a high penalty in the objective function, is either zero or a value corresponding with the excess of the maximum allowed value, while ˆu, with a low penalty, is a virtual bounded input without any physical mean-ing. Hence, ˆx is zero whenever x is inside its boundary, and ˆx becomes the deviation between x and its relative maximum value allowed when the violation occurs.

2.2. S

OLVING AN

M

ODEL

P

REDICTIVE

C

ONTROL

P

ROBLEM

2.2.1. T

HE

F

INAL

F

ORM AS A

Q

UADRATIC

P

ROGRAMMING

P

ROBLEM

Combining the linear internal model (2.8), the hard constraint (2.3) and soft con-straint (2.24), and the quadratic objective function (2.18), the MPC problem to be dis-cussed and applied in this thesis has a quadratic form as follows:

min ˜ u(k,t )J (k) = ( ˜x(k, t) − ˜r(k, t)) T_{Q ( ˜}_˜ _{x(k, t ) − ˜r(k, t)) + ˜}_{u(k, t )}T_{R ˜}_˜_{u(k, t )} _(2.25) subject to ˜ x(k, t ) = ˜A(k, t )x(k) + ˜Bu(k, t ) ˜u(k, t ) + ˜Bd(k, t ) ˜d (k, t ) (2.26) g ( ˜x(k, t ), ˜u(k, t )) ≤ ˜b(k, t) (2.27)

2.2.2. S

OLVING AN

U

NCONSTRAINED

M

ODEL

P

REDICTIVE

C

ONTROL

P

ROB

-LEM

An unconstrained MPC problem indicated that (2.27) can be excluded and the search-ing area isRm. Theoretically, by solving the equation ∇J = 0, the stationary point (also the optimum,∵Hessian matrix of J Â 0) can be obtained:

∇J = 0 ⇔ u˜∗(k, t ) = H−1F (2.28)

where u∗is the minimum of the optimization problem. Because Q is defined positive semi-definite and R is positive definite as mentioned in Remark 2, the existence of H−1 in (2.28) can be guaranteed. Besides, the Hessian matrix of J can be easily proved to be positive definite, which can guarantee the local minimum u∗is a global minimum as well.

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2

Remark 5 The above discussion only shows a solution from a theoretical point of view.

To numerically solve this problem, the optimization problem can be reformulated as a least-square form instead of making more effort to calculate the inverse matrix H−1

[Ma-ciejowski, 2002].

2.2.3. S

OLVING A

C

ONSTRAINED

M

ODEL

P

REDICTIVE

C

ONTROL

P

ROBLEM

Most actual control problems have constraints and solving a constrained MPC prob-lem usually results in substantial time and computational resources [Borrelli et al., 2014]. Two efficient on-line algorithms for solving a QP problem in MPC appear to have the best performance [Maciejowski, 2002; Lau et al., 2009]: the active-set method [Fletcher, 2000; Ferreau et al., 2008] and the interior-point method [Wright, 1997; Wang and Boyd, 2010; Domahidi et al., 2012; Borrelli et al., 2014].

For a standard MPC problem, the computational complexity by the interior-point method can be estimated as a polynomial function of the problem size (the state dimen-sion and the horizon length), whereas the complexity in the active-set method may be exponential to the problem size in the worst case [Maciejowski, 2002]. In this thesis, the interior-point method is chosen for solving the constrained optimization problem.

A wide range of approaches have been studied to address the issue of finding com-putationally efficient solutions of MPC. An accepted technique to speed up MPC is by using warm-starting, in which the result of the previous optimization is used as the ini-tial condition for the optimization at the present step [Maciejowski, 2002]. A significant improvement to implement efficient MPC is to exploit the structure of a optimization problem (with m control variables, n system states and the prediction horizon as Np)

takes O(Np(n + m)3) operations as opposed to O(Np3(n + m)3) if the structure is not

ex-ploited yet [Wang and Boyd, 2010]. In addition, a practical way to further reduce the number of operations is reformulating the system state vector in terms of the control variable vector [Wang and Boyd, 2010; Maciejowski, 2002], combining with a moving blocking strategy [Cagienard et al., 2007]. Finally, Wang and Boyd [Wang and Boyd, 2010] have proposed a fast method to early terminate an appropriate interior-point searching, which has the potential to solve an optimization in the order of 100 times faster. How-ever, the computational burden of the optimization in MPC is still heavy for large-scale problems, especially when considering a long-term prediction.

Moreover, when applying additional modules in MPC for more complicated WRM problems, for example, applying tree-based MPC to deal with an ensemble of possible evolution of system disturbances (hydrological forecast with uncertainties)[Raso et al., 2014] or applying distributed MPC to handle distributed management issues[Maestre et al., 2013], it accordingly leads to even more computation. Thus, there remains the challenge of how to obtain solutions efficiently in order to implement MPC in real-time on large-scale problems [Buerger, 2013], which is studied in Chapter 6 of this thesis.

2.3. C

ONCLUDING

R

EMARKS

In this chapter, the fundamental principles of MPC have been introduced, including the internal model, the objective function, and the constraint. By formulating an MPC problem in a QP form, efficient algorithms can be applied to solve the QP problem.

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3

M

ODEL

P

REDICTIVE

C

ONTROL FOR

W

ATER

R

ESOURCES

M

ANAGEMENT

Sensibility gives us access to the sensible world, while understanding enables us to grasp a distinct intelligible world. Immanuel Kant(1724 ∼ 1804)

Learning without contemplating is a waste of time, while contemplating without learning is rather perplexing.

学而不思则罔思而不学则殆

Confucius (551B.C. ∼ 479B.C.)

On the basis of the theory of MPC described in chapter 2, this chapter describes how MPC can be applied to general WRM problems.

To fit a linear internal model in MPC, Section 3.1 introduces the Saint-Venant equations and their linearized scheme. Next, section 3.2 gives details on how to link a WRM problem with a certain element of MPC. For further illustration, four typical WRM problems are presented. When applying MPC to WRM in time and space, we also encounter problems such as handling predictions, and managing a WRS in a centralized or distributed way, which is dealt with in Section 3.3.

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3

20 3.MODELPREDICTIVECONTROL FORWATERRESOURCESMANAGEMENT

3.1. M

ODELING

W

ATER

S

YSTEMS

3.1.1. M

ODELING

H

YDRODYNAMICS OF

R

IVERS AND

C

ANALS

I

n this section, the Saint-Venant equations are presented in order to describe the sys-tem dynamics of open water syssys-tems [van Overloop, 2006b; Litrico and Fromion, 2004] as follows: ∂Q ∂x + ∂Af ∂t = ql (3.1) ∂Q ∂t + ∂ ∂x µ_Q2 A_f ¶ + g · Af∂h ∂x+ g ·Q · |Q| C2_{· R} f· Af= 0 (3.2)

where Q is the flow (m3/s), h is the water level (m), Afis the wetted area of the flow (m2),

q is the lateral inflow (m3/s), g is the gravitational acceleration (m2/s), C is the Chézy friction coefficient (m1/2/s), Rfis hydraulic radius (m).

The Saint-Venant equations are based on the mass conservation (3.1) and momen-tum conservation (3.2) under the following assumptions [Litrico and Fromion, 2009]:

• The flow is one-dimensional, which means the velocity is uniform over the cross-section and the water level across the cross-section is horizontal.

• The streamline curvature is small and vertical accelerations are negligible, hence the pressure is hydrostatic.

• The effect of boundary friction and turbulence can be accounted for through re-sistance laws analogues as in those used for steady-state flow.

• The average channel bed slope is small so that the cosine of the angle it makes with the horizontal may be replaced by unity.

• The variation of channel width along x is small.

In order to manage water resources in real-time, the linearized De Saint-Venant equa-tions are proposed as follows for fitting a linear MPC problem, which is discretized in a staggered conservative implicit scheme [Xu et al., 2013; Stelling and Duinmeijer, 2003]:

dAf dt = Qi −1/2−Qi +1/2 ∆x + qi (3.3) dvi +1/2 dt + (Qi +1/2+Qi +3/2) · vi +1− (Qi +1/2+Qi −1/2) · vi 2A_{f,i +1/2}∆x (3.4) −vi +1/2 Q_{i +3/2}_−Q_{i −1/2} 2A_{f,i +1/2}_∆x + g h_{i +1}_{− h}i ∆x + g v_{i +1/2}_|v_{i +1/2}_| C2_{· R} f = 0

where the subscripts i , i +1/2 and i −1/2 denote the parameters or variables at staggered grid points i , i + 1/2 and i − 1/2 respectively. Note that the staggered scheme presented in (3.3) and (3.4) has unconditional stability [Stelling and Duinmeijer, 2003].

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3.1.MODELINGWATERSYSTEMS

3

21

h

down

h

up Qup Qdown

Q

s

Q

d

Figure 3.1: Schematic diagram of a canal-reach system.

Based on (3.3) and (3.4), the system dynamic function (2.2) of each section of a canal or a river, shown in Figure 3.1, can be described in a linear time-invariant state-space equation: e(k + 1) = e(k) +(Qs(k) + ∆Qs(k))∆t A + (Qup(k) −Qdown(k) +Qd(k))∆t A (3.5) Qup(k) = αup(hup− h) + βup (3.6)

Qdown(k) = αdown(h − hdown) + βdown (3.7)

where

∆Qs(k) = Qs(k + 1) −Qs(k) (3.8)

e(k) = h(k) − hr(k) (3.9)

and the state A (m2) is the surface area, e (m) is the deviation between the actual water level h (m) and the reference water level hr(m), Qs(m3/s) is the flow through the

struc-ture,∆Qs(m3/s) is the change of structure flow, Qd(m3/s) is the sum of disturbances and

other flows not related to structures, such as rainfall runoff and water abstractions, hup

and hdown are the upstream and downstream water levels respectively, Qupand Qdown

are the upstream and downstream flows respectively. For an overview of the coefficients α and β in (3.6) and (3.7), see [Xu, 2013, Appendix B].

3.1.2. M

ODELING

H

YDRAULIC

S

TRUCTURES

In this thesis, two types of hydraulic structures are used for the purpose of WRM: pumps and sluice gates (undershot gates). Pumps are operated in a straightforward way with given capacities.

Undershot gates have a gate that work from the top down, as shown in Figure 3.2. In particular, the structure flow via a free flowing undershot gate is given by:

Q(k) = Cg· Wg· µg·¡hg(k) − hcr¢ ·

q

2 · g ·¡h1(k) − hcr− µ¡hg(k) − hcr

¢¢

(3.10) and the structure flow via a submerged undershot gate is given by:

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3

Figure 3.2: Free-flowing and submerged undershot gates, source: [van Overloop, 2006b].

Q(k) = Cg· Wg· µg·¡hg(k) − hcr¢ ·

q

2 · g · (h1(k) − h2(k)) (3.11)

where Q is the flow through the gate (m3/s), k is time step (s), Cgis the calibration

co-efficient, Wgis the width of the gate (m),µ is the contraction coefficient, hgis the gate

height (m), h1is the upstream water level (m), h2is the downstream water level (m) and

hcris the crest level (m). Note that in the case of multiple gates in parallel, the total width

can be taken together and the gates can be considered as moving in a synchronized way. In order to reduce the calculation complexity and be able to fit the internal model, both (3.10) and (3.11) need to be linearized by applying a first-order Taylor expansion. In the following paragraphs, the linearization of the submerged flow (3.11) is shown. The free flow undershot gate can be treated very similarly.

The first-order Taylor series expansion on state x(k + 1) around x(k) is applied to (3.11). Then the structure flow is estimated as follows:

Q(k + 1)= f · (h. 1(k + 1) − h2(k + 1)) + c · ∆hg(k) + v (3.12) where f =g ·C · W · µ ·¡hg(k) − hcr ¢ p2 · g · (h1(k) − h2(k)) (3.13) c = C · W · µ · q 2 · g · (h1(k) − h2(k)) (3.14) v = Q(k) −g ·C · W · µ ·¡hg(k) − hcr ¢ p2 · g · (h1(k) − h2(k)) (h1(k) − h2(k)) (3.15)

where f , c and v are coefficients at step k, which can be obtained from previous simula-tion results (the last closed-loop result). For the upstream canal, the structure flow is the outflow which is negative in the internal model:

h1(k + 1) = h1(k) −∆t

A1

( f (h1(k + 1) − h2(k + 1)) + c∆hg(k) + v) +∆t

A1

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3.2.APPLYINGMODELPREDICTIVECONTROL TOWATERRESOURCESMANAGEMENT PROBLEMS

3

23 or: (1 +∆t A1 f )h1(k + 1) + (−∆t A1 f )h2(k + 1) = h1(k) −∆t A1 (c∆hg(k) + v −Qd,1) (3.17)

And for the downstream canal, the structure flow is just the inflow which is positive in the internal model:

h2(k + 1) = h2(k) +∆t A2 ( f (h1(k + 1) − h2(k + 1)) + c∆hg(k) + v) +∆t A2 Qd,2 (3.18) or: −∆t_A 2 f h1(k + 1) + (1 +∆t A2 f )h2(k + 1) = h2(k) +∆t A2 (c∆hg(k) + v +Qd,2) (3.19)

The matrix following from (3.17) and (3.19):

E · h1(k + 1) h2(k + 1) ¸ = · h1(k) h2(k) ¸ + Bu∆hg(k) + · Q_d,1 Q_d,2 ¸ (3.20) where E = " 1 +∆tA1f − ∆t A1f −∆tA2f 1 + ∆t A2f # , Bu= " −∆tA1c 0 0 ∆t_A 2c # (3.21) or: · h1(k + 1) h2(k + 1) ¸ = E−1 · h1(k) h2(k) ¸ + E−1Bu∆hg(k) + E−1 · Qd,1 Qd,2 ¸ (det (E ) > 0,∴E−1exists) (3.22)

Equation (3.22) indicates how the water level is controlled directly by the submerged undershot gate.

3.2. A

PPLYING

M

ODEL

P

REDICTIVE

C

ONTROL TO

W

ATER

R

E

-SOURCES

M

ANAGEMENT

P

ROBLEMS

I

n general, sub-objectives of WRM can be addressed by one of the elements of MPC, for example, the objective function, the constraint or the setpoint:

• Handled by setpoints: Navigation (to maintain the water level close to a setpoint, not too high or too low).

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3

• Handled by an objective function: Gate operations (not too frequent to save energy and avoid tear and wear, yet it has to meet the requirement).

• Handled by constraints: Flood management (to keep the water level in rivers and canals below the safety level), water supply (to assure the supply discharge not less than the demand discharge).

The stage cost function f in (2.1) can be expressed in a quadratic form as:

f (x(k + 1),u(k)) = · x(k + 1) − r (k + 1) u(k) ¸T· Q 0 0 R ¸ · x(k + 1) − r (k + 1) u(k) ¸ + · Ql Rl ¸T· x(k + 1) u(k) ¸ (3.23)

where Q ∈ Rn×n, R ∈ Rm×m, Ql∈ Rn and Rl∈ Rm are respectively quadratic or linear

penalties on states and control variables, r ∈ Rn is the reference vector with respect to the system states.

Constraints (2.3) are set as the physical limitations of the real water system [van Over-loop, 2006b; Tian et al., 2014]:

· xmax umax(k) ¸ ≤ · x(k + 1) u(k) ¸ ≤ · xmin umin(k) ¸ (3.24)

where xmin∈ Rnand xmax∈ Rnare the minimum and maximum allowed values, usually

as constants, on the state x, which are either the safety level or the maximum pump capacity or the highest and lowest gate positions, umin(t ) ∈ Rnand umax(t ) ∈ Rnare the

minimum and maximum allowed changes of pump flows or gate positions, usually as variables[Tian et al., 2014]: · umin(k) umax(k) ¸ = · xmin xmax ¸ − · x(k) x(k) ¸ (3.25)

Combining (3.5) - (3.24), standard MPC for WRM is formulated as:

min u(0),...,u(Np−1) Np−1 X k=0 Ã · x(k + 1) − r (k + 1) u(k) ¸T· Q 0 0 R ¸ · x(k + 1) − r (k + 1) u(k) ¸ + · Ql Rl ¸T· x(k + 1) u(k) ¸! (3.26) subject to x(k + 1) = Ax(k) + Buu(k) + Bdd (k) k = 0,1,..., Np− 1 · xmax umax(k) ¸ ≤ · x(k + 1) u(k) ¸ ≤ · xmin umin(k) ¸ k = 0,1,..., Np− 1

Remark 6 The objective function contains a linear term. Through defining xnew(k)=∆

x(k) +Ql

2, the objective function (3.26) can be expressed only with quadratic terms. For

the sake of a concise expression, the objective function is shown only with quadratic terms in following chapters.