Diﬀerent Model Predictive Control Approaches for ...

Different Model Predictive Control

Approaches

for Controlling Point Absorber Wave

Energy Converters

Diploma Thesis

by

cand. kyb. Markus Richter

Examiner: Prof. Dr.-Ing. O. Sawodny

Supervisor: Prof. Dr.-Ing. M. E. Magana

Institute for Systemdynamics, University Stuttgart

Prof. Dr.-Ing. O. Sawodny

October 5, 2011

Abstract

Ocean wave energy is a promising renewable energy source that can be converted

into useful electrical energy using wave energy converters (WECs). In order for

WECs to become a commercially viable alternative to established methods of en-

ergy generation, operating the WEC in an optimal fashion is a key task. Thus,

model predictive control (MPC) is an encouraging control method that exploits

the entire power potential of a WEC on the one hand, while respecting the con-

straints on motions and forces on the other hand. This work focuses on one

subclass of WECs, namely, buoy type point absorbers with a linear generator as

the power take-off (PTO) system and presents different MPCs whose goals are to

optimize the generator force. For a linear one-body model of the point absorber

it is shown that optimizing the power generation is possible without using an

optimal velocity reference trajectory. Therefore, the control of a linear two-body

model with a similar approach can be demonstrated, even though no adequate

reference trajectory can be calculated in this case. Moreover, possibilities to deal

with the occurance of infeasibility problems are introduced and implemented.

Due to possible nonlinear effects, such as the mooring forces, a nonlinear model

predictive controller is proposed, whose performance is compared to a linear

MPC, also controlling the nonlinear system. The proposed controllers are val-

idated and compared through simulation for regular and irregular sea states.

Declaration

Herewith I affirm that I have written this thesis on my own. I did not enlist unlawful

assistance of someone else. Cited sources of literature are perceptly marked and listed

at the end of this thesis.

Stuttgart, September 2011

Acknowledgement

I would like to ...

Contents

Nomenclature VII

1 Introduction 1

1.1 Ocean Energy Resources . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Wave Energy Converters . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Examples of Wave Energy Converters . . . . . . . . . . . . . . . 4

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Introduction to Linear Model Predictive Control 9

3 Linear Model Predictive Control of One-Body WEC 15

3.1 One-Body Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Optimal Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 MPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Optimal Trajectory Formulation . . . . . . . . . . . . . . . . . . 20

3.3.2 Direct Power Maximization Formulation . . . . . . . . . . . . . 23

3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Linear Model Predictive Control of Two-Body WEC 31

4.1 Two-Body Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 State Space Form and Discretization . . . . . . . . . . . . . . . 34

4.2 MPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Infeasibility handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Nonlinear Model Predictive Control of a Nonlinear Two-Body WEC

Model 49

II CONTENTS

5.1 Introduction to Nonlinear Model Predictive

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Nonlinear Two-Body Model . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Problem Formulation and Implementation . . . . . . . . . . . . . . . . 53

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Conclusions and Future Work 61

A Explantion: Direct Power Formulation is a Convex Problem i

List of Figures

1.1 Global distribution of approximate yearly average wave power in kW/m

crest length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Photograph of the Limpet [28] (a) and a schematic diagram of its prin-

ciple of operation (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Photographs of the Pelamis [43, 28] . . . . . . . . . . . . . . . . . . . . 5

1.4 Photograph of the Wave Dragon [28] (a) and a schematic diagram of its

principle of operating (b) [17] . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Point absorber Archimedes Wave Swing [28] . . . . . . . . . . . . . . . 6

1.6 Schematic sketch (a) and photograph (b) of the L10 point absorber [28] 7

2.1 Principle of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Usual structure of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Structure of MPC without reference trajectory . . . . . . . . . . . . . . 13

3.1 L10 Wave Energy Converter [38] . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Electrical equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 SIMULINK simulation model for one-body direct power formulation . . 24

3.4 Simulation results with the optimal control law and a monochromatic

sinusodial wave as input signal . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Phase relation between excitation force and actual velocity with the

optimal control law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Comparison between “MPC Traj” (T) and “MPC Power” (P) in case of

monochromatic sinusodial wave as input signal . . . . . . . . . . . . . . 28

3.7 Comparison between “MPC Traj” (T) and “MPC Power” (P) in case of

a superposition of sine waves as input signal . . . . . . . . . . . . . . . 29

4.1 L10 Wave Energy Coverter [38] (left) and schematic diagram (right) . . 32

4.2 Mooring configuration with three cables (left) and schematic diagram

for force derivation (right) . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 SIMULINK simulation model for two-body WEC control with MPC . . 39

4.4 Wave elevation, buoy position and spar position for the relaxed con-

straints formulation (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . 42

IV LIST OF FIGURES

4.5 Frequency of the infeasibility problem for Case 1 . . . . . . . . . . . . . 43

4.6 Comparison between relaxed (Case 1) and soft (Case 2) constraints for-

mulation of the MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.7 Comparison between the soft constraints formulation with the changed

parameter Thor = 6s (Case 3) and the same formulation with changed

parameter ∆t = 0.2s (Case 4) . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Comparison between the soft constraints formulation with mooring con-

stant Km = 300,000 N/m (Case 2) and the same formulation with

Km = 50,000 N/m (Case 5) . . . . . . . . . . . . . . . . . . . . . . . . 45

4.9 Comparison of Spar position between the soft constraints formulation

with mooring constant Km = 300,000 N/m (Case 2) and the same for-

mulation with Km = 50,000 N/m (Case 5) . . . . . . . . . . . . . . . . 46

4.10 Comparison between the original soft constraint formulation with orig-

inal WEC parameters of Table 4.1 (Case 2) and the same formulation

with the changed WEC parameters of Table 4.4 for the L10 (Case 6) . 47

5.1 Comparison between the extended and the reduced model . . . . . . . 52

5.2 Nonlinear and linear mooring forces for different choices of Km. . . . . 56

5.3 Comparison between the simulation results with the NMPC and the

linear MPC with Km = 100,000 N/m. The wave data is from the NDBC

Umpqua buoy 46229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


linear MPC with Km = 50,000 N/m and Km = 350,000 N/m . . . . . . 58


linear MPC with Km = 150,000 N/m. The wave data is from the NDBC

Umpqua buoy 46050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.6 Generated Power depending on Km compared to the generated power

with the NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

List of Tables

3.1 Hydrodynamic parameter of the point absorber . . . . . . . . . . . . . 25

3.2 Simulation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 System parameter of the two-body L10 WEC . . . . . . . . . . . . . . 40

4.2 MPC parameter values for the formulation with relaxed constraints

(Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 MPC parameter values for the formulation with soft constraints (Case 2) 41

4.4 Comparison of real plant and model parameters . . . . . . . . . . . . . 47

4.5 Overview Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 NMPC/MPC parameter values . . . . . . . . . . . . . . . . . . . . . . 55

VI LIST OF TABLES

Nomenclature

Abbreviations

AWS Archimedes Wave Swing

LWT Linear Wave Theory

MPC Model Predictive Control

NMPC Nonlinear Model Predictive Control

OWC Oscillation Water Column

PTO Power Take-Off

R&D Research and Development

Latin Letters

V Velocity phasor

F Augmented f -vector

T Augmented vector of optimal velocity

U Augmented u-vector

V Augmented v-vector

W Augmented w-vector

X Augmented state vector

A Standard system state space matrix

B1 System state space matrix of input v

B2 System state space matrix of input w

B System state space matrix of input u

A Vector field for nonlinear discretization method

f Input vector field nonlinear system

l Nonlinear vector field for mooring term

p Vector of optimization variables for NMPC

x State vector

y Output vector

A Added mass (kg)

B Damping of one-body WEC (N/(m/s))

b Damping (N/(m/s))

VIII Nomenclature

c Constraint

f System input excitation force one-body

Fe Excitation force (N)

Fr Radiation force (N)

fr Radiation force impulse response function

Fgen Generator force (N)

Fh Hydrostatic force (N)

Fm Mooring force (N)

g Acceleration of gravity (m/s2)

H Wave height (m)

h Discretization step time (s)

J Objective function

j Imaginary unit

K Mooring cable stiffness of one cable (N/m)

k Hydrostatic stiffness (N/m)

Km Overall mooring cable stiffness (N/m)

L Cable length (m)

M Discretization order

m Mass (kg)

me Equivalent mass (kg)

N Length of time horizon

Nr Relaxation number

P Active power (W)

Q Reactive power (W)

q Power, respectively, trajectory deviation weighting factor

r Input weighting factor

ru Relaxation factor generator force

rv Relaxation factor velocity

rbuoy Buoy radius (m)

S Overall power (W)

s Overall manipulable inputs

T Simulation time (s)

Tp Wave period Time (s)

Thor Time horizon (s)

u System input generator force

v System input excitation force buoy

w System input excitation force spar

w Weighting factor slack

x Vertical position spar (m)

Nomenclature IX

z Vertical position buoy (m)

Greek Letters

α Cable angle (◦)

∆t Optimization step time

ǫ Slack variable

η Wave surface elevation (m)

ω Wave frequency (1/s)

φ Phase angle (rad)

ρ Density of sea water (kg/m3)

Subscripts and Superscripts

(i) Denotes the i-th component of a vector

∗ Conjugated complex

1 Buoy parameter

12 Influence from spar on buoy

2 Spar parameter

21 Influence from buoy on spar

A Amplitude value

d Discrete matrices

gen Generator

hd hydrodynamik

k Discrete sample steps

mcl Meter crest length

nl Nonlinear

opt Optimal

p Position

T Transposed

u Generator force

v Velocity

ˆ Denotes phasor description

X Nomenclature

Chapter 1

Introduction

The idea of converting wave energy into usable electrical energy is very old. The first

patent for wave energy conversion was applied in 1799 by Gilard&Son in France. But

only in the 1960s, the first wave energy device, a self-recharging navigation light buoy,

was built and commercialized by Masuda in Japan [31]. This device was equipped

with an air turbine and today it would be considered as a (floating) oscillation water

column (OWC) [14]. Since then, many patents were registered, though little attention

was paid by the international scientific community for a long period of time. This

changed for the first time during the oil crisis of 1973, as the interest in renewable

energy technologies increased once again. After that, several research programs with

governmental or private support started in Europe, Japan and the USA.

One important step for wave energy conversion in Europe was taken in 1991 when

the European Commission decided to include wave energy in their official research and

developement (R&D) program on renewable energies [14]. Since then, the commission

supported many international wave energy conferences and funded several research

projects. So far, R&D in Europe is mainly being done in United Kingdom, Portugal,

Ireland, Norway, Sweden and Denmark.

In the USA, the Energy Act of 2005 helped support the developements in wave

energy conversion, since it allowed wave energy for the first time to receive governmental

support such as other renewable energy technologies [41].

Despite the large research efforts, WECs are still in the R&D stage but are closer to

commercialization than ever before [12]. Thus, immense work in R&D is still required,

where academic institutions make large contributions. Academic leaders in the field of

wave energy conversion are currently: University of Edinburgh, Uppsala University, the

Norwegian University of Science and Technology as well as Oregon State University.

Since 1998, Oregon State University (OSU) in Corvallis, OR, has established a

leading wave energy program for their students, where much research has been done

on innovative wave energy technologies [6]. Moreover, OSU collaborates with different

industrial partners such as Columbia Power Technologies. OSU also provided the

2 CHAPTER 1. INTRODUCTION

environment for creating this work.

1.1 Ocean Energy Resources

The ocean is an enormous, though so far almost untapped source of energy. Several

forms of ocean energy such as tidal and marine currents, thermal, salinity and wave

energy exist. This work exclusively focuses on wave energy.

Wave energy is basically concentrated solar energy. Wind which creates waves

by blowing across the surface of the ocean, is the result of the uneven heating of

the earth’s surface. Energy conversion attempts to convert the potential and kinetic

energy of the heaving water of a wave to usable energy. According to [6], the power

for a monochromatic wave can be calculated by

Pmcl =ρg2H2Tp

32π[P/m], (1.1)

where the wave height H is measured from trough to crest, ρ is the density of sea

water, g is the accaleration of gravity and Tp is the wave period. The power is usually

given in the unit of Power per meter crest length and is proportional to wave height

squared and linearly dependent on the wave period.

It is difficult to estimate the amount of exploitable wave energy in the world’s ocean.

According to [25], the ocean holds approximately 8, 000− 80, 000 TWh/year or 1− 10

TW, whereas Falnes in [32] quantifies the world’s exploitable wave power resource to

be of the order of 1 TW. Comparing this to the world’s annual energy consumption

of approximately 148, 000 TWh in 2008 [40] shows that wave energy could play an

important role in the world’s energy portfolio.

In comparison to solar and wind energy, wave energy stands out due to its low

hourly variation, high power density and its high availablity [6, 23]. Additionally,

wave’s motion can be forcasted very accurately. It should also be noted that average

wave power is cyclical, where the waves in the winter are larger than in the summer.

Moreover, the wave power decreases as the waves advance toward the shoreline on

account of frictional losses with the sea floor [25].

Due to the west-to-east winds, the waves on the west coasts of the continents contain

more energy as can be seen in Fig.1.1. Oregon with a yearly average power of about

40 kW/m is characterized by a similar average power level as, for instance, Portugal

or Norway. The British Isles, for example, provide an even higher power level of about

70 kW/m. The high power levels on these coasts may be a reason why wave energy

research has especially developed in these countries.

1.2. WAVE ENERGY CONVERTERS 3

Fig. 1.1: Global distribution of approximate yearly average wave power in kW/m

crest length [29].

1.2 Wave Energy Converters

In order to generate power from wind, the wind turbine is the only commercially viable

design. The situation for wave energy conversion is very different. There are various

concepts which differ in their power conversion principle, in their size or in their location

on the ocean. However, there is no concept which has really come out on top so far.

In the following, WECs are classified and currently promising WECs are described.

1.2.1 Classification

Converters are generally classified according to their location with respect to the shore

or to the applied conversion principle. Both ways of classification are discussed here

following the work in [31].

Shoreline devices can be located directly on the sea floor, in shallow water or can be

attached to a rock. Thus, they are easy to install and to maintain. Additionally, short

cables can be used and the environment for the device is very safe. As stated above,

waves loose energy when advancing toward the shoreline. Hence, the environment for

generating power near the shore is generally weak. However, in some areas the coastal

geometry helps focus the energy and thereby compensates for the weak environment.

Other disadvantages lie in the difficulty of finding suitable sites and in the impact on

coastal landscapes and environments.

Offshore devices are floating on the sea or are submerged in deep waters and need

to be moored to the sea floor. They can harness the high power potential of the open


sea and do not have an effect on the coastal landscapes and environments. However,

installation and maintainance costs are very high and harsh sea conditions give rise to

survivability and reliability problems.

According to [6], four different operating principles for WECs exist:

• Oscillating Water Column (OWC)

The water level in a chamber rises and falls with the incoming waves. Air rushes

in and out a small opening in the chamber driving a turbine which generates

electricity.

• Attenuator

It is a device with floaters, where the motion of the floaters relative to each other

causes energy generation by hydraulic pumps or other type of converters. It is

usually a rectangular device which can be oriented perpendicular or colinear with

the wave front.

• Overtopping

The device concentrates water with a ramp in a higher reservoir and releases the

water through a hydro turbine.

• Point Absorber

A relatively small device which captures energy from all directions at one point

by a floater which is moved by the waves. Usually the motion is only heave

motion.

1.2.2 Examples of Wave Energy Converters

In what follows, some WECs are briefly described. The first commercial scale wave

energy converter was the Limpet (Land Installed Marine Powered Energy Transformer)

[42] which is located on the shoreline of the Island of Islay, Scotland. The device was

built in 2000 and has produced power for the national grid since then. Its nominal

power is 500 kW. Limpet belongs to the class of OWC devices. Fig. 1.2 shows a

picture of the Limpet and a schematic diagram of its principle of operation. The water

level in the chamber rises and falls with the incoming waves causing air compressing

and decompressing which drives a turbine. Since the direction of the air flow changes

permanently, a special type of turbine, namely the Well’s turbine, is used to assure only

one direction of rotation. Limpet is also used as a testing facility for the development

of other turbines.

The Pelamis shown in Fig. 1.3 is floating offshore. It is an attenuator device with

four segments joined together by hinged joints [30]. By motion of the waves, the joints

flex relative to each other. The motion in the joints is resisted by hydraulic cylinders

1.2. WAVE ENERGY CONVERTERS 5

(a) (b)

Fig. 1.2: Photograph of the Limpet [28] (a) and a schematic diagram of its principle

of operation (b).

(a) (b)

Fig. 1.3: Photographs of the Pelamis [43, 28].

that pump fluid into high pressure accumulators. By means of a hydraulic engine power

can be generated [30]. The first full-scale prototype was completed 2004 and deployed

in Orkney, Scotland with the rated power of 750 kW. The world’s first small wave farm

with three devices was deployed in Portugal 2008. Additionally, further projects, such

as the wave farm with 14 devices in Shetland, Scotland, are in the development stage.

The Wave Dragon, Fig. 1.4, represents an overtopping device. The first prototype

was deployed at Nissum Bredning, Denmark in 2003. The device is floating on the

sea. By means of two “arms” (reflectors) and a submerged ramp, water is guided up

into a reservoir. The reservoir is higher than the sea level, thus potential energy is

stored which can be transformed into usable energy by letting the water run through

the turbine outlet. Further projects are planned in Wales and Portugal.

The Archimedes Wave Swing (AWS) belongs to the class of point absorbers. Two

photographs can be seen in Fig. 1.5. It is a submerged device and was first deployed

in Portugal in 2004 and successfully connected to the grid shortly afterwards. The


(a) (b)

Fig. 1.4: Photograph of the Wave Dragon [28] (a) and a schematic diagram of its

principle of operating (b) [17]

(a) Several AWS under water. (b) AWS plant before submersion.

Fig. 1.5: Point absorber Archimedes Wave Swing [28].

operating principle is based on pressure differences in a chamber when the wave rises

and falls. The chamber of the AWS is filled with air, where the bottom is fixed and the

lid moves up and down due to the pressure difference on top [31]. In the first prototype

the PTO system was a linear generator. Later, hydraulics were used as well. One

advantage of this type of point absorber is its completely submerged design which is in

general less vulnerable in harsh sea conditions and does not have an negative impact

on the landscape.

Fig. 1.6 shows the point absorber L10, developed at Oregon State University in col-

laboration with Columbia Power Technologies and the Navy, where a linear generator

as the direct-drive PTO system is used. This device was tested in September 2008 off

Newport, OR. Since then, Oregon State University is continuing the collaboration in

order to develope a full-scale WEC [6].

Various WECs have been deployed and tested in the last 10 years. However, there

is no concept or design which came out on top so far. Moreover, the design of wave

1.3. OUTLINE OF THE THESIS 7

(a) (b)

Fig. 1.6: Schematic sketch (a) and photograph (b) of the L10 point absorber [28]

farms is still in its early stages. It will be interesting to observe which concept will be

the most successful one in the future. Even though this work exclusively focuses on

the L10 point absorber, certain principles could also be applied to other designs.

1.3 Outline of the Thesis

Basically, controllers for WECs need to perform two main tasks, i.e., the generated

power needs to be maximized and the physical constraints on the machinery force

and on the WEC’s motion need to be respected. Additionally, controllers which can

benefit from prediction data of the wave motion are promising, since wave prediction

is generally possible. For this reason, MPC is a reasonable type of control algorithm,

since it provides a framework for optimal control and constraints. Moreover, prediction

data can be naturally included.

The objective of this work is to implement MPCs for controlling a point absorber

which is capable of operating it under optimal conditions, where the power generation

is maximized, while the constraints of the WEC are satisfied. Depending on the model

of the device, the MPCs differ.

The main part of this thesis is devided into four parts. The basic principles and

strategies of model predictive control (MPC) are briefly discussed in the beginning.

Chapter 3 discusses exclusively a linear one-body model of a point absorber WEC.

Based on the one-body model, an optimal control law for power optimization is in-

troduced and two different MPC formulations are implemented. The focus lies on

demonstrating that MPC of the WEC is possible without using an optimal velocity

trajectory as reference. The different controllers are validated and compared by means

of simulation results with a superposition of sine waves.

Based on linear wave theory (LWT), a linear two-body model for a point absorber


with sea floor mooring is derived and a MPC is implemented in Chapter 4. In MPC

applications, infeasibility problems can occur while solving the optimization problem.

Since this can cause big problems, two possibilities to handle the occurance of infeasi-

bilities are proposed and compared by simulation results for irregular sea. Additionally,

the influence of MPC parameters, mooring and model inaccuracies is discussed.

In Chapter 5, due to possible unmodelled nonlinear effects, a nonlinear model is

derived, where the mooring and radiation forces are assumed to be nonlinear. In order

to control the nonlinear system, two different controllers are proposed and compared.

First, the linear MPC from Chapter 4 and secondly a nonlinear MPC whose implemen-

tation is the focus in this chapter. By variation of the mooring constant of the linear

MPC, its performance is analyzed and compared to the nonlinear model predictive

controller (NMPC).

Chapter 6 concludes this thesis and suggests areas for future research.

Chapter 2

Introduction to Linear Model

Predictive Control

The first theories of model predictive control were introduced in the 70’s and were also

known as receding horizon control and open loop feedback optimization [3]. At that

time, most of the research was done in the area of process control engineering. Since

then, the theories have advanced and the computational power of control units has

increased tremendously so that today’s MPC applications are used in various techni-

cal control problems. One main reason for the extensive capabilities of MPC is its

flexibility, namely,

• System states constraints or actuator limits can be taken into account

• It can handle multivariable control

• It can handle input and output time delays

• It can handle unstable and non-minimal phase processes

• It can include predictions of external inputs naturally

Despite its flexibility, the core compenents of MPC are always the same, i.e.,

• Prediction model of the process

• Objective function

• Optimizing algorithm

In fact, many different MPC algorithms exist, where the model representation and the

form of the objective function, as well as the optimizing algorithm can vary.

Nevertheless, all MPC algorithms have in common the same control idea. The

current control signal is obtained by solving a finite horizon open loop optimal control

10 CHAPTER 2. LINEAR MPC

problem (mostly, minimizing an objective function) online at each time step [9]. Hence,

the current system state is used as the initial state for the model. This yields a sequence

of optimal control signals at every time step, where only the first signal is applied to

the process. Also, the prediction horizon is shifted one time step before the new

optimization starts. Fig. 2.1 shows this basic principle of MPC. In most cases, the

t−∆T t t+∆T t+k∆T t+N∆T. . . . . .

N

u(t)

y(t)

u(t+k∆T |t)

y(t+k∆T |t)

w(t+k∆T )

Fig. 2.1: Principle of MPC.

goal is that the system outputs y(t + k∆T | t)1 for k = 1, . . . , N follow the predefined

reference trajectory w(t+k∆T ) for k = 1, . . . , N . N denotes the number of time steps

within the prediction horizon and ∆t the optimization step time. A typical objective

function consists of two quadratic terms. One term evaluating the reference deviation

and the other term weighting the energy use. A possible objective function for a

single-input single-output system, similar to the one in [9], is given by

J(N) =N∑

j=1

δ(j)[y(t+ j∆T | t)−w(t+ j∆T )

]2+

N∑

j=1

λ(j)[∆u(t+(j− 1)∆T )

]2, (2.1)

where the coefficients δ(j) and λ(j) are sequences of mostly constant numbers weight-

ing the reference deviation and the energy use, respectively. If the objective function

is quadratic and there are no constraints, the control signals can be calculated ex-

plicitely [9]. Otherwise, iterative optimization algorithms have to be used. Other

types of objectice functions can be found in [3, 8], where an explicit terminal point

condition or separate control horizons for prediction and controlling are included. In

order to handle, for instance state transitions, the reference trajectory can be just a

step or a smooth approximation of that.

Other differences between MPC algorithms exist in the description of the model.

In general, most descriptions can be used, but it is of great importance to have an

appropriate model description in order to describe the dynamic behaviour of the pro-

1This notation denotes the variable values at the time t+ k∆T , calculated at the time t

11

cess adequately. In order to model the input/output behaviour of a process, system

identification by means of an impulse or a step response is often used. Also, transfer

functions are often used. As is mostly the case, state space models are applied to

characterize the system. A possible form for a continous, linear, time-invariant system

is

x = Ax+Bu, x(0) = x0, (2.2)

y = Cx, x ∈ Rn, u ∈ R

m, y ∈ Rr. (2.3)

In practice, there are constraints for most processes as well. Actuators have certain

physical or safety limits and also the states are often restricted, so that in most cases

constraints need to be considered in the optimization algorithm. Normally, the con-

straints are state and input constraints as

umin ≤ u(t) ≤ umax ∀t, (2.4)

∆umin ≤∆u(t) ≤ ∆umax ∀t, (2.5)

xmin ≤ x(t) ≤ xmax ∀t. (2.6)

The introduction of constraints yields more difficult optimization problems which are

only solvable with iterative methods, such as the active-set or the interior-point method,

as described in [27].

Now that the principles of MPC have been discussed, it has to be stated why MPC

is an adequate and promising control algorithm, especially in the field of point absorber

WECs, [7].

• The main goal of controlling point absorber is to maximize the power generation

in order to make WECs cost-effictive and able to compete with other methods

of energy conversion. At the same time, mechanical limits for PTO force and

system states need to be considered. For that, MPC provides a framework to

explicitely consider these constraints while maximizing the power generation.

• In general, having a prediction of the incoming waves yields benefits for optimal

energy capturing. However, many control algorithms are not capable of exploiting

prediction information. As stated above, predictions of external inputs can be

easily included in the MPC framework though.

• Traditionally, MPC has been applied to chemical or process control engineering

applications, since those are slower dynamic processes which do not overburden

the computational power for online optimization. Even though faster processes

are controlled with MPC nowadays, it is always easier to apply MPC if the process

is not too fast. Ocean wave applications are in general rather slow processes and

therefore fit well the MPC requirements.


In the following chapter, two different versions of MPC are introduced. As an outlook,

the two different MPC structures are briefly discussed here. The first approach requires

a reference, namely an optimal velocity reference trajectory which the actual velocity

has to follow in order to achieve optimal energy capturing. The calculation of this op-

timal trajectory is discussed in Section 3.2. In this case, the usual and above described

MPC structure can be used which is schematically shown in Fig. 2.2. The objective

Model+ −

Constraints

Optimizer

Reference Trajectory

Objective Function

Error

Current

States

Optimal Input

Future Inputs

Future Outputs

Predicted

Disturbances

Fig. 2.2: Usual structure of MPC.

function contains a term penalizing the error between reference trajectory and actual

velocity. The optimizer tries to find the optimum of the objective function with respect

to the constraints. Also, the MPC gets the current state information from measure-

ments and calculates with the aid of the process model the future outputs which are

compared to the reference. After optimization, the first control signal is applied to the

process.

This work shows that even in the case when no optimal trajectory can be calculated,

MPC can still be applied. Therefore, the objective function needs to be changed so that

one term directly indicates the generated power, as can be seen in Section 3.3.2 and

thereafter. The structure of this second version is shown in Fig. 2.3, where no reference

is used so that only the input and the system states are part of the objective function.

Here, future information is used as well. However, not by a reference trajectory but

only by the prediction of external disturbances which is also discussed in Section 3.3.2.

13

Model

Constraints

Optimizer

Future Inputs

Optimal Input

Objective Function

Current

States

Future States

Predicted

Disturbances

Fig. 2.3: Structure of MPC without reference trajectory.

Chapter 3

Linear Model Predictive Control of

One-Body WEC

This chapter mainly discusses the application of linear MPC to a relatively simple one-

body representation of a point absorber. At first, a state space model is derived and

brought into a discrete representation before different control methods are proposed.

The following two MPC formulations specifically differ in the use of prediction data

and in the formulation of the objective function. Also, the MPCs are compared with

each other and to an optimal control law via simulation.

3.1 One-Body Model

Fig. 3.1 shows a drawing of the point absorber L10, developed at Oregon State Uni-

versity. The L10 consists of two bodies, a buoy floating on the surface of the ocean

Fig. 3.1: L10 Wave Energy Converter [38]

16 CHAPTER 3. ONE-BODY WEC

and a damping body, namely, a combination of a spar and a ballast tank. The two

bodies are connected through a PTO system in order to convert the relative motion

into usable energy. The L10 is an example of a direct-drive WEC. In this case, the

wave energy is converted directly into electrical energy by means of a linear generator

without converting wave energy into mechanical energy as in hydraulic PTO systems

first. In what follows, the point absorber is simply considered as an axis-symmetric

float connected to a stationary reference body through an electric linear generator,

as considered in [7]. This is also comparable to the assumption of perfect mooring

of the spar as the reference body. Additionally, linear hydrodynamics and frequency

independent WEC parameters are assumed for modelling. These simplifications are

reasonable for simple body shapes and small ranges of motion. Based on Newton’s

law, the dynamic equations can be written as

mz(t) = Fe + Fgen + Fr + Fh, (3.1)

where z(t) is the buoy acceleration, m is the mass of the device and Fgen is the force

produced by the power take-off system. Fgen presents also the manipulable input to

control the system. According to [15], at least three other forces act on the body:

The radiation force, created by the moving of the float and thus radiating waves; the

hydrodynamic force which represents the restoring force of the water; and the excitation

force, which is the force the incoming wave exerts on the body. The excitation force

Fe represents an unmanipulable system disturbance. The radiation force Fr and the

hydrodynamic force Fh can be written simply as

Fr = −Az(t) + Bz(t), (3.2)

Fh = −gρπr2floatz(t) = −kz(t), (3.3)

where A is the added mass of the body, B is the viscous damping, g is the acceleration

of gravity, ρ is the density of seawater, rfloat is the radius of the float and k is then called

the hydrostatic stiffness. By means of the Morison approach [15], the wave motion can

be linearized and the excitation force can be described by

Fe = Aη(t) + Bη(t) + kη(t), (3.4)

where η is the wave elevation. Summarized can be obtained

Fe + Fgen = (A+m)z +Bz + kz. (3.5)

The system described by (3.5) can be written in state-space form as[

z

z

]

=

[

0 1

− km+A

− Bm+A

]

︸︷︷︸

A

[

z

z

]

+

[

01

m+A

]

︸︷︷︸

B

Fgen +

[

01

m+A

]

︸︷︷︸

B

Fe. (3.6)

3.2. OPTIMAL POWER CONTROL 17

The output of the system is equal to the states of the system

y = x =

[

z

z

]

. (3.7)

The MPC needs a discrete system representation whose sampling time h has to be the

same as the optimization sample time ∆t for the MPC. In what follows, the inputs Fgen

and Fe at the sample instant k are denoted by u and f , respectively. Let xk = x(kh).

Assuming a Zero-Order Hold for all inputs

u(t) = uk,

v(t) = vk, (3.8)

with kh ≤ t ≤ (k + 1)h, the following discrete system can be obtained [11]

xk+1 = Adxk +Bduk +Bdfk, x0 ∈ R2, (3.9)

where

Ad = eAh,

Bd =

∫ h

0

eAφ dφB,(3.10)

Many different ways of computing the matrix exponential eAh can be found in [24]. It

should also be noted that the discretization is exact for the input u, since it will be

calculated with the MPC at every sample time and is therefore a piecewise constant

signal as assumed in (3.8). For the excitation force f , which is in general not piecewise

constant, a small error can be expected which grows with larger sample times.

3.2 Optimal Power Control

In the past much work has been done on control algorithms, such as latching control

for WECs, and on optimizing the energy absorption [4, 18, 15]. Most of these stud-

ies assume, similar to here, one wave-absorbing semisubmerged heaving body. Often

mooring forces are not taken into account as well. For these assumptions, an optimal

control law can be derived. The next explanations follow the work in [6, 15].

Making the assumption that the wave climate is monochromatic and there is a

steady state wave input, phasors can be used, since all equations are linear. Then, the

position and the excitation force can be written as

η(t) = ηA cos(ωt+ φη) ⇒ H =ηA√2ejφη , (3.11)

Fe(t) = Fe,A cos(ωt+ φFe) ⇒ Fe =

Fe,A√2ejφFe , (3.12)

where ω = 2πTp, and ω and Tp denote the wave frequency and time period, respectively.

Also, φ is the phase angle and the subscript A denotes peak amplitude values. Thus,


(3.5) can be written in phasor form

Fe + Fgen = (jω(A+m) +B + k/(jw))jωH (3.13)

= (jω(A+m) +B + k/(jw))︸︷︷︸

Zhd

V , (3.14)

where V is the phasor of the velocity η and Zhd can be considered as hydrodynamic

impedance, analogous to an electric circuit. In order to find an optimal control law, the

linear generator is assumed to be equivalent to a spring-mass-damper system. Hence,

the generator produces a force proportional to acceleration, velocity and position of

the buoy.

Fgen = − (jωmgen +Bgen + kgen/(jw))︸︷︷︸

Zhd,gen

V . (3.15)

The minus sign is added in order to consider the generator as a resistance and not as

as source. Thus, (3.14) and (3.15) can be interpreted as the equivalent of the electric

circuit shown in Fig. 3.2. Voltage is equivalent to force and current is equivalent to the

Fig. 3.2: Electrical equivalent circuit

buoy velocity. Different types of power components need to be distinguished.

S = FeV∗ (3.16)

Shd = Phd + jQhd = (jω(A+m) + B + k/(jw))V · V ∗

= (B + j(ω(A+m)− k/ω))|V |2, (3.17)

Sgen = Pgen + jQgen = (jωmgen + Bgen + kgen/(jw))V · V ∗

= (Bgen + j(ωmgen − kgen/ω))|V |2, (3.18)

where S is overall power delivered to the WEC, Shd is the power delivered to the water

and Sgen is the power fed into the electrical generator. P and Q denote the active

and reactive power, respectively. If the imaginary component of the generator power

3.2. OPTIMAL POWER CONTROL 19

is nonzero, the generator is delivering power to the sea at some points. Thus, external

energy needs to be fed into the generator in order to source the reactive components for

several seconds. Therefore, the storage of energy is necessary which can be a significant

requirement for the WEC.

Maximizing power generation for the equivalent circuit is an impedance matching

problem, [2]. With

V =Fe

Zhd + Zhd,gen

=Fe

(jω(A+m) + B + k/(jw)) + (jωmgen + Bgen + kgen/(jw)),

(3.19)

it follows that the active generator power is given by

Pgen = Bgen|V |2 = Bgen

∣∣∣∣∣

Fe

Bgen + j(ωmgen − kgen/ω) + B + j(ω(A+m)− k/ω)

∣∣∣∣∣

2

.

(3.20)

By the following appropiate choice, the reactive components can be cancelled and the

power generation is maximized.

mgen = −(A+m), (3.21)

Bgen = B, (3.22)

kgen = −k. (3.23)

For the generated power yields

Pgen = B∣∣∣Fe

2B

∣∣∣

2

=|Fe|24B

(3.24)

and hence for the optimal velocity

V =Fe

2B. (3.25)

This means that the control law alters the resonant frequency of the system to resonate

with the dominant wave frequency, ω. This control law is also called reactive control

or phase and amplitude control [15], because condition (3.25) means that the velocity

V (or zcom) has to be in phase with the excitation force and the velocity amplitude |V |has to equal Fe

2B, [15].

Finally, the optimal control law can be written as

Fgen,opt = (A+m)z − Bz + kz. (3.26)

It can be observed that the theoretical optimal generated power is exactly one half of

the incoming power [6], where the other half diffuses back into the sea. In the case of

allowing surge motion, the theoretical limit of power generation is 1. Also, it has to

be mentioned that this control law requires significant energy storage because of the

reactive components.


3.3 MPC Formulation

In this section, two different MPC approaches are proposed. The first formulation

requires an optimal reference trajectory for the buoy velocity. This reference can be

calculated by using (3.25). Furthermore, constraints for the generator force, buoy po-

sition and buoy velocity are considered. The second formulation attempts to optimize

the power generation by directly including a power term into the objective function,

and thus no reference is neccessary.

3.3.1 Optimal Trajectory Formulation

The objective function in this formulation needs to contain one term which evalu-

ates the deviation of the actual buoy velocity from the optimal reference velocity and

another term which evaluates the energy use. Additionally, constraints need to be

considered. In the following, Fgen and Fe are denoted by u and f , respectively. Hence,

the optimization problen can be defined as

minxk,uk

J(xk, uk), (3.27)

where

J(xk, uk) =1

2

N∑

k=1

[(xk − xk,opt)

TQ(xk − xk,opt) + r u2k−1

], (3.28)

subject to

xk+1 = Adxk +Bduk +Bdfk (3.29)

|zk| ≤ cp, ∀k = 1, . . . , N (3.30)

|zk| ≤ cv, ∀k = 1, . . . , N (3.31)

|uk| ≤ cu, ∀k = 0, . . . , N − 1, (3.32)

where

Q =

[

0 0

0 q

]

, (3.33)

with weighting factors q ≥ 0, r > 0 and the symmetric relative position cp, velocity cv

and input cu constraint. N denotes the length of the time horizon. xk,opt denotes the

optimal state/output trajectory which the actual state/output has to follow. Only the

velocity values of the reference trajectory are weighted, since there is no information

about the optimal position trajectory. So, all optimal position values can be set to

zero.

To reformulate the optimization problem statement, define the following four aug-

3.3. MPC FORMULATION 21

mented vectors:

X =[

xT1 , x

T2 , ..., x

TN

]T

,

U =[

u0, u1, ..., uN−1

]T

,

F =[

f0, f1, ..., fN−1

]T

,

T =[

tT1 , tT2 , ..., t

TN

]T

,

(3.34)

with dim(X , T ) = 2N and dim(U ,F) = N , where T is the optimal output state

trajectory over the entire horizon with

tk =

[

012B

fk

]

. (3.35)

Thus, the objective function (3.28) can be formulated in matrix form as

J =1

2(X − T )T Q (X − T ) +

1

2UT RU , (3.36)

where Q und R are diagonal matrices

Q = diag(Q,Q, . . . ,Q), R = diag(r, r, . . . , r) (3.37)

with dim(Q) = (2N ×2N) and dim(R)(N ×N). Q has to be at least positive semidef-

inite and R positive definite.

In order to decrease the computational effort, the problem will be formulated only

in terms of the input variables, similar to the work in [44, 35]. A closed-form expression

for all states xk that only depends on uk and x0 can be formulated by solving the system

and substituting in the system equation (3.29)

x1 = Adx0 +Bdu0 +Bdf0,

x2 = Adx1 +Bdu1 +Bdf1 = Ad2x0 +AdBdu0 +Bdu1 +AdBdf0 +Bdf1,

...

xN = AdNx0 +Ad

N−1Bdu0 +AdN−2Bdu1 +Ad

N−1Bdf0 +AdN−2Bdf1 + . . .

+BduN−1 +BdfN−1.

(3.38)

In vector form,

x1

x2

x3...

xN

︸︷︷︸

X

=

Ad

Ad2

Ad3

...

AdN

︸︷︷︸

Jx

x0 +

Bd 0 0 . . . 0

AdBd Bd 0 . . . 0

Ad2Bd AdBd Bd . . . 0...

. . ....

AdN−1Bd Ad

N−2Bd . . . Bd

︸︷︷︸

Ju

u0

u1

u2

...

uN−1

︸︷︷︸

U

−

f0

f1

f2...

fN−1

︸︷︷︸

F

(3.39)


with

dim(x0) = 2, dim(X ) = 2N, dim(U) = N, (3.40)

dim(Jx) = (2N × 2), dim(Ju) = (2N ×N). (3.41)

By plugging (3.39) in the objective function (3.36), X is eliminated and one gets

J =1

2(Jxx0 + JuU + JuF − T )T Q(Jxx0 + JuU + JuF − T ) +

1

2UT RU . (3.42)

By expanding and neglecting the terms without dependence on U , the objective func-

tion can be written as

J =1

2UT (Ju

T QJu + R)U + UT (JuT QJxx0 − Ju

T QT + JuT QJuF), (3.43)

and is now only quadratic dependent on U .The constraints need to be reformulated as well. The constraints from (3.30) -

(3.32) can be also written as

Dkuk ≤ dk, for k = 0, . . . , N − 1, (3.44)

Ekxk ≤ ek, for k = 1, . . . , N. (3.45)

The input constraints are already only dependent on U and can be written in matrix

form as

DDU ≤ d, (3.46)

where DD = diag(D0, . . . ,DN−1) and d = [dT0 , . . . , dTN−1]

T .

The state constraints contain the state xk. To get rid of it, xk is replaced by (3.39). It

follows

E1Bdu0 ≤ e1 − E1Adx0 − E1Bdf0,

...

EN(AdN−1Bdu0 +Ad

N−2Bdu1 + . . .+BduN−1) ≤ eN − ENAdNx0 − EN(Ad

N−1Bdf0

+AdN−2Bdf1 + . . .+BdfN−1).

(3.47)

With the definitions ED = diag(E1, . . . ,EN) and e = [eT1 , . . . , eTN ]

T the constraints can

be written as

EDJuU ≤ e− EDJxx0 − EDJuF . (3.48)

Now, the MPC problem can be restated as

minU

J(U), (3.49)

where

J =1

2UT (Ju

T QJu + R)U + UT (JuT QJxx0 − Ju

T QT + JuT QJuF), (3.50)

subject to[

DD

EDJu

]

U ≤[

d

e− EDJxx0 − EDJuF

]

. (3.51)


Problems in this form are usually called Linear Quadratic Problems (LQP), because

the objective function is quadratic and all constraints are linear. There exist many

appropriate methods to solve problems of this class. The reader is referred to [27] and

[34].

3.3.2 Direct Power Maximization Formulation

The next MPC problem formulation is also done in terms of the input variables U ,which allows the constraints formulation to remain the same. The difference is that

this formulation can maximize the power directly, without the need of an optimimal

trajectory for the buoy velocity. This can be very advantageous in case no optimal

trajectory can be calculated. Also, the assumption of a sinusoidal monochromatic wave

in order to calculate the reference can be a problem in the previous formulation. For

irregular waves, it generally cannot be expected that the calculated optimal trajectory

be the optimal one. Moreover, this trajectory can be far away from the optimal power

generation under certain conditions.

The following formulation focuses on optimizing the generated power without using

a reference. In general it is defined by

Pgen = −Fgen(t)z(t). (3.52)

In matrix form over the complete horizon, the power is given by

Pgen = −X T(2)U (3.53)

where the subscript (2) denotes the second component of each xk in X . To extract the

second components of X , the matrix S is defined, i.e.

S =

0 1 0 0 . . . 0

0 0 0 1 . . . 0...

...

0 0 0 0 . . . 1

, (3.54)

with dim(S) = (N × 2N), so that the following holds:

Pgen = −(SX )TU . (3.55)

Plugging (3.55) in (3.39) yields

Pgen = −(S(Jxx0 + JuU + JuF))TU= −x0

TJxTSTU − UTJu

TSTU − FTJuTSTU

= −(UTJu

TSTU +(x0

TJxTST + FTJu

TST)U).

(3.56)

To make this problem fit the minimization problem, the sign is switched. Hence the

MPC formulation for generating power optimization is

minU

J(U), (3.57)


where

J(U) = 1

2UT

(

JuTST + R

)

U +1

2

(x0

TJxTST + FTJu

TST)U , (3.58)

(3.59)

subject to[

DD

EDJu

]

U ≤[

d

e− EDJxx0 − EDJuF

]

, (3.60)

where R is again the weighting matrix for the input.

The trajectory formulation is naturally convex, thus every local solution is the global

solution of the optimization problem. That is why convexity is a requirement for most

global optimization algorithm for LQP problems. It can also be shown that the power

formulation yields a convex problem which is no longer symmetric, see Appendix A.

However, this is only shown for the two-body case due to redundancy. Hence, the same

optimization methods used in the trajectory formulation can be applied.

3.4 Implementation

The simulation is implemented using Matlab/SIMULINK. The simulation model for

the one-body direct formulation is shown in Fig. 3.3. It mainly consists of four different

U Y

U Y

U Y

S−Function

generate _sinus

RealPlant

Fgen

Fe

pos

vel

Phase

(−pi/2 0 pi/2)

Offset

0

Level −2 M−fileS−Function

MPC

Iteratations firstoptimization

vel _buoy

pos_buoy

F_gen

F_e

Frequency

[1/(2*pi) 1/(2.5*pi) 1/(3*pi)]

vel _buoy

pos_buoy

pos_wave

vel _wave

F_gen

F_e computation

eta F_e

F_eExitflag Optimization

Exitflag Infeasibility catching

Amplitude

(0.1 0.2 0.3)

Fig. 3.3: SIMULINK simulation model for one-body direct power formulation.

blocks. The “generate sinus” block is able to generate a superposition of sine waves

as wave input signal with corresponding prediction over a certain horizon. The wave

signal consists of position, velocity and acceleration, and perfect prediction is assumed.

According to (3.4), a prediction for the excitation force can be calculated in the block

“Fe computation”. The actual MPC algorithm is implemented in the Level-2 M-file

s-function “MPC”. As stated earlier, the inputs of the “MPC” are the predictions of

the excitation force, as well as the current system state, which is the output of the

“RealPlant” block. It is assumed that the current system state can be measured. The

3.5. RESULTS 25

ouputs of the “MPC” block are the first generator force signal and certain signals for

the optimization status.

The hydrodynamic parameters of the point absorber used in the simulation are given

in Table 3.1. The parameters are taken from [7] and were determined from analysis

Variable Explanation Value

m Mass 1997 kg

A Added mass 5660 kg

B Viscous damping 11,400 N/(m/s)

k Hydrostatic stiffness 88,970 N/m

Tab. 3.1: Hydrodynamic parameter of the point absorber.

with ANSYS AQUA [1]. The simulation parameter used for the MPC simulations are

given in Table 3.2. The position constraint is equivalent to the stroke length of the


∆t Sample time for optimization 0.1 s

Thor Optimization horizon 3 s

N Number of values 30

T Simulation Time 50 s

cp Position constraint 1 m

cv Velocity constraint 1 m/s

cu Generator force constraint 80,000 N

Tab. 3.2: Simulation parameter.

generator and is taken from [33]. The generator constraint is chosen so that the MPC

control can be effectively tested.

3.5 Results

In this section, the two different implementations are verified and compared by means

of computer simulation. In the following, “MPC Traj” and “MPC Power” denote the

MPC with the trajectory formulation and the direct power formulation, respectively.

In all figures they are abbreviated by (T) and (P), respectively, and the optimal control

law by (O). First, a simple sinosodial wave with an amplitude of 0.6 m and a period

time of 7 s is used as the input wave signal. In Fig. 3.4, the simulation results with the

optimal control can be seen. On the top left, the wave elevation η and the actual buoy

displacement with optimal control are plotted. It can be noted that displacements

of 2.4 m are achieved which would be clearly above the actual requirements. The


0 10 20 30 40 50−3

−2

−1

0

1

2

3

Time [s]

Pos

ition

[m]

Displacement of Wave and Buoy

η

z (O)

0 10 20 30 40 50−3

−2

−1

0

1

2

3x 10

5

Time [s]

For

ce [N

]

Excitation and Generator Force

Fe

Fgen (O)

0 10 20 30 40 50−3

−2

−1

0

1

2

3

Time [s]

Vel

ocity

[m/s

]

Wave, Buoy and Commanded Velocity

η

z (O)zcom

0 10 20 30 40 50

−200

−100

0

100

200

300

Time [s]

Pow

er [k

W]

Power Generation

Pgen (O)

Fig. 3.4: Simulation results with the optimal control law and a monochromatic sinu-

sodial wave as input signal.

excitation force, seen on the top right, is in the range of ±50,000 N, whereas the

generator force with about 200,000 N would violate the constraints. On the bottom

left it can be seen that the actual velocity with this control law is exactly the same

as the earlier calculated commanded velocity which is later used as a reference. The

generated power, plotted on the bottom right, is within −210 kW and +260 kW and

the average generated power is 25.24 kW. It has to be noted that the amount of reactive

power at some points is enormous, so that a large energy storage is necessary.

Fig. 3.5 shows the phase relation between excitation force and actual velocity with

the optimal control law. It can be seen that the optimal velocity has to be in phase

with the excitation force, which was discussed in Section 3.2.

The next simulation compares the two different MPCs, where the same wave data as

in the previous simulation is used. The parameters and constraints of Table 3.2 are

applied, and Fig. 3.6 shows the results. The buoy positions for both controllers satisfy

the limits of ± 1 m well. Also, almost no difference between both trajectories can

be seen. The same holds for the generator force. Both trajectories are almost the

same and respect the generator limit as desired. On the bottom left, the buoy and the

commanded velocity are plotted. The MPC attempts to follow the commanded velocity,

but is restricted by the constraints. Even though, “MPC Power” is not designed to

follow the comanded velocity, the same results as for “MPC Traj” are obtained, and

3.5. RESULTS 27

0 10 20 30 40 50−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time [s]

For

ce [N

/10

5 ] / V

eloc

ity [m

/s]

Fe

z (O)

Fig. 3.5: Phase relation between excitation force and actual velocity with the optimal

control law

the generation power trajectories are almost the same. It turns out that the range here

is only between -23 kW and 58 kW, so significantly less energy storage is required.

However, the average generated power is less than with the optimal control law, namely

17.45 kW for “MPC Power” and 17.42 for “MPC Traj”. But that is still a large value

given that the constraints restrict the allowed ranges for position, velocity and generator

force by more than half. All in all it can be stated that both MPC formulations respect

the constraints as desired and that the power formulation yields almost the same results

as the trajectory formulation for the monochromatic wave data.

The wave data modelling the incoming waves is then modified. The new input

signal is a superposition of three sinewaves with amplitudes between 0.1 m and 0.3

m and period times between 1/2π s and 1/3π. Fig. 3.7 shows the simulation results

under the same conditions apart from the wave data change. Again, it can be seen

that all constraints are satisfied and that both formulations yield almost the same

results. There is one interesting time period between 33 s and 43 s. In that period, the

results of both formulations differ significantly. As can be seen on the bottom left, the

velocity trajectory of “MPC Power” does not follow the commanded one, even though

it is within the constraints. In this case, the power formulation calculates a completely

different solution. Nevertheless, the constraints are satisfied and the average power is

about the same. The average generated power for “MPC Traj” is 8.03 kW and for

“MPC Power” is 7.87 kW .

Interestingly, the optimal generated power with the optimal power law is only 3.72

kW. The wave signal in this simulation is a superposition of sine waves with different

frequencies and amplitudes, and certainly not a monochromatic wave. Also, the signal


0 10 20 30 40 50

−1

−0.5

0

0.5

1

Time [s]

Pos

ition

[m]

Position of Wave and Buoy

z (T)z (P)η

0 10 20 30 40 50

−1

−0.5

0

0.5

1

x 105

Time [s]

For

ce [N

]

Generator Force

Fgen (T)Fgen (P)

0 10 20 30 40 50

−2

−1

0

1

2

Time [s]

Vel

ocity

[m/s

]

Buoy and commanded Velocity

z (T)z (P)zcom

0 10 20 30 40 50

−20

0

20

40

60

80

Time [s]

Pow

er [k

W]

Power Generation

Pgen (T)Pgen (P)

Fig. 3.6: Comparison between “MPC Traj” (T) and “MPC Power” (P) in case of

monochromatic sinusodial wave as input signal.

is chosen so that there is no clear dominant wave frequency, making it one of the worst

cases for this control law. This is the reason why the optimal power law is not optimal

in this case. It is very interesting to see that the MPC algorithm of “MPC Traj”

finds almost the same optimum trajectory for Fgen as “MPC Power”, even though

“MPC Traj” tries to follow the “wrong” optimal reference trajectory.

In summary, two different observations can be made: First, the formulation by

means of the optimal trajectory yields good results even if the optimal trajetory is by

far not the optimal one. That is an intersting result which was not expected in the

first place. Secondly, the formulation by means of the direct power optimization leads

to almost the same results as the other formulation. The advantage here is that no

reference trajectory is necessary. Therefore, this approach can be easily applied to the

linear two-body case, since an optimal velocity can not be calculated to easily fit the

two-body case yet.

3.5. RESULTS 29

0 10 20 30 40 50

−1

−0.5

0

0.5

1

1.5

Time [s]

Pos

ition

[m]

Position of Wave and Buoy

z (T)z (P)η

0 10 20 30 40 50

−1

−0.5

0

0.5

1

x 105

Time [s]

For

ce [N

]

Generator Force

Fgen (T)Fgen (P)

0 10 20 30 40 50−2

−1

0

1

2

3

Time [s]

Vel

ocity

[m/s

]

Buoy and commanded Velocity

z (T)z (P)zcom

0 10 20 30 40 50−40

−20

0

20

40

60

Time [s]

Pow

er [k

W]

Power Generation

Pgen (T)Pgen (P)

Fig. 3.7: Comparison between “MPC Traj” (T) and “MPC Power” (P) in case of a

superposition of sine waves as input signal.

Chapter 4

Linear Model Predictive Control of

Two-Body WEC

This chapter presents a hydrodynamic two-body model of the point absorber in the

time domain, which emphasizes the special mooring configuration and the formula-

tion as a state space model. The next section deals with the model predictive control

algorithm and extents the work done in the previous chapter. Additionally, two dif-

ferent approaches to handle possible infeasibility problems are shown before giving an

insight into the implementation of the two-body MPC. Finally, simulation results of

both approaches with irregular wave data are compared. The influence of horizon and

step time of the control algorithm is also discussed by means of simulation results.

Finally the influence of the mooring parameter for power generation is shown before

the robustness behaviour is determined.

4.1 Two-Body Model

In this chapter, the point absorber is considered as a two-body device. This assumption

is more realistic and better reflects the actual physical device. The left side of Fig. 4.1

illustrates the L10 point absorber developed at the Oregon State University. The right

side shows a schematic two-body diagram which is used to derive the model. The

floating body is called float or buoy, and the damping body is a combination of a spar

and a ballast tank (in the following just called spar). Furthermore, the spar is moored

to the sea floor in order to dampen the spar’s motion. Also, the two-body model is

restricted to heave motion.

4.1.1 Equations of Motion

Much work has already been done on modelling two-body point absorbers. The herein

presented modelling follows the work of Eidsmoen [13] and Ruehl [37]. In order to derive

32 CHAPTER 4. TWO-BODY WEC

Sea Floor

mooring

Tank

Spar

PTO

x

zFloat

Fig. 4.1: L10 Wave Energy Coverter [38] (left) and schematic diagram (right).

the equations of motion for the two bodies in the time-domain, some assumptions are

made:

• The formulation is based on Linear Wave Theorie (LTW).

• Frequency dependent parameters of the L10 are assumed to be constant.

• The spar is taut-moored with three cables. The pretension of the cables is induced

by the buoyancy forces of the spar. Therefore, the spar’s buoyancy force can be

neglected.

• The radiation forces are assumed to be linear and no convolution terms are used

to calculate it.

As can be seen on the right side of Fig. 4.1, x and z denote the position of the spar,

respectively, of the buoy. Based on Newton’s law, the dynamic equations of the buoy

can be written as

Fgen + Fe1 − Fr1 − Fh1 − Fr12 = m1z, (4.1)

where z(t) is the buoy acceleration, m1 is the mass of the buoy, Fe1 is the excitaton

force induced by the incoming waves, and Fgen is , as in the one-body model, the force

produced by the power take-off system. Similar to the one-body case, the forces are

Fr1 = A1z(t) + b1z(t), (4.2)

Fh1 = gρπr2buoyz(t) = k1z(t), (4.3)

where A1 is the buoy’s added mass, b1 is its viscous damping and k1 is the hydrostatic

stiffness.

Furthermore, there is a coupled radiation force resulting from the interaction of the

two bodies. Here, the force Fr12 is acting from the spar on the buoy and can be written

as

Fr12 = A12x. (4.4)

4.1. TWO-BODY MODEL 33

The same procedure can be applied to the second body. It follows that

−Fgen + Fe2 − Fr2 − Fr21 − Fm = m2x (4.5)

Fr2 = A2x+ b2x (4.6)

Fr21 = A21z (4.7)

Fm = 3K(sinα)2x = Kmx. (4.8)

There is no restoring or buoyancy force, since it is assumed that this force causes the

pretension in the mooring cables.

The mooring force is dependent on the adopted configuration. In this thesis, a

taut-moored buoy with 3 cables is considered as shown in Fig. 4.2. The three mooring

Fig. 4.2: Mooring configuration with three cables (left) and schematic diagram for

force derivation (right).

cables are configured like a tripod and there is an angle α with respect to the sea floor.

According to Hook’s law, the mooring force in the direction of the cable can be written

as

F = −K(L′ − L), (4.9)

where K is the cable stiffness. The vertical mooring force is then

Fm = −3F sin(α). (4.10)

Using the cosine theorem yields

Fm = −3K(√

L2 + x2 − 2Lx cos(π

4+ α)− L

)sin(α). (4.11)

However, this is a nonlinear law for the mooring. It is expected that linearization

around x = 0 yields good results, since x << L.


The linearization gives

Fm,lin = Fm(x = 0) +δFm

δx(x = 0) x

= −3K sin(α)x− L cos(π

4+ α)

√L2 + x2 − 2Lx cos(π

4+ α)

(x = 0) x

= 3K sin(α) cos(π

4+ α)x

= −3K sin(α)2x = −Kmx

(4.12)

This holds for x > 0. The same result with the opposite sign follows for x < 0. Even

for large displacements such as 0.5 m, the difference between the nonlinear and the

linear mooring force calculation is only 0.04% for a cable length of 170 m. Hence, the

linear case is a very good approximation and the mooring force can be written as in

(4.8).

4.1.2 State Space Form and Discretization

In order to control the two-body point absorber via MPC, a discrete time state space

representation is necessary as well. Here, the formulation as a state space model is not

straightforward. The problem lies in the coupling terms within the coupled radiation

calculation. This coupling occurs in the second derivative, thus a state space model

can not be formed without reformulating first. Equations (4.1)-(4.8) can be rewritten

asFgen + Fe1 − b1z − k1z − A12x

m1 + A1

= z (4.13)

−Fgen + Fe2 − b2x−Kmx− A21z

m2 + A2

= x. (4.14)

Transforming these equations in state space form requires plugging (4.13) into (4.14)

and vice versa to get rid of the coupling terms A21z and A12x. The equations can be

restated as

z(m1 + A1 −

A12A21

m2 + A2︸︷︷︸

me1

)= Fgen + Fe1 − b1z − k1z

+A12

m2 + A2

Fgen −A12

m2 + A2

Fe2 +A12b2

m2 + A2

x+A12Km

m2 + A2

x

(4.15)

x(m2 + A2 −

A21A12

m1 + A1︸︷︷︸

me2

)= −Fgen + Fe2 − b2x− A21

m1 + A1

Fgen

− A21

m1 + A1

Fe1 +A21b1

m1 + A1

z +A21Km

m1 + A1

z −Kmx

(4.16)

Now, it is possible to find a state space formulation with the state vector

x =[

z z x x]T

and the initial conditions x0 =[

0 0 0 0]T

. Moreover, it is


assumed that the entire state is measurable. The system in state space form can be

written as

x = Ax+BFgen +B1Fe1 +B2Fe2 ,

y = x,(4.17)

where

A =

0 1 0 0

− k1me1

− b1me1

A12Km

me1 (m2+A2)A12b2

me1 (m2+A2)

0 0 0 1A21k1

me2 (m1+A1)A21b1

me2 (m1+A1)−Km

me2− b2

me2

, (4.18)

B =

01

me1+ A12

me1 (m2+A2)

0

− 1me2

− A21

me2 (m1+A1)

(4.19)

B1 =

01

me1

0

− A21

me2 (m1+A1)

, B2 =

0

− A12

me1 (m2+A2)

01

me2

. (4.20)

In what follows, the inputs Fgen, Fe1 and Fe2 are denoted by u, v and w, respectively.

With the same discretization approach as in Chapter 2, the following discrete-time

system can be obtained

xk+1 = Adxk +Bduk +Bd1vk +Bd2

wk, x0 ∈ R4,

yk= xk, (4.21)

where

Ad = eAh, Bd =

∫ h

0

eAφ dφB, (4.22)

Bd1=

∫ h

0

eAφ dφB1, Bd2=

∫ h

0

eAφ dφB2. (4.23)

4.2 MPC Formulation

The MPC formulation for the two-body case is very similar to the direct power for-

mulation in the one-body case. There is no information about an optimal reference

trajectory, so this formulation is also the appropriate way to deal with the two-body

problem.

The optimization problem for the two-body case can be defined as

minxk,uk

J(xk, uk), (4.24)


where

J(xk, uk) = −N∑

k=1

[−q

(zk − xk

)uk−1

︸︷︷︸

−Pgen

−r u2k−1

], (4.25)

subject to

xk+1 = Adxk +Bduk +Bd1vk +Bd2

wk (4.26)

|zk − xk| ≤ cp, ∀k = 1, . . . , N (4.27)

|zk − xk| ≤ cv, ∀k = 1, . . . , N (4.28)

|uk| ≤ cu, ∀k = 0, . . . , N − 1, (4.29)

One main difference is that the position and velocity constraints are now referred to the

relative position and relative velocity between buoy and spar, since the stroke length

is determined by the relative displacement. Also, the generated power is calculated by

the product of the generator force and the relative velocity.

Again, the following augmented vectors are used

X =[

xT1 , x

T2 , ..., x

TN

]T

, U =[

u0, u1, ..., uN−1

]T

, (4.30)

V =[

v0, v1, ..., vN−1

]T

, W =[

w0, w1, ..., wN−1

]T

. (4.31)

Solving system (4.26) and substituting in itself yields

X = Jxx0 + JuU + JvV + JwW , (4.32)

where

Jx =

Ad

Ad2

Ad3

...

AdN

, Ju =

Bd 0 0 . . . 0

AdBd Bd 0 . . . 0

Ad2Bd AdBd Bd . . . 0...

. . ....

AdN−1Bd Ad

N−2Bd . . . Bd

, (4.33)

Jv(w) =

Bd1(2)0 0 . . . 0

AdBd1(2)Bd1(2)

0 . . . 0

Ad2Bd1(2)

AdBd1(2)Bd1(2)

. . . 0...

. . ....

AdN−1Bd1(2)

AdN−2Bd1(2)

. . . Bd1(2)

, (4.34)

and

dim(Ju,Jv,Jw) = (4N ×N),

dim(Jx) = (4N × 4).(4.35)

4.3. INFEASIBILITY HANDLING 37

Now, the objective function (4.25) can be reformulated in vector form as

J(X ,U) = q(X(2) −X(4)

)TU + UT R U= q

(S1X − S2X

)TU + UT R U= qX T (S1

T − S2T )

︸︷︷︸

S

U + UT R U(4.36)

with the matrix R = diag(r) with dim(R) = (N × N) and the matrices S1, S2

with dim(S1,S2) = (N × 4N). S1 and S2 extract the second state (buoy velocity),

respectively, fourth state (spar velocity) from the augmented state vector X .

S1 =

0 1 0 0 0 0 . . . 0 0 0

0 0 0 0 0 1 . . . 0 0 0...

. . ....

0 0 0 0 0 0 . . . 1 0 0

, (4.37)

S2 =

0 0 0 1 0 0 0 . . . 0

0 0 0 0 0 0 0 1 . . . 0...

. . ....

0 0 0 0 0 0 0 0 . . . 1

. (4.38)

Using (4.32), the objective function can be formulated as a quadratic function that

depends on U , namely,

J(U) = UT(qJu

TS+ R)U + q

(xT0 Jx

T + VTJvT +WTJw

T)S U . (4.39)

The constraint formulation as well needs just a few adjustments based on the one-body

case. Provided that all constraints can be written as in (3.44) and (3.45), the MPC

can be written as

minU

J(U), (4.40)

where

J(U) = UT(qJu

TS+R)U + q

(xT0 Jx

T + VTJvT +WTJw

T)S U , (4.41)

subject to[

DD

EDJu

]

U ≤[

d

e− EDJxx0 − EDJvV − EDJwW

]

. (4.42)

This problem is also a convex optimization problem, as shown in Appendix A.

4.3 Infeasibility handling

In general, constrained optimization problems makes searching for the optimal solution

more difficult. Furthermore, a serious problem can arise: the optimization problem is

infeasible. That means there exists no solution within the boundaries. If this situation

occurs, the optimization stops without a solution and the control loop is open. This


breakdown of the control can cause serious damage to the device and should be avoided.

There are different possibilities to handle infeasibility problems. Most approaches

try to deal with the infeasibility problem after it has occured. A simple method consists

of setting the control signal equal to a constant value. Also, another type of control

law, for example a P, PI or PID controller can be used in the event that the MPC does

not find a solution. However, both implementations can not guarantee stability of the

system within the constraints, since the system was already in a critical state when the

breakdown occured. Another method prioritizes constraints and does not consider the

least important ones while solving the problem again. However, it can not be assured

how much the constraints get violated.

This thesis introduces two approaches regarding hard and soft constraints. The

first approach tries to find a feasible solution by relaxing some constraints after an

infeasibility condition occurs, and resolves the optimization problem for the expanded

region. Here the generator force limit cu and the velocity constraint cv can be relaxed,

since overspeed and over-rated generator conditions can be tolerated for the short term

[7]. The constraints for the new optimization can be defined by

|zk − xk| ≤ cp, ∀k = 1, . . . , N (4.43)

|zk − xk| ≤ rv cv, ∀k = 1, . . . , Nr (4.44)

|zk − xk| ≤ cv, ∀k = Nr + 1, . . . , N (4.45)

|uk| ≤ ru cu, ∀k = 0, . . . , Nr − 1 (4.46)

|uk| ≤ cu, ∀k = Nr, . . . , N (4.47)

(4.48)

where rv and ru are the relaxation factors for velocity and generator force, respectively,

and Nr denotes the number of time steps into the future until the non-relaxed con-

straints are valid again. This helps improve the stability attributes of the MPC, since

it allows to violate the limits only at the beginning of the horizon.

The second approach tries to prevent the occurance of infeasibility problems from

the start. The original hard constraints are replaced by soft constraints through the

introduction of positive slack variables ǫp, ǫv. In order to penalize violations of con-

straints, quadratic terms including the slack variables are inserted into the objective

function that yields the new optimization problem

minU ,ǫp,ǫv

Jnew(U , ǫp, ǫv), (4.49)

where

Jnew(U , ǫp, ǫv) = J(U) + wpǫ2p + wvǫ

2v (4.50)

4.4. IMPLEMENTATION 39

subject to

|zk − xk| − ǫp ≤ cp, ∀k = 1, . . . , N (4.51)

|zk − xk| − ǫv ≤ cv, ∀k = 1, . . . , N (4.52)

|uk| ≤ cu, ∀k = 0, . . . , N − 1 (4.53)

0 ≤ ǫp ≤ cǫp , 0 ≤ ǫv ≤ cǫv . (4.54)

The constraints cǫp , cǫv are hard upper bounds for the slack variables which are nec-

cessary, in particular, for the position. The weighting factors wp, wv can be arbitrarily

choosen and weight the penalty for violating the position and the velocity constraints,

respectively.

4.4 Implementation

The simulation is also implemented using Matlab/SIMULINK. A diagram of the simu-

lation model is shown in Fig. 4.3. Here, different types of irregular wave data obtained

from [36] can be loaded. The prediction for the incoming wave elevation is calculated

in the “Prediction calculation” block, where it is assumed that the prediction is per-

fect. In the next block, the excitation forces for both bodies are calculated by means

of causal and noncausal impulse response functions from [36] as well. In that work, the

impulse responses are obtained using frequency domain hydrodynamic analysis with

ANYSY/AQUA [1] for the L10 wave geometry and parameters. The actual MPC is

implemented similarly to the one-body case in a Level-2 Matlab S-Function, and ob-

tains the future predictions for the excitation forces of both bodies and the current

state of the device as inputs. The actual optimization problem is solved by means of

the Matlab QP-solver “quadprog”. In order to check the status of the optimization,

several other outputs are given to find, for instance, the points where the infeasibility

catching is active. Moreover, it is assumed that all states are measurable.

Simulation Time

12:34

UY

UY

RealPlant

Fgen

Fe_1

Fe_2

z

zdot

x

xdot

Prediction calculation

Prediction_2

PTO

−20000

MPC

TwoBody_diff

Iteratations firstoptimization

vel_buoy

pos_buoy

F_gen_opt

vel_plate

pos_plate

F_e_2

vel_plate

vel_buoy

F_e_1

vel_plate

pos_plate

vel_buoy

pos_buoy

F_gen

F_e_2

vel_diff

F_gen_optF_e_1

Exitflag Optimization

Exitflag Infeasibility catching

Excitation Force Determination

eta

eta1

Fe1

Fe2

Add

Fig. 4.3: SIMULINK simulation model for two-body WEC control with MPC.


For the system parameters of the L10, the results in [36] obtained by ANSYS AQWA

[1] analysis are used. An overview is shown in Table 4.1, where the mooring constant

Variable Value

m1 2625.3 kg

m2 2650.4 kg

A11 8866.7 kg

A22 361.99 kg

A12 361.99 kg

A21 361.99 kg

b1 5,000 N/(m/s)

b2 50,000 N/(m/s)

k1 96,743 N/m

Km 300,000 N/m

Tab. 4.1: System parameter of the two-body L10 WEC.

is based on an angle α = 60◦ and a cable stiffness K = 133,333Nm

which is within the

common range for the stiffness of mooring cables, according to [10].

4.5 Results

Computer simulations are used to validate the MPC algorithm and its performance

regarding the generated power. Furthermore, the formulation with relaxed constraints

is compared to the soft constraints formulation. Another focus is on the behaviour

when changing the parameters ∆t and Thor. In the end, the MPC is tested with

several changes of parameters only in the “RealPlant” block, in order to simulate

system uncertainties. A frequency spectrum from January 2009 of the NDBC Offshore

buoy 46050 which is deployed off the coast of Oregon west of Newport [26] is used to

generate a time series for the wave elevation. In all simulations this time series is used

as the input signal to calculate the excitation forces.

Six different cases are treated in this section. First, the controller behaviour of “Case

1” and “Case 2” are compared. The parameters for “Case 1” are listed in Table 4.2.

This MPC is implemented with the relaxed constraint formulation. This means that the

generator force constraint of 80,000 N can be violated in case an infeasibiliy condition

occurs up to a value of 120, 000 N and the velocity constraint up to 2 ms. In “Case

2”, the MPC is implemented by means of soft constraints. This formulation has no

parameter values for Nr, rv and ru. The changed and the new parameters are listed in

Table 4.3. To get comparable parameters, the hard constraint for the position is also

set to 1 m consisting of 0.9 m soft constraint and the slack variable constraint of 0.1 m,

4.5. RESULTS 41


∆t sample time for optimization 0.1 s




q Power weighting factor 1

r Input weighting factor 0

cp Relative position constraint 1 m

cv Relative velocity constraint 1 m/s


Nr Relaxation number 10

rv Relaxation factor velocity 2

ru Relaxation factor generator force 1.5

Tab. 4.2: MPC parameter values for the formulation with relaxed constraints (Case 1).


cp Relative position constraint 0.9 m

cǫp Slack variable position constraint 0.1 m

cǫv Slack variable velocity constraint 1 m/s

wp Weighting factor position slack 108

wv Weighting factor velocity slack 106

Tab. 4.3: MPC parameter values for the formulation with soft constraints (Case 2).

and the same holds for the velocity constraint. Furthermore, the weighting factor for

the position slack is much greater than for the velocity, in order to penalize violations

of the position constraint harder than violations of the velocity constraint.

To get a first insight into the motion of the incoming waves and of the device,

Fig. 4.4 shows wave, buoy and spar position for “Case 1”. The wave elevation η has a

significant wave height of 4 m. It can be seen that the buoy position slightly lags the

wave elevation and can not achieve the same amplitude as the wave elevation due to

the constraints and the induced generator force. In general the spar position is within

±0.3 m.

More interesting is how often an infeasibility problem occurs, and Fig. 4.5 shows

that the optimization problem is infeasible at several points. A consecutive time span

within which the optimization problem is not feasible is around 48 s. This is an

interesting point to look at in the next figure as well. Fig. 4.6 shows the comparison

between “Case 1” and “Case 2”. On the top left it can be seen that the hard position

constraint is never violated in either formulation. The trajectory in “Case 2” tries to

respect the soft constraint, since the penalty for violations is strong. The opposite can


0 20 40 60 80 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time [s]

Pos

ition

[m]

zxη

Fig. 4.4: Wave elevation, buoy position and spar position for the relaxed constraints

formulation (Case 1). The wave data is from January 2009 of the NDBC

Offshore buoy 46050.

be seen for the relative velocity on the bottom left. Since the penalty for violations

is less, “Case 2” tends to violate this soft constraint more often. The generator force

constraint in “Case 2” is never violated which is the purpose of the formulation. Only

in the relaxed formulation is this constraint allowed to relax and it seems to be used

at several instants in time, in particular around 48 s. At that time, the genarator force

almost reaches its hard limit.

On the bottom right, the power generation is shown and there is no noticeable

difference between both lines in the figure. However, the average power of “Case 1” is

26.13 kW and is only 24.62 kW for “Case 2”. A possible reason for this is, for example,

the possibility of having a larger generator force in “Case 1”. Moreover, in “Case 2”

the MPC tries to respect the soft constraint of 0.9 m and thus limits the usable stroke

displacement which leads to less power extraction. That can be reduced by lowering

the weighting factor for the position slack.

It should also be noted that the optimization problem in “Case 2” is always feasible,

which is a big advantage in regards to the computational effort of the MPC. Indeed,

the soft constraints formulation contains two more optimization variables (ǫp, ǫv) which

slightly increases the computation time, but in many cases it is possible to operate the

MPC without infeasibility problems. Since a MPC is supposed to run in real time, a

recalculation of the optimization problem as in “Case 1” is not always possible. So it

is adventageous to have a formulation which is in general able to calculate the solution

without recalculating as in the soft constraints formulation. However, it has to be

said that even the soft constraints formulation can yield infeasibility problems (though

4.5. RESULTS 43

0 20 40 60 80 100

−2

1

Time [s]

Infe

asib

ility

Cat

ch

Fig. 4.5: Frequency of the infeasibility problem for Case 1. 1 means original problem

is feasible. −2 means original problem is infeasible.

rarely), since the slack variables are also subject to constraints. So, for the worst case

scenario it makes sense to combine both formulations so that it is possible to relax the

generator force constraint, even in “Case 2” if an infeasibility problem occurs.

Next, two cases are compared in terms of the number of optimization variables. In

both cases the soft constraints formulation of “Case 2” is used. In “Case 3” the horizon

time is set to 6 s from 3 s and in “Case 4” the step time is doubled to 0.2 s. Therefore,

the weighting factors for the slack variables are set to half of the original value in “Case

3” and are doubled in “Case 4” in order to get comparable conditions. It is remarkable

that the results of “Case 3” are almost the same as in “Case 2”. Likewise, the average

generated power of 24.44 kW is similar as well. Therefore, the results of both cases are

compared directly, without comparison with “Case 2” which can be seen in Fig. 4.7.

Here as well, no noticeable difference can be seen and the average power in “Case 4”

is 24.34 kW. The advantage of the larger time step is that the number of optimization

variables decreases.

“Case 2” contains 32, “Case 3” 62 and “Case 4” only 17 optimization variables.

This makes a huge difference regarding the computation time, since the computational

effort increases with more optimization variables. “Case 4” yields almost same results,

even though the sample time is doubled. But one important difference can be seen

in Fig. 4.7, especially in the relative velocity plot. The trajectory calculted with the

larger time step tends to oscillate much more which can cause problems (particulary

in case of large inaccuracies between model and the actual device). However, it can be

seen that a horizon of 3 s is sufficient for this type of wave data.


0 20 40 60 80 100

−1

−0.5

0

0.5

1

Time [s]

Pos

ition

[m]

Position Difference of Buoy and Plate

Case 1Case 2

0 20 40 60 80 100

−1

−0.5

0

0.5

1

x 105

Time [s]

For

ce [N

]

Generator Force

Case 1Case 2

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]

Velocity Difference of Buoy and Plate

Case 1Case 2

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

120

Time [s]

Pow

er [k

W]

Power Generation

Case 1Case 2

Fig. 4.6: Comparison between relaxed (Case 1) and soft (Case 2) constraints formu-

lation of the MPC.

0 20 40 60 80 100

−1

−0.5

0

0.5

1

Time [s]

Pos

ition

[m]


Case 3Case 4

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


Case 3Case 4

Fig. 4.7: Comparison between the soft constraints formulation with the changed pa-

rameter Thor = 6s (Case 3) and the same formulation with changed param-

eter ∆t = 0.2s (Case 4).

4.5. RESULTS 45

Next, only the influence of the mooring parameter Km on the system peformance is

discussed. Therefore, the mooring constant is changed fromKm = 300,000 N/m (“Case

2”) toKm = 50,000 N/m (“Case 5”). Both cases are simulated with the soft constraints

formulation. The results are shown in Fig. 4.8. The position and velocity trajectories

0 20 40 60 80 100

−1

−0.5

0

0.5

1

Time [s]

Pos

ition

[m]


Case 2Case 5

0 20 40 60 80 100

−1

−0.5

0

0.5

1

x 105

Time [s]

For

ce [N

]

Generator Force

Case 2Case 5

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


Case 2Case 5

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

120

Time [s]

Pow

er [k

W]

Power Generation

Case 2Case 5

Fig. 4.8: Comparison between the soft constraints formulation with mooring constant

Km = 300,000 N/m (Case 2) and the same formulation with Km = 50,000

N/m (Case 5).

on the left do not differ significantly. However, a large difference can be seen in the

generator force. The generator force in “Case 5” is within its limits as well. But it is

in general far away from the limits and much smaller than in “Case 2”. Due to the

smaller generator force, it can be seen, that the generated power is also much smaller,

namely, only 11.91 kW in comparison to 24.62 kW on average. It stands out as well

that the required reactive power is less which is advantageous regarding the required

energy storage. Fig. 4.9 shows the spar position for both simulations. Because of the

extreme change of the mooring constant, the mooring force decreases significantly and

thus, the spar motion is within a larger range up to ±1.3 m in comparison to only ±0.3

m in “Case 2”. This is the reason why the relative position and velocity between buoy

and spar are naturally less with smaller mooring forces. So, the generator force does

not need to exceed its limits to respect position and velocity constraints. Hence, the

generated power decreases, since the relative motion difference decreases. Generally,

it can be stated that a stronger mooring increases the power generation. However, it


0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

Pos

ition

Spa

r [m

]

Case 2Case 5

Fig. 4.9: Comparison of Spar position between the soft constraints formulation with

mooring constant Km = 300,000 N/m (Case 2) and the same formulation

with Km = 50,000 N/m (Case 5).

has to be pointed out that a stronger mooring brings the generator faster to its limits.

Therefore, especially for rough wave conditions, it might be advantageous to have a

looser mooring to prevent violations of constraints and physical damage of the device.

Moreover, in case there is a chance to alter the mooring configuration during operation,

that can be used to improve the power generation in mild wave climates by increasing

the mooring constant and in rough conditions, the mooring constant could be reduced

to protect the device from damage.

Finally, the MPC is tested regarding inaccuracies between model and actual device.

Hence the parameters of the system model inside the MPC and the excitation force

calculation remain the same, whereas the parameters of the actual system change. This

definitely does not test the behaviour of every uncertainty. It is only a check regarding

parameter uncertainties and does not include unmodelled dynamic uncertainties or

non-linearities in the actual device. Nevertheless, it is a good first try to get an idea

of the robustness attributes of the proposed MPC.

The changed real plant parameters for “Case 6” and the unchanged model param-

eters are displayed in Table 4.4. Basically, the mooring and damping forces of the

assumed actual WEC are reduced and the added mass and restoring force of the buoy

are augmented. In order to compare the results with the unchanged model, “Case 2”

and “Case 6” are shown together in Fig. 4.10, where there are significant differences

in the velocity and position trajectories. But it is remarkable that the constraints are

not (or only slightly) violated. However the average generated power is only 18.59 kW.

Overall it can be stated that the MPC with the changed system parameters is still

4.5. RESULTS 47

Variable Real Plant Model

A11 9,500 kg 8866.7 kg

A22 320 kg 361.99 kg

A12 380 kg 361.99 kg

A21 340 kg 361.99 kg

b1 4,000 N/(m/s) 5,000 N/(m/s)

b2 42,000 N/(m/s) 50,000 N/(m/s)

k1 120,000 N/m 96,743 N/m

Km 210,000 N/m 300,000 N/m

Tab. 4.4: Comparison of real plant and model parameters.

0 20 40 60 80 100

−1

−0.5

0

0.5

1

Time [s]

Pos

ition

[m]


Case 2Case 6

0 20 40 60 80 100

−1

−0.5

0

0.5

1

x 105

Time [s]

For

ce [N

]

Generator Force

Case 2Case 6

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


Case 2Case 6

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

120

Time [s]

Pow

er [k

W]

Power Generation

Case 2Case 6

Fig. 4.10: Comparison between the original soft constraint formulation with original

WEC parameters of Table 4.1 (Case 2) and the same formulation with the

changed WEC parameters of Table 4.4 for the L10 (Case 6).

stable and remarkably robust with respect to the system constraints.

For a better overview, all discussed cases with explanation are shown in Table 4.5.


Case Explanation

1 Relaxed constraint formulation

2 Soft constraint formulation

3 Soft constraint formulation with larger step time (∆T = 0.2 s)

4 Soft constraint formulation with larger horizon (Thor = 6 s)

5 Soft constraint formulation with smaller mooring constant (Km = 50,000 N/m)

6 Changed system parameter for robustness check

Tab. 4.5: Overview Cases.

Chapter 5

Nonlinear Model Predictive

Control of a Nonlinear Two-Body

WEC Model

The goal of this chapter is to propose a nonlinear model predictive controller to deal

with a nonlinear WEC model. It presents first a brief introduction to nonlinear model

predictive control (NMPC). Following, the nonlinear model in [36] with convolution

terms and a nonlinear mooring law is discussed and compared to the model without

convolution terms. Also, the discretization of this nonlinear system and the imple-

mentation of the NMPC is a part of this chapter. By means of simulation results of

controlling the nonlinear model, the proposed NMPC is validated and compared to the

linear MPC from Chapter 4.

5.1 Introduction to Nonlinear Model Predictive

Control

Linear MPCs are well known and have been applied since the 1970’s, whereas the

attention to NMPC arises only in the 1990’s [16]. Linear MPC theory is quite mature

today and system theoretic attributes as stability and optimality are well addressed

[22]. Also, many different industrial MPC applications can be found. The case is

different with NMPC. While theoretical characteristics are well discussed, industrial

applications are difficult to find [5].

Linear MPCs and NMPCs have basically the same concepts. In general, the NMPC

problem is formulated as solving a finite horizon optimal control problem which is

subject to the constraints and to the system dynamics [16]. The basic principle is

the same as described in Chapter 2 and can also be seen in Fig. 2.1. Whereas only

linear models are used to predict future system behaviour in linear MPC techniques,

50 CHAPTER 5. NONLINEAR MPC

in NMPC, nonlinear models are used. Furthermore, even system constraints can be

considered nonlinear.

The motivation for NMPC arises, since there are systems which are inherently

nonlinear and can not be described adequately by a linear model. Also, control and

process requirements have increased. In these cases, the use of linear MPC is inadequate

and NMPC is a promising alternative to deal with inherently nonlinear systems.

A fundamental problem of NMPC schemes is that the constrained optimization

problem needs to be solved within a specified time limit. In the case of linear MPC,

the problem is convex and for the class of LQP, proven optimization algorithms exist

to solve the problem efficiently. NMPC requires the solution of a nonlinear problem,

though. In general, these problems are non-convex, thus it can not be assured to

find the global optimum. Additionally, the solution can be computationally expensive.

Therefore, it is important to exploit the special structure of each problem to obtain a

real-time feasible optimization problem.

This work does not focus on real-time applicability. The focus is on the quali-

tative performance of NMPC regarding a nonlinear WEC model whose nonlinearity

results from a nonlinear mooring force. So, this work attempts to establish if NMPC is

advantageous to use for controlling the nonlinear WEC compared to the linear MPC.

5.2 Nonlinear Two-Body Model

The proposed nonlinear two-body model follows the work in [36]. In addition to the

proposed two-body model in Chapter 4, impulse response functions are used in order

to calculate the radiation forces. These functions are obtained from [36]. For further

information regarding the impulse response functions, the readers are referred to [36]

as well. Using impulse response functions yields convolution terms in the expressions

for the radiation forces. Furthermore, a highly nonlinear mooring force law, [36, 21], is

used.

The equation of motions for the two bodies are:

Fgen + Fe1 − Fr1 − Fh1 − Fr12 = m1z (5.1)

−Fgen + Fe2 − Fr2 − Fr21 − Fm = m2x (5.2)

with the following terms for the forces

Fr1 = A1z(t) +

∫ t

−∞

fr1(t− τ)z(τ) dτ, (5.3)

Fh1 = gρπr2buoyz(t) = k1z(t), (5.4)

Fr12 = A12x(t) +

∫ t

−∞

fr12(t− τ)x(τ) dτ, (5.5)

5.2. NONLINEAR TWO-BODY MODEL 51

Fr2 = A2x(t) +

∫ t

−∞

fr2(t− τ)x(τ) dτ, (5.6)

Fr21 = A21z(t) +

∫ t

−∞

fr21(t− τ)z(τ) dτ, (5.7)

Fm = 8Knlx(t)(1− Lnl

√

L2nl + x(t)2

). (5.8)

The impulse response functions of the different radiation forces are denoted by fr. The

mooring system is based on the experimental mooring configuration in [21]. There, the

buoy is moored with 8 cables to a static reference around the buoy. No mooring to the

sea floor is assumed. Knl is the stiffness of one cable, and Lnl is the horizontal length

from the reference to the buoy.

The system expressed by (5.1)-(5.8) only differs from the proposed linear two-body

system in Chapter 4 in the use of the convolution terms and in the mooring force. So,

it is interesting to see how large the influence of the convolution terms is. Neglecting

these terms and assuming the mooring force (5.8) yields the nonlinear model

x = Anlx+ l(x) +Bu+B1v +B2w (5.9)

where

Anl =

0 1 0 0

− k1me1

− b1me1

0 A12b2me1 (m2+A2)

0 0 0 1A21k1

me2 (m1+A1)A21b1

me2 (m1+A1)0 − b2

me2

, (5.10)

l(x) =

08A12Knl

me1 (m2+A2)

(1− Lnl√

L2nl+x2

)x

0

−8Knl

me2

(1− Lnl√

L2nl+x2

)x

, (5.11)

and B,B1 and B are the same as in the linear two-body model from Chapter 4. Now

there are two different models to compare. The first model (5.1)-(5.8) with convolution

terms is called extended model, and the second model (5.9) is called reduced model.

Fig. 5.1 shows the relative position trajectories for simulations with both models. This

simulation uses wave data from buoy 40650 and applies a control law proportional to

the velocity difference. The control law was chosen simply, because only the influence

of the convolution terms should be determined. No difference between both results can

be seen on the left side, whereas zooming in shows a slight difference on the right side

which is only about 0.4%, even in the worst cases. Hence, it is reasonable to neglect the

convolution terms and using the reduced model for the NMPC approach. In case there

would be a larger influence of these terms, the convolution terms can be approximated

by a linear state space model by methods described in [19, 39]. However, this is not

necessary here.


0 20 40 60 80 100−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Rel

ativ

e po

sitio

n [m

]

extendedreduced

45.8 45.82 45.84 45.86 45.88 45.91.766

1.767

1.768

1.769

1.77

1.771

1.772

Time [s]

Rel

ativ

e po

sitio

n [m

]

extendedreduced

Fig. 5.1: Comparison between the extended and the reduced model. The wave data

is from January 2009 of the NDBC Offshore buoy 46050 [26]. Proportional

control is applied. On both sides, the relative position is shown. On the

right side only a small detail can be seen.

5.3 Discretization

In the following, a finite parameterization of the controls and constraints is used to

find a direct solution of the optimization problem. Therefore, the system is supposed

to be described as a discrete-time nonlinear state space model in the form

xk+1 = fk(xk, uk, sk) (5.12)

where the function fk(.) maps the current state xk, the manipulable input uk and the

unmanipulable inputs sk to the next state xk+1.

According to [20], a nonlinear system

x = f(x(t)) + g(x(t))u(t) (5.13)

with real analytic vector fields f(x(t)) and g(x(t)) can be discretized by an approximate

sampled-data representation under Zero-Order Hold assumption by

xk+1 = xk +M∑

l=1

A[l](xk, uk)hl

l!, (5.14)

where

A[l+1](x, u) =δA[l](x, u)

δx(f(x) + g(x)u) (5.15)

with l = 1, 2, 3, . . ..

Here M denotes the order of the discretization, and h is the sample time. In simula-

tion results it turns out that a discretization of order 1 (comparable to Euler forward

method) is not appropriate for the proposed NMPC approach. In fact, it yields unrea-

sonable results. Due to this fact, the discretization order is chosen to be M = 2 in the

5.4. PROBLEM FORMULATION AND IMPLEMENTATION 53

ensuing work. With the nonlinear system (5.9), it follows that

A[1] = Anlx+ l(x) +Bu+B1v +B2w, (5.16)

A[2] = [Anl + l′(x)] · [Anlx+ l(x) +Bu+B1v +B2w]. (5.17)

Using (5.14), after some manipulations the discrete-time system can be described by

xk+1 = Anl,d1xk +Anl,d2

l(xk) +h2

2l′(xk)Anlxk +

h2

2l′(xk)l(xk)

+Bdu+Bd1v +Bd2

w + l′(xk)(Bu+B1v +B2w),

(5.18)

where

Anl,d1= I+ hAnl +

h2

2Anl

2, Anl,d2= hI+

h2

2Anl, (5.19)

Bd = hB+h2

2AnlB, Bd1

= hB1 +h2

2AnlB1, (5.20)

Bd2= hB2 +

h2

2AnlB2, (5.21)

with the jacobian matrix

l′(xk) =

0 0 0 0

0 0 8A12Knl

me1 (m2+A2)·(

1− L3nl

(L2nl+x2

k,(3))1.5

)

0

0 0 0 0

0 0 −8Knl

me2·(

1− L3nl

(L2nl+x2

k,(3))1.5

)

0

. (5.22)

5.4 Problem Formulation and Implementation

The NMPC is also implemented using Matlab/SIMULINK. The simulation model has

the same structure as the linear MPC for the linear two-body model. The problem

solver “quadprog” is not appropriate for this case, since this solver can only handle

LQPs. Therefore, the solver “fmincon” of Matlab is used which can handle nonlinear

problems with nonlinear constraints as well. In the following, the solution of the

nonlinear optimization problem using Matlab is outlined.

First, the optimization problem can be stated as

minxk,uk,ǫ1,ǫ2

J(xk, uk, ǫ1, ǫ2), (5.23)

where

J(xk, uk, ǫ1, ǫ2) =N∑

k=1

[q(xk,(2) − xk,(4)

)uk−1 + r u2

k−1

]+ wpǫ

21 + wvǫ

22, (5.24)

subject to

xk+1 = fk(xk, uk, vk, wk) (5.25)


|xk,(1) − xk,(3)| − ǫ1 ≤ cp, ∀k = 1, . . . , N (5.26)

|xk,(2) − xk,(4)| − ǫ2 ≤ cv, ∀k = 1, . . . , N (5.27)

|uk| ≤ cu, ∀k = 0, . . . , N − 1 (5.28)

0 ≤ ǫp ≤ cǫp , 0 ≤ ǫv ≤ cǫv , (5.29)

with the same denotation as in Chapter 4.

The problem is implemented using the following optimization vector

p =[

xT1 u0 xT

2 u1 . . . xTN uN−1 ǫ1 ǫ2

]T

, (5.30)

with 5N + 2 variables and it is straightforward to express the objective function by

means of p.

The solver “fmincon” can handle box constraints, linear and nonlinear inequality and

equality constraints. The slack variables (5.29) and the input constraints (5.28) are

considered as box constraints which yield 2N + 4 equations. The position (5.26) and

velocity (5.27) limits are considered as linear inequality constraints which yield 4N

equations. Additionally, the system dynamics (5.25) (or rather (5.18)) need to be

included as constraints, here, as nonlinear equation constraints with 4N equations.

In the following, Thor = 3s and ∆T = 0.1s, and thus the horizon length is 30 steps.

In summary, the optimization problem consists of 152 optimization variables and 304

equations for the constraints. This is a large problem for online solving of the problem

within a step time of ∆T = 0.1s. However, as stated above, the goal of this work is to

determine the qualitive behaviour of the system controlled by the NMPC in comparison

with a linear MPC and not to discuss real-time applicability.

5.5 Results

In the next simulations the WEC parameters from Table 4.1 are used, and a cable

stiffness of Knl = 840,000 N/m and a cable length to the reference of Lnl = 1.7 m for

the mooring force calculation is assumed. The system which is controlled is always

the nonlinear system (5.1)-(5.8) with convolution terms. Two different controllers are

proposed to control this system: The NMPC as described in Section 5.4 and a linear

MPC which is implemented as shown in Chapter 4. The NMPC includes the non-

linear model without convolution terms as prediction model, whereas the linear MPC

is based on the linear model with linear moring force constant Km. Both controllers

are supposed to control the nonlinear system with nonlinear mooring force and with

convolution terms.

The results with the NMPC and the linear MPC are denoted by NL and LIN,

respectively. The simulations and control parameters are shown in Table 5.1. In what

follows, the NMPC is validated and compared to the linear MPC with different values

5.5. RESULTS 55


∆t sample time for optimization 0.1 s




q Power weighting factor 1

r Input weighting factor 0

cp Relative position constraint 1 m

cv Relative velocity constraint 0.9 m/s


cǫp Slack variable position constraint 0.1 m

cǫv Slack variable velocity constraint 1 m/s

wp Weighting factor position slack 108

wv Weighting factor velocity slack 106

Tab. 5.1: NMPC/MPC parameter values.

of Km through simulation.

Fig. 5.2 shows the nonlinear and linear mooring forces for certain values of Km for

spar displacements between ±0.6 m. It can be seen that the nonlinear law is highly

nonlinear and a linearization would not work. Still, there are many possibilities to

choose Km. In the following, three different cases are regarded. One (Km = 50,000

N/m) which underestimates the nonlinear mooring, another (Km = 350,000 N/m)

which assumes a large mooring force and a third choice of Km so that the linear MPC

generates the maximal power for the current wave data.

In the beginning, a time-series from the NDBC Umpqua buoy 46229 which is de-

ployed off the coast of Oregon north of Reedsport [26] is used as the wave data. In

Fig. 5.3 the comparison between the NMPC and the linear MPC with Km = 100,000

N/m is shown. First, it can be noted that the proposed NMPC yields reasonable re-

sults, the constraints are satisfied and a large amount of power is generated. For this

choice of Km, the linear MPC yields its best result. Obvious differences are between 15

s and 30 s. However, these differences in the trajectories are not significant and have

no effect on the generated power, since the NMPC yields a average power of 9.45 kW

and the linear MPC 9.41 kW.

More significant differences can be seen in Fig. 5.4, where the results with the NMPC

are compared to two different cases. The linear MPC which assumes a high mooring

force yields very different trajectories and worse results than the two cases. Due to

the assumption of high mooring forces, especially for small displacements, the MPC

attempts to prevent the bouy’s large displacements, in order to satisfy the constraints.


−0.6 −0.4 −0.2 0 0.2 0.4 0.6−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

5

Position of Spar [m]

Moo

ring

For

ce [N

]

NLK=50000K=125000K=350000

Fig. 5.2: Nonlinear and linear mooring forces for different choices of Km.

Since the actual mooring force is smaller, the relative motion between spar and buoy

is reduced. Thus, the average generated power for this case is only 5.42 kW.

The MPC with a smaller mooring constant yields clearly better results. The average

generated power is 8.39 kW which is less than in the optimal case. However, this choice

has also certain advantages. As can be seen, the position and generator force constraints

are very well satisfied, where the trajectories remain well within the constraint bounds.

Also, the required reactive power is much less. Regardless of the values of Km the

simulation with NMPC produces the best power generation for this wave data.

In the following, the controller performance with different wave data, where the

generator force limit is also increased to 80,000 N, is tested. The wave data is from

the NDBC Offshore buoy 46050 with a little rougher wave climate. Even though

the dominant frequency is smaller, the wave amplitudes are significantly larger. The

optimal value of Km for this wave data is 150,000 N/m. Fig. 5.5 shows the comparison

between NMPC and the linear MPC for the new wave data, where some differences can

be seen. The NMPC trajectories run far away from their constraints, thus, the entire

power potential is not used. Hence, the average generated power for the NMPC is

only 17.98 kW, compared to 22.8 kW for the linear MPC. Even though the parameter

Km is perfectly chosen and there exist many values which do not yield such results,

it is remarkable that the results with the linear MPC are significantly better. It can

thus be concluded that the NMPC has probably certain weak points. One reason

might be the discretization method of only order of 2, whereas the linear discretization

is exact. Furthermore, in some cases the nonlinear solver might only find the local

optimum which yields non-optimal results. In future research, these problems need to

be given attention. Nevertheless, the results with NMPC satisfy the constraints and

5.5. RESULTS 57

0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

Time [s]

Pos

ition

[m]


NLK

m=100000

0 10 20 30 40 50−6

−4

−2

0

2

4

6

8

x 104

Time [s]

For

ce [N

]

Generator Force

NLK

m=100000

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


NLK

m=100000

0 10 20 30 40 50−40

−20

0

20

40

60

80

Time [s]

Pow

er [k

W]

Power Generation

NLK

m=100000

Fig. 5.3: Comparison between the simulation results with the NMPC and the linear

MPC with Km = 100,000 N/m. The wave data is from the NDBC Umpqua

buoy 46229.

are reasonable, even though they are not optimal.

It is interesting to see, for which values of Km the linear MPC yields the best

results and how sensitive this controller is with respect to changes of Km. In Fig. 5.6,

the generated power depending on the value of Km for both wave datas is shown.

As stated above, the result with the NMPC is the best for the milder wave climate.

However, with a mooring constant of about Km = 100,000 N/m the results with the

linear MPC are almost the same. Especially for higher mooring constants, the average

generated power decreases significantly. For the rougher wave climate, there is a large

range of values of Km, where the results with the linear MPC are even better than

with the LMPC. There is a considerable range of values for Km in both cases, where

changes of Km do not affect the generated power very much. These regions are near

the optimum value.

It can be observed that the optimum performance for the rougher wave climate is

achieved by a larger value of Km, namely, 150,000 N/m. This can be explained by

the spar position. The spar position for the milder wave climate is around ±0.35 m,

whereas it is ±0.42 m for the rougher climate, since the wave exerts more force to

the bodies. Due to the larger displacements, the mooring force increases significantly

because of the highly nonlinear mooring force as seen in Fig. 5.2. In this plot it can


0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

Time [s]

Pos

ition

[m]


NLK

m=50000

Km

=350000

0 10 20 30 40 50

−6

−4

−2

0

2

4

6

8

x 104

Time [s]

For

ce [N

]

Generator Force

NLK

m=50000

Km

=350000

0 10 20 30 40 50

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


NLK

m=50000

Km

=350000

0 10 20 30 40 50

−40

−20

0

20

40

60

80

Time [s]

Pow

er [k

W]

Power Generation

NLK

m=50000

Km

=350000


MPC with Km = 50,000 N/m and Km = 350,000 N/m. The wave data is

from the NDBC Umpqua buoy 46229.

be seen that with the optimal value of Km the linear mooring force is similar to a

best-fit line for the nonlinear mooring force. In this case, higher nonlinear mooring

forces occur, thus the best-fit line is naturally a line with a larger slope, which results

in a larger mooring constant Km.

5.5. RESULTS 59

0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

Time [s]

Pos

ition

[m]


NLK

m=150000

0 10 20 30 40 50

−5

0

5

10

x 104

Time [s]

For

ce [N

]

Generator Force

NLK

m=150000

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Vel

ocity

[m/s

]


NLK

m=150000

0 10 20 30 40 50−50

0

50

100

Time [s]

Pow

er [k

W]

Power Generation

NLK

m=150000


MPC with Km = 150,000 N/m. The wave data is from the NDBC Umpqua

buoy 46050.

0 0.5 1 1.5 2 2.5 3 3.5 40

2.5

5

7.5

10

12.5

15

Assumed Mooring Stiffness K [N/10 5]

Ave

rage

Gen

erat

ed P

ower

[kW

]

LIN(K

m)

NL

0 0.5 1 1.5 2 2.5 3 3.5 45

7.5

10

12.5

15

17.5

20

22.5

25

Assumed Mooring Stiffness K [N/10 5]

Ave

rage

Gen

erat

ed P

ower

[kW

]

LIN(K

m)

NL

Fig. 5.6: Generated Power depending on Km compared to the generated power with

the NMPC. On the left side with wave data from buoy 46050 and on the

right side with data from buoy 46229.

Chapter 6

Conclusions and Future Work

This thesis presented several different model predictive control approaches for con-

trolling wave energy converters. It exclusively discussed point absorbers, though the

general concepts could be applied to other types of WECs as well. In all types of control

approaches the goal was to maximize the generated power while satisfying generator

force as well as position and velocity constraints for the buoy. Also, different state

space models and a nonlinear model for the point absorber has been derived.

Two different MPC formulations were proposed to control a WEC modelled as a

one-body device. Both formulations have been compared with each other and to a

well-known optimal control law. It was successfully demonstrated that it is possible to

control the one-body WEC efficiently without using an optimal velocity trajectory for

the buoy velocity. Moreover, both formulations yielded almost the same results. Since

the calculation of an optimal velocity trajectory is not always possible, especially in

the two-body case, this was an important achievement in order to contol a two-body

device.

Thus, it could be shown that a two-body model can be controlled by a MPC scheme

as well. This fact has been demonstrated by simulation results with real wave data

from an offshore buoy. Additionally, the infeasibility problem of optimization prob-

lems was discussed and two different approaches to handle this problem were pro-

posed, namely, by relaxing the constraints and by introducing soft constraints. Both

approaches yielded good results, where the generated power with the proposed soft

constraint formulation was in general a little less than with the relaxed constraint for-

mulation. However, it could be shown that the soft constraint formulation handles

the infeasibility problem better and avoids its occurance from the start which is also

computationally cheaper.

Also, it has been shown that the proposed horizon and step time is a good choice to

deal with the applied wave data and that the MPC is quite robust against parameter

changes of the WEC.

Additionally, the influence of the mooring of the spar on the system performance

62 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

was under examination. The results showed that a better mooring of the spar leads in

general to a higher power generation. However, the danger that the constraints can not

be maintained increases with better mooring, since the relative motion between buoy

and spar increases. Thus, the mooring configuration has an essential influence on the

WEC performance. So the idea arises that variable mooring, which means adjusting

the mooring stiffness during operation, could be used to improve the system’s safety

and performance, if variable mooring is technically possible.

Many assumptions for the linear models has been made that are not always ap-

propriate. That is why a nonlinear MPC was also proposed in this work to deal with

mooring and radiation force nonlinearities. It has been demonstated that the NMPC

is able to keep the point absorber states within its limits. In order to evaluate the per-

formance, the results were compared to a linear MPC with varying mooring constant.

It was shown that the performance with the linear MPC is even better under certain

wave conditions and for certain values of the mooring constant. It has been concluded

that the performance of the NMPC could be improved by a better discretization and

a specialized optimization method.

Overall, it can be stated that MPC is a very promising and adequate approach to

control point absorber wave energy converters, since MPCs are able to maximize the

generated power while taking physical constraints into account, which is exactly the

challenge for controlling WECs.

However, much research still remains be done in the future to bring ocean wave

energy applications to commercialication and expanded implementation. Therefore,

this work suggests several areas for future research.

In all simulations, a perfect prediction of the incoming wave elevation is assumed

and hence a perfect prediction of the excitation forces. The implementation of predic-

tion algorithms such as a Kalman filter or auto-regression methods can enhance this

work and can identify the influence of prediction errors. Additionally, this will impact

optimal energy capture performance.

Furthermore, it was assumed that the entire state of the device is measured which

is not always possible. If not, an observer has to be implemented to identify all states,

since the MPC needs the full state information.

The proposed two-body model is still not validated against experimental data and

all parameters arise only from AQUA analysis. In case the model structure is not

appropriate to describe the point absorber well enough, more precice hydrodynamics

including frequency dependence or other non-linearities have to be introduced. The

proposed NMPC sets a framework for dealing with these nonlinearities. However,

some improvements regarding discretization and solver need to be considered in future

research.

Additionally, the field of real-time applicability needs to get center stage for all

63

types of proposed MPCs. The real-time implementation of the linear MPCs with two,

respectively, four states and 30 instants of time does not cause large difficulties regard-

ing online computation time. More work regarding solver and suitable implementation

platform needs to be done in the nonlinear case.

64 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

Appendix A

Explantion: Direct Power

Formulation is a Convex Problem

Optimization problems can have many different structures. The problem (4.41,4.42)

which needs to be solved in the Two-Body MPC algorithm is a QLP. For this class of

optimization problems there exist well known and suitable methods to solve it [27], like

the interior-point or active-set method. However, finding the global optimum requires

convex problems. Thus, the local optimum is the global optimum. The usual tracking

formulation of MPC is inherently convex. In the following, it is explained why the

power formulation as well yields a convex problem.

For a general QLP with objective function

J =1

2xTQx+ fTx, (A.1)

if Q is a positive definite matrix then the optimization problem is convex. Moreover,

if all eigenvalues of Q are positive, then Q is positive definite. For problem (4.41,4.42)

Q = qJuTS + R. Since q > 0 and R is a diagonal matrix with elements r ≥ 0, it

is sufficient to consider only JuTS. This matrix is an upper triangular matrix and

therefore, it is positive definite if all values on the diagonal are positive.

JuTS =

Bd(2) −Bd

(4) ⋆ . . . ⋆

Bd(2) −Bd

(4)... ⋆. . .

...

∅ Bd(2) −Bd

(4)

(A.2)

Since the optimization problem is convex if Bd(2) − Bd

(4) > 0, where (2), (4) denote

the second and forth component of the vector Bd, respectively. The continuous state

ii APPENDIX A. CONVEX PROBLEM

space vector B has always the form

B =

0

+

0

−

. (A.3)

After discretization,

Bd =∞∑

i=1

Ai−1Bhi

i!. (A.4)

Therefore, Bd has at least the form

Bd

⋆

+

⋆

−

, (A.5)

even if the discretization is not exact, since the first component Bh of the sum is the

dominant factor. It follows that Bd(2) − Bd

(4) is always positive and hence the opti-

mization problem is convex and can be solved, for example, using Matlab’s quadprog

solver which deals with convex quadratic problems.

Bibliography

[1] ANSYS Inc. ANSYS AQWA 13.0. 275 Technology Drive, Canonsburg, PA.

[2] C. K. Alexander and M. N. O. Sadiku. Fundamentals of Electric Circuits. Tata

McGraw-Hill Publishing Company Limited, 2007.

[3] E. Arnold. Numerische Methoden der Optimierung und Optimalen Steuerung.

Institute for Systemdynamics, University of Stuttgart, 2010. script.

[4] A. Babarit and A. Clement. Optimal latching control of a wave energy device in

regular and irregular waves. Applied Ocean Research, 28(2):77 – 91, 2006.

[5] T. Badgwell and S. Qin. Nonlinear predictive control - theory and practice. The

Institution of Electrical Engineers, 2001.

[6] T. Brekken, A. von Jouanne, and H. Y. Han. Ocean wave energy overview and

research at oregon state university. In Power Electronics and Machines in Wind

Applications, 2009. PEMWA 2009. IEEE, pages 1 –7, jun. 2009.

[7] T. K. Brekken. On model predictive control for a point absorber wave energy

converter. IEEE.

[8] J. Buijs, J. Ludlage, W. V. Brempt, and B. D. Moor. Quadratic programming

in model predictive control for large scale systems. Technical report, University

Leuven, Department of Electrical Engineering, 2002.

[9] E. Camacho and C. Bordons. Model Predictive Control. Springer, 2003.

[10] S. K. Chakrabarti. Handbook of Offshore Engineering. Elsevier, 2005.

[11] C. Chen. Linear System Theorie and Design. Oxford University Press, 1999.

[12] A. Clement et al. Wave energy in europe: current status and perspectives. Re-

newable and Sustainable Energy Reviews, 6(5):405 – 431, 2002.

[13] H. Eidsmoen. Simulation of a slack-moored heaving-buoy wave-energy converter

with phase control. PhD thesis, Norwegian University of Science and Technology,

Trondheim, Norway, 1996.

iv BIBLIOGRAPHY

[14] A. F. O. Falcao. Wave energy utilization: A review of the technologies. Renewable

and Sustainable Energy Reviews, 14(3):899 – 918, 2010.

[15] J. Falnes. Ocean Waves and Oscillating Systems, Linear Interaction Including

Wave-Energy Extraction. Cambridge University Press, 2002.

[16] R. Findeisen and F. Allgower. An introduction to nonlinear predictive control. In

21st Benelux Meeting on Systems and Control, Veldhoven, 2002.

[17] Global Greenhouse Warming. Wave Dragon. http://www.

global-greenhouse-warming.com/wave-dragon.html, sep. 2011.

[18] J. Hals, T. Bjarte-Larsson, and J. Falnes. Optimum reactive control and control by

latching of a wave-absorbing semisubmerged heaving sphere. ASME Conference

Proceedings, 2002(36142):415–423, 2002.

[19] E. Jefferys. Simulation of wave power devices. Applied Ocean Research, 6(1):31 –

39, 1984.

[20] N. Kazantzis, K. T. Chong, J. H. Park, and A. G. Parlos. Control-relevant dis-

cretization of nonlinear systems with time-delay using taylor-lie series. Journal of

Dynamic Systems, Measurement, and Control, 127(1):153–159, 2005.

[21] Y. Li, Y. Yu, M. Previsic, E. Nelson, and R. Thresher. Numerical and experimental

investigation of a foating point absorber wave energy converter under extreme wave

condition.

[22] D. Mayne, J. Rawlings, C. Rao, and P. Scokaert. Constrained model predictive

control: Stability and optimality. Automatica, 36(6):789–814, 2000.

[23] S. McArthur and T. Brekken. Ocean wave power data generation for grid integra-

tion studies. In Power and Energy Society General Meeting, 2010 IEEE, pages 1

–6, jul. 2010.

[24] C. Moler and C. V. Loan. Nineteen dubious ways to compute the exponential of

a matrix, twenty-five years later. Siam Review, 45:3–49, 2003.

[25] A. Muetze and J. Vining. Ocean wave energy conversion - a survey. In Industry

Applications Conference, 2006. 41st IAS Annual Meeting. Conference Record of

the 2006 IEEE, volume 3, pages 1410 –1417, oct. 2006.

[26] National Data Buoy Center. http://www.ndbc.noaa.gov/, sep. 2011.

[27] J. Nocedal and S. J. Wright. Numerical Optimization. Springer Verlag, 2006.

BIBLIOGRAPHY v

[28] Northwest National Marine Renewable Energy Center (NNMREC). The Limpet

OWC by Wavegen. http://nnmrec.oregonstate.edu/wavegen-owc, sep. 2011.

[29] Ocean Power Technologies (OPT). Global Resources, sep. 2011.

[30] Pelamis Wave Power. http://www.pelamiswave.com, sep. 2011.

[31] H. Polinder and M. Scuotto. Wave energy converters and their impact on power

systems. In Future Power Systems, 2005 International Conference on, pages 9 pp.

–9, nov. 2005.

[32] T. Pontes et al. The european wave energy resource. In Proceedings of the 3rd

European Wave Energy Conference, Patras, Greece, 1998.

[33] J. Prudell, M. Stoddard, E. Amon, T. Brekken, and A. von Jouanne. A permanent-

magnet tubular linear generator for ocean wave energy conversion. Industry Ap-

plications, IEEE Transactions on, 46(6):2392 –2400, nov.-dec. 2010.

[34] C. V. Rao, S. J. Wright, and J. B. Rawlings. Application of interior-point methods

to model predictive control. Journal of Optimization Theory and Applications,

99:723–757, 1998.

[35] M. Richter. Modell-praediktive Trajektorienplanung zur aktiven Seegangskom-

pensation. Student thesis, Institute for Systemdynamics, 2010.

[36] K. Ruehl. Time-domain modeling of heaving point absorber wave energy convert-

ers, including power take-off and mooring. Master’s thesis, Oregon State Univer-

sity, Mechanical Engineering, 2011.

[37] K. Ruehl, T. Brekken, B. Bosma, and R. Paasch. Large-scale ocean wave energy

plant modeling. In Innovative Technologies for an Efficient and Reliable Electricity

Supply (CITRES), 2010 IEEE Conference on, pages 379 –386, sep. 2010.

[38] E. Rusch. Catching a wave, powering an electrical grid? In Smithsonian Magazine,

2009.

[39] R. Taghipour, T. Perez, and T. Moan. Hybrid frequency–time domain models for

dynamic response analysis of marine structures. Ocean Engineering, 35(7):685 –

705, 2008.

[40] U.S. Energy Information Administraion. International Energy Outlook 2011.

http://www.eia.gov/, sep. 2011.

[41] J. Vining and A. Muetze. Economic factors and incentives for ocean wave energy

conversion. Industry Applications, IEEE Transactions on, 45(2):547 –554, mar.-

apr. 2009.

vi BIBLIOGRAPHY

[42] Voith Hydro Wavegen Limited. Wavegen brochure. http://www.wavegen.co.

uk/, sep. 2011.

[43] Wave Dragon. http://www.wavedragon.net, sep. 2011.

[44] A. Wills and W. Heath. EE03016 - interior-point methods for linear model pre-

dictive control. Technical report, University of Newcastle, Australia, 2003.

Diﬀerent Model Predictive Control Approaches for ...

Documents

Transcript of Diﬀerent Model Predictive Control Approaches for ...