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Control of the Wave Energy Converter
ISWEC in Simulation
Ricardo António Vaz Mendes Laranjeira
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Supervisors: Prof. Duarte Pedro Mata de Oliveira Valério
Prof. Pedro Jorge Borges Fontes Negrão Beirão
Examination Committee
Chairperson: Prof. João Rogério Caldas Pinto
Supervisor: Prof. Duarte Pedro Mata de Oliveira Valério
Member of the Committee: Dr. José Gaspar
November 2016
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ACKNOWLEDGMENTS
I would first like to thank my thesis advisor Professor Duarte Valério for his valuable and constructive
suggestions during development of this research work. His willingness to give his time so generously
has been very much appreciated.
I would also like to give my thanks to Giacomo Vissio for sharing his knowledge and insight related to
this problem.
Finally, I must express my very profound gratitude to my parents and to my friends for providing me with
unfailing support and continuous encouragement throughout my years of study.
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ABSTRACT
The control of a Wave Energy Converter is a key factor for its power extraction capabilities. This work
is focused on the control of the ISWEC (Inertial Sea Wave Energy Converter), a wave energy converter
that uses its gyroscopic properties to extract sea wave energy.
In this work, the model of the device is presented as well as the control technique currently used. Three
control strategies are developed with the objective of improving the amount of energy extracted by the
device.
Fractional Control, Internal Model Control and Feedback Linearization control are implemented resorting
to MATLAB and Simulink. Irregular waves are considered and the performance of these control
strategies is evaluated and compared to the currently in use Proportional Derivative controller.
The simulations performed in this work show that the use of these controllers leads to similar power
extraction. However, the use of Internal Model Control or Feedback Linearization control is
advantageous as these controllers have less parameters to tune once deployed in the real environment.
Keywords: Wave Energy Converter; Fractional Control; Internal Model Control, Feedback Linearization
Control
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RESUMO
O controlo de um conversor de energia de ondas é um factor crucial nas suas capacidades de extracção
de energia. Este trabalho foca-se no ISWEC (Inertial Sea Wave Energy Converter), um conversor de
energia que recorre às suas propriedades giroscópicas para extrair energia das ondas do mar.
Neste trabalho é apresentado o modelo do sistema assim como a estratégia de controlo actualmente
em uso. Três estratégias de controlo são desenvolvidas no âmbito deste trabalho com o objectivo de
aumentar a quantidade de energia extraída.
Fractional Control, Internal Model Control e Feedback Linearization Control são implementados
recorrendo ao MATLAB e ao Simulink. Ondas irregulares são consideradas e a performance destas
estratégias de controlo é avaliada e comparada com o controlador Proporcional Derivativo actual. As
simulações feitas no decorrer deste trabalho mostram que os controladores considerados apresentam
resultados semelhantes. No entanto o uso de Internal Model Control ou Feedback Linearization Control
é vantajoso no sentido que estes controladores possuem menos parâmetros a afinar quando aplicados
no ambiente real.
Palavras-chave: Conversor de Energia de Ondas; Fractional Control; Internal Model Control, Feedback
Linearization Control
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CONTENTS Acknowledgments ................................................................................................................................... iv
Abstract.....................................................................................................................................................v
Resumo ................................................................................................................................................... vi
Contents ................................................................................................................................................. vii
1. Introduction ....................................................................................................................................... 1
1.1 Motivations and Objectives ............................................................................................................ 1
1.2 Main Contributions ......................................................................................................................... 2
1.3 Structure of the Dissertation .......................................................................................................... 2
2. Wave Energy Converters ................................................................................................................. 4
2.1 Oscillating water column ................................................................................................................ 4
2.2 Oscillating bodies ........................................................................................................................... 5
2.3 Overtopping devices ...................................................................................................................... 6
2.4 Inertial Sea Wave Energy Converter ............................................................................................. 6
3. ISWEC Model and Existing Controller ............................................................................................. 8
3.1 Float dynamics ............................................................................................................................... 8
3.2 Gyroscope Dynamics ..................................................................................................................... 9
3.3 Nonlinear Model ........................................................................................................................... 10
3.4 Wave Data ................................................................................................................................... 11
3.4.1 Measured wave ..................................................................................................................... 11
3.4.2 Computer generated waves .................................................................................................. 12
3.5 Existing PD controller................................................................................................................... 13
3.6 Linearized State Space Model ..................................................................................................... 15
3.7 Proposed models ......................................................................................................................... 18
3.7.1 Output combination model .................................................................................................... 18
3.7.2 Variable State Matrix model .................................................................................................. 21
3.7.3 Evaluation of the proposed models: ...................................................................................... 21
4. Controller Design ............................................................................................................................ 23
4.1 Fractional PD controller ............................................................................................................... 23
4.2 Internal Model Control.................................................................................................................. 24
4.2.1 Implementation of IMC to the ISWEC ................................................................................... 26
4.2.2 Internal Model Control with the developed models ............................................................... 30
4.3 Feedback Linearization ................................................................................................................ 32
4.3.1 Implementation of Feedback Linearization to the ISWEC system ........................................ 33
5. Power Extraction ............................................................................................................................ 37
5.1 Fractional PD controller ............................................................................................................... 37
5.2 Reference Definition .................................................................................................................... 38
5.2.1 Internal Model Control ........................................................................................................... 40
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5.2.2 Feedback Linearization Controller ........................................................................................ 41
5.2 Power extraction Comparison ...................................................................................................... 42
6. Conclusion ...................................................................................................................................... 44
Bibliography ........................................................................................................................................... 45
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LIST OF FIGURES
Figure 1.1 - Approximate wave power levels given as kW/m of wave front [3] 1
Figure 2.1 - Various wave energy technologies [4] 4
Figure 2.2 - Schematic representation of (a) fixed-structure OWC (b) floating OWC [5] 5
Figure 2.3 - Schematic representation of (a) Wavebob WEC [6] and (b) AWS [4] 5
Figure 2.4 - Schematic representation of (a) Wave Dragon WEC [7] and (b) TAPCHAN WEC [3] 6
Figure 2.5 - Float body (concept) [9] 6
Figure 2.6 - Gyroscopic System [10] 7
Figure 3.1 - Model Layout 8
Figure 3.3 - Nonlinear model 11
Figure 3.4 - Measured wave data 12
Figure 3.5 - Bretchneider Spectrum and wave surface elevation for Hs=0.5 and T=5.5 13
Figure 3.6 - Nonlinear model with PTO control law 14
Figure 3.7 - State Space Model 16
Figure 3.8 - PTO angle output from the State Space model 17
Figure 3.9 - Hull pitch output from the State-Space model 17
Figure 3.10 - PTO mechanical power from the State-space model 17
Figure 3.11 - Membership Functions 19
Figure 3.12 - Output combination model 20
Figure 3.13 - Variable state matrix model 21
Figure 4.1 - Fractional PD Controller in Simulink 24
Figure 4.2 - Internal model control structure 25
Figure 4.3 - IMC structure for the ISWEC 26
Figure 4.4 - Pole-Zero map for � 28
Figure 4.5 - Closed loop step response of IMC controlled ISWEC 29
Figure 4.6 - Disturbance Force applied 30
Figure 4.7 - Inverse model for: (a) Output combination model (b) Variable state matrix model 30
Figure 4.8 - Unstable output for IMC with variable matrix model 31
Figure 4.9 - IMC combination structure 31
Figure 4.10 - Feedback Linearization 33
Figure 4.11 - Feedback linearization implementation 34
Figure 4.12 - Feedback Controller 34
Figure 4.13 - Step response of the controlled system 35
Figure 4.14 - System Response (overlaping Setpoint and Response curves) 35
Figure 5.1 - ISWEC system dynamics 39
Figure 5.2 - PTO speed and (a) �� (b) ����� 40
Figure 5.3 - Dimensionless mean absorbed power 42
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LIST OF TABLES
Table 2.1 Main system parameters 7
Table 3.1 Radiation force model parameters 9
Table 3.2 Significant Height and period of generated waves 13
Table 3.3 Optimized PD controller parameters 14
Table 3.4 Sets of linear models 20
Table 3.5 Performance parameters 21
Table 3.6 Model evaluation 22
Table 4.1 Poles and Zeros of � 28
Table 4.2 Controller performance for different filter parameters 29
Table 4.3 IMC controller Performance 32
Table 4.4 Parameters considered for variation 35
Table 4.5 Sensitivity analysis on the model parameters 36
Table 5.1 Extracted Power for Fractional PD controller 37
Table 5.2 Controller parameters optimized for measured wave 38
Table 5.3 Extracted power considering non optimized parameters 38
Table 5.4 Reference Gains considering IMC controller 40
Table 5.5 Extracted Power with IMC controller 41
Table 5.6 Extracted power (kW) by each controller for each wave 41
Table 5.7 Reference Gain considering FL controller 41
Table 5.8 Extracted Power with FL controller 41
Table 5.9 Mean absorbed Power 42
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NOTATION
The following notation is used throughout this work
Acronyms
CRONE Commande Robuste d’Ordre Non Entier
DOF Degrees Of Freedom
FL Feedback Linearization
FPD Fractional Proportional Derivative
IMC Internal Model Control
ISWEC Inertial Sea Wave Energy Converter
MD Maximum Deviation
MSE Mean Squared Error
PD Proportional Derivative
PTO Power Take-Off
RAO Response Amplitude Operator
VAF Variance Accounted for
WEC Wave Energy Converter
Symbols
� Added mass Damping parameter � Disturbance �̅ Feedback signal ℎ� Response function of the radiation forces �� Significant wave height ��� Equivalent moment of inertia of the float �� Moment of inertia of the float around � �� Flywheel moment of inertia around � � Flywheel moment of inertia around �� � Hydrostatic stiffness � Stiffness parameter ��� Feedback controller gain �� Reference gain ℒ Laplace transform � Linearized model i � Plant �̅ Plant model
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! IMC controller � Reference �" Reference in phase with �# �$ Reference in phase with �� � Total torque �% Pitch control torque �& Control PTO torque �' Mooring torque ��() Radiation Torque �# Wave excitation torque ��*�+ Torque on the gyroscope by the hull �,-.. Torque on the hull by the gyroscope / Process input 0 Feedback linearization control action � Degree of Membership of model 1 � process output
� Hull pitch angle � PTO angle �2 Linearization point 3 IMC filter parameter �� Flywheel speed 4 Angular Frequency 4' Modal frequency
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1
1. INTRODUCTION
1.1 Motivations and Objectives
With the oil crisis of 1973, the industrialized countries, with economies largely dependent in the use of
oil and its derivatives, were alerted to the necessity of diversification of their energy sources. Since then,
there has been an increase in funding for research and development for renewable energy technologies.
It is easy to think of renewable energy sources as solar, wind and hydraulic energy for having the largest
market exposure and implementation. However, the ocean offers wave energy, a highly energetic
source that could be a major source of renewable energy. The worldwide wave energy source is
estimated to be around 2TW which is in the magnitude of the world’s yearly power demand [1].
Wave energy is essentially a concentrated form of solar energy. The heating of air masses caused by
solar incidence creates wind which in turn causes waves due to drag on the water surface. These waves
carry both kinetic energy (water motion) and potential gravitational energy (due to the different heights
experienced in relation to the sea level) [2]. As the conversion from solar energy to wave energy takes
place, there is an increase in power density attributed to the time integration of the primary driving
source. Solar power density is of the order of 100W/m2 and can be eventually converted in wave power
densities of over 100kW per meter of wave length. Waves are a very efficient way to transport energy,
as ocean waves can travel thousands of kilometers with a small loss of energy [1].
The distribution of wave energy worldwide is represented in Figure 1.1:
Figure 1.1 - Approximate wave power levels given as kW/m of wave front [3]
The wave power density is very variable around the world and its highest values are detected in the
oceans between latitudes of ~30º and ~60º. In Europe, the coast of the U.K. and Ireland along with
Norway and Portugal present the highest power densities.
2
The potential of wave energy has been known for a long time and the first industrial applications appear
in the mid-20th century when Japanese navy commander Yoshio Masuda developed a navigation buoy
powered by an air turbine [4]. When compared to other renewable energy sources, such as wind energy,
wave energy experienced a slower development, explained mainly by the increased complexity and an
aggressive and destructive environment.
A device that harnesses wave energy and transforms it in to electric energy is called a Wave Energy
Converter (WEC). The control of the motion of a WEC is a key factor for its power extraction capabilities.
The goal of this work is to study such a device and to improve its performance by implementing a more
efficient control strategy. The device in consideration is the Inertial Sea Wave Energy Converter
(ISWEC), a device that uses the gyroscopic reactions provided from a spinning flywheel to extract sea
wave energy.
To achieve the goal of improving its performance, the existing ISWEC model and control strategy are
studied and alternatives are proposed. In this work two new models of the ISWEC are considered and
three control strategies are proposed.
First a Fractional Proportional Derivative (FPD) controller is considered, followed by an Internal Model
Controller (IMC) and a Feedback Linearization controller (FL). These controllers are implemented and
compared to the existing controller by the use of simulation using MATLAB.
1.2 Main Contributions
These are in summary the main contributions of this work:
• Models obtained by combination of linear models around different working points (Section 3.7);
• Implementation of a FPD control strategy (Section 4.1);
• Implementation of an IMC controller (Section 4.2);
• Implementation of a FL Controller (Section 4.3);
• Choice of a suitable reference to provide good power extraction (Section 5.2).
1.3 Structure of the Dissertation
This work is divided into six chapters in the following way:
Chapter 1: Introduction
This first chapter provides an introduction to the wave energy resource and presents the motivations for
the development of this work, and its main contributions.
3
Chapter 2: Wave Energy Converters
This chapter gives a brief overview of existing WEC technologies and presents the ISWEC system. In
this section the main system parameters are given.
Chapter 3: ISWEC model and Existing Controller
In this chapter, the mathematical model of the ISWEC is presented as well as the currently implemented
PD controller. The proposed ISWEC models are presented and their performance is evaluated.
Chapter 4: Controller design
This chapter describes the control strategies implemented in this work: FPD; IMC and FL. The structure
of the controllers is described and the IMC and FL controllers are tuned to track a given reference.
Chapter 5: Power Extraction
In this chapter, a reference for the IMC and FL is proposed and the power extraction capabilities of each
of the developed controllers is evaluated.
Chapter 6: Conclusion
In this chapter the main conclusions are drawn and some possible improvements on the system are
presented for future work.
4
2. WAVE ENERGY CONVERTERS
A WEC is a device that generates useful energy from the energy of the waves. Most concepts are based
on harvesting energy by placing buoyant bodies in the sea. As the waves force the bodies to oscillate,
this motion is converted into electricity by a system referred to as the Power Take-Off (PTO).
Nowadays there is a wide variety of wave energy conversion systems, which can be classified according
to working principle, to size, and deployment location (onshore, near-shore, offshore). Figure 2.1 shows
some technologies classified mostly on working principle:
Figure 2.1 - Various wave energy technologies [4]
A brief description of some existent WEC technologies is given.
2.1 Oscillating water column
The oscillating water column (OWC) device [5] is composed mainly of an air chamber and a turbine
(Figure 2.2). The air chamber is partially submerged and open bellow the waterline. The heave (vertical)
motion of the sea surface drives the air in the chamber through the turbine. Due to the oscillatory nature
of waves, the air is pushed or pulled, resulting in an alternate flow through the turbine. To use this
alternate flow, the Wells turbine is usually used because of its capacity to rotate continually in one
direction.
5
(a)
(b)
Figure 2.2 - Schematic representation of (a) fixed-structure OWC (b) floating OWC [5]
2.2 Oscillating bodies
The simplest oscillating body device consists of a heaving buoy reacting against a frame of reference
(the sea bottom or a reference structure). The relative motion between the heaving buoy and the
reference structure is used to run a PTO system. The reference body can be a bottom-fixed structure or
another oscillating body. Two examples of Oscillating body devices are represented in Figure 2.3: the
Wavebob WEC [6] and the Archimedes Wave Swing (AWS) [4].
(a)
(b)
Figure 2.3 - Schematic representation of (a) Wavebob WEC [6] and (b) AWS [4]
6
2.3 Overtopping devices
Overtopping devices work by capturing the water close to the wave crest and introducing it into a
reservoir at a higher level than the average level of the sea. The potential energy of the stored water is
then harnessed in a manner similar to a conventional hydroelectric system. Examples of overtopping
devices are the Wave Dragon [7], a floating WEC and the TAPCHAN [3], an on-shore fixed device.
These systems are represented in Figure 2.4:
(a) (b)
Figure 2.4 - Schematic representation of (a) Wave Dragon WEC [7] and (b) TAPCHAN WEC [3]
For a more comprehensive review of WEC technologies see [4], [5] and [9].
2.4 Inertial Sea Wave Energy Converter
The ISWEC can be classified as an Oscillating body WEC. It is a floating system moored to the seafloor
converting mechanical wave energy to electrical energy using a gyroscopic system. A full-scale
prototype has been successfully launched in 2015 and it has been deployed on the shore of Pantelleria
island in Italy. Its floating body is represented in Figure 2.5, and the gyroscopic system is represented
in Figure 2.6.
Figure 2.5 - Float body (concept) [9]
7
As a result of the interaction with the waves, the float rotates with a pitching motion�. Due to the
conservation of angular momentum on the flywheel, the combination of the flywheel speed φ� and the
pitch speed �� generates a torque along the � axis that can be used to generate electrical power using
a PTO system.
Figure 2.6 - Gyroscopic System [10]
The main advantage of the ISWEC over most of the other wave energy converters is that externally it is
composed only of a floating body without moving parts exposed to sea water or spray, thus achieving a
better reliability and lower maintenance costs. Additionally, in the presence of wave conditions too
dangerous for safe operation, the device can be switched off by stopping the flywheel, in which case the
device floats as a dead body (like a wave measurement buoy) [11].
The main system parameters considered are shown in Table 2.1:
Table 2.1 Main system parameters Symbol Parameter Value �� Flywheel speed 40 rad/s � Hydrostatic Stiffness 1.9×107 Nm/rad ��� Equivalent moment of inertia of the Float 1.2×107 kg·m2 � Flywheel moment of inertia around rotation axis 15000 kg·m2 �� Flywheel moment of inertia perpendicular to rotation axis 45000 kg·m2
The ISWEC device presented here is the subject of study of this dissertation. Its mathematical model
is described in further detail in the next chapter.
8
3. ISWEC MODEL AND EXISTING CONTROLLER
This chapter describes the ISWEC mathematical model and the PD controller currently implemented.
The ISWEC system involves two main phenomena: the float hydrodynamics and the gyroscope
mechanics. Essentiality the float hydrodynamics describes de pitch dynamics of the float when subject
to a wave force and the gyroscope dynamics describes how the pitch speed combined with the flywheel
speed generate a precession torque on the PTO axis. The system is modeled by combining these
phenomena as shown in Figure 3.1.
Figure 3.1 - Model Layout
3.1 Float dynamics
Float dynamics describes the interaction between the waves and the ISWEC body. According to [12]
and [10] the pitch motion of the float can be described as:
7�� + �9�: + ; ℎ�(=>2 − @)���@ + �� = � (3.1)
Equation (3.1) assumes a rigid structure with zero forward speed, negligible coupling with the other
degrees of freedom and small pitch oscillations. In the expression above � represents the pitch angle, �� the float moment of inertia, � the added mass due to the water dragged as the float moves, � the
hydrostatic stiffness, ℎ�the response function of the radiation forces, and � the excitation torques on
the float. The torques on the float can be decomposed as:
� = �# − �% − �' (3.2)
Where �# represents the pitch torque due to the incoming wave, �% pitch control torque and �'the
pitch torque due to the mooring forces acting on the float. Under the assumption that the effect of
mooring forces on the pitch motion of the float is small, �'is neglected.
The second term in equation (3.1), C ℎ�(=>2 − @)���@, represents the radiation torques experienced by
the float. The numerical computation of the convolution integral may be quite time-consuming and can
prove to be an inconvenient for simulation, control, and analysis. Since the convolution is a dynamic
linear operation it can be approximated by a state-space model [13].
Gyroscope Dynamics
Wave Force Float Dynamics
Force Exchange
Power Generation
9
��() = ; ℎ�(=>2 − @)���@ ≈ F� = �F + G����() = HF (3.3)
Where the state vectorF, input vector/ and output vector� are defined as:
F = I�J0"�J0$�J0K�J0LM , / = �, � = ��()
and the state matrix�, input matrixG, Output matrixH and Feed-forward matrixN are defined as:
� = OP"" P"$ P"K P"L1 0 0 00 1 0 00 0 1 0 S , G = I1000M
H = T" $ K LU, N = 0
With the model parameters presented in Table 3.1.
Table 3.1 Radiation force model parameters
Parameter Value P"" -4.52 P"$ -10.53 P"K -10.78 P"L -8.03 " 5.14 x 106 $ 9.29 x 106 K 7.01 x 106 L 0
3.2 Gyroscope Dynamics
Gyroscope dynamics describes the relationship between the PTO angle (�) and the torque on the PTO.
From the time derivation of the flywheel angular momentum, the torque around the PTO (�V) is described
by equation (3.4). [10]
�V = ��ε: + 7�� − �9��$ sin ε cos ε − �φ� δ� cos ε (3.4)
10
The pitching torque, �% is obtained by projecting the torques created by the gyroscope along the pitch
direction [10]:
�% = 7� sin$ ε + �� cos$ ε9�: + �φ: sin � + ��� �� cos � + 27� − ��9���� sin ε cos ε (3.5)
In the expressions above � represents the moment of inertia of the flywheel around its spinning axis and �� the moment of inertia around the axis perpendicular to the spinning motion.
Equation (3.4) is simplified by neglecting its second term as it is two orders of magnitude smaller than
the others.
�V = ��ε: − �φ� δ� cos ε (3.6)
Equation (3.5) is simplified by considering small pitch accelerations, constant flywheel speed and
neglecting the higher order term.
�_ = ��� �� cos � (3.7)
3.3 Nonlinear Model
Taking equation (3.1), replacing �% with (3.7) and the convolution integral with the state space
approximation (3.3) and considering�`! = �J + �∞, the equivalent moment of inertia of the float a
nonlinear model is obtained:
�# = ����: + ��() + �� + �φ� ��cosε (3.8)
�V = ��ε: − �φ� δ� cos ε (3.9)
Equations (3.8) and (3.9) are modeled in Simulink and presented in Figure 3.2:
11
Figure 3.2 - Nonlinear model
This model has been verified experimentally and is assumed to be an acceptable model of the system
[9]. As such this model is considered as the process model of the ISWEC system and is used as
comparison for the design of the proposed models.
3.4 Wave Data
To evaluate the performance of these models, the power extracted by the ISWEC system is analyzed
under different wave conditions.
Real on-site wave data is used to evaluate the system as well as computer generated irregular waves
due to limited access to wave data and to ensure that the system experiences different sea states.
3.4.1 Measured wave
The measured wave is obtained from the coast of Pantelleria, Italy where the ISWEC prototype
is installed. The measured wave height data is represented in Figure 3.3 as well as the corresponding
wave excitation force.
� �� �:
� �� �:
�V
��� ���
�# Hull Dynamics
Force Exchange
Gyroscope Dynamics
Radiation Force
12
Figure 3.3 - Measured wave data
3.4.2 Computer generated waves
Irregular sea waves can be described by two parameters: the significant wave height�b, defined
as the mean wave height of the highest third of waves, and the modal (most likely) frequency4'. The computer generated waves are obtained following linear wave theory using the Bretschneider
Spectrum. The formula for the Bretschneider one-sided ocean wave spectrum is [14]:
d(4) = 5164'L4g �b$`hgijk /Lik (3.10)
where 4 is the angular frequency. From the spectrum a finite number of sinusoidal waves are created,
each with its own amplitude and frequency characterized by the spectrum. Each individual harmonic is
assigned with a random phase.
Following the linear superposition principle the total excitation force acting on the system is computed
as the sum of the excitation force of each harmonic wave. Each excitation force is computed referring
to the response amplitude operator (RAO) of the ISWEC system.
Figure 3.4 shows an example of a Bretchneider spectrum and the corresponding wave elevation profile
generated for�b=0.5m and T=5.5 s
Wav
e H
eigh
t (m
) W
ave
Tor
que
(Nm
)
13
Figure 3.4 - Bretchneider Spectrum and wave surface elevation for Hs=0.5 and T=5.5
Eight waves are created with the following significant heights and periods chosen as representative of
several different sea states:
Table 3.2 Significant Height and period of generated waves Wave 1 2 3 4 5 6 7 8 ��(m) 0.5 0.5 1.5 1.5 1.5 2.5 2.5 3.25 �(�) 5.5 6.5 5.5 6.5 7.5 6.5 7.5 8.0
These generated waves are used to evaluate the performance of the various models and controllers
developed in this work.
3.5 Existing PD controller
The idea behind this controller is to control the PTO to behave as a spring-damper system with stiffness � and damping. This means applying the following control law:
�V = −�� − �� (3.11)
This can be modeled in Simulink using a feedback loop as shown in Figure 3.5:
14
Figure 3.5 - Nonlinear model with PTO control law
The design of the PD controller is made by searching for the pair of parameters (, �) that maximizes
the average power absorption for a given wave. Simulations of 600 seconds are carried, considering
the waves described in section 3.4.2. This optimization is done resorting to the fminsearch MATLAB
function. fminsearch uses the Nelder-Mead algorithm to find the minimum of a multivariable function. It
is a simplex-based direct search method as it uses only function values, without any derivative
information. The algorithm is described in detail in [15].
To evaluate the power extracted by the ISWEC, no losses are taken into consideration, and so the
negative mechanical power on the PTO is considered:
m = −�&�� (3.12)
Note that positive values of mechanical power represent an energy transfer from the ISWEC to the sea
so the negative power is considered.
Using this optimization the, � parameters found for each wave are shown in Table 3.3:
Table 3.3 Optimized PD controller parameters
Wave ID k / 10LNm c / 10LNms Power / Kw
1 11.17 3.83 5.99 2 10.49 6.42 5.74 3 6.61 7.14 47.80 4 6.91 7.51 47.90 5 6.49 5.95 52.93 6 5.97 10.75 115.25 7 5.82 9.84 121.66 8 5.30 11.24 150.21
After optimizing the control law through numerical simulation, the PD parameters are stored for each
sea state condition. These control parameters are then changed in the real environment according to
the sea state forecast. Because this change is expected to happen sporadically, and the system
stabilizes easily, there is no consideration taken in to the stability of this adaptive controller [16].
ISWEC Model
� �� �&
�
�#
Controller
15
3.6 Linearized State Space Model
In order to apply conventional control methodologies to the system, the nonlinear model has to be
linearized. The nonlinearity present in the model is the ��� cos � term that appears in both equations
(3.8) and (3.9). In addition that term is multiplied by �� or �� representing another nonlinearity. This term
describes the torque exchange between the hull and the PTO.
The linear approximation of a function J(F) at point F2 is given by its 1st order Taylor series expansion:
�(F) = J(F2) + Jn(F2)(F − F2) (3.13)
Considering the nonlinear term ��� cos � in the model and linearizing around �2 the following expression
is obtained:
�(�) = ��� cos �2 − ��� sin �2 (� − �2) (3.14)
considering �2 = 0the expression above is reduced to:
�(�) = ��� cos0 − ��� sin 0 (� − 0)�(�) = ���
��� cos(�) ≅ ��� (3.15)
Applying the linearization described by equation (3.15) to the nonlinear model equations (3.8) and (3.9)
the following set of equations modeling the open loop behavior of the ISWEC system are obtained:
�# = ����: + ��() + �� + �φ� �� (3.16)
�V = ��ε: − �φ� δ� (3.17)
Introducing the control law defined by equation (3.11) the closed loop model of the ISWEC system is
obtained:
�# = ����: + ��() + �� + �φ� �� (3.18)
��ε: + �� + �� = �φ� δ� (3.19)
These equations are reordered to the following form:
�: = − ������ �� − ���� � − ��()��� + �#��� (3.20)
ε: = − �� �� − ��� � + ��� ���� (3.21)
16
The linearized equations (3.20) and (3.21) can be represented by a state space model defined as:
F� = �F + G/
� = HF + N/ (3.22)
Where the state vectorF, the input vector/ and the output vector� are defined as:
F =pqqqqqqr �������J0"�J0$�J0K�J0Lst
tttttu, / = �# , � = ��
The state matrix�, the input matrixG, the Output matrix H and the Feed-forward matrixN are defined
as:
� =
pqqqqqqqqqr − �� − ��� ����� 0 0 0 0 01 0 0 0 0 0 0 0− ������ 0 0 −�#��� − "��� − $��� − K��� − L���0 0 1 0 0 0 0 00 0 1 0 P"" P"$ P"K P"L0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 0 stt
tttttttu, G =
pqqqqqqqr 001���00000 st
ttttttu
H = T0 1 0 0 0 0 0 0U, N = 0
The state-space model can be represented by the block diagram bellow:
Figure 3.6 - State Space Model
The PTO angle�, obtained with the state-space model can be seen in Figure 3.7 and compared with
the output from the non-linear model:
B C
A
�# F� F �
17
Figure 3.7 - PTO angle output from the State Space model
As the state space model is linearized around�2 = 0, it is expected that it presents the greater deviation
from the non-linear model for values of � distant to 0. When comparing the hull pitch angle �and the
PTO power P, obtained using the state space model with the reference values obtained from the non-
linear model, it is easy to see that the state-space model is a good approximation of the system
considering small deviations. These comparisons are represented below in Figure 3.8 and Figure 3.9:
Figure 3.8 - Hull pitch output from the State-Space model
Figure 3.9 - PTO mechanical power from the State-space model
18
Figure 3.7 to Figure 3.9 show that the state-space model provides a good approximation of the non-
linear model for small deviations from the point of operation (� = 0). For greater deviations from � = 0
the output of the model starts to deviate from the non-linear model.
3.7 Proposed models
The deviation between the non-linear model and state space model is expected as the state-space
model does not take into account the system’s non-linearity in the coupling of the float and gyroscope
dynamics. To provide a more accurate model for greater deviation from the operation point two models
are proposed in this work: First, a linear model combination where the output of this model is the result
of the linear combination of the output of several linear models (linearized around different points).
Second, a variable state-space model using a time variant state-matrix�.
3.7.1 Output combination model
This model is the result of the combination of v multiple linear models in the following way:
�(F) = ∑� � (F)∑� ,for1 = 1,2,3…v (3.23)
� represents the 1>,linear model, obtained by linearizing the nonlinear model described by equations
(3.8) and (3.9) around the operation point�1 using the linearization described by equation (3.14) the
following set of equations is obtained:
�# = ����: + ��() + �� + ��� �� cos �2 − ��� ��sin �2 (� − �2) (3.24)
�V = ��ε: − ��� �� cos �2 + ��� ��sin �2 (� − �2) (3.25)
The nonlinear terms still present in these expressions are ignored, because they provide a small
contribution. The equations for the linear model considered are the following:
�# = ����: + ��() + �� + ��� �� cos �2 (3.26)
�V = ��ε: − ��� �� cos �2 (3.27)
The weights � are obtained using trapezoidal membership functions [17] for 1 = 1 and 1 = v and
triangular membership function for every other value of 1:
19
�" = | 1, � ≤ �"�$ − ��$ − �" , �" ≤ � ≤ �$0, � ≥ �$
(3.28)
� =����� 0, � < � h"� − � h"� − � h" , � h" ≤ � ≤ � � �" − �� �" − � , � ≤ � ≤ � �"0, � �" ≤ �
(3.29)
�� = | 0, � ≤ ��h"� − ��h"�� − ��h" , ��h" ≤ � ≤ ��1, � ≥ �� (3.30)
The membership functions described above in equations (3.28), (3.29) and (3.30) can be visualized
below in Figure 3.10:
�" � ��
Figure 3.10 - Membership Functions
This model is presented in Figure 3.11 for three linear models linearized around three working points:
�" �$ � � h" � �" �� ��h"
20
Figure 3.11 - Output combination model
The results obtained using this scheme will change depending on the number of linear models used.
For this work five models obtained with different sets of linear models are used. To differentiate them,
the models will be named model (a) to (e).
Table 3.4 Sets of linear models
Model Points of linearization, �2
(a) [0,90] (b) [0,45,90] (c) [0,15,30,45,60,75,90] (d) [0,10,20,30,40,50] (e) [0,5,10,15,20,25,35,45]
The performance of these models is evaluated in section 3.7.3.
�"
�
��
�"n
� n
��n
�"n
� n
��n
Membership Function
Membership Function
Membership Function
Σ Σ
�′
�"
�
��
�#
21
3.7.2 Variable State Matrix model
This model is created by replacing the state matrix � with a variable matrix�′(�):
F� = �n(�)F + G/ (3.31)
Figure 3.12 - Variable state matrix model
The variable state matrix �′(�)is obtained combining v matrices in a similar way as with the previous
model:
�′(�) = ∑� � ∑� ,for1 = 1,2,3…v (3.32)
where � represents the state matrix obtained from the model linearized around � and � the weight
of matrix � obtained using the same membership functions considered for the output combination
model.
Like the previous model, the results vary depending on the point of linearization chosen. The same
naming structure is chosen and the linearization points are the same.
3.7.3 Evaluation of the proposed models:
Simulations of 900 seconds (15 min) are carried out, using both Simulink implementations of the models
proposed above. To carry out these simulations the measured wave described in section 3.4.1 is used
as input. The performance parameters considered are shown in Table 3.5.
Table 3.5 Performance parameters
Mean squared error Variance accounted for Maximum deviation
��� = � (� − �� )$�� �" ��� = �1 − �$(� − ��)�$(�) � �� = max|� − �� |
B C
A
�# F� F �
22
where � is the output of the non-linear model and �� is the output of the evaluated model. Table 3.6
shows the results for PTO angle and pitch angle.
Table 3.6 Model evaluation
Output Model Performance Parameter MSE VAF MD
PTO angle, � State space 0.8947 0.9954 6.4381
Output combination (a) 0,8962 0,9954 6,4408
(b) 1,1727 0,9940 6,9360
(c) 0,4031 0,9979 5,2474
(d) 0,4077 0,9979 4,2679
(e) 0,3922 0,9980 3,9460
State Matrix combination (a) 2,7490 0,9859 6,3653
(b) 0,4244 0,9978 1,8409
(c) 0,0049 1,0000 0,2017
(d) 0,0009 1,0000 0,0896
(e) 0,0002 1,0000 0,0649
Pitch angle, � State space 0.0713 0.9979 1.5591
Output combination (a) 0,0713 0,9979 1,5581
(b) 0,3067 0,9910 3,1662
(c) 0,0904 0,9973 2,6688
(d) 0,0927 0,9973 2,6593
(e) 0,0874 0,9974 2,2834
State Matrix combination (a) 0,0405 0,9988 0,8944
(b) 0,0215 0,9994 0,9044
(c) 0,0438 0,9987 1,3089
(d) 0,0493 0,9986 1,3518
(e) 0,0530 0,9984 1,3790
Table 3.6 shows that the Output combination model presents an improvement over the state space
model for modeling the PTO angle, but it shows a slightly worse model for the pitch angle. Meanwhile,
the State Matrix combination model shows a much greater increase in performance when modeling the
PTO angle while also showing an improvement modeling the pitch angle. It can be concluded that this
model is preferable, even because it is not significantly more complex.
23
4. CONTROLLER DESIGN
In this chapter three controllers are proposed as alternatives to the existing PD controller currently in
use with the ISWEC system:
• Fractional controller
• Internal Model Controller
• Feedback Linearization Controller
The design of the FPD controller consists of a parameter optimization problem to maximize the extracted
power, like the PD controller. In contrast, to design the IMC and FL controllers a two-step approach is
followed:
1. The controllers are tuned to follow an arbitrary reference;
2. An adequate reference is defined to increase power extraction.
In this chapter the FPD controller is presented, and the IMC and FL are tuned and evaluated on their
ability to track a reference. According to [18], wave energy is captured most efficiently when the WEC
speed is in resonance with the wave excitation force, and so it is natural to consider such a reference:
This reference is chosen with no consideration for extracted power and the gain is chosen to ensure
only that its amplitude stays within reasonable PTO speed values. This reference will be used in this
chapter for the tuning, and evaluation of the IMC and FL controllers.
The power extraction analysis in presented in Chapter 5.
4.1 Fractional PD controller
The idea behind the Fractional PD controller is to change the order of the damping of the PD controller.
This means implementing the following control law:
���b = − ����=� − �� (4.2)
where the term )�&)>� represents the fractional derivative of order � of �.
The Simulink implementation of this controller is similar to that of the integer PD controller. Figure 4.1
shows the fractional PD controller implementation:
�(=) = �180 2 × 10hg�#(=) (4.1)
24
Figure 4.1 - Fractional PD Controller in Simulink
In the Simulink implementation )�¢£)>�¢£ �� is used to preserve the similarities with the PD controller. Note
that )�¢£)>�¢£ �� = )�)>� �.
The implementation of the fractional derivative is done using a CRONE (Commande Robuste d’Ordre
Non-Entier) 7th order approximation. The expected wave frequency range is around 1 rad/s. A
conservative frequency range is considered for the poles and zeros of the CRONE approximation of
[0.001, 1000] rad/s. More information on fractional control and the CRONE approximation can be found
in [19].
The choice of parameters will be done in chapter 5 using the same optimization technique used in
section 3.5 to find the PD controller parameters.
4.2 Internal Model Control
The internal model control methodology ([16], [20]) uses the control structure represented in Figure 4.2.
In that control loop � represents the model of the process to control (�) and ! represents the IMC
controller. � represents the reference, / the control action, � the controlled variable, � the output
disturbance, and � is the feedback signal that represents all that is unknown about the process �.
ISWEC Model �# �&
�� �
�
��h"�=�h"
Fractional Controller
25
Figure 4.2 - Internal model control structure
This control structure is referred to as Internal Model Control or IMC, because the process model � is
explicitly an internal part of the controller.
The relationship between the controlled variable � and the reference � and output disturbance � is [16]:
� = �!1 + !(� − �) � + 1 − �!1 + !(� − �) � (4.3)
In the absence of plant/model mismatch (� = �), this equation becomes:
� = �!� + (1 −�!)� (4.4)
which leads to the following relationship for a controller defined as! = �−1:
� = �
This means that the in the absence of modeling uncertainties the IMC strategy ensures perfect reference
tracking and total disturbance rejection.
The IMC design procedure is a two-step approach that, although sub-optimal in a general sense,
provides a reasonable tradeoff between performance and robustness. The first step will insure that ! is
stable and causal and the second step will require ! to be proper.
Step 1: Factor the model � into:
� = ���h (4.5)
where �� contains all nonminimum phase elements in the plant model, while �h is minimum phase and
invertible. An IMC controller defined as
is stable and causal.
! = �hh" (4.6)
� !
�
� / � �
�
26
Step 2: Augment ! with a filter J to make sure that the final IMC controller ! = !J is, in addiction to
stable and causal, proper.
Some common filter choices are:
J(�) = 1(3� + 1)� (4.7)
J(�) = v3� + 1(3� + 1)� (4.8)
where 3 is an adjustable parameter that determines the speed-of-response and the filter order v is
selected large enough to make ! proper.
4.2.1 Implementation of IMC to the ISWEC
To implement IMC to the ISWEC the control structure represented in Figure 4.2 is used and � becomes
the non-linear ISWEC model, � the state-space ISWEC model, and ! is the IMC controller obtained
following the steps described above. In addition, the controlled variable is�� , the PTO angular velocity,
the control action is��, the PTO torque and the disturbance �, is replaced by �#, the wave excitation
force.
Figure 4.3 - IMC structure for the ISWEC
To implement the control scheme described above it is necessary to obtain the open loop model of the
ISWEC system. This is described by equations (3.8) and (3.9). Considering�# as an unmeasurable
disturbance the following set of model equations is obtained:
����: = −��() − �� − �φ� ��cosε (4.9)
�V = ��ε: − �φ� δ� cos ε (4.10)
ISWEC IMC controller
Linear model
� �& ��
�
�#
�� ̅
27
Like in section 3.6 these equations can be described by a state space model. The following State-
Space model is obtained:
F� = �F + G/ � = HF + N/ (4.11)
Where,
� =
pqqqqqqqqqr 0 0 ����� 0 0 0 0 01 0 0 0 0 0 0 0− ������ 0 0 − �#��� − "��� − $��� − K��� − L���0 0 1 0 0 0 0 00 0 1 0 P"" P"$ P"K P"L0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 0 stt
tttttttu
, G =pqqqqqqqr 1��0000000 st
ttttttu
H = T1 0 0 0 0 0 0 0U, N = 0
And the state, input and output vectors are:
F =pqqqqqqr �������J0"�J0$�J0K�J0Lst
tttttu
, / = �& , � = ��
This model describes the open-loop behavior of the PTO velocity of the ISWEC system subject to control
action ���b. As part of the IMC design procedure an inverse model of the system is required. To achieve this goal,
the state-space model is converted to a transfer function representation. Using MATLAB, a 7th order
transfer function is obtained:
� = 0.22�¤ + 1.0�g + 2.8�L + 4.2�K + 5.6�$ + 3.8� + 2.8�¦ + 4.5�¤ + 13.2 �g + 21.7 �L + 32.3 �K + 24.3 �$ + 18.1�" + 5.8 ∙ 10h"g ∙ 10hL
(4.12)
28
Table 4.1 presents the poles and zeros of transfer function p and Figure 4.4 shows the corresponding
Pole-Zero map:
Figure 4.4 - Pole-Zero map for �
Table 4.1 Poles and Zeros of �
Poles Zeros 0 —
-0.084±1.456i -0.051±1.193i -0.508±1.148i -0.537±1.162i -1.667±1.616i -1.671±1.627i
The model � has no zeros or poles in the right-half s plane, and so it is minimum phase, and following
the IMC design procedure previously described, an intermediate controller ! is defined:
! = �h" (4.13)
And to obtain the IMC controller
The filter J(s) has to be defined. This filter is chosen by simulation and evaluating the performance of
the controller using the same performance parameters defined in Table 3.5. Both types of filter described
by (4.7) and (4.8) are considered and several filter parameters 3 are tested.
Using the measured wave as disturbance, the performance obtained for each considered filter is
presented in Table 4.2:
! = J(�) ∙ !(�) (4.14)
29
Table 4.2 Controller performance for different filter parameters
Filter 3 MSE VAF MD
a) 1(3� + 1)$
10 8.47 × 10K -0.0049 278.0826 1 5.33 × 10L -0.0116 996.8283
0.1 107.6043 0.5912 32.2889 0.01 1.0941 0.9887 3.2884
0.001 0.0114 0.9999 0.3432 0.0001 0.0003 1.0000 0.0666
b) 23� + 1(3� + 1)$
10 4.08 × 10K -0.0221 286.9043 1 1.12 × 10L 0.0303 983.1729
0.1 0.5087 0.9942 2.3203 0.01 0.0001 1.0000 0.0636
0.001 0.0001 1.0000 0.0446 0.0001 0.0001 1.0000 0.0435
From the results of Table 4.2, it is concluded that the best candidates are filter a) with 3 = 0.0001 and filter b) with 3 = 0.01, as decreasing this parameter does not contribute significantly to
an improvement on the performance of the controller. Filter b) with 3 = 0.01 is chosen because it is less
computationally expensive that filter a) with 3 = 0.0001.
The IMC controller is defined as:
! = (0.02� + 1) ∙ 10L(0.01� + 1)$ ∙ �¦ + 4.5�¤ + 13.2 �g + 21.7 �L + 32.3 �K + 24.3 �$ + 18.1�" + 5.8 ∙ 10h"g0.22�¤ + 1.0�g + 2.8�L + 4.2�K + 5.6�$ + 3.8� + 2.8
(4.15)
Figure 4.5 shows the closed loop response of the system for a unit step reference while considering a �# disturbance, represented in Figure 4.6, as a step with amplitude 107Nm at time 0.2s (an order of
magnitude greater than the wave torques considered).
Figure 4.5 - Closed loop step response of IMC controlled ISWEC
PT
O S
peed
, �� (rad/
s)
30
Figure 4.6 - Disturbance Force applied
Figure 4.5 shows that the system has good disturbance rejection, meaning that the developed controller
can force the PTO speed to follow the desired reference whenever there is a load on the system caused
by the incoming wave forces. This enables the design of a reference signal that would maximize power
extraction.
4.2.2 Internal Model Control with the developed models
An attempt to implement the IMC strategy using the models developed in section 3.7 is presented. To
design the IMC controller, following the procedure described in the beginning of the chapter, the inverse
model of the process is necessary.
For the Output combination model the inverse was defined as the linear combination of the inverse of
each model as shown in Figure 4.7.a. For the Variable state matrix model, the inverse was define using
a variable state space model shown in Figure 4.7.b
(a) (b)
Figure 4.7 - Inverse model for: (a) Output combination model (b) Variable state matrix model
In Figure 4.7.a L«h" represents the inverse model of each linear model; w« represents the weight of each
model calculated using the same membership functions shown in Figure 3.10. In Figure 4.7.b, the
variable state space model defined by the matrices (A(ε), B(ε), C(ε), D(ε)) is obtained by combining
the inverse model of each linear model with a filter to ensure properness as part of the IMC control
�"h"J"
�$h"J$
�Kh"JK
�"
�$
�K
Σ
Σ
�(�)
G(�) H(�)
N(�)
Dis
turb
ance
, � # (N
m)
31
methodology. The matrix combination method is the same used in Section 3.7.2 to define the Variable
Matrix Model.
Additionally to the inverse models described above, it was attempted to define the inverse of each model
as the inverse of the model linearized aroundε = 0º.
These systems were simulated for linearization point shown in Table 3.4 with the same inputs as the
IMC described before, and led to unstable outputs. Other linearization points were tested but no stable
controller was found. Figure 4.8 shows and example of a simulation made for a variable matrix model
linearized around ε = [0 30 60 90 120 150 180].
Figure 4.8 - Unstable output for IMC with variable matrix model
The proposed alternative is to implement IMC using each linearized model and its corresponding inverse
model augmented with a filter, and use the same linear combination method on the control action of
each IMC:
/ = ∑� / ∑� ,for1 = 1,2,3…v
Where u« is the control action given by the IMC controller designed with linear model � and the
corresponding inverse�1−1. Figure 4.9 shows the IMC combination structure:
Figure 4.9 - IMC combination structure
The performance of this controller is evaluated using the same performance parameters as before.
Simulations are run with the same inputs and the linearization points of Table 3.4 and compared with
the IMC controller developed using the linear state-space model:
ISWEC � h"J �
∑� / ∑� / �
�̅
/ �
32
Table 4.3 IMC controller Performance
Controller MSE(× 10K) VAF MD IMC combination
Points of linearization, �2
a) [0,90] 0.6805 1.0000 0.0626 b) [0,45,90] 0.6913 1.0000 0.0620 c) [0,15,30,45,60,75,90] 0.6951 1.0000 0.0618 d) [0,10,20,30,40,50] 0.6953 1.0000 0.0618 e) [0,5,10,15,20,25,35,45] 0.6955 1.0000 0.0618 f) [0 30 60 90 120 150 180] 0.6941 1.0000 0.0618
State space model IMC 0.6724 1.0000 0.0636
The controllers that show the best results are controller a) which has the lowest MSE and controller f)
which shows the smallest MSE when compared with the other models with the same MD. These
controllers will be considered when evaluating the power extraction capabilities of the ISWEC. However
when comparing the IMC combination with the State-space model IMC there is a small decrease in the
maximum deviation observed. A deviation of this magnitude (10h$) can be ignored as the magnitude of
the output is several orders of magnitude greater (10"). It can be concluded that there is no significant
improvement with this control structure. It is expected that this structure should not present any
significant variation over the IMC with state-space model in regard to power extraction.
4.3 Feedback Linearization
Feedback Linearization is a nonlinear control strategy. In its simplest form it amounts to use feedback
to cancel the nonlinearities of a nonlinear system so that the closed-loop dynamics become linear [16],
[21]. This allows for the use of linear control techniques. Considering a class of nonlinear systems of the
form:
F� = J(F) + �(F)/ (4.16)
� = ℎ(F) (4.17)
The objective is to find a control law
/ = �(F) + ³(F)0 (4.18)
Such that the closed-loop system
F� = J(F) + �(F)�(F) + �(F)³(F)0 (4.19)
� = ℎ(F) (4.20)
shown in Figure 4.10, has the behavior of a linear system between the new input 0 and the system
output�.
33
Figure 4.10 - Feedback Linearization
It is assumed that the entire state vector F is available for feedback.
4.3.1 Implementation of Feedback Linearization to the ISWEC system
The ISWEC nonlinear model is described by:
�� = �`!�: + ��P� + �� + �φ� ��cosε (4.21)
�ε = ��ε: − �φ� δ� cos ε (4.22)
To control the PTO speed �� using the PTO torque �& as control action equation (4.21) is considered and
reordered to take the form of:
�: = ��� �� �� ��� + ���� (4.23)
To cancel the nonlinearity the following control action is considered:
�� = −��� �� �� � + ��0 (4.24)
By replacing (4.24) in (4.23) the following expression is obtained:
�: = (��0 − ��� �� �� � + ��� �� �� �) 1�� (4.25)
�: = 0 (4.26)
The PTO speed is obtained by integration:
�� = ; 0>2 �= (4.27)
And by applying the Laplace transform the following transfer function for the linearized ISWEC model is
obtained:
³(F) F� = J(F) + �(F)/ � = ℎ(F)
�(F) F
� 0 /
´(�)
34
´(�) = ℒ(��)ℒ(0) = µ� (�)¶(�) = 1� (4.28)
The implementation of feedback linearization described above is represented in Figure 4.11.
Figure 4.11 - Feedback linearization implementation
A proportional feedback controller is designed with the following configuration:
Figure 4.12 - Feedback Controller
The transfer function of the closed-loop system described above is:
H(�) = µ� (�)·(�) = ���´(�)1 + ���´(�) (4.29)
Replacing the transfer function of G(s) in (4.29):
H(�) = ��� �⁄ 1 + ��� �⁄ = 1���� + 1 (4.30)
Choosing a gain ��� of 100, the response of the system to a unit step input is represented in Figure
4.13:
ISWEC
��� cos �
�� 0 �&
�� � �
ISWEC Linearized Model ���·(�) ¶(�) µ� (�)
´(�)
35
Figure 4.13 - Step response of the controlled system
As expected the controlled system has the behavior of a first order system with time constant1 ���⁄ . As
the�# disturbance is completely canceled by the feedback linearization and so the system presents
perfect disturbance rejection.
The response of the system for the reference defined by (4.1) is shown in Figure 4.14:
Figure 4.14 - System Response (overlaping Setpoint and Response curves)
This ideal behavior is observed because perfect non-linearity canceling is considered in (4.25). In
practice this is not possible as perfect knowledge of the model is not available. To verify the viability of
the controller in face of model uncertainties a sensibility analysis on the model parameters is performed.
The considered parameters are listed in Table 4.4:
Table 4.4 Parameters considered for variation
Parameter Symbol Flywheel speed ��
Hydrostatic stiffness � Equivalent moment of inertia of the float ���
Flywheel moment of inertia around rotation axis �� Flywheel moment of inertia perpendicular to rotation axis �
Looking at the model equation considered for the development of the feedback linearization it can be
seen that only the parameters ��� and �� should affect the system performance:
�ε = ��ε: − �φ� δ� cos ε
36
It can then be expected that a variation of the other model parameters should have no effect on the
system response. Simulations are made considering up to 10% increase and decrease of these
parameters considering a reference in phase with the wave torque, and the output is compared with the
system with no parameter variation. To account for the worst case, a variation of 10% in all the
parameters is also considered. The results are shown in Table 4.5:
Table 4.5 Sensitivity analysis on the model parameters
Parameter Variation MSE VAF MD �� -10 1.0x10-4 1.000 0.0355 -5 2.6x10-5 1.000 0.0178 +5 3.7x10-5 1.000 0.0213 +10 1.3x10-4 1.000 0.0391 �� | � -10 0.011 0.999 0.2971 -5 0.003 1.000 0.1466 +5 0.003 1.000 0.1566 +10 0.011 0.999 0.2948 � -10 2.4x10-31 1.000 7.1x10-15 +10 2.2x10-31 1.000 7.1x10-15 ��� -10 2.2x10-31 1.000 7.1x10-15 +10 2.3x10-31 1.000 7.1x10-15
All Parameters -10 0.0423 0.9995 0.5751 +10 0.0407 0.9995 0.5692
The table above shows that, as expected, uncertainties on �# and ��� have no effect on the results. The
parameters � and �� are grouped together as they appear on the same term of the model equation, so a
variation of these parameters produces the same results.
It can be concluded that the controller is robust to model uncertainty. Table 4.5 shows that even the
worst case (considering a 10% variation in all the model parameters) produces a maximum deviation of
around 0.6º/s. This amounts to a small variation in the system output as it is two orders of magnitude
smaller than the output.
37
5. POWER EXTRACTION
The purpose of the controllers developed in this work is to improve the power extraction capabilities of
the ISWEC. As stated in Chapter 4 the design of both IMC and FL controllers consists of a Reference
following problem. In this chapter the reference to track will be defined and the power extracting
capabilities of both the IMC and FL controllers are presented. Finally all controllers will be compared to
the original PD controller.
5.1 Fractional PD controller
The control law that defines the Fractional PD controller is presented again:
���b = − ����=� − �� (5.1)
The design process of this controller is essentially the same as that of the integer PD controller with an
additional parameter to optimize. Again fminsearch is used to find the parameters (, �, �) that produce
the greatest average power extraction over simulations of 600 seconds.
The results obtained are shown below as well as the integer PD results for comparison:
Table 5.1 Extracted Power for Fractional PD controller
Wave ID k / × 10L c / × 10L � [-] Power
Fractional kW
Power PD kW
PFractionalPPD %
1 24.60 12.43 1.81 6.47 5.99 108.01 2 20.55 10.63 1.62 5.84 5.74 101.76 3 19.65 12.55 1.63 49.34 47.80 103.23 4 14.99 9.56 1.50 48.62 47.90 101.51 5 8.79 6.24 1.21 53.06 52.93 100.25 6 9.53 11.22 1.18 115.38 115.25 100.11 7 6.21 9.85 1.02 121.66 121.66 100.00 8 3.20 11.43 0.89 150.38 150.21 100.11
Additional analysis is performed where controller parameters are optimized for the measured wave and
the results are evaluated when used with the eight irregular waves. The motivation behind this analysis
is to evaluate if there is an advantage to use the fractional controller over the PD controller when the
control parameters are not optimized to the current sea state.
The control parameters obtained for the measured wave are listed in Table 5.2 for both the PD controller
and the fractional controller:
38
Table 5.2 Controller parameters optimized for measured wave
PD controller Fractional controller
k × 10L c × 10L k × 10L c × 10L � 8.87 8.35 12.89 9.03 1.28
When these controller parameters are used the following results are obtained:
Table 5.3 Extracted power considering non optimized parameters
Wave ID Power
Fractional × 10K
Power PD × 10K PFractional
PPD %
1 5.63 5.54 101.54 2 5.69 5.64 100.98 3 47.71 46.41 102.80 4 47.93 47.24 101.46 5 51.77 51.40 100.72 6 107.20 105.66 101.46 7 109.66 108.58 100.99 8 128.97 131.20 98.30
While these results show an increased power extraction using the fractional controller over the integer
PD controller it is a small improvement in the order of 1%. The fractional controller is a more complex
controller that requires the selection of an additional parameter over the PD controller. It can then be
concluded that is not worth it to consider the fractional controller as a viable solution for this problem.
5.2 Reference Definition
According to [18], wave energy is captured most efficiently when the WEC speed is in resonance with
the wave excitation force. This statement is valid assuming absorption by a single body oscillating in
one mode of motion. This is not true for the ISWEC system as there are two modes of motion. A pitch
oscillation � produced directly by the incoming wave and a precession mode � on the PTO due to the
gyroscopic effect of the system.
Looking at the system model, it is essentially described by the hull dynamics that determine the pitch
oscillation and gyro dynamics that determine the precession oscillation. These dynamics are connected
by the torques exchanged between the gyroscope and the hull. Figure 5.1 shows a scheme of this
concept where �# represents the wave excitation torque, ���b represents the control torque applied by
the PTO, �,-.. and ��*�+represent the torques exchanged by the hull and the gyroscope.
39
Figure 5.1 - ISWEC system dynamics
Looking at the Gyroscope dynamics described in equation:
�V = ��ε: − �φ� δ� cos ε (5.2)
The term �φ� δ� cos ε represents the torque applied by the hull on the gyroscopic system�����:
Looking at this representation of the system it can be seen as a single body excited by ��*�+ and
controlled by�V. So instead of considering the ISWEC as being excited by �¹ we can consider the hull
dynamics as a filter acting on the wave excitation torque transforming it in��*�+. Then by making it so
that the PTO speed is in phase with this excitation torque a greater power extraction should be
obtained.
This idea is supported by looking at the behavior of the system when using the PD controller. The and � parameters for this simulation are chosen in a way to produce close to maximum power extraction
according to section 3.5. Figure 5.2.(a) shows dimensionless scaled PTO speed and �# and Figure
5.2.(b) shows dimensionless scaled PTO speed and ��*�+.
�V + ��*�+ = ��ε: (5.3)
�� [-] �#(º� [-]
��*�+ �V Gyroscope
Dynamics
�# Hull Dynamics
�,-..
40
(a)
(b)
Figure 5.2 - PTO speed and (a) �# (b) ��*�+
Figure 5.2 show that for a PD controller designed to maximize power by optimization of its
parameters(, �), the resulting PTO speed presents a behavior in phase with the torque exchanged
between the hull and gyroscope ��*�+ as hypothesized before. To further validate this hypothesis both
a reference in phase with �# and �, will be considered. These references are defined as:
�# = � ��������� �#(º� (5.4)
�� = � ������� (5.5)
The� gain parameters are chosen through simulation in order to achieve good power extraction. Like
the design of the PD controller, this parameter is obtained through the use of the fminsearch optimization
function. This analysis is made considering both the IMC and FL controllers.
5.2.1 Internal Model Control
Table 5.4 shows the values for the Gains found when considering the IMC controller:
Table 5.4 Reference Gains considering IMC controller
Reference Gain�
Wave ID 1 2 3 4 5 6 7 8 �# 0.76 0.77 0.84 0.96 0.73 0.72 0.54 0.43 �� 0.40 0.41 0.53 0.53 0.69 1.01 1.04 0.99
Table 5.5 shows the respective Power extracted when considering the gains above:
�� [-] ��*�+ [-]
41
Table 5.5 Extracted Power with IMC controller
Absorbed power / Kw
Wave ID 1 2 3 4 5 6 7 8 �# 2.31 2.34 20.54 21.42 18.25 50.68 40.41 36.30 �� 4.92 5.05 43.64 44.82 47.96 116.48 121.99 156.09
A significant increase in extracted power can be obtained by following the second reference over the
first.
The power extraction capabilities of the IMC controller when considering the models developed in this
work was measured and the results are shown Table 5.6 for 3 different linearization configurations
considering�$ as reference:
Table 5.6 Extracted power (kW) by each controller for each wave
Controller Wave
1 2 3 4 5 6 7 8 IMC 4.92 5.05 43.64 44.82 47.96 116.48 121.99 156.09
IMC Combination
a) 4.91 5.05 43.60 44.84 48.10 116.96 122.71 160.41 b) 4.91 5.05 43.60 44.84 48.13 116.98 122.71 160.51 f) 4.91 5.05 43.61 44.84 48.15 116.99 122.71 160.56
This table shows that for most waves the implementation of the IMC combination does not improve the
power extraction capabilities of the ISWEC significantly. However for waves 6, 7, and 8 there is a slight
increase in performance over the linear model. This can be explained as these waves produce the
greatest oscillations and thus produce the greatest effect on the nonlinear term of the model.
5.2.2 Feedback Linearization Controller
The same procedure is repeated for the FL controller. Table 5.7 shows the Gains K obtained and Table
5.8 the respective Power extracted:
Table 5.7 Reference Gain considering FL controller
Reference Gain�
Wave ID 1 2 3 4 5 6 7 8 �# 0.76 0.76 0.84 0.96 0.73 0.72 0.54 0.43 �� 0.40 0.41 0.52 0.53 0.67 0.67 0.74 0.78
Table 5.8 Extracted Power with FL controller
Absorbed power / Kw
Wave ID 1 2 3 4 5 6 7 8 �# 2.31 2.33 20.54 21.38 18.20 50.62 40.34 36.22 �� 4.92 5.05 43.61 44.78 47.87 116.16 120.14 155.92
42
Again a significant increase in performance can be seen from the results of the FL controller when
choosing�� over�#. It can also be seen that both controllers show similar power extracting capabilities,
with the choice of reference having a much greater impact than the choice of controller. In fact it can be
seen that the gains obtained are very similar except for the last three waves with the �� reference. These
differences are not present in the �# reference as this reference is only depenent on the wave force.
The second reference is obtained through feedback of the forces exchanged in the ISWEC system and
as such it is more sensible to slight variations in system behavior. Waves 6, 7 and 8 where the
differences are greater, are the waves with the greatest wave amplitudes and so these produce the
greatest oscillations on the system and so the small variations of behavior between the two controllers
propagate through the reference. Since it leads to a large improvement in power extraction, �� is
considered as the reference used.
5.2 Power extraction Comparison
In this section the Power extraction capabilities of all the controllers present in this work will be
presented. Table 5.9 shows the power extracted for all the controllers considering the 8 irregular waves:
Table 5.9 Mean absorbed Power
Extracted Power (kW) Wave ID 1 2 3 4 5 6 7 8
PD 5.99 5.74 47.80 47.90 52.93 115.25 121.66 150.21 FPD 6.47 5.84 49.34 48.62 53.06 115.38 121.66 150.38 IMC 4.92 5.05 43.64 44.82 47.96 116.48 121.99 156.09 FL 4.92 5.05 43.61 44.78 47.87 116.16 120.14 155.92
These results are presented in graphical form in scaled power absorbed by the PD controller for each
wave:
Figure 5.3 - Dimensionless mean absorbed power
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8
Power Extracion
PD FPD IMC FL
43
Looking at Table 5.9 Figure 5.3 it can be concluded that the FPD controller is the controller that provides
the best power extraction to the ISWEC, followed by the PD controller. The reference based controllers
(IMC and FL) show a small decrease in extracted power. This can be explained because the design of
the PD and FPD controller is fundamentally different than the design of the IMC and FL controllers.
The first set of controllers are tuned resorting to extensive simulation to find the appropriate set of
parameters that achieve the maximum power while the second set of controllers are tuned to follow a
desired PTO speed reference.
It is expected that when implemented in the real environment, the performance of every controller is
worse than the one obtained through simulation. Because the parameters of the PD and FPD controllers
are obtained through optimization with no physical meaning, these controllers should be more
susceptible to performance deterioration in the real environment. Additionally, these controllers, due to
the fact of having more parameters, are harder to tune.
44
6. CONCLUSION
From the performed simulations, some conclusions can be drawn about the designed controllers. First
it can be concluded that all three designed controllers provide the ISWEC system with similar power
extraction capabilities.
Second it can be concluded that IMC and FL controllers are able to control the ISWEC to system to
follow a desired reference. Aside from good reference tracking they present the system with good
disturbance rejection. These two controllers show similar power extraction, with the correct reference
definition showing a much greater impact in extracted power over the controller choice.
The models developed in section 3.7 through linearization over various working points led to a small
increase in performance over the nonlinear zone of the system. This leads to a small increase of power
extracted for the waves with greater significant height.
The FPD shows an improvement over the current PD controller as expected, as the PD controller can
be seen as a particular case of the FPD with one fixed parameter. The observed improvement is not
significant enough to justify the increase in controller complexity while keeping the same shortcomings
of the PD controller in the need of extensive simulation to obtain the right parameters.
The main advantage of the IMC and FL controllers over the PD controller is the presence of fewer tuning
parameters, as after the design of any controller through simulation, there must be some tuning of the
control parameters in the real environment. This tuning procedure can be very difficult and time
consuming as every change in parameters as to be evaluated, and during this time the sea state is
changing.
To conclude, the final choice of control strategy should be made from experimental results, as the
performance of the controllers is expected to be worse in the real environment. Because the PD and
FPD controllers are tuned by the use of simulation (without any physical meaning), it is expected that
these controllers deteriorate more than the reference based controllers. For the reference based
controllers it is expected that the IMC would outperform the FL controller as the nonlinearities can never
be so accurately canceled in the real environment as in simulation.
For future work, it is recommended that the controllers developed are tested in the real device, or first
considering a more complex model, without ignoring the lower magnitude terms. The developed
controllers should also be tested for different flywheel speeds. Additionally a more suitable reference
can be proposed that could lead to better power extraction than the one obtained in this work.
45
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