Different Model Predictive Control Approaches for Controlling Point Absorber Wave Energy ...

86 Pages·2011·3.13 MB·English

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. Magan˜a
Institute for Systemdynamics, University Stuttgart
Prof. Dr.-Ing. O. Sawodny
October 5, 2011

Abstract
Oceanwave energyisapromisingrenewable energysource that canbe converted
into useful electrical energy using wave energy converters (WECs). In order for
WECstobecomeacommerciallyviablealternativetoestablishedmethodsofen-
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
thepowertake-off(PTO)systemandpresentsdifferentMPCswhosegoalsareto
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 T = 6s (Case 3) and the same formulation with changed
hor
parameter ∆t = 0.2s (Case 4) . . . . . . . . . . . . . . . . . . . . . . . 44
4.8 Comparison between the soft constraints formulation with mooring con-
stant K = 300,000 N/m (Case 2) and the same formulation with
m
K = 50,000 N/m (Case 5) . . . . . . . . . . . . . . . . . . . . . . . . 45
m
4.9 Comparison of Spar position between the soft constraints formulation
with mooring constant K = 300,000 N/m (Case 2) and the same for-
m
mulation with K = 50,000 N/m (Case 5) . . . . . . . . . . . . . . . . 46
m
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 K . . . . . 56
m
5.3 Comparison between the simulation results with the NMPC and the
linear MPC with K = 100,000 N/m. The wave data is from the NDBC
m
Umpqua buoy 46229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Comparison between the simulation results with the NMPC and the
linear MPC with K = 50,000 N/m and K = 350,000 N/m . . . . . . 58
m m
5.5 Comparison between the simulation results with the NMPC and the
linear MPC with K = 150,000 N/m. The wave data is from the NDBC
m
Umpqua buoy 46050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.6 Generated Power depending on K compared to the generated power
m
with the NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59