118
IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013
Nonlinear Model Predictive Control of a Point
Absorber Wave Energy Converter
Markus Richter, Mario E. Magaña,
Senior Member, IEEE,
Oliver Sawodny, and Ted K. A. Brekken,
Member, IEEE
Abstract—This
paper presents the application of nonlinear
model predictive control (NMPC) to a point absorber wave energy
converter (WEC). Model predictive control (MPC) is generally
a promising approach for WECs, since system constraints and
actuator limits can be taken into account. Moreover, it provides a
framework for defining optimal energy capture and it can benefit
from predictions. Due to possible nonlinear effects, such as the
mooring forces, an NMPC is proposed in this paper, whose perfor-
mance is compared to that of a linear MPC. Both controllers are
supposed to control a nonlinear point absorber model. Computer
simulations show that the proposed NMPC is able to optimize the
energy capture while satisfying system limits.
Index Terms—Nonlinear
model predictive control, point ab-
sorber, wave energy converter.
I. I
NTRODUCTION
CEAN wave energy is a promising renewable energy
source that can be converted into useful electrical energy
using WECs. In general, it is difficult to estimate the amount
of exploitable wave energy in the world’s ocean. According
to [1], the ocean holds approximately 8000–80 000 TWh/year
or 1–10 TW, whereas Falnes in [2] quantifies the world’s ex-
ploitable wave power resource to be of the order of 1 TW. Com-
paring this to the world’s annual energy consumption of approx-
imately 148 000 TWh in 2008, [3] shows that wave energy could
play an important role in the world’s energy portfolio. Further-
more, most renewable energy resources are variable and nondis-
patchable and thus have a significant impact on the utility re-
serve requirements. Halamay
et al.
showed in [4] that this im-
pact can be reduced by a diversified energy mix, where WECs
could play an important role.
In order for WECs to become a commercially viable alter-
native, operating the WEC in an optimal fashion is a key task.
Much work on optimizing the energy generation of WECs has
been done which leads to control laws such as latching control
and phase and amplitude control [5]–[7].
O
Manuscript received November 08, 2011; revised May 11, 2012; accepted
May 21, 2012. Date of publication July 31, 2012; date of current version De-
cember 12, 2012.
M. Richter and O. Sawodny are with the Institute for System Dy-
namics, University of Stuttgart, Stuttgart 70049, Germany (e-mail:
markus-c.richter@arcor.de; sawodny@isys.uni-stuttgart.de).
M. E. Magaña and T. K. A. Brekken are with Oregon State University, Cor-
vallis, OR 97331 USA (e-mail: magana@eecs.oregonstate.edu; brekken@eecs.
oregonstate.edu).
Color versions of one or more of the
figures
in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSTE.2012.2202929
Furthermore, Schoen
et al.
[8] have proposed a hybrid con-
trol strategy in order to increase the energy conversion while
being robust to modeling errors. The short-term tuning of the
converter is done by a fuzzy logic controller, while the robust
controller attempts to minimize the modeling errors.
Recently, researchers increasingly deal with MPC algorithms
in order to control point absorbers. They have shown that MPC
is a very promising control method, since it is able to exploit
the entire power potential of a WEC on the one hand, while
respecting the constraints on motions and forces on the other
hand.
All the above-mentioned control approaches require predic-
tion data of the wave’s motion or at least they can benefit from
predictions. The problem of short-term wave forecasting has al-
ready been studied by many researchers. Fusco and Ringwood
[9], for example, focused on wave forecasting and prediction
requirements for unconstrained phase and amplitude control. In
that work, the required forecasting horizon and the achievable
performance of wave predictions for different properties of the
floating
system has been quantified. The same authors presented
several wave prediction algorithms in [10]. They implemented
cyclical models, autoregressive models, and neural networks in
order to predict the wave’s motion and validated these models
against real observations from data buoys. It was shown that
an accurate prediction for up to two typical wave periods into
the future can be calculated. Different prediction models have
been presented in [8]. There, a predictive Kautz model and a
combination of a Kautz and an autoregressive model have been
proposed, where both
filters
have reduced orders compared to
conventional prediction
filters.
Hence, the proposed
filters
can
lower the computational effort in real-time applications.
This paper deals with model predictive control, where a pre-
diction of the wave’s motion is required. However, it does not
focus on wave prediction and an ideal prediction is assumed.
However, the above-mentioned work on wave prediction could
be used to estimate the wave horizon for our model predictive
controller as well.
Current research into MPC for wave energy applications ex-
clusively focuses on linear MPC. Brekken in [11] and Hals
et
al.
in [12] successfully applied linear MPC to one-body WEC
models. Moreover, the application to a linear two-body model
with mooring was demonstrated in [13].
In order to deal with possible nonlinear effects, such as the
mooring forces, a nonlinear model predictive controller is pro-
posed in this paper. Its performance is compared to that of a
linear MPC, also controlling the nonlinear system. The pro-
posed controllers are validated and compared through simula-
tion for irregular sea states.
1949-3029/$31.00 © 2012 IEEE
RICHTER
et al.:
NONLINEAR MODEL PREDICTIVE CONTROL OF A POINT ABSORBER WAVE ENERGY CONVERTER
119
culate the radiation forces. Using impulse response functions
yields convolution terms in the expressions for the radiation
forces. Furthermore, a highly nonlinear mooring force law [17]
is used.
The motion of the buoy and the spar is denoted by and ,
respectively. The equation of motions (EOMs) can be derived
using Newton’s second law. The formulation is based on linear
wave theory (LWT) and the frequency-dependent parameters
of the L10 are assumed to be constant. The EOMs for the two
bodies are
(1)
(2)
, the hydrodynamic force
where the radiation forces
are described by
the mooring force
, and
(3)
(4)
(5)
(6)
(7)
Fig. 1. L10 Wave Energy Coverter [14] (top) and schematic diagram (bottom).
(8)
The paper is organized as follows. In Section II, a nonlinear
two-body model in the time-domain is presented, where the dis-
cretization of the nonlinear model is also discussed. The imple-
mentation of the NMPC is described in Section III. By means of
computer simulation, the performance of the proposed NMPC
is validated and compared to that of the linear MPC from the
work in [13].
II. P
OINT
A
BSORBER
M
ODEL
This paper exclusively focuses on one subclass of WECs,
namely, point absorbers. The point absorber L10, developed at
Oregon State University is considered in this work and is shown
on the left side of Fig. 1. It consists of a
float,
also called buoy,
floating
on the ocean surface and a second body consisting of
a spar and a ballast tank (in what follows, just spar), where the
spar’s motion is damped through mooring. The relative motion
of the two bodies can be converted into usable energy through
a power takeoff (PTO) system. A schematic diagram of the L10
is shown at the top of Fig. 1, where and denote the positions
of the spar and the
float,
respectively. The readers are referred
to [15] for detailed information about the L10.
A. Equations of Motion
The proposed nonlinear two-body model follows the work in
[16]. Thus, impulse response functions are used in order to cal-
where the impulse response functions of the different radiation
forces are denoted by
. The hydrodynamic parameters
are the added masses,
are the viscous damping factors, and
are the hydrostatic stiffnesses. Also, there are coupled ra-
resulting from the interaction of the spar
diation forces
and the buoy.
denotes the PTO force which is the control
and
input for the system, whereas the excitation forces
are the system disturbances.
The mooring system is based on the experimental mooring
configuration in [17]. A top view schematic diagram is shown in
Fig. 2. There, the buoy is moored to a static reference around the
buoy. There are two layers of cables, where each layer consists
of four cables as can be seen in Fig. 2. Here, no mooring to the
is the stiffness of one cable, and
sea
floor
is assumed.
is the horizontal length from the reference to the buoy.
Table I lists the parameters used for modeling, where a cable
N m and a cable length to the
stiffness of
reference of
m is assumed for the mooring force
calculation.
In what follows, we neglect the convolution terms because the
inertia and the damping term are the dominating terms for the ra-
diation forces. The convolution terms are fairly small compared
to other external forces. Simulation results confirm that assump-
tion. In case there was a larger influence of these terms, the
120
IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013
can be formulated. Thereby, it is as-
sumed that the entire state is measurable. The model can be
written as
(13)
where
Fig. 2. Mooring configuration (top view).
(14)
(15)
TABLE I
S
YSTEM
P
ARAMETER OF THE
L10 WEC
(16)
(17)
convolution terms can be approximated by a linear state space
model by methods described in [18] and [19]. However, this is
not necessary here.
For model predictive control in general, using a model in state
space form is convenient. Equations (1)–(8) can be reformulated
as
(9)
(10)
These equations can be transformed in state space form by sub-
stituting (9) into (10) and vice versa to get rid of the coupling
and
. The equations can be restated as
terms
where the PTO force is now denoted by and the excitation
and
by and , respectively.
forces
In our work, we consider three different models. The
first
model (1)–(8) with convolution terms is called the extended
model. The plant model in the simulations is always this model.
The second model (13)–(17) without convolution terms is
called the reduced model. The nonlinear MPC approach is
based on this one. The only nonlinearity in the reduced model
is the mooring force. Replacing the nonlinear mooring force by
the linear law
(18)
yields the linear model, which the linear MPC is based on in this
paper.
B. Discretization
(11)
A
finite
parameterization of the controls and constraints is
used to
find
a direct solution of the optimization problem. There-
fore, the system must be described as a discrete-time nonlinear
state space model in the form
(19)
where the function
maps the current state , the control-
lable input , and the uncontrollable input
to the next state
.
According to [20], a nonlinear system
(20)
(12)
Now, the nonlinear state space model with the state
vector
and the initial conditions
RICHTER
et al.:
NONLINEAR MODEL PREDICTIVE CONTROL OF A POINT ABSORBER WAVE ENERGY CONVERTER
121
with real analytic vector
fields
and
can be dis-
cretized by an approximate sampled-data representation under
zero-order hold assumption by
(21)
with the Jacobian matrix, shown in (29) at the bottom of the
page.
III. P
ROBLEM
F
ORMULATION AND
I
MPLEMENTATION OF
NMPC
Linear MPCs are well known and have been applied since the
1970s, whereas NMPC’s have been used since the 1990s [21].
Linear MPC theory is quite mature today and system theoretic at-
tributes such 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 character-
istics are well discussed, industrial applications are difficult to
find
[23].
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 constraints
and to system dynamics [21]. The readers are referred to [21]
and [24] regarding general information about MPC.
A fundamental problem germane to 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 algo-
rithms exist to solve the problem efficiently. NMPC requires the
solution of a nonlinear problem, though. In general, these prob-
lems are nonconvex, thus it cannot be assured to
find
the global
optimum. Additionally, the solution can be computationally ex-
pensive. Therefore, it is important to exploit the special struc-
ture 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 qualitative performance of NMPC regarding a non-
linear 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 selected nonlinear WEC
compared to the linear MPC. The NMPC and the linear MPC
are implemented using Matlab/SIMULINK. The linear MPC is
based on the work in [13] and uses the reduced system with the
linear mooring law (18). The NMPC uses the solver “fmincon”
of Matlab which can handle nonlinear problems with nonlinear
constraints as well. Here, the interior-point method is used as
the optimization algorithm of “fmincon.” In the following, the
solution of the nonlinear optimization problem using Matlab is
outlined.
The objective function is formulated to include one term ex-
pressing the generated power and another term presenting the
energy use. In general, the generated power for point absorbers
is the product of relative velocity and PTO force. Furthermore,
the optimization problem includes slack variables and as
where
(22)
.
with
Here
denotes the order of the discretization, and is the
sampling interval.
Simulation shows that a discretization of order 1 (comparable
to Euler forward method) is not appropriate for the proposed
NMPC approach. In fact, it yields unreasonable results. Due to
this fact, the discretization order is chosen to be
in the
ensuing work. With the nonlinear reduced system (13)–(17), it
follows that
(23)
(24)
Using (21), after some manipulations the discrete-time system
can be described by
(25)
where
(26)
(27)
(28)
(29)
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IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013
in [13] to avoid infeasibilities. The optimization problem can be
stated as
(30)
where
TABLE II
NMPC/MPC P
ARAMETER
V
ALUES
(31)
subject to
(32)
(33)
(34)
(35)
(36)
are the relative position, velocity, and the
where , , and
generator constraints, respectively.
and
are the weighting
factors that penalize the slack variables and
and
are the
constraints for them.
The problem is implemented using the following optimiza-
tion vector:
(37)
Fig. 3. Nonlinear and linear mooring forces for different choices of
.
variables. It is straightforward to express the ob-
with
jective function by means of .
The solver “fmincon” can handle box constraints, linear, and
nonlinear inequality and equality constraints. The slack vari-
ables (36) and the input constraints (35) are considered as box
constraints which yield
. The position (33) and velocity
(34) limits are considered as linear inequality constraints which
yield
. Additionally, the system dynamics (32) need to be
included as constraints, here as nonlinear equation constraints
with
equations.
According to [10], the wave’s motion can be accurately pre-
dicted for up to two typical wave periods into the future. Thus,
a prediction time of at least 10 s is realistic. However, it should
be noted that a large horizon time normally improves the perfor-
mance, but at the same time, the computational effort increases
since the optimization problem increases. Due to this fact,
is chosen through simulation results. Since the results with a
larger horizon time are not significantly better, a horizon time
of 3 s is used which normally contains an half wave period and
thus, the dominant dynamics. In the following,
s and
s, and thus the horizon length is 30 steps.
In summary, the optimization problem consists of 152 opti-
mization variables and 304 equations for the constraints. This is
a large problem for online solving of the problem within a step
time of
s. However, as stated above, the goal of this
work is to determine the qualitive behavior of the system con-
trolled by the NMPC in comparison with a linear MPC and not
to discuss real-time applicability.
IV. R
ESULTS
Two different controllers are proposed to control the extended
nonlinear model (1)–(8): The NMPC as described in Section III
and a linear MPC which is implemented as in [13]. The NMPC
is based on the reduced nonlinear model without convolution
terms as prediction model, whereas the linear MPC is based on
the linear model with linear mooring force constant
.
The results with the NMPC and the linear MPC are denoted
by NMPC and MPC, respectively. The simulations and control
parameters are shown in Table II. In what follows, the NMPC is
validated and compared to the linear MPC with different values
of
through simulation.
Fig. 3 shows the nonlinear and linear mooring forces for cer-
tain values of
for spar displacements between 0.6 m. It
can be seen that the nonlinear mooring law is highly nonlinear
and a linearization would not work. Still, there are many possi-
bilities to choose
. In the following, three different cases are
considered: One
N m which underestimates
the nonlinear mooring, another
N m which
assumes a large mooring force, and a third choice of
so that
the linear MPC generates the maximum power for the current
wave data.
In the beginning, a time-series from the National Data Buoy
Center (NDBC) Umpqua buoy 46 229 which is deployed off the
coast of Oregon north of Reedsport [25] is used as the wave
data with the wave elevation . The wave is characterized by
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