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\confshortname{OMAE 2014 \vspace{2pt}}
\conffullname{the 33rd International Conference on Ocean, Offshore and Arctic Engineering}

\confdate{8-13}
\confmonth{June}
\confyear{2014}
\confcity{San Francisco}
\confcountry{USA}

\papernum{DRAFT OMAE 2014-24132}

\title{Results from numerical simulation and field tests of an oceanographic buoy powered by sea waves}

\author{J. A. Hamilton$^{(1)}$\thanks{Address all correspondence to this author.}, J. C. C. Henriques$^{(2)}$, F. Cazenave$^{(1)}$, 
P. McGill$^{(1)}$, \\
\textbf{W. Radachonski$^{(1)}$, R. P. F. Gomes$^{(2)}$, L. M. C. Gato$^{(2)}$ and A. F. O. Falcão$^{(2)}$}
%\textbf{W. Radachonski$^{(1)}$, R. P. F. Gomes$^{(2)}$, L. M. C. Gato$^{(2)}$ and A. F. O. Falcao$^{(2)}$}
 %
\affiliation{$^{(1)}$Monterey Bay Aquarium Research Institute\\
7700 Sandholdt Road, Moss Landing, CA, USA}
\affiliation{$^{(2)}$ IDMEC, Instituto Superior T\'ecnico,\\Universidade de Lisboa,\\
Av. Rovisco Pais, 1049-001 Lisboa, Portugal}
}

\begin{document}

\maketitle
%
\begin{abstract}
%
The paper presents analysis of the MBARI wave-energy converter designed for oceanographic applications and 
is intended to generate 300-500 W of electrical power on average. The prototype consists of a small 
draft cylindrical buoy connected to a submerged anti-heave plate by a hydraulic power take-off system 
and cable. Engineering details and design of the system has been based on the results of a detailed 
time-domain hydrodynamic analysis of the system, discussed here. The performance of the prototype 
is assessed with respect to the submerged plate size/added-mass and the effectiveness of latching 
schemes. Additionally, results for the power extraction are statistically compared with 
measurements performed during field tests in Monterey Bay, California.  During these tests performance 
measurements were logged by the buoy and wave data was recorded by a measurement buoy positioned 1 km 
from the deployment site. This allows the comparison of actual performance to model predictions run 
for the sea state present during testing. In general, the results of the numerical model match fairly well 
the data acquired in the MBARI buoy field testing.  Latching schemes have been show to be ineffective for two-body systems of this size.
%
\end{abstract}
%
\section{Introduction}

MBARI has been developing a wave-energy conversion buoy specifically for powering oceanographic instrumentation, the resulting device is a  two-body point absorber in which a power take-off device connects a surface buoy and a submerged anti-heave plate (see figure \ref{fig:MBARI_WEC}).  The relatively low average power level requirements (300-500 Watts) and small size make this system distinct from the larger wave-energy conversion devices currently being explored for utility power applications.  Nevertheless it is still important to develop as much power as possible from these systems with respect to their size and weight.  The work presented here is the result of a collaboration between MBARI and IDMEC intended to utilize the computational tools present at IDMEC to explore advanced control schemes (latching in particular) to increase the amount of power captured by the MBARI system.  Up to the point of this work the MBARI system had been deployed only with a linear damping scheme in which the power take-off machinery exerts a force that is proportional to the relative speed between the surface buoy and submerged plate.  Previous work at IDMEC \cite{Henriques201231} had explored the utility of latching in larger two-body systems and shown that an increase in total power capture can be obtained when the inertia of the submerged element is significantly larger than the mass of the surface expression.  This work was undertaken to determine specifically if a similar result applied to the small scale system.  

The analysis undertaken and presented here has shown that at the small scale of this system, latching techniques only start to increase the power capture at inertia ratios (submerged element inertia to buoy mass) that are above 25:1, ratios that are impractically large for this system.  This is in contrast to the findings of \cite{Henriques201231} that show an increase in power capture for inertia ratios of 5:1 for the larger systems.  As such the latching scheme was not in implemented in the MBARI device but instead a test deployment was done with a larger plate added-mass than had been previously tested since the analysis suggested a strong relationship between that parameter and total energy capture.  


\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{./Figs/MBARI_dimensions.png}
\caption{View of the MBARI's Wave-Power Buoy. Dimensions are in meters.}
\label{fig:MBARI_WEC}
\end{figure}

\section{System Description}
The wave-energy conversion device considered here consists of a cylindrical surface buoy connected to a submerged plate, 30\,m below the surface, by an electro-hydraulic
power take-off system, and is illustrated in figure \ref{fig:MBARI_WEC}.  

The surface expression is typical of oceanographic systems and houses batteries, system control and power electronics, batteries, and other instrumentation.  
The physical parameters of the surface buoy are:

\begin{itemize}
  \item  diameter: 2.5\,m;
  \item  water-plane area: 4.9\,m$^2$;
  \item  mass (with power take-off system attached): 1520\,kg;
  \item  displacement (with PTO system and plate attached): 2.29\,m$^3$;
\end{itemize}

The submerged plate is a welded steel frame with aluminum plates that can be arranged to provide different drag and added mass characteristics without significantly changing the submerged plates mass.  The system has been deployed with several configurations of the submerged plate, allowing performance sensitivity to various parameters to be evaluated.  The drag coefficient and added mass values are obtained as estimates from both numerical analysis using CFD and WAMIT as well as full-scale lab tests.  For extended deployments the submerged plate is anchored to the sea-floor with a chain-catenary mooring, it is not believed this mooring is a significant factor in the dynamics of the system because the mean tensions in the system vary little with changing sea, wind, and current conditions. i.e. the mooring loads are small compared to the inertial and drag forces the submerged plate exerts on the surface buoy through the power take-off system. The physical properties of the submerged plate used in the field test presented here are:

\begin{itemize}
  \item  area: 13.7\,m$^2$;
  \item  mass: 972\,kg;
  \item  added mass: 10140\,kg;
  \item  drag coefficient: 1.9\, 
\end{itemize}

The power-take-off device is the most complicated part of the system and consists of an electro-hydraulic device as a damping element and a pneumatic spring element arranged in parallel.  The damping component is a hydraulic cylinder plumbed directly to a hydraulic motor (bent-axis piston type).  The hydraulic motor is coupled directly without a gearbox to a permanent magnet motor operating as a generator.  There is no accumulator in the system and the generator and hydraulic motor stop and reverse direction with each oscillation of the power take-off ram.  The small size and low inertia of the system allows this and while some oscillations are present due to the compressibility of the hydraulic oil reacting against the rotational inertia when the system is lightly loaded, these oscillations are heavily damped at all loadings that are used in the system.  The absence of an  accumulator simplifies the system and eliminates a loss element, but does result in a wide range of voltage and power levels that the power electronics must convert to battery bus voltages.  Presently this is accomplished by DC-DC converters in the system, with any excess power being shunted to a resistive load dump to maintain a load on the system at all times.  This arrangement allows for simple electronic control of the load presented to the wave-energy converter.  The mechanical-to-electrical efficiency of this power take-off device has been measured to be between 65 and 75 percent, depending on the condition of the seals and the care with which the long hydraulic rams are aligned.  Like all hydraulic systems, efficiency is better at high damping levels (and subsequent high oil pressures).  The numerical modeling below characterizes the systems ability to capture wave energy as a function of this damping force and it is seen to be fairly insensitive to this value, allowing the hydraulic system to be operated at it's most efficient point in almost all conditions.  The hydraulic PTO system has the following characteristics:
\begin{itemize}
  \item bore diameter 6.35\,cm (2.5\,in);
  \item rod diameter 3.81\,cm (1.5\,in);
  \item total stroke: 2\,m;
  \item maximum damping force is approximately 20000\,N (set by a pressure relief valve).
\end{itemize}

The spring element in the system consists of a second piston working against two volumes of compressible gas, the gas pressures and volumes are selected to create a nearly linear spring of appropriate spring constant which is not adjustable.  The characteristics of this element are:
\begin{itemize}
\item \underline{Upper chamber}
	\begin{itemize}
	  \item absolute pressure: 5.170\,bar;
	  \item chamber volume: 0.036\,m$^3$.
	\end{itemize}
\item \underline{Lower chamber}
	\begin{itemize}
	  \item absolute pressure: 8.690\,bar;
	  \item chamber volume: 0.062\,m$^3$.
	\end{itemize}
\end{itemize}

The control system on board the buoy measures a range of parameters at 10Hz throughout the deployments.  In addition to logging this data and transmitting it to shore in near real-time, the damping force exerted is controlled electronically as a function of piston speed (or any other parameter) at a rate of 10Hz.  The following data is recorded:
\begin{itemize}
  \item  piston position;
  \item  force between buoy and PTO system;
  \item  pressure and temperature sensors to monitor spring gas conditions;
  \item  pressure sensor to monitor sea pressure at top of the PTO (buoy draft);
  \item  6 DOF GPS aided IMU to measure the buoy motions and location;
  \item  generator voltage;
  \item  generator current;
  \item  battery bus voltage and current.
\end{itemize}


\section{Numerical Modeling}
%
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{./Figs/MBARI_WEC_cap.pdf}
\caption{Conceptual model of the MBARI ocean's wave energy converter. $x_1$, $x_2$ and $x_3$ are local coordinate systems
and the zero is at the equilibrium positions without waves.}
\label{fig:MBARI_conptual}
\end{figure}
%
The numerical modeling takes place in the time domain and solves the equations of motion of four elements simultaneously; the buoy, the piston, the hydraulic motor/generator (rotational motion), and the submerged plate.  The equations of motion for the system are written in terms of the vertical coordinates $x_i$ 
defining the position of body $i$ $(i = 1, 2, 3)$, with $x_i = 0$ in the absence of waves and $x_i$ increasing upwards. 
The floater is denoted as 1, the piston as 2 and the submerged plate as 3, see Fig.~\ref{fig:MBARI_conptual}.
%
\subsection{The floater}
%
\par
The equation of motion for the floater is
%
\begin{equation}
\label{eq:floater}
	( m_1 + A_1^\infty ) \ddot{x}_1
	=
	- \rho g S x_1 + F_\text{ext} - F_\text{rad} +  F_\text{gas} + F_\text{M},
\end{equation}
%
where a dot represents time derivative. Here, $g$ is the acceleration of gravity, $\rho$ is water density,  $A_i^\infty = \lim\limits_{\omega \rightarrow\infty} A_i(\omega)$, where $A_i(\omega)$ is the frequency dependent added mass coefficient for body~$i$
%
and $F_\text{ext}$ is the excitation force modeled as in \cite{Henriques201231}.
%
The radiation force $F_\text{rad}$ is computed in the time domain as
%
\begin{equation}
\label{eq:F_rad}
  F_\text{rad} = \int\limits_{-\infty}^t K_1 ( t- \tau ) \, \dot{x}_1 (\tau) \, d\tau,
\end{equation}
%
see \cite{Henriques201301} for further details. 
%
\par The force acting on the piston is modeled assuming an isentropic compression/expansion
$(p V^\gamma = p_0 V_0^\gamma)$ resulting, after some manipulation, in
%
\begin{equation}
  F_\text{gas} =
  	  F_\text{pu0} \left( 1 - \frac{x_2-x_1}{l_\text{pu}} \right)^{-\gamma}
  	- F_\text{pl0} \left( 1 + \frac{x_2-x_1}{l_\text{pl}} \right)^{-\gamma},
\end{equation}
%
where $F_\text{pu0} = p_\text{pu0} A_\text{pu}$ and $F_\text{pl0} = p_\text{pl0} A_\text{pl}$.
%
As in Fig.~\ref{fig:MBARI_conptual}, subscripts pu, pl refer to the pneumatic upper chamber and the pneumatic lower chamber, respectively. The subscript 0 denotes mid-stroke conditions.
%
\par
Assuming oil as an incompressible fluid, the force on the hydraulic cylinder is given by
%
\begin{equation}
F_\text{M} = A_\text{oil} \, \Delta p_\text{oil},
\end{equation}
%
where $A_\text{oil} = A_\text{piston}-A_\text{bore}$ is the wet piston area.
%
\subsection{Hydraulic piston} 
%
The equation of motion for the piston is
%
\begin{equation}
\label{eq:piston}
	m_2\, \ddot{x}_2
	=
	- F_\text{gas} - F_\text{M} - F_\text{mg,2} - F_\text{cable}.
\end{equation}
%
$F_\text{mg,2} = m_2\,g$ is the weight of the piston and rods. The tether force is given by
%
\begin{equation}
\label{eq:cable}
F_\text{cable} = k_c \max \! \left( x_2 - x_3, 0 \right).
\end{equation}
%
\subsection{Hydraulic motor}
%
The theoretical volume flow rate and torque of the motor are given by
%
\begin{equation}
Q_m = D_m \, \omega_m,
\end{equation}
%
\begin{equation}
T_m = D_m \, \Delta p_\text{oil},
\end{equation}
%
where $D_m$ is the volumetric displacement of the motor.
%
The theoretical volume flow rate of the motor is related to the cylinder motion (actual volume flow rate) through
%
\begin{equation}
D_m \, \omega_m = A_\text{oil} \, \left( \dot{x}_2 - \dot{x}_1 \right).
\end{equation}
%
The generator motion is modeled using
%
\begin{equation}
J \dot{\omega}_m = T_m - T_\text{gen},
\end{equation}
%
which yields
%
\begin{equation}
\label{eq:generator_1}
\frac{J A_\text{oil}}{D_m} \, \left( \ddot{x}_2 - \ddot{x}_1 \right) = D_m \Delta p_\text{oil} - T_\text{gen},
\end{equation}
%
where the generator torque is given by the control relationship
%
\begin{equation}
T_\text{gen} =  C_\text{gen} \, \left( \dot{x}_2 - \dot{x}_1 \right),
\end{equation}
%
and $C_\text{gen}$ is a user defined constant.
%
\par
Eq. \eqref{eq:generator_1} can be written in a more appropriate form for the computations
%
\begin{equation}
\label{eq:motor}
F_\text{M} = m_J \, \ddot{x}_2 - m_J \, \ddot{x}_1 + F_\text{gen},
\end{equation}
%
where $m_J = J A_\text{oil}^2 D_m^{-2}$ and $F_\text{gen} = A_\text{oil} T_\text{gen} D_m^{-1}$.
%
\subsection{Submerged plate}
%
The motion of the submerged plate is given by
\begin{equation}
\label{eq:plate}
	\left( m_3 + A_3^\infty \right) \ddot{x}_3
	=
	F_\text{cable} - F_\text{mg,3} - F_\text{drag},
\end{equation}
%
where $F_\text{mg,3} = m_3\,g$, $F_\text{drag} = \frac{1}{2} \rho \, \dot{x}_3 |\dot{x}_3| A_p C_D$ and $A_p$ is the plate area.
%
\subsection{Numerical solution}
%
The numerical model comprises the system of equations \eqref{eq:floater}, \eqref{eq:piston}, \eqref{eq:motor} and \eqref{eq:plate}.
The system is solved using the numerical method described in \cite{Henriques201231}. A distinction must be made when the cable is in tension $(F_\text{cable}>0)$ or when it is slack $(F_\text{cable}=0)$, see Eq. \eqref{eq:cable}. 
%
























\begin{comment}


\section{Numerical Modeling}

\subsection{The buoy}

\par
The equation of motion for the buoy, $x_1$ is, see Fig.~\ref{fig:MBARI_conptual},
%
\begin{equation}
\label{eq:body1:1}
	m_1\, \ddot{x}_1(t)
	=
	F_\text{ext}(t) - F_\text{imp}(x_1) - F_\text{r}(t,\dot{x}_1,\ddot{x}_1)
	+ F_\text{PTO},
\end{equation}
%
where $F_\text{ext}$ is the excitation force computed using WAMIT as function of the wave frequencies
and $F_\text{imp}$ is the impulsion force
%
\begin{equation}
  F_\text{imp} = \rho g S x_1.
\end{equation}
%
The radiation force $F_\text{rad}$ is computed in the time domain as
%
\begin{equation}
F_\text{rad} =  A_1^\infty \, \ddot{x}_1 + F_\text{cnv},
\end{equation}
%
where
%
\begin{equation}
\label{eq:F_c}
  F_\text{cnv} = \int\limits_{-\infty}^t K_1 ( t- \tau ) \, \dot{x}_1 (\tau) \, d\tau,
\end{equation}
%
and $A_1^\infty = \lim\limits_{\omega \rightarrow\infty} A_1(\omega)$, where $A_1(\omega)$ is the frequency dependent added mass coefficient for body~1.
%
The impulse response function is given in Appendix~\ref{chap:MBARI_Coefs}.
%
$F_\text{PTO}$ is the force of the PTO on the joint linked to the floater
%
\begin{equation}
  F_\text{PTO} = F_\text{gas}(x_1,x_2) + F_\text{oil}(\dot{x}_1,\dot{x}_2,T_\text{gen}).
\end{equation}
%
\par The force acting on each gas piston chamber is modeled as an isentropic compression/expansion
%
\begin{equation}
  p V^\gamma = p_0 V_0^\gamma,
\end{equation}
%
resulting, after some manipulation, in\footnote[2]{Referring to Fig.~\ref{fig:MBARI_conptual}, subscripts pu, pl are, respectively, pneumatic upper and pneumatic lower. The ``0''  subscript denotes mid-stroke conditions.}
%
\begin{equation}
  F_\text{gas} =
  	  F_\text{pu0} \left( 1 - \frac{x_2-x_1}{l_\text{pu}} \right)^{-\gamma}
  	- F_\text{pl0} \left( 1 + \frac{x_2-x_1}{l_\text{pl}} \right)^{-\gamma},
\end{equation}
%
where
%
\begin{equation}
F_\text{pu0} = p_\text{pu0} A_\text{pu},
\end{equation}
%
\begin{equation}
F_\text{pl0} = p_\text{pl0} A_\text{pl}.
\end{equation}
%
\begin{figure}[h]
\centering
\includegraphics[width=0.65\linewidth]{./Figs/MBARI_WEC_cap.pdf}
\caption{Conceptual model of the MBARI ocean's wave energy converter. $x_1$, $x_2$ and $x_3$ are local coordinate systems
and the zero is at the equilibrium positions without waves.}
\label{fig:MBARI_conptual}
\end{figure}
%
\par
Oil has been modeled as an incompressible fluid. The force on the hydraulic cylinder is given by
%
\begin{equation}
F_\text{oil} = A_\text{oil} \, \Delta p_\text{oil},
\end{equation}
%
where
%
\begin{equation}
A_\text{oil} = A_\text{piston}-A_\text{bore},
\end{equation}
%
is the wet piston area.
\par
Introducing the linear terms in Eq.~\eqref{eq:body1:1}, we get
%
\begin{equation}
\label{eq:buoy_motion_1}
    \left( m_1 + A_1^\infty \right) \ddot{x}_1 + \rho g S x_1 - A_\text{oil} \, \Delta p_\text{oil}
    =
    F_\text{exc} - F_\text{cnv} + F_\text{gas}.
\end{equation}
%
\subsection{Piston and hydraulic motor}
%
The equation of motion for the piston is
%
\begin{equation}
\label{eq:cylinders_1}
	m_2\, \ddot{x}_2(t)
	=
	- F_\text{gas}(x_1,x_2) - F_\text{oil}\left(\dot{x}_1,\dot{x}_2,T_\text{gen}\right) - F_\text{mg,2} - F_\text{cable}(x_2,x_3).
\end{equation}
%
$F_\text{mg,2} = m_2\,g$ is the weight of the piston and rods. The tether force is given by
%
\begin{equation}
F_\text{cable} = k_c \max \! \left( x_2 - x_3, 0 \right).
\label{eq:force_cable}
\end{equation}
%
The theoretical motor volume flow rate and torque are given by
%
\begin{equation}
Q_m = D_m \, \omega_m,
\end{equation}
%
\begin{equation}
T_m = D_m \, \Delta p_\text{oil},
\end{equation}
%
where $D_m$ is the volumetric displacement of the motor.
%
The theoretical motor volume flow rate is related to the cylinder motion (actual volume flow rate) through
%
\begin{equation}
D_m \, \omega_m = A_\text{oil} \, \left( \dot{x}_2 - \dot{x}_1 \right).
\end{equation}
%
The generator motion is modeled using
%
\begin{equation}
J \dot{\omega}_m = T_m - T_\text{gen},
\end{equation}
%
resulting
%
\begin{equation}
\label{eq:generator_1}
\frac{J A_\text{oil}}{D_m} \, \left( \ddot{x}_2 - \ddot{x}_1 \right) = D_m \Delta p_\text{oil} - T_\text{gen},
\end{equation}
%
where the generator torque is given by the control relationship
%
\begin{equation}
T_\text{gen} =  C_\text{gen} \, \left( \dot{x}_2 - \dot{x}_1 \right),
\end{equation}
%
where
%
\begin{equation}
C_\text{gen} =
\left\{
\begin{array}{cl}
0,
& \text{if\ } \left( \dot{x}_2 - \dot{x}_1 \right) \Delta p_\text{oil} \le 0,\\
%
\max \bigl( D_m \Delta p_\text{oil} \left( \dot{x}_2 - \dot{x}_1 \right)^{-1}, C_\text{gen}^{\max}\bigr),
& \text{if\ } \left( \dot{x}_2 - \dot{x}_1 \right) \Delta p_\text{oil} > 0.
\end{array}
\right.
\end{equation}
%
and $C_\text{gen}^{\max}$ is a user defined constant.
\par
%
\par
Eq. \eqref{eq:generator_1} can be written in a more appropriate form for our computations
%
\begin{equation}
\label{eq:generator_2}
A_\text{oil} \, \Delta p_\text{oil} = m_J \, \ddot{x}_2 - m_J \, \ddot{x}_1 + F_\text{gen},
\end{equation}
%
where
%
\begin{equation}
m_J = \frac{J A_\text{oil}^2}{D_m^2},
\end{equation}
%
and
%
\begin{equation}
F_\text{gen} = \frac{A_\text{oil}}{D_m} \, T_\text{gen}.
\end{equation}
%
\par
Replacing Eq. \eqref{eq:generator_2} in Eq. \eqref{eq:buoy_motion_1} results
%
\begin{equation}
\label{eq:buoy_motion_2}
    \left( m_1 + A_1^\infty + m_J \right) \ddot{x}_1 + \rho g S x_1 - m_J \, \ddot{x}_2
    =
    F_\text{exc} - F_\text{cnv} + F_\text{gas} + F_\text{gen},
\end{equation}
%
and analogously for Eq.~\eqref{eq:cylinders_1}
%
\begin{equation}
\label{eq:cylinders_2}
    - m_J \, \ddot{x}_1 + \left( m_2 + m_J \right) \ddot{x}_2 + k_c \, x_2 - k_c \, x_3
	=
	 - F_\text{mg,2} - F_\text{gas} - F_\text{gen}.
\end{equation}
%
\subsection{Generator averaged output power}

The generator averaged output power is given by:
%
\begin{equation}
\bar{P}_\text{gen} =  \frac{A_\text{oil}}{\left(t_1-t_0\right) D_m} \, \int_{t_0}^{t_1} \, C_\text{gen} \left( \dot{x}_2 - \dot{x}_1 \right)^2 \, d t.
\end{equation}

\subsection{Submerged plate}
%
The motion of the submerged plate is given by
\begin{equation}
	\left( m_3 + A_3^\infty \right) \ddot{x}_3
	=
	F_\text{cable}(x_2,x_3) - F_\text{mg,3} - F_\text{drag}(\dot{x}_3),
\end{equation}
%
where $F_\text{mg,3} = m_3\,g$, $A_p$ is the plate area and
%
\begin{equation}
F_\text{drag} = \frac{1}{2} \rho \, \dot{x}_3 |\dot{x}_3| A_p C_D,
\end{equation}
%
resulting
%
\begin{equation}
\label{eq:plate}
	 - k_c \, x_2 + \left( m_3 + A_3^\infty \right) \ddot{x}_3 + k_c \, x_3
	=
	- F_\text{mg,3} - F_\text{drag}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Numerical continuous solution of the system of equations}
%
\subsection{Polynomial approximation of the solution}
\par

From this point onwards the calculations will refer to a relative time $\tau$ starting at the beginning of the
current absolute time step, $t_n$, such that
%
\begin{equation}
\tau = t - t_n.
\end{equation}
The time step will be denoted by $\Delta \tau$.
%
Approximating $x_i$ by $n^\text{th}$ order polynomials we get
%
\begin{equation}
\label{eq:poly_x}
x_i = x_{i,0} + \dot{x}_{i,0} \, \tau + \mathbf{t}_n \, \mathbf{a}_i,
\end{equation}
%
where $\mathbf{a}_i = ( a_2, a_3, \ldots, a_n )_i^T$
are column vectors of the unknown coefficients and
%
\begin{equation}
\label{eq:tx_polynomial}
\begin{split}
\mathbf{t}_n &= ( \tau^2, \tau^3, \ldots, \tau^n), \\
\end{split}
\end{equation}
%
is a line vector with the monomial coefficients.
%
Consequently,
%
\begin{equation}
\label{eq:poly_x_deriv}
\dot{x}_i = \dot{x}_{i,0} + \dot{\mathbf{t}}_n \, \mathbf{a}_i,
\end{equation}
%
and
%
\begin{equation}
\ddot{x}_i = \ddot{\mathbf{t}}_n \, \mathbf{a}_i.
\end{equation}
%
The use of the relative time allows the introduction of initial conditions
directly in the polynomial approximations \eqref{eq:poly_x} and \eqref{eq:poly_x_deriv}.
%
\par
Introducing this approximating polynomials in Eqs.~\eqref{eq:cylinders_2},
\eqref{eq:buoy_motion_2} and \eqref{eq:plate}, the resulting system of equations can be written in a block matrix form
%
\begin{equation}
\label{eq:sys_eq_global_1}
\mathbf{A} \, \mathbf{y} = \mathbf{f},
\end{equation}
where
%
\begin{equation}
\label{eq:f}
\mathbf{f}
=
\mathbf{f}_\text{sys} + \mathbf{f}_0.
\end{equation}
The force $\mathbf{f}$ is decomposed in two components,
the forces applied to the system $\mathbf{f}_\text{sys}$
and a component associated with the initial conditions $\mathbf{f}_0$.
%
\subsection{Matricial form of the system of equations}
%
\subsubsection{Cable in tensile stress}
%
When the cable is subjected to a tensile stress, Eq.~\eqref{eq:sys_eq_global_1} can be written in the following block matrix form
%
\begin{equation}
\label{eq:matrix_A_kx}
  \mathbf{A}
  =
  \begin{pmatrix}
	\left( m_1 + A^\infty_{11} + m_J \right) \ddot{\mathbf{t}}_n +
	\rho g S \, \mathbf{t}_n & -m_J \, \ddot{\mathbf{t}}_n  & 0 \\
    -m_J \, \ddot{\mathbf{t}}_n & \left( m_2 + m_J \right) \, \ddot{\mathbf{t}}_n + k_c \, \mathbf{t}_n & -k_c \, \mathbf{t}_n \\
    0 & -k_c \, \mathbf{t}_n & \left( m_3 + A^\infty_{33} \right) \ddot{\mathbf{t}}_n + k_c \, \mathbf{t}_n \\
      \end{pmatrix},
\end{equation}
\begin{equation}
  \mathbf{y} =
  \begin{pmatrix}
  \mathbf{a}_1 &
  \mathbf{a}_2 &
  \mathbf{a}_3
\end{pmatrix}^T,
\end{equation}
\begin{equation}
\label{eq:f_sys_kx}
  \mathbf{f}_\text{sys} =
  \begin{pmatrix}
    F_\text{exc} - F_\text{cnv} + F_\text{gas} + F_\text{gen} \\
    - F_\text{mg,2} - F_\text{gas} - F_\text{gen} \\
  - F_\text{mg,3} - F_\text{drag}
  \end{pmatrix},
\end{equation}
\begin{equation}
\label{eq:f_0_kx}
  \mathbf{f}_0 =
  \begin{pmatrix}
    - \rho g S \left( x_{1,0} + \dot{x}_{1,0} \, \tau \right)	\\
    - k_c \left( x_{2,0} + \dot{x}_{2,0} \, \tau \right)
    + k_c \left( x_{3,0} + \dot{x}_{3,0} \, \tau \right) \\
    \phantom{+}  k_c \left( x_{2,0} + \dot{x}_{2,0} \, \tau \right)
    - k_c \left( x_{3,0} + \dot{x}_{3,0} \, \tau \right)
  \end{pmatrix}.
\end{equation}

\subsubsection{Slack cable}
%
If the cable is slack, $k_c = 0$, matrix \eqref{eq:matrix_A_kx} and vector \eqref{eq:f_0_kx} are given by
%
\begin{equation}
\label{eq:matrix_A_no_kx}
  \mathbf{A}
  =
  \begin{pmatrix}
	\left( m_1 + A^\infty_{11} + m_J \right) \ddot{\mathbf{t}}_n +
	\rho g S \, \mathbf{t}_n & -m_J \, \ddot{\mathbf{t}}_n  & 0 \\
    -m_J \, \ddot{\mathbf{t}}_n & \left( m_2 + m_J \right) \, \ddot{\mathbf{t}}_n & 0 \\
    0 & 0 & \left( m_3 + A^\infty_{33} \right) \ddot{\mathbf{t}}_n
      \end{pmatrix},
\end{equation}

%

\begin{equation}
\label{eq:f_0_no_kx}
  \mathbf{f}_0 =
  \begin{pmatrix}
    - \rho g S \left( x_{1,0} + \dot{x}_{1,0} \, \tau \right)	\\
      0\\
      0
  \end{pmatrix}.
\end{equation}
%
\subsection{Numerical solution of the resulting algebraic system of equations}
\par
The vector of the polynomial coefficients $\mathbf{x}$ is computed using the normal system of equations
associated with the continuous least squares problem
%
\begin{equation}
  \label{eq:sys_eq_global_A}
  \int_0^{\Delta \tau} \mathbf{A}^T \mathbf{A} \, d \tau \, \mathbf{x} = \int_0^{\Delta \tau} \mathbf{A}^T \mathbf{f} \, d \tau,
\end{equation}
%
This continuous least squares formulation is similar to the one presented in \cite{Henriques201231}.
%
The matrix $\mathbf{A}^T \mathbf{A}$ is a symmetric matrix factorized using the Cholesky decomposition
$\mathbf{LDL}^T$ \cite{Watkins1991}.
The matrix $\mathbf{A}^T \mathbf{A}$ is decomposed only once  before the computations.
%
\par
To simplify the integration of the $\mathbf{A}^T \mathbf{f}$, the entries of $\mathbf{f}$ are approximated
Eq.~\eqref{eq:f} by interpolating polynomials of degree $n$ at $n+1$ equally
spaced points in $\Delta \tau$.
%
\par
Since the resulting system of equations \eqref{eq:sys_eq_global_1} is non-linear,
the system of Eqs.~\eqref{eq:sys_eq_global_1} must be solved iteratively until convergence.
%
In this iterative process the following stopping criterion has been used
%
\begin{equation}
\max \! \left(
\varepsilon_1, \varepsilon_2, \varepsilon_3 \right) < 10^{-4},
\end{equation}
%
where $m=\dim(\mathbf{x})$ and
%
\begin{equation}
\varepsilon_k = \frac{1}{\Delta \tau} \int_0^{\Delta \tau} \left( x_k^{(n)} - x_k^{(n-1)} \right)^2 d \tau,
\end{equation}
%
and $(n)$ denotes the $n^\text{th}$ iteration.
%
\begin{table}[ht]
\caption{MBARI's Wave-Power Buoy system characteristics used in the simulations}
\label{tab:MBARI_chars}
\centering
\begin{tabular}{|c|c|}
\hline
\multirow{2}{*}{Buoy} & $D = 2.5$\,m\\
\hhline{~-}
 & $A_\infty^1 = 2804.87$\,kg \\
\hline
\multirow{6}{*}{\centering PTO} & $p_\text{pu} = 8.690$\,bar \\
\hhline{~-}
& $l_\text{pu} = 2.868$\,m \\
\hhline{~-}
& $S_\text{pu} = 0.0126677$\,m$^2$ \\
\hhline{~-}
& $p_\text{pl} = 8.69$\,bar \\
\hhline{~-}
& $l_\text{pl} = 0.012677$\,m \\
\hhline{~-}
& $S_\text{pl} = 0.0115276$\, m$^2$ \\
\hline
\multirow{2}{*}{Hydraulics} & $A_\text{oil} = 0.000887$\,m$^2$\\
\hhline{~-}
 & $D_m = 0.77986 \times 10^{-6}$\,m$^3$/rad\\
 \hline
\multirow{1}{*}{Generator} & $J_\text{gen} = 1.13 \times 10^{-3}$\,kg/m$^3$\\
\hline
\end{tabular}
%
\end{table}
%

\end{comment}

\begin{table}[ht]
\caption{MBARI's Wave-Power Buoy system characteristics used in the simulations}
\label{tab:MBARI_chars}
\centering
\begin{tabular}{|c|c|}
\hline
\multirow{2}{*}{Buoy} & $D = 2.5$\,m\\
\hhline{~-}
 & $A_\infty^1 = 2804.87$\,kg \\
\hline
\multirow{6}{*}{\centering PTO} & $p_\text{pu} = 8.690$\,bar \\
\hhline{~-}
& $l_\text{pu} = 2.868$\,m \\
\hhline{~-}
& $S_\text{pu} = 0.0126677$\,m$^2$ \\
\hhline{~-}
& $p_\text{pl} = 8.69$\,bar \\
\hhline{~-}
& $l_\text{pl} = 0.012677$\,m \\
\hhline{~-}
& $S_\text{pl} = 0.0115276$\, m$^2$ \\
\hline
\multirow{2}{*}{Hydraulics} & $A_\text{oil} = 0.000887$\,m$^2$\\
\hhline{~-}
 & $D_m = 0.77986 \times 10^{-6}$\,m$^3$/rad\\
 \hline
\multirow{1}{*}{Generator} & $J_\text{gen} = 1.13 \times 10^{-3}$\,kg/m$^3$\\
\hline
\end{tabular}
%
\end{table}

\section{Numerical and Field Test Results}
Two types of numerical studies have been performed, the first to evaluate the systems performance with various values of linear damping coefficient and submerged plate added mass.  The second explores the effects of latching control schemes for this system at various values of submerged plate added mass.  The submerged plate added mass is found to be an important parameter in both cases.  Additionally, results from field tests performed in May 2013 in Monterey bay are compared to the numerical predictions.

For the system characteristics shown in table \ref{tab:MBARI_chars}, the extracted power from the wave energy converter was studied as a function of two main parameters:
the system damping coefficient, defined as
%
\begin{equation}
C_\text{sys} = \frac {C_\text{gen} A_\text{oil}}{D_m},
\end{equation}
%
and the dimensionless plate added mass
%
\begin{equation}
\lambda = \frac{m_3 + A_3^\infty  }{m_1}.
\end{equation}
%
\par
For regular waves, the available power is given by
%
\begin{equation}
  P_\text{avai} = \frac{\rho g^2 H^2}{16 \omega}.
\end{equation}
%
\begin{comment}
\par
Under linear water wave theory, the resulting diffraction force for a spectrum is obtained as a superposition of $N$ frequency components
%
\begin{equation}
\label{eq5000kk}
F_\text{exc}(t) = \sum_{j=1}^N \Gamma(\omega_j) \, A_{\omega_j} \cos( \omega_j t +  \phi_{j} ),
\end{equation}
%
with
%
\begin{equation}
A_{\omega_j} = \sqrt{ 2 \, \Delta \omega_j \, S_{\omega,i} },
\end{equation}
%
\begin{equation}
\label{eq:delta_omega_j}
\Delta \omega_j = \left( 1 + \varpi_j \right) \Delta \omega ,
\end{equation}
%
\begin{equation}
\label{eq:omega_j}
\omega_j = \omega_{j-1} + 0.5 \left( \Delta \omega_j + \Delta \omega_{j-1} \right), \quad j=2,\ldots,N\,,
\end{equation}
%
where $\phi_{i}$ is a phase constant equal to a random number between 0 and $2\pi$, $\varpi_j$ is a random number between 0 and 0.2 and $S_{\omega,i}$  is the frequency spectral density function.
%
\par

\end{comment}
\par
Under linear water wave theory, the resulting free surface elevation time history for a spectrum is obtained as a superposition of $N$ frequency components
%
\begin{equation}
\label{eq5000kk}
\eta(t) = \sum_{j=1}^N \Gamma(\omega_j) \, A_{\omega_j} \cos( \omega_j t +  \phi_{j} ),
\end{equation}
%
with
%
\begin{equation}
A_{\omega_j} = \sqrt{ 2 \, \Delta \omega_j \, S_{\omega,i} },
\end{equation}
%
\begin{equation}
\label{eq:delta_omega_j}
\Delta \omega_j = \left( 1 + \varpi_j \right) \Delta \omega ,
\end{equation}
%
\begin{equation}
\label{eq:omega_j}
\omega_j = \omega_{j-1} + 0.5 \left( \Delta \omega_j + \Delta \omega_{j-1} \right), \quad j=2,\ldots,N\,,
\end{equation}
%
where $\phi_{i}$ is a phase constant equal to a random number between 0 and $2\pi$, $\varpi_j$ is a random number between 0 and 0.2 and $S_{\omega,i}$  is the frequency spectral density function.  The inclusion of the perturbation $\varpi_j$ has been shown to avoid repeating patterns appearing in the time records constructed from the superposition of frequency components \ref{?}.
%


The available power for a spectra is given by the weighted sum of the power of each sinusoidal wave
%
\begin{equation}
  P_\text{avai} = \frac{\rho g^2}{4} \sum^N_{\substack{i=1}} \frac{A_{\omega,i}^2}{\omega_i}.
\end{equation}


\subsection{Linear Damping Behaviors}
Figure \ref{fig:reg} shows the numerically predicted power across a range of damping coefficients, inertia ratios ($\lambda$), and wave periods. The mean power conversion under regular waves was computed for several wave periods $T=6, 8, 10, 12, 14$ and $16$\,s and a wave height of $H = 1.0$\,m.  The total simulation time considered was 1200\,s with a time step of $\Delta t = 0.1$\,s.

The reported average powers represent mechanical power and don't include inefficiencies in the mechanical to electrical conversion process. These basic results illustrate a strong dependence on the submerged plate added-mass, and a weak dependence on system damping coefficient.  Even at the smaller inertia ratios, the predicted performance degrades only slightly for high values of the damping coefficient.  This is important because as described above the hydraulic power-take-off device is most efficient at high loadings.

To estimate the power capture capabilities of the system in irregular waves, the numerical model was run with a range of spectrum's that represented the expected conditions at the test deployment site. Figure \ref{fig:May_Spectra} shows the characteristic spectra's used, these were derived from 5 years of measured data at the deployment site.  All the calculations were performed assuming $N=300$, $\omega_1 = 0.63$ rad/s and $\Delta \omega = 2.46 / (N-1)$ rad/s, see Eqs.~\eqref{eq5000kk}-\eqref{eq:omega_j}.   Each measured spectra during the 5 years of May measurements represents the conditions during a half-hour period, and each has an associated significant wave height.  All the measured spectra's for each range of significant wave-heights were averaged to create a representative spectra.  Figure \ref{fig:res_may} shows the model predictions and includes a data point that shows the actual average power production measured during the at-sea tests performed in May 2013 with a non-dimensional plate added mass of $\lambda = 5$.  Because the at-sea system logs electrical power, the data points shown on the plots are the measured electrical power divided by the power-take-off device efficiency factor that was measured in the lab, allowing comparison between the numerical and measured results.  The agreement is quite good in the milder sea-states but degrades in larger waves, due possibly to a discrepancy between the sea states actually present during the test for each wave-height range and the representative spectra used to drive the model.  A more accurate comparison would re-run the model for each and every sea-state encountered during the test and compare the predicted versus measured power production for each half-hour spectra integration period.  This has not been done at this point but the efficiency of the numerical scheme presented above makes this possible. 


\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{./Figs/Spectra_May.png}
\caption{May sea spectra.}
\label{fig:May_Spectra}
\end{figure}


\begin{figure*}
\centering
\subfloat[$T=6$\,s] {%
\includegraphics[scale=.32]{./Figs/Results//Imagens_regular/H1_Te6_Pmax14708.pdf}
}%
\subfloat[$T=8$\,s] {%
\includegraphics[scale=.32]{./Figs/Results//Imagens_regular/H1_Te8_Pmax19610.pdf}
}
\\
\subfloat[$T=10$\,s] {%
\includegraphics[scale=0.32]{./Figs/Results//Imagens_regular/H1_Te10_Pmax24513.pdf}
}%
\subfloat[$T=12$\,s] {%
\includegraphics[scale=.32]{./Figs/Results//Imagens_regular/H1_Te12_Pmax29416.pdf}
}
\\
\subfloat[$T=14$\,s] {%
\includegraphics[scale=0.32]{./Figs/Results//Imagens_regular/H1_Te14_Pmax34318.pdf}
}%
\subfloat[$T=16$\,s] {
\includegraphics[scale=.32]{./Figs/Results//Imagens_regular/H1_Te16_Pmax39221.pdf}
}
\caption{Mean power conversion, $P_\text{mean}$, as function of $\lambda$ and $C_\text{sys}$, for regular waves with $H=1.0$\,m and different periods.}
\label{fig:reg}
\end{figure*}

\begin{figure*}
\centering
%%	\subfloat[$H=0.0-0.5$\,m] {%
%%	\includegraphics[scale=.32]{./Figs/Results//Imagens_May_power_wFieldResults/H0-0_Pmax2599.pdf}
%%	}

	\subfloat[$H=0.5-1.0$\,m] {
	\includegraphics[scale=.32]{./Figs/Results//Imagens_May_power_wFieldResults/H0-1_Pmax6323.pdf}
	}
	\subfloat[$H=1.0-1.5$\,m] {
	\includegraphics[scale=0.32]{./Figs/Results//Imagens_May_power_wFieldResults/H1-1_Pmax14991.pdf}
	}
\\
	\subfloat[$H=1.5-2.0$\,m] {
	\includegraphics[scale=.32]{./Figs/Results//Imagens_May_power_wFieldResults/H1-2_Pmax28691.pdf}
	}
	\subfloat[$H=2.0-2.5$\,m] {
	\includegraphics[scale=0.32]{./Figs/Results//Imagens_May_power_wFieldResults/H2-2_Pmax48348.pdf}
	}
\\
	\subfloat[$H=2.5-3.0$\,m] {
	\includegraphics[scale=.32]{./Figs/Results//Imagens_May_power_wFieldResults/H2-3_Pmax73737.pdf}
        }
	\subfloat[$H=3.0-3.5$\,m] {
	\includegraphics[scale=0.32]{./Figs/Results//Imagens_May_power_wFieldResults/H3-3_Pmax106240.pdf}
	}

%%	\subfloat[$H=3.5-4.0$\,m] {%
%%	\includegraphics[scale=.32]{./Figs/Results//Imagens_May_power_wFieldResults/H3-4_Pmax137261.pdf}
%%	}
\caption{Mean power conversion, $P_\text{mean}$, as function of $\lambda$ and $C_\text{sys}$, for each of the May spectra.}
\label{fig:res_may}
\end{figure*}



\subsection{Latching Control Behaviors}
In an attempt to increase the amount of energy captured by the device, a latching control scheme was investigated as this would be one of the simplest advanced control schemes that could be implemented in the hardware.  Latching control is a technique of phase control developed by Budal and Falnes~\cite{BudalFalnes78} and consists in latching the device in a fixed position, during certain intervals of the wave cycle, to approximately bring into phase the body velocity and the excitation force on the body.  The advantage of this technique is that it does not require that the system power electronics return energy to the hydraulic system and could therefore be directly applied to the existing system.  The disadvantage of this approach is that it is suboptimal and has only been demonstrated to be effective in certain cases of two-body point absorbers.

Applying phase control by latching to the present system faces some difficulties:
%
\begin{itemize}
\item During the latched period, the pair floater/plate cannot be kept fixed, unlike the case of a single body reacting against a fixed reference, reducing latching performance. This situation would be expected to significantly improve if the inertia of the second body (plate mass plus added mass) increases.
%
\item The calculation of the latched period has to be performed in real-time, as function of the current sea wave conditions.
%
\item No latching while the cable is under compression forces (slack cable).
%
\item In order to limit brake forces on the system, the braking should occur approximately when the relative velocity between body 1 and body 2 is zero ($\dot x_r$ = 0)
%
\end{itemize}
%
\par
Taking this into account the implemented latching algorithm is shown in Fig.~\ref{fig:latch_algo} and it is based on \cite{Henriques201231}. The algorithm detects when the oil pressure crosses zero falling to start a timer that waits for a time period of $t_\text{unlatch}$ to unlatch the system. Then, when the system has zero relative velocity between the buoy and the plate is latched again. To avoid spurious latching orders due to cable oscillations a secondary timer is triggered which disables latching orders immediately after unlatching.
%
\begin{figure}
\begin{scriptsize}
\begin{verbatim}
switch (latch_stage) {
    case 0:
      // If oil pressure crosses zero falling then initiate timers
      //
      if( dp_oil(0) > dp_oil(Delta_tau) and dp_oil(Delta_tau) < 0.0 ) {
        UnlatchTimer.Trigger(t, sys.t_unl);
        LatchTimer.Trigger(t, sys.t_lat);
        LatchStage = 1;
      }
      break;
    case 1:
      // If the unlatched timer has elapsed then the system 
      // is unlatched
      if( UnlatchTimer.Elapsed(t) ) {
        LatchStage = 2;
        IsLatched = false;
      }
      break;
    case 2:
      // If relative velocity is zero and the secondary timer has 
      // elapsed then
      // latch the system.
      //
      // The secondary timer is used to avoid spurious 
      // latching orders due to oscillations in the cable.
      //
      if( dx_r(0) * dx_r(n_tau - 1) <= 0.0 and dx_r(n_tau - 1) < 0.0 
          and LatchTimer.Elapsed(t) ) {
        LatchStage = 0;
        IsLatched = true;
      }
  }
\end{verbatim}
\end{scriptsize}
\caption{Stages of the latching algorithm written in C++.
The code is executed once at the end of each time step $\Delta \tau$.}
\label{fig:latch_algo}
\end{figure}


Utilizing the numerical tools described above and the latching algorithm described, numerical results have been computed for regular waves and the mean power conversion results using phase control by latching are presented in Fig.~\ref{fig:res_latch} for regular waves with periods of $T = 8.0, 9.0, 10.0$ and $12.0$\,s.
The total simulation time was 1200\,s with a time step of $\Delta t = 0.005$\,s.

Results for the system without latching are shown in Fig.~\ref{fig:res_may} and demonstrate this scheme to be ineffective for this wave-energy converter, i.e. the latching algorithm tends to lower the average extracted power as the submerged plate movement.  Only for very high submerged plate added mass (very high $\lambda$) would latching start to be successful and these ratios are beyond what is practical to field.  The results shown are computed for a damping coefficient of $C_\text{sys}=20000$\,Ns/m and similar behaviors were seen for other values of damping coefficient.


\begin{figure*}
\centering
%
\subfloat[$T=8$\,s and $C_\text{sys}=20000$\,Ns/m] {
  \includegraphics[scale=.32]{./Figs/Results//Latching/H1_Te8_D_20000.pdf}}
\subfloat[$T=9$\,s and $C_\text{sys}=20000$\,Ns/m] {
  \includegraphics[scale=.32]{./Figs/Results//Latching/H1_Te9_D_20000.pdf}}\\
%
\subfloat[$T=10$\,s and $C_\text{sys}=20000$\,Ns/m] {
  \includegraphics[scale=.32]{./Figs/Results//Latching/H1_Te10_D_20000.pdf}}
\subfloat[$T=12$\,s and $C_\text{sys}=20000$\,Ns/m] {
  \includegraphics[scale=.32]{./Figs/Results//Latching/H1_Te12_D_20000.pdf}}\\
%
\caption{
Mean power conversion, $P_\text{mean}$, using latching control as function of $\lambda$ and unlatched time interval, $t_\text{unlatch}$, for regular waves with $H=1.0$\,m and different periods.  $t_\text{unlatch} = 0$ corresponds to linear damping with no latching effects.}  
\label{fig:res_latch}
\end{figure*}


\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{./Figs/Results/MBARI/RectifiedPowerVsWaveHeight.pdf}
\caption{Hourly electrical power averages throughout five test deployments.}
\label{fig:DeploymentComparisons}
\end{figure}

\section{Conclusions}

It is evident that the mean power extracted increases substantially with the $\lambda$ coefficient, i.e. with the submerged steel plate added mass.
For wave periods between 8\,s and 16\,s, the mean power curves present an optimal damping value, $C_\text{sys}$, although the total energy capture is not very sensitive to this parameter. The same analysis was performed for the spectra that represent the wave climate of May at the test site, see Fig.~\ref{fig:res_may}.
As in regular wave results, the mean power extracted increases with plate added mass ($\lambda$).  For the various sea states and $\lambda$ curves, $P_\text{mean}$ exhibits a ``plateau'' like behavior when increasing the damping coefficient, i.e. after a given threshold  $P_\text{mean}$ is weakly dependent  of the damping coefficient.  

Although the numerical simulations of latching did not lead to an increase in power capture in the prototype system, this work highlighted the importance of maximizing the submerged plate added mass as much as possible.  Figure \ref{fig:DeploymentComparisons} shows hourly averages of the power captured for five different test deployments, in different system configurations and sea conditions.  The May 2013 deployment had the largest submerged plate added mass and subsequently performed the best, especially at lower sea states.    For this application of wave-energy, the low sea-state performance is particularly important to maintain high average powers, i.e. it is not hard to meet the power goals in higher sea-states and the total energy capture quickly becomes limited by on-board energy storage limitations.  For this reason the low sea-state performance is the biggest factor limiting the average power this system can make available to systems it hosts. The lack of an increase in the measured results at higher sea states in figure \ref{fig:DeploymentComparisons}  is likely due to the fact that the system sheds peak pressures in the hydraulic system through a pressure relief valve.

The future holds many ambitious projects for oceanographic data-acquisition systems. At-sea charging stations for autonomous underwater vehicles (AUVs) and autonomous un-moored platforms for extended deployments are examples of such projects that are underway and that we will likely see come to fruition in the coming years. The development of energy harvesting techniques for these systems is important and we aim to advance those efforts.  In particular, this work has illustrated that for small wave-energy harvesting systems of this type, the most expedient way to increase the power capture is to increase the size, mass, and drag of the submerged element of the system and utilize simple linear-damping power take-off devices.  The limits to this approach become the large peak loads that result and the implications for storm survivability.  The natural development path is towards compliant sub-surface heave plates that can exert large inertial and drag forces in moderate sea states but that can shed power in larger sea states.



\begin{comment}
\section*{Nomenclature}
%
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\noindent \textbf{Abbreviations}
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    MBARI    \> = \> Monterey Bay Aquarium Research Institute \\
    DOF      \> = \> Degrees of Freedom \\
    IMU      \> = \> Inertial Measurement Unit \\
    PTO      \> = \> Power Take-Off \\
    WEC      \> = \> Wave Energy Converter \\
\end{tabbing}
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\noindent \textbf{Uppercase Latin}

\begin{tabbing}
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    $A_{ij}$             \> = \> added mass coefficients \\
    $A_{ij}^\infty$      \> = \> $\lim\limits_{\omega \rightarrow\infty}A_{ij}(\omega$) \\
    $A_\text{oil}$            \> = \> wet piston area \\
    $A_{\omega}$         \> = \> wave amplitude \\
    $B_{ij}$             \> = \> radiation damping coefficients \\
    $C_{D}$              \> = \> drag coefficient \\
    $C_\text{gen}^{\max}$     \> = \> user defined maximum damping\\
    $C_\text{sys}$            \> = \> damping coefficient \\
    $D_{m}$              \> = \> hydraulic motor volumetric displacement \\
    $F_{\text{ext}}$     \> = \> excitation or diffraction force\\
    $F_{\text{gas}}$     \> = \> pneumatic force\\
    $F_{\text{oil}}$     \> = \> hydraulic force\\
    $F_{\text{imp}}$     \> = \> impulsion force\\
    $F_{\text{ine}}$     \> = \> inertial force \\
    $F_{\text{PTO}}$     \> = \> power take-off force\\
    $F_{\text{r}}$       \> = \> radiation force\\
    $H$                  \> = \> wave height \\
    $J$                  \> = \> generator moment of polar inertia \\
    $K$                  \> = \> impulse response function \\
    $P_{\text{gen}}$     \> = \> generator mean output power \\
    $P_{\text{mean}}$    \> = \> mean output power\\
    $Q_{\text{m}}$       \> = \> hydraulic motor theoretical volume flow rate \\
    $S$                  \> = \> buoy water plane area\\
    $S_{\omega}$         \> = \> frequency spectral density function \\
    $T$                  \> = \> wave period \\
    $T_{\text{gen}}$     \> = \> generator torque \\
    $T_{\text{m}}$       \> = \> hydraulic motor torque \\ 
    $V$                  \> = \> volume \\
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\noindent \textbf{Lowercase Latin}

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    $a_i$                \> = \> column vector of unknown coefficients \\
    $f$                  \> = \> force vector \\
    $g$                  \> = \> acceleration of gravity \\
    $k_c$                \> = \> tether stiffness \\
    $m$                  \> = \> body mass \\
    $m_i$                \> = \> mass of body $i$ \\
    $t$                  \> = \> absolute time\\
    $p$                  \> = \> pressure \\
    $t_i$                \> = \> line vector with monomial coefficients \\
    $t^*$                \> = \> dimensionless time \\
    $x_i$                \> = \> vertical displacement of body $i$ \\
    $\dot{x}_i$          \> = \> velocity of body $i$ \\
    $\ddot{x}_i$         \> = \> acceleration of body $i$ \\
\end{tabbing}
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\noindent \textbf{Uppercase Greek}
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    $\Gamma(\omega)$     \> = \> excitation force coefficient \\    
    $\Delta p_\text{oil}$     \> = \> pressure loss through the hydraulic motor  \\    
    $\Delta \tau$        \> = \> time step \\    
\end{tabbing}

\noindent \textbf{Lowercase Greek}

\begin{tabbing}
    \hspace{40pt} \= \hspace{20pt} \=\kill    $\gamma$             \> = \>specific heat ratio \\
    $\varepsilon$        \> = \> mean error \\
    $\eta_m$             \> = \> hydraulic motor mechanical efficiency \\
    $\lambda$            \> = \> dimensionless plate added mass \\
    $\eta_v$             \> = \> hydraulic motor volumetric efficiency\\
    $\rho$               \> = \> water density\\
    $\tau$               \> = \> relative time\\
    $\phi_i$             \> = \> phase response\\
    $\psi$               \> = \> line vector with monomial coefficients \\
    $\omega$             \> = \> angular frequency \\
    $\omega_m$           \> = \> hydraulic motor angular velocity \\
    $\omega^*$           \> = \> dimensionless angular frequency \\
    
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