Nielsen, Kirsten M.

Pedersen, Tom S.

    Aalborg UniversitetA Dynamic Model of a Buoyancy System in a Wave Energy Power PlantPedersen, Tom Søndergård; Nielsen, Kirsten MølgaardPublished in:Proceedings of the 5th International Conference on Informatics in Control, Automation and Robotics - ICINCO2008Publication date:2008Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Pedersen, T. S., & Nielsen, K. M. (2008). A Dynamic Model of a Buoyancy System in a Wave Energy PowerPlant. In Proceedings of the 5th International Conference on Informatics in Control, Automation and Robotics -ICINCO 2008 Institute for Systems and Technologies of Information, Control and Communication.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: May 01, 2017A DYNAMIC MODEL OF A BUOYANCY SYSTEM IN A WAVE ENERGY POWER PLANT  Tom S. Pedersen, Kirsten M. Nielsen Department of Automation and Control Aalborg University, Fr. Bajersvej 7 Aalborg, Denmark tom@es.aau.dk, kmn@es.aau.dk  Keywords: Dynamic model, wave energy, simulation, buoyancy control, verification, renewable energy  Abstract: A nonlinear dynamic model of the buoyancy system in a wave energy power plant is presented. The plant (“Wave Dragon”) is a floating device using the potential energy in overtopping waves to produce power. A water reservoir is placed on top of the WD, and hydro turbines lead the water to the sea producing electrical power. Through air chambers it is possible to control the level, the trim and the heel of the WD. It is important to control the level (and trim, heel) of the WD in order to maximize the power production in proportion to the wave height, here the amount of overtopping water and the amount of potential energy is conflicting. Five separate air chambers, all open to the sea, makes the device float. The pressures in the air chambers may be individually controlled by an air fan through an array of valves. In order to make a model-based control system, this paper presents a model describing the dynamics from the air inlet to the level, trim and heel. The model is derived from first principles and is characterized by physical parameters. Results from validation of the model against plant data are presented.    1  INTRODUCTION  Renewable energy is an important issue due to the global warming problem and utilisation of wave power is one of the energy resources to be exploited.  The wave power system “Wave Dragon”, on which this paper focuses, was invented by Erik Friis Madsen, Löwenmark and tested at Aalborg University and University of Cork. An EU based European consortium has been involved in the construction and implementation of a 1:4 scaled test site - 57x27 m wide and with a weight of 237 tonnes- which is placed in Nissum Bredning in Denmark. Large numbers of tests have been carried out during a two years operating period. One goal for energy production improvement is a better control of the Wave Dragon buoyancy. Wave Dragon (WD) is an offshore wave energy converter of the overtopping type, a description is found in (Kofoed, 2006) and (W.D.Aps, 2006). The main structure consists of a ramp where the waves are overtopping and led to a reservoir (basin). Two reflectors are focusing the waves towards the ramp as seen on figure 1. WD is fastened to an anchor making it possible to turn the ramp towards the dominant wave direction.     Figure 1: Main components of the Wave Dragon (Kofoed, 2006).  The WD use the potential energy of the waves, meaning that for a given wave type there exist an optimal level of the reservoir. As shown on figure 2 the reservoir water is led through a turbine. The WD floats on open air chambers used to adjust the floating level. Control of the floating level is a part of optimizing the overtopping and a dynamic model for a model-based control system is the topic of this paper. It should be noted that the wave conditions are measured online and may be used as reference to the level control system.    Figure 2: The basic principle of WD showing waves loading the reservoir via the ramp. (Kofoed, 2006). First the wave dragon buoyancy system is presented. A dynamic model of the air supply system controlling the pressure in the air chambers is set up. The model parameters have been adjusted to the test site Wave Dragon. Finally the buoyancy model is verified by comparing simulation results and measurements from the test site Wave Dragon.  2 DYNAMIC MODEL OF THE BUOYANCY SYSTEM  The buoyancy system consists of five air chambers all open to the water surface. The five chambers are shown on figure 3 (21,22,23,24,25,26,27), (15,16,18,19,20), (3,4,5,10,11), (1,2,8,9,14) and (6,7,12,13,17). Furthermore 9 small chambers contain a constant amount of air. The air pressures in the chambers are controlled by an air supply system using an on/off driven air fan and input/output valves to each chamber. The valves are operated as on/off valves and only one valve is allowed to be active at the time in order to prevent pressure equalizing in the chambers. A PWM scheme is in (Andersen, 2007) proposed to handle this problem.  The air supply system model consist of two parts, one describing air inlet to the chamber and one describing air flow out of the chamber. In both models tube pressure drops are ignored. The inlet air mass flow, mai, is given by     pbacpaai KppKm +−−= )(   which is an approximation to the fan characteristic.  pc is the chamber pressure, pa is the inlet pressure to the fan (atmospheric pressure) and  the two constants Kpa, Kpb  are from the fan data sheet.  Figure 3: The air chambers in the Wave Dragon.   Air outlet mass flow, mao, is given by the Bernoulli equation:  acvoao ppKm −=           where the constant Kvo depend on the outlet tube dimension and the air density. Each input/output valve pair is controlled by a signal u, where u=1 allows an airflow into the chamber, u=0 closes both valves and u=-1 open the outlet valve. This gives the air mass flow equation:  )()( 21 ugmugmm aoaia −=                     (1)  where          Figure 4: Fan/valve system.   ⎩⎨⎧<≥=1011)(1 uifuifug                ⎩⎨⎧−>−≤=1011)(2 uifuifug            Eq. (1) describes the air mass flow to the chamber and is valid when only one chamber is operated at a time.  3 ONE CHAMBER DYNAMIC MODEL  This section outlines the dynamic describing equations for a single chamber. The equations assumes that the air density is constant (variations could be included using the ideal gas equation), the chamber is only moving along a vertical axis perpendicular to the water surface, the cross section area is constant along this axis and there is only one rigid moving body. The model then consists of a mass balance equation describing the air in the chamber and a Newton equation describing the motion of the chamber.       Figure 5: One chamber model variables.   The mass balance equation for the chamber is  dthhdAmdtdMaaa )( 11111 +==Δρ         (2)  where  Ma1 is the air mass in the chamber, ma1 is the mass flow to the chamber, A1 is the cross section area, ρa is the air density,  is the distance from the chamber water surface to the ambient water surface and hΔ1h1 is the distance from the ambient water surface to the top of the chamber.  The pressure force from the chamber is assumed to equal the buoyancy force. This implies that the acceleration of the water volume in the chamber is small.  1111111cwcw pghApgAh ρρ =⇒=ΔΔ                                                   (3)  where ρw is the water density.  Insertion of Eq. (2) in Eq. (3) gives  dtdhgmAgdtdpwaawc 1111 ρρρ −=                                                    (4)  The other main equation for describing the dynamics of the chamber is Newton’s 2’th law used on the free body with the mass Mc1.  Pressure forces, gravity and a friction force proportional with the chamber velocity is assumed to act on the body.  dtdhKgMAppdthdM fcacc 111112121 )( −−−=    (5)  In order to use an ODE-solver to simulate the one chamber model, the three states , 11 cpx = 12 hx =  and dtdhx 13 =  may be selected resulting in the differential equation system  ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−−−−+−−+−=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡311113211113321)())()())(((xKgMApxxugpxKugKpxKAggxdtdxdtdxdtdxfcaavopbapaaww ρρρ                                                                              (6)  4 FIVE-CHAMBER DYNAMIC MODEL  In this section a 5 chamber dynamic model is described. Although the WD has a complex geometry the model assumes that each of the five chambers may be regarded as a control volume with position independent internal variables and that each chambers pressure force actuate the WD bode in a single point.  The model consist of 8 differential equations, 5 equations describing the pressures in the five chambers derived from mass balance equations, and 3 equations describing height, trim and heel derived from Newton’s law.  The five chambers are placed in a “wave dragon” coordinate system {WD} with the coordinates ),,( nnn zyx . An inertial coordinate system {I} is placed in the water level as shown on the figure. The states in the model are the five chamber pressures (pc1, pc2, pc3, pc4, pc5), the trim angle (θ), the heel angle (γ) and the level (h) of the wave dragon.    Figure 6: Wave dragon coordinate systems.   The mass balance for the n’th chamber is  dtdhgmAgdtdp nwananwcn ρρρ −=   hn is the height in the chamber and not a model state. Using a rotation matrix (see (Craig, 1989)), h),,( αγθRIWDn and the states are related through  hzyxhnnnn+++−=)cos()cos()sin()cos()sin(θγθγγ  The time derivative of hn is a tedious equation but using the approximations cos(θ)= 1, cos(γ) = 1, sin(θ) = θ, sin(γ)= γ assuming small heal and trim angles (all zn’s are 0) gives the simple relation  dtdhdtdydtdxdtdhnnn ++= θγ   which inserted gives   )(dtdhdtdydtdxgmAgdtdpnnwananwcn ++−= θλρρρ                                              (7)  Newton’s 2’th law for the translational system is  2251)(1dthdgApMdtdhKApgMhKpAwwwdfwwwdflhncnn+=−−−+∑=       (8)   where the sum is chamber pressure forces acting on the wave dragon body. The second term is the forces from the air chambers containing a constant air mass (see figure 3). In the model these are approximated with two chambers. They may be modelled using the mass balance Eq. (4) but in order to keep the model order low they are modelled using a static balance.  Kflh may be found using the ideal gas law and the geometrics of the chambers.  Mwd is the total mass of the wave dragon. The term pwAw represents the force from the water reservoir on top the Wave Dragon. It may be noted that the pressure pw is an input to the model.  The rotational trim equation is   22251 dtdJdtdKhKypA flnncnnθθθ θθθ =−−∑=                                                                     (9)  where the sum is the torque from the chambers. The second term is the torque from the constant air mass chambers modelled as two symmetrical chambers. It should be noted that Kflh as well as Kflθ are very dependent on the total air mass in the chambers. The moment of inertia is calculated as a constant although it depends on the water level in a very complex manner. In the simulation a situation with low water level has been used.  The heel equation is   2251 dtdJdtdKKypA flnncnnγγγ γγγ =−−∑=                                                                       (10)  This linear second order equation captures the gross behaviour of the heel dynamics.  The total model now consist of eleven equations, equation (7) used 5 times for the five chambers giving the five states , 11 cpx = 22 cpx = , , ,  , equation (8) with the states 33 cpx = 44 cpx = 55 cpx =hx =6 ,  dtdhx =7  and the equation 76 xdtdx =  gives 2 equations, equation (9) with the states θ=8x , dtdx θ=9  and the equation 98 xdtdx =  gives 2 equations, and finally equation (10) with the states γ=10x , dtdx γ=11  and the equation 1110 xdtdx =  gives 2 equations.  Inserting the states ( ) gives eleven nonlinear first order equations. The equations are solved using an ODE-solver. 1121 ,...,, xxx 5 SIMULATION  The model is tested using measured data from the wave dragon. The inputs to the model are the chamber pressures as well as the water level in the dragon represented by the pressure measurement pw. The reason for not using the valve signals is that these were not recorded in the data acquisition equipment. As seen there is a good agreement between level in the model and the experimental data. (The agreements are found on the measurements from the same day. Measurements on different days are based on different initial pressures causing identification of slightly different model parameters)   (This is also found on data measured on the same day.) Because the pressures in the constant air chambers are not measured these have been estimated from steady state observations.    Figure 7: Level simulated and measured.    Figure 8: Heel simulated and measured (Andersen, 2007).      Figure 9: Trim simulated and measured (Andersen, 2007).   The variations on the heel and trim angles are small. As seen on the figures the behaviour is captures by the model. All the data were recording prior to this project and not prepared for this study, unfortunately the WD run severely aground during the project meaning that controlled input signal could not be tested. Regardless of this the model performed well on the recorded data.   6  CONCLUSION  A nonlinear physical model with a complexity that is suitable for model based control has been presented. The model is partly based on physical parameters for the Wave Dragon and may be scaled to a future larger version. The model has four main equations, one describing the state of the air in a chamber, and three accounting for the level, trim and heel motion of the WD. The model has been validated against measured WD data, where it captures the gross behaviour of the Wave Dragon. In particular it describes the response of the level very well. The model does, however, have one serious deficiency because it does not capture the distribution and movement of water in the water reservoir. A comprehensive study of this is outside the scope of this paper.   REFERENCES  Andersen, J., Hundsdahl, M.Y., Jensen, P.K.,  Vilbergsson, K.S., Vidarsson, O., Skagestad, R., 2007, Control of Wave Dragon Buoyancy, master thesis, Aalborg University. Craig, J.J., 1989, Introduction to Robotics, mechanics and control,  ISBN 0-201-09528-9. Kofoed, J.P., Frigaard, P. Friis-Madsen, E. Sørensen, H.C. 2006, Prototype testing of the wave energy converter wave dragon, Renewable Energy 31  181-189.  W.D. Aps , 2006, Wave Dragon – principles, http://www.wavedragon.net  

A Dynamic Model of a Buoyancy System in a Wave Energy Power Plant

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A Dynamic Model of a Buoyancy System in a Wave Energy Power Plant

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