**ABSTRACT**

The objective of this project is to develop a model that will integrate multiple buoys to a power take-off hub. The model will be derived using a time domain analysis and will consider the hydraulic coupling of the buoys and the power take-off. The derived model is reproduced in MATLAB in order to run simulations. This will give possibility to conduct a parameter study and evaluate the performance of the system.

The buoy simulation model is provided by Wave4Power (W4P). It consists of a floater that is rigidly connected to a fully submerged vertical (acceleration) tube open at both ends. The tube contains a piston whose motion relative to the floater-tube system drives a power take-off mechanism. The power take-off model is provided by Ocean Harvesting Technologies AB (OHT). It comprises a mechanical gearbox and a gravity accumulator. The system is utilized to transform the irregular wave energy into a smooth electrical power output. OHT’s simulation model needs to be extended with a hydraulic motor at the input shaft. There are control features in both systems, that need to be connected and synchronized with each other.

Another major goal within the thesis is to test different online control techniques. A simple control strategy to optimize power capture is called sea-state tuning and it can be achieved by using a mechanical gearbox with several discrete gear ratios or with a variable displacement pump. The gear ratio of the gear box can be regulated according to a 2D look up table based on the average wave amplitude and frequency over a defined time frame. The OHT power take-off utilizes a control strategy, called spill function, to limit the excess power capture and keep the weight accumulator within a span by disengaging the input shaft from the power take-off. This is to be modified to implement power limitation with regulation of the gear ratio of the gearbox.

**PROJECT GOAL**

Currently, OHT uses a generic buoy model, which was designed by Norwegian University of Science and Technology (NTNU). In this thesis, this buoy model is replaced with the W4P buoy model to benchmark them against each other. OHT’s model is extended with a hydraulic motor as an interface between the input shaft and hydraulic circuit. The hydraulic circuit is located between the buoy and the hub before the hydraulic motor as shown in the Figure 2.1.

The dynamics of the OHT hub should not prevent the piston of the W4P buoy from moving, i.e the piston must not get stuck in the wider area of the tube. In order to avoid that, a control function should be implemented which will make the piston return into the narrowing by releasing the pressure in the cylinder. Unlike the W4P model, the hydraulic interface will include the inertances and the friction of the fluid both in the high-pressure and low-pressure pipes from the buoy to the power take-off and the return pipe from the power take-off to the buoy respectively as illustrated in the Figure 2.2.

**BACKGROUND**

A check valve is a valve that allows fluid to flow through it in only one direction. A check valve can be modelled as an orifice with variable and saturated cross-section area. The cross-section area of the orifice is presented as a function of the valve pressure drop in the Figure 3.2.

A gas-charged accumulator is shown in the Figure 3.3. The main equation, that is used to analyse the gas characteristics in the accumulator, is the ideal gas law given by

where P is pressure, V-volume, T-temperature, R is universal time constant, n is the number of moles, and k is the ratio of the specific heat at constant pressure and specific heat at constant volume, k = c_{p}/c_{v}. If we assume there is no heat transfer to the environment, the process is reversible, adiabatic and is represented by

where subscripts refer to states 1 and 2, respectively.

**HYDRAULIC INTERFACE MODEL**

Waves4Power buoy, contains a hydraulic cylinder with a piston which pumps fluid bidirectionally. The piston is driven by the motion of the water piston, relative to the floater-tube system. The flow, generated by the hydraulic actuator, is rectified by a Graetz bridge and smoothed by a bladder accumulator prior to driving a hydraulic motor. The hydraulic motor is loaded by a non-linear generator. The hydraulic system of the Waves4Power model is shown in the Figure 4.1.

In this section the the connection of multiple Waves4Power buoys to the OHT hub is modelled. The model of the hydraulic interface is shown in the Figure 4.4. The hydraulic interface is extended with two more bladder accumulators. One of them is placed after the junction, where the flows in the high pressure links flow in, and the other one is placed before the junction, where the flows to the low pressure links are distrinuted.

This section describes the derivation of the model for connecting multiple wave buoys to the OHT collection hub system. In the model of a single buoy, connected to a power take off with a planetary gearbox and gravity accumulator, is presented. In order to connect multiple buoys, the previous model is extended with a hydraulic interface shown in the Figure 4.5.

**MODELLING LOSSES IN THE HYDRAULIC INTERFACE**

The oil assumed in the motor has viscosity of 40 cSt/187 SSU. The volumetric losses for the stated oil viscosity can be derived from the Figure 5.3. The figure presents leakage of flow at the operating pressure. To derive the volumetric losses as percentage it is enough to map the motor on the figure and use. Considering motor with zero leakage leads to η_{mech} = η_{total}, having in mind.

**FORCED OSCILLATION AND BACK-STOP FUNCTIONALITY**

The buoy can be represented by a simple mechanical oscillator in the form of a mass-spring-damper system. A mass m is suspended through a spring S and a mechanical damper R, as shown in the Figure 6.1.

Because of the application of an external force F the mass has a position displacement x from its equilibrium position. Newton’s law gives

where the spring force and the damper force are F_{R} = − S_{x } and F_{S} = − R_{x}, respectively.

The need for back-stop functionality is more evident in mild sea-states with relatively high load damping force on the buoy. The Figure 6.2 demonstarates the need for back-stop at H_{s }= 2 . 25, T_{e} = 6 . 5 and F_{PTO} = 150 KN. It can be seen that without back-stop functionality the energy flows back from the machinery and pushes the buoy in the opposite direction and tries to damp it with the same force. And the Figure 6.4 shows the control signal to stop the energy from flowing back. As it shows, it is quite often in this interval of 50 seconds.

**WAVE-TO-BUOY DISTANCE AND DELAY**

A MATLAB function is created to calculate the distances between the wave and each buoy. Inputs to the function are the radius of the buoy circle (distance between the hub and each buoy), number of buoys and angle (direction) of the wave, defined with respect to a reference line. The placement of the buoys and the hub, along with the wave and the reference line are shown on the Figure 7.2. There are several conventions, used to calculate the distances between each buoy and the wave. Polar coordinates are used to find the placement of each buoy. The hub is placed in the origin of a polar coordinate system. All the buoys are at a distance r far from the hub. The first buoy is assumed to be at an angle − 90^{◦}.

**IMPLEMENTATION IN SIMULINK**

Input to the Heaving Buoy block is the loaded excitation force and the load force from the rack and pinion that connects the buoy to the hydraulic pump. The Hydraulic System block takes as input the buoy velocity, which is an output from the Heaving Buoy block, and the pressures of hydraulic accumulators around the motor. Output from the Hydraulic System block are the flows in the high and low pressure pipes and the load force on the buoy. The content of the Heaving Buoy block is shown in the Figure 8.4.

The pipe blocks (low and high pressure) models flow of fluid though the pipes and check valves, shown in the Figure 8.9. The inertia of the fluid and the losses inside the pipe are modelled. The check valve is modelled as a discrete event system with an SR flip-flop. If the flow gets equal to 0, the output of the flip-flop Q is set to 1, which resets the integral that has accumulated flow and sets the flow inside the pipe to 0. There is no flow until the sum of the torques gets greater than 0. The SR flip-flop is used to memorize the state of the check valve.

In order to prevent the water piston from getting stuck in the wide area of the tube a flow-control valve is implemented in SIMULINK as a discrete event system with two states. The controller checks the leakage area between the water piston and the tube. If the area exceeds a threshold, the state of the SR flip-flop, shown in the Figure 8.20, is set to 1 and the load force on the hydraulic piston equal is set to 0. The state of the flip-flop resets when the leakage area drops below some threshhold and the load force on the piston gets different than 0. This makes the water piston to move into the narrowing that facilitates power capture.

**CONTROL STRATEGIES**

In order to prevent that an electronically controlled valve is added after the rectifier that connects the high pressure side of the hydraulic circuit to the low pressure one once the valve is opened. This enables the fluid to flow from the high pressure accumulator to the low pressure accumulator. That makes the pressure difference reduce and the water piston return back to the narrowing. The valve is opened once the leakage area between the tube and the piston exceeds some threshhold and it is closed again when the leakage area is equal to the leakage area of the narrowing, i.e. once the piston is in the narrowing again. This strategy that prevents the water piston from getting stuck is called bypass control. The location of the valve is shown in the Figure 9.1.

**VALIDATION OF THE PLANETARY GEARBOX MODEL**

The sign in front of each term can be obtained by looking at the equations that describe the dynamics of each shaft, defined. Figure 10.3, obtained by simulating the frictionless model of the OHT’s hub with five NTNU buoys, displays the carrier torque calculated in three different ways using the torque ratios. From the Figure it can be seen that there is a perfect match.

**RESULTS**

To elaborate how the power is lost in the hydraulic hosing, the difference in average power on hydraulic pump and hydraulic motor is shown on the second column of table 11.1. It shows the captured power on the pump is always bigger than the motor which demonstrates the flow in the system. It can be also seen, as the length of the pipe increases linearly, the lost of power also increases linearly. This matches the linear dependency of length for pressure drop too.

Source: Blekinge Institute of Technology

Authors: Ashkan Ghodrati | Ahmed Rashid

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