Hadi Mirab, Reza Fathi, Vahid Jahangiri, Mir Mohammad Ettefaghand Reza Hassannejad

Energy Harvesting From Sea Waves With Consideration of Airy and JONSWAP Theory and Optimization of Energy Harvester Parameters

Hadi Mirab, Reza Fathi, Vahid Jahangiri, Mir Mohammad Ettefagh*and Reza Hassannejad

Faculty of Mechanical Engineering, University of Tabriz, 5166616471, Tabriz, Iran

One of the new methods for powering low-power electronic devices at sea is a wave energy harvesting system. In this method, piezoelectric material is employed to convert the mechanical energy of sea waves into electrical energy. The advantage of this method is based on avoiding a battery charging system. Studies have been done on energy harvesting from sea waves, however, considering energy harvesting with random JONSWAP wave theory, then determining the optimum values of energy harvested is new. This paper does that by implementing the JONSWAP wave model, calculating produced power, and realistically showing that output power is decreased in comparison with the more simple Airy wave model. In addition, parameters of the energy harvester system are optimized using a simulated annealing algorithm, yielding increased produced power.

energy harvesting; sea waves; JONSWAP; Airy wave model; piezoelectric material; beam vibration; simulated annealing algorithm

1 Introduction1

In recent years, application of low power electricity sensors and wireless communication systems has increased in various systems such as structural health monitoring (SHM), alarming devices and the marine industry. Floating radars and submarine sensors are among the most prominent examples of these sensors in naval usage. Energy supply for wireless sensors with separate sources, such as batteries, is undesirable for many reasons, e.g., they are quite bulky, have a limited life and store a finite quantity of energy.

Nowadays, progress in compact circuit technologies such as sensors and wireless systems have made it possible to use autonomous energy systems. These systems do not require an independent source to supply required energy, instead they acquire energy from environmental resources such as solar energy, wind energy and thermal gradients (Nagayamaet al., 2010; Parket al., 2010; Farinholtet al., 2010). But, each of the mentioned environmental resources has its own problems. For example, in the case of solar energy, insufficient sunlight is one of the disadvantages, so other sources of energy should be considered. In this regard, researchers proposed new methods of supplying energy which can convert mechanical vibration to electrical power. Among the various mechanisms which are used to convert vibrational energy to electrical energy are electromagnetic, piezoelectric, and electrostatic. Piezoelectric material has been noticed more than the others, due to the simple structure and convenient usage. Many studies have considered harvesting energy using piezoelectric converters.

In one study (Williams and Yates, 1996) the possibility of energy harvesting through bridge vibrationwas investigated. They measured bridge vibrations induced by passing vehicles. Furthermore, they assumed vibration of the bridge as an input excitation of a discrete 1 DOF model energy harvester and calculated the output power. Erturk (2011) simulated a bridge-vehicle system as a beam with moving load. In first step of their study, transverse displacement for each point of the beam was calculated by analytical solution of the beam governing equations. Then, the output voltage of the piezoelectric harvester as a result of moving load excitationwinvestigated with two different approaches. Kimet al. (2011) studied the possibility of energy harvesting from bridge vibration and converting this energy to electrical energy by piezoelectric materials. By considering a wide variety of traffic conditions such as vehicle speed and weight, tests were performed with changing load amplitude and frequency, and the output voltage was measured. Aliet al. (2011) investigated the possibility of piezoelectric energy harvesters as energy scavenging devices in bridges, and considered bridge vibration from the movement of vehicles as the source of energy generation for the harvester. They determined the optimum locations for the harvester along the length of the bridge in order to maximize the scavenged energy. Wu and Lee (2014) introduced a miniature windmill-structured energy harvester that converted ambient wind energy into electricity to supply power for sensing and signal transmission in a forest fire monitoring application. Wang and Zou (2013) analyzed the effect of the interfacial properties on mechanical behaviors of a piezoelectric beam. They developed a model for experimental analyzing andcompared it with simulation of finite element method in ANSYS software. Liang and Liao (2012) provided impedance modeling and analysis for a piezoelectric energy harvesting system with different interface circuits, and they also proposed an equivalent impedance network and corresponding mechanical schematics of a general piezoelectric system. In addition, they discussed the differences between these two systems. Tayloret al. (2001) designed an energy harvesting system which used piezoelectric polymers to convert river mechanical flow energy to electrical power. Zurkindenet al. (2007) designed some similar wave energy converter devices from piezoelectric material, which were forced by the wave action at a characteristic wave frequency to harvest energy from ocean surface waves. Moreover, a multi-physics simulation was used to focus on various aspects, namely free surface waves, fluid-structure-interactions, mechanical energy input to the piezoelectric material, and the electric power output using an equivalent open circuit model. Murray and Rastegar (2009) presented a new two stage electrical energy generator for buoyant structures. The generator used the buoy’s interaction with the sea waves as a low speed input to an initial system, which excited an array of vibratory elements (secondary system) into resonance and harvests energy from the piezoelectric elements. The advantage of this system is that by having two decoupled systems, the low frequency and varying buoy movement are changed to more useful constant frequency or extremely higher frequency mechanical vibration. Xieet al. (2014a) studied energy harvesting from transverse ocean waves by a piezoelectric plate. Xieet al. (2014b) considered a sea wave energy harvester from longitudinal wave motion of water particles. They studied the effect of harvester and wave parameters on output electrical power generated from a piezoelectric energy harvester according to the Airy linear wave theory. Wuet al. (2015) investigated ocean wave energy harvesting with a piezoelectric coupled buoy structure, in which piezoelectric coupled cantilevers are attached directly to the buoy close to the ocean surface to absorb the transverse ocean wave energy.

In previous research, energy harvesting from sea waves by considering an exact model of wave theory such as JONSWAP random wave theory as well as optimizing harvester parameters, has not been studied. The JONSWAP wave model includes random characteristics of sea waves, so this model is an exact model of waves, in comparison to the linear Airy wave theory. This is beneficial because, in order to obtain more power from a harvester it is necessary torealistically optimize its parameters.

In this paper, an anchored energy harvester is considered according to both a regular Airy wave model and an irregular JONSWAP wave model. The vibration response and produced power, due to wave forces acting on the beam, are calculated. Parameters of the energy harvester are optimized via a simulated annealing algorithm for maximizing produced power. Additionally, to avoid fracture of the beam and damaged piezoelectric patches, some constraints are added to the object function.

2 Modeling of energy harvester system using Airy and JONSWAP wave theory

2.1 Vibration equation of beam under excitation of sea waves

A cantilever beam with lengthl, and piezoelectric patches which are attached to the beam, is the most current device for harvesting energy from vibration energy (Fig. 1). The beam is fixed to the seabed in depthdand the beam starts to vibrate when it is subjected to sea waves. This causes dynamical strain in piezoelectric layers which produces electrical power. Transverse motion of the beam can be expressed as:

wherem′ is the mass per unit length of the beam,EIandAare the bending rigidity and cross section area of the beam respectively,ρis the density,w（z,t） is the longitudinal displacement of the beam at positionzandfH（t,z） is the force resulted from sea waves. In fluid dynamics the Morison equation is used to estimate the loads from sea waves on offshore structures.

Fig. 1 Set up of the piezoelectric energy harvester (Xieet al., 2014b)

The Morison equation is used to calculate the horizontal force applied to the beam caused by sea waves, as follows (Morisonet al., 1950):

wherecD,cMandcmare the coefficients of the drag, inertia forces of the beam and the added mass respectively,ρwis the material density of the beam,handbare the thickness and width of beam respectively. Also,uxandaxare the longitudinal velocity and acceleration of the water particles in the sea respectively. Here, in order to determine velocity and acceleration for the Morrison equation, Airy regular and JONSWAP irregular wave models are considered.

2.2 Airy wave theory

Airy wave theory, which is also known as small amplitude wave theory or sinusoidal shaped wave theory, is the well-known theory of regular surface waves. In this theory assumptions are: sinusoidal shaped waves, and wave amplitude smaller compared to wavelength and water depth. According to this theory the surface elevation, longitudinal velocity and acceleration of water particles are computed from Eqs. (3)-(5) (Kumari Ramachandran, 2013).

whereH,T,kandω′ are wave height, time period of wave, wave number, and angular wave frequency respectively. Also for determining angular wave frequency and wave number in shallow water, Eq. (6) is used:

By assuming that longitudinal velocity of water particles is much bigger than longitudinal velocity of beam,ux??w/?t, atx=0, Eq. (2) is changed to Eq. (7) as follows:

where

Additionally, by assumingH=2 m andT=15 s, the Airy wave model shape which is determined from Eq. (3), is shown in Fig. 2.

Fig. 2 Regular Airy wave model shape

2.3 JONSWAP irregular random wave theory

Waves from wind have irregular and random characteristics. In other words, regular wave theories are imprecise approximations of waves but irregular wave theories are an exact approximation of waves .

JONSWAP theory is one of these irregular wave theories. Spectral density of these waves is introduced in Eq. (9), as can be seen in the following equation (Liu and Frigaard, 1999):

Irregular wave shape is determined from Eq. (12) and velocity and acceleration of water particles are computed by Eqs. (13) and (14) (Han and Benaroya, 2000; Haritos, 2007; Adrezin and Haym, 1999; Seyul, 2006; Carter, 1982):

Fig. 3 Irregular wave model shape

By considering the force which is applied to the beam,fH（t,z）, the response of Eq. (1) can be determined by applying mode summation and the variable separation method, using Eq. (15):

whereWi（z） is thei-th mode shape function andqi（t） is thei-th generalized coordinate. The mode shape function for transverse vibration of the beam is expressed as:

By consideration of the mentioned boundary conditions, Eq. (19) is determined as:

Determinant of coefficient matrix should be zero.

Characterization equation of the system is determined by expansion of Eq. (20) where,i-th roots of the equation are thei-th natural frequencies of the system. Also, thei-th generalized coordinate for Airy wave model is determined by considering forced vibration of the beam affected by the wave force as follows:

For the JONSWAP model, response in the time domain is determined by substituting Eq. (15) in Eq. (1), and also by the usage of mode shape or thogonality and discretization method. Now by solving the extracted discrete equations numerically for the wave forces, response in the time domain and vibration response of the energy harvester are computed. With knowledge of the displacement function of the beam affected by the wave force, the electrical charge and voltage produced by piezoelectric patches in timetare described in Eqs. (24) and (25) (Lee and Moon, 1990) as follows:

wherea,e31,c′vandN′（1≤pp≤N′） are piezoelectric length, constant, electrical capacity per unit width of the piezoelectric patches, and number of the patches. The generated output power in timetcan be written as:

Finally, the average output power is calculated from Eq. (27) whereTis the total simulation time.

2.5 Parameters optimization of energy harvesting system

2.5.1 Determining constraints of object function

Given known displacement of the beam, bending moment can be computed from Eq. (28) as follows:

Axial stress of the beam due to bending moment is determined with Eq. (29) which can be expressed as:

Loads applied to the beam are dynamic, therefore mean and alternating stresses are computed by Eqs. (30) and (31) as follows:

To avoid fracture due to fatigue, the Soderberg equation is used as follows (Shigleyet al., 2004):

where, distance from the neutral axis for the beam ish/2 and for the patches ish/2+h1. Additionally, by considering Fig. 4 which shows change of the bending moment in time domain, it can be seen that in the fixed end, bending moment is maximum, thus another constraint is added as Eq. (34):

Fig. 4 Change of bending moment in time domain

2.5.2 Implementation of annealing algorithm

Simulated annealing is a simple and effective optimizing algorithm for solving optimization problems. Gradual annealing is used by metallurgists to attain a solid condition by which energy is minimized. This technique includes heating the material to high temperature then cooling it gradually. The annealing algorithm is dependent on initial point, so a genetic algorithm is used to define this point. A genetic algorithm is a method for searching and optimizing,based on natural selection. This method improves population under specified selective rules. In order to optimize the object function, an initial population should be produced at first. In the current study the initial population has 100 members. In later stages, the population will be organized according to the object function. After the organization, half of the inappropriate population is omitted. Then by choosing parents among the remaining population, new data will be produced. In order to select parents, cumulative probability is used. By the use of a continuous crossover operator, each pair of parents produces two children, and half of the population is omitted at a later stage, so the population is constant. To prevent convergence of the algorithm to a local minimum, a mutation operator is used with a rate of mutation of 0.02 (Haupt and Haupt, 2004). After this stage, values of the object function are calculated for each of these parameters, and the stages are continued until convergence to an optimized answer.

The optimized answer is used as initial condition in the annealing algorithm. So start point, initial temperature, final temperature, number of iterations at each temperature, and also number of cyclic coolings are needed. Initial temperature in the first cycle is introduced in Eq. (38), and afterN′ times trial and error, the final temperature is determined from Eq. (39) as follows:

Neighboring points are created by determining random numbers on the right side and left side of a specific point from Eq. (41) as follows:

wherexc（j） is the specific point, Maxxi（j）is the end and Minxi（j）is the first point of each variable. Also by using a clipping function, if the neighboring points are bigger than an upper bound, they are shifted to the right bound and if they are smaller than lower bound, they are shifted to the left bound. As can be seen from Eq. (42) the magnitude of the object function in the specific point and neighboring point, is computed in temperature which is equal to initial temperature and the differences are calculated as:

If these differences are positive, the probability is calculated by Eq. (44) and it is compared to a random number with uniform distribution between zero and one. If the probability is bigger than the random number, a neighbor point is accepted, otherwise it is rejected and a new neighbor point is produced:

where DeltaEavgis Boltzmann coefficient and is introduced in Eq. (45) as follows:

Fig. 5 Flowchart of optimizing by annealing algorithm

3. Simulation and results

Properties of the energy harvester beam and characteristics of the waves, which are used in simulation, are given in Tables 1 and 2 respectively. By considering these properties, equations of the energy harvester system are solved by programming in MATLAB, and vibration response is determined. Then the output power produced from the energy harvester is computed. For validation of the system, parameters of the energy harvester for the Airy wave model (Xieet al., 2014b) are given in Table 3. Additionally, output power in this paper is compared with that obtained from (Xieet al., 2014b). As it can be seen from Table 3, differences between output power produced in this paper and Ref (Xieet al., 2014b) are small.

Table 1 Properties of the beam

Table 2 Properties of the sea and wave

Table 3 Comparison of output power results of this paper with Xieet al. (2014b) for validation purpose

Vibration response of the beam is simulated in time domain as illustrated in Fig. 7 and the corresponding spectrum is shown in Fig. 8. As can be seen from Fig. 8, dominant frequencies represent natural frequencies of the energy harvester system (2.04, 21.48, and 59.21 Hz) and wave frequency (0.123 6 Hz). It should be noted that natural frequencies of the system are found by Eq. (20), they are 2.03, 20.35 and 63.52 Hz, and also as was mentioned, the wave frequency is 0.14 Hz. Therefore, the simulation of the energy harvester system with consideration of JONSWAP wave model can be validated.

Now by consideration of parameters given in Table 3, simulated plots of generated electrical power, （）Pe tby Airy and JOWNSWAP wave models are shown in Figs. 9 and 10, respectively.

Fig. 6 Irregular JONSWAP wave spectrum shape

Fig. 7 Vibration response of the beam in time domain

Fig. 8 Vibration response of the beam in frequency domain

Fig. 9 Generated electrical power by considering nonoptimized properties and Airy wave theory

Fig. 10 Generated electrical power by considering nonoptimized properties and JONSWAP wave theory

Average output power using the Airy wave model is 52 W,for JONSWAP model this value is reduced to 21 W. It is clear that by considering a realistic model for waves output power is reduced. This shows the importance of precisely modelling sea waves.

Now, the optimized parameters of the converter are determined by the annealing algorithm, to increase the output power. As it was described before, values of the optimized parameters are computed by continuous genetic algorithm in order to find the initial point of the annealing algorithm. Then by using the annealing algorithm over ten annealing cycles, with 50 trials and errors in each cycle,p1=0.8 andp50=0.001, the optimized value for thickness, width, tip mass of the beam, length, and number of patches, are computed in the Airy wave model in order to increase the output power. Also values of the optimized parameters are given in Table 4. Fig. 11 illustrates convergence of the object function. By consideration of optimized parameters, given in Table 4, the generated power is illustrated for the Airy and JONSWAP wave models in Figs. 12 and 13 respectively.

By comparing Figs. 12 and 13, optimized parameters, with Figs. 9 and 10, related to non-optimized parameters, it can be seen that the optimized output power is higher.

To verify the results of optimized parameters, produced output power for optimized parameters and for those given by Xie et al. (2014b), the two wave models are compared in Table 5.

Table 4 Optimized parameters of harvesting system

Fig. 11 Variation of object function in annealing algorithm

Fig. 12 Generated electrical power by considering optimized properties and Airy wave theory

Fig. 13 Generated electrical power by considering optimized properties and JONSWAP wave theory

Table 5 Output power comparison with two wave models

Table 5 shows that the value of output power with optimized parameters, compared with the non-optimized parameters, is increased for both wave models. It should be noted that in (Xieet al., 2014b), the output power is 145 Watts whereas the length of the beam is 6 m and width of the beam is 3 m. However by increasing length and width of the beam, the expenditure on piezoelectric materials will increase which will cause cost increase. In this paper output power is increased without changing length of the beam. It is increased solely by optimizing the dimensions of the piezoelectric converter, without violation of fatigue constraints.

4 Conclusions

One of the new methods for providing electrical power for low power electronic devices at sea is energy harvesting systems. In this method the energy harvesting system is vibrated by the force of sea waves. This vibration causes dynamic strain in piezoelectric patches, and so electrical power is produced. In this paper, after modeling the energy harvesting system and deriving governing equations for the system, the output power is computed for both Airy and JONSWAP wave models. Then the output power is considered as an object function and, in order to avoid fracture of the beam and damaged piezoelectric patches, some constraints are added to the object function. Results show that, for optimized parameters of the energy harvesting system, the produced power increased. The results also show that the more realistic JONSWAP wave model yields lower output power than the more simple Airy wave model.

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The 12th ISOPE PACOMS Symposium (PACOMS-2016) will be held at Crowne Plaza Hotel in Gold Coast, Australia. The Symposium is being organized by the ISOPE PACOMS-2016 International Organizing Committee (IOC), and Local Organizing Committee (LOC). It will be held in Gold Coast, Australia hosted by Griffith University. Previous 11 ISOPE PACOMS symposia have successfully been held: Seoul 1990, San Francisco 1992, Beijing 1994, Busan 1996, Daejeon 2002, Vladivostok 2004, Dalian 2006, Bangkok 2008, Busan 2010, Vladivostok. 2012 and Shanghai 2014.

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The topics include, but are not limited to:

◆ Challenges in Surface and Subsea Design and Installation Technology and Simulation

◆ Hydrodynamics & CFD

◆ Coastal Engineering, Integrity Management

◆ Offshore Mechanics, Offshore & Arctic Technology

◆ Offshore Wind and Ocean Energy

◆ Port, Harbour & Climate Change Adaptation

◆ Environment: Polar/Arctic and Ocean

Deadlines

Abstract submission: February 1, 2016

Tentative abstract acceptance notice: February 15, 2016

Manuscript for review: April 20, 2016

Contact

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meetings@isope.org

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Website: http://www.isope.org/call4papers/call4papers.htm

10.1007/s11804-015-1327-5

1671-9433(2015)04-0440-10

Received date: 2015-04-25.

Accepted date: 2015-07-17.

*Corresponding author Email: ettefagh@tabrizu.ac.ir

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015