Yang Li, Xiantao Zhang , Longfei Xiao

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, China

ABSTRACT The power capture performance of an adaptive bistable point absorber wave energy converter (WEC) in irregular waves is investigated in this paper.Based on the linear potential flow theory and Cummins equations, the equation of motion for the WEC is established in time domain.Then a parametric study is performed on the power capture performance of the WEC by considering the influences of different wave and system parameters such as the spring combination parameter, the stiffness of auxiliary springs and main springs, the original length value of the main spring, damping coefficient of the PTO systems and the significant wave height.The results show that in an irregular wave, each parameter will have a certain impact on the performance of the device.Appropriate device parameters need to be selected according to actual environmental parameters to ensure that the adaptive bistable WEC has better power capture performance than its linear and conventional bistable counterparts.

Keywords:Adaptive bistable Point absorber Power capture Parametric study

1.Introduction

The world’s consumption of electricity will increase by about 40% in 2040 [10] .Considering the limited and non-renewable characteristics of traditional energy sources such as petroleum, the future human demand for energy will no longer be able to rely solely on traditional energy sources, and it is necessary to find new energy supplements to meet demand.In addition, the environmental pollution caused by the mining and use of traditional energy sources has also attracted widespread attention [11] .Therefore, in order to effectively solve the problem of human potential huge energy demand, finding and developing an efficient, clean and environmentally friendly new energy source has become a hot topic for scientists.As a clean, environmentally friendly and renewable ocean new energy, wave energy has broad application prospects in the future.

The earliest research to explore the use of wave energy for power generation can be traced back to more than 200 years ago.As early as 1799, a research scholar in Paris, France, developed the first wave energy conversion device to successfully convert wave energy and applied for an invention patent [3] .In recent years,wave energy power generation devices represented by the oscillating water column type and the oscillating float type have attracted widespread attention.The power generation principle of the oscillating water column (OWC) wave energy power generation device is that the water column in the power generation device can reciprocate up and down under the action of waves.The continuous movement of the water column will cause the air column above the free surface of the water column to oscillate.The air vent above the chamber flows out, thereby smoothly converting the kinetic energy of the high-speed air into electrical energy[4] .Research has been carried out to improve the performance of the OWC device.In addition, He [5] also proposed to install OWC on the floating breakwater and conducted a series of wave flume experiments in regular waves to check the wave power extraction of the floating box breakwater with double aerodynamic chambers.He [6] also proposed a pile-supported OWC breakwater, which proved to be a promising multifunctional marine structure.The oscillating buoy-type wave energy generating device is a buoy-type device lurking under the water surface.The floating body (single point absorber) or two oscillating bodies (two-body system) are excited by ocean waves to produce relative motion,which is converted into electricity by the power-takeoff(PTO) systems.The power efficiency of the wave energy converter is a key issue for the oscillating float type device.One of the main disadvantages of traditional linear WEC is its poor performance when the wave period deviates from resonance.

In recent years, domestic and foreign research hotspots aimed at improving the power capture efficiency of wave energy devices have mainly focused on "using nonlinear power capture mechanisms." [20] Among them, a prominent advantage of the nonlinear system is that it may have a relatively high power capture efficiency under the random excitation of the broadband spectrum(that is, the energy distribution is more dispersed) [21 , 12].Among the different nonlinear mechanisms, the bistable mechanism (also known as the snap through mechanism or the negative stiffness mechanism) has attracted more attention in the field of vibration power capture, such as micro-electromechanical and other smallscale energy systems [7] .The bistable system has an important feature: when the external excitation is relatively small, periodic in-well motion (under periodic external excitation) will occur, but when the external excitation is relatively large, cross-well motion will occur [9] .The amplitude of cross-well movement is relatively large, which corresponds to a form of movement with high power capture efficiency.In the field of vibration energy, many studies have shown that when the bistable system can move across the well under external excitation, its power capture performance is significantly better than that of linear devices [8 , 15 , 22 , 23 , 14].

The application of bistable mechanism in the wave energy research is still relatively limited, although it has attracted a lot of attention.Zhang introduced the bistable mechanism into the float-type wave energy device, and used the frequency domain method to roughly explore the power capture performance of the device under regular waves [20] .Studies have shown that when the wave frequency is relatively low, the bistable float-type wave energy device captures more energy than the linear device, and the higher the barrier of the bistable mechanism, the greater the efficiency improvement [21] .At present, some scholars have carried out related research, mainly: Zhang et al.explored the power capture performance of bistable float-type wave energy devices (including regular and irregular waves), further confirming that the bistable mechanism can improve the energy capture efficiency of the device when the external wave frequency is relatively low [19] .Todalshaug et al.used two air pressure devices to form a negative stiffness mechanism (named "WaveSpring")and applied it to a float-type wave energy device.Their irregular wave test results showed that WaveSpring can make the device capture about three times more waves energy [13] , the"WaveSpring" technology was adopted by a float-type wave energy device designed by CorPower Ocean, a Swedish wave energy company ( http://www.corpowerocean.com/corpower-technology/wave- spring- technology/ ).Younesian and Alam use two inclined springs but different arrangements to achieve bistable and tristable mechanisms, and the following numerical research results show that this mechanism can improve the power capture efficiency of the float-type wave energy device while increasing the wave energy conversion Module equivalent damping bandwidth [17] .Numerical studies by Wang et al.showed that the bistable mechanism can improve the performance of submerged wave energy devices [14] .In addition to using springs and pneumatic devices to achieve bistable (or multi-stable), there are also other ways such as using a series of magnet devices to achieve bistable (or multi-stable) mechanisms [2 , 16].Zhang et al.proposed a new adaptive bistable mechanism to be applied to the float-type wave energy device [1888] .

In this article, the research is focused on the new adaptive bistable point absorber WEC proposed by Zhang et al.in 2018.Existing work is based on regular wave analysis to get the conclusion that the adaptive bistable mechanism improves the efficiency and frequency bandwidth of the device.However, the performance of the research device in irregular waves is more valuable for practical applications.For nonlinear power capture mechanisms, the results under regular waves cannot be directly linearly superimposed to obtain power capture under irregular waves.This subject will study the power capture performance of adaptive bistable wave energy devices under the action of irregular waves, and explore the effects of different device parameters on the capture capabilities.Based on the established time-domain equation, the dynamic response of WEC in irregular waves is numerically calculated.The power capture performance of adaptive bistable WEC is compared with its linear and conventional bistable WEC.It focuses on the influence of different system parameters on the power capture performance of adaptive bistable WEC, such as the spring combination parameter (which is defined as the ratio of half of the distance between two sliders at a time instant t to the original length of main springs) [18] , PTO damping, stiffness and length of auxiliary spring,stiffness of main spring, significant wave height, etc.

2.Motion analysis of the device in irregular waves

2.1.Description of the wave energy device

The direct-drive float type wave energy power generation device consists of a float and an energy output system.The energy output system consists of a linear generator and a connecting spring.The linear generator consists of a stator (induction coil winding) and a mover (permanent magnet shaft).This part can be regarded as a PTO system.The adaptive bistable wave energy capture device has been described in detail in Zhang’s paper [18] .The device is composed of two main springs and two auxiliary springs,one end of the auxiliary spring is fixed, and the other end is connected with the main spring through pulleys.The schematic diagram of the device is shown in Fig.1 .

2.2.Mathematical model for the wave energy device

For the dynamic response of nonlinear systems in irregular waves, the frequency domain method is no longer applicable.Therefore, Cummins proposed in 1962 the theory of time-domain equations applied to the motion response of ships on waves to solve the response of nonlinear wave energy generating devices in regular waves [1] .The motion control equation of the point absorber WEC with adaptive bistable power capture mechanism is as follows [18] :

Where ρis the liquid density, g is the acceleration of gravity; R is the radius of the floating body, M is the mass of the floating body, K is the main spring stiffness, l0is the initial length of the spring, l(t) is the horizontal distance between the slider and the rigid rod, and C is PTO damping, z(t) is the vertical displacement of the floating body, ˙ z (t) is the vertical speed of the floating body,is the acceleration of the floating body in the vertical direction; A(∞ ) is the added mass produced when ω = ∞ (for a vertical hemisphere, A(∞ ) = 0 .5M , Γ(t) is the delay function related to radiation damping B(ω) , which, for a vertical hemisphere, can be expressed as

fE(t) is the force received by the floating body; for the irregular wave studied in this article, the time-varying force acting on the hemispherical floating body can be expressed by the following formula:

Fig.1.Schematic of a point absorber WEC with a novel adaptive bistable power capture mechanism.

After receiving the movement brought by the floating body, the total amount of power captured by the PTO in a period of time can be expressed by the average captured power.

The power calculation formula is P = F ·is the motor damping force generated by the PTO system, so a certain instantaneous power is

Among them, C is the motor damping,τ) is the speed of the floating body moving in the vertical direction.Since the speed ˙ z (τ)of the appendage movement is a physical quantity that changes with time, if the average power over a period of time is calculated,it can be integrated and divided by the total length of this period of time, as follows:

Where T is the calculated duration.

Next, we will normalize the above equation.First, we define the following dimensionless variables (represented by the symbol *)

This paper studies the response of a bistable oscillatory wave energy generating device under irregular waves.Here, the JONSWAP spectrum recommended by the 17th ITTC meeting is selected to generate the random irregular wave time history, the energy density equation of the JONSWAP spectrum is

It is known that the relationship between the amplitude H and the energy density Sζ(ω) is

so

The motion Eq.(1) of the bistable oscillating wave energy power generation device can be obtained in the following form after dimensionless calculation:

The displacement z(t) and velocity ˙ z (t) corresponding to the equation can be solved by the fourth-order Runge-Kutta method, and then the velocity can be substituted into Eq.(6) to calculate the corresponding average power

3.Results

It should be noted that in this article, for different types of WEC, the comparison of power capture capabilities is only valid for the selected parameters.The choice of device parameters comes from the research of the device in regular waves.

The Fig.2 shows the power capture performance of the linear WEC, the conventional bistable WEC, and the adaptive bistable WEC.As shown in this Figure, for a relatively large significant wave height (i.e.= 0 .5 or 0 .8 ), the conventional bistable WEC’s performance in power capture is comparable to the adaptive bistable WEC, and are both far better than the linear one.However, in the case of a relatively small significant wave height(i.e.= 0 .1 or 0 .3 ), the superiority of the adaptive bistable WEC is very obvious.Especially when= 0 .1 , the adaptive bistable WEC has the largest averaged power= 4 .4 , which performed much better than the linear WEC= 0 .9 ) and the conventional bistable WEC (= 0 .46 ).This results are consistent with the performance of the devices in a regular wave.

Three motion characteristics of three WECs at= 0 .2 and= 0 .3 are presented in Fig.3 , to show the reasons for their different energy capture capabilities more intuitively, and to explain the phenomenon of violent oscillations of conventional bistable WEC in the low spectral peak frequency region(see Fig.2 (b)).It can be seen from Fig.3 that the amplitude and speed of the adaptive bistable WEC are significantly better than the other two, while the conventional bistable WEC has intermittent intra-well motion (see Fig.3 (b)), which leads to its energy capture efficiency is extremely unstable, and it will oscillate violently.

Since the adaptive bistable WEC reduces the potential barrier near the unstable equilibrium position, it can still achieve interwell oscillation.Therefore, it has better power capture performance.For the case where the significant wave height is relatively small, the conventional bistable WEC cannot overcome the potential barrier, so it can only perform in-well motion (smaller amplitude of motion).Therefore, as the significant wave height decreases, the adaptive bistable WEC performs better and better,while the conventional bistable WEC shows worse performance in power capture.

It can be seen from Fig.2 that the optimal peak frequencies of conventional bistable WEC and adaptive bistable WEC are offset compared to the linear one, which can be clearly shown in Fig.4 .For relatively large wave amplitude≥0 .3 , both the conventional and adaptive bistable WECs have a lower optimal peak frequency than the linear.This shows that the phenomenon of reducing the resonance (or natural) frequency of the conventional (or adaptive)bistable WEC still exists because of the negative stiffness in irregular waves [18] .But for= 0 .1 , the conventional bistable WEC has intra-well oscillation, which will bring extra positive stiffness to the system, so its optimal peak frequency is higher than the linear WEC.The adaptive bistable WEC has cross-well oscillations,which can cause negative stiffness, so the optimal peak frequency is lower than the linear WEC.

Fig.3.The motion characteristics for three types of WEC at = 0 .2 .= 0 .5 , = 0 .5 , = 0 .5 , = 0 .1 , C *= 0 .25 .(A)(B) the linear WEC, (C)(D) the conventional bistable WEC, (E)(F) the adaptive bistable WEC.

Fig.4.The optimal peak frequency for linear WEC, conventional bistable WEC and adaptive bistable WEC.

3.1.The effect of the spring combination parameterthe averaged power

This section investigates the effect of parameteron the power capture performance of adaptive bistable WEC.As mentioned earlier,is the ratio of half of the distance between the two sliders when both the main and auxiliary springs are at their original length to the original length of the main spring.

3.2.The effect of the stiffness of auxiliary springs on the averaged power

As indicated by Zhang et al., the stiffness of auxiliary springshas a significant effect on the potential function of the system for the adaptive bistable WEC [18] , and thus also affects the power capture performance of adaptive bistable WEC.The fixed dimensionless parameters for an adaptive bistable WEC are given as follows:= 0 .1 , K*= 0 .5 ,= 0 .5 , C*= 0 .25 .Three different significant wave amplitudes are adopted, i.e.= 0 .1 , 0 .3 , 0 .5 .A wide range of the stiffness of auxiliary springs arechosen withvarying from 0.05 to 20.

Fig.6.The power captured performance forunder different values of= 0 .1 ).

Fig.7.The optimal peak frequency at different

It can be seen from Fig.8 that the power capture performance of adaptive bistable WEC is significantly affected by the stiffness of the auxiliary springWhenis very small (such as=0 .1 ), the power capture performance curve of adaptive bistable WEC is very close to that of its linear device.Whenis very large(such as= 20 ), the power capture performance curve of adaptive bistable WEC is very close to that of conventional bistable device.Whenchanges from small to large, the power capture performance curve of the adaptive bistable WEC gradually changes from a linear device curve to a conventional bistable device curve.This phenomenon is consistent with the performance of the device in regular waves, and the relevant explanation is mentioned in Zhang’s article [18] .

It can be found from the curves of different significant wave heightthat whenis small (such as= 0 .1 ), the curve shape and optimal value change significantly with, indicating that the power capture performance of the adaptive bistable WEC is greatly affected by the stiffness of the auxiliary springWhenincreases, the curve shape and optimal value gradually decrease with the change of, which indicates that the power capture performance of the adaptive bistable WEC will gradually decrease under the influence of the stiffness of the auxiliary springwhen the high sense wave Hs is large.

Fig.9.The optimal peak frequency at different = 0 .1 .

Fig.10.The power captured performance for under different values of= 0 .1 ).

3.3.The effect of the stiffness of main springs K *on the averaged power

3.4.The effect of l0 *on the averaged power

As indicated by Zhang et al., the value of the non-linear parameter r determines the strength of the negative stiffness and the area where the negative stiffness exists [21] , thus affecting the power capture performance of adaptive bistable WEC.For a given non-linear parametervalue, the original length value of the main springwill also affect the negative stiffness, thus affecting the power capture performance.We choose a different set of valuesthe other geometrical parameters of the WEC are taken as follows:= 0 .5 , K*= 0 .5 ,= 0 .1 , C*= 0 .25 .As the adaptive bistable WEC performs well in low amplitude waves, we choose the amplitude of significant wave incident waves as= 0 .1 , andis inthe rangeof 0.02 ～2.Itcan beseen fromFig.12 that for all the values ofthe curve shape of power capture performance is similar with the change of peak frequency of irregular wave, it increases first and then decreases, which indicates that the device has an optimal peak frequency.However, it can be seen from Fig.12 that whenchanges, the optimal peak frequency of the device will change, and the specific changing trend is shown in Fig.13 .As shown in the figure, when= 0 .1 , the increase ofreduces the optimal peak frequency of the device, but there may be abnormal situations in some cases, such as= 0 .6 .The best peak frequency is basically maintained at about 0.5.

Fig.12.The effects of the parameter on the power capture performance of the adaptive bistable WEC. 0 .5 , K *= 0 .5 , = 0 .1 , C *= 0 .25 , H *s = 0 .1 .

However, the power capture performance of adaptive bistable WEC is affected by the original length of the main springin different ways in terms of the peak frequency of the irregular waves.In order to show more clearly the influence of the original length of the main springon the power capture performance, under different peak frequencies of irregular waves,Fig.14 shows the trend of the power capture performance of the adaptive bistable WEC withchanges under 10 groups of peak frequencies.It can be seen from the figure that for the optimal peak frequency (= 0 .5 ), the capture performance first increases and then decreases with the increase ofIn this case, it is found that the optimal original length of the main springis 1.When the peak frequencyis less than the optimal peak frequency (i.e.= 0 .1 ～0 .4 ), the capture performance shows an increasing trend asincreases.For high peak frequencies (i.e.= 0 .6 ～1 )), the capture performance decreases asincreases.

Fig.14.The power captured performance forunder different values of = 0 .1 ).

3.5.The influence of damping coefficient of the pto systems C *on the averaged power

The damping coefficient of the nonlinear WEC PTO system C*will affect the power capture performance of the adaptive bistable WEC, which may be different from the linear WEC.In the above analysis, we chose the best value of linear WEC (i.e.C*= 0 .25 )as the PTO damping value, but this is most likely not the optimum value of adaptive bistable WEC.Thus, we choose a set of different values of damping coefficient to investigate its effects on the power capture performance of an adaptive bistable WEC.The other geometrical parameters of the WEC are taken as follows:= 0 .5 , K*= 0.5 ,= 0 .5 ,= 0.1 .The results are given in Fig.15 .

It can be seen that, for different significant wave heightsthe variation of the powerwith PTO damping C*is quite similar.When the PTO damping gradually increased from a small value (i.e.C*= 0 .05 ), the power shows a significant increase trend.When the peak frequency is relatively small, the power also increases with the increase of the damping, but the effect is not such obvious.When the peak frequency is relatively large, especially when＞1 .2 , the powerhardly changes with the change in damping.In other words, when the damping is increased, the increase in ˉP*in the high-frequency region is not significant.

The optimum peak frequency is also influenced by the damping coefficient.Results are given in Fig.16 .It shows that the effect of the damping coefficient of the PTO system on the optimum peak frequency of the adaptive bistable WEC.Overall, as the damping coefficient increases, the optimal peak frequency appears to decrease.The effect of significant wave heighton the optimal peak frequency is also presented.There are also cases where the optimal peak frequency remains the same when the damping coefficient increases above a certain value (when0 .3 or 0 .5 ).

3.6.The effect of the significant wave height H s *on the averaged power

It is well known that the amplitude of the regular wave component is in proportional to the significant wave height of the JONSWAP spectrum, and the power obtained by the linear WEC varies in proportion to the square of the significant wave height.However, things might be different for the conventional bistable WEC.In the previous discussion, we have all considered the effect of changes inon the results.In order to further explore the effect of Hs on power capture performance, we choose a set of different values of significant wave heightto investigate its effects on the power capture performance.The other geometrical parameters of the WEC are taken as follows:= 0.5 , K*= 0.5 ,= 0.5 ,=0 .1 , C*= 0 .25 .The results are given in Fig.17 .

Fig.16.The optimal peak frequency at different C*.

It can be clearly seen from the figure that within the selectedrange ( 0 .05 -1 .5 ), whenincreases gradually from a small value (= 0 .05 ), both the maximum value of the power curve and the area under the curve show a relatively obvious increasing trend.Whenis constant, the power curve increases first and then decreases with the increase of the peak frequency.In other words, when the parameters of the device are all constant,the change ofdoes not affect the trend of the device’s power capture performance with the peak frequency, but the increase ofwill enhance the overall capture performance (the area under the curve) of the device.

However, it can be seen from Fig.17 that whenchanges,the optimal peak frequency of the device will change, and the specific changing trend is shown in Fig.18 .As shown in the figure,the increase inmakes the optimal peak frequency of the device appear to increase, but abnormal conditions may occur at some points, such as= 0.6, 0.7or 0.85.This is because the peak frequencycorresponding to the maximum power value is selected during the analysis, and the peak frequencyis calculated at intervals (every 0.02) during the calculation, which may cause deviations in the results.

Fig.17.The effects of the parameter on the power capture performance of the adaptive bistable WEC.= 0 .5 , K *= 0 .5 , 0 .5 , = 0 .1 , C *= 0 .25 .

Fig.18.The optimal peak frequency at different

We also selected some peak frequencies (= 0 .2 ～1 .2 ) to explore the effect of the significance wave heighton the power capture performance of the device under different peak frequencies.It can be seen from Fig.19 that the performance of an oscillating WEC with adaptive bistable power capture mechanism PTO systems is affected by the significance wave heightin different ways in terms of the peak frequencyof the irregular waves.The increase of the value ofleads to a rise in the powerFor the low or high peak frequency (= 0 .2 or 1 .2 ),the trend of increasing power with increasingvalue is not particularly obvious, and the increasing rate is also very slow.Whenis near the optimum frequency, the increase inwill greatly improve the power capture performance of the adaptive bistable WEC.

Fig.19.The power captured performance for under different values of

4.Conclusions

In this paper, the power capture performance of an adaptive bistable WEC in irregular waves is considered.The influences of wave parameters such as peak frequency, significant wave height,damping coefficient of the PTO system and the spring combination parameter of the adaptive bistable mechanism are discussed in detail.Conclusions can be made as follows:

a) The adaptive bistable WEC performs better than its linear and conventional bistable counterparts for a relatively small significant wave height, but this advantage is not obvious for large wave height.Considering the wave characteristics of real sea conditions, this research further proves the application value of the adaptive bistable device in practice, and demonstrates that the device is more adaptable to real sea conditions, especially in sea areas with low energy density and variable sea conditions.

c) When K*is constant, there is avalue in a certain interval.In this interval, the performance of the device is significantly improved, and there is a certainin the interval to make the power capture performance of the device optimal,and when it is larger or smaller than this value, the power capture performance of the device will be lower.As K*increases, the upper and lower limits of the interval will become smaller, and the interval width will also become smaller.In short, for a given value of the main spring stiffness K*,there exists a proper range of auxiliary spring stiffnessthat corresponds to a good power capture performance of the WEC.

d) If the PTO damping of the adaptive bistable WEC C*is increased on the basis of its linear optimal value, the performance of the device will be significantly improved.

e) When the parameters of the device are all constant, the change ofdoes not affect the trend of the device’s power capture performance with the peak frequency.When ωis near the optimum frequency, the increase in H*s will greatly improve the power capture performance of the adaptive bistable WEC.

f) The research on the influence of parameters in this paper will help the development of follow-up experiments.Firstly, it is verified that the adaptive device under irregular waves can also have certain advantages.Secondly, it clarified the environmental parameters in which the adaptive device can best exert its advantages.Thirdly, for the design of the device, especially the selection of the main spring and the auxiliary spring, a relatively clear optimal value interval is given, which will provide a selection basis for future experimental design.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work is supported by Shanghai Pujiang Program (Grant No.19PJ1405400), the 2020 Research Program of Sanya Yazhou Bay Science and Technology City (Grant No.SKJC-2020-01-006), and Hainan Provincial Natural Science Foundation of China (Grant no.520QN290).