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Practical considerations on nonlinear stiffness system for wave energy converter
Applied Ocean Research 92 (2019) 101935
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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Practical considerations on nonlinear stiffness system for wave energy converter Zhijia Wua,b, Carlos Levia, Segen F. Estefena, a b
T
⁎
Ocean Engineering Department, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil China Ship Scientific Research Center, Wuxi, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlinear stiffness system Mechanical compression spring Wave energy converter Pneumatic cylinder
Mechanical restoration of a point absorber (PA) wave energy converter (WEC) featured a nonlinear stiffness system built by conventional mechanical compression springs (NSMech). Numerical simulations conducted here considered irregular waves in a specific sea site. Results showed considerable improvement for a given suitable spring configuration (length and stiffness parameters). However, practical implementation of NSMech imposed limiting constraints on the geometrical and physical characteristics of the mechanical compression springs. Such constraints limited very much the feasible region of NSMech configurations, restricting significantly applications for wave energy converter. One alternative approach using pneumatic cylinder springs proved to overcome such a limitation aggregating obvious advantages with fewer elements and bringing more enhancement to the WEC performance.
1. Introduction The nonlinear stiffness system (NSS) is one structure which can provide nonlinear restoring force through some specially configured mechanical compression springs, buckling beams, or magnets. By adjusting its positive or negative contribution to the restoring effects on a given dynamic system, it can shift the corresponding natural frequency and/or broaden the response bandwidth to weaken or strengthen the response. Benefiting from such characteristics, NSS has been widely applied on vibration isolation [1–4] and vibration energy harvesting [5–9] engineering solutions. Recently, NSS also attracts the attention of wave energy converter (WEC) researchers because its natural frequency-shifting characteristic may increase the efficiency of point absorber (PA) type WEC. Both numerical and experimental investigations have been carried out to explore the implementation of NSS in PA. Zhang et al. [10, 11] study the application of snap-through power take-off (PTO) system in WEC and present extensive analysis and comparisons featuring the influences due to PTO damping, buoy geometry and dimensions, and wave frequency on wave energy conversion, in both regular and irregular waves. They have shown that if compared with linear PTO system, the nonlinear snap-through PTO system may improve the point absorber (PA) wave energy harvesting, especially for lower frequency waves. Almost at the same time, Todalshaug [12]
patented a wave energy converter which can perform better with the negative stiffness device. The device could feature either mechanical springs or pneumatic cylinders. The pneumatic configuration was patented as “WaveSpring” technique and used in the CorPower buoy prototype [13]. From there on, more and more studies on similar principles have been conducted, mostly numerically. Younesian and Alam [14] expanded the numerical analysis of the system working with multi-stable characteristics for different geometric parameters. Zhang et al. [15] proposed a novel adaptive bi-stable mechanism which could adjust the potential function automatically to lower the potential barrier near the unstable equilibrium position, and hence helping to solve the low-energy-absorption problem of conventional bi-stable wave energy converter. Wang, Tang and Wu [16] investigated the performance of bistable snap-through PTO implemented in a submerged surging WEC. Li et al. [17] proposed a system with four inclined mechanical springs based on the multi-stable mechanism and demonstrated its benefits on the wave energy conversion. Instead of controlling the PTO mechanism with the mechanical NSS directly, the WETFEET project featuring an oscillating water column (OWC) spar buoy [18] proposed two alternative negative stiffness concepts: (a) immersed varying volume (IVV) [19]; and (b) hydrodynamic negative stiffness (HNS) [20] to adjust the hydrostatic restoring stiffness. Harne et al. [21], Xiao et al. [22], Zhang et al. [23]
⁎ Corresponding author at: Federal University of Rio de Janeiro, COPPE, Ocean Engineering Department, P.O. Box 68508 – CEP 21945-970 - Rio de Janeiro, RJ, Brazil. E-mail address: [email protected] (S.F. Estefen).
https://doi.org/10.1016/j.apor.2019.101935 Received 13 April 2019; Received in revised form 21 July 2019; Accepted 9 September 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
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WEC with NSMech in the time domain. The linear potential flow theory was adopted to evaluate the system hydrodynamic characteristicsadded mass, radiation damping and excitation force response in the frequency domain. The PTO system was assumed to be linear. The only remaining nonlinear contribution to the problem will be due to the NSMech. In recent work, the effect of viscous damping was ignored by Penalba et al. [30] as they mentioned that the viscous effect on small heaving point absorber (HPA), in general, was low. Eq. (1) shows the general expression of the time domain equation of motion (EOM),
used the bi-stable mechanism with magnets. Besides most of the above analysis featured numerical aspects, some experimental analysis have also been carried out. A 1/16 scale tank test [24] was conducted to verify such an quite innovative concept. Experimental results have shown that it could be tuned to provide both resonant behavior and broadening response bandwidth. Compared with the latching control strategy, for instance, the “WaveSpring” produces similar power capture with a small fraction of the transmission force [25]. Therefore, it enables smaller and less costly PTO, and no need to predict the incident waves as it is required in the latching control. Recently, a similar concept was also tested using a 1/20 scale model (named “WaveStar”) by Têtu et al. [26]. “WEPTOS”, a pitching WEC with NSS, was tested at the Aalborg University wave basin tests [27]. The experimental results did not show any noticeable improvements, probably due to not accounted for friction effects. However, numerical simulations considering prototype scale, with no friction effects, indeed revealed quite positive benefits to increase the device efficiency. During this review work of the NSS implementations in WEC, all concepts were still at an early stage of development: either numerical simulation or small scale test. An exception is the “WaveSpring” applied on a 1/2 model scale of CorPower buoy with a dry test (2017) and sea trail (2018) but, in the authors acknowledgement with no further detail or results about its performance published yet. Most of the applications feature mechanical compression springs. However, to produce significant improvement for the WEC performance, the NSS parameter related to the spring lengths should be assigned as quite small values [11,17], requiring both long free length (exceeding several meters) and extremely large compressible length. It is very difficult to accomplish with such geometry characteristics in practical applications of mechanical compression spring. Not to mention that high stiffness should be simultaneously satisfied. Some other factors, e.g., buckling, fatigue load, also bring unfavorable effects on the behavior of the mechanical compression spring. Despite of existing case of successful implementation, as reported in Ref. [26], the scale-up of length from the 1/20 model to the prototype size is unfeasible for the corresponding stiffness. It means that it is very difficult, not to say impossible, to find suitable springs in the typical prototype scale. Motivated by such a practical limitation, the authors, in previous work – Ref. [28], reached some initial results related to the practical application of the NSS on point absorber. The present article is a further contribution to the subject aiming to identify eventual practical restrictions and investigate the potential ways to overcome those limitations. The paper organization presents the following sequence: first, the fundamental mathematical model of the combination of WEC and NSMech in a single degree-of-freedom is described in Section 2. According to the preliminary simulations in irregular waves in Section 3, the potential ranges of the NSMech parameters are determined. Then, the geometrical and physical constraints of mechanical compression spring are investigated in Section 4, followed by the performance analysis of feasible NSMech. In Section 5, the benefits from nonlinear stiffness system with the pneumatic cylinder (NSPneu) are explored and preliminarily proved. Finally, some primary conclusions are drawn, together with some suggestions for future work.
m ·z¨ (t ) = fR (t ) + fW (t ) + fH (t ) + fPTO (t ) + fNSMech (t )
(1)
where: m – total mass of buoy; z¨ (t ) – vertical acceleration. The terms on the right hand side of equation are: the radiation force – fR(t); wave excitation force – fW(t); hydrostatic restoring force – fH(t); linear PTO force – fPTO(t); and nonlinear restoring force – fNSMech(t) from NSMech respectively. The derivation of fR(t), fH(t), and fPTO(t) expressions can be found in Ref. [28]. Irregular wave excitation force fW(t) may be defined as a linear superposition of a finite number of frequency components – Eq. (2): n
fW (t ) =
∑ Fai·ζai cos(ωi t + εi + εpi)
(2)
i=1
ζai =
2Sζ (ωi )Δωi
(3)
where: Fai and ɛpi are the ith frequency component amplitude and phase shift of the wave excitation force transfer function; ζai is the ith component of wave amplitude (here, related with the area under the associated segment of a given sea spectrum, Sζ(ωi)Δωi); and ɛi is the corresponding random phase angle (between 0 and 2π). The JONSWAP spectrum with peak enhancement factor 3.3 [31] is selected as the approximation of the local sea spectrum Sζ(ω), as shown in Fig. 2, together with fW(t). The remaining nonlinear part is the force fNSMech(t) from NSMech, as depicted in Fig. 1. The NSMech is composed of several mechanical compression springs in an axisymmetric arrangement. One end of the spring is hinged at the support structure on the concrete base fixed on the seabed. The free end is connected to the vertical rod following the heaving motion of the buoy. The mechanical compression spring is characterized by its free length L0 and stiffness K0. At equilibrium position of NSMech, the spring is fully compressed to its shortest length L, with no vertical force component acting on the buoy. Leaving from its equilibrium position, a net vertical force due to the NSMech starts acting on the buoy along the same direction of buoy displacement up to the springs recover their free length. Such displacement excursions correspond to the horizontal position (red lines in Fig. 1) and can be defined as:
z = ± zf 0 = ± L02 − L2
(4)
Here, one implicit essential condition is L < L0. Exceeding these limits, the NSMech will disconnect from the column, due to the inextensibility of the mechanical compression spring. Actually, such limit situations can be avoided with suitable system damping or end-stop system. Through straight forward derivation, one can obtain the vertical nonlinear restoring force fNSMech(t) at displacement z(t) – Eq. (5),
2. Mathematical model of WEC with NSMech
fNSMech (t ) =
One small buoy (diameter = 4.0 m, draft = 5.0 m and natural period = 5.0 s) is preliminarily designed to be located in the near-shore region of Rio de Janeiro - Brazil [29], with 50 m water depth. The buoy is constrained to oscillate only in the vertical direction by a concrete base fixed structure on the seabed. The power take-off (PTO) system is arranged on the above water upper platform. One nonlinear stiffness system with mechanical compression spring (NSMech) works between the buoy and PTO system through a rod, as shown in Fig. 1. Cummins’ equation was used to set up the mathematical model of
⎧− nK 0·z (t )·(1 − ⎨0 ⎩
L0 z 2 + L2
) z ≤ zf 0 z > zf 0
(5)
where: n - number of the parallel springs. Two convenient dimensionless parameters of NSMech may be defined as stiffness ratio α (=Cwl/ nK 0) and geometric ratio γ (=L/ L0) for the expression of NSMech. It should be also noticed that there exists a critical value of γ: γcr = 1/(1 + α ) , characterizing the system behavior as mono-stable, bi-stable and quasi-zero stiffness (QZS) – Fig. 3. The black curve in Fig. 3 depicts the condition when Cγ (=γ / γcr ) is equal to 1.0, 2
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Fig. 1. Schematic view of the WEC with NSMech (adpted from [28]). (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
corresponding to the QZS behavior (red potential energy curve). In the region above it, where Cγ > 1.0, the system presents a mono-stable behavior; and if Cγ < 1.0, it defines a bi-stable behavior. Further details including a comprehensive discussion about the static characteristics of the system can be found in Ref. [28]. Then, the state-space model of the whole system may be set up as Eq. (6),
⎡ {A′}n × n {0}n × 1 {B′}n × 1⎤ {X} ⎡ {0}n × 1 ⎤ ⎡ {X˙ }n × 1⎤ ⎢ ⎥ ⎡ n × 1⎤ ⎢ 0 { } 0 1 ⎥ 0 1 × n ⎢ z˙ ⎥ = ⎢ ⎥·⎢ z ⎥ + ⎢ fW + fNSMech ⎥ −BPTO ⎢ v˙ ⎥ ⎢ −{C′}1 × n −Cwl v ⎥ ⎣ ⎦ ⎢ ⎣ ⎦ ⎣ m + A∞ ⎥ ⎦ ⎣ m + A∞ m + A∞ m + A∞ ⎦
(6)
where: ( · ) denotes derivative associated with time t; z – vertical displacement; and v – velocity of the PA buoy; {X}n × 1 defines the nth order state vector and {A′}n × n,{B′}n × 1, {C′}1 × n are the constant state-space matrices in terms of fR(t); A∞ is the infinite frequency added mass; Cwl is the constant hydrostatic restoring coefficient; BPTO is the linear PTO damping. Finally, the classical 4th order Runge–Kutta algorithm is utilized to solve the ordinary differential equation (ODE) and Eq. (7) calculates the mean power Pm absorbed by PA.
1 Pm = T
∫0
T
(BPTO ·v )·v·dt
Fig. 3. System behaviors in the (α, γ) domain. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
CWR =
Pm = Pw D
Pm ρg 2 2 H T ·D 64π S e
(8)
where: Pw – wave energy flux in irregular waves; Te – energy period (assumed as TP/1.12 for the standard JONSWAP spectrum [32]); D – PA buoy diameter. To evaluate the WEC performance in a given sea site, the annual energy production (AEP) is also adopted, as in Eq. (9),
(7)
The capture width ratio (CWR) is defined by the Eq. (8):
Fig. 2. Sea spectrum and the corresponding wave excitation force time record. 3
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Table 2 Irregular wave conditions.
Fig. 4. Bivariate distributions of occurrence and energy in terms of HS and TP in the near-shore region of Rio de Janeiro [29]. The color scale works as percentage values, representing the contribution of the sea state to the total wave energy, while the black numbers indicate the occurrence of sea states in the number of hours. The average wave energy flux in this region is about 10.5 kW/ m, with black curves depicting the wave flux distribution.
Variables
Units
Values
HS Tp
[m] [s]
/0.5, 1.0, 1.5/ /5.0, 7.0, 9.0, 11.0, 13.0, 15.0/
•
N
AEP = 8670· ∑ Pmi·fi
(9)
i=1
N
∑ fi
=1
groups of random phase angles, while the shaded areas define the range within one standard deviation on each side of solid curves. The peaks of the solid curves, 3 red points, are the representative points for different NSMech configurations. Note that as α increasing, Pmmax decreases, however, the corresponding optimal BPTO increases. Besides, the changes of BPTO around the optimal value have weaker influences on the performance of the configuration with higher α, which is a positive characteristic for PTO system designs; (b) Presents the convergence curves of Pmmax (solid curves) and the corresponding coefficient of variation (COV, dashed curves) with respect to T. It is shown that Pmmax is relatively stable with respect to T, while the COV decreases as increasing T. To meet the balance of accuracy and efficiency of simulation, T = 2000 s is taken as the effective simulation length removing the first 300s’ transient part from the total simulation length 2300 s. In addition, the COV of bistable system (as blue dashed curve) is larger than that of monostable system (as orange and yellow dashed curves), which means that the behavior of mono-stable system brings lower dispersion.
(10)
i=1
In Fig. 6, groups of CWR patterns with respect to different NSMech configurations under groups of irregular wave conditions are presented. The significant wave height HS and peak period Tp are shown on the left and bottom axis respectively. In each sub-graph, the horizontal axis defines Cγ, while the vertical axis is the CWR corresponding to the optimal BPTO. Red dots depict the CWR of the linear model without NSMech in each sea state. The blue and red dash curves divide each subgraph at Cγ = 1 and Cγ = 2 . The configurations with α from 1.0 to 7.0 correspond to other color curves following the color scale on the rightbottom of the figure. An obvious pattern is that,
where: Pmi – mean power in the ith sea state; fi – frequency of occurrence of the corresponding sea state; N – total number of sea states appearing in the sea site; and one year time length – 8760 h. Fig. 4 depicts the distributions of occurrence and energy in terms of HS and TP in the near-shore region of Rio de Janeiro [29]. 3. Performance of WEC with NSMech in irregular waves To obtain the WEC performance with NSMech in irregular waves, the ranges of NSMech dimensions have to be defined first. For sake of convenience, the horizontal length L of NSMech is taken shorter than the buoy radius. Once L < L0, then γ should be always smaller than 1.0. The NSMech configurations are presented in Table 1. Since most of the energy content of ocean waves is composed by waves with period between 5.0 s and 15.0 s [33], and the significant wave heights (HS) with larger occurrence are below 2.0 m (see Fig. 4), 18 sea states with 3 groups of different HS and 6 groups of different peak periods (Tp) (Table 2) are selected. Considering the effect of the random phase in the irregular wave records, each sea state is repeated with 50 distinct groups of random phase angles. In the present article, the focus is the influences of NSMech. Pmmax is defined as the optimal mean value of Pm with respect to BPTO in each sea state, taking into account 50 groups of random phase angles. As well, the CWR is derived from Pmmax corresponding to optimal BPTO, based on Eq. (8), if not specially mentioned. Fig. 5 depicts the convergence analysis due to the simulation length (T):
• When T •
Such a phenomenon is caused by the change of the natural period of WEC. As the natural period of the linear model without NSMech is around 5.0 s, the system perform best around Tp = 5.0 s ; while with NSMech, the natural period is increased, then the better performance appears at somewhere Tp > 5.0 s. Additionally, the improvements with respect to the best performance of linear model at Tp = 5.0 s appear at a wider range between Tp = 7.0 s and Tp = 11.0 s which could be attributed to the broadened bandwidth brought by NSMech. Furtherly, as the wave height decreases, the range can be broadened more significantly, in together with that the CWR is higher in low wave height conditon than that in high wave height condition. Then it can be concluded that in low wave height condition, NSMech can play a better role in improving the performance of WEC. It should be noted that the corresponding Cγ in which significant improvement can be supplied usualy exists between 0.5 and 2.0, especially around Cγ = 1. In other words, when the values of γ and α are too high, the effect of the corresponding NSMech is weakened. Consequently, the following ananlysis will focus more on this region. Further results about the influence of L will be shown in Fig. 7.
• (a) Defines P
m patterns of 3 groups of NSMech configurations with respect to varied BPTO through 2000s effective simulation length. The solid curves represent the mean values, taking into account 50
Table 1 Configurations of NSMech. Variables
Units/Expressions
Values
L α γ
[m] Cwl/nK0 L/L0
/0.5, 1.0, 1.5/ 1.0 ∼ 7.0 0.25 ∼ 0.95
p = 5.0 s , the maximum CWR seems to approximate to the value of the linear model without NSMech, and there exists a local minimum point at around Cγ = 1.0 ; When Tp > 5.0 s, the optimal CWR are always higher than the corresponding values of the linear model, and appear in the region around Cγ = 1.0 , or within the range between Cγ = 1 and Cγ = 2.0 .
4
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Fig. 5. Convergence analysis on the simulation length (T). Sea state: HS = 1.5 m, Tp = 9.0 s ; 3 groups of NSMech configurations: I (bi-stable): L = 1.5 m, γ = 0.4, α = 1.0 ; II (mono-stable): L = 1.5 m, γ = 0.4, α = 2.0 ; III (mono-stable): L = 1.5 m, γ = 0.4, α = 3.0 . COV defines the ratio of standard deviation to the mean value. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
difference. The present case supplies further evidences that the two factors, L and HS, influence the performance of NSMech throughout the range of L/HS. In summary, in order to enhance significantly WEC power capture using NSMech, the configurations with high value of L and small γ, and small α perform better. Such requirements correspond to the mechanical compression spring with both large compressible length and high stiffness. Then, it brings about discussion about exploring feasible configurations based on the mechanical compression spring design principles.
Fig. 7 presents patterns of CWR depending on Tp and L featuring four groups with different (α, γ) configurations in irregular waves. From (a) to (c), results corresponding to Cγ equal to 0.8, 1.0 and 1.2 respectively, characterizing bi-stable, QZS and mono-stable behaviors, as defined in Fig. 3. The NSMech shown in Fig. 7(d) are same as that in Fig. 7(c) but with lower wave height. Each figure corresponds to different Tp for each one of the six groups . Each group is composed by t results from three NSMech configurations with different L and one linear model without NSMech taken as reference. In sea states with higher Tp than the natural period (5.0 s) of PA, larger L can better enhance the power capture. One exception exists when Tp = 7.0 s in Fig. 7(d) due to the natural period increase to near 9.0 s when L = 1.5 m , while it increases only to 7.0 s when L = 1.0 m . Consequently, the NSMech with L = 1.0 m presents a better performance at Tp = 7.0 s . Based on this analysis, one may conclude that larger L increases the natural period to a higher level, which is also influenced by the wave height. Noting that the case – L = 1.5 m , shown in Fig. 7(c) and the case – L = 0.5 m , shown in Fig. 7(d) are characterized by the same value of L/ HS = 1.0 , the corresponding CWRs are almost same with very little
4. Constraints of mechanical compression spring 4.1. Determination of the feasible NSMech The conventional mechanical spring design procedure finds a group of suitable variables (both geometrical and physical) according to some special requirements. The present section indicates the set of constraints to be taken into account in the mechanical compression spring when exploring the feasible region of geometrical and physical
Fig. 6. CWR of WEC with NSMech (L = 1.5 m ) in irregular waves. (For interpretation of the references to color in text, the reader is referred to the web version of this article.) 5
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Fig. 7. CWR of WEC with NSMech in irregular waves. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
Fig. 8. Schematic main dimensions of the mechanical compression spring.
variables associated with NSMech. The schematic main dimensions of the compression spring are presented in Fig. 8. The procedures to determine the feasible region considered the following steps:
Table 3 Variables for mechanical compression spring design.
Step 1: Define the set of parameters to be explored – horizontal length of NSMech L, the number of adopted springs n, geometric ratio γ and stiffness ratio α, wire diameter d and spring index C, as in Table 3. Here, the ranges of values of L, γ and α are adjusted according to the analysis presented in the previous section. Due to the oscillation of PA in waves, the ranges of d and C are determined based on the characteristics of the material (e.g. 6
Parameters
Units/Expressions
Values
L n γ α d C
[m] [–] L/L0 Cwl/nK0 [mm] D/d
/0.5, 1.0, 1.5/ /3, 6, 9, 12/ 0.02 ∼ 0.98 1.0 ∼ 7.0 3.2 ∼ 10.0 4.0 ∼ 12.0
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hard drawn wire A227) associated to fatigue loading [34]; Step 2: Derive the basic parameters – stiffness K0 (Unit: N/m), free length L0 (Unit: m), coil diameter D (Unit: mm), and the curvature correction factor – Bergsträsser factor KB [34];
K0 =
Cwl L 4C + 2 , L0 = , D = C·d, KB = n·α γ 4C − 3
Step 3: Derive axial load amplitude Fa (Unit: kN), and the number of active coils Na. Fa is the average value of the maximum and minimum values of axial load [34]. As the maximum load in NSMech appears when the spring in the horizontal equilibrium position and the minimum load is zero, it can be expressed as,
Fa =
10−3K 0 (L0 − L) 2
Fig. 9. Feasible region derived by the constraints of mechanical compression spring design.
The number of active coils Na can be defined in according to the stiffness by the expression,
K0 =
can be satisfied simultaneously, the corresponding design parameters will be feasible. Following the above steps, the feasible regions of (α, γ) with different (L, n) configurations are investigated and shown in Fig. 9. The bottom and left axis define the number of springs adopted (n) and the horizontal length (L) for each sub-graph. The black curve divides the (α, γ) domain into feasible (green) and infeasible (white) region. The blue and red dashed curves, corresponding to the reference curves in Fig. 7, divide each sub-graph at Cγ = 1 and Cγ = 2 showing that more CWR can be obtained if the parameter is near Cγ = 1 or within the range between Cγ = 1 and Cγ < 2. The results discussed above give grounds for the following additional comments:
103Gd 8C 3Na
where: shear modulus: G = 78603.0 MPa . Step 4: Three parameters – shear stress amplitude τa (Unit: MPa), solid length LS (Unit: mm), and pitch p (Unit: mm) for squared and ground end are obtained as above to play the role of constraints. Because buckling can be avoided for long spring if guided by sleeves or over an arbor [35], the conditions of stability and critical frequency were not considered in the present discussion. 8F C Maximum stress: τa = KB · a2 ·103 πd Solid length: LS = (Na + 2) d
• Adopting more springs, keeping the same L, can broaden the feasible region for lower values of α and γ; • Keeping the same n, shortening the horizontal length can lower the
103L − 2d
0 pitch: p = Na Step 5: Constraints. First, the maximum stress τa should satisfy the strength condition
τa ≤
Ssa nf
bound of γ and broaden the feasible region as well.
In other words, more feasible configurations can be found when more springs are adopted and shorter horizontal length applied. Another interesting finding shows that the same feasible regions exist if L/n, for different configurations are kept equal, such as in Fig. 9 subgraphs 9, 6 and 3 or 10 and 8,
(11)
where: Ssa – torsional endurance strength for infinite life. For the peened spring, Ssa = 398 MPa [34], and the safety factor nf = 1.5. Accordingly, the minimum total axial gap should be taken as 15% (cg) of maximum deflection (δmax) [35],
L9 L L 1 L L 1 = 6 = 3 = or 10 = 8 = n9 n6 n3 6 n10 n8 12
total gap = cg·δmax
where: the subscript identifies the different sub-graphs. The practical selection can benefit from such a relationship. Besides, the feasible parameters (α, γ) occupy the top right region in each sub-graph, meaning that,
Free length L0 is also the summation of solid length LS, maximum deflection δmax and total gap,
L0 = LS × 10−3 + total gap + δmax
• The constraints on the design of spring indeed exist when determining (α, γ); • It is easier to obtain the feasible configuration for higher α and γ
In the structure of NSMech, the allowable deflection of spring should satisfy,
values.
L0 − L ≤ δmax The present distribution characteristics of the feasible NSMech configurations will result in a significant constraint on the enhancement of the power capture with NSMech. This effect will be investigated in the next sub-section.
Then the following expression is obtained,
LS + (1 + cg )·(L0 − L)·103 ≤ L0 ·103
(12)
4.2. WEC performance with feasible NSMech
Finally, the pitch p is preferred to be less than half of the coil diameter
d < p ≤ 0.5D
Noting the existence of constraints on the NSMech configurations, the operational performance (AEP) in the near-shore region of Rio de Janeiro, will be investigated considering the feasible configurations. In Fig. 10, the configurations in the sub-graphs (a), (b), (c) and (d) corresponding to that the same configurations taken in Fig. 9 sub-9,
(13)
When all constraints, defined in Step 5 – Eqs. (11), (12) and (13), 7
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Fig. 10. AEP of WEC with NSMech in the near-shore region of Rio de Janeiro.
Table 4. Based on the data presented in Table 4, one can conclude that,
sub-6, sub-3 and sub-12 respectively. The bar height in each sub-graphs depicts the resulting AEP values. With respect to each (α, γ) configuration, the corresponding AEP is calculated with an optimal BPTO for all the sea states in the given sea site. Obviously, the better performance in each sub-graph concentrates itself in the region with lower values of α and γ, satisfying Cγ around 1 or, at least, Cγ < 2 keeping fully consistency with the characteristics presented in Fig. 6. In such region, larger L leads to greater enhancement. As the values of α and γ increase, the AEPs approach stable values, whatever would be L and n. Actually, the reference value around 33.17 MWh was obtained from PA without NSMech. However, when considering the constraints, the feasible AEPs are described by color scale. The optimal performance is indicated by red arrow in each sub-graph. Though the feasible regions in Fig. 10(a), (b) and (c) are the same, the optimal (α, γ) are different due to the L variations. Compared with Fig. 10(a), Fig. 10(d) features a broader feasible region through adopting more springs. Furthermore, the optimal AEP and the increase with respect to the reference value are presented in
• With the same feasible region (as the blue cells), the higher value of L induces higher performance enhancement; • If the same L is selected, better performance is obtained by adopting more springs.
Moreover, lower values of L also lead to better enhancement by broadening the feasible region with more springs (orange cell). In other words, sometimes the feasible region has more significant effect on the performance of WEC with NSMech than L. However, it should be noted that the relative increase is really small, around 20% at most, when the constraints of NSMech are considered. Not to be mentioned if compared with other enhancement approaches, such as latching control or model predictive control (MPC). The results also suggest the need to investigate other approaches with less constraints to substitute NSMech. The pneumatic cylinder is one alternative approach to be introduced in the next section.
Table 4 WEC Performance with NSMech in the near-shore region of Rio de Janeiro.
5. Nonlinear stiffness system with pneumatic cylinder (NSPneu) The former section indicated the limited performance of WEC with NSMech due to the physical constraints of the mechanical compression spring. To overcome the weakness of NSMech, one alternative approach substitutes the mechanical compression springs by pneumatic cylinders, as shown in Fig. 11. The rods of cylinders are hinged to follow the buoy vertical motion. The pressurized gas (blue) fills in the cylinder chambers. An additional gas reservoir is necessary to link cylinders all together. First, it keeps the same gas pressure in all cylinders, satisfying the horizontal force balance; second, the additional volume broadens the range of pressure and volume variation. Other schematic explanations follow the same description as shown in Fig. 1. The objective of 8
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Fig. 11. Schematic view of the WEC with NSPneu. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
Fig. 12. NSPneu force and total restoring force for different NSPneu configurations (L = 2.0 m ).
the present section is to investigate the practical possibilities of the nonlinear stiffness system using the pneumatic cylinder (NSPneu). The determination of optimal parameters of NSPneu will not be addressed here. In the NSPneu device, n cylinders (each piston area: A0) are axisymmetrically connected. At equilibrium position, the cylinder length is L, the total gas volume is V0, and the internal pressure is P0. At a given displacement z(t), the gas volume increases by V:
Table 5 Ranges of primary parameters of NSPneu. Parameters
Unit
Min
Max
D0 P0 V0
cm bar (1 bar = 105 Pa) m3
2 1 0.001
20 20 1
Note: D0 - piston internal diameter ( A0 = πD02 /4 ).
V = V0 + nA0 ·( z 2 + L2 − L)
(14)
Assuming adiabatic process, the gas pressure can be calculated by:
PV μ
= P0 V0μ
(15)
where: μ is the specific heat ratio (1.4 for nitrogen). Thus, the net vertical force supplied by the NSPneu device will be:
fNSPneu (t ) = nP0 A0 ·
V μ ·⎛ 0 ⎞ z 2 + L2 ⎝ V ⎠ z
(16)
For the sake of simplicity, two new variables are adopted: equivalent stiffness – nK0 and equivalent length – L0,
nK 0 =
V nP0 A0 , L0 = 0 L nA0
(17)
and the dimensionless variables are:
Fig. 13. Feasible region of NSPneu.
α= 9
Cwl L·Cwl L nA0 L , γ= = = nK 0 nP0 A0 L0 V0
(18)
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Fig. 14. WEC with NSPneu: AEP in near-shore region of Rio de Janeiro.
• α < 1.0, bi-stable behavior, with three zero values of restoring force; • α = 1.0, QZS behavior, with only one zero and zero stiffness at equilibrium position; • α > 1.0, mono-stable behavior, with only one zero but non-zero
Table 6 Performance comparison between WEC with NSMech and with NSPneu.
stiffness at equilibrium position.
Another aspect which deserves mentioning is if the buoy displacement is around the equilibrium position (|z| < 0.8 m, in the present case), the effect of γ on the NSPneu force will be small. Therefore, the total restoring force does not change dramatically. In other words, the influence of α will be more significant in the performance of NSPneu around the equilibrium position. As α and γ increase, it reduces the NSPneu force, requiring smaller values which are preferable to obtain more significant influence from NSPneu. Once understood the basic static characteristics of the system, the feasibility of NSPneu will be investigated. First, the primary parameters ranges of NSPneu are presented in Table 5. The ranges may be not quite convenient, but are in according to the industry applications [12]. A group of (α, γ) may be considered feasible if suitable parameters exist in the space (D0, P0, V0) as shown in Table 5. Thus the feasible regions (α, γ) of NSPneu under different combinations groups (L, n) were investigated and the results are presented in Fig. 13. The most obvious feature is that feasible region occupies a broader area, meaning that the influence of constraints are quite limited. Similar results as those presented in Fig. 9 show that decreasing L or increasing n induces the feasible region to cover range of smaller α and γ where better performances are expected. However, if the L/n are equal, the feasible regions are no longer the same. Consequently, Fig. 14 presents the AEP of WEC taking NSPneu with some typical (L, n) combinations. The unfeasible region only appears if α was small. Therefore, to some extent, it will be easier to obtain better
Therefore, the expression of fNSPneu is reduced from the dimension of (n, L, P0, A0, V0) in Eq. (16) to the dimension of (n, L, L0, K0) as:
fNSPneu (t ) = nK 0·
z
·L·⎛⎜ z 2 + L2 ⎝ L0 +
L0
μ
⎞
⎟
z 2 + L2 − L ⎠
(19)
Furthermore, similar approach as used before in the NSMech case, will be adopted in the analysis of the influence of different parameters on the performance of NSPneu. It should be noticed that setting L0 together with suitable piston area A0 and gas volume V0, the value of γ can be either larger or smaller than 1.0, which makes an important practical difference if compared with NSMech system. Fig. 12(a) and (b) present patterns of NSPneu force and total restoring force as function of displacement (z), considering different NSPneu configurations. The NSPneu nonlinearity features negative stiffness near the equilibrium position (z = 0 ) (Fig. 12(a)). Adding the hydrostatic restoring force (fH), Fig. 12(b) allows for obtaining the total restoring force. Similarly, different values of α define three distinct behaviors: 10
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Supplementary materials
enhancement with NSPneu. Furthermore, the performance comparisons between WEC with NSMech and with NSPneu are listed in Table 6. The results show great advantage from using NSPneu that less elements are required to achieve better performance (around 1-fold to 2-fold increases). The feasible region features a varied optimal L for each n. Therefore, a balance between L and n should be carefully determined.
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apor.2019.101935. References [1] A.A. Sarlis, D.T.R. Pasala, M.C. Constantinou, A.M. Reinhorn, S. Nagarajaiah, D.P. Taylor, Negative stiffness device for seismic protection of structures, J. Struct. Eng. 139 (2013) 25–28, https://doi.org/10.1061/(ASCE)ST.1943-541X.0000616. [2] D.T.R. Pasala, A.A. Sarlis, S. Nagarajaiah, A.M. Reinhorn, M.C. Constantinou, D. Taylor, Adaptive negative stiffness : new structural modification approach for seismic protection, J. Struct. Eng. 139 (2013) 1112–1123, https://doi.org/10.1061/ (ASCE)ST.1943-541X.0000615. [3] C.B. Churchill, D.W. Shahan, S.P. Smith, A.C. Keefe, G.P. Mcknight, Dynamically variable negative stiffness structures, Sci. Adv. 2 (2016) 1–7, https://doi.org/10. 1126/sciadv.1500778. [4] E. Palomares, A.J. Nieto, A.L. Morales, J.M. Chicharro, P. Pintado, Numerical and experimental analysis of a vibration isolator equipped with a negative stiffness system, J. Sound Vib. 414 (2018) 31–42, https://doi.org/10.1016/j.jsv.2017.11. 006. [5] R. Ramlan, M.J. Brennan, B.R. MacE, I. Kovacic, Potential benefits of a non-linear stiffness in an energy harvesting device, Nonlinear Dyn. 59 (2010) 545–558, https://doi.org/10.1007/s11071-009-9561-5. [6] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy harvesting via bistable systems, Smart Mater. Struct. (2013) 22, https://doi.org/10. 1088/0964-1726/22/2/023001. [7] M.F. Daqaq, R. Masana, A. Erturk, D. Dane Quinn, On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion, Appl. Mech. Rev. 66 (2014) 040801, , https://doi.org/10.1115/1.4026278. [8] A.H. Hosseinloo, K. Turitsyn, Non-resonant energy harvesting via an adaptive bistable potential, Smart Mater. Struct. (2015) 25, https://doi.org/10.1088/09641726/25/1/015010. [9] G. Shan, D.F. Wang, J. Song, Y. Fu, X. Yang, A spring-assisted adaptive bistable energy harvester for high output in low-excitation, Microsyst. Technol. 5 (2018) 1–10, https://doi.org/10.1007/s00542-018-3778-5. [10] X. Zhang, J. Yang, L. Xiao, Numerical study of an oscillating wave energy converter with nonlinear snap-through power-take-off systems in regular waves, TwentyFourth Int. Ocean Polar Eng. Conf. 2014, pp. 225–230, , https://doi.org/10.13140/ 2.1.4414.4000. [11] X. Zhang, J. Yang, Power capture performance of an oscillating-body WEC with nonlinear snap through PTO systems in irregular waves, Appl. Ocean Res. 52 (2015) 261–273, https://doi.org/10.1016/j.apor.2015.06.012. [12] J.H. Todalshaug, Wave Energy Convertor, (2015) WO 2015/107158. [13] CorPower, (2018). http://www.corpowerocean.com/(accessed September 24, 2018). [14] D. Younesian, M.R. Alam, Multi-stable mechanisms for high-efficiency and broadband ocean wave energy harvesting, Appl. Energy 197 (2017) 292–302, https:// doi.org/10.1016/j.apenergy.2017.04.019. [15] X. Zhang, X. Tian, L. Xiao, X. Li, L. Chen, Application of an adaptive bistable power capture mechanism to a point absorber wave energy converter, Appl. Energy 228 (2018) 450–467, https://doi.org/10.1016/j.apenergy.2018.06.100. [16] L. Wang, H. Tang, Y. Wu, On a submerged wave energy converter with snapthrough power take-off, Appl. Ocean Res. 80 (2018) 24–36, https://doi.org/10. 1016/j.apor.2018.08.005. [17] L. Li, X. Zhang, Z. Yuan, Y. Gao, Multi-stable mechanism of an oscillating-body wave energy converter, IEEE Trans. Sustain. Energy (2019), https://doi.org/10. 1109/tste.2019.2896991 1–1. [18] B. Teillant, K. Krügel, M. Guérinel, M. Vicente, Y. Debruyne, F. Malerba, M. Gradowski, S. Roveda, F. Neumann, H. Von Noorloos, R. Schuitema, R. Gomes, J. Henriques, L. Gato, A. Combourieu, A. Neau, B. Borgarino, J. Doussal, M. Philippe, G. Moretti, M. Fontana, D2.3 - Engineering Challenges Related to Full Scale and Large Deployment Implementation of the Proposed Breakthroughs, 2016. [19] S. Roveda, Numerical Assessment of Negative Spring on Spar OWC Based in the IVV Method, Instituto Superior Técnico, 2016. [20] M. Gradowski, M. Alves, R.P.F. Gomes, J.C.C. Henriques, Integration of a hydrodynamic negative spring concept into the OWC spar buoy, 12th EWTEC, Cork, Ireland, 2017. [21] R.L. Harne, M.E. Schoemaker, B.E. Dussault, K.W. Wang, Wave heave energy eonversion using modular multistability, Appl. Energy 130 (2014) 148–156, https:// doi.org/10.1016/j.apenergy.2014.05.038. [22] X. Xiao, L. Xiao, T. Peng, Comparative study on power capture performance of oscillating-body wave energy converters with three novel power take-off systems, Renew. Energy 103 (2017) 94–105, https://doi.org/10.1016/j.renene.2016.11. 030. [23] H. Zhang, R. Xi, D. Xu, K. Wang, J. Zhou, Q. Shi, Efficiency enhancement of a point ocean wave energy converter by using a magnetic bistable mechanism, Energy 181 (2019) 1152–1165, https://doi.org/10.1016/j.energy.2019.06.008. [24] J.H. Todalshaug, G.S. Ásgeirsson, E. Hjálmarsson, J. Maillet, P. Möller, P. Pires, M. Guérinel, M. Lopes, Tank testing of an inherently phase-controlled wave energy converter, Int. J. Mar. Energy 15 (2016) 68–84, https://doi.org/10.1016/j.ijome. 2016.04.007. [25] J. Josefsson, Numerical Study of Resonant Wave Energy Converters in Selected Sites, KTH, 2015, https://www.diva-portal.org/smash/get/diva2:871459/
6. Conclusions The conventional time domain approach applied to obtain a series simulations featuring NSMech in irregular wave conditions produced good and reliable results to fill in the lack of researches on constraints and limitations concerning applications on wave energy conversion. An innovative modification on the expression of NSMech restoring force discussed here considered the pure compressed characteristic of mechanical compression spring. To deal with the typical randomness of the real sea conditions and the system nonlinearities, the analysis assumed the average performance of a great number of random phases grouped for every sea state. Preliminary simulations showed that the enhancement due to NSMech application on PA type WEC was very much dependent on the suitable L, γ, α (or Cγ) configurations. The optimal performance appeared near Cγ = 1.0 . In addition, large compressible length (or long L and small γ) and high stiffness (small α) should be preferred in practical design of NSMech. Such results can be used to determine the ranges of L, γ, α (or Cγ) when investigating the possible NSMech configurations. Procedures aiming at defining the feasible region of NSMech should be based on the mechanical compression spring design process. The results provided the feasible region of (α, γ) for different (L, n). There exists one single positive correlation between L and n leading to the feasible region. Within the same feasible region, the higher L and more springs adopted induce higher improvement. However, the relative AEP increase is really small, around 20% at most, within the feasible region. In other words, the NSMech constraints indeed weaken the WEC with NSMech performance. Adopting a large number of springs poses a great challenge to the design of the whole system. To overcome the weakness and practical difficulties of the NSMech, one alternative approach would be to substitute the mechanical compression spring by pneumatic cylinder. The preliminary studies succeeded to demonstrate advantageous towards the NSPneu. It requires fewer elements to improve performance (around 1-fold to 2-fold increases). Simultaneously, a careful balance between L and n should be considered. It should also be noticed that the dynamic model of NSPneu presented here was a simplified model, assuming an adiabatic process. Other factors, such as piston seal friction, turbulent losses due to the air transmission, and the non-adiabatic compression, should be incorporated carefully to improve the dynamic model. Besides, a much more realistic model should also include an end-stop mechanism together with feasible buoy excursion and NSPneu stroke.
Acknowledgement The first author would like to acknowledge the D.Sc. Scholarship from CAPES, Ministry of Education/Brazil, and the support from China Ship Scientific Research Center. Special thanks to FURNAS through ANEEL (Brazilian Electrical Energy Agency) R&D Program for the financial support of the research in progress at Subsea Technology Laboratory (COPPE - Federal University of Rio de Janeiro) on wave energy (contract number PD-0394-1248/2012). The third author would like to thank INOEF for the support to research activities on wave energy. 11
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Energy 115 (2018) 533–546, https://doi.org/10.1016/j.renene.2017.08.055. [30] M. Penalba, G. Giorgi, J.V. Ringwood, Mathematical modelling of wave energy converters: a review of nonlinear approaches, Renew. Sustain. Energy Rev. 78 (2017) 1188–1207, https://doi.org/10.1016/j.rser.2016.11.137. [31] J.M.J. Journée, W.W. Massie, Offshore Hydromechanics, Delft University of Technology, 2001. [32] A. Pecher, J.P. Kofoed, Handbook of Ocean Wave Energy, University of Edinburgh, 2017. [33] J. Falnes, Ocean Waves and Oscillating Systems, Cambridge University Press, 2002. [34] R.G. Budynas, J.K. Nisbett, Shigley's Mechanical Engineering Design, ninth ed., McGraw Hill, 2009. [35] V.B. Bhandari, Design of Machine Elements, third ed., McGraw Hill, 2010.
FULLTEXT01.pdf. [26] A. Têtu, F. Ferri, M. Kramer, J. Todalshaug, Physical and mathematical modeling of a wave energy converter equipped with a negative spring mechanism for phase control, Energies 11 (9) (2018) 2362, https://doi.org/10.3390/en11092362. [27] S. Peretta, P. Ruol, L. Martinelli, A. Tetu, J.P. Kofoed, Effect of a negative stiffness mechanism on the performance of the Weptos rotors, 6th Int. Conf. Comput. Methods Mar. Eng. 2015. [28] Z. Wu, C. Levi, S.F. Estefen, Wave energy harvesting using nonlinear stiffness system, Appl. Ocean Res. 74 (2018) 102–116, https://doi.org/10.1016/j.apor. 2018.02.009. [29] M. Shadman, S.F. Estefen, C.A. Rodriguez, I.C.M. Nogueira, A geometrical optimization method applied to a heaving point absorber wave energy converter, Renew.
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Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Practical considerations on nonlinear stiffness system for wave energy converter Zhijia Wua,b, Carlos Levia, Segen F. Estefena, a b
T
⁎
Ocean Engineering Department, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil China Ship Scientific Research Center, Wuxi, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlinear stiffness system Mechanical compression spring Wave energy converter Pneumatic cylinder
Mechanical restoration of a point absorber (PA) wave energy converter (WEC) featured a nonlinear stiffness system built by conventional mechanical compression springs (NSMech). Numerical simulations conducted here considered irregular waves in a specific sea site. Results showed considerable improvement for a given suitable spring configuration (length and stiffness parameters). However, practical implementation of NSMech imposed limiting constraints on the geometrical and physical characteristics of the mechanical compression springs. Such constraints limited very much the feasible region of NSMech configurations, restricting significantly applications for wave energy converter. One alternative approach using pneumatic cylinder springs proved to overcome such a limitation aggregating obvious advantages with fewer elements and bringing more enhancement to the WEC performance.
1. Introduction The nonlinear stiffness system (NSS) is one structure which can provide nonlinear restoring force through some specially configured mechanical compression springs, buckling beams, or magnets. By adjusting its positive or negative contribution to the restoring effects on a given dynamic system, it can shift the corresponding natural frequency and/or broaden the response bandwidth to weaken or strengthen the response. Benefiting from such characteristics, NSS has been widely applied on vibration isolation [1–4] and vibration energy harvesting [5–9] engineering solutions. Recently, NSS also attracts the attention of wave energy converter (WEC) researchers because its natural frequency-shifting characteristic may increase the efficiency of point absorber (PA) type WEC. Both numerical and experimental investigations have been carried out to explore the implementation of NSS in PA. Zhang et al. [10, 11] study the application of snap-through power take-off (PTO) system in WEC and present extensive analysis and comparisons featuring the influences due to PTO damping, buoy geometry and dimensions, and wave frequency on wave energy conversion, in both regular and irregular waves. They have shown that if compared with linear PTO system, the nonlinear snap-through PTO system may improve the point absorber (PA) wave energy harvesting, especially for lower frequency waves. Almost at the same time, Todalshaug [12]
patented a wave energy converter which can perform better with the negative stiffness device. The device could feature either mechanical springs or pneumatic cylinders. The pneumatic configuration was patented as “WaveSpring” technique and used in the CorPower buoy prototype [13]. From there on, more and more studies on similar principles have been conducted, mostly numerically. Younesian and Alam [14] expanded the numerical analysis of the system working with multi-stable characteristics for different geometric parameters. Zhang et al. [15] proposed a novel adaptive bi-stable mechanism which could adjust the potential function automatically to lower the potential barrier near the unstable equilibrium position, and hence helping to solve the low-energy-absorption problem of conventional bi-stable wave energy converter. Wang, Tang and Wu [16] investigated the performance of bistable snap-through PTO implemented in a submerged surging WEC. Li et al. [17] proposed a system with four inclined mechanical springs based on the multi-stable mechanism and demonstrated its benefits on the wave energy conversion. Instead of controlling the PTO mechanism with the mechanical NSS directly, the WETFEET project featuring an oscillating water column (OWC) spar buoy [18] proposed two alternative negative stiffness concepts: (a) immersed varying volume (IVV) [19]; and (b) hydrodynamic negative stiffness (HNS) [20] to adjust the hydrostatic restoring stiffness. Harne et al. [21], Xiao et al. [22], Zhang et al. [23]
⁎ Corresponding author at: Federal University of Rio de Janeiro, COPPE, Ocean Engineering Department, P.O. Box 68508 – CEP 21945-970 - Rio de Janeiro, RJ, Brazil. E-mail address: [email protected] (S.F. Estefen).
https://doi.org/10.1016/j.apor.2019.101935 Received 13 April 2019; Received in revised form 21 July 2019; Accepted 9 September 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
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WEC with NSMech in the time domain. The linear potential flow theory was adopted to evaluate the system hydrodynamic characteristicsadded mass, radiation damping and excitation force response in the frequency domain. The PTO system was assumed to be linear. The only remaining nonlinear contribution to the problem will be due to the NSMech. In recent work, the effect of viscous damping was ignored by Penalba et al. [30] as they mentioned that the viscous effect on small heaving point absorber (HPA), in general, was low. Eq. (1) shows the general expression of the time domain equation of motion (EOM),
used the bi-stable mechanism with magnets. Besides most of the above analysis featured numerical aspects, some experimental analysis have also been carried out. A 1/16 scale tank test [24] was conducted to verify such an quite innovative concept. Experimental results have shown that it could be tuned to provide both resonant behavior and broadening response bandwidth. Compared with the latching control strategy, for instance, the “WaveSpring” produces similar power capture with a small fraction of the transmission force [25]. Therefore, it enables smaller and less costly PTO, and no need to predict the incident waves as it is required in the latching control. Recently, a similar concept was also tested using a 1/20 scale model (named “WaveStar”) by Têtu et al. [26]. “WEPTOS”, a pitching WEC with NSS, was tested at the Aalborg University wave basin tests [27]. The experimental results did not show any noticeable improvements, probably due to not accounted for friction effects. However, numerical simulations considering prototype scale, with no friction effects, indeed revealed quite positive benefits to increase the device efficiency. During this review work of the NSS implementations in WEC, all concepts were still at an early stage of development: either numerical simulation or small scale test. An exception is the “WaveSpring” applied on a 1/2 model scale of CorPower buoy with a dry test (2017) and sea trail (2018) but, in the authors acknowledgement with no further detail or results about its performance published yet. Most of the applications feature mechanical compression springs. However, to produce significant improvement for the WEC performance, the NSS parameter related to the spring lengths should be assigned as quite small values [11,17], requiring both long free length (exceeding several meters) and extremely large compressible length. It is very difficult to accomplish with such geometry characteristics in practical applications of mechanical compression spring. Not to mention that high stiffness should be simultaneously satisfied. Some other factors, e.g., buckling, fatigue load, also bring unfavorable effects on the behavior of the mechanical compression spring. Despite of existing case of successful implementation, as reported in Ref. [26], the scale-up of length from the 1/20 model to the prototype size is unfeasible for the corresponding stiffness. It means that it is very difficult, not to say impossible, to find suitable springs in the typical prototype scale. Motivated by such a practical limitation, the authors, in previous work – Ref. [28], reached some initial results related to the practical application of the NSS on point absorber. The present article is a further contribution to the subject aiming to identify eventual practical restrictions and investigate the potential ways to overcome those limitations. The paper organization presents the following sequence: first, the fundamental mathematical model of the combination of WEC and NSMech in a single degree-of-freedom is described in Section 2. According to the preliminary simulations in irregular waves in Section 3, the potential ranges of the NSMech parameters are determined. Then, the geometrical and physical constraints of mechanical compression spring are investigated in Section 4, followed by the performance analysis of feasible NSMech. In Section 5, the benefits from nonlinear stiffness system with the pneumatic cylinder (NSPneu) are explored and preliminarily proved. Finally, some primary conclusions are drawn, together with some suggestions for future work.
m ·z¨ (t ) = fR (t ) + fW (t ) + fH (t ) + fPTO (t ) + fNSMech (t )
(1)
where: m – total mass of buoy; z¨ (t ) – vertical acceleration. The terms on the right hand side of equation are: the radiation force – fR(t); wave excitation force – fW(t); hydrostatic restoring force – fH(t); linear PTO force – fPTO(t); and nonlinear restoring force – fNSMech(t) from NSMech respectively. The derivation of fR(t), fH(t), and fPTO(t) expressions can be found in Ref. [28]. Irregular wave excitation force fW(t) may be defined as a linear superposition of a finite number of frequency components – Eq. (2): n
fW (t ) =
∑ Fai·ζai cos(ωi t + εi + εpi)
(2)
i=1
ζai =
2Sζ (ωi )Δωi
(3)
where: Fai and ɛpi are the ith frequency component amplitude and phase shift of the wave excitation force transfer function; ζai is the ith component of wave amplitude (here, related with the area under the associated segment of a given sea spectrum, Sζ(ωi)Δωi); and ɛi is the corresponding random phase angle (between 0 and 2π). The JONSWAP spectrum with peak enhancement factor 3.3 [31] is selected as the approximation of the local sea spectrum Sζ(ω), as shown in Fig. 2, together with fW(t). The remaining nonlinear part is the force fNSMech(t) from NSMech, as depicted in Fig. 1. The NSMech is composed of several mechanical compression springs in an axisymmetric arrangement. One end of the spring is hinged at the support structure on the concrete base fixed on the seabed. The free end is connected to the vertical rod following the heaving motion of the buoy. The mechanical compression spring is characterized by its free length L0 and stiffness K0. At equilibrium position of NSMech, the spring is fully compressed to its shortest length L, with no vertical force component acting on the buoy. Leaving from its equilibrium position, a net vertical force due to the NSMech starts acting on the buoy along the same direction of buoy displacement up to the springs recover their free length. Such displacement excursions correspond to the horizontal position (red lines in Fig. 1) and can be defined as:
z = ± zf 0 = ± L02 − L2
(4)
Here, one implicit essential condition is L < L0. Exceeding these limits, the NSMech will disconnect from the column, due to the inextensibility of the mechanical compression spring. Actually, such limit situations can be avoided with suitable system damping or end-stop system. Through straight forward derivation, one can obtain the vertical nonlinear restoring force fNSMech(t) at displacement z(t) – Eq. (5),
2. Mathematical model of WEC with NSMech
fNSMech (t ) =
One small buoy (diameter = 4.0 m, draft = 5.0 m and natural period = 5.0 s) is preliminarily designed to be located in the near-shore region of Rio de Janeiro - Brazil [29], with 50 m water depth. The buoy is constrained to oscillate only in the vertical direction by a concrete base fixed structure on the seabed. The power take-off (PTO) system is arranged on the above water upper platform. One nonlinear stiffness system with mechanical compression spring (NSMech) works between the buoy and PTO system through a rod, as shown in Fig. 1. Cummins’ equation was used to set up the mathematical model of
⎧− nK 0·z (t )·(1 − ⎨0 ⎩
L0 z 2 + L2
) z ≤ zf 0 z > zf 0
(5)
where: n - number of the parallel springs. Two convenient dimensionless parameters of NSMech may be defined as stiffness ratio α (=Cwl/ nK 0) and geometric ratio γ (=L/ L0) for the expression of NSMech. It should be also noticed that there exists a critical value of γ: γcr = 1/(1 + α ) , characterizing the system behavior as mono-stable, bi-stable and quasi-zero stiffness (QZS) – Fig. 3. The black curve in Fig. 3 depicts the condition when Cγ (=γ / γcr ) is equal to 1.0, 2
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Fig. 1. Schematic view of the WEC with NSMech (adpted from [28]). (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
corresponding to the QZS behavior (red potential energy curve). In the region above it, where Cγ > 1.0, the system presents a mono-stable behavior; and if Cγ < 1.0, it defines a bi-stable behavior. Further details including a comprehensive discussion about the static characteristics of the system can be found in Ref. [28]. Then, the state-space model of the whole system may be set up as Eq. (6),
⎡ {A′}n × n {0}n × 1 {B′}n × 1⎤ {X} ⎡ {0}n × 1 ⎤ ⎡ {X˙ }n × 1⎤ ⎢ ⎥ ⎡ n × 1⎤ ⎢ 0 { } 0 1 ⎥ 0 1 × n ⎢ z˙ ⎥ = ⎢ ⎥·⎢ z ⎥ + ⎢ fW + fNSMech ⎥ −BPTO ⎢ v˙ ⎥ ⎢ −{C′}1 × n −Cwl v ⎥ ⎣ ⎦ ⎢ ⎣ ⎦ ⎣ m + A∞ ⎥ ⎦ ⎣ m + A∞ m + A∞ m + A∞ ⎦
(6)
where: ( · ) denotes derivative associated with time t; z – vertical displacement; and v – velocity of the PA buoy; {X}n × 1 defines the nth order state vector and {A′}n × n,{B′}n × 1, {C′}1 × n are the constant state-space matrices in terms of fR(t); A∞ is the infinite frequency added mass; Cwl is the constant hydrostatic restoring coefficient; BPTO is the linear PTO damping. Finally, the classical 4th order Runge–Kutta algorithm is utilized to solve the ordinary differential equation (ODE) and Eq. (7) calculates the mean power Pm absorbed by PA.
1 Pm = T
∫0
T
(BPTO ·v )·v·dt
Fig. 3. System behaviors in the (α, γ) domain. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
CWR =
Pm = Pw D
Pm ρg 2 2 H T ·D 64π S e
(8)
where: Pw – wave energy flux in irregular waves; Te – energy period (assumed as TP/1.12 for the standard JONSWAP spectrum [32]); D – PA buoy diameter. To evaluate the WEC performance in a given sea site, the annual energy production (AEP) is also adopted, as in Eq. (9),
(7)
The capture width ratio (CWR) is defined by the Eq. (8):
Fig. 2. Sea spectrum and the corresponding wave excitation force time record. 3
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Table 2 Irregular wave conditions.
Fig. 4. Bivariate distributions of occurrence and energy in terms of HS and TP in the near-shore region of Rio de Janeiro [29]. The color scale works as percentage values, representing the contribution of the sea state to the total wave energy, while the black numbers indicate the occurrence of sea states in the number of hours. The average wave energy flux in this region is about 10.5 kW/ m, with black curves depicting the wave flux distribution.
Variables
Units
Values
HS Tp
[m] [s]
/0.5, 1.0, 1.5/ /5.0, 7.0, 9.0, 11.0, 13.0, 15.0/
•
N
AEP = 8670· ∑ Pmi·fi
(9)
i=1
N
∑ fi
=1
groups of random phase angles, while the shaded areas define the range within one standard deviation on each side of solid curves. The peaks of the solid curves, 3 red points, are the representative points for different NSMech configurations. Note that as α increasing, Pmmax decreases, however, the corresponding optimal BPTO increases. Besides, the changes of BPTO around the optimal value have weaker influences on the performance of the configuration with higher α, which is a positive characteristic for PTO system designs; (b) Presents the convergence curves of Pmmax (solid curves) and the corresponding coefficient of variation (COV, dashed curves) with respect to T. It is shown that Pmmax is relatively stable with respect to T, while the COV decreases as increasing T. To meet the balance of accuracy and efficiency of simulation, T = 2000 s is taken as the effective simulation length removing the first 300s’ transient part from the total simulation length 2300 s. In addition, the COV of bistable system (as blue dashed curve) is larger than that of monostable system (as orange and yellow dashed curves), which means that the behavior of mono-stable system brings lower dispersion.
(10)
i=1
In Fig. 6, groups of CWR patterns with respect to different NSMech configurations under groups of irregular wave conditions are presented. The significant wave height HS and peak period Tp are shown on the left and bottom axis respectively. In each sub-graph, the horizontal axis defines Cγ, while the vertical axis is the CWR corresponding to the optimal BPTO. Red dots depict the CWR of the linear model without NSMech in each sea state. The blue and red dash curves divide each subgraph at Cγ = 1 and Cγ = 2 . The configurations with α from 1.0 to 7.0 correspond to other color curves following the color scale on the rightbottom of the figure. An obvious pattern is that,
where: Pmi – mean power in the ith sea state; fi – frequency of occurrence of the corresponding sea state; N – total number of sea states appearing in the sea site; and one year time length – 8760 h. Fig. 4 depicts the distributions of occurrence and energy in terms of HS and TP in the near-shore region of Rio de Janeiro [29]. 3. Performance of WEC with NSMech in irregular waves To obtain the WEC performance with NSMech in irregular waves, the ranges of NSMech dimensions have to be defined first. For sake of convenience, the horizontal length L of NSMech is taken shorter than the buoy radius. Once L < L0, then γ should be always smaller than 1.0. The NSMech configurations are presented in Table 1. Since most of the energy content of ocean waves is composed by waves with period between 5.0 s and 15.0 s [33], and the significant wave heights (HS) with larger occurrence are below 2.0 m (see Fig. 4), 18 sea states with 3 groups of different HS and 6 groups of different peak periods (Tp) (Table 2) are selected. Considering the effect of the random phase in the irregular wave records, each sea state is repeated with 50 distinct groups of random phase angles. In the present article, the focus is the influences of NSMech. Pmmax is defined as the optimal mean value of Pm with respect to BPTO in each sea state, taking into account 50 groups of random phase angles. As well, the CWR is derived from Pmmax corresponding to optimal BPTO, based on Eq. (8), if not specially mentioned. Fig. 5 depicts the convergence analysis due to the simulation length (T):
• When T •
Such a phenomenon is caused by the change of the natural period of WEC. As the natural period of the linear model without NSMech is around 5.0 s, the system perform best around Tp = 5.0 s ; while with NSMech, the natural period is increased, then the better performance appears at somewhere Tp > 5.0 s. Additionally, the improvements with respect to the best performance of linear model at Tp = 5.0 s appear at a wider range between Tp = 7.0 s and Tp = 11.0 s which could be attributed to the broadened bandwidth brought by NSMech. Furtherly, as the wave height decreases, the range can be broadened more significantly, in together with that the CWR is higher in low wave height conditon than that in high wave height condition. Then it can be concluded that in low wave height condition, NSMech can play a better role in improving the performance of WEC. It should be noted that the corresponding Cγ in which significant improvement can be supplied usualy exists between 0.5 and 2.0, especially around Cγ = 1. In other words, when the values of γ and α are too high, the effect of the corresponding NSMech is weakened. Consequently, the following ananlysis will focus more on this region. Further results about the influence of L will be shown in Fig. 7.
• (a) Defines P
m patterns of 3 groups of NSMech configurations with respect to varied BPTO through 2000s effective simulation length. The solid curves represent the mean values, taking into account 50
Table 1 Configurations of NSMech. Variables
Units/Expressions
Values
L α γ
[m] Cwl/nK0 L/L0
/0.5, 1.0, 1.5/ 1.0 ∼ 7.0 0.25 ∼ 0.95
p = 5.0 s , the maximum CWR seems to approximate to the value of the linear model without NSMech, and there exists a local minimum point at around Cγ = 1.0 ; When Tp > 5.0 s, the optimal CWR are always higher than the corresponding values of the linear model, and appear in the region around Cγ = 1.0 , or within the range between Cγ = 1 and Cγ = 2.0 .
4
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Fig. 5. Convergence analysis on the simulation length (T). Sea state: HS = 1.5 m, Tp = 9.0 s ; 3 groups of NSMech configurations: I (bi-stable): L = 1.5 m, γ = 0.4, α = 1.0 ; II (mono-stable): L = 1.5 m, γ = 0.4, α = 2.0 ; III (mono-stable): L = 1.5 m, γ = 0.4, α = 3.0 . COV defines the ratio of standard deviation to the mean value. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
difference. The present case supplies further evidences that the two factors, L and HS, influence the performance of NSMech throughout the range of L/HS. In summary, in order to enhance significantly WEC power capture using NSMech, the configurations with high value of L and small γ, and small α perform better. Such requirements correspond to the mechanical compression spring with both large compressible length and high stiffness. Then, it brings about discussion about exploring feasible configurations based on the mechanical compression spring design principles.
Fig. 7 presents patterns of CWR depending on Tp and L featuring four groups with different (α, γ) configurations in irregular waves. From (a) to (c), results corresponding to Cγ equal to 0.8, 1.0 and 1.2 respectively, characterizing bi-stable, QZS and mono-stable behaviors, as defined in Fig. 3. The NSMech shown in Fig. 7(d) are same as that in Fig. 7(c) but with lower wave height. Each figure corresponds to different Tp for each one of the six groups . Each group is composed by t results from three NSMech configurations with different L and one linear model without NSMech taken as reference. In sea states with higher Tp than the natural period (5.0 s) of PA, larger L can better enhance the power capture. One exception exists when Tp = 7.0 s in Fig. 7(d) due to the natural period increase to near 9.0 s when L = 1.5 m , while it increases only to 7.0 s when L = 1.0 m . Consequently, the NSMech with L = 1.0 m presents a better performance at Tp = 7.0 s . Based on this analysis, one may conclude that larger L increases the natural period to a higher level, which is also influenced by the wave height. Noting that the case – L = 1.5 m , shown in Fig. 7(c) and the case – L = 0.5 m , shown in Fig. 7(d) are characterized by the same value of L/ HS = 1.0 , the corresponding CWRs are almost same with very little
4. Constraints of mechanical compression spring 4.1. Determination of the feasible NSMech The conventional mechanical spring design procedure finds a group of suitable variables (both geometrical and physical) according to some special requirements. The present section indicates the set of constraints to be taken into account in the mechanical compression spring when exploring the feasible region of geometrical and physical
Fig. 6. CWR of WEC with NSMech (L = 1.5 m ) in irregular waves. (For interpretation of the references to color in text, the reader is referred to the web version of this article.) 5
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Fig. 7. CWR of WEC with NSMech in irregular waves. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
Fig. 8. Schematic main dimensions of the mechanical compression spring.
variables associated with NSMech. The schematic main dimensions of the compression spring are presented in Fig. 8. The procedures to determine the feasible region considered the following steps:
Table 3 Variables for mechanical compression spring design.
Step 1: Define the set of parameters to be explored – horizontal length of NSMech L, the number of adopted springs n, geometric ratio γ and stiffness ratio α, wire diameter d and spring index C, as in Table 3. Here, the ranges of values of L, γ and α are adjusted according to the analysis presented in the previous section. Due to the oscillation of PA in waves, the ranges of d and C are determined based on the characteristics of the material (e.g. 6
Parameters
Units/Expressions
Values
L n γ α d C
[m] [–] L/L0 Cwl/nK0 [mm] D/d
/0.5, 1.0, 1.5/ /3, 6, 9, 12/ 0.02 ∼ 0.98 1.0 ∼ 7.0 3.2 ∼ 10.0 4.0 ∼ 12.0
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hard drawn wire A227) associated to fatigue loading [34]; Step 2: Derive the basic parameters – stiffness K0 (Unit: N/m), free length L0 (Unit: m), coil diameter D (Unit: mm), and the curvature correction factor – Bergsträsser factor KB [34];
K0 =
Cwl L 4C + 2 , L0 = , D = C·d, KB = n·α γ 4C − 3
Step 3: Derive axial load amplitude Fa (Unit: kN), and the number of active coils Na. Fa is the average value of the maximum and minimum values of axial load [34]. As the maximum load in NSMech appears when the spring in the horizontal equilibrium position and the minimum load is zero, it can be expressed as,
Fa =
10−3K 0 (L0 − L) 2
Fig. 9. Feasible region derived by the constraints of mechanical compression spring design.
The number of active coils Na can be defined in according to the stiffness by the expression,
K0 =
can be satisfied simultaneously, the corresponding design parameters will be feasible. Following the above steps, the feasible regions of (α, γ) with different (L, n) configurations are investigated and shown in Fig. 9. The bottom and left axis define the number of springs adopted (n) and the horizontal length (L) for each sub-graph. The black curve divides the (α, γ) domain into feasible (green) and infeasible (white) region. The blue and red dashed curves, corresponding to the reference curves in Fig. 7, divide each sub-graph at Cγ = 1 and Cγ = 2 showing that more CWR can be obtained if the parameter is near Cγ = 1 or within the range between Cγ = 1 and Cγ < 2. The results discussed above give grounds for the following additional comments:
103Gd 8C 3Na
where: shear modulus: G = 78603.0 MPa . Step 4: Three parameters – shear stress amplitude τa (Unit: MPa), solid length LS (Unit: mm), and pitch p (Unit: mm) for squared and ground end are obtained as above to play the role of constraints. Because buckling can be avoided for long spring if guided by sleeves or over an arbor [35], the conditions of stability and critical frequency were not considered in the present discussion. 8F C Maximum stress: τa = KB · a2 ·103 πd Solid length: LS = (Na + 2) d
• Adopting more springs, keeping the same L, can broaden the feasible region for lower values of α and γ; • Keeping the same n, shortening the horizontal length can lower the
103L − 2d
0 pitch: p = Na Step 5: Constraints. First, the maximum stress τa should satisfy the strength condition
τa ≤
Ssa nf
bound of γ and broaden the feasible region as well.
In other words, more feasible configurations can be found when more springs are adopted and shorter horizontal length applied. Another interesting finding shows that the same feasible regions exist if L/n, for different configurations are kept equal, such as in Fig. 9 subgraphs 9, 6 and 3 or 10 and 8,
(11)
where: Ssa – torsional endurance strength for infinite life. For the peened spring, Ssa = 398 MPa [34], and the safety factor nf = 1.5. Accordingly, the minimum total axial gap should be taken as 15% (cg) of maximum deflection (δmax) [35],
L9 L L 1 L L 1 = 6 = 3 = or 10 = 8 = n9 n6 n3 6 n10 n8 12
total gap = cg·δmax
where: the subscript identifies the different sub-graphs. The practical selection can benefit from such a relationship. Besides, the feasible parameters (α, γ) occupy the top right region in each sub-graph, meaning that,
Free length L0 is also the summation of solid length LS, maximum deflection δmax and total gap,
L0 = LS × 10−3 + total gap + δmax
• The constraints on the design of spring indeed exist when determining (α, γ); • It is easier to obtain the feasible configuration for higher α and γ
In the structure of NSMech, the allowable deflection of spring should satisfy,
values.
L0 − L ≤ δmax The present distribution characteristics of the feasible NSMech configurations will result in a significant constraint on the enhancement of the power capture with NSMech. This effect will be investigated in the next sub-section.
Then the following expression is obtained,
LS + (1 + cg )·(L0 − L)·103 ≤ L0 ·103
(12)
4.2. WEC performance with feasible NSMech
Finally, the pitch p is preferred to be less than half of the coil diameter
d < p ≤ 0.5D
Noting the existence of constraints on the NSMech configurations, the operational performance (AEP) in the near-shore region of Rio de Janeiro, will be investigated considering the feasible configurations. In Fig. 10, the configurations in the sub-graphs (a), (b), (c) and (d) corresponding to that the same configurations taken in Fig. 9 sub-9,
(13)
When all constraints, defined in Step 5 – Eqs. (11), (12) and (13), 7
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Fig. 10. AEP of WEC with NSMech in the near-shore region of Rio de Janeiro.
Table 4. Based on the data presented in Table 4, one can conclude that,
sub-6, sub-3 and sub-12 respectively. The bar height in each sub-graphs depicts the resulting AEP values. With respect to each (α, γ) configuration, the corresponding AEP is calculated with an optimal BPTO for all the sea states in the given sea site. Obviously, the better performance in each sub-graph concentrates itself in the region with lower values of α and γ, satisfying Cγ around 1 or, at least, Cγ < 2 keeping fully consistency with the characteristics presented in Fig. 6. In such region, larger L leads to greater enhancement. As the values of α and γ increase, the AEPs approach stable values, whatever would be L and n. Actually, the reference value around 33.17 MWh was obtained from PA without NSMech. However, when considering the constraints, the feasible AEPs are described by color scale. The optimal performance is indicated by red arrow in each sub-graph. Though the feasible regions in Fig. 10(a), (b) and (c) are the same, the optimal (α, γ) are different due to the L variations. Compared with Fig. 10(a), Fig. 10(d) features a broader feasible region through adopting more springs. Furthermore, the optimal AEP and the increase with respect to the reference value are presented in
• With the same feasible region (as the blue cells), the higher value of L induces higher performance enhancement; • If the same L is selected, better performance is obtained by adopting more springs.
Moreover, lower values of L also lead to better enhancement by broadening the feasible region with more springs (orange cell). In other words, sometimes the feasible region has more significant effect on the performance of WEC with NSMech than L. However, it should be noted that the relative increase is really small, around 20% at most, when the constraints of NSMech are considered. Not to be mentioned if compared with other enhancement approaches, such as latching control or model predictive control (MPC). The results also suggest the need to investigate other approaches with less constraints to substitute NSMech. The pneumatic cylinder is one alternative approach to be introduced in the next section.
Table 4 WEC Performance with NSMech in the near-shore region of Rio de Janeiro.
5. Nonlinear stiffness system with pneumatic cylinder (NSPneu) The former section indicated the limited performance of WEC with NSMech due to the physical constraints of the mechanical compression spring. To overcome the weakness of NSMech, one alternative approach substitutes the mechanical compression springs by pneumatic cylinders, as shown in Fig. 11. The rods of cylinders are hinged to follow the buoy vertical motion. The pressurized gas (blue) fills in the cylinder chambers. An additional gas reservoir is necessary to link cylinders all together. First, it keeps the same gas pressure in all cylinders, satisfying the horizontal force balance; second, the additional volume broadens the range of pressure and volume variation. Other schematic explanations follow the same description as shown in Fig. 1. The objective of 8
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Fig. 11. Schematic view of the WEC with NSPneu. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)
Fig. 12. NSPneu force and total restoring force for different NSPneu configurations (L = 2.0 m ).
the present section is to investigate the practical possibilities of the nonlinear stiffness system using the pneumatic cylinder (NSPneu). The determination of optimal parameters of NSPneu will not be addressed here. In the NSPneu device, n cylinders (each piston area: A0) are axisymmetrically connected. At equilibrium position, the cylinder length is L, the total gas volume is V0, and the internal pressure is P0. At a given displacement z(t), the gas volume increases by V:
Table 5 Ranges of primary parameters of NSPneu. Parameters
Unit
Min
Max
D0 P0 V0
cm bar (1 bar = 105 Pa) m3
2 1 0.001
20 20 1
Note: D0 - piston internal diameter ( A0 = πD02 /4 ).
V = V0 + nA0 ·( z 2 + L2 − L)
(14)
Assuming adiabatic process, the gas pressure can be calculated by:
PV μ
= P0 V0μ
(15)
where: μ is the specific heat ratio (1.4 for nitrogen). Thus, the net vertical force supplied by the NSPneu device will be:
fNSPneu (t ) = nP0 A0 ·
V μ ·⎛ 0 ⎞ z 2 + L2 ⎝ V ⎠ z
(16)
For the sake of simplicity, two new variables are adopted: equivalent stiffness – nK0 and equivalent length – L0,
nK 0 =
V nP0 A0 , L0 = 0 L nA0
(17)
and the dimensionless variables are:
Fig. 13. Feasible region of NSPneu.
α= 9
Cwl L·Cwl L nA0 L , γ= = = nK 0 nP0 A0 L0 V0
(18)
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Fig. 14. WEC with NSPneu: AEP in near-shore region of Rio de Janeiro.
• α < 1.0, bi-stable behavior, with three zero values of restoring force; • α = 1.0, QZS behavior, with only one zero and zero stiffness at equilibrium position; • α > 1.0, mono-stable behavior, with only one zero but non-zero
Table 6 Performance comparison between WEC with NSMech and with NSPneu.
stiffness at equilibrium position.
Another aspect which deserves mentioning is if the buoy displacement is around the equilibrium position (|z| < 0.8 m, in the present case), the effect of γ on the NSPneu force will be small. Therefore, the total restoring force does not change dramatically. In other words, the influence of α will be more significant in the performance of NSPneu around the equilibrium position. As α and γ increase, it reduces the NSPneu force, requiring smaller values which are preferable to obtain more significant influence from NSPneu. Once understood the basic static characteristics of the system, the feasibility of NSPneu will be investigated. First, the primary parameters ranges of NSPneu are presented in Table 5. The ranges may be not quite convenient, but are in according to the industry applications [12]. A group of (α, γ) may be considered feasible if suitable parameters exist in the space (D0, P0, V0) as shown in Table 5. Thus the feasible regions (α, γ) of NSPneu under different combinations groups (L, n) were investigated and the results are presented in Fig. 13. The most obvious feature is that feasible region occupies a broader area, meaning that the influence of constraints are quite limited. Similar results as those presented in Fig. 9 show that decreasing L or increasing n induces the feasible region to cover range of smaller α and γ where better performances are expected. However, if the L/n are equal, the feasible regions are no longer the same. Consequently, Fig. 14 presents the AEP of WEC taking NSPneu with some typical (L, n) combinations. The unfeasible region only appears if α was small. Therefore, to some extent, it will be easier to obtain better
Therefore, the expression of fNSPneu is reduced from the dimension of (n, L, P0, A0, V0) in Eq. (16) to the dimension of (n, L, L0, K0) as:
fNSPneu (t ) = nK 0·
z
·L·⎛⎜ z 2 + L2 ⎝ L0 +
L0
μ
⎞
⎟
z 2 + L2 − L ⎠
(19)
Furthermore, similar approach as used before in the NSMech case, will be adopted in the analysis of the influence of different parameters on the performance of NSPneu. It should be noticed that setting L0 together with suitable piston area A0 and gas volume V0, the value of γ can be either larger or smaller than 1.0, which makes an important practical difference if compared with NSMech system. Fig. 12(a) and (b) present patterns of NSPneu force and total restoring force as function of displacement (z), considering different NSPneu configurations. The NSPneu nonlinearity features negative stiffness near the equilibrium position (z = 0 ) (Fig. 12(a)). Adding the hydrostatic restoring force (fH), Fig. 12(b) allows for obtaining the total restoring force. Similarly, different values of α define three distinct behaviors: 10
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Supplementary materials
enhancement with NSPneu. Furthermore, the performance comparisons between WEC with NSMech and with NSPneu are listed in Table 6. The results show great advantage from using NSPneu that less elements are required to achieve better performance (around 1-fold to 2-fold increases). The feasible region features a varied optimal L for each n. Therefore, a balance between L and n should be carefully determined.
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6. Conclusions The conventional time domain approach applied to obtain a series simulations featuring NSMech in irregular wave conditions produced good and reliable results to fill in the lack of researches on constraints and limitations concerning applications on wave energy conversion. An innovative modification on the expression of NSMech restoring force discussed here considered the pure compressed characteristic of mechanical compression spring. To deal with the typical randomness of the real sea conditions and the system nonlinearities, the analysis assumed the average performance of a great number of random phases grouped for every sea state. Preliminary simulations showed that the enhancement due to NSMech application on PA type WEC was very much dependent on the suitable L, γ, α (or Cγ) configurations. The optimal performance appeared near Cγ = 1.0 . In addition, large compressible length (or long L and small γ) and high stiffness (small α) should be preferred in practical design of NSMech. Such results can be used to determine the ranges of L, γ, α (or Cγ) when investigating the possible NSMech configurations. Procedures aiming at defining the feasible region of NSMech should be based on the mechanical compression spring design process. The results provided the feasible region of (α, γ) for different (L, n). There exists one single positive correlation between L and n leading to the feasible region. Within the same feasible region, the higher L and more springs adopted induce higher improvement. However, the relative AEP increase is really small, around 20% at most, within the feasible region. In other words, the NSMech constraints indeed weaken the WEC with NSMech performance. Adopting a large number of springs poses a great challenge to the design of the whole system. To overcome the weakness and practical difficulties of the NSMech, one alternative approach would be to substitute the mechanical compression spring by pneumatic cylinder. The preliminary studies succeeded to demonstrate advantageous towards the NSPneu. It requires fewer elements to improve performance (around 1-fold to 2-fold increases). Simultaneously, a careful balance between L and n should be considered. It should also be noticed that the dynamic model of NSPneu presented here was a simplified model, assuming an adiabatic process. Other factors, such as piston seal friction, turbulent losses due to the air transmission, and the non-adiabatic compression, should be incorporated carefully to improve the dynamic model. Besides, a much more realistic model should also include an end-stop mechanism together with feasible buoy excursion and NSPneu stroke.
Acknowledgement The first author would like to acknowledge the D.Sc. Scholarship from CAPES, Ministry of Education/Brazil, and the support from China Ship Scientific Research Center. Special thanks to FURNAS through ANEEL (Brazilian Electrical Energy Agency) R&D Program for the financial support of the research in progress at Subsea Technology Laboratory (COPPE - Federal University of Rio de Janeiro) on wave energy (contract number PD-0394-1248/2012). The third author would like to thank INOEF for the support to research activities on wave energy. 11
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