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An approach for non-linear stochastic analysis of U-shaped OWC wave energy converters
Accepted Manuscript An approach for non-linear stochastic analysis of U-shaped OWC wave energy converters Pol D. Spanos, Federica Maria Strati, Giovanni Malara, Felice Arena
PII: DOI: Reference:
S0266-8920(17)30113-3 http://dx.doi.org/10.1016/j.probengmech.2017.07.001 PREM 2931
To appear in:
Probabilistic Engineering Mechanics
Received date : 4 June 2017 Accepted date : 12 July 2017 Please cite this article as: P.D. Spanos, F.M. Strati, G. Malara, F. Arena, An approach for non-linear stochastic analysis of U-shaped OWC wave energy converters, Probabilistic Engineering Mechanics (2017), http://dx.doi.org/10.1016/j.probengmech.2017.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
AN APPROACH FOR NON-LINEAR STOCHASTIC ANALYSIS OF U-SHAPED OWC WAVE
2
ENERGY CONVERTERS
3 4
Pol D. Spanos(1), Federica Maria Strati(2), Giovanni Malara(3), Felice Arena(4)*
5 6
(1) L. B. Ryon Chair in Engrg., G. R. Brown School of Engrg., Rice University, Houston, TX
7
77005. Email: [email protected].
8
(2) Ph. D. student, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea” University of
9
Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email: [email protected]
10
(3) Post-Doctoral research fellow, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea”
11
University of Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email:
12
[email protected]
13
(4) Professor, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea” University of
14
Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email: [email protected]
15
* Corresponding author: [email protected]
16
1
17
Abstract
18
This paper analyzes the dynamics of a U-Oscillating Water Column (U-OWC) wave energy
19
harvester. The geometrical configuration of this device comprises: an air mass enclosed into a
20
pneumatic chamber and connected to the atmosphere through a duct containing the Power-Take-Off
21
device; and a U-shaped water column connecting the air mass to the open wave field. The dynamics
22
of the U-OWC is described by a set of two coupled nonlinear integro-differential equations which
23
have no obvious exact analytical solution. In this context, the paper addresses the problem of
24
estimating efficiently and reliably, albeit approximately, the dynamic response of the U-OWC
25
system given a certain power spectrum-compatible wave excitation. For this purpose, an equivalent
26
linear model describing the dynamic response of the system is derived by the technique of statistical
27
linearization, and used to determine iteratively the statistics of the harvester response. Further, two
28
specific U-OWC configurations are considered: the full-scale U-OWC prototype equipped with a
29
linear Wells turbine in the port of Civitavecchia (Rome, Italy); and a U-OWC small scale model
30
which comprises a non-linear orifice, only, and is installed in the benign basin of the NOEL
31
laboratory (Reggio Calabria, Italy). Relevant Monte Carlo data are used to demonstrate the
32
efficiency and reliability of the proposed approach.
33 34
Keywords: Wave energy converter; U-OWC; statistical linearization; Monte Carlo data.
2
35
1. Introduction
36
In the recent decades the increasing demand of energy and the progressive depletion of traditional
37
sources has generated a significant challenge to the scientific community. That is, to develop novel
38
energy harvesting technologies exploiting renewable resources. In this context, the energy
39
associated with ocean waves is a promising source whose potential global power is estimated to
40
have a magnitude of the order of 1013 W [1]. Within this field, a large number of wave energy
41
converters (WECs) have been investigated, and some of them have reached the stage of a full-scale
42
prototype [2]. Their structures can be either floating or fixed, with their configurations properly
43
designed to harvest the kinetic or potential energy of ocean waves. Typically, nonlinear differential
44
or integral equations in conjunction with a statistically described wave excitation are required to
45
capture the dynamic behavior of WECs. Thus, the application of computationally costly numerical
46
algorithms is often necessary. Alternatively, the possibility to generate fast and reliable approximate
47
solutions of the dynamic response of a non-linear system can be desirable. In this regard, various
48
approaches have been developed in the open literature; for nonlinear stochastic dynamics problems
49
an extensive description can be found in references such as Roberts and Spanos [3].
50
One of the most versatile tools is the statistical linearization technique which is applied herein for
51
the determination of the dynamic response of a specific wave-energy harvester, named U-shaped
52
Oscillating Water Column (U-OWC).
53
Succinctly, the implementation of the statistical linearization technique can generate an approximate
54
solution for the response of a multi-degree of freedom dynamic system subjected to a random
55
stationary excitation. The technique involves the replacement of the nonlinear terms in the
56
equations governing the system dynamics by equivalent (in a specified sense) linear terms, so that
57
the exact solution for the equivalent linear model can be determined by the classical input-output
58
relationship. Applications to marine and ocean engineering can be found in references such as
59
Spanos and Agarwal [4], Low [5], and Spanos, et al. [6].
3
60
Within the field of wave energy converters, oscillating water columns (OWC) devices have reached
61
the most advanced development stage. Their dynamics and energy conversion mechanism have
62
been extensively investigated [7-10]. The basic geometrical configuration of an OWC comprises a
63
pneumatic chamber which contains an air mass in its upper part and water in the lower part
64
connected to the open sea. The fluctuations of the water alternately compress and expand the air,
65
which is forced to flow through an air duct connected to the atmosphere. Thus, the rotor of a self-
66
rectifying turbine located in the air duct is activated, and the mechanical torque inducing the
67
production of electrical energy is generated. Starting from the 80s, a number of OWCs were build
68
and tested. Example of operative OWCs are [11]: the five chambered plant in the port of Sakata
69
(1989), Japan, with an original rating of 60kW provided by a single Wells turbine; the PICO plant
70
in the Azores Island (1999) with an installed capacity of 400kW; the LIMPET plant (2000) in Islay
71
island, Portugal, originally rated at 500kW, presently used as test site for various turbine designs;
72
and the Mutriku plant (2011), Spain, representing the first multiple chamber OWC plant with a total
73
power of 296 kW.
74
In this framework, the U-OWC represents a modification of the basic OWC geometrical
75
configuration [12]. Its basic structure is quite similar to the one characterizing the conventional
76
OWC. However, it includes an additional vertical element on the wave-beaten side of the plant
77
forming a typical U-shaped duct. This small element allows introducing an additional mass that can
78
be used for tuning the water column natural period to a desired value. This device has recently
79
reached the stage of full-scale prototype; such a configuration is presently under construction in the
80
port of Civitavecchia (Rome) [13] and has a (potential) estimated installed power of 2.7 MW.
81
The first theoretical model for the U-OWC dynamics was proposed by Boccotti [14]. Next, Malara
82
and Arena [15] developed a consistent representation of the diffracted/radiated wave field in front
83
of the plant. Small-scale field experiments validated the theoretical model and the absorption
84
capacity of such device [12, 16-18]. The mathematical description of the U-OWC dynamics
4
85
involves two coupled non-linear integro-differential equations. This feature points out a significant
86
difference with respect to conventional OWCs, for which a linear analysis is reliable as long as the
87
linear water wave theory, and the assumption of an isentropic thermodynamic process for the air
88
pocket hold [1, 8, 19-21]. Further, the complexity of the U-OWC theoretical model introduces
89
significant difficulties when dealing with the optimal-design of the system and with control related
90
aspects.
91
In this framework, an optimal approximate linear model of the U-OWC system is pursued by
92
implementing the technique of statistical linearization. Analyses are conducted by considering two
93
U-OWC configurations. The first comprises a linear Wells turbine located in the air duct. In this
94
context, a case study pertains to the full-scale prototype located in the port of Civitavecchia (Rome,
95
Italy) [22]. The second configuration is a U-OWC comprising an orifice without a turbine. For such
96
a case, an additional non-linear term in the dynamic modelling of the pressure fluctuations inside
97
the pneumatic chamber must be included. In view of a full-size realization, the modelling of a U-
98
OWC equipped with an orifice is of particular interest. Indeed, such a configuration either can
99
simulate the presence of valves in the chamber, or can be interpreted as the approximation of a
100
nonlinear PTO mechanism, such as the case of an Impulse turbine [11]. A small-scale U-OWC
101
model with such configuration is currently tested in the benign basin of the NOEL laboratory
102
(http://noel.unirc.it/) (Reggio Calabria, Italy).
103
Solutions for realistic sea states compatible with typical power spectral density functions of sea
104
waves are derived. To assess the efficiency and reliability of the proposed approach, responses
105
computed from the equivalent linear system are compared with relevant Monte Carlo data derived
106
by numerical integration of the original nonlinear system dynamics equations.
107
108
2. Physical U-OWC system description
109
Next the two different U-OWC plants considered in this study are discussed. The first configuration 5
110
refers to a U-OWC equipped with w a lineaar Wells tu urbine locaated in the air duct. The T secondd
111
configuratiion is a U-O OWC which h comprise a non-linearr orifice, without the prresence of th he turbine.
112
2.1 U-OWC C equippedd with a linea ar turbine
113
The schem matic view of o a U-OWC C is shownn in Fig. 1. Here, dimensions of thhe pneumattic chamberr
114
are definedd by the heeight hc, and d the width b2 and b3 (not ( shown in the figuure) in the longitudinal l l
115
and transveerse directioon, respectiively. The ssymbols li and a b1 defin ne the length th and the width w of thee
116
external vertical ductt with the opening hhaving subm mergence dh. The tim me-dependen nt variabless
117
describing the U-OW WC dynamics are x annd pc, deno oting the water w level inside the pneumaticc
118
chamber m measured froom the meaan water levvel (positivee downward ds), and the air pressuree inside thee
119
pneumatic chamber, respectively.
120 121
Fig gure 1. Schem matic view of th he U-OWC plant.
122
In this worrk, the plannt configuraation of the full-scale U-OWC U pro ototype undder construcction in thee
123
port of Civvitavecchiaa (Rome, Italy) [22] iss considereed (Fig. 2). Tab. 1 sum mmarizes the relevantt
124
geometricaal parameterrs of the deevice. Furtheer, a Wells turbine witth an outer ddiameter D = 0.738 m,,
125
a hub diam meter Dhub = 0.5 m, an nd a dampinng ratio K = 2.9 is considered. FFor the purp pose of thee 6
126
presented aanalysis, a constant c rotational speeed equal to 4000 rpm iss assumed.
127 128
Figure 2. Civitavecchia U-OWC U caissoons. Photo takeen on May 2015 (Source: w wavenergy.it).
129
d [m]
hc [m]
b2 [m]
b3 [m]
dh [m]
b1 [m]
D [m]
14.20
9.40
3.20
3.87
2.00
1.60
0.738
130 131
Table 1. Geoometrical charracteristics off the Civitaveccchia U-OWC C plant. The syymbols refer too the scheme shown in Fig..
132
1, while b3 iss the width of the t pneumaticc chamber in tthe transversee direction; an nd D is the diaameter of the air a duct.
133
2.2 U-OWC C with a noon-linear oriifice (only)
134
A U-OWC C small scalle model is presently ttested in thee natural beenign basinn of the Nattural Oceann
135
Engineerinng Laborattory (NOEL, www.nnoel.unirc.it,, Reggio Calabria, IItaly), wheere a new w
136
experimenttal activityy started in 2014 [18]] (Fig. 3). The peculiarity of thhis site relates to thee
137
possibility to conduct field experriments direectly in a beenign sea by y the implem mentations of commonn
138
methodoloogies used in i large arttificial wavve tanks [23 3]. The mo odel compriises three in ndependentt
139
chambers w whose totall width is 3..79 m in thee transversee direction. The width oof a single cell is 1 m..
140
The cross-sectional viiew of the small-scale s U-OWC model m is shown in Fig. 44. The three chamberss
141
t orifice, which is eq qual to 0.32 m for the ccentral cell, and 0.15 m differ withh regards to the size of the
142
for the lateeral cells. Further, F pro oper instrum ments for th he measurem ment of botth pressuress and waterr
143
levels are located insside the pn neumatic chhamber and d in the U-shaped ducct. Their po ositions aree
144
shown in F Fig. 4. In thee present ap pplication, tthe geometrry of the lateeral cell is uused. For th his, relevantt
145
geometricaal parameterrs are shown in Tab. 2..
7
146 Figure 3. Exp perimental sett-up at NOEL laboratory (R Reggio Calabrria, Italy).
147
148
149
Figure 4. Schhematic view of the centrall (left) and latteral (right) chamber of thee NOEL U-OW WC small-scale model. Thee
150
numbers in bbrackets indicaate the positio on of pressuree transducers.
d [m]
hc [m]
b 2 [m]
b3 [m]
dh [m m]
b1 [m m]
1.67
1.90
1.00
1.20
0.50
0.555
151
Table 2. Geoometrical paraameters descriibing the U-O OWC configura ation. The sym mbols refer to Fig. 4. Further, b3 is
152
the width of the chamber in the longitu udinal directioon. Furthermo ore, the diameeter of the circcular orifice is i equal
153
to 0.15 m forr the central cell, and 0.33 m for the laterral cells.
154
155
3. Equatioons of motioon of the U-OWC U
156
The set of two coupleed integro-d differential eequations required to capture c the hydrodynaamics of thee 8
157
water column oscillations jointly with the aerodynamics of the pneumatic chamber are next
158
presented. The theoretical model of a U-OWC equipped with a linear turbine is firstly presented.
159
Next, the model is revised for the case of the U-OWC comprising a non-linear orifice (only).
160
3.1 Equations of motion of the U-OWC equipped with a linear turbine
161
The equation of motion of the water displacement along the U-duct is derived [24] from the
162
unsteady Bernoulli equation by balancing the total heads at the U-OWC opening (point (1) of Fig.
163
1), and at the inner free surface (point (2) of Fig. 1). In this context, the wave pressure applied at the
164
external opening is determined according to the linear water wave theory [25]. Hence, the equation
165
governing the dynamics of the water column oscillations is
166
M ( x ) x C ( x ) x
167
where dot denotes differentiation with respect to time; ρ is the water density; g is the acceleration
168
due to gravity; patm is the atmospheric pressure; K0 is the retardation function accounting for the
169
memory effects; and p ( s ) is the wave pressure excitation calculated in a scattered wave field at the
170
center of the outer opening of the external vertical duct (point (1) of Fig. 1). Such a process is
171
mathematically described as a zero-mean ergodic stochastic process, characterized by a Gaussian
172
probability density function, consistently with common sea state representation approaches [26].
173
Further, the non-linear mass and damping terms are defined as
174
M ( x)
175
and
176
C ( x )
177
where H∞ denotes a length representing the infinite frequency added mass, and Kw is the head loss
178
coefficient.
179
The equation governing the time variations of the pressure into the air mass is derived by the
1 2g
1 b2 g b1
t
0
x ( t )K 0 (0, d h , ) d x
p c p atm p (s) , g g
(1)
b2 1 b2 li li d h H x , g b1 b1
(2)
2 x b2 K w x , 2 b1
(3)
9
180
approach of Falcão and Rodrigues [21] and Falcão [27]. In this context, the influence of the Wells
181
turbine is accounted for by a linear relationship between the air flow rate across the turbine and the
182
pressure drop (pc-patm) between the chamber and the atmospheric conditions. Thus, the equation
183
governing the aerodynamic process is
184
p c
185
where c is the speed of sound in air; γ is the ratio of specific heat at constant pressure over specific
186
heat at constant volume; A is the net flow area of the air duct accounting for the diameter of the duct
187
and of the rotor; R is the external radius of the air duct; V is the volume of the air mass inside the
188
chamber, and K is the turbine damping coefficient.
189
In a matrix notation, eq. (1) and (4) are concisely written as
190
CL q K L q Τ Ψ (q, q , q ) Q , ML q
191
where ML, CL, and KL denote the linear contribution of mass, damping, and stiffness matrices,
192
respectively. Further, q is the generalized coordinate vector; Q is the time-dependent vector
193
representing the wave excitation; Ψ (q, q , q) is a non-linear function of q and its derivatives; and the
194
constant vector T is introduced to take into account terms which are linear, but independent from
195
the generalized coordinate q. Specifically,
196
1 b2 b2 li li dh H x 0 ΜL g b1 b1 , 0 0
(6)
197
0 0 CL , 0 hc
(7)
198
1 1 g , KL 2 0 A c 1 KNR b2b3
(8)
A c2 1 V ( pc patm ) pc , KNR V V
(4)
(5)
10
199
p ( s ) Q g , 0
200
x q , pc
201
1 b2 g b 1 T
202
and
(9)
(10)
t
0
x (t )K 0 (0, d h , )d A c2 1 patm KNR b2b3
xx 1 b2 x 2 K w x x Ψ (q, q, q) 2g g 2 g b1 xp c xp c
203
patm g ,
.
(11)
(12)
204
3.2 Equations of motion of the U-OWC with a non-linear orifice (only)
205
The theoretical model described in the previous section is revised herein for the case of the small-
206
scale U-OWC model tested in the natural basin of NOEL laboratory. For such a system, the
207
equation of motion is
208
M ( x ) x C ( x , x ) x x
209
where p m 1 is the time-dependent wave pressure excitation at the outer opening of the vertical
210
duct,
211
M ( x)
212
C( x, x)
p c p atm p m 1 , g g
1 b2 li li dh x Cin x Cin 2 , g b1
1 x C x x dg 1 , 2 g Rh,2
(13)
(14)
(15)
11
213
with Rh,1 and Rh,2 being the hydraulic radii of the small vertical duct and of the inner water column,
214
respectively; and
215
l b l dh , 1 i 2 i Rh ,1 b1 Rh ,2
216
and
217
2
218
In this case it is noted that that this model assumes that the process p m 1 is directly derived from
219
data measured from the pressure transducer (1) of Fig. 4. Hence, eq. (13) implicitly includes the
220
convolution integral and the infinite frequency added mass H appearing in eq. (1), that are
221
commonly derived by the decomposition of the diffracted and radiated wave field in front of the
222
plant (see Malara, et al. [24]). Further, note that the mass and damping terms are represented
223
according to the results of Malara, et al. [18] concerning the small scale model investigated. The
224
mass flow rate across the orifice depends on the square root of the air pressure drop p between the
225
chamber and atmosphere [28]. Under this assumption the resulting governing equation is
2
li b2 li d h , g b1 g
p c
226
c2Cd A0 pc x sign(p) 2air p , b2b3 hc x hc x
(16)
(17)
(18)
227
where A0 is the orifice area; and Cd is the coefficient of discharge, that is experimentally estimated
228
to be equal to 0.652.
229
The mechanical analogy for a non-linear oscillator of eq. (5) is employed to recast eq. (13) and (18).
230
In this sense, the vector of the generalized coordinates
12
231
x q p
232
is introduced. Note that the utilization of the pressure drop in place of the dynamic air pressure to
233
define vector
234
constant vector
235
the wave excitation is
236
pm1 Q g , 0
237
Finally, the terms of eq. (5) are defined as
238
1 b2 li li d h Cin2 0 ΜL g b1 , 0 0
(21)
239
0 0 CL , patm hc
(22)
240
1 KL 0
(23)
241
and
242
Cdg x x Cdg 1 x x x x x x Cin x 2g Rh,2 , q , q g Ψq , hc p x p K 2 air p x p
243
where the constant K is defined as
(19)
q , as well as the absence of the convolution integral, leads to a null value of the Τ of eq. (5). Further, the time-dependent vector representing the random process of
1 g , 0
(20)
(24)
13
c 2 C d A0 . R 2
244
K
245
4. Equivalent linear system of U-OWC devices
246
The mathematical background underlying the application of the statistical linearization technique is
247
outlined herein. Next, the technique is employed to derive approximate linear models for the two U-
248
OWC configurations considered in the study.
249
4.1 The statistical linearization formalism
250
The exact analytical solution of the non-linear eq. (5) is currently unavailable. Thus, commonly
251
computationally demanding numerical algorithms are employed for determining the system
252
response in the time domain. In this context, an approximate solution is sought by the technique of
253
statistical linearization [3]. Such technique relies on the replacement of the original non-linear
254
system by an equivalent (in a specified sense) linear system for which the exact solution is known.
255
Thus, the generic model given by the eq. (5) is replaced by the linear system
256
M
257
E E ) . Further, where M , CE , and K denote the linear substitutes of the non-linear vector Ψ (q, q , q
258
) to be symmetric is defined as [29] a condition for the function Ψ (q, q , q
259
) Ψ (q, q , q ) . Ψ (q, q , q
260
The decomposition
261
q q0 w ,
262
of the general solution of eq. (5) is implemented since the condition given by eq. (27) is not
263
satisfied [29]. In eq. (28) q 0 is a constant real vector, and w a time-dependent vector. With the
264
employment of eq. (28), the original system must still be satisfied in a specified sense. In this
265
context, eq. (5) can be written in terms of the vector w as
L
(25)
C L C E q K L K E q Q ME q
,
(26)
(27)
(28)
14
C L w K w
) Q , w β ( w q 0 , w , w
(29)
266
M
267
where
268
) K L q 0 Ψ ( w q 0 , w , w ) . β ( w q 0 , w , w
269
E E The coefficients M , CE , and K of eq. (26) are determined by a minimization procedure on the
270
vector difference ε between the original (eq. 5) and the equivalent-linear system (eq. 26). The
271
minimization procedure is performed according to the criterion
272
g ( ε T ε ) min ,
273
for every w solution of eq. (26). In eq. (31), the function g () is an averaging operator which
274
ensures defined characteristics of the equivalent linear system. Specifically, the criterion
275
E ε T ε min ,
276
is adopted, where E[·] denotes the operator of mathematical expectation. From eq. (32), the
277
approximation of a Gaussian distribution for the response and for its derivative, leads to the
278
equations
279
i m e i, j E q j
280
where k e i , j , c e i , j , and m e i , j are the coefficients of the matrices KE, CE and ME; see [3, 29].
281
4.2 Statistical linearization solution: U-OWC equipped with a linear turbine
282
In this section the statistical linearization method is applied to derive an approximate linear model
283
for eq. (5), with matrices defined by eq. (6) – (12). Further, the excitation is described by a
284
Gaussian random process in agreement with the common sea states approximation for ocean waves
285
[26].
286
Due to the non-symmetric form of the non-linear terms of eq. (5) the decomposition given by eq.
287
(28) must be employed. In this context, the components of the vector w are the zero-mean
L
L
(30)
(31)
(32)
i c ei, j E q j
i k e i, j E q j
i , j 1, 2
,
(33)
15
288
stationary stochastic processes z(t) and p(t). Further, the components of the vector q 0 are the means
289
of the processes z and p, named x0 and p0 respectively. Under this assumption, the two average
290
restoring force conditions
291
p p0 z2 1 E z z x 0 atm 0, g g 2g
292
and
293
E z p E z p
294
are derived, where z is the standard deviation of the derivative of the water column displacement.
295
Finally, the equivalent matrices
E
x0 g 0
(34)
Ac 2 1 ( p0 p atm ) 0 , KNR b2 b3
0 ; 0
CL
E
1 b2 3 K g 2 b 3 w z 1 p0
(35)
0 ; x0
E
. 1 z 2 0
(36)
296
ML
297
are derived as the solution of eq. (33), where z , z , p are the standard deviation of the responses
298
z and p, and of their derivatives, respectively.
299
Clearly, the computations involved in eq. (36) depend on the unknown statistics of the response.
300
Thus, an iterative procedure based on the stationary solution of the U-OWC governing equations is
301
adopted. Such a step is pursued consistently with the common practice used in the frequency-
302
domain analysis, based on input-output relationships. Specifically, the spectral density matrix of the
303
response S q is computed as
304
Sq () α() SQ () αT* () ,
305
where
306
S zz S q ( ) S pz
307
z and p indicate the hydrodynamic and aerodynamic response, respectively; and the frequency
308
response function α(ω) is expressed as
S zp ; S pp
KL
1 g 2 z p 2
(37)
(38)
16
1
,
309
α(ω) 2 M L M L E i C L C L E K L K L E
310
with T * denoting conjugate transposition of a matrix.
311
Then, the variances of the response are computed as
312
(39)
2 z
S d ,
(40)
z
313
2 z
314
and
315
2 z
Sz d
S () d , 2
z
S d S () d 4
z
z
(41)
(42)
316
Clearly, equations similar to eq. (37) - (39) can be derived for the process p. Further, the mean
317
values of the response are computed using eq. (34) and eq. (35), where
318
Ez z 2 Szzd ,
(43)
319
E z p i S pz d ,
(44)
320
and
321
E z p i S zp d ,
(45)
322
with i
1 . The iterative scheme is implemented by first neglecting the non-linearities and then
323
proceeding to calculate iteratively the U-OWC response statistics and account for the nonlinearities.
324
4.3 Statistical linearization solution: U-OWC with a non-linear orifice (only)
325
The statistical linearization technique is applied in this section to derive an approximate linear
326
model of eq. (5), with the matrices defined by eq. (19)-(24). For this, the decompositions of eq. (28)
327
is employed to account for the non-symmetric form of the non-linear terms of eq. (5). In this
328
context, the symbols z and h indicate the zero-mean processes for the water column oscillations, 17
329
and air pressure drop inside the chamber, respectively; the corresponding offsets are denoted by the
330
symbols x0 and p0. Thus, the two averaging restoring force conditions
331
332
and
333
b2 b3 E z h b2 b3 E z h K E sign ( p )
334
must be satisfied. Using again eq. (33) the terms of the equivalent-linear model are found to be
335
2 1 1 Cin Ez z z x0 p0 0 , g 2g g
ML
E
E
(46)
p 0 ,
x0 (1 Cin ) 0 , g 0 0
(47)
(48)
x 2 Cdg 1 0 z 0 g 2 Rh,2 , b3b3 x0 b3b3p0
336
CL
337
and
338
0 0 E KL 0 K sign(p0 h)
339
As previously mentioned, an iterative procedure must be implemented to compute the parameters of
340
matrices ME, CE, and KE, and to derive the stationary solution of the response. For this purpose, the
341
input-output relationships given in eq. (40)-(42) are employed. In this task, the term
342
E sign(p) p of eq. (50) is computed numerically.
p0 h .
(49)
(50)
18
343
6. Reliability of the statistical linearization solution: approximate solution vis-à-vis Monte
344
Carlo data
345
The efficiency and reliability of the proposed approximate solutions is next assessed. System
346
responses computed by the equivalent linear model are compared vis-à-vis the solution obtained via
347
the numerical integration of the original non-linear model (Monte Carlo study).
348
6.1 Solution for the U-OWC equipped with a linear turbine
349
The dynamic response of the U-OWC plant described in Tab. 1 equipped with a linear turbine is
350
analyzed. Specifically, the stationary response computed in the frequency-domain by the
351
equivalent-linear system is compared to relevant Monte Carlo data. Typical waves representative of
352
realistic conditions occurring in the Civitavecchia site are employed [13]. Specifically, sea states
353
compatible with mean JONSWAP spectra [30] and having significant wave heights Hs between 2 m
354
and 3.5 m are assumed.
355
Time histories of the excitation are synthesized by the spectral method in conjunction with the Fast
356
Fourier Transform algorithm [31]. The resulting time history involves about 106 samples. Next, the
357
system response is computed in the time domain using the constant acceleration method. The
358
response statistics are determined by considering the response computed after 10Tp (Tp being the
359
peak spectral period) to exclude the transient part of the response. Quiescent initial conditions for
360
the water column are considered in conjunction with atmospheric pressure into the air chamber.
361
The response statistics of the water-column displacement and of the air pressure fluctuations
362
compared with the Monte Carlo data are shown in Fig. 5. It is seen that the response statistical
363
moments are captured with reasonable reliability by the approximate linear solution derived from
364
the equivalent-linear model. Nevertheless, a discrepancy in the computed means of the response is
365
observed. However, such a result does not affect the overall reliability of the adopted approach,
366
since the magnitude of the differences is small compared to the absolute values of the response, and
367
to the size of the full-scale prototype plant. 19
368 369
Figure 5. Staatistics of the response for a U-OWC equuipped with a linear turbin ne. Left panelss illustrate thee mean valuess
370
(x0) and stanndard deviatioon (σz) of the water colum n displacements. Right pan nels illustratee the mean va alues (p0) andd
371
standard devviation (σp) off the air-pressu ure fluctuationns
372
Next, the eenergy convversion perfformances dderived from m the solutio on of the eqquivalent lin near system m
373
are comparred versus the t solution ns obtained from the original o non-linear systtem. In this regard, thee
374
power outpput estimated from thee non-lineaar system reelies on thee numericall integration n of the U--
375
OWC govverning equations. On the other hhand, the identificatio on of the appproximate equivalentt
376
linear soluution allows the direct computation c n of the aveeraged turbine power ooutput as a function off
377
the root-m mean-squaree value off the air ppressure osscillations inside the pneumaticc chamber..
378
Specificallly, under thhe approxim mation of a Gaussian distribution d p thee for the airr pressure process,
379
averaged cconverted poower is estim mated as
380
a N 3 D5 p2 p Pt exp 2 f p 2 2 dp, 2 p 2 p a N D
381
where ρa iss the air dennsity of the atmospherre, and fp(·) is a functio on of the airr pressure p and of thee
382
instantaneoous power output o of th he turbine P t depending g on the turrbine characcteristics. For F the casee
(51)
20 0
383
of the Wellls turbine, the description of the function fp can be found in Falcãão and Rodrrigues [21]..
384
Fig. 6 show ws that thee power outtput is reasoonably well estimated in all casee studies, allbeit with a
385
systematic under estim mations of th he values.
386 387
Figure 6. Averaged turbbine power ou utput computted from the equivalent-linear system vis-à-vis resu ults from thee
388
numerical integration of thhe U-OWC go overning equaations.
389
Next, the spectral diistributions of the ressponse deriived from the equivallent-linear system aree
390
c with w the speectra compu uted by the time-domain t n y computed. These are compared n solution numerically
391
computed for the origginal non-lin near system m (Fig. 7). In I this conteext, spectraal characteriistics of thee
392
design seaa state are assumed a (H Hs = 2.5 m aand Tp = 6.74 s). For the t case off the equivaalent linearr
393
model, thee spectral distribution d of the wateer column displacemen d nt and of aair pressure fluctuationn
394
inside the ppneumatic chamber c aree computedd by the inpu ut-output reelationshipss (eq. 37). On O the otherr
395
hand, specctral distribuutions for th he original non-linear system are computed from the time-domainn
396
numerical determinatiion of the U-OWC U dyynamics. Fo or this, the Welch W Meth thod [32] baased on thee
397
time averaaging of a number off periodogrrams in Fig g. 7 compu uted over sshort segmeents of thee
398
processes iis employedd.
399
The resultts show thaat the specctrum derivved from th he equivaleent linear ssystem yiellds a goodd
400
estimate, oover the entiire frequenccy spectrum m, of the num merically co omputed speectrum.
21 1
401 402
Figure 7. P Power spectraal density fun nction of thee water colum mn displacem ments (left paanel) and of air pressuree
403
fluctuations ((right panel) inside i the pneumatic chambber, for a sea state with Hs equal to 2.5 m and Tp equal to 6 s.
404
6.2 Solutioon for a U-O OWC with a non-linearr orifice (only)
405
The reliabiility of the linearization l n technique for a U-OW WC plant with a non-linnear orifice is assessedd
406
next. Thuss, the dynam mic responsse computeed from thee equivalentt-linear systtem is com mpared withh
407
relevant nuumerical daata. For this, the signaal captured d from the pressure p traansducer (1) of Fig. 4
408
provides thhe time histtories of thee excitation to the systeem ( pm1 ).. For the puurpose of the presentedd
409
analyses, sspectral charracteristics of record li sted in Tab. 3 are considered.
410
Starting frrom measurred time hiistories of tthe excitatiion of the system, s thee dynamic response r iss
411
computed numericallyy in the tim me domain. In this con ntext, a con nstant time step of 0.1 1 s is used..
412
Quiescent initial condditions are consideredd in conjuncction with atmospheric a c pressure into i the airr
413
chamber. A Also, ergodiic features are a assumedd for the U-OWC respo onse.
414
Fig. 8 show ws the respoonse statistiics obtainedd from the equivalent e liinear system m compared d versus thee
415
results of tthe numericcal solutionss of the origginal non-lin near system m. Specificallly, the statiistics of thee
416
water-coluumn displaccement and of the preessure drop fluctuation ns inside thhe U-OWC pneumaticc
417
chamber arre shown. With W referen nce to the zzero-mean processes p z (water coluumn oscillattions) and h
418
(pressure ddrop inside the chamb ber), the re sults pertaiin to the mean m values (x0 and p 0), and thee
419
standard deeviations (σσz and σh).
420 22 2
421
Recorrd
Hs [m]
Tp [s]
1
0.5012
3.33032
2
0.5737
3.44133
3
0.5382
3.33032
4
0.5448
3.55310
5
0.5237
3.44133
6
0.5017
3.33332
422
Table 3. Speectral characteristics of th he recorded w wave data ela aborated for the purpose oof the presen nted analyses..
423
Symbols Hs aand Tp indicatee the significa ant wave heighht and the pea ak period of the sea state, reespectively.
424
It is furthher seen thaat the response statisttical momeents are relliably captuured by the proposedd
425
approach. In this regaard, note thaat the best aagreement is i observed d for the seccond order statistics off
426
the responsse. This is a desirable feature f of thhe approach h, because itt yields a relliable prediction of thee
427
power production in case the U-OWC U is iimplementeed in conjun nction withh a Power – Take Offf
428
system.
429 430
Figure 8. Staatistical charaacterization of o the dynamiic response off the system for f records lis isted in Tab. 3. Due to thee
431
closer valuess of peak periiods of record ds listed in Taab. 3, the resu ults are shown with respecct to the numb ber of record.
432
Top panels rrepresent the mean m for the processes p of tthe water colu umn oscillatio on (x0) and off the pressure drop (Δp0)
23 3
433
inside the pneumatic chamber. Bottom panels represent the standard deviation for the zero-mean processes the water
434
column oscillation (z) and of the pressure drop (h) inside the pneumatic chamber.
435
7. Concluding remarks
436
In the paper an approach for an approximate linear solution of the equation of motion of a non-
437
linear U-Oscillating Water Column wave energy converter has been developed. The dynamic
438
behavior of such a system is captured by a set of non-linear integro-differential equations
439
characterized by non-symmetric nonlinearities. The approximate solution has been sought using the
440
technique of statistical linearization. That is, a surrogate linear system approximating the original
441
one in a specified sense has been determined via an iterative procedure involving the computation
442
of the U-OWC response statistics.
443
The reliability of the technique has been demonstrated for the case of two specific U-OWC plant.
444
First, the configuration of a U-OWC equipped with a linear Wells turbine has been investigated.
445
This application has relied on the full-scale U-OWC prototype in the port of Civitavecchia (Rome,
446
Italy). Next, the dynamics of a U-OWC comprising a non-linear orifice without the inclusion of a
447
turbine has been examined. A practical case study has been discussed, that is the small-scale U-
448
OWC model tested in the benign basin of the NOEL laboratory (Reggio Calabria, Italy).
449
Comparisons between the results obtained by the statistical linearization versus Monte Carlo data
450
have been used to validate the proposed approach. In this regard it is notes that the results have
451
shown that the approximate solution can be used for estimating reliably the response statistics, for
452
both of the investigated U-OWC configurations. It is further noted that the proposed approach
453
compared to Monte Carlo simulations has been found to be of the order 10-2 more computationally
454
efficient in predicting the statistics of the U-shaped OWC response.
455
Future developments of this technique may be envisaged in the context of control theory, where it
456
could be used for maximizing the performance of the PTO, or in the context of optimal U-OWC
457
design where it can be adopted for determining efficiently the optimal value of the frequency-
24
458
dependent coefficients of the system.
459 460
Acknowledgements
461
This paper has been developed during the Marie Curie IRSES project “Large Multi- Purpose
462
Platforms for Exploiting Renewable Energy in Open Seas (PLENOSE)” funded by the European
463
Union (Grant Agreement Number: PIRSES-GA-2013-612581).
464
G. Malara is grateful to ENEA (Agenzia nazionale per le nuove tecnologie, l'energia e lo sviluppo
465
economico sostenibile) for supporting his postdoctoral fellowship “Experimental and full scale
466
analysis of wave energy devices” funded by the Italian Ministry of Economic Development for
467
‘Ricerca di Sistema Elettrico’.
468
Both G. Malara and F.M. Strati acknowledge with pleasure their research sojourns at Rice
469
University, USA.
470 471
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Ocean, Offshore and Arctic Engineering, OMAE2013, Nantes, France, 2013.
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Ocean Engineering, 34 (2007) 806-819.
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[15] G. Malara, F. Arena, Analytical modelling of an U-Oscillating Water Column and performance
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U-OWC energy harvesters, Renewable Energy, 101 (2017) 964-972.
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Applied Ocean Research, 17 (1995) 155-164.
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[31] M. Shinozuka, G. Deodatis, Simulation of Stochastic Processes by Spectral Representation,
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[32] P.D. Welch, The use of fast Fourier transform for the estimation of power spectra: A method
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Transactions on, 15 (1967) 70-73.
28
Highlights •
The random vibration analysis of a non-linear mechanical oscillator is presented.
•
Case study are a small-scale and a full-scale U-OWC wave energy converter.
•
An equivalent linear model is derived by the technique of statistical linearization.
•
An exact solution can be determined by the classical input-output relationship.
•
Solutions derived from the novel approach are validated against Monte Carlo data.
PII: DOI: Reference:
S0266-8920(17)30113-3 http://dx.doi.org/10.1016/j.probengmech.2017.07.001 PREM 2931
To appear in:
Probabilistic Engineering Mechanics
Received date : 4 June 2017 Accepted date : 12 July 2017 Please cite this article as: P.D. Spanos, F.M. Strati, G. Malara, F. Arena, An approach for non-linear stochastic analysis of U-shaped OWC wave energy converters, Probabilistic Engineering Mechanics (2017), http://dx.doi.org/10.1016/j.probengmech.2017.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
AN APPROACH FOR NON-LINEAR STOCHASTIC ANALYSIS OF U-SHAPED OWC WAVE
2
ENERGY CONVERTERS
3 4
Pol D. Spanos(1), Federica Maria Strati(2), Giovanni Malara(3), Felice Arena(4)*
5 6
(1) L. B. Ryon Chair in Engrg., G. R. Brown School of Engrg., Rice University, Houston, TX
7
77005. Email: [email protected].
8
(2) Ph. D. student, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea” University of
9
Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email: [email protected]
10
(3) Post-Doctoral research fellow, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea”
11
University of Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email:
12
[email protected]
13
(4) Professor, Natural Ocean Engineering Laboratory (NOEL), “Mediterranea” University of
14
Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy. Email: [email protected]
15
* Corresponding author: [email protected]
16
1
17
Abstract
18
This paper analyzes the dynamics of a U-Oscillating Water Column (U-OWC) wave energy
19
harvester. The geometrical configuration of this device comprises: an air mass enclosed into a
20
pneumatic chamber and connected to the atmosphere through a duct containing the Power-Take-Off
21
device; and a U-shaped water column connecting the air mass to the open wave field. The dynamics
22
of the U-OWC is described by a set of two coupled nonlinear integro-differential equations which
23
have no obvious exact analytical solution. In this context, the paper addresses the problem of
24
estimating efficiently and reliably, albeit approximately, the dynamic response of the U-OWC
25
system given a certain power spectrum-compatible wave excitation. For this purpose, an equivalent
26
linear model describing the dynamic response of the system is derived by the technique of statistical
27
linearization, and used to determine iteratively the statistics of the harvester response. Further, two
28
specific U-OWC configurations are considered: the full-scale U-OWC prototype equipped with a
29
linear Wells turbine in the port of Civitavecchia (Rome, Italy); and a U-OWC small scale model
30
which comprises a non-linear orifice, only, and is installed in the benign basin of the NOEL
31
laboratory (Reggio Calabria, Italy). Relevant Monte Carlo data are used to demonstrate the
32
efficiency and reliability of the proposed approach.
33 34
Keywords: Wave energy converter; U-OWC; statistical linearization; Monte Carlo data.
2
35
1. Introduction
36
In the recent decades the increasing demand of energy and the progressive depletion of traditional
37
sources has generated a significant challenge to the scientific community. That is, to develop novel
38
energy harvesting technologies exploiting renewable resources. In this context, the energy
39
associated with ocean waves is a promising source whose potential global power is estimated to
40
have a magnitude of the order of 1013 W [1]. Within this field, a large number of wave energy
41
converters (WECs) have been investigated, and some of them have reached the stage of a full-scale
42
prototype [2]. Their structures can be either floating or fixed, with their configurations properly
43
designed to harvest the kinetic or potential energy of ocean waves. Typically, nonlinear differential
44
or integral equations in conjunction with a statistically described wave excitation are required to
45
capture the dynamic behavior of WECs. Thus, the application of computationally costly numerical
46
algorithms is often necessary. Alternatively, the possibility to generate fast and reliable approximate
47
solutions of the dynamic response of a non-linear system can be desirable. In this regard, various
48
approaches have been developed in the open literature; for nonlinear stochastic dynamics problems
49
an extensive description can be found in references such as Roberts and Spanos [3].
50
One of the most versatile tools is the statistical linearization technique which is applied herein for
51
the determination of the dynamic response of a specific wave-energy harvester, named U-shaped
52
Oscillating Water Column (U-OWC).
53
Succinctly, the implementation of the statistical linearization technique can generate an approximate
54
solution for the response of a multi-degree of freedom dynamic system subjected to a random
55
stationary excitation. The technique involves the replacement of the nonlinear terms in the
56
equations governing the system dynamics by equivalent (in a specified sense) linear terms, so that
57
the exact solution for the equivalent linear model can be determined by the classical input-output
58
relationship. Applications to marine and ocean engineering can be found in references such as
59
Spanos and Agarwal [4], Low [5], and Spanos, et al. [6].
3
60
Within the field of wave energy converters, oscillating water columns (OWC) devices have reached
61
the most advanced development stage. Their dynamics and energy conversion mechanism have
62
been extensively investigated [7-10]. The basic geometrical configuration of an OWC comprises a
63
pneumatic chamber which contains an air mass in its upper part and water in the lower part
64
connected to the open sea. The fluctuations of the water alternately compress and expand the air,
65
which is forced to flow through an air duct connected to the atmosphere. Thus, the rotor of a self-
66
rectifying turbine located in the air duct is activated, and the mechanical torque inducing the
67
production of electrical energy is generated. Starting from the 80s, a number of OWCs were build
68
and tested. Example of operative OWCs are [11]: the five chambered plant in the port of Sakata
69
(1989), Japan, with an original rating of 60kW provided by a single Wells turbine; the PICO plant
70
in the Azores Island (1999) with an installed capacity of 400kW; the LIMPET plant (2000) in Islay
71
island, Portugal, originally rated at 500kW, presently used as test site for various turbine designs;
72
and the Mutriku plant (2011), Spain, representing the first multiple chamber OWC plant with a total
73
power of 296 kW.
74
In this framework, the U-OWC represents a modification of the basic OWC geometrical
75
configuration [12]. Its basic structure is quite similar to the one characterizing the conventional
76
OWC. However, it includes an additional vertical element on the wave-beaten side of the plant
77
forming a typical U-shaped duct. This small element allows introducing an additional mass that can
78
be used for tuning the water column natural period to a desired value. This device has recently
79
reached the stage of full-scale prototype; such a configuration is presently under construction in the
80
port of Civitavecchia (Rome) [13] and has a (potential) estimated installed power of 2.7 MW.
81
The first theoretical model for the U-OWC dynamics was proposed by Boccotti [14]. Next, Malara
82
and Arena [15] developed a consistent representation of the diffracted/radiated wave field in front
83
of the plant. Small-scale field experiments validated the theoretical model and the absorption
84
capacity of such device [12, 16-18]. The mathematical description of the U-OWC dynamics
4
85
involves two coupled non-linear integro-differential equations. This feature points out a significant
86
difference with respect to conventional OWCs, for which a linear analysis is reliable as long as the
87
linear water wave theory, and the assumption of an isentropic thermodynamic process for the air
88
pocket hold [1, 8, 19-21]. Further, the complexity of the U-OWC theoretical model introduces
89
significant difficulties when dealing with the optimal-design of the system and with control related
90
aspects.
91
In this framework, an optimal approximate linear model of the U-OWC system is pursued by
92
implementing the technique of statistical linearization. Analyses are conducted by considering two
93
U-OWC configurations. The first comprises a linear Wells turbine located in the air duct. In this
94
context, a case study pertains to the full-scale prototype located in the port of Civitavecchia (Rome,
95
Italy) [22]. The second configuration is a U-OWC comprising an orifice without a turbine. For such
96
a case, an additional non-linear term in the dynamic modelling of the pressure fluctuations inside
97
the pneumatic chamber must be included. In view of a full-size realization, the modelling of a U-
98
OWC equipped with an orifice is of particular interest. Indeed, such a configuration either can
99
simulate the presence of valves in the chamber, or can be interpreted as the approximation of a
100
nonlinear PTO mechanism, such as the case of an Impulse turbine [11]. A small-scale U-OWC
101
model with such configuration is currently tested in the benign basin of the NOEL laboratory
102
(http://noel.unirc.it/) (Reggio Calabria, Italy).
103
Solutions for realistic sea states compatible with typical power spectral density functions of sea
104
waves are derived. To assess the efficiency and reliability of the proposed approach, responses
105
computed from the equivalent linear system are compared with relevant Monte Carlo data derived
106
by numerical integration of the original nonlinear system dynamics equations.
107
108
2. Physical U-OWC system description
109
Next the two different U-OWC plants considered in this study are discussed. The first configuration 5
110
refers to a U-OWC equipped with w a lineaar Wells tu urbine locaated in the air duct. The T secondd
111
configuratiion is a U-O OWC which h comprise a non-linearr orifice, without the prresence of th he turbine.
112
2.1 U-OWC C equippedd with a linea ar turbine
113
The schem matic view of o a U-OWC C is shownn in Fig. 1. Here, dimensions of thhe pneumattic chamberr
114
are definedd by the heeight hc, and d the width b2 and b3 (not ( shown in the figuure) in the longitudinal l l
115
and transveerse directioon, respectiively. The ssymbols li and a b1 defin ne the length th and the width w of thee
116
external vertical ductt with the opening hhaving subm mergence dh. The tim me-dependen nt variabless
117
describing the U-OW WC dynamics are x annd pc, deno oting the water w level inside the pneumaticc
118
chamber m measured froom the meaan water levvel (positivee downward ds), and the air pressuree inside thee
119
pneumatic chamber, respectively.
120 121
Fig gure 1. Schem matic view of th he U-OWC plant.
122
In this worrk, the plannt configuraation of the full-scale U-OWC U pro ototype undder construcction in thee
123
port of Civvitavecchiaa (Rome, Italy) [22] iss considereed (Fig. 2). Tab. 1 sum mmarizes the relevantt
124
geometricaal parameterrs of the deevice. Furtheer, a Wells turbine witth an outer ddiameter D = 0.738 m,,
125
a hub diam meter Dhub = 0.5 m, an nd a dampinng ratio K = 2.9 is considered. FFor the purp pose of thee 6
126
presented aanalysis, a constant c rotational speeed equal to 4000 rpm iss assumed.
127 128
Figure 2. Civitavecchia U-OWC U caissoons. Photo takeen on May 2015 (Source: w wavenergy.it).
129
d [m]
hc [m]
b2 [m]
b3 [m]
dh [m]
b1 [m]
D [m]
14.20
9.40
3.20
3.87
2.00
1.60
0.738
130 131
Table 1. Geoometrical charracteristics off the Civitaveccchia U-OWC C plant. The syymbols refer too the scheme shown in Fig..
132
1, while b3 iss the width of the t pneumaticc chamber in tthe transversee direction; an nd D is the diaameter of the air a duct.
133
2.2 U-OWC C with a noon-linear oriifice (only)
134
A U-OWC C small scalle model is presently ttested in thee natural beenign basinn of the Nattural Oceann
135
Engineerinng Laborattory (NOEL, www.nnoel.unirc.it,, Reggio Calabria, IItaly), wheere a new w
136
experimenttal activityy started in 2014 [18]] (Fig. 3). The peculiarity of thhis site relates to thee
137
possibility to conduct field experriments direectly in a beenign sea by y the implem mentations of commonn
138
methodoloogies used in i large arttificial wavve tanks [23 3]. The mo odel compriises three in ndependentt
139
chambers w whose totall width is 3..79 m in thee transversee direction. The width oof a single cell is 1 m..
140
The cross-sectional viiew of the small-scale s U-OWC model m is shown in Fig. 44. The three chamberss
141
t orifice, which is eq qual to 0.32 m for the ccentral cell, and 0.15 m differ withh regards to the size of the
142
for the lateeral cells. Further, F pro oper instrum ments for th he measurem ment of botth pressuress and waterr
143
levels are located insside the pn neumatic chhamber and d in the U-shaped ducct. Their po ositions aree
144
shown in F Fig. 4. In thee present ap pplication, tthe geometrry of the lateeral cell is uused. For th his, relevantt
145
geometricaal parameterrs are shown in Tab. 2..
7
146 Figure 3. Exp perimental sett-up at NOEL laboratory (R Reggio Calabrria, Italy).
147
148
149
Figure 4. Schhematic view of the centrall (left) and latteral (right) chamber of thee NOEL U-OW WC small-scale model. Thee
150
numbers in bbrackets indicaate the positio on of pressuree transducers.
d [m]
hc [m]
b 2 [m]
b3 [m]
dh [m m]
b1 [m m]
1.67
1.90
1.00
1.20
0.50
0.555
151
Table 2. Geoometrical paraameters descriibing the U-O OWC configura ation. The sym mbols refer to Fig. 4. Further, b3 is
152
the width of the chamber in the longitu udinal directioon. Furthermo ore, the diameeter of the circcular orifice is i equal
153
to 0.15 m forr the central cell, and 0.33 m for the laterral cells.
154
155
3. Equatioons of motioon of the U-OWC U
156
The set of two coupleed integro-d differential eequations required to capture c the hydrodynaamics of thee 8
157
water column oscillations jointly with the aerodynamics of the pneumatic chamber are next
158
presented. The theoretical model of a U-OWC equipped with a linear turbine is firstly presented.
159
Next, the model is revised for the case of the U-OWC comprising a non-linear orifice (only).
160
3.1 Equations of motion of the U-OWC equipped with a linear turbine
161
The equation of motion of the water displacement along the U-duct is derived [24] from the
162
unsteady Bernoulli equation by balancing the total heads at the U-OWC opening (point (1) of Fig.
163
1), and at the inner free surface (point (2) of Fig. 1). In this context, the wave pressure applied at the
164
external opening is determined according to the linear water wave theory [25]. Hence, the equation
165
governing the dynamics of the water column oscillations is
166
M ( x ) x C ( x ) x
167
where dot denotes differentiation with respect to time; ρ is the water density; g is the acceleration
168
due to gravity; patm is the atmospheric pressure; K0 is the retardation function accounting for the
169
memory effects; and p ( s ) is the wave pressure excitation calculated in a scattered wave field at the
170
center of the outer opening of the external vertical duct (point (1) of Fig. 1). Such a process is
171
mathematically described as a zero-mean ergodic stochastic process, characterized by a Gaussian
172
probability density function, consistently with common sea state representation approaches [26].
173
Further, the non-linear mass and damping terms are defined as
174
M ( x)
175
and
176
C ( x )
177
where H∞ denotes a length representing the infinite frequency added mass, and Kw is the head loss
178
coefficient.
179
The equation governing the time variations of the pressure into the air mass is derived by the
1 2g
1 b2 g b1
t
0
x ( t )K 0 (0, d h , ) d x
p c p atm p (s) , g g
(1)
b2 1 b2 li li d h H x , g b1 b1
(2)
2 x b2 K w x , 2 b1
(3)
9
180
approach of Falcão and Rodrigues [21] and Falcão [27]. In this context, the influence of the Wells
181
turbine is accounted for by a linear relationship between the air flow rate across the turbine and the
182
pressure drop (pc-patm) between the chamber and the atmospheric conditions. Thus, the equation
183
governing the aerodynamic process is
184
p c
185
where c is the speed of sound in air; γ is the ratio of specific heat at constant pressure over specific
186
heat at constant volume; A is the net flow area of the air duct accounting for the diameter of the duct
187
and of the rotor; R is the external radius of the air duct; V is the volume of the air mass inside the
188
chamber, and K is the turbine damping coefficient.
189
In a matrix notation, eq. (1) and (4) are concisely written as
190
CL q K L q Τ Ψ (q, q , q ) Q , ML q
191
where ML, CL, and KL denote the linear contribution of mass, damping, and stiffness matrices,
192
respectively. Further, q is the generalized coordinate vector; Q is the time-dependent vector
193
representing the wave excitation; Ψ (q, q , q) is a non-linear function of q and its derivatives; and the
194
constant vector T is introduced to take into account terms which are linear, but independent from
195
the generalized coordinate q. Specifically,
196
1 b2 b2 li li dh H x 0 ΜL g b1 b1 , 0 0
(6)
197
0 0 CL , 0 hc
(7)
198
1 1 g , KL 2 0 A c 1 KNR b2b3
(8)
A c2 1 V ( pc patm ) pc , KNR V V
(4)
(5)
10
199
p ( s ) Q g , 0
200
x q , pc
201
1 b2 g b 1 T
202
and
(9)
(10)
t
0
x (t )K 0 (0, d h , )d A c2 1 patm KNR b2b3
xx 1 b2 x 2 K w x x Ψ (q, q, q) 2g g 2 g b1 xp c xp c
203
patm g ,
.
(11)
(12)
204
3.2 Equations of motion of the U-OWC with a non-linear orifice (only)
205
The theoretical model described in the previous section is revised herein for the case of the small-
206
scale U-OWC model tested in the natural basin of NOEL laboratory. For such a system, the
207
equation of motion is
208
M ( x ) x C ( x , x ) x x
209
where p m 1 is the time-dependent wave pressure excitation at the outer opening of the vertical
210
duct,
211
M ( x)
212
C( x, x)
p c p atm p m 1 , g g
1 b2 li li dh x Cin x Cin 2 , g b1
1 x C x x dg 1 , 2 g Rh,2
(13)
(14)
(15)
11
213
with Rh,1 and Rh,2 being the hydraulic radii of the small vertical duct and of the inner water column,
214
respectively; and
215
l b l dh , 1 i 2 i Rh ,1 b1 Rh ,2
216
and
217
2
218
In this case it is noted that that this model assumes that the process p m 1 is directly derived from
219
data measured from the pressure transducer (1) of Fig. 4. Hence, eq. (13) implicitly includes the
220
convolution integral and the infinite frequency added mass H appearing in eq. (1), that are
221
commonly derived by the decomposition of the diffracted and radiated wave field in front of the
222
plant (see Malara, et al. [24]). Further, note that the mass and damping terms are represented
223
according to the results of Malara, et al. [18] concerning the small scale model investigated. The
224
mass flow rate across the orifice depends on the square root of the air pressure drop p between the
225
chamber and atmosphere [28]. Under this assumption the resulting governing equation is
2
li b2 li d h , g b1 g
p c
226
c2Cd A0 pc x sign(p) 2air p , b2b3 hc x hc x
(16)
(17)
(18)
227
where A0 is the orifice area; and Cd is the coefficient of discharge, that is experimentally estimated
228
to be equal to 0.652.
229
The mechanical analogy for a non-linear oscillator of eq. (5) is employed to recast eq. (13) and (18).
230
In this sense, the vector of the generalized coordinates
12
231
x q p
232
is introduced. Note that the utilization of the pressure drop in place of the dynamic air pressure to
233
define vector
234
constant vector
235
the wave excitation is
236
pm1 Q g , 0
237
Finally, the terms of eq. (5) are defined as
238
1 b2 li li d h Cin2 0 ΜL g b1 , 0 0
(21)
239
0 0 CL , patm hc
(22)
240
1 KL 0
(23)
241
and
242
Cdg x x Cdg 1 x x x x x x Cin x 2g Rh,2 , q , q g Ψq , hc p x p K 2 air p x p
243
where the constant K is defined as
(19)
q , as well as the absence of the convolution integral, leads to a null value of the Τ of eq. (5). Further, the time-dependent vector representing the random process of
1 g , 0
(20)
(24)
13
c 2 C d A0 . R 2
244
K
245
4. Equivalent linear system of U-OWC devices
246
The mathematical background underlying the application of the statistical linearization technique is
247
outlined herein. Next, the technique is employed to derive approximate linear models for the two U-
248
OWC configurations considered in the study.
249
4.1 The statistical linearization formalism
250
The exact analytical solution of the non-linear eq. (5) is currently unavailable. Thus, commonly
251
computationally demanding numerical algorithms are employed for determining the system
252
response in the time domain. In this context, an approximate solution is sought by the technique of
253
statistical linearization [3]. Such technique relies on the replacement of the original non-linear
254
system by an equivalent (in a specified sense) linear system for which the exact solution is known.
255
Thus, the generic model given by the eq. (5) is replaced by the linear system
256
M
257
E E ) . Further, where M , CE , and K denote the linear substitutes of the non-linear vector Ψ (q, q , q
258
) to be symmetric is defined as [29] a condition for the function Ψ (q, q , q
259
) Ψ (q, q , q ) . Ψ (q, q , q
260
The decomposition
261
q q0 w ,
262
of the general solution of eq. (5) is implemented since the condition given by eq. (27) is not
263
satisfied [29]. In eq. (28) q 0 is a constant real vector, and w a time-dependent vector. With the
264
employment of eq. (28), the original system must still be satisfied in a specified sense. In this
265
context, eq. (5) can be written in terms of the vector w as
L
(25)
C L C E q K L K E q Q ME q
,
(26)
(27)
(28)
14
C L w K w
) Q , w β ( w q 0 , w , w
(29)
266
M
267
where
268
) K L q 0 Ψ ( w q 0 , w , w ) . β ( w q 0 , w , w
269
E E The coefficients M , CE , and K of eq. (26) are determined by a minimization procedure on the
270
vector difference ε between the original (eq. 5) and the equivalent-linear system (eq. 26). The
271
minimization procedure is performed according to the criterion
272
g ( ε T ε ) min ,
273
for every w solution of eq. (26). In eq. (31), the function g () is an averaging operator which
274
ensures defined characteristics of the equivalent linear system. Specifically, the criterion
275
E ε T ε min ,
276
is adopted, where E[·] denotes the operator of mathematical expectation. From eq. (32), the
277
approximation of a Gaussian distribution for the response and for its derivative, leads to the
278
equations
279
i m e i, j E q j
280
where k e i , j , c e i , j , and m e i , j are the coefficients of the matrices KE, CE and ME; see [3, 29].
281
4.2 Statistical linearization solution: U-OWC equipped with a linear turbine
282
In this section the statistical linearization method is applied to derive an approximate linear model
283
for eq. (5), with matrices defined by eq. (6) – (12). Further, the excitation is described by a
284
Gaussian random process in agreement with the common sea states approximation for ocean waves
285
[26].
286
Due to the non-symmetric form of the non-linear terms of eq. (5) the decomposition given by eq.
287
(28) must be employed. In this context, the components of the vector w are the zero-mean
L
L
(30)
(31)
(32)
i c ei, j E q j
i k e i, j E q j
i , j 1, 2
,
(33)
15
288
stationary stochastic processes z(t) and p(t). Further, the components of the vector q 0 are the means
289
of the processes z and p, named x0 and p0 respectively. Under this assumption, the two average
290
restoring force conditions
291
p p0 z2 1 E z z x 0 atm 0, g g 2g
292
and
293
E z p E z p
294
are derived, where z is the standard deviation of the derivative of the water column displacement.
295
Finally, the equivalent matrices
E
x0 g 0
(34)
Ac 2 1 ( p0 p atm ) 0 , KNR b2 b3
0 ; 0
CL
E
1 b2 3 K g 2 b 3 w z 1 p0
(35)
0 ; x0
E
. 1 z 2 0
(36)
296
ML
297
are derived as the solution of eq. (33), where z , z , p are the standard deviation of the responses
298
z and p, and of their derivatives, respectively.
299
Clearly, the computations involved in eq. (36) depend on the unknown statistics of the response.
300
Thus, an iterative procedure based on the stationary solution of the U-OWC governing equations is
301
adopted. Such a step is pursued consistently with the common practice used in the frequency-
302
domain analysis, based on input-output relationships. Specifically, the spectral density matrix of the
303
response S q is computed as
304
Sq () α() SQ () αT* () ,
305
where
306
S zz S q ( ) S pz
307
z and p indicate the hydrodynamic and aerodynamic response, respectively; and the frequency
308
response function α(ω) is expressed as
S zp ; S pp
KL
1 g 2 z p 2
(37)
(38)
16
1
,
309
α(ω) 2 M L M L E i C L C L E K L K L E
310
with T * denoting conjugate transposition of a matrix.
311
Then, the variances of the response are computed as
312
(39)
2 z
S d ,
(40)
z
313
2 z
314
and
315
2 z
Sz d
S () d , 2
z
S d S () d 4
z
z
(41)
(42)
316
Clearly, equations similar to eq. (37) - (39) can be derived for the process p. Further, the mean
317
values of the response are computed using eq. (34) and eq. (35), where
318
Ez z 2 Szzd ,
(43)
319
E z p i S pz d ,
(44)
320
and
321
E z p i S zp d ,
(45)
322
with i
1 . The iterative scheme is implemented by first neglecting the non-linearities and then
323
proceeding to calculate iteratively the U-OWC response statistics and account for the nonlinearities.
324
4.3 Statistical linearization solution: U-OWC with a non-linear orifice (only)
325
The statistical linearization technique is applied in this section to derive an approximate linear
326
model of eq. (5), with the matrices defined by eq. (19)-(24). For this, the decompositions of eq. (28)
327
is employed to account for the non-symmetric form of the non-linear terms of eq. (5). In this
328
context, the symbols z and h indicate the zero-mean processes for the water column oscillations, 17
329
and air pressure drop inside the chamber, respectively; the corresponding offsets are denoted by the
330
symbols x0 and p0. Thus, the two averaging restoring force conditions
331
332
and
333
b2 b3 E z h b2 b3 E z h K E sign ( p )
334
must be satisfied. Using again eq. (33) the terms of the equivalent-linear model are found to be
335
2 1 1 Cin Ez z z x0 p0 0 , g 2g g
ML
E
E
(46)
p 0 ,
x0 (1 Cin ) 0 , g 0 0
(47)
(48)
x 2 Cdg 1 0 z 0 g 2 Rh,2 , b3b3 x0 b3b3p0
336
CL
337
and
338
0 0 E KL 0 K sign(p0 h)
339
As previously mentioned, an iterative procedure must be implemented to compute the parameters of
340
matrices ME, CE, and KE, and to derive the stationary solution of the response. For this purpose, the
341
input-output relationships given in eq. (40)-(42) are employed. In this task, the term
342
E sign(p) p of eq. (50) is computed numerically.
p0 h .
(49)
(50)
18
343
6. Reliability of the statistical linearization solution: approximate solution vis-à-vis Monte
344
Carlo data
345
The efficiency and reliability of the proposed approximate solutions is next assessed. System
346
responses computed by the equivalent linear model are compared vis-à-vis the solution obtained via
347
the numerical integration of the original non-linear model (Monte Carlo study).
348
6.1 Solution for the U-OWC equipped with a linear turbine
349
The dynamic response of the U-OWC plant described in Tab. 1 equipped with a linear turbine is
350
analyzed. Specifically, the stationary response computed in the frequency-domain by the
351
equivalent-linear system is compared to relevant Monte Carlo data. Typical waves representative of
352
realistic conditions occurring in the Civitavecchia site are employed [13]. Specifically, sea states
353
compatible with mean JONSWAP spectra [30] and having significant wave heights Hs between 2 m
354
and 3.5 m are assumed.
355
Time histories of the excitation are synthesized by the spectral method in conjunction with the Fast
356
Fourier Transform algorithm [31]. The resulting time history involves about 106 samples. Next, the
357
system response is computed in the time domain using the constant acceleration method. The
358
response statistics are determined by considering the response computed after 10Tp (Tp being the
359
peak spectral period) to exclude the transient part of the response. Quiescent initial conditions for
360
the water column are considered in conjunction with atmospheric pressure into the air chamber.
361
The response statistics of the water-column displacement and of the air pressure fluctuations
362
compared with the Monte Carlo data are shown in Fig. 5. It is seen that the response statistical
363
moments are captured with reasonable reliability by the approximate linear solution derived from
364
the equivalent-linear model. Nevertheless, a discrepancy in the computed means of the response is
365
observed. However, such a result does not affect the overall reliability of the adopted approach,
366
since the magnitude of the differences is small compared to the absolute values of the response, and
367
to the size of the full-scale prototype plant. 19
368 369
Figure 5. Staatistics of the response for a U-OWC equuipped with a linear turbin ne. Left panelss illustrate thee mean valuess
370
(x0) and stanndard deviatioon (σz) of the water colum n displacements. Right pan nels illustratee the mean va alues (p0) andd
371
standard devviation (σp) off the air-pressu ure fluctuationns
372
Next, the eenergy convversion perfformances dderived from m the solutio on of the eqquivalent lin near system m
373
are comparred versus the t solution ns obtained from the original o non-linear systtem. In this regard, thee
374
power outpput estimated from thee non-lineaar system reelies on thee numericall integration n of the U--
375
OWC govverning equations. On the other hhand, the identificatio on of the appproximate equivalentt
376
linear soluution allows the direct computation c n of the aveeraged turbine power ooutput as a function off
377
the root-m mean-squaree value off the air ppressure osscillations inside the pneumaticc chamber..
378
Specificallly, under thhe approxim mation of a Gaussian distribution d p thee for the airr pressure process,
379
averaged cconverted poower is estim mated as
380
a N 3 D5 p2 p Pt exp 2 f p 2 2 dp, 2 p 2 p a N D
381
where ρa iss the air dennsity of the atmospherre, and fp(·) is a functio on of the airr pressure p and of thee
382
instantaneoous power output o of th he turbine P t depending g on the turrbine characcteristics. For F the casee
(51)
20 0
383
of the Wellls turbine, the description of the function fp can be found in Falcãão and Rodrrigues [21]..
384
Fig. 6 show ws that thee power outtput is reasoonably well estimated in all casee studies, allbeit with a
385
systematic under estim mations of th he values.
386 387
Figure 6. Averaged turbbine power ou utput computted from the equivalent-linear system vis-à-vis resu ults from thee
388
numerical integration of thhe U-OWC go overning equaations.
389
Next, the spectral diistributions of the ressponse deriived from the equivallent-linear system aree
390
c with w the speectra compu uted by the time-domain t n y computed. These are compared n solution numerically
391
computed for the origginal non-lin near system m (Fig. 7). In I this conteext, spectraal characteriistics of thee
392
design seaa state are assumed a (H Hs = 2.5 m aand Tp = 6.74 s). For the t case off the equivaalent linearr
393
model, thee spectral distribution d of the wateer column displacemen d nt and of aair pressure fluctuationn
394
inside the ppneumatic chamber c aree computedd by the inpu ut-output reelationshipss (eq. 37). On O the otherr
395
hand, specctral distribuutions for th he original non-linear system are computed from the time-domainn
396
numerical determinatiion of the U-OWC U dyynamics. Fo or this, the Welch W Meth thod [32] baased on thee
397
time averaaging of a number off periodogrrams in Fig g. 7 compu uted over sshort segmeents of thee
398
processes iis employedd.
399
The resultts show thaat the specctrum derivved from th he equivaleent linear ssystem yiellds a goodd
400
estimate, oover the entiire frequenccy spectrum m, of the num merically co omputed speectrum.
21 1
401 402
Figure 7. P Power spectraal density fun nction of thee water colum mn displacem ments (left paanel) and of air pressuree
403
fluctuations ((right panel) inside i the pneumatic chambber, for a sea state with Hs equal to 2.5 m and Tp equal to 6 s.
404
6.2 Solutioon for a U-O OWC with a non-linearr orifice (only)
405
The reliabiility of the linearization l n technique for a U-OW WC plant with a non-linnear orifice is assessedd
406
next. Thuss, the dynam mic responsse computeed from thee equivalentt-linear systtem is com mpared withh
407
relevant nuumerical daata. For this, the signaal captured d from the pressure p traansducer (1) of Fig. 4
408
provides thhe time histtories of thee excitation to the systeem ( pm1 ).. For the puurpose of the presentedd
409
analyses, sspectral charracteristics of record li sted in Tab. 3 are considered.
410
Starting frrom measurred time hiistories of tthe excitatiion of the system, s thee dynamic response r iss
411
computed numericallyy in the tim me domain. In this con ntext, a con nstant time step of 0.1 1 s is used..
412
Quiescent initial condditions are consideredd in conjuncction with atmospheric a c pressure into i the airr
413
chamber. A Also, ergodiic features are a assumedd for the U-OWC respo onse.
414
Fig. 8 show ws the respoonse statistiics obtainedd from the equivalent e liinear system m compared d versus thee
415
results of tthe numericcal solutionss of the origginal non-lin near system m. Specificallly, the statiistics of thee
416
water-coluumn displaccement and of the preessure drop fluctuation ns inside thhe U-OWC pneumaticc
417
chamber arre shown. With W referen nce to the zzero-mean processes p z (water coluumn oscillattions) and h
418
(pressure ddrop inside the chamb ber), the re sults pertaiin to the mean m values (x0 and p 0), and thee
419
standard deeviations (σσz and σh).
420 22 2
421
Recorrd
Hs [m]
Tp [s]
1
0.5012
3.33032
2
0.5737
3.44133
3
0.5382
3.33032
4
0.5448
3.55310
5
0.5237
3.44133
6
0.5017
3.33332
422
Table 3. Speectral characteristics of th he recorded w wave data ela aborated for the purpose oof the presen nted analyses..
423
Symbols Hs aand Tp indicatee the significa ant wave heighht and the pea ak period of the sea state, reespectively.
424
It is furthher seen thaat the response statisttical momeents are relliably captuured by the proposedd
425
approach. In this regaard, note thaat the best aagreement is i observed d for the seccond order statistics off
426
the responsse. This is a desirable feature f of thhe approach h, because itt yields a relliable prediction of thee
427
power production in case the U-OWC U is iimplementeed in conjun nction withh a Power – Take Offf
428
system.
429 430
Figure 8. Staatistical charaacterization of o the dynamiic response off the system for f records lis isted in Tab. 3. Due to thee
431
closer valuess of peak periiods of record ds listed in Taab. 3, the resu ults are shown with respecct to the numb ber of record.
432
Top panels rrepresent the mean m for the processes p of tthe water colu umn oscillatio on (x0) and off the pressure drop (Δp0)
23 3
433
inside the pneumatic chamber. Bottom panels represent the standard deviation for the zero-mean processes the water
434
column oscillation (z) and of the pressure drop (h) inside the pneumatic chamber.
435
7. Concluding remarks
436
In the paper an approach for an approximate linear solution of the equation of motion of a non-
437
linear U-Oscillating Water Column wave energy converter has been developed. The dynamic
438
behavior of such a system is captured by a set of non-linear integro-differential equations
439
characterized by non-symmetric nonlinearities. The approximate solution has been sought using the
440
technique of statistical linearization. That is, a surrogate linear system approximating the original
441
one in a specified sense has been determined via an iterative procedure involving the computation
442
of the U-OWC response statistics.
443
The reliability of the technique has been demonstrated for the case of two specific U-OWC plant.
444
First, the configuration of a U-OWC equipped with a linear Wells turbine has been investigated.
445
This application has relied on the full-scale U-OWC prototype in the port of Civitavecchia (Rome,
446
Italy). Next, the dynamics of a U-OWC comprising a non-linear orifice without the inclusion of a
447
turbine has been examined. A practical case study has been discussed, that is the small-scale U-
448
OWC model tested in the benign basin of the NOEL laboratory (Reggio Calabria, Italy).
449
Comparisons between the results obtained by the statistical linearization versus Monte Carlo data
450
have been used to validate the proposed approach. In this regard it is notes that the results have
451
shown that the approximate solution can be used for estimating reliably the response statistics, for
452
both of the investigated U-OWC configurations. It is further noted that the proposed approach
453
compared to Monte Carlo simulations has been found to be of the order 10-2 more computationally
454
efficient in predicting the statistics of the U-shaped OWC response.
455
Future developments of this technique may be envisaged in the context of control theory, where it
456
could be used for maximizing the performance of the PTO, or in the context of optimal U-OWC
457
design where it can be adopted for determining efficiently the optimal value of the frequency-
24
458
dependent coefficients of the system.
459 460
Acknowledgements
461
This paper has been developed during the Marie Curie IRSES project “Large Multi- Purpose
462
Platforms for Exploiting Renewable Energy in Open Seas (PLENOSE)” funded by the European
463
Union (Grant Agreement Number: PIRSES-GA-2013-612581).
464
G. Malara is grateful to ENEA (Agenzia nazionale per le nuove tecnologie, l'energia e lo sviluppo
465
economico sostenibile) for supporting his postdoctoral fellowship “Experimental and full scale
466
analysis of wave energy devices” funded by the Italian Ministry of Economic Development for
467
‘Ricerca di Sistema Elettrico’.
468
Both G. Malara and F.M. Strati acknowledge with pleasure their research sojourns at Rice
469
University, USA.
470 471
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472
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28
Highlights •
The random vibration analysis of a non-linear mechanical oscillator is presented.
•
Case study are a small-scale and a full-scale U-OWC wave energy converter.
•
An equivalent linear model is derived by the technique of statistical linearization.
•
An exact solution can be determined by the classical input-output relationship.
•
Solutions derived from the novel approach are validated against Monte Carlo data.