# The SSG Wave Energy Converter: Performance, Status and Recent Developments

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## Abstract

**:**

## 1. Introduction

- -
- Oscillating Water Columns (OWC; Figure 1a) can be described as a caisson breakwater with a gap on the seaside face, which encloses a mass of water. Waves cause the water to rise and fall and this alternately compresses and depressurizes an air column. The energy is extracted from the oscillating air flow by using a Wells turbine;
- -
- OverTopping Devices (OTD; Figure 1b) use a sloping plate that leads the waves to overtop into a reservoir located immediately behind it. The energy is extracted via low head turbines, using the difference in water levels between the reservoir and the average sea water level;
- -
- In the Wave Activated Bodies (WAB; Figure 1c), waves cause the body parts of a device to oscillate relative to each other; alternatively, the whole body may oscillate against a fixed reference. The oscillatory motion can be heave, pitch or roll. Hydraulic systems are generally employed to compress oil, air or water, which is then used to drive a generator.

- –
- Sharing structure costs;
- –
- Availability of grid connections and infrastructures;
- –
- Recirculation of water inside the harbors as the outlet of the turbines is on the rear part of the system;
- –
- Easy installation and maintenance;
- –
- No deep-water moorings or long lengths of underwater electrical cable.

## 2. The Hydraulic Response

#### 2.1. Overtopping Performance

_{ov}

_{,j}. The latter represents a leading variable in the functional design of SSG as well as of many other maritime structures.

_{ov}

_{,j}, extensive experimental work has been performed between 2004 and 2007, at the Hydraulic and Coastal Engineering Laboratory of the Department of Civil Engineering of Aalborg University.

- √
- Reservoir crest levels, R
_{cj} - √
- Ramp angle, α
_{r}; - √
- Ramp draught, d
_{r}; - √
- Front angles, θ
_{j}; - √
- Horizontal distance between the reservoir crests, HD
_{j}beside the role of wave height and period.

#### 2.1.1. 2D Waves

_{m}

_{0,t}, affects q

_{ov}

_{,j}in a system of three reservoirs [12]. Experimental points refer to different SSG layouts having the same crest levels R

_{c,j}; incident waves have a peak wave steepness, s

_{0p}:

**Figure 4.**Overtopping discharge vs. wave height. Redrawn from [12]. Data at prototype scale [SR 1:15].

_{ov}

_{,j}to increase with H

_{m}

_{0,t}. However, the trend-lines have different shapes: for the lowest reservoir, Res.1, the curve is convex, while for the highest one, Res.3, the flow rate seems to increase, on average, more than linearly. It is of interest that similar trends have also been observed under 3D conditions [14]. The reason of this behavior is rather uncertain: for reservoirs 1 and 2, the distance between the lower and the upper front forms a gap which seems to set an upper limit to the entering volume flux. In other words the presence of the fronts seems to cut the amount of water the lower reservoirs can capture. In the limit, an asymptotic value of q

_{ov}

_{,1}and q

_{ov}

_{,2}might be attained when H

_{m}

_{0,t}becomes very large. This wouldn’t happen for the highest reservoir, Res.3, as there is no structure above it (Figure 3). More details on this point can be found in [8]. Figure 5 displays the effect of the peak wave steepness.

**Figure 5.**Effect of peak wave steepness on the overtopping discharge [13]. SR = 1:30.

_{ov}

_{,1}, is made non-dimensional by the significant wave height; all the incident waves were driven by mean JONSWAP spectra.

_{c}. A great deal of literature studies (e.g., [15,16]) have revealed that the mean overtopping discharge reduces more than linearly when the height of crest increases; moreover, the rate of reduction increases with increasing wave height. This is partly shown in the top panel of Figure 6; the latter reports the data from two structures where the sole difference is in the height of the lowest reservoir (R

_{c,1}= 2.25 m in the structure “D” and 1.5 m in the structure “E”). Note that the distance between the curves representing q

_{ov,1}increases with wave height. At the same time, the overtopping discharge in the other reservoirs remains basically the same. However it is clear that any reduction of crest freeboard reduces the hydraulic head of the entering flows, besides increasing the discharge. Consequently, its effect in terms of hydraulic efficiency derives from a balance between those two variables (see Appendix I).

_{ov,1}in “E” is nearly twice than in “D” (upper panel). The reason why this would occur is due to the fact that for high H

_{m}

_{0,t}the power related to the upper reservoirs should tend to dominate the value of the overall efficiency, because the amount water they capture becomes large and with a high hydraulic head.

**Figure 6.**Top panel: Effect of the reservoir crest on wave overtopping. Lower panel: Effect of the reservoir crest on the sea state hydraulic efficiency. Data in prototype scale [SR = 1:15].

_{r}below the mean water level (see Figure 3 for reference). Data are presented for a single sea state (H

_{m}

_{0,t}= 3.5 m, T

_{e}= 11.66 s; SR = 1:15) and refer to four structures, which are identical (three reservoirs) but for the value of the draught.

**Figure 7.**Effect of the draught on the overtopping rate and on the hydraulic efficiency; [SR = 1:15].

_{r}/h = 1), apart from the lowest reservoir, where a constant value seems to be reached between d

_{r}/h = 0.625 and d

_{r}/h = 1. The dotted line in the graph represents the hydraulic efficiency; a gain of 5% is progressively achieved going from d

_{r}/h = 0.375 to d

_{r}/h = 1.

_{r}, is explained in Figure 8. To facilitate the comprehension, data are presented as (sea state) hydraulic efficiency vs. wave height to depth ratio; in the graph three structure layouts which differ only by α

_{r}have been considered, being all the other parameters the same. Accordingly, the results depend uniquely on the overtopping response. Note that all the curves show an efficiency reduction at H

_{s}/h = 0.56, due to some breaking occurrence on the foreshore. The graph suggests 19° is slightly better performing (maximum gain about 4%), while little difference is detected, on average, between 30° and 35°.

_{j}= 35°; j = 1,…,n) has been proposed.

_{r}/h = 1) and an angle of 35° was used for both α

_{r}and θ

_{j}. As already mentioned, wave attacks were driven by mean JONSWAP spectra; peak wave steepness (Equation (1)) has been varied between 0.005 and 0.05. It has been observed that when HD1 is small compared to the wave height (say, HD1/H

_{m}

_{0,t}< 2), the upper level has a significant influence on the water storage in the level below. Figure 9 displays the overtopping discharge in the lower reservoir increases with increasing HD1, whereas the opposite trend is observed for the upper reservoir.

_{c,1}; in this case the value of q

_{ov,1}might be calculated by ordinary overtopping formulae for sloping face breakwaters, such as the van der Meer and Janssen equation [16]. The latter is plotted as a solid blue line in the graph. It is also noted that when HD1 becomes small q

_{ov,2}appears well predicted by the van der Meer and Janssen formula, although the mean discharge in the lower reservoir does not vanish.

**Figure 9.**Non dimensional overtopping rates in the reservoirs as function of HD1 [13]. SR = 1:30.

#### 2.1.2. Effect of Wave Directionality

_{0}is the mean wave direction. D and D

_{0}are taken from the normal to the SSG. n is a spreading index: the larger n, the lower the directional spreading of the waves about D

_{0}. The analysis of data revealed that the general effect of both short-crestedness and obliquity is limiting the amount of overtopping discharge (e.g., Figure 10). The reduction is relatively small for the first two levels (q

_{ov,1}and q

_{ov,2}), but is noticeable for the upper one (q

_{ov,3}), especially under high waves. Altogether, it has been found out that short-crested seas with high spreading (n < 100) decrease the overtopping rate at the lower reservoirs as much as 10%, while a cut by 35% occurs at the top. As far as the role of obliquity is concerned, a similar behavior has been observed.

**Figure 10.**Effect of directional spreading (up) and obliquity (low) on wave overtopping. Data at model scale. SR = 1:60.

_{ov,3}becomes larger than q

_{ov,1}; this should be due to the presence of the fronts that, as commented earlier, “confine” the entering water flow at the lower levels.

**Figure 11.**Effect of short-crestedness on the hydraulic efficiency. Each data series refers to a given sea state. Re-drawn from [14]; SR = 1:60.

#### 2.1.3. Design Equations

_{j}. Note that, according to the previous discussion, neither wave period nor wave steepness are explicitly included in the formula. Finally we can estimate the amount of water entering the j-th reservoir, by integrating between the crest levels R

_{c,j}and R

_{c,j+1}. We get:

_{c,j+1}can be set equal to infinite or to some high value, e.g., 2 times R

_{c,j}.

_{r}/h =1 and α

_{r}= θ

_{j}= 30°–35°:

_{j}are concerned, it should be emphasized that in most of cases the effects of deviating from the reference configuration have been evaluated “one by one”. That is, in a given test series a single parameter has been varied, whereas the others have been kept constant. Accordingly, there are only few cases where the “cross effect” of two or more λ

_{j}has been really verified. Then one may conclude that the product at the right hand side of Equations (3) and (4) has been simply “postulated” from the experimental results and that further analyses are needed to investigate the relationships which possibly link the geometric correction indexes. On the other hand, it is also worth noticing that the “one by one” approach described above is a standard in the experimental research on wave run-up and overtopping. In the worldwide known work by van der Meer and Janssen [16], a general run-up formula for sloping structures is established, where a number of correction factors are multiplied by each other to account for the effects of roughness (γ

_{f}), shallow foreshore (γ

_{h}), wave obliquity (γ

_{β}), presence of berms, etc. In the study, the expressions for calculating those indexes have been derived from different experimental studies with no care about “cross effects”. For example the calibration of γ

_{β}comes from 3D tests, whereas the role of roughness and shallow foreshore has been studied through 2D experiments; no results on the “cross effect” of roughness and shallow foreshore in directional seas are presented. Hence one may reason that Equations (3) and (4) are as reliable as most of the practical overtopping formulae of coastal engineering. As a compromise between the two previous views of the problem, it might be suggested that in practical applications the correction coefficients λ

_{j}are applied within the experimental frame where they have been derived. Some indication is given below.

_{p}is the wave number based on the peak local wavelength L

_{p}. It is of interest that the expression above has been originally formulated for floating devices, when waves were allowed to pass under the structure [6]. This may be not really surprising as in presence of a vertical front in the approaching slope, a portion of the incoming wave energy is however lost, although the loss is caused by reflection instead of under-passing.

_{r}= 19° and θ

_{j}= 35°; the ratio HD1/R

_{c,1}(Figure 3) was about 1.25 and d

_{r}/d ranged from 0.375 and 1. As shown in Figure 12, λ

_{dr}increases from 0.6 to 1, when the draught ratio varies from 0 (fully vertical front) to 1 (sloping front).

_{dr}is closer to 1) as the wave energy tends to vanish near the bottom. In [13] the following factors have been introduced to account the effect of HD1 in a two reservoirs system (α

_{r}= θ

_{j}= 35° and d

_{r}/d = 1, see above discussion):

_{1,HD}applies to the first reservoir (the lower one) and λ

_{2,HD}applies to the second reservoir. The graph of Figure 13 shows that Equation (12) is rather consistent to the experimental findings displayed in Figure 9. No “ad hoc” functional form has been found to account for wave directionality, although it has been suggested [8] that a first estimation of the effect of wave obliquity might come from the van der Meer and Janssen formula [16]:

_{0}is in degree; for an attack of 45° λ

_{D0}= 0.85.

- the draught coefficient λ
_{dr}given in Equation (11); - a ramp factor λ
_{αr}given by:$${\lambda}_{\alpha r}={\mathrm{cos}}^{3}\left({\alpha}_{r}-30\xb0\right)$$ - a low crest factor:$${\lambda}_{s}=\{\begin{array}{c}0.4\cdot \mathrm{sin}\left(\frac{2\pi}{3}\frac{{R}_{c,1}}{{H}_{m0,t}}\right)+0.6\text{\hspace{0.17em}}for\text{\hspace{0.17em}}\frac{{R}_{c,1}}{{H}_{m0,t}}<0.75\\ \\ 1\text{\hspace{0.17em}}\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& & \end{array}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}\frac{{R}_{c,1}}{{H}_{m0,t}}\ge 0.75\end{array}$$

_{j}seems plainly justified in this case. It is also noteworthy that λ

_{αr}is maximum for a ramp angle of 30°, in contrast with the experimental findings discussed above. λ

_{s}increases from 0.6 to 1, when the relative crest freeboard grows from 0 to 0.75. Equation (9) holds provided that waves do not overpass the device (no overtopping water behind the structure is allowed).

#### 2.2. Two Further Items of the Hydraulic Design: Tide and Number of Reservoirs

#### 2.2.1. The Role of Tide

_{ov,j}, as well on the sea-state hydraulic efficiency ${\eta}_{Hyd}^{ss}$. In each set of experiments, the water depth has been kept constant and accordingly no information has been obtained about the role of tide. To fill this gap, a number of numerical simulations have been performed by means of the program WOPsim 3.01 [19,20], which is described in detail in Appendix II at the end of the paper. The main inputs to the software are water levels, crest levels, wave conditions and turbine strategy; the output are, among others, power production and overall efficiency (output power to wave power ratio).

_{c,j}) have been optimized by maximizing the global hydraulic efficiency, ${\eta}_{Hyd}^{G}$ (see Appendix I), under the assumption of no tide (reference has been made to the chart datum). Once R

_{c,j}have been obtained, the structure has been subjected to the same climates including tides and ${\eta}_{Hyd}^{G}$ has been re-calculated. The ratio between the latter and former value of the efficiency has been considered as an indicator of the influence of tide on the SSG performances.

**Figure 14.**Decrease of hydraulic efficiency with increasing tidal ranges [19].

_{c,j}are optimized taking into account of both wave climate and tidal variations, a larger value of ${\eta}_{Hyd}^{G}$ is obtained (gain up to 3%) compared to the case where the optimization is performed only with respect to the wave climate. In conclusion we may state the time variation of water levels to significantly affect the hydraulic response of SSG.

#### 2.2.2. Adding Further Reservoirs

_{c,j}(j = 1,…,5) have been optimized with respect to ${\eta}_{Hyd}^{G}$ for different values of TR (Figure 15, T1 corresponds to 0.8 m TR, T3 is TR = 4.8 m, T6 is TR = 9.6 m).

**Figure 15.**Effect of number of reservoirs on the hydraulic efficiency [19].

#### 2.3. Reflection Performance

_{r}, is needed. The latter is defined as the reflected to incident significant wave height ratio.

_{r}is in the range 30–40%, design formulae for rubble mound breakwaters will be employed [21], whereas, for values in the range 50%–100%, an expression valid for vertical breakwaters will be the most adequate [22].

_{e}, referred to as s

_{0e}hereafter, ranged between 0.008 and 0.058.

_{r}was also suggested. The latter reads:

_{0}is the surf similarity parameter:

_{fs}is the front slope angle.

_{eq}, has to be used. In [24] the latter has been calculated as the weighted average of the mean slope in the run-up/run-down area (α

_{incl}) and the slope of the approach ramp, α

_{r}(Figure 16).

_{SSG}accounts for the effect of the water volume captured in the lowest reservoir, which is always placed in the run-up/run-down area. It is calculated as:

_{r}and that predicted by the formulae) is about 6%. This means the actual (measured) reflection coefficient is in a range of ±0.1 around the predicted one with a 90% probability.

## 3. Structure Response

**Figure 17.**View of pressure cell positions, after [11].

- Under front attacks, surging breakers rapidly rise along the three front plates, originating quasi-static pressure paths
**.**The massive wave overtopping causes a water jet to hit the vertical rear wall in the upper reservoir, which has no roof (position 14 of Figure 17). Figure 18 shows the pressure chronograms at four transducers along the SSG cross section; it is clear that the slamming of the impinging jet at the position 14 induces a quasi-impulsive loading, with a rise time rather short, compared to the other positions, and a magnitude which is about twice the pressure at the front face. - Under side attacks, the wave experiences a rotation due to refraction. At the structure, only one part of the front climbs the plates with a shape similar to the case 1 (Figure 19a); another part hits the side wall producing a partially damped plunging breaker, which again leads the pressure to get an impulsive or “quasi-impulsive” nature (Figure 20).

**Figure 18.**Pressure signals at the front of SSG [11]. Green curve: transducer # 24; red curve: transducer # 21; blue curve: transducer # 18; yellow curve: transducer # 14.

**Figure 19.**Wave shape under side attacks [11].

**Figure 20.**Quasi impulsive event at position 12 under a side attack [11].

_{m}

_{0},

_{t}was measured. It is worth to notice that plunging breakers impacting vertical-face breakwaters usually produce pressures of order of 10 times the wave height (a factor of 50 times was measured in [27] and this justifies the term “quasi-impulsive” reported above).

## 4. Efficiency and Energy Production

- Wave to crests, i.e., where the different waves are captured at the crest heights of the reservoirs, R
_{c}_{,j}(j = 1,2…n, n = number of reservoirs). During the of laboratory tests described in the previous sections, it has been measured that around 40% of the available sea state energy (Appendix I) is captured; - Crests to reservoirs, i.e., where the potential energy relative to the specific crest heights is reduced by falling into the reservoir at a lower height. It is estimated that 75% of the energy from the previous step is maintained;
- Low head water turbines, i.e., where the water in the reservoirs is utilized by the hydraulic turbines with 90% efficiency;
- Electrical generator and electrical equipment, 95% efficiency.

## 5. Power Take-off

**Figure 22.**Three-levels Multi-stage Turbine [32].

## 6. Feasibility Studies

- (1)
- sharing of construction costs,
- (2)
- access and therefore operation and maintenance are easier compared to an offshore situation,
- (3)
- sharing of infrastructures.

- (1)
- Recirculation of the water inside the harbor, i.e., improvement of water quality as the outlet of the turbines would be in the rear part of the breakwater,
- (2)
- Potential lower visual impact as a consequence of a lower crest level,
- (3)
- Clean electricity generation.

- (1)
- Local wave and tide climate (determines the number and size of reservoirs, in average passing from three to four reservoirs will see an increase of construction cost of 4%),
- (2)
- Design wave height (determines ballast and size of the structure),
- (3)
- Water depth (determines the construction method and overall size of the caisson).

Annual wave energy = 15.7 kW/m | |

H_{s} = 7.9 m | H_{max} = 9.9 m |

Water depth = 11.3 m | Tidal range = 1.6 (± 0.8) m |

Capture crest levels: R_{c}_{1} = 1 m, R_{c}_{2} = 2.5 m, R_{c}_{3} = 4 m | |

Crest level: 8 m | Base width: 28 m |

Installed capacity: 12.8 kW/m | |

Expected power production: 18,000 kWh/y/m | |

Construction costs inclusive of turbines and generators: 150,700 €/m |

Annual wave energy = 14.4 kW/m | |

H_{s} = 13.9 m | H_{max} = 18.1 m |

Water depth = 18 m | Tidal range = 3.37 (±1.68) m |

Capture crest levels: R_{c}_{1} = 0.75 m, R_{c}_{2} =2.05 m, R_{c}_{3} =3.35 m, R_{c}_{4} = 4.65 m | |

Crest level: 15 m | Base width: 45 m |

Installed capacity: 12 kW/m | |

Expected power production: 12,000 kWh/y/m | |

Construction costs inclusive of turbines and generators: 285,800 €/m |

Location | Rubble mound | Traditional caisson | SSG-breakwater | Additional costs |
---|---|---|---|---|

Swakopmund | 67,200 €/m | 124,500 €/m | 150,700 €/m | 83,500 €/m–26,200 €/m |

Sines | - | 231,000 €/m | 285,800 €/m | 54,800 €/m |

## 7. Conclusions

_{r}, which is never lower than 40%, can rise up to 90%. Therefore, it is a design issue to construct a proper toe protection layer to avoid scour holes or a berm to reduce the reflection.

## Acknowledgements

## Nomenclature

A, B, C | experimental coefficients for overtopping prediction |

B_{u} | Hydraulic Head (Bernoulli Trinomial) (m) |

C_{g} | group velocity (m/s) |

D | direction (relative to the orthogonal to the structure) of a single Fourier wave component in the directional power spectrum (deg.) |

D_{0} | mean wave direction (deg.) |

d_{r} | distance between the mean water level and the lower edge of the run-up ramp (draught, (m)) |

g | gravity acceleration (m/s ^{2}) |

k_{p} | $\frac{2\pi}{{L}_{p}}$
wave number associated with the peak wavelength L _{p} (rad/m) |

K_{r} | reflection coefficient (-) |

HD | horizontal distance between the opening of two consecutive reservoir levels |

f_{j} | R _{c}_{,j} − H_{h}_{,j}. “freespace” for the j-th reservoir (m) |

h | water depth at the toe of the structure (m) |

H_{h}_{,j} | hydraulic head at the j-th reservoir (m) |

H_{s} | significant wave height (m) |

H_{m}_{0,t} | spectral estimate of H _{s} at toe of the structure (m) |

H_{rms} | root mean square wave height (m) |

L_{0e} | $\frac{g{T}_{e}^{2}}{2\pi}$
deep water wave length based on the mean period T _{e} (m) |

L_{0P} | $\frac{g{T}_{p}^{2}}{2\pi}$
deep water peak wave length (m) |

L_{p(e)} | ${L}_{0p(e)}\cdot \mathrm{tanh}\left(\frac{2\pi h}{{L}_{p(e)}}\right)$
= peak (mean) wave length at the depth of placement of the structure (m) |

m_{n} | n ^{th} spectral moment |

MWL | mean water level (m) |

N_{w} | number of waves in a sea state (-) |

OTD | OverTopping Devices |

OWC | Oscillating Water Columns |

P_{Hyd} | mean potential power of the overtopping water per unit of width (W/m) |

P_{Res} | mean power in the reservoirs per unit of width (W/m) |

P_{P} | mean power production per unit of width (W/m) |

P_{wave} | mean power of the waves per unit of width (W/m) |

Pr_{occ.} | Probability of occurrence (-) |

q_{ov}_{,j} | sea-state averaged overtopping discharge to the j-th reservoir per unit of width (m ^{3}/s/m) |

Q_{in}_{,j} | individual overtopping discharge to the j-th reservoir per unit of width, averaged over a wave cycle (m ^{3}/s/m) |

Q_{over}_{,j} | rate of overflow at the j-th reservoir (m ^{3}/s/m) |

Q_{turb}_{,j} | flow through the turbine at the j-th reservoir (m ^{3}/s/m) |

Q_{Res}_{,j} | volume of water stored in the j-th reservoir during an unitary time-step (m ^{3}/s/m) |

Q_{over}_{-upper,j} | overflow discharge at the (j + 1)-th reservoir, which is re-used at the j-th reservoir (m ^{3}/s/m) |

R_{c}_{,j} | crest height of the j-th reservoir (m) |

s_{0p} | $\frac{{H}_{m0,t}}{{L}_{0p}}$
= peak wave steepness (-) |

s_{0e} | $\frac{{H}_{m0,t}}{{L}_{0e}}$
= mean wave steepness (-) |

SSG | Seawave Slot-cone Generator |

T_{e} | energy wave period (in (s)) calculated as
$\frac{{m}_{-1}}{{m}_{0}}$ |

T_{m} | time domain mean wave period (s) |

T_{p} | peak wave period (s) |

WAB | Wave Activated Bodies |

WEC | Wave Energy Converter |

## Greek Letters:

α_{r} | front ramp angle on the horizontal (deg.) |

α_{eq}_{.} | equivalent front angle for reflection analysis |

α_{incl} | mean front slope in the run-up area |

Δ | duration of a sea-state (s or hr.) |

${\eta}_{v}^{ss}$ | SSG efficiency in a sea state (-). ν = (Hyd, Res, P) |

${\eta}_{\nu}^{G}=$ | SSG efficiency for a given wave climate (-). ν = (Hyd, Res, P) |

${\eta}_{turb}$ | turbine efficiency (-) |

θ_{j} | angle of the front of j-th reservoir (deg.) |

λ_{j} | correction factors (-) |

ξ_{0} | $\frac{\mathrm{tan}(\alpha )}{{s}_{0e}}$
= surf similarity parameter (-) |

ρ | sea water density (kg/m ^{3}) |

## Appendix I: Efficiency of SSG

#### A-I.1. The Wave Power (Mean Sea-State Power)

_{rms,0}represents the root mean square wave height and ${\left[{C}_{g0}\right]}_{\stackrel{\_}{T}}$ is the offshore group velocity calculated at the mean period $\stackrel{\_}{T}$. The latter equals:

_{e}is employed for $\stackrel{\_}{T}$; the latter is about 1.1 the peak period T

_{p}and is calculated as the ratio between the spectral moment of order −1 and the area of the power spectrum . Now, by noting that the deep water significant wave height H

_{s}

_{,0}, whether spectrally or statistically defined, is approximately √2 H

_{rms}

_{,0}, we get:

#### A-I.2. The Mean Overtopping Power in a Sea-State

_{u}is the hydraulic head (Bernoulli trinomial).

_{ov}

_{,j}is the mean overtopping discharge, per unit of width, which enters the jth-reservoir; the reservoir has a crest height R

_{c}

_{,j}from the mean water level (see Figure 3 and Figure A1). The total number of reservoirs in the device is N

_{Res}

_{.}

#### A-I.3. The Mean Available Power in a Sea-State

_{w}waves. Then a time-domain average wave period, T

_{m}, can be defined as:

_{h}

_{,j}(Figure A1), is generally less than R

_{c}

_{,j}. Secondly, if the water level in the reservoir equals R

_{c}

_{,j}, some water will be lost by overflow, because the reservoir is full. Thus, a more realistic estimate of the potential energy theoretically available for electricity production is given by:

#### A-I.4. Mean Produced Power

_{j}, between the crest of a reservoir and the water surface (see Figure A1) is less than a threshold value, say F

_{on}, then the turbine starts. Otherwise, when the freespace becomes larger than a second threshold, F

_{off}, then the turbine stops. Both F

_{on}and F

_{off}may be function of the incoming significant wave height.

_{turb}

_{,j.}, and its efficiency, η

_{turb}

_{,j.}, are function of the hydraulic head, via the so-called turbine characteristics. As H

_{h}

_{,j}varies in the time, so do Q

_{turb}

_{,j}and η

_{turb}

_{,j}. Thus, the mean power production related to the j-th reservoir can be calculated as:

_{turb}

_{,j}equals 0 when the turbine is turned off.

#### A-I.5. Sea-State and Global Efficiencies

## Appendix II: The WOPSim Program

_{Res}reservoirs, each reservoir with an independent turbine set-up. It produces a time series of overtopping discharges and calculates the power produced by the turbines. The intention of WOPSim is to allow the user to optimize both the structure layout and the turbine strategy. Once the SSG geometry and the sea-state duration have been inputted, the software works on the mass balance, which must be satisfied at each time-step, Δt, and for each separate reservoir, j:

_{in}

_{,j}is the volume of overtopping water during the time Δt, Q

_{turb}

_{, j}is the flow through the turbine for the same time interval, Q

_{res}

_{,j}is the amount of water stored in the reservoir and Q

_{over}

_{,j}is the water overflow if the reservoir will be full. Furthermore, an option called “overflow to next reservoir” can be enabled, where the overflow at the j-th reservoir can be re-used (spillage) by the lower reservoirs (1 to j − 1). In that case Equation (A.14) is re-formulated as:

_{m}. As far as the volume influx Q

_{in}

_{,j}is concerned, it is supposed to be constant over a “wave cycle” of duration T

_{m}and to vary wave by wave. Thus, a random series of overtopping discharges is simulated, using the approach described in [34], which is based on a three parameters Weibull distribution. One of these parameters is the mean overtopping discharge, q

_{ov}

_{,j}, whose estimation is discussed in Section 2.

_{turb}

_{, j}is computed at each time step based on both the turbine strategy and the turbine characteristics:

_{h}and the freespace f

_{j}(which rules the ON/OFF periods) come from the preceding calculation step. Analogously, the power production during Δt can be evaluated as [Equation (A.10)]:

_{in}

_{,j}and Q

_{turb}

_{, j}allows calculating, by difference, the amount of water stored in the reservoir and, accordingly, the new values of the hydraulic head H

_{h}and of the freespace f

_{j}. If the water level in the reservoir overcomes the crest level R

_{c}

_{,j}, then overflow takes place. Once the simulation has been completed, the program computes all the efficiency indexes defined in Appendix I.

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## Share and Cite

**MDPI and ACS Style**

Vicinanza, D.; Margheritini, L.; Kofoed, J.P.; Buccino, M.
The SSG Wave Energy Converter: Performance, Status and Recent Developments. *Energies* **2012**, *5*, 193-226.
https://doi.org/10.3390/en5020193

**AMA Style**

Vicinanza D, Margheritini L, Kofoed JP, Buccino M.
The SSG Wave Energy Converter: Performance, Status and Recent Developments. *Energies*. 2012; 5(2):193-226.
https://doi.org/10.3390/en5020193

**Chicago/Turabian Style**

Vicinanza, Diego, Lucia Margheritini, Jens Peter Kofoed, and Mariano Buccino.
2012. "The SSG Wave Energy Converter: Performance, Status and Recent Developments" *Energies* 5, no. 2: 193-226.
https://doi.org/10.3390/en5020193