Performance of arrays of direct-driven wave energy converters under optimal power take-off damping

It is well known that the total power converted by a wave energy farm is influenced by the hydrodynamic interactions between wave energy converters, especially when they are close to each other. Th ...


I. INTRODUCTION
Ocean wave is an important potential source of renewable energy.3][4][5][6] However, only a minority of them have been really tested at sea. [7][8][9] The results from long-time test indicate that, survival under harsh sea conditions is a key issue for WECs. 10 To improve the survivability, it is important to avoid the destruction of the structure by a large striking force or an unexpected displacement amplitude.
For several WECs deployed in close proximity, resulting in a wave energy farm (WEF), another important issue to be considered is the hydrodynamic interaction between converters.This results in a coupling which is directly related to diffraction of the waves and the radiation properties of the WECs.This coupling presents another challenge: how to capture the energy efficiently by a WEF taking into account the destructive or constructive interaction between radiated waves. 10,113][14][15][16] This project develops a WEC of point absorber type with a direct-driven linear generator as power take-off (PTO).A buoy located at the ocean surface is connected to a linear generator placed on the seabed via a line.The translator of the generator is driven by the buoy's motion transferred from the wave motion.The first full-scale prototype was installed in spring 2006 off the Swedish west coast, for initial test of this WEC concept. 17revious experiments were performed with WECs connected to several different resistive loads, and the electrical damping of the generator was increased by decreasing the value of the resistive load. 18Results indicate that the PTO damping can influence the power production of the WEC, as well as its motion.The motion of the WEC varies in velocity and displacement amplitude with different PTO damping, resulting in different radiation influence to nearby WECs as well as different striking end stop forces.Another result for a single WEC show that the proper constraint on the displacement amplitude or the damping coefficients can be employed to avoid large striking force to the hull, and this will not decrease the energy production significantly under proper constraints. 19,20t is important to study the performance of a wave energy farm under the influence of PTO damping and constraints.For a wave energy farm analyzed in the frequency domain, different selection strategies of PTO damping are indicated in the literature.These strategies fall into three main categories.In the first category, the same PTO damping is selected for all WECs in a farm, and the value selected is determined by designers or researchers, based on their experience.2][23] The second strategy is to find the optimal value to maximize the power production from the farm. 24,25The improvement of the mean power is due to the optimal match of PTO damping.In those two methods, different WECs in the wave energy farm are matched with same PTO damping.However, this cannot guarantee that their motions and the resulting hydrodynamic interaction are optimal.To achieve the highest total power production, the motion of individual WECs must be coordinated.In this study, a different approach has been explored, optimal PTO damping are selected for each WEC in the given sea states, resulting in the third category.Here, optimal PTO damping means the damping values leading to the maximum mean power production from the whole farm when all WECs are subject to motion constraints.The control method also serves the purpose of reducing the end stop force by restricting the amplitude of motion of the translator, contributing to improved survivability of the WEF.
The rest of this paper is organized as follows.The hydrodynamic theory is introduced in Section II A. The algorithms to find optimal PTO damping of individual WECs are introduced in II B, where the optimal control problem is solved numerically.Parameters used in the case study are defined in Section III.Results are presented and discussed in section IV.

A. Hydrodynamic theory
For a wave energy farm consisting of N WECs, and if only the heave motion is considered, the motion of each WEC can be expressed as follows: where w is the frequency of the incident wave, R j , K j , S j , M j , x j , Fe, j the PTO damping, spring coefficient, buoyancy stiffness coefficient, mass, complex displacement amplitude, complex excitation force amplitude of WEC j in the heave mode, respectively.The displacement of WEC j, as a function of time, can be expressed as 26 x j (t) = Re( x j exp(iwt)). ( Z j j ′ = R j j ′ + iwm j j ′ is the complex radiation coefficient with R j j ′ being the radiation damping coefficients, m j j ′ being the hydrodynamic added mass, j = 1, 2, . . ., N, and j ′ = 1, 2, . . ., N. Using potential flow theory, the complex radiation coefficient can be expressed as where φj ′ is the velocity potential of radiated wave at mode j ′ , n j is a component of the generalized normal, and ρ is the density of ocean water.The radiation force of WEC j at heave mode is influenced by the motion of all WECs, considering the hydrodynamic interaction between converters.It is written as Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.The excitation force for WEC j, represented by Fe, j , is written as where φ0 is the velocity potential of the incoming wave, φd the velocity potential of diffracted wave.The total mean power converted by all WECs is given by: where the complex velocity ûj = iw x j , * represents the complex conjugate.R j is the PTO damping of WEC j, and its optimal value R j,opt is calculated as follows where x j,max is the maximum value of displacement amplitude of WEC j, and j = 1, 2, . . ., N.

B. Algorithm
As indicated in equation ( 6), the total mean power is a function of optimal PTO damping R j,opt and velocity amplitude ûj , and they are mathematically coupled by equation (1).Inserting equation ( 1) into equation ( 6), the total mean power becomes a function of the complex displacements Now, equation ( 7) is equivalently expressed as: where Equation ( 9) is a constrained optimization problem, where s is the vector of optimization variables and s opt the local optimum.Different methods are available to solve this problem with nonlinear objective function and constraints.The active set algorithm is one of these, and equation ( 9) is written in the standard form: where s is the vector of length 2N design parameters, f (s) the objective function, which returns a scalar value, and the vector function G(s) returns a vector containing the values of inequality constraints evaluated at s.
The efficiency and accuracy of the solution to the constrained optimization problem depends on the number of constraints and design variables, and on the characteristics of the objective function and constraints.When the constraints are linear and the objective function is quadratic or linear, reliable solutions are readily available.When both the objective function and the constraints are nonlinear functions, as the case shown in equation (10), the problem can be transformed into a subproblem that can be solved and used as the core of an iterative process.The active set method transforms the nonlinear problem into quadratic programming by using a penalty function for constraints near the constraint boundary.This can be implemented in MATLAB by the fmincon function, where the quadratic programming subproblem is solved and the search direction is updated using a merit function in each iteration.Equation ( 9) can be solved using another algorithm for engineering applications.Considering the physical limit of PTO damping R j , it is only tuned in a proper range [R j,min , R j,max ] in the physical world.Assuming that the PTO damping is tuned discontinuously under different sea states, and the optimal value for WEC j is chosen from discrete set R j = R j,min + n j, R dr with dr being the step.For a given sea state, the excitation force of each WEC can be calculated, then with trail values R j for WEC j with j = 1, 2, . . ., N, equation (1) becomes a system of linear equations in N variables with x j being the variables, and can be solved analytically.By repeating this for all the trial values and for all the WECs, a database is compiled without considering the constraints, from which the corresponding displacement amplitudes and total output power can be calculated.The set, where the corresponding mean power is maximal and the displacement amplitudes do not exceed the limit, will be chosen as the final optimal values from the data base.Therefore, the procedure involves three phases, and can be structured as follows: Phase one: Choose a trial value of R j from [R j,min , R j,max ], calculate the displacements of all WECs by solving equation ( 1), and calculate the total mean power by equation (6).
Phase two: Choose a different combination of R j , and repeat phase one to formulate a database of mean power.
Phase three: Compare with the motion constraints, select the maximum mean power with displacement amplitudes satisfying the motion constraints, and output the corresponding parameters.
The hydrodynamic parameters can be calculated analytically or solved numerically using commercial software.The accuracy of the second algorithm depends on the step dr.A comparison between the two algorithms is not made in this paper, and only the first algorithm is used in the following sections.

III. MODEL AND PARAMETERS
The layout of the farm is shown in figure 1.Eight cases, listed in table I, consisting of WECs with different quantities or gap distances, are studied.The individual WECs all have same physical parameters, given in table II.
To evaluate the performance of the control method, the total mean power produced from the farm using coordinated control method is compared with that using same PTO damping for all WECs, and the efficiency improvement (represented by EI) is defined as,  where R sm is the optimal value in the case of all the WECs using same PTO damping (belong to the second category introduced previously).
Since the power production from a WEF using same PTO damping for all WECs varies with the damping values, as well as sea states, the optimal damping value R sm will be calculated for each sea state and used for all WECs, in order to quantify the benefits of the control method.Here the optimal value also means the damping leading to the maximum mean power production from the WEF while all the WECs are subject to motion constraints.
The radiation damping coefficients, hydrodynamic added mass, and excitation force are calculated in WAMIT, other calculations are implemented in MATLAB.

A. Influence of gap distance
Results for case 1-4 under different sea states with a wave height step of 0.2 m and an angular frequency step of 0.2 rad/s are presented in figure 2. Results indicate that, for the farm with fixed quantity of WECs, the power production can be improved by the coordinated control method, especially for the case with small gap distance.It is obvious that the gap distance can influence the efficiency improvement, which can be found from the comparison of the four subfigures of figure 2. The peak value of the efficiency improvement is 15% for D = 10 m, decreasing to 11% for D = 20 m, 7% for D = 40 m, 2% for D = 300 m.This is due to the fact that the efficiency improvement caused by the coordinated control is correlated to the hydrodynamic interaction between WECs.Therefore, the level of the hydrodynamic interaction will influence the efficiency improvement.Since the hydrodynamic interaction becomes weaker when the WECs are far away from each other, 27 the power production improvement caused by the coordinated control will diminish with increasing WEC distance.It can be noted that the corresponding angular frequencies for the peak value are all around the angular frequency of 1.4 rad/s, the natural frequency of the WEC.This matches the well-known fact that the most efficient conversion will be achieved when the incident wave frequency is close to the natural frequency of the WEC, 28 and the natural frequency is decided by the physical characteristics of the system.As shown in figure 2(a), generally, starting around the angular frequency of 0.2 rad/s, the efficiency improvement increases and reaches a peak value, where the corresponding angular frequency is around 1.4 rad/s, and then decreases.The efficiency improvement for different wave heights larger than 0.6 m are almost the same when frequencies are less than the natural frequency, while it has a sharp change for frequencies near the natural frequency.For large angular frequencies, corresponding to short-period waves, the performance of the wave energy farm cannot be improved, in this way.

B. Influence of WEC quantities
Figure 3 shows results from the WEF of different scales.A gap distance of D = 10 m is used for each scale.The peak values of the efficiency improvement in the four subfigures are 11% for the farm with 4 WECs, 18% for the farm with 8 WECs, 18% for the farm with 16 WECs, and 16% for the farm with 32 WECs, respectively.The trend lines for the efficiency improvement are almost the same as that for case 1, which means they increase with angular frequency at first, then decrease.For this small gap distance, the power production of the wave energy farm can be strongly improved using the coordinated control, especially for the large scale farm.It should be noted that, under sea states with small angular frequency, this improvement is very stable especially when the WECs are subjected to wave heights larger than 0.6 m, which can be found in each subfigure of figure 3.
Besides the influence of the number of WECs in the farm, comparison between case 1 and case 6, corresponding to the results in figure 2(a) and figure 3(b), respectively, indicate that, for the farm with given quantity of WECs and given gap distance between WECs, the efficiency improvement is also influenced by the layout of the farm.The maximum improvement in case 1 is about 15%, while it is about 18% in case 6.

C. Performance of WECs under optimal damping
The performance in terms of mean power production, PTO damping, and displacement amplitudes for a large array, corresponding to the case 7 in table I and figure 1, are shown in figure 4. It presents the results of WECs with optimal PTO damping in a sea state with wave height of 2 m and angular frequency of 1 rad/s.As shown in figure 4(a), the mean power produced by the WEC ranges from 38.9 kW to 51.6 kW.If evaluated in terms of power, WEC 5 and 8 have the best performance, then WEC 2 and 3, while WEC 14 and 15 have the worst performance.It is obvious to see that, the shadowed WECs, located in the most right two columns (called left column 3 and 4 in this paper), have worse performance compared to that of other two columns.However, the shadowing effect is not obvious for the second left column (called left column 2 in this paper).It can also be found that the WECs located at the outside of left column 2, 3 and 4, produce more mean power than that located at the inside of same columns.
Figure 4(b) shows the corresponding PTO damping for each individual WEC.The optimal PTO damping for each WEC ranges from 311.1 kNs/m to 412.8 kNs/m, and is proportional to the mean power of each WEC. Figure 4(c) presents the displacement amplitudes of WECs.It indicates that, for this given sea state, all the WECs in this farm have same displacement amplitude, 0.5 m, which is also the maximum value of displacement amplitude used in the calculation.It should be noted that those optimal PTO damping, mean power and motion amplitudes, resulted from coordinated control method, varies over sea states.

V. CONCLUSION
By coordinating the PTO damping of each WEC, a preferred hydrodynamic interaction is obtained, which increases the power production of a wave energy farm.This performance enhancement will be larger for wave energy farms with WECs in close proximity.The power production is also improved significantly for a farm with a large number of WECs, and this improvement can reach the high value of 18% in the case used in our simulation.Since different motion constraints can be considered for each WEC in this method, excessive motion amplitudes of the translator can be avoided to improve the survivability of the WEC.

AUTHOR CONTRIBUTIONS
Liguo Wang coordinated the main theme of this paper, conducted the calculation, prepared the manuscript, and contributed to the revision.Jan Isberg and Mats Leijon supervised the wave power project.Jan Isberg and Jens Engström commented on the manuscript, and contributed to the revision.

FIG. 1 .
FIG.1.Layout (top view) of the wave energy farm used in the calculation.X is the direction of ocean wave propagation, and Y direction is perpendicular to that.

FIG. 2 .
FIG.2.Results from the wave energy farm consisting of 8 wave energy converters.The x axis is angular frequency with unit of rad/s.There are 150 sea states for each subfigure.

FIG. 4 .
FIG. 4. Performance of individual wave energy converters of case 7 in sea state with wave height of 2.0 m and angular frequency of 1 rad/s.(a) is the performance in terms of mean power.(b) is the optimal PTO damping calculated from coordinated control.(c) is the amplitude of displacement of wave energy converter.

TABLE I .
Different cases of wave energy farm layout.

TABLE II .
Main parameters of wave energy converters.