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In order to reduce the cost of electricity produced by wave energy converters (WECs), the benefit of selling electricity as well as the investment costs of the structure has to be considered. This paper presents a methodology for assessing the control strategy for a WEC with respect to both energy output and structural fatigue loads. Different active and passive control strategies are implemented (proportional (P) controller, proportional-integral (PI) controller, proportional-integral-derivative with memory compensation (PID) controller, model predictive control (MPC) and maximum energy controller (MEC)), and load time-series resulting from numerical simulations are used to design structural parts based on fatigue analysis using rain-flow counting, Stress-Number (SN) curves and Miner's rule. The objective of the methodology is to obtain a cost-effective WEC with a more comprehensive analysis of a WEC based on a combination of well known control strategies and standardised fatigue methods. The presented method is then applied to a particular case study, the Wavestar WEC, for a specific location in the North Sea. Results, which are based on numerical simulations, show the importance of balancing the gained power against structural fatigue. Based on a simple cost model, the PI controller is shown as a viable solution.

Wave energy converters (WECs) have a high potential to contribute significantly to the world energy mix in the future. The practically exploitable wave power potential has been assessed to be up to 3.7 TW, which is about a fourth of the global demand and roughly a double of the global electrical consumption [

In terms of maximum absorbed power namely reactive controller or maximum energy controller (MEC), the condition for optimality was firstly derived in the 1970s [

Overall, much effort has been put into finding the optimal control strategies where the main focus is the maximisation of the gained mechanical or electrical energy. Albeit maximising the absorbed energy is an important task, it has not been proven whether this is also optimal from an overall cost point of view. Since the global aim of the sector is the commercialisation of WECs, the economic quota needs to be considered too. For offshore structure, due to the high ratio between extreme and operational loads, the structural cost is expected to lie in the range 30%–50% of the capital cost [

The work shown in [

The present article shows a simple non-recursive methodology to select the best control strategy for a given WEC where both energy absorption and structural design are considered. The methodology flowchart is presented in

The paper is organised in five sections: Section 1 introduces the problem, together with the previous and proposed solution; Section 2 gives a general introduction of the different models used, namely WEC modelling, control problem and fatigue assessment; Section 3 specifies the equations presented in Section 2 for a particular WEC in a definite location; Section 4 contains a detailed discussion around the take-home messages embedded in the results; and Section 5 gives a brief recap of the work done besides the main outcome of the article.

A numerical model is used to describe the dynamical behaviour of a given WEC due to its flexibility. Despite the fact that numerical models are only an approximate representation of the corresponding physical models, once their accuracy is proven they represent a cheap and a fast tool for analysing a WEC, compared to physical modelling. In particular, numerical models based on linear potential theory coefficients require low computation cost whilst outputting accurate results if the motion of the WEC is kept bounded around the linearisation point [

It is important to bear in mind that the proposed numerical model is specified for WECs of the point absorber type due to the problem formulation simplicity, but the same approach can be used for a generic WEC of the activated body type after providing a representative model for the wave-body interaction. In the following, the definition of the used numerical model, control strategies and fatigue model composing the main algorithm are presented. The equations are expressed in the international system of units (SI) system of measurement and each parameter is expressed in accordance with the SI base.

A WEC can be defined as a dynamic system with one or more degrees of freedom, used to transform the wave energy content into useful—typically electrical—energy. In this work, only the absorbed mechanical power will be considered. The transfer from mechanical to electrical power is not taken into account. A point absorber is a particular type of WEC where the characteristic length of the device is small compared with the wavelength. A single body floating system, like a point absorber WEC, presents a resonant-like behaviour similar to a mechanical oscillator. The dynamical model of system can be obtained using the Newton-Euler equation. If the system has only one degree of freedom, then the Newton-Euler equation is simplified to [_{i}_{ex}_{RAD}_{hy}

The wave excitation force is defined as an external force acting on the fixed structure subjected to a wave field and is defined by a complex function associated with the frequency of the incoming wave. It accounts for two distinct contributions namely the diffraction force plus the first order Froude-Krylov force evaluated for fixed wetted surface. The radiation force describes the loads acting on the structure due to its motion in still water and is defined as [_{∞} represents the limit value of the added mass when the frequency tends to infinity; B(ω) is the radiation damping coefficient; and _{RAD}_{hy}_{hy}_{hy}

_{i}_{u}_{rad}_{u}_{ex}

The average mechanical power (_{u}

The absorbed mean power is often used as a performance indicator for a generic WEC. The dynamic response of a WEC, and its production capabilities, is affected by the applied control scheme. Because the scope of the work is the balanced power optimisation of a WEC against structural fatigue, it is important to define both non-aggressive and aggressive controllers. In the context of this work, the term “aggressiveness of a controller” is meant in a qualitative sense. It refers to the actuator duty cycle.

Five different control strategies will be introduced and used in the case study. Three of them, P, PI and PIDc, are applied for the true comparison while the other two controllers (MEC and MPC) are employed as benchmarks. The P, PI and PIDc controllers have been chosen for their implementation simplicity and efficacy. They are listed in increasing aggressiveness order. The power performance of the three controllers will be compared with the maximum achievable mean power. The theoretical superior limit of

For a simple oscillator, the optimal energy transfer between external source and the moving body happens when the system is in resonance with the input's frequency. The analytical solution of the maximum absorbed power problem is [

Some considerations can be drawn from

Since

The power maximisation is an impedance matching problem [

Some energy needs to be fed back to the floater during a wave cycle. Therefore, the actuator needs to be reversible, and the instantaneous power will be both positive and negative at different positions of the wave cycle.

The actuator is considered ideal,

In addition to the inability to handle physical constraints and the unrealistic large amplitude of motion, the practical implementation of this type of controller is crippled by the non-causality of the control law. The definitions introduced in

Among the two methods introduced in

Another issue arises when the reference velocity needs to be evaluated: the transfer function (TF) between the excitation force and the optimal reference is frequency dependent. This type of problem can be solved using an observer of the excitation force state, in the form of an Extended Kalman Filter [

MPC refers to an advanced, digital control technique. Its basic principle is to calculate optimal values for the control signals over a certain time horizon in the future. To this end, it uses a dynamic model of the plant, current measurements to update the dynamic states of this model, and the future course of external inputs; the latter being the wave excitation force and the control force in this paper. Given that the dynamic model sufficiently represents the real system and that measurements for updating the states and a prediction of the future wave excitation force are available, the WEC's behaviour over the time horizon can be calculated depending on the control force. This is used to determine the optimal trajectory of the control force by solving a possibly constrained optimisation problem. This optimisation problem includes a cost function that reflects the control objectives. The optimal control force is then applied to the point absorber until the new measurement and predictions samples are available. After that, the whole procedure is repeated in order to account for unforeseen disturbances. The ability of directly incorporating constraints in the optimisation problem makes MPC a straightforward choice for control problems with constraints on the control input or the plant states.

Several applications of MPC for the control of WEC are reported in the literature. Some of them directly incorporate the objective of maximum average power into the cost function [

The general formulation of the applied cost function _{p}

If feasible, the solution of this problem yields an optimal control sequence over the prediction horizon. Using the receding horizon control principle, only the current sample _{u}

When the WEC actuator only works in an unidirectional mode, no energy can be fed back to the oscillator. This type of controller commonly known as passive damper has the great advantage of requiring simple machinery and having a smaller ratio between peak average power. Otherwise, since the body reactance will not be compensated, the WEC will mainly work in a tight range around the natural frequency of the oscillator, limiting the overall system efficiency. The general form of the control force is as follows [_{c}

The PI controller is also denoted as spring-damper controller because an additional term proportional to the body displacement is used to compensate the intrinsic reactance of the body. In theory, the PD controller can also achieve similar results, but as shown in [_{c}

The PID controller is a mass-spring-damper controller with the introduction of an additional convolution term. The general form of the controller is as follows [_{c}

Fatigue failure is an important failure mode of offshore structures and expectably even more for WECs where the resonance condition is sought. For estimating the fatigue of a structural part, the Stress-Number (SN) of cycles curve together with Miner's rule [_{i}_{i}_{i}_{i}_{i}_{i}

The bilinear SN curve has a slope change at Δσ_{D}_{D}^{6}:
_{1,2} are the stress intensity factors; and _{1,2} are the crack growth parameters. A design equation can be used to calculate the design equation parameter ^{c}_{ijk}_{m}_{0}_{i}_{Pj}_{ijk}_{ijk}/z_{m}_{0}_{i}_{Pj}_{fat}_{L}_{L}_{m}_{0}_{i}_{Pj}_{m}_{0}_{i},T_{Pj}_{m}_{0}_{i}_{Pj}_{m}_{0}_{i}

This section focuses on bringing together the influence of a certain control strategy on harvested energy, which will define the income during lifetime, and the resulting structural design which drives the investment costs. In order to compare different control strategies, a simple economical model is used.

From a certain structural detail, one can hardly comment on the control strategy's overall cost impact. But one can, based on some simple assumptions get an idea of the relative impact of the different control strategies on the overall cost.

It is assumed that the total lifetime costs consists of one part, which is dependent on the control strategy (called _{1}, mainly cost for PTO and structure of PTO arm) and other investment costs (called _{2}, e.g., platform, or electricity connection to shore), which are assumed to be constant for all control strategies. The control dependent costs are assumed to be proportional to the cross sectional area of a certain critical structural component. The cost factor _{c}_{c}_{ref}_{tot}_{ref}_{c}

This section presents a case study focused on the Wavestar WEC. The particular WEC has been chosen mainly because its numerical model already has been compared with experimental data from a scaled physical model of the device [

Sea state measurements over a period of six years are provided by [_{m}_{0}) as well as the peak period (_{P}

The incoming wave direction is not considered in this case study and is assumed to be of minor importance for the load and power output assessment, as a consequence of the symmetry of the system.

Since the Wavestar machine belongs to the point absorber WEC class, the model introduced in

For each sea state described in

The linear hydrodynamic coefficients as well as the hydrostatic stiffness coefficients have been evaluated using a commercial Boundary Element Method (BEM) solver [

The power performance of the WEC is evaluated for the unconstrained model and for two other cases, all defined in

Case 1—Unconstrained case. Neither the saturation of the maximum PTO force, nor the maximum allowed PTO stroke, nor the additional damping is included.

Case 2—Unconstrained case with linearised viscous drag moment implemented as additional damping. Neither the saturation of the maximum PTO force, nor the maximum allowed PTO stroke is included.

Case 3—PTO constraint and linearised viscous drag moment implemented as additional damping. The PTO constraint is implemented as a saturation function of the full control force.

Hereafter, the rationale behind these choices is briefly given. The linearised viscous drag moment has been inserted in the WEC model in order to account for a viscous dissipative effect. This type of effect is normally negligible when the system operates in a non-resonant condition due to the relative low body velocity. On the other hand, when the resonance condition is achieved, a significant part of the energy is dissipated in turbulence [

The PTO constraint is introduced to simulate a model as close as possible to the device deployed at DanWEC in Hanstholm (Denmark). This type of constraint reflects a realistic feature of many types of actuators, to be used in a WEC, and it is worth investigating its effect in both power and fatigue assessment. The PTO constraint is taken from [

In order to simulate a realistic PTO system, two other parameters were identified besides the saturation function applied in Case 3:

Case 4—End-Stop of the PTO actuator. Another external force is added in the force summation in the right hand side of

Case 5—PTO delay [_{u}

In both cases, the PTO was saturated as in Case 3. The sensitivity analysis of the WEC model response with respect to these two additional cases shows that the PTO constraint is the most critical parameter. In fact, in Case 4, the End-Stop system is rarely exerted as a result of the non-optimality of the controller induced by the limited PTO capacity. Further, in Case 5, the PTO delay is small compared with the system dynamic. Therefore, the induced phase shift accounts for only a small reduction of the power performances. Since the contribution of Cases 4 and 5 is below 1% in both AEP and cross section area, these cases will be omitted in the following.

The controller schemes presented in Section 2.2 have been introduced into the WEC model in order to estimate its average power performances. As shown in

P—proportional or passive controller (Section 2.2.3.);

PI—proportional-integral or spring-damper controller (Section 2.2.4.);

PIDc—proportional-integral-derivative with memory compensation or sub-optimal controller (Section 2.2.5.);

MEC—maximum energy controller (Section 2.2.1.);

MPC—model predictive controller (Section 2.2.2.).

The MEC is considered a benchmark for Cases 1 and 2 since it defines the theoretical upper bound of the absorbed energy. For Case 3, the MPC will be considered as reference point; the MEC is indeed unable to deal with constraints. The remaining controllers are then used to assess the effect of control aggressiveness into the absorbed power.

The MPC cannot be considered an applicable controller in the present formulation, because in the simulations the control algorithm was assumed to have perfect knowledge of the future excitation moment over the full prediction horizon. Therefore, uncertainties related to the excitation moment prediction are discarded with a reasonable increment of the power performance. Furthermore, the current state of the plant is assumed to be perfectly known. Hence, the MPC results are considered an upper bound in terms of the objective function rather than a practicable solution in the present formulation. The implementation of the different controllers is described in detail in the following.

The MEC is implemented using the velocity tracking logic where the velocity signal is generated in two steps: first, the actual wave frequency and amplitude are observed using the Hilbert transform [

The MPC model is based on the state-space approximation of the WEC model summarised in _{p}_{p}_{p}_{p}_{p}_{p}

Controllers P, PI and PIDc have already been defined up to the establishment of the control parameters (_{c}_{c}_{c}

A special attention needs to be given to the PI and PIDc controllers when the PTO moment is constrained. In order to reduce the error accumulated in the integrator introduced by the abrupt saturation, an integral windup with back-calculation scheme was used [

The absolute performance of the different controllers is quantified using the power matrix, which defines the average power production for each simulated sea-state. _{p}_{n}_{p}_{n}

Moreover, when the PTO moment is bounded, most of the energy in the high energetic sea-states cannot be absorbed (

Using the power results of the Wavestar WEC and the scatter diagram shown in

The energy performance of the P controller can be increased reasonably by a factor of two when an active controller is adopted into the WEC model. The five-fold increase of the unconstrained case is mainly linked to the unrealistic motion amplitude achieved in the high-energetic sea states. In those conditions, the linear theory assumptions are heavily violated, and the simulation results become unreliable. The viscous dissipative term accounts for a global performance reduction of up to 15% if an active controller is used. For the passive controller, the small body velocity induced by the non-resonant WEC makes the turbulence effect negligible [

Both active controllers used (PI and PIDc) show similar results. In addition, their operation is close to the benchmark capability. Given that the performance gap does not seem to be a function of the applied constraint, it is reasonable to assume that both PI and PIDc controllers have a flat power performance compared with the optimal controller. This trend is also visualised in

For the fatigue assessment, the focus is put on different components of the floater arm. Due to the fact that the floater can be taken out of the water during storm events, extreme events are of minor importance for the structural design of the floaters. Critical subcomponents whose failure may lead to an overall breakdown are bolted as well as welded connections. Therefore, the focus here is on the two welded details and one bolted connection shown in

_{1} and _{2}, which are equal to the negative inverse slope of the SN curves and log(_{1}) as well as log(_{2}), which show the intercept of log(

The life time _{L}

_{1},_{i}_{i}_{2},_{i}

In the following, the discussion of the results for the specific case study elaborated in Section 3 is given. It is important to bear in mind that whilst the proposed method can be extended, prior to modification, to other WECs of the activated body type, any extrapolation of the results of Section 3, for any other device, should be considered inaccurate. Different conclusion should also be expected if different connection points are chosen for a given WEC.

Based on the cost factor, CF (see

A sensitivity analysis has been carried out in order to investigate the impact of detuned control parameters on AEP and cross section area. _{c}_{c}

As expected, the AEP is reduced in all cases, while the fatigue loads acting on the structure can be reduced by choosing the right detuning combination. From an economic point of view, only those cases where the reduction of power output comes along with reduced fatigue loads are of potential interest.

This is clear in the case of the P controller where the amount of damping is positively correlated to the structural load, and only the reduced damping coefficient (0.8_{c}

For the PI controller, _{c}_{c}

In addition, there is a clear lack of specificity in the wave energy sector which makes finding an economical optimum point an ill-conditioned problem.

Although the proposed methodology presents a novel and alternative description of the power optimised WEC taking into account structural fatigue several assumption has been adopted. The economical model assumes a linear dependence between structural dimensions and cost, while for a more comprehensive analysis a detailed cost model of the system should be created. Other factors as systems' complexity, operational over rated generator power, commissioning, decommissioning and operational and maintenance costs, PTO efficiency, downtime periods, components reliability,

Given that the exploitation of the potential energy embedded in ocean waves can play an important role in the future energy mix, so far much effort has been put in finding the best energy configuration for a WEC. But other effects like the influence of the control strategy on the structural fatigue should not be disregarded. This might lead to designs of WECs that are not cost effective.

The methodology presented in this article aims at selecting a controller that balances energy yield and structural fatigue in an economic sense. To this end, well known control strategies, standard fatigue calculations and a simple cost model are brought together (

A case study using a numerical model of the Wavestar WEC including a non-ideal PTO has been carried out to exemplarily demonstrate the method (Section 3). Even if it is applied for a specific case study, it can easily be adopted for other WEC of the wave activated body type. In contrast, the results of the case study should only very cautiously be generalised, because a specific device at a specific location has been considered.

These particular results are summarised as follows. Both energy as well as fatigue is governed by the constraint of the PTO moment for all considered controllers rather than by the position constraint or the PTO delay (Section 3.2). Two main technical conclusions can be drawn from the comparison of the passive controller (P controller) with the two active sub-optimal controllers (PI and PIDc controller) in the case with constrained PTO moment (

harvest 80% of the maximum achievable energy and twice as much energy as the passive controller, and;

need roughly 50% more material at the three considered structural details in order to reach the same life-time as the passive controller.

Feeding the proposed economic model with these results reveals that the best choice for the selected WEC is an active sub-optimal controller (

Altogether, this paper indicates the importance of balancing power output and structural fatigue for the choice of an economically optimal controller. A number of other factors such as operational vs. rated generator power or system complexity have not been considered here. Future work should aim to include these other factors, and more sophisticated cost models have to be developed. This would provide a basis for a certification guideline for WEC. Further steps could also be to implement the overall cost consideration already in the optimisation of the control algorithm.

The authors gratefully acknowledge the financial support from the Danish Council for Strategic Research under the Programme Commission on Sustainable Energy and Environment (Contract 09-067257, Structural Design of Wave Energy Devices) which made this work possible.

The authors declare no conflicts of interest.

Work flow diagram of the proposed methodology. Subscript “

Hydrodynamic frequency response functions for the Wavestar model: (

AEP of the WEC as a function of the applied controller for each simulated case. Control type proportional (P) controller, proportional-integral (PI) controller, proportional-integral-derivative with memory compensation (PID) controller, maximum energy controller (MEC) and model predictive control (MPC).

Comparison of P and PI controller power performance normalised by the MPC results, for constrained case (Case 3): (

Comparison of P and PI controller power performance normalised by the MEC results, for unconstrained case (Case 1): (

Position of the two welded details as well as the bolted connection between the arm and the power take off (PTO) of the Wavestar device.

Considered SN curves for the Wavestar case study. The SN curves are taken from [_{1}, _{2}, _{1} and K_{2} (see

Design cross section areas for the three different connections defined in

Expected number of cycles for a given load range during life-time of 20 years: (

Comparison of AEP (blue) and Cross Section Area (green) between different control strategies, for (

Comparison of Cost Factor CF between different control strategies and

Variation of the cost factor with respect to the percentage of the cost depending on the control strategy for the P controller with reduced damping coefficient 0.8 _{c}

Comparison of Cost Factor _{opt}_{opt}

Relative occurrence of different wave states from six years, buoy measurements ([_{m}_{0}] = m, [_{P}_{p}_{m}_{0}.

_{m}_{0}/_{P} |
||||||||
---|---|---|---|---|---|---|---|---|

0.25 | - | - | - | 0.04 | 0.04 | 0.02 | 0.01 | - |

0.75 | - | - | - | 0.07 | 0.17 | 0.11 | 0.05 | 0.01 |

1.25 | - | - | - | - | 0.06 | 0.11 | 0.05 | 0.01 |

1.75 | - | - | - | - | - | 0.06 | 0.05 | 0.02 |

2.25 | - | - | - | - | - | 0.01 | 0.05 | 0.02 |

2.75 | - | - | - | - | - | - | 0.01 | 0.02 |

3.25 | - | - | - | - | - | - | - | 0.01 |

Model parameters for the Wavestar WEC [

Moment of Inertia | _{st} |
2.45 × 10^{6} |
kg·m^{2} |

Hydrostatic Stiffness | _{hy} |
14.0 × 10^{6} |
N·m/rad |

Maximum exerted moment | _{max} |
1.0 × 10^{6} |
N·m |

Drag Coefficient | _{D} |
0.25 | - |

PTO stroke | _{pto} |
2.0 | m |

Natural Period in Pitch | _{n} |
∼3.5 | s |

| |||

| |||

Added mass at infinity frequency | _{∞} |
1.32 × 10^{6} |
kg·m^{2} |

Radiation Moment TF numerator | _{RAD} |
[4.93, 1.08] × 10^{6} |
- |

Radiation Moment TF denominator | _{RAD} |
[1, 2.56, 5.16] | - |

Excitation Moment TF numerator | _{EX} |
[5.4 × 10^{10}, 2.7 × 10^{12}] |
- |

Excitation Moment TF denominator | _{EX} |
[3.6 × 10^{4}, 3.9 × 10^{5}, 1.5 × 10^{6} 2.6 × 10^{6}, 1.6 × 10^{6}] |
- |

Parameters of Stress-Number (SN) curves shown in

_{1} (-) |
log(_{1}) (-) |
_{2} (-) |
log(_{2}) (-) | |
---|---|---|---|---|

Weld 1 | 3 | 11.455 | 5 | 15.091 |

Weld 2 | 3 | 11.455 | 5 | 15.091 |

Bolt 1 | 5 | 16.301 | - | - |

Results of the sensitivity analysis for P and PI controller. Variation of the cross section area (red text) of Weld 1 and annual energy product (blue text), normalised by the controller where AEP is maximised.

| ||||
---|---|---|---|---|

_{c} |
_{c} |
_{c} | ||

1.2_{c} |
+5% | +1% | −3% | −10% |

−1% | −3% | −3% | −8% | |

_{c} |
0% | +4% | 0% | −8% |

0% | −2% | 0% | −5% | |

0.8 _{c} |
−6% | −5% | +3% | −5% |

−1% | −5% | −1% | −4% |