Modelling and analysis of inertia self‐tuning phase control strategy for a floating multi‐body wave energy converter

Funding information National Natural Science Foundation of China, Grant/Award Number: 51779104; Natural Science Foundation of Fujian Province, China, Grant/Award Number: 2020J01694; Foreign Cooperation Program of Fujian Province, China, Grant/Award Number: 2020I0021 Abstract Wave energy converters (WECs) exhibit higher energy capture capacity when the wave frequency is close to the nature frequency of the device. A possible way to enhance energy capture efficiency is to modify the nature frequency of a WEC to match the incoming wave frequency. This paper presents an inertia self-tuning phase control strategy which can optimize the wave energy capture efficiency of a floating multi-body (FMB) WEC by regulating automatically the total mass of the capture energy part according to the incoming wave conditions. A variable-frequency inertia self-tuning control system is designed to conduct the proposed control strategy. The mathematical model of rotational motion of the FMB WEC is built and the dynamic performance of the control system is analysed in detail. Simulation results illustrate the proposed control strategy can achieve a phase optimization and hence significantly improve wave energy extraction of the FMB WEC in both regular and irregular waves. For regular waves, the maximum capture width ratio)CWR( under the control strategy is 7.3 times higher than that no control strategy is used. For irregular waves, the maximum CWR under the control strategy is 4.4 times that without the control strategy at the same wave frequency.


INTRODUCTION
Oceans cover two thirds of the earth's surface and large amount of ocean energy is contained within the motions of the waves. Energy extraction and conversion from ocean waves are now gaining more and more attention as the years pass by. With the increasing attention to harnessing the wave energy, many control and optimal technologies have been proposed and developed, aiming at increasing the wave energy conversion efficiency [1]. It is well-known that optimizing the geometric structure or shape of a WEC can achieve higher energy capture efficiency. A famous case is the nodding Duck WEC [2]. It was reported that the optimized hydrodynamic structure made the wave energy capture efficiency of the nodding Duck WEC reach 90% under This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. IET Renewable Power Generation published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology ideal operating conditions [3]. The sharp eagle WEC proposed by the Guangzhou Institute of Energy conversion in China has an excellent wave energy absorbing capacity because the shape of the capturing energy buoy is designed to be like an eagle's beak [4,5]. SEAREV WEC has been in development for many years in France. Three main generations of the technology have been defined and studied in detail since 2002. It has been proved that each new generation brought remarkable improvements over the previous generation because of the shape optimization [6]. Goggins and Finnegan [7] investigated the effects about optimizing the draft and the diameter of a double-body point absorbing WEC. The results showed that the optimization of the geometry parameters could clearly help to maximize the WEC's efficiency when considering specific wave conditions.
The purpose of optimization design is for matching the nature frequency of a WEC with the incoming wave frequency [8], that is the WEC should operate at the optimal phase condition (resonance condition) for improving wave energy capture capacity. When the optimal phase condition is satisfied, the heave motion velocity of the capture energy part is in phase with the wave excitation force. However, as real waves are not singlefrequency and always vary with the time, the static shape and geometry optimization generally taking place in the design and manufacturing stage may get low energy conversion efficiency under the real sea conditions. Advanced dynamic control technologies must be employed to realize the automatic tuning of the WEC's nature frequency according to the sea states. Many phase control technologies for achieving optimal phase condition have been proposed and studied widely since 1970s [1,9].
Latching control is one of the phase control technologies. It has been validated by some numerical analyses and physical experiments [10,11]. In the latching control strategy, the control unit latches the capture energy device when its motion velocity reaches zero or is very small. In the process, the WEC will be halted for a certain period (i.e. the latching duration). When the WEC is released at the instant of de-latching, it is supposed to heave, and the heaving velocity is in-phase with the wave excitation force [10]. How to decide the latching duration is considered to be one of the challenges due to the requirement of detailed future wave information [12]. Another challenge for latching control strategy is the design of a mechanism or method to carry out the latching [13]. Besides, the optimal phase condition just is partially fulfilled in the latching control. Thus, latching control is often called the sub-optimal control [10,14].
For satisfying the full phase control, the WECs must have the ability to regulate the internal dynamic parameters such as the elastic coefficient, the wet surface area or the inertia term of the WECs. In [15], a reactive control strategy has been proposed which can adjust the Power Take-off (PTO) elastic coefficient to counteract the intrinsic reactance of the WEC. In the reactive control strategy, a part of the extracted energy must be effectively fed back into the waves through the PTO and the elastic coefficient of the PTO must take negative value [8]. Tuning the wetted surface area can also fulfil phase optimum because the hydrodynamic elastic coefficient of the energy absorber is changed with the incident waves. As indicated in [16], the wetted surface area of an oscillating surge WEC is modified according to the real-time the sea states. That matches the natural period of the device with the peak excitation period of the sea spectrum to improve energy conversion efficiency. In [17], a mechanical controller is designed to tune the buoy-shaft titling angle so that the wet surface of hemispherical buoy can be adjusted. The flume experimental results indicate that the overall efficiency of the system can be improved by 15%.
Another approach utilized to satisfy the optimal phase condition is to tune the inertia of the WEC. In [18], the energy capture of a bottom-hinged pitching point absorber is improved by modifying the inertia of the device. The experimental results show that this method of inertia modification could result in an increase of capture factor by 70-100% for larger regular waves when compared to a constant inertia configuration. An analytical phase optimal procedure is presented for a double-body point absorbing WEC in [19]. Through invoking resonant conditions in the wave energy device, the authors illustrate how the phase optimal method has significant influence on the power capture characteristics of the WEC device. In [20], a fully submerged mass is added to a point-absorption WEC to properly shift the device's heave natural frequency and gain resonance with the most energetic waves to maximize the energy absorption. In [21], the inertia of a floating absorber is regulated for satisfying the optimal phase condition by changing the position of a sliding mass on the lever arm. The amount of extracted energy from the controlled system increases from about 1.1 to 2.9 times in irregular waves.
In this study, a novel inertia self-tuning phase control technology is proposed for maximizing the wave energy absorption of the FMB WEC which was validated by the numerical model and the real sea test in the previous works [22,23]. A close-loop phase controller is designed to adjust the inertia of the energy capture part according to the sea states. The natural frequency of the FMB WEC is then altered and reaches automatically a value closer to the frequency of the incident wave so that the phase condition is satisfied. Simulation results in regular and irregular waves illustrate the performance and feasibility of the proposed control strategy. Figure 1 shows the schematic of the FMB WEC which mainly consists of a floating platform, multiple oscillating buoys, actuating arms and PTOs. As shown in Figure 1, the oscillating buoys are distributed symmetrically around the floating platform and are linked with the floating platform through the actuating arms. For each actuating arm, there are three hinge axes designated as A, B and C, respectively. The hinge axis A allows rotation of the oscillating buoy respect with the floating platform; the hinge axis B links the PTO to the actuating arm and permits relative rotation of its anchor points, keeping the PTO perpendicular to the actuating arm; the hinge axis C joins the PTO to the supporting frame fixed on the floating platform. A damping plate is mounted under the floating platform. Under the operating mode, the damping plate is submerged into the seawater, reducing the heave amplitude of the floating platform and improving the stability of the whole system. Due to the wave forces, the relative heave motion between the oscillating buoy and the floating platform compresses or stretches the PTO on the actuating arm, thus converting wave energy to hydraulic energy. A hydrodynamic mathematical model of the FMB WEC was built in [22]. Numerical analysis indicates that the dynamical property and energy conversion efficiency of the FMB WEC are related not only to the sea states but also to the geometric dimensions and the PTO coefficients of the FMB WEC. In [23], a 10 kW sea test prototype was developed and manufactured using a grant from the State Ocean Administration of China. A three-month real sea trial was carried out in the Taiwan Strait, China. The experimental results confirmed that the FMB WEC had higher energy capture capacity when the wave periods were 3-5 s and the significant wave heights were 0.6-1.3 m. The increase in number of oscillating buoys used around the floating platform can increase the absorption of wave energy and improve the power output quality. Both the numerical analysis and sea test results proved that the heave amplitude of the floating platform is much less than the oscillating buoys because of their differences such as the hydrodynamic coefficient, mass property, geometric shape and so on. Therefore, in this study we assume that the floating platform is motionless for simplifying the analysis of the proposed control strategy. Figure 2 shows the prototype of the FMB WEC in real sea condition. Figure 2(a) is off-working state, Figure 2(b) is working state.

MATHEMATICAL MODELING OF THE FMB WEC
According to the theory of fixed-point rotation for rigid bodies, the equilibrium equation of the moments around the hinge axis A of the FMB WEC can be expressed as: where J is the moment of inertia, J = mL 2 , m is the mass of each oscillating buoy, L is the length of each actuating arm; θ is the rotation angle displacement of the actuating arm; T hyd is the hydrodynamic torque, T hyd = F hyd L, F hyd refers to the hydrodynamic force produced by the incoming wave; T pto is the PTO torque, T pto = F pto L 1 , F pto is the force produced by the PTO; L 1 is the distance between the hinge A and the PTO. The hydrodynamic force, F hyd , can be calculated as [15], where F e is the wave excitation force (due to the incident waves), F r is the wave radiation force (due to the buoy's motion) and F b is the hydrostatic restoring force (connected with the instantaneous position of the buoy with respect to the undisturbed free surface) [8]. F r can be decomposed into components in phase with the buoy's acceleration and velocity [8,15] so that where x is the heave displacement of the buoy, m a is the added mass coefficient and c w is the radiation damping coefficient. Both the added mass coefficient and the radiation damping    [10,15]. As shown in Figures 3 and 4, it can be seen that the buoy has a minimum added mass at wave frequency ω = 1.96 rad/s, and its added mass approached a constant value at large frequencies. The radiation damping coefficient has a peak value at ω = 1.51 rad/s. After reaching the peak value, the radiation coefficient begins to decrease as the wave frequency increasing.
When the heave motion of the buoy is small, that means the angle displacement θ of the actuating arm is small [28], so that Substituting Equation (4) into Equation (3), we can get And the hydrostatic restoring force F b can be calculated as [15], Where is the seawater density, g is the acceleration due to gravity, A w is the cross-sectional area of the buoy by the unperturbed free surface plane, A w = πR b 2 , R b is the radius of each buoy.
If a full linear PTO is applied in further analysis, the PTO force can be expressed as: where F c is the PTO damping force and F s is the PTO elastic force which are proportional to the velocity and the displacement of the buoy, respectively [24]. The expressions are: where c p and k is the PTO damping coefficient and elastic coefficient, respectively.
Substituting Equation (2), (5)-(9) into Equations (1), the equilibrium equation of the moments becomes: The corresponding frequency-domain equation of Equation (10) has a form as where ω is the incoming wave frequency. According to Equation (11), the solution of the angle displacement of the actuating arm can be obtained as: Equation (12) can be also re-written in terms of angle velocity, Looking at the transfer of power in the FMB WEC, the time averaged power captured by the PTO is calculated as: The average capture power can be obtained in the analytical form as: where |F e (i )| is the module of the wave excitation force.
According to Equation (16), the nature frequency n of the FMB WEC can be obtained as: It is obvious that the angular velocity of the actuating arm is in phase with the wave excitation force and the optimal phase condition is satisfied when the incident wave frequency equals to the nature frequency of the FMB WEC. At the same time, it can be found we can change the mass of each buoy to ensure Equation (16) when the incident wave frequency varies.

4.1
Control system structure Figure 5 shows the structure of the proposed inertia self-tuning control system which is composed of a close-loop phase controller, a frequency converter, a variable-frequency injection pump, a variable-frequency suction pump and a water tank. It should be noticed that the altitude difference between the water tank and the buoys is about 7 m. According to the incoming wave frequency, the close-loop phase controller gives the control signal to change the inertia (i.e. the mass) of the oscillating buoy. For increasing the inertia, the injection pump is started and the fresh water in the water tanker is filled into the oscillating buoy. For decreasing the inertia, the suction pump is started and the fresh water is discharged from the oscillating buoy and returns to the water tank. The frequency converter gives the corresponding frequency signal to adjust the rotate speed of the variable-frequency pumps so that the water flow rate in the pipes can be controlled by the control system.

Desired increment of the buoy mass
Since the total mass of each oscillating buoy is controlled to achieve the optimal phase condition, the corresponding nature frequency of the FMB WEC will vary with the buoy mass. The relation between the new nature frequency of the WEC and the increment of the buoy mass can be described as: where Δm represents the increment of the buoy mass. According to Equation (18), when the incident wave frequency is ω, the desired increment of the mass Δm d should be: The mass of the buoy itself, m, is constant and its added mass, m a , is frequency-dependent. Therefore, according to Equation (19), the desired increment of the buoy mass can be obtained if the incident wave frequency is given.

Close-loop phase controller
In the inertia self-tuning control system, a close-loop phase controller is used to obtain the input frequency signal of the variable-frequency pumps according to the incident wave conditions. Figure 6 is the block diagram of the phase controller.
It is assumed that the incident wave frequency has been measured and the desired value of Δm d is determined according to Equation (19). Then, an error signal e(t) is defined as the difference between the actual Δm and the desired Δm d , and the input frequency signal f i (t) obtained by a proportional integral differentiation (PID) controller is sent to the injection/suction pumps. The pumps tune the total mass of the oscillating buoy by water suction or water drainage which consequently matches the natural frequency of the FMB WEC with the incoming wave frequency. The adjustment of the proportional gains k p , the integral gain k i and the differential gain k d is based on pole placement method [25].

Mathematical model of the pumps
The mathematical models of the injection pump and the suction pump are built as a second order transfer function.
where ΔM(s) and F i (s) are the Laplace transforms of the increment of the buoy mass and the pumps' input frequency signal, respectively. k 1 is the scale factor and T is the time constant. The instantaneous power P c consumed by the pumps can be calculated as: where f i is the input frequency of the pumps, f r and P r are the rated input frequency and rated power of the pumps, respectively.

RESULTS AND ANALYSIS
In this section, the dynamic performances of the inertia selftuning phase control strategy are investigated in regular and irregular wave conditions. The related hydrodynamic coefficients of the FMB WEC used in the simulation analysis are obtained via ANSIS AQWA software. Tables 1-4 show the values of the other related parameters. In the hydrodynamic numerical calculation, the geometry of the FMB WEC model is

Variation of buoy mass
Due to the limitation of the internal volume of the buoy, the maximum mass of the fresh water that can be filled into the   buoy is about 18.0t. Therefore, the variation range of the total mass of each buoy is from 2.0t (the mass of each buoy itself) to 20.0t. Figure 8 depicts the relation between the desired increment of the mass and the incident wave frequency when the initial value of the buoy mass is set to 11.0t (i.e. 9.0t fresh water is filled into the buoy in the initial state). As shown in Figure 8, Δm d decreases as the increase of the incident wave frequency when the incident wave frequency is between 1.78 and 3.62 rad/s. When the wave frequency is less than 2.32 rad/s, Δm d is positive number that means the water in the water tank needs to be filled into the buoy. And when the wave frequency is larger than 2.32 rad/s, Δm d is negative number that means the water needs to be discharged from the buoy. If the wave frequency is less than 1.78 rad/s, Δm d is constant and equals to 9.0t. In this case, the total mass of the buoy is maximized and always equals to 20.0t. If the wave frequency is larger than 3.62 rad/s, Δm d equals to −9.0t. The total mass of the buoy is minimized and always equals to 2.0t.

Regular and irregular waves
In the study, a regular wave with a wave height of H = 1.0 m and a frequency of ω = 2.1 rad/s is adopted to analyse the dynamic performance of the proposed control strategy as illustrated in Figure 10(a). For irregular wave, it can be represented Pierson-Moskowitz wave spectrum when the significant wave height is 1.0 m and peak frequency is 2.1 rad/s as a superposition of a set of regular waves. A conventional method is to describe an irregular wave by the wave spectrum with the corresponding significant wave height and peak period. The wave spectrum is a statistical feature in describing the distribution of the wave components. The conventional wave spectra used for wave energy extraction generally include the Pierson-Moskowitz spectrum, the Bretschneider spectrum and the JONSWAP spectrum. In this paper, we utilizes the Pierson-Moskowitz spectrum to describe the irregular wave which is defined as [26]: with ω p = 2π/T p , where T p is the peak period, ω p is the peak frequency and H s is the significant wave height. From the wave spectrum, a finite number of sinusoidal waves can be created. Each individual wave component is created with its own amplitude and frequency characterized by the spectrum. Each sinusoidal wave is assigned with a random phase and the wave elevation time series is generated as a sum of the individual components according to: where ω i and φ i are the frequency and random phase component of the ith wave, and Δω is the frequency band calculated from: where ω max is the maximum frequency of the spectrum and n is the number of wave components. Figure 9 presents an example of a Pierson-Moskowitz spectrum and the corresponding H s and ω p are set as 1 m and 2.1 rad/s for the comparison with the regular wave in the following analysis. Figure 10(b) is the wave elevation time series of the irregular wave when H s = 1.0 m and ω p = 2.1 rad/s. Figure 11 shows the comparison of the angular displacements without and with the inertia self-tuning control strategy in the regular wave when H = 1.0 m and ω = 2.1 rad/s. When no control strategy is applied to the FMB WEC, the angular displacement amplitude remains unchanged and is about 0.084. Under the control strategy, the angular displacement amplitude gradually increases with the simulation time and finally reaches to 0.126 as shown in Figure 11(b). The angular displacement amplitude has been significantly increased to about 1.5 times by the inertia self-tuning control strategy in the regular wave. As shown in Figure 12, the power captured by the buoy with the control strategy also increases obviously when compared with the absorbed power by the uncontrolled buoy. As a result of the inertia self-tuning control strategy, the average capture power increases from 5.1 to 11.3 kW, an increase by 122%. Figure 13 shows the variations of the input frequency of the injection pump and the power consumed by the injection pump for filling the water into the buoy. In the first 70 s of the simulation process, the input frequency and the consumed power equal to the rated frequency and the rated power of the injection pump, respectively. And then, the input frequency and the consumed power gradually decrease under the action of the phase controller. After about 140 s, the consumed power reduces to zero. As indicated in Figure 14, in accordance with the change of the consumed power of the pump, the increment of the buoy mass increases rapidly in the first 70 s. And at 140 s, the increment of the buoy mass reaches to the desired value, 3.41t, namely that the corresponding total mass of the buoy is 14.41t at this moment.

Dynamic performance in irregular wave
One must note that the implementation of control strategies in irregular waves requires the prediction of the incoming waves. Nevertheless, the topic of wave prediction is beyond the scope of this paper. It is assumed that the sea state can be predicted for a specific time period in the near future. The power consumed by the pump and the desired increment of the buoy mass in irregular waves are calculated according to the peak frequency ω p . When the peak frequency of the irregular wave is 2.1 rad/s, the variation tendencies of the control frequency, the consumed power of the pumps and the desired buoy mass increment are the same with that shown in Figures 13 and 14. Figure 15 shows the comparison of angular displacement with and without control strategy in the irregular wave of H s = 1.0 m and ω p = 2.1 rad/s. In the initial stage of the simulation, the angular displacements with and without control are roughly equal. In Figure 15(a), the average amplitude of the angular displacement is only 0.025 when no control strategy is applied in the FMB WEC. But under the control strategy, the angular displacement amplitude becomes larger with the increase of the total buoy mass as shown in Figure 15(b), and the average amplitude of the angular displacement reaches   Figure 16 shows the comparison of capture power with and without control strategy in the irregular wave of H s = 1.0 m and ω p = 2.1 rad/s. It is very obvious that the absorbed power under the control strategy is larger than that without the con-trol strategy. If the FMB WEC is left uncontrolled, the average capture power during the simulation process is 1.35 kW which is only 47% of the average capture power (2.86 kW) if the proposed control strategy is applied. The maximum instantaneous capture power without and with control are 5.43 and 11.07 kW, respectively.

Energy conversion efficiency
In this paper, the wave energy conversion efficiency is quantified in terms of capture width ratio (CWR) which is a key indicator utilized to estimate the energy capture capacity of a WEC [28]. In regular waves, the average wave power P r per unit width can be calculated by this formula [15,27]: The CWR of the FMB WEC in regular waves can be defined as: For irregular waves, the average wave powerP ir per unit width can be calculated by this formula [15,26]: The CWR of the FMB WEC in irregular waves is: Figure 17 shows the comparison of CWRs at different wave frequencies without and with control strategy when the wave heights equal to 1 m. It can be seen the inertia self-tuning control strategy obviously enhances the CWRs and widens the response frequency band whether in regular waves or in irregular waves. If the FMB WEC is left uncontrolled, the maximum CWR is only 0.52 in regular waves and 0.27 in irregular waves. The wave frequency corresponding to the maximum CWR is 2.32 rad/s (i.e. the initial nature frequency). If the proposed control strategy is applied to the FMB WEC, the wave frequency corresponding to the maximum CWR shifts left to 1.78 rad/s and the maximum value of the CWRs rises to 3.14 in regular waves and 1.23 in irregular waves, respectively. For regular waves, the maximum CWR under the control strategy is 7.3 times higher than that no control strategy is used when ω = 1.78 rad/s. However, for irregular waves, the maximum CWRs under the control strategy is 4.4 times that without the control strategy at the same wave frequency. It can be noted that the increase of the CWR is higher in regular waves than in irregular waves. Figure 18 depicts the comparison between the extra energy extracted from the regular and irregular waves and the energy consumed by the pumps under the proposed control strategy when the wave heights are 1.0 m and the simulated time is set to 3600 s. The consumed energy equals to zero at the initial nature frequency (2.32 rad/s). But away from the initial nature frequency, the energy consumed by the control strategy increases on both sides for longer and shorter waves. When the incident wave frequency is greater than 3.5 rad/s or less than 1.8 rad/s, the consumed energy reaches the maximum value (3876 kJ). For the extra energy extracted from the waves under the control strategy, it can be seen that the proposed control strategy can extracted more energy in regular waves than in irregular waves. Take an example of a wave of ω/ω p = 1.8 rad/s (wave height H/H s = 1.0 m), the extra energy extracted from the regular wave is about 5.5 times that from the irregular wave. Moreover, it must be noted that the inertia self-tuning control strategy is more efficient in the lower wave frequencies (i.e. longer wave periods). When the wave frequency is greater than 2.4 rad/s, the consumed energy by the control strategy is more than the extra energy extracted from the waves because the wave energy per unit width is too small in such situation.

CONCLUSIONS
In this study, an inertia self-tuning phase control strategy was proposed to increase the wave energy absorption of the FMB WEC. The proposed control strategy is based on the adjustment of the buoy mass to change the inertia of capture energy part. Then, the nature frequency of the FMB WEC reaches a value closer to the incident wave frequency and the phase

FIGURE 18
Comparison between extra energy extracted from regular and irregular waves under the control strategy and energy consumed by the control strategy when the simulated time is 3600 s optimization condition is satisfied. An inertia control system and a phase controller were designed to perform the proposed control strategy. The dynamic performance and energy conversion efficiency of the proposed control strategy were analysed for both regular and irregular waves. In the case of regular waves, the gain in the level of absorbed energy is higher, as the waves are characterized by a single frequency. Whether in regular waves or irregular waves, the proposed control strategy can improve significantly wave energy capture capacity of the FMB WEC. On the other hand, it should be noted that the proposed control strategy needs to consume some energy to change the inertia of the capture energy part. The consumed energy cannot be neglected especially when the wave frequency is very high.