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

2021; Institution of Engineering and Technology; Volume: 15; Issue: 14 Linguagem: Inglês

10.1049/rpg2.12208

1752-1424

Shaohui Yang, Hu Chen, Zhifei Ji, Hui Li, Xiang Xu,

HVDC Systems and Fault Protection

IET Renewable Power GenerationVolume 15, Issue 14 p. 3126-3137 ORIGINAL RESEARCH PAPEROpen Access Modelling and analysis of inertia self-tuning phase control strategy for a floating multi-body wave energy converter Shaohui Yang, Shaohui Yang orcid.org/0000-0002-7076-8598 College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Energy Cleaning Utilization and Development of Fujian Province, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorHu Chen, Corresponding Author Hu Chen 13163996278g@163.com College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, China Correspondence Hu Chen, College of Mechanical and Energy Engineering, Jimei University, Xiamen City, Fujian Province, 361021, China. Email: 13163996278g@163.comSearch for more papers by this authorZhifei Ji, Zhifei Ji College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorHui Li, Hui Li College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorXu Xiang, Xu Xiang Norwegian Public Roads Administration, Complex constructions Brynsengfaret 6a, Oslo, NorwaySearch for more papers by this author Shaohui Yang, Shaohui Yang orcid.org/0000-0002-7076-8598 College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Energy Cleaning Utilization and Development of Fujian Province, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorHu Chen, Corresponding Author Hu Chen 13163996278g@163.com College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, China Correspondence Hu Chen, College of Mechanical and Energy Engineering, Jimei University, Xiamen City, Fujian Province, 361021, China. Email: 13163996278g@163.comSearch for more papers by this authorZhifei Ji, Zhifei Ji College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorHui Li, Hui Li College of Mechanical and Energy Engineering, Jimei University, Xiamen, China Key Laboratory of Ocean Renewable Energy Equipment, Fujian Province University, Xiamen, ChinaSearch for more papers by this authorXu Xiang, Xu Xiang Norwegian Public Roads Administration, Complex constructions Brynsengfaret 6a, Oslo, NorwaySearch for more papers by this author First published: 15 May 2021 https://doi.org/10.1049/rpg2.12208Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat 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. 1 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 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 single-frequency 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. 2 CONCEPT OF THE FMB WEC 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. FIGURE 1Open in figure viewerPowerPoint Schematic of the FMB WEC. (a) Side view. (b) Axonometric view. (c) Top view 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. FIGURE 2Open in figure viewerPowerPoint Prototype of FMB WEC in real sea condition. (a) Off-working state. (b) Working state 3 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: J θ ̈ = T h y d − T p t o (1)where J is the moment of inertia, J = mL2, 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; Thyd is the hydrodynamic torque, Thyd = FhydL, Fhyd refers to the hydrodynamic force produced by the incoming wave; Tpto is the PTO torque, Tpto = FptoL1, Fpto is the force produced by the PTO; L1 is the distance between the hinge A and the PTO. The hydrodynamic force, Fhyd, can be calculated as [15], F h y d = F e − F r − F b (2)where Fe is the wave excitation force (due to the incident waves), Fr is the wave radiation force (due to the buoy's motion) and Fb is the hydrostatic restoring force (connected with the instantaneous position of the buoy with respect to the undisturbed free surface) [8]. Fr can be decomposed into components in phase with the buoy's acceleration and velocity [8, 15] so that F r = m a x ̈ + c w x ̇ (3)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 coefficient are the functions of the incident wave frequency [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. FIGURE 3Open in figure viewerPowerPoint Added mass coefficient of the buoy FIGURE 4Open in figure viewerPowerPoint Radiation damping coefficient of the buoy When the heave motion of the buoy is small, that means the angle displacement θ of the actuating arm is small [28], so that x ≈ L θ (4) Substituting Equation (4) into Equation (3), we can get F r = m a L θ ̈ + c w L θ ̇ (5) And the hydrostatic restoring force Fb can be calculated as [15], F b = ρ g A w x = ρ g A w L θ (6)Where ρ 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, Aw = πRb2, Rb is the radius of each buoy. If a full linear PTO is applied in further analysis, the PTO force can be expressed as: F p t o = F c + F s (7)where Fc is the PTO damping force and Fs is the PTO elastic force which are proportional to the velocity and the displacement of the buoy, respectively [24]. The expressions are: F c = c p L 1 L x ̇ = c p L 1 θ ̇ (8) F s = k L 1 L x = k L 1 θ (9)where cp 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: m L 2 θ ̈ = F e L − m a L 2 θ ̈ − c w L 2 θ ̇ − ρ g A w L 2 θ − c p L 1 2 θ ̇ − k L 1 2 θ (10) The corresponding frequency-domain equation of Equation (10) has a form as − ω 2 ( m + m a ) + i ω c w + c p L 1 L 2 + ρ g A w + k L 1 L 2 θ i ω = F e ( i ω ) L (11)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: θ ( i ω ) = F e ( i ω ) / L [ ρ g A w + k ( L 1 / L ) 2 ] − ω 2 ( m + m a ) + i ω [ c w + c p ( L 1 / L ) 2 ] (12) Equation (12) can be also re-written in terms of angle velocity, V ( i ω ) = F e ( i ω ) / L i ω ( m + m a ) + [ c w + c p ( L 1 / L ) 2 ] + [ ρ g A w + k ( L 1 / L ) 2 ] / i ω (13) Looking at the transfer of power in the FMB WEC, the time averaged power captured by the PTO is calculated as: P ¯ = 1 2 c p L 1 V i ω ¯ 2 (14)The average capture power can be obtained in the analytical form as: P ¯ = 1 2 c p L 1 2 | F e ( i ω ) | 2 [ c w + c p ( L 1 / L ) 2 ] 2 + [ ω ( m + m a ) − ( ρ g A w + k ( L 1 / L ) 2 ) / ω ] 2 (15)where | F e ( i ω ) | is the module of the wave excitation force. From the look of the Equation (15), there exists a maximum value of the average captured energy if: ω ( m + m a ) − ( ρ g A w + k ( L 1 / L ) 2 ) ω = 0 (16) According to Equation (16), the nature frequency ω n of the FMB WEC can be obtained as: ω n = ρ g A w + k L 1 L 2 m + m a (17) 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 DESCRIPTION OF THE INERTIA SELF-TUNING CONTROL STRATEGY 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. FIGURE 5Open in figure viewerPowerPoint Structure diagram of the inertia self-tuning control system 4.2 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: ω n ′ = ρ g A w + k L 1 L 2 m + Δ m + m a (18)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 Δmd should be: Δ m d = ρ g A w + k L 1 L 2 ω 2 − m − m a (19) The mass of the buoy itself, m, is constant and its added mass, ma, 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. 4.3 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 Δmd is determined according to Equation (19). Then, an error signal e(t) is defined as the difference between the actual Δm and the desired Δmd, and the input frequency signal fi(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 kp, the integral gain ki and the differential gain kd is based on pole placement method [25]. FIGURE 6Open in figure viewerPowerPoint Block diagram of the close-loop phase controller. M(ω) represents the function relationship between Δmd and ω. T(ω) represents the function relationship between the Thyd and ω 4.4 Mathematical model of the pumps The mathematical models of the injection pump and the suction pump are built as a second order transfer function. G s = Δ M s F i s = k 1 s T s + 1 (20)where ΔM(s) and Fi(s) are the Laplace transforms of the increment of the buoy mass and the pumps' input frequency signal, respectively. k1 is the scale factor and T is the time constant. The instantaneous power Pc consumed by the pumps can be calculated as: P c = f i 3 f r 3 P r (21)where fi is the input frequency of the pumps, fr and Pr are the rated input frequency and rated power of the pumps, respectively. 5 RESULTS AND ANALYSIS In this section, the dynamic performances of the inertia self-tuning 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 divided into meshes, as shown in Figure 7. The sea state parameters are set to the water depth of 15 m and the sea area of 100 m2. The maximum grid size of the central platform and the buoys are 0.6 and 0.2 m, respectively. Meanwhile, the control strategy of the close-loop phase controller and the mathematical models of the pumps were analysed in the Matlab Simulink software. These hydrodynamic coefficients obtaining from the AQWA software are used in the programming codes of Matlab. TABLE 1. Geometrical parameters of the FMB WEC Dimension parameters Value Radius of each buoy, Rb 1.6 m Height of each buoy, Hb 2.2 m Draft of each buoy, Db 1.0 m Radius of floating platform, Rb 4.0 m Radius of damping plate, Rd 7.6 m Depth of damping plate from still water level, Ds 6.0 m Length of each actuating arm, L 6.8 m Distance between the hinge A and the PTO, L1 3.4 m TABLE 2. PTO parameters used in the simulation PTO Parameters Value Mass of each buoy, m 2000 kg PTO damping coefficient, cp 28 kN/(m/s) PTO elastic coefficient, k 3.0 kN/ m TABLE 3. Pumps parameters used in the simulation Pumps parameters Value scale factor, k1 3.33 Time constant, T 0.5s Rated frequency, fr 50 Hz Rated power, Pr 7.5 kW Rated displacement, Vr 72m3/h Rated speed, Pr 2900 rpm TABLE 4. PID controller parameters used in the simulation Controller parameters Value proportional gain, kp 1.2 integral gain, ki 0.001 differential gain, kd 40 FIGURE 7Open in figure viewerPowerPoint Hydrodynamic model of the FMB WEC 5.1 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, Δmd 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, Δmd 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, Δmd 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, Δmd 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, Δmd equals to −9.0t. The total mass of the buoy is minimized and always equals to 2.0t. FIGURE 8Open in figure viewerPowerPoint Relation curve between the desired increment of the buoy mass and the incident wave frequency 5.2 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 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]: S n ω = 5 π 4 H s 2 T p 4 ω 5 exp − 20 π 4 4 T p 4 ω 4 (22)with ωp = 2π/Tp, where Tp is the peak period, ωp is the peak frequency and Hs 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: η t = ∑ i = 1 n 2 s n ω i Δ ω sin ω i t + φ i (23)where ωi and φi are the frequency and random phase component of the ith wave, and Δω is the frequency band calculated from: Δ ω = ω m a x n (24)where ωmax 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 Hs and ωp 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 Hs = 1.0 m and ωp = 2.1 rad/s. FIGURE 9Open in figure viewerPowerPoint Pierson–Moskowitz wave spectrum when the significant wave height is 1.0 m and peak frequency is 2.1 rad/s FIGURE 10Open in figure viewerPowerPoint Comparison of regular wave and irregular wave of H/Hs = 1.0 m and ω/ωp = 2.1 rad/. (a) Regular wave (b) Irregular wave 5.3 Dynamic performance in regular wave 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 ab