**1. Introduction**

Ocean waves are one of the renewable energy resources that potentially can be exploited to produce usable electricity due to their excellent features of predictability, high energy density and high source availability [1,2]. Currently, numerous wave energy converters (WECs) have been designed, developed, tested and patented through a variety of harnessing techniques that are subjected to the characteristics of the target location such as shoreline, nearshore and offshore, as reported in [3–6]. In general, WECs are a combination of three main parts, such as a wave energy converter (WEC) device, power take-off (PTO) unit and control system unit. Recently, various types of PTO units have been developed for WEC devices based on different working principles, such as the air and water turbinebased, direct-electrical drive-based, direct-mechanical drive-based and hydraulic-based, as reported in [7,8]. A hydraulic PTO (HPTO) is considered to be the most effective PTO for wave-activated-body (WAB) or point-absorber based WECs due to the outstanding

**Citation:** Jusoh, M.A.; Ibrahim, M.Z.; Daud, M.Z.; Yusop, Z.M.; Albani, A. An Estimation of Hydraulic Power Take-off Unit Parameters for Wave Energy Converter Device Using Non-Evolutionary NLPQL and Evolutionary GA Approaches †. *Energies* **2021**, *14*, 79. https:// dx.doi.org/10.3390/en14010079

Received: 2 December 2020 Accepted: 20 December 2020 Published: 25 December 2020

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features, including high-efficiency, high-controllability, well-adapted to the large power density ocean waves and low-frequency [9]. It has been reported in the literature that this type of PTO system's efficiency could be achieved up to 90% [10]. Furthermore, the HPTO unit is also easily constructed using standard hydraulic components, which are commonly used in other applications. Due to such a promising characteristic, the HPTO system finds its application in the majority of the WAB-WECs field.

Recently, many HPTO unit applications in various WECs have been published [11–17]. From the preliminary survey, most of the studies concentrated on the HPTO unit's efficiency without taking into account the optimal parameters of the HPTO model, such as hydraulic cylinder size, hydraulic accumulator capacity and pre-charge pressure and hydraulic motor displacement. The optimal configuration parameters of the HPTO unit is a crucial issue as it can affect the system's efficiency and the amount of output to be produced [13]. Only a few reports, for example, in [13–15], have considered this critical problem. However, the optimal parameters setting has been obtained by manually tuning these configuration parameters [13]. This method is usually prone to human error and easy to cause a non-optimal selection of the HPTO system's parameters. In addition, this approach also frequently involves a long-time process in order to obtain the optimal configuration parameters.

Presently, the optimisation of design parameters using a mathematical algorithm is an attractive method to estimate the accurate parameters during the design phase. It is due to the advancement of fast computing technologies that can be reliably used. A variety of studies were performed using different types of mathematical algorithms, such as non-linear programming by quadratic Lagrangian (NLPQL) and genetic algorithm (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Tabu Search (TS), et cetera, in order to obtain the best parameters for the design model [18–23]. For example, GA has been used to optimise the parameters of state-of-charge (SOC) controllers for battery energy storage in photovoltaic device applications [21]. The authors emphasized that the GA-based optimisation method has accelerated the optimisation process of the considered design parameters and effectively improves the design model's performance. Similarly, in [22], different types of heuristic optimisation approaches were applied, including Gravitational Search Algorithm (GSA) and PSO, to conduct model optimization-based studies for improving the efficiency of developed power converter units. The authors had concluded that GSA-based optimisation provides the highest convergence speed and best fitness value compared to the other algorithms. Motivated by the studies presented in [21,22], an optimized new design of WEC with an HPTO unit is presented in this study.

From the WECs point of view, a similar optimisation approach has been implemented in optimising the performance of the WECs. From the preliminary survey, several studies of the GA applications for WECs optimisation have been done [24–28]. For example, in [24], GA has been used to obtain some WECs parameters, such as buoy radius, draft, generator damping and the optimal spatial layout of a WECs park. In [25], GA has been adopted to optimally design the shape and dimensions of a WEC and also the PTO and other subsystems parameters. The techno-economic aspects of energy productivity and WECs device cost have also been considered in the study. In [28], GA has been used to obtain the optimal HPTO parameters of WECs, such as hydraulic cylinder size, hydraulic accumulator capacity and pre-charge pressure, and hydraulic motor displacement without considering the hydrodynamic effects of the floater. Since the hydrodynamic effects of the floater are the vital factors in WECs design, the optimal HPTO obtained in the study is inapplicable for real wave application. Therefore, the HPTO optimisation with the consideration of the floater's hydrodynamic effects using two different types of algorithm, such as NLPQL and GA, were investigated in this present study. The optimisation approaches presented in this study can be a useful reference to other researchers and engineers of WECs in order to design an accurate and reliable HPTO unit for the future WECs application.

The paper is organised as follows. The technical descriptions of the considered WEC with HPTO unit and its important configuration parameters are given in Section 2. The simulation studies of the HPTO unit, which includes the simulation set-up process, optimisation process and evaluation of HPTO unit performance, are described in Section 3. Results and discussion are provided in Section 4, and finally, the Conclusions are given in Section 5.

#### **2. Mathematical Modelling of WEC with HPTO Unit**

The design of the WECs depends on the characteristics of the installed location. In the present study, the rotation-based WEC attached to the fixed body concept was considered, as illustrated in Figure 1. This WECs concept has been implemented in numerous studies for various investigation goals, for example, in [29–32]. This concept is suitable to be installed at shoreline, nearshore and offshore locations. In this concept, the WEC device consists of a single or multiple floating buoy or floater attached to the rotatable arm and connected to the fixed body directed towards the dominant wave direction, as depicted in Figure 1. The multi-design of floater can be used, which is dependent on the direction of the ocean wave, either single or multi-direction. Usually, the semisphere-shaped and boatshaped floater have been considered for the offshore and nearshore location, as investigated in [29,30,33–35]. In this concept, the HPTO unit is utilised to convert the absorbed energy by the WEC device from the ocean wave to become usable electricity. A hydraulic actuator module of the HPTO unit is attached to the rotatable floater arm in order to absorb the mechanical energy produced by the WEC device, as presented in Figure 1. Meanwhile, the rest of the HPTO unit components are placed in the PTO house located on the top of the fixed-structure. In the present study, the model of WECs with a capacity of 0.1 kW was considered.

**Figure 1.** Conceptual design of future wave energy converter (WEC) with the hydraulic power take-off (HPTO) unit.

#### *2.1. Hydrodynamic Motion of the Floater*

In general, the hydrodynamic motion of the WEC device in real waves can be formulated in the time domain using the linear wave theory, as described in Equation (1). *MA* is the D'Alembert moment of inertia, *Mex* is the moment due to the diffracted waves, *Mrad* is the moment due to radiated waves, *Mres* is the hydrostatic restoring moment and *MPTO* is the moment due to the HPTO unit.

$$M\_A = M\_{\rm cx} - M\_{\rm rad} - M\_{\rm res} - M\_{\rm PTO} \tag{1}$$

The equation of the hydrodynamic motion in Equation (1) can be expended as given in Equation (2). Here, *JWEC* is the inertia moment of the floater and arm. Whereas, *Jadd*, <sup>∞</sup> is the added mass at the infinite frequency and .. *θarm* is the angular acceleration of a WEC device during the pitch motion. Then, *krad*(*t*) is the radiation impulse response function, *<sup>τ</sup>* is the time delay and . *θarm* is the angular velocity of the floater's arm. Other variables such as *kres* is the hydrostatic restoring coefficient and *θarm* is the angular of the floater's arm during the pitch motion. Finally, *hex*(*t* − *τ*) is the impulse response function of the excitation moment and *η<sup>W</sup>* is the undisturbed wave elevation at the floater center point.

$$(\mathrm{J}\_{\mathrm{WEC}} + \mathrm{J}\_{\mathrm{add,\infty}})\ddot{\theta}\_{\mathrm{arm}}(\mathrm{t}) + \int\_{0}^{\mathrm{t}} k\_{\mathrm{rad}}(\mathrm{t} - \tau)\,\dot{\theta}\_{\mathrm{arm}}(\mathrm{t}) + k\_{\mathrm{res}}\,\theta\_{\mathrm{arm}}(\mathrm{t}) + M\_{\mathrm{PTO}}(\mathrm{t}) \,\ = \int\_{-\infty}^{\infty} h\_{\mathrm{cr}}(\mathrm{t} - \tau)\eta\_{\mathrm{W}}(\tau)d\tau \tag{2}$$

The impulse response function in Equation (2) can be obtained from the hydrodynamic diffraction analysis using Computational Fluid Dynamics (CFD) software. In the present study, the hydrodynamic diffraction analysis of the WEC model was performed using ANSYS/AQWA software.

In addition, the moment due to the HPTO unit, *MPTO* can be defined using Equations (3)–(5), where *FPTO* is the feedback force from the HPTO unit applied to the WEC device. The variables *L*1, *L*2, *L*<sup>3</sup> and *L*<sup>4</sup> are the lengths between point *a*, *b*, *c* and *d*, as illustrated in Figure 2 [30,36–38]. *xp* is the displacement of hydraulic cylinder piston and *L*3,0 is the initial stroke of the hydraulic cylinder.

$$M\_{\rm PTO} = F\_{\rm PTO} L\_4 \tag{3}$$

$$L\_4 = \frac{L\_1 L\_2 \sin\left(\theta\_{arm,0} - \theta\_{arm}\right)}{L\_{3,0} + \mathbf{x}\_c} \tag{4}$$

$$\mathbf{x}\_p = L\_{3,0} - \sqrt{L\_1^2 + L\_2^2 - 2L\_1L\_2\cos(\theta\_{arm,0} - \theta\_{arm})} \tag{5}$$

**Figure 2.** (**A**) Illustration of WEC with HPTO unit and (**B**) Configuration of hydraulic cylinder motion.
