3.3.1. Non-Evolutionary NLPQL-Based Optimisation

The NLPQL algorithm was a local optimiser and has the advantages of fast convergence and high-stability [42]. In several studies, the NLPQL-based optimisation was applied to solve and optimise various non-linear problems during the design stage [42–45]. Figure 7A shows the flowchart of the NLPQL-based optimisation technique. Initially, the NLPQL-based optimisation process was started by randomly generating the guest point of each study parameter (*dp*, *dr*, *p*0*,HPA*, *Vcap,HPA*, *p*0*,LPA*, *Vcap,LPA*, *DHM*). Then, in the first iteration, the generated random guest point was chosen for each study parameter, and the HPTO model was then evaluated based on the objective function in Equation (22). The linear search calculation method was then implemented in order to determine the convergence satisfaction of the objective function. As presented in Figure 7A, the new iteration will be started if the objective function does not meet the convergence criterion. A new iteration was initially started to determine the new search direction and step size using the sequential quadratic programming (SQP) method. Then, the variables for each study parameters were determined based on the new search direction and step size. Finally, the Hessian approximation was updated by the modified BFGS-formula, as described in [42]. The parameters setting of the NLPQL-based optimisation is listed in Table 4. This process was repeated until the NLPQL algorithm met the termination accuracy.

**Table 4.** Parameters setting of NLPQL.

### 3.3.2. Evolutionary GA-Based Optimisation

In contrast to the NLPQL, GA was an evolutionary algorithm that was inspired by the natural evolution process. GA has been effectively applied to a wide range of realworld problems. In this algorithm, the variables of the optimisation problem were coded in chromosomes. Figure 7B presents the flowchart of the GA pseudo-code. The GAbased optimisation process was initially started by randomly generating a population of chromosomes (study parameters: *dp*, *dr*, *p*0*,HPA*, *Vcap,HPA*, *p*0*,LPA*, *Vcap,LPA*, *DHM*), as presented in Figure 7B. Thereafter, for the first iteration, the random values from the generated population were chosen for each study parameter. The HPTO model was then evaluated based on the objective function in Equation (22). The chromosomes of the population were then sorted according to the least cost or highest fitness. Some percentages of the best chromosomes were transferred directly to the next generation based on their merit. Then, three GA operators named as selection, crossover and mutation were implemented to manipulate the rest of the chromosomes for the next generation. During the selection rule, the parent's chromosome that contributed to the current population was selected for the next generation process. Then, pairs of selected parents were recombined by a crossover operator to produce new chromosomes. A mutation rule was then applied to the new

chromosomes to avoid the GA converging to the local optimum. Finally, this process was iterated until the satisfactory fitness level was reached. The parameters setting of GA was gathered in Table 5.

**Setting Value** Population size 50 Reproduction ratio (%) 80 Maximum number of generations 100 Mutation probability (%) 10 Mutation amplitude 0.1 Seed 1 Final accuracy 0.0001

**Table 5.** Parameters setting of GA.

**Figure 7.** Optimisation procedures using (**A**) Non-Linear Programming by Quadratic Lagrangian (NLPQL) and (**B**) Genetic Algorithm (GA).

#### **4. Results and Discussion**

#### *4.1. Comparisons between NLPQL and GA Optimisation of HPTO Unit*

In order to evaluate the best optimisation approaches for the HPTO unit, a critical comparison analysis was performed. The comparison in terms of the final objective function, the best-estimated parameter values and the HPTO unit's performance were considered.

4.1.1. Chronological Variation of the Objective Function and Parameters Variables

Figures 8 and 9 depicted the chronological variation of the objective function and parameters variables with respect to the number of generations of the optimisation processes done by NLPQL and GA operators. The red vertical line in both figures indicated the optimisation process's termination at the lowest objective function value. The lowest objective function value was of interest for optimisation purposes in a feasible solution framework. Both of the optimisation processes were terminated after the algorithms

met the optimum point, which was determined based on the termination criterion (final accuracy), as previously mentioned in Tables 4 and 5.

**Figure 8.** Chronological variation of (**A**) objective function, (**B**) diameter of piston and rod, (**C**) pre-charge gas pressure and capacity of HPA, (**D**) pre-charge gas pressure and capacity of LPA, and (**E**) displacement of HM for NLPQL algorithm.

**Figure 9.** Chronological variation of (**A**) objective function, (**B**) diameter of piston and rod, (**C**) pre-charge gas pressure and capacity of HPA, (**D**) pre-charge gas pressure and capacity of LPA, and (**E**) displacement of HM for GA algorithm.

Figure 8A showed that the estimation process of the best configuration parameters was completed after the 22 number of iterations since the NLPQL operator had satisfied its accuracy requirement. The overall simulation–optimisation process using the NLPQL algorithm was carried out for 3237 s (approximately 53 m 57 s). As shown in the figure, the lowest objective function value at 22 iterations was obtained at 0.0492. Meanwhile, 56 numbers of iterations were needed to find the optimum case by the GA operator, as exhibited in Figure 9A. A complete simulation–optimisation process by the GA operator was performed for 7 h and 32 min. The lowest objective function was obtained equal to 0.0375, as illustrated in Figure 9A. Besides that, Figure 8B–E showed the chronological variation of the HPTO parameters throughout the optimisation process by the NLPQL algorithm. From

these figures, the HPTO parameters seemed to approach the optimum conditions starting from 14 number of iterations. On the other hand, for the GA optimisation, Figure 9B–E showed that some of the HPTO parameters reached the best condition after 5 iterations.

In summary, the comparison of the chronological results of both optimisation approaches in Figure 8, and Figure 9 found that the optimisation using the NLPQL algorithm was much faster than the GA optimisation case. The reason was that since the NLPQL was the local optimisation approach, this algorithm depended on the initial point of each HPTO parameter. As reported in [42], the numerical test showed that different initial points required different time consumed and would give different optimal results. In contrast to the NLPQL algorithm, since the GA is a global algorithm, it takes more time in its exploration and exploitation processes that consider more points in search space in order to find the optimum condition. Thus, it returns more accurate and reliable results as depicted in Figure 9. In order to improve the performance of the NLPQL algorithm, the hybridisation of the NLPQL algorithm with the other global optimisation operators can be considered, as presented in [46]. In [46], the optimal starting points of the NLPQL algorithm were set by GA, and better optimum results have turned up.

#### 4.1.2. Best Estimated Parameters

Table 6 presents the best configuration parameters sets of the HPTO unit that were successfully estimated using NLPQL and GA optimisation approaches. As shown in the table, the *dp* and *dr* parameters of the hydraulic cylinder were estimated at 3% and 12.8% smaller than their initial values for the NLPQL case, which equaled to 34.9 mm and 21.8 mm, respectively. For the GA case, the *dp* and *dr* were estimated closely to their minimum constraints, which equal to 37.6 mm and 10 mm. Apart from that, the data in Table 6 reported that the best-estimated values of the *p*0*,HPA,* and *Vcap,HPA* from the NLPQL, and GA optimisation were significantly different from their initial condition. The optimal value of *p*0*,HPA* was estimated larger than its initial value for both cases. While, for the *Vcap,HPA*, Table 6 clearly shows that the best values of *Vcap,HPA* were estimated 65% lower and 275% larger than its initial value for the NLPQL and GA cases. For the *p*0*,LPA* and *Vcap,LPA*, the best-estimated values were not too significantly different from their initial values for both cases. Furthermore, it can be found in Table 6 that the best values of *DHM* were significantly different between both optimisation cases. The result from the table shows that the GA operator successfully estimated a smaller value of *DHM* compared to the NLPQL case.

In summary, based on the comparison of best configuration parameters estimated from both optimisation approaches, a few preliminary conclusions in terms of physical size, cost of the HPTO unit and others can be drawn. Practically, the physical size, weight and cost of the HPTO unit depend on its configuration parameters. Based on the results in Table 6, it can be preliminarily concluded that the physical size and weight of the HPTO unit for the NLPQL approach were much smaller than the GA approach case. This was due to the larger capacity of HPA as estimated by the GA approach. The physical size and weight of the HPTO unit were vital to being reduced since they can influence the complete design of the WECs, as reported in [13]. Moreover, the configuration parameters also influenced the total cost of the HPTO unit. As reported in [10], the hydraulic accumulator and the hydraulic motor were the most expensive HPTO unit components. Thus, from Table 6, it can be concluded that the overall cost of the WECs from the NLPQL optimisation approach was much lower than the GA case.

**Table 6.** Best configuration parameters from NLPQL and GA parameter estimation approaches.

#### 4.1.3. Operational Behaviour of the HPTO Unit

Table 7 compares the operational behaviour of the HPTO unit for the non-optimal, NLPQL-optimal, and GA-optimal cases. By comparing the data in Table 7 and Appendix A, the HPTO unit's operations were satisfied with its operational constraints for all cases. As reported in Table 7, the overall operating pressure of the HPTO unit increased for both optimal cases. For example, the operating pressures of the hydraulic cylinder chambers for the NLPQL-optimal case increased by 8.6% (side A) and 8.8% (side B). While, for the GAoptimal case, the operating pressures of the hydraulic cylinder increased by 56.6% (side A) and 56.8% (side B) from the non-optimal case, respectively. Besides that, the pressures in the HPA for both cases also increased up to 10.1% and 60.8%, respectively. The hydraulic motor pressure also significantly increased by 9.4% and 60% for both cases, up to 47.5 bar and 69.8 bar.

**Table 7.** Comparison of the operational behaviour of the HPTO unit for non-optimal, NLPQL-optimal, and GA-optimal cases.

The increasing pressure in the HPTO unit significantly increased the speed and torque of the hydraulic motor. As depicted in Table 7, the hydraulic motor speed and torque increased to its rated (200 rpm, 6 Nm) for both cases. In short, the results in Table 7 clearly show that the operational speed and torque of the hydraulic were influenced by the pressure of the other components in the HPTO unit. Thus, the presented results in Table 7 proved that the optimisation process by NLPQL and GA were highly effective in estimating the best component parameters of the HPTO unit.

#### 4.1.4. Performance of the WECs

Technically, the force of the HPTO unit was directly proportional to its operational pressure [15,37]. Since the HPTO unit's pressure significantly increased, the HPTO force applied to the WEC device also increased, as depicted in Figure 10. Comparing Figure 10 with Figure 6B showed that the HPTO force applied to the WEC obviously increased for both cases. As depicted in Figure 10A, the HPTO forces applied to the WEC for the NLPQL case can be reached up to 1.65 kN (upward) and 0.78 kN (downward). While, for the GA-optimal case, the HPTO forces applied to the WEC can be reached up to 2.3 kN (upward) and 1.2 kN (downward), respectively. The results clearly showed that the overall HPTO force applied to the WEC device for the GA-optimal case was significantly larger than the HPTO force in the NLPQL case. This significant difference was attributed due to the larger pre-charge gas pressure and volume capacity of the HPA in the HPTO unit for the GA-optimal case, as depicted in Table 6. A larger pre-charge gas pressure required a more massive flow of high-pressure fluid [10]. In addition, from both figures, it can be seen that the HPTO forces applied to the WEC device during the upward movement were larger compared to the downward movement for both cases. This was due to the unsymmetrical double-acting hydraulic cylinder used in the HPTO unit. Since the hydraulic cylinder chambers were unsymmetrical, the fluid pressure in the chamber, which comprises a large effective piston area, was higher than the fluid pressure in the small effective area chamber, as clearly described in [10].

**Figure 10.** HPTO force applied to the WEC device (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

Furthermore, the HPTO force's effect on the displacement of the WEC device and hydraulic cylinder piston can be seen in Figure 11. Figure 11A,B illustrated the displacement of the wave, WEC, and hydraulic cylinder piston during the HPTO unit operation for both optimal cases. In Figure 11A, it was depicted that the average displacements of the WEC device and hydraulic cylinder piston for the NLPQL-optimal were 77.5% and 19.3% of the average wave elevation. Meanwhile, for the GA-optimal case, the average displacement of

the WEC device and hydraulic cylinder piston was 65% and 16% of average wave elevation. The comparison of the results in Figures 6A and 11 showed that the displacement of the WEC device and hydraulic cylinder piston was slightly reduced for the NLPQL-optimal and GA-optimal cases. The reduction was due to the larger HPTO force applied to the WEC device in both cases. In addition, the comparison of Figure 11A,B showed that the average displacement of the WEC device and hydraulic cylinder piston for GA-optimal was less than the NLPQL-optimal case.

**Figure 11.** Displacement of the wave, WEC, and piston for three different cases (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

Apart from that, Figure 12 illustrated the comparison of the electrical power generation profiles of WECs for both optimal cases. Comparing the results in Figure 12 with Figure 6C, the overall electrical power generated from the HPTO unit optimised by NLPQL and GA approaches was successfully enhanced. For the non-optimal case, the electrical power profile in Figure 6C clearly indicated that the electrical power generated from the HPTO unit was lower than its rated capacity. Figure 6C showed the electrical power generated from the non-optimal HPTO was up to 71% (71 W) of its rated capacity. In contrast to both optimal cases. From Figure 12, the result showed the electrical generated output of HPTO was close to its rated capacity. The average electrical power generated from the HPTO unit for the NLPQL-optimal and GA-optimal cases was calculated up to 96% (96 W) and 97% (97 W) rated capacity, respectively. The comparison results in Figure 12 also showed that the electrical power generated from the HPTO unit of the GA-optimal case fluctuated less compared to the NLPQL-optimal case. This was due to the larger HPA used in the HPTO unit of the GA-optimal case. In addition, both of the optimal HPTO units reached their steady-state condition around 80 s.

**Figure 12.** Comparison of electrical power generated from the best HPTO unit estimated by NLPQL and GA optimisations (*HW* = 0.8, *TW* = 2.5 s).

#### *4.2. Evaluations of Optimal WECs Using Irregular Wave Data*

The optimal HPTO units obtained from the optimisation processes were evaluated using irregular wave elevation input in order to evaluate their performance in generating the electricity in irregular wave circumstances. The results in Figure 13 provided the hydraulic cylinder piston responses for both cases. The figure showed that the displacement of the hydraulic cylinder piston for GA-optimal was smaller than the NLPQL-optimal case. This was due to the different pressures in the hydraulic cylinder chambers, as shown in Figure 14. Figure 14A showed that the reciprocating motions of the piston for the NLQPL case produced high-pressure liquid in both hydraulic cylinder chambers that reached up to 54 Bar. At the same time, the pressure of the hydraulic cylinder chamber for the GA case reached up to 75 Bar. The pressure difference for both cases was due to the difference in the HPTO unit's parameter design.

The high-pressure liquid produced in the hydraulic cylinder chamber then flowed to HPA and hydraulic motor. The HPA was used as liquid energy storage to smooth out pressure fluctuation in the HPTO unit. Thus, the liquid's pressure, which exceeded the HPA pre-charge pressure setting, was accumulated in the HPA ballast. In contrast, the HPA released the high-pressure liquid stored in its ballast when the HPTO system's pressure was lower than its pre-charge pressure setting. Figure 15 showed the pressure of the HPA for both optimal cases. For both cases, the pre-charge pressures of HPA were set to 46.9 Bar and 68.9 Bar, as previously given in Table 6. In Figure 15A, the pressure of the HPA reached up to 49 Bar, which was 4.5% higher than its pre-charge pressure setting several times. For the GA case, the highest pressure of the HPA can be reached up to 69.01 Bar, which was 0.16% higher than its pre-charge pressure setting, as depicted in Figure 15B. The difference in the pressure variation rate of HPA in both cases was due to the different HPA volume capacity. As given in Table 6, the volume capacity of HPA for the GA-optimal case was larger than the NLPQL-optimal case. In addition, the comparison of results in Figure 15A,B showed that the high-pressure liquid accumulation was more often for the GA-optimal case. This was due to a larger volume capacity of HPA used in the HPTO unit.

The smoothing effects of the HPA unit on the pressure in the HPTO unit for both optimal cases can be clearly seen in Figure 16. Figure 16A,B showed the smoothed pressure of the hydraulic motor for both cases. The comparison results in Figure 16A,B showed that the smoothing effect of the hydraulic motor pressure for the GA-optimal case was higher than the NLPQL-optimal case. It can be seen from the figures, the pressure of the hydraulic motor fluctuated less for the GA-optimal case compared to the NLPQLoptimal case. However, at the initial state of both cases, the hydraulic motor's pressure was more fluctuating due to insufficient energy stored in the HPA, as depicted in Figure 15. In addition, Figure 17 illustrated the comparison of the hydraulic motor speed for both cases. From the figure, the average speed of the hydraulic motor for the GA-optimal case was higher than the NLQPL-optimal case, which was 163 rpm instead of 137 rpm.

**Figure 13.** Displacement of hydraulic cylinder piston of HPTO unit (*HW* = 0.8, *TW* = 2.5 s).

**Figure 14.** The pressure of the hydraulic cylinder chamber of HPTO unit (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

**Figure 15.** Pressure of high-pressure accumulator of HPTO unit (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

**Figure 16.** The pressure of hydraulic motor of HPTO unit (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

**Figure 17.** Speed of hydraulic motor of HPTO unit (*HW* = 0.8, *TW* = 2.5 s), (**A**) NLPQL and (**B**) GA cases.

Figure 18A–C presented the electrical power profiles of the HPTO unit for the nonoptimal, NLPQL-optimal and GA-optimal cases. For the non-optimal case, the average electrical power generated from the PMSG generator in the HPTO unit was equal to 55 W, which was only 55% of its rated capacity, as shown in Figure 18A. For this case, the highest electrical power that was generated only reached up to 71 W. This was significantly different for the cases of the optimal HPTO unit optimised by NLPQL and GA approaches. It can be seen in Figure 18B,C, both optimal HPTO units capable of generating electricity of up to an average of 62 W and 77 W, which was 62% and 77% of its rated capacity, respectively.

**Figure 18.** Comparison of electrical power generated from the best HPTO unit in irregular wave condition (*HW* = 0.8, *TW* = 2.5 s), (**A**) Non-optimal, (**B**) NLPQL-optimal, and (**C**) GA-optimal cases.

#### **5. Conclusions**

A comprehensive study was conducted to estimate the configuration parameters of the HPTO unit for a wave energy conversion device using a non-evolutionary NLPQL and evolutionary genetic algorithm. Seven important configuration parameters of the HPTO unit were considered in this optimisation study. The simulation–optimisation of HPTO model parameters was performed using MATLAB®®/Simulink software. The optimisation function problem was designed to maximise the output power generated from the HPTO unit. The optimal HPTO unit was then evaluated using irregular wave input to evaluate its performance in irregular circumstances. From the simulation studies, the key results can be listed as follows:

approach is more relevant. While, for the sake of effectiveness, the GA approach is more recommended.

The simulation–optimisation framework presented may help the engineers and researchers of WECs to design a reliable and high-efficiency HPTO unit for wave energy converter devices. It is suggested that further researches should be conducted in the following areas:

**Author Contributions:** M.A.J., conceptualisation, methodology, software, data curation, analysis, writing—original draft; M.Z.I., writing—review and editing and supervision, project administration, funding acquisition; M.Z.D., conceptualisation, methodology, writing—review and editing and supervision; Z.M.Y., software, data curation, analysis and writing—original draft; A.A. data curation, analysis and writing—review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This project was funded by the Ministry of Higher Education (MOHE) under Fundamental Research Grant Scheme (FRGS/1/2019/TK07/UMT/01/1).

**Acknowledgments:** The authors would like to thank the Ministry of Higher Education (MOHE) and Universiti Malaysia Terengganu (UMT) for financial support for this research.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Abbreviations**

#### **Appendix A**

**Table A1.** Specifications of hydraulic components from Parker Hannifin Manufacturer.

<sup>a</sup> Heavy Duty Roundline Welded Series, <sup>b</sup> High-Pressure Bladder Accumulator Series, <sup>c</sup> Low-Pressure Bladder Accumulator Series, <sup>d</sup> High Torque Radial Piston Motors Series.

#### **References**