A Hybrid Evolutionary Algorithm Framework for Optimising Power Take Off and Placements of Wave Energy Converters

TL;DR

This paper presents a hybrid evolutionary algorithm framework that optimizes the placement and power-take-off settings of wave energy converters to maximize energy output, demonstrating improved performance over existing methods.

Contribution

It introduces a novel hybrid heuristic search approach combining local and direct search methods for wave farm optimization, addressing hydrodynamic complexity constraints.

Findings

Hybrid approach outperforms previous techniques by up to 3%

Effective in two real wave scenarios (Sydney and Perth)

Optimizes both WEC placement and PTO settings efficiently

Abstract

Ocean wave energy is a source of renewable energy that has gained much attention for its potential to contribute significantly to meeting the global energy demand. In this research, we investigate the problem of maximising the energy delivered by farms of wave energy converters (WEC's). We consider state-of-the-art fully submerged three-tether converters deployed in arrays. The goal of this work is to use heuristic search to optimise the power output of arrays in a size-constrained environment by configuring WEC locations and the power-take-off (PTO) settings for each WEC. Modelling the complex hydrodynamic interactions in wave farms is expensive, which constrains search to only a few thousand model evaluations. We explore a variety of heuristic approaches including cooperative and hybrid methods. The effectiveness of these approaches is assessed in two real wave scenarios (Sydney and Perth) with farms of two different scales. We find that a combination of symmetric local search with Nelder-Mead Simplex direct search combined with a back-tracking optimization strategy is able to outperform previously defined search techniques by up to 3\%.

Tables

Table 1. Table 1: The performance comparison of various heuristics for the 16-buoy case, based on maximum, median and mean power output layout of the best solution per experiment.

\hlineB4                                                                                        Perth wave scenario (16-buoy)
\hlineB2 Methods	DE	CMA-ES	1+1EA	PSO	NM-M	DE-NM	CMAES-NM	1+1EA-NM	Dual-DE	LS-N$M_{16s}$	LS-N$M_{32s}$	LS-N$M_{64s}$	SLS-NM(BR)	SLS-NM(r)	SLS-NM(C)	SLS-NM-B1	SLS-NM-B2
\hlineB2 Max	2652393	2680843	2644987	2289764	1893411	1845065	2059607	2125726	2453857	2554865	2613619	2626506	2723676	2716463	2709385	2739658	2741489
\hlineB2 Min	2582793	2603920	2263180	1935340	1561609	1829109	1816940	1790521	2399372	2384981	2481663	2482512	2669097	2540090	2635628	2723886	2723470
\hlineB2 Mean	2613938	2657924	2476649	2034625	1709664	1839680	1917947	1930481	2442276	2449269	2547633	2570651	2708267	2677821	2691542	2733105	2735345
\hlineB2 Median	2609441	2661285	2476649	2011311	1696728	1840299	1902074	1902254	2453857	2442901	2545870	2584010	2711875	2692056	2701771	2733962	2736453
\hlineB2 Std	21601.36	20844.29	109986.19	90666.26	96667.21	4261.50	76927.84	96648.77	20511.38	53689.15	40651.08	49948.44	14434.14	48718.95	24252.10	4426.12	4986.80
\hlineB4 \hlineB4                                                                                        Sydney wave scenario (16-buoy)
\hlineB2 Methods	DE	CMA-ES	1+1EA	PSO	NM-M	DE-NM	CMAES-NM	1+1EA-NM	Dual-DE	LS-N$M_{16s}$	LS-N$M_{32s}$	LS-N$M_{64s}$	SLS-NM(BR)	SLS-NM(r)	SLS-NM(C)	SLS-NM-B1	SLS-NM-B2
\hlineB2 Max	1544911	1551852	1550820	1498996	1393383	1372431	1524002	1541064	1488451	1525789	1542636	1551640	1556956	1550054	1534157	1559578	1564334
\hlineB2 Min	1525043	1533453	1461996	1396223	1256857	1363834	1392057	1414872	1420995	1507479	1523444	1518276	1526266	1489493	1465638	1546369	1529929
\hlineB2 Mean	1536324	1547951	1526867	1438377	1337175	1367502	1454505	1467659	1462382	1514404	1532215	1535923	1544706	1525152	1512476	1553629	1556447
\hlineB2 Median	1538708	1549616	1531683	1435726	1338054	1367767	1441785	1467420	1465419	1513593	1528728	1535516	1548100	1523762	1518423	1553779	1558319
\hlineB2 Std	6559.22	4996.61	25962.37	31262	41794.00	2508.76	47091.11	32623.75	14999.60	5125.37	7224.27	12944.20	10965.95	17681.23	18379.27	3293.68	8931.08
\hlineB4

Table 2. Table 2: The performance comparison of various heuristics for the 4-buoy case, based on maximum, median and mean power output layout of the best solution per experiment (Std = standard deviation). In SLS-NM, the first buoy location in the search space is investigated and three options are evaluated including: Bottom right (BR), Bottom Center (C) and random (r).

\hlineB4                                                                                   Perth wave scenario (4-buoy)
\hlineB2 Methods	DE	CMA-ES	1+1EA	PSO	NM-M	DE-NM	CMAES-NM	1+1EA-NM	Dual-DE	LS-N$M_{16s}$	LS-N$M_{32s}$	LS-N$M_{64s}$	SLS-NM(BR)	SLS-NM(r)	SLS-NM(C)	SLS-NM-B1
\hlineB2 Max	719978	719879	719851	719913	719845	718321	718418	719049	719915	629667	633448	635676	713573	714041	703908	719663
\hlineB2 Min	719878	708731	708731	708445	708690	713598	706583	717363	719851	546821	600825	615328	710449	694667	701964	719143
\hlineB2 Mean	719921	718005	718491	715730	718914	717041	715364	718500	719882	599239	617694	622393	711976	704714	702821	719495
\hlineB2 Median	719914	719851	719850	719107	719844	717380	716988	718653	719879	599921	617716	621512	711877	705196	702835	719554
\hlineB2 Std	27.78	4331.96	3170.29	5078.80	3219.83	1509.80	3925.23	478.99	28.92	24069.76	9739.71	5585.69	835.78	6707.32	563.52	172.24
\hlineB4                                                                                   Sydney wave scenario (4-buoy)
\hlineB2 Methods	DE	CMA-ES	1+1EA	PSO	NM-M	DE-NM	CMAES-NM	1+1EA-NM	Dual-DE	LS-N$M_{16s}$	LS-N$M_{32s}$	LS-N$M_{64s}$	SLS-NM(BR)	SLS-NM(r)	SLS-NM(C)	SLS-NM-B1
\hlineB2 Max	423898	423878	423847	423872	423806	423628	423485	423775	423899	419504	420549	420850	422619	422906	422878	422866
\hlineB2 Min	423489	422046	422784	420883	423392	423255	422464	423397	423789	386137	415848	413949	420667	401907	420125	420724
\hlineB2 Mean	423767	423516	423579	423218	423703	423406	423006	423602	423844	411155	418305	418210	421665	414943	421368	422335
\hlineB2 Median	423808	423646	423636	423564	423710	423352	422988	423625	423840	415909	418262	418607	421798	416878	421421	422660
\hlineB2 Std	140.96	492.19	285.52	859.18	119.82	125.47	294.73	132.68	37.9	11436.02	1523.88	2104.40	624.80	7100.91	771.31	638.80
\hlineB4

Equations

$$M_p{\ddot{X}}_p(t)=F_{exc,p}(t)+F_{rad,p}(t)+F_{pto,p}(t)$$

$${\hat{F}}_{exc,\Sigma }=((M_{\Sigma }+A_{\sigma }(\omega ))j\omega +B_{\sigma }(\omega )-\frac{K_{pto,\Sigma }}{\omega }j+D_{pto,\Sigma }){\ddot{X}}_{\Sigma }$$

$$P_{\Sigma }=\frac{1}{4}({\widehat{F^{\ast }}}_{exc,\Sigma }{\ddot{X}}_{\Sigma }+{\ddot{X^{\ast }}}_{\Sigma }{\hat{F}}_{exc,\Sigma })-\frac{1}{2}{\ddot{X^{\ast }}}_{\Sigma }B{\ddot{X^{\ast }}}_{\Sigma }$$

$$q=\frac{P_{\sum }}{{\sum }_{i=1}^N{P}_i}.$$

$$P_{\Sigma }^{\ast }=\text{mbox}{\text{em}argmax}_{X,Y,K_{pto},D_{pto}}{P}_{\Sigma }(X,Y,K_{pto},D_{pto})$$

Full text

K****eywords Renewable energy $\cdot$ Evolutionary Algorithms $\cdot$ Position Optimisation $\cdot$ Power Take Off system $\cdot$ Wave Energy Converters

1 Introduction

Environmental concerns and declining costs are favouring the widespread deployment of renewable electricity generation. Wave energy converters (WECs), in particular, offer strong potential for growth because of their high capacity factors and energy densities compared to other renewable energy technologies [1]. However, WECs are relatively new technology, which presents design challenges in the development of individual converters and in the configuration of farms consisting of arrays of WECs. The WEC model considered in this research is similar to a new generation of CETO systems that were introduced and developed by the Carnegie Clean Energy company [2]. The CETO system is composed of an array of fully submerged three-tether converters (buoys) [3]. The aim of this research is to maximise the absorbed power of an array (farm) of these buoys. In maximising the power produced by such an array the key factors are [4]: (1) the layout of WECs in the sea, (2) the power-takeoff (PTO) parameters for each WEC, (3) wave climate (wave frequencies and directions) of a specific test site, and (4) the number of WECs.

The combined search space for optimising WECs placements and PTO settings is non-linear and multi-modal. Furthermore, because of complicated and extensive hydrodynamic interactions among generators, the evaluation of each farm configuration is expensive, taking several minutes in larger farms. These factors make the use of smart and specialised meta-heuristics attractive for this problem.

One early work [5] used a simple uni-directional wave model to compare a custom GA with an iterative Parabolic Intersection (PI) method for placing 5 buoys. Both of these search methods deployed a high number of evaluations (37000). A recent study by Ruiz et al. [6] used another simple wave model to compare a specialised GA, CMA-ES [7], and glow-worm optimisation [8] in placing buoys at positions in a discrete grid. The study found that CMA-ES converged faster than the other two methods, but ultimately produced poorer-performing layouts. In other recent work, Wu et al. [9] studied two EAs: a 1+1EA and CMA-ES for optimising buoy’s positions in an array of fully submerged three-tether WECs using a simplified uni-directional irregular wave model. That work found that the 1+1EA with a simple mutation operator performed better than CMA-ES. More recently, Neshat et al. [10] applied a more detailed wave scenario (seven wave directions and 50 wave frequencies) to evaluate a wide range of generic and custom EAs for the buoy placement. This study found that a hybrid approach (local search + Nelder-Mead) achieved better 4 and 16-buoy arrangements in terms of power produced. However, the model used by that work still embedded an artificial wave scenario. Moreover, the optimisation did not attempt to tune buoy PTO parameters to maximise the power produced by each buoy. The optimisation of PTO parameters presents another dimension for WEC farm optimisation. PTO parameters control how WECs oscillate with the frequency of incoming waves. Maximum efficiency is achieved when converters resonate with the sea waves. However, maintaining a resonant condition is not easy because real sea waves consist of multiple different frequencies [11]. In work optimising the PTO damping of one converter (CETO 6), Ding et al. [12] applied the maximum power point tracking (MPPT) control method which is a simple gradient-ascent algorithm for the online-optimisation of the deployed WEC. The results show that the MPPT damping controller can be more effective and robust than a fixed-damping system. However, when the buoy number is increased the optimisation process becomes more complicated because of the hydrodynamic interactions between buoys. In later work Abdelkhalik et al. [13] used a version of the hidden genes genetic algorithm (HGGA) to control PTO parameters. While this work raised the effective energy harvested the algorithm was not compared to other methods.

In this paper, we develop a new hybrid Evolutionary framework for simultaneously optimising both placement and PTO parameters of a wave farm. We study a broad range of meta-heuristic approaches: (1) five well-known off-the-shelf EAs, (2) four alternating optimisation ideas, and (3) three hybrid optimisation algorithms. Additionally, two new real wave scenarios from the southern coast of Australia (Perth and Sydney) with a high granularity of wave direction is used (Figure 1) to evaluate and compare the performance of the proposed methods. According to our optimisation results, a new hybrid search heuristic combining symmetric local search with Nelder-Mead simplex direct search, coupled with a backtracking strategy outperforms other proposed optimisation methods in terms of the power output and computational time.

The rest of this paper is arranged as follows. Section 2 formulates the WEC model. Section 3 gives the details of the optimisation problem. The search methods are explained in Section 4 and a brief characterisation of the fitness landscape is given. We present our comparative studies and experimental results in Section 5. Finally, Section 6 concludes this paper.

2 Model for wave energy converters

In this paper, we consider a fully submerged three-tether buoy model with each tether fastened to a converter installed on the seabed. We assume an optimal tether angle of 55 degrees, which was previously observed to maximise the extraction of energy from heave and surge motions [14]. Other features of the wave energy converters (WECs) used in this investigation, such as physical dimensions and submergence depth, can be found in [10].

2.1 Power Model

In the WEC model used here, linear wave theory is used to calculate the system dynamics [15]. This model includes three different key forces:

The wave excitation force ($F_{exc,p}(t)$) combines the incident and diffracted waves forces from generators in a fixed location. 2. 2.

The radiation force ($F_{rad,p}(t)$), derived by the oscillating body due to their motion independent of incident waves. 3. 3.

Power take-off (PTO) force ($F_{pto,p}(t)$) is the control force applied to the buoy from the PTO machinery.

Through these forces, the buoys can affect each other’s output through hydrodynamic interactions. The complex nature of these interactions, which can either be constructive or destructive, makes the calculation of farm layout and PTO parameter settings a challenging optimisation problem. The dynamic equation that describes a buoy motion in ocean waves has the form:

[TABLE]

where $M_{p}$ is the mass matrix of a $p_{th}$ buoy, ${X}_{p}(t)$ is the buoy displacement expressed as surge, heave and sway. Finally, the power take-off system is modeled as a linear spring-damper system. For each mooring line two control factors are involved: the damping $D_{pto}$ and stiffness $K_{pto}$ coefficients. Therefore, Equation (1) can be written in a frequency domain for all WECs in a farm as:

[TABLE]

The hydrodynamic parameters ($A_{\Sigma}(\omega))$ and $B_{\Sigma}(\omega)$ ) are calculated from the semi-analytical model described in [16]. In addition, $K_{pto,\Sigma}$ and $D_{pto,\Sigma}$ are control factors, described above, which can be adjusted to maximise the power output of each buoy. The total power output of the layout is computed by Equation (3):

[TABLE]

Additionally, the q-factor ($q$) of the array measures the efficiency of a entire wave farm as compared to the power output from $N$ isolated WECs. For a given layout, the $q$-factor can be calculated as:

[TABLE]

$q>1$ indicates constructive interference between WECs. The main purpose of this study is maximising the total power output: $P_{\Sigma}$ for $N$ buoys within a constrained farm area.

3 Optimisation problem formulation

The formulation of the optimisation problem in this paper can be declared as:

[TABLE]

where $P_{\Sigma}(\mathit{X,Y,K_{pto},D_{pto}})$ is the mean power obtained by placements and PTO parameters of the buoys in a 2-D coordinate system at $x$-positions: $\mathit{X}=[x_{1},\ldots,x_{N}]$, $y$-positions: $\mathit{Y}=[y_{1},\ldots,y_{N}]$ and corresponding Power Take-off parameters including $\mathit{K_{pto}}=[k_{1},\ldots,k_{N}]$ and $\mathit{D_{pto}}=[d_{1},\ldots,d_{N}]$ . In the experiments here $N\in\{4,16\}$.

Constraints

All buoy locations $(x_{i},y_{i})$ are constrained to a square search space $S=[x_{l},x_{u}]\times[y_{l},y_{u}]$: where $x_{l}=y_{l}=0~{}and~{}x_{u}=y_{u}=\sqrt{N*20000}\,m$. This allocates $20000m^{2}$ of farm-area per-buoy. Moreover, a safety distance for maintenance vessels must be maintained between buoys of at least 50 meters. For spring and damper coefficients the boundary constraints are $d_{l}=5\times 10^{4},d_{u}=4\times 10^{5}$ and $k_{l}=1,k_{u}=5.5\times 10^{5}$. For any array $\mathit{X,Y}$ the sum-total violations of the inter-buoy distance calculated in meters, is:

$\mbox{\em{Sum}}_{\mbox{\em dist}}=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}(\mbox{\em{dist}}((x_{i},y_{i}),(x_{j},y_{j}))-50),$

$\mbox{if }\mbox{\em{dist}}((x_{i},y_{i}),(x_{j},y_{j}))<50$ else 0

where $dist((x_{i},y_{i}),(x_{j},y_{j}))$ is the Euclidean distance between buoys $i$ and $j$. The penalty function of the power output (in Watts) is computed by $(\mbox{\em{Sum}}_{\mbox{\em{dist}}}+1)^{20}$. The penalty strongly encourages feasible buoy placements. This penalty is also used to handle farm-boundary constraints. For the $D_{pto}$ and $K_{pto}$ parameters, we handle constraint violations by setting the parameter to the nearest valid value.

Computational Resources

In this paper, we aim to compare a various heuristic search methods, for 4 and 16 buoy arrays, in two realistic wave scenarios. We allocate a time budget for each optimization run of three days on dedicated platform with a 2.4GHz Intel 6148 processor running 12 processes in parallel with 128GB of RAM. Note, that where the search heuristic allows, we tune algorithm settings to utilise this time budget. The software environment running the function evaluations and the search algorithm is MATLAB R2017. On this platform, parallelisation provides up to 10 times speedup.

4 Optimisation Methods

In this research, our search methods employ three broad strategies. The first strategy is to optimise all decision variables at once. This means that for a 16-buoy farm we search in $16\times 4$ dimensions simultaneously. Here, we test five heuristics that apply this strategy. The second strategy is to optimise the positions and PTO parameters of all buoys in an alternating cooperative algorithm [17]. We test four different methods that apply this strategy. Finally, the third strategy, used in [10] is to place and optimise each buoy in sequence. Here, we deploy this strategy for three hybrid EAs. Details of the algorithms tested for each strategy follow.

4.1 Evolutionary Algorithms (All-at-once)

For the first strategy, five well-known off-the-shelf EAs are deployed to simultaneously optimise all problem dimensions. (Positions+PTOs). These EAs are: (1) covariance matrix adaptation evolutionary-strategy (CMA-ES) [7] with the default $\lambda=12$, for 4-buoy layouts and and $\lambda=16$ for 16-buoy layouts; (2) Differential Evolution (DE) [18], with parameter settings of $\lambda=50,30$, respectively for 4 and 16-buoy layouts, and $F=0.5$, $P_{cr}=0.5$; (3) a (1+1)EA [19] that mutates buoys’ location and PTO parameters with a probability of $1/N$ using a normal distribution ($\sigma=0.1\times(U_{b}-L_{b})$); (4) Particle Swarm optimisation (PSO) [20], with $\lambda$= DE settings, $c_{1}=1.5,c_{2}=2,\omega=1$ (linearly decreased); (5) Nelder-Mead simplex direct search (NM) [21] is combined with a mutation operator (Nelder-Mead+Mutation or NM-M). The mutation operation is applied when the NM has converged to a solution before exhausting its computational budget, so that it can explore other parts of the solution-space (Algorithm 1).

4.2 Alternating optimisation methods (Cooperative ideas)

Optimising both positions and PTO parameters of a WEC array simultaneously can be challenging because of the high number of dimensions and heterogeneous kinds of variables. There is a natural division of variables into two subsets which might, at least in part, be optimised separately. In this section, we describe a set of alternating optimisation techniques which combine one evolutionary algorithm idea such as CMA-ES, DE, and 1+1EA, with Nelder-Mead. In addition, a cooperative, Dual-DE (DE+DE), algorithm is also described. The details of each are given next.

4.2.1 (2+2)CMA-ES + Nelder-Mead

This alternating strategy applies CMA-ES with $\mu=\lambda=2$ for $\mathit{iter}=25$ iterations to optimise buoy positions. Then the best solution is selected and NM is applied to PTO settings for $\mathit{iter}*\lambda$ iterations. This improved setting is then given to the CMA-ES population for another round of optimisation. The CMA-ES and NM optimisation processes are alternated until the time budget expires. Algorithm 2 shows the process of the CMAES-NM approach.

4.2.2 DE + Nelder-Mead

(DE-NM) This method alternates DE, for buoy-positions, and NM for PTO parameters, using the same iteration settings as above until the time budget runs out.

4.2.3 1+1EA + Nelder-Mead

(1+1EA-NM) This method alternates a 1+1 EA, for buoy positions, and NM, for PTO parameters until the time budget runs out. The iteration settings for the 1+1EA are, respectively, 200 and 50 times, for 4 and 16-buoy layouts. The same limits are also used for the NM optimisation rounds.

4.2.4 Dual-DE

This method uses the same parameter settings as described for DE in subsection 4.1 to optimise both buoy positions and PTO parameters in parallel. After $\mathit{iter}$ iterations the improved values from the positional and PTO optimisations are exchanged. This iterative pattern continues until the time budget runs out.

4.3 Hybrid optimisation algorithms

In other WEC-related research [10], it was found that applying local search around the neighborhood of previously placed buoys could help exploit constructive interactions between buoys. The following methods exploit this observation by placing and optimising the position and PTO parameters of one buoy at a time.

4.3.1 Local Search + Nelder-Mead(LS-NM)

This method places buoys sequentially. The position of each buoy placement is optimised by sampling at a normally-distributed random offset ($\sigma=70m$) from the previous buoy position. The sampled location giving the highest output is chosen. In our experiments we try three different numbers of samples: ($N_{s}=2^{4},2^{5}~{}and~{}2^{6}$). After the best position is selected, we optimise the PTO parameters of the last placed buoy using $N_{s}$ iterations of Nelder-Mead search. This process is repeated until all buoys are placed. Note that, the $\mathit{Eval}$ function of LS-NM is parallelised on a per-wave-frequency basis. An example of 16-buoy layout that is built by LS-NM(16s) and the sampling process used to build it, is shown in Figure 4(a). The details of the proposed method can be seen in Algorithm 3.

4.3.2 Symmetric Local Search + Nelder-Mead (SLS+NM(2D))

This method also places one buoy at a time, but performs a more systematic local search. The search starts by placing the first buoy in the middle of the bottom of the field and then uses NM to optimise the PTO parameters for $25$ iterations.

For each subsequent buoy placement, eight local samples are made in different sectors starting at angles: $\{angles=[0,45,90,...,315]\}$ and bounded by a radial distance of between $50$ (safe distance) and $50+R^{\prime}$. Within each sector a buoy position is sampled uniformly. Our strategy for handling infeasible solutions is that we refuse them and if all symmetric solutions are infeasible, a feasible layout is produced using uniform random sampling.

After finding the best sample among the eight local samples, two extra samples are done for increasing the resolution of the search direction. The angles of these two samples are $\pm~{}15^{o}$ plus the best angle sample. The candidate position is then selected from the 8 original samples plus these two extra samples based on the buoy’s energy output.

In the next step a check is done to see if the PTO optimisation process for the previously placed buoy (using NM) had a high percentage improvement in its last step. A large improvement indicates that there is scope to improve energy production, in this environment, by giving priority to PTO optimisation. Thus, if the last PTO search step for the last buoy is greater than $0.01\%$ then we optimise PTO parameters for $25$ iterations using NM. Otherwise we check to see if the last position optimisation converged to within $0.01\%$ and if so, we optimise position instead. Otherwise we choose between optimising PTO or position parameters for this buoy at random.

Note that this design assigns optimisation resources to PTO parameters as a first priority because we have observed stronger gains in output from tuning PTO parameters. Position parameters are given priority only when the PTO parameters for the last buoy were observed to be close to a local optimum. Algorithm 4 describes this method in detail. In addition, experiments were run with different starting buoy positions of were run with bottom center (C), bottom right (BR) and a uniform random position (r).

4.3.3 Symmetric Local Search + Nelder-Mead + Backtracking (SLS-NM-B)

The general idea of SLS-NM-B is like SLS-NM but with two differences. The first difference is optimising the initial buoy PTO settings by Nelder-Mead and then to share this configuration with the next placed buoys for speeding up the search process and saving computational time. Therefore, after applying symmetric local sampling and finding the best position, Nelder-Mead search tries to improve just the position (2D) of the new buoy.

The second contribution is applying a backtracking optimisation idea (described in Algorithm 5). As the search process of SLS is based on the greedy selection, we never come back to enhance previous buoys’ attributes, so introducing backtracking can be effective for maximising total power output. Among all placed buoys in the array, the worst $round(N\times 0.25)$ buoys in terms of power are chosen and Nelder-Mead search is then used to optimise the position (2D) and PTO settings (2D) of these buoys in a bi-level optimisation process. This procedure is called SLS-NM-B1. We can observe the performance of SLS-NM-B1 in Figure 4(b,c). This shows how the eight symmetric samples are done and the effect of the later backtracking process which refines buoy placements. A second version of this algorithm is proposed (SLS-NM-B2) to evaluate the effectiveness of optimising both position and PTOs of each buoy (4D) simultaneously instead of in a bi-level search. Other details of the backtracking algorithm are the same.

5 Experiments

This section first presents a brief landscape analysis for PTO parameters for two wave scenarios (Perth and Sydney). We then present detailed results comparing the different search heuristics outlined in the previous section.

5.1 Landscape analysis

For visualising the impact of PTO parameter optimisation, a simple experiment was done. First of all, we optimised the buoy positions for a 4-buoy layout using a manufacturer’s PTOs defaults ($k=407510$ and $d=97412$) for all converters for both the Perth and Sydney test sites. The black circle in Figure 2 marks this default PTO configuration. The energy produced by this layout is $402$ kW and $703$ kW, respectively, for the Sydney and Perth wave climates. Next, this obtained layout is evaluated where the buoy positions are fixed and we grid-sample the energy produced when all four buoys are assigned the same PTO parameters. This process produces the contoured backgrounds shown in Figure 2. Finally, we optimise the PTO parameters for each buoy independently and plot a marker for each of the four buoys. These markers are roughly, but not completely, coincident with the peak in the background power landscape produced by optimising buoys’ PTO parameters in unison. These markers are also at a different point to that produced by the default setting. The best energy produced after optimisation has improved to $420$ kW and $720$ kW respectively for Sydney and Perth. Another observation from Figure 2 is that the best PTO configurations of the 4-buoy layouts are relatively alike in both wave scenarios.

5.2 Layout evaluations

In order to evaluate the effectiveness the proposed algorithms in Sections 4.1, 4.2, and 4.3, we performed a systematic comparison of the best layouts produced by each in two different real wave scenarios (Perth and Sydney), and for two different numbers of buoys ($N=4$ and $N=16$). Ten runs were performed for each optimisation method and the best solutions were collected for each.