Investigating a Renewable-Resource-Targeting Mobile Aquaculture System Using Route Optimization Based on Optimal Foraging Theory

: Aquaculture systems require careful consideration of location, which determines water conditions, pollution impacts, and hazardous conditions. Mobility may be able to address these factors while also supporting the targeting of renewable energy sources such as wind, wave, and solar power throughout the year. In this paper, a purpose-built mobile aquaculture ship is identiﬁed and modeled with a combination of renewable energy harvesting capabilities as a case study with the objective of assessing the potential beneﬁts of targeting high renewable energy potentials to power aquaculture operations. A route optimization algorithm is created and tuned to simulate the mobility of the aquaculture platform and cost-basis comparisons are made to a stationary system. The small spatial variability in renewable energy potential when combining multiple resources signiﬁcantly limits the beneﬁts of a mobile, renewable-targeting aquaculture system. On the other hand, the consistent energy harvest from a blend of renewable energy types (13 kW installed wind capacity, 661 m 2 installed solar, and 1 m characteristic width wave-energy converter) suggests that the potential beneﬁts of a mobile platform for offshore aquaculture (mitigation of environmental and social concerns, any potential positive impact on yields, hazard avoidance, etc.) can likely be pursued without signiﬁcant increases in energy harvester costs.


Introduction
The concept of aquaculture has been around for thousands of years.Also known as fish culture or fish farming, both the Chinese and Roman civilizations utilized the practice of farming fish and shellfish in ponds for centuries.Similarly, fish farming was popularized in the US over a century ago through the development of hatcheries and fisheries in the 19th century [1].Within the last few decades, aquaculture has quickly increased in popularity.Specifically, finfish farming has seen a rapid growth in both volume and economic yield [2].
With this large increase comes a greater concern for the environmental impact of aquaculture installations, including water conditions, light intensity, and pollutants [3].For instance, a framework for the increased monitoring and control of environmental conditions was proposed to mitigate some of these issues in [2].Additionally, there are a host of social concerns: widespread use of stationary offshore aquaculture systems may rely on aggressive marine spatial planning policy amounting to privatization of coastal seas and displacing local fishers.Finfish farming utilizes lower-cost fish as feed which risks exacerbating food inequity by increasing market demands on lower-cost food stocks to generate high-cost products for a premium consumer [4,5].Mobile aquaculture is a potential solution pathway that can alleviate some of these practical, environmental, and social concerns.
Mobile aquaculture is a novel concept in which the fish farming system can relocate itself.These systems eschew moored offshore infrastructure for purpose-built vessels which include the pens, feed, and processing systems needed for aquaculture.Under their own power, these vessels can avoid hazards, seek optimal conditions for aquaculture, and avoid point-generation of polluting fish waste.While mobilizing aquacultural platforms may additionally potentially simplify issues related to marine spatial planning and area use conflicts, it may also exacerbate environmental and other regulatory concerns.Nonetheless, two mobile aquaculture systems have been designed and are currently under development in [6,7], indicating the importance of understanding the capabilities of such technology.
Aquaculture requires energy, which is typically derived from the onshore grid or diesel sources.Fuel sources are consistently cited as the largest contributor to operational expenditures for offshore aquaculture systems [8][9][10], and fuel demand is likely to be exacerbated by the transportation needs of a mobile system.Displacing traditional fossil fuels for such systems could be cheaper and have less environmental impact.Renewable resources have already been evaluated for some stationary aquaculture applications.Ref. [11] indicates the feasibility of an aquaculture system powered by a 5 MW wind turbine and the importance of location selection for future studies.Ref. [12] suggests the potential for a hybrid system including wind turbines, solar panels, and a diesel generator for salmon farming in Norway.Ref. [13] investigates wave energy's feasibility for powering aquaculture but cites low wave-energy potential in the summer as a limitation.In a more general sense, floating wind, wave, and solar-energy platforms are gaining traction as feasible offshore renewable energy sources [14,15].Together, these studies suggest that powering aquaculture with multiple renewable energy sources may be able to mitigate the spatial and temporal variations in a single resource potential (such as increased wave potential in winter and solar potential in summer), but none address the potential synergy with a mobile system.
Powering a mobile aquaculture system with renewable energy may be ideal to both target locations with more energy potential throughout the year and consider conditions affecting aquaculture.The study presented here is particularly valuable due to its unique application of renewable energy to mobile aquaculture.The route optimization method used in this study to identify optimal energy locations can also be relevant to any applications that require weighing movement costs against potential benefits.
A region offshore of southern Alaska is chosen as the focus of this study.Although there are some regulatory concerns [16] that likely preclude this location from actual development, it is a suitable location for this theoretical study as it is known to support salmon production and, while there are substantial renewable resources, they exhibit significant seasonal variability.The Ocean Arks [6] mobile aquaculture ship is used to estimate the power needs and production potential from available data.In short, the selection of the study area/resource was made on the basis of readily available data: while regulatory issues likely preclude its physical implementation, useful conclusions can be drawn from a theoretical study of this early-stage technology.
In this study, the aquaculture system, renewable harvesters, and vessel configurations (including cost) are modeled in Section 2.1.The route optimization method along with various methods for detailed analysis (energy storage, region restriction) are presented in Section 2.2.Next, utilizing the aquaculture and energy models and optimization and analysis methods, the results are shown in Section 3. Section 4 discusses the relevance of these results and their implications for the industry and future study.Note that the details of this study are confined to an examination of the feasibility of powering such a platform with renewable resources.While a comprehensive consideration of the potential environmental impacts of mobilizing aquaculture prior to its adoption is absolutely critical, this is beyond the scope of this work.

Materials and Methods
The modeling and methods for completing this study include the aquaculture system model (Ocean Arks [6]), the renewable energy harvesters (wind, wave, and solar), and the route optimization algorithm.The energy harvesters are defined based on their ability to capture energy from the available resource data.The route optimization algorithm tests varying combinations of energy harvesters to determine an optimal route and the feasibility of supporting the Ocean Arks aquaculture system based on its modeled energy needs.

System Modeling
The two main design components for this project are the mobile aquaculture platform and the renewable energy harvesting systems.The mobile aquaculture system is modeled based on the energy needs for Norwegian aquaculture (chosen due to the similarity with the Alaskan climate) and the Ocean Arks vessel, a proposed mobile aquaculture platform [6].The renewable energy harvesting capabilities are modeled as a combination of wave, wind, and solar-energy harvesters and their respective energy potential throughout the year within the selected region.

Aquaculture Modeling
As such a novel technology, mobile aquaculture has extremely limited public design data.For this study, the Ocean Arks ship [6] was used as a generic model for a mobile aquaculture system.The Ocean Arks vessel is designed to utilize sensing technology to seek out and travel to the best possible water conditions for fish production while also being able to avoid harsh weather when necessary.The ship is 170 m long and 60 m wide and has 8 cages that are each estimated to be 35 m long, 28 m wide, and 20 m deep.The vessel supports up to 3900 metric tons of commercial salmon production in each cycle.Three 4000 kW engines can support velocities of up to 4 knots (∼2 m/s).These parameters support an understanding of the energy requirements for both the aquaculture production and the transportation between locations.
For salmon production energy requirements, a similar location can be studied.Multiple sources on the energy requirements for salmon production in Norway are found in [12,17].Although the energy requirements in Norway would likely be slightly different than for Alaska, the similar climate and fish production is assumed to make it a reasonable representation.The Norwegian aquaculture sources both suggest relatively similar energy requirements throughout the year.Although there is some variation based on the seasons and production cycle, 680 kWh was used as an estimate of the daily energy requirements for 3120 tons of salmon production [17].Since the Ocean Arks can support 3900 tons of salmon product, the power is scaled up linearly to 850 kWh/day to provide an estimate of the power requirements to support an aquaculture system of this scale.Note that because renewable energy sources, even in combination, may exhibit intermittent availability on shorter timescales than these data sources are able to resolve, this analysis presumes sufficient energy storage to support this average daily power requirement in the event that instantaneous renewable energy availability is insufficient (except where additional energy storage is explicitly noted).
For assessing the transportation energy requirements, the power required for motion can be simply assumed as equal to the product of force and velocity.The velocity at which the ship is traveling and the drag force can be multiplied to calculate the power, where ρ is the density of water, v is the ship velocity, A is the ship's frontal area (60 × 20 m), and C d is the drag coefficient, which is estimated at 0.82 based on a long cylindrical shape.Although potential flow theory could be used to solve for the force acting on the vessel more accurately, the current fidelity of the estimated ship hull is insufficient for such a calculation.Thus, although not entirely accurate for ships, the drag force was deemed sufficient for this study.
The power requirements and sources are presented in units of kWh/day, so it is necessary to multiply by the number of hours per day for a 1:1 comparison.This conversion yields Equation (2) for calculation of the transportation power requirements in kWh/day.
The energy required for travel (kWh) will also be considered in this study and is simply the power in kWh/day multiplied by the travel time in days in Equation (3).

Renewable Energy Modeling
This paper considers three sources of renewable energy: solar, wind, and wave.Each of these sources are significant in the study area, but also experience seasonal variation.Wave-resource data can be found through the Marine Energy Atlas [18], data from which is shown in Figure 1.The wind and solar data can be accessed using NASA's POWER (prediction of worldwide energy resource) tool [19].The specific region chosen for this analysis is shown by the data in Figure 1 which encompasses latitudes from 56.W. This region was chosen because it is potentially viable for offshore aquaculture operations, and, due to its variation in wave, wind, and solar resources both spatially and temporally, it is a region in which a mobile aquaculture system that can proactively seek out optimal conditions may be of interest.In order to analyze the selected region, the wind-, wave-, and solar-resource data for a 10 × 10 grid of points within the region were downloaded for the years 2001 to 2010 and the results for each month were averaged.The downloaded wave data are in units of watts per meter in 180 min increments, which was averaged over each timestep of the route-finding optimization and divided by 1000 to convert to kW/m.The wind data are in units of m/s and available in average monthly increments (interpolated for smaller timesteps).Each wind speed v was converted to power density (kW/m 2 ) using Equation (4) with the density of air (ρ) equal to 1.225 kg/m 3 .
The available solar data are in units of kW/m 2 /day and also available in average monthly increments.
The resource potential data for each of the 100 points was compiled from the respective data sources [18,19], averaged for each month, and plotted for the month of July as an example in Figure 2. Within these plots, the spatial energy trends are apparent.Wind and wave energy are both greater in the southeast portion of the region, while solar energy is greater in the northwest.The spatial energy differences suggest that the ability to move within the region may be beneficial for energy harvesting.It is necessary to convert these resource-specific power densities to kWh/day to be useful for the comparison between the harvested and required power established in Section 2.1.1.The conversions for wave, wind, and solar power are shown in Equations ( 5)- (7), respectively.
where CW (m) is the capture width of a wave-energy converter, P dens,wave is the power per crest length (kW/m), and η wave (unitless) is the wave-energy converter efficiency.
P wind [kWh/day] = P dens,wind where A s (m 2 ) is the swept area, P dens,wind is the power per area (kW/m 2 ), and η wind is the balance of turbine efficiencies.
P solar [kWh/day] = P dens,solar where A (m 2 ) is the total area of the panels, P dens,solar (kWh/m 2 /day) is the estimated daily power capture per unit area, and η solar is the panel efficiency.

Renewable Harvester Configurations
In order to apply the route optimization and assess the potential for mobile aquaculture, the energy harvesting capabilities of the ship need to be specified.The capabilities can be defined as the solar panel area, the number of 5 kW wind turbines, and the effective WEC capture width, each of which can be input into Equations ( 5)- (7), respectively, to calculate the potential power in units of kWh/day.

Wave-Energy Converters
The Wavestar wave-energy converter is used as a reference for the wave-energy harvesting capabilities [20], which can support a capture-width ratio (CWR) of about 40% in irregular waves.The WaveStar is an articulating point absorber attached to a floating hull.While it is not suggested as a turn-key solution for this application, the potential mounting options on the mobile aquaculture vessel are attractive and the published performance information is available for reasonable energy capture estimates.The CWR only accounts for the mechanical power capture, so η wave was set to 0.4 to account for mechanical to electrical power conversion efficiency [21].Although much higher efficiencies can be theoretically achieved, these can usually only be achieved over a narrow range of frequencies: the WaveStar performance model utilized herein does not depend on wave frequency.With the average wave period varying from about 4.5 s to 8 s throughout the region and the year, the WEC would often be required to operate in a wave period for which it is not designed.By assuming the efficiency to be low, the steep drop off in efficiency that would take place at any off-design frequency is taken into account [22].
With this CWR and efficiency, it was found that a capture width of less than 4 m is consistently sufficient for supporting aquaculture when combined with wind and solar harvesters.By narrowing down the range of CW, the values could be discretized further to identify optimal combinations.Thus, the runs detailed in this paper consider WEC characteristic widths between 0 and 4 m in 0.1 m intervals, creating 41 different potential WEC area values to test.
In order to assess the cost of the wave-energy harvesting, a relationship between the installation cost and the characteristic width is required.No publicly available data exists for the Wavestar, and, in the current nascent position of the industry, limited public data exists for wave-energy cost.Ref. [23] studies the cost of multiple WEC designs, but the results vary significantly based on design and scale.Although EUR 5 million per MW is listed, this estimate is for an installed capacity of 0.25 MW which is significantly larger than the Wavestar (30 kW) device.The Pelamis wave-energy converter, although not a great representation of the Wavestar, has some publicly available cost data and is estimated to cost around USD 6 million [24].Considering the Pelamis' installed capacity of 750 kW, the installation cost corresponds to USD 8000 per kW.Although Pelamis is very different to the Wavestar, USD 8000 per kW was used for this study due to the limited available data.USD 8000 is also a high estimate as compared to [23], which is appropriate considering the expected complications of installing on a mobile platform.
Since the Wavestar's (single oscillating body) capacity is equal to 30 kW and the characteristic width is 5 m, the cost with respect to characteristic width can be assumed as USD 48,000 per m.Clearly, this should be regarded as an approximation made due to the highly limited cost information available for contemporary WECs.

Wind Turbines
For the wind turbine swept area, the number of turbines was considered as the defining factor.The maximum number for a reasonable configuration was determined to be 8, the configuration for which can be seen in Figure 3, with the turbines (red X's) slightly staggered to avoid wake effects.Assuming two different rotor elevations can be utilized, downwind rotors can, thus, be nearly 5 diameters downstream of a rotor with an intersecting projected rotor plane.Other turbine designs may be more suitable for this application (such as vertical-axis), but a three-blade horizontal-axis wind turbine was used based on available data.However, at this level of fidelity, the only rotor-specific information utilized for optimization is an efficiency and a per-rotor cost: no other characteristics specific to the turbine morphology are assumed, and, thus, the 3-bladed horizontal morphology is largely arbitrary.Based on a 5 kW turbine from Aeolos [25] with a diameter of 6.4 m, the swept area could be calculated: The total swept area can be found by multiplying by the number of turbines, which was varied from 0 to 8 (0.2-turbine interval) for 41 different potential turbine configurations.Although 0.2 wind turbines is obviously not possible, this finer discretization is informative and can be considered an interpolation between the 5 kW turbines used as a cost basis.The turbine efficiency is estimated at η = 40%, which is lower than what might be expected for a grid-scale commercial turbine [26], to account for potential disturbances between turbines.For simplicity in this study, wind direction is not considered.
Unfortunately, no cost metrics are available for the Aeolos wind turbine specifically, but a range of between USD 3000 and USD 8000 per kW is estimated by [27] for small-scale wind turbines.USD 8000 per kW, or USD 40,000 for the entire Aeolos 5 kW turbine, is used in order to account for the increased cost of installation on a mobile vessel.This should also be regarded as an estimate due to the uncertainty in small-scale wind turbines and their installation on a mobile vessel.Additionally, while the total thrust load exerted on the relatively low maximum total turbine swept area (257.6 m 2 ) is unlikely to cause a static instability in a vessel with an approximate water-plane area of 10,200 m 2 , an explicit evaluation of the vessel stability in dynamic conditions is not carried out here.

Solar Panels
For the solar panel area, the area of the ship at the water's surface (excluding cage area) was used as a maximum value in Equation (9).
The solar area was varied from 0 to 100% of the maximum value in 2% intervals to create 51 different potential solar panel area values.A solar panel efficiency of η = 25% is used as an estimate for high-quality panels [28].A 25% solar panel efficiency implies a rating of 250 W per m 2 [29].For commercial solar panels, a cost of USD 1.50 per kW is a reasonable estimate [30].Thus, the cost of solar power is taken as USD 375 per m 2 of installed capacity.In marine environments, solar panel efficiency is known to degrade rapidly due to the effect of salt spray: thus, maintaining these efficiencies throughout the study period implies that the panels are regularly maintained.

Applying Harvester Configurations
To limit considerations for robust solutions, the starting locations within the study area were varied for each configuration.For each combination of wind, solar, and wave harvesters, 10 different starting locations were tested.The starting locations were randomized within each 10-location interval on the 100-point grid (e.g., 1-10, 11-20,. . ., 91-100) to ensure diverse starting locations.
Ultimately, by looping through all of the installed wind, wave, and solar harvesting capacities outlined in the following sections, 85,731 different combinations were produced and tested with 10 different starting locations each.The testing was completed using the route optimization algorithms identified in Section 2.2, the results of which are discussed in Section 3. The feasibility of each combination was defined as the ability to meet minimal energy requirements during the completion of the respective route optimization algorithm.

Costs
This section addresses the cost of the energy harvesting and storage components, and a comparison is made to diesel engines and fuel.Table 1 shows the installation and operations and maintenance (O&M) costs of renewable sources, energy storage, and diesel systems.The sources from which the cost metrics are obtained are shown in the last column.These values should be regarded only as estimates due to uncertainties in cost data, scale, and installation of a mobile vessel.
In order to realistically compare multiple potential systems, a net present cost comparison model is developed.The net present cost is calculated by scaling the yearly costs based on a discount rate and adding it to the installation costs to represent an aggregate cost of the system at the onset of installation.Equation ( 10) is used to calculate the net present cost, with IC being the installation cost, R the yearly costs, i the discount rate (assumed 4%), and n the project duration (assumed 20 years).
The cost metrics and net present cost calculation are to be used for system comparison in Section 3.4.As each cost component considered has significant associated uncertainty owing to the novelty of this field and technology, more complex cost models may have more substantial uncertainties, though in form they represent a more realistic and comprehensive comparison.

Route Optimization Methods
The concept of optimal foraging theory [37] is used as inspiration for a route optimization algorithm in this paper.The general theory comes from an ecological model for the prediction of animal feeding behavior and assumes the feeding strategy of a species will be optimized through natural selection.In optimal foraging theory, the decision to feed on a specific prey is dependent on an optimization of the energy expended with respect to the energy gained.With this idea in mind, the route optimization problem at hand can be achieved with a focus on renewable energy availability.With the ship able to move anywhere within the region each month, the difference between the energy potential at each location and the transportation energy needs can be used to assess the location feasibility at each subsequent timestep.

Optimal Foraging Theory Route Optimization
To complete the route optimization, optimal foraging theory was applied in multiple ways.Initially, a distance minimization approach was used to select the closest location which can support the energy needed for aquaculture and transportation at each timestep.Limitations on minimizing the distance include rewarding low movement and low velocity, which may satisfy current energy requirements but may not place the ship in the best position to maximize future energy returns, or frankly, be tractable in an actual ocean.While perhaps of some academic interest, this approach is not discussed further here due to the limited relevance of these findings for future investigations.
A contrasting objective is the maximization of the energy return.Although still unable to discretely consider future energy harvesting potential, which would rely upon ever-bolder assumptions of future forecast accuracy, this optimization incentivizes larger energy potentials that can accommodate forecasting errors, rather than seeking only to meet operating energy requirements.
A flowchart outlining the route optimization procedure for energy return maximization is shown in Figure 4. First, the aquaculture system and travel energy requirements are assessed and the ship configuration is specified.Then, the energy potential at and travel energy to all locations is assessed to determine an optimal location to move to at each timestep.The travel energy is assumed to be optimal by minimizing the velocity to reach the location within the specified timestep (with max of 2 m/s).There is instead a trade-off: a faster moving vessel may reach energy-rich locations further from its current position, but it does so at the cost of increased energy expended during travel.If no locations satisfy the energy needs, including aquaculture and travel, at any timestep during the simulation, the solution is deemed infeasible for that vessel configuration.
According to the above optimization procedure, the objective function to maximize is the energy return (Equation ( 11)), also known as the difference between the energy potential at the new location and the travel energy.The two constraints which need to be satisfied are to ensure the power potential at both the new location and the previous location satisfy the power requirements to run the aquaculture system.The power required to complete travel is included in the power requirements for both the new and previous location to ensure sufficient power is available to start and finish travel at each location (sufficient power between locations is assumed if each meets the power requirements).maxE return = E newLoc − E travel (11) subject to: P return = P newLoc − P travel ≥ P required P prev = P prevLoc − P travel ≥ P required (12) First, the energy needs and ship configurations are defined.The configuration (Section 2.1.3)includes the number and type of energy harvesters and starting location of the vessel.Starting locations were randomly seeded within each 10-location interval on the 100-point grid (e.g., 1-10, 11-20,. . ., 91-100), which meant each combination of harvesters was tested with 10 different starting locations to ensure diverse initialization.By evaluating Equation (11) to determine the optimal location for each timestep throughout one year, an optimal route can be determined that maximizes the energy potential while sustaining the aquaculture energy needs (or the solution is deemed infeasible).The algorithm results depend on the timestep.By discretizing the energy needs and energy potentials, the energy maximization routine will be carried out for monthly, weekly, and daily timesteps: these imply increasing travel velocities.The wind-and solar-resource data were available in monthly increments and interpolated for smaller timesteps, while the wave-resource data were available in 180 min increments and simply averaged over the respective timestep.Based on the Ocean Arks vessel, the algorithm applies a constraint on the ship velocity of 2 m/s.
For the initial employment of the route optimization routine, the combinations are classified by their feasibility as a mobile system and as a stationary system.Any combination of renewable harvesters that can feasibly support energy needs as a mobile system but does not harvest enough energy as a stationary system is deemed favorable, because it shows potential benefits for the use of mobility for resource targeting.

Mobile System with Reduced Transportation Costs
The sensitivity of favorable combinations to the presumed travel energy costs was investigated by presuming that travel energy requirements could be reduced significantly, Realistically, the transportation energy costs would be heavily dependent on wind speed and direction, and the sea state: this is beyond the fidelity of the available resource data over the broad study area.In order to simplify the concept, a 50% reduction in transportation energy was considered to understand the sensitivity of the study to transportation energy.

Energy Storage
The spatially and temporally averaged energy estimates available in the resource data do not consider short-timescale variations: in order to maintain uninterrupted operations, a real mobile aquaculture system will require at least some onboard energy storage.The required energy storage consists of two components: a baseline storage need, and a variable storage need based on expected energy shortages during the year.The baseline energy storage need accounts for short time-scale (e.g., daily) energy fluctuations and was set equal to the energy requirements for three days of aquaculture operation, including the largest travel energy required throughout the year.The variable storage need was included by calculating the difference between the power at the current location and power needed during each month to establish an energy surplus or deficit.Throughout the course of the year-long simulation, consecutive energy deficits were recorded.The total required energy storage capacity in Equation ( 13) is equal to the sum of the baseline need and the largest recorded consecutive energy deficit.
Energy storage was considered only for the energy maximization algorithm for simplicity.Without energy storage, the algorithm requires the aquaculture operation energy to be met during each month throughout the year to maintain feasibility.On the other hand, energy storage only requires the total yearly energy harvest to meet the total yearly aquaculture operation energy to be feasible.This means that the energy harvested during individual timesteps may be less than the aquaculture operational energy as long as surpluses in energy harvest during other months can make up for the deficits.A greater energy storage capacity can allow for larger energy-harvest fluctuations, but will also cost more.The integration of energy storage will support the identification of an optimal combination of harvesters that can support the aquaculture operation energy needs while balancing the costs of energy harvester and storage capacities.
The modified flow diagram for the energy maximization algorithm with the energy storage tracking is shown in Figure 5.The described implementation assumes perfect (100% efficient) energy storage, which is not realistic but sufficient for this study.Another limitation of the current implementation is that the energy storage needs may not be fully taken into account if extreme changes in energy harvest occur between two consecutive locations.Because the energy harvesting is split between two locations (previous and current) for each timestep, if one location has significantly less energy but the other makes up for it, the energy storage need may not be accurately calculated; however, over the examined area and timescales, the significance of any associated errors are likely to be minimal, owing to the generally shallow energy gradients in space and time.The consideration of energy storage is limited by the timescale (month) used by the route optimization algorithm, and more generally limited by the temporal discretization of the energy resource data that is available: this study presumes that the 3-day baseline energy storage needs are sufficient for the uncaptured shorter-term variations in energy availability on timescale of days.However, validation of this assumption is not available given the coarseness of the utilized data, and should be considered necessary future work for any higher-fidelity attempts at modeling these platforms.

Non-Energy-Related Required Move
A mobile aquaculture system may be motivated to move for other reasons than just energy gradients.For example, targeting healthy water conditions, mitigating environmental and pollution impacts, supporting harvesting or resupply operations, and avoiding dangerous conditions.While we do not presume an impetus for the action, a required movement to point 100, which is located close to Chignik, Alaska, at the extreme northwest of the study region, is intended to model some balance of these motives.
The required move takes two additional inputs: month to move and required speed of movement.When the required move month is reached during the route optimization process, the algorithm forces the vessel to travel to location 100 while assessing the feasibility.The effects of the required move are studied with and without energy storage.To eliminate the dependence on the move month, a two-year simulation that forces the vessel to move during odd months during year one and even months during year two is implemented for an assessment of the move velocity in the full and restricted region.This ensures selection of a vessel that can support a non-energy-related movement during any part of the year: the vessel's harvester needs to account for required movement during each month separately but also to consider the benefits of energy storage by moving back to energy-rich waters between the required moves.

Required Move with Restricted Region
When implementing a required move into the route optimization algorithm, a primary concern is the algorithm's ability to account for the future travel requirements associated with the required move.Due to the time-stepping nature of this route optimization, the algorithm does not take into account future energy potentials or required movements more than one timestep in advance when selecting the current/move location for each month.Location 100 is on the far north side of the region while the largest energy potentials are on the south side of the region for most systems.This location discrepancy means that the vessel will need to move very far from its current location to reach location 100, a movement which may be sub-optimal when considering the entire year's route.In other words, a vessel in June may make different route decisions if it is aware of an impending move in September, but in this model it lacks this foresight.Given the relatively small gradients in energy availability between adjacent points in a given month, this is not generally predicted to be a significant effect.It may be more optimal to remain closer to location 100 throughout the whole year to limit the energy required for travel, especially if the move is repeated often enough.Although less energy may be harvested while staying closer to location 100, this may be outweighed by the lower travel energy/energy storage requirements.A system that spends time in energy-rich areas, but needs to move a significant distance at a time may be able to get away with having less energy harvesters, but needs more energy storage capability for the increased movement.On the other hand, a system which spends time closer to location 100 may require more energy harvesters due to the lower energy potentials but would likely require less energy storage, because it is already prepared for the less energy-dense areas and does not need to save up as much energy for travel.
In order to assess these competing objectives, the required movement can be applied to the platform in a restricted region.Restricting the region to only locations near Chignik, Alaska means that the system will likely require an increase in energy harvesters with a trade-off of lower energy storage requirements.The restriction of the region was relatively simple to complete by selecting the energy data from the northwest quarter of the region, as shown in Figure 6.

Initial Results (Stationary Offshore Aquaculture)
An initial baseline for comparison can be established by investigating the combinations of harvesters that are necessary to support a stationary platform (3900 tons of salmon product) at 80% of the tested locations throughout the entire year (Table 2).Outfitting the vessel with a sufficient number of harvesters for a single resource is prohibitively expensive due to substantial resource variability throughout the year.Combinations of different harvesters provide a much more cost-effective solution: where solar is reduced in the winter, wind/wave availability generally increases.Because wind-and waveenergy resources exhibit similar spatiotemporal trends, augmenting one or both with solar harvesters is necessary to meaningfully reduce the overall cost.Ultimately, a combination of multiple renewable harvester types, although potentially adding complexity, leads to extremely significant savings over individual harvester types from a basic installation cost perspective.

Route Optimization Results
The optimal foraging theory route optimization with the energy maximization objective was tested by inputting the combinations of harvesters and starting locations while varying the timestep.The favorable combinations of renewable harvesters are ones that are unable to support a stationary aquaculture system but are feasible for a mobile system (Table 3).There are few favorable combinations (out of almost 100,000 tested), and the change in timestep duration does not seem to significantly affect this quantity.A daily timestep was also examined but implied a transit velocity that was infeasibly high.
The optimized route for a monthly timestep with the first combination listed in the bottom section of Table 3 is shown in Figure 7.Although the energy return maximization provides an advantageous way of rewarding high energy harvesting potentials to encourage movement, the range of favorable combinations is very limited.The narrow range of favorable combinations suggests that within this region, energy resources alone do not strongly motivate movement of renewable-harvesting aquaculture systems.As a basic sensitivity analysis, this study was repeated with travel energy costs reduced by 50%: while the range of favorable conditions were slightly expanded, there were no significant differences in the conclusions drawn.This indicates that even when associated costs are significantly reduced, there is still little motivation for a renewable-powered aquaculture vessel to move. Figure 8 shows the energy captured from each harvester type during each month of the simulation.It is clear that more wind and wave energy is harvested in the winter while more solar energy is harvested in the summer.While the mobility of a renewable-powered system did not show substantive impact, the benefits of a blend of renewable-harvesting technologies are clearly demonstrated for this application.

Mobile vs. Stationary Cost
Despite the small number of favorable combinations, the cost could still be an important factor in determining any potential benefits of an energy-resource-targeting mobile aquaculture vessel.If any favorable combination of harvesters costs significantly less than the cheapest stationary system, resource-targeting mobile aquaculture may be financially worth pursuing.Based on the cost of each renewable energy harvester type detailed in Section 2.1.4,the cost of each combination can be calculated and examined.The analysis in this section only takes into account installation costs while ignoring operational costs due to high associated uncertainties.
Table 4 compares the minimum cost of renewable-harvester combinations for a stationary versus a mobile system for each of the discussed route optimization algorithms, including combinations that are favorable with a 50% reduction in the transportation energy needs.Similarly to the identification of favorable combinations, the cost comparison does not suggest significant benefits of a mobile system.A few of the algorithms, such as with a weekly timestep and with velocities of 0.02 and 0.05 m/s, indicate some cost savings, but these are limited to less than 0.5%.Any stated cost savings would be likely overshadowed by the expenses of fixing harvesters to a mobile platform.Even with a reduction in transportation energy (right column), the cost of required harvesters is not significantly reduced.

Energy Storage
Table 5 details the results of the energy maximization route optimization algorithm with included energy storage.The most inexpensive yet feasible stationary and mobile systems were identified based on harvester and storage installation costs.When comparing the renewable-harvester combinations with energy storage to without energy storage, the results are actually very similar.In fact, the cheapest combination with energy storage is the same as one of the favorable combinations without energy storage (Table 3).Of course, the consideration of energy storage also increases the installation cost of the system.Since perfect energy storage is not realistic, the increased cost when including energy storage is not indicative of a different, more expensive system.Rather, it is simply an improved estimate under a more realistic set of assumptions that allows for short-term variability in energy resources.
The similar results from the inclusion of energy storage suggest that the ability to store energy and use it at different times during the year does not significantly outweigh the installation cost of energy storage.A system with less harvesters may be able to function with more energy storage, but the cost savings in renewable harvesters are not worth it when compared to the cost of extra energy storage.

Required Move to Simulate Aquaculture Needs
The required move serves as a representation of a non-energy-related movement of the ship, and can represent several potential aquaculture needs.The minimum cost of harvesters required to complete the route optimization with the required move can be compared to the baseline result.Because of the temporal variation in energy, the number of necessary extra harvesters depends significantly on the month of the move.Table 6 and Figure 9 show the cost of systems with a required move to location 100 during different months throughout the year.Overall, the required move almost doubles the cost of the system.This is mostly because location 100 has significantly less wind and wave energy available due to its nearshore location.In the winter months, more harvesters are required because of the low energy potentials at location 100 during those months.The addition of energy storage (ES) makes a much larger impact for a system with a required move.This is not unexpected.Ideally, a ship that must move to a less energydense area can benefit greatly from storing up energy while in the energy-rich areas.The trends, as shown in Table 7, are still the same, where a winter move is more costly due to the lower energy potential at location 100 during the winter, but the aggregate number of harvesters needed when allowing for energy storage is considerably less than when energy storage is not considered.When comparing the cost of the energy harvesters by themselves, it is clear that the number of required harvesters is reduced substantially when able to store energy.The total cost is, of course, larger with the inclusion of energy storage, reflective of more realistic energy storage assumptions.

Restricted Region
By restricting the region of interest to only the northwest corner, the trade-off between remaining close to the required move location and moving to energy-rich waters can be assessed.The results of this restricted region are shown in Table 8.The red boxes indicate where total system costs (harvester + storage) were increased by restricting the region while the green boxes show where the costs were decreased.As expected, the cost of renewable energy harvesters increases for all cases when restricting the region.This occurs because there is significantly less wind-and wave-energy potential in this quarter of the region throughout the entire year.Although the solar-energy potential is greater in this region than other regions during the summer, the increase is small relative to the decrease in waveand wind-energy potentials.Essentially, the region restriction means the ship spends more time in less energy-dense areas, meaning more harvesters are required to meet operational energy needs.Despite (or because of) the increase in energy harvesters, some move months require less energy storage, which leads to lower overall system costs.Remaining in the less energy-dense waters rather than moving across significant energy gradients means the system has a more consistent energy flow and less of a need to store energy.Less energy is also required for travel by being close to the required move location which further supports a consistent energy flow.Figure 10 compares the cost of a system which can complete a required move during the different months for the full and restricted regions.The resulting cost is very similar between the two regions regardless of month.Despite having consistently larger harvester costs, the restricted region exhibits smaller energy storage costs, which leads to only slight differences in total costs depending on the move month.When comparing back to Figure 9, it is clear that the cost of a required move is more consistent across the different months due to energy storage for both regions.

Net Present Cost Assessment
Next, various systems with and without a required move are investigated more thoroughly according to their net present cost (NPC) in Table 9.The first two rows show the optimal solutions for the basic route optimization algorithm in which the mobile system offers no net present cost savings over the stationary system.
The last three rows in Table 9 compare the costs of vessels able to complete the required move at various velocities.While the previous approach allows the system an entire month to move to the required move location, the move time or velocity in Table 9 specifies the amount of time required to complete the move or the travel speed, respectively.The lower move times and faster travel speeds account for scenarios in which the vessel may need to quickly change locations, such as for storm avoidance.These results are based on a two-year simulation that forces the vessel to move to Chignik, AK during odd months during year one and even months during year two, and is intended to consider a reliable system capable of supporting required movement during any month of the year.The increase in cost with respect to velocity is relatively large, because the system costs become more and more dependent on the travel energy for the required move as the velocity increases.As shown in Equation ( 2), the transportation energy is proportional to the cube of velocity.The significant increases in costs associated with higher velocities occur for both restricted and full region systems but are more extreme for the full region.The decrease in required move time affects the full region simulation more because of the further distances being traveled.Thus, the restricted-region system needs significantly less energy harvesters and energy storage than the full region system for shorter (faster velocity) move times.Lastly, when the move time is reduced to 3 days or less, the system (regardless of region size) requires significant energy storage to make such a quick move and it would actually be cheaper (although potentially less desirable) to have a stationary system located at location 100.In summary, the result is intuitive: restricting operations to a region proximal to the destination of a required move is advantageous from an energy perspective if those moves are required to be fast and/or frequent.More practically, a mobile aquaculture vessel should be proactive and consider known future movements (such as for harvesting, regulations, etc.) when determining its route.
Because of the different operations and maintenance costs (Table 1), the optimal harvester combinations are slightly different when calculating net present cost versus installation costs.For example, for the full region required move case (move time = 7 days), a solution only considering installation cost has 0.4 wind turbines, 40% solar area, and 10 m WEC CW, while the NPC solution has 3.2 wind turbines, 32% solar area, and 4.8 m WEC CW.The larger O&M costs of wave-energy converters leads to a solution with more wind turbines instead, suggesting the impact that O&M costs can have on the optimal solution.
The net present cost also presents the opportunity for a basic comparison to a dieselpowered system.The Ocean Arks vessel has three 4000 kW engines, which would cost an estimated USD 6 million to install, already much more than the cost of any renewable system with a slow required move time (>3 days).The diesel system is designed for movement speeds up to 2 m/s, and (including O&M and fuel costs) has an NPC of only USD 15.5 million for the 2 m/s required move speed.Thus, a diesel-powered system is likely cheaper for a high travel speed while a renewable-powered system may still offer benefits for low travel speeds.To properly compare the diesel and renewable system, they would both need to be designed for the same operation conditions.Still, it is clear that proper design (and comparison) for both a renewable and diesel system would need to factor in the necessary move speed to complete required tasks (such as storm avoidance) and maintain a reliable aquaculture system.

Basic Route Optimization
The initial modeling shows that there is potential for a cost-efficient combination of various renewable energy harvesters.Wind and wave energy have similar temporal trends (large energy in winter and small in summer), while solar exhibits the opposite trend.The opposing trends suggest a desirable, consistent source of energy when combined, and this is demonstrated by the relative cost of single-source and combined-source systems in Table 2.It is clear that a combination of multiple energy harvester types presents the opportunity for a cost-efficient energy system and was considered for mobile aquaculture using a route optimization algorithm based on optimal foraging theory.
The selected method emphasizes maximization of the energy return, which allows for automatic velocity selection and better rewards for higher energy potentials to encourage future success.Less than 10 favorable combinations (feasible for mobile but not stationary) were identified out of almost 100,000 tested.The identification of favorable combinations at all is somewhat optimistic for a mobile renewable-powered system, but the small number indicates that only a narrow band of favorable conditions exists, meaning any small perturbations or imperfections in the model may cause significant variation and eliminate the potential benefits of a renewable-targeting mobile system for the studied region.The consideration of a 50% reduction in travel energy requirements leads to little increase in the diversity of favorable concepts, demonstrating strongly that the spatiotemporal variation in the renewable resource impacts the relative viability of a mobile system more significantly than transportation costs.The more consistent energy harvest as a result of the combination of multiple harvester types leads to a more cost-efficient system while also smoothing the total (harvestable) energy gradient both spatially and temporally, effectively creating less incentive for mobility from an energy perspective.The cost assessment also confirms minimal cost savings associated with the favorable combinations over a stationary platform, even without considering the increased costs of fixing harvesters to a mobile system.
The present conclusions are confined to a strictly energy-based consideration for a particular region and do not undermine the potential economic benefits of a mobile platform under more holistic considerations (e.g., benefits of mobility related to aquaculture, like seeking improved water quality, storm avoidance, etc.) or deployed in a different region.Both of these are areas of planned future study.In the case of moving for aquaculture reasons, extra harvesters would likely be required in order to accomplish these other tasks in excess of the number noted in this study because of the transportation costs and because the most energetic sites may not be co-located with those best suited for offshore aquaculture.Ultimately, the analysis shows that strictly energy advantages of mobility are only likely to be significant in an area with substantial spatial and temporal gradients in energy availability.While other studies have not explicitly considered a mobile platform to expand the regions of viability [13], the feasibility of renewable-powered stationary platforms in some circumstances aligns with these conclusions.

Required Move
The results of the required move, as presented in Section 3.3, represent a first effort at incorporating a non-energy-related motive for movement into the algorithm.Because the cost of the required move in terms of the number of harvesters required depends so significantly on the month of the move, and that it may not be possible to plan such a maneuver well in advance, a robust vessel must be prepared for the worst, most energyintensive case.This prioritizes energy storage both because the favorable system costs are more consistent across different months of the year, and because it will generally improve the consistency and reliability of the vessel.
With a required move, the region was restricted to consider the trade-off between energy harvesters and energy storage.This generally leads to higher energy harvester costs but lower energy storage costs, while the total cost remains similar to systems optimized for the full restricted region (though this is sensitive to the component cost estimates utilized).Again, this may be due to the lack of steep energy gradients in the region.However, if the required travel time for any move becomes short, the high velocities and, thus, travel energy costs implied for vessels otherwise operating optimally in the unrestricted domain can mean significant cost benefits of a restricted region (i.e., remaining closer to the required move location).This discrepancy indicates that it is necessary to consider external, non-energy factors when considering a domain of operation.This analysis identifies a potential weakness within the algorithm: the route optimization does not take into account future energy potentials, gradients, or required moves.The route optimization is greedy in that it moves to the location of greatest energy return at the current timestep, even if that location means significantly more movement (or less available energy) later in the simulation.This sub-optimality is especially present with a required move and large travel speeds, where the travel energy is significant and unavoidable.It is likely that large temporal energy gradients would also significantly hinder the optimality of the route optimization, as an optimal location for one month may be significantly different than the next, meaning energy returns of future timesteps may be significantly impacted by decisions made in the current timestep.With respect to future knowledge of energy potentials, this limitation is reflective of reality: resource forecasts decrease in accuracy for more distant time periods.However, in the case of a scheduled activity, the algorithm's inability to incorporate this future requirement into present decisions is less realistic.

Economics
The net present costs of various systems are presented in Section 3.4.The cost of the monthly, route-optimized mobile solution is still very close to the cost of a stationary system, suggesting little benefit of energy-potential targeting in the studied location.When considering various required move velocities in both the full and restricted regions, it is clear that the required move velocity has a large impact on the optimal solution.A diesel-powered vessel (as designed for Ocean Arks) is likely cost-efficient when higher travel velocities are required but a more in-depth study is needed to compare at different velocities.Notably, diesel hybrid systems, which may be attractive for their reliability offshore, energy storage, and (relatively) higher travel velocities, were not considered here.Inclusion of a propulsion-system cost model in future work will allow the equitable analysis of such systems.
Additionally, the harvester cost models employed herein, particularly for wave energy, are highly approximate, with the uncertainty elevated further from the unknown installation costs of these devices on a mobile aquaculture platform.There is little open-source cost and performance data available for WECs at this scale.In the absence of this data, a broad sensitivity study is necessary to better understand the impact of harvester cost on the study conclusions: this computationally intensive work is to be included in a future study.

Environmental Considerations
At present, regulations for stationary aquaculture systems are highly region-specific to facilitate the understanding and potential mitigation of environmental risks [13].Existing regulations may already geographically restrict the regions in which a mobile aquaculture system can operate, and mobilizing such a platform in general introduces serious complications and potentially exacerbates environmental risk.An examination of these risks is beyond the scope of the presented work, but it is imperative that these implications are understood, both to evaluate the overall practicality of mobile aquaculture and to ensure these systems are beneficial to society and the environment.Because existing development pursuits [6,7] emphasize the mobility of these platforms more so than their renewable operation/optimization, this may be an urgently necessary area of inquiry.

Conclusions
Energy-optimal routing of a renewable-powered mobile aquaculture platform was considered.Multiple route optimization algorithms based on the concept of optimal foraging theory were employed to identify optimal energy harvesting routes and favorable combinations of wind, wave, and solar-energy harvesters.Individual energy resources exhibit significant spatial and temporal variation over the studied coastal waters of southcentral Alaska: the most cost-effective solutions consisted of multiple energy types (wave, wind, solar) which leads to a smoothing in both space and time of the total (harvestable) energy gradients.This smoothing generally decreased the need for the platform to travel in order to capture sufficient energy.Reducing the assumed cost of transit by 50% did not substantially alter these conclusions.In summary, over the studied area, utilizing a blend of wind, solar, and wave-energy harvesters has a more significant impact on reducing the overall installed harvester capacity than allowing the system to move to seek more energy-rich conditions.
When the platform does move, either as a result of route optimization or an externally imposed requirement, the transit velocity is found to strongly drive system cost: if a relatively large transit velocity is required, restricting operation to locations proximal to the required travel destination may be advantageous even if this restricted region has lower renewable energy potential.
The completed study is of relatively low fidelity and has several significant sources of uncertainty.Firstly, a coarse model of the energy harvesters was utilized, due in part to the limited published cost/performance data for small-scale wind and WEC converters.Because the cost-optimal blend of energy harvesters depends upon these cost models, and the overall cost depends additionally on accurate performance data, both must be refined for the accuracy of an optimal system design and cost to be improved.In particular, a known power matrix for a WEC could be coupled with local frequency-resolved sea-state information, to replace the current estimate of a static efficiency.Further, solar panels especially are known to rapidly degrade in a marine environment: while operations and maintenance costs are largely unknown for energy harvesters installed on a mobile platform, they are likely to be significant and may affect the optimal system design.In the absence of more refined models, future work will perform a sensitivity analysis in converter efficiencies and cost per capacity to understand the implication of this uncertainty on the optimal system design and cost.
Secondly, energy feasibility is only one facet of the optimal operation of a renewablepowered mobile aquaculture vessel.In fact, existing proposed vessels are powered traditionally, and will select their routes based mostly on the implications for aquaculture yields and the overall cost of operation.Future work aims to modify the energy optimization cost function presented in this work to capture this more holistic picture of relevant operational considerations: ultimately, this cost function is likely to have units of dollars-per pound of yield, rather than the dollars per energy to which the present study was restricted.It is also critically important that developers and regulators pursue a mutual understanding of the potential environmental impacts of mobile aquaculture and the permitting and regulatory environment in which these vessels will operate, as the implications on operating requirements and route optimization (and thereby, optimal design) are likely to be significant.

Figure 1 .
Figure 1.Map of wave-energy potential off the coast of Alaska in selected region.

Figure 3 .
Figure 3. Diagram showing turbine configuration for maximum of 8 wind turbines mounted on Ocean Arks ship.

Figure 4 .
Figure 4. Route optimization procedure for maximizing energy return.

Figure 5 .
Figure 5. Flow diagram of energy maximization algorithm with energy storage tracking.

Figure 6 .
Figure 6.Restricted region showing locations with available wave resource for month of July.

Figure 9 .
Figure 9. System cost for required move during respective months.

Figure 10 .
Figure 10.System cost for required move during respective months with energy storage, comparing the full region to restricted region results.

Table 1 .
Energy harvesting and storage costs.

Table 2 .
Minimum harvester combinations and costs to support stationary aquaculture at 80% of the locations tested.

Table 4 .
Optimal harvester combination cost comparison for different algorithms capable of supporting Ocean Arks vessel and aquaculture pens.

Table 5 .
Least expensive energy harvester combinations to support mobile and stationary aquaculture with energy storage based on installation costs (energy maximization with monthly timestep).

Table 6 .
Best required move combinations without energy storage (full region).

Table 7 .
Best required move combinations with energy storage (full region).

Table 8 .
Best required move combinations with energy storage in the restricted region.Red boxes indicate cost increases while green boxes indicate cost savings over the full region.

Table 9 .
Net present cost (million USD) of solutions with various required move times and velocities.For each column, the net present costs are compared and colored in green, yellow, and red from least to most expensive, respectively.The cost of a stationary system at the required move location is also shown for comparison.