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Aquaculture Economics & Management

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Efficient and economical way of operating arecirculation aquaculture system in an aquaponicsfarm

Divas Karimanzira, Karel Keesman, Werner Kloas, Daniela Baganz & ThomasRauschenbach

To cite this article: Divas Karimanzira, Karel Keesman, Werner Kloas, Daniela Baganz &Thomas Rauschenbach (2016): Efficient and economical way of operating a recirculationaquaculture system in an aquaponics farm, Aquaculture Economics & Management, DOI:10.1080/13657305.2016.1259368

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AQUACULTURE ECONOMICS & MANAGEMENT http://dx.doi.org/10.1080/13657305.2016.1259368

Efficient and economical way of operating a recirculation aquaculture system in an aquaponics farm Divas Karimanziraa, Karel Keesmanb, Werner Kloasc, Daniela Baganzd, and Thomas Rauschenbacha

aDepartment of Surface Water and Maritime Systems, Fraunhofer I0SB-AST, Ilmenau, Germany; bDepartment of Biobased Chemistry & Technology, Wageningen University, Wageningen, The Netherlands; cDepartment of Endocrinology, Institute of Biology, Humboldt University Berlin, Berlin, Germany; dDepartment of Biology and Ecology of Fishes, Leibniz–Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany

ABSTRACT In this article, optimal control methods based on a metabolite- constrained fish growth model are applied to the operation of fish production in an aquaponic system. The system is formulated for the twin objective of fish growth and plant fertilization to maximize the benefits by optimal and efficient use of resources from aquaculture. The state equations, basically mass balances, required by the optimization algorithms are given in the form of differential equations for the number of fish in the stock, their average weight as mediated through metabolism and appetite, the water recirculation and waste treatment, hydroponic nutrient requirements and their loss functions. Six parameters, that is, water temperature, flow rate, stock density, feed ration size per fish, energy consumption rate and the quality of food (percentage of digestible proteins) are used to control the system under dynamic conditions. The time to harvest is treated as a static decision variable that is repeatedly adjusted to find the profit-maximizing solution. By modeling the complex interactions between the economic and biological systems, it is possible to obtain the most efficient decisions with respect to diet composition, feeding rates, harvesting time and nutrient releases. Some sample numerical results using data from a tilapia-tomato farm are presented and discussed.

KEYWORDS Aquaponics; dynamic modeling; economics; optimization

Introduction

Aquaponics is gaining increased attention as a bio-integrated food production system and belongs to the class of large-scale systems, which can be viewed as a network of interconnected subsystems. A common feature of these systems is that subsystems must make control decisions with limited information. The hope is that despite the decentralized nature of the system, global performance criteria can be optimized. There are mainly two objectives in

CONTACT Divas Karimanzira divas.karimanzira@iosb-ast.fraunhofer.de Department of Surface Water and Maritime Systems, Fraunhofer I0SB-AST, Ilmenau, Germany. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uaqm. © 2016 Taylor & Francis

aquaponics. The first and most important objective is the biophysical objective, which is raising the fish to maturity with low mortality while producing enough waste for the crops. The second objective is a commercial one, which forces the fish and the plants to be raised in an economic way. Unfortunately, the nature of recirculating system operation dictates that producers stock their systems at high densities to overcome the higher fixed and variable costs normally associated with closed system operation (Kazmierczak, 1996).

However, as the use of resources in aquaponic systems gets intensified, the role of management ability becomes critical to their economic success. This fact is confirmed in Tokunaga, Tamaru, Ako, and Leung (2015), who collected economic and production information from three aquaponic farms to investigate the economic feasibility of the aquaponics industry in Hawaii. They conducted sensitivity and decision reversal analysis to inves-tigate how output prices and operational cost parameters affect the overall economic outcome. At this point, it can be noted that the combination of aquaponic dynamics and the economics of producer decision making is scarce in the literature.

In Cacho, Kinnucan, and Hatch (1991) a bioeconometric model of pond catfish production was developed and used for the determination of cost- effective feeding regimes. Some researchers have also used bioeconometric models of varying degrees of sophistication to examine open system rearing of fish (Karmierczak, 1996; Liu & Chang, 1992; Love, Uhl, & Genello, 2015). However, no study has examined the optimization of recirculating aquaponics systems incorporating not only realistic metabolite-constrained fish growth over time, but also the economic and nutrient requirement constraints for crop growth. Metabolite-constrained fish growth model describes a model that considers the effects of water quality on fish growth and mortality.

In this article, a model for determining cost-effective feeding regimes and dietary protein composition for fish reared in aquaponic recirculation aquaculture (RAS) and maximizing the fertirrigation releases is developed, and the interplay among ration size, feed quality (protein percent), water quality and harvest date are quantified.

System description

The features of the “Innovative Aquaponics for Professional Application” (INAPRO) aquaponics system can be seen in Figure 1. It is based on the ASTAF-PRO technology as described in Kloas et al. (2015). In contrast to conventional aquaponics, the INAPRO system is a double recirculation system whose two subsystems, a recirculating aquaculture system (RAS) and a recirculating hydroponics unit within a greenhouse, are mutually unidirectional connected to (a) deliver fish water containing nutrients into

2 D. KARIMANZIRA ET AL.

the hydroponic reservoir as fertilizer and (b) return the condensed plant evapo-transpirated water into the aquaculture part to minimize the overall water consumption of the system and to ensure that optimum conditions can be set up in each subsystem independently from the other.

Practical experience on the INAPRO system and discussions with several aquaponics stakeholders have shown that, for optimal operational manage-ment of aquaponics, trying to find a holistic criterion for optimizing both parts at the same time is not necessary if the nutrient buffer tank between the fish and the crops is well dimensioned in the design phase. Therefore, this article deals with the optimization of the RAS using information about nutrient uptake from an optimized hydroponic system to achieve the holistic objective of the aquaponic system.

Methods

Here we describe the optimization framework and present the general management problem faced by system managers. In aquaponics, management primarily affects the variable costs associated with short-term decision making over a fish growth-cycle, which is assumed to be significant for the crop growth cycle as well. Along with the direct financial costs related to stocking, feeding, and electrical power use, indirect costs can arise when the waste filtering devices do not completely remove metabolic waste material (Rackocy, Masser, & Losordo, 2006). These indirect costs appear in the form of retarded fish and crop growth and increased fish mortality. Considering that RAS managers seek to maximize returns above variable costs and objectives are conflicting, a multi-objective approach is put forward, which reduces to the maximization of discounted profits. The cost function to be maximized is the profit J(tf), which is the difference between the wholesale purchase price

Figure 1. Structure of an aquaponics management system as a two-player system.

AQUACULTURE ECONOMICS & MANAGEMENT 3

per unit weight Cw times the average weight W times the number of fish N and the cost spent throughout the growout cycle t ∈ [t0, tf] of the fish L (tf), i.e.,.

J tf� �¼ CwW tf

� �N tf� �� L tf� �

ð1Þ

This problem falls in the category of optimal control problems with general path and boundary constraints, free final time tf and with some unknown parameters.

maxu tð Þ;tf ;p;x0

J x tð Þ; u tð Þ; d tð Þ; t; p; tf� �

subject to _x tð Þ ¼ f t; x tð Þ; u tð Þ; d tð Þ; p½ �; x 0ð Þ ¼ x0 ð2Þh t; x tð Þ; u tð Þ; d tð Þ; p½ � ¼ 0 ð3Þ

g t; x tð Þ; u tð Þ; d tð Þ½ � � 0 ð4Þ

x tð ÞLB� x tð Þ � x tð ÞUB

; u tð ÞLB� u tð Þ � u tð ÞUB

; and pLB � p � pUB

where f is a vector of differential equations describing the system state dynamics, h is a vector of equality constraints, g is a vector of inequality constraints, the state vector x tð Þ ¼ W tð Þ N tð Þ L tð Þ½ �

T , x0 comprises the process initial con-ditions, the control vector u tð Þ ¼ T tð Þ R tð Þ Re tð Þ S tð ÞDC tð Þ tf

� �T , the superscripts LB, UB indicate the lower and upper bounds for the parameters p, states and the controls, and d(t) are disturbances. The state variables are the average weight of the fish stock W, the average number of fish in the stock N, and the average cost of the fish stock L (the costs incurred by aeration, heating, electricity, extra fertilizers for the crops, etc.).

The variables applied in this article as control variables are mainly the tem-perature T, the water recirculation and waste treatment Re accounting for how much water is allowed to be recirculated without violating the metabolite critical values, the stocking density S or the amount of space per fish Sp, the ration size R, and the dietary protein composition DC. The three variables R, Re, and Sp are dimensionless with a value domain between zero and one. For example, the ration size R is defined in relation to the maximum ration which would lead to satiation. R is equal to 0 when there is no feeding and approaches 1 at satiation level. The rate of aeration (At) is automatically adjusted in response to the fish oxygen demand to keep the dissolved oxygen level at 5 ppm. The initial stocking density is fixed for any given set of simulation optimizations over, but allowed to vary between optimizations. The results of optimizations of different stocking densities are compared and analyzed for the impact of S on the potential returns. Because initial stocking density S0 and to are fixed, i.e., a given number of fish/l are given at the beginning of the cyclogram, the control problem is reduced to finding the optimal combination of R, DC, Re and tf. Furthermore, an external

4 D. KARIMANZIRA ET AL.

variable, considered as a disturbance to take care of the crop nutrient requirements ; is also included.

Additionally, the technical and biological relationships embedded in Equation 2 can be defined by the following equations:

Wt ¼ w W t � 1ð Þ; Ft tð Þ; Bt;i tð Þ; R tð Þ; DC tð Þ;T;BE� �

; ð5ÞNt ¼ n S0;Mtf g; ð6Þ

Lt ¼ l Nt;Wt;Tt; Sp;Q tð Þ;G; ;t tð Þ;KL� �

ð7ÞAt ¼ afWðtÞ;TðtÞ;RðtÞ;DCðtÞ; SðtÞ;QðtÞg; ð8Þ

Bt;i ¼ b W tð Þ; S tð Þ; Ft tð Þ;R tð Þ;Re tð Þ;DC tð Þ;T tð Þ;BE;MEf g; i ¼ 1 . . . N; ð9ÞEt ¼ e At;W tð Þ;T tð Þ;R tð Þ;DC tð Þ; S tð Þ;Q tð Þf g; ð10Þ

Ft ¼ r W tð Þ;T tð Þf g; ð11ÞMt ¼ m S tð Þ;W tð Þ;DC tð Þ;Bt;i tð Þ

� �ð12Þ

where Bt, i is the production rate of metabolite i at time t, BE is the biological filter efficiency, ME is the mechanical filter efficiency, G is uneaten feed, Q is the water flow, Et is the rate of energy use, Mt is the rate of mortality, ;t ¼ ω{Pd, Nd} is the crop nutrient requirements for phoshorus P and nitrate N and KL are diverse costs contributing to the discount function. Furthermore, P and N releases from the RAS (;s) should be sufficient to meet ;d at least by a fraction e van Straten, van Willigenburg, van Henten, and van Ooteghem (2010), i.e.,

;s � E � ;d ð13Þ

Equations (1)–(13) describing the optimization model require that the technical and biological relations embedded in the model be fully described and must be empirical. The bioenergetic fish growth model simulates protein and fat dynamics as affected by temperature, body weight, ration size, and diet quality. Besides these factors, fish growth is also a function of other environ-mental factors, such as dissolved oxygen demand (DO), unionized ammonia (UAN), space or stocking density (S), and biochemical oxygen demand (BOD) FAO (2014). The differential equation describing the fish growth including all important limiting environmental factors and assuming a multiplicative influence may be expressed as follows:

dWdt¼ S�WD

�YN

i¼1Bi ð14Þ

where N is the number of metabolites and Bi is the metabolite and WD is the theoretical growth rate. Detailed description of the effects of metabolites on fish feed consumption, mortality and reduced growth and how they are implemented can be obtained upon request. The effects of space, Sp, UAN and DO on growth and mortality of the fish are shown in the following as examples. An inverse relationship between survival rate and stocking density

AQUACULTURE ECONOMICS & MANAGEMENT 5

has been found in several studies (Winberg, 1971; Abdel-Fattah, 2002; Osofero, Otubusin, & Daramola, 2009, etc.). The function of the limiting relation assumed here is similar to limiting-growth models for other species under other conditions (Winberg, 1971)

dW=dt ¼WDSp; ð15Þ

where Sp ¼ 1 − (W/Wmax)μ, Wmax is the limiting weight which is a function of space, and μ is an empirically determined constant. The relationship between Sp, and the mortality rate was determined using the anti-logistic model (Chachuat, Srinivasan, & Bonvin, 2009) expressed as

S ¼ 1 � Mt; and Mt ¼ M1= 1þ ðM1 � M0Þ=M0½ �e� ztð Þ ð16Þ

where Mt, is the cumulative mortality rate (%) at the experimental period t, M∞ is the theoretical asymptotic cumulative mortality rate, M0 is the hypothetical cumulative mortality rate at the beginning and z is the mortality coefficient.

Each food composition j is assumed to have a separate growth rate constant and a separate mortality MQ(%) with mortality rate constant Pj. The optimization problem involves deciding when to switch from one food to another. Consequently, we can derive the number of surviving fish N from the mortality rate constant Pj given the initial number of fish in stock to

MQ ¼ � PjS ð17Þ

Furthermore, mortality is also induced by the water quality. While no complete information exists on tilapia mortality for various levels of UAN over time, the acute toxic UAN levels for channel catfish (Ictaluras punctatus) closely resemble those for tilapia. Using catfish data from Tomasso, Simco, & Davis (1979), an expression for daily percent UAN-induced mortality (MUAN (%)) could be derived:

MUAN ¼ � 100� 0:51

aþbe� cUAN tð Þ � 1:0� �

ð18Þ

where a, b and γ are parameters with corresponding values of 0.16, 1420.5, and −1.97, respectively. Given (Equation 18), mean daily mortality increases gradually as UAN concentrations rise to 3 mg/liter and then increases rapidly for UAN concentrations between 3 and 5 mg/liter.

There is data for DO-induced mortality in tilapia (Drummond, Murgas, & Vincetini, 2009). With this information, mean daily mortality relative to DO level MDO (%) was calibrated as

MDO ¼ 100� 0:51

e� cðDO tð Þ� DOsat � 1:0� �

ð19Þ

where γ is a parameter equals to 19 L/mg. DO equation (19) provides for subtle mortality effects just below the maximum lethal DO concentration of

6 D. KARIMANZIRA ET AL.

1.5 mg/liter. These mortality effects increase gradually down to DO concentrations of 0.8 mg/liter and then increase rapidly as DO concentration falls below 0.7 mg/liter.

To be able to calculate these effects of the environmental parameters, their production rates are required. The general metabolite production Equation (9) is a function of the fish weight W, the stocking density (S), the individual fish appetite at time t (Ft), the number of fish/liter (D), the feed ration size (R) and the water treatment efficiency (E). For example the production of UAN at time t is calculated as a fraction of the TAN in the system (Emerson, Russo, Lund, & Thurston, 1975), i.e.,

UANt ¼ 0:008� explog 9:375ð Þ

30�T

� �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Conversion

� S�DC�W

1000�D�Ft

�R|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

TAN Production

; ð20Þ

and finally, due to recirculation through the biological filter and the residual UAN from the previous time step,

UANt ¼ 1 � BEð Þ UANt þ UAN t � 1ð Þð Þ ð21Þ

Exchange of water (WT ¼ 1 − Re(t)) was allowed, so that more fresh water is added to the system at higher limiting metabolite concentrations.

The consumption of oxygen (BOD), arises from three sources; fish respiration (BODfish ¼ KM1 * S * Wσ), oxidation of ammonia compounds by autotrophic bacteria (BODox ¼ KM2 * D * S * Ft * R), and the decompo-sition of organic solids by heterotrophic bacteria ðBODox

ME Þ (Wheaton, Hochhei-mer, & Kaiser, 1991), where ME is the mechanical filter efficiency, KM and σ are constants associated with each oxygen demand.

Sub-models for energy use in the aquaponic system are also required. The amount of pumped water Q is proportional to the energy Ep[kWh] used for this purpose, i.e., Q ¼ Ep �

gpsq�g�h where ρ is the water density [kgm−3], g is

the gravity constant acceleration [ms−2]; ηps is the pumping system efficiency and is the height of pumping. Energy used for aeration is estimated by At ¼

DOdOTp

, with OTp ¼ OTsbDOs� DOp

9:07 1.024T − 20a and DOd is oxygen demand (kg/l/day) and OTp represents oxygen transfer rate into the rearing basin (kg/kWh) and is defined as in (Boyd et al., 1986), where OTs is standard oxygen transfer rate (kg/kWh) (provided by the aerator manufacturer); DOs is oxygen saturation level at temperature T; DOp is the dissolved oxygen level in the rearing basin; a ¼ 0.77 and ß ¼ 0.94 L/mg are correction factors.

The last functional constraint is related to the greenhouse crop nutrient requirements ;t. Real data for a good year from a tilapia-tomato farm was used as in Mischan, Passos, Pinho, and Carvalho (2015) to fit a logistic function to determine the model for N&P uptake (;t) by the plants during the growth cycle.

AQUACULTURE ECONOMICS & MANAGEMENT 7

The models of the discount function L are described in the following. The loss function directly associated with raising the fish to a certain weight can be written as

L1 ¼ K1WD T � TNð Þ þ K2WD þ K3; 8T > TN ð22Þ

where T0 � T � Tmax and zero otherwise, WD is the actual average weight of the individual fish, is the water temperature; T0 is the empirically determined zero growth temperature (zero appetite); Tmax is the maximum temperature for growing the fish, Ki are estimates of the costs, and TN is the temperature of the source water. The first term accounts for the costs of heating replaced water and the second term accounts for other costs which depend on fish weight WD. The third term K3 is the cost for fingerlings.

The loss function L2 associated with the feed depends on both the cost of the food and the cost of waste treatment for the uneaten food.

L2 ¼ K5WDGþ K24WD G � Vð Þ ð23Þ

where V and G are dimensionless food variables which are proportional to food eaten and the food fed, respectively. K24 is the cost of waste treatment of the uneaten food and K5 is the cost of food ($/kg), which is calculated as a function of the protein content using the function K5 ¼ δ1 þ δ2 · DC, where and DC is the dietary protein (kg protein /kg feed). In the equation, the intercept represents the non-protein part of the feed and is affected by the changes in the cost of marketing and processing, as well as by the prices of other ingredients of the feed. Therefore, the protein cost of the feed is represented by the slope δ2. The coefficients in the equation were estimated from prices of tilapia feeds of different protein contents and correspond to the values of δ1 ¼ 0.112 and δ2 ¼ 0.543 as also estimated in Hicks (2015).

The discount function for the recirculation and water treatment is mainly a function of the cost allocated to metabolite-induced reduction in fish growth, heating the water, waste treatment and water pumping. It can be expressed as follows:

L3 ¼ K11Q 1 � Rð Þ T � TNð Þ þ K12Wa T � TAð Þ þ K21Q 1 � Rð Þ þ K22 QRð24Þ

where Kij are estimates of costs, TA is the ambient temperature of the atmos-phere (as contrasted to TN, which is the natural temperature of source water), and a is an empirically determined exponent. The first term (K11) represents the cost of heating the incoming source water; the second term (K12) repre-sents heat loss to the atmosphere from holding tanks; the fourth term (K21) represents costs of pumping water from the source, pre- and post-treatment: and the last term (K22) represents costs of pumping water through the

8 D. KARIMANZIRA ET AL.

recirculating system and treating it. It is assumed pump operation and maintenance costs are proportional to flow rate Q.

A preliminary estimate of the costs allocated to space and maintenance indicates the cost L4, can be represented as

L4 ¼ K23Wc 1 � Sp� �� c=l

ð25Þ

where K23 is an estimate of space and maintenance related costs and γ is an empirically determined exponent. From (Equation 13), the loss function due to the nutrient supply deficit can be expressed as

L5 ¼ Kfert � E� ;d � ;sð Þ ð26Þ

This loss function can be easily implemented as a benefit function in which the nutrients produced by the fish are considered as an extra revenue Cf,j ¼ ;j � Pf, where Pf is the price for (P,N) fertilizer and Cf,j is the cost of pro-duced fertilizer at time j.

Solution strategies

Having specified the dynamic constraints embedded in equations (5)-(26), the solution of the optimization problem requires a strategy for dealing with the large number of potential control variables and free final time. Two solution strategies were implemented: 1. A multidisciplinary feasible approach (MDF) is applied to solve the

differential equations within the objective function. This approach implicitly satisfies the constraints, so there is no need to add explicit equality constraints to the optimization problem. A terminal individual fish weight WF is used as a boundary condition.

2. Another optimizing technique also implemented is an indirect method, which is based on choosing adjoint variables to satisfy the necessary conditions for an optimal trajectory as suggested by the Pontryagin’s maximum principle. The Hamiltonian H and the differential equations for the adjoint variables (costates) k tð Þ ¼ kW tð Þ kN tð Þ kL tð Þ½ �

T can be written directly.

H ¼ kWdWdtþ kN

dNdtþ kL

dLdt

ð27Þ

The required conditions for optimality are:

ddt

x tð Þ ¼@H@k

x tð Þ; u tð Þ; k tð Þ; tð Þ ð28Þ

ddt

k tð Þ ¼@H@x

x tð Þ; u tð Þ; k tð Þ; tð Þ ð29Þ

AQUACULTURE ECONOMICS & MANAGEMENT 9

@H@u

x tð Þ; u tð Þ; k tð Þ; tð Þ ¼ 0 ð30Þ

and the boundary conditions at t ¼ 0 are: xðtÞ ¼ x0; it will be assumed that the final weight is fixed at w(tf) ¼WF grams. The desired value of the control variables will maximize the Hamiltonian, H, as determined by the partial derivatives @H

du . If the control variable is not limited by constraints, the partial derivative of the Hamiltonian with respect to u will be zero. As described previously regarding induced mortality, the feed quality uses the subscript j to distinguish between possible food candidates. At each instant in time the value of j is chosen in a second step which maximizes H(j).

The algorithm has been implemented in Matlab1� environment to obtain the optimal control for the recirculation aquaculture system. For numerical integration of the states and costates differential equations, a Runge–Kutta method was used. The free final time tf, was determined based on a two-level search procedure. The upper-level determines the final time by the method of “divide and conquer” and the lower level optimizes the dynamic system with the final time decided from the upper level.

Mechanical and biological filter efficiencies were fixed for any given set of simulation optimizations over tf, but allowed to vary between optimizations. The results of different sets of simulation optimizations could then be compared and analyzed for the impact of technology on potential returns. Feed protein composition (DC) and initial stocking density (S0) were similarly treated.

Results

In this study, tilapia production was staggered in different rearing tanks, where the optimization was conducted in 100 g increments. The advantage here was that, for each group, individual optimum stocking density and temperature could be set. The tilapias were started at 10 g and grow up to 700 g under different conditions as described by different scenarios here. Prices were obtained from surveys of major suppliers, budget-based analyses of recirculating systems and trade journals. In the base scenario, we assumed perfect biological filter (BE ¼ 100%) and mechanical filter (ME ¼ 100%) operation for maximum system performance regardless of feed quality and optimized the system for net returns to raise the tilapia to 700 g considering all costs (e.g., food, heating, pumping, water use, etc.). In the second scenario, we investigated the effect of changes in the cost of water treatment. In further scenarios, we examined the sensitivity of the aquaculture system to the waste-water treatment and stocking density.

Thereby, we varied the efficiency parameters of the mechanical and the bio-logical filters and every time re-optimized the system to get the 700-g tilapia. With changes in the efficiency of the biological filter more/less ammonia is

10 D. KARIMANZIRA ET AL.

nitrified. This leads to an adjustment of optimal control strategy and another growth rate. Thus, in general, with a better biological filter a larger ration size and a higher protein content of the feed is possible, the raising period is thus

Table 1. Fish growth model coefficients. Constant Value Description Unit Source

DE 2.800 Digestible energy content of ration

kcal/g

Ω1 2.763 West (1966) Ω2 1.037 West (1966) Ω3 4.123e12 West (1966) Ω4 31.639 West (1966) aR 0.426 – [kcal/(g day)] Cacho (1984) bR 0.685 Appetite

exponent Cacho (1997)

γ1 0.020 Coefficient estimated algebraically γ2 0.147 – 1/°C Coefficient estimated algebraically aP 0.007 – kcal/(g day) Gatlin, Poe, and Wilson (1986) ψP 0.006 – kcal/(g day) Gatlin et al. (1986) µP 3.128 Gatlin et al. (1986) τP 1.419 Gatlin et al. (1986) aF 0.009 – kcal/(kcal day) Gatlin et al. (1986) ψF 0.027 – kcal/(kcal day) Gatlin et al. (1986) µF 0.043 – kcal/(kcal day) Gatlin et al. (1986) a 0.77 Reis (1987) b 0.94 Reis (1987) DOs [0.0146 0.0113

0.0091 0.0076 0.0065]

Oxygen saturation level for 0°C, 10°C,

20°C, 30°C, 40°C

g/l

Table 2. Parameters used in the simulation. Parameter Value Description Unit

t0 1 Initial time 0 < t_0 < 365 days, 1.jan ¼ 1

tf 350 End time t_H > t_0 days, 1.jan ¼ 1

Wf 700 Weight at harvest time per fish g W0 10 Initial weight g S0 0.1 Initial stocking rate fish/l phi 0.55 Converts energy into weight units 0.33:0.86

depending on proportion of water, protein, fat in the fish body

KP 5.650 Energy content of fish protein kcal/g KF 9.450 Energy content of fish fat kcal/g P 0.2 Protein content per unit of fish weight F 0.1 Fat content per unit of fish weight OTs 1900 Standard oxygen transfer rate

(depends on the aerator) g/kWh

DOp 0.005 Dissolved oxygen level to be maintained g/l UANmax 0.0001 Maximum un-ionized ammonia level g/l BF 0.90 Biological filter (0 � BF < 1) MF 0.5 Mechanical filter (0.25 �MF � 1) D 0.26 Percentage protein 0.26 … 0.42

AQUACULTURE ECONOMICS & MANAGEMENT 11

shorter and the profit greater. The mechanical filter is used in the model only in the calculation of oxygen consumption. With lower filter efficiency, more solids remain in the water, and thus more oxygen is consumed by the decomposition of the solids. Thus, the mechanical filter has no large effects on the course of the control and state variables. However, the profit is higher with a better filter. Table 3 summarizes the effects of these parameters on the

Table 3. Scenario specifications.

Scenario BBE MME Stocking denstiy

So [g/liter] Net Return

[cents/liter/day] Days to

Harvest [days]

Base 0 1 1 0.13 0.034 265 1 1 0.5 0.13 −3% þ2 2 1 0.25 0.13 −15% þ15 3 0.95 1.0 0.13 −12% þ14 4 0.80 1.0 0.13 −80% þ86 5 1 0.5 0.07 −3% þ2 6 1 0.5 0.13 þ79% þ2 7 0.95 0.5 0.13 þ44% þ20 8 0.90 0.5 0.07 −44% þ55 9 0.85 0.5 0.13 −70% þ77

Figure 2. Scenario 7 (Fish growth, Stocking density and Controls).

12 D. KARIMANZIRA ET AL.

profit and the duration of raising the fish. It shows optimal simulated returns/ liter, time-to-harvest, and returns/liter/day for varying BE, ME, and DC. Some of the important parameters used in the models are listed in Tables 1 and 2.

Table 3 shows that under the base conditions, a 700-g tilapia can be pro-duced in 265 days with net return of 0.034 cents/liter/day using a 20% dietary protein. Scenarios 1–4 show the increase in days to harvest and declination in net return if the efficiency of the wastewater treatment units decrease with effects of the biological filter being predominant. The returns fall by over 80% from optimal baseline levels for BE below 80%, irrespective of the ME level. The drop due to the ME is caused by the solids removal inefficiency and the need for increased aeration.

In Scenarios 5–8, the sensitivity of the system to stocking density was analyzed. Hereby, the experiments were repeated, but fixing the ME at 50%, and allowing a range of stocking densities and biofilter efficiency levels. The results show that increases in stocking density produce no additional increases in the optimal 267 days required to produce a 700-g tilapia. We

Figure 3. Scenario 7 (P, N, SS, UAN and Water exchange).

AQUACULTURE ECONOMICS & MANAGEMENT 13

expected this result, because of the relative absence of metabolic feedbacks in the biological filter. The low biochemical oxygen demand in the mechanical filter allows returns to increase as stocking density increases, up to a maximum of 0.061 cents/liter/day for an initial density of 0.13 fish/liter, which is the recommended maximum stocking density in real systems (Osofero, Otubusin, & Daramola, 2009). Further numerical experiments show that the decrease in efficiency of the BF has significant effects on returns and time to harvest. For example, a decrease in BE from 1.0 to 0.95 leads to an approxi-mate 20-day increase in the time to harvest for all levels of stocking. However, decreasing BE to 0.85 causes a 55-day harvest time increase for a density of 0.07 fish/liter and increases 77 days for the densities of 0.13.

Figures 2 and 3 show, as an example, a detailed description of the optimization results for Scenario 7 in which the biological filter and mechan-ical filter efficiencies were chosen to 0.95 and 0.5, respectively. Under optimal control conditions, both the ration size R as well as the protein content DC were always as high as possible, but as soon as the ammonia concentration

Figure 4. Scenario 8 (Fish growth, Stocking density and Controls).

14 D. KARIMANZIRA ET AL.

approached the critical limit of 0.2 mg/l, the values went down again due to water exchange (fresh water is supplied into the system). The results of Scenario 8 in Figures 4 and 5 show the retarded fish growth and greater requirement for fresh water supply.

Discussion

The results achieved indicate that suboptimal management can eradicate the advantages related to using high protein feed. While higher protein compo-sition produces faster growth, and are consequently used in the industry, the increased feed costs and indirect costs due to metabolic feedback produce lower returns. This study also showed that inadequate removal of solids nega-tively affects revenue and that declines in biological filter efficiency have large impacts. With the decline in biological filter efficiency, the time-to-harvest increases. Results also show that, if the stocking density is increased, the increase in revenue can only be assured if water exchange is increased. If the filters are inefficient, higher stocking density may actually lead to economic

Figure 5. Scenario 8 (P, N, SS, UAN and Water exchange).

AQUACULTURE ECONOMICS & MANAGEMENT 15

failure. Thus, a trade-off exists between management ability and stocking density, with the trade-off significantly affected by levels of protein in the feed. Essentially, economically viable trade-offs between the protein in the feed and stocking density occur over relatively narrow ranges of management ability. Without experienced and competent management, the biological realities of RAS systems may hinder beneficial system operation. These simulated observations may in part explain why RAS systems have yet to demonstrate widespread success on a commercial scale.

In this article, it is assumed that the plant buffer tank is dimensioned so that no nutrients are discarded at any time of the growout cycle. The model is designed on a per liter for RAS or per m2 for greenhouse basis to avoid the need for explicit description of the sizes, types and configuration of various physical system components.

Conclusions

In this article, mathematical models for some of the components of a recircu-lation aquaculture system were presented. Optimal control methods have been applied to these models to show how to operate the recirculation aqua-culture system in a more efficient and economical way. A numerical example is presented to show the properties of the optimal control. It could be shown that the systems model and the related optimal control are very useful tools for aquaponic systems in the research and verification of design and operation of the INAPRO aquaponic system based on the ASTAF-PRO technology as described in Kloas et al. (2015). The models were also used to determine the sensitivity of final cost to changes in parameter values. Therefore, the results from the models can indicate the operating domain for the control variables of the commercial INAPRO aquaponic system. Further tests need to be done on other commercial systems to verify the generalizability of the system.

Funding

The authors of this study acknowledge the financial support of the EU FP7-ENV-2013- WATER-INNO-DEMO and the technical support of all the INAPRO project partners.

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