## 2 Methodology of Research
### Mathematical Model  
As discussed earlier, the main objective function is to minimize the total energy cost of cooling systems within a typical day based on the forecasted electricity prices. This can be formulated as follows:  
\[\min F=\sum_{t=1}^{24}(\pi_{t}^{dec}+\pi^{ObM})[P_{A_{t}}^{CS}+P_{B_{t}}^{CS}+ P_{C_{t}}^{CS}] \tag{1}\]  
where \(\pi_{t}^{dec}\), \(\pi^{ObM}\), \(P_{A_{t}}^{CS}\), \(P_{B_{t}}^{CS}\), and \(P_{C_{t}}^{CS}\) show the electricity price, operating cost of the cooling systems, and electric power consumption of the cooling system A, B, and C, respectively. This optimization model incorporates key technical constraints, including the calculation of electrical power consumption for the pre-cooling system (plant A) and the primary cooling systems (plants B and C) as well as limitations related to maintaining the water temperature in the breeding pools within acceptable bounds. In relation to the precooling system, the amount of heat power absorbed from the water intake from the water tank can be calculated using Equation (2):  
\[Q_{t}^{ACS}=m_{H_{2}O}C_{H_{2}O}(T_{t}^{in}-T_{t}^{out}) \tag{2}\]  
Figure 1: Typical schematic of cooling system configuration in aquaculture industry. where \(m_{H_{2}O}\), \(C_{H_{2}O}\), \(T_{t}^{in}\), and \(T_{t}^{out}\) stand for the mass flow rate of water intake from the tank (m\({}^{3}\)/h), specific heat capacity of water (4.186 MJ/m\({}^{3}\)C), water intake temperature from seawater, and temperature of the water discharged into the breeding pools, respectively. Then, based on the Coefficient Of Performance (COP) of this system, its electrical power consumption at time t is calculated as follows:  
\[P_{j,t}^{CS}=\frac{Q_{t}^{A,CS}}{COP^{A}}\quad\forall j\in[A] \tag{3}\]  
where \(COP^{A}\) stands for the COP of the pre-cooling system. There is a technical constraint that concerns the extent to which the pre-cooling system can reduce the water temperature. This reduction is limited by the system's capacity along with other operational considerations such as equipment longevity and the need to avoid excessive cooling. As a result, the temperature of the discharged water must remain within a predefined acceptable range, represented by the following constraint:  
\[T_{t}^{min}\leq T_{t}^{out}\leq T_{t}^{in} \tag{4}\]  
where \(T_{t}^{min}\) represents the minimum allowable temperature of the water discharged into the breeding pool. In addition, the water temperature of the each breeding pool must be kept within the acceptable range regardless of the temperature of the discharging water, and the primary cooling systems associated with each of them must cool the water when the water temperature exceeds the threshold. Therefore, based on the temperature requirements of the breeding pools, the temperature of the water \(T^{pool}\) in a typical breeding pool (e.g., \(k\)) must be less than the prespecified value \(T^{SP}\), as follows:  
\[T_{k,t}^{pool}\leq T^{SP}. \tag{5}\]  
As mentioned, the primary CO\({}_{2}\) and ammonia cooling systems (those highlighted by plant B and C in Figure 1) are designed to lower the temperatures of their respective pools when exceeding the specific threshold, which can happen due to fish metabolism and machinery processes as well as to discharges from the water tank. Hence, the temperature of the \(k^{\text{th}}\) breeding pool at the next time step can be calculated as follows:  
\[T_{k,t+1}^{pool}=\frac{T_{k,t}^{pool}(V_{k,t}^{pool}-Q_{in,t}^{pool})+T_{in,t} ^{pool}Q_{in,t}^{pool}\Delta t+Q_{fish,t}+Q_{machinery,t}}{V_{k,t}^{pool}} \tag{6}\]  
where \(T_{k,t+1}^{pool}\), \(T_{k,t}^{pool}\), \(V_{k,t}^{pool}\), \(Q_{in,t}^{pool}\), \(Q_{fish,t}\), and \(Q_{machinery,t}\) refer to the next temperature of the breeding pool cooled by the k\({}^{\text{th}}\) cooling system, the current temperature of the breeding pool, the total volume of the breeding pool, the water inflow rate, the heat produced from fish metabolism, and the heat produced by machinery processes, respectively.