## 3 Methodology  
The proposed system focuses on height control in a system of interconnected tanks. The modeling is grounded in [PERSON]'s equations for laminar flow, accounting for flow rate and balance associated with the energy conservation of mass flow at various points. Figure 2 illustrates the plant model.The parameters used in the mathematical model of the interconnected tank system are summarized in Table 3. This table lists the symbols and their corresponding descriptions, including flow rates (\(U_{in}\), \(U_{12}\), \(U_{out}\)), tank heights (\(H_{1}\), \(H_{2}\)), cross-sectional area (\(A\)), and coefficients (\(c_{1}\), \(c_{2}\)). The redefined variables (\(Z_{1}\), \(Z_{2}\)) simplify the system equations, while \(K_{1}\) and \(a_{1}\) represent the gain and valve opening for the inlet flow. These parameters are crucial for understanding the system dynamics and deriving the transfer functions for control purposes. \begin{table}
\begin{tabular}{l l c c} \hline \hline
**Main Author** & **Reference** & **Citations** & **Methodology Summary** \\ \hline [PERSON] & [13] & 214 & Risk-sensitive RL for constrained control \\ [PERSON] & [14] & 203 & ACO for optimal pump control in water networks \\ [PERSON] & [15] & 147 & Optimal allocation of empty containers between ports \\ [PERSON] & [16] & 93 & Design hybrid renewable energy systems in buildings \\ [PERSON] & [17] & 85 & Reliability-based design of pumping systems \\ [PERSON] & [18] & 84 & Shipments planning in marine inventory routing \\ [PERSON] & [19] & 80 & Model-on-demand identification for \\ [PERSON] & [20] & 67 & Demand response using batch RL in water heaters \\ [PERSON] & [21] & 65 & Real-time cost optimization for HVAC equipment \\ [PERSON] & [22] & 61 & Rule-based control strategy for ice storage systems \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Top 10 papers with methodologies. Figure 1: Number of citations of the top 10 most cited articles. The percentage change in liquid volume in the tank is determined by the inlet flow rate from the pump minus the outlet flow rate through the orifice, as described by Equations (1) and (2). The velocity (\(v\)) of liquid passing through an orifice is akin to that of an object falling freely from a height \(h\), represented by [PERSON]'s formula (Equation (3)). Redefining [PERSON]'s formula and flow balance equations, the intermediate flow (\(U_{12}\)) and outflow (\(U_{out}\)) are given by Equations (4) and (5), respectively. The inlet flow (\(U_{in}\)) is modeled with gain \(K_{1}\) and valve opening \(a_{1}\) as shown in Equation (6). Through variable changes, \(z_{1}\) is defined as \(H_{2}\) with a coefficient \(c_{1}\), in Equation (7), and \(z_{2}\) as \(H_{1}-H_{2}\) with \(c_{2}\), in Equation (8). These transformations simplify the complex dynamics of the tank system. \[U_{in}-U_{12}=A\frac{dH_{1}}{dt} \tag{1}\]  
\[U_{12}-U_{out}=A\frac{dH_{2}}{dt} \tag{2}\]  
\[v=\sqrt{2 gh} \tag{3}\]  
\[U_{12}=A_{12}\sqrt{2g(H_{1}-H_{2})} \tag{4}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(U_{in}\) & Inlet flow rate from the pump \\ \(U_{12}\) & Intermediate flow between tanks \\ \(U_{out}\) & Outlet flow rate through the orifice \\ \(A\) & Cross-sectional area of the tank \\ \(H_{1}\) & Height of liquid in tank 1 \\ \(H_{2}\) & Height of liquid in tank 2 \\ \(v\) & Velocity of liquid through an orifice \\ \(K_{1}\) & Gain for the inlet flow rate \\ \(a_{1}\) & Valve opening for the inlet flow \\ \(Z_{1}\) & Redefined variable for height of tank 2 \\ \(Z_{2}\) & Redefined variable for height difference between tanks \\ \(c_{1}\) & Coefficient related to tank 2 outflow \\ \(c_{2}\) & Coefficient related to intermediate flow \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for the mathematical model of the interconnected tank system. Figure 2: Diagram of the interconnected tank system with flow and height parameters.