## 2 Analysis and Literature Review of Clearance in Port Channels and Basins
### Mathematical Model  
The results of the literature review, the results of simulators and the results of the real ship tests when different clearances were used between the ship's hull and the bottom of the channels were used to develop mathematical models of shipping navigational safety, energy (fuel) consumption and emission generation at shallow depths [1; 9; 15; 37; 41; 42; 43; 44]. When conducting research and creating mathematical models, it was assumed that ships in navigational channels sail independently, and the controllability of the ship is ensured by the ship's own steering equipment. In channel bends and turning basins, ships turn with the help of their control equipment (propulsion complex), and if necessary, they can use the help of ship steering devices and tugboats [5; 7]. It is also assumed that when ships move straight through channels due to the effect of low depth, only the longitudinal resistance of the ship changes, and when ships sail in bends of navigation channels and when turning in turning basins, the resistance of the ship's lateral movement additionally increases [43; 45]. The safe depth of the port channels and port water areas, so that the largest ship in the port does not touch the bottom of the navigation channels and basins with its hull, can be calculated with the help of the following formula [1; 4]:  
\[H_{\min}=T+\Delta T_{\vartheta}+\Delta T_{\vartheta}+\Delta T_{\Psi}+\Delta H_{ m}+\Delta H_{V.L}+\Delta H_{\Delta V.L}+\Delta H_{n}, \tag{1}\]  
where \(T\)--the maximum draught of the calculated vessel; \(\Delta T_{V}\)--the increase in draught due to settlement (speed), sometimes named squat [4; 5]; \(\Delta T_{\vartheta}\)--the increase in draught due to heeling [4; 5]; \(\Delta T_{\vartheta}\)--the increase in draught due to the effect of swell (change in the difference) [5]; \(\Delta H_{m}\)--the accuracy of the depth measurement, depending on the port depth measurement technique used; \(\Delta H_{V.L}\)--the level of the water in the particular port; \(\Delta H_{\Delta V.L}\)--the accuracy of the measurement of the water level, depending on the port water level measurement technique used; \(\Delta H_{n.}\)--navigational margin, which can be decomposed into a direct navigational margin, which is assumed to be about 2-3% of the ship's draught, by means of accurate bottom depth measurements (using modern depth measurement techniques), and a layer of sediment, which has to be periodically removed (cleaning). The above elements of Formula (1) can be calculated using the methodology presented in [4; 5]. The increase in draught (\(\Delta T_{\vartheta}\)) due to low-depth effects and the ship's speed can be calculated using the following formula and graph (Figure 4) to estimate the added water mass coefficients [1; 4; 5]:  
\[\Delta T_{V}=\frac{\rho LB}{\delta}\sqrt{\frac{1+k_{11}^{\prime}}{1+k_{11}}} \tag{2}\]  
where \(\rho\)--water density; \(L\)--ship's length between perpendiculars; \(B\)--ship's width; \(\delta\)--overall hull fullness factor; \(k_{11}\) and \(k_{11}^{\prime}\)--added water mass coefficients at high depth and in shallow water, depending on the ship's speed and \(T/H\) ratio (Figure 4) [5]. The added water mass depends on the speed of the ship and the draft of the ship, as well as the depths of the navigation channels and the port water area ratio (\(T\)/\(H\)). Added water mass coefficients, which are presented in Figure 4 when the ship is moving in the longitudinal direction, are usually used for calculation [4; 5]. In the article, the water mass coefficients presented in Figure 4 are used to calculate the change in the ship's draft and the increase in the ship's draft, as well as the correspondence of the specified coefficients, verified by experiments by measuring clearance on real ships [4; 5].