## 2 Analysis and Literature Review of Clearance in Port Channels and Basins
### Mathematical Model
Formulas can be used to calculate the trajectory of a ship at shallow depth and in the presence of wind and current, which is a characteristic of ports [4, 40, 43, 45]:  
\[X_{0i(s)}=\int v_{i(s)}\cdot\cos(\int\left(\omega_{i(s)}dt-\beta_{i(s)}\right)) dt+\int v_{cr}dt\cdot\cos q_{cr}+\int v_{d}dt\cos q_{a}; \tag{3}\]  
\[Y_{0i(s)}=\int v_{i(s)}\cdot\sin(\int\left(\omega_{i(s)}dt-\beta_{i(s)}\right) )dt+\int v_{cr}dt\cdot\sin q_{cr}+\int v_{d}dt\cdot\sin q_{a}, \tag{4}\]  
where \(v_{i(s)}\)--ship's speed at low depths; \(\omega_{i(s)}\)--turning velocity at low depths; \(\beta_{i(s)}\)--drift angle at low depths; \(v_{cr}\)--current velocity; \(q_{sr}\)--current course angle during the start of the maneuver; \(v_{d}\)--ship's drift speed; \(q_{a}\)--wind course angle during the start of the maneuver. The above elements of Formulas (2) and (3) can be calculated using the methodology presented in [5]. When examining the passage of ships in ports, it is assumed that the movement of ships in the port approach and internal navigation channels lead to the fact that ships independently sail on a straight or almost straight trajectory (turning up to 30-40 degrees). The research also assumes that the change in the power used by the ship's main engine due to the effect of shallow water is basically related to the change in the resistance of the ship's longitudinal movement [3, 10, 11]. Experiments with real ships were carried out in the open sea and in ports (shallow waters) for the power factor, maintaining a set constant speed and accurately recording the main engine power, the speed of the ship with the help of electronic lag and DGPS and the depth with the help of echo sounder. On the basis of studies carried out on real ships, a dependence of the calculation of the power factor of the ship's main propulsion system (\(\Delta N\)) on the _T/H_ ratio (ship's draught/depth ratio) and the overall hull fullness factor is obtained as follows:  
\[\Delta N=(1+1.25(\frac{T}{H})^{2})\sqrt{\frac{\delta}{0.65}}, \tag{5}\]  
Calculation of the power factor of the ship's main engine due to the influence of shallow water according to Formula (5) and experimental tests with real ships showed that the difference between the calculations and the results of experimental tests is no more than 10%. Thus, Formula (5) can be successfully used in practice. The limitations of Formula (5) evaluated during the experiments are as follows: the speed of the ships should be between  
Figure 4: Dependence of the added water mass coefficients (\(k_{11}\)—deep water, _T/H_ ratio is zero); \(k^{\prime}_{11}\)—shallow water) on the _T/H_ ratio and the vessel’s sealing speed \(v\). 6 knots and 10 knots, and the overall fullness factor of the ship's hull should be between 0.65 and 0.90. If the overall fullness ratio of the ship's hull is less than 0.65, Formula (5) can be used when the ratio of the ship's draft to the depth of the navigation channel (_T/H_) is greater than 0.3. As hull resistance increases, the power of the ship's engine must be increased to maintain the ship's target or planned speed. The relative power of the ship's engine (\(N^{\prime}\)) and the relative speed of the ship (\(v^{\prime}\)) can be expressed in the following formula:  
\[N^{\prime}=\frac{N}{N_{0}}; \tag{6}\]  
\[v^{\prime}=\frac{v}{v_{0}}, \tag{7}\]  
where \(N_{0}\)--nominal power of the main engine(s) of the vessel; \(v\)--speed of the vessel at the power of the vessel's engine(s) \(N\); \(v_{0}\)--speed of the vessel at the nominal power of the vessel's engine(s) \(N_{0}\). The relationship between the ship's relative speed and the relative power of the ship's main engine(s), presented in the literature [3,10,11], was verified by the authors through experiments on real ships of various types and sizes.