1   Introduction

Floating docks play an important role in shipyards, serving as essential equipment for various marine operations, such as ship construction, maintenance and repair. They are U-shaped floating structures, with a pontoon at the bottom and two wing walls at two sides. A number of ballast tanks are equipped inside the pontoon and the wind walls. The control on the floating position of the docks is achieved by filling and emptying the ballast tanks.

Floating docks are more flexible and efficient than graving docks for vessels' maintenance. A vessel docking process as shown in Figure 1 was demonstrated by Zhang et al. (2022, 2023a). In Figures 1(a)‒(b), the floating dock is initially ballasted to allow the vessel entering into the floating dock and float above the pontoon deck. In Figures 1(b)‒(c), the floating dock is de-ballasted to bring the vessel to rest on the dry pontoon deck. The entire docking operation takes hours. It's essential to meticulously manage the dock's heeling and trimming within small ranges to ensure its stability. However, accidents can still occur during the operations. A list of accidents of floating dock operations reported from 2012 was summarized by Wen et al. (2024) and shown in Table 1. These accidents can bring significant stability loss and structural damage. These accidents highlight the potential operational risks in vessel docking, often related to the malfunctions of the ballast water system (Insurance Marine News, 2019).

Figure  1  Illustration of a vessel docking operation (Zhang et al., 2022)

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The ballast water system includes ballast pumps, valves, pipes, and ballast tanks. Its sound operations are vital for ensuring the docks' adequate buoyancy and stability during operations. The ballast pumps are important for emptying and filling the ballast tanks and balancing the dock during most operational conditions. Therefore, the safety of the floating docks highly relies on the functionality of the ballast pumps (Kimera and Nangolo, 2020). The malfunctioning ballast pumps can cause imbalanced weight distribution of the dock and increase the risk of capsizing. Since the ballast pumps normally rely on electrical power during operations, the fault in electrical supply, such as power outages (Rainsford, 2018) and overloaded motors (Kimera and Nangolo, 2020), can cause the failure of pump operations. Moreover, the solid particulates in the sea water passing through the pumps can damage the mechanical system of the pumps.

It is crucial to implement mitigation measures for malfunctions of ballast pumps due to the severe consequences mentioned above. In normal operations, each pump independently manages a specific ballast tank group. When a malfunction of a ballast pump occurs, a connection valve between the affected and adjacent groups of ballast water systems is opened, enabling the adjacent pump to serve as a helper pump for the ballast water systems of both groups. To assess the effectiveness of this mitigation measure, conducting comprehensive numerical studies on the floating dock's dynamic responses to malfunctioning pumps is essential.

Motivated by this, an in-house code for simulating the floating dock operations has been developed by Zhang et al. (2023a), which aims to provide comprehensive simulations of floating dock operations in time-domain. This code contains a six-degree-of-freedom (6-DOF) model, a hydrostatic force model, a hydrodynamic force model, a mooring force model, a hydraulic model, and a contact force model. Additionally, an automatic ballast control algorithm is incorporated to regulate the opening angles of the ballast valves, ensuring that the dock's roll and pitch motions remain within allowable ranges. The hydrostatic, hydrodynamic, and mooring force models were verified against their corresponding results from various software and models, including theoretical models, HydroD, Autodesk Inventor, lumped-mass method and Code-Aster. This code has been successfully applied to simulate normal ballasting and de-ballasting operations by Zhang et al. (2023a) and Wen et al. (2023a, 2024, 2023b). The effect of malfunctioning ballast valves on the dock's responses was investigated by Wen et al. (2024) based on this numerical platform. Validations of the numerical code were conducted by Zhang et al. (2024) on a model-scale floating dock in a model basin. Zhang et al. (2023b) also employed this method to conduct a primitive study of the dock's dynamic response when the ballast pump at the dock's aft malfunctions. The ballast system of the malfunctioning and helper pumps was modelled together. However, the connection valve, which will reduce the flow rate from the pipe network of the malfunctioning pump to the help pump, was not applied. Only one scenario was discussed, and the malfunctioning pump scenarios required complete examinations to find the best solution of the remedial measures for the accidents.

This work studies the dock's responses and evaluate the robustness of an automatic ballast control algorithm during de-ballasting operations with malfunctioning ballast pumps, which continues the work on the automatic ballast control conducted by Wen et al. (2024, 2023b) and the digital floating dock model developed by Zhang et al. (2023a). The application of the numerical model in de-ballasting operations, including scenarios with a malfunctioning ballast pump is examined. A new mitigation measure for the accident of a malfunctioning ballast pump by using a helper pump is proposed. The hydraulic model is modified accordingly to model the ballast piping network of both the malfunctioning pump and the helper pump. The effectiveness of the mitigation measure alongside the automatic ballast control algorithm is evaluated. This work is outlined below. In Section 2, the methodology is described, including a hydrostatic force model, a hydrodynamic force model, a mooring force model, a 6-DOF model, a hydraulic model and an automatic ballast control algorithm. In Section 3, the dynamic analysis of the de-ballasting operations involving a malfunctioning pump is performed for a single dock or the dock with a vessel onboard. Finally, the conclusions are summarized in Section 4.

2   Methodology

During docking operations, the varying draughts of the dock and vessel pose challenges in the modelling of hydrostatic and hydrodynamic forces using classical equations of motion (Zhang et al., 2022). To establish the governing equations of the dock-vessel system, a quasi-static assumption is proposed. This assumption is based on the extended timescale of the floating dock's motions during operations, which can take hours. Moreover, the floating docks are usually operated in sheltered area with minimal wave and current loads, allowing for neglecting the hydrodynamic forces applied to the dock and vessel due to waves and currents. Under the quasi-static assumption, hydrostatic forces, including the dock and vessel's buoyancy, and the ballast water's gravitational force, govern the dock's motions during docking operations. The forces due to the radiation of the movements of the dock and the mooring loads should also be incorporated in the numerical model. For simulating floating dock operations, it is important to model the dock's ballast system and propose an appropriate ballast control algorithm since the motions of the dock-vessel system are controlled by the variation in ballast water distribution during docking operations.

An in-house code is developed based on the abovementioned quasi-static assumption. The framework of this code is illustrated in Figure 2. The development of the hydrostatic and hydrodynamic force models, and the mooring force model is shown in Sections 2.1 to 2.3. The 6-DOF model is described in Section 2.4. Moreover, the hydraulic calculation for the ballast water system with a malfunctioning pump is detailed in Section 2.5, and an automatic ballast control strategy is enabled using a modified P-controller in Section 2.6.

Figure  2  Diagram of the proposed numerical model for a dock-vessel system (Zhang et al., 2023b)

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2.1   Hydrostatic force model

Traditional hydrostatic calculations typically focus on the restoring force resulting from buoyancy and the gravitational force acting on a floating structure. However, this approach presents limitations in floating dock operations due to the dock's changing draught, trim, and heel over time. Therefore, it is necessary to develop a hydrostatic force model that can accurately provide the dock's buoyancy, and the ballast water's gravitational force at any floating position of the dock. A strip theory and Archimedes'principle are adopted in this model. The outer surfaces of the floating dock and ballast tanks are sliced into 2D polygonal sections along the direction of length. The boundary of each section is represented by the polygon's vertices, as illustrated in Figure 3. Hydrostatic forces applied to these 2D sections are computed using Archimedes' principle. The hydrostatic force and moment of the entire dock are determined by integrating these hydrostatic loads along the x-direction.

Figure  3  Floating dock sections

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2.2   Hydrodynamic force model

The hydrodynamic forces applied to the floating dock arise from the radiation of waves induced by their motions. These forces involve the dock's added mass, added mass moments of inertia and damping coefficients. Since the hydrodynamic forces are relatively small compared with the hydrostatic forces described in Section 2.1, the hydrodynamic forces can be approximated to reduce computational resources. Only the added mass and mass moments of inertia in heave, roll, and pitch, denoted as m_A, I_xxA, I_yyA, are considered. The surge, sway, and yaw components are neglected since they are relatively small compared to those of heave, roll, and pitch.

The added mass as well as mass moments of inertia are determined using a strip theory. This involves integrating the 2D results of the added mass and mass moments of inertia of a plate spanning along the dock's length. The final expressions of these forces are shown in Equation (1), where B and L are the width and length of the dock, and λ = B/L.

The dock's damping coefficients in heave, roll and pitch are determined using their mass matrices, a damping ratio of 5%, and the corresponding natural frequencies. The natural frequencies are calculated using Equation (2). The damping ratio of 5% is chosen according to the DNV recommended practice (DNV, 2012).

where m₃₃, I₁₁, and I₂₂ represent the summation of the dock's mass and added mass, and ballast water mass. C₃₃, C₄₄, and C₅₅ are the dock's hydrostatic restoring coefficients in heave, roll, and pitch.

2.3   Mooring force model

The floating dock is secured by 12 mooring lines, as shown in Figure 4. The mooring line forces acting at the fairleads are calculated using catenary equations described by Faltinsen (1993) and Masciola (2018). These equations are solved using the Newton-Raphson method.

Figure  4  Mooring system of the floating dock in top view

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2.4   6-DOF model

The floating dock is described as a rigid body with six degrees of freedom. The translational motions of the floating dock are represented by its CoG position in the global coordinate system, denoted as X_CG = (X_CG, Y_CG, Z_CG), and can be calculated using Equation (3) (Schaub and Junkins, 2003).

where m is the mass matrix, and C_m is the damping coefficient matrix. These matrices contain the values in surge, sway and heave. The mass matrix m for the dock represent the summation of the mass and added mass of the empty floating dock, along with the ballast water mass. The damping coefficient matrix C_m is obtained from the hydrodynamic model described in Section 2.2. G_d is the empty dock's gravitational vector. G_w is the weight of the ballast water. F_b and F_m represent the dock's buoyancy and mooring force, respectively. Both G_w and F_b are obtained from the hydrostatic force model presented in Section 2.1.

As for the rotational motions, Equation (4) can be used to obtain the angular velocity vector ω_B = (ω_B1, ω_B2, ω_B3) in the body-fixed coordinate system (Schaub and Junkins, 2003). The subscript "B" stands for the body-fixed coordinate system.

where I and C_I are the matrices of the mass moment of inertia and the damping coefficient in roll, pitch and yaw. Like the mass matrix of the dock, the mass moment of inertia matrix also includes the contributions of the empty dock, its added mass, and ballast water, as described in Section 2.2. The matrix of C_I can be obtained using the hydrodynamic model given in Section 2.2. M_w and M_b are the moment vectors resulting from the ballast water's weight and the dock's buoyancy, respectively. They are calculated about the dock's CoG in the body-fixed coordinate system.

Based on the angular velocity, the yaw, pitch, and roll (φ, ψ, and γ) in the global coordinate system can be determined using Equation (5) (Schaub and Junkins, 2003).

In this work, in the vessel-docking cases, the dock and vessel's motions during the vessel-docking cases are calculated similarly to the single dock case. Initially, before the vessel contacts the docking blocks of the dock, the calculation of the dock's motions follows the same procedures in single dock operations. Once the vessel touches the blocks, the dock-vessel system are treated together as one rigid body, and the parameters related to the dock (such as mass (m), mass moment of inertia (I), gravitational force (G_d), buoyancy (F_b)) are replaced by the values corresponding to the floating dock with a vessel onboard.

2.5   Hydraulic model of the ballast water system

The hydraulic model is developed to calculate the flow rates into or out of the ballast tanks and update ballast water volumes using Equation (6) (Wen et al., 2023a; 2023b).

where V_j is the ballast water volume, Q_j is the flow rate of Tank No.j, and t is the time.

Figure 5 illustrates the ballast piping network with two neighbouring pumps in the de-ballasting operation. During normal de-ballasting operations, the outlet, port, centre, and starboard valves are open, but the inlet and connection valves are closed. When Pump 1 malfunctions, Outlet valve 1 should be closed and Connection valve 1 is opened. This allows Pump 2 to serve as a 'Helper' pump to de-ballast two groups of ballast tanks simultaneously.

Figure  5  Schematic of the ballast piping network for two neighbouring pumps in the de-ballasting operation (Zhang et al., 2023b)

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Flow rate calculations for normal de-ballasting operations have been detailed by Wen et al. (2023a; 2023b), while the present study focuses on determining flow rate calculations in scenarios involving malfunctioning pumps. The flow rates through two groups of main pipes and ballast tanks are calculated based on the differences of water head during the flow is going through the pump and valves, while neglecting the difference of the water head along the pipes. The relationship between the difference of the water head and the flow rates during de-ballasting operations is represented using Equations (7)‒(17). The continuity equation among the main and branch pipes are represented in Equations (16) and (17).

λ_M, λ_P, λ_C, and λ_S denote the valve opening coefficients of the outlet and tank valves, and are calculated using Equation (18) (Wen et al., 2023a; 2023b):

where K_V is the water volume in cubic metre going through a butterfly valve within one hour when the pressure drop is one bar. It depends on the valve opening angles, and its values are determined in experimental tests (Eurotorc, 2022).

Equations (7)‒(17) are solved numerically using an iterative method (Wen et al., 2023a; 2023b). Convergence is typically achieved within 50 steps, ensuring high computational efficiency. Upon updating the volumes of ballast water in all 18 tanks, the corresponding mass, gravitational force, and moments of the ballast water are updated using the hydrostatic force model detailed in Section 2.1. These updated parameters are then used in the hydrodynamic force model described in Section 2.2 to compute the hydrodynamic forces.

2.6   Automatic ballast control using a modified P-controller

As described in Section 2.5, the valve status in the ballast system directly influences the flow rates through the ballast valves, change the ballast water distribution, and further affect the floating dock's motions. Thus, controlling the valve opening angles is essential for ensuring the floating dock's stability. To maintain the desired positions of the floating dock, a modified P-controller is proposed to ensure the dock's roll and pitch within acceptable ranges during operational conditions (Wen et al., 2024).

In this automatic control algorithm, the ballast tank valves are separated into six groups, i.e., No.1‒3, No.4‒6, No.7‒9, No.10‒12, No.13‒15, and No.16‒18, and each group has the same valve opening angle to reduce the system complexity. The control output is the target valve opening angle of the i-th group of tanks at the n-th time step, $\theta_{\text {target }, i}^{(n)}$, calculated using Equation (19). The value of $\theta_{\text {target }, i}^{(n)}$ needs to be within the range of [0, 90 deg].

Here θ_max is the maximum valve opening angle. c_{p, i} and c_{r, i} are pitch and roll control coefficients of the i-th group of tank valves. Their values for six groups during de-ballasting operations are shown in Table 2.

$K_p^{(n)}$ and $K_r^{(n)}$ are the pitch and roll proportional coefficients at the n-th time step. Their values are shown in Equations (20)‒(21), where ψ_upper and γ_upper are determined using the equation L_ψupper = B_γupper = ΔD. L and B are the length and width of the dock, and ΔD is a constant control range. The initial proportional coefficients $K_p^{(1)}$ and $K_r^{(1)}$ are equal to zero.

Moreover, the target valve opening angles will be updated with a time interval of ΔT, where ΔT = N_TΔt, and Δt is the simulation time step. The values of the control parameters, including θ_max, K, ΔD, and ΔT are shown in Table 3, as optimized by Wen et al. (2023b).

The delay between the target and actual opening angles due to the valve discs' angular velocity is considered. The actual valve opening angle $\theta_{\text {present }, i}^{(n+1)}$ is updated based on Equation (22).

where Δt denotes time step; ω_valve represents the valve disc's angular velocity, equal to 1.5 deg/s.

The updated actual valve opening angles are imported to the hydraulic model. The dock's roll and pitch are determined accordingly and then fed back to the controller.

3   Results and discussion

The de-ballasting operations of a full-scale floating dock equipped with a malfunctioning pump are simulated using the numerical tools described in Section 2. The investigated floating dock is simplified based on an actual floating dock. Its specifications are presented in Section 3.1, and the case descriptions of the numerical simulations are shown in Section 3.2. The time-step sensitivity study is performed in Section 3.3. The de-ballasting operations of a single dock and with a vessel onboard are simulated and analysed in Sections 3.4 and 3.5, respectively.

3.1   Specifications of the floating dock

Figures 6‒7 show the geometry and initial positions of the floating dock and the docked vessel in the global coordinate system. Details about their specifications are provided in Table 4. The distance between the vessel bottom and the pontoon deck of the dock is 6.2 m in the z-direction.

Figure  6  Geometry of a full-scale floating dock

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Figure  7  Geometry of the docked vessel

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18 ballast tanks are located beneath the dock's pontoon deck. Their inner surfaces are presented in Figure 8. Six ballast pumps are installed in the flaoting dock, each de-ballasting three tanks.

Figure  8  Inner surfaces of 18 ballast tanks (Zhang et al., 2023b)

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The ballast piping network in the 18 ballast tanks is illustrated in Figure 9. The main and branch pipes are marked in red and green, respectively. Cconnection pipes, marked black, connect one pump's piping network to another. Butterfly valves are installed on these pipes. Inlet and outlet valves are installed on the main pipes for ballasting and de-ballasting, respectively. The discs of these valves have a diameter of 600 mm. The disc diameter of the other valves is 400 mm.

Figure  9  Ballast pumps, valves, and pipes of the floating dock (Zhang et al., 2023b)

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3.2   Case description

Four sets of cases, shown in Tables 5 and 6, are explored in this work. Cases A and C represent normal de-ballasting operations of a single floating dock and the dock with an onboard vessel, respectively. Cases B1 – B10 and Cases D1–D10 are conducted to study scenarios involving single malfunctioning pumps for a single floating dock and the floating dock with the docked vessel onboard, respectively. The initial draught for Case sets A and B is 12 m, while that for Case sets C and D is 9.885 m, where the vessel starts to contact the docking blocks. The initial roll and pitch are both zero for all cases. The numbering of the ballast pumps is shown in Figure 9. In all cases, the opening angles of the malfunctioning pumps' connection valves are set as 90 deg. All simulations have a duration of 5 000 s.

The time-step sensitivity study is performed using Case B1 in Section 3.3. The numerical results for the de-ballasting operations of a single dock and the dock with an onboard vessel involving one malfunctioning pump are shown in Sections 3.4 and 3.5, respectively.

3.3   Sensitivity study of time step

The sensitivity study of the section numbers of the floating dock (as presented in Figure 3), the ballast tanks and the docked vessel was performed by Zhang et al. (2023a) and Wen et al. (2024). 150 sections of the dock, 20 sections of each ballast tank, and 100 sections of the vessel are adopted. The time-step sensitivity study is performed to find the proper temporal resolution for simulating de-ballasting operations. Three different time steps (0.25 s, 0.5 s, and 1.0 s) are adopted for Case B1 where Pump No.1 is malfunctioning and Pump No. 2 serves as the 'helper', as shown in Figure 5. Figure 10 shows the time histories of draught, roll, and pitch using different time steps and a good agreement is achieved between them. The simulations using the time steps of 0.25 s, 0.5 s, and 1.0 s take 654 s, 333 s, and 165 s on a desktop with 12 Advanced Micro Devices (AMD) cores and 32 GB Random Access Memory (RAM), respectively. Therefore, the time step of 0.5 s is selected for the time-domain simulations to balance the computational efficiency and the temporal resolution.

Figure  10  Time histories of the dock's (a) draught, (b) roll, and (c) pitch for Case B1

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3.4   De-ballasting operations of a single floating dock

Figure 11 presents the comparison of the dock's draught between a normal de-ballasting operation (Case A) and the de-ballasting operation with malfunctioning Pump No.1 (Case B1). For this case, only Pump No.2 is available to serve as the helper pump. The floating dock rises from the draught of 12 m to 1.787 m and 3.496 m at 5 000 s for Cases A and B1, respectively. Additionally, the dock's pontoon deck emerges from sea water level at 1 981 s and 3 187 s for Cases A and B1, respectively, indicating a decreased rising speed of the floating dock in Case B1 due to the malfunctioning pump. This reduced efficiency can also be shown in Figure 12, illustrating a decrease in the total de-ballasting efficiency of all pumps. At 5 000 s, much less ballast water is pumped out of 18 ballast tanks for Case B1 than those for Case A1. Moreover, the roll and pitch fluctuations throughout the simulations can be observed in Figure 10, with maximum values of 0.048 1 deg and 0.031 4 deg. The draught differences caused by the maximum roll and pitch reach 0.033 4 m and 0.092 4 m, respectively. These observations indicate that the proposed ballast water control algorithm is effective to minimize the dock's roll and pitch, during both normal de-ballasting operations and scenarios of the malfunctioning Pump No.1.

Figure  11  Time histories of the draughts for Case B1 and Case A

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Figure  12  Initial and final tank fractions V_b/V_t for Case B1 and Case A (V_b: ballast water volume, V_t: maximum ballast tank volume)

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Figure 13 presents the valve opening angles of Tank No.13 to Tank No.18 for Cases A and B1. The difference of valve opening angle between No.13‒15 and No.16‒18 contributes to the control in pitch motion. In Figure 13(a), as the sea water level exceeds the pontoon deck in both cases, the valve opening angles of aft tanks (from No.13 to No.15) remain at 70°. Once the pontoon deck emerges from mean sea water level, the opening angles rise to 90°. However, the fore tanks' valve opening angles (from No.16 to No.18) differ significantly between the two cases. In normal de-ballasting operation, the valve opening angles of the fore tanks are close to those of the aft tanks. When Pump No.1 is malfunctioning, the fore tanks' valve opening angles range from 25 deg to 53.25 deg. The reason for the discrepancy is that the malfunction of Pump No.1 reduces the aft side's de-ballasting capacity. Therefore, the fore side's valve opening angles should be smaller than the aft ones to minimize the pitch of the dock.

Figure  13  Valve opening angles of Tanks No.13‒15 (a) and No.16‒18 (b) for Case A and B1

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Figure 14 shows the valve opening angles of Tanks No.4‒6 and 16‒18 for Case A and Case B1. The discrepancy of valve opening angle between Tanks No.4‒6 and 16‒18 indicates the control in roll motion. This discrepancy observed in Case A (the normal de-ballasting operation) is much smaller than in Case B1 (malfunctioning Pump No.1), indicating that controlling the roll during malfunction is more challenging than that during normal operations.

Figure  14  Valve opening angles of Tanks No.4‒6 and 16‒18 for Case A (a) and Case B1 (b)

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The variation in tank valve opening angles among different tank groups affects the flow rates through the pumps, as shown in Figure 15. In Case A (the normal de-ballasting operation), the flow rates through six pumps are close to each other. In Case B1 (malfunctioning Pump No.1), the flow rates through Pumps No.4‒6 are smaller than those through Pumps No.2 and No.3. These differences are primarily attributed to asymmetric aft and fore tank valve opening angles.

Figure  15  Flow rates through Pumps No.1–6 for Case A (a) and Case B1 (b)

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In addition to the case of malfunctioning pump No.1, other cases of one malfunctioning pump are performed. For cases B2‒B9, where Pumps No.2‒4 are malfunctioning, each malfunctioning pump has two adjacent pumps available to serve as the helper pump. However, in Case B10, only Pump No. 5 is available to assist the malfunctioning Pump No.6. Figure 16 presents the dock's draught for Cases B1‒B10, and the final draughts for these cases are given in Figure 17(a). These figures show that the dock's rising speed is reduced due to the malfunctioning pumps, and the final draughts for Cases B1 ‒ B10 are all larger than the draught of 1.787 m in Case A.

Figure  16  Time histories of the draughts for Cases B1–B5 (a) and Cases B6–B10 (b)

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Figure  17  Final draughts (a) and maximum draught differences resulting from the roll and pitch motions, B|γ|_max (b) and L|ψ|_max (c) for Cases B1–B10

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Figures 17(b)–(c) summarize the maximum draught differences resulting from the roll and pitch motions for Cases B1‒B10. The maximum draught differences resulting from the roll motions (B|γ|_max) for these cases are similar, ranging from 0.031 09 m to 0.038 51 m. However, the range of maximum draught differences resulting from the pitch motions (L|ψ|_max) is wider, ranging from 0.029 31 m to 0.095 23 m, compared to that in the roll motions. It can be concluded from Figure 17(c) that the malfunctioning pump close to the aft or fore of the floating dock leads to a larger L|ψ|_maxthan that close to the centre. Among Cases B1‒B10, scenarios involving the malfunction of Pumps No.1 (Case B1) and No.6 (Case B10) are considered as the most severe cases. On the other hand, cases involving the malfunctioning Pumps No.3 or No.4, where the malfunctioning pump is closest to the centre, are considered as the safest cases.

Figure 18 shows the comparisons of the dock's roll and pitch between Cases B6 (malfunctioning pump No.4, helper pump No.3) and Case B10 (malfunctioning pump No.6, helper pump No. 5). It can be observed that both draught differences (B|γ|_max and L|ψ|_max) remain negligible relative to the dock's width, even in these most severe scenarios. This indicates the effectiveness of the present automatic ballast control strategy in both the normal and malfunctioning case.

Figure  18  Time histories of the dock's roll (a) and pitch (b) for Cases B6 and B10

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In Cases B2‒B9, when one of Pumps No.2‒5 malfunctions, each malfunctioning pump has two helper pump options. Based on the analysis on the time histories of the draught and draught differences L|ψ|_max, a better helper pump can be determined from the two options of helper pumps. For instance, in the case of malfunctioning Pump No.2, selecting Pump No.3 as the helper leads to a smaller L|ψ|_max than choosing Pump No.1. It is because the variation in ballast water volumes in the tanks closer to the dock's centre has less influence on the dock's trimming moment compared to those closer to the fore or aft. Moreover, the final draught of the dock is lower when Pump No.3 acts as the helper, providing higher de-ballasting efficiency.

3.5   De-ballasting operations of the floating dock with a vessel onboard

Figure 19 shows the comparisons of the dock's draught, roll, and pitch during the normal de-ballasting (Case C) and the de-ballasting with malfunctioning Pump No.1 (Case D1). The dock's de-ballasting efficiency in Case D1 decreases, as observed from the draught histories in Figure 19(a). In both cases, the dock's rising speeds exhibit sudden increases and decreases at draughts of 6.2 m and 4.5 m, corresponding to the instants when the vessel bottom and the deck emerge from the water, respectively. The dock's roll and pitch time histories are shown in Figures 19(b)‒(c). The dock's maximum roll and pitch are larger when Pump No.1 is malfunctioning compared with the normal de-ballasting scenario. However, these values remain within acceptable ranges, not exceeding 0.071 5 deg and 0.031 3 deg, respectively, showing the effectiveness of the modified P-controller when Pump No.1 malfunctions.

Figure  19  Time histories of the dock's draught, roll, pitch for Cases C and D1

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To study the effect of different malfunctioning pumps on the dock's dynamic response when a vessel is on board, the dock's final draughts at 5 000 s, and the maximum draught differences induced by the roll and pitch for Cases D1–D10 are summarized in Figure 20. In scenarios involving two available helper pumps (malfunctions of Pumps No.2‒5), the final draught and the maximum draught differences can be used to determine the optimal helper pumps in terms of the total de-ballasting efficiency and stability of the floating dock, respectively. As discussed in Section 3.4, Pumps No.3, 4 are recommended as helper pumps to address the malfunctions of Pumps No.2, 3, 4, and 5, respectively. These helper pumps result in smaller final draughts and reduced B|γ|_max and L|ψ|_max. Among Cases D1–D10, the results of B|γ|_max are all close to 0.05 m, while those of L|ψ|_max are within a slightly wider range than that of B|γ|_max, from 0.055 94 m to 0.091 9 m. The most severe case occurs at the malfunction of Pump No.1, with the largest L|ψ|_max and final draught. Conversely, situations involving malfunctioning pumps closer to the dock's centre are safer. Even in the most severe case, the results of L|ψ|_max are still small compared with the dock's length. Thus, the effectiveness of the proposed automatic ballast control strategy is proved for the scenarios of the floating dock with a vessel onboard. Additionally, B|γ|_max remain stable across all cases, whereas L|ψ|_max exhibits more significant variations with different malfunctioning pumps. This observation highlights that the position of the malfunctioning pump has a more significant effect on the dock's pitch rather than its roll.

Figure  20  (a) Final draughts and draught differences resulting from maximum (b) roll B|γ|_max and (c) pitch motions L|ψ|_max for Cases D1–D10

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4   Conclusions

The de-ballasting operations of a floating dock with a malfunctioning ballast pump are numerically studied using an in-house code. In this numerical model, the dock is described as a 6-DOF rigid body, subjected to the hydrostatic, hydrodynamic, and mooring loads. The hydraulic model of the ballast water system is modified for malfunctioning pumps and their helper pumps. A modified P-controller is adopted for automatic ballast control. The de-ballasting operations involving malfunctions of single ballast pumps are simulated, and scenarios of a single floating dock and the dock with an onboard vessel are both considered. The proposed controller's control performance in these scenarios is investigated. The main results are summarized as follows:

1) The adoption of a helper pump and the present automatic ballast control algorithm can effectively address onepump malfunctions during de-ballasting operations, ensuring the floating dock with minimal heel and trim, even for the case of the floating dock with a vessel onboard.

2) In scenarios with two available helper pumps, the optimal helper pump is determined according to the final draught and maximum draught differences induced by roll and pitch motions.

3) Selecting a helper pump closer to the dock's centre is generally more favourable than near the aft or fore. The malfunctions of the pumps at the aft or fore end are usually considered the most severe situation.