Particle Swarm-Based Translation Control for Immersed Tunnel Element in the Hong Kong–Zhuhai–Macao Bridge Project

Immersed tunnel is an important part of the Hong Kong–Zhuhai–Macao Bridge (HZMB) project. In immersed tunnel floating, translation which includes straight and transverse movements is the main working mode. To decide the magnitude and direction of the towing force for each tug, a particle swarm-based translation control method is presented for non-power immersed tunnel element. A sort of linear weighted logarithmic function is exploited to avoid weak subgoals. In simulation, the particle swarm-based control method is evaluated and compared with traditional empirical method in the case of the HZMB project. Simulation results show that the presented method delivers performance improvement in terms of the enhanced surplus towing force.


Introduction
An immersed tunnel is a kind of underwater tunnel composed of elements prefabricated elsewhere in a manageable length. Each tunnel element is transported to the tunnel siteusually floating, sunk into place, and then linked together. For the transportation of elements from the flooded casting basin or dock to the tunnel trench, conventional towage, in which several tugs assist the tunnel element in floating, is normally used. Straight movement (moving forward or backward) and transverse movement (moving left or right), which are collectively called "translation" in this work, are normally the main working conditions of immersed tunnel element floating. To follow a control strategy with satisfactory efficiency and stability of tunnel element, the translation control optimization for tugs coordination in an efficient way should be studied.
Some studies have discussed the planning and operational challenges that authorities and operators have to contend with in immersed tunnel element floating (Xiao et al., 2010;Chen et al., 2012;Wu et al., 2013). Nonaka et al. (1996) developed an estimation method of hydrodynamic forces acting on a ship in manoeuvring motion, and found that there was clear difference in the center of sway force that had a serious effect on ship manoeuvrability. By using the numerical simulations, Kishimoto and Kijima (2001) estimated the manoeuvring performance of the tug-towed ship systems and examined the course stability of the tug and towed ship taking into account some factors such as the length of towing ropes, the location of towing points and the condition of the disabled ship. Fitriadhy et al. (2015) presented a fundamental investigation for a more comprehensive insight into the basic mechanism of the slack towline condition identified via several towing parameters affecting its occurrence. However, there are few researches on the coordination of tugs in the floating of immersed tunnel element.
Similar to the floating of immersed tunnel element, tugs should collaborate with each other in towing, berthing and unberthing of large non-power vessels. Varela and Soares (2013) described the conceptual overview, interface specification and object models of a high level architecture framework for real-time simulation of ship towing operations in virtual environments. Couce et al. (2015) developed automatic maneuvering systems, which consisted of information technology applications that helped the skipper with maneuvering the tug. But these towing control methods cannot be directly applied to the immersed tunnel element which is also unpowered, because the tunnel element's configuration is completely different from vessel.
In this work, taking the Hong Kong-Zhuhai-Macao Bridge (HZMB) project as an example, the application of particle swarm-based control into immersed tunnel element translation optimization is investigated. First, immersed tunnel element translation is mathematically described in the aspects of tugs' velocity, resistance, resultant force and moment. Second, the immersed tunnel element translation control model is built through analyzing resultant forces and moments of tugs. Third, a kind of linear weighted logarithmic objective function is presented aiming at the multiobjective characteristics of translation control model. Finally, the case of the HZMB project to test the proposed approach is demonstrated by the translation velocity and surplus towing force. And a comparison between the results of the particle swarm-based control method and traditional empirical method is presented.

Description of immersed tunnel element translation
Immersed tunnel element translates with a certain velocity which leads to resistance. Resistance needs to be got over by towing forces. There are certain requirements on resultant forces and the corresponding resultant moment to promote immersed tunnel element translate reposefully. This section unfolds gradually in these parts: velocity, resistance, towing force, resultant force and resultant moment. At last, the angle difference between the tugs' resultant force and the tunnel element velocity relative to current flow is pointed out.

Translation velocity
Here, a tunnel element and its pontoons are viewed as a whole with a cuboid structure. Its length, width and draft are L, B, and d, respectively. The number of tugs working collaboratively is N. For the convenience of analysis, a 2D axis (shown in Fig. 1) is drawn in the horizontal plane. In Fig. 1, the element center is the origin of the coordinate, the x-axis is set in the longitudinal direction of tunnel element, and the y-axis is perpendicular to the x-axis.
The current velocity is set as V 0 . The element's velocity relative to ground is set as V 1 , while the element's velocity relative to current flow is set as V. The angle between V 0 and the positive direction of the x-axis is θ 0 ; the angle between V 1 and the positive direction of the x-axis is θ 1 ; the angle between V and the positive direction of the x-axis is θ. The relation among V 0 , V 1 , and V is shown in Fig. 1. Components V in the x and y directions are V x and V y respectively: (1) 2.2 Resistance of immersed tunnel element translation According to "Guidelines for Marine Towage (2011)" (China Classification Society, 2011), the total marine towage resistance R T can be calculated by Eq. (3). Where, R f and R B are the friction resistance and residual resistance of the immersed tunnel element respectively. R f is represented in Eq. (4) (China Classification Society, 2011). In consideration of the non-streamline structure of immersed tunnel element, R B is calculated according to Eq. (5) (Shen, 2011). (3) In Eqs. (4) and (5), a 1 is the wetted surface area under water line of tunnel element, is the block coefficient, and a 2 is the submerged part of transverse section area in tunnel element. If the element is parallel to current flow, a 1 =L(B+2d), a 2 =B·d; if the tunnel element is perpendicular to current flow, a 1 =B(L+2d), a 2 =Ld. From Section 2.1, it is known that velocity V may be positive or negative. So an absolute value symbol is added on V in Eq. (4). Units of R T , R f and R B are all "kN". θ = kπ/2 Eqs. (3)-(5) are only suitable when (k is an integer), it means that the element's movement is parallel or perpendicular to current flow. Here, R Tx and R Ty denote components of R T in the directions of the x and y axes, respectively. f denotes the hydrodynamic resistance of the immersed tunnel element with pontoons. f x and f y denote components of f in the directions of the x and y axes, respectively. In this work, R Tx and R Ty are calculated based on V x and V y , respectively. R Tx and R Ty are all non-negative real numbers, and then the relationships among f x , f y , R Tx and R Ty are shown in Eqs. (6) and (7).
2.3 Towing force It is set that the securing point of the i-th tug G i at the tunnel element is A i (i=1, 2, …, N). The coordinate of A i is (x i , y i ). The towing force of tug G i is F i . The angle of the xaxis's positive direction counter clockwise to F i is set as α i , which is called the angle of F i . The scopes of towing forces' LI Jun-jun et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 32-40 33 magnitude and angle are represented in Eqs. (8) and (9), where, i=1, 2, …, N.
2.4 Resultant force and resultant moment of tugs The tunnel element moves very slow in the floating process. It is assumed that the translation velocity of the element is constant in this work. Then, F=-f. The mass of the tunnel element are basically symmetrical. It is thought that the hydrodynamic resistance f goes through the center of the tunnel element. To avoid the tunnel element rotating in translation process, resultant moment T should meet Eq. (10): (10)

Direction inconsistence between the resultant force f and velocity
The angle of the x-axis's positive direction counter clockwise to F is set as . Components F in the x and y directions are set as F x and F y , respectively. Then, , . From Eqs. (3)- (7), it can be known that (11) When , it is known that , , and . The absolute values of and are represented in Eqs. (13) and (14), respectively.
In Eq. (13), , , and the exponents of V x and V y are 1.83 and 2, respectively. So " " does not hold. Therefore, " " does not hold if . Accordingly, there is angle difference between F and V. From Eqs. (13) and (14), it can be known that this angle difference is not fixed. Meanwhile there is no obvious change law in this angle difference. These characteristics, which make the magnitudes and angles of towing forces difficult to be chosen, bring complexity to translation control of immersed tunnel element.

Translation control model of immersed tunnel element
The translation control model of immersed tunnel element is developed based on the analysis of the resultant force and resultant moment. To bypass the problem men-tioned in Section 2.5, the resultant force is analyzed in both the x and y axes on the basis of V x and V y .

Resultant force in the x and y axes
and are just positive numbers without direction. Meanwhile, and are in the same direction, while and are in the same direction. Then and can be calculated according to Eqs. (15) and (16). Where, "sgn" is sign function.
Components in the x and y directions are and , respectively. The sum of and is and , respectively:

Resultant moment in four quadrants
To obtain the resultant moment of tugs, moment arm and moment of each towing force should be calculated. In Fig. 1, different towing forces may be in different quadrants. The moment calculation methods in different quadrants are different. Here the moment in the first quadrant is calculated firstly, and then extended to other quadrants.
In the first quadrant, . It is denoted that β i = , . , so . The calculations of the moment arm and moment are different when is in different ranges. In Fig. 2, moment arms are plotted when is in different ranges.

When
, through geometrical analysis of Fig. 2a, it can be known that The clockwise direction is set as the reference direction. It can be derived that T i in the second quadrant also obeys Eq. (20), and T i in the third and fourth quadrants both obey Eq. (21).
It is found that the signs ahead of cos in Eqs. (20) and (21) . For the convenience of analysis, variable is set in Eq. (23). Then, the moment of the towing force for the i-th tug G i is . With consideration of Eq. (10), the resultant moment will meet Eq. (24).
3.3 Translation control model for immersed tunnel element 3.3.1 Objective function Floating velocity should be as fast as possible from the aspect of efficiency, and surplus towing forces are supposed to be as large as possible to enhance the ability to deal with the uncertainty of wind and current flow. Therefore, Eqs. (25) and (26)

Restrictions
Through increasing restriction in Eq. (28), some surplus towing capacity is ensured to enhance the ability to deal with uncertainty. Where, .
is the minimum surplus towing capacity for the i-th tug. Eqs. (8) and (28) are united as Eq. (29).

Particle swarm-based translation control optimization
Because of many decision variables and restrictions, it is difficult for traditional optimization method to solve translation control optimization problem of immersed tunnel element. Particle swarm optimization (PSO) algorithm is a famous intelligent optimization algorithm, which is simple and practical and has strong ability to solve nonlinear optimization problems Eberhart, 1995, 2001;Clerc, 2006). Therefore, PSO is adopted to optimize the translation control problem of immersed tunnel element in this work.

PSO Algorithm
PSO algorithm firstly initializes a group of particles. These particles find the optimal solution by iterations. Recurrence formulas are: where, , D denoting the dimension of particles; , T denoting the maximum iterations; , M denoting the number of particles; de- notes inertia weight while and are cognitive and social parameters, respectively; R denotes a random number in the range of [0, 1]; denotes the optimal position of a particle reached in the search process; denotes the optimal position of which all the particles reached. Particle velocity V id is not larger than the maximum velocity V max, d .

Fitness function
From Eqs. (25) and (27), ranges of two subgoals in the translation control optimization model are obviously different. In order to treat these two subgoals equally, normalization is done firstly: (32) Then, , . and both seek the maximum value. In order to give one control strategy to constructor, linear weighted method is adopted to integrate with . However, and conflict with each other. If the traditional linear weighted method is used directly, the phenomena that one subgoal is too strong while the other subgoal is too weak tends to occur. This does not conform to the original intention of taking these two subgoals into account. To avoid this problem, natural logarithms of are used to replace . A sort of linear weighted logarithmic function is exploited as the fitness function here: where and are weighting coefficients, , . If a subgoal is small in a control scheme, is a negative number with a large absolute value. This control scheme is easy to be washed out in iterations. Then the phenomena that one subgoal is too strong while the other goal is too weak can be avoided.

Decision variable setting and restriction handling
There are 2N decision variables in the translation control problem of immersed tunnel element, such as , . It is easy to calculate the resistance by the velocity, while it is difficult to calculate the velocity by the resistance. Decision variables of the particle swarm-based control method in this work are set as: . Eqs. .
, let be a very small positive real number to avoid non-solution to Eq. (39). If and exceed the range of Eq. (8), the punish function is adopted. If and exceed the ranges of Eqs. (8) and (9) in iterations, they are limited to their boundary values.

Algorithm flow
In Eqs. (32) and (33), , and . But cannot be obtained directly. Here, PSO algorithm is used to seek whose fitness function is Eq. (25), without regard to restriction in Eq. (28). Calculation processes of (or ) are as follows: Step 1. Setting model and algorithm parameters, initiation of the original position and velocity for particles.
Step 2. Calculate according to Section 4.3.
Step 3. Calculate the fitness value (or ) for each particle. Step 4. Seek the historically optimal solution for each particle, and let the optimal be .
Step 5. If the end condition is fitted, output the result; else, turn to Step 6.
Step 6. Create a new generation of particles by adjusting particles' position and velocity based on Eqs. (30) and (31), and then turn to Step 3.

Translation control simulation for immersed tunnel element in the HZMB project
The performance of the proposed method is validated through translation control simulation of immersed tunnel element in the HZMB project, and compared with traditional empirical method.
The length, width and draft of the immersed tunnel are 180 m, 56.4 m and 11.1 m, respectively. Square coefficient . Take Rongshutou sea-route as an example. V 0 =2 kn. The moving direction of immersed tunnel element is 12°, which is the same as the positive direction of the x-axis. Then, θ 1 =0°. The flow directions of tide rise and retreat are 355° and 175°, respectively. It can be calculated that θ 0 = (12+360-355)°=17° in tide rise while θ 0 =(12+360-175)°= 197° in tide retreat.

Traditional empirical method
The control program designed according to traditional empirical method is described here.
Four main tugs (G 1 -G 4 ) and two assisted tugs (G 5 and G 6 ) are employed. The deployment of six tugs around the tunnel element is shown in Fig. 3. G 1 -G 4 drag the tunnel element to move ahead. G 1 and G 2 control the floating direction. G 5 and G 6 push or pull the tunnel element to overcome the transverse current flow, which ensures that the floating track is a line approximately.

Analysis of empirical method
Obviously, the utilization rates of F 1 -F 4 reaches their maximum when . In this situation, . From Eqs. (15) and (16), it can be known that Average distribution of the surplus towing forces is most appropriate because of uncertainty wind and current flow. Then Eq. (42) can be derived. And can be calculated through Eqs. (43) and (44). Based on the relationship between the velocity and resistance in Section 2.2, the six towing forces can be obtained according to Eqs. (41), (43) and (44) if the velocity is known.

Particle swarm-based control method
F sc i = 10 tons i = 1, 2, · · · , N In the particle swarm-based control method, the same six tugs as in empirical method are employed. Maximum towing forces, towing points and ranges of are the same as empirical method. and are not limited to or because G 5 and G 6 are not limited to overcoming the transverse current flow. The ranges of them are all . Besides, ( ).
As mentioned in Section 4.3, a one-dimensional vector is coded as a particle. M=20, T=500. (Shi and Eberhart, 1998). According to algorithm flow in Section 4.4, in tide rise and tide retreat are 6.675 kn and 2.816 kn, respectively.

Simulation results
To fully verify the model and algorithm presented in this work, 12 situations with different λ 1 (λ 1 =1, 0.98, …, 0.78) are calculated in tide rise and tide retreat. The variation of the surplus towing force with the velocity in traditional empirical method and the particle swarm-based control method is drawn in Figs. 5 and 6, which are in tide rise and retreat, respectively. The minimum and mean surplus towing forces of the two methods in tide rise and tide retreat are shown in Tables 1 and 2, respectively.
In Fig. 5, the two heavy lines are variations of the surplus towing force in traditional empirical method. The horizontal one indicates surplus towing forces of G 5 and G 6 , while the other one indicates surplus towing forces of G 1 -G 4 . The fine lines are surplus towing forces of the control scheme calculated by particle swarm-based control  LI Jun-jun et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 32-40 37 method. The points in fine lines are surplus towing forces of 12 situations. The indications of different lines in Fig. 6 are the same as those in Fig. 5.
In Tables 1 and 2, "EM" and "PS" are the abbreviations of "empirical method" and "particle swarm-based control method", respectively. Data (such as 75.03%, 9.87%) in column "+%" denote the minimum (or mean) surplus towing force of particle swarm-based control method is 75.03% (or 9.87%) more than that of empirical method. In empiric-al method, it can be calculated that V 1 <6.050 kn in tide rise, and V 1 <2.362 kn in tide retreat.
The sketch maps of tunnel element towing are drawn taking two situations for example. In Fig. 7, V 1 =5.544 kn in tide rise. In Fig. 8, V 1 =1.715 kn in tide retreat. To compare the empirical method with the particle swarm-based control method, the schemes acquired by these two methods are adjacent to each other. In Figs. 7 and 8, ↑ denotes north while * shows the towing point. The length of the directed line is  proportional to the magnitude of each towing force.

Discussion of simulation results
From Figs. 5 and 6, Tables 1 and 2, the ranges of the velocities in tide rise and retreat are 3.592-6.193 kn, 1.250-2.400 kn, respectively. The actual floating velocities of immersed tunnel element are also in these ranges.
When λ 1 is 0.88 or 0.86 in tide rise, the minimal surplus towing forces of the particle swarm-based control method is 1.25% or 3.25% smaller than those of empirical method, respectively. However, the mean surplus towing force of particle swarm-based control method is 5.62% or 5.27% more than that of empirical method, respectively. Therefore, the particle swarm-based control method is still better than the empirical method. In other 10 situations of tide rise and all 12 situations of tide retreat, the minimum and mean surplus towing forces of particle swarm-based control method are all better than those of empirical method.
From Fig. 5 and Table 1, the velocity of empirical method in tide rise cannot exceed 6.050 kn. And when the velocity reaches its maximum, the surplus towing forces of G 1 -G 4 will be zero together, which is not suitable for floating. However, the velocity of particle swarm-based control method can exceed 6.050 kn. And when the velocity is between 6.050 and 6.193 kn, the surplus towing force of each tug is not less than 10 tons.
Being similar to tide rise, from Fig. 6 and Table 2, the velocity of empirical method in tide retreat cannot exceed 2.362 kn, while the surplus towing forces of G 1 -G 4 will be zero together when the velocity reaches its maximum. And the velocity of the particle swarm-based control method can exceed 2.362 kn, while the surplus towing force of each tug is not less than 10 tons if the velocity is between 2.362 and 2.400 kn.
It can also be seen in Figs. 7 and 8, F 1 -F 4 are larger in empirical method, which makes the surplus towing forces of them smaller than those of the particle swarm-based control method.
As mentioned in Section 5.1, the role of G 5 and G 6 in empirical method is to overcome the transverse current flow. So the towing forces and surplus capacities of G 5 and G 6 depend on the transverse current flow, and have nothing to do with the element velocity and the towing forces of G 1 -G 4 . In particle swarm-based control method, the element is towed ahead by these six tugs jointly. G 5 and G 6 also drag the tunnel element to float ahead. Then the surplus towing forces of G 1 -G 4 can be larger than those of empirical method. And the multiplication and division method is used in the subgoal of surplus towing forces. The surplus towing forces of tugs in particle swarm-based control method are more reasonable than those in empirical method. In summary, the particle swarm-based control method outperforms empirical method in translation control of immersed tunnel element.

Conclusions
The translation control and optimization of immersed tunnel element in the HZMB project is discussed in this work. It proposes a particle swarm-based control method which is validated through the translation control simulation of immersed tunnel element in the HZMB project. The simulation results indicate that the particle swarm-based control method is better than empirical method in the aspect of surplus towing forces.
The method proposed in this work is applicable to either straight movement or transverse movement, regardless of any current velocity and tunnel element velocity. This research is helpful to towing of immersed tunnel element, caisson, and non-power vessel.
In future, the dynamics in the process of immersed tunnel element translation and steering control will be studied further.