Modified quantum particle swarm optimization for translation control of immersed tunnel element with pontoons

Abstract

Immersed tube tunnel serves as a preferred method of construction in large underwater tunnel engineering. In this work, modified quantum particle swarm optimization for translation control of immersed tunnel element with pontoons is studied aiming at its specific configuration. The translation control model is built based on the resistances of immersed tunnel element and two floating pontoons. To expand the search space, particles are coded according to Bloch coordinates. To make full use of three positions in each particle, they are selected with certain probabilities in accordance with the corresponding fitness values. Main dimension change and phase shift are implemented to improve the efficiency of velocity update for particles. Simulation results of Hong Kong-Zhuhai-Macao Bridge project delivers performance improvement of the proposed method.

Keywords

Immersed tunnel element with pontoons translation control quantum particle swarm optimization Bloch coordinates of qubits

1 Introduction

The importance of immersed tunnel in the construction of roads and railways crossing a narrow waterway has been well documented during the past several decades [1, 2]. An immersed tunnel is a kind of underwater tunnel composed of elements in a manageable length. Immersed tunnel elements are prefabricated in the flooded casting basin or dock, floated to the tunnel site, and then connected together. Generally there’re two pontoons helping the tunnel element floating because the tunnel element’s density is much higher than that of water. Conventional towage, in which several tugs assist the tunnel element transportation, is normally used in floating.

Straight movement (moving forward or backward) and transverse movement (moving left or right), which are collectively called “translation”, are usually the main working modes of immersed tunnel element floating [3]. To follow a control strategy with satisfactory efficiency and stability of tunnel element translation, all tugs should coordinate in an efficient way. Varela and Soares [4] described the conceptual overview, interface specification and object models for real-time simulation of ship towing operations in virtual environments. Luis, et al. [5] developed automatic maneuvering systems, which consisted of information technology applications helping the skipper maneuvering the tug. However, the tunnel element and floating pontoons are all cuboids approximately, and their bow constructions are completely different from that of vessel. Therefore, the calculation method of vessel resistance can’t be directly applied to the tunnel element towage. Li, et al. [3] built the immersed tunnel element translation control model and proposed a particle swarm-based translation control method. However, the tunnel element and floating pontoons are simply viewed as a whole with a cuboid structure. This simplification may reduce the feasibility and effectiveness of the towing control.

To determine the precise translation control of immersed tunnel element with pontoons, the authors investigated the modeling and optimization for translation control of immersed tunnel element with pontoons in this work. First, a translation control model was built based on the comprehensive resistance analysis of the immersed tunnel element and floating pontoons. Second, a modified quantum particle swarm optimization based on Bloch spherical search (MBQPSO) was designed. Specifically, one of the three positions for each particle, which obtains the best solution, was set as the main basis for searching, while the other two positions were set as assistant bases for searching; the best dimensions of the personal and general historically best particles were changed to be consistent with the selected dimension of the current particle; the phases of the personal and general historically best particle were shifted to the same monotonic interval with the phase of the current particle. Third, the case of the Hong Kong-Zhuhai-Macao Bridge project was demonstrated to verify the performance of the proposed approach by comparing it with particle swarm optimization (PSO) [3], quantum particle swarm optimization with Bloch sphere (BQPSO) [6], and quantum ant colony optimization algorithm based on Bloch spherical search (BQACO) [7].

The main novel contributions of this work are as follows: (1) building an accurate translation control model for immersed tunnel element and floating pontoons; (2) designing the probability selection mechanism for particles with Bloch spherical description to fully utilize three-dimensional positions in expanding the search space; (3) changing the main dimension and shifting the phase for the personal and general historically best particles to enhance the effectiveness of particles’ velocities update in iterations; and (4) demonstrating improved performance using the proposed method through simulations of the Hong Kong-Zhuhai-Macao Bridge project.

2 Translation control of immersed tunnel element with pontoons

2.1 Resistance and towing forces

The top view, front view and side view of the tunnel element with pontoons are illustrated in Fig. 1(a-c), respectively. And a 2D axis is drawn in Fig. 1(a). Where, the element center is the origin of coordinate, x-axis is set in the longitudinal direction of tunnel element, and y-axis is perpendicular to x-axis.

Fig.1

Immersed tunnel element with two floating pontoons.

The current velocity, the tunnel element velocity relative to shore and the tunnel element velocity relative to current flow are set as V₀, V₁ and V respectively. θ₀/θ₁/θ denotes the angle between V₀/V₁/V and the positive direction of x-axis. The relation among V₀, V₁, and V is shown in Fig. 1(a). V_x and V_y denote components of V in the positive directions of x and y axes, respectively. Then, V_x = V₁ cos θ₁ - V₀ cos θ₀, and V_y = V₁ sin θ₁ - V₀ sin θ₀ [3].

Different from reference [3], the resistances of immersed tunnel element with two floating pontoons are analyzed separately in this work. The marine towage resistance R_T of the tunnel element can be calculated by Equation (1) [3 , 9]. Similarly, the towage resistance of a floating pontoon, R_Tp, is represented in Equation (2). Then, the total towage resistance of towing system, $R_{T}^{'}$ , is represented in Equation (3).

$R_{T} = 1.15 (1.67 A_{e, 1} | V |^{1.83} \times 10^{- 3} + 0.62 C_{b} A_{e, 2} V^{2})$ (1) $R_{Tp} = 1.15 (1.67 A_{p, 1} | V |^{1.83} \times 10^{- 3} + 0.62 C_{b} A_{p, 2} V^{2})$ (2) $R_{T}^{'} = R_{T} + 2 R_{Tp}$ (3)

In Equations (1 and 2), A_e,1 and A_p,1 are the wetted surface areas under water line of tunnel element and floating pontoon respectively; C_b is block coefficient; A_e,2 and A_p,2 are the submerged parts of transverse section area in tunnel element and floating pontoon respectively. Units of R_T, R_Tp, and $R_{T}^{'}$ are all “kN”.

If the tunnel element is parallel to current flow, then A_e,1 = L (B + 2d), A_e,2 = B · d, and A_p,2 = B_p · d_p. The wetted surface area of bottom surface is: L_p · B_p, while the wetted surface area of side surfaces is: 4·L_p · d_p. Then, A_p,1 = L_p (B_p + 4d_p). If the tunnel element is perpendicular to current flow, then A_e,1 = B (L + 2d), A_p,1 = B_p (L_p + 2d_p), A_e,2 = L · d, and A_p,2 = L_p · d_p. Where L, B and d are the length, width and draft of the tunnel element respectively; L_p, B_p, $B_{p}^{'}$ and d_p are the length, width under water line, width above water line, and draft of the floating pontoon respectively. L, B, L_p, B_p and $B_{p}^{'}$ are illustrated in Fig. 1(b) and (c).

Considering the balance between towing forces and resistance in x and y axes, Equations (4) and (5) can be deduced. In Equations (4∼7), F_i denotes the towing force of tug G_i; α_i, which is called angle of towing force F_i, denotes the angle of x-axis’s positive direction counter clockwise to F_i; $R_{Tx}^{'}$ and $R_{Ty}^{'}$ denote components of $R_{T}^{'}$ in the directions of x and y axes, respectively; i = 1, 2, …, N, and N is the number of tugs. $\sum_{i = 1}^{N} F_{i} cos α_{i} = R_{Tx}^{'} \cdot sgn (V_{x})$ (4) $\sum_{i = 1}^{N} F_{i} sin α_{i} = R_{Ty}^{'} \cdot sgn (V_{y})$ (5)

In order to avoid tunnel element rotating in translation process, the resultant moment should meet Equation (6). In Equation (6), $L_{i}^{'}$ is set in Equation (7). In Equation (7), β_i= arctan (x_i/y_i); x_i and y_i are the horizontal and vertical coordinates of A_i, respectively; A_i is the towing point of F_i. $T = \sum_{i = 1}^{N} F_{i} L_{i}^{'} = 0$ (6) $L_{i}^{'} = sgn (y_{i}) \cdot \sqrt{x_{i}^{2} + y_{i}^{2}} cos (α_{i} + β_{i})$ (7)

2.2 Translation control model

Equations (8) and (9) are set as objective functions. In Equation (9), if the multiplication and division method is used directly, g₂ would easily become much larger than g₁. It is unreasonable to put them together in calculation. Therefore, it is more apposite that g₂ is modified to be the N-th root of $\prod_{i = 1}^{N} (F_{i}^{max} - F_{i})$ .

$max g_{1} = V_{1}$ (8)

$max g_{2} = {[\prod_{i = 1}^{N} (F_{i}^{max} - F_{i})]}^{raise 0.7 ex 1 / - lower 0.7 ex N}$ (9)

Formulas (4∼6) represent translation control restrictions. Besides, $F_{i} \in [0, F_{i}^{max} - F_{i}^{sc}]$ , and $α_{i} \in [α_{i}^{min}, α_{i}^{max}]$ , where, $F_{i}^{max}$ and $F_{i}^{sc}$ are the maximum towing force and the minimum surplus towing force for the ith tug, respectively, and $F_{i}^{max} > F_{i}^{sc} > 0$ ; $α_{i}^{min}$ and $α_{i}^{max}$ are the minimum and maximum angles of α_i, respectively.

3 Modified quantum particle swarm optimization

3.1 Bloch spherical description of particles

In the Bloch spherical coordinate system, the state of a qubit can be expressed as follows [10]: $| ψ 〉 = cos (θ / 2) | 1 〉 + e^{i \emptyset} sin (θ / 2) | 0 〉$ (10)

A point U on Bloch sphere can be determined by angles θ and φ, as shown in Fig. 2. A qubit can be described by three-dimensional coordinates of Bloch sphere: |ψ〉 = [cos φ sin θ, sin φ sin θ, cos θ] ^T, where φ ∈ [0, 2π) and φ ∈ [0, π].

Fig.2

Bloch sphere representation of a qubit.

In this work, particles are coded with Bloch coordinates. The maximum and minimum values of the jth dimension (j = 1, 2, …, n, and n is the number of dimensions.) gene in a particle are denoted as $p_{j}^{min}$ and $p_{j}^{max}$ , respectively. Bloch coordinates of qubits are mapped from the hypercube space Iⁿ = [-1, 1] ⁿ to the gene space. If the coordinates of the jth qubit in the ith particle are [x_ij, y_ij, z_ij] ^T, then the corresponding dimension genes in this particle are [X_ij, Y_ij, Z_ij] ^T. Here, x_ij = cos φ_ij sin θ_ij, y_ij = sin φ_ij sin θ_ij, z_ij = cos θ_ij, i = 1, 2, …, m, and m is the number of particles; X_j, Y_j and Z_j are described in Equations (11–13), respectively. $X_{ij} = \frac{1}{2} [p_{j}^{min} (1 - x_{ij}) + p_{j}^{max} (1 + x_{ij})]$ (11) $Y_{ij} = \frac{1}{2} [p_{j}^{min} (1 - y_{ij}) + p_{j}^{max} (1 + y_{ij})]$ (12) $Z_{ij} = \frac{1}{2} [p_{j}^{min} (1 - z_{ij}) + p_{j}^{max} (1 + z_{ij})]$ (13)

And the ith particle is: $p_{i} = [\begin{matrix} X_{i 1} \\ Y_{i 1} \\ Z_{i 1} \end{matrix} | \begin{matrix} X_{i 2} \\ Y_{i 2} \\ Z_{i 2} \end{matrix} | \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} | \begin{matrix} X_{ij} \\ Y_{ij} \\ Z_{ij} \end{matrix} | \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} | \begin{matrix} X_{in} \\ Y_{in} \\ Z_{in} \end{matrix}]$ (14)

It can be expressed as: $p_{i} = [X_{i}, Y_{i}, Z_{i}]^{T}$ (15)

where, X_i = [X_i1, X_i2, ⋯ , X_in], Y_i = [Y_i1, Y_i2, ⋯, Y_in], and Z_i = [Z_i1, Z_i2, ⋯ , Z_in]. Thus, a particle represents three positions in the solution space, which can extend the number of optimization solutions and enhance the search ability of the algorithm.

3.2 Selection probabilities of three - dimensional positions

Generally, three genes represented by a qubit with Bloch coordinates are employed simultaneously to search the optimal solution [6 , 11]. On the one hand, this measure would result in three times computation amount compared with the traditional PSO; on the other hand, the best position in one particle deserves more attention than the other two positions.

To make better use of the three-dimensional Bloch coordinates, positions X_i, Y_i and Z_i are selected with certain probabilities. If the solution obtained based on a position is better, this position would have a higher probability to be selected. Details are as follows:

In the first iteration, positions X_i, Y_i and Z_i are selected simultaneously. At the end of the first generation, let the optimal solutions based on X_i, Y_i and Z_i be R_x, R_y and R_z, respectively, and then sort X_i, Y_i and Z_i according to R_x, R_y and R_z. The best dimension is called the main dimension, while the other two dimensions are called the non-main dimensions. From the second iteration, the selection probabilities of positions are calculated by Equations (16∼18), where $f (t) = \frac{1 - exp [- c_{p} (t - t_{c})]}{1 + exp [- c_{p} (t - t_{c})]}$ , c_p > 0, t_c = 1, and P_min = (1 - P_max)/3. Here, c_p is the probability constant which is used to adjust the changing rate of P₁, P₂ and P₃. It can be seen that P₁ + P₂ + P₃ = 1. Therefore, with increasing the number of iterations, P₁ gradually increases from 1/3 to P_max, while P₂ gradually reduces from 1/3 to 2P_min, and P₃ gradually reduces from 1/3 to P_min. $P_{1} = 1 / 3 + (P_{max} - 1 / 3) f (t)$ (16) $P_{2} = 1 / 3 - (1 / 3 - 2 P_{min}) f (t)$ (17) $P_{3} = 1 / 3 - (1 / 3 - P_{min}) f (t)$ (18)

If the sequence of X_i, Y_i and Z_i changes in the τth iteration, let t_c = τ, and recalculate the selection probabilities of positions according to the new sequence of X_i, Y_i and Z_i.

For example, the maximum iteration number N_t = 40, P_max= 0.9, c_p= 0.3, and the sequences of X_i, Y_i and Z_i change at the 8th and 20th iterations. The sequence of X_j, Y_j and Z_j is “X_i, Y_i and Z_i” in the 1st∼7th iterations, “Z_i, X_i and Y_i” in the 8th∼19th iterations, and “Y_i, Z_i and X_i” in the 20th∼40th iterations. Let P_X, P_Y and P_Z denote the selection probabilities of positions X_i, Y_i and Z_i, respectively. P_X, P_Y and P_Z in 1st∼40th iterations are shown in Fig. 3.

When P₁, P₂ and P₃ don’t differ much, the qubit encoding’s advantage of expanding the search space can be fully utilized. When P₁, P₂ and P₃ differ much, particles search solutions mainly base on the main dimension position. The other two dimension positions with selection probabilities at least P_min play the role of mutation.

Fig.3

Selection probabilities of positions.

3.3 Iterations of particles

In each iteration, particles are updated based on the quantum rotation gate in Equation (19) [3].

$\begin{matrix} U = \\ [\begin{matrix} cos Δ φ cos Δ θ & - sin Δ φ cos Δ θ & sin Δ θ cos (φ + Δ φ) \\ sin Δ φ cos Δ θ & cos Δ φ cos Δ θ & sin Δ θ sin (φ + Δ φ) \\ - sin Δ θ & - tan (φ / 2) sin Δ θ & cos Δ θ \end{matrix}] \end{matrix}$ (19)

In the (t + 1) ^th iteration, Δφ and Δθ are replaced with Δφ_ij (t + 1) and Δθ_ij (t + 1) in Equations (20 and 21), respectively: $Δ φ_{ij} (t + 1) = ω φ_{ij} (t) + c_{1} r Δ φ_{l} + c_{2} r Δ φ_{g}$ (20) $Δ θ_{ij} (t + 1) = ω θ_{ij} (t) + c_{1} r Δ θ_{l} + c_{2} r Δ θ_{g}$ (21)

where ω denotes the inertia weight while c₁ and c₂ are the cognitive and social parameters, respectively; Δφ_l = φ_l - φ_ij (t), Δφ_g = φ_g - φ_ij (t), Δθ_l = θ_l - θ_ij (t), and Δθ_g = θ_g - θ_ij (t); φ_l and θ_l are angles of the personal historically best particle $p_{ij}^{pb} (t)$ ; φ_g and θ_g are angles of the general historically best particle $p_{j}^{gb} (t)$ ; φ_ij (t) and θ_ij (t) are angles of the current particle p_ij (t) in the tth iteration. The new angles in the (t + 1) th iteration are described in Equations (22 and 23) as: $φ_{ij} (t + 1) = φ_{ij} (t) + Δ φ_{ij} (t + 1)$ (22) $θ_{ij} (t + 1) = θ_{ij} (t) + Δ θ_{ij} (t + 1)$ (23)

3.4 Main dimension change and phase shift

The main dimension of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) and the selected dimension of p_ij (t) may not be the same, and the genes of the particles are not monotonic relative to the angles. Therefore, it is not suitable to calculate angles directly according to Section 3.3. To improve the efficiency of particles updating, the main dimension change and phase shift were executed in this work.

(1) Main dimension change

If the main dimension of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) is different from the selected dimension of p_ij (t), then the main dimension of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) changes to the selected dimension of p_ij (t), and the angles of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) change according to Table 1.

Table 1
Main dimension change of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ )

d _cc d _mp/g $θ_{p / g}^{'}$ $φ_{p / g}^{'}$

x y θ _p/g π/2 - φ_p/g

y x

x z arccos x $arcsin (y / \sqrt{1 - x^{2}})$

z x

y z arccos y $arcsin (z / \sqrt{1 - y^{2}})$

z y

d _cc	d _mp/g	$θ_{p / g}^{'}$	$φ_{p / g}^{'}$
x	y	θ _p/g	π/2 - φ_p/g
y	x
x	z	arccos x	$arcsin (y / \sqrt{1 - x^{2}})$
z	x
y	z	arccos y	$arcsin (z / \sqrt{1 - y^{2}})$
z	y

In Table 1, d_cc denotes the selected dimension of p_ij (t), and d_mp/g denotes the main dimension of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ); θ_p/g and φ_p/g denote the original angles θ and φ of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ); $θ_{p / g}^{'}$ and $φ_{p / g}^{'}$ denote the new angles θ and φ of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ).

(2) Phase shift of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ )

If the angles of p_ij (t) and $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) are not in the same monotonic interval, then the calculations of Δφ_l and Δθ_l directly according to “Δφ_l = φ_p - φ_ij (t), Δφ_g = φ_g - φ_ij (t), Δθ_l = θ_p - θ_ij (t), and Δθ_g = θ_g - θ_ij (t)” are unreasonable. Here, the phase of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) is shifted to solve this problem.

The scope of θ is [0, π), in which cos θ is monotonic, so that phase shift is not needed once the selected dimension of p_ij (t) is z.

After the main dimension change of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ), angles φ and θ of $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ ) shift according to Tables 2 and 3 , respectively.

Table 2

Phase shift of φ in $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ )

d _cc	φ _ij	φ _p/g	$φ_{p / g}^{'}$
x	[0, π)	[π, 2π]	2π - φ_p/g
	[π, 2π]	[0, π)
y	[0, π/2)	[π/2, 3π/2)	π - φ_p/g
	[π/2, 3π/2)	[0, π/2)
	[0, π/2)	[3π/2, 2π)	$\begin{matrix} φ_{p / g} + 2 π \cdot \\ sgn (φ_{ij} - φ_{p / g}) \end{matrix}$
	[3π/2, 2π)	[0, π/2)
	[π/2, 3π/2)	[3π/2, 2π)	3π - φ_p/g
	[3π/2, 2π)	[π/2, 3π/2)

Table 3

Phase shift of θ in $p_{ij}^{pb} (t)$ (or $p_{j}^{gb} (t)$ )

d _cc	θ _ij	θ _p/g	$θ_{p / g}^{'}$
x/y	[0, π/2)	[π/2, π]	π - θ_p/g
	[π/2, π]	[0, π/2)

(3) Phase shifts of φ_ij (t + 1) and θ_ij (t + 1)

If φ_ij (t + 1) and θ_ij (t + 1) exceed their scope, they should be shifted. φ_ij (t + 1) is shifted to $φ_{ij}^{'} (t + 1)$ according to Equation (24), and θ_ij (t + 1) is shifted to $θ_{ij}^{″} (t + 1)$ according to Equations (25 and 26). $φ_{ij}^{'} (t + 1) = φ_{ij} (t + 1) - ⌊ φ_{ij} (t + 1) / (2 π) ⌋ \cdot 2 π$ (24) $θ_{ij}^{'} (t + 1) = θ_{ij} (t + 1) - ⌊ θ_{ij} (t + 1) / (2 π) ⌋ \cdot 2 π$ (25) $θ_{ij}^{″} (t + 1) = {\begin{matrix} θ_{ij}^{'} (t + 1), & θ_{ij}^{'} (t + 1) \in [0, π] \\ 2 π - θ_{ij}^{'} (t + 1), & θ_{ij}^{'} (t + 1) \in (π, 2 π) \end{matrix}$ (26)

3.5 Fitness function

The fitness function is set in Equation (27), where h₁ and h₂ are set in Equations (28 and 29) [3], and λ denotes the importance of h₂. $max h = ln h_{1} + λ ln h_{2}$ (27) $h_{1} = (g_{1} - g_{1}^{min}) / (g_{1}^{max} - g_{1}^{min})$ (28) $h_{2} = (g_{2} - g_{2}^{min}) / (g_{2}^{max} - g_{2}^{min})$ (29)

Here, $g_{1}^{min}$ and $g_{1}^{max}$ are the minimum and maximum values of g₁ respectively; $g_{2}^{min}$ and $g_{2}^{max}$ are the minimum and maximum values of g₂ respectively.

Decision variables of particle swarm-based control method are set as “V₁, F₁, ⋯ , F_N-3, α₁, ⋯ , α_N” [3]. Punish function will be adopted if F_N-2, F_N-1 and F_N exceed their ranges.

4 Simulations

4.1 Problem description

The performance of the proposed algorithm is validated through translation control simulations of immersed tunnel element with pontoons in Hong Kong-Zhuhai-Macao Bridge project [3], and compared with other methods.

Hong Kong-Zhuhai-Macao Bridge, which is about 50 km long, is the world’s largest cross-sea bridge. There is a 5.99 km immersed tunnel consisted of 28 line tunnel elements and 5 curve tunnel elements in this project. Here the line tunnel element is simulated. L = 180 m, B = 37.95 m, d = 11.1 m, L_p= 40.2 m, B_p= 14.4 m, $B_{p}^{'}$ = 56.4 m, d_p= 6.2 m, and C_b= 1. V₀= 2 knots in both tide rise and retreat. Take Rongshutou sea-route for example. The direction of immersed tunnel element moves is 12°, which is the same as the positive direction of x-axis. It means θ₁= 0°. The flow directions of tide rise and retreat are 355° and 175°, respectively. It can be calculated that θ₀= (12+360-355)°= 17° in tide rise while θ₀= (12+360-175)°= 197° in tide retreat.

Four main tugs (G₁∼G₄) and two assisted tugs (G₅ and G₆) are employed in the transportation process [3]. The sketch map of their towing forces F₁∼F₆ is shown in Fig. 4.

N = 6; $F_{1}^{max} = F_{2}^{max} = 56$ tons, $F_{3}^{max} = F_{4}^{max} = 50$ tons, and $F_{5}^{max} = F_{6}^{max} = 43$ tons; $F_{i}^{sc} = 10 t$ [3]. The ranges of α₁, α₂, α₃ and α₄ are [–75°, 165°], [–165°, 75°], [35°, 255°] and [105°, 325°], respectively; while the ranges of α₅ and α₆ are both [15°, 345°].

Fig.4

The towing force directions and towing points’ coordinates.

4.2 Simulation comparison

The performance of MBQPSO is verified by comparing it with three other methods: PSO [3], BQPSO [6] and BQACO [7]. In MBQPSO, M = 20, N_t= 100, P_max= 0.9, c_p= 0.3. In order to ensure the convergent trajectories of particles [12], ω= 0.7298 and c₁=c₂= 1.49618 [13]. Five cases with different λ (λ= 0.2, 0.4, 0.6, 0.8 and 1.0) are calculated both in tide rise and tide retreat.

The results comparisons among these methods are shown in Tables 4 and 5. In Tables 4 and 5, g₁ and g₂ are two sub-objects described in Equations (8 and 9), respectively. They denote the velocity of immersed tunnel element with pontoons and the surplus towing forces of tugs, respectively.

Table 4
Results comparison in tide rise

λ Method g₁ / knot g₂ / ton fitness

0.2 PSO 5.229 17.918 –0.439

BQPSO 5.286 18.357 –0.423

BQACO 5.482 17.975 –0.391

MBQPSO 5.656 18.883 –0.350

0.4 PSO 4.669 26.525 –0.598

BQPSO 4.831 25.753 –0.576

BQACO 4.901 27.291 –0.538

MBQPSO 5.057 28.022 –0.496

0.6 PSO 4.489 30.735 –0.673

BQPSO 4.566 31.268 –0.646

BQACO 4.520 32.744 –0.629

MBQPSO 4.647 33.313 –0.591

0.8 PSO 4.121 35.569 –0.737

BQPSO 4.147 36.120 –0.718

BQACO 4.224 36.793 –0.685

MBQPSO 4.335 36.808 –0.659

1.0 PSO 3.862 35.458 –0.870

BQPSO 3.837 37.932 –0.810

BQACO 4.092 37.211 –0.764

MBQPSO 4.133 38.829 –0.712

λ	Method	g₁ / knot	g₂ / ton	fitness
0.2	PSO	5.229	17.918	–0.439
	BQPSO	5.286	18.357	–0.423
	BQACO	5.482	17.975	–0.391
	MBQPSO	5.656	18.883	–0.350
0.4	PSO	4.669	26.525	–0.598
	BQPSO	4.831	25.753	–0.576
	BQACO	4.901	27.291	–0.538
	MBQPSO	5.057	28.022	–0.496
0.6	PSO	4.489	30.735	–0.673
	BQPSO	4.566	31.268	–0.646
	BQACO	4.520	32.744	–0.629
	MBQPSO	4.647	33.313	–0.591
0.8	PSO	4.121	35.569	–0.737
	BQPSO	4.147	36.120	–0.718
	BQACO	4.224	36.793	–0.685
	MBQPSO	4.335	36.808	–0.659
1.0	PSO	3.862	35.458	–0.870
	BQPSO	3.837	37.932	–0.810
	BQACO	4.092	37.211	–0.764
	MBQPSO	4.133	38.829	–0.712

The fitness value comparisons among these four methods when λ= 0.4 and 0.8 in tide rise and retreat are shown in Fig. 5. Some fitness values in some early iterations are out of the figure because these values are smaller than –2 in Fig. 5(a) and (b), –4 in (c) or –7 in (d). This phenomenon results from the punish function for punishing the solutions in which F_N-2, F_N-1 and F_N exceed their range. This phenomenon, which only appears before the 2nd iteration in Fig. 5(a), the 5th iteration in (b), the 7th iteration in (c), and the 14th iteration in (d), has no effect on the average fitness value comparisons.

Table 5

Results comparison in tide retreat

λ	Method	g₁ / knot	g₂ / ton	fitness
0.2	PSO	2.001	11.132	–0.604
	BQPSO	1.999	11.819	–0.593
	BQACO	2.041	11.308	–0.581
	MBQPSO	2.223	11.847	–0.487
0.4	PSO	1.851	14.087	–0.886
	BQPSO	1.945	14.279	–0.831
	BQACO	2.010	13.821	–0.811
	MBQPSO	2.060	15.034	–0.753
0.6	PSO	1.751	16.612	–1.093
	BQPSO	1.715	18.686	–1.044
	BQACO	1.840	18.109	–0.992
	MBQPSO	1.850	18.703	–0.967
0.8	PSO	1.254	20.028	–1.496
	BQPSO	1.459	21.354	–1.293
	BQACO	1.652	20.730	–1.192
	MBQPSO	1.684	21.433	–1.147
1.0	PSO	1.5012	21.587	–1.421
	BQPSO	1.519	21.589	–1.409
	BQACO	1.481	22.879	–1.377
	MBQPSO	1.547	23.555	–1.304

Fig.5

Changes of fitness value with iterations.

From Tables 4 and 5, g₁ and g₂ of MBQPSO are always larger than those of the other three methods in the five cases of both tide rise and retreat. In some cases, g₁ or g₂ of some other methods is close to that of MBQPSO (for example, g₂ of BQACO and MBQPSO when λ= 0.8 in tide rise, g₁ of BQACO and MBQPSO when λ= 0.6 in tide retreat), but the fitness values of MBQACO are always obviously optimal. From Fig. 5, MBQPSO is obviously better than the other three methods from the 2nd, 35th, 54th, and 45th iterations in Fig. 5(a-d), respectively. From the subgraphs in Fig. 5, the advantage of MBQPSO is evident.

Although the performances of PSO, BQACO, and BQACO have been validated in previous literatures, the translation control problem of immersed tunnel element with pontoons is a complex optimization problem with many variables and strict constraints. In contrast, the probability selection mechanism, main dimension change, and phase shift have enhanced the efficiency of MBQACO evidently. Therefore, compared with the other three methods, the performance of MBQPSO is better for translation control of immersed tunnel element with pontoons.

5 Conclusion

This work mainly discussed the optimization for translation control of immersed tunnel element with pontoons. The translation control model considered the resistances of immersed tunnel element and two floating pontoons. A modified quantum particle swarm optimization based on Bloch spherical description of particles was presented. To enhance the search capability, the three-dimension positions of particles were selected according to the sorting of corresponding solutions. Main dimension change and phase shift were executed to make the velocity of particles update efficiently. Simulation results showed that MBQPSO could provide better translation control solutions with faster velocity and more surplus towing forces.

Footnotes

Acknowledgments

This work is supported by the Ministry of Education of Humanities and Social Science project (No. 15YJC630145, 15YJC630059), Natural Science Foundation supported by Shanghai Science and Technology Committee (No. 15ZR1420200). Here we would like to express our gratitude to them.

References

Grantz

W.C.

, Steel-shell immersed tunnels – forty years of experience, Tunnelling and Underground Space Technology 12(1) (1997), 23–31.

, Fang

Y.G.

, Mo

H.H.

, Gu

R.G.

, Chen

J.S.

, Wang

Y.Z.

and Feng

D.L.

, Model test of immersed tube tunnel foundation treated by sand-flow method, Tunnelling and Underground Space Technology 40 (2014), 102–108.

J.J.

, Yang

X.J.

, Xiao

Y.J.

, Xu

B.W.

and Wu

H.F.

, Particle swarm-based translation control for immersed tunnel element in Hong Kong-Zhuhai-Macao Bridge project, China Ocean Eng 32(1) (2018), 1–9.

Varela

J.M.

, Soares

C.G.

A high level architecture framework for real-time simulation of ship towing operations in virtual environments, 15th International Congress of the International-Maritime-Association- of-the-Mediterranean (IMAM), A Coruña, SPAIN 1, 2013, pp. 135–146.

Luis

C.C.

, Juan

C.C.C.

and Jose

A.F.F.

, Operation and handling in escort tugboat manoeuvres with the aid of automatic towing winch systems, Journal of Navigation 68(1) (2015), 71–88.

, Duan

, Liao

and Li

, Improved quantum particle swarm optimization by bloch sphere, International Conference on Advances in Swarm Intelligence 6145 (2010), 135–143.

, Zhao

and Lu

, Quantum ant colony optimization algorithm based on Bloch spherical search, Int J Eng Sci 4(4) (2015), 41–51.

China Classification Society. Guidelines for Towage at Sea

2011. http://www.ccs.org.cn/ccsewwms/displayNews.do?id=ffde736ae39c000b. (in Chinese).

Shen

P.G.

, Estimation of towing resistance, Marine Technology 5 (2011), 9–12. (in Chinese)

10.

and Li

, Quantum-inspired evolutionary algorithm for continuous space optimization based on Bloch coordinates of qubits, Neurocomputing 72(1) (2008), 581–591.

11.

Chen

, Xia

and Yu

, Quantum ant colony algorithm based on bloch coordinates, Springer Berlin Heidelberg 7473 (2012), 405–412.

12.

Van

D.B.F.

and Engelbrecht

A.P.

, A study of particle swarm optimization particle trajectories, Information Sciences 176 (2006), 937–971.

13.

Shi

Y.H.

, Eberhart

A modified particle swarm optimizer, Proc IEEE International Conference on Evolutionary Computation, Anchorage, 1998, pp. 69–73.