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An energy optimal thrust allocation method for the marine dynamic positioning system based on adaptive hybrid artificial bee colony algorithm
Ocean Engineering 118 (2016) 216–226
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An energy optimal thrust allocation method for the marine dynamic positioning system based on adaptive hybrid artificial bee colony algorithm Defeng Wu a,b,n, Fengkun Ren a,b, Weidong Zhang c,d a
School of Marine Engineering, Jimei University, Xiamen 361021, PR China Fujian Provincial Key Laboratory of Naval Architecture and Ocean Engineering, Xiamen 361021, PR China c Department of Automation, Shanghai Jiaotong University, Shanghai 200240, PR China d Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai 200240, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 May 2015 Received in revised form 21 March 2016 Accepted 5 April 2016
Thrust allocation (TA) is an important part in dynamic positioning systems (DPS). The function of TA is to allocate the thrust and angle of each thruster so that the desired force and moment can be achieved. Based on our previous work, an adaptive hybrid artificial bee colony algorithm with chaotic search (AHABCC) is proposed in this study. This algorithm introduced a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) to the honeybee and chaotic search strategies to scouts searching. The proportion of each search strategy selected is dynamically adjusted to achieve the optimization. Therefore, the AHABCC can automatically switch the search strategy for different bee colonies. The optimal search of AHABCC is faster compared to HABCC, and the probability of obtaining optimal results and avoiding local optimums is significantly increased. In addition, the power consumption of AHABCC is less than that of HABCC. The effectiveness of the AHABCC algorithm is demonstrated using simulations. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Thrust allocation Adaptive hybrid artificial bee colony algorithm Particle swarm optimization Chaotic search Differential evolution
1. Introduction Marine dynamic positioning system (DPS) has been widely employed in many floating vessels/platforms in the sea and is an important support technology for exploration and exploitation of oceanic resources. A DPS mainly consists of a position measurement system, control system, thrust allocation (TA) system and propulsion system (Sorensen, 2011). It is known that vessels/ platforms equipped with a DPS use thrusters and main propellers to produce a desired thrust via azimuth/tunnel thrusters, as well as to control maneuvering. Thrust can compensate for environmental forces acting on the vessel/platform to maintain position and head as closely as required to some desired position in the horizontal plane. There are several types of thrusters, namely tunnel thrusters, azimuth thrusters, aft rudders, stabilizing fins, control surfaces and so on (Fossen, 2002). Generally speaking, tunnel and azimuth thrusters are the most widely used thrusters in DPS. However, DPS have three equations that include a longitudinal force, a lateral force, and a moment requirement from the n Corresponding author at: School of Marine Engineering, Jimei University, Xiamen 361021, PR China. E-mail address: [email protected] (D. Wu).
http://dx.doi.org/10.1016/j.oceaneng.2016.04.004 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
control system to be satisfied. The task of a thrust system is to allocate the force/angle commands to each thruster so that the desired force and moment from control system can be achieved. Therefore, there have three different situations when comparing the control input and vessel degree-of-freedom, namely underactuated, fully-actuated and over-actuated. It was noted in (Rindarøy and Johansen, 2013) that most of the dynamic positioning (DP) vessels are over-actuated. Therefore, the over-actuated vessel with DPS is considered in this study. Then, the TA problem is usually formulated as an optimization problem with some constraints. The main constraints (Johansen et al., 2004), or optimization objectives, of DPS are the handling and compensating thruster failures, minimizing fuel consumption, minimizing thruster wear, accounting for thruster losses of various types, and handling problems when required power is not available. The solutions for TA optimization problems were widely studied. When the desired forces are provided by thrusters in fixed directions alone or in combination with rudders and control surfaces, then the TA problem can be considered as an unconstrained least-squares optimization problem and an explicit solution can be obtained (Fossen and Sagatun, 1991; Fossen, 1994). If the thruster limitations are taken into account, then the TA problem becomes a constrained optimization problem, several methods are proposed to solve this type of problem. For instance, Johansen et al. (2005,
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2003) proposed an explicit solution for parametric quadratic programming (QP) to marine vessels and an extension of the mpQP algorithm respectively. Sørdalen (1997) proposed an iterative solution to solve the QP problem. When the vessel is equipped with azimuth thrusters, then the TA problem generally becomes a non-convex nonlinear optimization problem. Liang and Cheng (2004) proposed an optimum control of a thruster system using the sequential quadratic method. Wit (2009) selected the thrust and azimuth direction of the thrusters as design variables for the optimization problem, and the QP algorithm was used to solve the problem. It should be noted that there is a tradeoff between energy consumption and maneuverability of dynamically positioned vessels. Johansen et al. (2004) proposed a TA optimization objective function via adding a term to avoid singular configurations and Sequential Quadratic Programming (SQP) was employed to find the TA solution. The advantages of aforementioned QP and SQP methods are real-time implementation and strong local optimization ability. However, the global optimization ability is comparatively weak. Therefore, several TA solutions based on intelligent computation techniques were presented. For instance, Dawei et al. (2010) proposed a TA solution based on a genetic algorithm (GA) (Yanduo and Jing, 2007). The presented method is effective and validated via simulation results. Parikshit (2013) also proposed a method based on ITHS (Intelligent Tuned Harmony Search) to solve TA problem. Zhao and Myung-IlRoh (2015) proposed a hybrid optimization algorithm based on GA and SQP algorithm to solve TA problem. However, the convergence speed of such methods still needs to be further improved when it is employed in DPS. As an intelligent computation technique, an artificial bee colony (ABC) (Karaboga, 2005) algorithm simulates the intelligent foraging behavior of honey bee swarms. It is a very simple, robust and population-based stochastic optimization algorithm. ABC has been employed to solve many problems (Karaboga et al., 2007; Karaboga, 2009; Apalak et al., 2014). However, ABC algorithms are premature and their speed of convergence needs to be further improved. For instance, Singh (2009) presented a new ABC algorithm to the leaf-constrained minimum spanning tree problem. Pei-Wei et al. (2009) presented an enhanced ABC optimization with the formula of gravitation. However, when an ABC is applied to a TA problem, the improved algorithm should consider the unique problem characteristics. Besides aforementioned GA and ABC, there are several intelligent optimization algorithms such as Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Differential Evolution (DE) and Biogeography-based Optimization (BBO) and so on. Hybrid intelligent optimization algorithms have the advantages of avoiding local minimums and speeding up the convergence rate compared with single intelligent optimization method, for instance hybrid PSO and DE (Liu et al., 2010), hybrid ACO and DE (Ying-Pin, 2010), hybrid BBO and PSO (Guo et al., 2014). However, the adaptive switch of different search mechanisms in different algorithms is rarely raised. In our previous work (Wu et al., 2014), a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) were introduced to the honeybee search strategy. Therefore, the probability of finding optimal results and avoiding local optimums is significantly increased. The proposed algorithm is called the hybrid artificial bee colony with chaotic search (HABCC). However, changing different search strategies for bees in HABCC is based on the manual setting of the iteration number. Therefore, it is necessary to further develop HABCC into an adaptive hybrid artificial bee colony with chaotic search (AHABCC) algorithm. The AHABCC can automatically switch the search strategy for different bees. One of the advantages for AHABCC is that it does not require a manual setting of the iteration number, making it much more
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practical for engineering application. Another advantage is that the AHABCC can fully utilize the merits of DE and PSO, as the switch search strategy can switch automatically based on the performances of different search strategies. The rest of this paper is organized as follows. In Section 2, the TA mathematical model for one type of pipe-laying crane vessels is presented to lay a basis for the following development. In Section 3, the basic artificial bee colony algorithm is introduced and analysed for further development. In Section 4, an adaptive hybrid artificial bee colony algorithm is proposed and the procedure is also given. In Section 5, a comparison experiment is conducted to verify the feasibility and effectiveness of AHABCC over the other three algorithms including HABCC, ABC and SQP. Our concluding remarks and future work are contained in the final section.
2. Thruster allocation mathematical model In this study, the mathematical model of thruster allocation is based on a pipe-laying crane vessel with a DPS equipped with seven azimuth thrusters. The arrangement of the propellers is shown in Fig. 1. In this section, the mathematical model of thruster allocation in DPS is introduced to lay a basis for finding the solution. Due to the capacity constraints of the propulsion system, the optimization objective function (Johansen et al., 2004) of thrust allocation can be formulated as follows:
min J (α, F , s ) = PW + s T Qs + (α − α0 )T Ω (α − α0 ) +
δ ε + det (B (α ) B′(α ))
(1)
Subject to the constraints (Rindarøy and Johansen, 2013; Johansen et al., 2004):
s = τ − B (α ) F
(2)
Fmin ≤ F ≤ Fmax
(3)
ΔFmin ≤ F − F0 ≤ ΔFmax
(4)
αmin ≤ α ≤ αmax
(5)
Δαmin ≤ α − α0 ≤ Δαmax
(6)
where W is the total power consumption in the first term, combining the power consumption of the individual thrusters and P is the weight factor. In the second term, s is the error between the commanded and achieved generalized force. Q is the matrix of the diagonal weights. The sT Qs penalizes the error s between the commanded and achieved generalized force. τ is commanded generalized force. B (α ) is the control allocation matrix. F is the thrust force provided by thrusters. Fmin and Fmax are the minimum
Fig. 1. Thruster layout of the vessel with DPS.
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and maximum forces produced by the thrusters, respectively. The rate of thrust change is limited via constraint (4). F0 is the provided thrust in previous step, and ΔFmin and ΔFmax are the minimum and maximum thrust change from one time-step to the next, respectively. α is the rotation angle of thrusters. αmin and αmax are the lower and upper bounds for rotation angle. α0 is the previous thruster angle vector, and Δαmin and Δαmax are the minimum and maximum angle change from one time-step to the next. The weight matrix, Ω , is used to adjust the optimization goal. The last part of Eq. (1) is used to avoid singularity. Setting ε > 0is used to avoid the denominator becoming zero. δ > 0 is the weight coefficient. A larger δ results in better vessel maneuverability and more energy consumption of the thrusters. Based on this pipe-laying crane vessel, B (α ) F in Eq. (2) represents the force of the X and Y axes and the total torque. The force and moment equations can be formulated as follows: ⎧ Fx = F1 cos α1 + F2 cos α 2 + F3 cos α 3 + F4 cos α4 ⎪ + F5 cos α5 + F6 cos α 6 + F7 cos α 7 ⎪ ⎪ F = F sin α + F sin α + F sin α + F sin α y 1 1 2 2 3 3 4 4 ⎪ ⎪ + F5 sin α5 + F6 sin α 6 + F7 sin α 7 ⎨ ⎪ M = F1 sin α1l x1 + F2 sin α 2 l x2 + F3 sin α 3 l x3 + F4 sin α4 l x4 ⎪ − F5 sin α5 l x5 − F6 sin α 6 l x6 − F7 sin α 7 l x7 + F1 cos α1l y1 ⎪ ⎪ − F2 cos α 2 l y2 + F3 cos α 3 l y3 − F4 cos α4 l y4 + F5 cos α5 l y5 ⎪ − F6 cos α 6 l y6 + F7 cos α 7 l y7 ⎩
⎧ 1, f (vi ) ≥ f (xi ) P=⎨ ⎩ 0, f (vi ) < f (xi )
(7)
3. Artificial bee colony (ABC) algorithm The minimal model of swarm-intelligent forage selection in a honey bee colony simulated with the ABC algorithm consists of three types of bees (Karaboga et al., 2007): employed bees, onlooker bees and scout bees. The number of employed bees is equal to the onlooker bees, and the sum of the two types represents the whole colony. Employed bees are responsible for exploiting the food sources explored before and giving information to the waiting bees (onlooker bees) in the hive about the quality of the food source sites that they are exploiting. Onlooker bees wait in the hive and decide on a food source to exploit based on the information shared by the employed bees. Scouts either randomly search the environment to find a new food source depending on an internal motivation or based on possible external clues. Thus, this search system combines the local search strategy, managed by employed and onlooker bees, with global search methods, carried out by onlookers and scouts, attempting to balance the exploitation and exploration processes. There are three important steps in the basic ABC algorithm (Karaboga and Basturk, 2008): moving the employed and onlooker bees to find the food source and evaluating their nectar amounts, determining the scouts, and then finding the food source randomly. In detail, the first step for the basic ABC algorithm is generating the initial food source positions randomly. Then, to generate feasible solutions, all employed bees select a new candidate food source position, different from the previous positions, with the following equation.
(8)
In the above equation, vij is the new food source position modified from the previous position x ij based on the comparison with a random neighboring position xkj . φ is a random number
(9)
The greedy selection operator guarantees that the algorithm can retain the elite individual and not retreat due to the evolution of the process of optimization. Thus, the distribution of P has no relation with the iteration. In the second step, each onlooker selects one food source using “roulette wheel selection” (Goldberg, 1989) and searches for new solutions around its neighborhood. The probability value of one employed bee being chosen is calculated by the following equation:
Pi =
where lx and ly are the distances between each thruster and the xand y-axis, respectively. F1–F7 are the forces provided by thruster #1–#7 respectively. ɑ1–ɑ7 are the angles of thruster #1–#7 respectively.
vi j = xi j + φ (xi j − xkj )
generated in [ 1,1]. k ∈ {1, 2, 3, ... , SN}, j ∈ {1, 2, 3, ... , D} and SN are the number of food sources. D is the dimension of the problem that requires a solution. A greedy selection operator is usually employed to choose the better solution between the searched new vector vi and the original vector xi into the next generation. The probability distribution of vi being selected can be described as follows:
fiti n ∑i = 1 fitn
(10)
where fiti is the fitness value (profitability) of the food source xi . After selecting the food source, the onlooker goes to obtain the new food source position with Eq. (8). In the basic ABC algorithm, if a solution represented by a food source is not changed by a predetermined number ‘limit’ of iterations, the food source is abandoned by its employed bee and then the employed bee associated with that food source becomes a scout. The scout is the colony's explorer. The explorer does not have any knowledge while searching for a food source. They begin to search for a new food source randomly with the following equation: j j j xi j = x min + rand (0, 1)(x max − x min )
(11)
j j where x min and x max represent the lower bound and the upper bound of the search space of food source in dimension j, respectively. And rand (0, 1) denotes a random number generated in [0,1]. It should be noted that the ABC algorithm is inspired from bee colony, therefore the ABC theory seems to be somewhat “bee” language. For those who are not familiar with the ABC algorithm, there are two points that may help understand it easier: (1). the food source in ABC is actually the feasible solution for the optimized problem; (2). the optimized solution will be obtained via several steps which are inspired from bee colony behavior.
4. Proposed AHABCC algorithm and its variants In analysing the process and strategy, the basic artificial bee colony algorithm has the advantage of global convergence, a simple structure, and is easily realized. However, the algorithm also has the disadvantage of premature and slow convergence speed. Therefore, the ABC algorithm needs improvement for applications in solving thrust allocation problems of dynamic positioning systems. Because of that, in our previous work, we proposed an enhanced artificial bee colony (EABC) (Ren et al., 2014) algorithm that introduced chaotic search to scout searching and a hybrid artificial bee colony algorithm with chaotic search (HABCC) (Wu et al., 2014) that introduced a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) to the honeybee search strategy based on EABC to solve TA problems of DPS. They both work better than the basic ABC algorithm. However, in the HABCC algorithm, we
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introduced a new parameter, k, selected from [0,1] by trial and error to address the conversion from one method to another in the employed bee and onlooker search strategies. Despite improvement of the global search performance and convergence speed of the algorithm, the HABCC algorithm has lower efficiency and availability because of the increasing number of user-defined parameters. Therefore, we introduce an adaptive strategy for the conversion of employed bee and onlooker search methods and propose an adaptive hybrid artificial bee colony algorithm with chaotic search (AHABCC). The details of the adaptive strategy are as follows. 4.1. Modified foraging process of employed bees
vi j = x rj1 + Fch (x rj2 − x rj3 )
being selected is equal. Then, the value of s1 adaptively changes based on performance of problem solving with the following rule: If the first search method with Eq. (8) works well and obtains the best solution of the whole colony, then the value of s1 is increased with the following equation (the new s′1 is denoted by s1, and s1 is updated using the following equation):
s1′ = s1 + λ1 (1 − s1)
(12)
where r1, r2 and r3 are three mutually different random individuals. Fch is the scaling factor generated via Fch = rand × exp ( − T /Tmax ), T is the number of cycles, and Tmax is the maximum number of cycles. Fch will be a small value when T is small. Therefore, in the earlier stage of the optimization program, the search strategy pays attention to the local search around each individual. When T increases, the value of Fch changes dynamically over a wider range. This will not only speed up the convergence rate, but also take the individual optimum into consideration. The search process of the employed bee with an adaptive hybrid search strategy can be summarized as follows: If (rando s1) Find a neighbouring food source using Eq. (8); Otherwise Find a new food source using Eq. (12); End where s1 is the control parameter lying in [0,1]. The values of s1 represent the probability of being selected to optimize the problem. The strategy with Eq. (8) is selected when “rand” lies in the region of [0, s1]; otherwise, another method with Eq. (12) is selected. The initialization of s1 is 0.5 because of no knowledge about the performance of the two strategies. Thus, the probability of
(13)
where λ1 is a parameter which influences the change speed of s1. When the first search method does not optimize the problem or contribute to the further optimization of the current best solution, the value of s1is decreased with
s1′ = s1 − λ2 (1 − s1)
An employed bee updates the position of the food source (solution) in its memory via searching its local space and finding a neighboring food source and then evaluates its quality. In ABC, the employed bee finds a neighboring food source using Eq. (8). However, according to “No free lunch theorems for optimization” (Wolpert and Macready, 1997), one algorithm or search strategy does not always work well in dealing with all problems. For example, in the ABC algorithm, the employed bee strategy with Eq. (8) also does not work well in all stages of the optimization in handling TA problems of DPS. At the late stage of optimization, the convergence speed of the local search strategy is slow because of the multi-dimensional variables. To solve this predicament caused by a single strategy approach, the employed bee should have more selections in terms of search strategies that are different from the local search strategy and have enhanced capability, such as in the form of a global search or faster convergence speed. If the adaptive hybrid strategy proposed in this paper is applied, the employed bee could switch search strategies adaptively based on each strategy's performance to improve the search ability. Therefore, in this section, we introduce a mutation operator from differential evolution (DE) to a new strategy and propose an adaptive hybrid strategy to achieve the ability to switch from one search method to another during the optimizing procedure based on performance. The new search strategy is described as follows:
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(14)
where λ2 is a parameter which influences the change speed of s1.In addition, the value of s1 is not less than 0.5 when it is decreased with (14). When the value of s1 approaches 0.5 because of falling into a local or global optimum, the probability of being selected is approximately equal in both methods. Therefore, the minimum of s1 is 0.5 in this part. If the second search method with Eq. (12) obtains the best solution, the value of s1 is decreased with
s1′ = s1 − λ1s1
(15)
When the second search method does not optimize or contribute to the further optimization of the current best solution, the value of s1 is increased with
s1′ = s1 + λ2 s1
(16)
The value of s1 is no more than 0.5 when it is increased with (16). 4.2. Modified foraging process of onlooker bees Different from the search method with Eq. (12), which enhanced the global exploration ability, a new method that could increase the convergence speed to obtain the global optimum quickly is necessary for onlooker searching when the local search strategy is unsuccessfully working. Therefore, a new search strategy combined with the social cognitive part of particle swarm optimization for onlookers is followed.
vi j = xi j + φ (xi j − xkj ) + ψ (Pgj − xi j )
(17)
In the above equation, ψ is a randomly generated number that lies in [0,1], and Pg is the best individual selected from all population produced in one iteration. Similar to the employed bee, the main procedure of the onlooker bee is as follows: If (rando s2) Find a neighbouring food source using Eq. (8); Otherwise Find a new food source using Eq. (17); End where s2 is the control parameter lying in [0,1] and its changes are the same as s1 in the previous section. 4.3. Chaotic search strategy In AHABCC, if the time of an employed bee searching exceeds a certain threshold limit and does not find better solutions (Karaboga, 2009), the greedy selection operator will be employed to choose a better solution between this position and the new
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positions generated by a chaotic search (Alatas, 2010). The process of chaotic search is given as follows: 4.3.1). This paper assumes that the search stagnation solution is j j x ij = (x i1, x i2, ⋅⋅⋅, x iD ) , x ij ∈ [x min , x max ]. The stagnation x ij should be mapped to the definition domain [0,1] via the following equation: j
Zi =
(18)
where i = 1, 2, ⋅⋅⋅, SN ; j = 1, 2, ⋅⋅⋅, D , Zij is the mapped value of x ij . Take the constraints (4) and (6) into consideration,
j x max = α0 + Δαmax ,
j x min = F0 + ΔFmin or
j x min
= α0 + Δαmin . 4.3.2). Chaotic variable sequences Zn (n = 1, 2, ⋅⋅⋅, Cmax ) can be generated by a Logistic equation (Yang et al., 2014):
Z n + 1 = μ × Z n (1 − Z n )
j xi j − x min j j − x min x max
j x max = F0 + ΔFmax or
(19)
where Cmax is the maximum number of the chaotic search and μ is a control parameter. When μ = 4 , the sequences Zn is chaotic. 4.3.3). The new variable sequences Zij are mapped to the original solution space, and the new position (solution) is generated
Fig. 2. Flow chart of AHABCC employed for TA.
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by the following equation:
vi j = xi j + 2 × (Zi j − 0.5) ×
j x max
− 2
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Table 1 The size parameters of ‘HYSY 201’.
j x min
(20)
Then, the fitness value of the new position (solution) is calculated, and the best solution with a greedy selection operator is obtained compared with the original solution. 4.3.4). If the cycle number of the chaotic search is more than Cmax , the process ends. If not, then step 4.3.2) is repeated. Finally, the main steps of the AHABCC algorithm in solving the TA problem of DPSs are presented below, and the corresponding flow chart is given in Fig. 2. Step 1: Initialize the population of solutions ( xi , i= 1, 2, 3, ... , SN ) for a TA problem of a DPS. Each solution consists of fourteen parameters from seven thrusters; Step 2: Evaluate the value of each solution with Eq. (1); Step 3: Repeat; Step 4: Produce new solutions, vi , for the employed bees via an adaptive search strategy consisting of Eqs. (8) and (12) and evaluate them; Step 5: Apply the greedy selection process for the employed bees with Eq. (9) to keep the better one; Step 6: Calculate the probability values, Pi , for the solutions, xi , using Eq. (10); Step 7: Produce and evaluate the new solutions, vi , for the onlookers via an adaptive search strategy consisting of Eqs. (8) and (17) from the solutions, xi , selected depending on Pi ; Step 8: Apply the greedy selection process for the onlookers with Eq. (9); Step 9: Determine the abandoned solution for the scout, if one exists, and replace it with a new solution produced using the chaotic search strategy described in Section 4.3; Step 10: Update s1 and s2 ; Step 11: Memorize the best solution achieved thus far; Step 12: Continue until the requirements are met. In a searching iteration, the processes of exploration and exploitation must be conducted together. In the basic ABC algorithm, while the employed bees carry out the exploration process in the search space, the onlookers and scouts control the exploration process. However, this is insufficient when ABC is applied in realworld optimization problems. It is easy for the ABC algorithm to fall into the local optimum when the algorithm is running to the global optimum. Therefore, the exploration ability needs to be enhanced. In the previous HABCC algorithm, we enhanced the employed bee searching for a new source using Eq. (12) and improved onlooker bee exploitation ability via Eq. (17). With that, the application of the adaptive hybrid search strategy, which combines two search methods based on different situations, can also enhance the algorithm's capability and increase its efficiency and availability. In addition, the chaotic search strategy with characteristics of randomness and ergodicity can improve the probability of finding the optimal result and avoiding local optimums. Those functions are verified in the next section.
Length overall L oa/m
Length between Displacement V /t perpendiculars Lpp/m
Designed draft d/m
Beam /m Molded depth /m
204.65
185
7–9.54
39.2
59,000
5.2. Parameter setting The ‘HYSY 201’ is used as the case study vessel. The vessel moves from initial state [ 0 m 0 m 0° ] to desired position and orientation [10 m 10 m 20° ] and keeps the desired position with desired heading. The desired longitudinal resultant thrust, the lateral resultant thrust, and the moment of 250 s are shown in Figs. 3–5. The mission of the HABCC or AHABCC algorithm is to address the thruster allocation problem to make the seven actuators produce various thrusts at accurate angles. Thus, the ship can move to and also keep desired position with desired heading under the oceanic environmental conditions. In addition, SQP and ABC algorithms are also included to make a comparative study. They are all employed to solve TA problems in DPS for ships with seven thrusters for positioning. One thruster has two parameters: thrust and angle (i.e., the dimension of a solution is 14). To make the comparative study fair, for the four algorithms, the initial values of first second are chosen randomly within thruster limits. The (N 1)th (NZ3) second TA results are chosen as the initial values of Nth second respectively. For ABC, HABCCA and AHABCC these three swarm algorithms, the population size is set as 32, and the maximum number of iterations is 1500 for each second of the optimization process. The value of the limit is set to be 200, and the value of Cmax is chosen as limit/2, as suggested in (Alatas, 2010). In addition, in terms of the constraints of the objective function, the value of Fmin is zero and that of Fmax is described in Table 2. The maximum change of the thrust angle in one second is 20% of the maximum thrust ( ΔFmax or ΔFmin = Fmax × 20%). Δαmin and Δαmax are 0.314 rad and 0.314 rad, respectively. Furthermore, as to make the comparative study fair, the control parameter k in HABCC is varied from 0 to 1 with step width 0.1 and the best one is chosen via trial and error. In this study when k¼0.7, the performance of HABCC is best thus k is set to be 0.7 to make a fair comparative study. 5.3. Results and analysis The simulation has run 30 times repeatedly, and the average results are shown in the following figures and tables. Firstly, the simulation results will be given from Figs. 6–13 and Tables 3 and 4. Secondly, Fig. 14 is given to demonstrate that: (1).The optimal search of AHABCC is faster compared to HABCC; (2).the probability of obtaining optimal results and avoiding local optimums is Table 2 The parameters of the thruster system. Thruster No. Rotate speed N /(r / min)
Power P/kW
Maximum thrust Fmax/kN
5. Experimental results 5.1. Simulation plant In this paper, the simulation plant is a deepwater pipe-laying crane vessel named ‘HYSY 201’ with a DPS. Its major size is given in Table 1. This DP vessel realizes its positioning ability with 7 thrusters whose information is described in Table 2.
14
1# 2# 3# 4# 5# 6# 7#
181.4 181.4 192.3 192.3 192.3 192.3 192.3
4500 4500 3200 3200 3200 3200 3200
680 680 540 540 540 540 540
Installation site
X /m
Y /m
93.8 93.8 12.55 12.55 37.85 37.85 82.65
9.45 9.45 15.4 15.4 14 14 0
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Fig. 3. Desired longitudinal resultant thrust. Fig. 5. Desired moment.
Fig. 6. Results of the thrust and the angle of thruster #1 from the HABCC and AHABCC methods.
Fig. 4. Desired lateral resultant thrust.
increased. Finally, Fig. 15 and Table 5 are shown to indicate that: (3) the power consumption of AHABCC is less than that of HABCC. Figs. 6–12 show the individual thrust and angle for each thruster, and Fig. 13 shows the changes in s1 and s2. Fig. 14 illustrates the Convergence of the HABCC and AHABCC method. Fig. 15 shows the difference of total power consumption between HABCC and AHABCC during TA for 250 s.
In addition, sensitivity tests of λ1 and λ2 were conducted, and the results are described in Table 3. Each result in this table represents the value of the sum of errors in 250 s of the X-axis, Y-axis, direction force and moment. There are approximately 12 combinations when λ1 = [1/2, 1/3, 1/4] and λ2 = [1/8, 1/9, 1/10, 1/11]. It shows that when λ1 = 1/3 λ2 = 1/10, AHABCC obtains the best result. However, the rest of the combinations are also acceptable because of the small difference from the best. Therefore, Table 4 is provided for the comparisons. In Table 4, the corresponding error results of four algorithms when they are employed for TA in DPS are recorded. The header of the table, Best, refers to the minimum value of the sum of errors in 250 s found in 30 times. Average1 refers to the average value among 30 times, and Average2 is the average value of each second. The Worst indicates the maximum error found in 30 times. Std is the standard deviation of errors.
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Fig. 7. Results of the thrust and the angle of thruster #2 from the HABCC and AHABCC methods.
Fig. 10. Results of the thrust and the angle of thruster #5 from the HABCC and AHABCC methods.
Fig. 8. Results of the thrust and the angle of thruster #3 from the HABCC and AHABCC methods.
Fig. 11. Results of the thrust and the angle of thruster #6 from the HABCC and AHABCC methods.
Fig. 9. Results of the thrust and the angle of thruster #4 from the HABCC and the AHABCC methods.
Fig. 12. Results of the thrust and the angle of thruster #7 from the HABCC and AHABCC methods.
From Table 4, we can see that the Average error of AHABCC is better than that of HABCC, ABC and SQP. Correspondingly, other errors, such as Worst, Best and Std, of AHABCC are similar to or
better than that of HABCC. The experimental results also show that the performance of the AHABCC algorithm integrated with an adaptive hybrid search strategy outperforms HABCC, ABC and SQP
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Fig. 13. The changes in s1 and s2.
Fig. 14. Convergence of the HABCC and AHABCC method.
Table 3 The sensitivity test results of λ1 and λ2 .
Table 5 Total power consumption of different algorithms. Algorithm
Total power conventional thrust allocation (MW)
SQP ABC HABCC AHABCC
305.611 237.389 214.235 188.895
Table 4 Comparisons of different algorithms.
SQP
ABC
HABCC
AHABCC
Fx Fy N Fx Fy N Fx Fy N Fx Fy N
Average1
Average2
Worst
Best
Std
1292.9 1824.94 1221.39 317.326 359.471 442.416 249.490 276.246 246.656 233.459 247.775 244.649
5.1716 7.2998 4.8855 1.2693 1.4378 1.7696 0.9980 1.1050 0.9866 0.9338 0.9911 0.9786
4193.67 3755.89 3775.91 428.161 450.334 521.952 290.023 355.588 298.517 286.763 323.878 296.200
182.769 335.912 180.451 251.346 283.983 348.636 202.661 207.714 216.077 188.482 195.915 214.392
1029.01 1104.80 990.888 47.9211 46.1647 59.5018 23.0075 37.2987 23.2812 22.4375 27.8681 22.8282
methods. The results of SQP also show that SQP is very sensitive to its initial values. It also can be seen that there exists some differences between Worst and Best, this is because swarm intelligence based algorithms have randomness characteristic. It should be note that the results are recorded during 250 seconds. It is obvious when simulation time increases, the sum of errors such as Worst and Best increase. However, when we look at the average error of one second for Worst and Best, take Fy for instance, that is 355.588/ 250 ¼ 1.42 and 207.714/250 ¼ 0.83 respectively for HABCC and 323.878/250 ¼1.29 and 195.915/250 ¼0.78 respectively for
Fig. 15. The difference of power consumption between HABCC and AHABCC methods.
AHABCC. It can be seen that the average error for one second is acceptable for TA solution even though in Worst case which is 1.42 and 1.29 respectively as there are seven thrusters in DPS. From Figs. 6–12, it can be observed that the thrust force curves produced by seven thrusters based on AHABCC are similar to that of HABCC. As far as thrust force is concerned, the results from both AHABCC and HABCC are reasonable, and the change in each second does not exceed the limited boundary. In addition, the objective of optimization is not to obtain the value of the thruster's angle but rather to determine the change in the angle. From these
D. Wu et al. / Ocean Engineering 118 (2016) 216–226
figures, we can conclude that the angle changes are frequently within the limit of each thruster and that the results from both HABCC and AHABCC are similar, indicating satisfactory results. Finally, Fig. 13 shows the changes in the control parameters, s1 and s2, in the iteration of optimization. It describes the changes in proportion, which is the possibility of being selected to address the problem in each cycle. From Fig. 13, it can be observed that the AHABCC algorithm can adaptively adjust the value of s1 and s2 to control each method's action time and proportion. This is to show that our proposed adaptive hybrid strategy is effective and useful. Fig. 14 shows the convergence of the HABCC and AHABCC method. And it is observed that AHABCC has faster convergence rate than HABCC. In addition, it is demonstrated that the probability of obtaining optimal results and avoiding local optimums is increased. The difference of total power consumption (Yadav et al., 2014) using the seven thrusters during 250 s between HABCC and AHABCC is shown in Fig. 15. It is evident from Fig. 15 that the power consumption of the AHABCC algorithm is lower than the power consumption of the HABCC algorithm as a whole. It is also observed that when desired thrusts increase sharply (see Fig. 3), the corresponding power-saving performance of AHABCC is obvious. When desired thrusts change not so much (see Figs. 3–5), the corresponding power-saving performance of AHABCC is not as obvious as before. When desired thrusts change not so much, the (N 1)th TA results which are chosen as the initial values of Nth second are closer to the Nth TA results compared with thrusts increasing sharply, therefore the superiority of AHABCC becomes not so obvious compared with HABCC. This is why the powersaving performance has become not obvious as the time goes by in Fig. 15. The total power consumptions of seven thrusters during 250 s are listed in Table 5 for SQP, ABC, HABCC and AHABCC algorithms. It can be seen that consumed power are 305.611, 237.389, 214.235 and 188.895 (MW) for the four algorithms respectively. It is evident that AHABCC performs best for these four algorithms. All of the simulations are completed via the Matlabs platform on a personal computer. The configurations of the computer are CPU 2.9 GHz, RAM 6.0 GB, and 32-bit operation system. It should be noted that the AHABCC algorithm for TA in this study is implemented using a serial computing technique and serves as a theoretical basis for future industrial application. The calculation of TA for 1 s requires approximately 8 s. In Rui (2009), it is shown that the running time of FPGA (Field Programmable Gate Array)based PSO achieves approximately 0.032% of that in Matlabs, as a parallel computing technique is employed in FPGA. Thus, the calculation time can be shortened within 1 second to achieve realtime TA when the algorithm is implemented based on FPGA, even though the AHABCC algorithm involves ABC, PSO and DE. This is also our future work.
6. Conclusions This paper has presented a thrust allocation method based on the AHABCC algorithm. The introduction of a mutation operator from DE and the social cognitive part from PSO to honeybee search strategy, a chaotic search strategy to the scout searching component, and an adaptive hybrid strategy contribute to both the exploration and exploitation abilities in AHABCC. The simulation results show that both the power consumption and error of the AHABCC algorithm are less than the previous HABCC algorithm in solving TA problems. Furthermore, the AHABCC algorithm has the control parameter, s, which is adaptively changed. However, there is a control parameter, k, in HABCC that is required to be set by trial and error. This all shows that AHABCC is better than HABCC
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and that our presented adaptive strategy is successful and effective. However, it should be noted that like HABCC, AHABCC has only been tested with simulated data and the combination of search methods hybridized to a search strategy is not perfect. Thus, more intensive bench tests with real data should be conducted and more new search mechanisms should be introduced to achieve better performances for various optimization problems.
Acknowledgments The authors would like to thank the financial support from the National Natural Science Foundation of China (51249006, 61473183, and U1509211), the Fujian University Outstanding Young Scientific Talent Cultivation Plan (JA13169), the Natural Science Foundation of Fujian Province, China (2015J05104) and the Program of Shanghai Subject Chief Scientist (14XD1402400) for this project.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.oceaneng.2016.04.004.
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Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An energy optimal thrust allocation method for the marine dynamic positioning system based on adaptive hybrid artificial bee colony algorithm Defeng Wu a,b,n, Fengkun Ren a,b, Weidong Zhang c,d a
School of Marine Engineering, Jimei University, Xiamen 361021, PR China Fujian Provincial Key Laboratory of Naval Architecture and Ocean Engineering, Xiamen 361021, PR China c Department of Automation, Shanghai Jiaotong University, Shanghai 200240, PR China d Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai 200240, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 May 2015 Received in revised form 21 March 2016 Accepted 5 April 2016
Thrust allocation (TA) is an important part in dynamic positioning systems (DPS). The function of TA is to allocate the thrust and angle of each thruster so that the desired force and moment can be achieved. Based on our previous work, an adaptive hybrid artificial bee colony algorithm with chaotic search (AHABCC) is proposed in this study. This algorithm introduced a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) to the honeybee and chaotic search strategies to scouts searching. The proportion of each search strategy selected is dynamically adjusted to achieve the optimization. Therefore, the AHABCC can automatically switch the search strategy for different bee colonies. The optimal search of AHABCC is faster compared to HABCC, and the probability of obtaining optimal results and avoiding local optimums is significantly increased. In addition, the power consumption of AHABCC is less than that of HABCC. The effectiveness of the AHABCC algorithm is demonstrated using simulations. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Thrust allocation Adaptive hybrid artificial bee colony algorithm Particle swarm optimization Chaotic search Differential evolution
1. Introduction Marine dynamic positioning system (DPS) has been widely employed in many floating vessels/platforms in the sea and is an important support technology for exploration and exploitation of oceanic resources. A DPS mainly consists of a position measurement system, control system, thrust allocation (TA) system and propulsion system (Sorensen, 2011). It is known that vessels/ platforms equipped with a DPS use thrusters and main propellers to produce a desired thrust via azimuth/tunnel thrusters, as well as to control maneuvering. Thrust can compensate for environmental forces acting on the vessel/platform to maintain position and head as closely as required to some desired position in the horizontal plane. There are several types of thrusters, namely tunnel thrusters, azimuth thrusters, aft rudders, stabilizing fins, control surfaces and so on (Fossen, 2002). Generally speaking, tunnel and azimuth thrusters are the most widely used thrusters in DPS. However, DPS have three equations that include a longitudinal force, a lateral force, and a moment requirement from the n Corresponding author at: School of Marine Engineering, Jimei University, Xiamen 361021, PR China. E-mail address: [email protected] (D. Wu).
http://dx.doi.org/10.1016/j.oceaneng.2016.04.004 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
control system to be satisfied. The task of a thrust system is to allocate the force/angle commands to each thruster so that the desired force and moment from control system can be achieved. Therefore, there have three different situations when comparing the control input and vessel degree-of-freedom, namely underactuated, fully-actuated and over-actuated. It was noted in (Rindarøy and Johansen, 2013) that most of the dynamic positioning (DP) vessels are over-actuated. Therefore, the over-actuated vessel with DPS is considered in this study. Then, the TA problem is usually formulated as an optimization problem with some constraints. The main constraints (Johansen et al., 2004), or optimization objectives, of DPS are the handling and compensating thruster failures, minimizing fuel consumption, minimizing thruster wear, accounting for thruster losses of various types, and handling problems when required power is not available. The solutions for TA optimization problems were widely studied. When the desired forces are provided by thrusters in fixed directions alone or in combination with rudders and control surfaces, then the TA problem can be considered as an unconstrained least-squares optimization problem and an explicit solution can be obtained (Fossen and Sagatun, 1991; Fossen, 1994). If the thruster limitations are taken into account, then the TA problem becomes a constrained optimization problem, several methods are proposed to solve this type of problem. For instance, Johansen et al. (2005,
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2003) proposed an explicit solution for parametric quadratic programming (QP) to marine vessels and an extension of the mpQP algorithm respectively. Sørdalen (1997) proposed an iterative solution to solve the QP problem. When the vessel is equipped with azimuth thrusters, then the TA problem generally becomes a non-convex nonlinear optimization problem. Liang and Cheng (2004) proposed an optimum control of a thruster system using the sequential quadratic method. Wit (2009) selected the thrust and azimuth direction of the thrusters as design variables for the optimization problem, and the QP algorithm was used to solve the problem. It should be noted that there is a tradeoff between energy consumption and maneuverability of dynamically positioned vessels. Johansen et al. (2004) proposed a TA optimization objective function via adding a term to avoid singular configurations and Sequential Quadratic Programming (SQP) was employed to find the TA solution. The advantages of aforementioned QP and SQP methods are real-time implementation and strong local optimization ability. However, the global optimization ability is comparatively weak. Therefore, several TA solutions based on intelligent computation techniques were presented. For instance, Dawei et al. (2010) proposed a TA solution based on a genetic algorithm (GA) (Yanduo and Jing, 2007). The presented method is effective and validated via simulation results. Parikshit (2013) also proposed a method based on ITHS (Intelligent Tuned Harmony Search) to solve TA problem. Zhao and Myung-IlRoh (2015) proposed a hybrid optimization algorithm based on GA and SQP algorithm to solve TA problem. However, the convergence speed of such methods still needs to be further improved when it is employed in DPS. As an intelligent computation technique, an artificial bee colony (ABC) (Karaboga, 2005) algorithm simulates the intelligent foraging behavior of honey bee swarms. It is a very simple, robust and population-based stochastic optimization algorithm. ABC has been employed to solve many problems (Karaboga et al., 2007; Karaboga, 2009; Apalak et al., 2014). However, ABC algorithms are premature and their speed of convergence needs to be further improved. For instance, Singh (2009) presented a new ABC algorithm to the leaf-constrained minimum spanning tree problem. Pei-Wei et al. (2009) presented an enhanced ABC optimization with the formula of gravitation. However, when an ABC is applied to a TA problem, the improved algorithm should consider the unique problem characteristics. Besides aforementioned GA and ABC, there are several intelligent optimization algorithms such as Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Differential Evolution (DE) and Biogeography-based Optimization (BBO) and so on. Hybrid intelligent optimization algorithms have the advantages of avoiding local minimums and speeding up the convergence rate compared with single intelligent optimization method, for instance hybrid PSO and DE (Liu et al., 2010), hybrid ACO and DE (Ying-Pin, 2010), hybrid BBO and PSO (Guo et al., 2014). However, the adaptive switch of different search mechanisms in different algorithms is rarely raised. In our previous work (Wu et al., 2014), a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) were introduced to the honeybee search strategy. Therefore, the probability of finding optimal results and avoiding local optimums is significantly increased. The proposed algorithm is called the hybrid artificial bee colony with chaotic search (HABCC). However, changing different search strategies for bees in HABCC is based on the manual setting of the iteration number. Therefore, it is necessary to further develop HABCC into an adaptive hybrid artificial bee colony with chaotic search (AHABCC) algorithm. The AHABCC can automatically switch the search strategy for different bees. One of the advantages for AHABCC is that it does not require a manual setting of the iteration number, making it much more
217
practical for engineering application. Another advantage is that the AHABCC can fully utilize the merits of DE and PSO, as the switch search strategy can switch automatically based on the performances of different search strategies. The rest of this paper is organized as follows. In Section 2, the TA mathematical model for one type of pipe-laying crane vessels is presented to lay a basis for the following development. In Section 3, the basic artificial bee colony algorithm is introduced and analysed for further development. In Section 4, an adaptive hybrid artificial bee colony algorithm is proposed and the procedure is also given. In Section 5, a comparison experiment is conducted to verify the feasibility and effectiveness of AHABCC over the other three algorithms including HABCC, ABC and SQP. Our concluding remarks and future work are contained in the final section.
2. Thruster allocation mathematical model In this study, the mathematical model of thruster allocation is based on a pipe-laying crane vessel with a DPS equipped with seven azimuth thrusters. The arrangement of the propellers is shown in Fig. 1. In this section, the mathematical model of thruster allocation in DPS is introduced to lay a basis for finding the solution. Due to the capacity constraints of the propulsion system, the optimization objective function (Johansen et al., 2004) of thrust allocation can be formulated as follows:
min J (α, F , s ) = PW + s T Qs + (α − α0 )T Ω (α − α0 ) +
δ ε + det (B (α ) B′(α ))
(1)
Subject to the constraints (Rindarøy and Johansen, 2013; Johansen et al., 2004):
s = τ − B (α ) F
(2)
Fmin ≤ F ≤ Fmax
(3)
ΔFmin ≤ F − F0 ≤ ΔFmax
(4)
αmin ≤ α ≤ αmax
(5)
Δαmin ≤ α − α0 ≤ Δαmax
(6)
where W is the total power consumption in the first term, combining the power consumption of the individual thrusters and P is the weight factor. In the second term, s is the error between the commanded and achieved generalized force. Q is the matrix of the diagonal weights. The sT Qs penalizes the error s between the commanded and achieved generalized force. τ is commanded generalized force. B (α ) is the control allocation matrix. F is the thrust force provided by thrusters. Fmin and Fmax are the minimum
Fig. 1. Thruster layout of the vessel with DPS.
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and maximum forces produced by the thrusters, respectively. The rate of thrust change is limited via constraint (4). F0 is the provided thrust in previous step, and ΔFmin and ΔFmax are the minimum and maximum thrust change from one time-step to the next, respectively. α is the rotation angle of thrusters. αmin and αmax are the lower and upper bounds for rotation angle. α0 is the previous thruster angle vector, and Δαmin and Δαmax are the minimum and maximum angle change from one time-step to the next. The weight matrix, Ω , is used to adjust the optimization goal. The last part of Eq. (1) is used to avoid singularity. Setting ε > 0is used to avoid the denominator becoming zero. δ > 0 is the weight coefficient. A larger δ results in better vessel maneuverability and more energy consumption of the thrusters. Based on this pipe-laying crane vessel, B (α ) F in Eq. (2) represents the force of the X and Y axes and the total torque. The force and moment equations can be formulated as follows: ⎧ Fx = F1 cos α1 + F2 cos α 2 + F3 cos α 3 + F4 cos α4 ⎪ + F5 cos α5 + F6 cos α 6 + F7 cos α 7 ⎪ ⎪ F = F sin α + F sin α + F sin α + F sin α y 1 1 2 2 3 3 4 4 ⎪ ⎪ + F5 sin α5 + F6 sin α 6 + F7 sin α 7 ⎨ ⎪ M = F1 sin α1l x1 + F2 sin α 2 l x2 + F3 sin α 3 l x3 + F4 sin α4 l x4 ⎪ − F5 sin α5 l x5 − F6 sin α 6 l x6 − F7 sin α 7 l x7 + F1 cos α1l y1 ⎪ ⎪ − F2 cos α 2 l y2 + F3 cos α 3 l y3 − F4 cos α4 l y4 + F5 cos α5 l y5 ⎪ − F6 cos α 6 l y6 + F7 cos α 7 l y7 ⎩
⎧ 1, f (vi ) ≥ f (xi ) P=⎨ ⎩ 0, f (vi ) < f (xi )
(7)
3. Artificial bee colony (ABC) algorithm The minimal model of swarm-intelligent forage selection in a honey bee colony simulated with the ABC algorithm consists of three types of bees (Karaboga et al., 2007): employed bees, onlooker bees and scout bees. The number of employed bees is equal to the onlooker bees, and the sum of the two types represents the whole colony. Employed bees are responsible for exploiting the food sources explored before and giving information to the waiting bees (onlooker bees) in the hive about the quality of the food source sites that they are exploiting. Onlooker bees wait in the hive and decide on a food source to exploit based on the information shared by the employed bees. Scouts either randomly search the environment to find a new food source depending on an internal motivation or based on possible external clues. Thus, this search system combines the local search strategy, managed by employed and onlooker bees, with global search methods, carried out by onlookers and scouts, attempting to balance the exploitation and exploration processes. There are three important steps in the basic ABC algorithm (Karaboga and Basturk, 2008): moving the employed and onlooker bees to find the food source and evaluating their nectar amounts, determining the scouts, and then finding the food source randomly. In detail, the first step for the basic ABC algorithm is generating the initial food source positions randomly. Then, to generate feasible solutions, all employed bees select a new candidate food source position, different from the previous positions, with the following equation.
(8)
In the above equation, vij is the new food source position modified from the previous position x ij based on the comparison with a random neighboring position xkj . φ is a random number
(9)
The greedy selection operator guarantees that the algorithm can retain the elite individual and not retreat due to the evolution of the process of optimization. Thus, the distribution of P has no relation with the iteration. In the second step, each onlooker selects one food source using “roulette wheel selection” (Goldberg, 1989) and searches for new solutions around its neighborhood. The probability value of one employed bee being chosen is calculated by the following equation:
Pi =
where lx and ly are the distances between each thruster and the xand y-axis, respectively. F1–F7 are the forces provided by thruster #1–#7 respectively. ɑ1–ɑ7 are the angles of thruster #1–#7 respectively.
vi j = xi j + φ (xi j − xkj )
generated in [ 1,1]. k ∈ {1, 2, 3, ... , SN}, j ∈ {1, 2, 3, ... , D} and SN are the number of food sources. D is the dimension of the problem that requires a solution. A greedy selection operator is usually employed to choose the better solution between the searched new vector vi and the original vector xi into the next generation. The probability distribution of vi being selected can be described as follows:
fiti n ∑i = 1 fitn
(10)
where fiti is the fitness value (profitability) of the food source xi . After selecting the food source, the onlooker goes to obtain the new food source position with Eq. (8). In the basic ABC algorithm, if a solution represented by a food source is not changed by a predetermined number ‘limit’ of iterations, the food source is abandoned by its employed bee and then the employed bee associated with that food source becomes a scout. The scout is the colony's explorer. The explorer does not have any knowledge while searching for a food source. They begin to search for a new food source randomly with the following equation: j j j xi j = x min + rand (0, 1)(x max − x min )
(11)
j j where x min and x max represent the lower bound and the upper bound of the search space of food source in dimension j, respectively. And rand (0, 1) denotes a random number generated in [0,1]. It should be noted that the ABC algorithm is inspired from bee colony, therefore the ABC theory seems to be somewhat “bee” language. For those who are not familiar with the ABC algorithm, there are two points that may help understand it easier: (1). the food source in ABC is actually the feasible solution for the optimized problem; (2). the optimized solution will be obtained via several steps which are inspired from bee colony behavior.
4. Proposed AHABCC algorithm and its variants In analysing the process and strategy, the basic artificial bee colony algorithm has the advantage of global convergence, a simple structure, and is easily realized. However, the algorithm also has the disadvantage of premature and slow convergence speed. Therefore, the ABC algorithm needs improvement for applications in solving thrust allocation problems of dynamic positioning systems. Because of that, in our previous work, we proposed an enhanced artificial bee colony (EABC) (Ren et al., 2014) algorithm that introduced chaotic search to scout searching and a hybrid artificial bee colony algorithm with chaotic search (HABCC) (Wu et al., 2014) that introduced a mutation operator from differential evolution (DE) and the social cognitive part of particle swarm optimization (PSO) to the honeybee search strategy based on EABC to solve TA problems of DPS. They both work better than the basic ABC algorithm. However, in the HABCC algorithm, we
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introduced a new parameter, k, selected from [0,1] by trial and error to address the conversion from one method to another in the employed bee and onlooker search strategies. Despite improvement of the global search performance and convergence speed of the algorithm, the HABCC algorithm has lower efficiency and availability because of the increasing number of user-defined parameters. Therefore, we introduce an adaptive strategy for the conversion of employed bee and onlooker search methods and propose an adaptive hybrid artificial bee colony algorithm with chaotic search (AHABCC). The details of the adaptive strategy are as follows. 4.1. Modified foraging process of employed bees
vi j = x rj1 + Fch (x rj2 − x rj3 )
being selected is equal. Then, the value of s1 adaptively changes based on performance of problem solving with the following rule: If the first search method with Eq. (8) works well and obtains the best solution of the whole colony, then the value of s1 is increased with the following equation (the new s′1 is denoted by s1, and s1 is updated using the following equation):
s1′ = s1 + λ1 (1 − s1)
(12)
where r1, r2 and r3 are three mutually different random individuals. Fch is the scaling factor generated via Fch = rand × exp ( − T /Tmax ), T is the number of cycles, and Tmax is the maximum number of cycles. Fch will be a small value when T is small. Therefore, in the earlier stage of the optimization program, the search strategy pays attention to the local search around each individual. When T increases, the value of Fch changes dynamically over a wider range. This will not only speed up the convergence rate, but also take the individual optimum into consideration. The search process of the employed bee with an adaptive hybrid search strategy can be summarized as follows: If (rando s1) Find a neighbouring food source using Eq. (8); Otherwise Find a new food source using Eq. (12); End where s1 is the control parameter lying in [0,1]. The values of s1 represent the probability of being selected to optimize the problem. The strategy with Eq. (8) is selected when “rand” lies in the region of [0, s1]; otherwise, another method with Eq. (12) is selected. The initialization of s1 is 0.5 because of no knowledge about the performance of the two strategies. Thus, the probability of
(13)
where λ1 is a parameter which influences the change speed of s1. When the first search method does not optimize the problem or contribute to the further optimization of the current best solution, the value of s1is decreased with
s1′ = s1 − λ2 (1 − s1)
An employed bee updates the position of the food source (solution) in its memory via searching its local space and finding a neighboring food source and then evaluates its quality. In ABC, the employed bee finds a neighboring food source using Eq. (8). However, according to “No free lunch theorems for optimization” (Wolpert and Macready, 1997), one algorithm or search strategy does not always work well in dealing with all problems. For example, in the ABC algorithm, the employed bee strategy with Eq. (8) also does not work well in all stages of the optimization in handling TA problems of DPS. At the late stage of optimization, the convergence speed of the local search strategy is slow because of the multi-dimensional variables. To solve this predicament caused by a single strategy approach, the employed bee should have more selections in terms of search strategies that are different from the local search strategy and have enhanced capability, such as in the form of a global search or faster convergence speed. If the adaptive hybrid strategy proposed in this paper is applied, the employed bee could switch search strategies adaptively based on each strategy's performance to improve the search ability. Therefore, in this section, we introduce a mutation operator from differential evolution (DE) to a new strategy and propose an adaptive hybrid strategy to achieve the ability to switch from one search method to another during the optimizing procedure based on performance. The new search strategy is described as follows:
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(14)
where λ2 is a parameter which influences the change speed of s1.In addition, the value of s1 is not less than 0.5 when it is decreased with (14). When the value of s1 approaches 0.5 because of falling into a local or global optimum, the probability of being selected is approximately equal in both methods. Therefore, the minimum of s1 is 0.5 in this part. If the second search method with Eq. (12) obtains the best solution, the value of s1 is decreased with
s1′ = s1 − λ1s1
(15)
When the second search method does not optimize or contribute to the further optimization of the current best solution, the value of s1 is increased with
s1′ = s1 + λ2 s1
(16)
The value of s1 is no more than 0.5 when it is increased with (16). 4.2. Modified foraging process of onlooker bees Different from the search method with Eq. (12), which enhanced the global exploration ability, a new method that could increase the convergence speed to obtain the global optimum quickly is necessary for onlooker searching when the local search strategy is unsuccessfully working. Therefore, a new search strategy combined with the social cognitive part of particle swarm optimization for onlookers is followed.
vi j = xi j + φ (xi j − xkj ) + ψ (Pgj − xi j )
(17)
In the above equation, ψ is a randomly generated number that lies in [0,1], and Pg is the best individual selected from all population produced in one iteration. Similar to the employed bee, the main procedure of the onlooker bee is as follows: If (rando s2) Find a neighbouring food source using Eq. (8); Otherwise Find a new food source using Eq. (17); End where s2 is the control parameter lying in [0,1] and its changes are the same as s1 in the previous section. 4.3. Chaotic search strategy In AHABCC, if the time of an employed bee searching exceeds a certain threshold limit and does not find better solutions (Karaboga, 2009), the greedy selection operator will be employed to choose a better solution between this position and the new
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positions generated by a chaotic search (Alatas, 2010). The process of chaotic search is given as follows: 4.3.1). This paper assumes that the search stagnation solution is j j x ij = (x i1, x i2, ⋅⋅⋅, x iD ) , x ij ∈ [x min , x max ]. The stagnation x ij should be mapped to the definition domain [0,1] via the following equation: j
Zi =
(18)
where i = 1, 2, ⋅⋅⋅, SN ; j = 1, 2, ⋅⋅⋅, D , Zij is the mapped value of x ij . Take the constraints (4) and (6) into consideration,
j x max = α0 + Δαmax ,
j x min = F0 + ΔFmin or
j x min
= α0 + Δαmin . 4.3.2). Chaotic variable sequences Zn (n = 1, 2, ⋅⋅⋅, Cmax ) can be generated by a Logistic equation (Yang et al., 2014):
Z n + 1 = μ × Z n (1 − Z n )
j xi j − x min j j − x min x max
j x max = F0 + ΔFmax or
(19)
where Cmax is the maximum number of the chaotic search and μ is a control parameter. When μ = 4 , the sequences Zn is chaotic. 4.3.3). The new variable sequences Zij are mapped to the original solution space, and the new position (solution) is generated
Fig. 2. Flow chart of AHABCC employed for TA.
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by the following equation:
vi j = xi j + 2 × (Zi j − 0.5) ×
j x max
− 2
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Table 1 The size parameters of ‘HYSY 201’.
j x min
(20)
Then, the fitness value of the new position (solution) is calculated, and the best solution with a greedy selection operator is obtained compared with the original solution. 4.3.4). If the cycle number of the chaotic search is more than Cmax , the process ends. If not, then step 4.3.2) is repeated. Finally, the main steps of the AHABCC algorithm in solving the TA problem of DPSs are presented below, and the corresponding flow chart is given in Fig. 2. Step 1: Initialize the population of solutions ( xi , i= 1, 2, 3, ... , SN ) for a TA problem of a DPS. Each solution consists of fourteen parameters from seven thrusters; Step 2: Evaluate the value of each solution with Eq. (1); Step 3: Repeat; Step 4: Produce new solutions, vi , for the employed bees via an adaptive search strategy consisting of Eqs. (8) and (12) and evaluate them; Step 5: Apply the greedy selection process for the employed bees with Eq. (9) to keep the better one; Step 6: Calculate the probability values, Pi , for the solutions, xi , using Eq. (10); Step 7: Produce and evaluate the new solutions, vi , for the onlookers via an adaptive search strategy consisting of Eqs. (8) and (17) from the solutions, xi , selected depending on Pi ; Step 8: Apply the greedy selection process for the onlookers with Eq. (9); Step 9: Determine the abandoned solution for the scout, if one exists, and replace it with a new solution produced using the chaotic search strategy described in Section 4.3; Step 10: Update s1 and s2 ; Step 11: Memorize the best solution achieved thus far; Step 12: Continue until the requirements are met. In a searching iteration, the processes of exploration and exploitation must be conducted together. In the basic ABC algorithm, while the employed bees carry out the exploration process in the search space, the onlookers and scouts control the exploration process. However, this is insufficient when ABC is applied in realworld optimization problems. It is easy for the ABC algorithm to fall into the local optimum when the algorithm is running to the global optimum. Therefore, the exploration ability needs to be enhanced. In the previous HABCC algorithm, we enhanced the employed bee searching for a new source using Eq. (12) and improved onlooker bee exploitation ability via Eq. (17). With that, the application of the adaptive hybrid search strategy, which combines two search methods based on different situations, can also enhance the algorithm's capability and increase its efficiency and availability. In addition, the chaotic search strategy with characteristics of randomness and ergodicity can improve the probability of finding the optimal result and avoiding local optimums. Those functions are verified in the next section.
Length overall L oa/m
Length between Displacement V /t perpendiculars Lpp/m
Designed draft d/m
Beam /m Molded depth /m
204.65
185
7–9.54
39.2
59,000
5.2. Parameter setting The ‘HYSY 201’ is used as the case study vessel. The vessel moves from initial state [ 0 m 0 m 0° ] to desired position and orientation [10 m 10 m 20° ] and keeps the desired position with desired heading. The desired longitudinal resultant thrust, the lateral resultant thrust, and the moment of 250 s are shown in Figs. 3–5. The mission of the HABCC or AHABCC algorithm is to address the thruster allocation problem to make the seven actuators produce various thrusts at accurate angles. Thus, the ship can move to and also keep desired position with desired heading under the oceanic environmental conditions. In addition, SQP and ABC algorithms are also included to make a comparative study. They are all employed to solve TA problems in DPS for ships with seven thrusters for positioning. One thruster has two parameters: thrust and angle (i.e., the dimension of a solution is 14). To make the comparative study fair, for the four algorithms, the initial values of first second are chosen randomly within thruster limits. The (N 1)th (NZ3) second TA results are chosen as the initial values of Nth second respectively. For ABC, HABCCA and AHABCC these three swarm algorithms, the population size is set as 32, and the maximum number of iterations is 1500 for each second of the optimization process. The value of the limit is set to be 200, and the value of Cmax is chosen as limit/2, as suggested in (Alatas, 2010). In addition, in terms of the constraints of the objective function, the value of Fmin is zero and that of Fmax is described in Table 2. The maximum change of the thrust angle in one second is 20% of the maximum thrust ( ΔFmax or ΔFmin = Fmax × 20%). Δαmin and Δαmax are 0.314 rad and 0.314 rad, respectively. Furthermore, as to make the comparative study fair, the control parameter k in HABCC is varied from 0 to 1 with step width 0.1 and the best one is chosen via trial and error. In this study when k¼0.7, the performance of HABCC is best thus k is set to be 0.7 to make a fair comparative study. 5.3. Results and analysis The simulation has run 30 times repeatedly, and the average results are shown in the following figures and tables. Firstly, the simulation results will be given from Figs. 6–13 and Tables 3 and 4. Secondly, Fig. 14 is given to demonstrate that: (1).The optimal search of AHABCC is faster compared to HABCC; (2).the probability of obtaining optimal results and avoiding local optimums is Table 2 The parameters of the thruster system. Thruster No. Rotate speed N /(r / min)
Power P/kW
Maximum thrust Fmax/kN
5. Experimental results 5.1. Simulation plant In this paper, the simulation plant is a deepwater pipe-laying crane vessel named ‘HYSY 201’ with a DPS. Its major size is given in Table 1. This DP vessel realizes its positioning ability with 7 thrusters whose information is described in Table 2.
14
1# 2# 3# 4# 5# 6# 7#
181.4 181.4 192.3 192.3 192.3 192.3 192.3
4500 4500 3200 3200 3200 3200 3200
680 680 540 540 540 540 540
Installation site
X /m
Y /m
93.8 93.8 12.55 12.55 37.85 37.85 82.65
9.45 9.45 15.4 15.4 14 14 0
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Fig. 3. Desired longitudinal resultant thrust. Fig. 5. Desired moment.
Fig. 6. Results of the thrust and the angle of thruster #1 from the HABCC and AHABCC methods.
Fig. 4. Desired lateral resultant thrust.
increased. Finally, Fig. 15 and Table 5 are shown to indicate that: (3) the power consumption of AHABCC is less than that of HABCC. Figs. 6–12 show the individual thrust and angle for each thruster, and Fig. 13 shows the changes in s1 and s2. Fig. 14 illustrates the Convergence of the HABCC and AHABCC method. Fig. 15 shows the difference of total power consumption between HABCC and AHABCC during TA for 250 s.
In addition, sensitivity tests of λ1 and λ2 were conducted, and the results are described in Table 3. Each result in this table represents the value of the sum of errors in 250 s of the X-axis, Y-axis, direction force and moment. There are approximately 12 combinations when λ1 = [1/2, 1/3, 1/4] and λ2 = [1/8, 1/9, 1/10, 1/11]. It shows that when λ1 = 1/3 λ2 = 1/10, AHABCC obtains the best result. However, the rest of the combinations are also acceptable because of the small difference from the best. Therefore, Table 4 is provided for the comparisons. In Table 4, the corresponding error results of four algorithms when they are employed for TA in DPS are recorded. The header of the table, Best, refers to the minimum value of the sum of errors in 250 s found in 30 times. Average1 refers to the average value among 30 times, and Average2 is the average value of each second. The Worst indicates the maximum error found in 30 times. Std is the standard deviation of errors.
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Fig. 7. Results of the thrust and the angle of thruster #2 from the HABCC and AHABCC methods.
Fig. 10. Results of the thrust and the angle of thruster #5 from the HABCC and AHABCC methods.
Fig. 8. Results of the thrust and the angle of thruster #3 from the HABCC and AHABCC methods.
Fig. 11. Results of the thrust and the angle of thruster #6 from the HABCC and AHABCC methods.
Fig. 9. Results of the thrust and the angle of thruster #4 from the HABCC and the AHABCC methods.
Fig. 12. Results of the thrust and the angle of thruster #7 from the HABCC and AHABCC methods.
From Table 4, we can see that the Average error of AHABCC is better than that of HABCC, ABC and SQP. Correspondingly, other errors, such as Worst, Best and Std, of AHABCC are similar to or
better than that of HABCC. The experimental results also show that the performance of the AHABCC algorithm integrated with an adaptive hybrid search strategy outperforms HABCC, ABC and SQP
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Fig. 13. The changes in s1 and s2.
Fig. 14. Convergence of the HABCC and AHABCC method.
Table 3 The sensitivity test results of λ1 and λ2 .
Table 5 Total power consumption of different algorithms. Algorithm
Total power conventional thrust allocation (MW)
SQP ABC HABCC AHABCC
305.611 237.389 214.235 188.895
Table 4 Comparisons of different algorithms.
SQP
ABC
HABCC
AHABCC
Fx Fy N Fx Fy N Fx Fy N Fx Fy N
Average1
Average2
Worst
Best
Std
1292.9 1824.94 1221.39 317.326 359.471 442.416 249.490 276.246 246.656 233.459 247.775 244.649
5.1716 7.2998 4.8855 1.2693 1.4378 1.7696 0.9980 1.1050 0.9866 0.9338 0.9911 0.9786
4193.67 3755.89 3775.91 428.161 450.334 521.952 290.023 355.588 298.517 286.763 323.878 296.200
182.769 335.912 180.451 251.346 283.983 348.636 202.661 207.714 216.077 188.482 195.915 214.392
1029.01 1104.80 990.888 47.9211 46.1647 59.5018 23.0075 37.2987 23.2812 22.4375 27.8681 22.8282
methods. The results of SQP also show that SQP is very sensitive to its initial values. It also can be seen that there exists some differences between Worst and Best, this is because swarm intelligence based algorithms have randomness characteristic. It should be note that the results are recorded during 250 seconds. It is obvious when simulation time increases, the sum of errors such as Worst and Best increase. However, when we look at the average error of one second for Worst and Best, take Fy for instance, that is 355.588/ 250 ¼ 1.42 and 207.714/250 ¼ 0.83 respectively for HABCC and 323.878/250 ¼1.29 and 195.915/250 ¼0.78 respectively for
Fig. 15. The difference of power consumption between HABCC and AHABCC methods.
AHABCC. It can be seen that the average error for one second is acceptable for TA solution even though in Worst case which is 1.42 and 1.29 respectively as there are seven thrusters in DPS. From Figs. 6–12, it can be observed that the thrust force curves produced by seven thrusters based on AHABCC are similar to that of HABCC. As far as thrust force is concerned, the results from both AHABCC and HABCC are reasonable, and the change in each second does not exceed the limited boundary. In addition, the objective of optimization is not to obtain the value of the thruster's angle but rather to determine the change in the angle. From these
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figures, we can conclude that the angle changes are frequently within the limit of each thruster and that the results from both HABCC and AHABCC are similar, indicating satisfactory results. Finally, Fig. 13 shows the changes in the control parameters, s1 and s2, in the iteration of optimization. It describes the changes in proportion, which is the possibility of being selected to address the problem in each cycle. From Fig. 13, it can be observed that the AHABCC algorithm can adaptively adjust the value of s1 and s2 to control each method's action time and proportion. This is to show that our proposed adaptive hybrid strategy is effective and useful. Fig. 14 shows the convergence of the HABCC and AHABCC method. And it is observed that AHABCC has faster convergence rate than HABCC. In addition, it is demonstrated that the probability of obtaining optimal results and avoiding local optimums is increased. The difference of total power consumption (Yadav et al., 2014) using the seven thrusters during 250 s between HABCC and AHABCC is shown in Fig. 15. It is evident from Fig. 15 that the power consumption of the AHABCC algorithm is lower than the power consumption of the HABCC algorithm as a whole. It is also observed that when desired thrusts increase sharply (see Fig. 3), the corresponding power-saving performance of AHABCC is obvious. When desired thrusts change not so much (see Figs. 3–5), the corresponding power-saving performance of AHABCC is not as obvious as before. When desired thrusts change not so much, the (N 1)th TA results which are chosen as the initial values of Nth second are closer to the Nth TA results compared with thrusts increasing sharply, therefore the superiority of AHABCC becomes not so obvious compared with HABCC. This is why the powersaving performance has become not obvious as the time goes by in Fig. 15. The total power consumptions of seven thrusters during 250 s are listed in Table 5 for SQP, ABC, HABCC and AHABCC algorithms. It can be seen that consumed power are 305.611, 237.389, 214.235 and 188.895 (MW) for the four algorithms respectively. It is evident that AHABCC performs best for these four algorithms. All of the simulations are completed via the Matlabs platform on a personal computer. The configurations of the computer are CPU 2.9 GHz, RAM 6.0 GB, and 32-bit operation system. It should be noted that the AHABCC algorithm for TA in this study is implemented using a serial computing technique and serves as a theoretical basis for future industrial application. The calculation of TA for 1 s requires approximately 8 s. In Rui (2009), it is shown that the running time of FPGA (Field Programmable Gate Array)based PSO achieves approximately 0.032% of that in Matlabs, as a parallel computing technique is employed in FPGA. Thus, the calculation time can be shortened within 1 second to achieve realtime TA when the algorithm is implemented based on FPGA, even though the AHABCC algorithm involves ABC, PSO and DE. This is also our future work.
6. Conclusions This paper has presented a thrust allocation method based on the AHABCC algorithm. The introduction of a mutation operator from DE and the social cognitive part from PSO to honeybee search strategy, a chaotic search strategy to the scout searching component, and an adaptive hybrid strategy contribute to both the exploration and exploitation abilities in AHABCC. The simulation results show that both the power consumption and error of the AHABCC algorithm are less than the previous HABCC algorithm in solving TA problems. Furthermore, the AHABCC algorithm has the control parameter, s, which is adaptively changed. However, there is a control parameter, k, in HABCC that is required to be set by trial and error. This all shows that AHABCC is better than HABCC
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and that our presented adaptive strategy is successful and effective. However, it should be noted that like HABCC, AHABCC has only been tested with simulated data and the combination of search methods hybridized to a search strategy is not perfect. Thus, more intensive bench tests with real data should be conducted and more new search mechanisms should be introduced to achieve better performances for various optimization problems.
Acknowledgments The authors would like to thank the financial support from the National Natural Science Foundation of China (51249006, 61473183, and U1509211), the Fujian University Outstanding Young Scientific Talent Cultivation Plan (JA13169), the Natural Science Foundation of Fujian Province, China (2015J05104) and the Program of Shanghai Subject Chief Scientist (14XD1402400) for this project.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.oceaneng.2016.04.004.
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