Hull form uncertainty optimization design for minimum EEOI with influence of different speed perturbation types

Ocean Engineering 140 (2017) 66–72

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Hull form uncertainty optimization design for minimum EEOI with influence of different speed perturbation types

MARK

⁎

Yuan Hang Hou

Transportation Equipment and Ocean Engineering College, Dalian Maritime University, Linghai Road No. 1, High tech Zone, Dalian City, Liaoning Province 116026, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Uncertainty optimization EEOI Speed perturbation Hull form

The paper presents the uncertainty hull form optimization design method for minimum EEOI, considering the influence of travelling speed perturbation. Four types of perturbation parameters are addressed and applicable methods for each type are given. Firstly, uncertainty optimization methods: Interval optimization (IO) and robust optimization under moment uncertainty (DRO-MU) are introduced. Then, formula and parameters of EEOI with the numerical method for calculating resistance are described. Finally, hull form optimization model of minimum EEOI is given and four case studies are conducted to verify the feasibility and superiority of the novel approach. Results show that the uncertainty optimization design is of excellent adaptability and reliability in minimum EEOI ship hull lines design.

1. Introduction Energy Efficiency Operational Indicator (EEOI), proposed by IMO in 2009 (MEPC, 2009), expresses the environmental costs, CO2 discharge namely, which is generated by social benefit of ship transportation of each tonnage unit (quantity of shipments). EEOI is closely related to numbers of internal or external parameters, such as ship resistance, service speed, some turbine factors, and so on. However, relevant studies are mostly carried out based on simplified environment, thus some important environmental parameters, such as service speed of the vehicle, are usually assumed as deterministic or constant in design process (Acomi and Acomi, 2014; Cheng et al., 2014). Neglect of influence of uncertainty would lead to a low robustness result which would have specific limitations in practical application. In the real travelling process, disturbance of certain parameters are unavoidable, which reflects the uncertainty of environment. Because of the accuracy of navigation state and resistance evaluation, service/ operational speed, as one of the most important navigation parameters, plays a vital role in minimum EEOI optimization design mode. By monitoring the ship speed in a period of time, which is shown in Fig. 1, it is obviously find that the speed perturbation around a sailing ship is universal existence. Although the perturbation uncertainty magnitudes are relatively small in most cases, but their continuous calculation and coupling effects with other parameters will cause the system response to a large deviation in the optimization design process.

⁎

Generally, perturbation parameter falls into 4 types according to its included uncertainty information:

• • • •

Constant: Certainty without any perturbation; Interval: An interval number with midpoint and radius; Probability: Number with an explicit probability distribution density; Interval probability: Probability distribution density with interval moments (such as expectation, variance and covariance).

As is listed, each type of perturbation parameter has the different characteristic properties, such as information amount, solving complexity, accuracy and robustness of optimal result. Four types of perturbation parameter are shown in Fig. 2. It is revealed from the famous “No free lunch theorem”, that there is no one method can solve all the optimization problems with different types of parameter. Therefore, it's necessary to optimize separately pointing at the different types of perturbation parameter. Prior information of interval parameter is the upper and lower bounds, and many researches indicated that the interval method based on two layers nesting system can provide an effective way to do the optimization. Interval multi-objective evolutionary optimized method was researched by (Gong and Sun, 2013); and numerical solution method of interval optimization was proposed by (Li and Tian, 2008); problems of interpretation of inequality constraints involving interval coefficients were studied by (Sengupta et al., 2001; Kim et al., 2014)

Corresponding author. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.oceaneng.2017.05.018 Received 29 November 2016; Received in revised form 10 April 2017; Accepted 13 May 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.

Ocean Engineering 140 (2017) 66–72

Travelling Speed

Y.H. Hou

Loeve expansion; Xi et al. (2015) (Chen et al., 2015) researched ship hull form shape design problem by High-fidelity global particle swarm optimization; Probability distribution of travelling speed and its moments, namely expectation, variance and covariance, are mainly got through statistical analysis of historical data. In a long period of time, travelling speed or its measuring error follows a certain distribution. But in a short period of time, the probability distribution and its moments are uncertainty, which are similar to the long term certain distribution. The distributional robust optimization under moment uncertainty method (DRO-MU) could both consider the long term probability information and short term uncertainty properties (Delage and Ye Yiny, 2010; Jianqiang et al., 2013), and would provide an attemptable way for the hull form design under uncertainty travelling speed. The purpose of this study is to provide guidance for designers on hull form optimization with different types of influence of travelling speed perturbation. Firstly, uncertainty optimization methods: Interval optimization (IO) and robust optimization under moment uncertainty (DRO-MU) are introduced. Then, formula and parameters of EEOI with the numerical method for calculating resistance are described. Finally, hull form optimization model of minimum EEOI is given and four case studies are conducted to verify the feasibility and superiority of the novel approach.

Monitor speed Design Speed

Time Node of Monitor Fig. 1. Monitor and design operational speed for contrast.

applied interval optimization research on molecular communication with drift and proved its effectiveness. As is revealed, key point to do the interval optimization is searching for suitable internal algorithm and its control parameters. Probability optimization problem with parameter's probability distribution information is derived from chance constrained optimization. Robust optimization method, which uses “min-max” object pointing at the worst situation, becomes an effective method to solve this kind of problem. Recently it was widely used in ship principal dimensions and hull form optimization design. Shari et al. (2010) (Hannapel and Nickolas, 2010) introduced uncertainty and its optimization methods in multidiscipline ship design, and proved its fine feasibility. Taking variable regular wave and geometry into account, Wei et al. (2013) (He et al., 2013) studied delft catamaran total/added resistance, motions and slamming loads in head sea; then Matteo et al. (2014) (Diez et al., 2014) extended the above model for variable Froude number and geometry using meta-models, quadrate and Karhunen–

2. The uncertainty optimization method According to the different types of uncertainty parameters in optimization, uncertainty optimization system needs to be established respectively. Typical methods include interval optimization and robust optimization, which reflect thoughts of “interval-oriented” and “worstcase-oriented”. 2.1. Interval optimization (IO) Interval number is a type of number expressed by an interval, as is shown in Eq. (1):

a I = [a L , a R ]

(1)

(2):

(1):

Midpoint

0

0

Constant

Lower bounds

upper bounds

SD σ1

SD σ2

-SD

0

Probability density

(4):

Probability density

(3):

Radius

+SD

SD Interval [σ1,σ2] 0

E

E Interval

Fig. 2. Four types of parameter. (1): Constant. (2): Interval. (3): Probability. (4): Interval probability. SD: standard deviation, E: expectation.

67

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Y.H. Hou

Fig. 3. Double nested architecture for Interval optimization (IO).

Where: aL , aR ∈ R , and aL ≤ aR ; aL and aR is the upper and lower bound of interval number aI . When aL = aR , aI turns out to be a real with deterministic value. For interval numbers AI and BI, interval order relationship, expressed by ≤mw , can be used to evaluate their degrees, as in Eqs. (2)–(3):

⎧ AI ≤mw BI , onlyifm (AI ) ≥ m (BI )andw (AI ) ≥ w (BI ) ⎨ I ⎩ A
AL + AR , 2

w (AI ) =

AR − AL 2

BL + BR , 2

w (BI ) =

BR − BL 2

Where: min f (x , u) and max f (x , u) are lower and upper bounds of target when x is deterministic and u changes in interval range, and are obtained by inner layer optimization. Eq. (5) shows that, the main idea of the transformation: from uncertainty optimization to certainty optimization, is to evaluate the design variables by midpoint and radius of interval objective function, thus certainty objective function is obtained. Afterwards, weights of the two objectives: midpoint and radius, are given in order to achieve the single objective optimization model, whose objective function is as in Eq. (6):

(2)

opt (3)

⎧ P (C I ≥ D I ) ≥ λ ⎨ I L R I L R ⎩ C = [g (x), g (x)], D = [b , b ] ⎪ ⎪

(4)

Where: x is design variable with n dimensions, and its range is ; u is uncertain vector with q dimensions, and its uncertainty is described by an interval number uI . f and g is objective function and restrictive condition respectively, which are related to x and u; b I is the allowable interval of uncertain restrictive condition. Objective function in Eq. (4) can be transformed to certainty based on interval order relationship, as in Eq. (5):

Ωn

⎧ min {m ( f (x , u)), w ( f (x , u))} ⎪ 1 ⎪ m ( f (x , u)) = 2 ( f L (x) + f R (x)) ⎨ 1 ⎪ w ( f (x , u)) = 2 ( f R (x) − f L (x)) ⎪ ⎪ f L (x ) = min f (x , u), f R (x ) = max f (x , u) ⎩

(6)

Where: β is the weight, 0 ≤ β ≤ 1, because the midpoint and radius of interval objective function are usually seen as equally important, thus is taken 0.5 here. Uncertainty restrictive condition in Eq. (4) can be converted to certainty restrictive via Eq. (7):

Where m is the midpoint of interval, and w is the radius. After expressing the uncertainty number by interval number, uncertainty optimization problem can be described as in Eq. (4):

⎧ opt min { f (x , u)} ⎪ ⎨ s. t . g (x , u) ≤ b I = [b L , b R ] ⎪ I L R ⎩ x ∈ Ω n , u ∈ ui = [ui , ui ], i = 1, 2, ... ,q

min {(1 − β ) m ( f (x , u)) + βw ( f (x , u))}

(7)

Where: CI is the possible range of uncertainty restrictive function at x, D I is the permissible restrictive interval number. P (CI ≥ D I ) means the probability of CI is greater than or equal to D I , λ is the probability threshold given in advance. g L (x) and g R (x) is the upper and lower bounds of restrictive interval, which is defined as in Eq. (8):

g L (x) = min g (x , u), g R (x) = max g (x , u)

(8)

Where: min g (x , u) and max g (x , u) are lower and upper bounds of constraint when x is deterministic and u changes in interval range. Restrictive condition (as in Eq. (7)) is integrated into the objective function (as in Eq. (8)), as a form of penalty function, thus a certainty unconstrained optimization model is established as in Eq. (9):

(5) 68

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⎧ opt min fp (x , u) ⎪ ⎨ fp (x , u) = (1 − β ) m ( f (x , u)) + βw ( f (x , u)) + σϕ (P (CI ≥ D I ) − λ ) ⎪ ⎩ ϕ (P (CI ≥ D I ) − λ ) = (max(0, −[P (CI ≥ D I ) − λ]))2

g is constraints. Robust optimization method describes the uncertainty parameter in the function as a box or ellipsoid form of uncertainty set. DRO-MU method combines the robust optimization and stochastic programming, based on the uncertainty set of two important moments (expectation and variance) of random variables, a Min-Max optimization can be carried out, the model is as following:

(9) where: σ is penalty factor with a large value, ϕ is penalty function. Double-levels nested optimization architecture is normally used to solving the interval optimization problem: the outer layer is used for generating the design variables; and inner layer is used for calculating the interval objective function. Namely, mounts of design variables are generated via the outer layer, and inner layer algorithms is called to obtain the uncertainty interval of objective function and restrictive, and then converting them to the certainty function. Fig. 3 is an illustration of the interval optimization process. Mission of outer layer is to find better design variable individuals according to current optimal objective value, and then set the conditions for convergence, when the iteration step arriving the threshold limit or upper limit, and 500 in this study. Thus the optimization of the outer layer is deterministic and without any interval calculation. Inner layer expands the new generation design variables to interval number, and evaluates the superior degree of objective function to decide whether to accept the new design variables. Then the optimization model is transformed to the deterministic without constraint, which is easily for solving. Interval optimization problem is usually non-continuous and nondifferentiable, that would disenable the application of traditional optimization algorithms based on gradient. Two optimization algorithms based on Monte-Carlo modern bionic theory is adopted here: particle swarm optimization with learning factor improvement strategy (IPSO) (Haipeng et al., 2012) and very fast simulated annealing with “Annealing-Tempering” mode (MVFSA) (Huagen et al., 2006). Purpose of outer layer optimization is to generate design variable individuals with wide coverage among the global scope, and this requires strong search ability for the algorithm. PSO (Particle Swarm Optimization) algorithm has the advantage that optimization process has a wide range, direction, and high group collaboration. Learning factor improvement strategy of IPSO can make the particle swarm has a larger cognition part in the early iterations and a larger social part in the later stage (Haipeng et al., 2012). As the core of uncertainty optimization, inner layer has a high requirement for local search ability and efficiency of the algorithm. Through long-term research and application, SA (Simulated Annealing) proves to be strict and effective. However, SA needs enough model perturbation and iteration, and a suitable annealing plan. To overcome this shortcoming, MVFSA introduces two new ideas based on SA: for high temperature, global random generator, which has a stronger ability than the traditional perturbation of SA and has nothing to do with initial temperature, is adopted; for low temperature, in order for the decrease of perturbation space, a certain restriction of perturbation is made, thus the optimum solution can be found quickly, and its acceptance probability can also be increased.

min max E [ f (x , ξ )] x∈R F∈D

Where: F is distribution of random variable ξ, D is uncertainty set of ξ, E represents the expectation. Taking the hull shape design optimization into consideration, service speed is expressed as random variable, of which the expectation and variance are seen as uncertainty. Ellipsoid uncertainty set of expectation and semi-definite cone uncertainty set of variance are established, as following:

⎧ P (v ∈ S ) = 1 ⎪ D = ⎨[E (v ) − μi ]T σi−1 [E (v ) − μi ] ≤ γ1 ⎪ ⎩ E [(v − μi )(v − μi )T ]≺γ2 σi

(12)

Where: μi is expectation column vector of incoming flow velocity random variable v , and σi is variance vector of v . γ1 is control parameters of expectation ellipsoid uncertainty set, which meets γ1 ≥ 0 ; γ2 is control parameters of variance semi-definite cone uncertainty set, which meets γ2 ≥ 1. Eq. (11) is a typical NP (Non-Deterministic Polynomial) hard MinMax problem, which is hard to solve directly, and can be transformed into a deterministic semi-definite programming one by Lagrange duality principle as an effective solution way (Zhou et al., 2015):

⎧ min r + t ⎪ ⎪ Q, q, r , t ⎨ s. t .r ≥ U0 (x , v ) − v T Qv − v T q, ∀ Li ∈ S ⎪ ⎪ t ≥ (γ2 σi + μi μiT)⋅Q + μiT q + γ1 σ1/2 i (q + 2Qμi) ⎩

(13)

Where: Q、q are dual variables, which meet Q≻0 ; r、t are slack variables; indicates Frobenius product; ≻ indicates semi-definite. Namely, via duality principle, robust optimization with random variables is transformed to a minimum problem of slack variables r and t, which can be easily calculated and solved.

3. Formulas for EEOI 3.1. EEOI According to MEPC.1/Circ.684 resolution form IMO, formula of EEOI is expressed as (MEPC, 2009):

EEOI =

∑j FCj × CFj mcargo × d

(14)

Where: ∑j FCj is the mass of consumed fuel at voyage j; CFj is the fuel mass to CO2 mass conversion factor at voyage j; mcargo is cargo carried (tonnes) or work done (number of TEU or passengers) or gross tonnes for passenger ships; d is the distance in nautical miles corresponding to the cargo carried or work done. Take one typical voyage for example, and mcargo and d are both uncontrollable, then Eq. (14) becomes to Eq. (15), in which the formula form is similar to EEDI's except the perturbation service speed, (for EEDI's is the constant design speed). For calculation and analysis, EEOI can be expressed by the ratio of CO2 emissions and quantity of shipments, and is related to the ship's fuel consumption, engine power, auxiliary power, energy-saving equipment, service speed, tonnage and other factors (Acomi and Acomi, 2014). Its calculation formula is shown as:

2.2. Robust optimization under moment uncertainty (DRO-MU) Robust optimization (RO) is a method to solve the internal structure and the external environment with uncertainty, in which, the uncertainty of parameters in constraints or objective functions is usually oriented. Conventional robust optimization model is (Dimitris et al., 2004):

⎧ min sup f (x , ξ ) ⎪ x∈R ξ∈S ⎨ ⎪ s. t .sup gk (x , ξ ) ≤ 0, (k = 1, ... ,m ) ξ∈S ⎩

(11)

(10)

Where: x is design variable, ξ is random variable, S is uncertainty set (distribution space of random variable, namely); f is objective function, 69

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EEOI =

EME + EAE + EPTI + Eeff FC⋅CF = DWT ⋅vser fi ⋅Capacity⋅vser ⋅fw

Table 2 Parameter magnitude in EEOI formula.

(15)

Where:

Parameters

Magnitude

Parameters

Magnitude

⎧ ⎞ ⎛ ⎪ EME = ⎜∏nj =1 f j ⎟ (∑inME P ⋅C ⋅SFCME (i ) ) =1 ME (i ) FME (i ) ⎠ ⎝ ⎪ ⎪ ⎪ EAE = PAE⋅CFAE⋅SFCAE ⎨ ⎞ ⎛ ⎪ EPTI = ⎜∏n f ∑nPTI PPTI (i ) − ∑neff f ⋅P C ⋅SFCME j =1 j i =1 i =1 eff (i ) AEeff (i ) ⎟ FME ⎪ ⎠ ⎝ ⎪ ⎪ Eeff = −(∑neff feff (i ) ⋅Peff (i )⋅CFME⋅SFCME ) ⎩ i =1

SFCME PME PAE

190 g/(kWh) 75%MCRME 5%MCRME

SFCAE fi、fj、fw PAEeff、PPTI、Peff

215 g/(kwh) 1.0 0

consideration of fluid motion. In this study, Reynolds Average N-S Formula (RANS) are selected in the CFD solver, and k-ε viscous model are used for the multiphase calculation. Because of the lengthy iterative process of CFD calculation, approximate model is necessary for the optimization, which has strict requirement for the length of each step, and is also the typical way in the ship Simulation Based Design (SBD) (Kandasamy et al., 2013) process. BP (Back Propagation) neural network is a kind of multilayer feed forward network with error back propagation algorithm (Li and Liu, 2015), and becomes one of typical approximation technologies because of its excellent ability to approximate nonlinear function, Eq. (18) represents a three-layer BP neural network model which using tangent sigmoid as transfer function of neurons:

(16)

And where: EME is CO2 discharge of main engine, and EAE is of auxiliary engine, EPTI is shaft belt device, Eeff is energy-saving equipment.CF is carbon conversion coefficient, vser is service/operational ship speed on deep and shallow water in variable load conditions, at variable engine shaft power and actual environmental conditions (wind, waves, current, etc.), kn; SFC is specific fuel consumption in 75% rated power, g /(kW ⋅h ); Capacity is deadweight tonnage, t; P is power for main or auxiliary engine, kw; f j is modifying factor for ship special design, fi is modifying factor for ice strengthened ship, and is taken 1.0 for non ice strengthening; feff is innovation factor, and is taken 1.0 for waste heat recovery unit, and for other energy or technology, it should be evaluated by classification society; fw is wind wave correction factor. Take the mathematical ship form for example, Wigley hull is used here for the EEOI computation within the resistance approximate model mentioned later. As the displacement is sum of deadweight DWT and hull weight, meanwhile, hull weight can be approximate set as a specific percentage of displacement in the preliminary design stage. Thus in this study, DWT is also setup as a percentage of displacement, and the ratio is set as 50%. Main particulars of hull are showed in Table 1. As case study, engine parameters, such as Specific Fuel Consumption (SFC) and Engine Power (P), for the subject ship should be defined in advance. Take the conventional marine cargo ship as a reference, the SFC and P for main and auxiliary engine are defined as certified values. Parameters in this research are taken as Table 2. Upon substitution of magnitudes in Table 1 to Eq. (3), the simplified EEOI formula can be got as Eq. (17):

⎛

J

BPNN3: Oi =

⎛

K

N

⎞

⎞

⎠

⎠

∑ Wij tanh ⎜⎜∑ Wjk tanh ⎜⎜ ∑ Wkn ξn + b1k ⎟⎟ + b2j⎟⎟ + b3i ⎝ k =1

j =1

⎝ n =1

(18) Where: ξn is input variable, Oi is output variable, Wkn ,Wjk ,Wij are the weights of the layers between neurons, b1k ,b 2j ,b3i are thresholds of neuron unit in each layer. BP neural network approximate model needs mounts of simulation results as inputs, and is very sensitive to the internal parameters, thus errors of output will occur due to some uncontrollable causes, Although this error or uncertainty has a small value in most cases, large deviation of the whole system could also be generated by continuous iterative computation. Therefore, considering the uncertainty of approximate model has an important significance. A successfully trained neural network output is expressed as interval number, as is shown in Eq. (19):

BPNN I = Rt I = [Rt L , Rt R] = [Rt (1 − γ ), Rt (1 + γ )]

142.5MCRME + 215(0.05MCRME ) 153.25MCRME = CF × Capacity × vser DWT × vser 153.25Rt = CF × (17) DWT × PC × 9.8 × 103

EEOI = CF ×

(19)

Where: Rt L and Rt R is the lower and upper bound of interval number Rt I ; γ represents the uncertainty level of Rt I , which is usually in terms of percentages. The larger γ is, the greater uncertainty degree of interval number will be.

Where: MCRME is power rating of main engine, kW; Rt is travelling resistance, N; PC is propulsive coefficient, take routine ship hull with similar dimension as reference, 0.6 is set here; DWT is deadweight, has the same meaning with Capacity, t.

4. Hull form uncertainty optimization design for minimum EEOI 4.1. Optimization model

3.2. Travelling resistance Rt Taking Wigley hull as an example, the design variables are identified by whole ships’ principal dimensions and the overall shape of a ship, in which the principal dimensions are represented by the waterline length L, waterline width B, draft T. The modification of the hull shape can represent by the original data points multiplied hull modification function, as shown in Eqs. (20)–(21):

Formula based on the slender body theory such as Michell, is frequently-used in the ship resistance estimation because of its simple assumptions and fast capabilities. However, such approach is considered out-dated and probably not accurate enough to capture the effect of small hull form modifications on the vessel resistance. Recently, Computational Fluid Dynamics (CFD) approach is commonly used for hull form optimization studies because of its high precision and

⎧ yf (x, z ) = yf 0 (x, z )⋅ω (x , z ) ⎨ ⎩ ya (x, z ) = ya0 (x, z )⋅ω (x, z )

Table 1 Main particulars of Wigley hull.

w (x , z ) = 1 −

L/m

B/m

T/m

DWT/ t

Cb

Cp

2.0

0.2

0.125

0.0125

0.48

0.53

(20)

⎡ ⎛ x − x ⎞m +2 ⎤ ⎡ ⎛ ⎞n +2 ⎤ 0 ⎥⋅ sin ⎢π ⎜ z 0 − z ⎟ ⎥ ⎟ ⎥ ⎢ ⎠ ⎝ ⎣ x max − x 0 ⎦ ⎣ z 0 + T ⎠ ⎥⎦

∑ ∑ Amn sin ⎢⎢π ⎜⎝ m

n

(21) Where: yf (a) (x, z ) represents before (after) half of the lateral data points 70

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size 100, iteration number 600, convergence threshold 1e-5. After calculation of every case, optimization design results are obtained as in Table 5: As shown in Table 5, all the optimization method (Case1#, 2#, 3# and 4#) could obtain the optimal EEOI. However, there are some differences: 1# result reflects the best of all while 2# represents the worst, which is attributed to interval number in the calculation and influence of uncertainty travelling speed, is considered. Numbers of probability and interval probability have more probability distribution information in the calculation, therefore, compared to 2#, results of 3# and 4# are better optimized, and differences between 3# and 4# represent not much. Although results of uncertainty optimization are slightly worse than the deterministic, that's because the former considers the uncertainty influence of perturbation parameters, which are closer to reality in engineering. Meanwhile, calculation time of uncertainty optimization is more than deterministic, which is obvious, and that can attribute to the double-levels nested and complex analysis and optimization architecture, but the extra calculation time is also reasonable and acceptable. For further discussion, uncertainty optimization method has embed the mathematical modeling of uncertain environmental parameters to optimization process, and consider it as part of the optimization parameter, consequentially, the results has higher adaptability and robustness to the uncertainty environmental. Therefore, in the marine environment with full of uncertainties and random factors, optimization result has superior navigation capability and output less EEOI than the initial scheme. In order to further improve the optimization result, detailed design of ship hull lines seems to be necessary, which would pay close attention to more detailed ship performance, and this will also be the next work plan. Also, as is shown in Table 5, DWT directly affects the EEOI values where DWT figures for all optimization cases are different: when other factors remain unchanged, EEOI is inversely proportional to DWT, although this regular pattern is not clearly demonstrated, that's because in Eq. (17), EEOI are also closely related to other factors, such as hull resistance Rt, which directly depends on hull lines. Therefore, ranking of min. EEOI considering different DWT and Rt, or other important factors, would be more scientific and can be made in the next work plan.

Table 3 Initial value and range of design variables. Design variables

L/m

B/m

T/m

Amn (m,n=1,2,3)

Initial value Changing range

2 1.6–2.4

0.2 0.16–0.24

0.125 0.1–0.15

0 0–0.15

of the hull after changed, both in the mid ship-section of the interface; ω (x, z ) is modification function of hull form; Amn is to characterize the magnitude of the control variables, as is represented in the equation, the more numbers of Amn existence the more accurate it could express the hull lines. In this paper m, n=1,2,3, thus the matrix scale of A is 3*3, and 9 numbers existence here as design variables. Initial value and range of design variables is shown as in Table 3. Non-significant changing of displacement is taking as constraint condition:

∇0 − ∇ ≤ε ∇0

(22)

Where: ∇ and ∇0 are optimal and initial hull form's displacement, which can be calculated utilizing Simpson method; ε is a small value, to ensure that the displacement volume of optimized ship is not below the lower limit. After integrating the constraint condition into the optimization objective, minimum EEOI optimization model is obtained, as in shown in Eq. (23):

⎧ opt min fp (x) ⎪ ⎪ 2 ⎨ fp (x) = f (x) + σ (max(0, −[P ((∇0 − ∇)/∇0 − ε ≥ 0) − λ])) ⎪ 153.25 × Rt (x ) ⎪ f (x) = EEOI (x) = CF × ⎩ D × PC × 9.8 × 103

(23)

Where: x is vector of design variables, x = {L , B, T , Amn }(m, n = 1, 2, 3). 4.2. Case study and results For this study, both the deterministic optimization and uncertainty optimization are conducted to handle four types of parameter, which are mentioned in Fig. 3. Pointing at hull form optimization design for minimum EEOI, four cases in Table 4 are researched: As is shown, for γ1 is control parameters of expectation ellipsoid uncertainty set, and γ2 is control parameters of variance semi-definite cone uncertainty set, and they are set as 0.5 and 1.2 to ensure building a suitable computing space for the current problem. Improve particle swarm optimization (IPSO), which introduces the learning factor improvement strategy, has high applicability for this kind of nonlinear complex optimization problem, and is adopted in this study. If, on one hand, different algorithm would not likely lead to an opposite conclusion in the study, on the other hand, this does not represent a limitation for the application of the present uncertainty optimization. Furthermore, assessing different algorithms and inner parameters of one is outside the scopes of the present work, and, therefore, no further addressed. Inner parameters in IPSO of each case are mainly set as: population

5. Conclusions Pointing at ship hull form optimization design, a minimum EEOI optimization model is established with the design space constituted by principal dimensions and ship form coefficients. Influence of travelling speed perturbation is considered in this study, four parameter types: constant, interval, probability and interval probability are participated in the optimization, respectively. Some conclusions can be got: (1) Uncertainty optimization method with interval or probability can both obtain the optimum result with an acceptable calculation time, thus this method has the effectiveness. (2) Under the same conditions, the result of uncertainty optimization is slightly worse than the certainty optimization results, but former considers the uncertainty influence of travelling speed, so it can better reflect the reality and have the superiority.

Table 4 Cases of hull form optimization design for minimum EEOI. Case 1# 2# 3#

Perturbation Parameter vser vser vser

Perturbation type Constant Interval Probability

4#

vser

Interval probability

Value (m/s) 1.5 [1.2,1.8] Normal, μ=1.5, σ=0.1μ Normal, μ=1.5, σ=0.1μ, γ1=0.5, γ2=1.2

71

Optimization method Deterministic IO RO

Algorithm IPSO IPSO+MVFSA IPSO

Objective Min EEOI Min EEOI Min EEOI

DRO-MU

IPSO

Min EEOI

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Y.H. Hou

Table 5 Optimization design results of each case. Case

Optimization method

perturbation type of vser

L/m

B/m

T/m

DWT/t

EEOI

Time consuming (s)

Initial 1# 2# 3# 4#

– Deterministic IO RO DRO-MU

– Constant Interval Probability Interval probability

2.000 2.241 1.820 1.590 1.799

0.200 0.210 0.190 0.231 0.213

0.125 0.100 0.105 0.119 0.100

0.0125 0.0128 0.0121 0.0129 0.0126

134.66 122.02 131.91 128.65 131.59

– 41 223 278 381

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(3) With probability and interval probability numbers, EEOI are better optimized, because of there are more probability distribution information in the calculation. In order to further improve the optimization result, detailed design of ship hull lines can be conducted synthetically. (4) Further studies can be carried out from researching the properties of the uncertain parameters and applicability of optimization algorithms. As is discussed in this study, uncertainty optimization method considers the different type of environmental parameters’ mathematical modeling, and got the superior results. Thus for the uncertainty and randomness of marine environment, uncertainty optimization method could also be suitable for other marine design field, and more different type of uncertainty parameters and optimization method are waiting for development and research. Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 51609030) and Fundamental Research Funds for the Central Universities of China (Grant No. 3132016339, 3132016358, 3132017017). References Acomi, N., Acomi, O.C., 2014. Improving the voyage energy efficiency by using EEOI. Procedia - Social. Behav. Sci. 138, 531–536. Cheng, H.R., Liu, X.D., Feng, B.W., 2014. Study on multidisciplinary optimization method for hull forms design. Shipbuild. China 55 (1), 76–82. Chen, Xi, Diez, Matteo, Manivannan, K., et al., 2015. High-fidelity global optimization of shape design by dimensionality reduction, metamodels and deterministic particle

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