Uncertain Multidisciplinary Design Optimization on Next Generation Subsea Production System by Using Surrogate Model and Interval Method

The innovative Next Generation Subsea Production System (NextGen SPS) concept is a newly proposed petroleum development solution in ultra-deep water areas. The definition of NextGen SPS involves several disciplines, which makes the design process difficult. In this paper, the definition of NextGen SPS is modeled as an uncertain multidisciplinary design optimization (MDO) problem. The deterministic optimization model is formulated, and three concerning disciplines—cost calculation, hydrodynamic analysis and global performance analysis are presented. Surrogate model technique is applied in the latter two disciplines. Collaborative optimization (CO) architecture is utilized to organize the concerning disciplines. A deterministic CO framework with two discipline-level optimizations is proposed firstly. Then the uncertainties of design parameters and surrogate models are incorporated by using interval method, and uncertain CO frameworks with triple loop and double loop optimization structure are established respectively. The optimization results illustrate that, although the deterministic MDO result achieves higher reduction in objective function than the uncertain MDO result, the latter is more reliable than the former.


Introduction
Nowadays offshore petroleum in deep and ultra-deep water is developed by dry tree or subsea tree. However, both of them have limitations: the drawbacks of dry tree development lie on fabrication cost of hull, development flexibility, riser/vessel interfaces and so on; on the other hand, the subsea tree developments lack efficient drilling and completion capability (Lim, 2009). Table 1 summarizes the limitations of subsea and dry tree developments.
To overcome these drawbacks, the NextGen SPS concept (Zhen and Huang, 2017;Zhen et al., 2018bZhen et al., , 2020Wu et al., 2019) has been proposed as a new petroleum production solution in ultra-deep water areas. As shown in Fig. 1 and Fig. 2, the artificial seabed (AS) is located below the mean sea level and positioned by vertical mooring lines; the rigid risers and lower-rated subsurface well completion equipment are supported by the AS; the flexible jumpers bridge the AS and the floating production unit (FPU). Util- izing the lower-rated subsea well completion equipment in ultra-deep water and detaching the huge riser weight from FPU are essential features of NextGen SPS. The AS, mooring lines, rigid risers and flexible jumpers constitute an integrated system that jointly responds to the environmental loadings. Compared with the traditional sequence approach that defines these components individually, the integrated design approach that treats them as a whole and defines them simultaneously is more reasonable (Girón et al., 2014). However, the integrated design of Nex-tGen SPS is a complicated multidisciplinary problem that involves consideration of several disciplines, such as weight estimations, hydrodynamic analysis, and global performance analysis. The interaction between these disciplines further increases the design complexity (Zhang et al., 2017;Chen et al., 2018). The challenges of the integrated design lie on the long design cycle and how to organize these disciplines.
MDO is a field that applies optimization algorithm for the design of complicated systems that involve several disciplines. The MDO concept came from the integrated design practice of aerospace vehicles, and its applications have been extended to other fields (Sobieszczanski-Sobieski and Haftka, 1997;Pan et al., 2009;Martins and Lambe, 2013;Zhao et al., 2015;Liu et al., 2017;Bidoki et al., 2018). The concerning disciplines are organized and coordinated during the process of implementing MDO. The designer can improve the design and shorten the design cycle simultaneously by solving the MDO problem. Thus, the integrated design of NextGen SPS can be formulated as an MDO problem to overcome the above mentioned design challenges. During the optimization process, the drag force of AS and global performance parameters of NextGen SPS need to be calculated, through the time consuming computational fluid dynamics (CFD) simulation and time domain analysis respectively. To improve the design efficiency, some surrogate models are developed to approximate the CFD simulation and time domain analysis. According to the input parameters, these surrogate models could predict output parameters rapidly, with sufficient accuracy.
MDO architecture refers to the way to formulate MDO problem and organize the concerning disciplines in accordance with the problem formulation. Collaborative optimization (CO) architecture (Braun, 1996) that characterizes a two-level hierarchical structure is adopted in this work. It is a widely used MDO architecture which has been successfully applied in the design of aircraft (Kroo et al., 1994), container ship (Fu et al., 2012), underwater vehicle (Gou and Cui, 2010;Luo and Lyu, 2015;Li et al., 2018) and so on. By using the CO architecture, the complicated MDO problem is decomposed into several simpler, more manageable discipline-level optimization problems, and there are no interactions between these discipline-level optimization problems. Compatibility between these discipline-level optimization problems is ensured by the system-level optimization. CO architecture provides much more flexibility and modularity for the complicated problem, and the parallel and distributed process is available as well (Pan et al., 2009;Luo and Lyu, 2015;Meng et al., 2019). The above MDO on NextGen SPS is a deterministic optimization. It should be noted that, the deterministic optimization result tends to barely meet the design constraints, and leave little or no tolerance for uncertainties. In practice, there are uncertainties in design parameters due to manufacturing imperfections (Wu et al., 2014). Moreover, the output values obtained from the surrogate models are just an approximation of the true values, thus there are uncertainties in surrogate models (Sun et al., 2020). The optimization result under the deterministic assumption might become unfeasible when considering the uncertainties (Wu et al., 2015). Therefore, the uncertainties should be incorporated into the design optimization to get a more reliable design.
In uncertain optimization, fuzzy set method (Zhu et al., 2020a), probabilistic method (Zhu et al., 2020b) and interval method (Li et al., 2013) are usually used to consider the uncertainties. The exact probability distributions of uncertain parameters are needed for the probabilistic method (Sun et al., 2020;Zhu et al., 2021), and the exact membership functions of fuzzy set are needed for the fuzzy set method (Chalco and Román-Flores, 2008). However, it is expensive or even impossible to acquire complete information to define the exact probability distributions or membership functions (Peng et al., 2019). With respect to interval method, each uncertain parameter is treated as an interval number which contains all possible values. It is relatively easier to define upper and lower bounds of the uncertain parameters (Wu et al., 2014), compared to define the exact probability distributions or membership functions. Owing to this advantage, the interval method has a wide application in uncertain optimization (Wu et al., 2014;Peng et al., 2019;Faes and Moens, 2020). The interval method is adopted here to incorporate the uncertainties. This paper aims to develop an uncertain MDO approach for the integrated design of NextGen SPS. The determinist- ic optimization model is proposed firstly. Followed by the presentation of three concerning disciplines: cost calculation, hydrodynamic analysis for AS and global performance analysis for NextGen SPS. Some surrogate models are developed in the latter two disciplines. Based on the CO architecture, the deterministic MDO framework is established.
Then the triple loop and double loop uncertain CO frameworks are developed respectively, through incorporating interval uncertainty model into the deterministic MDO. Finally, all the optimization problems are solved by using particle swarm optimization (PSO) algorithm. The flowchart in Fig. 3 depicts the general view of this work. The organization of this paper is as follows. Section 2 presents the general description and case data of the Next-Gen SPS as well as the formulation of deterministic optimization. In Section 3, the concerning disciplines are introduced and some surrogate models are developed. The deterministic and uncertain MDO frameworks are developed in Section 4, and the optimization problems are solved in Section 5. Finally, Section 6 summarizes the work in this paper.

NextGen SPS concept and its deterministic optimization formulation
2.1 General description As depicted in Fig. 1 and Fig. 2, NextGen SPS mainly consists of the AS, riser system and mooring lines.
The AS is positioned at an appropriate depth to weaken the effect of surface current and wave. It contains one outer platform (OP) and four internal buoyancy cans (IBC). The main hull of OP is a flat column with four channels. An extended bilge box is attached at the bottom of the flat column, and the chain stoppers are accommodated at the bilge box. The manifold and flexible jumpers are supported by the OP. The IBC characterizes a slender column inside the OP, and it is the same height as the OP. In addition, the IBC supports the subsurface christmas tree and provides buoyancy to tension the rigid riser. The IBC and OP both are divided into multiple compartments to minimize the effect of damage in any one compartment. Compliant guide (Karayaka et al., 2004) is adopted to provide the contact between the IBC and the OP. Consequently, the horizontal motion of the OP can be transferred to the IBC, meanwhile their vertical motions are independent. This configuration will improve the mechanical properties of the rigid riser which is sensitive to vertical motion .
The riser system comprises rigid risers and flexible jumpers. The petroleum is transmitted from the subsea wellhead to the AS through the rigid risers, then transmitted from the AS to the FPU through the flexible jumpers. The rigid riser is a single casing system which comprises an outer casing and an internal production tubing. The flexible jumpers are in a slack shape to isolate the AS from FPU motions.
The OP is positioned by four groups of two vertical mooring lines. Each mooring line is a combination of chain and tether.
The NextGen SPS concept provides the following benefits: unrestricted FPU selection and reduced fabrication cost of FPU as the tremendous riser weight is removed from it; simpler riser/FPU interaction; assured flow near the seabed; improved mechanical properties of rigid riser; improved commercial and technical performance in ultra-deep water due to the adoption of shallow-watered rated subsurface well completion equipment. Thus, NextGen SPS can be deemed as a promising alternative ultra-deep water development solution.

Environmental conditions
NextGen SPS in a depth of 3000 meters is adopted as the study case. The environmental load conditions are listed in Table 2 and Table 3. The 100-yr typhoon and 10-yr mon-soon are taken as the extreme storm and operational conditions respectively.

Optimization problem 2.3.1 Design variables
The parameters concerning the AS, rigid risers and mooring lines are taken as the design variables. For the AS, geometric parameters are considered: the OP height H and diameter D 1 , the IBC diameter D 2 , the bilge box height h b and width w b , as depicted in Fig. 4. Besides, diving depth s of AS is involved. The ratio (a 1 ) of the bilge box height h b to the OP height H is used to replace the bilge box height h b , and the ratio a 1 is set to keep only one decimal place. Design variables of the rigid risers are top tension factor F TT as well as outer diameter D o and wall thickness T o of the outer casing. With respect to the mooring lines, tether diameter d and pretension ratio a 2 are considered. All the design variables as well as their boundaries are listed in Table 4.

Design constraints
The design constraints cover the sizing criteria of AS and the design criteria of NextGen SPS. The sizing criteria of AS is twofold: (1) enough top area of AS is required to place the subsurface well completion equipment; (2) in case one compartment buoyancy is lost, the residual buoyancy of AS is required to support the subsurface equipment and provide sufficient top tension. σ y The design criteria are summarized in Table 5. Vertical distance ΔZ between the OP and IBC is limited to keep the integrity of AS. Strength of rigid riser and mooring line is compliant with the rules (American Petroleum Institute, 2005;Exploration and Production Department, 2006). denotes the yield strength of rigid riser and it is 551 MPa in this paper; T represents the minimum breaking load of tether and it is a function of the tether diameter d. The effective tension of rigid riser is checked to prevent global buckling (Det Norske Veritas, 2010). As for the flexible jumper, its operating radius curvature is restricted (Qin et al., 2011).

Objective function
The cost of NextGen SPS, which includes the cost of rigid risers, flexible jumpers, mooring lines and AS, is taken as the objective function, and it is given by where represents the weight of rigid riser, represents the length of flexible jumper, represents the weight of tether and represents the weight of the AS; the numbers 4, 2 and 8 represent the number of rigid risers, flexible jumpers and tethers respectively; , , , and are cost weight corresponding to each part. In this paper, these cost weights are set as 5400, 26400, 12000 and 4700 respectively, which reflect the unit price of each part.   M AS l f way to obtain the AS weight is detailed in Section 3.2. The flexible jumper length is determined using the criterion presented in Zhen et al. (2018a), as shown in Eq. (2): s d where represents the horizontal span of flexible jumper, and represents the diving depth of the AS.

Concerning disciplines
The MDO of NextGen SPS involves consideration of the following three disciplines: cost calculation, hydrodynamic analysis for AS and global performance analysis for NextGen SPS. The input and output parameters of these disciplines are summarized in Table 6, where N represents the product of AS drag coefficient C d and drag area A.
The cost calculation discipline shares part of input parameters with the other two disciplines respectively, and the cost of NextGen SPS which represents objective function f, is calculated in this discipline. The hydrodynamic analysis discipline outputs the parameter N, which is taken as input parameter in performance analysis discipline. The global performance parameters (T 1 , T 2 , T 3 , σ 1 , σ 2 , k, ΔZ) are calculated in performance analysis discipline and the parameter ΔZ is delivered into hydrodynamic analysis discipline as the input parameter.

Cost calculation
The cost of NextGen SPS is calculated according to Eq.
(1). As noted in Section 2.3.2, the weight of rigid riser , the length of flexible jumper and the weight of tether can be obtained easily based on related design parameters (D o , T o , d, s). However, to calculate the AS weight , structural design for the AS should be implemented firstly according to the AS geometrical parameters (H, w b , D 1 , D 2 , a 1 ).
The layout of plates, stiffeners and girders for AS is assumed constant. As shown in Figs. 5a and 5b, the IBC is divided into ten identical compartments; the horizontal plate is stiffened by eight radial stiffeners, at 45° angle. Figs. 5c−5e show the OP structure: a watertight bulkhead is arranged in the half height; the vertical shell is stiffened by horizontal girders, vertical stiffeners and girders; the space between vertical stiffeners is specified as 3° angle. The definition of plate thickness, stiffener and girder sizes is in accordance with the scantling requirements in Mobile Offshore Drilling Unit rule (American Bureau of Shipping, 2013). Based on the structural design and AS geometric parameters, the weight of the obtained AS structure could be calculated.

Hydrodynamic analysis for AS
Through the commercial software STAR-CCM+, the CFD numerical simulation is employed to calculate the drag coefficient C d of AS. The CFD model which involves the AS geometry, mesh generation and physic model is presented. After that a gird independence analysis is carried out to check the model accuracy and provide guidance for selecting appropriate mesh size.

CFD model
The AS geometry is built in accordance with its geometric parameters (H, w b , D 1 , D 2 , a 1 ). To reduce the modelling difficulty, production equipment is replaced by equivalent cuboid which has the same drag area with the equipment. The simulation domain characterizes a cuboid whose length is 7D 1 , width is 10D 1 and height is 10H, as shown in Fig. 6. The inlet velocity is set as 1 m/s while the outlet pressure is set as 0 Pa. The AS is located nearer to the velocity inlet to capture the wake from AS body. The meshes in the domain are generated by using trimmed mesher and prism layer mesher. The trimmed mesher cuts a hexahedral template mesh in the AS surface to generate volume meshes, and the prism layer mesher generates transitional prismatic mesh layers near the AS boundaries. The height of the first prismatic mesh layer is calculated by Eq. (3), ∆y where y + is specified as 40, is the height of the first layer, L is the AS diameter, Re is the Reynolds number.
Five regions with different mesh sizes are defined in the domain. Ratio between the mesh sizes on neighboring regions is set as 2. The generated meshes are shown in Fig. 7.
The realizable two-layer k−ε model is taken as turbulence model and the all y + is taken as the wall treatment approach. The fluid material is water-liquid and the flow type is segregated flow. Second-order upwind convection scheme is adopted. The under-relaxation factors remain the default values.

Grid independence analysis
Grid independence analysis is a crucial step in CFD simulation (Esfeh et al., 2017;Chen et al., 2018). It is carried out to ensure the independence between the mesh size and the solution. Three cases with different mesh sizes are defined for this analysis. Table 7 summarizes the solution and cell number of each case. There is a deviation of 0.751% in drag coefficient for the coarse mesh and the fine mesh. It indicates that the solution is stable and accurate. The medium mesh size is adopted to balance the accuracy and computational efficiency.

Global performance analysis for NextGen SPS
Time domain analysis method is employed to calculate the global performance parameters (T 1 , T 2 , T 3 , σ 1 , σ 2 , k, ΔZ), through the commercial software ORCAFLEX. Fig. 8 presents the NextGen SPS model in ORCAFLEX. Both the OP and IBC are modeled as the 6-degree of freedom buoys. The line elements with different line-types are used to model the rigid risers, flexible jumpers and mooring lines. It should be added that, through the equivalent method (Bai and Bai, 2005), the tubing and outer casing are translated into an "equivalent riser" which can represent the actual rigid riser behavior.
The link element that has no mass or hydrodynamic loading is utilized to simulate the contact between the OP

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Surrogate model
Both the above CFD simulation and time domain analysis are time consuming, and the computational cost of uncertain optimization would be unaffordable (Sun et al., 2020). To overcome this difficulty, some surrogate models are built to approximate the CFD simulation and time domain analysis.

Development method of surrogate model
The flowchart in Fig. 10 depicts the process of developing surrogate model: First, generating the reasonable sample points in the design space; second, building and running the sample cases, as well as extracting the output parameters; followed by the training of surrogate model using a portion of sample cases (training sample); finally, evaluating the model accuracy utilizing the rest of sample cases (testing sample). In case the model accuracy does not meet the requirement, it is advisable to change the parameters of surrogate model firstly; if the former does not work, one should expand the sample size. Latin hypercube sampling method (McKay et al., 1979) and back propagation neural network (Mouazen et al., 2010) are adopted to generate sample points and train surrogate model respectively. The operations that generating sample points, training and testing surrogate model are implemented through the MATLAB language.

Sample cases
The training sample and testing sample are generated independently. The training sample is used to train the surrogate model and the testing sample is used for evaluating the trained surrogate model accuracy. In the hydrodynamic analysis discipline, training sample contains 400 cases and testing sample contains 40 cases. The 440 cases are run on the computer cluster of Dalian University of Technology. It includes 229 blade nodes, and each node has two Intel Xeon E5-2600 v3 CPU. In the global performance analysis discipline, training sample and testing sample contain 600 cases and 60 cases respectively. The 660 cases are run on a single desktop computer with an Intel Core i7-7700 CPU.

Performance
Before the training process, input and output parameters are normalized into [−1, 1] to improve the training performance. Correspondingly, the input parameters for the surrogate model should also be normalized into [−1, 1], and the output parameters produced by the surrogate model should be reversed to their true values.
The Bayesian regularization training method is adopted during the training process. It only takes a minute to accomplish the training process.
After the training process, the surrogate model accuracy is evaluated using the testing sample, by calculating the Mean Absolute Percentage Error (MAPE), which is defined as:

CO architecture
The CO architecture consists of one system-level optimization problem and several discipline-level optimization problems. One of the discipline-level optimization problems can be stated as: where is the locally calculated state variable that might be the input parameter for other discipline-level optimization, is the state variable that specified by system-level optimization, is the shared design variable that is shared with other discipline-level optimization (also called as global design variable), is the local design variable which is unique to this discipline and is the local constraint. In the discipline-level optimization, and are taken as fixed values.
The system-level optimization problem can be stated as: where is the objective function, is obtained from discipline-level optimization and it is used to estimate objective function or constraints, is the systemic constraints, denotes the compatibility constraint which ensure the consistency between disciplines and is the relaxation factor. The systemic design variables consist of global design variable and state variables .
An MDO problem with two discipline-level optimizations is adopted to illustrate the structure and running process of the CO architecture, which characterizes a nested double loop optimization, as presented in Fig. 11. The discipline-level optimizations must be completed in each iteration of the system-level optimization. CO framework with more discipline-level optimizations has the similar structure. The application process of CO architecture could be outlined below: (1) the system-level optimizer assigns the systemic design variables and into each discipline-level optimization; (2) in each discipline-level optimization, the optimization aiming to minimize the is performed under the local constraints ; (3) the state variables and are delivered into system-level optimization after this discipline-level optimization; (4) the system constraint g, compatibility constraint and the objective function are estimated in the system-level optimization. The above steps would be repeated until the optimal solution is found.
In CO architecture, there is no dependency between the discipline-level problems so that the parallel and distributed process is available. The CO architecture is especially attractive for the MDO problem with a relatively weak interdisciplinary interaction.

PSO algorithm
All the optimization problems are solved by using PSO algorithm (Shi and Eberhart, 1999). It is a widely applied global algorithm and regarded as efficient to obtain a global optimal solution.
In PSO algorithm, a collection of particles moves in steps inside the design space. Each particle has its unique position and velocity vector. The positions represent the values of design variables. As a particle arrives at a new position, the fitness of it is evaluated. Based on this evaluation, the velocity vector and position of this particle are updated as follows ] ; (7) where n denotes the total number of particles; d denotes the total number of design variables; t denotes the current step number; denotes the velocity vector; denotes the position; denotes the individual best fitness value; denotes the global best fitness value; and are random numbers in the range [0, 1]; and are acceleration factors; is the inertia weight. In this work, the inertia w c 1 c 2 weight is limited in [0.5, 1], and are set as 2. And 10*m (m is the number of design variables) particles are taken in each optimization problem.

Framework of deterministic MDO
As shown in Table 6, the MDO of NextGen SPS covers three disciplines. Without loss of generality, the deterministic CO framework for NextGen SPS includes three discipline-level optimization problems and nine shared design variables (D o , T o , d, s, H, w b , D 1 , D 2 , a 1 ). In application of CO, reducing the numbers of disciplines and system-level design variables will simplify the optimization process and decrease computational cost. As the cost calculation discipline contains all the shared design variables and has no dependency with other disciplines, this discipline is placed in system-level. A deterministic CO framework involving two discipline-level optimization problems is proposed, as presented in Fig. 12.

N
There is no constraint in the first discipline-level optimization. According to the local design variables (H, w b , D 1 , D 2 , a 1 ), (product of AS drag coefficient and drag area) is calculated through the surrogate model. The purpose of the first discipline-level optimization is to minimize , where is specified by the system-level optimization.
The second discipline-level optimization is constrained by the design criteria which detailed in Table 5. In this part, the global performance parameters (T 1 , T 2 , T 3 , σ 1 , σ 2 , k, ΔZ) are calculated according to the six local design variables (D o , T o , F TT , d, s, a 2 ), through the surrogate models. Similarly, the second discipline-level optimization aims to minimize , where is assigned by the systemlevel optimization.
After completing of the two discipline-level optimizations, the values of local design variables ((D o , T o , d, s, H, w b , D 1 , D 2 , a 1 ) and state variables (ΔZ, N) are delivered into system-level optimization where they are used to estimate the design constraints and objective function f. A part of design constraints are the sizing criteria of AS mentioned in Section 2.3.2, and the other constraints are the compatibility constraints and . The relaxation factor of is taken, where k represents or . In this framework, only two state variables and are considered as systemic design variables.

Interval uncertainty model
In the interval method, an uncertain parameter is considered as an interval number which contains all the possible values. The interval number is defined as: where denotes the interval number, and are the lower and upper bounds of respectively. The midpoint , width , and radius of are defined as: (10) The interval radii of each design variable in this work are presented in Section 5.2. The parameter with superscript c denotes the midpoint of interval number.

y I y y
Besides the design parameters, the uncertainties of surrogate models are also considered using interval number . The lower bound and upper bound are defined as follows, y c where denotes the predictive value from the surrogate model, a, b, c and d are calculated according to the testing sample in Section 3.5: where and represent the output value and predictive value of the testing sample, respectively. a is the minimal percentage deviation and b is the minimal deviation. The predictive value is modified by the two values individually and the larger value is taken as the lower bound . The upper bound is calculated in the same way.

Framework of uncertain MDO
In the optimization model of NextGen SPS, the objective function represents the system cost and the design constraints are related to the system safety. The uncertainties in the design constraints must be incorporated, as it might lead to the system failure. Compared with the system safety, the uncertainties in the system cost is relatively unimportant so that it is ignored for the purpose of improving the optimization efficiency.
Under the circumstance of the interval uncertain design parameters and surrogate models, in general, the corres- ponding response parameters also characterize interval number, denoted by . The deterministic constraint is translated into the interval uncertain constraint , which is equal to , where is the upper bound of . The calculation of the upper bound could be treated as an unconstraint optimization, which aims at searching the maximal response values within the ranges of interval design parameters. Based on the deterministic CO framework with double loop optimization, the interval uncertain CO framework with the nested triple loop optimization is proposed, as shown in Fig. 13. It should be noted that for the sizing criteria in system-level optimization, the interval values of constrained parameters could be easily calculated by relevant interval design parameters. Thus, there is no need to utilize the interval-level optimization in system-level optimization. There are four systemic design variables ( ) that describe two interval numbers, where and are lower bounds, and are interval widths. The parameter with superscript c is the midpoint of interval number. The and represent upper and lower bounds of interval response values, respectively. The uncertain framework includes three types of optimizations: the system-level optimization, discipline-level op-g i g i timization and interval-level optimization. The most striking difference with the deterministic framework is the existence of the unconstraint interval-level optimization, which is responsible for finding the maximal (or minimal) response values ( or ) within the ranges of interval design parameters. The system-level optimization and disciplinelevel optimization have similar functions in the two frameworks.
However, the uncertain framework with triple loop would be quite inefficient. Utilizing the interval-level optimization to estimate the interval response value contributes the inefficiency. To solve this problem, some surrogate models are developed to replace the interval-level optimization. Consequently, the triple loop optimization framework is transformed into double loop, as shown in Fig. 14. In this framework, surrogate models are responsible for estimating the interval response values. The method in Section 3.5.1 is employed to develop these surrogate models. In discipline-level optimization 1, training sample contains 400 cases and testing sample contains 40 cases; while in discipline-level optimization 2, training sample and testing sample contain 600 cases and 60 cases respectively. All the sample cases are solved by using the PSO algorithm. Table 9 summarizes the accuracy of Similarly, uncertainties of these surrogate models are incorporated during the optimization process.

Implementation
The flowchart in Fig. 15 illustrates the way to treat the constrained optimization problem. The design constraints are estimated firstly and the objective function is calculated only if the design constraints are fulfilled. It avoids calculating the objective function of the unfeasible designs so that the optimization efficiency would be improved. In addition, the design variables are always discontinuous in practical engineering. For example, the diameter of rigid riser (D o ) characters integer (in millimeter) and the geometric parameters of AS have only one decimal place (in meters). Nevertheless, the PSO algorithm is not competent for solving the optimization with discontinuous design variables.
10 a i i = 1, 2, · · · , n a i Consequently, the original PSO algorithm is adjusted as follows: first of all, the lower and upper bounds of each design variable are multiplied by ( , is the required number of decimal places for this design variable); in the search process, Eq. (8) is adjusted as: 10 a i where round is a function that can round object to the nearest integer; finally, the rounded value is divided by and the obtained value is used to estimate the design constraints and objective function. The adjusted PSO is capable of solving the optimization with discontinuous design variables, as only the parameters which satisfy the practical requirement are generated.

Results and discussions
All the optimization problems are solved in MATLAB software and the results and interval radii are presented in Table 10. Compared with the initial design, the deterministic MDO result achieves a reduction of 19.3% in the objective function, while the uncertain MDO results under triple loop and double loop framework achieve 17.4% and 17.2% respectively. For the computational time, the deterministic MDO takes about one and a half hours, the uncertain MDO under double loop structure takes almost four hours (including the time spent in the development of surrogate models of the interval response values), while the uncertain MDO under triple loop structure takes more than twelve hours.  Besides that, the interval responses of each optimization result are calculated and listed in Table 11. The values highlighted on bold format denote the violation of design constraints. For example, the maximum vertical distance (ΔZ) of deterministic MDO result is 7.3 m, which violates the constraint value 7.2 m. It demonstrates that the deterministic MDO result is not absolutely reliable. Meanwhile, the interval responses of the uncertain MDO results meet the design constraints absolutely, and it indicates that the uncertain MDO results are reliable. Therefore, a more reliable result can be obtained from uncertain MDO than that from deterministic MDO. Although the deterministic MDO takes fewer computational cost and its result achieves higher reduction in objective function, it is recommended to conduct the uncertain MDO on NextGen SPS. The uncertain MDO result under double loop structure is very close to the uncertain MDO result under triple loop structure, due to the high accuracy of surrogate models of the interval response values. However, the double loop structure has distinct advantage in efficiency, as the number of optimization loops is reduced. Thus, it is significant to replace the interval-level optimization with the surrogate model and transform the triple loop optimization structure into double loop.
In addition, the uncertain MDO approach can obtain a reliable result within four days (including the time spent in the calculation of sample cases), as against the traditional sequence design approach that consumes several weeks.

Conclusions
This paper presents an uncertain MDO on the innovative NextGen SPS. The MDO of NextGen SPS is decomposed into three disciplines: cost calculation, hydrodynamic analysis and global performance analysis, and the CO architecture is adopted to organize the concerning disciplines. Some surrogate models are developed to replace the time domain analysis and CFD simulation in the latter two disciplines. A deterministic CO framework involving two discipline-level optimization problems is established. On the basis of this framework, the uncertainties of design parameters and surrogate models are incorporated through the interval method. The uncertain CO frameworks featuring with triple loop and double loop structure are proposed individually. The interval-level optimization and the surrogate model are responsible for calculating the interval responses values respectively.
Based on the above work, the following conclusions could be obtained.
(1) Although the deterministic MDO takes fewer computational cost and its result achieves higher reduction in objective function, the uncertain MDO results are more reliable than the deterministic MDO result.
(2) The triple loop optimization and the double loop optimization have close results, while the latter has distinct advantage in efficiency.
(3) Compared with the traditional sequence design approach that consumes several weeks, the uncertain MDO approach shortens the design cycle.
This work would provide reference for integrated design of the TLP, SPAR and hybrid riser system, which are also complex multidisciplinary system in offshore oil industry. To further improve the design efficiency, future work would focus on developing uncertain MDO approach based on the other MDO architectures.