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Optimization of a FloatingPlatform Mooring System Based on a Genetic Algorithm Aidin Rezvani Sarabi Nelson Szilard Galgoul NSG Engenharia, Projetos e Representacao Comercial Ltda.Sesam Users Conference – 03.Dec.2012 1
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Objective Optimization of the platform heading Optimization of the mooring pattern Searching for the tension or length of the mooring lines Choosing the optimum line material and sizeSesam Users Conference – 03.Dec.2012 2
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Choosing an Optimization Method Many optimization problems in practical engineering are quite hard to be solved by conventional optimization techniques. So there has been an increasing interest in solving such hard optimization problems by imitating the behavior of living beings.Sesam Users Conference – 03.Dec.2012 3
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Choosing Optimization Method Simulating the natural evolutionary process of living beings results in stochastic optimization techniques called evolutionary algorithms. The most widely developed type of evolutionary algorithms are known today as Genetic Algorithms (GAs).Sesam Users Conference – 03.Dec.2012 4
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Genetic Algorithm Fundamentals GAs work with a coding of the solution set, not the solutions themselves GAs search for a population of solutions, not a single solution Genetic Algorithms use payoff information (Fitness Functions), not derivatives or other auxiliary knowledge GAs use probabilistic transition rules, not deterministic rulesSesam Users Conference – 03.Dec.2012 5
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Analysis of Mooring System using Mimosa First, Mimosa determines an equilibrium position by applying a numerical procedure that solves the equation below:F ( mo 1 2 6 cu 1 wi 2 ) x ,x ,x +F x +F x + f x =0 wa 6 () ( ) ( ) The solution to this equation is the equilibrium position that defines the platform coordinates and heading under static loadsSesam Users Conference – 03.Dec.2012 6
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Analysis of the Mooring System using Mimosa The actual platform motions are computed by performing a dynamic analysis, where the corresponding responses are categorized as high frequency (HF) and low frequency (LF) motions The HF responses are calculated using a linear spectral analysis.Sesam Users Conference – 03.Dec.2012 7
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Analysis of Mooring System using Mimosa The LF responses are horizontal motions (Surge, Sway and Yaw) which result from the solution of equation below: M x LF + C x LF + Kx LF = FLFSesam Users Conference – 03.Dec.2012 8
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Analysis of the Mooring System using Mimosa In order to calculate the extreme values for the combinations of HF and LF motions, Mimosa uses a heuristic equation which is based on model tests and simulation studies as given in the equation below for one variable x LF + x HF tot Sign ext x = max ext HF LF x Sign + x ext Sesam Users Conference – 03.Dec.2012 9
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Objective Function Formulation Each floating unit has six degrees of freedom (DOF) which include surge, sway, yaw, roll, pitch and heave. The mooring system is only capable of controlling the surge, sway and yaw responses i.e. horizontal responses. To reduce roll, pitch and heave, i.e. vertical responses, the vessel shape and dimensions may be optimized.Sesam Users Conference – 03.Dec.2012 10
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Objective Function Formulation Here optimization of the mooring design, means to minimize the surge and sway responses. Surge and sway are platform longitudinal and transverse displacementsSesam Users Conference – 03.Dec.2012 11
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Objective Function Formulation Minimizing the horizontal translational response (Platform Offset) is our optimization problem objective function. The objective function of the mooring design optimization problem, the optimization parameter boundaries and the problem constraints could be defined as: m i =1 i 2 m [( Minimize : ∑ a .∆ i (α ) = ∑ a ∆x i (α ) + ∆y i (α ) i =1 i 2 2 )] α j min ≤ α j ≤ α j max , j = 1,...n Subjected to : g (α ) ≥ 1.67 , k = 1,... pSesam Users Conference – 03.Dec.2012 k 12
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Penalty Function Formulation The sequence presented in the previous slide has led to a constrained optimization problem which now must be solved The penalizing strategy is chosen to handle the constraints. So a constrained problem is transformed into an unconstrained problem by penalizing unfeasible solutions. The penalty function is described as below: Pi = 1 if g i, k (α ) ≥ 1.67 , i = 1...m , k = 1,..., p 0.5 ≤ P ≤ 0.9 if g < 1.67 , i = 1...m , k = 1,..., p i i, kSesam Users Conference – 03.Dec.2012 13
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Fitness Function Formulation Fitness is a quality value that is a measure of the reproducing efficiency of individuals in a population. A potential solution with a higher fitness value will have greater probability of being selected as a parent in the reproduction process. Therefore, the minimization problem must be transformed into a maximization problem of a fitness function, using the following expressions: ∆ 2 ϕ ϕi = 2 i Fi = 1 − i ϕ .Pi ∆ avg max Sesam Users Conference – 03.Dec.2012 14
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Genetic Algorithms As already mentioned above GAs differ from conventional optimization methods and search procedures in several fundamental ways. A GA’s basic execution cycle can be described by the following steps: Step 1: Reproduction Step 2: Recombination Step 3: Replacement If some convergence criteria is satisfied, Stop Otherwise, go to step 1Sesam Users Conference – 03.Dec.2012 15
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Implementation Details 1- Coding Design Variable Design variables were coded using a fixed-length binary-digit {0,1} string 2- Decoding To obtain the real values of the design variables in the domain region, each chromosome must be decoded 3- Offset Computation Dynamic analyses are carried out in the frequency domain using Mimosa 4- Fitness Function Calculation The fitness value of each chromosome is computed by considering offset values obtained from MimosaSesam Users Conference – 03.Dec.2012 16
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Implementation Details 5- Selection Chromosomes are selected as parents to produce children and this selection depends on fitness values 6- Crossover Operator The two-point crossover operator (2X), has been adopted herein, for example “000000” and “111111” make “001100” and “110011” 7- Mutation Operator This operator changes the bit from 1 to 0 or vice versa 8- Generation Gap It is a parameter that controls percentage of the population that will be replaced in each generationSesam Users Conference – 03.Dec.2012 17
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Computational Procedure 1- Start Initialize parameters: population size, crossover and mutation probabilities 2- Seeding Initial population is generated randomly Initial population is decoded Fitness value of each individual is computed by using the Mimosa software applying fitness equation 3- Reproduction Chromosomes are selected as parents Application of the crossover operator Application of the mutation operatorSesam Users Conference – 03.Dec.2012 18
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Computational Procedure 4- Updating The new offspring chromosomes substitute the worst chromosomes of the current population 5- Evaluation The new chromosomes are decoded Fitness of the new chromosomes is computed 6- Stopping Criterion Satisfied If so, then go to step 7; else, go back to step 3 (Here maximum number of 8000 iterations is considered 7- Repeat 8- EndSesam Users Conference – 03.Dec.2012 19
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Case Study The procedure described in the previous sections, has been implemented in a computer program (based on Matlab) which has been written to solve a mooring design optimization problem using Mimosa. As a case study, a floating unit anchored by 10 mooring lines, was considered.Sesam Users Conference – 03.Dec.2012 20
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Case Study The 10 lines were divided into 4 groups with side constraints, as given in the Table below: The floating unit is subjected to a set of environmental conditions that are combined according to a collinear approach, i.e. with currents, winds and waves acting simultaneously in the same direction.Sesam Users Conference – 03.Dec.2012 21
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Case Study In this case study, eight combinations have been considered. The JONSWAP spectrum for the Caspian Sea conditions was used to calculate wave HF responses, while the API spectrum was used for determining the time varying part of the wind forces.Sesam Users Conference – 03.Dec.2012 22
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Case Study The total number of iterations considered here was 8000 and the minimum value of the objective function was reached at the 43rd generation in offspring 4269. The next table presents the final results of the mooring design optimization problem including azimuths of each line, anchor position, line length, line size and line material. Also the platform heading is 180 deg. relative to true North.Sesam Users Conference – 03.Dec.2012 23
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Case StudySesam Users Conference – 03.Dec.2012
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Case Study The optimized mooring pattern is illustrated below. Line size and material in the ordinary case is 3.5” chain and 4” chain in the optimized design.Sesam Users Conference – 03.Dec.2012 25
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Case Study The responses are reduced by the optimized mooring design to up to 3.5 times less than in an ordinary design. This matter has a great effect on platform workability because of the reduction of down-time. Sesam Users Conference – 03.Dec.2012 26
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