Optimal Design of Multi-stage Flash (MSF) Seawater Desalination Processes
2.7 Simulation methodology
Overall solution strategy 2.7.1
The overall solution methodology for the optimization study is presented in Fig. 2.5. As shown, the methodology involves the regulation of MSF simulation model using input, parameter and other design specifications (in the form model equations) and optimization algorithm using its algorithm parameters. Based on in the input, design and other specifications, the MSF model would solve and obtain the dependent variables and cost function values. Eventually, the optimization algorithm would facilitate the identification of the best combinations of independent variables for the chosen MSF process model. While deterministic optimization algorithms have relatively lesser degrees of freedom to explore upon their performance, the performance of evolutionary optimization algorithms can be improved by targeting the optimal optimization algorithm parameters. Further details with respect to these will be elaborated in the later sections of the chapter.
Further, it shall be noted that in addition to the Differential Evolution (DE) algorithm, the Ph.D. thesis attempts to address the competence of other evolutionary and deterministic algorithms. In this regard, MATLAB built in functions and tools for deterministic and non- deterministic optimization were explored. Thus, the deterministic optimization studies were targeted using sequential quadratic programming (SQP), multi-start based SQP (MS-SQP).
Further, non-deterministic optimization was targeted using MATLAB based Genetic Algorithm (GA) and Simulated Annealing (SA) algorithms. Further DE-SQP optimization approach was also targeted where the best solution(s) obtained from DE were evaluated further using SQP method to judge upon the possibility to further improve the obtained solution locally. In this regard, it can be noted that the CPU time increases significantly for
Optimal Design of MSF Desalination Processes
DE-SQP optimization approach but solutions quality can be increased to obtain the best optimal solution.
The primary objective of the thesis is to take up various alternate optimization techniques for MSF plant optimization. This is due to the following reasons:
a) Deterministic techniques (SQP and MS-SQP) may provide global optimal solutions but confidence levels could not be ensured upon their solution quality. However, the same can be defined better using the solutions obtained from non-deterministic optimization techniques, where in the standard deviation of the optimal solutions can be evaluated to infer upon the quality of the obtained solutions.
b) The efficacy of DE with respect to other deterministic (SQP and MS-SQP) and non- deterministic (GA and SA) optimization methods be explored for the optimality of MSF plants.
c) The optimality of various MSF plant configurations (MSF-M, MSF-OT and MSF- BR) be explored from the perspective of global optimization techniques.
d) Quality of the solutions obtained from DE could be judged based on the results reported in the literature and results obtained with other techniques.
e) The necessity to explore DE-SQP optimization methodology could be judged in improving and obtaining the best solutions close to the global optimal domain.
Chapter 2
Figure 2.5: Overall solution methodology for MSF process optimization.
Algorithm and solution Hierarchy 2.7.2
An insight into the implementation issues related to MSF simulation model can be obtained from Fig. 2.6 in which the solution hierarchy of the MSF plant model has been presented. As shown, the solution methodology does involve iterative approaches for the evaluation of dependent variables as functions of independent variables and system specifications. Further insights with respect to the optimization methodology are presented in the next sub-section.
Input Design and problem Specifications
MSF Simulation
model Optimization model
Optimization Parameters
Optimal solution
Optimal Design of MSF Desalination Processes
Figure 2.6: Schematic of MSF process simulation model.
Start
Define Input variables (Table 2.1)
Initialize: independent variables Assume initial values for
Adjustable input: Unknowns:
Calculate
from equation (2.3) from equation (2.82) from equation (2.65) Calculate
from equation (2.73) from equation (2.81) from equation (2.5)
Calculate
from equation (2.66) from equation (2.68) from equation (2.69)
Calculate
from equation (2.76) from equation (2.70) from equation (2.72)
Calculate
from equation (2.71) from equation (2.79) from equation (2.16) Calculate
from equation (2.78) from equation (2.16) from equation (2.80) from equation (2.16)
Calculate Cost
from equation (2.83) from equation (2.25) from equation (2.26)
Calculate
from equation (2.27) from equation (2.33)
from equation (2.34) from equation (2.35)
Stop Evaluate penalties using inequality constraints (Table 2.4)
Objective function from equation (2.92)
Chapter 2
Figure 2.7: DE algorithm for the optimal design of MSF desalination process.
Optimization methods 2.7.3
As discussed previously, optimization methodology involves applying either one of the following: DE, GA, SA, SQP, MS-SQP, DE-SQP and GA-SQP. In this sub-section, optimization methodology has been presented for all cases.
Set NP, NG
IGEN=1
Is IGEN > NG Solve MSF Model (Fig. 2.6) Evaluate cost function and total penalties
Apply DE Optimization algorithm
For IPOP, Generate independent variables for next generation
No
Random Initialization of independent variables
IGEN = IGEN + 1
Yes
Terminate and analyze results for penalties (if any)
For IPOP = 1….NP
Set DE algorithm parameters
(F, CR)
Optimal Design of MSF Desalination Processes 2.7.3.1 Differential Evolution (DE) Algorithm
Introduced by Storn and Price (1995), the DE optimization algorithm is a stochastic population based direct search optimization method. Among various evolutionary algorithms that target non-linear and non-differentiable optimization problems, DE is one among the best known techniques. The basic principle of the DE algorithm refers to the creation of new candidate solutions by combining the parent individual and several other individuals of the same population. This is facilitated by adding the weighted difference between any two population vectors to a third population vector (Shaheen et al. 2011). Further, the parent vector is replaced with the mutant vector only when the mutant vector provides better fitness value (Ramirez et al. 2011). Thus, DE is an effective, fast, simple, robust, inherently parallel technique that has few control parameters and needs little tuning. DE has been evaluated to be effective to handle noisy, flat, multidimensional, and time-dependent objective functions and constraint optimization problems by adopting penalty function method (Shaheen et al.
2011).
Fig. 2.7 presents the generalized DE algorithm for the design of MSF desalination plant. For a chosen set of population size, the design variables have been randomly initialized as a population of vectors. Eventually, MSF model was solved using the model described previously in the chapter to obtain dependent variables, cost function with penalties.
Eventually for the vectors, DE algorithm is applied as a sequence of mutation, cross-over and selection operations for all populations (Fig. 2.9). The termination criteria that was adopted in this work refers to the achievement of maximum number of permitted generations. Further details with respect to mutation, cross-over and selection operations of the DE algorithm are presented as follows:
Chapter 2 Step I. Initialize cost data, and DE related parameters such as the size of population (NP), the maximum number of iterations or generations (Gmax), the number of variables (D) to be optimized, cross over Ratio (CR) and mutation factor (F).