Impact of LPSPs on Pareto Fronts

This case study has been conducted to investigate the impact of LPSPs on the Pareto fronts. In this case, the battery bank autonomy hour is considered twelve hours. The lower and upper boundaries of the decision variable vector, [Pw

rat P pv

rat Eratb Pbrat Sdirat Sconrat]T, for this case study

are taken [0, 0, 600, 50, 80, 0]Tand [900, 450, 3000, 250, 480, 480 ]T, respectively. Figure 5.5 shows three Pareto fronts for LPSP of 0%, 2%, and 4%. As can be seen in Figure 5.5, the more reliable the power supply system (i.e., low LPSP) is requested the more the Pareto front moves up due to the high value of LCC. At low REP, the Pareto fronts produced by numerous requested LPSPs are very near to each other due to the sharing operation of the components of the IMG and the sharing operation is performed among the DGS, RERs and BESS. However, Figure 5.5 demonstrates that the Pareto front solutions at high REP and at high LPSPs (i.e., at low reliability) stay at the bottom due to the low value of LCC. To achieve a high REP at low LPSP, more RERs and a large battery bank, which consequently moves up the LCC, are required. Figure 5.5 further demonstrates that the LCC increases sharply at above 80% of REP and for a low LPSP (e.g., LPSP=0%).

55 60 65 70 75 80 85 6.5 7 7.5 8 8.5 9x 10 6

L

C

C

($

)

γre(%)

118 Chapter 5. A Bi-objective Optimal Sizing Approach

5.5

Conclusions

This chapter has proposed a bi-objective optimization approach for generating Pareto fronts and Pareto optimal solutions of the IMG in a discrete decision variable environment. The mathe- matical model of the bi-objective optimization and the algorithmic flowchart of the approach have also been presented in this chapter. The results of the case studies have indicated that the approach is capable of generating many solution points on the Pareto optimal front. The more Pareto fronts indicate more options available for the decision maker. The comparisons between the traditional WS method and the AWS method have also been demonstrated in this chapter. It is observed that the AWS method has included many solution points that cannot be iden- tified by the WS method and the approach has identified almost evenly distributed solutions. One of the studies has indicated that the global optimum value of Chapter 3 coincides with a solution in the Pareto optimal front of the AWS method. This chapter has further investigated the sensitivity of battery bank autonomy hour and LPSP on Pareto optimal fronts. When two objective functions are involved, the proposed approach can easily be utilized for designing an IMG, along with, optimizing the PMSs based on a Pareto front set.

Chapter 6

A Multiobjective Optimal Sizing

Approach

6.1

Introduction

Two single objective optimal sizing approaches for IMGs were discussed in Chapter 3 and Chapter 4, respectively. Subsequently, one bi-objective optimal sizing approach was intro- duced in Chapter 5. Owing to increased social awareness, real-life engineering problems are encountering pressure to increase the number of objective functions in the optimization prob- lems. The criteria of objective functions include economic criteria (e.g., LCC), reliability (e.g., LPSP), environmental criteria (e.g., greenhouse gas emission/REP), and social criteria (e.g., so- cial acceptance) [85]. An intensive investigation of the IMG reveals that the optimization of the IMG is complex as the problem is composed of several competing objective functions, multiple variables, and a high degree of nonlinearity. When an optimization problem includes (a) multi- ple objectives, (b) multiple constraints, (c) multiple variables, and (d) a high degree of nonlin- earity, the MOP becomes challenging to solve [130]. Generally, the MOO approaches provide the best possible trade-offsolutions [131] for a decision maker (DM). The AWS method, which is employed in the bi-objective optimization of Chapter 5, is based on the WS method, a clas- sical technique that converts an MOP into an SOP for each weight. The AWS method imposes

120 Chapter 6. A Multiobjective Optimal Sizing Approach

constraints in smaller regions for further searching. In the AWS method, the computation of surfaces, i.e., the determination of Pareto patches, becomes difficult with the increased number of objective functions. Many studies [71], [72] support the idea that evolutionary algorithms (EAs) are effective for finding a global optimum solution and also for performing MOO re- gardless of the nature of the objective function, decision variable, modality, and constraint. Moreover, all the GAs have the ability to deal with non-convex optimization problems, non- differentiable functions, parallel functions, and noisy environments [131], [132]. This chapter proposes a fundamentally robust MOO approach for optimal sizing of the IMG. The proposed approach takes benefit of the non-dominated sorting genetic algorithm-II (NSGA-II) [133] method, as it provides less computational complexity than that of any other GAs. Additionally, the NSGA-II method not only extracts the better fitness of chromosome/ individual but also increases the diversity of the individual in the Pareto optimal set. The approach of this chap- ter systematically evolves efficient solutions and successively converges to the global Pareto optimal solution.

6.2

Problem Statement

This chapter has considered Figure 2.1 of Section 2.2 as the study system (i.e., the IMG) in order to determine the optimal sizes utilizing the MOO approach. The mathematical models of the subsystems of Figure 2.1 are taken from Section 3.2. Those models include the renew- able resources, WPS, PVS, BESS, active and reactive powers of primary load, dump load, and the BESS converter. In order to simultaneously optimize the sizes and PMSs of the IMG, the concept of collaborative optimization of Figure 6.1 is adopted for which one new gene (deci- sion variable) is introduced in each individual (decision variable vector) to represent the PMSs. The value of the new gene determines the PMS and thus only one PMS is required to be em- ployed for simulating the IMG. This scheme is implemented incorporating a few conditional (IF-ELSE) statements in the pseudo code of the algorithm. The MOO is generally utilized for determining a set of trade-off solutions when the optimization problem contains multiple

6.2. Problem Statement 121

IMG sizing and PMSs optimizer optimizer upon IMG simulation using PMS-A system of PMS-A analysis system of PMS-B analysis system of PMS-D analysis …. optimizer upon IMG simulation using PMS-B optimizer upon IMG simulation using PMS-D

Figure 6.1: Simultaneous optimization of PMSs and sizes

objective functions that usually conflict with each other. The multiobjective optimization prob- lem (MOP) of the study system is formulated considering the following objective functions. The three equations that are described in (3.24), (3.25), and (3.43) are taken as three objective functions, expressed as,

f1(X)=−γre, f2(X)=LCC, f3(X)= LPS P (6.1)

whereXis an individual/decision variable vector in a population set. The details of the decision variable vector and the MOO formulation are provided in the following subsection.