generalized normal distribution optimization and its applications in parameter extraction of photovoltaic models

Our research contributes to advancing the understanding of evolutionary optimization algorithms and provides valuable guidance for algorithm selection and parameter tuning in practical a

VIETNAM NATIONAL UNIVERSITY — HOCHIMINH CITY

INTERNATIONAL UNIVERSITY DEPARTMENT OF INDUSTRIAL ENGINEERING &

MANAGEMENT

GENERALIZED NORMAL DISTRIBUTION OPTIMIZATION AND ITS APPLICATIONS IN PARAMETER EXTRACTION OF

PHOTOVOLTAIC MODELS

INVENTORY MANAGEMENT COURSE

Lecturer: PhD Dao Vu Truong Son

Group Number: 11 Class: G02_Sat_Mor_456

IELSIUxxxx Tran Thi Anh Thu 100%%

February/2024

ABSTRACT Optimization algorithms based on evolutionary principles play a crucial role in solving complex problems across various domains The stochastic nature of these algorithms necessitates thorough evaluation to ascertain their effectiveness and reliability Central to this evaluation is the selection of appropriate test cases, which should represent the challenges encountered in real-world applications However, the absence of standardized guidelines for benchmark selection presents a significant challenge in algorithmic evaluation

In this study, we present a comprehensive framework for assessing the performance of evolutionary optimization algorithms Our framework addresses the need for a diverse set

of test functions, encompassing various characteristics and complexities commonly encountered in optimization problems By incorporating a wide range of test families, including unimodal and multi-modal functions, we aim to provide a holistic assessment of algorithmic capabilities in exploring and exploiting search spaces efficiently

Through meticulous experimentation and analysis, we evaluate the algorithms’ performance across multiple dimensions, including convergence speed, solution quality, and robustness Furthermore, we investigate the algorithms’ sensitivity to different problem types and parameter settings to gain deeper insights into their behavior Our research contributes to advancing the understanding of evolutionary optimization algorithms and provides valuable guidance for algorithm selection and parameter tuning in practical applications By establishing a standardized evaluation framework, we aim to foster transparency, reproducibility, and comparability in the assessment of optimization algorithms, ultimately facilitating their adoption in solving real-world optimization problems

H

CHAPTER 1 INTRODUCTION

L1 Problem Statement

The field of optimization using evolutionary algorithms is constantly evolving, with new algorithms emerging and vying for superiority However, assessing their performance remains a challenge due to their stochastic nature While testing them on every possible instance is impractical, a well-defined set of benchmarks is crucial for reliable comparisons

This project aims to contribute to the continuous search for superior approaches by benchmarking several promising optimization algorithms against an established one - Particle Swarm Optimization (PSO) and Gaussian Noise Differential Opposition (GNDO) We will utilize a diverse set of test functions encompassing both unimodal (single optimum) and multimodal (multiple optimum) landscapes

Our key objectives encompass three areas: performance comparison, algorithm improvement, and real-world application

Firstly, we will comprehensively compare several proposed algorithms against a well- established benchmark — Particle Swarm Optimization (PSO) Utilizing the diverse test functions, we will analyze their convergence speed, global optimum achievement, and local optima avoidance capabilities, providing valuable insights into their relative strengths and weaknesses

Secondly, we will embark on an iterative improvement process for the proposed algorithms By exploring and implementing potential modifications, we will meticulously evaluate each enhancement through comprehensive re-benchmarking on the entire suite of test functions This process aims to continuously refine and elevate the performance of the algorithms

Thirdly, to demonstrate the practical value of the best-performing improved algorithm,

we will apply it to tackle a chosen real-world optimization problem This final stage will showcase the algorithm's ability to handle practical challenges and its potential for real-world application

Through meticulous execution, rigorous analysis, and insightful comparison, this project aims to achieve several expected outcomes: an in-depth comparative analysis of

the algorithms, identification and implementation of effective improvements, and successful application to a real-world problem Ultimately, this project's findings aspire

to guide future advancements in this vital field by informing decision-making and facilitating further development of superior optimization algorithms

1.2 Scope and Limitations

This project evaluates and improves evolutionary optimization algorithms, with a focus

on Particle Swarm Optimization (PSO) and Gaussian Noise Differential Opposition (GNDO), using 13 different test functions We evaluate key performance metrics such

as convergence speed and global optimal achievement before exploring and implementing potential algorithm improvements, which we then evaluate using re- benchmarking However, the test procedure has some limitations For example, not having access to copyrighted Matlab influences the run time and speed of each test Furthermore, while constraining limitations such as the choice of test functions not being exhaustive and parameter tuning focusing on standard settings, the best- performing improved algorithm will be applied to a selected real-world problem to demonstrate its practical value Recognizing these limitations motivates future research

to investigate broader test functions, delve deeper into improvements, and apply findings to a wide range of real-world scenarios, ultimately contributing to the ongoing advancement of evolutionary optimization algorithms and their applications

CHAPTER 2

2.1 Mathematical model fornutlation

METHODOLOGY

2.1.1 Notations

Nomenclature i the lower boundary of variables

“ the upper boundary of variables Ion photo generated current (A) Trax the maximum number of iterations

Rsh shunt resistance (2) t the current number of iterations

R Series resistance (@) NFEs the number of function evaluations

ty output current (A) 4,.42,a,b,8,45 random number between 0 and 1

I4 diode current (A) Asha random number subject to standard normal distribution

la reverse saturation current (A) pl,p2,p3 random integers between 1 and N

Vo output voltage (V) Xhet the position of the optimal individual at time r

q electronic charge (1.60217646 x 107!°C) x the position of the ith individual at time ¢

n,m,na diode ideality factor vt the trail vector of the ith individual at time cell temperature in kelvin x Population at time ¢

k Boltzmann constant (1.3806503 x 10°*9J/k) y trial individual

Ton shunt resistance current (A) RMSE root mean square error

lại the first điode current (A) ụ mean value of normal distribution

Tag the second diode current (A) 6 standard variance of normal distribution

Tear diffusion current (A) M Mean position of the current population

Tea2 saturation current (A) y; and v2 trail vectors

Np the number of solar cells in parallel 0 penalty factor

Ne the number of solar cells in series cs cuckoo search

m the number of experimental data NNA neural network algorithm

Ip Benchmark current (A) GWO grey wolf optimizer

Vg Benchmark voltage (V) WOA whale optimization algorithm

Py Benchmark power (W) SCA sine cosine algorithm

lay absolute error of current (A) SSA slap swarm algorithm

Is simulation current (A) BSA backtracking search algorithm

Ps simulation power (w) TLBO Teaching-learning-based optimization

Par absolute error of power (w) GNDO generalized normal distribution optimization

N population size

D the number of variables

In the field of optimization with evolutionary algorithms, several test cases should be used to ensure that an algorithm's performance is not random In this project, we also use several test functions with different characteristics based on a diverse set of 13 test functions commonly used in evolutionary algorithm benchmarking, which are classified as unimodal and multimodal:

Function Dimension Range finin Unimodal

F,(x) = max,{|x,|,1 < í < n} 30 {-100 100] 0

F;(x) = Ef} [100(x; — x?)Ï + (xị — 1)? 30 [-30.30] 0 Fe(x) = 7, ([x; + 0.5])? 30 [-100,100] 0

F,(x) = D2, ix} + random{0,1] 30 [-1.28,1.28] 0 Multimodal

—20 exp (~o2 fxn 1 +) —exp (227.,cos (2mx) ) +20+e

=(10 sin sin (Ty;) + #J=‡Œ¡ — 1)? [1 + 10sin?(3xy,„:)] +

Om — 1)? + SL, u(y, 10,100,4)

x+1

y= t+

u(x;,a,m) = {k(x; — a)™ 0 k(-x; —a)™ x, >a -a<xj<

ax,;<-a

F,3(x) = 0.1{(sin?(3mxị) + D/L, — 1)? [1 + sin? (3x; + 1)) + 30 [-50.50] 0 (x, — 1)2[1 + sin?(2mx„)]) + 7s; u(x¿,5,100,4)

Figure: Unimodal Benchmark Functions

2.2 Algorithm

2.2.1 Particle Swarm Optimization (PSO) Implementation

Particle Swarm Optimization (PSO) is a powerful and popular optimization algorithm inspired by the collective behavior of natural organisms like bird flocks or fish schools

It mimics how these creatures move around searching for food, adapting their positions based on their own individual experience and the knowledge of their neighbors Standard PSO with velocity clamping will be implemented

Parameters:

Number of particles = 40

Maximum iterations = 5000

Cognitive coefficient (cl) = 2

Social coefficient (c2) = 2

Inertia weight (w) linearly decreasing from 0.9 to 0.2

Applying the standard Matlab Code for the test functions:

% Particle Swarm Optimization

function [Y, X,cg_curve]=PSO(N,Max_iteration,lb,ub,dim, fhandle,fnonlin)

%PSO Information

Vmax=6;

noP=N;

wMax=0.9;

wMin=0.2;

cl=2;

c2=2;

% Initializations

fork=1:N

end

2.2.2 GNDO Implementation

Gaussian Noise Differential Opposition (GNDO) is a population-based evolutionary algorithm inspired by Differential Opposition (DO) and incorporating Gaussian noise for enhanced exploration and diversity

Parameters:

Number of populations = 40

Maximum iterations = 5000

Scaling factor (F) = 0.5

Crossover rate (CR) = 0.9

Proposed Information Sharing Strategies Between Solutions

Ce ESE lay

CE no)

i

COO}

Ou ey

"

t=t+1

Fig 3 The framework of the proposed GNDO.

Input: population size NV , the upper limits of variables u, the lower limits of variables / , the current number of iterations ¢ = 0 and the maximum number of iterations 7

* Initialization

Ol: Initialize population by Eq (26)

02: Calculate the fitness value of every individual and achieve the optimal solution x

03: Update the current number of iterations ¢ by =1+1

04: whilers 7 do

05 fori=1:N

06 Generate a random number @ between 0 and |

07 if a>05

* Local exploitation strategy *

08 Select the current optimal solution x„ and calculate the mean position Ä# by Eq (22)

09 Compute generalized mean position /, generalized standard variance d and penalty factor by Eq (19), Eq (20) and Eq (21), respectively

10 Perform the local exploitation strategy by Eq (18) and Eq (27)

II else

* Global exploration strategy *

12 Perform the global exploration strategy by Eq (23), Eq 24), Eq (25) and Eq (27)

13 end if

14 end for

15 Update the current number of iterations ¢ by f=*+

16: end while

Output: the optimal solution x,

2.2.3 Benchmarking Procedure

- Individual Test Function Runs: Both PSO and GNDO will be executed 20 times for each test function and 5000 iterations each

- Performance Metrics: For each run, record:

e Best fitness value achieved

e Average fitness value across iterations

e Number of iterations required to reach convergence

e Standard deviation of fitness values

- Data Storage: Results for each test function and algorithm will be saved in separate spreadsheets for analysis

- Propose and implement algorithmic improvements, re-evaluating them through the benchmarking process

- Apply the best-performing improved algorithm to a chosen real-world optimization problem

This methodology outlines a rigorous and transparent approach to benchmarking, refining, and applying both PSO and GNDO algorithms By comparing their performance and exploring potential improvements, the project aims to gain valuable insights for future optimization algorithm development

CHAPTER 3 RESULTS

3.1 Results presentation

In this section, we embark on an extensive comparative investigation to ascertain the efficacy and prowess of our newly introduced Generalized Normal Distribution Optimization (GNDO) algorithm Our primary objective is to evaluate its performance against a formidable lineup comprising ten established algorithms, all vying for supremacy in the realm of parameter extraction within photovoltaic (PV) models Specifically, we target three pivotal PV models: the single diode model, the double diode model, and the PV module model

To rigorously assess the competitive edge of GNDO, we subject it to a head-to-head confrontation with a diverse array of metaheuristic methodologies This illustrious lineup includes renowned algorithms such as Particle Swarm Optimization (PSO), Teaching-Leaming-Based Optimization (TLBO), Cuckoo Search (CS), Jaya Algorithm (JAYA), Grey Wolf Optimizer (GWO), Salp Swarm Algorithm (SSA), Sine Cosine Algorithm (SCA), Whale Optimization Algorithm (WOA), Neural Network Algorithm (NNA), and Backtracking Search Algorithm (BSA)

Our comparative analysis delineates these algorithms into two distinct cohorts: those endowed with specialized parameters (e.g., PSO, CS, GWO, SSA, SCA, WOA, and BSA) and those operating parameter-free (e.g., JAYA, NNA, and TLBO) To ensure equitable grounds for comparison, we meticulously standardize the population size across all algorithms to 50 individuals Furthermore, we enforce a uniform termination criterion based on the maximum number of function evaluations, setting it at 35,000, 45,000, and 35,000 for the single diode, double diode, and PV module models, respectively Additionally, we diligently source control parameters for the compared algorithms from reputable references, ensuring methodological consistency

Our study heralds the advent of GNDO, an innovative metaheuristic algorithm ingeniously crafted upon the bedrock of normal distribution theory One of the hallmark attributes distinguishing GNDO from its counterparts lies in its innate capacity to operate seamlessly sans the need for intricate fine-tuning of initial parameters Through a series of meticulously designed experiments, we put GNDO through its paces, tasking it with the formidable challenge of parameter extraction across the three distinct PV models The empirical findings gleaned from our exhaustive analyses unequivocally underscore GNDO's ascendancy, boasting superior efficiency and precision when juxtaposed against its peers

Crucially, the overarching objective of parameter extraction within PV models lies in its pivotal role in enhancing the optimization and control of practical PV systems In light of our empirical revelations, solutions derived using GNDO emerge as not merely competitive but as unequivocally superior options for real-world deployment The inherent simplicity and efficiency inherent in GNDO, coupled with its remarkable parameter-independence, render it eminently suited for a diverse array of optimization challenges across various domains Looking ahead, our future research endeavors will center on expanding the horizons of GNDO's applicability, encompassing diverse

energy optimization realms such as economic load dispatch for power systems and wind farm layout optimization Furthermore, we are committed to exploring avenues for developing multi-objective and binary variants of the algorithm, thereby augmenting its versatility and utility in addressing multifaceted optimization

challenges

Table 1

The statistical results obtained by the applied algorithms for single diode model

Algorithm Worst Mean Median Best Sed

ca 1.6549E -03 1.2932 - 03 1.2785E-03 1.0768E-03 1.4742E-04

to 1749603 — 1agnE-0$ 12690804

GNo 45I3đ8-02 6aza0m-0 sơse-09 148378-08

NNA 4.8836E- 2.1810E-03 22472E-03 1.0406E

SSA 1.0339E-02 2.7847E 03 NE 9.9098E

SCA 5.4111E-02 4.2809E-02 4.3391E-02 2.1923E-02

WOA 5.0973E-02 1.4733E -02 55432E 03 1.2340E-03

SA 1.58088 -0: 1.4146 - 03 1.4246E - 03 1.1263E-03

JAYA 1.5832E-03 1.3976E -0° 1 4E—' 1.1489E-03

TLBO 1.5367E-03 1.1692E-03 1.1034E-03 9.9267E-04

GNDO 9.8602E-04 J2E-04 9 9.8 04

Table 2

The best solutions obtained bythe applied algorithms for single diode model

Algorithm: Ign (A) g(a) R,(0) Rạ(Q) n RMSE

cs 0.76071 0.38612 0.03581 62 44536 1.49931 1.0768F 03 PsO 0.76078 0.31906 0.03643 40907 1.47994 9.8630E-04 GWO 0.76116 0.41861 0.09541 57.13802 1.1597E-093 NNA 0.76085 0.03568 S6.42245 1.0406E-03 SSA 0.76078 0.03617 54.70626 9098E — SCA 0.77814 0.03084 100.00000 21923E-02

WOA BẠA 076001 076098 008669 00868 6231742 67 ass68 12340603 11269803

JAYA 0.76075 0.03586 5S8.20751 1.1489E -03 TLRO 0.76073 0.03625 55.36187 99267E-04 GNDO 0.76078 0.03638 53.71852 1.48118

7

results obtained by the applied algorithms for double diode model

Worst Mean Best Std

cs 7E 1.7887E-03 1.1714E-03 3.1573E-04

mo 170865-08 11890803 886g9-04 2.586

Gwo 226E-02 6.4731E-03 1.0038E-03 1.0229E-02 NNA 3.83528 03 2.4791E 03 1.1294F 03 6.1567E (04 SSA 9.73568 03 2.93! 03 1768 1.72038 SCA 2287E-(01 4.5453E-02 1.3745E-02 3.4900E-02 WOA 4.9473F - (2 1.2591E-02 1.1046E -03 1.4568E -02 BSA 3961E-03 1.4923E-03 1/0799E-03 2.8213E-04 JAYA 23529E-03 1.5190E-03 1.1398E-03 2.4882E—04 TLBO 1.7247E-03 1.1536E-03 9.8406E~\ 1.7644E-~\ GNDO 9.8398E 04 9.8248E 04 1.4417E 06

Table 4

The best solutions obtained by the applied algorithms for double diode model

Algorithm Tạ(A) lại (MÀ) R(0) Ruy (2) m mm nạ RMSE

cs 0.76040 044135 4937162 1.92163 0.21049 1.4431 1.1714E-03 PSO ) 76079 0.88429 56.02421 1.89986 0.16267 1.42552 9.8563E-04 GWO 0.76108 011171 5297190 1.41428 1.61847 1.0038E-03 NNA 0.76067 0.56109 65.08110 1.55541 1.00007 1.1294E - 03 SSA 0.76031 0.40553 3694 239025 1.68997 1.42139 1.0476E-03 SCA 0.77850 0.97412 0.03278 67 93661 1.59976 1.3745E-02 WOA 0.76147 0.13062 0.03588 51.15127 1.81450 0.32 1.48583 1.1046E-03 BSA 0.76034 0.36483 03608 63.56984 1.96800 0.31589 1.48189 1.07998 03 JAYA 0.76035 0.00000 4 60.30758 1.86654 0.29106 1.47064 1.1398E-03 TLhO 76080 039780 003659 5431022 2.00000 0.26775 1.46517 9.8406E 04 GNDO 0.76078 0.74935 0.03674 55 2.00000 0.22597 1.45102 9.8248E-04

Table 5

The statistical results obtained bythe applied algorithms for PV module model

Algorithm Worst Mean Median Best Std

cs 255775F-03 2.49100F -03 2.489208 03 243826F 03 3.3708E 05 PsO 3.46542F -03 2.59052E-03 254424F-03 2 42508F-03 223891E-04 Gwo 6.79852E—03 3.45865E-03 2.76756E-03 1.17692E-03 NNA 9.93471E-03 2.8078&8E—03 2.58140E-03 1.34914E-03 SSA 7.13378E 03 3.06402E - 03 2.70224E-03 9.79741E-04

&CA 27435&F 01 1.64222F -01 1.83550E -Ø1 1.13594E -01

WOA 2755298-0I 2.062E4ã-02 577256803 115749601

SA 280815609 3⁄4806-09 Hay 341871E-g6

JAYA 260715E-03 2.4 1E-03 2.47649E-03 4.47505E-05

7120 2894788-09 2/40127-03 4.as-ta 2240198-06

GNDO 2.42507E 03 2.42507E 03 32.42507E

3.2 Compdrisons

The compared algorithms can be divided into the following two categories: (1) metaheuristic algorithms with special parameters (1.e PSO, CS, GWO, SSA, SCA, WOA and BSA); (2) metaheuristic algorithms without special parameters (i.e JAY A,