# Enhanced Evolutionary Sizing Algorithms for Optimal Sizing of a Stand-Alone PV-WT-Battery Hybrid System

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## Abstract

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## 1. Introduction

- A model based on HRESs is proposed, where a PV-WT-battery system and its components are formulated and elaborated.
- Meta-heuristic algorithms: teaching-learning based optimization (TLBO), enhanced differential evolution (EDE), and salp swarm algorithm (SSA) are firstly used to find the optimal number of HRESs with an objective function to minimize the user’s TAC in an SA environment.
- Two novel hybrid approaches based on combining (TLBO + EDE and TLBO + SSA) are also proposed for the better exploitation of the search space. These hybrid approaches are called enhanced evolutionary sizing algorithms (EESAs).
- The results obtained by EESAs are compared to their ancestor schemes in three different scenarios: PV-WT-battery, PV-battery, and WT-battery systems for a yearly user’s load profile. Further, the real solar irradiation and wind speed data are used, which are obtained from Rafsanjan, Iran.
- The reliability of HRESs is considered using four maximum allowable LPSP ($LPS{P}^{max}$) values: 0%, 0.5%, 1%, and 3%, which are provided by the user. The TACs at different $LPS{P}^{max}$ values are presented and analyzed.

## 2. System Model

## 3. Formulation of HRESs

#### 3.1. Formulation of the PV System

#### 3.2. Formulation of the WT System

#### 3.3. Formulation of User’s Load

#### 3.4. Excess and Deficit Cases of HRESs and Sizing of the Batteries

#### 3.5. Formulation of the System’s Reliability

#### 3.6. Total Annual Cost Modeling and Constraints

#### 3.6.1. Objective Function Formulation

#### 3.6.2. Constraints

## 4. Proposed Algorithms for the Unit Sizing Problem

#### 4.1. TLBO

#### 4.2. EDE

#### 4.3. SSA

#### 4.4. EESAs

- (i)
- The first step includes initialization of parameters: hourly input solar irradiation, ambient temperature, the speed of the wind, and user’s load profile data.
- (ii)
- (iii)
- In the third step, a solution space of two decision variables is randomly generated within the upper and lower bounds as given below.$$Ge{n}^{i}=\left[\begin{array}{cc}{s}_{1}^{1}& {s}_{2}^{1}\\ {s}_{1}^{2}& {s}_{2}^{2}\\ \vdots & \vdots \\ {s}_{1}^{j}& {s}_{2}^{j}\end{array}\right]$$In Equation (34), the first and second columns are associated with the number of PVs and WTs, respectively.
- (iv)
- In this step, based on the RESs’ generation and user’s load, the total number of batteries required for each j solution is calculated using Equation (8). Thus, the cluster of configurations showing the solution space is depicted as:$$Ge{n}^{i}=\left[\begin{array}{ccc}{s}_{1}^{1}& {s}_{2}^{1}& {s}_{3}^{1}\\ {s}_{1}^{2}& {s}_{2}^{2}& {s}_{3}^{2}\\ \vdots & \vdots & \vdots \\ {s}_{1}^{j}& {s}_{2}^{j}& {s}_{3}^{j}\end{array}\right]=\left[\begin{array}{c}{S}^{1}\\ {S}^{2}\\ \vdots \\ {S}^{j}\end{array}\right].$$Here, the third column shows the total number of batteries ${N}^{b}$ in the battery bank. In Equation (35), for the ${i}^{\mathrm{th}}$ population generation, ${S}^{i}$ represents j possible configurations. Each configuration represents a possible solution competing to fulfill the EESA objective.
- (v)
- (vi)
- Here, each configuration is evaluated using fitness criteria as depicted in Equation (10).$$F\left(Ge{n}^{i}\right)=F\left[\begin{array}{c}{S}^{1}\\ {S}^{2}\\ \vdots \\ {S}^{j}\end{array}\right]=\left[\begin{array}{c}F\left({S}^{1}\right)\\ F\left({S}^{2}\right)\\ \vdots \\ F\left({S}^{j}\right)\end{array}\right].$$The fitness value of each configuration $F\left({S}^{j}\right)$ shows the respective TAC, which is obtained by the summation of capital and maintenance costs.
- (vii)
- Here, TLBO steps are applied to update the population. First, the mean M of the learners is calculated subject wise. The best learner based on $F\left({S}^{j}\right)$ is chosen as a teacher. The mean of learners is shifted toward the teacher via Equation (23). In the learner phase, the population is updated using Equation (25). The new solution is accepted only if it gives a better TAC value. The new population is called ${X}_{new}$.
- (viii)
- (ix)
- Steps (iv)–(viii) are repeated by the EESA process until the global termination criterion of 100 generations is satisfied.
- (x)
- Lastly, the global best solution among 100 generations based on the TAC value is returned. The global best solution contains the respective number of ${N}^{pv}$, ${N}^{wt}$, ${N}^{b}$, TAC, and LPSP values.

## 5. Results and Discussion

- (i)
- PV-WT-battery: ($S=\left[{s}_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$),
- (ii)
- PV-battery: ($S=\left[{s}_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$), and
- (iii)
- WT-battery: ($S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$).

#### 5.1. Scenario 1: PV-WT-Battery Hybrid System ($S=\left[{s}_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$)

- $S=\left[111\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}17\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1753\right]$ at $LPS{P}^{max}=0\%$,
- $S=\left[117\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}15\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1685\right]$ at $LPS{P}^{max}=0.5\%$,
- $S=\left[127\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1612\right]$ at $LPS{P}^{max}=1\%$, and
- $S=\left[126\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1458\right]$ at $LPS{P}^{max}=3\%$.

#### 5.2. Scenario 2: PV-Battery System ($S=\left[{s}_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$)

- $S=\left[199\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3150\right]$ at $LPS{P}^{max}=0\%$,
- $S=\left[194\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2898\right]$ at $LPS{P}^{max}=0.5\%$,
- $S=\left[191\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2746\right]$ at $LPS{P}^{max}=1\%$, and
- $S=\left[178\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2090\right]$ at $LPS{P}^{max}=3\%$.

#### 5.3. Scenario 3: WT-Battery System ($S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{3}\right]$)

- $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}50\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3552\right]$ at $LPS{P}^{max}=0\%$,
- $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}50\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3552\right]$ at $LPS{P}^{max}=0.5\%$,
- $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}49\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3362\right]$ at $LPS{P}^{max}=1\%$, and
- $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}49\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3362\right]$ at $LPS{P}^{max}=3\%$.

#### 5.4. Convergence Process of the EESA Algorithm for Three Scenarios

## 6. Conclusions and Future Work

- (i)
- Hybrid EESAs were developed for better exploration and exploitation of the search space. EESAs accepted inputs, including solar irradiation, ambient temperature, wind speed, and user’s load data.
- (ii)
- The PV-WT-battery hybrid system was found with the best optimal configuration of RESs with reduced TACs as compared to the PV-battery and WT-battery systems. The TACs achieved by EESA were $64,430, $61,970, $59,200, and $54,171 at $LPS{P}^{max}$ values of 0%, 0.5%, 1%, and 3%, respectively. In the PV-WT-battery hybrid system, the algorithms EESA, TLBO, EDE, and SSA were ranked as 1
^{st}, 2^{nd}, 3^{rd}, and 4^{th}, respectively, based on their average TAC values. EESA performed better than other algorithms due to the better search on more promising areas of the solution space. On the other hand, TLBO’s performance was found better compared to the EDE and SSA schemes because it neither required any algorithm specific parameter, nor its calibration to obtain the optimal results. - (iii)
- The PV-battery system provided the second most economical results. The TACs achieved were $104,640, $96,840, $92,120, and $71,790 at $LPS{P}^{max}$ values of 0%, 0.5%, 1%, and 3%, respectively. In this scenario, EESA and TLBO performed equally and were placed in the first category. EDE and SSA achieved second and third rankings based on their average TACs, respectively.
- (iv)
- The third scenario: The WT-battery system was the most expensive case, due to the high price of WTs. The TACs $124,750 and $118,690 were achieved by EESA at $LPS{P}^{max}$ values of (0%, 0.5%) and (1%, 3%), respectively. Here, all algorithms achieved the same optimal results due to a fewer number of decision variables compared to the PV-WT-battery hybrid system, thus being ranked equally.
- (v)
- The trade-off analysis between TAC and $LPS{P}^{max}$ was also evaluated. It was found that when the $LPS{P}^{max}$ values were increased from 0%, TACs were minimized and vice versa.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CRF | Capital recovery factor | ||

DE | Differential evolution | ||

DoD | Depth of discharge | ||

EESAs | Enhanced evolutionary sizing algorithms | ||

EDE | Enhanced differential evolution | ||

ESSs | Energy storage systems | ||

FF | Fossil fuel | ||

GA | Genetic algorithm | ||

HOMER | Hybrid optimization model for electric renewables | ||

HRESs | Hybrid renewable energy sources | ||

HT | Hybrid technique | ||

LPSP | Loss of power supply probability | ||

NSGA-II | Non-dominated sorting genetic algorithm II | ||

PSO | Particle swarm optimization | ||

PV | Photovoltaic | ||

RESs | Renewable energy sources | ||

SA | Stand-alone | ||

SSA | Salp swarm algorithm | ||

SoC | State of charge | ||

TAC | Total annual cost | ||

TLBO | Teaching-learning based optimization | ||

TS | Tabu search | ||

WT | Wind turbine | ||

Acronyms | |||

${\zeta}^{tot}$ | Total cost | ${\zeta}^{cap}$ | Capital cost |

${\zeta}^{mtn}$ | Maintenance cost | ${\zeta}^{bat}$ | Present battery worth |

${\zeta}^{ic}$ | Present worth of the inverter/converter | ${\zeta}^{wt}$ | Unit cost of WT |

${\zeta}^{pv}$ | Unit cost of the PV panel | ${\zeta}^{bat}$ | Unit cost of the battery unit |

${\zeta}^{ic}$ | Unit cost of the inverter/converter | ${\zeta}_{mtn}^{pv}$ | Annual maintenance costs of PV panels |

${\zeta}_{mtn}^{wt}$ | Annual maintenance costs of WTs | ${\xi}_{tot}^{pv}$ | Total PV generated power |

${\xi}^{wt}$ | Overall produced wind power | ${\xi}^{ld}$ | User’s load |

${\xi}^{str}\left(t\right)$ | Energy stored at time slots t | ||

${\xi}^{str}(t-1)$ | Energy stored at time slots $t-1$ | ${G}_{new}$ | Salp new population |

i | Appliance | ||

${\iota}^{sdr}$ | Self-discharging state | ${i}^{rat}$ | Interest rate |

${I}^{rad}$ | Solar radiation | ${I}_{ref}^{rad}$ | Solar radiation at reference conditions |

$LPSP$ | Loss of power supply probability | $LPS{P}^{max}$ | Maximum allowable LPSP value |

$max\left(point\right)$ | Maximum generation point | $min\left(point\right)$ | Minimum generation point |

${M}^{l}$ | Mean of the learners | ${M}_{i}^{vec}$ | Mutant vector |

${\eta}_{i}$ | Efficiency of the inverter | ${\eta}^{bat}$ | Battery bank charging efficiency |

n | System’s life span in years | ${N}^{b}$ | Total number of batteries in the battery bank |

${N}^{wt}$ | Number of WTs | ${N}_{max}^{pv}$ | Maximum number of PV panels |

${N}_{max}^{b}$ | Maximum number of batteries | ${N}_{max}^{wt}$ | Maximum number of WTs |

${\rho}^{ic}$ | Present price of the inverter/converter | ${\rho}^{bat}$ | Present battery price |

${p}_{i}^{rat}$ | Power rating | ${P}^{pv}$ | Total hourly PV panel power output |

${P}_{rat}^{pv}$ | Rated PV power | ${P}^{wt}$ | Output power generated by the WT |

${P}_{nom}^{wt}$ | Nominal rated WT power | ${r}^{rnd}$ or r | Random number |

${R}_{i}^{vec}$ | Target vector | s | Number of decision variables |

S | Row vector of positive integers | ${S}_{new}^{l}$ or ${S}_{new}$ | New vector |

${S}_{old}^{l}$ or ${S}_{old}$ | Old vector | ${S}_{teacher}^{l}$ | Teacher in the TLBO process |

${S}_{x1}^{l}$ or ${S}_{m}$ | Learner 1 | ${S}_{x2}^{l}$ or ${S}_{n}$ | Learner 2 |

t | Time slot | ${T}^{cof}$ | Temperature coefficient of PV panels |

${T}^{cel}$ | Cell temperature | ${T}^{amb}$ | Ambient air temperature |

${T}_{noct}^{cel}$ | Normal operating cell temperature | ${T}_{fcr}$ | Teaching factor |

${T}_{i}^{vec}$ or ${T}_{vec}$ | Trial vector | v | Wind speed |

${v}^{r}$ | Rated wind speed | ${v}^{co}$ | Cut-out wind speed |

${v}^{ci}$ | Cut-in wind speed | $\chi $ | Boolean integer |

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**Figure 6.**Hourly PVs power produced, WTs power produced, and energy storage level of the PV-WT-battery system by EESA during a year at various $LPS{P}^{max}$ values.

**Figure 7.**Breakdown of the total annual cost values of the PV-WT-battery system at various $LPS{P}^{max}$.

**Figure 8.**Hourly PVs’ power produced and energy storage level of the PV-battery system by EESA during a year at various $LPS{P}^{max}$ values.

**Figure 10.**Hourly WTs’ power produced and energy storage level of the WT-battery system by EESA during a year at various $LPS{P}^{max}$ values.

**Figure 12.**Convergence process of the EESA algorithm for three scenarios at different $LPS{P}^{max}$ values.

**Table 1.**PV-WT-battery hybrid system components and parameters [29].

Hybrid System Components | Parameters | Value |
---|---|---|

PV panel | ${P}_{rat}^{pv}$ | 120 W |

${\zeta}^{pv}$ | $614 | |

${\zeta}_{mtn}^{pv}$ | $0 | |

${A}^{pv}$ | 1.07 m${}^{2}$ | |

${\eta}^{pv}$ | 12% | |

${T}_{noct}^{cel}$ | 33 °C | |

WT | ${P}_{nom}^{wt}$ | 1 kW |

${v}^{ci}$ | 2.5 m/s | |

${v}^{r}$ | 11 m/s | |

${v}^{co}$ | 13 m/s | |

${\zeta}^{wt}$ | $3200 | |

${\zeta}_{mtn}^{wt}$ | $100 | |

Battery | $Voltage$ | 12 V |

Battery nominal capacity | 1.3 kWh | |

Life span | 5 years | |

${\eta}^{bat}$ | 85% | |

${\rho}^{bat}$ | $130 | |

$DoD$ | 0.8 | |

${\iota}^{sdr}$ | 0.0002 | |

Power inv/conv | ${P}_{rat}^{ic}$ | 3 kW |

Life span | 10 years | |

${\eta}^{i}$ | 95% | |

${\rho}^{ic}$ | $2000 | |

Other parameters | ${i}^{rat}$ | 5% |

n | 20 years |

**Table 2.**Summary of TLBO, EDE, SSA, and EESA results for the PV-WT-battery hybrid system at various $LPS{P}^{max}$ values.

System | TLBO | EDE | SSA | EESA | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{LPSP}$${}^{\mathit{max}}$ (%) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | |

PV-WT-Battery | 0 | 0 | 111 | 17 | 1753 | 64,430 | 0 | 126 | 14 | 1837 | 66,621 | 0 | 112 | 17 | 1794 | 65,710 | 0 | 111 | 17 | 1753 | 64,430 |

0.5 | 0.2762 | 113 | 16 | 1688 | 62,220 | 0.3641 | 122 | 14 | 1711 | 62,640 | 0.4778 | 136 | 11 | 1771 | 64,061 | 0.3779 | 117 | 15 | 1685 | 61,970 | |

1 | 0.6543 | 116 | 15 | 1654 | 60,990 | 0.8524 | 133 | 11 | 1673 | 60,971 | 0.9725 | 132 | 11 | 1640 | 59,931 | 0.9645 | 127 | 12 | 1612 | 59,200 | |

3 | 2.7274 | 122 | 12 | 1461 | 54,420 | 2.9859 | 142 | 7 | 1539 | 55,964 | 1.6359 | 125 | 12 | 1552 | 57,300 | 2.8168 | 126 | 11 | 1458 | 54,171 | |

Average rank | 60,515 | 61,549 | 61,750 | 59,943 | |||||||||||||||||

Final rank | 2 | 3 | 4 | 1 |

**Table 3.**Breakdown of TAC by EESA for the PV-WT-battery hybrid system at various $LPS{P}^{max}$ values.

System | EESA | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{LPSP}}^{\mathit{max}}$ (%) | Configuration | ${\mathit{\zeta}}^{\mathit{pv}}$ ($) | ${\mathit{\zeta}}^{\mathit{wt}}$ ($) | ${\mathit{\zeta}}^{\mathit{bat}}$ ($) | ${\mathit{\zeta}}^{\mathit{ic}}$ ($) | ${\mathit{\zeta}}^{\mathit{mtn}}$ ($) | TAC ($) | |

PV-WT-battery | 0 | $S=\left[111\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}17\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1753\right]$ | 5469 | 4365 | 52,637 | 259 | 1700 | 64,430 |

0.5 | $S=\left[117\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}15\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1685\right]$ | 5764 | 3852 | 50,595 | 259 | 1500 | 61,970 | |

1 | $S=\left[127\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1612\right]$ | 6257 | 3081 | 48,403 | 259 | 1200 | 59,200 | |

3 | $S=\left[126\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1458\right]$ | 6208 | 2825 | 43,779 | 259 | 1100 | 54,171 |

**Table 4.**Summary of TLBO, EDE, SSA, and EESA results for the PV-battery hybrid system at various $LPS{P}^{max}$ values.

System | TLBO | EDE | SSA | EESA | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{LPSP}$${}^{\mathit{max}}$ (%) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | |

PV-Battery | 0 | 0 | 199 | 0 | 3150 | 104,640 | 0 | 199 | 0 | 3150 | 104,640 | 0 | 200 | 0 | 3200 | 106,200 | 0 | 199 | 0 | 3150 | 104,640 |

0.5 | 0.4932 | 194 | 0 | 2898 | 96,840 | 0.4932 | 194 | 0 | 2898 | 96,840 | 0.4932 | 194 | 0 | 2898 | 96,840 | 0.4932 | 194 | 0 | 2898 | 96,840 | |

1 | 0.8820 | 191 | 0 | 2746 | 92,120 | 0.7526 | 192 | 0 | 2797 | 93,700 | 0.6160 | 193 | 0 | 2847 | 95,250 | 0.8820 | 191 | 0 | 2746 | 92,120 | |

3 | 2.8942 | 178 | 0 | 2090 | 71,790 | 2.8942 | 178 | 0 | 2090 | 71,790 | 2.7459 | 179 | 0 | 2141 | 73,370 | 2.8942 | 178 | 0 | 2090 | 71,790 | |

Average rank | 91,348 | 91,743 | 92,915 | 91,348 | |||||||||||||||||

Final rank | 1 | 2 | 3 | 1 |

**Table 5.**Breakdown of TAC by EESA for the PV-battery hybrid system at various $LPS{P}^{max}$ values.

System | EESA | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{LPSP}}^{\mathit{max}}$ (%) | Configuration | ${\mathit{\zeta}}^{\mathit{pv}}$ ($) | ${\mathit{\zeta}}^{\mathit{wt}}$ ($) | ${\mathit{\zeta}}^{\mathit{bat}}$ ($) | ${\mathit{\zeta}}^{\mathit{ic}}$ ($) | ${\mathit{\zeta}}^{\mathit{mtn}}$ ($) | TAC ($) | |

PV-battery | 0 | $S=\left[199\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3150\right]$ | 9800 | 0 | 94,580 | 260 | 0 | 10,4640 |

0.5 | $S=\left[194\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2898\right]$ | 9560 | 0 | 87,020 | 260 | 0 | 96,840 | |

1 | $S=\left[191\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2746\right]$ | 9410 | 0 | 82,450 | 260 | 0 | 92,120 | |

3 | $S=\left[178\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2090\right]$ | 8770 | 0 | 62,760 | 260 | 0 | 71,790 |

**Table 6.**Summary of TLBO, EDE, SSA, and EESA results for the WT-battery hybrid system at various $LPS{P}^{max}$ values.

System | TLBO | EDE | SSA | EESA | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{LPSP}$${}^{\mathit{max}}$ (%) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | $\mathit{LPSP}$ (%) | ${\mathit{N}}^{\mathit{pv}}$ | ${\mathit{N}}^{\mathit{wt}}$ | ${\mathit{N}}^{\mathit{b}}$ | TAC ($) | |

PV-Battery | 0 and 0.5 | 0 | 0 | 50 | 3552 | 124,750 | 0 | 0 | 50 | 3552 | 124,750 | 0 | 0 | 50 | 3552 | 124,750 | 0 | 0 | 50 | 3552 | 124,750 |

1 and 3 | 0.5503 | 0 | 49 | 3362 | 118,690 | 0.5503 | 0 | 49 | 3362 | 118,690 | 0.5503 | 0 | 49 | 3362 | 118,690 | 0.5503 | 0 | 49 | 3362 | 118,690 | |

Average rank | 121,720 | 121,720 | 121,720 | 121,720 | |||||||||||||||||

Final rank | 1 | 1 | 1 | 1 |

**Table 7.**Breakdown of TAC by EESA for the WT-battery hybrid system at various $LPS{P}^{max}$ values.

System | EESA | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{LPSP}}^{\mathit{max}}$ (%) | Configuration | ${\mathit{\zeta}}^{\mathit{pv}}$ ($) | ${\mathit{\zeta}}^{\mathit{wt}}$ ($) | ${\mathit{\zeta}}^{\mathit{bat}}$ ($) | ${\mathit{\zeta}}^{\mathit{ic}}$ ($) | ${\mathit{\zeta}}^{\mathit{mtn}}$ ($) | TAC ($) | |

WT-Bat. | 0 and 0.5 | $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}50\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3552\right]$ | 0 | 12,840 | 106,650 | 260 | 5000 | 124,750 |

1 and 3 | $S=\left[0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}49\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3362\right]$ | 0 | 12,580 | 100,950 | 260 | 4900 | 118,690 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khan, A.; Alghamdi, T.A.; Khan, Z.A.; Fatima, A.; Abid, S.; Khalid, A.; Javaid, N.
Enhanced Evolutionary Sizing Algorithms for Optimal Sizing of a Stand-Alone PV-WT-Battery Hybrid System. *Appl. Sci.* **2019**, *9*, 5197.
https://doi.org/10.3390/app9235197

**AMA Style**

Khan A, Alghamdi TA, Khan ZA, Fatima A, Abid S, Khalid A, Javaid N.
Enhanced Evolutionary Sizing Algorithms for Optimal Sizing of a Stand-Alone PV-WT-Battery Hybrid System. *Applied Sciences*. 2019; 9(23):5197.
https://doi.org/10.3390/app9235197

**Chicago/Turabian Style**

Khan, Asif, Turki Ali Alghamdi, Zahoor Ali Khan, Aisha Fatima, Samia Abid, Adia Khalid, and Nadeem Javaid.
2019. "Enhanced Evolutionary Sizing Algorithms for Optimal Sizing of a Stand-Alone PV-WT-Battery Hybrid System" *Applied Sciences* 9, no. 23: 5197.
https://doi.org/10.3390/app9235197