# An Effective Metaheuristic Approach for Building Energy Optimization Problems

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## Abstract

**:**

## 1. Introduction

- The development of POSCO, a powerful hybrid metaheuristic method based on single candidate and pelican optimization, has been made.
- Thirteen popular benchmarking functions are used to evaluate POSCO’s performance for numerical function optimization, and the findings are contrasted with those of other widely used optimization techniques.
- To show how well the suggested method works when used on real-life issues, the new method is used to build the energy optimization problem.
- The efficiency of the proposed POSCO for BEO is investigated and the results acquired are contrasted with those previously assessed by the other procedures.

## 2. Pelican Optimization Algorithm

- a.
- Moving Approaching the Prey or Exploration Phase:

_{j}is the prey’s position in the jth dimension, and F

_{p}is the value of its objective function.

- b.
- Exploitation Phase or Water Surface Winging:

_{ij}, while t is the counter for iterations, and T is the most iterations allowed. Efficient updating is also employed at this stage to accept or reject the new pelican location, which is described in Equation (7):

Algorithm 1: Pseudo-code of the pelican optimization algorithm |

Determine the POA population size (N) and the number of iterations (T) Initialization of the position of pelicans randomly based on Equation (1) Calculate the objective function of the population For t = 1:TGenerate the position of the prey at random For I = 1:NPhase 1: Moving towards prey (exploration phase) For j = 1:mCalculate new status of the jth dimension using Equation (4) EndUpdate the ith population member using Equation (5) Phase 2: Winging on the water surface (exploitation phase) For j = 1:m.Calculate new status of the jth dimension using Equation (6) EndUpdate the ith population member using Equation (7) EndUpdate best candidate solution EndOutput best solution obtained by POA |

## 3. Single Candidate Optimizer

_{max}function evaluations or iterations that make up the overall optimization process are split into two phases, with the candidate solution updating its position in each phase in a different way. In order to create a single, robust algorithm, the SCO approach combines the single candidate technique and the two-phase strategy. The algorithm, most importantly, uses a special set of equations to update the candidate solution’s position exclusively on the basis of its information, i.e., its location at the time. When T

_{1}function evaluations are completed, the first phase of SCO comes to an end, and T

_{2}function evaluations are undertaken in the second phase, where T

_{1}+ T

_{2}= T

_{max}. The candidate solution adjusts its places as follows throughout the first stage of SCO:

_{1}is a random number in interval [0, 1]. Here is how w is defined mathematically:

## 4. Hybrid Pelican and Single Candidate Optimizer

## 5. Building Energy Optimization Problems

#### 5.1. Simple Office Building

_{1}–X

_{4}are shown in Table 1.

^{2}K). Carpet, 5 cm of concrete, padding, and concrete (18 cm) make up the floor and ceiling. Bricks used for the internal walls have a 12 cm thickness. There is an outside shading mechanism, and the double-panel windows are Krypton gas-filled and low-emissivity. The objective function is the sum of the energy consumption of a chiller, a boiler, and lighting as presented in the following equation:

_{h}(.), Q

_{c}(.), and E(.) stand for the yearly energy usage for heating, cooling, and zone lighting electricity consumption in kWh/a, respectively.

_{h}= 0.44 and η

_{c}= 0.77 were utilized, respectively. In addition, the primary energy factor (PEF) for electricity is set to 3.0 to convert site electricity to source fuel energy consumption.

^{2}a.

#### 5.2. Detailed Office Building

^{2}a). Energy use for cooling coils, fans, heating, and zone lighting is included in the energy consumption. The BOP can be written as follows:

_{h}), cooling coil (E

_{c}), fans and zone lighting energy consumption (E

_{el}) were considered. Here, the primary energy factors for electricity (PEF

_{el}= 3) and gas (PEF

_{gas}= 1) are also taken into consideration.

#### 5.3. Simulation Software for Building Energy Consumption

#### 5.4. Combining the POSCO Algorithm with EP

## 6. Verification of the POSCO

## 7. Results and Comparison

#### 7.1. Simple Office Building Results

_{unt}and GA perform the poorest in this task in terms of median and spread, but other algorithms regularly outperform them with low median and spread. As can be observed, when compared to the algorithms used in the literature, the POSCO worst case might still produce usable results. Additionally, POSCO’s best-case scenario might deliver exceptional performance in locating the optimal solution. The PSO technique, which provides the best solution for the basic office in Seattle in the range of 132.9–133.5, is one of the finest approaches. The POSCO offers the best option, which falls between 132.6 and 133, with a few points outside of that range. As a result, the worst possible outcome for PSO may be calculated as 133.5, while POSCO results in 133. For the other area, the results are almost the same. The best values of the annual energy usage for Chicago and Houston are 152.2 and 185.5, respectively, which are lower than those evaluated by PO and other techniques. The algorithm was run 20 times, as indicated. The optimum control settings attained for the straightforward office are shown in Table 7.

#### 7.2. Detailed Office Building Results

_{unt}and GA have the worst median and spread values.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Building model of BEO problem for a simple office building [34].

**Figure 5.**Building model of BEO problem for detailed office building [34].

Variables | X1 | X2 | X3 | X4 |
---|---|---|---|---|

Description | Building orientation | Window width West | Window width East | Shading transmit-tance |

Bounds | [−180, 180] | [0.1, 5.9] | [0.1, 5.9] | [0.2, 0.8] |

Units | ◦ | m | m | - |

Variables | Description | Bounds | Units |
---|---|---|---|

X1 | Window width North | [1.224, 5.8321] | m |

X2 | Window width West | [7.344, 25.668] | m |

X3 | Window width East | [7.344, 25.668] | m |

X4 | Window width South | [1.224, 5.8321] | m |

X5 | Overhang depth West | [0.05, 1.05] | m |

X6 | Overhang depth East | [0.05, 1.05] | m |

X7 | Overhang depth South | [0.05, 1.05] | m |

X8 | Shading set point West | [100, 600] | W/m^{2} |

X9 | Shading set point East | [100, 600] | W/m^{2} |

X10 | Shading set point South | [100, 600] | W/m^{2} |

X11 | Night cooling summer, set point | [20, 25] | °C |

X12 | Night cooling winter, set point | [20, 25] | °C |

X13 | Supply air temperature cooling | [12, 18] | °C |

Function | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | n (Dim) |
---|---|---|---|

${F}_{1}\left(X\right)={\displaystyle \sum}_{i=1}^{n}{x}_{i}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{2}\left(X\right)={\displaystyle \sum}_{i=1}^{n}\left|{x}_{i}\right|+{\displaystyle \prod}_{i=1}^{n}\left|{x}_{i}\right|$ | ${\left[-10,10\right]}^{n}$ | 0 | 30 |

${F}_{3}\left(X\right)={\displaystyle \sum}_{i=1}^{n}{\left({\displaystyle \sum}_{j=1}^{i}{x}_{j}\right)}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{4}\left(X\right)=\underset{i}{\mathrm{max}}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{5}\left(X\right)={\displaystyle \sum}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | ${\left[-30,30\right]}^{n}$ | 0 | 30 |

${F}_{6}\left(X\right)={\displaystyle \sum}_{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | ${\left[-100,100\right]}^{n}$ | 0 | 30 |

${F}_{7}\left(X\right)={\displaystyle \sum}_{i=1}^{n}i{x}_{i}^{4}+random\left[0,1\right)$ | ${\left[-1.28,1.28\right]}^{n}$ | 0 | 30 |

Function | Range | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | n (Dim) |
---|---|---|---|

${F}_{8}\left(X\right)={\displaystyle \sum}_{i=1}^{n}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | ${\left[-500,500\right]}^{n}$ | 428.9829 × n | 30 |

${F}_{9}\left(X\right)={\displaystyle \sum}_{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | ${\left[-5.12,5.12\right]}^{n}$ | 0 | 30 |

${F}_{10}\left(X\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{x}_{i}^{2}}\right)-exp\left(\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | ${\left[-32,32\right]}^{n}$ | 0 | 30 |

${F}_{11}\left(X\right)=\frac{1}{4000}{\displaystyle \sum}_{i=1}^{n}{x}_{i}^{2}-{\displaystyle \prod}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | ${\left[-600,600\right]}^{n}$ | 0 | 30 |

${F}_{12}\left(X\right)=$ $\frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle \sum}_{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+{\displaystyle \sum}_{i=1}^{n}u\left({x}_{i},10,100,4\right)$${y}_{i}=1+\frac{{x}_{i+4}}{4}$ $u\left({x}_{i},a,k,m\right)=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}$ | ${\left[-50,50\right]}^{n}$ | 0 | 30 |

${F}_{13}\left(X\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum}_{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+{\displaystyle \sum}_{i=1}^{n}u\left({x}_{i},5,100,4\right)$ | ${\left[-50,50\right]}^{n}$ | 0 | 30 |

F | Index | POSCO | POA | PSO | FA | MVO | SSA | TSA |
---|---|---|---|---|---|---|---|---|

F_{1} | Mean | 0.00 | 2.42 × 10^{−97} | 4.98 × 10^{−9} | 7.11 × 10^{−3} | 2.81 × 10^{−1} | 3.29 × 10^{−7} | 8.31 × 10^{−56} |

Std. | 0.00 | 7.22 × 10^{−97} | 1.40 × 10^{−8} | 3.21 × 10^{−3} | 1.11 × 10^{−1} | 5.92 × 10^{−7} | 1.02 × 10^{−58} | |

F_{2} | Mean | 0.00 | 1.16 × 10^{−52} | 7.29 × 10^{−4} | 4.34 × 10^{−1} | 3.96 × 10^{−1} | 1.9111 | 8.36 × 10^{−35} |

Std. | 0.00 | 2.55 × 10^{−52} | 1.84 × 10^{−3} | 1.84 × 10^{−1} | 1.41 × 10^{−1} | 1.6142 | 9.86 × 10^{−35} | |

F_{3} | Mean | 4.37 × 10^{−178} | 7.84 × 10^{−81} | 1.40 × 10 | 1.66 × 10^{3} | 4.31 × 10 | 1.50 × 10^{3} | 1.51 × 10^{−14} |

Std. | 5.76 × 10^{−181} | 3.49 × 10^{−80} | 7.13 | 6.72 × 10^{2} | 8.97 | 707.05 | 6.55 × 10^{−14} | |

F_{4} | Mean | 2.58 × 10^{−106} | 4.57 × 10^{−46} | 6.00 × 10^{−1} | 1.11 × 10^{−1} | 8.80 × 10^{−1} | 2.44 × 10^{−5} | 1.95 × 10^{−5} |

Std. | 4.49 × 10^{−108} | 9.98 × 10^{−46} | 1.72 × 10^{−1} | 4.75 × 10^{−2} | 2.50 × 10^{−1} | 1.89 × 10^{−5} | 4.49 × 10^{−4} | |

F_{5} | Mean | 2.71 × 10^{−1} | 2.80 × 10 | 4.93 × 10 | 7.97 × 10 | 1.18 × 10^{2} | 136.56 | 28.4 |

Std. | 5.68 × 10^{−1} | 8.73 × 10^{−1} | 3.89 × 10 | 7.39 × 10 | 1.43 × 10^{2} | 154.00 | 0.842 | |

F_{6} | Mean | 4.77 × 10^{−17} | 2.15 | 6.92 × 10^{−2} | 6.94 × 10^{−3} | 2.02 × 10^{−2} | 5.72 × 10^{−7} | 3.67 |

Std. | 2.25 × 10^{−7} | 4.47 × 10^{−1} | 2.87 × 10^{−2} | 3.61 × 10^{−3} | 7.43 × 10^{−3} | 2.44 × 10^{−7} | 0.3353 | |

F_{7} | Mean | 3.73 × 10^{−6} | 1.51 × 10^{−4} | 8.94 × 10^{−2} | 6.62 × 10^{−2} | 5.24 × 10^{−2} | 8.82 × 10^{−5} | 0.0018 |

Std. | 3.36 × 10^{−6} | 1.33 × 10^{−4} | 0.0206 | 4.23 × 10^{−2} | 1.37 × 10^{−2} | 6.94 × 10^{−5} | 4.62 × 10^{−4} |

F | Index | POSCO | POA | PSO | FA | MVO | SSA | TSA |
---|---|---|---|---|---|---|---|---|

F_{8} | Mean | –1.22 × 10^{4} | −1.01 × 10^{4} | −6.01 × 10^{3} | −5.85 × 10^{3} | −6.92 × 10^{3} | −7.46 × 10^{3} | −7.89 × 10^{3} |

Std. | 5.21 × 10^{2} | 1.70 × 10^{3} | 1.30 × 10^{3} | 1.61 × 10^{3} | 9.19 × 10^{2} | 634.67 | 599.26 | |

F_{9} | Mean | 0.00 | 0.00 | 4.72 × 10 | 1.51 × 10 | 1.01 × 10^{2} | 55.45 | 151.45 |

Std. | 0.00 | 0.00 | 1.03 × 10 | 1.25 × 10 | 1.89 × 10 | 18.27 | 35.87 | |

F_{10} | Mean | 8.88 × 10^{−16} | 8.77 × 10^{−16} | 3.86 × 10^{−2} | 4.58 × 10^{−2} | 1.15 | 2.84 | 2.409 |

Std. | 0.00 | 0.00 | 2.11 × 10^{−1} | 1.20 × 10^{−2} | 7.87 × 10^{−1} | 6.58 × 10^{−1} | 1.392 | |

F_{11} | Mean | 0.00 | 0.00 | 5.50 × 10^{−3} | 4.23 × 10^{−3} | 5.74 × 10^{−1} | 2.29 × 10^{−1} | 0.0077 |

Std. | 0.00 | 0.00 | 7.39 × 10^{−3} | 1.29 × 10^{−3} | 1.12 × 10^{−1} | 1.29 × 10^{−1} | 0.0057 | |

F_{12} | Mean | 1.35 × 10^{−5} | 1.25 × 10^{−1} | 1.05 × 10^{−2} | 3.13 × 10^{−4} | 1.27 | 6.82 | 6.373 |

Std. | 1.48 × 10^{−5} | 5.41 × 10^{−2} | 2.06 × 10^{−2} | 1.76 × 10^{−4} | 1.02 | 2.72 | 3.458 | |

F_{13} | Mean | 2.46 × 10^{−4} | 1.99 | 4.03 × 10^{−1} | 2.08 × 10^{−3} | 6.60 × 10^{−2} | 21.31 | 2.897 |

Std. | 2.92 × 10^{−4} | 2.51 × 10^{−1} | 5.39 × 10^{−1} | 9.62 × 10^{−4} | 4.33 × 10^{−2} | 16.99 | 0.643 |

Variables | X1 | X2 | X3 | X4 | F(x) |
---|---|---|---|---|---|

Description | Building orientation | Window width West | Window width East | Shading transmittance | Objective function |

Units | ◦ | m | m | - | |

Optimum value (Seattle) | 71.924 | 5.9 | 5.9 | 0.2876 | 132.6 |

Optimum value(Chicago) | 70.342 | 4.1 | 5.9 | 0.3126 | 152.2 |

Optimum value (Houston) | 75.564 | 5.1 | 3.5 | 0.4873 | 185.5 |

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## Share and Cite

**MDPI and ACS Style**

Yuan, X.; Karbasforoushha, M.A.; Syah, R.B.Y.; Khajehzadeh, M.; Keawsawasvong, S.; Nehdi, M.L.
An Effective Metaheuristic Approach for Building Energy Optimization Problems. *Buildings* **2023**, *13*, 80.
https://doi.org/10.3390/buildings13010080

**AMA Style**

Yuan X, Karbasforoushha MA, Syah RBY, Khajehzadeh M, Keawsawasvong S, Nehdi ML.
An Effective Metaheuristic Approach for Building Energy Optimization Problems. *Buildings*. 2023; 13(1):80.
https://doi.org/10.3390/buildings13010080

**Chicago/Turabian Style**

Yuan, Xinzhe, Mohammad Ali Karbasforoushha, Rahmad B. Y. Syah, Mohammad Khajehzadeh, Suraparb Keawsawasvong, and Moncef L. Nehdi.
2023. "An Effective Metaheuristic Approach for Building Energy Optimization Problems" *Buildings* 13, no. 1: 80.
https://doi.org/10.3390/buildings13010080