# A Metaheuristic Hybrid of Double-Target Multi-Layer Perceptron for Energy Performance Analysis in Residential Buildings

^{1}

^{2}

^{*}

## Abstract

**:**

_{A}) and weighted average discomfort degree-hours (DD

_{A}), for a residential building. For this purpose, a double-target multi-layer perceptron (2TMLP) model is created to establish the connections between the TD

_{A}and DD

_{A}with the geometry and architecture of the building. These connections are then processed and optimized by the WCA using 80% of the data. Next, the applicability of the model is examined using the residual 20%. According to the results, the goodness-of-fit for the TD

_{A}and DD

_{A}was 98.67% and 99.74%, respectively, in terms of the Pearson correlation index. Moreover, a comparison between WCA-2TMLP and other hybrid models revealed that this model enjoys the highest accuracy of prediction. However, the shuffled complex evolution (SCE) optimizer has a better convergence rate. Hence, the final mathematical equation of the SCE-2TMLP is derived for directly predicting the TD

_{A}and DD

_{A}without the need of using programming environments. Altogether, this study may shed light on the applications of artificial intelligence for optimizing building energy performance and related components (e.g., heating, ventilation, and air conditioning systems) in new construction projects.

## 1. Introduction

^{2}of more than 0.991. Zhang et al. [12] utilized the support vector regression (SVR) method for the mentioned prediction tasks and stated that the accuracy of this method is highly based on household behavior variability. Olu-Ajayi et al. [13] utilized many different ML techniques, including ANN, gradient boosting (GB), DNN, RF, stacking, K-nearest neighbor (KNN), SVM, decision tree (DT), and linear regression. They showed that DT presented the best outcome, with a 1.2 s training time. Banik et al. [14] also utilized the RF and extreme gradient boosting (XGBoost) ensemble and stated that the precision of this method was 15–29%. They also found that the machine learning approaches can be very impressive for forecasting the energy consumption of buildings.

_{A}) and annual weighted average discomfort degree-hours (DD

_{A}). For this purpose, the algorithm should be coupled with a framework that supports dual-prediction. A double-target MLP (2TMLP) plays the role of this framework. As such, the proposed model is named WCA-2TMLP hereafter. Many studies have previously confirmed the suitable optimization competency of WCA when incorporated with ANN techniques for various purposes [30,31], especially building energy assessment [32].

## 2. Materials and Methods

#### 2.1. Data Provision

^{2}and a height of 3 m, and the building is divided into a total of 13 thermal zones. The floors have different plans, and each is designated to four people. In the reference paper [33], the simulation conditions and building characteristics are fully presented. Interested readers are recommended to refer to [33] for further details.

_{A}and DD

_{A}, whose values were listed versus eleven characteristics of the buildings, including: transmission coefficient of the external walls (U

_{M}), transmission coefficient of the roof (U

_{T}), transmission coefficient of the floor (U

_{P}), solar radiation absorption coefficient of the exterior walls (α

_{M}), solar radiation absorption coefficient of the roof (α

_{T}), linear coefficient of thermal bridges (Pt), air change rate (ACH), shading coefficient of north-facing windows (Scw-N), shading coefficient of south-facing windows (Scw-S), shading coefficient of east-facing windows (Scw-E), and glazing (Glz). To calculate the TD

_{A}, the thermal loads (i.e., the heating and cooling loads) were summed and divided by the total conditioned area of the building, while the DD

_{A}was simply the representative of the annual weighted average of degree-hours when the occupants are not comfortable (i.e., times that they are supposed to be in more comfort).

_{A}and DD

_{A}, respectively. Moreover, the relationship between these two parameters is presented in Figure 1n. It can be seen from these three charts that the trends of TD

_{A}and DD

_{A}did not follow a meaningful correlation, and the TD

_{A}was more compatible with the input trends.

_{A}and DD

_{A}. These values confirm the earlier analysis as the correlation values corresponding to TD

_{A}were much larger and indicate a direct proportionality, while those corresponding to DD

_{A}were around 30% lower and indicate an adverse proportionality for U

_{M}, U

_{T}, U

_{P}, α

_{M}, α

_{T}, Pt, ACH, Scw-N, Scw-S, and Scw-E. Not surprisingly, the behavior of Glz was in contrast to the other inputs, due to the correlation values of −0.83 and 0.47 obtained in correspondence with TD

_{A}and DD

_{A}, respectively.

#### 2.2. Overview of WCA

_{b}and L

_{b}), as follows:

#### 2.3. Benchmark Strategies

_{A}and DD

_{A}to the input factors (i.e., U

_{M}, U

_{T}, U

_{P}, α

_{M}, α

_{T}, Pt, ACH, Scw-N, Scw-S, Scw-E, and Glz). The optimization process will be explained in detail in the following sections.

#### 2.4. Data Division

_{A}and DD

_{A}with reference to the behavior of the input parameters. Since these data are dedicated to training the models, they compose the major portion of the dataset. For this study, out of the 35 samples, 28 samples were used as the training data. The second group was used afterward for evaluating the goodness of the acquired knowledge when it is applied to new conditions. These are called testing data, and here contained the remaining seven samples. Note that in this step, a random division was considered to allow the training and testing datasets to have samples from throughout the original dataset.

#### 2.5. Goodness-of-Fit Equations

## 3. Results and Discussion

#### 3.1. Hybrid Creation

_{M}, U

_{T}, U

_{P}, α

_{M}, α

_{T}, Pt, ACH, Scw-N, Scw-S, Scw-E, and Glz, (ii) using 11 × 9 = 99 weights and 9 biases, the neurons in the middle layer performed the first level of calculations and sent the results to the output layer, and (iii) using 9 × 2 = 18 weights and 2 biases, the neurons in the output layer calculated the TD

_{A}and DD

_{A}.

#### 3.2. Training 2TMLP Using the Metaheuristic Algorithm

_{A}and DD

_{A}was considered as the cost function.

#### 3.3. Training Assessment

_{A}. This chart shows how the target values were hit by the predictions of the four models. The graphical interpretations indicated a satisfying goodness-of-fit for all models as the general patterns (i.e., significant ups and downs) were well-followed. However, a noticeable distinction could be seen between the HBO-2TMLP and the three other models. There were some overestimations and underestimations by the line of HBO-2TMLP that were more accurately modeled by SCE-2TMLP, SSA-2TMLP, and WCA-2TMLP.

_{A}. The prediction results were quite promising because both small and large fluctuations were nicely recognized and followed by all models. However, similar to the DD

_{A}results, there were some weaknesses that were more tangible for the HBO-2TMLP relative to the other algorithms.

#### 3.4. Testing Assessment

_{A}with excellent accuracy. Figure 5 shows the correlation between the real and predicted TD

_{A}values. The CR values demonstrated 97.33%, 90.88%, 95.55%, and 98.67% agreement for the results. However, similar to the previous phase, the WCA-2TMLP achieved the most accurate results in this step.

_{A}, the RMSE values were 91.52, 68.74, 90.15, and 96.10, associated with the MAEs of 44.01, 51.89, 41.25, and 40.07. The correlations of the DD

_{A}testing results are shown in Figure 6. Referring to the CR values of 99.60%, 93.82%, 99.47%, and 99.74%, the products of all models were in good harmony with the real values. In this phase, the WCA-2TMLP, despite achieving the smallest MAE and the largest CR, obtained the largest RMSE. However, based on the better performance in terms of two indices (out of three), the superiority of the WCA-2TMLP was evident here, too.

#### 3.5. Comparison

_{A}and DD

_{A}. From the overall comparison, it was found that the WCA-2TMLP had the largest accuracy in most stages. More clearly, for the TD

_{A}analysis (in both phases), the order of the algorithms from strongest to weakest was: (1) WCA-2TMLP, (2) SCE-2TMLP, (3) SSA-2TMLP, and (4) HBO-2TMLP. However, the outcome was different for the DD

_{A}analysis. While the training RMSE demonstrated the lowest error for WCA-2TMLP and the highest for HBO-2TMLP, the ranking was adverse in the testing phase. In both phases, the SSA-2TMLP captured the second position, followed by SCE-2TMLP. However, the MAE consistently suggested the following ranking: (1) WCA-2TMLP, (2) SSA-2TMLP, (3) SCE-2TMLP, and (4) HBO-2TMLP. As for the CR, the WCA-2TMLP and HBO-2TMLP were the strongest and the weakest predictor in both phases, respectively, while the SSA-2TMLP and SCE-2TMLP shared the second and third positions interchangeably in the training and testing phases.

#### 3.6. Discussion

_{A}prediction using EO and Harris hawks optimization (HHO), the best model was ANFIS-400-EO, which achieved a training MAE = 1.87, while in this work the lowest MAE was 0.81. As for testing, the lowest MAE of the cited study was 5.74, obtained by ANFIS-100-HHO, while this study reduced it to 4.12 using WCA-2TMLP. Hence, significant improvements can be detected.

#### 3.7. TD_{A} and DD_{A} Formula

_{A}and DD

_{A}. Based on the architecture of the 2TMLP (i.e., 11 input neurons, 9 hidden neurons with Tansig activation, and 2 output neurons with Purelin activation function), the procedure of calculating the TD

_{A}and DD

_{A}requires producing 9 outputs from the middle layer. As expressed by Equations (12) and (13), as well as Table 6, each of these outputs (represented by O

_{1}, O

_{2}, …, O

_{9}) is a non-linear function of the input parameters (i.e., U

_{M}, U

_{T}, U

_{P}, α

_{M}, α

_{T}, Pt, ACH, Scw-N, Scw-S, Scw-E, and Glz).

_{i1}, W

_{i2}, …, W

_{i11}, as well as the biases, b

_{i}, are presented in Table 3.

_{1}, O

_{2}, …, O

_{9}are calculated, Equations (14) and (15) yield the TD

_{A}and DD

_{A}, respectively. The reason for the linear calculations in these two equations lies in the Purelin function, which is described as f(x) = x.

_{A}= −0.198 × O1 − 0.874 × O2 + 0.940 × O3 + 0.315 × O4 + 0.422 × O5 + 0.666 × O6 − 0.307 × O7 − 0.517 × O8 − 0.852 × O9 − 0.193,

_{A}= 0.961 × O1 − 0.979 × O2 − 0.964 × O3 − 0.523 × O4 − 0.644 × O5 + 0.862 × O6 + 0.612 × O7 + 0.370 × O8 + 0.051 × O9 + 0.638

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

ANN | Artificial neural network |

ANFIS | Adaptive neuro-fuzzy inference system |

WCA | Water cycle algorithm |

2TMLP | Double-target multi-layer perceptron |

SCE | Shuffled complex evolution |

EFA | Electromagnetism-based firefly algorithm |

MR | Multiple regression |

GP | Genetic programming |

DNN | Deep neural network |

SVM | Support vector machine |

SVR | Support vector regression |

GB | Gradient boosting |

KNN | K-nearest neighbor |

DT | Decision tree |

XGBoost | Extreme gradient boosting |

DELM | Deep extreme learning |

LSTM | Long short-term memory |

KF | Kalman filter |

DCNN | Convolutional neural network |

BiLSTM | Bidirectional long short-term memory |

GRU | Gated recurrent unit |

BIM | Building information modeling |

HBO | Heap-based optimizer |

SSA | Salp swarm algorithm |

EO | Equilibrium optimizer |

MVO | Multi-verse optimizer |

MTOA | Multi-tracker optimization algorithm |

EFO | Electromagnetic field optimization |

SMA | Slime mold algorithm |

HHO | Harris hawks optimization |

U_{M} | Transmission coefficient of the external walls |

U_{T} | Transmission coefficient of the roof |

U_{P} | Transmission coefficient of the floor |

α_{M} | Solar radiation absorption coefficient of the exterior walls |

α_{T} | Solar radiation absorption coefficient of the roof |

Pt | Linear coefficient of thermal bridges |

ACH | Air change rate |

Scw-N | Shading coefficient of north-facing windows |

Scw-S | Shading coefficient of south-facing windows |

Scw-E | Shading coefficient of east-facing windows |

Glz | Glazing |

MAE | Mean absolute error |

RMSE | Root mean square error |

CR | Pearson correlation index |

DD_{A} | Weighted average discomfort degree-hours |

TD_{A} | Thermal energy demand |

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**Figure 2.**Averaged cost functions obtained for optimizing 2TMLP using (

**a**) SCE, (

**b**) HBO, (

**c**) SSA, and (

**d**) WCA.

**Figure 5.**Correlation of the TD

_{A}testing results obtained for: (

**a**) SCE-2TMLP, (

**b**) HBO-2TMLP, (

**c**) SSA-2TMLP, and (

**d**) WCA-2TMLP.

**Figure 6.**Correlation of the DD

_{A}testing results obtained for: (

**a**) SCE-2TMLP, (

**b**) HBO-2TMLP, (

**c**) SSA-2TMLP, and (

**d**) WCA-2TMLP.

U_{M} | U_{T} | U_{P} | α_{M} | α_{T} | Pt | ACH | Scw-N | Scw-S | Scw-E | Glz | TD_{A} | DD_{A} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

TD_{A} | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | 0.90 | −0.83 | 1.00 | −0.26 |

DD_{A} | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | −0.61 | 0.47 | −0.26 | 1.00 |

Cost Function | SCE-2TMLP | HBO-2TMLP | SSA-2TMLP | WCA-2TMLP |
---|---|---|---|---|

Initial | 56.3373759727784 | 53.0278785825291 | 52.8183542220224 | 31.9380917181655 |

Final | 8.83468212953907 | 27.8041304186964 | 6.72926499469168 | 3.35913787449231 |

Target | Hybrid Model | Training | Testing |
---|---|---|---|

TD_{A} | SCE-2TMLP | 3.01 | 9.14 |

HBO-2TMLP | 12.71 | 14.66 | |

SSA-2TMLP | 3.68 | 10.51 | |

WCA-2TMLP | 1.17 | 7.39 | |

DD_{A} | SCE-2TMLP | 14.65 | 91.52 |

HBO-2TMLP | 42.89 | 68.74 | |

SSA-2TMLP | 9.77 | 90.15 | |

WCA-2TMLP | 5.54 | 96.10 |

Target | Hybrid Model | Training | Testing |
---|---|---|---|

TD_{A} | SCE-2TMLP | 2.66 | 6.83 |

HBO-2TMLP | 9.28 | 11.70 | |

SSA-2TMLP | 2.94 | 7.17 | |

WCA-2TMLP | 0.81 | 4.12 | |

DD_{A} | SCE-2TMLP | 9.84 | 44.01 |

HBO-2TMLP | 29.69 | 51.89 | |

SSA-2TMLP | 6.42 | 41.25 | |

WCA-2TMLP | 3.63 | 40.07 |

Target | Hybrid Model | Training | Testing |
---|---|---|---|

TD_{A} | SCE-2TMLP | 99.40 | 97.33 |

HBO-2TMLP | 88.43 | 90.88 | |

SSA-2TMLP | 99.05 | 95.55 | |

WCA-2TMLP | 99.90 | 98.67 | |

DD_{A} | SCE-2TMLP | 98.92 | 99.60 |

HBO-2TMLP | 89.45 | 93.82 | |

SSA-2TMLP | 99.45 | 99.47 | |

WCA-2TMLP | 99.82 | 99.74 |

i | W_{i1} | W_{i2} | W_{i3} | W_{i4} | W_{i5} | W_{i6} | W_{i7} | W_{i8} | W_{i9} | W_{i10} | W_{i11} | b_{i} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | −0.807 | 0.811 | 0.594 | 0.456 | −0.807 | −0.157 | 0.113 | −0.344 | 0.473 | −0.146 | −0.019 | 1.710 |

2 | 0.543 | −0.497 | 0.186 | −0.725 | 0.190 | 0.535 | −0.055 | −0.082 | −0.823 | 0.836 | 0.335 | −1.282 |

3 | 1.134 | 0.136 | 0.157 | −0.148 | −0.700 | −0.524 | 0.338 | 0.307 | −0.080 | 0.630 | −0.441 | −0.855 |

4 | 0.384 | −0.464 | −0.503 | −0.104 | 0.875 | −0.582 | −0.061 | −0.526 | −0.601 | −0.721 | −0.173 | −0.427 |

5 | 0.775 | 0.251 | −0.195 | 0.900 | 0.241 | −0.403 | −0.546 | 0.416 | 0.844 | 0.012 | −0.080 | 0.000 |

6 | 0.181 | 0.141 | 0.636 | 0.099 | 0.812 | −0.528 | −0.568 | −0.713 | 0.159 | 0.286 | −0.761 | 0.427 |

7 | 0.863 | 0.778 | 0.214 | 0.441 | −0.309 | 0.656 | −0.377 | −0.375 | −0.025 | 0.540 | −0.480 | 0.855 |

8 | 0.316 | −0.790 | −0.014 | −0.516 | 0.716 | 0.542 | 0.426 | −0.640 | −0.281 | −0.675 | 0.006 | 1.282 |

9 | 0.823 | 0.689 | 0.596 | −0.305 | −0.702 | −0.274 | −0.011 | −0.163 | 0.233 | 0.136 | 0.810 | 1.710 |

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## Share and Cite

**MDPI and ACS Style**

Lin, C.; Lin, Y.
A Metaheuristic Hybrid of Double-Target Multi-Layer Perceptron for Energy Performance Analysis in Residential Buildings. *Buildings* **2023**, *13*, 1086.
https://doi.org/10.3390/buildings13041086

**AMA Style**

Lin C, Lin Y.
A Metaheuristic Hybrid of Double-Target Multi-Layer Perceptron for Energy Performance Analysis in Residential Buildings. *Buildings*. 2023; 13(4):1086.
https://doi.org/10.3390/buildings13041086

**Chicago/Turabian Style**

Lin, Cheng, and Yunting Lin.
2023. "A Metaheuristic Hybrid of Double-Target Multi-Layer Perceptron for Energy Performance Analysis in Residential Buildings" *Buildings* 13, no. 4: 1086.
https://doi.org/10.3390/buildings13041086