# A Dynamic Model for Indoor Temperature Prediction in Buildings

^{*}

## Abstract

**:**

## 1. Introduction

- To combine easily available, existing real measurements, building information and tabular values while minimizing the number of model parameters and inputs,
- To apply the modelling approach for multiple buildings to validate the generalizability.

## 2. Building Characteristics and Measurement Data

#### 2.1. Buildings

#### 2.2. Measurements

## 3. Modelling and Analysis Methods

#### 3.1. General Structure of the Model

^{2}·K)), A is the surface area through which the heat is being transferred (m

^{2}), T is the temperature of the object (K) and T

_{env}is the temperature of the surrounding environment (K).

_{in}is the indoor temperature (K) and T

_{out}is the outdoor temperature (K). Including heating power P (W) into Equation (2) gives:

_{in}, P and T

_{out}as:

_{in}(t) and T

_{in}(t − 1). Equation (4) now presents the general model structure for the indoor temperature prediction. In this work, time step of 3600 s (one hour) has been used. When predicting the future values of T

_{in}for longer time horizons, the previous model predictions of T

_{in}are used as inputs as can be seen from Equation (5), where ${\widehat{T}}_{in}$ is the predicted indoor temperature:

_{1}, x

_{2}and x

_{3}are parameters and 1 ≤ ϴ

_{1}< ϴ

_{2}< ϴ

_{3}are delay numbers. As the dynamics of the indoor temperature in buildings can be different, several different model structures were analysed to find one generalizable model structure that can be applied to all buildings. In this article, the variants of the general model structure (Equation (4)) are named according to the applied heating power values used in the model. Then, for example P3P8 indicates that previous heating power values only at delays t − 3 and t − 8 were applied.

#### 3.2. Model Performance Analysis

_{t}and y

_{t}are the predicted and the measured values of variable y at time instance t respectively as:

^{®}environment (version 2013b).

## 4. Model Identification and Validation

#### 4.1. Physical Parameters

^{2}·K) in all buildings. The values in Table 3 were multiplied with the appropriate surface areas (Table 1) and then summed together to get the final U value presented in Table 4.

^{2}·K) respectively. For the apartment buildings (buildings B, D and E) the characteristic heat capacity values were 252, 396 and 576 kJ/(m

^{2}·K) for light, medium and heavy structures respectively. As instructed in the National Building Code of Finland [55], these characteristic heat capacity values were multiplied with the floor area of the buildings to get the C value for the building interior. Results are summarised in Table 4.

#### 4.2. Cross-Validation

^{®}’s Global Optimization Toolbox and fminunc algorithm from MATLAB

^{®}’s Optimization Toolbox were used to identify model parameters from Equation (8) by minimizing RMSE between the measured indoor temperature and the model predictions. All algorithms were used with their default parameter settings except for fminunc for which maximum function evaluations were set to 10,000 and maximum iterations to 1000. Algorithms were applied in sequence: First, pattern search was used to find a rough estimate for global minimum, then this estimate served as a starting point for simulated annealing to further refine the result, and finally, the identified parameter values from simulated annealing were set as a new starting point for fminunc to find local minimum.

#### 4.3. Generalizability Assessment

_{1}, x

_{2}and x

_{3}are the mean values of the corresponding parameter with four identification data sets. For most of the best performing model structures the C value corresponds to light building structure (see Table 4). For the parameter a, the standard deviation is low among all the best performing models. Considering the model structures with the fewest parameters (P3 and P1), the standard deviation of the parameter a is six or seven times greater than the lowest standard deviation achieved. The smallest standard deviation for the parameter a can be achieved by the model structures with two heating power values, namely P1P3 and P1P8. When it comes to parameter b, the standard deviation as well as the value of the parameter varies more among the different model structures. Still, one of the smallest standard deviations is achieved with model P1P3.

_{1}, x

_{2}and x

_{3}vary depending on the model structure. However, model structures with two heating power values produce smaller standard deviation for these parameters in general compared with model structures with three heating power values.

#### 4.4. Error Analysis

^{®}was applied to check if the modelling error can be considered as a white noise indicating appropriate complexity level of model structure.

#### 4.5. Uncertainty of the Outdoor Temperature Forecast

## 5. Applying the Model in Optimization of the Heat Demand

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Measured indoor temperature (T

_{in}), heating power (P) and outdoor temperature (T

_{out}) for the five different buildings. Two measurements from different locations are presented (one in black and the other in grey) for indoor temperature except for building B where only one measurement was available.

**Figure 2.**The principle of the applied 4-fold cross-validation method. Measurement data is divided into four folds of equal lengths and one is used for the validation while the other ones are merged as an identification data for the parameter estimation.

**Figure 3.**An overview of the model identification and validation process. Cross-validation of the model structures is carried out for each building with all the combinations of indoor temperature measurements and C values and RMSE, NRMSE and r are calculated. For each model structures, the representative indoor temperature and C value combination is determined based on NRMSE and r. Generalizability is studied after all model structures are cross-validated in all of the buildings. The best model structures are determined based on RMSE and r.

**Figure 4.**Cross-validation results with data from building A with model structure P1P3. Black line is the measured indoor temperature and grey line is the estimated indoor temperature. For validation data sets, a 28-h ahead prediction was simulated with measured data. The RMSE for the identification data sets 1, 2, 3 and 4 was 0.09 °C, 0.13 °C, 0.11 °C and 0.09 °C, respectively, and for the validation data sets 1, 2, 3 and 4 it was 0.16 °C, 0.07 °C, 0.18 °C and 0.18 °C, respectively. Correlation coefficients for the identification data sets 1, 2, 3 and 4 were 0.97, 0.96, 0.98 and 0.96, respectively, and for the validation data sets 1, 2, 3 and 4 they were 0.84, 0.89, 0.94 and 0.79, respectively.

**Figure 5.**Ten-day indoor temperature simulation for building A with model structure P1P3 utilizing the same four sets of parameters that were estimated in the 4-fold cross-validation. The measured (black line) and the estimated (grey lines) indoor temperatures are presented in the top. Error plots show the evolution of the modelling error during the ten-day simulation period.

**Figure 6.**Analysis of the modelling error in case of model structure P1P3 with data from building A for four validation runs. Validation data set 1, 2, 3 and 4 are presented from the up to bottom row respectively. From left to right column: Modelling error (model output–measured); Histogram of the modelling error (grey bars) with normal distribution fit (black line); Normal probability plot for modelling errors; the sample autocorrelation function of the modelling error with 95% confidence bounds (grey lines).

**Figure 7.**Sensitivity analysis of the indoor temperature prediction with simulated bias to the weather forecast in case of the model structure P1P3 with measured data from building A and four validation data sets. Black line is the measured indoor temperature and dark grey line is the indoor temperature prediction with measured outdoor temperature. Light grey line with circles is a representative example of the indoor temperature prediction with a normally distributed weather forecast. Light grey lines with upward and downward triangles are the simulated indoor temperature predictions using a weather forecast that contains a bias of 3 °C below or above the measured outdoor temperature respectively.

**Figure 8.**An example on the optimization of the heat demand in building A for a 24-h period. Upper: Measured heating powers for simulation day (black) and reference day (grey). Middle: Measured outdoor temperatures for simulation day (black) and reference day (grey). Lower: Measured (solid black) and simulated (dashed black) indoor temperature for simulation day and measured indoor temperature for reference day (grey). Black line with circles is the simulated indoor temperature for simulation day using reference day heating power.

Building | Basal Area (m^{2}) | Wall Area (m^{2}) | Window Area (m^{2}) | Floor Area (m^{2}) | Volume (m^{3}) | Construction Year | Building Type |
---|---|---|---|---|---|---|---|

A | 6700 | 4979 | 2418 | 13,400 | 79,781 | 2003–2004 | School |

B | 633 | 1153 | 352 | 1903 | 8300 | 1982 | Apartment building |

C | 1510 | 1274 | 142 | 3020 | 10,250 | 1983 | Municipal hall |

D | 545 | 2000 | 400 | 3703 | 12,400 | 1972 | Apartment building |

E | 993 | 2900 | 510 | 4200 | 15,617 | 2011 | Apartment building |

**Table 2.**Measurement period, the number of different locations for indoor temperature sensors and the range of power and outdoor and indoor temperature measurement.

Building | Measurement Period (mm/dd/yyyy) | Number of Sensor Locations | Range of Measurement Data | ||
---|---|---|---|---|---|

Heating Power (kW) | Outdoor Temperature (°C) | Indoor Temperature (°C) | |||

A | 1/30/2014–2/5/2014 | 6 | 140–790 | −20.8–+0.5 | +16.1–+22.7 |

B | 10/14/2014–10/20/2014 | 1 | 30–53 | −6.5–+3.8 | +14.3–+17.9 |

C | 4/8/2014–4/14/2014 | 5 | 20–70 | −2.7–+9.0 | +19.3–+24.4 |

D | 1/3/2015–1/8/2015 | 8 | 60–160 | −24.0–+0.4 | +20.2–+24.0 |

E | 1/3/2015–1/8/2015 | 6 | 30–160 | −24.0–+0.4 | +21.2–+24.0 |

**Table 3.**The overall heat loss coefficient values for the walls, roof and floor of the building in different years based on the National Building Code of Finland [55].

Year | Walls (W/(m^{2}·K)) | Roof (W/(m^{2}·K)) | Floor (W/(m^{2}·K)) | Building(s) Where Used |
---|---|---|---|---|

1978 | 0.35 | 0.29 | 0.29 | B, C and D |

2003 | 0.25 | 0.16 | 0.20 | A |

2010 | 0.17 | 0.09 | 0.14 | E |

Building | U (kW/K) | C (kJ/K) | ||
---|---|---|---|---|

Light | Medium | Heavy | ||

A | 6.58 | 3,376,800 | 5,306,400 | 7,718,400 |

B | 1.19 | 274,032 | 1,096,128 | 1,507,176 |

C | 1.49 | 761,040 | 1,195,920 | 1,739,520 |

D | 1.50 | 533,232 | 2,132,928 | 2,932,776 |

E | 1.33 | 604,800 | 2,419,200 | 3,326,400 |

**Table 5.**Cross-validation results with mean RMSE and r and their standard deviation. RMSE and r values were calculated by averaging the values of four identification/validation data sets. Parameters are presented as average values with their standard deviation over four identification data sets.

Building (Model Structure) | Identification Data | Validation Data | Parameters ^{1} | |||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE ± std_{RMSE} (°C) | corr ± std_{corr} | RMSE ± std_{RMSE} (°C) | corr ± std_{corr} | C | a | b | x_{1} | x_{2} | x_{3} | |

A (P2P3) | 0.12 ± 0.03 | 0.96 ± 0.01 | 0.14 ± 0.06 | 0.87 ± 0.07 | L | 0.92 ± 0.01 | 1.65 ± 0.15 | −0.64 ± 0.29 | 0.82 ± 0.31 | - |

B (P2P3P7) | 0.48 ± 0.10 | 0.63 ± 0.14 | 0.42 ± 0.44 | 0.59 ± 0.29 | L | 0.96 ± 0.07 | 0.91 ± 1.53 | −0.15 ± 0.83 | 1.12 ± 1.25 | −0.85 ± 1.26 |

C (P8) | 0.28 ± 0.12 | 0.45 ± 0.16 | 0.23 ± 0.20 | 0.67 ± 0.33 | L | 0.65 ± 0.10 | 7.06 ± 1.98 | - | - | - |

D (P1P2) | 0.21 ± 0.01 | 0.74 ± 0.04 | 0.23 ± 0.06 | 0.68 ± 0.14 | M | 0.79 ± 0.06 | 4.50 ± 1.39 | 2.61 ± 1.92 | 0.06 ± 1.67 | - |

E (P3) | 0.25 ± 0.01 | 0.69 ± 0.11 | 0.27 ± 0.11 | 0.65 ± 0.28 | M | 0.90 ± 0.03 | 2.09 ± 0.56 | - | - | - |

^{1}For C values, light (L), medium (M) or heavy (H) is used and the reader is referred to Table 4 for numerical values as well as for U values.

**Table 6.**Modelling results with mean RMSE and r and their standard deviation with five buildings. RMSE and r values were calculated by averaging the values of the four identification/validation data sets for every building and then calculating the mean using values of all five buildings.

Model Structure (Rank No.) | Identification Data | Validation Data | Parameters ^{1} | |||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE ± std_{RMSE} (°C) | corr ± std_{corr} | RMSE ± std_{RMSE} (°C) | corr ± std_{corr} | C | a | b | x_{1} | x_{2} | x_{3} | |

P1P3 (1) | 0.25 ± 0.13 | 0.68 ± 0.18 | 0.27 ± 0.14 | 0.59 ± 0.19 | L | 0.92 ± 0.01 | 1.62 ± 0.12 | −0.32 ± 0.12 | 0.53 ± 0.09 | - |

P3P6 (1) | 0.25 ± 0.13 | 0.69 ± 0.17 | 0.27 ± 0.11 | 0.59 ± 0.12 | L | 0.91 ± 0.02 | 1.86 ± 0.31 | −0.32 ± 0.08 | 0.44 ± 0.07 | - |

P3 (2) | 0.28 ± 0.13 | 0.61 ± 0.17 | 0.28 ± 0.10 | 0.59 ± 0.15 | L | 0.51 ± 0.06 | 10.11 ± 1.29 | - | - | - |

P1P8 (2) | 0.25 ± 0.13 | 0.69 ± 0.18 | 0.27 ± 0.13 | 0.58 ± 0.17 | L | 0.91 ± 0.01 | 1.76 ± 0.21 | −0.09 ± 0.06 | 0.28 ± 0.02 | - |

P1P4P7 (2) | 0.24 ± 0.13 | 0.69 ± 0.18 | 0.29 ± 0.13 | 0.60 ± 0.18 | M | 0.95 ± 0.01 | 1.08 ± 0.11 | 0.13 ± 0.22 | −0.50 ± 0.22 | 0.65 ± 0.20 |

P2P6P7 (2) | 0.24 ± 0.14 | 0.68 ± 0.15 | 0.26 ± 0.10 | 0.57 ± 0.12 | L | 0.80 ± 0.11 | 4.34 ± 2.18 | 0.63 ± 0.22 | −0.85 ± 0.29 | 0.67 ± 0.10 |

P1P4 (3) | 0.24 ± 0.14 | 0.67 ± 0.17 | 0.28 ± 0.15 | 0.58 ± 0.15 | M | 0.88 ± 0.07 | 2.65 ± 1.36 | −0.25 ± 0.14 | 0.70 ± 0.02 | - |

P2P3P6 (3) | 0.25 ± 0.12 | 0.73 ± 0.15 | 0.29 ± 0.12 | 0.59 ± 0.17 | L | 0.92 ± 0.01 | 1.63 ± 0.14 | 0.08 ± 0.25 | −1.04 ± 0.62 | 1.19 ± 0.45 |

P3P7P8 (3) | 0.27 ± 0.13 | 0.69 ± 0.19 | 0.31 ± 0.13 | 0.61 ± 0.12 | L | 0.92 ± 0.01 | 1.69 ± 0.28 | 0.84 ± 0.34 | −1.18 ± 0.33 | 0.54 ± 0.05 |

P1 (17) | 0.28 ± 0.14 | 0.61 ± 0.19 | 0.31 ± 0.15 | 0.46 ± 0.28 | L | 0.56 ± 0.07 | 9.23 ± 1.40 | - | - | - |

^{1}Parameters are presented for building A. For C values, light (L), medium (M) or heavy (H) is used and the reader is referred to Table 4 for numerical values as well as for U values.

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**MDPI and ACS Style**

Hietaharju, P.; Ruusunen, M.; Leiviskä, K.
A Dynamic Model for Indoor Temperature Prediction in Buildings. *Energies* **2018**, *11*, 1477.
https://doi.org/10.3390/en11061477

**AMA Style**

Hietaharju P, Ruusunen M, Leiviskä K.
A Dynamic Model for Indoor Temperature Prediction in Buildings. *Energies*. 2018; 11(6):1477.
https://doi.org/10.3390/en11061477

**Chicago/Turabian Style**

Hietaharju, Petri, Mika Ruusunen, and Kauko Leiviskä.
2018. "A Dynamic Model for Indoor Temperature Prediction in Buildings" *Energies* 11, no. 6: 1477.
https://doi.org/10.3390/en11061477