# A Novel Temperature-Independent Model for Estimating the Cooling Energy in Residential Homes for Pre-Cooling and Solar Pre-Cooling

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## Abstract

**:**

## 1. Introduction

#### 1.1. Existing Research on Pre-Cooling

#### 1.2. Existing Research on Solar Pre-Cooling

#### 1.3. Paper Contribution

## 2. AccuRate

- T
_{FR}(°C) is the free running temperature, i.e., the indoor temperature with no AC; - T
_{AC}(°C) is the air conditioned temperature, i.e., the indoor temperature where AC is controlled to maintain thermal comfort using a temperature setpoint; and - EE
_{in}(kWh) is the electrical energy consumed by the AC unit.

_{FR}, T

_{AC}) and air conditioning data (EE

_{in}) were generated for each template home for four bedrooms (thermal zones). Assumed occupancy profiles, thermostat settings, and window, curtain, and external shading operations used by AccuRate when generating the simulated measurements are given in [37], Appendix B.

## 3. Method

_{FR}, T

_{AC}, and EE

_{in}, and resulted in the derivation of four linear equations, where their respective slope intercepts represent a thermal efficiency metric of a thermal zone in the template residential home.

_{in}and two derived variables: T

_{in}and T

_{tm}. Where T

_{in}represents the temperature difference due to cooling energy injected into the thermal zone by the AC unit and T

_{tm}represents the temperature difference due to the cooling energy stored in the thermal mass.

_{FR}and T

_{AC}, defined as T

_{FRAC}, which is equivalent to the cooling energy in the thermal zone.

_{FR}, T

_{AC}, T

_{in}, and T

_{tm}, a time-series (hourly) plot illustrating their typical behaviour during the cooling period is presented in Figure 1. The cooling period consists of two phases, the charging phase and the discharging phase, where h

_{ch}is the first hour of the charging phase and h

_{dc}is the first hour of the discharging phase. During the charging phase, electrical energy (EE

_{in}) is consumed by the AC unit and cooling energy is injected into the thermal zone. During the discharging phase, no new cooling energy is injected and T

_{tm}decays over time. T

_{in}is represented by the green arrows and T

_{tm}by the orange bars.

#### 3.1. Linear Equations

^{F}), and the thermal mass slow discharging rate (d

^{S}).

**AC cooling efficiency (e).**There is a linear relationship at h = h

_{ch}between EE

_{in}and T

_{in}, as defined by Equation (1):

_{m}and e

_{i}are the slope and intercept, respectively, of the AC cooling efficiency (e), CoP is the coefficient of performance of the AC unit, and h is the hour. As T

_{in}= T

_{FRAC}at h = h

_{ch}, this linear relationship indicates that T

_{FRAC}is a measure of the cooling energy in the thermal zone. The average R

^{2}value of this linear relationship for all template homes is 0.93. Table A2 in Appendix A gives the AC cooling efficiency R

^{2}values for all template homes. As an example, Figure 2 gives the scatter plot of EE

_{in}versus T

_{in}for Bedroom 1 in template home M2L.

**Thermal mass charging rate (c).**There is a second linear relationship at h = h

_{ch}between T

_{in}and T

_{tm}, as defined by Equation (2):

_{m}and c

_{i}are the slope and intercept, respectively, of the thermal mass charging rate (c). Equation (2) represents the rate in which the thermal mass is charged with cooling energy. The average R

^{2}value of this linear relationship for all template homes is 0.9. Table A3 in Appendix A gives the thermal mass charging rate R

^{2}values for all template homes. As an example, Figure 3 gives the scatter plot of T

_{in}[h

_{ch}] versus T

_{tm}[h

_{ch}+ 1] for Bedroom 1 in template home M2L.

**Thermal mass fast discharging rate (d**There is a linear relationship at h = h

^{F})._{dc}between T

_{FRAC}and T

_{tm}, as defined by Equation (3):

^{F}). Equation (3) represents the rate at which cooling energy is discharged from the thermal mass at h = h

_{dc}. The average R

^{2}value of this linear relationship for all template homes is 0.91. Table A4 in Appendix A gives the thermal mass fast discharging rate R

^{2}values for all template homes. As an example, Figure 4 gives the scatter plot of T

_{FRAC}[h

_{dc}] versus T

_{tm}[h

_{dc}+ 1] for Bedroom 1 in template home M2L.

**Thermal mass slow discharging rate (d**There is a linear relationship for h > h

^{S})._{dc}between T

_{tm}[h + 1] and T

_{tm}[h], as defined by Equation (4):

^{S}). Equation (4) represents the rate at which cooling energy is discharged from the thermal mass for h > h

_{dc}. The average R

^{2}value of this linear relationship for all template homes is 0.99. Table A5 in Appendix A gives the thermal mass slow discharging rate R

^{2}values for all template homes. As an example, Figure 5 gives the scatter plot of T

_{tm}[h] versus T

_{tm}[h + 1] for h > h

_{dc}for Bedroom 1 in template home M2L.

_{dc}was calculated. It was discovered that the discharging rate for the first hour was clearly faster than later hours, but then remained relatively unchanged, similar to what was discovered in [38]. Therefore, it was decided that two rates, one for the first hour (fast), and a second for the hours following (slow), were sufficient to accurately model thermal discharging.

#### 3.2. Implementation of the Model

- (1)
- The sum of T
_{in}and T_{tm}equals T_{FRAC}, as defined by Equation (5):

- (2)
- Equation (1) can be used to calculate T
_{in}at any hour h. - (3)
- During the charging phase, the thermal dynamics of the home are assumed to operate in accordance with Equation (6):

_{in}from the previous hour) and undischarged energy preserved in the thermal mass from the previous hour.

_{tm}and T

_{FRAC}at any hour of the charging or discharging phase. A flowchart describing the process for implementing the model is given in Figure 6. The flowchart shows that EE

_{in}is the only input from the AccuRate dataset used to implement the model. Neither T

_{FR}nor T

_{AC}are required, which is evidence that the model is temperature-independent.

## 4. Results

_{AC}), and the AccuRate air conditioning temperature (T

_{AC}) during the charging and discharging period for 15 of the 24 template homes, where eT

_{AC}is calculated using Equation (7):

_{FRAC}is calculated using the process described in the flowchart in Figure 6.

_{FR}), air conditioned temperature (T

_{AC}), and the estimated air conditioned temperature (eT

_{AC}) for the first four days for template home M2L. The light orange boxes highlight the periods of charging and discharging, which is also the period over which the CV-RMSE and MAE are calculated. The MAE during the charging and discharging periods is less than 0.5 °C at all hours, demonstrating the accuracy of the proposed model.

## 5. Model Significance

## 6. Conclusions

- Extending the model to solar pre-heating;
- An analysis of the potential of solar pre-cooling using measured residential energy data to examine the impact of build type, climate, and different solar pre-cooling control and optimisation algorithms to reduce peak demand, increase minimum demand, and reduce electricity costs;
- The development of proof to explain the strong R
^{2}values for the thermal metrics; and - The development of a model to derive the thermal metrics of a building from its energy data.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c | is the thermal mass charging rate |

c_{m} | is the slope of the thermal mass charging rate (c) |

c_{i} (°C) | is the intercept of the thermal mass charging rate (c) |

CoP | is the coefficient of performance of the AC unit |

CV-RMSE (%) | is the Coefficient of the Variation of the Root Mean Square Error |

d^{F} | is the thermal mass fast discharging rate |

${d}_{m}^{F}$ | is the slope of the thermal mass fast discharging rate (d^{F}) |

${d}_{i}^{F}$ (°C) | is the intercept of the thermal mass fast discharging rate (d^{F}) |

d^{S} | is the thermal mass slow charging rate |

${d}_{m}^{S}$ | is the slope of the thermal mass slow discharging rate (d^{S}) |

${d}_{i}^{S}$ (°C) | is the intercept of the thermal mass slow discharging rate (d^{S}) |

e | is the AC cooling efficiency |

e_{m} (°C/kWh) | is the slope of the AC cooling efficiency (e) |

e_{i} (°C) | is the intercept of the AC cooling efficiency (e) |

EE_{in} (kWh) | is the electrical energy consumed by the AC unit |

$E{E}_{in}^{*}$ (kWh) | is the equivalent electrical energy consumed by the AC unit to attain T_{tm} |

h_{ch} | is the first hour of the charging phase |

h_{dc} | is the first hour of the discharging phase |

MAE (°C) | is the Mean Absolute Error |

R^{2} | is the coefficient of determination |

T_{AC} (°C) | is the air conditioned temperature. |

T_{FR} (°C) | is the free running temperature, the indoor temperature with no air conditioning. |

T_{FRAC} (°C) | is the difference between T_{FR} and T_{AC}. |

T_{in} (°C) | represents the temperature difference due to cooling energy injected into the thermal zone by the AC unit |

T_{tm} (°C) | represents the temperature difference due to the cooling energy stored in the thermal mass |

## Appendix A

#### Solar Pre-Cooling Example

_{solar}) at hours h

_{ch}= 13 and h

_{dc}= 14 is diverted to the AC unit (EE

_{in}). In this scenario, the existing (peak) AC consumption (EE

_{peak}) at h = 16 is to be reduced through solar pre-cooling. Figure A1b shows the resultant calculated temperatures T

_{tm}, T

_{in}, and T

_{peak}, where T

_{peak}is the existing temperature due to EE

_{peak}. In this example the home is assumed to have thermal metrics e

_{m}= CoP = 1, c

_{m}= 0.5, ${d}_{m}^{F}$ = 0.75, ${d}_{m}^{s}$ = 0.85, and e

_{i}= c

_{i}= ${d}_{i}^{F}$ = ${d}_{i}^{S}$ = 0. Table A1 gives the calculation of T

_{in}[h], T

_{tm}[h], T

_{FRAC}[h], and T

_{tm}[h + 1] for the example, with the relevant equations also included.

_{tm}= 3.2 °C, is deducted from T

_{peak}. Re-arranging Equation (1), the reduction to EE

_{peak}is calculated to be 3.2 kWh, where T

_{tm}= 3.2 °C, e

_{m}= CoP = 1 and e

_{i}= 0. This example shows that no temperature inputs (T

_{FR}, T

_{AC}, or outdoor temperature) are required by the model to simulate solar pre-cooling.

**Figure A1.**Illustration of example (

**a**) gives the energy values (EE

_{solar}, EE

_{in}, and EE

_{peak}) and (

**b**) gives the temperature values (T

_{in}, T

_{tm}, and T

_{peak}).

**Table A1.**Calculation of T

_{in}[h], T

_{tm}[h], T

_{FRAC}[h], and T

_{tm}[h + 1] for the example; the equation used to make each calculation is included.

h = h_{ch} | h = h_{dc} | |

T_{in}[h]
| $\left(1\right){e}_{m}\times CoP\times E{E}_{in}\left[h\right]=5$ | $\left(1\right){e}_{m}\times CoP\times E{E}_{in}\left[h\right]=2.5$ |

T_{tm}[h]
| 0 | 2.5 |

T_{FRAC}[h]
| $\left(5\right){T}_{in}\left[h\right]+{T}_{tm}\left[h\right]=5$ | $\left(5\right){T}_{in}\left[h\right]+{T}_{tm}\left[h\right]=5$ |

T_{tm}[h + 1]
| $\left(6\right){c}_{m}\times {T}_{in}\left[h\right]+{d}_{m}^{f}\times {T}_{tm}\left[h\right]=2.5$ | $\left(3\right){d}_{m}^{f}\times {T}_{FRAC}\left[h\right]=3.75$ |

h = 15 | h = 16 | |

T_{in}[h]
| 0 | 0 |

T_{tm}[h]
| 3.75 | 3.2 |

T_{FRAC}[h]
| $\left(5\right){T}_{in}\left[h\right]+{T}_{tm}\left[h\right]=3.75$ | |

T_{tm}[h + 1]
| $\left(4\right){d}_{m}^{s}\times {T}_{tm}\left[h\right]=3.2$ |

Template Home | City | AC Cooling Efficiency (e) | ||
---|---|---|---|---|

e_{m} (°C/kWh) | e_{i} (°C) | R^{2} | ||

B2L | Brisbane | 1.38 | −0.15 | 0.91 |

B2M | 1.24 | −0.11 | 0.94 | |

B2H | 1.03 | −0.01 | 0.95 | |

B6L | 1.40 | −0.08 | 0.93 | |

B6M | 1.39 | −0.08 | 0.91 | |

B6H | 0.90 | −0.05 | 0.88 | |

S2L | Sydney | 1.45 | −0.46 | 0.90 |

S2M | 1.24 | −0.08 | 0.96 | |

S2H | 0.80 | −0.02 | 0.93 | |

S6L | 1.39 | −0.11 | 0.94 | |

S6M | 1.36 | −0.14 | 0.89 | |

S6H | 0.94 | −0.12 | 0.83 | |

M2L | Melbourne | 1.22 | −0.31 | 0.98 |

M2M | 1.22 | −0.28 | 0.98 | |

M2H | 0.81 | −0.11 | 0.98 | |

M6L | 1.36 | −0.41 | 0.97 | |

M6M | 1.31 | −0.22 | 0.97 | |

M6H | 0.89 | −0.17 | 0.93 | |

A2L | Adelaide | 1.41 | −1.10 | 0.93 |

A2M | 1.16 | −0.25 | 0.97 | |

A2H | 0.81 | 0.02 | 0.99 | |

A6L | 1.45 | −0.65 | 0.96 | |

A6M | 1.37 | −0.46 | 0.94 | |

A6H | 0.85 | −0.18 | 0.89 |

Template Home | City | Thermal Mass Charging Rate (c) | ||
---|---|---|---|---|

c_{m} | c_{i} (°C) | R^{2} | ||

B2L | Brisbane | 0.78 | 0.06 | 0.91 |

B2M | 0.32 | 0.39 | 0.82 | |

B2H | 0.40 | 0.15 | 0.86 | |

B6L | 0.90 | 0.07 | 0.99 | |

B6M | 0.46 | 0.44 | 0.83 | |

B6H | 0.60 | 0.20 | 0.88 | |

S2L | Sydney | 0.50 | 0.59 | 0.95 |

S2M | 0.40 | 0.31 | 0.94 | |

S2H | 0.35 | 0.27 | 0.69 | |

S6L | 0.76 | 0.32 | 0.93 | |

S6M | 0.48 | 0.54 | 0.81 | |

S6H | 0.53 | 0.30 | 0.81 | |

M2L | Melbourne | 0.53 | 0.38 | 0.99 |

M2M | 0.41 | 0.14 | 0.96 | |

M2H | 0.36 | 0.09 | 0.87 | |

M6L | 0.62 | 0.39 | 0.93 | |

M6M | 0.50 | 0.32 | 0.94 | |

M6H | 0.27 | 0.37 | 0.80 | |

A2L | Adelaide | 0.55 | 0.70 | 0.97 |

A2M | 0.50 | −0.34 | 0.95 | |

A2H | 0.63 | −0.70 | 0.91 | |

A6L | 0.77 | −0.61 | 0.94 | |

A6M | 0.62 | −0.27 | 0.90 | |

A6H | 0.82 | −1.52 | 0.90 |

Template Home | City | Thermal Mass Fast Discharging Rate (d^{F}) | ||
---|---|---|---|---|

${\mathit{d}}_{\mathit{m}}^{\mathit{F}}$ | ${\mathit{d}}_{\mathit{i}}^{\mathit{F}}$ (°C) | R^{2} | ||

B2L | Brisbane | 0.53 | 0.24 | 0.86 |

B2M | 0.58 | 0.10 | 0.84 | |

B2H | 0.67 | 0.07 | 0.81 | |

B6L | 0.74 | 0.12 | 0.90 | |

B6M | 0.68 | 0.10 | 0.87 | |

B6H | 0.67 | 0.06 | 0.88 | |

S2L | Sydney | 0.62 | 0.02 | 0.92 |

S2M | 0.63 | 0.09 | 0.95 | |

S2H | 0.67 | 0.07 | 0.87 | |

S6L | 0.67 | 0.29 | 0.90 | |

S6M | 0.75 | 0.06 | 0.93 | |

S6H | 0.74 | 0.00 | 0.88 | |

M2L | Melbourne | 0.76 | −0.21 | 0.97 |

M2M | 0.63 | −0.09 | 0.95 | |

M2H | 0.75 | −0.15 | 0.97 | |

M6L | 0.87 | −0.31 | 0.97 | |

M6M | 0.81 | −0.14 | 0.97 | |

M6H | 0.73 | −0.12 | 0.85 | |

A2L | Adelaide | 0.86 | −1.05 | 0.93 |

A2M | 0.71 | −0.44 | 0.90 | |

A2H | 0.79 | −0.41 | 0.97 | |

A6L | 0.95 | −1.11 | 0.96 | |

A6M | 0.80 | −0.45 | 0.92 | |

A6H | 0.85 | −0.73 | 0.87 |

Template Home | City | Thermal Mass Slow Discharging Rate (d^{S}) | ||
---|---|---|---|---|

${\mathit{d}}_{\mathit{m}}^{\mathit{S}}$ | ${\mathit{d}}_{\mathit{i}}^{\mathit{S}}$ (°C) | R^{2} | ||

B2L | Brisbane | 0.92 | −0.04 | 0.99 |

B2M | 0.85 | 0.00 | 0.97 | |

B2H | 0.90 | −0.03 | 0.97 | |

B6L | 0.96 | −0.04 | 0.99 | |

B6M | 0.92 | −0.01 | 0.98 | |

B6H | 0.95 | −0.04 | 0.98 | |

S2L | Sydney | 0.91 | −0.01 | 0.99 |

S2M | 0.82 | 0.03 | 0.98 | |

S2H | 0.94 | −0.04 | 0.98 | |

S6L | 0.96 | −0.03 | 1.00 | |

S6M | 0.92 | 0.00 | 0.99 | |

S6H | 0.98 | −0.06 | 0.99 | |

M2L | Melbourne | 0.86 | 0.00 | 0.99 |

M2M | 0.84 | −0.04 | 0.98 | |

M2H | 0.94 | −0.05 | 1.00 | |

M6L | 0.93 | 0.02 | 1.00 | |

M6M | 0.92 | −0.04 | 1.00 | |

M6H | 0.97 | −0.06 | 1.00 | |

A2L | Adelaide | 0.88 | 0.05 | 0.99 |

A2M | 0.87 | −0.04 | 0.98 | |

A2H | 0.96 | −0.08 | 0.99 | |

A6L | 0.95 | 0.01 | 1.00 | |

A6M | 0.90 | −0.02 | 0.99 | |

A6H | 0.97 | −0.06 | 1.00 |

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**Figure 1.**Time-series plot illustrating the typical behaviour of temperatures T

_{FR}, T

_{AC}, T

_{tm}, and T

_{in}during the charging and discharging phase.

**Figure 2.**Scatter plot of EE

_{in}[h

_{ch}] versus T

_{in}[h

_{ch}], defining the AC cooling efficiency (e) for Bedroom 1 in template home M2L.

**Figure 3.**Scatter plot of T

_{in}[h

_{ch}] versus T

_{tm}[h

_{ch}+ 1], defining the thermal mass charging rate (c) for Bedroom 1 in template home M2L.

**Figure 4.**Scatter plot of T

_{FRAC}[h

_{dc}] versus T

_{tm}[h

_{dc}+ 1], defining the thermal mass fast charging rate d

^{F}, for Bedroom 1 in template home M2L.

**Figure 5.**Scatter plot of T

_{tm}[h] versus T

_{tm}[h + 1] for h > h

_{dc}, defining the thermal mass slow charging rate d

^{S}for Bedroom 1 in template home M2L.

**Figure 7.**Time series of the free-running temperature (T

_{FR}), air conditioned temperature (T

_{AC}), and the estimated air conditioned temperature (eT

_{AC}) for four days for template home M2L. The light orange boxes highlight the periods of charging and discharging.

**Table 1.**Build characteristics for the twenty-four template homes created in AccuRate. Rx = the R-value of the construction material (K.m

^{2}/W), SG = Single glazing and DG = Double glazing.

Template Home | City | Climate | Build Type | Walls | Windows | Floors | Ceilings | |
---|---|---|---|---|---|---|---|---|

Star Rating | Build Weight | |||||||

B2L | Brisbane | Humid Subtropical | 2 | Light | R0 | SG | R0 | R0 |

B2M | Medium | R0 | SG | R0 | R0.1 | |||

B2H | Heavy | R0 | SG | R0 | R0.4 | |||

B6L | 6 | Light | R0 | SG | R0 | R4 | ||

B6M | Medium | R0.7 | DG | R0 | R4 | |||

B6H | Heavy | R2.3 | DG | R2.5 | R5 | |||

S2L | Sydney | Humid Subtropical | 2 | Light | R0 | SG | R0 | R0 |

S2M | Medium | R0 | SG | R0 | R0 | |||

S2H | Heavy | R0 | SG | R0 | R0.5 | |||

S6L | 6 | Light | R1.3 | SG | R0 | R4 | ||

S6M | Medium | R0.9 | DG | R0 | R4 | |||

S6H | Heavy | R2.5 | DG | R3 | R5 | |||

M2L | Melbourne | Temperate | 2 | Light | R0 | SG | R0 | R0.1 |

M2M | Medium | R0 | SG | R0 | R0.1 | |||

M2H | Heavy | R0 | SG | R0 | R0 | |||

M6L | 6 | Light | R2 | SG | R0 | R4 | ||

M6M | Medium | R1.1 | DG | R0 | R4 | |||

M6H | Heavy | R1.4 | DG | R3 | R4 | |||

A2L | Adelaide | Mediterranean | 2 | Light | R0 | SG | R0 | R0 |

A2M | Medium | R0 | SG | R0 | R0.1 | |||

A2H | Heavy | R0 | SG | R0 | R0.3 | |||

A6L | 6 | Light | R0.4 | SG | R0 | R2.5 | ||

A6M | Medium | R0.7 | DG | R0 | R4 | |||

A6H | Heavy | R2 | DG | R3 | R4 |

Template Home | City | CV-RMSE (%) | MAE (°C) | Air Conditioning Days |
---|---|---|---|---|

B2L | Brisbane | 19.18 | 0.21 | 63 |

B2M | 19.74 | 0.22 | 62 | |

B2H | 22.08 | 0.24 | 91 | |

B6L | 14.99 | 0.15 | 29 | |

B6M | 22.33 | 0.21 | 52 | |

B6H | N/A | N/A | 3 | |

S2L | Sydney | 20.43 | 0.25 | 21 |

S2M | 21.91 | 0.23 | 27 | |

S2H | N/A | N/A | 8 | |

S6L | N/A | N/A | 7 | |

S6M | 21.82 | 0.25 | 15 | |

S6H | N/A | N/A | 5 | |

M2L | Melbourne | 23.1 | 0.3 | 13 |

M2M | 16.76 | 0.28 | 13 | |

M2H | N/A | N/A | 3 | |

M6L | 18.79 | 0.31 | 12 | |

M6M | N/A | N/A | 5 | |

M6H | N/A | N/A | 2 | |

A2L | Adelaide | 26.53 | 0.44 | 20 |

A2M | 25.77 | 0.43 | 24 | |

A2H | N/A | N/A | 7 | |

A6L | 28.12 | 0.57 | 12 | |

A6M | 26.65 | 0.42 | 18 | |

A6H | N/A | N/A | 8 |

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## Share and Cite

**MDPI and ACS Style**

Heslop, S.; Yildiz, B.; Roberts, M.; Chen, D.; Lau, T.; Naderi, S.; Bruce, A.; MacGill, I.; Egan, R.
A Novel Temperature-Independent Model for Estimating the Cooling Energy in Residential Homes for Pre-Cooling and Solar Pre-Cooling. *Energies* **2022**, *15*, 9257.
https://doi.org/10.3390/en15239257

**AMA Style**

Heslop S, Yildiz B, Roberts M, Chen D, Lau T, Naderi S, Bruce A, MacGill I, Egan R.
A Novel Temperature-Independent Model for Estimating the Cooling Energy in Residential Homes for Pre-Cooling and Solar Pre-Cooling. *Energies*. 2022; 15(23):9257.
https://doi.org/10.3390/en15239257

**Chicago/Turabian Style**

Heslop, Simon, Baran Yildiz, Mike Roberts, Dong Chen, Tim Lau, Shayan Naderi, Anna Bruce, Iain MacGill, and Renate Egan.
2022. "A Novel Temperature-Independent Model for Estimating the Cooling Energy in Residential Homes for Pre-Cooling and Solar Pre-Cooling" *Energies* 15, no. 23: 9257.
https://doi.org/10.3390/en15239257