#
Control of Heat Pumps with CO_{2} Emission Intensity Forecasts

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Model

#### 3.1. Assumptions

#### 3.2. State Space Equations

#### 3.3. MPC

**A**states the dynamic behavior of the system, whereas matrix

**B**specifies how the controllable input signals enter the system, and

**E**specifies the uncontrollable input signals. Furthermore, $\mathit{C}$ is a constant matrix that specifies the controllable state(s), in this case, $\mathit{C}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$. For a deeper explanation, please refer to References [6,24].

_{max}is the maximum power input signal the heat pump can receive. As it may not always be possible to meet the temperature demand, a slack variable ${v}_{k}$ is introduced and connected to the violation penalty ${p}_{v}$. This value is set relatively high to avoid temperature violations.

## 4. Inputs for the Model

_{year}. The minimum requirements for outer walls, roof, windows and doors are listed by year in Table 1.

#### Forecasts

**Case**: This takes the exact value of a future CO${}_{2}$ emission intensity as prediction hence, a perfect forecast. This provides an upper limit of CO${}_{2}$ savings._{Ideal}**Case**: This takes the CO${}_{2}$ emission forecast developed in Reference [18] and represents the performance of the MPC with real forecasts._{Real}**Case**: This makes no use of forecasts and will thus result in a non predictive controller that simply controls the heat pump keeping the temperature at the lower limit if possible._{Trivial}

## 5. Results and Discussion

_{2018}, see Table 1.

**Heating system and varying set points:**Both radiator and floor heating are considered and the use of varying set points (lower temperature during the night).**Horizon of forecasts:**To get an idea about how long horizons actually are needed to get a well performing MPC.**Size of heat pumps:**Essential for comparing the buildings and to know whether the potential is reached. Also economically, this is important because as the price increases with larger heat pumps. This will become a compromise between price and CO${}_{2}$ emission. The default values for the family house and office are the minimum sizes required to meet the heat demand on the coldest day (−12 ${}^{\circ}$C); 3 and 13 kW_{heat}respectively Appendix (see Appendix C for calculations). Requirements are thus a 1 and 4.3 kW input signal respectively according to Equation (1).**Insulation and concrete thicknesses:**These will be adjusted to see the impact of levels of insulation and heat capacity. The default thicknesses and material properties are shown in Appendix C.

_{Trivial}and Case

_{Real}on a four day period for the family house is shown. The result of Case

_{Trivial}is slightly different for the two heating systems. With radiators, it needs to heat more continuously than the floor heating throughout the day. This is because the radiators transfer the heat directly to the internal air, and not through the large heat capacity in the floor, resulting in a much faster response. In both cases, the heating is switched off during the night time to reach the lower set point. However, the floor heating violates the temperature restrictions more during the morning while heating the house, which is due to its slow response. Case

_{Real}seeks to only switch on the heat pump during low emission periods. The radiator system does this well, but it is clearly limited by the maximum indoor temperature limit and the power input decays immediately to avoid temperature violation. In the floor heating system, the heat pump can operate at full load for a longer time using the floor as storage. An interesting point is that using day and night profiles, Case

_{Real}has no benefits of letting the temperature drop during the night because of: (i) the temperature response is too slow and (ii) the emissions are usually lowest during the night, so this is the best time to use the heat pump. Contrarily, the indoor temperature in the radiator system occasionally drops during the night if there is no significant drop in CO${}_{2}$ emissions.

_{Real}.

_{Trivial}switches on at six AM every morning and the emission peak is happening already at four AM (Figure 7). For example, when using a two-hour control horizon, the heat pump will be forced to switch on at four AM instead and thus increase the emissions.

_{2}emission savings will increase with more insulation.

#### Model Simplification

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

CHP | Combined Heat and Power |

MPC | Model Predictive Control |

## Appendix A. Kernel Smoothing

**Figure A1.**Solar irradiation 6 h horizon forecasts for the 21st of June 2017 (updated at 2 a.m., 8 a.m., 14 p.m. and 20 p.m.) and estimates of the real time irradiation (solid line) w/o splines (left). Estimates of the real-time irradiation using splines and the kernel smoothing approach (right)

## Appendix B. Building Parts

**Figure A3.**Definition of model parameters (heat capacities and thermal resistances) in the roof (left), floor (middle) and wall (right).

## Appendix C. Parameter Estimation and Heat Pump Dimensions

**Table A1.**Thickness ($\zeta $), density ($\rho $), heat capacity ($\mathbf{C}$), thermal conductivity ($\mathit{k}$) and thermal resistance ($\mathit{R}$) of all the building parts. The thermal resistance is; $\mathit{R}=\frac{\zeta}{\mathit{k}}$. Thicknesses are examples for a BR18 building.

#### Heat Pump Dimensions

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**Figure 1.**The figure is the merit order illustrated with a supply and demand curve. The x-axis is the accumulated generators in the power system and the y-axis is their corresponding costs. The highest generator in the merit order is the one crossing with the demand curve—a coal Combined Heat and Power (CHP) plant in this example. The average emissions are a weighted average from all activated generators. The

**marginal generator**is the generator that will be activated by moving the demand line slightly to the right (dashed blue line). Data source: Nord Pool AS.

**Figure 2.**Marginal CO${}_{2}$ emissions (real-time), Temperature and solar irradiation (forecasts) plotted for the evaluated period.

**Figure 3.**Solar irradiation 6-h horizon forecasts (left) for the 21st of June 2017 (updated at 2 a.m., 8 a.m., 2 p.m. and 8 p.m.). Model of the real-time irradiation derived from a kernel smoothing approach on the updated forecasts (right).

**Figure 5.**Varying temperature constraints–night [11pm:5am]; 18 ${}^{\circ}$C. Note the CO${}_{2}$ emission intensity and heating does not follow the Y-axis range, but rather the specified range in the colour legend.

**Figure 6.**Savings with respect to the forecast horizon. Shown for both an ideal (perfect) and the Real forecasts.

**Figure 7.**Average heat pump load in a family house with floor heating (200 mm floor concrete) versus the hour of the day for both Case

_{Triviall}and Case

_{Real}. Hourly average marginal CO${}_{2}$ emissions are shown in green.

**Figure 8.**CO

_{2}emission reduction as a function of the size of the heat pump in kW shown for the real forecasts.

**Figure 9.**Savings with respect to the building code year and concrete thickness in the floor. Refer to Table 1.

**Table 1.**Building code requirements to insulation properties by year. Windows requirements are speficied with both U-values, glazing (g) and the estimated net heat transfer through the window into the room, ${E}_{\mathrm{ref}}$ (follows; ${E}_{\mathrm{ref}}=194.4$ g$-90.36$ U ).

BC | Walls | Roof | Doors | Windows | ||
---|---|---|---|---|---|---|

Year | $\mathit{U}\left[\frac{\mathbf{W}}{{\mathbf{m}}^{2}\mathbf{K}}\right]$ | $\mathit{U}\left[\frac{\mathbf{W}}{{\mathbf{m}}^{2}\mathbf{K}}\right]$ | $\mathit{U}\left[\frac{\mathbf{W}}{{\mathbf{m}}^{2}\mathbf{K}}\right]$ | $\mathit{U}\left[\frac{\mathbf{W}}{{\mathbf{m}}^{2}\mathbf{K}}\right]$ | ${\mathit{E}}_{\mathit{r}\mathit{e}\mathit{f}}\left[\frac{\mathbf{kWh}}{{\mathbf{m}}^{2}}\right]$ | $\mathit{g}[-]$ |

1977 | 1 | 0.45 | 3.6 | 3.6 | −174.314 ** | 0.777 ** |

1979–1985 | 0.4 | 0.2 | 2 | 2.9 | −117.378 ** | 0.744 ** |

1995–1998 | 0.4 | 0.2 | 2 | 2.3 | −69.934 ** | 0.709 ** |

2008 | 0.4 | 0.2 | 2 | 2 | −46.909 ** | 0.688 ** |

2010 | 0.3 | 0.2 | 2 | 1.8 ** | −33 | 0.673 ** |

2015–2018 | 0.14 * | 0.1 * | 2 | 1.6 ** | −17 | 0.654 ** |

**Table 2.**Building dimensions and model parameters for both buildings based on BC${}_{2018}$—calculations are as specified in Appendix C.

Family House | Office Building | ||
---|---|---|---|

${A}_{f}$ | ${\mathrm{m}}^{2}$ | 156 | 1250 |

${A}_{w}$ | ${\mathrm{m}}^{2}$ | 107 | 302 |

${A}_{\mathrm{doors}}$ | ${\mathrm{m}}^{2}$ | 4 | 13 |

${A}_{\mathrm{windows}}$ | ${\mathrm{m}}^{2}$ | 14 | 39 |

${R}_{\mathrm{e},\mathrm{a}}$ | $\frac{\mathrm{K}}{\mathrm{kW}}$ | 10.398 | 2.379 |

${R}_{\mathrm{i},\mathrm{e}}$ | $\frac{\mathrm{K}}{\mathrm{kW}}$ | 1.190 | 0.269 |

${R}_{\mathrm{f},\mathrm{i}}$ | $\frac{\mathrm{K}}{\mathrm{kW}}$ | 1.442 | 0.180 |

${C}_{\mathrm{e}}$ | $\frac{\mathrm{kWh}}{\mathrm{K}}$ | 7.508 | 39.527 |

${C}_{\mathrm{f}}$ | $\frac{\mathrm{kWh}}{\mathrm{K}}$ | 3.198 | 25.623 |

${C}_{\mathrm{i}}$ | $\frac{\mathrm{kWh}}{\mathrm{K}}$ | 0.876 | 6.944 |

**Table 3.**Min. savings are derived from a concrete thickness of 10 mm. Max. savings are derived from concrete thicknesses of 200 mm by default.

Floor | Radiator | |||
---|---|---|---|---|

BC${}_{\mathbf{1977}}$ | BC${}_{\mathbf{2015}:\mathbf{2018}}$ | BC${}_{\mathbf{1977}}$ | BC${}_{\mathbf{2015}:\mathbf{2018}}$ | |

Family house | ||||

Min. savings | 0.5% | 9.4% | 1.8% | 9.9% |

Max. savings | 2.7% | 17.4% | 2.8% | 12.4% |

Office building | ||||

Min. savings | 1.4% | 7.8% | 2.4% | 8.1% |

Max. savings | 9.2% | 16.0% | 4.1% | 12.3% |

© 2020 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**

Leerbeck, K.; Bacher, P.; Junker, R.G.; Tveit, A.; Corradi, O.; Madsen, H.; Ebrahimy, R. Control of Heat Pumps with CO_{2} Emission Intensity Forecasts. *Energies* **2020**, *13*, 2851.
https://doi.org/10.3390/en13112851

**AMA Style**

Leerbeck K, Bacher P, Junker RG, Tveit A, Corradi O, Madsen H, Ebrahimy R. Control of Heat Pumps with CO_{2} Emission Intensity Forecasts. *Energies*. 2020; 13(11):2851.
https://doi.org/10.3390/en13112851

**Chicago/Turabian Style**

Leerbeck, Kenneth, Peder Bacher, Rune Grønborg Junker, Anna Tveit, Olivier Corradi, Henrik Madsen, and Razgar Ebrahimy. 2020. "Control of Heat Pumps with CO_{2} Emission Intensity Forecasts" *Energies* 13, no. 11: 2851.
https://doi.org/10.3390/en13112851