# Integration of Flow Temperatures in Unit Commitment Models of Future District Heating Systems

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## Abstract

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## 1. Introduction

- How can varying temperatures be integrated in linear optimization models of future DH systems?
- How does the flow temperature level influence the operational results?
- How significant are the deviations between results modeled with varying and constant temperatures?

## 2. State of Research and Scientific Contribution

_{2}emissions. A stochastic programming approach to address parametric uncertainty considering a broad range of technologies such as extraction turbines, peak load boilers (PLB), and TES can be found in Reference [5]. Other models include distributed combined heat and power generation in virtual power plants [6], the assessment of TES in centralized systems [7], and the analysis of DH systems in renewable energy-based power systems [8]. Although all of these models differ with respect to their research questions, many mathematical formulations are similar for the modeled components [2].

## 3. Model Description

#### 3.1. General Overview

#### 3.2. Technology Representation

#### 3.2.1. Combined Heat and Power

#### 3.2.2. Power to Heat

#### 3.2.3. Thermal Energy Storage

#### 3.2.4. Peak Load Boiler

#### 3.3. Formulation as a Mixed-Integer Linear Program

_{2}certificates, variable costs of electricity generation, and fuel-dependent start costs. In contrast, all revenues for delivered heat and power are expressed in Equation (5). Costs and revenues for other technologies are expressed in the same way and are always related to the specific input and output quantities. If not mentioned explicitly, all (non-heat) capacities are assumed to be electrical capacities in the following.

#### 3.3.1. System Heat Balance

#### 3.3.2. Combined Heat and Power

#### 3.3.3. Power to Heat

#### 3.3.4. Thermal Energy Storage

#### 3.3.5. Peak Load Boiler

#### 3.3.6. Additional Formulations

## 4. Case Study of a District Heating System in Germany

#### 4.1. Setup

_{2}-certificates is here assumed with 50 EUR/t CO

_{2}in contrast to 21 EUR/t CO

_{2}in the BAU-scenario. The optimization against these significantly different market conditions ensures the general validity of conclusions for temperature integration methods in unit commitment models.

#### 4.2. Optimization Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

$abs$ | Absolute |

$cond$ | Condenser |

$dh$ | District heating |

$er$ | Edge region |

$el$ | Electrical |

$elm$ | Electricity market |

$emi$ | Emissions |

$fg$ | Flue gas |

$fuel$ | Fuel related |

$init$ | Initial value |

$rel$ | Relative |

$start$ | Start related |

$tmp$ | Temporary |

$var$ | Variable |

$wodh$ | Without district heating |

Indices and sets | |

$c\in C$ | Index and set of all CHP units |

$d\in D$ | Index and set of all heat demands |

$h\in H$ | Index and set of all heat pumps |

$e\in E$ | Index and set of all electric boilers |

$b\in B$ | Index and set of all peak load boilers |

$s\in S$ | Index and set of all thermal storages |

$u\in U$ | Index and set of all production units defined as $(U=C\cup H\cup E\cup B)$ |

$t\in T$ | Index and set of all time steps |

${T}_{u}^{er}$ | Set of timesteps in edge region |

Parameters | |

${\alpha}_{c,t}^{i}$ | Alpha coefficient of CHP unit |

${\beta}_{c,t}$ | Power loss factor of CHP unit |

${\u03f5}_{h,t}^{m}$ | Slope of heat pump unit COP |

${\u03f5}_{h,t}^{b}$ | y-intercept of heat pump unit COP |

${{\displaystyle \underline{\eta}}}_{c,t}^{wodh}$ | Max. electrical efficiency of CHP unit |

${\overline{\eta}}_{c,t}^{wodh}$ | Min. electrical efficiency of CHP unit |

${\eta}_{b,t}^{th}$ | Thermal efficiency of boiler unit |

${\eta}_{s,t}^{tmp}$ | Temporal efficiency of storage unit |

${\eta}_{s,t}^{dh,in}$ | Efficiency of storage unit charging |

${\eta}_{s,t}^{dh,out}$ | Efficiency of storage unit discharging |

$\tau $ | Time increment |

${b}_{u,t}$ | y-intercept of linear function |

${c}_{u,t}^{cmp,var}$ | Additional variable costs |

${c}_{t}^{dh}$ | District heating price |

${c}_{t}^{elm}$ | Electricity price |

${c}_{u,t}^{emi}$ | CO_{2} related costs |

${c}_{u,t}^{fuel}$ | Fuel related costs |

${c}_{u,t}^{start}$ | Start related costs |

${{\displaystyle \underline{p}}}_{c,t}^{wodh}$ | Min. electrical load of CHP unit |

${\overline{p}}_{c,t}^{wodh}$ | Max. electrical load of CHP unit |

${{\displaystyle \underline{q}}}_{s,t}^{dh,in}$ | Min. input heat flow of storage unit |

${\overline{q}}_{s,t}^{dh,in}$ | Max. input heat flow of storage unit |

${{\displaystyle \underline{q}}}_{s,t}^{dh,out}$ | Min. output heat flow of storage unit |

${\overline{q}}_{s,t}^{dh,out}$ | Max. output heat flow of storage unit |

${{\displaystyle \underline{q}}}_{b,t}^{dh}$ | Min. heat flow of boiler unit |

${\overline{q}}_{b,t}^{dh}$ | Max. heat flow of boiler unit |

${{\displaystyle \underline{q}}}_{c,t}^{cond}$ | Min. heat flow in condenser |

${{\displaystyle \underline{l}}}_{c,t}^{fg,rel}$ | Rel. min. flue gas losses of CHP unit |

${\overline{l}}_{c,t}^{fg,rel}$ | Rel. max. flue gas losses of CHP unit |

${m}_{u,t}$ | Slope of linear function |

${n}_{u}^{up}$ | Min. uptime of unit |

${n}_{u}^{down}$ | Min. downtime of unit |

${t}^{1}$ | First time step of optimization period |

${t}^{max}$ | Last time step of optimization period |

Variables | |

${{\displaystyle \underline{L}}}_{c,t}^{fg,abs}$ | Abs. min. flue gas losses of CHP unit |

${\overline{L}}_{c,t}^{fg,abs}$ | Abs. max. flue gas losses of CHP unit |

${P}_{u,t}$ | Electrical power of unit |

${\dot{Q}}_{u,t}^{fuel}$ | Fuel supply of unit |

${\dot{Q}}_{u,t}^{dh}$ | Heat flow of unit into grid |

${\dot{Q}}_{d,t}^{dh}$ | Aggregated heat load |

${\dot{Q}}_{s,t}^{dh,in}$ | Heat flow from grid into storage unit |

${\dot{Q}}_{s,t}^{dh,out}$ | Heat flow from storage unit into grid |

${Q}_{s,t}$ | Amount of heat in storage |

${Y}_{u,t}$ | Status variable of unit |

${Y}_{u,t}^{start}$ | Start variable of unit |

${Y}_{u,t}^{stop}$ | Stop variable of unit |

Other | |

${C}_{u,t}$ | Cost term of heat production unit |

${R}_{u,t}$ | Revenue term of heat production unit |

${T}_{t}^{st}$ | Supply temperature |

## Appendix A

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**Figure 2.**Modeled technologies with related energy flows. (HLD—heat load, CHP—combined heat and power plant, HP—heat pump, EHB—electric heating boiler, PLB—peak load boiler, TES—thermal energy storage)

**Figure 4.**Validation of the resulting extraction steam turbine dispatch (diamond symbols) with the pre-calculated operational ranges in the business-as-usual (BAU) high scenario.

**Figure 5.**Load duration curves for variable (solid) and constant (dashed) medium supply temperature setups in the PTH (power to heat) scenario.

**Figure 8.**Analysis of annual heat supply for additional PTH scenarios and different sensitivities (MRT—minimum runtimes; SC—start costs).

Parameter | High | Medium | Low |
---|---|---|---|

Supply temperature mean | 91 | 70 | 50 |

Supply temperature standard deviation | 10 | 8 | 6 |

Supply temperature maximum | 124 | 95 | 68 |

Supply temperature minimum | 66 | 51 | 36 |

Return temperature | 60 | 40 | 20 |

**Table 2.**Case study technology data (Heat output expressed as share of maximum heat load; capacity given in h of maximum annual heat load; BPT—back pressure turbine, CET—combined cycle extraction turbine, ICE—internal combustion turbine, COP—coefficient of performance).

Technology | Nominal Heat Output | Thermal Efficiency | Electrical Efficiency | Efficiency | COP |
---|---|---|---|---|---|

BPT | 25% | 62–77% | 15–28% | - | - |

CET | 25% | 0–37% | 37–54% | - | - |

ICE | 25% | 16–46% | 40–44% | - | - |

EHB | 25% | - | - | 99% | - |

HP | 25% | - | - | - | 1.32–3.18 |

PLB | 25% | - | - | 95% | - |

Capacity | Efficiency | ||||

TES | 2 h | 99% (in/out) |

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**MDPI and ACS Style**

Boysen, C.; Kaldemeyer, C.; Hilpert, S.; Tuschy, I.
Integration of Flow Temperatures in Unit Commitment Models of Future District Heating Systems. *Energies* **2019**, *12*, 1061.
https://doi.org/10.3390/en12061061

**AMA Style**

Boysen C, Kaldemeyer C, Hilpert S, Tuschy I.
Integration of Flow Temperatures in Unit Commitment Models of Future District Heating Systems. *Energies*. 2019; 12(6):1061.
https://doi.org/10.3390/en12061061

**Chicago/Turabian Style**

Boysen, Cynthia, Cord Kaldemeyer, Simon Hilpert, and Ilja Tuschy.
2019. "Integration of Flow Temperatures in Unit Commitment Models of Future District Heating Systems" *Energies* 12, no. 6: 1061.
https://doi.org/10.3390/en12061061