# Reducing Computational Load for Mixed Integer Linear Programming: An Example for a District and an Island Energy System

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## Abstract

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## 1. Introduction

#### 1.1. The Complexity of Microgrid Supply System Optimization

#### 1.2. State of the Art of Complexity Reduction

^{3}method, which includes one strategy for determining an upper level and two methods for determining the lower level of the overall cost minimization of a design and operational optimization MILP. The upper bound is determined by making design decisions with an aggregated time series and checking the system for operational feasibility with the full time series. For maintaining feasibility, so-called feasibility time steps including the non-feasible time-steps and recalculating the operational optimization. The aggregation used is thereby two-fold: First, the time series are aggregated to typical periods, and then the typical periods are aggregated to a smaller number of segments. In this way, the problem size is further decreased. However, it can only account for storages within each typical period, which means that seasonal storage is not accounted for, as mentioned by Bahl et al. [32]. The lower bound of the problem is determined in two ways: Either by aggregating the input time series to typical periods with typical sequences and simultaneously over- and underestimating supply and demand for each segment while stepwise increasing the number of periods, or by using a common branch-and-bound procedure, which is used as a benchmark. With these formulations and a stepwise increase of the temporal resolution, the proposed algorithm tries to increase the gap between these bounds under a predefined optimality gap.

#### 1.3. Research Objective

## 2. Methodology

#### 2.1. Energy System Optimization Model Formulation

#### 2.2. 2-Level Optimization Approach

#### 2.3. Scenario Definition

## 3. First Case Study—Urban Energy System

#### 3.1. Data Basis

#### 3.2. Technology Portfolio

#### 3.3. Results of the First Case Study

#### 3.3.1. Investigation of Total Annual Costs

#### 3.3.2. Investigation of Computing Time

#### 3.3.3. Investigation of the Supply Technologies

_{th}in 40 typical days and a minimum of −0.12 kW

_{th}in 5 typical days. In the time series aggregation, the mean deviation decreases from −51.8 kW

_{th}in 5 typical days to −17.57 kW

_{th}in 40 typical days. Thus, the mean results show a significantly lower deviation in the 2-Level Approach than in the time series aggregation.

_{th}for all typical days. The optimization with the time series aggregation results in a minimal deviation of −0.02 kW

_{th}for 5 typical days and a maximum of 0.39 kW

_{th}for 40 typical days. A possible reason for the building shifting the CHP installation between the different typical days is that the difference in the TAC is not significant. Hence, its location does not affect the TAC. A further analysis is performed in order to investigate the TAC in relation to the location of the CHP in Section 3.3.5.

#### 3.3.4. Investigation of Peak Load

#### 3.3.5. Impact Analysis of Fixed CHP Position

## 4. Second Case Study—Island System

#### 4.1. Data Basis

_{2}reduction costs grow exponentially with decreasing CO

_{2}emissions. In order to avoid unrealistic surplus capacities of the renewable energy and storage components, and in order to make the hydrogen storages a competitive solution in that system, a maximum total feed-in of 10% by the power plant turned out to be an appropriate percentage. Moreover, this highlights the influence of the presented method on the storage dimensioning if a non-sufficient number of typical days is chosen in the first level. The system is shown in Figure 11. Moreover, the modeling of seasonal storages as proposed by Kotzur et al. [37] is also taken into account, which is especially important for the hydrogen storage.

#### 4.2. Technology Portfolio

_{Fix}in the tables) and fixed operation costs (OPEX

_{Fix}) depending on the decision as to whether these units are chosen (1) or not (0). In contrast to that, the backup plant, the battery and the hydrogen storage are modeled linearly, since their overall costs only depend on their consumed commodity (gas with 20 ct/kWh in case of the backup plant) and their capacities, respectively. This fairly simple layout is chosen to emphasize the big impact of the proposed 2-Level Approach on the computing time even for small systems while maintaining good results. Additionally, a single-node model with a high share of renewable energy was chosen to highlight the impact of the proposed method on seasonal storages while neglecting balancing effects of multi-regional distribution grid modeling.

#### 4.3. Results of the Second Case Study

#### 4.3.1. Investigation of Total Annual Costs

#### 4.3.2. Investigation of Computing Time

#### 4.3.3. Investigation of the Different Technology Capacities

#### 4.3.4. Investigation of the Connection between the Storages

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Load Profiletable

## Appendix B. Demand Characterization

Buildings | Bd1 | Bd2 | Bd3 | Bd4 | Bd5 | Bd6 |
---|---|---|---|---|---|---|

Electricity demand [kWh] | 24,718 | 19,409 | 24,892 | 17,439 | 23,310 | 19,718 |

Heating demand [kWh] | 213,008 | 211,961 | 212,160 | 212,865 | 212,987 | 213,360 |

PV potential roof 1 [kWp] | 16.42 | 16.45 | 17.88 | 17.27 | 16.16 | 16.63 |

PV potential roof 2 [kWp] | 16.42 | 16.45 | 17.88 | 17.27 | 16.16 | 16.63 |

Construction Year | 1965 | 1965 | 1965 | 1965 | 1965 | 1965 |

Building Type | Multi-Family House | Multi-Family House | Multi-Family House | Multi-Family House | Multi-Family House | Multi-Family House |

## References

- Hirsch, A.; Parag, Y.; Guerrero, J. Microgrids: A review of technologies, key drivers, and outstanding issues. Renew. Sustain. Energy Rev.
**2018**, 90, 402–411. [Google Scholar] [CrossRef] - Schütz, T.; Hu, X.; Fuchs, M.; Müller, D. Optimal design of decentralized energy conversion systems for smart microgrids using decomposition methods. Energy
**2018**, 156, 250–263. [Google Scholar] [CrossRef] - Obara, S.Y.; Sato, K.; Utsugi, Y. Study on the operation optimization of an isolated island microgrid with renewable energy layout planning. Energy
**2018**, 161, 1211–1225. [Google Scholar] [CrossRef] - Vafaei, M.; Kazerani, M. Optimal Unit-Sizing of a Wind-Hydrogen-Diesel Microgrid System for a Remote Community. In Proceedings of the IEEE Trondheim PowerTech, Trondheim, Norway, 19–23 June 2011; pp. 1–7. [Google Scholar]
- Mashayekh, S.; Stadler, M.; Cardoso, G.; Heleno, M. A mixed integer linear programming approach for optimal DER portfolio, sizing, and placement in multi-energy microgrids. Appl. Energy
**2017**, 187, 154–168. [Google Scholar] [CrossRef] [Green Version] - Mehleri, E.D.; Sarimveis, H.; Markatos, N.C.; Papageorgiou, L.G. Optimal design and operation of distributed energy systems: Application to Greek residential sector. Renew. Energy
**2013**, 51, 331–342. [Google Scholar] [CrossRef] - Akbari, K.; Jolai, F.; Ghaderi, S.F. Optimal design of distributed energy system in a neighborhood under uncertainty. Energy
**2016**, 116, 567–582. [Google Scholar] [CrossRef] - Basu, A.K.; Chowdhury, S.P.; Chowdhury, S.; Paul, S. Microgrids: Energy management by strategic deployment of DERs-A comprehensive survey. Renew. Sustain. Energy Rev.
**2011**, 15, 4348–4356. [Google Scholar] [CrossRef] - Morvaj, B.; Evins, R.; Carmeliet, J. Optimization framework for distributed energy systems with integrated electrical grid constraints. Appl. Energy
**2016**, 171, 296–313. [Google Scholar] [CrossRef] - Morvaj, B.; Evins, R.; Carmeliet, J. Decarbonizing the electricity grid: The impact on urban energy systems, distribution grids and district heating potential. Appl. Energy
**2017**, 191, 125–140. [Google Scholar] [CrossRef] - Omu, A.; Choudhary, R.; Boies, A. Distributed energy resource system optimisation using mixed integer linear programming. Energy Policy
**2013**, 61, 249–266. [Google Scholar] [CrossRef] - Yang, Y.; Zhang, S.; Xiao, Y. Optimal design of distributed energy resource systems coupled with energy distribution networks. Energy
**2015**, 85, 433–448. [Google Scholar] [CrossRef] - Gabrielli, P.; Gazzani, M.; Martelli, E.; Mazzotti, M. Optimal design of multi-energy systems with seasonal storage. Appl. Energy
**2018**, 219, 408–424. [Google Scholar] [CrossRef] - Helal, S.A.; Najee, R.J.; Hanna, M.O.; Shaaban, M.F.; Osman, A.H.; Hassan, M.S. An energy management system for hybrid microgrids in remote communities. In Proceedings of the IEEE 30th Canadian Conference on Electrical and Computer Engineering (CCECE), Windsor, ON, Canada, 30 April–3 May 2017; pp. 1–4. [Google Scholar]
- Olivares, D.E.; Cañizares, C.A.; Kazerani, M. A Centralized Energy Management System for Isolated Microgrids. IEEE Trans. Smart Grid
**2014**, 5, 1864–1875. [Google Scholar] [CrossRef] - Falke, T.; Krengel, S.; Meinerzhagen, A.-K.; Schnettler, A. Multi-objective optimization and simulation model for the design of distributed energy systems. Appl. Energy
**2016**, 184, 1508–1516. [Google Scholar] [CrossRef] - Prousch, S.; Breuer, C.; Moser, A. Optimization of decentralized energy supply systems. In Proceedings of the 7th International Conference on the European Energy Market, Madrid, Spain, 23–25 June 2010; pp. 1–6. [Google Scholar]
- Fazlollahi, S.; Girardin, L.; Maréchal, F. Clustering Urban Areas for Optimizing the Design and the Operation of District Energy Systems. Comput. Aided Chem. Eng.
**2014**, 33, 1291–1296. [Google Scholar] [CrossRef] - Stadler, P.; Girardin, L.; Ashouri, A.; Maréchal, F. Contribution of Model Predictive Control in the Integration of Renewable Energy Sources within the Built Environment. Front. Energy Res.
**2018**, 6, 22. [Google Scholar] [CrossRef] - Unternährer, J.; Moret, S.; Joost, S.; Maréchal, F. Spatial clustering for district heating integration in urban energy systems: Application to geothermal energy. Appl. Energy
**2017**, 190, 749–763. [Google Scholar] [CrossRef] [Green Version] - Geidl, M.; Andersson, G. A modeling and optimization approach for multiple energy carrier power flow. In Proceedings of the IEEE Russia Power Tech, St. Petersburg, Russia, 27–30 June 2005; pp. 1–7. [Google Scholar]
- Bracco, S.; Dentici, G.; Siri, S. DESOD: A mathematical programming tool to optimally design a distributed energy system. Energy
**2016**, 100, 298–309. [Google Scholar] [CrossRef] - Haikarainen, C.; Pettersson, F.; Saxén, H. A model for structural and operational optimization of distributed energy systems. Appl. Therm. Eng.
**2014**, 70, 211–218. [Google Scholar] [CrossRef] - Harb, H.; Schwager, C.; Streblow, R.; Müller, D. Optimal Design of Energy Systems in Residential Districts with Interconnected Local Heating and Electrical Networks. In Proceedings of the Building Simulation Conference, Hyderabad, India, 7–9 December 2015. [Google Scholar]
- Jennings, M.; Fisk, D.; Shah, N. Modelling and optimization of retrofitting residential energy systems at the urban scale. Energy
**2014**, 64, 220–233. [Google Scholar] [CrossRef] - Zhou, Z.; Liu, P.; Li, Z.; Ni, W. Economic assessment of a distributed energy system in a new residential area with existing grid coverage in China. Comput. Chem. Eng.
**2013**, 48, 165–174. [Google Scholar] [CrossRef] - Domínguez-Muñoz, F.; Cejudo-López, J.M.; Carrillo-Andrés, A.; Gallardo-Salazar, M. Selection of typical demand days for CHP optimization. Energy Build.
**2011**, 43, 3036–3043. [Google Scholar] [CrossRef] - Kotzur, L.; Markewitz, P.; Robinius, M.; Stolten, D. Impact of different time series aggregation methods on optimal energy system design. Renew. Energy
**2018**, 117, 474–487. [Google Scholar] [CrossRef] [Green Version] - Pfenninger, S. Dealing with multiple decades of hourly wind and PV time series in energy models: A comparison of methods to reduce time resolution and the planning implications of inter-annual variability. Appl. Energy
**2017**, 197, 1–13. [Google Scholar] [CrossRef] - Bahl, B.; Kümpel, A.; Seele, H.; Lampe, M.; Bardow, A. Time-series aggregation for synthesis problems by bounding error in the objective function. Energy
**2017**, 135, 900–912. [Google Scholar] [CrossRef] - Baumgärtner, N.; Bahl, B.; Hennen, M.; Bardow, A. RiSES3: Rigorous Synthesis of Energy Supply and Storage Systems via time-series relaxation and aggregation. Comput. Chem. Eng.
**2019**, 127, 127–139. [Google Scholar] [CrossRef] - Bahl, B.; Söhler, T.; Hennen, M.; Bardow, A. Typical Periods for Two-Stage Synthesis by Time-Series Aggregation with Bounded Error in Objective Function. Front. Energy Res.
**2018**, 5, 35. [Google Scholar] [CrossRef] - Yokoyama, R.; Shinano, Y.; Wakayama, Y.; Wakui, T. Model reduction by time aggregation for optimal design of energy supply systems by an MILP hierarchical branch and bound method. Energy
**2019**, 181, 782–792. [Google Scholar] [CrossRef] - Welder, L.; Ryberg, D.S.; Kotzur, L.; Grube, T.; Robinius, M.; Stolten, D. Spatio-temporal optimization of a future energy system for power-to-hydrogen applications in Germany. Energy
**2018**, 158, 1130–1149. [Google Scholar] [CrossRef] - Welder, L.; Linssen, J.; Robinius, M.; Stolten, D. FINE—Framework for Integrated Energy System Assessment 2018. Available online: https://github.com/FZJ-IEK3-VSA/FINE (accessed on 1 July 2019).
- Kotzur, L.; Markewitz, P.; Robinius, M.; Stolten, D. tsam—Time Series Aggregation Module 2017. Available online: https://github.com/FZJ-IEK3-VSA/tsam (accessed on 1 July 2019).
- Kotzur, L.; Markewitz, P.; Robinius, M.; Stolten, D. Time series aggregation for energy system design: Modeling seasonal storage. Appl. Energy
**2018**, 213, 123–135. [Google Scholar] [CrossRef] [Green Version] - Nahmmacher, P.; Schmid, E.; Hirth, L.; Knopf, B. Carpe diem: A novel approach to select representative days for long-term power system modeling. Energy
**2016**, 112, 430–442. [Google Scholar] [CrossRef] - Ward, J.H., Jr. Hierarchical Grouping to Optimize an Objective Function. J. Am. Stat. Assoc.
**1963**, 5, 236–244. [Google Scholar] [CrossRef] - Richardson, I.; Thomson, M.; Infield, D. A high-resolution domestic building occupancy model for energy demand simulations. Energy Build.
**2008**, 40, 1560–1566. [Google Scholar] [CrossRef] [Green Version] - Richardson, I.; Thomson, M.; Infield, D. Domestic electricity use: A high-resolution energy demand model. Energy Build.
**2010**, 42, 1878–1887. [Google Scholar] [CrossRef] [Green Version] - Richardson, I.; Thomson, M.; Infield, D.; Delahunty, A. Domestic lighting: A high-resolution energy demand model. Energy Build.
**2009**, 41, 781–789. [Google Scholar] [CrossRef] [Green Version] - Kemmler, A.; Straßburg, S.; Seefeldt, F.; Anders, N.; Rohde, C.; Fleiter, T.; Aydemir, A.; Kleeberger, H.; Hardi, L.; Geiger, B. Datenbasis Zur Bewertung Von Energieeffizienzmaßnahmen in Der Zeitreihe 2005–2014; Umweltbundesamt: Berlin, Germany, 2017. [Google Scholar]
- Holmgren, W.F.; Hansen, C.W.; Mikofski, M.A. pvlib python: A python package for modeling solar energy systems. J. Open Sour. Softw.
**2018**, 3, 884. [Google Scholar] [CrossRef] - Lindberg, K.B.; Doorman, G.; Fischer, D.; Korpås, M.; Ånestad, A.; Sartori, I. Methodology for optimal energy system design of Zero Energy Buildings using mixed-integer linear programming. Energy Build.
**2016**, 127, 194–205. [Google Scholar] [CrossRef] [Green Version] - Frauenhofer ISE. Current and Future Cost of Photovoltaics. Long-term Scenarios for Market Development, System Prices and LcoE of Utility-Scale Pv Systems 2015. Available online: https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/AgoraEnergiewende_Current_and_Future_Cost_of_PV_Feb2015_web.pdf (accessed on 1 July 2019).
- Sterchele, P.; Kalz, D.; Palzer, A. Technisch-ökonomische Analyse von Maßnahmen und Potentialen zur energetischen Sanierung im Wohngebäudesektor heute und für das Jahr 2050. Bauphysik
**2016**, 38, 193–211. [Google Scholar] [CrossRef] - Streblow, R.; Ansorge, K. Genetischer Algorithmuszur kombinatorischen Optimierung von Gebäudehülle und Anlagentechnik; Gebäude-Energiewende: Berlin, Germany, 2017. [Google Scholar]
- Lauinger, D.; Caliandro, P.; Van herle, J.; Kuhn, D. A linear programming approach to the optimization of residential energy systems. J. Energy Storage
**2016**, 7, 24–37. [Google Scholar] [CrossRef] - ASUE. BHKW-Kenndaten 2014/2015–Module, Anbieter, Kosten; Arbeitsgemeinschaft für sparsamen und umweltfreundlichen Energieverbrauch e.V.: Berlin, Germany, 2015. [Google Scholar]
- Rager, J.M.F.; Maréchal, F. Urban Energy System Design from the Heat Perspective Using Mathematical Programming Including Thermal Storage; Thèse École polytechnique fédérale de Lausanne EPFL: Lausanne, Switzerland, 2015. [Google Scholar]
- Klingler, A.-L. Self-consumption with PV+Battery systems: A market diffusion model considering individual consumer behaviour and preferences. Appl. Energy
**2017**, 205, 1560–1570. [Google Scholar] [CrossRef] - Linssen, J.; Stenzel, P.; Fleer, J. Techno-economic analysis of photovoltaic battery systems and the influence of different consumer load profiles. Appl. Energy
**2017**, 185, 2019–2025. [Google Scholar] [CrossRef] - Figgener, J.; Haberschusz, D.; Kairies, P.K.; Wessels, O.; Tepe, B.; Sauer, U.D. Wissenschaftliches Mess-und Evaluierungsprogramm Solarstromspeicher 2.0—Jahresbericht 2017; Institut für Stromrichtertechnik und Elektrische Antriebe der RWTH Aachen: Aachen, Germany, 2017. [Google Scholar]
- Lindberg, K.B.; Fischer, D.; Doorman, G.; Korpås, M.; Sartori, I. Cost-optimal energy system design in Zero Energy Buildings with resulting grid impact: A case study of a German multi-family house. Energy Build.
**2016**, 127, 830–845. [Google Scholar] [CrossRef] [Green Version] - Bundesnetzagentur Deutschland, Haushaltskundenpreis Strom und Gas/Entwicklungen Beschaffungskosten, Netzentgelte und EEG-Umlage (Stichtag 1. April 2017), 2017. Available online: https://www.bundesnetzagentur.de/SharedDocs/Downloads/DE/Sachgebiete/Energie/Unternehmen_Institutionen/DatenaustauschUndMonitoring/Monitoring/Monitoring2017_Kapitel/E_Einzelhandel2017.pdf?__blob=publicationFile&v=1 (accessed on 1 July 2019).
- Bundesnetzagentur Deutschland. EEG-Registerdaten und EEG-Fördersätze; Publications Office of the European Union: Luxembourg, 2017. [Google Scholar]
- Bundesministerium der Justiz und für Verbraucherschutz KWKG 2016. Available online: https://www.gesetze-im-internet.de/kwkg_2016/ (accessed on 1 July 2019).
- European Energy Exchange AG. Üblicher Strompreis gemäß KWK-Gesetz. Available online: https://www.eex.com/de/marktdaten/strom/spotmarkt/kwk-index/kwk-index-download (accessed on 1 July 2019).
- Schiebahn, S.; Grube, T.; Robinius, M.; Tietze, V.; Kumar, B.; Stolten, D. Power to gas: Technological overview, systems analysis and economic assessment for a case study in Germany. Int. J. Hydrogen Energy
**2015**, 40, 4285–4294. [Google Scholar] [CrossRef] - Robinius, M.; Stein, F.T.; Schwane, A.; Stolten, D. A Top-Down Spatially Resolved Electrical Load Model. Energies
**2017**, 10, 361. [Google Scholar] [CrossRef] - Andrews, R.W.; Stein, J.S.; Hansen, C.; Riley, D. Introduction to the open source PV LIB for python Photovoltaic system modelling package. In Proceedings of the IEEE 40th Photovoltaic Specialist Conference (PVSC), Denver, CO, USA, 8–13 June 2014; pp. 170–174. [Google Scholar]

**Figure 6.**Deviation of installed boiler capacities with the 2-Level Approach and time series aggregation in buildings (bd) compared to full time series.

**Figure 7.**Deviation of installed CHP capacities with 2-Level Approach and time series aggregation in buildings (bd) compared to full time series.

**Figure 8.**Deviation of installed heat storage capacities with the 2-Level Approach and time series aggregation.

**Figure 9.**Top-down (

**a**) and bottom-up (

**b**) peak load flow of time series aggregation and 2-Level Approach compared to the reference case with full time series.

**Figure 14.**Deviation of installed photovoltaic capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 15.**Deviation of installed wind energy capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 16.**Deviation of installed backup plant capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 17.**Deviation of installed electrolyzer capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 18.**Deviation of installed fuel cell capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 19.**Deviation of installed hydrogen storage capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Figure 20.**Deviation of installed battery capacities with the 2-Level Approach and time series aggregation compared to full time series.

**Table 1.**Overview of optimization approach and considered time series in 2-Level Approach, time series aggregation and the reference case.

Reference | Time Series Aggregation | 2-Level Approach | |
---|---|---|---|

1st Level | MILP | MILP | MILP |

Full-Time Series | Typical Periods | Typical Periods | |

2nd Level | - | - | LP |

- | - | Full-Time Series |

Technologies | CAPEX_{Cap} | CAPEX_{Fix} | OPEX | Lifetime | Efficiency | Based on Literature |
---|---|---|---|---|---|---|

PV | 1400 €/kW_{p} | 1000 € | 1% of CAPEX | 25 | - | [45,46] |

Condensing Boiler | 100 €/kW_{th} | 2800 € | 1.5% of CAPEX | 20 | 90% | [47,48,49] |

Heat Pump | 600 €/kW_{th} | 5000 € | 1% of CAPEX | 20 | COP = 2 | [45,47,48] |

ICE-CHP | 1000 €/kW_{el} | 15,000 € | 3% of CAPEX | 20 | 25%_{el}/60%_{th} | [48,50] |

Heat Storage | 34 €/kWh_{th} | 23 € | - | 25 | 0,1%/h | [48,51] |

Battery Storage | 700 €/kWh_{el} | 2000 € | - | 15 | 0,01%/h | [52,53,54] |

Interest Rate | 4% | [55] |

Household Electricity Price | 0.2986 €/kWh | [56] |

Household Natural Gas Price | 0.0615 €/kWh | [56] |

PV Feed-In Tariff | 0.1245 €/kWh | [57] |

CHP Premium | 0.08 €/kWh | [58] |

CHP Index | 0.0342 €/kWh | [59] |

CHP Fix Location | ||||||
---|---|---|---|---|---|---|

Deviation TAC in % | Bd1 | Bd2 | Bd3 | Bd4 | Bd5 | Bd6 |

5 Typical Days | 0.013 | 0.092 | 0.075 | 0.121 | 0.083 | 0.059 |

10 Typical Days | 0.072 | 0.151 | 0.053 | 0.099 | 0.142 | 0.041 |

20 Typical Days | 0.013 | 0.092 | 0.053 | 0.099 | 0.083 | 0.041 |

40 Typical Days | 0.013 | 0.091 | 0.053 | 0.099 | 0.083 | 0.041 |

**Table 5.**Unit parameters of the island system derived from Kotzur et al. [37].

CAPEX_{Cap} [€/kW_{p}] | CAPEX_{Fix} [€] | OPEX_{Cap} [€/kWp] | OPEX_{Fix} [€] | OPEX_{Var} [€/kWh] | Efficiency [%] | Charge Efficiency [%] | Discharge Efficiency [%] | Self-Discharge [%/h] | Lifetime [a] | |
---|---|---|---|---|---|---|---|---|---|---|

Photovoltaic | 800 | 1000 | 8 | 100 | 0 | 20 | ||||

Wind Energy | 1000 | 100,000 | 20 | 2000 | 0 | 20 | ||||

Backup Plant | 1000 | 0 | 30 | 0 | 0.2 | 25 | ||||

Electrolyzer | 500 | 100,000 | 15 | 3000 | 0 | 70 | 15 | |||

Fuel Cell | 1100 | 100,000 | 33 | 3000 | 0 | 50 | 15 | |||

Battery | 300 | 0 | 3 | 0 | 0 | 96 | 96 | 0.05 | 15 | |

Hydrogen Storage | 15 | 0 | 0 | 0 | 0 | 90 | 1 | 0 | 25 |

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## Share and Cite

**MDPI and ACS Style**

Kannengießer, T.; Hoffmann, M.; Kotzur, L.; Stenzel, P.; Schuetz, F.; Peters, K.; Nykamp, S.; Stolten, D.; Robinius, M.
Reducing Computational Load for Mixed Integer Linear Programming: An Example for a District and an Island Energy System. *Energies* **2019**, *12*, 2825.
https://doi.org/10.3390/en12142825

**AMA Style**

Kannengießer T, Hoffmann M, Kotzur L, Stenzel P, Schuetz F, Peters K, Nykamp S, Stolten D, Robinius M.
Reducing Computational Load for Mixed Integer Linear Programming: An Example for a District and an Island Energy System. *Energies*. 2019; 12(14):2825.
https://doi.org/10.3390/en12142825

**Chicago/Turabian Style**

Kannengießer, Timo, Maximilian Hoffmann, Leander Kotzur, Peter Stenzel, Fabian Schuetz, Klaus Peters, Stefan Nykamp, Detlef Stolten, and Martin Robinius.
2019. "Reducing Computational Load for Mixed Integer Linear Programming: An Example for a District and an Island Energy System" *Energies* 12, no. 14: 2825.
https://doi.org/10.3390/en12142825