# 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 |

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**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