# Effect of Different Interval Lengths in a Rolling Horizon MILP Unit Commitment with Non-Linear Control Model for a Small Energy System

^{*}

## Abstract

**:**

^{®}as the problem solver. The resulting unit commitment is input into a non-linear control model (NLC), which tries to match the plan of the UCO as closely as possible. Using the approach of a rolling horizon the result of the NLC is fed back to the interval of the next optimisation run. The problem’s objective is set to minimise CO

_{2}emissions of the whole electricity producer park. Different interval lengths are tested with perfect foresight. The results gained with different interval lengths are compared to each other and to a simple heuristic approach. As non-linear control model a characteristic line model is used. The results show that the influence of the interval length is rather small, which leads to the conclusion that realistic forecast lengths of two days can be used to achieve not only a sufficient quality of solutions, but shorter computational times as well.

## 1. Introduction

_{each}, meaning the installed power each generation type has in a given configuration. As demand for the energy system, a constant power of 1000 MW is defined.

_{2}emissions of the electricity producer park are the minimisation objective of the MILP, which is why the specific CO

_{2}emissions of the electricity producer park are used as the primary quality measure of the UC results. Additionally, the share of storages, meaning the amount of energy delivered by the energy storage devices in respect to total demand, acts as an indicator of difference between UCO and UCH.

## 2. Model Structure and Materials

_{2}at any given point of time in the simulation.

#### 2.1. Program Flow of CharL

#### 2.2. Renewable Energy Modelling

#### 2.3. Modelling Residual Power Generators

_{th}as an additional parameter. The NLC considers start-up times as the corresponding functions are implemented. For this paper, the residual power generators are parameterised so that the start-up times have no influence on the UC. That way the MILP model (see Section 2.5) can resemble the NLC more closely.

#### 2.4. Heuristic Unit Commitment

#### 2.5. MILP Formulation

#### 2.5.1. Constraints to Model Thermal Power Plants

#### 2.5.2. Constraints to Model Energy Storage Devices

#### 2.5.3. Constraints to Model the Energy System

#### 2.5.4. Objective Function

#### 2.6. Temporal and Technical Scenario Configuration

#### 2.6.1. The Actors

_{net}is chosen. A long-term energy storage device consists of an electrolyser for conversion of power to hydrogen and its own CCGT plant for reconversion of hydrogen to power. As storage capacity 240,000 MWh

_{net}is chosen. The storage capacities translate to their respective storage type. Both storages are rated at 1000 MW, which for the electrolyser results in a higher maximum load of around 1600 MW due to its overload capacities [45]. The installed rated powers of the VRE types solar and wind are set between 1000 and 5000 MW each, referred to as MW

_{each}in the following (see Table 2), since both technologies being equally sized in all configurations. The installed VRE power in the following graphs is to be understood as the value both units each are rated at.

#### 2.6.2. Parameterisation of Initial Storage Levels

#### 2.6.3. Parameterisation of Efficiency Characteristics

## 3. Results

_{2}emissions (dots) of the plants of the copper plate island and the storage share (square) in dependence on installed VRE power. The results show that the UCO model (dotted lines) leads to slightly lower CO

_{2}emissions than the UCH (solid lines) for all configurations.

_{each}configuration shows a lower absolute improvement than the 2000 MW

_{each}configuration, with the storage share of the UCH. Since the UCH only integrates surplus of VRE power supply, it is obvious that the 1000 MW

_{each}configuration has negligible surplus of VRE to be integrated, thus the UCO has only the CCGT left to optimise. In practice, this is done using the short-term storage.

_{2}emissions due to optimisation in comparison to the UCH is shown in absolute and relative terms in Figure 7 also for the interval length of two days. The figure shows that the relative CO

_{2}reduction through UCO for installed VRE power below 3000 MW

_{each}is generally below 10%. Between 3000 and 3750 MW

_{each}the absolute reduction is rather constant across VRE configurations while the relative reduction shows increasing values. For 4000 MW

_{each}both absolute and relative reduction show lower values, which can be explained with the maximum share of storage to be integrated as seen in Figure 6 and discussed above. For higher installed capacities, the denominator becomes so insignificant that the relative reduction is amplified while the absolute CO

_{2}reduction becomes remarkably small.

_{2}emissions through UCO relative to the heuristic approach (UCH) is shown in relation to the interval length, which in this context implies the corresponding period lengths of Table 1. The configurations with 1000 MW

_{each}(circles) and 2000 MW

_{each}(squares) VRE power show no significant influence of the interval length across the scenarios (interval lengths). The configuration with 3000 MW

_{each}shows a slight advantage for a whole year optimisation run, which is explicable with the maximum possible integration of storage power (see Figure 6) for the assumed residual power units’ configuration. For the 4000 and 5000 MW

_{each}configurations the whole year optimisation run shows, against expectations, negative emission reduction, which can be explained by the linearisation effects, that are discussed in Section 3.2 and visible in Figure 10.

_{each}configuration, as discussed above.

#### 3.1. Result of Unit Commitment

_{2}emissions and the efficiency spread of the CCGT between minimum and maximum operating point is 20%-rel. and the losses due to battery usage lie around 10%.

#### 3.2. Results for Different Interval Lengths

_{each}. The dashed lines represent the MILP results, while the solid lines show the storage level after the NLC. The time series are presented for a 5 days interval (scenario c, blue lines) and a 365 days optimization (scenario f, green lines).

## 4. Discussion

## 5. Conclusions

_{2}emissions of the energy park, with this approach, even though, the detailed unit commitment can vary significantly (see Figure 10)). However, the difference is mainly to be found in computational effort.

_{2}emissions in this study) than the UCO. Considering the massive computation time of a UCO, it is suggested to explore configuration variations using a simple heuristic and to optimise a small set of resulting configurations with a UCO only afterwards.

#### Outlook

_{2}emissions of the underlying energy system. Another common approach is the minimisation of (operational) costs, which is currently being implemented allowing multi-objective optimisations (costs and CO

_{2}emissions).

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Unit | Description |

$a$ | - | Efficiency parameter |

$b$ | - | Efficiency parameter |

${C}_{\mathrm{net}}$ | MWh_{net} | Net capacity of storage |

$E$ | MWh | Storage level |

$e$ | kg/MWh_{th} | Specific CO_{2} emissions |

$G$ | - | Set of conventional generation units |

$g$ | - | Index of conventional generation unit in set $G$ |

$I$ | - | Set of points in efficiency curve |

$i$ | - | Index of points in set $I$ |

$k$ | - | Index of optimisation period |

$K$ | - | Set of optimisation periods |

$P$ | MW | Power of unit on grid side |

${P}_{u}$ | MW | Power of unit on unit side |

$p$ | - | Relative load |

$R$ | MW/min | Maximum power gradient of unit |

$S$ | - | Set of storage units |

$s$ | - | Index of storage unit in set $S$ |

$T$ | - | Set of timesteps |

${T}_{k}$ | - | Set of timesteps of interval for period $k$ |

${T}_{k,0}$ | - | First timestep of interval for period $k$ |

$t$ | - | Timestep in Set $T$ |

${t}_{\mathrm{period}}$ | - | Number of timesteps in a period |

$\delta $ | - | Deviation in parameterisation problem |

$\eta $ | - | Efficiency |

$\xi $ | %/h | Self-discharge rate of storage |

$\tau $ | h | Length of timestep |

$\psi $ | - | Status variable for unit (0 = off, 1 = on) |

## Terms and Abbreviations

Term/Abbreviation | Description |

Configuration | Types, powers and capacities of installed units |

Scenario | Set of interval and period length |

MW_{each} | Installed power for each solar and wind generation in MW |

CCGT | Combined Cycle Gas Turbine |

CharL | Characteristic Line Model |

FlopC++ | Formulation of Linear Optimization Problems in C++ |

MILP | Mixed Integer Linear Problem |

OSI | Open Solver Interface |

SOC | State of Charge |

UCH | Unit Commitment Heuristic |

UCO | Unit Commitment Optimisation |

NLC | Non-linear control (model) |

TSO | Transmission System Operator |

VRE | Volatile Renewable Energy |

## Appendix A

Setup Name | Processor | RAM |
---|---|---|

PC 1 | Intel Xeon E5-1620 @3.6 GHz | 64 GB RAM UDIMM |

PC 2 | Intel i7-3770 @3.4 GHz | 16 GB RAM UDIMM |

PC 3 | Intel i7-4790 @3.6 GHz | 32 GB RAM UDIMM |

PC 4 | Intel i7-6700 @3.4 GHz | 32 GB RAM UDIMM |

Parameter | Unit | Value | Explanation/Source |
---|---|---|---|

CCGT Ramp Rate | %/min | 100 | Beyond of scope |

CCGT Start-up Times | h | 0 | Beyond of scope |

Specific Fuel Emission | t_{CO2}/MWh_{th} | 0.202 | [47] |

Ramp Rate Storages (all) | %/min | 100 | Beyond of scope |

Start-up Times Storages (all) | h | 0 | Beyond of scope |

Self-discharge Rate Short-Term | %/h | 0.01 | [48] (Middle of spread) |

Self-discharge Rate Long-Term | %/h | 0.0006875 | [46] (Middle of spread) |

Virtual positive penalty emission | t_{CO2}/MWh | 10^{6} | - |

Virtual negative penalty emission | t_{CO2}/MWh | 10^{2} | - |

Virtual storage emission | t_{CO2}/MWh | 10^{−3} | - |

Installed Power Wind and Solar in MW Each | Specific CO_{2} Emissions in g/kWh | ||||||
---|---|---|---|---|---|---|---|

Heuristic | 2 Days | 3 Days | 5 Days | 14 Days | 30 Days | Year | |

1000 | 236.77 | 230.06 | 230.06 | 230.07 | 230.10 | 230.10 | 230.11 |

1250 | 214.13 | 202.76 | - | - | - | - | - |

1500 | 187.69 | 175.55 | 175.53 | - | - | - | - |

1750 | 161.06 | 149.72 | - | - | - | - | - |

2000 | 136.29 | 126.16 | 126.13 | 126.13 | 126.17 | 126.19 | 126.39 |

2250 | 113.34 | 104.43 | - | - | - | - | - |

2500 | 92.26 | 84.51 | 84.48 | - | - | - | - |

2750 | 72.71 | 65.95 | - | - | - | - | - |

3000 | 54.64 | 49.52 | 49.48 | 49.47 | 49.46 | 49.46 | 48.88 |

3250 | 40.72 | 35.74 | - | - | - | - | - |

3500 | 28.87 | 23.70 | 23.65 | - | - | - | - |

3750 | 18.08 | 12.77 | - | - | - | - | - |

4000 | 10.31 | 8.35 | 8.34 | 8.39 | 8.50 | 8.71 | 12.48 |

4250 | 7.74 | 5.67 | - | - | - | - | - |

4500 | 5.24 | 3.22 | 3.20 | - | - | - | - |

4750 | 2.92 | 1.00 | - | - | - | - | - |

5000 | 0.74 | 0.00 | 0.00 | 0.00 | 0.02 | 0.03 | 8.59 |

Installed Power Wind and Solar in MW Each | Share of Storages in % | ||||||
---|---|---|---|---|---|---|---|

Heuristic | 2 Days | 3 Days | 5 Days | 14 Days | 30 Days | Year | |

1000 | 0.01 | 12.50 | 12.56 | 12.62 | 12.59 | 12.62 | 12.53 |

1250 | 0.26 | 12.37 | - | - | - | - | - |

1500 | 1.17 | 12.11 | 12.11 | - | - | - | - |

1750 | 2.72 | 12.16 | - | - | - | - | - |

2000 | 4.55 | 12.61 | 12.59 | 12.63 | 12.61 | 12.61 | 12.50 |

2250 | 6.61 | 13.40 | - | - | - | - | - |

2500 | 8.88 | 14.40 | 14.35 | - | - | - | - |

2750 | 11.13 | 15.67 | - | - | - | - | - |

3000 | 13.42 | 16.89 | 16.86 | 16.85 | 16.76 | 16.69 | 16.92 |

3250 | 15.03 | 17.75 | - | - | - | - | - |

3500 | 16.29 | 18.50 | 18.50 | - | - | - | - |

3750 | 17.43 | 19.26 | - | - | - | - | - |

4000 | 17.91 | 18.67 | 18.66 | 18.68 | 18.57 | 18.62 | 17.96 |

4250 | 17.18 | 17.88 | - | - | - | - | - |

4500 | 16.57 | 17.18 | 17.19 | - | - | - | - |

4750 | 16.04 | 16.55 | - | - | - | - | - |

5000 | 15.52 | 15.72 | 15.72 | 15.72 | 15.63 | 15.63 | 13.85 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding the Relative Gap of the Solutions All in - | |||||
---|---|---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Min. | Max. | Average | Standard Deviation | # of Times No Solution Found | |

1000 | 255.22 | 68 | 0 | 2.81 × 10^{−5} | 8.30 × 10^{−5} | 7.54 × 10^{−4} | 0 |

1250 | 296.50 | 81 | 0 | 7.53 × 10^{−5} | 3.35 × 10^{−4} | 4.14 × 10^{−3} | 0 |

1500 | 232.86 | 61 | 0 | 6.01 × 10^{−5} | 2.28 × 10^{−4} | 2.74 × 10^{−3} | 0 |

1750 | 249.27 | 65 | 0 | 1.13 × 10^{−4} | 7.34 × 10^{−4} | 1.26 × 10^{−2} | 0 |

2000 | 298.44 | 80 | 0 | 2.08 × 10^{−4} | 1.47 × 10^{−3} | 2.61 × 10^{−2} | 1 |

2250 | 350.70 | 96 | 0 | 1.36 × 10^{−4} | 6.60 × 10^{−4} | 1.00 × 10^{−2} | 0 |

2500 | 373.67 | 102 | 0 | 2.79 × 10^{−4} | 1.56 × 10^{−3} | 2.25 × 10^{−2} | 0 |

2750 | 331.37 | 91 | 0 | 1.39 × 10^{−4} | 4.88 × 10^{−4} | 4.60 × 10^{−3} | 4 |

3000 | 357.32 | 98 | 0 | 3.48 × 10^{−4} | 3.67 × 10^{−3} | 6.99 × 10^{−2} | 4 |

3250 | 361.49 | 99 | −4.00 × 10^{−6} | 3.47 × 10^{−4} | 1.65 × 10^{−3} | 2.23 × 10^{−2} | 4 |

3500 | 306.08 | 79 | 0 | 5.56 × 10^{−4} | 3.56 × 10^{−3} | 4.40 × 10^{−2} | 2 |

3750 | 303.41 | 83 | −3.31 × 10^{−3} | 4.83 × 10^{−4} | 2.32 × 10^{−3} | 2.76 × 10^{−2} | 2 |

4000 | 372.83 | 103 | 0 | 4.84 × 10^{−4} | 1.52 × 10^{−3} | 1.21 × 10^{−2} | 2 |

4250 | 385.11 | 109 | −5.83 × 10^{−4} | 7.64 × 10^{−4} | 3.08 × 10^{−3} | 4.95 × 10^{−2} | 1 |

4500 | 416.41 | 116 | −1.78 × 10^{−4} | 7.68 × 10^{−4} | 2.41 × 10^{−3} | 1.99 × 10^{−2} | 3 |

4750 | 396.97 | 114 | 0 | 8.16 × 10^{−4} | 3.28 × 10^{−3} | 4.81 × 10^{−2} | 1 |

5000 | 474.40 | 135 | 0 | 1.06 × 10^{−3} | 3.79 × 10^{−3} | 5.75 × 10^{−2} | 1 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding the Relative Gap of the Solutions all in - | |||||
---|---|---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Min. | Max. | Average | Standard Deviation | # of Times No Solution Found | |

1000 | 625.84 | 89 | 0 | 2.07 × 10^{−5} | 5.58 × 10^{−5} | 5.55 × 10^{−4} | 0 |

1500 | 696.96 | 101 | 0 | 5.91 × 10^{−5} | 2.19 × 10^{−4} | 3.03 × 10^{−3} | 0 |

2000 | 936.51 | 138 | 0 | 1.79 × 10^{−4} | 1.16 × 10^{−3} | 2.07 × 10^{−2} | 0 |

2500 | 1125.33 | 158 | 0 | 2.78 × 10^{−4} | 1.97 × 10^{−3} | 3.29 × 10^{−2} | 0 |

3000 | 1064.41 | 151 | −1.00 × 10^{−6} | 2.55 × 10^{−4} | 1.58 × 10^{−3} | 2.91 × 10^{−2} | 1 |

3500 | 900.47 | 125 | 0 | 1.01 × 10^{−3} | 7.11 × 10^{−3} | 8.81 × 10^{−2} | 2 |

4000 | 931.97 | 131 | 0 | 7.44 × 10^{−4} | 3.31 × 10^{−3} | 5.32 × 10^{−2} | 1 |

4500 | 1126.80 | 165 | 0 | 1.18 × 10^{−3} | 3.60 × 10^{−3} | 3.54 × 10^{−2} | 0 |

5000 | 1136.87 | 164 | −2.73 × 10^{−3} | 1.63 × 10^{−3} | 5.42 × 10^{−3} | 4.91 × 10^{−2} | 4 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding the Relative Gap of the Solutions All in - | |||||
---|---|---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Min. | Max. | Average | Standard Deviation | # of Times No Solution Found | |

1000 | 1105.31 | 104 | 1.00 × 10^{−6} | 2.37 × 10^{−5} | 3.78 × 10^{−5} | 2.55 × 10^{−4} | 0 |

2000 | 2217.89 | 107.5 | 0 | 1.41 × 10^{−4} | 3.98 × 10^{−4} | 2.87 × 10^{−3} | 0 |

3000 | 2445.49 | 120.5 | 0 | 2.03 × 10^{−4} | 4.67 × 10^{−4} | 5.54 × 10^{−3} | 0 |

4000 | 2205.14 | 106.5 | 0 | 9.62 × 10^{−4} | 2.82 × 10^{−3} | 2.26 × 10^{−2} | 0 |

5000 | 2229.26 | 109.5 | 0 | 1.26 × 10^{−3} | 3.21 × 10^{−3} | 2.66 × 10^{−2} | 0 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding the Relative Gap of the Solutions All in - | |||||
---|---|---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Min. | Max. | Average | Standard Deviation | # of Times No Solution Found | |

1000 | 1975.74 | 13 | 7.00 × 10^{−6} | 3.16 × 10^{−5} | 5.46 × 10^{−5} | 2.41 × 10^{−4} | 0 |

2000 | 6005.35 | 38 | 1.00 × 10^{−5} | 1.16 × 10^{−4} | 2.14 × 10^{−4} | 1.51 × 10^{−3} | 0 |

3000 | 6487.44 | 43 | 0 | 2.51 × 10^{−4} | 3.97 × 10^{−4} | 2.29 × 10^{−3} | 0 |

4000 | 3013.66 | 18 | 0 | 7.99 × 10^{−4} | 2.80 × 10^{−3} | 2.00 × 10^{−2} | 0 |

5000 | 3940.38 | 25 | −1.30 × 10^{−5} | 2.40 × 10^{−3} | 6.98 × 10^{−3} | 4.33 × 10^{−2} | 0 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding The Relative Gap of the Solutions All in - | |||||
---|---|---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Min. | Max. | Average | Standard Deviation | # of Times No Solution Found | |

1000 | 11,190.58 | 16.5 | 1.30 × 10^{−5} | 5.79 × 10^{−5} | 7.75 × 10^{−5} | 3.42 × 10^{−4} | 0 |

2000 | 21,609.54 | 36.5 | 4.50 × 10^{−5} | 2.08 × 10^{−4} | 1.54 × 10^{−4} | 6.99 × 10^{−4} | 0 |

3000 | 20,163.36 | 34 | 1.50 × 10^{−5} | 7.14 × 10^{−4} | 6.50 × 10^{−4} | 2.52 × 10^{−3} | 0 |

4000 | 7155.09 | 12 | 0 | 2.77 × 10^{−4} | 6.53 × 10^{−4} | 3.39 × 10^{−3} | 0 |

5000 | 9712.99 | 15.5 | 0 | 2.56 × 10^{−3} | 9.59 × 10^{−3} | 5.88 × 10^{−2} | 0 |

Installed Power Each Wind and Solar in MW | Statistics of Rolling Horizon Regarding the Time of the Solution | Statistics of Rolling Horizon Regarding the Relative Gap of the Solutions All in - | |||
---|---|---|---|---|---|

Average Time in s | # of Times Time Limit Reached | Relative Gap | Standard Deviation | # of Times No Solution Found | |

1000 | 100,036.54 | 1 | 1.77 × 10^{−4} | 0 | 0 |

2000 | 100,047.74 | 1 | 4.43 × 10^{−4} | 0 | 0 |

3000 | 100,076.56 | 1 | 1.04 × 10^{−3} | 0 | 0 |

4000 | 100,092.65 | 1 | 3.19 × 10^{−2} | 0 | 0 |

5000 | 100,053.86 | 1 | 6.59 × 10^{−3} | 0 | 0 |

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**Figure 1.**Relative efficiency curves for combined cycle gas turbines (solid line) and open cycle gas turbines (dashed line) according to [27].

**Figure 3.**Simplified diagram of a rolling horizon with a ratio of period length to interval length of 1/3.

**Figure 4.**Input data for the model: solar (yellow) and wind (blue) power generation. The capacity factor represents the ratio of generated power to installed capacity.

**Figure 5.**Efficiency curves as used in algorithm (dashed lines) and their linearised counter parts (dotted lines) as used for MILP for CCGT (which is also long-term storage discharge) (circles), and long-term storage charge unit (squares). The linearised counterparts are generated using Equation (15) with the parameters of Table 3.

**Figure 6.**Specific CO

_{2}emissions (circles) and share of storages of supplied demand (squares) for the heuristic approach (solid lines) and a UCO with a 2 days interval (dotted lines).

**Figure 7.**Absolute (circles) and relative (squares) reductions in CO

_{2}emissions as a result of UCO in comparison to UCH for scenario a (2 days interval).

**Figure 8.**Reduction of specific CO

_{2}emissions through UCO compared to heuristic approach for different installed VRE powers (expressed in MW

_{each}) and interval lengths.

**Figure 9.**Time series excerpt of power of storages according to heuristic approach and UCO with 5 days interval on 17 January. 3000 MW

_{each}of VRE are installed. Short-term storage power of UCH is not shown, since it is zero.

**Figure 10.**Time series of the storage level of a long-term storage for MILP result (dashed) and after control calculation (solid) for 3000 MW

_{each}VRE power, using 5 days interval (scenario c, blue) and 365 days interval (scenario f, green).

Scenario | Hardware Used | Interval Length | Period Length | Abort Criteria | |
---|---|---|---|---|---|

(see Table A1) | Max. Time in s | Rel. Gap | |||

a | PC 1 | 2 days | 1 day | 1200 | 1.0 × 10^{−6} |

b | PC 2 | 3 days | 1 day | 2400 | 1.0 × 10^{−6} |

c | PC 3 | 5 days | 2 days | 3600 | 1.0 × 10^{−5} |

d | PC 4 | 14 days | 5 days | 10,800 | 1.0 × 10^{−5} |

e | PC 3 | 30 days | 10 days | 21,600 | 1.5 × 10^{−5} |

f | PC 1 | 365 days | 365 days | 100,000 | 1.0 × 10^{−4} |

VRE Configuration in MW_{each} | Initial SOC of Long-Term Storage in - |
---|---|

1000 … 2250 | 0.000 |

2500 | 0.056 |

2750 | 0.182 |

3000 | 0.417 |

3250 | 0.911 |

3500 … 5000 | 1.000 |

Unit | a | b |
---|---|---|

CCGT/Long-term discharge | 0.696639 | 0.2044030 |

Long-term charge | 0.670219 | 0.0283414 |

Short-term (dis)charge | 0.920500 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Erichsen, G.; Zimmermann, T.; Kather, A.
Effect of Different Interval Lengths in a Rolling Horizon MILP Unit Commitment with Non-Linear Control Model for a Small Energy System. *Energies* **2019**, *12*, 1003.
https://doi.org/10.3390/en12061003

**AMA Style**

Erichsen G, Zimmermann T, Kather A.
Effect of Different Interval Lengths in a Rolling Horizon MILP Unit Commitment with Non-Linear Control Model for a Small Energy System. *Energies*. 2019; 12(6):1003.
https://doi.org/10.3390/en12061003

**Chicago/Turabian Style**

Erichsen, Gerrit, Tobias Zimmermann, and Alfons Kather.
2019. "Effect of Different Interval Lengths in a Rolling Horizon MILP Unit Commitment with Non-Linear Control Model for a Small Energy System" *Energies* 12, no. 6: 1003.
https://doi.org/10.3390/en12061003