# Optimization Model for Biogas Power Plant Feedstock Mixture Considering Feedstock and Transportation Costs Using a Differential Evolution Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{3}in 1 MWe biogas power plant. The results show the needed yearly amounts and the maximum transportation distance of each feedstock.

## 1. Introduction

_{2}and technical feasibility of each scenario’s implementation. Here, the distances of feedstock are defined as input data for different scenarios and given in three values: 7.5, 15 and 22.5 km.

## 2. Model Description

#### 2.1. Mathematical Modeling

- m
_{tfs}—total mass of the feedstock (t), - m
_{i}—mass of the i-th feedstock (t), - n—number of different feedstocks.

- V
_{tfs}—total volume of the feedstock (m^{3}), - ρ
_{i}—density of the i-th feedstock (t/m^{3}).

- V
_{methane}—total volume of methane (Nm^{3}), - v
_{i}—biogas yield of the i-th feedstock (m^{3}/t of input), - x
_{i}—share of methane in 1 m^{3}of biogas expressed as a dimensionless relative number corresponding to the given percentage value.

^{3}of methane is calculated as follows:

- c
_{methane}—cost of 1 m^{3}of methane (EUR/m^{3}), - c
_{i}—cost of the i-th feedstock (EUR/t), - c
_{ti}—cost of the transport of the i-th feedstock (EUR/m^{3}, km), - V
_{i}—volume of the i-th feedstock (m^{3}), - d
_{i}—distance of the i-th feedstock for transportation (km).

#### 2.2. Optimization Problem Definition and Objective Functions

_{methane,max}is the upper limit of the methane production cost and d

_{i,max}is the upper limit for transportation distance.

#### 2.3. Optimization Problem Constraints

_{ds. lower}and HRT

_{ds. upper}are the lower and upper limits for HRT (usually 50–60 days) and HRT

_{s}is the HRT value for a possible solution to the problem. This constraint ensures that the reactor is not emptied during the year, as it must remain full throughout the entire period.

- y—content of the dry matter expressed as a dimensionless relative number,
- y
_{i}—content of the dry matter of the i-th feedstock, expressed as a dimensionless relative number corresponding to the given percentage value.

- V
_{reactor}—available volume of the reactor (m^{3}), - V
_{H2O}—volume of water needed to achieve the maximum allowed content of the dry matter (m^{3}).

- W
_{Y}—electric energy per year (kWh), - Pn—nominal plant power (electrical) (kW),
- CF—capacity factor,
- h
_{Y}—planned number of working hours per year (h).

- η—efficiency of biogas power plant.

- H
_{u}—heat value of methane (kWh/m^{3}).

#### 2.4. Optimization Method

## 3. Results of the Case Study

- Purple circles represent the maximum distances for the collections of the cow’s manure using local roads,
- Red circles represent the maximum distances for the collections of the cow’s slurry using local roads,
- Yellow circles represent the maximum distances for the collections of the pig’s slurry using local roads,
- Blue circles represent the maximum distances for the collections of the millet silage using local roads,
- Orange circles represent the maximum distances for the collections of the corn silage using local roads.

^{3}thanks to lower prices of the feedstock and transportation cost. In case of Equation (7), considering the bio methane production cost, only the upper limit distances for pig slurry and millet silage are lower than the upper limit and have values up to 56 and 24 km, respectively. For other feedstock, there are solutions including the highest distances.

^{3}, Figure 12).

## 4. Discussion

- There are more different combinations of the feedstock volumes and distances which give very close values of the objective function values (for all three used objective formulations).
- Using Equation (6) (including maximization of the distances only) enables us to find the upper limit of profitable feedstock distances considering the methane production cost constraint (Equation (23)) (profitable limit of the methane production cost).
- The proposed procedure enables us to study the impact of different feedstock combinations and transportation prices according to the existing market prices and to determine the upper limit of the feedstock distances projecting the expected methane production cost.
- The optimal mixture of feedstock can be found and used instead of only one feedstock type for minimization of the methane production cost.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Value range for Equation (5) and methane production cost for base input data and optimization, repeated 50 times: (

**a**) objective function value; (

**b**): methane production cost (the part of the objective function).

**Figure 4.**Value range of the feedstock volumes for Equation (5) for base input data and optimization, repeated 50 times.

**Figure 5.**Value range of the feedstock distances for Equation (5) for base input data and optimization, repeated 50 times.

**Figure 6.**Visualization example of the results from Table 6 (solution 1). 3.1. Impact of Objective Function Formulation, Feedstock Price, Transport Price and Upper Limit of the Methane Production Cost on Optimization Results.

**Figure 8.**Change of the feedstock volumes depending on the objective function formulation: (

**a**) for cow manure; (

**b**) for cow slurry; (

**c**) for pig slurry; (

**d**) for millet silage; (

**e**) for corn silage.

**Figure 9.**Change of the transportation distances depending on the objective function formulation: (

**a**) for cow manure; (

**b**) for cow slurry; (

**c**) for pig slurry; (

**d**) for millet silage; (

**e**): for corn silage.

**Figure 10.**Change of the methane production cost depending on the feedstock and transportation costs (b.cst = base cost, 2b.cst = 2 x base cost, 3b.cst = 3 x base cost).

**Figure 11.**Change of the feedstock volumes depending on the feedstock and transportation costs (b.cst = base cost, 2b.cst = 2 x base cost, 3b.cst = 3 x base cost: (

**a**) for cow manure; (

**b**) for cow slurry; (

**c**) for pig slurry; (

**d**) for millet silage; (

**e**) for corn silage.

**Figure 12.**Change of the transportation distances depending on the feedstock and transportation costs (b.cst = base cost, 2b.cst = 2 x base cost, 3b.cst = 3 x base cost: (

**a**) for cow manure; (

**b**) for cow slurry; (

**c**) for pig slurry; (

**d**) for millet silage; (

**e**) for corn silage).

**Figure 13.**Value range of Equation (5) and methane production cost for the increased costs and limited feedstock volume (optimization repeated 50 times).

**Figure 14.**Value range of the feedstock volumes for Equation (5), increased costs and limited feedstock volume (optimization repeated 50 times).

**Figure 15.**Value range of the feedstock distances for Equation (5), increased costs and limited feedstock volume (optimization repeated 50 times).

**Figure 16.**The methane production cost for Equation (7) in case of increased costs (feedstock cost 2 x baseline and transportation cost 3 x baseline, optimization repeated 50 times).

**Figure 17.**The feedstock volumes for Equation (7) in case of increased costs (feedstock cost 2 x baseline and transportation cost 3 x baseline, optimization repeated 50 times).

**Figure 18.**The feedstock distances for Equation (7) in case of increased costs (feedstock cost 2 x baseline and transportation cost 3 x baseline, optimization repeated 50 times).

Constraint | Expression | Lower Value | Upper Value |
---|---|---|---|

HRT^{1}_{s} | (12) | 50 days | 60 days |

y_{max} | (14) | - | 20% |

M_{i,max} | (17) | - | 20,000 tons |

d_{i,max} | (18) | - | 100 km |

V_{methane, req} | (22) | 2,212,121 m^{3} | 2,212,121 m^{3} |

c_{methane, max} | (23) | - | 0.21–0.75 EUR/m^{3} [58] |

r_{min} | (24) | 0.1 | - |

r_{max} | (24) | - | 0.5 |

^{1}Hydraulic Retention Time.

Type of the Feedstock | Dry Matter (%)—A | Organic Matter (% of Dry Matter)—B | Specific Biogas Yield from Organic Matter (m^{3}/t)—C | Biogas Yield of the Feedstock (m^{3}/t of Input)—A/100 x B/100 x C | |
---|---|---|---|---|---|

cow’s manure | 25.0 | 78.0 | 260 | 50.07 | |

cow’s slurry | 8.0 | 78.0 | 450 | 28.08 | |

pig’s slurry | 7.0 | 80.0 | 550 | 30.08 | |

millet silage | 29.4 | 93.0 | 560 | 153.12 | |

corn silage | 35.0 | 90.0 | 650 | 204.75 |

Type of the Feedstock | Biogas Yield of the Feedstock (m^{3}/t of Input)—v_{i} | Share of Methane in Biogas—x_{i} | Specific Density (t/m^{3})—ρ_{i} | Cost of the Feedstock (EUR/t)—c_{i} | Cost of the Feedstock Transport (EUR/m^{3})—c_{ti} |
---|---|---|---|---|---|

cow’s manure | 50.07 | 60.0% | 0.6 | 3.00 | 0.0094·d ^{1} + 2.4 |

cow’s slurry | 28.08 | 60.0% | 1.0 | 2.00 | 0.0214·d + 3.0 |

pig’s slurry | 30.08 | 65.0% | 1.0 | 1.20 | 0.0187·d + 2.8 |

millet silage | 153.12 | 54.0% | 0.7 | 17.00 | 0.04·d + 3.5 |

corn silage | 204.75 | 55.0% | 0.75 | 34.00 | 0.04·d + 3.5 |

^{1}Transportation distance.

**Table 4.**Optimal transportation distances and yearly needed feedstock masses for methane production cost below 0.35 EUR/m

^{3}(Equation (5)).

Type of the Feedstock | Amounts (t/a) | Maximum Distance (km) | Objective Function Value | Cost of Methane (EUR/m^{3}) | Possible Solutions |
---|---|---|---|---|---|

cow’s manure | 9737 | 89.5 | 6.49 | 0.311 | Solution 1 |

cow’s slurry | 11,727 | 96.2 | |||

pig’s slurry | 18,143 | 96.0 | |||

millet silage | 8827 | 68.9 | |||

corn silage | 5653 | 97.5 | |||

cow’s manure | 15,407 | 82.3 | 6.47 | 0.291 | Solution 2 |

cow’s slurry | 3945 | 95.5 | |||

pig’s slurry | 15,349 | 88.1 | |||

millet silage | 11,153 | 85.0 | |||

corn silage | 4099 | 88.5 |

**Table 5.**Optimal transportation distances and yearly needed feedstock masses for methane production cost below 0.35 EUR/m

^{3}(Equation (6)).

Type of the Feedstock | Amounts (t/a) | Maximum Distance (km) | Objective Function Value | Cost of Methane (EUR/m^{3}) | Possible Solutions |
---|---|---|---|---|---|

cow’s manure | 15,282 | 98.0 | 5.27 | 0.300 | Solution 1 |

cow’s slurry | 8223 | 88.3 | |||

pig’s slurry | 11,561 | 95.0 | |||

millet silage | 10,892 | 97.0 | |||

corn silage | 4317 | 98.0 | |||

cow’s manure | 7124 | 91.3 | 5.26 | 0.304 | Solution 2 |

cow’s slurry | 13,525 | 88.7 | |||

pig’s slurry | 16,890 | 99.5 | |||

millet silage | 15,684 | 95.3 | |||

corn silage | 1272 | 99.5 |

**Table 6.**Optimal transportation distances and yearly needed feedstock masses for methane production cost below 0.35 EUR/m

^{3}(Equation (7)).

Type of the Feedstock | Amounts (t/a) | Maximum Distance (km) | Objective Function Value = Cost of Methane (EUR/m^{3}) | Possible Solutions |
---|---|---|---|---|

cow’s manure | 12,137 | 4.7 | 0.235 | Solution 1 |

cow’s slurry | 2500 | 14.9 | ||

pig’s slurry | 19,186 | 21.5 | ||

millet silage | 17,245 | 1.4 | ||

corn silage | 20 | 24.0 | ||

cow’s manure | 13,266 | 41.0 | 0.235 | Solution 2 |

cow’s slurry | 1397 | 45.5 | ||

pig’s slurry | 17,722 | 0.7 | ||

millet silage | 17,299 | 6.6 | ||

corn silage | 117 | 37.8 |

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**MDPI and ACS Style**

Topić, D.; Barukčić, M.; Mandžukić, D.; Mezei, C.
Optimization Model for Biogas Power Plant Feedstock Mixture Considering Feedstock and Transportation Costs Using a Differential Evolution Algorithm. *Energies* **2020**, *13*, 1610.
https://doi.org/10.3390/en13071610

**AMA Style**

Topić D, Barukčić M, Mandžukić D, Mezei C.
Optimization Model for Biogas Power Plant Feedstock Mixture Considering Feedstock and Transportation Costs Using a Differential Evolution Algorithm. *Energies*. 2020; 13(7):1610.
https://doi.org/10.3390/en13071610

**Chicago/Turabian Style**

Topić, Danijel, Marinko Barukčić, Dražen Mandžukić, and Cecilia Mezei.
2020. "Optimization Model for Biogas Power Plant Feedstock Mixture Considering Feedstock and Transportation Costs Using a Differential Evolution Algorithm" *Energies* 13, no. 7: 1610.
https://doi.org/10.3390/en13071610