# Modelling Energy Transition in Germany: An Analysis through Ordinary Differential Equations and System Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Background: Innovation Diffusion Models

#### 2.1. Ordinary Differential Equations Approach

#### 2.2. System Dynamics Approach

## 3. Materials and Methods

#### 3.1. Model Definition with ODE

#### 3.2. Model Definition with System Dynamics

## 4. Model Application and Results

#### 4.1. UCTT Model Application

#### 4.2. UCTT—SFN Model Application

## 5. Simulation Study for Alternative Scenarios

#### 5.1. Policy 1: Stimulate Consumption of Renewable Energy

#### 5.2. Policy 2: Limit Consumption of Coal and Natural Gas

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EEG | Renewable Energy Act |

EJ | Exajoule |

IRENA | International Renewable Energy Agency |

MTI | Massachusetts Institute of Technology |

NLS | Nonlinear Least Square |

ODE | Ordinary Differential Equations |

RETs | Renewable Energy Technologies |

SFN | Stock-Flow Networks |

UCTT | Unbalanced Competition among Three Technologies |

UCRCD | Unbalanced Competition Regime Change Diachronic |

## Appendix A

- 3PM model uses ${\delta}_{\beta}$ as a constant hyperparameter to determine the cross-influence in all the equations;
- UCTT model uses $\zeta ,\rho $, and $\xi $ to distinguish the effects of the cross-influence effect on each of the three technologies.

## Appendix B

## References

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**Figure 2.**UCTT-SFN model. Cause–effect relationships affecting market potential (yellow), market share (brown), coefficients of the technologies diffusion dynamics (green). Cause–effect relationships determined by market share (purple), residual market (light blue), cumulative consumptions on the same (internal-influence, in grey), as well as on different (cross-influence, in black) technologies.

**Figure 4.**UCTT-SFN relevant feedback loops. Cause–effect relationships affecting market potential (yellow), market share (brown). Cause–effect relationships determined by market share (purple), cumulative consumptions on the same (internal-influence, in gray), as well as on different (cross-influence, positive in blue and negative in red) technologies.

**Figure 5.**Simulation of coal, natural gas, and renewable energy consumption in the observed scenario.

Competition Phase | |
---|---|

${p}_{1c}$ | Seed coefficient of technology 1 |

${q}_{1c}+\delta $ | Internal-influence of technology 1 |

${q}_{1c}$ | Cross-influence of technology 2 on technology 1 |

${p}_{2c}$ | Seed coefficient of technology 2 |

${q}_{2c}$ | Internal-influence of technology 2 |

${q}_{2c}-\gamma $ | Cross-influence of technology 1 on technology 2 |

Double-competition phase | |

${p}_{1d}$ | Seed coefficient of technology 1 |

${q}_{1d}+\zeta $ | Internal-influence of technology 1 |

${q}_{1d}$ | Cross-influence of technologies 2 and 3 on technology 1 |

${p}_{2d}$ | Seed coefficient of technology 2 |

${q}_{2d}$ | Internal-influence of technology 2 |

${q}_{2d}-\rho $ | Cross-influence of technologies 1 and 3 on technology 2 |

${p}_{3d}$ | Seed coefficient of technology 3 |

${q}_{3d}$ | Internal-influence of technology 3 |

${q}_{3d}-\xi $ | Cross-influence of technologies 1 and 2 on technology 3 |

Parameter | Estimate | Std.Error | Lower c.i. | Upper c.i. | p-Value |
---|---|---|---|---|---|

${m}_{c}$ | $483.5858$ | $116.9050$ | $254.4561$ | $712.7154$ | $0.0002$ |

${p}_{1c}$ | $0.0142$ | $0.0036$ | $0.0072$ | $0.0213$ | $0.0003$ |

${p}_{2c}$ | $-0.0009$ | $0.0004$ | $-0.0018$ | $-0.0001$ | $0.0266$ |

${q}_{1c}$ | $0.1500$ | $0.0417$ | $0.0683$ | $0.2317$ | $0.0008$ |

${q}_{2c}$ | $-0.0389$ | $0.0279$ | $-0.0936$ | $0.0157$ | $0.1690$ |

$\delta $ | $-0.1694$ | $0.0454$ | $-0.2584$ | $-0.0804$ | $0.0006$ |

$\gamma $ | $-0.0775$ | $0.0331$ | $-0.1424$ | $-0.0126$ | $0.0240$ |

${m}_{d}$ | $370.7024$ | $23.0794$ | $325.4677$ | $415.9372$ | $0.0000$ |

${p}_{1d}$ | $0.0135$ | $0.0011$ | $0.0113$ | $0.0157$ | $0.0000$ |

${p}_{2d}$ | $0.0048$ | $0.0007$ | $0.0035$ | $0.0061$ | $0.0000$ |

${p}_{3d}$ | $0.0002$ | $0.0003$ | $-0.0005$ | $0.0009$ | $0.5960$ |

${q}_{1d}$ | $0.0707$ | $0.0213$ | $0.0290$ | $0.1124$ | $0.0013$ |

${q}_{2d}$ | $-0.0708$ | $0.0510$ | $-0.1707$ | $0.0292$ | $0.1690$ |

${q}_{3d}$ | $0.3123$ | $0.0423$ | $0.2294$ | $0.3952$ | $0.0000$ |

$\zeta $ | $-0.1260$ | $0.0391$ | $-0.2027$ | $-0.0494$ | $0.0018$ |

$\rho $ | $-0.1643$ | $0.0915$ | $-0.3436$ | $0.0149$ | $0.0759$ |

$\xi $ | $0.3137$ | $0.0438$ | $0.2279$ | $0.3995$ | $0.0000$ |

${R}^{2}=0.9903495$ | |||||

$SSE=6.622326$ |

Estimate | Description | |
---|---|---|

${m}_{d}$ | $370.70$ | Market Potential |

${p}_{1d}$ | $0.0135$ | Coal—Seed |

${q}_{1d}+\zeta $ | $-0.0553$ | Coal—Internal-influence |

${q}_{1d}$ | $0.0707$ | Coal—Cross-influence |

${p}_{2d}$ | $0.0048$ | Natural Gas—Seed |

${q}_{2d}$ | $-0.0708$ | Natural Gas—Internal-influence |

${q}_{2d}-\rho $ | $0.0936$ | Natural Gas—Cross-influence |

${p}_{3d}$ | $0.0002$ | Renewables—Seed |

${q}_{3d}$ | $0.3123$ | Renewables—Internal-influence |

${q}_{3d}-\xi $ | $-0.0014$ | Renewables—Cross-influence |

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## Share and Cite

**MDPI and ACS Style**

Savio, A.; De Giovanni, L.; Guidolin, M.
Modelling Energy Transition in Germany: An Analysis through Ordinary Differential Equations and System Dynamics. *Forecasting* **2022**, *4*, 438-455.
https://doi.org/10.3390/forecast4020025

**AMA Style**

Savio A, De Giovanni L, Guidolin M.
Modelling Energy Transition in Germany: An Analysis through Ordinary Differential Equations and System Dynamics. *Forecasting*. 2022; 4(2):438-455.
https://doi.org/10.3390/forecast4020025

**Chicago/Turabian Style**

Savio, Andrea, Luigi De Giovanni, and Mariangela Guidolin.
2022. "Modelling Energy Transition in Germany: An Analysis through Ordinary Differential Equations and System Dynamics" *Forecasting* 4, no. 2: 438-455.
https://doi.org/10.3390/forecast4020025