# Uncertainty Quantification in Energy Management Procedures

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. ANOVA and Polynomial Chaos Expansion

## 3. Model of Cogeneration Plant

## 4. Sensitivity Analysis

- The cost of natural gas is multiplied by ${p}_{1}$;
- The heating load is multiplied by ${p}_{2}$;
- The selling price of electricity is multiplied by ${p}_{3}$.

- The cost function is mostly affected by the natural gas cost as the sensitivity function ${S}_{1}$ reaches about 49% of the global variance in the case of both uniform and normal distributions;
- In terms of importance, the second parameter affecting the variance is the electricity price, whose influence ${S}_{3}$ is about 39% of the total variance;
- A lower importance parameter is the thermal load, whose effect ${S}_{2}$ is around 12%;
- Cross effects due to interaction of the parameters are not meaningful (${S}_{12}$, ${S}_{13}$, ${S}_{23}$ and ${S}_{123}$ are negligible).

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ANOVA | Analysis of variance |

KPI | Key performance index |

MC | Monte-Carlo |

PCE | Polynomial chaos expansion |

SA | Sensitivity analysis |

UC | Unit commitment |

UQ | Uncertainty quantification |

## References

- Grumm, F.; Schumann, M.; Cosse, C.; Plenz, M.; Lücken, A.; Schulz, D. Short Circuit Characteristics of PEM Fuel Cells for Grid Integration Applications. Electronics
**2020**, 9, 602. [Google Scholar] [CrossRef] [Green Version] - Vivas, F.J.; Segura, F.; Andújar, J.M.; Palacio, A.; Saenz, J.L.; Isorna, F.; López, E. Multi-Objective Fuzzy Logic-Based Energy Management System for Microgrids with Battery and Hydrogen Energy Storage System. Electronics
**2020**, 9, 1074. [Google Scholar] [CrossRef] - Rodriguez del Nozal, A.; Gutierrez Reina, D.; Alvarado-Barrios, L.; Tapia, A.; Escaño, J.M. A MPC Strategy for the Optimal Management of Microgrids Based on Evolutionary Optimization. Electronics
**2019**, 8, 1371. [Google Scholar] [CrossRef] [Green Version] - Hybrid Energy Networks: District Heating and Cooling Networks in an Integrated Energy System Context. Available online: https://www.iea-dhc.org/the-research/annexes/2017-2021-annex-ts3-draft/ (accessed on 4 September 2020).
- Maidment, G.G.; Zhao, X.; Riffat, S.B. Combined cooling and heating using a gas engine in a supermarket. Appl. Energy
**2001**, 68, 321–335. [Google Scholar] [CrossRef] - Khan, K.H.; Rasul, M.G.; Khan, M.M.K. Energy conservation in buildings: Cogeneration and cogeneration coupled with thermal-energy storage. Appl. Energy
**2004**, 77, 15–34. [Google Scholar] [CrossRef] - Shah, A.; Krishnan, N. Life Cycle evaluation of combined heat and power alternatives in data centers. In Proceedings of the 2005 IEEE International Symposium on Electronics and the Environment, New Orleans, LA, USA, 16–19 May 2005; pp. 19–24. [Google Scholar] [CrossRef]
- Kosugi, T.; Tokimatsu, K.; Zhou, W. An economic analysis of a clean-development mechanism project: A case introducing a natural gas-fired combined heat-and-power facility in a Chinese industrial area. Appl. Energy
**2005**, 80, 197–212. [Google Scholar] [CrossRef] - Chicco, G.; Mancarella, P. From cogeneration to trigeneration: Profitable alternatives in a competitive market. IEEE Trans. Energy Convers.
**2006**, 21, 265–272. [Google Scholar] [CrossRef] - Lund, H.; Werner, S.; Wiltshire, R.; Svendsen, S.; Thorsen, J.E.; Hvelplund, F.; Mathiesen, B.V. 4th Generation District Heating (4GDH): Integrating smart thermal grids into future sustainable energy systems. Energy
**2014**, 68, 1–11. [Google Scholar] [CrossRef] - Arcuri, P.; Florio, G.; Fragiacomo, P. A mixed integer programming model for optimal design of trigeneration in a hospital complex. Energy
**2007**, 32, 1430–1447. [Google Scholar] [CrossRef] - Canova, A.; Cavallero, C.; Freschi, F.; Giaccone, L.; Repetto, M.; Tartaglia, M. Optimal energy management. IEEE Ind. Appl. Mag.
**2009**, 15, 62–65. [Google Scholar] [CrossRef] - Cardona, E.; Piacentino, A. Optimal design of CHCP plants in the civil sector by thermoeconomics. Appl. Energy
**2007**, 84, 729–748. [Google Scholar] [CrossRef] - Giaccone, L.; Canova, A. Economical comparison of CHP systems for industrial user with large steam demand. Appl. Energy
**2009**, 86, 904–914. [Google Scholar] [CrossRef] [Green Version] - Freschi, F.; Giaccone, L.; Lazzeroni, P.; Repetto, M. Economic and environmental analysis of a trigeneration system for food-industry: A case study. Appl. Energy
**2013**, 107, 157–172. [Google Scholar] [CrossRef] - Medrano, M.; Brouwer, J.; McDonnel, V.; Mauzey, J.; Samuelsen, S. Integration of distributed generation systems into generic types of commercial buildings in California. Energy Build.
**2008**, 40, 537–548. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Rezgui, Y.; Zhu, H. District heating and cooling optimization and enhancement—Towards integration of renewables, storage and smart grid. Renew. Sustain. Energy Rev.
**2017**, 72, 281–294. [Google Scholar] [CrossRef] - Olsthoorn, D.; Haghighat, F.; Mirzaei, P.A. Integration of storage and renewable energy into district heating systems: A review of modelling and optimization. Sol. Energy
**2016**, 136, 49–64. [Google Scholar] [CrossRef] - Zakaria, A.; Ismail, F.B.; Lipu, M.H.; Hannani, M. Uncertainty models for stochastic optimization in renewable energy applications. Renew. Energy
**2020**, 145, 1543–1571. [Google Scholar] [CrossRef] - Urbanucci, L.; Testi, D. Optimal integrated sizing and operation of a CHP system with Monte Carlo risk analysis for long-term uncertainty in energy demands. Energy Convers. Manag.
**2018**, 157, 307–316. [Google Scholar] [CrossRef] - Kaintura, A.; Dhaene, T.; Spina, D. Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits. Electronics
**2018**, 7, 30. [Google Scholar] [CrossRef] [Green Version] - Blatman, G.; Sudret, B. Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf.
**2010**, 95, 1216–1229. [Google Scholar] [CrossRef] - Sobol, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul.
**2001**, 55, 271–2080. [Google Scholar] [CrossRef] - PCE Python Package. Available online: https://github.com/giaccone/pce (accessed on 2 September 2020).
- Pérez-Mora, N.; Lazzeroni, P.; Martínez-Moll, V.; Repetto, M. Optimal management of a complex DHC plant. Energy Convers. Manag.
**2017**, 145, 386–397. [Google Scholar] [CrossRef] - Crestaux, T.; Maitre, O.L.; Martinez, J.M. Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf.
**2009**, 94, 1161–1172. [Google Scholar] [CrossRef] - Sudret, B. Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf.
**2008**, 93, 964–979. [Google Scholar] [CrossRef] - Barthelmann, V.; Novak, E.; Ritter, K. High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math.
**2000**, 12, 273–288. [Google Scholar] [CrossRef] - Smith, A.H.C.; Monti, A.; Ponci, F. Uncertainty and Worst-Case Analysis in Electrical Measurements Using Polynomial Chaos Theory. IEEE Trans. Instrum. Meas.
**2009**, 58, 58–67. [Google Scholar] [CrossRef]

**Figure 5.**Mean and standard deviation of the cost function by the PCE and MC methods in the case of uniform distribution; results in €/day.

**Figure 6.**Mean and standard deviation of the cost function by the PCE and MC methods in the case of normal distrinution, results in €/day.

Variation | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{12}$ | ${\mathit{S}}_{13}$ | ${\mathit{S}}_{23}$ | ${\mathit{S}}_{123}$ |
---|---|---|---|---|---|---|---|

Uniform | 0.4931 | 0.1172 | 0.3891 | 0.0002 | 0.0002 | 0.0002 | 0.0000 |

Normal | 0.4933 | 0.1177 | 0.3885 | 0.0002 | 0.0002 | 0.0001 | 0.0000 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Giaccone, L.; Lazzeroni, P.; Repetto, M.
Uncertainty Quantification in Energy Management Procedures. *Electronics* **2020**, *9*, 1471.
https://doi.org/10.3390/electronics9091471

**AMA Style**

Giaccone L, Lazzeroni P, Repetto M.
Uncertainty Quantification in Energy Management Procedures. *Electronics*. 2020; 9(9):1471.
https://doi.org/10.3390/electronics9091471

**Chicago/Turabian Style**

Giaccone, Luca, Paolo Lazzeroni, and Maurizio Repetto.
2020. "Uncertainty Quantification in Energy Management Procedures" *Electronics* 9, no. 9: 1471.
https://doi.org/10.3390/electronics9091471