# Electrical Power Diversification: An Approach Based on the Method of Maximum Entropy in the Mean

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statements

**Problem**

**1**

**Comments:**

- (
**i**) - Production per unit time refers to an average produced or required during some standardized time interval (one hour, for example).
- (
**ii**) - These constraints can be replaced by intervals to allow for uncertainty in the demand or uncertainty in the supply. We describe this further below.
- (
**iii**) - Note that we are taking into account possible nonlinear constraints, resulting from the actual physical transport of energy through the network, in which the losses could depend on the amount of energy being transported.

**Problem**

**2**

**Problem**

**3**

**point constraints**.

#### 1.2. Contents of the Paper

## 2. Mathematical Model Derivation

#### 2.1. MEM for Point Data

**Definition**

**1.**

**Problem**

**4.**

#### 2.2. MEM for Data in Ranges

**Problem**

**5.**

#### 2.3. Computation of Z($\mathbf{\lambda}$)

#### 2.3.1. The Case of Point Constraints

#### 2.3.2. The Case of Data in Ranges

## 3. The Maxentropic Solution to the Energy Diversification Problem

#### 3.1. Statement of the Problems and Representation of the Solution

#### Relabeling the Problem

**Problem**

**6**

#### 3.2. The Energy Diversification According to Stirling’s Measure

## 4. Numerical Examples

#### 4.1. Case 1: Aggregated Demand

**The diversification measure**

#### 4.2. Case 2: Disaggregated Demand

${x}_{i,j}$ | 1 | 2 | 3 | 4 |

1 | 0.300 | 0.178 | 0.301 | 0.106 |

2 | 0.369 | 0.221 | 0.327 | 0.119 |

**The diversification measure**

## 5. Conclusions and Policy Implications

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ku, A. Modeling Uncertainty in Electricity Capacity Planning, Thesis, 1995. Available online: www.s3.amazonaws.com/academia.edu.documents/52911593/thesis_aku.pdf (accessed on 13 December 2020).
- Jaynes, E. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Kapur, J.N. Maximum Entropy Models in Science and Engineering; Wiley: New York, NY, USA, 1998. [Google Scholar]
- Wilson, A.G. Entropy in Urban and Regional Modeling; Pion: London, UK, 1970. [Google Scholar]
- Xiao, X.Y.; Chao, M.A.; Yang, H.G.; Li, H.Q. Maximum entropy probability method applied to assess voltage sag frequency due to transmission line fault in the electric power system. Appl. Stoch. Models Bus. Ind.
**2009**, 26, 595–608. [Google Scholar] [CrossRef] - Stirling, A. Multicriteria diversity analysis. A novel heuristic framework for appraising energy portfolios. Energy Policy
**2010**, 38, 1622–1634. [Google Scholar] [CrossRef] - Stirling, A. Diversity and ignorance in electricity supply investment. Energy Policy
**1994**, 22, 195–216. [Google Scholar] [CrossRef] - Kullback, S. Information Theory and Statistics, 2nd ed.; Dover Publications: New York, NY, USA, 1955. [Google Scholar]
- Golan, A.; Gzyl, H. A generalized maxentropic inversion procedure for noisy data. Appl. Math. Comput.
**2002**, 127, 249–260. [Google Scholar] [CrossRef] - Gzyl, H.; Velásquez, Y. Linear Inverse Problems: The Maximum Entropy Connection; World Scientific Publishers: Singapore, 2011. [Google Scholar]
- Borwein, J.; Lewis, A. Convex Analysis and Nonlinear Optimization; CMS Books; Springer: New York, NY, USA, 2000. [Google Scholar]
- Mead, L.R.; Papanicolau, N. Maximum entropy in the problem of moments. J. Math. Phys.
**1984**, 25, 2404–2417. [Google Scholar] [CrossRef] [Green Version]

Type | Source | Lower Bd. (a) | Upper Bd. (b) | Cost |
---|---|---|---|---|

1 | Hydraulic | 0.01 | 0.8 | 0.00 |

2 | Gas | 0.01 | 0.5 | 0.37 |

3 | Fuel | 0.01 | 0.7 | 0.76 |

4 | Coal | 0.01 | 0.4 | 0.96 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Bautista, R.; Gzyl, H.; ter Horst, E.; Molina, G.
Electrical Power Diversification: An Approach Based on the Method of Maximum Entropy in the Mean. *Entropy* **2021**, *23*, 281.
https://doi.org/10.3390/e23030281

**AMA Style**

Bautista R, Gzyl H, ter Horst E, Molina G.
Electrical Power Diversification: An Approach Based on the Method of Maximum Entropy in the Mean. *Entropy*. 2021; 23(3):281.
https://doi.org/10.3390/e23030281

**Chicago/Turabian Style**

Bautista, Rafael, Henryk Gzyl, Enrique ter Horst, and Germán Molina.
2021. "Electrical Power Diversification: An Approach Based on the Method of Maximum Entropy in the Mean" *Entropy* 23, no. 3: 281.
https://doi.org/10.3390/e23030281