# Insurance Contracts for Hedging Wind Power Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. The Stochastic Models

#### 2.1. The WISMC Model of Wind Power Production

**Definition**

**1.**

**Assumption**

**1.**

**Lemma**

**1.**

#### 2.2. Joint Model of Electricity Price and Wind Power Production

**Assumption**

**2.**

**Assumption**

**3.**

## 3. The Insurance Problem

- if $r\left(s\right)<K$, he/she gets from the insurer the benefit $K-r\left(s\right)$;
- if $r\left(s\right)\ge K$, no money transfer from the insurer to the WPP occurs;
- at any time during the validity period of the contract, he/she pays a fixed premium to the insurer equal to $U\in \mathbb{R}$.
- money amounts are discounted with fixed discount factor v; accordingly, ${v}^{s}$ denotes the discount factor for s periods of time.

**Lemma**

**2.**

**Proof.**

**A2**, we have that:

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Materials and Methods

- -
- geographical coordinates: 39.5 N (latitude) and 8.75 E (longitude);
- -
- hub height of the turbine: 95 m;
- -
- rated power of the turbine: 2 MW;
- -
- cut-in wind speed: 4 m/s;
- -
- rated wind speed: 13 m/s;
- -
- cut-out wind speed: 25 m/s.

## 5. Results on the Insurance Problem

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Empirical pdf of wind speed fitted with a Weibull distribution and prices fitted with a lognormal distribution. The fits are performed by using an MLE algorithm.

**Figure 5.**Fair premium for real (continuous) and simulated (dashed) data as a function of the revenue K.

**Figure 6.**Differences between benefits and the periodic premium as a function of time and revenue K.

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**MDPI and ACS Style**

D’Amico, G.; Gismondi, F.; Petroni, F.
Insurance Contracts for Hedging Wind Power Uncertainty. *Mathematics* **2020**, *8*, 1376.
https://doi.org/10.3390/math8081376

**AMA Style**

D’Amico G, Gismondi F, Petroni F.
Insurance Contracts for Hedging Wind Power Uncertainty. *Mathematics*. 2020; 8(8):1376.
https://doi.org/10.3390/math8081376

**Chicago/Turabian Style**

D’Amico, Guglielmo, Fulvio Gismondi, and Filippo Petroni.
2020. "Insurance Contracts for Hedging Wind Power Uncertainty" *Mathematics* 8, no. 8: 1376.
https://doi.org/10.3390/math8081376