# Measuring the Risk of Supply and Demand Imbalance at the Monthly to Seasonal Scale in France

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Risk Measures

- the upper-left quadrant, where $Co>\overline{Co}$ and $Pr<\overline{Pr}$, corresponds to a risk of high consumption and low production. It is referred in the following as ${R}_{c+p-}$.
- the lower-right quandrant, where $Co<\overline{Co}$ and $Pr>\overline{Pr}$, orresponds to a risk of low consumption and high production. It is referred in the following as ${R}_{c-p+}$.

- a seasonal climatology estimated over a long period of observed consumption and production (typically 30 years or more)
- a ‘real’ seasonal joint distribution estimated on the sample of observed or inferred consumption and production for the specific season
- a reconstructed (or forecasted) seasonal joint distribution using the model detailed in Section 2.2

#### 2.2. Modelling the Joint PDF of Consumption and Production

#### 2.2.1. Explanatory Variables

#### 2.2.2. Consumption ($Co$) and production ($Pr$)

#### 2.3. Calibration of the Model

## 3. Model Performance Assessment

#### 3.1. Modelling Results

#### 3.2. Explanatory Value of the First PCs

## 4. Risk Forecasting

#### 4.1. Integration over the Seasonal Ensemble Forecasts of the ECMWF

#### 4.2. Monthly and Seasonal Forecasts

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sensitivity of the Model to the Number of PCs Used to Fit the Indexes

**Figure A1.**Mean absolute error (MAE) between the real and reconstructed risks normalized by the climatological risk as a function of the number of PCs used to fit the indexes ${I}_{Pr}$ and ${I}_{Co}$ in Equation (4).

## Appendix B. Significance Levels for the Risk Measures

**Figure A2.**p-value resulting from the 2 dimensional Kolmogorov-Smirnov test performed between the climatological sample and the seasonal samples as a function of the real computed risk measure normalized by climatology. Black dashed line represents the 10% significance level for which the hypothesis of 2 samples coming from the same distribution cannot be rejected; Red dashed lines represent the threshold defined; (

**a**) for winter and risk of high consumption and low production ${R}_{c+p-}$, (

**b**) for winter and risk of low consumption and high production ${R}_{c-p+}$, (

**c**) for fall and risk of high consumption and low production ${R}_{c+p-}$, (

**d**) for fall and risk of low consumption and high production ${R}_{c-p+}$.

**Figure A3.**p-value resulting from the 2 dimensional Kolmogorov-Smirnov test performed between the climatological sample and the seasonal samples as a function of the absolute difference between the $Va{R}_{95}^{real}$ and $Va{R}_{95}^{clim}$. Black dashed line represents the 10% significance level for which the hypothesis of 2 samples coming from the same distribution cannot be rejected; Red dashed lines represent the threshold defined; (

**a**) for winter, (

**b**) for fall.

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**Figure 1.**(

**a**) Schematic of the joint distribution $P(Co,Pr)$ for a given season overlaid with the climatological production and consumption averages $\overline{Pr}$ and $\overline{Co}$ displayed with the vertical and horizontal lines. The quantities $C+$, $C-$, $P+$ and $P-$ correspond to $Co>\overline{Co}$, $Co<\overline{Co}$, $Pr>\overline{Pr}$ and $Pr<\overline{Pr}$, respectively. The quantities ${R}_{c+p-}=P(Pr<\overline{Pr},Co>\overline{Co})$ and ${R}_{c-p+}=P(Pr>\overline{Pr},Co<\overline{Co})$ correspond to risk of low production and high consumption and to risk of low consumption and high production, respectively. (

**b**) Schematic of the joint distribution $P(Co,Pr)$ for a given season overlaid with the line of constant $Dcp=Co-Pr$ which separate the probability into a part where $P(Co,Pr)=0.05$ and another where $P(Co,Pr)=0.95$. The corresponding quantity $Dcp={R}_{95}$ is the Value at Risk 95 ($Va{R}_{95}$) of $Dcp$

**Figure 2.**(

**a**) North Atlantic/European domain from where the 500-hPa geopotential height $Z500$ is retrieved. The red box corresponds to the domain in panel b. (

**b**) Domain covering France and part of its neighbouring countries. The colors represents the topography and bathymetry.

**Figure 3.**First EOFs (left) and first five PCs (right) of the PCA performed on the fitting and validation period on the North Atlantic/Europe domain of ERA-Interim Z500 dataset. (

**a**) EOF and (

**b**) PC corresponding to the seasonal pattern. (

**c**) EOF and (

**d**) PC corresponding to North Atlantic Oscillation pattern. (

**e**) EOF and (

**f**) PC corresponding to Scandinavian pattern.

**Figure 4.**French daily wind energy production (

**a**) in winter and (

**b**) in fall 2015 from RTE data (in black) and computed using ERA-I surface wind speed (in green); and French daily maximum consumption (

**c**) in winter and (

**d**) in fall 2015 from RTE data (in black) and computed using ERA-I surface temperature (in green).

**Figure 5.**Blue contour lines show (

**a**) the real and (

**b**) the reconstructed joint PDF for winter 2010. Red shading show the winter climatological joint PDF. Red vertical and horizontal lines correspond respectively to the mean production and mean consumption given by climatology. Red dashed line shows the $Va{R}_{95}$ of $Co-Pr$ given by the climatological joint PDF, black dashed lines show the $Va{R}_{95}$ of $Co-Pr$ given by (

**a**) the real and (

**b**) the reconstructed joint PDF. (

**c**) Time series of the wind energy production (in blue) and the temperature driven consumption (in red) during winter 2010. Dashed lines show the winter mean climatological value of the wind energy production (in blue) and of the consumption (in red).

**Figure 6.**Time series of the real (black curves) and reconstructed (blue curves) measures risk compared with the climatological level of risk (in red). (

**a**) Risk ${R}_{c+p-}$ in winter; (

**b**) Risk ${R}_{c+p-}$ in fall; (

**c**) Risk ${R}_{c-p+}$ in winter; (

**d**) Risk ${R}_{c-p+}$ in fall; (

**e**) Risk $Va{R}_{95}$ in winter; (

**f**) Risk $Va{R}_{95}$ in winter. For risks ${R}_{c+p-}$ (panels

**a**,

**b**) and ${R}_{c-p+}$ (panels

**c**,

**d**) the risk measure is normalized by the climatological risk so that a value above (below) one highlight a higher (smaller) risk than normal., while for the risks $Va{R}_{95}$, the risk is measured in GW. Dashed dotted red lines correspond significance levels at 10% as defined in the Appendix B.

**Figure 7.**Winter distributions of the 3 first Principal Components corresponding to Figure 3. In each panel 3 distributions are plotted, the distribution of all winters (in black); the distribution of risk higher than normal (in red); the distribution of risk lower than normal (in blue). (

**a**) distributions of the seasonal pattern PC corresponding to a risk of high consumption and low production (${R}_{c+p-}$); (

**b**) distributions of the seasonal pattern PC corresponding to a risk of a risk of low consumption and high production higher than normal (${R}_{c-p+}$). (

**c**,

**d**) same as (

**a**,

**b**) for the NAO pattern. (

**e**,

**f**) same as (

**a**,

**b**) for the SCA pattern.

**Figure 8.**Forecast of January 2014 at the monthly horizon. (

**a**) Reconstructed joint PDF (blue contour), real daily data (yellow dots) and the climatological PDF for January (red contour). (

**b**) time series of the ensemble forecast of production (in blue contour) and of the real production (in yellow). (

**c**) time series of the ensemble forecast of consumption (in blue contour) and of the real consumption (in yellow). The red line indicates the climatological value of production and consumption. The red, blue and yellow dashed lines indicate the extreme risk $Va{R}_{95}$ given by the climatology, the forecasted and the real PDFs respectively.

**Figure 9.**Monthly forecasted (blue) and real (black) risk relative to the climatology for winter months for (

**a**) high consumption and low production (${R}_{c+p-}$) and (

**b**) low consumption and high production (${R}_{c-p+}$) and (

**c**) Extreme risk $Va{R}_{95}$. Dashed red lines indicate significance levels at 10%.

**Figure 10.**Same as Figure 8 for winter 2014 and the 3-month forecast horizon (January to March).

**Figure 11.**Same as Figure 9 for winters at the 3-month forecast horizon (January to March).

Variable | Data Source | Time Period | Time Resolution | Computation | Result |
---|---|---|---|---|---|

10 m wind speed | ERA-I | 01-01-1979 31-12-2015 | 6-hourly | - Averaged daily - Compute Pr | Pr (1979 to 2015) |

French Production | RTE | 01-01-2015 31-12-2015 | hourly | ||

2 m temperature | ERA-I | 01-01-1979 31-12-2015 | 6-hourly | - Averaged daily - Average in France - Compute Max Co | Co (1979 to 2015) |

French consumption | RTE | 01-01-2015 31-12-2015 | hourly | ||

Z500 | ERA-I | 01-01-1979 31-12-2015 | 6-hourly | - Averaged daily - PCA - Select several PCs | ${X}_{1},\dots ,{X}_{n}$ (1979 to 2015) |

Z500 | ECMWF seasonal ensemble forecasts | 01-01-2012 31-12-2015 | 6-hourly Available every month | - Averaged daily - Apply PCA - Select horizon | Ensemble of ${X}_{n}$ (2012 to 2015) |

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**MDPI and ACS Style**

Alonzo, B.; Drobinski, P.; Plougonven, R.; Tankov, P.
Measuring the Risk of Supply and Demand Imbalance at the Monthly to Seasonal Scale in France. *Energies* **2020**, *13*, 4888.
https://doi.org/10.3390/en13184888

**AMA Style**

Alonzo B, Drobinski P, Plougonven R, Tankov P.
Measuring the Risk of Supply and Demand Imbalance at the Monthly to Seasonal Scale in France. *Energies*. 2020; 13(18):4888.
https://doi.org/10.3390/en13184888

**Chicago/Turabian Style**

Alonzo, Bastien, Philippe Drobinski, Riwal Plougonven, and Peter Tankov.
2020. "Measuring the Risk of Supply and Demand Imbalance at the Monthly to Seasonal Scale in France" *Energies* 13, no. 18: 4888.
https://doi.org/10.3390/en13184888