#
A Probabilistically Constrained Approach for the Energy Procurement Problem^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Formulation

#### 2.1. Constraints

#### 2.2. Objective Function

## 3. Dealing with Joint Chance Constraints and Risk

## 4. Computational Results

#### 4.1. The Impact of the Reliability Level

#### 4.2. The Effect of the Risk Modeling

#### 4.3. Out-of-Sample Analysis

## 5. Discussion

## 6. Conclusions and Future Research Directions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Nomenclature

Nomenclature | |
---|---|

Sets | |

$\mathcal{T}$ | set of elementary time periods (months) |

F | set of time-of-use blocks in which hours are divided |

$\mathcal{N}$ | set of potential bilateral contracts |

S | set of scenarios used for representing evolution of uncertain parameters |

Deterministic Parameters | |

${B}_{itf}$ | unit energy price of bilateral contract i for time-of-use block f of time period t (€/MWh) |

$F{B}_{i}$ | fixed cost for selection of bilateal contrat i (€) |

$\alpha $ | confidence level for chance constraints |

$\beta $ | confidence level for Conditional Value at Risk |

K | maximum number of active bilateral contracts |

${Q}_{tf}$ | upper bound on the energy amount that can be produced in time-of-use block f |

of time period t (MWh) | |

$L{B}_{itf},U{B}_{itf}$ | lower and upper bound on the quantity that can be bought from bilateral contract i |

in time-of-use block f of time period t (MWh) | |

$P{C}_{tf}$ | production cost for time-of-use block f of time period t (€/MWh) |

${\pi}^{s}$ | probability of occurrance of scenario s |

Stochastic Parameters | |

${d}_{tf}^{s}$ | overall energy demand for time-of-use block f of time period t under scenario s ([MWh]) |

${P}_{tf}^{s}$ | unit purchasing price on day-ahead market for time-of-use block f of time period t |

under scenario s (€/MWh) | |

${R}_{tf}^{s}$ | unit selling price on day-ahead market for time-of-use block f of time period t |

under scenario s (€/MWh) | |

Decision Variables | |

${z}_{i}$ | selection of contract i |

${x}_{itf}$ | amount of energy to purchase by bilateral contract i in time-of-use block f |

of time period t (MWh) | |

${y}_{tf}$ | amount of energy to buy from the day-ahead market in time-of-use block f |

of time period t (MWh) | |

${w}_{tf}$ | amount of energy to sell on the day-ahead market in time-of-use block f |

of time period t (MWh) | |

${q}_{tf}$ | amount of energy to produce in time-of-use block f of time period t (MWh) |

Auxiliary Variables | |

${\eta}^{s}$ | auxiliary variable for CVaR linearization |

${\delta}_{tf}$ | auxiliary variable for chance constraints linearization |

${\gamma}^{s}$ | scenario dependent binary support variables |

Dependent Variables | |

${C}^{s}$ | overall cost under scenario s |

$Va{R}_{\beta}$ | Value at Risk for a certain confidence level $\beta $ (€) |

$CVa{R}_{\beta}$ | Conditional Value at Risk for a certain confidence level $\beta $ (€) |

## Appendix B. Bilateral contracts

Bilateral Contracts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

January | F1 | 42.27 | 44.38 | 35.51 | 41.90 | 41.42 | 39.35 | 39.35 | 41.32 | 43.39 | 49.89 |

F2 | 43.27 | 34.62 | 36.35 | 42.89 | 42.41 | 46.22 | 48.53 | 55.81 | 44.65 | 35.72 | |

F3 | 35.77 | 42.21 | 33.77 | 35.46 | 35.06 | 41.02 | 43.07 | 34.45 | 39.62 | 40.81 | |

February | F1 | 42.27 | 40.16 | 44.17 | 48.59 | 41.42 | 45.57 | 50.12 | 47.62 | 45.24 | 49.76 |

F2 | 43.27 | 47.60 | 45.22 | 49.74 | 42.41 | 46.65 | 44.31 | 48.75 | 53.62 | 58.98 | |

F3 | 35.77 | 39.35 | 43.28 | 41.12 | 35.06 | 38.56 | 36.63 | 40.30 | 44.33 | 47.43 | |

March | F1 | 42.27 | 41.42 | 41.42 | 37.28 | 41.42 | 41.42 | 37.28 | 36.54 | 35.80 | 32.22 |

F2 | 43.27 | 43.27 | 42.41 | 38.17 | 42.41 | 38.17 | 37.40 | 33.66 | 33.66 | 33.66 | |

F3 | 35.77 | 32.20 | 32.20 | 31.55 | 35.06 | 31.55 | 30.92 | 30.92 | 27.83 | 24.49 | |

April | F1 | 42.27 | 41.85 | 43.94 | 43.94 | 41.42 | 43.49 | 43.49 | 43.06 | 42.63 | 42.63 |

F2 | 43.27 | 45.44 | 44.98 | 44.98 | 42.41 | 42.41 | 41.98 | 41.98 | 44.08 | 46.29 | |

F3 | 35.77 | 35.77 | 37.56 | 37.19 | 35.06 | 35.06 | 34.71 | 36.44 | 36.44 | 34.98 | |

May | F1 | 42.27 | 43.54 | 44.84 | 47.08 | 41.42 | 42.67 | 44.80 | 46.14 | 47.53 | 49.90 |

F2 | 43.27 | 44.57 | 45.91 | 48.20 | 42.41 | 44.53 | 45.86 | 48.16 | 49.60 | 51.09 | |

F3 | 35.77 | 37.56 | 38.69 | 39.85 | 35.06 | 36.81 | 37.91 | 39.05 | 41.00 | 43.46 | |

June | F1 | 42.27 | 43.11 | 42.25 | 42.25 | 41.42 | 40.59 | 40.59 | 41.41 | 42.23 | 42.23 |

F2 | 43.27 | 42.41 | 43.25 | 43.25 | 42.41 | 42.41 | 43.25 | 43.25 | 42.39 | 41.54 | |

F3 | 35.77 | 35.77 | 35.06 | 35.76 | 35.06 | 35.06 | 35.76 | 35.04 | 35.04 | 35.74 | |

July | F1 | 42.27 | 42.27 | 35.93 | 28.74 | 41.42 | 35.21 | 28.17 | 28.17 | 28.17 | 22.53 |

F2 | 43.27 | 36.78 | 36.78 | 29.42 | 42.41 | 33.93 | 33.93 | 27.14 | 23.07 | 19.61 | |

F3 | 35.77 | 28.62 | 24.33 | 24.33 | 35.06 | 28.05 | 28.05 | 23.84 | 19.07 | 17.55 | |

August | F1 | 42.27 | 44.38 | 53.26 | 55.92 | 41.42 | 49.71 | 52.19 | 54.80 | 57.54 | 60.42 |

F2 | 43.27 | 51.93 | 54.52 | 57.25 | 42.41 | 44.53 | 46.75 | 49.09 | 58.91 | 70.69 | |

F3 | 35.77 | 37.56 | 45.07 | 47.33 | 35.06 | 36.81 | 38.65 | 46.38 | 48.70 | 51.62 | |

September | F1 | 42.27 | 45.65 | 50.22 | 37.66 | 41.42 | 45.57 | 34.17 | 36.91 | 39.86 | 29.90 |

F2 | 43.27 | 47.60 | 51.41 | 48.84 | 42.41 | 31.80 | 34.35 | 25.76 | 28.34 | 31.17 | |

F3 | 35.77 | 26.83 | 29.51 | 31.87 | 35.06 | 26.29 | 28.40 | 31.24 | 23.43 | 20.85 | |

October | F1 | 42.27 | 38.89 | 34.61 | 38.07 | 41.42 | 36.87 | 41.29 | 37.99 | 34.95 | 39.14 |

F2 | 43.27 | 38.51 | 35.43 | 42.52 | 42.41 | 44.53 | 40.96 | 43.01 | 38.28 | 34.07 | |

F3 | 35.77 | 44.72 | 39.80 | 36.61 | 35.06 | 38.91 | 35.80 | 31.86 | 35.69 | 39.25 | |

November | F1 | 42.27 | 41.42 | 39.35 | 38.57 | 41.42 | 39.35 | 38.57 | 37.79 | 37.04 | 36.30 |

F2 | 43.27 | 41.11 | 40.29 | 39.48 | 42.41 | 41.56 | 40.73 | 39.91 | 37.92 | 36.02 | |

F3 | 35.77 | 35.06 | 33.30 | 32.64 | 35.06 | 34.36 | 33.67 | 31.99 | 31.35 | 30.41 | |

December | F1 | 42.27 | 40.16 | 42.56 | 39.16 | 41.42 | 43.91 | 40.40 | 38.38 | 36.46 | 34.63 |

F2 | 43.27 | 45.87 | 43.57 | 40.09 | 42.41 | 39.01 | 37.06 | 34.10 | 36.14 | 38.31 | |

F3 | 35.77 | 32.91 | 34.89 | 33.14 | 35.06 | 32.25 | 30.64 | 32.48 | 29.88 | 29.28 | |

Fixed cost (€) | 650 | 675 | 575 | 375 | 725 | 650 | 625 | 500 | 550 | 850 |

Bilateral Contracts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

January | F1 | 9.87 | 15.38 | 8.74 | 8.55 | 11.01 | 14.81 | 9.49 | 11.39 | 8.36 | 19.37 |

F2 | 5.29 | 8.25 | 4.68 | 4.58 | 5.90 | 7.94 | 5.09 | 6.11 | 4.48 | 10.38 | |

F3 | 9.80 | 15.27 | 8.67 | 8.48 | 10.93 | 14.70 | 9.42 | 11.31 | 8.29 | 19.22 | |

February | F1 | 11.10 | 17.29 | 9.82 | 9.60 | 12.38 | 16.65 | 10.67 | 12.81 | 9.39 | 21.77 |

F2 | 4.87 | 7.59 | 4.31 | 4.22 | 5.43 | 7.31 | 4.68 | 5.62 | 4.12 | 9.55 | |

F3 | 7.85 | 12.23 | 6.95 | 6.80 | 8.76 | 11.78 | 7.55 | 9.06 | 6.64 | 15.40 | |

March | F1 | 9.74 | 15.17 | 8.62 | 8.43 | 10.86 | 14.61 | 9.37 | 11.24 | 8.24 | 19.11 |

F2 | 5.04 | 7.84 | 4.45 | 4.36 | 5.62 | 7.55 | 4.84 | 5.81 | 4.26 | 9.88 | |

F3 | 7.53 | 11.74 | 6.67 | 6.52 | 8.40 | 11.30 | 7.24 | 8.69 | 6.38 | 14.78 | |

April | F1 | 4.49 | 6.99 | 3.97 | 3.88 | 5.01 | 6.73 | 4.32 | 5.18 | 3.80 | 8.80 |

F2 | 3.80 | 5.92 | 3.36 | 3.29 | 4.24 | 5.70 | 3.66 | 4.39 | 3.22 | 7.46 | |

F3 | 7.68 | 11.96 | 6.79 | 6.64 | 8.56 | 11.52 | 7.38 | 8.86 | 6.50 | 15.06 | |

May | F1 | 6.68 | 10.40 | 5.91 | 5.78 | 7.45 | 10.02 | 6.42 | 7.71 | 5.65 | 13.10 |

F2 | 4.34 | 6.77 | 3.84 | 3.76 | 4.85 | 6.52 | 4.18 | 5.01 | 3.68 | 8.52 | |

F3 | 9.30 | 14.49 | 8.23 | 8.05 | 10.37 | 13.95 | 8.94 | 10.73 | 7.87 | 18.24 | |

June | F1 | 7.93 | 12.35 | 7.01 | 6.86 | 8.84 | 11.89 | 7.62 | 9.15 | 6.71 | 15.55 |

F2 | 5.21 | 8.12 | 4.61 | 4.51 | 5.81 | 7.82 | 5.01 | 6.02 | 4.41 | 10.23 | |

F3 | 9.01 | 14.03 | 7.97 | 7.80 | 10.05 | 13.51 | 8.66 | 10.40 | 7.62 | 17.67 | |

July | F1 | 9.65 | 15.04 | 8.54 | 8.35 | 10.77 | 14.48 | 9.28 | 11.14 | 8.17 | 18.93 |

F2 | 5.76 | 8.97 | 5.09 | 4.98 | 6.42 | 8.64 | 5.54 | 6.64 | 4.87 | 11.29 | |

F3 | 9.31 | 14.50 | 8.24 | 8.06 | 10.38 | 13.96 | 8.95 | 10.74 | 7.88 | 18.26 | |

August | F1 | 2.63 | 4.09 | 2.32 | 2.27 | 2.93 | 3.94 | 2.53 | 3.03 | 2.22 | 5.15 |

F2 | 3.64 | 5.67 | 3.22 | 3.15 | 4.06 | 5.46 | 3.50 | 4.20 | 3.08 | 7.14 | |

F3 | 8.19 | 12.76 | 7.25 | 7.09 | 9.14 | 12.29 | 7.88 | 9.45 | 6.93 | 16.07 | |

September | F1 | 6.89 | 10.73 | 6.10 | 5.96 | 7.69 | 10.34 | 6.63 | 7.95 | 5.83 | 13.52 |

F2 | 4.66 | 7.26 | 4.12 | 4.03 | 5.20 | 6.99 | 4.48 | 5.38 | 3.94 | 9.14 | |

F3 | 8.14 | 12.67 | 7.20 | 7.04 | 9.07 | 12.20 | 7.82 | 9.39 | 6.88 | 15.96 | |

October | F1 | 7.50 | 11.68 | 6.64 | 6.49 | 8.37 | 11.25 | 7.21 | 8.65 | 6.35 | 14.71 |

F2 | 4.96 | 7.72 | 4.38 | 4.29 | 5.53 | 7.43 | 4.76 | 5.72 | 4.19 | 9.72 | |

F3 | 8.62 | 13.43 | 7.63 | 7.46 | 9.62 | 12.94 | 8.29 | 9.95 | 7.30 | 16.92 | |

November | F1 | 9.14 | 14.23 | 8.08 | 7.91 | 10.19 | 13.71 | 8.79 | 10.54 | 7.73 | 17.92 |

F2 | 4.97 | 7.74 | 4.39 | 4.30 | 5.54 | 7.45 | 4.78 | 5.73 | 4.20 | 9.74 | |

F3 | 8.40 | 13.09 | 7.43 | 7.27 | 9.37 | 12.61 | 8.08 | 9.70 | 7.11 | 16.48 | |

December | F1 | 8.05 | 12.53 | 7.12 | 6.96 | 8.97 | 12.07 | 7.74 | 9.28 | 6.81 | 15.78 |

F2 | 4.61 | 7.18 | 4.08 | 3.99 | 5.14 | 6.92 | 4.43 | 5.32 | 3.90 | 9.04 | |

F3 | 10.14 | 15.80 | 8.97 | 8.78 | 11.31 | 15.21 | 9.75 | 11.70 | 8.58 | 19.89 |

Bilateral Contracts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

January | F1 | 98.75 | 102.54 | 87.35 | 56.97 | 110.14 | 98.75 | 94.95 | 75.96 | 83.55 | 129.13 |

F2 | 52.94 | 54.97 | 46.83 | 30.54 | 59.05 | 52.94 | 50.90 | 40.72 | 44.79 | 69.23 | |

F3 | 98.00 | 101.77 | 86.69 | 56.54 | 109.31 | 98.00 | 94.23 | 75.39 | 82.93 | 128.16 | |

February | F1 | 110.99 | 115.25 | 98.18 | 64.03 | 123.79 | 110.99 | 106.72 | 85.37 | 93.91 | 145.14 |

F2 | 48.71 | 50.58 | 43.09 | 28.10 | 54.33 | 48.71 | 46.84 | 37.47 | 41.22 | 63.70 | |

F3 | 78.52 | 81.54 | 69.46 | 45.30 | 87.58 | 78.52 | 75.50 | 60.40 | 66.44 | 102.69 | |

March | F1 | 97.41 | 101.15 | 86.17 | 56.20 | 108.65 | 97.41 | 93.66 | 74.93 | 82.42 | 127.38 |

F2 | 50.36 | 52.30 | 44.55 | 29.05 | 56.17 | 50.36 | 48.42 | 38.74 | 42.61 | 65.86 | |

F3 | 75.35 | 78.24 | 66.65 | 43.47 | 84.04 | 75.35 | 72.45 | 57.96 | 63.76 | 98.53 | |

April | F1 | 44.89 | 46.61 | 39.71 | 25.90 | 50.07 | 44.89 | 43.16 | 34.53 | 37.98 | 58.70 |

F2 | 38.02 | 39.49 | 33.64 | 21.94 | 42.41 | 38.02 | 36.56 | 29.25 | 32.17 | 49.72 | |

F3 | 76.78 | 79.73 | 67.92 | 44.30 | 85.64 | 76.78 | 73.83 | 59.06 | 64.97 | 100.41 | |

May | F1 | 66.79 | 69.36 | 59.08 | 38.53 | 74.50 | 66.79 | 64.22 | 51.38 | 56.52 | 87.34 |

F2 | 43.44 | 45.11 | 38.43 | 25.06 | 48.45 | 43.44 | 41.77 | 33.42 | 36.76 | 56.81 | |

F3 | 93.00 | 96.57 | 82.27 | 53.65 | 103.73 | 93.00 | 89.42 | 71.54 | 78.69 | 121.61 | |

June | F1 | 79.27 | 82.32 | 70.12 | 45.73 | 88.42 | 79.27 | 76.22 | 60.98 | 67.08 | 103.66 |

F2 | 52.13 | 54.14 | 46.12 | 30.08 | 58.15 | 52.13 | 50.13 | 40.10 | 44.11 | 68.17 | |

F3 | 90.09 | 93.56 | 79.70 | 51.98 | 100.49 | 90.09 | 86.63 | 69.30 | 76.23 | 117.81 | |

July | F1 | 96.53 | 100.24 | 85.39 | 55.69 | 107.67 | 96.53 | 92.82 | 74.25 | 81.68 | 126.23 |

F2 | 57.58 | 59.79 | 50.94 | 33.22 | 64.22 | 57.58 | 55.36 | 44.29 | 48.72 | 75.30 | |

F3 | 93.10 | 96.68 | 82.35 | 53.71 | 103.84 | 93.10 | 89.52 | 71.61 | 78.77 | 121.74 | |

August | F1 | 26.27 | 27.28 | 23.24 | 15.15 | 29.30 | 26.27 | 25.26 | 20.21 | 22.23 | 34.35 |

F2 | 36.41 | 37.81 | 32.21 | 21.01 | 40.61 | 36.41 | 35.01 | 28.01 | 30.81 | 47.61 | |

F3 | 81.92 | 85.07 | 72.47 | 47.26 | 91.37 | 81.92 | 78.77 | 63.02 | 69.32 | 107.13 | |

September | F1 | 68.90 | 71.55 | 60.95 | 39.75 | 76.85 | 68.90 | 66.25 | 53.00 | 58.30 | 90.10 |

F2 | 46.61 | 48.40 | 41.23 | 26.89 | 51.99 | 46.61 | 44.82 | 35.85 | 39.44 | 60.95 | |

F3 | 81.36 | 84.49 | 71.97 | 46.94 | 90.74 | 81.36 | 78.23 | 62.58 | 68.84 | 106.39 | |

October | F1 | 75.01 | 77.89 | 66.35 | 43.27 | 83.66 | 75.01 | 72.12 | 57.70 | 63.47 | 98.08 |

F2 | 49.55 | 51.46 | 43.83 | 28.59 | 55.27 | 49.55 | 47.65 | 38.12 | 41.93 | 64.80 | |

F3 | 86.24 | 89.56 | 76.29 | 49.75 | 96.19 | 86.24 | 82.92 | 66.34 | 72.97 | 112.77 | |

November | F1 | 91.38 | 94.89 | 80.83 | 52.72 | 101.92 | 91.38 | 87.86 | 70.29 | 77.32 | 119.49 |

F2 | 49.66 | 51.57 | 43.93 | 28.65 | 55.39 | 49.66 | 47.75 | 38.20 | 42.02 | 64.94 | |

F3 | 84.04 | 87.27 | 74.34 | 48.48 | 93.74 | 84.04 | 80.81 | 64.65 | 71.11 | 109.90 | |

December | F1 | 80.45 | 83.55 | 71.17 | 46.41 | 89.73 | 80.45 | 77.36 | 61.89 | 68.07 | 105.21 |

F2 | 46.11 | 47.88 | 40.79 | 26.60 | 51.43 | 46.11 | 44.33 | 35.47 | 39.01 | 60.29 | |

F3 | 101.42 | 105.32 | 89.72 | 58.51 | 113.13 | 101.42 | 97.52 | 78.02 | 85.82 | 132.63 |

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Bilateral Contracts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Average unit price (€/MWh) | F1 | 42.3 | 42.3 | 42.3 | 41.6 | 41.4 | 42.0 | 40.9 | 40.8 | 40.9 | 40.8 |

F2 | 43.3 | 43.3 | 43.3 | 43.7 | 42.4 | 41.3 | 41.3 | 40.9 | 40.9 | 41.4 | |

F3 | 35.8 | 35.7 | 35.6 | 35.6 | 35.1 | 34.6 | 34.5 | 34.5 | 34.4 | 34.7 | |

Fixed cost (€) | 650 | 675 | 575 | 375 | 725 | 650 | 625 | 500 | 550 | 850 |

$\mathit{\alpha}$ | Expected Cost (k€) | ${\mathit{CVaR}}_{\mathit{\beta}}$ (k€) |
---|---|---|

0.85 | 457.48 | 465.89 |

0.90 | 462.83 | 471.42 |

0.95 | 470.84 | 479.61 |

0.99 | 481.03 | 490.20 |

1 | 485.45 | 494.58 |

$\mathit{\lambda}$ | Bil. Contract Cost (k€) | Market Cost (k€) | Production Cost (k€) | Total Cost (k€) |
---|---|---|---|---|

0 | 322.16 | −2.27 | 18.70 | 338.55 |

0.5 | 314.88 | 5.87 | 16.64 | 337.39 |

1 | 287.79 | 35.76 | 14.58 | 338.13 |

$\mathit{\lambda}$ | 2S Model | CC Approach |
---|---|---|

0 | 365.64 | 338.55 |

0.5 | 364.69 | 337.39 |

1 | 364.99 | 338.13 |

© 2017 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**

Beraldi, P.; Violi, A.; Bruni, M.E.; Carrozzino, G.
A Probabilistically Constrained Approach for the Energy Procurement Problem. *Energies* **2017**, *10*, 2179.
https://doi.org/10.3390/en10122179

**AMA Style**

Beraldi P, Violi A, Bruni ME, Carrozzino G.
A Probabilistically Constrained Approach for the Energy Procurement Problem. *Energies*. 2017; 10(12):2179.
https://doi.org/10.3390/en10122179

**Chicago/Turabian Style**

Beraldi, Patrizia, Antonio Violi, Maria Elena Bruni, and Gianluca Carrozzino.
2017. "A Probabilistically Constrained Approach for the Energy Procurement Problem" *Energies* 10, no. 12: 2179.
https://doi.org/10.3390/en10122179