# Duality Based Risk Mitigation Method for Construction of Joint Hydro-Wind Coordination Short-Run Marginal Cost Curves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of the Joint Short-Run Marginal Cost Curve

## 3. Case Study and Results

^{3}) and is used for frequency regulation ancillary services [29]. A total of 10 electricity price events is generated with the random number generator based on the statistical analyses of the Croatian nodal electricity prices of the first week of December 2016 with the same probability of each event equal to $P\left(A\right)=\frac{1}{10},\forall t\in T\text{}\forall A\in {\mathcal{F}}_{t}$. The CVaR percentile $\alpha $ is 85%.

#### 3.1. Results

#### 3.2. Discussion

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

A | An event from the family of all events ${\mathcal{F}}_{t}$, $\text{}A\text{}\in {\mathcal{F}}_{t}$. |

${c}^{\prime}$ | Short-run marginal cost function (€/MW∙h). |

$d$ | Difference between forecasted $Y$ and actual wind generation ${y}_{w}$ (MW). |

${d}^{+}$ | Positive wind difference (MW). |

${d}^{-}$ | Negative wind difference (MW). |

$e$ | Natural water inflow (MW). |

${F}_{\alpha}$ | Special function used for risk shaping of CVaR (€). |

${\mathcal{F}}_{t}$ | Sigma algebra which defines all possible events in hour $t$. |

$g$ | Net outflow of hydro generation (MW) |

$I$ | Hourly revenue (€). |

${k}_{St}$ | Maximal capacity of reservoir (MW∙h). |

${k}_{Rm}$ | Parameter used for risk exposure reduction in risk shaping procedure (€). |

${k}_{Tu}$ | Hydro turbine maximal capacity (MW). |

${k}_{Tu}{}^{w}$ | Wind turbine maximal capacity (MW). |

${n}_{St}$ | Minimal capacity of reservoir (MW∙h). |

${n}_{Tu}$ | Hydro turbine minimal capacity (MW). |

${n}_{Tu}{}^{w}$ | Wind turbine minimal capacity (MW). |

$P\left(A\right)$ | Probability of the event $A$ and consequently probability of realization of the price $\pi \left(A\right)$ |

$s$ | Energy stock, amount of water in reservoir in t, (MW∙h). |

${s}_{0}$ | Energy stock at the beginning of planning interval (MW∙h) |

${s}_{T}$ | Energy stock surplus or deficit at the end of planning interval (MWh) |

$y$ | Hydro generation (MW). |

Y | Contracted wind generation (MW). |

${y}_{w}$ | Actual wind generation (MW). |

Greek | |

$\alpha $ | Percentile used for the CVaR where 1-$\alpha $ defines the worst events (%). |

$\delta $ | Shadow prices associated with constraint of CVaR’s helping variable $\eta $ (€). |

$\epsilon $ | Shadow prices associated with CVaR’s hourly revenue constraint. |

$\zeta $ | The decision variable which defines the Value at Risk (€). |

$\eta $ | Variable used for obtainment of the CVaR (€). |

${\kappa}^{Tu}$ | Shadow price of hydro generation maximum capacity (€/MW∙h). |

${\kappa}^{St}$ | Shadow price of reservoir maximum capacity (€/MW∙h). |

$\mathsf{\lambda}$ | Shadow price of energy stock surplus/deficit at the end of operation(€/MW∙h). |

${\nu}^{Tu}$ | Shadow price of hydro generation minimum capacity (€/MW∙h). |

${\nu}^{St}$ | Shadow price of reservoir minimum capacity (€/MW∙h). |

$\xi $ | Shadow price of risk mitigation capability (dimensionless). |

$\pi $ | Price of electricity (€/MW∙h). |

$\psi $ | Shadow price of water (€/MW∙h). |

Spaces | |

$\left[0,T\right]$ | Planning interval, subspace of the real line $t\in \left[0,T\right]\subset \mathbb{R}$. |

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**Figure 1.**The example of construction of the joint SRMC curves: (

**a**) hydro generation SRMC; (

**b**) vertical shift due to wind power differences; (

**c**) horizontal shift due to forecasted wind generation; (

**d**) final hydro-wind coordination SRMC.

**Figure 4.**Part of the 16th hour SRMC curves showing vertical upward shift due to increase in ${k}_{Rm}$.

**Figure 5.**The SRMC curves for ${k}_{Rm}$ = 1380 € and 5 €/MW∙h electricity generation cost of a Vinodol and Vrataruša hydro-wind coordination from (

**a**) 1st hour to (

**g**) 24th hour for the 5th December in 2016.

Reservoir | k_{St} (GWh) | Power Plant | k_{Tu}/n_{Tu} (m^{3}/s) | k_{Tu}/n_{Tu} (MW) |
---|---|---|---|---|

Lokve | 52 | PSP Fužine | 10/9 | 4.6/4.8 |

Bajer | 1.9 | HP Vinodol | 18.6 | 94.5 |

Lepenica | 5.9 | PSP Lepenica | 6.2/5.3 | 1.14/1.25 |

**Table 2.**Energy stock in percent of maximum reservoir capacity ${k}_{St}$ at the beginning of and the end of planning interval.

Reservoir | ${\mathit{s}}_{0}$ | ${\mathit{s}}_{\mathit{T}}$ | e (m3/s) |
---|---|---|---|

Lokve | 66 | 46 | 0.68 |

Bajer | 64 | 64 | 0.89 |

Lepenica | 58 | 58 | 0.21 |

**Table 3.**Comparison of the expected daily revenue for the coordinated case with proposed method and an uncoordinated case without the proposed method for variations in ${k}_{Rm},\text{}\forall t$.

Revenue with the Proposed Method (€) | Revenue in “Bussines as Usual” Case (€) | ${\mathit{k}}_{\mathit{R}\mathit{m}}$(€) |
---|---|---|

98,885 | 89,351 | 0 |

97,995 | 88,461 | 400 |

97,100 | 87,566 | 800 |

96,332 | 86,798 | 1380 |

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**MDPI and ACS Style**

Ilak, P.; Rajšl, I.; Đaković, J.; Delimar, M. Duality Based Risk Mitigation Method for Construction of Joint Hydro-Wind Coordination Short-Run Marginal Cost Curves. *Energies* **2018**, *11*, 1254.
https://doi.org/10.3390/en11051254

**AMA Style**

Ilak P, Rajšl I, Đaković J, Delimar M. Duality Based Risk Mitigation Method for Construction of Joint Hydro-Wind Coordination Short-Run Marginal Cost Curves. *Energies*. 2018; 11(5):1254.
https://doi.org/10.3390/en11051254

**Chicago/Turabian Style**

Ilak, Perica, Ivan Rajšl, Josip Đaković, and Marko Delimar. 2018. "Duality Based Risk Mitigation Method for Construction of Joint Hydro-Wind Coordination Short-Run Marginal Cost Curves" *Energies* 11, no. 5: 1254.
https://doi.org/10.3390/en11051254