# Hydro Power Reservoir Aggregation via Genetic Algorithms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Implications of Reservoir Capacities

## 3. Deterministic Hydropower Scheduling Equivalent

## 4. Scaleable Deterministic Model

## 5. Parameter Fitting

## 6. Genetic Algorithm

**mutation**: changes in decision variables based on random variables drawn from predetermined possibility distributions,**crossover (”mating”)**: merger of decision variables of “parents” to establish new “children” that are added to the population,**selection**: removal of members of P by comparing their fitness values.

#### 6.1. Individual

#### 6.2. Feedback Function

#### 6.3. Algorithm Parameters

- -
- Due to crossover, the increase is exponential (with every generation, the size of the population increases by 100%) and the model requires large quantities of scaling; the population was collapsed to 67% of its size after each generation (via tournament selection as denoted above).
- -
- To eventually collapse the population to a single individual, a second set ${\Omega}_{2}$ is required. On those scenarios $\omega \in {\Omega}_{2}$, no mutations and crossovers are applied. The rate of collapse per generation can be calculated as $\sqrt[\mathrm{card}({\Omega}_{2})]{\frac{1}{\mathrm{card}(P)}}$.

## 7. Case Study

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Indexes: | |

$\omega $ | scenario |

$i,j$ | generation unit/reservoir |

${b}_{i}$ | discharge steps of generation unit i |

t | time period |

$\tau $ | alternative time period |

Variables: | |

${q}_{i,t}^{b}$ | generation $[\mathrm{MWh}/\mathrm{week}]$ |

${s}_{i,t}$ | spillage $[{\mathrm{Mm}}^{3}/\mathrm{week}]$ |

${C}_{i}$ | station cut $[0,1]$ |

${v}_{i}^{q}$ | generation leverage variable $\mathbb{R}$ |

${v}_{i}^{r}$ | inventory leverage variable $\mathbb{R}$ |

${v}_{i,j}^{l}$ | inflow leverage variable $\mathbb{R}$ |

Parameters: | |

${p}_{t}^{\overline{\omega}}$ | market price $[\u20ac/\mathrm{MWh}]$ |

${q}_{i}^{max}$ | maximum generation capacity of generator i $\left[\mathrm{MW}\right]$ |

${q}_{i}^{b,max}$ | maximum generation capacity of discarche step b $\left[\mathrm{MW}\right]$ |

${l}_{i,t}^{\overline{\omega}}$ | hydrological inflow $[{\mathrm{Mm}}^{3}/\mathrm{week}]$ |

${\eta}^{b}$ | conversion rate $[{\mathrm{Mm}}^{3}/\mathrm{MWh}]$ |

${r}_{i}^{max}$ | maximum reservoir capacity $\left[{\mathrm{Mm}}^{3}\right]$ |

${W}_{i,t}^{\overline{\omega}}$ | water value $[\u20ac/{\mathrm{Mm}}^{3}]$ |

${F}_{j,i}$ | water course matrix $[\%]$ |

${r}_{i}^{\mathrm{start}}$ | start period hydrological inventory $\left[{\mathrm{Mm}}^{3}\right]$ |

${r}_{i}^{\mathrm{end}}$ | end period hydrological inventory $\left[{\mathrm{Mm}}^{3}\right]$ |

${s}_{i}^{max}$ | maximum spillage $\left[{\mathrm{Mm}}^{3}\right]$ |

w | weight factor $\mathbb{R}$ |

${c}^{max}$ | maximum number of power stations/reservoirs in the system ${\mathbb{Z}}^{+}$ |

Functions: | |

$f(\xb7)$ | defines a function |

$V(\xb7)$ | operation function of a system $[\u20ac]$ |

${r}_{i,t}$ | reservoir level $[{\mathrm{Mm}}^{3}/\mathrm{week}]$ |

${\mathsf{\Pi}}_{i,t}^{b}$ | profit function $[\u20ac]$ |

$c{(q)}_{i,t}^{\overline{\omega}}$ | generation cost function $\left[\frac{\u20ac/\mathrm{MWh}}{\mathrm{week}}\right]$ |

${r}_{i,t}^{\mathrm{in}}$ | periodic inflow $[{\mathrm{Mm}}^{3}/\mathrm{week}]$ |

${r}_{i,t}^{\mathrm{out}}$ | periodic loss $[{\mathrm{Mm}}^{3}/\mathrm{week}]$ |

$F{(C)}_{j,i}$ | water course matrix with cuts C applied $[\%]$ |

${V}^{\ast}(\xb7)$ | optimal operation $[\u20ac]$ |

Sets: | |

$\Omega $ | scenario set |

P | population |

Tuples: | |

p | individual |

## Appendix A. Maximum Spillage Constraint

## Appendix B. Scenario Generation

## Appendix C. Network Matrix Rerouting

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Löschenbrand, M.; Korpås, M.
Hydro Power Reservoir Aggregation via Genetic Algorithms. *Energies* **2017**, *10*, 2165.
https://doi.org/10.3390/en10122165

**AMA Style**

Löschenbrand M, Korpås M.
Hydro Power Reservoir Aggregation via Genetic Algorithms. *Energies*. 2017; 10(12):2165.
https://doi.org/10.3390/en10122165

**Chicago/Turabian Style**

Löschenbrand, Markus, and Magnus Korpås.
2017. "Hydro Power Reservoir Aggregation via Genetic Algorithms" *Energies* 10, no. 12: 2165.
https://doi.org/10.3390/en10122165