# Tuning Model Predictive Control for Rigorous Operation of the Dalsfoss Hydropower Plant

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description

#### 2.1. System Model

#### 2.2. Operational Constraints

- The sudden change in the flowrate at downstream, ${\dot{V}}_{\mathrm{o}}$, can endanger the properties and ecosystem along the watercourse. Therefore, the flowrate at downstream must be kept as constant as possible. The change of the flowrate at downstream is desired to keep less than 50 ${\mathrm{m}}^{3}/\mathrm{s}$ on every time step.
- The flowrate at downstream, ${\dot{V}}_{\mathrm{o}}$, must be more than $4\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{s}$. This constraint is to ensure that fishes in downstream can move freely along the water course.
- The water level at Merkebekk, ${x}_{\mathrm{M}}$, must be maintained within a certain range as:$${x}_{\mathrm{M}}\in [{x}_{\mathrm{LRV}},{x}_{\mathrm{HRV}}]$$
- The maximum flowrate through the turbine, ${V}_{\mathrm{t}}$, is limited up to 36 ${\mathrm{m}}^{3}/\mathrm{s}$.

#### 2.3. Optimal Control Problem

## 3. Simulation Setting

#### 3.1. Test Sets of Parameters

#### 3.2. Simulation Condition

## 4. Simulation Result

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of lake Toke [10].

**Figure 2.**Structure of floodgate [10].

**Figure 6.**Change of the water level at Merkebekk during the simulations with parameter test set 1 and test set 4.

**Figure 7.**Flowrate of water at downstream, ${V}_{\mathrm{O}}$, with different sets of parameters when the length of prediction horizon, ${N}_{\mathrm{p}}$, is set as 3 days.

**Figure 8.**Average computational time taken depending on the length of the prediction horizon when the test set 3 is used.

Parameter | Value | Unit | Comment |
---|---|---|---|

$\alpha $ | 0.05 | - | Fraction of surface area in compartment 2 |

$\beta $ | 0.02 | - | Fraction of inflow to compartment 2 |

${K}_{12}$ | 800 | m${}^{\frac{3}{2}}$/s | Flow coefficient |

${C}_{\mathrm{d}}$ | 0.7 | - | Discharge coefficient, Dalsfoss gate |

${w}_{1}$ | 11.6 | m | Width of Dalsfoss gate 1 |

${w}_{2}$ | 11.0 | m | Width of Dalsfoss gate 2 |

${x}_{\mathrm{LRV}}^{\mathrm{min}}$ | 55.75 | m MSL | Minimal low regulated level value |

g | 9.81 | m/s${}^{2}$ | Acceleration of gravity |

a | 124.69 | Pa${}^{-1}$ | Coefficient in Equation (6) |

b | 3.161 | m${}^{3}$/s | Coefficient in Equation (6) |

${c}_{1}$ | 0.13152 | W/m${}^{-3}$ | Polynomial coefficient in Equation (7) |

${c}_{2}$ | −9.5241 | W/m${}^{2}$ | Polynomial coefficient in Equation (7) |

${c}_{3}$ | 1.7234 · 10${}^{2}$ | W/m | Polynomial coefficient in Equation (7) |

${c}_{4}$ | −7.7045 · 10${}^{-3}$ | Pa/m | Polynomial coefficient in Equation (7) |

${c}_{5}$ | −8.7359 · 10${}^{-1}$ | W | Polynomial coefficient in Equation (7) |

Date | ${\mathit{x}}_{\mathbf{LRV}}\left[\mathbf{m}\right]$ | ${\mathit{x}}_{\mathbf{HRV}}\left[\mathbf{m}\right]$ |
---|---|---|

1 January–30 April | 55.75 | 60.35 |

1 May–30 August | 58.85 | 59.85 |

1 September–14 September | 55.75 | 59.35 |

28 October–11 November | 55.75 | 59.85 |

12 November–31 December | 55.75 | 60.35 |

Parameter | Test Set 1 | Test Set 2 | Test Set 3 | Test Set 4 |
---|---|---|---|---|

${\omega}_{R}$ | 10 | 10 | 10 | 10 |

${\omega}_{\Delta u}$ | 1 | 10 | 100 | 1000 |

${\omega}_{u}$ | 1 | 1 | 1 | 1 |

${\omega}_{p}$ | 10,000 | 10,000 | 10,000 | 10,000 |

${\mathit{N}}_{\mathbf{p}}$ [d] | Testset 1 | Testset 2 | Testset 3 | Test Set 4 |
---|---|---|---|---|

1 | 4.2637 | 4.2625 | 4.2595 | 4.1644 |

3 | 4.2554 | 4.2537 | 4.2472 | 4.2314 |

5 | 4.2545 | 4.2528 | 4.2463 | 4.2290 |

7 | 4.2542 | 4.2526 | 4.2462 | 4.2294 |

9 | 4.2542 | 4.2526 | 4.2462 | 4.2294 |

11 | 4.2542 | 4.2526 | 4.2462 | 4.2294 |

13 | 4.2542 | 4.2526 | 4.2462 | 4.2294 |

**Table 5.**Maximum change of water amount discharged from the plant throughout simulations [${\mathrm{m}}^{3}$].

${\mathit{N}}_{\mathit{p}}$ [d] | Testset 1 | Testset 2 | Testset 3 | Test Set 4 |
---|---|---|---|---|

1 | 139.1463 | 71.9443 | 38.7613 | 31.4636 |

3 | 57.2664 | 33.3261 | 18.6869 | 17.2223 |

5 | 57.0969 | 33.1774 | 19.7069 | 14.8696 |

7 | 57.0943 | 33.1746 | 19.6943 | 16.4958 |

9 | 57.0941 | 33.1746 | 19.6950 | 16.3057 |

11 | 57.0941 | 33.1746 | 19.6935 | 16.3650 |

13 | 57.0941 | 33.1746 | 19.6934 | 16.3708 |

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**MDPI and ACS Style**

Jeong, C.; Sharma, R.
Tuning Model Predictive Control for Rigorous Operation of the Dalsfoss Hydropower Plant. *Energies* **2022**, *15*, 8678.
https://doi.org/10.3390/en15228678

**AMA Style**

Jeong C, Sharma R.
Tuning Model Predictive Control for Rigorous Operation of the Dalsfoss Hydropower Plant. *Energies*. 2022; 15(22):8678.
https://doi.org/10.3390/en15228678

**Chicago/Turabian Style**

Jeong, Changhun, and Roshan Sharma.
2022. "Tuning Model Predictive Control for Rigorous Operation of the Dalsfoss Hydropower Plant" *Energies* 15, no. 22: 8678.
https://doi.org/10.3390/en15228678