# Optimisation of Small Hydropower Units in Water Distribution Systems by Demand Forecasting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}, H

_{2}, and H

_{3}), which serve as operating levels for control actions:

- Maximum flow operation: When the storage tank is approximately full and the water level is above H
_{2}, the SHPU operates at maximum flow. - Normal flow operation: When the water level is between H
_{1}and H_{2}, the turbine operates with the current settings. - Flow reduction: If the water level falls between H
_{1}and H_{3}, the control strategy involves gradually reducing the turbine’s flow rate by 1/8 of the maximum flow rate. This reduction in flow rate aims to restore the water level in the subsequent simulation time step. However, if the water level in the storage tank continues to decrease despite the flow rate reduction, the turbine’s flow rate is further reduced. - SHPU shutdown: If the water level falls below H
_{3}, the control strategy involves completely shutting off the turbine to ensure an adequate water supply is reserved for firefighting. - Resuming maximum flow operation: Once the water level rises above H
_{2}again, the control strategy sets the turbine back to maximum flow.

_{1}, H

_{2}, and H

_{3}are determined through a previously optimised process for each control category. A detailed explanation of this procedure is provided in the following subchapter.

#### 2.1. Optimisation of the Control Strategy

#### 2.1.1. Control Categories

#### 2.1.2. Optimisation of Operating Levels

_{1}, H

_{2}, and H

_{3}are optimised for each category. The aim of this process is to maximise electrical energy production by determining the optimal operating levels using a simplified evolutionary optimisation technique, where the next generation is created with a Monte Carlo sampling technique [31] within the best solutions of the previous generation. Thereby, the best solutions are selected based on the electrical energy generation, and the operation levels H

_{1}, H

_{2}, and H

_{3}are determined. In the next step, the operation levels for the offspring are randomly selected between these values, meaning that all H

_{1}of the new generation are between H

_{1},min and H

_{1},max of the previous generation. The simplified evolutionary optimisation technique is developed and described in more detail in Sitzenfrei and Rauch [32], whereas in the following, the process is briefly explained.

_{2}to be greater than H

_{1}and H

_{1}to be greater than H

_{3}, are defined, and random values are assigned to them. Specifically, H

_{1}is selected randomly within the range of 1.5 m and 3.5 m using a uniform distribution. Next, H

_{2}is chosen randomly between H

_{1}and the maximum water level of the storage tank. H

_{3}is selected randomly between the minimum water level in the storage tank and 2.5 m using a uniform distribution. If H

_{3}is larger than H

_{1}, H

_{3}is set to H

_{1}minus 0.1 m.

_{1}, H

_{2}, and H

_{3}are restricted to fall within the range of the previous best 10%. If any random selection violates the defined dependencies, the values are adjusted to be 0.01 m below or above the neighbouring level. This process is repeated for each subsequent generation, resulting in the convergence of the operating levels towards their respective optima for maximum electrical energy production.

#### 2.1.3. Forecasting of Control Categories

- Perfect forecast: Assumes a perfect forecast where the predicted control categories match the actual conditions with 100% accuracy, representing the maximum potential of the forecast.
- Tomorrow like today: Assumes that the spring discharge and total water demand will be the same as the current day.
- Tomorrow like last week: Assumes that the spring discharge and total water demand will be the same as the corresponding weekday last week.
- False forecast: Examines the effect of an incorrect forecast as the worst-case performance scenario. The correct control category is disregarded, and the control category for the next day is randomly selected from the remaining categories.

#### 2.2. Profitability Analysis

_{G}is the efficiency of the generator and other plant components (-), η

_{T}

_{,t}is the efficiency of the Pelton turbine (-), Q

_{t}is the flow (m

^{3}/s), H

_{T}is the hydraulic head (m) at the installation place of the Pelton turbine, ρ is the density of water (kg/m

^{3}), g is the earth acceleration (m/s

^{2}), t is the simulation time step (h), t

_{sim}is the total simulation period (a), and p is the energy tariff (€/kWh).

_{Pelton}is the investment costs for the SHPU (€/kW), DP is the design performance, and I

_{Additional}is the additional costs for installation. Based on the investment costs, the operational costs OC (€/a) and capital costs CC (€/a) are estimated with Equations (4) and (5):

#### 2.3. Case Study

^{3}to balance daily water fluctuations and to provide the necessary demand for firefighting. The storage tank is filled by a hillside spring, and it is positioned approximately 30 to 90 m above the supply area (Figure 2). The water distribution network is fully gravity driven, ensuring sufficient hydraulic pressures even during peak flow periods. For security reasons, the real layout is not depicted here in Euclidean space; instead, a distorting function from NetworkX [33] is utilised for illustration.

#### 2.3.1. Water Surplus

^{3}/day. Similarly, total water demand experiences seasonal variations throughout the year with peak demand occurring during the summer due to irrigation. The average daily total water demand is approximately 704 m

^{3}/day. The difference between the spring discharge and total water demand, which is represented by the grey area in Figure 3a, results in a water surplus. The water surplus has an average value of 476 m

^{3}/day. For more information, Figure 3b provides the cumulative distribution function of the water surplus over the investigation period. Currently, the water surplus is discharged into a nearby receiving water in close proximity to the tank. However, there is the potential to utilise the water surplus for the implementation of an SHPU and discharge it further downstream to the receiving water, thereby utilising the additional height difference for energy production.

#### 2.3.2. Numerical Model

#### 2.3.3. Small Hydropower Unit

_{G}is assumed to be 0.95, and the device efficiency curve for the selected Pelton turbine is shown in Figure 4. The energy tariff is 0.1055 €/kWh, while the investment costs for the Pelton turbine and the additional expenses are assumed to be 3123 €/kW and 11,000 €, respectively. Furthermore, an annuity factor of 0.084 is applied, assuming an amortisation period of 15 years and a real interest rate of 3%.

## 3. Results and Discussion

#### 3.1. Optimisation with One Control Category

_{1}ranges from 1.5 to 3.5 m, h

_{2}ranges from 1.5 to 3.63 m, and h

_{3}ranges from 1.0 to 2.5 m. As observed in Figure 6b,c, a relatively high profit can be achieved without precise knowledge of the optimal ranges for H

_{2}and H

_{3}. In contrast, H

_{1}shows a tighter clustering of profitable settings between 2.0 and 2.7 m (Figure 6a) compared to the broader range of values for H

_{2}and H

_{3}. Consequently, the selection of the filling depth from where the turbine inflow is gradually reduced has a greater impact on the results than the switch-off point or the switch-on point after water level regeneration.

#### 3.2. Optimisation with Six Control Categories

- Category 1: Low water demand with low spring discharge;
- Category 2: Low water demand with high spring discharge;
- Category 3: Medium water demand with low spring discharge;
- Category 4: Medium water demand with high spring discharge;
- Category 5: High water demand with low spring discharge;
- Category 6: High water demand with high spring discharge.

_{1}, H

_{2}, and H

_{3}, respectively (primary y-axis). The boxplots show the results of the best 100 different triples, whereas the values for H

_{1}vary between 1.55 and 3.25 m, for H

_{2}vary between 2.10 and 3.30 m, and for H

_{3}vary between 1.05 and 2.10 m within these triples. For more details, the correlations of the different operating levels for the best 100 results per category are illustrated in Figure 9. Figure 9a for H

_{1}and H

_{2}; Figure 9b for H

_{1}and H

_{3}; and Figure 9c for H

_{2}and H

_{3}).

_{1}has the highest values in the categories 1, 3, and 5, which are associated with a low discharge flow, and is just below the operating level H

_{2}. This indicates that the SHPU flow is reduced very early within these categories, and it is mainly operated with a reduced flow rate to achieve the highest electrical energy production. Furthermore, there is no correlation between H

_{3}and H

_{1}or H

_{2}for these categories, and the switch-off point (H

_{3}) has a strong variation between the optimised trigger level, showing no correlation to the other two operating levels, H

_{1}and H

_{2}.

_{1}and H

_{2}for the top 100 solutions are clearly distinguishable for high spring discharge (categories 2, 4, and 6). For example, category 2, representing optimal turbine operation with a low water demand and high spring discharge, has the lowest values for all three operating levels. The turbine operates at maximum flow already at a water level of 2.2 m (H

_{2}), whereas the turbine flow is continuously reduced only below a water level of approximately 1.6 m (H

_{1}). Therefore, the influence of the two operating levels H

_{1}and H

_{2}on the switch-off point is small, as H

_{3}shows a small bandwidth.

_{3}among the three operating levels. This suggests that the operating levels for reduction (H

_{1}) and full load (H

_{2}) have a greater impact on the produced electrical energy than the switch-off point H

_{3}.

#### 3.3. Economic Evaluation

#### 3.4. Limitations and Future Research Directions

#### 3.5. Further Discussion

## 4. Conclusions

- Incorporating demand forecasts and adjusting controls for different flow conditions can improve the electrical energy potential of an SHPU;
- However, it is worth noting that the controls in the reference state were already based on well-reasoned expert knowledge, making improvements marginal compared to the effort required for more complex control strategies in this specific case study;
- The prediction approach shows potential when dealing with devices that have a steep and narrow device efficiency curve, such as pump as turbines, or when considering fluctuating electricity prices;
- Additionally, an SHPU can significantly improve the quality of drinking water due to higher abstraction volumes, and if the generated electrical energy is directly used to operate the network, it also increases the resilience of the water supply system against outages of the power grid.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The hydraulic model showing the pressure at maximum flow according to [20]. The position of the hydropower station and the control points are highlighted with red and orange circles.

**Figure 3.**(

**a**) Daily spring discharge and total water demand of the case study over the investigation period according to [21] and (

**b**) cumulative distribution function of the water surplus usable for the SHPU.

**Figure 4.**The device efficiency curve for the considered Pelton turbine with a design performance of 3 kW according to [20].

**Figure 6.**Annual profits depending on different parameter settings of the TWKW control system in comparison with the reference scenario (blue markers).

**Figure 7.**Distribution of total water demand from 10 years with a subdivision into the corresponding control category.

**Figure 8.**Results of the optimisation of the operating levels H

_{1}(green), H

_{2}(blue), and H

_{3}(black) in comparison with the spring discharge (inflow, light blue) and total water demand (demand, light red) for the six control categories.

**Figure 9.**Correlations of the different water level heights of the best 100 results of the optimisation for each of the 6 categories.

Scenario | Profit (€/a) | Change (%) |
---|---|---|

Reference state ^{1} | 807.64 | - |

One category—Without forecast | 812.42 | +0.6 |

Six categories—Perfect forecast | 816.78 | +1.1 |

Six categories—Tomorrow as today | 816.19 | +0.9 |

Six categories—Tomorrow as last week | 811.31 | +0.5 |

Six categories—False forecast | 782.65 | −3.1 |

^{1}The profit of the SHPU with operating levels based on local experiences and without forecast [20] was assumed to be the reference state.

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## Share and Cite

**MDPI and ACS Style**

Oberascher, M.; Schartner, L.; Sitzenfrei, R.
Optimisation of Small Hydropower Units in Water Distribution Systems by Demand Forecasting. *Water* **2023**, *15*, 3998.
https://doi.org/10.3390/w15223998

**AMA Style**

Oberascher M, Schartner L, Sitzenfrei R.
Optimisation of Small Hydropower Units in Water Distribution Systems by Demand Forecasting. *Water*. 2023; 15(22):3998.
https://doi.org/10.3390/w15223998

**Chicago/Turabian Style**

Oberascher, Martin, Lukas Schartner, and Robert Sitzenfrei.
2023. "Optimisation of Small Hydropower Units in Water Distribution Systems by Demand Forecasting" *Water* 15, no. 22: 3998.
https://doi.org/10.3390/w15223998