# Robust Scheduling for Pumping in a Water Distribution System under the Uncertainty of Activating Regulation Reserves

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## Abstract

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## 1. Introduction

- Robust operational planning model under the uncertainty of activating regulation reserves: The proposed model is sufficiently robust to maintain the water volume in the reservoir within the constraints and to maintain a stable water supply when there is uncertainty about whether the regulation reserves provided to the power system will be activated. Because the proposed optimal operational planning model is formulated as an MILP, it can be solved by many commercial solvers and is highly practical.
- Determination of water pump operation based on electrical energy prices and incentive prices: In the proposed model, it is possible to optimize the water pump operation plan according to the electrical energy prices and the incentive prices for each time slot to provide regulation reserves presented in advance by the retail electric utilities and aggregators.

## 2. Assumed Environment

#### 2.1. Water Distribution System

#### 2.2. Price Information

#### 2.3. Regulation Reserves

- Compensation is paid by the contractor based on the incentive price for the regulation reserve capacity provided during each time slot. In this case, the occurrence of the consideration does not depend on whether the regulation reserves are triggered during actual operation.
- In actual operation, the final volume of provision is determined through transactions with retail electricity providers and aggregators. In this paper, the amount that can be provided in the plan is treated as the amount provided.
- The time and amount of the activation of the provided regulation reserves are random.
- The imbalance from the demand plan caused by activating regulation reserves is settled between transmission and distribution companies based on the imbalanced price. The imbalance is not treated in the optimal operation planning model.

## 3. Optimal Operation Planning Model for Water Pumping

#### 3.1. Power Consumption Characteristic Model of Water Pumps

#### 3.2. Approximation of the Power Consumption Characteristics of the Water Pumps

#### 3.3. Optimal Operational Planning Model

## 4. Numerical Experiment

#### 4.1. Experimental Conditions

- Case 1: The electrical energy price is fixed at 10 yen/kWh, and the price of regulation reserves is 0 yen/kW.
- Case 2: The electrical energy price is fixed at 10 yen/kWh; the price of upward regulation reserves is 20 yen/kW ($t=11,\dots ,14$); and the price of downward regulation reserves is 0 yen/kW.
- Case 3: The electrical energy price is fixed at 10 yen/kWh; the price of upward regulation reserves is 0 yen/kW; and the price of downward regulation reserves is 20 yen/kW ($t=18,\dots ,20$).
- Case 4: The electrical energy price is variable, and the price of regulation reserves is 0 yen/kW.
- Case 5: The electrical energy price is variable; the price of upward regulation reserves is 20 yen/kW ($t=11,\dots ,14$); and the price of downward regulation reserves is 0 yen/kW.
- Case 6: The electrical energy price is variable; the price of upward regulation reserves is 0 yen/kW; and the price of downward regulation reserves is 20 yen/kW ($t=18,\dots ,20$).

#### 4.2. Results

#### 4.2.1. In the Case of Fixed Electrical Energy Prices

#### 4.2.2. Variable Electrical Energy Price

#### 4.2.3. Comparison of Operational Costs

## 5. Conclusions

- For the optimal operation planning problem of the water pumps that provides regulation reserves, a method to ensure that the water storage volume in the reservoir is within the constraints and to maintain a stable water distribution under the uncertainty of the activation of the provided regulation reserves was proposed.
- Based on electrical energy and incentive prices for the provision of regulation reserves, a model was proposed that allows drawing an operation plan for water pumps that minimizes the net energy cost.
- As a case study, a numerical experiment was conducted on changing the electrical energy and incentive prices. The results confirmed the time shift in power consumption with the electrical energy price as a control signal. To maximize the revenues from the provision of regulation reserves, an operational plan was obtained that adjusts the planned water storage volume for the time slots when regulation reserves were required.
- Through a comparison of the objective function values obtained as a result of the numerical experiments, a net operating cost reduction in water pump operation was achieved by establishing a new source of income: the use of water pumps to provide regulation reserves to the power system.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Profile of hourly variable electrical energy prices and incentive price for the provided regulation reserves.

**Figure 7.**Results for the electrical power consumption of water pumps under the fixed electrical energy price. The solid red, dashed blue, and dashed green lines indicate the planned power consumption of the water pump, when upward regulation reserve capacity provided to the power system is fully activated and when downward regulation reserve capacity provided to the power system is fully activated, respectively.

**Figure 8.**Results for the volume of the reservoir under the fixed electrical energy price. The solid red line shows the planned water storage volume of the reservoir. The blue and green dashed lines show the change in water storage volume when the provided regulation reserve capacities are fully activated. The yellow dotted line represents a changed scenario in the water storage volume when the provided regulation reserves are activated for only half the time in each time slot.

**Figure 9.**Results for the electrical load of water pumps under the variable electrical energy prices. The solid red, dashed blue, and dashed green lines indicate the planned power consumption of the water pump, when upward regulation reserve capacity provided to the power system is fully activated and when downward regulation reserve capacity provided to the power system is fully activated, respectively

**Figure 10.**Results for the volume of the reservoir under the variable electrical energy prices. The blue and green dashed lines show the change in water storage volume when the provided regulation reserve capacities are fully activated. The yellow dotted line represents a changed scenario in the water storage volume when the provided regulation reserves are activated for only half the time in each time slot.

Sum of time slots T | 24 |

Time step ${t}_{ref}$ (h) | 1.0 |

Sum of pump patterns ${N}_{l}$ | 6 |

Upper bound of volume of the reservoir ${V}^{UB}$ (m${}^{3}$) | 1600 |

Lower bound of volume of the reservoir ${V}^{LB}$ (m${}^{3}$) | 600 |

Initial volume of the reservoir ${V}^{INI}$ (m${}^{3}$) | 800 |

Parameter of pumps a | −0.005 |

Parameter of pumps b | 0.064 |

Parameter of pumps c | 50 |

Parameter of pumps d | −0.0002 |

Parameter of pumps e | 0.0286 |

Parameter of pumps f | −0.03 |

Pipeline resistance r (h${}^{2}$/m${}^{5}$) | 0.00005 |

Gross pump head H (m) | 30 |

**Table 2.**Results of estimation parameters: Regression coefficients and higher/lower bounds of the flow rate.

Pump Patterns l | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Regression coefficient ${\alpha}_{l}$ | 0.0778 | 0.0803 | 0.0934 | 0.1097 | 0.1281 | 0.1705 |

Regression coefficient ${\beta}_{l}$ | 0.6938 | 0.9017 | −0.8533 | −4.4538 | −9.953 | −26.6 |

Upper bound of flow rate ${Q}_{l}^{UB}$ (m${}^{3}$/h) | 95.37 | 168.64 | 241.46 | 313.54 | 383.91 | 496.80 |

Lower bound of flow rate ${Q}_{l}^{LB}$ (m${}^{3}$/h) | 34.79 | 95.37 | 168.64 | 241.46 | 313.54 | 383.91 |

Case | Objective Function Value (yen) |
---|---|

Case 1 | 3835 |

Case 2 | 2686 |

Case 3 | 3388 |

Case 4 | 2806 |

Case 5 | 1795 |

Case 6 | 2679 |

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**MDPI and ACS Style**

Negishi, S.; Ikegami, T.
Robust Scheduling for Pumping in a Water Distribution System under the Uncertainty of Activating Regulation Reserves. *Energies* **2021**, *14*, 302.
https://doi.org/10.3390/en14020302

**AMA Style**

Negishi S, Ikegami T.
Robust Scheduling for Pumping in a Water Distribution System under the Uncertainty of Activating Regulation Reserves. *Energies*. 2021; 14(2):302.
https://doi.org/10.3390/en14020302

**Chicago/Turabian Style**

Negishi, Shintaro, and Takashi Ikegami.
2021. "Robust Scheduling for Pumping in a Water Distribution System under the Uncertainty of Activating Regulation Reserves" *Energies* 14, no. 2: 302.
https://doi.org/10.3390/en14020302