# Optimal Pump Scheduling for Urban Drainage under Variable Flow Conditions

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## Abstract

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## 1. Introduction

#### Aim of the Paper and Methodology

- Formulating a complete mathematical model of a drainage pumping system
- Optimizing the pumping system aiming at reducing the required energy
- Showing the benefit of such an optimization comparing the results with a classical CS plant.

## 2. Pumping Station Description and Case Study

#### 2.1. Classical Design

#### 2.2. Experimental Investigation of the Behaviour of Two Submersible Pumps

#### 2.3. Input Discharge Pattern

#### 2.4. Plant Behaviour

## 3. Optimization Model

#### Resolution of the Optimization Model

## 4. Application and Results

## 5. Conclusions

- An experimental campaign is undertaken to explore the effects of variable speed on the pumping efficiency. Specifically, in accordance with the affinity laws, an empirical equation is provided to compute the pumping head (Equation (7)), whereas a novel approach based on the concept of relative efficiency (Equation (10)) is provided to compute the pumping power under variable speed conditions (Equation (12)).
- On the basis of the above-mentioned theoretical framework, a mixed-integer optimization problem (Equation (30)) is built that is made up of an objective function (the overall pumping energy) to be minimized and a set of constraints for the variables of interest. The model is also able to comply with the ON/OFF switch of the pump, and two parameters ($\alpha $ and $\beta $) are introduced to simulate different scenarios for the inflow discharge and the plant configuration. The influence of the time window and step for computations is also discussed.
- The model is solved for a case study (a literature sewage daily pattern provided for the City of Naples, Italy) relying on a literature algorithm, and some indicators are analyzed to test the computational performance of the algorithm and the overall energy savings given by the optimal solution for the different scenarios. The case study was developed assuming a known inflow pattern and a constant energy cost during the day.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

$\alpha $ | Ratio between ${Q}_{{N}_{max}}^{BEP}$ and ${Q}_{max}^{in}$ |

$\beta $ | Ratio between static head and ${H}_{{N}_{max}}^{BEP}$ |

$\Delta t$ | Length of the timestep |

${\eta}^{BEP}$ | Efficiency of the pump at its BEP |

${\eta}_{{N}_{max}}^{BEP}$ | Efficiency of the pump at its BEP for the maximum rotational speed |

${\eta}_{N}^{BEP}$ | Efficiency of the pump at its BEP at N rotational speed |

${\eta}_{CS}$ | Efficiency of the CS operation |

${\eta}_{opt}$ | Efficiency of optimal scheduling |

${\eta}_{N}$ | Efficiency of the pump at N rotational speed |

$\tau $ | Time interval between two consecutive starts of the pump |

$\theta $ | Ratio between ${Q}_{in}$ and ${Q}_{{N}_{max}}^{BEP}$ |

$\epsilon $ | Benefit of the optimal operation, when compared to the CS mode |

$BEP$ | Best Efficiency Point |

${c}_{h}^{2},{c}_{h}^{1},{c}_{h}^{0}$ | Regression coefficients of the head curve |

${c}_{p}^{3},{c}_{p}^{2},{c}_{p}^{1},{c}_{p}^{0}$ | Regression coefficients of the power curve |

${c}_{\eta}^{3},{c}_{\eta}^{2},{c}_{\eta}^{1},{c}_{\eta}^{0}$ | Regression coefficients of the efficiency curve |

${c}_{e}^{2},{c}_{e}^{1},{c}_{e}^{0}$ | Regression coefficients of the relative efficiency curve |

$C.T.$ | Computational time |

$CS$ | Constant Speed operation |

e | Relative efficiency |

E | Required pumping energy |

${E}_{CS}$ | Daily required pumping energy for the CS operation |

${E}_{opt}$ | Daily required pumping energy resulting from the optimization |

${E}_{ref}$ | Daily reference energy |

${E}_{T}$ | Required pumping energy for each time window |

f | Electrical frequency |

H | Pumped head |

${H}_{{N}_{max}}^{BEP}$ | Pumped head at ${N}_{max}$ rotational speed at the BEP of the pump |

${H}_{man}$ | Required pumping head |

${H}_{{N}_{max}}$ | Pumped head at ${N}_{max}$ rotational speed |

${H}_{0}$ | Static head |

${H}_{N}$ | Pumped head at N rotational speed |

${H}_{w}$ | Water level in the wet well |

${H}_{w}^{max}$ | Maximum allowable water level in the wet well |

${H}_{w}^{min}$ | Minimum allowable water level in the wet well |

${}_{i}$ | Subscript indicating the i-th timestep |

I | Switch of the pump |

K | Head loss coefficient |

${n}_{t}$ | Number of timesteps in the time window |

${n}_{w}$ | Number of time windows within the whole day |

N | rotational speed of the pump |

${N}_{max}$ | Maximum rotational speed of the pump |

P | Pumped power |

${P}_{{N}_{max}}$ | Pumped power at ${N}_{max}$ rotational speed |

${P}_{N}$ | Pumped power at N rotational speed |

${p}_{p}$ | Number of pole pairs of the motor |

$p{N}_{max}$ | Ratio between ${P}_{{N}_{max}}$ and ${{N}_{max}}^{3}$ |

Q | Pumped discharge |

$q(t)$ | Non dimensional inflow pattern |

${Q}_{{N}_{max}}^{BEP}$ | Discharge of the pump at its BEP for the maximum rotational speed |

${Q}_{max}^{in}$ | Maximum inflow discharge |

${Q}_{exp}$ | Measured drainage discharge |

${Q}_{in}$ | Inflow discharge |

${Q}_{p}$ | Outflow discharge |

S | Cross section of the wet well |

${S}_{max}^{h}$ | Maximum allowable number of starts per hour |

${S}_{h}$ | Number of starts per hour |

t | Time |

T | Time window |

W | Storage volume of the wet well |

${W}_{w}$ | Volume of water inside the wet well |

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**Figure 2.**Performance curves of the two tested machines (Machine 1 on the left column, Machine 2 on the right column).

**Figure 3.**Experimental points of $Q/N$, $H/{N}^{2}$ and $P/{N}^{3}$ for the two machines (Machine 1 on the left column, Machine 2 on the right column).

**Figure 4.**Best efficiency variation with the rotational speed (Machine 1 on the left column, Machine 2 on the right column).

**Figure 5.**Relative efficiency of both machines (Machine 1 on the left column, Machine 2 on the right column).

**Figure 8.**Efficiency of optimal (${\eta}_{opt}$) and classical (${\eta}_{cl}$) sytems with $\alpha $ and $\beta $.

**Table 1.**Parameters of 30 the different studied scenarios, resulting from the choice of the machine and different values of $\alpha $ and $\beta $.

Machine | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{Q}}_{\mathit{max}}$ | ${\mathit{H}}_{0}$ | ${\mathit{E}}_{\mathit{ref}}$ | ${\mathit{E}}_{\mathit{cs}}$ |
---|---|---|---|---|---|---|

[-] | [-] | [-] | [L/s] | [m] | kWh/day | kWh/day |

1 | 1.00 | 0.00 | 236.66 | 0.00 | 616.9 | 1960.8 |

1 | 1.00 | 0.25 | 236.66 | 9.71 | 790.4 | 1956.7 |

1 | 1.00 | 0.50 | 236.66 | 19.42 | 963.9 | 1951.0 |

1 | 1.00 | 0.75 | 236.66 | 29.13 | 1137.4 | 1939.4 |

1 | 1.00 | 1.00 | 236.66 | 38.84 | 1310.8 | 1914.4 |

1 | 1.50 | 0.00 | 157.78 | 0.00 | 182.8 | 1305.5 |

1 | 1.50 | 0.25 | 157.78 | 9.71 | 355.6 | 1306.0 |

1 | 1.50 | 0.50 | 157.78 | 19.42 | 528.3 | 1298.8 |

1 | 1.50 | 0.75 | 157.78 | 29.13 | 701.1 | 1291.5 |

1 | 1.50 | 1.00 | 157.78 | 38.84 | 873.9 | 1274.7 |

1 | 2.00 | 0.00 | 118.33 | 0.00 | 77.1 | 978.1 |

1 | 2.00 | 0.25 | 118.33 | 9.71 | 221.7 | 979.5 |

1 | 2.00 | 0.50 | 118.33 | 19.42 | 366.3 | 975.6 |

1 | 2.00 | 0.75 | 118.33 | 29.13 | 510.8 | 969.0 |

1 | 2.00 | 1.00 | 118.33 | 38.84 | 655.4 | 957.2 |

2 | 1.00 | 0.00 | 135.04 | 0.00 | 356.5 | 1112.8 |

2 | 1.00 | 0.25 | 135.04 | 9.83 | 456.7 | 1113.7 |

2 | 1.00 | 0.50 | 135.04 | 19.67 | 557.0 | 1110.7 |

2 | 1.00 | 0.75 | 135.04 | 29.50 | 657.2 | 1106.2 |

2 | 1.00 | 1.00 | 135.04 | 39.33 | 757.5 | 1098.3 |

2 | 1.50 | 0.00 | 90.03 | 0.00 | 105.6 | 743.7 |

2 | 1.50 | 0.25 | 90.03 | 9.83 | 205.5 | 741.7 |

2 | 1.50 | 0.50 | 90.03 | 19.67 | 305.3 | 739.3 |

2 | 1.50 | 0.75 | 90.03 | 29.50 | 405.2 | 738.7 |

2 | 1.50 | 1.00 | 90.03 | 39.33 | 505.0 | 732.2 |

2 | 2.00 | 0.00 | 67.52 | 0.00 | 44.6 | 557.0 |

2 | 2.00 | 0.25 | 67.52 | 9.83 | 128.1 | 556.3 |

2 | 2.00 | 0.50 | 67.52 | 19.67 | 211.7 | 554.5 |

2 | 2.00 | 0.75 | 67.52 | 29.50 | 295.2 | 554.7 |

2 | 2.00 | 1.00 | 67.52 | 39.33 | 378.7 | 549.5 |

Machine | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{E}}_{\mathit{ref}}$ | ${\mathit{E}}_{\mathit{cs}}$ | ${\mathit{E}}_{\mathit{opt}}$ | ${\mathit{\eta}}_{\mathit{cs}}$ | ${\mathit{\eta}}_{\mathit{opt}}$ | $\mathit{\epsilon}$ | C.T. |
---|---|---|---|---|---|---|---|---|---|

[-] | [-] | [-] | [kWh/day] | [kWh/day] | [kWh/day] | [-] | [-] | [-] | [min] |

1 | 1.0 | 0.00 | 616.9 | 1960.8 | 924.5 | 0.315 | 0.667 | 2.121 | 1.8 |

1 | 1.0 | 0.25 | 790.4 | 1956.7 | 1132.3 | 0.404 | 0.698 | 1.728 | 1.3 |

1 | 1.0 | 0.50 | 963.9 | 1951.0 | 1404.9 | 0.494 | 0.686 | 1.389 | 1.6 |

1 | 1.0 | 0.75 | 1137.4 | 1939.4 | 1684.4 | 0.586 | 0.675 | 1.151 | 1.4 |

1 | 1.0 | 1.00 | 1310.8 | 1914.4 | 1924.8 | 0.685 | 0.681 | 0.995 | 2.4 |

1 | 1.5 | 0.00 | 182.8 | 1305.5 | 384.2 | 0.140 | 0.476 | 3.398 | 6.6 |

1 | 1.5 | 0.25 | 355.6 | 1306.0 | 541.9 | 0.272 | 0.656 | 2.410 | 1.6 |

1 | 1.5 | 0.50 | 528.3 | 1298.8 | 829.3 | 0.407 | 0.637 | 1.566 | 21.2 |

1 | 1.5 | 0.75 | 701.1 | 1291.5 | 1129.8 | 0.543 | 0.621 | 1.143 | 904.6 |

1 | 1.5 | 1.00 | 873.9 | 1274.7 | 1321.8 | 0.686 | 0.661 | 0.964 | 3.0 |

1 | 2.0 | 0.00 | 77.1 | 978.1 | 284.5 | 0.079 | 0.271 | 3.438 | 5.4 |

1 | 2.0 | 0.25 | 221.7 | 979.5 | 373.5 | 0.226 | 0.594 | 2.623 | 1.4 |

1 | 2.0 | 0.50 | 366.3 | 975.6 | 622.6 | 0.375 | 0.588 | 1.567 | 2906.8 |

1 | 2.0 | 0.75 | 510.8 | 969.0 | 867.2 | 0.527 | 0.589 | 1.117 | 383.3 |

1 | 2.0 | 1.00 | 655.4 | 957.2 | 1004.3 | 0.685 | 0.653 | 0.953 | 3.5 |

2 | 1.0 | 0.00 | 356.5 | 1112.8 | 538.5 | 0.320 | 0.662 | 2.067 | 2.3 |

2 | 1.0 | 0.25 | 456.7 | 1113.7 | 660.5 | 0.410 | 0.692 | 1.686 | 1.2 |

2 | 1.0 | 0.50 | 557.0 | 1110.7 | 817.6 | 0.501 | 0.681 | 1.358 | 28.6 |

2 | 1.0 | 0.75 | 657.2 | 1106.2 | 975.0 | 0.594 | 0.674 | 1.135 | 1.4 |

2 | 1.0 | 1.00 | 757.5 | 1098.3 | 1093.7 | 0.690 | 0.693 | 1.004 | 2.5 |

2 | 1.5 | 0.00 | 105.6 | 743.7 | 218.8 | 0.142 | 0.483 | 3.399 | 2.2 |

2 | 1.5 | 0.25 | 205.5 | 741.7 | 316.9 | 0.277 | 0.648 | 2.340 | 1.6 |

2 | 1.5 | 0.50 | 305.3 | 739.3 | 484.7 | 0.413 | 0.630 | 1.525 | 16,559.1 |

2 | 1.5 | 0.75 | 405.2 | 738.7 | 643.4 | 0.548 | 0.630 | 1.148 | 1.6 |

2 | 1.5 | 1.00 | 505.0 | 732.2 | 740.0 | 0.690 | 0.682 | 0.990 | 2.6 |

2 | 2.0 | 0.00 | 44.6 | 557.0 | 159.3 | 0.080 | 0.280 | 3.496 | 2.5 |

2 | 2.0 | 0.25 | 128.1 | 556.3 | 214.4 | 0.230 | 0.598 | 2.595 | 1.3 |

2 | 2.0 | 0.50 | 211.7 | 554.5 | 362.8 | 0.382 | 0.583 | 1.529 | 977.0 |

2 | 2.0 | 0.75 | 295.2 | 554.7 | 486.9 | 0.532 | 0.606 | 1.139 | 2.1 |

2 | 2.0 | 1.00 | 378.7 | 549.5 | 562.0 | 0.689 | 0.674 | 0.978 | 2.5 |

T | ${\mathit{E}}_{\mathit{opt}}$ | ${\mathit{\eta}}_{\mathit{opt}}$ | $\mathit{\epsilon}$ | C.T. |
---|---|---|---|---|

[min] | [kWh/day] | [-] | [-] | [min] |

30 | 924.5 | 0.667 | 2.121 | 1.8 |

30 | 1132.3 | 0.698 | 1.728 | 1.3 |

30 | 1404.9 | 0.686 | 1.389 | 1.6 |

30 | 1684.4 | 0.675 | 1.151 | 1.4 |

30 | 1924.8 | 0.681 | 0.995 | 2.4 |

60 | 920.8 | 0.670 | 2.129 | 1.8 |

60 | 1127.9 | 0.701 | 1.735 | 1.5 |

60 | 1399.3 | 0.689 | 1.394 | 2.0 |

60 | 1676.7 | 0.678 | 1.157 | 1.8 |

60 | 1910.2 | 0.686 | 1.002 | 2.2 |

120 | 918.6 | 0.672 | 2.135 | 6.6 |

120 | 1125.7 | 0.702 | 1.738 | 2.4 |

120 | 1396.5 | 0.690 | 1.397 | 3.4 |

120 | 1672.6 | 0.680 | 1.159 | 2.0 |

120 | 1902.0 | 0.689 | 1.007 | 3.5 |

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## Share and Cite

**MDPI and ACS Style**

Fecarotta, O.; Carravetta, A.; Morani, M.C.; Padulano, R.
Optimal Pump Scheduling for Urban Drainage under Variable Flow Conditions. *Resources* **2018**, *7*, 73.
https://doi.org/10.3390/resources7040073

**AMA Style**

Fecarotta O, Carravetta A, Morani MC, Padulano R.
Optimal Pump Scheduling for Urban Drainage under Variable Flow Conditions. *Resources*. 2018; 7(4):73.
https://doi.org/10.3390/resources7040073

**Chicago/Turabian Style**

Fecarotta, Oreste, Armando Carravetta, Maria Cristina Morani, and Roberta Padulano.
2018. "Optimal Pump Scheduling for Urban Drainage under Variable Flow Conditions" *Resources* 7, no. 4: 73.
https://doi.org/10.3390/resources7040073