# Definition of the Operational Curves by Modification of the Affinity Laws to Improve the Simulation of PATs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Hydraulic Mathematical Development

^{3}/s; ${\eta}_{0}$ is the efficiency of the machine; ${P}_{0}$ is the generated power in kW. The rest of the coefficients (A, B, C, ${E}_{4}$,${E}_{3}$,${E}_{2}$, ${E}_{1}$, ${E}_{0}$, ${P}_{4}$, ${P}_{3}$,${P}_{2}$, ${P}_{1}$ and ${P}_{5}$) are the coefficients, which define the HC, EC and PC of the pump working as the turbine.

^{3}/s. This value is inside of the VOS; $H$ is the recovered head for this flow in m w.c. when the machine operates at the ratio of rotational speed equal to $\alpha $; $\eta $ is the efficiency for values of $Q$, $H$ and $\alpha ;$ $P$ is the shaft power in this operation point in kW; ${Q}_{0}$, ${H}_{0}$, ${P}_{0}$ and ${\eta}_{0}$ are referred to any point of the characteristic curve of the machine when it is operating at nominal rotational speed (i.e., α = 1).

#### 2.2. Methodology

#### 2.3. Materials

## 3. Results

#### 3.1. Definition of the Regression Expression to Define ${F}_{i}$

_{BEP}. The ‘$\alpha $’ parameter also did other formulations in the published literature. Besides, this study adds another parameter, Q/Q

_{BEP}. It considers the distance to the working zone of the maximum efficiencies).

^{2}is calculated. In this case, ${F}_{6}$ was the expression, in which the errors were low when BEH and BPH are observed. In all cases, BIAS values were low, and this index was between 0 and $\pm 3\%$. These values are no significant because they show average errors, which are less than 3% when the head curve and power curve are estimated.

#### 3.2. Comparison of the Proposal Study vs. Other Published Methods Applied to BEH, BPH and BPF

^{2}and he/q

^{2}) showed the minor error values of BEH, BPH and BPF.

^{2}), is observed similar results were obtained. The proposed functions were the best compared to the other five methods. RMSE, MAD and MRD were 0.067, 0.051 and 0.054, respectively. When these error indexes were compared with the second-best value (affinity laws or Carravetta’s method), the error values decreased between 26.5, 28.8 and 25.4%. For BPH, the trend of the results was similar. The proposed method was the most accurate, decreasing the errors between 25 and 32% compared with the second-best-option functions (Carravetta’s). When BPF was analyzed, RMSE, MAD and MRD decreased 14.3, 12.4 and 13.4%, respectively. The use of MOALs demonstrated better accuracy in the operational curves when these results are compared with the affinity laws, which are simpler, but they showed higher errors [11,17,19,22]. Figure 4 shows the decrease of the error in the different curves. Therefore, although the expressions use high degree equations, they improve the simulation of the PATs when they operate under variable rotational speeds.

^{2}(Figure 4a) is considered for BEH, the RMSE, MAD and MRS errors were 0.1077, 0.0755 and 0.622, respectively. These errors decreased between 17.4, 16 and 0.7% respectively, compared to the second-best method (affinity laws). If the BPH is analyzed similar results were obtained. The proposed functions were the best and the error indexes were reduced between 9 and 16% compared to the second−best method. When the error indexes were calculated for the BPF, the proposed study was the best, and the error values decreased 20.1, 17.5 and 14.9% for RMSE, MAD and MRD respectively.

^{2}is analysed in Figure 4a, the proposed regression expression showed errors between 8.6% and 49.8% less than the obtained errors by application of the affinity laws, which was the second−best method for BEH and BPH, being the third when BPF was calculated. RMSE value was 0.1 for BEH and BPF, while it was 0.148 when BPH was estimated. If the MAD and MRD were analysed in this table, these values were around 0.07 and 0.06 respectively when BEH was compared. If BPH is compared, the MAD and MRD errors 0.1 and 0.08 respectively when they were 0.12 and 0.09 when the second−best method (affinity laws) is compared.

^{2}and he/q

^{2}showed the proposed regression using F

_{6}enabled to estimate the best efficiency head, best power head and best power flow. Three lines are strategic considerations when water managers define operations rules to maximize the energy recovery using micro hydropower systems.

#### 3.3. Application of the Proposed Curves to Experimental Machine

_{st}when Figure 4 and Figure 5 are considered and the errors are evaluated.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Da Silveira, A.P.P.; Mata-Lima, H. Energy audit in water supply systems: A proposal of integrated approach towards energy efficiency. Water Policy
**2020**, 22, 1126–1141. [Google Scholar] [CrossRef] - Postacchini, M.; Darvini, G.; Finizio, F.; Pelagalli, L.; Soldini, L.; Di Giuseppe, E. Hydropower generation through pump as turbine: Experimental study and potential application to small-scale WDN. Water
**2020**, 12, 958. [Google Scholar] [CrossRef][Green Version] - Pugliese, F.; De Paola, F.; Fontana, N.; Giugni, M.; Marini, G. Experimental characterization of two Pumps As Turbines for hydropower generation. Renew. Energy
**2016**, 99, 180–187. [Google Scholar] [CrossRef] - Kim, J.W.; Suh, J.W.; Choi, Y.S.; Lee, K.Y.; Kim, J.H.; Kanemoto, T.; Kim, J.H. Simultaneous efficiency improvement of pump and turbine modes for a counter-rotating type pump-turbine. Adv. Mech. Eng.
**2016**, 8, 1–14. [Google Scholar] [CrossRef][Green Version] - Binama, M.; Su, W.T.; Li, X.B.; Li, F.C.; Wei, X.Z.; An, S. Investigation on pump as turbine (PAT) technical aspects for micro hydropower schemes: A state-of-the-art review. Renew. Sustain. Energy Rev.
**2017**, 79, 148–179. [Google Scholar] [CrossRef] - Ortiz Flórez, R.; Abella Jiménez, J. Máquinas Hidráulicas Reversibles Aplicadas a Micro Centrales Hidroeléctricas. IEEE Lat. Am. Trans.
**2008**, 6, 170–175. [Google Scholar] - Sharma, K. Small Hydroelectric Project-Use of Centrifugal Pumps as Turbines; Kirloskar: Bangalore, India, 1985. [Google Scholar]
- Derakhshan, S.; Nourbakhsh, A. Experimental study of characteristic curves of centrifugal pumps working as turbines in different specific speeds. Exp. Therm. Fluid Sci.
**2008**, 32, 800–807. [Google Scholar] [CrossRef] - Pérez-Sánchez, M.; Sánchez-Romero, F.J.; Ramos, H.M.; López-Jiménez, P.A. Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves. Water
**2020**, 12, 468. [Google Scholar] [CrossRef][Green Version] - Plua, F.A.; Sánchez-Romero, F.J.; Hidalgo, V.; López-Jiménez, P.A.; Pérez-Sánchez, M. New expressions to apply the variation operation strategy in engineering tools using pumps working as turbines. Mathematics
**2021**, 9, 860. [Google Scholar] [CrossRef] - Tahani, M.; Kandi, A.; Moghimi, M.; Houreh, S.D. Rotational speed variation assessment of centrifugal pump-as-turbine as an energy utilization device under water distribution network condition. Energy
**2020**, 213, 118502. [Google Scholar] [CrossRef] - Chen, S.; Chen, B. Urban energy–water nexus: A network perspective. Appl. Energy
**2016**, 184, 905–914. [Google Scholar] [CrossRef] - Carravetta, A.; Del Giudice, G.; Fecarotta, O.; Ramos, H. Pump as Turbine (PAT) Design in Water Distribution Network by System Effectiveness. Water
**2013**, 5, 1211–1225. [Google Scholar] [CrossRef][Green Version] - Ebrahimi, S.; Riasi, A.; Kandi, A. Selection optimization of variable speed pump as turbine (PAT) for energy recovery and pressure management. Energy Convers. Manag.
**2021**, 227, 113586. [Google Scholar] [CrossRef] - Man, Y.; Han, Y.; Liu, Y.; Lin, R.; Ren, J. Multi-criteria decision making for sustainability assessment of boxboard production: A life cycle perspective considering water consumption, energy consumption, GHG emissions, and internal costs. J. Environ. Manag.
**2020**, 255, 109860. [Google Scholar] [CrossRef] - Niet, T.; Arianpoo, N.; Kuling, K.; Wright, A.S. Embedding the United Nations sustainable development goals into energy systems analysis: Expanding the food–energy–water nexus. Energy Sustain. Soc.
**2021**, 11, 1–12. [Google Scholar] [CrossRef] - Pérez-Sánchez, M.; López-Jiménez, P.A.; Ramos, H.M. Modified Affinity Laws in Hydraulic Machines towards the Best Efficiency Line. Water Resour. Manag.
**2018**, 32, 829–844. [Google Scholar] [CrossRef][Green Version] - Mataix, C. Turbomáquinas Hidráulicas; Universidad Pontificia Comillas: Madrid, Spain, 2009; ISBN 978-84-8468-252-3. [Google Scholar]
- Fecarotta, O.; Carravetta, A.; Ramos, H.M.; Martino, R. An improved affinity model to enhance variable operating strategy for pumps used as turbines. J. Hydraul. Res.
**2016**, 54, 332–341. [Google Scholar] [CrossRef] - Hyypiä, J. Hydraulic energy recovery by replacing a control valve with a centrifugal pump used as a turbine. Ph.D. Thesis, Lappeenranta University of Technology, Lappeenranta, Finland, 2016. [Google Scholar]
- Nygren, L. Hydraulic Energy Harvesting with Variable-Speed-Driven Centrifugal Pump as Turbine. Master’s Thesis, Lappeenranta University of Technology, Lappeenranta, Finland, 2017. [Google Scholar]
- Carravetta, A.; Conte, M.C.; Fecarotta, O.; Ramos, H.M. Evaluation of PAT performances by modified affinity law. Procedia Eng.
**2014**, 89, 581–587. [Google Scholar] [CrossRef][Green Version] - KSB. “PATs Curves,” Catalogue. 2019. Available online: https://www.ksb.com/es-es/productos/catalogo-de-productos (accessed on 5 July 2021).
- Jain, S.V.; Swarnkar, A.; Motwani, K.H.; Patel, R.N. Effects of impeller diameter and rotational speed on performance of pump running in turbine mode. Energy Convers. Manag.
**2015**, 89, 808–824. [Google Scholar] [CrossRef] - Abazariyan, S.; Rafee, R.; Derakhshan, S. Experimental study of viscosity effects on a pump as turbine performance. Renew. Energy
**2018**, 127, 539–547. [Google Scholar] [CrossRef] - Kramer, M.; Terheiden, K.; Wieprecht, S. Pumps as turbines for efficient energy recovery in water supply networks. Renew. Energy
**2018**, 122, 17–25. [Google Scholar] [CrossRef][Green Version] - Delgado, J.; Ferreira, J.P.; Covas, D.I.C.; Avellan, F. Variable speed operation of centrifugal pumps running as turbines. Experimental investigation. Renew. Energy
**2019**, 142, 437–450. [Google Scholar] [CrossRef] - Stefanizzi, M.; Torresi, M.; Fortunato, B.; Camporeale, S.M. Experimental investigation and performance prediction modeling of a single stage centrifugal pump operating as turbine. Energy Procedia
**2017**, 126, 589–596. [Google Scholar] [CrossRef]

**Figure 4.**(

**a**) Error indexes for the different methods applied to h/q

^{2}; (

**b**) error indexes for the different methods applied to he/q

^{2}.

**Figure 5.**(

**a**) H−Q for BEH; (

**b**) P−Q for BEH; (

**c**) H−Q for BPH; (

**d**) P−Q for BPH; (

**e**) H−Q for BEH, BPH and BPF; (

**f**) P−Q for BEH, BPH and BPF.

Curve | Curve Type | AL | MOAL |
---|---|---|---|

Head | $H={k}_{H}{Q}^{2}$ | ${k}_{H,AL}=\frac{{H}_{0}}{{Q}_{0}^{2}}=\left(A{m}^{2}+Bm+C\right)$ | ${k}_{H,MOAL}=\frac{h}{{q}^{2}}{k}_{H,AL}$ |

Efficiency | $\eta =e{\eta}_{0}$ | ${\eta}_{AL}={\eta}_{0}$ | ${\eta}_{MOAL}=e\xb7{\eta}_{AL}$ |

Power | $P={k}_{P}{Q}^{3}$ | Using head and efficiency curves ${k}_{P,AL}=9.81\xb7{k}_{H,AL}\xb7{\eta}_{AL}$ | ${k}_{P,MOAL}=\frac{he}{{q}^{2}}{k}_{P,AL}$ |

Using power expression ${k}_{P,AL}=\frac{{P}_{0}}{{Q}_{0}^{3}}$ ${k}_{P,AL}=\left(\frac{{P}_{4}}{m}+{P}_{3}+{P}_{2}m+{P}_{1}{m}^{2}+{P}_{5}{m}^{3}\right)$ | ${k}_{P,MOAL}=\frac{p}{{q}^{3}}{k}_{P,AL}=\frac{he}{{q}^{2}}{k}_{P,AL}$ |

Curve | BEH | BPH | BPF |
---|---|---|---|

Restriction | $\frac{d\eta}{d\alpha}=0$ | $\frac{dP}{d\alpha}=0$ | $\frac{dP}{dQ}=0$ |

${\gamma}_{6}$ | 0 | $2{E}_{0}A$ | ${E}_{0}A$ |

${\gamma}_{5}$ | 0 | ${E}_{0}B+{E}_{1}A$ | $2\left({E}_{0}B+{E}_{1}A\right)$ |

${\gamma}_{4}$ | 0 | $0$ | $3\left({E}_{0}C+{E}_{1}B+{E}_{2}A\right)$ |

${\gamma}_{3}$ | ${E}_{1}$ | $-{E}_{1}C-{E}_{2}B-{E}_{3}A$ | $4\left({E}_{1}C+{E}_{2}B+{E}_{3}A\right)$ |

${\gamma}_{2}$ | $2{E}_{2}$ | $-2{E}_{2}C-2{E}_{3}B-2{E}_{4}A$ | $5\left({E}_{2}C+{E}_{3}B+{E}_{4}A\right)$ |

${\gamma}_{1}$ | $3{E}_{3}$ | $-3{E}_{3}C-3{E}_{4}B$ | $6\left({E}_{3}C+{E}_{4}B\right)$ |

${\gamma}_{0}$ | $4{E}_{4}$ | $-4{E}_{4}C$ | $7{E}_{4}C$ |

Solution | ${m}_{BEH}$ | ${m}_{BPH}$ | ${m}_{BPF}$ |

Function Model (FM) | $\mathbf{Polynomial}\text{}\mathbf{function}\text{}(\mathbf{From}\text{}{\mathit{F}}_{1}$$\mathbf{to}\text{}{\mathit{F}}_{6})\text{:}$ $\mathit{N}\mathit{P}={\mathit{\beta}}_{1}\left(\mathit{\alpha}\frac{\mathit{Q}}{{\mathit{Q}}_{\mathit{B}\mathit{E}\mathit{P}}}\right)+{\mathit{\beta}}_{2}{\left(\frac{\mathit{Q}}{{\mathit{Q}}_{\mathit{B}\mathit{E}\mathit{P}}}\right)}^{2}+{\mathit{\beta}}_{3}\left(\frac{\mathit{Q}}{{\mathit{Q}}_{\mathit{B}\mathit{E}\mathit{P}}}\right)+{\mathit{\beta}}_{4}{\mathit{\alpha}}^{2}+{\mathit{\beta}}_{5}\mathit{\alpha}+{\mathit{\beta}}_{6}$ $\mathbf{Potential}\text{}\mathbf{function}\text{}(\mathbf{From}\text{}{\mathit{F}}_{7}$$\text{}\mathbf{to}\text{}{\mathit{F}}_{10})\text{:}$ $\mathit{N}\mathit{P}={\left(\frac{\mathit{Q}}{{\mathit{Q}}_{\mathit{B}\mathit{E}\mathit{P}}}\right)}^{{\mathit{\beta}}_{3}}{\mathit{\alpha}}^{{\mathit{\beta}}_{5}}\xb7\mathit{e}\mathit{x}{\mathit{p}}^{{\mathit{\beta}}_{6}}$ |
---|---|

${F}_{1}$ | $NP={\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha $ |

${F}_{2}$ | $NP={\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha +{\beta}_{6}$ |

${F}_{3}$ | $NP={\beta}_{2}{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+{\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha $ |

${F}_{4}$ | $NP={\beta}_{2}{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+{\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha +{\beta}_{6}$ |

${F}_{5}$ | $NP={\beta}_{1}\left(\alpha \frac{Q}{{Q}_{BEP}}\right)+{\beta}_{2}{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+{\beta}_{3}\left(\frac{Q}{{Q}_{BEP}}\right)+{\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha $ |

${F}_{6}$ | $NP={\beta}_{1}\left(\alpha \frac{Q}{{Q}_{BEP}}\right)+{\beta}_{2}{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+{\beta}_{3}\left(\frac{Q}{{Q}_{BEP}}\right)+{\beta}_{4}{\alpha}^{2}+{\beta}_{5}\alpha +{\beta}_{6}$ |

${F}_{7}$ | $NP={\alpha}^{{\beta}_{5}}$ |

${F}_{8}$ | $NP={\alpha}^{{\beta}_{5}}\xb7ex{p}^{{\beta}_{6}}$ |

${F}_{9}$ | $NP={\left(\frac{Q}{{Q}_{BEP}}\right)}^{{\beta}_{3}}{\alpha}^{{\beta}_{5}}$ |

${F}_{10}$ | $NP={\left(\frac{Q}{{Q}_{BEP}}\right)}^{{\beta}_{3}}{\alpha}^{{\beta}_{5}}\xb7ex{p}^{{\beta}_{6}}$ |

Error Index | Equation | Variable | Accuracy |
---|---|---|---|

Root Mean Square Error (RMSE) | $RMSE=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{s}{\left[{O}_{i}-{P}_{i}\right]}^{2}}{s}}$ | ${O}_{i}$ are the estimated values; ${P}_{i}$ the experimental values and s the number of observations | Perfect fit when RMSE is zero |

Mean Absolute Deviation (MAD) | $MAD={\displaystyle {\displaystyle \sum}_{1}^{s}}\frac{1}{\mathrm{s}}\left|{O}_{i}-{P}_{i}\right|$ | ${O}_{i}$ are the estimated values; ${P}_{i}$ the experimental values and s the number of observations | Perfect fit when MAD is zero |

Mean Relative Deviation (MRD) | $MRD={\displaystyle {\displaystyle \sum}_{1}^{s}}\frac{\left|{O}_{i}-{P}_{i}\right|/{P}_{i}}{\mathrm{s}}$ | ${O}_{i}$ are the estimated values; ${P}_{i}$ the experimental values and s the number of observations | |

BIAS | $BIAS=\frac{{{\displaystyle \sum}}_{i=1}^{s}{\left[{O}_{i}-{P}_{i}\right]}^{}}{s}$ | ${O}_{i}$ are the estimated values; ${P}_{i}$ the experimental values and s the number of observations | Perfect fit is zero |

Function Model (FM) | Expressions |
---|---|

${F}_{1}$ | $\frac{h}{{q}^{2}}=-0.512{\alpha}^{2}+1.63\alpha \left({R}^{2}=0.978\right)$ $\frac{he}{{q}^{2}}=-0.826{\alpha}^{2}+1.843\alpha \left({R}^{2}=0.979\right)$ |

${F}_{2}$ | $\frac{h}{{q}^{2}}=1.048{\alpha}^{2}-1.487\alpha +1.469$ $\left({R}^{2}=0.738\right)$ $\frac{he}{{q}^{2}}=-0.1706{\alpha}^{2}+0.5336\alpha +0.617$ $\left({R}^{2}=0.151\right)$ |

${F}_{3}$ | $\frac{h}{{q}^{2}}=-0.235{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}-0.5196{\alpha}^{2}+1.789\alpha $ $\left({R}^{2}=0.981\right)$ $\frac{he}{{q}^{2}}=-0.024{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}-0.826{\alpha}^{2}+1.858\alpha $ $\left({R}^{2}=0.979\right)$ |

${F}_{4}$ | $\frac{h}{{q}^{2}}=-0.1139{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+0.965{\alpha}^{2}-1.254\alpha +1.395\text{}\left({R}^{2}=0.764\right)$ $\frac{he}{{q}^{2}}=0.032{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}-0.148{\alpha}^{2}+0.469\alpha +0.637\text{}\left({R}^{2}=0.154\right)$ |

${F}_{5}$ | $\frac{h}{{q}^{2}}=-1.508\left(\alpha \frac{Q}{{Q}_{BEP}}\right)-0.471{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+1.93\left(\frac{Q}{{Q}_{BEP}}\right)+0.714{\alpha}^{2}+0.342\alpha \left({R}^{2}=0.986\right)$ $\frac{he}{{q}^{2}}=0.532\left(\alpha \frac{Q}{{Q}_{BEP}}\right)-0.828{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+0.757\left(\frac{Q}{{Q}_{BEP}}\right)-0.757{\alpha}^{2}+1.287\alpha \left({R}^{2}=0.98\right)$ |

${F}_{6}$ | $\frac{h}{{q}^{2}}=-0.69\left(\alpha \frac{Q}{{Q}_{BEP}}\right)+0.122{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}+0.313\left(\frac{Q}{{Q}_{BEP}}\right)+1.222{\alpha}^{2}-1.267\alpha +1.294\text{}\left({R}^{2}=0.778\right)$ $\frac{he}{{q}^{2}}=0.993\left(\alpha \frac{Q}{{Q}_{BEP}}\right)-0.494{\left(\frac{Q}{{Q}_{BEP}}\right)}^{2}-0.156\left(\frac{Q}{{Q}_{BEP}}\right)-0.471{\alpha}^{2}+0.38\alpha +0.73\text{}\left({R}^{2}=0.191\right)$ |

${F}_{7}$ | $\frac{h}{{q}^{2}}={\alpha}^{0.214}\left({R}^{2}=0.207\right)$ $\frac{he}{{q}^{2}}={\alpha}^{0.245}\left({R}^{2}=0.175\right)$ |

${F}_{8}$ | $\frac{h}{{q}^{2}}={\alpha}^{0.305}\xb7ex{p}^{0.077}\left({R}^{2}=0.402\right)$ $\frac{he}{{q}^{2}}={\alpha}^{0.202}\xb7ex{p}^{-0.036}\left({R}^{2}=0.116\right)$ |

${F}_{9}$ | $\frac{h}{{q}^{2}}={\left(\frac{Q}{{Q}_{BEP}}\right)}^{-0.23}{\alpha}^{0.451}\left({R}^{2}=0.506\right)$ $\frac{he}{{q}^{2}}={\left(\frac{Q}{{Q}_{BEP}}\right)}^{0.11}{\alpha}^{0.132}\left({R}^{2}=0.22\right)$ |

${F}_{10}$ | $\frac{h}{{q}^{2}}={\left(\frac{Q}{{Q}_{BEP}}\right)}^{-0.166}{\alpha}^{0.422}\xb7ex{p}^{0.032}\left({R}^{2}=0.478\right)$ $\frac{he}{{q}^{2}}={\left(\frac{Q}{{Q}_{BEP}}\right)}^{0.083}{\alpha}^{0.143}\xb7ex{p}^{-0.013}\left({R}^{2}=0.129\right)$ |

$\mathit{B}\mathit{E}\mathit{H}$ | $\mathit{B}\mathit{P}\mathit{H}$ | $\mathit{B}\mathit{P}\mathit{F}$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

FM | RMSE | MAD | MRD | BIAS | FM | RMSE | MAD | MRD | BIAS | FM | RMSE | MAD | MRD | BIAS | |

${F}_{1}$ | 0.1515 (9) | 0.1144 (9) | 0.1023 (10) | −0.04 (3) | ${F}_{1}$ | 0.1863 (8) | 0.1354 (10) | 0.1136 (10) | −0.0356 (5) | ${F}_{1}$ | 0.1393 (7) | 0.1059 (7) | 0.0964 (8) | −0.0238 (3) | |

${F}_{2}$ | 0.1035 (1) | 0.0735 (1) | 0.0628 (2) | −0.0172 (1) | ${F}_{2}$ | 0.1537 (3) | 0.1107 (3) | 0.0917 (7) | −0.0128 (1) | ${F}_{2}$ | 0.0959 (1) | 0.0708 (1) | 0.0629 (1) | −0.001 (1) | |

${F}_{3}$ | 0.1636 (10) | 0.1154 (10) | 0.0999 (9) | −0.0966 (10) | ${F}_{3}$ | 0.1882 (9) | 0.1321 (9) | 0.1055 (9) | −0.0585 (8) | ${F}_{3}$ | 0.2115 (9) | 0.1709 (9) | 0.1515 (9) | −0.1634 (9) | |

${F}_{4}$ | 0.1134 (3) | 0.0819 (3) | 0.0695 (3) | −0.0441 (5) | ${F}_{4}$ | 0.1531 (2) | 0.1106 (2) | 0.0898 (6) | −0.0249 (2) | ${F}_{4}$ | 0.119 (4) | 0.0957 (6) | 0.0838 (6) | −0.0635 (5) | |

${F}_{5}$ | 0.1398 (7) | 0.0974 (7) | 0.0778 (7) | −0.0755 (7) | ${F}_{5}$ | 0.1924 (10) | 0.1275 (8) | 0.0954 (8) | −0.0754 (9) | ${F}_{5}$ | 0.3517 (10) | 0.257 (10) | 0.2114 (10) | −0.2302 (10) | |

${F}_{6}$ | 0.1077 (2) | 0.0755 (2) | 0.0622 (1) | −0.0408 (4) | ${F}_{6}$ | 0.1479 (1) | 0.1043 (1) | 0.0819 (2) | −0.0313 (4) | ${F}_{6}$ | 0.1056 (2) | 0.079 (2) | 0.0664 (2) | −0.0432 (4) | |

${F}_{7}$ | 0.1341 (6) | 0.0927 (6) | 0.074 (6) | −0.0865 (8) | ${F}_{7}$ | 0.1641 (5) | 0.1116 (4) | 0.0813 (1) | −0.082 (10) | ${F}_{7}$ | 0.1213 (5) | 0.0844 (4) | 0.0676 (3) | −0.0703 (6) | |

${F}_{8}$ | 0.1194 (4) | 0.0865 (4) | 0.0727 (5) | −0.0308 (2) | ${F}_{8}$ | 0.1596 (4) | 0.1127 (5) | 0.0887 (4) | −0.0263 (3) | ${F}_{8}$ | 0.1146 (3) | 0.0828 (3) | 0.0712 (4) | −0.0146 (2) | |

${F}_{9}$ | 0.1418 (8) | 0.0998 (8) | 0.0819 (8) | −0.0929 (9) | ${F}_{9}$ | 0.1719 (7) | 0.1171 (7) | 0.0895 (5) | −0.0543 (7) | ${F}_{9}$ | 0.149 (8) | 0.1128 (8) | 0.0951 (7) | −0.1052 (8) | |

${F}_{10}$ | 0.1289 (5) | 0.0885 (5) | 0.0724 (4) | −0.0673 (6) | ${F}_{10}$ | 0.1654 (6) | 0.1138 (6) | 0.088 (3) | −0.0386 (6) | ${F}_{10}$ | 0.1265 (6) | 0.0879 (5) | 0.0725 (5) | −0.0721 (7) |

$\mathit{B}\mathit{E}\mathit{H}$ | $\mathit{B}\mathit{P}\mathit{H}$ | $\mathit{B}\mathit{P}\mathit{F}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

FM | RMSE | MAD | MRD | BIAS | FM | RMSE | MAD | MRD | BIAS | FM | RMSE | MAD | MRD | BIAS |

${F}_{1}$ | 0.1 (9) | 0.0741 (9) | 0.0766 (9) | −0.0269 (8) | ${F}_{1}$ | 0.0754 (8) | 0.0529 (8) | 0.0549 (8) | −0.0241 (8) | ${F}_{1}$ | 0.1502 (8) | 0.1162 (7) | 0.1206 (6) | −0.0139 (5) |

${F}_{2}$ | 0.073 (6) | 0.056 (6) | 0.0582 (6) | −0.0173 (6) | ${F}_{2}$ | 0.0523 (3) | 0.0386 (3) | 0.0398 (2) | −0.0145 (5) | ${F}_{2}$ | 0.1438 (6) | 0.1111 (4) | 0.1183 (3) | −0.0043 (1) |

${F}_{3}$ | 0.1017 (10) | 0.075 (10) | 0.0771 (10) | −0.0326 (10) | ${F}_{3}$ | 0.0763 (9) | 0.0536 (9) | 0.0553 (9) | −0.0264 (9) | ${F}_{3}$ | 0.1529 (9) | 0.1171 (9) | 0.1193 (5) | −0.0279 (7) |

${F}_{4}$ | 0.0702 (5) | 0.054 (5) | 0.0568 (5) | −0.0099 (4) | ${F}_{4}$ | 0.0534 (4) | 0.0407 (4) | 0.0424 (4) | −0.0111 (4) | ${F}_{4}$ | 0.1415 (2) | 0.1111 (3) | 0.1213 (7) | 0.013 (4) |

${F}_{5}$ | 0.0972 (8) | 0.0723 (8) | 0.0737 (8) | −0.0307 (9) | ${F}_{5}$ | 0.0849 (10) | 0.0631 (10) | 0.0649 (10) | −0.0301 (10) | ${F}_{5}$ | 0.2814 (10) | 0.2218 (10) | 0.2061 (10) | −0.1608 (10) |

${F}_{6}$ | 0.0677 (3) | 0.0513 (1) | 0.0539 (1) | −0.0138 (5) | ${F}_{6}$ | 0.0467 (1) | 0.0356 (1) | 0.0371 (1) | −0.0054 (2) | ${F}_{6}$ | 0.1433 (4) | 0.1129 (5) | 0.1164 (2) | −0.0407 (9) |

${F}_{7}$ | 0.0669 (2) | 0.0517 (3) | 0.0552 (4) | −0.0002 (1) | ${F}_{7}$ | 0.0504 (2) | 0.038 (2) | 0.0399 (3) | 0.0026 (1) | ${F}_{7}$ | 0.1388 (1) | 0.1089 (2) | 0.1192 (4) | 0.0128 (3) |

${F}_{8}$ | 0.0744 (7) | 0.0572 (7) | 0.0589 (7) | −0.0248 (7) | ${F}_{8}$ | 0.0554 (5) | 0.0414 (5) | 0.0424 (5) | −0.022 (7) | ${F}_{8}$ | 0.1422 (3) | 0.1087 (1) | 0.1147 (1) | −0.0118 (2) |

${F}_{9}$ | 0.0659 (1) | 0.0515 (2) | 0.0551 (3) | 0.003 (2) | ${F}_{9}$ | 0.0584 (7) | 0.0451 (7) | 0.0473 (7) | −0.0098 (3) | ${F}_{9}$ | 0.1451 (7) | 0.1168 (8) | 0.1307 (9) | 0.0302 (8) |

${F}_{10}$ | 0.0679 (4) | 0.0522 (4) | 0.055 (2) | −0.0076 (3) | ${F}_{10}$ | 0.0581 (6) | 0.0441 (6) | 0.0459 (6) | −0.0162 (6) | ${F}_{10}$ | 0.1437 (5) | 0.1135 (6) | 0.1247 (8) | 0.0157 (6) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Macías Ávila, C.A.; Sánchez-Romero, F.-J.; López-Jiménez, P.A.; Pérez-Sánchez, M. Definition of the Operational Curves by Modification of the Affinity Laws to Improve the Simulation of PATs. *Water* **2021**, *13*, 1880.
https://doi.org/10.3390/w13141880

**AMA Style**

Macías Ávila CA, Sánchez-Romero F-J, López-Jiménez PA, Pérez-Sánchez M. Definition of the Operational Curves by Modification of the Affinity Laws to Improve the Simulation of PATs. *Water*. 2021; 13(14):1880.
https://doi.org/10.3390/w13141880

**Chicago/Turabian Style**

Macías Ávila, Carlos Andrés, Francisco-Javier Sánchez-Romero, P. Amparo López-Jiménez, and Modesto Pérez-Sánchez. 2021. "Definition of the Operational Curves by Modification of the Affinity Laws to Improve the Simulation of PATs" *Water* 13, no. 14: 1880.
https://doi.org/10.3390/w13141880