# Modeling and Simulation of Multipumping Photovoltaic Irrigation Systems

^{*}

## Abstract

**:**

## 1. Introduction

#### Objectives and Organization of the Article

## 2. Materials and Methods

- The active pumps of the first group operate synchronized with each other at a fixed frequency (typically, it corresponds to the nominal frequency or the frequency at which maximum efficiency is achieved), and the active pumps of the second group operate synchronized with each other at a variable frequency;
- The pumps in both groups operate at a variable frequency, and this frequency may be different in each group but equal within the same group.

## 3. Modeling of the Multipumping PV Irrigation Systems

#### 3.1. Configuration 1: Multipump PVIS Made up of a Group of N Equal Motor Pumps, in Parallel and Synchronized in Frequency

_{2}is the demanded power at the pump input (or motor output), Q

_{1}is the water flow, and the parameters ${k}_{\mathrm{b}0,1}$, ${k}_{\mathrm{b}1,1}$, ${k}_{\mathrm{b}2,1}$, ${k}_{\mathrm{p}0,1}$, ${k}_{\mathrm{p}1,1}$, and ${k}_{\mathrm{p}2,1}$ are the parameters that fit the curves of the individual pump.

_{N}corresponding to the sum of the flow rates Q

_{1}that each individual pump elevates at the same head; therefore, from [1], it is simple to deduce that:

_{2,N}necessary for the equivalent pump to pump a flow Q

_{N}corresponds to the sum of the power P

_{2}of the N individual pumps when they pump a flow ${Q}_{1}$:

_{DC,mpp}, it is necessary to deduce the DC power, P

_{DC}, needed to satisfy the one demanded by the pump, P

_{2}, at all possible Q-H working points within the range of frequencies at which it can operate. This calculation is made from the equivalent pump curves in three consecutive steps:

- Determine the power P
_{2}demanded by the equivalent pump at a certain working point Q-H; - Obtain P
_{DC}at the input of the frequency converter corresponding to that P_{2}, assuming that the motors of all pumps are identical; - Select the combination of active pumps that maximizes the flow rate pumped given the available PV power P
_{DC,mpp}.

#### 3.1.1. Calculation of P_{2} for Any Q-H Working Point

_{2}required by the equivalent pump working at a certain point (Q

_{B}, H

_{B}) not included in the Q-H curve at nominal frequency, the affinity laws are used following the procedure described in [41] but varying the head instead of the flow (Figure 5). Given a certain H

_{B}:

- 1.
- Calculate the required flow Q
_{B}as the real positive root for Equation (3): $0=\left({k}_{\mathrm{b}0,N}-{H}_{\mathrm{B}}\right)+{k}_{\mathrm{b}1,N}{Q}_{\mathrm{B}}+{k}_{\mathrm{b}2,N}{Q}_{\mathrm{B}}^{2}$ obtaining point B (Q_{B}, H_{B}) at frequency $\omega $. - 2.
- Determine the affinity parabola (H = ${k}_{0}$ + ${k}_{2}$Q
^{2}) that passes through point B, where ${k}_{0}$ = 0, ${k}_{2}={H}_{\mathrm{B}}/{Q}_{\mathrm{B}}^{2}$. The affinity parabola connects the points of equal efficiency. - 3.
- Calculate the intersection of the affinity parabola and Q-H characteristic curve of the pump at rated frequency, ${\omega}_{\mathrm{rated}}$, to obtain point C (Q
_{C}, H_{C}). - 4.
- Determine the hydraulic power at point C, P
_{HC}= $\theta $·H_{C}·Q_{C}, where $\theta $ is a constant value that depends on the water density and gravity. - 5.
- Determine P
_{2}at point C, P_{2C}(Q_{C}) using Equation (4). - 6.
- Calculate pump efficiency at point C, η
_{PC}= P_{HC}/P_{2}(Q_{C}), which is equal to the efficiency at point B, η_{PB}= η_{PC}. - 7.
- Calculate the hydraulic power at point B, P
_{HB}. - 8.
- Finally, determine P
_{2}at point B as P_{2B}= P_{HB}/η_{PB}.

#### 3.1.2. Calculation of P_{DC} at the Input of the Frequency Converter

_{DC}calculation depends on other elements present in the real system in which the pump operates, such as the pump motor, frequency converter, and the losses in the AC wiring between them. It is deduced from the value of P

_{2B}according to the following algorithm:

- Determine the power at the input of the motor (P
_{1}) at point B: P_{1B}= P_{2B}/η_{M}(P_{2B}), η_{M}(P_{2B}) being the efficiency of the motor at P_{2B}. - Determine the power at the output of the frequency converter (P
_{AC}), given the AC wiring losses W_{AC}at P_{1}: P_{AC,B}= P_{1B}/W_{AC}(P_{1B}). - Determine P
_{DC}: P_{DC,B}= P_{AC,B/}η_{FC}(P_{AC,B}), η_{FC}(P_{AC,B}) being the efficiency of the frequency converter at the load P_{AC,B}.

#### 3.1.3. Selection of the Combination of Active Pumps That Maximizes the Pumped Water Flow

_{DC,mpp}, a procedure is necessary to select the number of operating pumps and the value of their synchronized frequency.

_{2}, and P

_{DC}for each of the Z equivalent pumps resulting from the possible combinations of N pumps, where $Z=N$ as the N pumps are equal. Note that we will finally calculate S sets of Z values of Q, P

_{2}, and P

_{DC}.

_{DC}-Q curve of each of the Z equivalent pumps may be fitted with a third-degree polynomial:

_{DC}, it can be observed how it is possible to choose the highest flow rate as the maximum value of the different Q = f

_{z}(P

_{DC}), where z = 1..Z.

_{DC,mpp}, not all the Z equivalent pumps are eligible because each one of them works in a certain P

_{DC}range; therefore, the selection procedure applies a series of restrictions:

- The minimum DC power of the equivalent pump (P
_{DCmin}) that is determined by the minimum cooling water flow of the pumps (Q_{min}) that its manufacturer requires. This restriction disables some combinations of pumps. It should be noted that Q_{min}of the equivalent pump is different from N times the minimum one of the individual pump due to the higher working head in the system curve. - The maximum DC power of the equivalent pump (P
_{DCmax}), which is determined by the maximum operating frequency of each individual pump (${\omega}_{\mathrm{max}}$). The equivalent pump is still eligible but is limited to this maximum operating frequency even if more power is available (P_{DC,mpp}> P_{DCmax}). Again, note that the power demanded at this ${\omega}_{\mathrm{max}}$ by the equivalent pump is different from N times that of an individual pump, also due to the higher working head in the system curve.

#### 3.2. Configuration 2: Group of N Equal Pumps + Group of M Equal Pumps

#### 3.2.1. Selection of the Combination of Pumps

_{1}(x) = g

_{1}(x) x > 0.

_{s}, that all z active pumps must meet; ${g}_{z}\left(x\right)$ is the state cost function, for this case, the demanded DC power by a single pump while pumping a water flow ${Q}_{z}$ at H

_{s}, ${P}_{\mathrm{DC},z}$; ${f}_{z-1}\left(x\right)$ is the cost of the rest of the path, hence the demanded DC power of the rest of z−1 active pumps, ${P}_{\mathrm{DC},z-1}$, pumping the rest of water flow, ${Q}_{s}-{Q}_{z}$ at H

_{s}. Therefore,

#### 3.2.2. Subconfiguration 2.1: A First Group of N Pumps at Fixed Frequency and a Second Group of M Pumps at Variable Frequency

_{n}, Q

_{n}) and (H

_{m}, Q

_{m}) where each group of pumps operates are not on the system curve.

_{s}, H

_{s}) where 1 < s < S, for each H

_{s}, the second group of pumps must pump the difference between the flow of the first group, Q

_{n}, and the total flow Q

_{s}at the height H

_{s}(see Figure 7). The first group operates at the point (Q

_{n}, H

_{s}) of the equivalent Q-H nominal curve of n active pumps. The second group operates at the point (Q

_{s}− Q

_{n}, H

_{s}) on the equivalent Q-H curve of m active pumps at the frequency necessary to contain said point.

_{s}.

_{n}locus (H, Q) of the second group of pumps can be modeled with third-degree polynomials that would constitute z

_{n}“pseudo system curves” for the second group of pumps (Equation (9)).

_{s}determines the frequency of the second group of m active pumps.

_{DC}-Q curves for each combination of (n, m) active pumps for the case N = 2 and M = 2, where Q is normalized to its maximum value. For each combination, the flat part of the curve corresponds to all active pumps operating at their maximum (nominal) frequency.

_{DC}, (a) the rate flow normalized to its maximum value (blue line in both Figure 10a,b), which is the envelope curve of Figure 9; (b) the chosen combination of (n, m) active pumps (dashed blue line in both Figure 10a,b); (c) the working frequency of the n-pump group normalized to its maximum value (orange line in Figure 10a); (d) the working frequency of the m-pump group normalized to its maximum value (yellow line in Figure 10a); (e) the ratio of P

_{DC}demanded by the n-pump group, normalized to P

_{DC}value (orange line in Figure 10b), i.e., 1 when the n-pump group is the only one operating and consuming all the power, 0.5 when both groups of pumps demand an equal power, and 0 when the n-pump group is stopped; (f) the power not used by the system, P

_{DC,mpp}− P

_{DC}(yellow line in Figure 10b); and the efficiency of each group of pumps (dotted purple and green lines in Figure 10b) in terms of hydraulic power (P

_{H}) regarding the DC power consumed by the group of pumps (P

_{DC,n or m-pumps}).

#### 3.2.3. Subconfiguration 2.2: Two (or More) Groups of Pumps Working at Variable Frequency

_{DC}.

_{n}, at the head H

_{s}, but there is a range of Q

_{n}values since it can work at different frequencies and not only at the nominal frequency.

## 4. Results and Discussion

#### 4.1. Independent PV Generators vs. Shared PV Generator

_{nom1}(hereafter referred to as the base case, Figure 14a). Therefore, to simulate a group of N pumps synchronized at a variable frequency and sharing a PV generator of rated power N × P

_{nom1}(hereafter referred to as the N-case, Figure 14b), the volume pumped simulated in the base case was multiplied by N. It was known that this assumption is somewhat erroneous because, in conditions of low irradiance (early and late morning or cloudy days), the volume pumped in the N case should improve the volume pumped in the base case. It was because, at certain times, the base case PV generator may not generate enough power to activate the pump but N times this power (N case) could be enough to activate a subset of the n pumps, 0 < n < N, even if it was not enough to activate all N pumps.

#### 4.2. Performance vs. Type of Frequency Control

- In control (b), there is a larger range of P
_{DC,mpp}values that cannot be fully exploited, mainly because there is not enough power available yet to start the large pump at its nominal frequency while the small pump is already operating at its nominal frequency. - The efficiency of the converter, wiring, motor, and pump assembly is different for each duty point. In case (c), there are duty points corresponding to two pumps working at different frequencies that provide more total water flow than the other duty points with frequency restrictions (either one pump is forced to nominal frequency or both pumps are forced to equal frequencies). In other words, the higher efficiency of one pump can compensate for the worse efficiency of the other pump and overcome the combination where both pumps must operate at the same efficiency. The final gain is small compared to control (a) but appreciable. This gain improves if the number of pumps is increased. In the literature, some studies can be found in which this gain is neglected and the P
_{DC,mpp}is assumed to be equally distributed between both pumps, perhaps due to the impossibility of predicting a different distribution, as this requires a solution such as the one proposed in this work, using dynamic programming.

#### 4.3. Performance of a Study Case N = 5 and M = 2

_{DC}at any given time. This control (a), therefore, protects more the reliability of the large pumps, which operate at nominal power without a significant loss of productivity, and simplifies the control of both groups of pumps.

^{3}/kWp compared to 28.1 m

^{3}for control (a) (Figure 18a), i.e., 6.4% more. Part of this difference is due to the higher efficiency of the pump at working points in control (b) (dotted blue and red lines in Figure 18), but another part is due to the fact that 1.67% of the available daily energy could not be used because there was no possible working point for the system in control (a) (yellow line at design view in Figure 17), compared to 1.0% of the daily energy not used in control (b), mainly at the beginning and end of the day. In contrast, in high-irradiance conditions, such as day 188 of 365, control (b) (Figure 18d) manages to pump 45.9 m

^{3}/kWp vs. 45.1 m

^{3}/kWp of control (a), only 1.7% more. This better result in pumps synchronized in frequency with respect to some pumps working at nominal frequency coincides with what has been stated by other authors for the case of grid-connected systems [44].

## 5. Conclusions

- Better selecting the large and small pumps in each group to seek greater complementarity in order to minimize the number of pump starts/stops;
- Better withstanding fades in irradiance due to the passage of clouds;
- Better adapting to the climate of the site;
- Taking advantage of all the energy that the PV generator can produce at its maximum power point;
- Limiting the working point of the pumps to efficiency ranges close to their best efficiency point;
- Reducing the number of frequency variators required;
- Combining several of these objectives, sometimes conflicting, as long as the required pumped water flow during the irrigation period can be met.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

DP | Dynamic programming | P_{DC,B} | DC power at point B |

E_{1} | AC energy yield at the motor-pump input | P_{DC,z} | Demanded DC power by a single pump while pumping a water flow Q_{z} at H_{s} |

E_{DC} | DC energy yield | P_{DC,z−1} | Demanded DC power of the rest of z − 1 active pumps, P_{DC,z−1}, pumping the rest of water flow, Q_{s} − Q_{z} at H_{s} |

E_{DC,MPP} | Potential DC energy yield (at MPP) | P_{DC,mpp} | Available PV power |

E_{H} | Hydraulic energy yield | P_{DCmax} | Maximum DC power of the equivalent pump |

f, ω | Frequency | P_{DCmin} | Minimum DC power of the equivalent pump |

f_{z}_{−1}(x) | Cost of the rest of the path, in this case, the demanded DC power of the rest of z − 1 active pumps, P_{DC,z−1}, pumping the rest of water flow, Q_{s} − Q_{z} at H_{s} | P_{HB} | Hydraulic power at point B |

g_{z}(x) | State cost function, for this case, the demanded DC power by a single pump while pumping a water flow Q_{z} at H_{s} | P_{HC} | Hydraulic power at point C |

H | Pumping head | PV | Photovoltaic |

H_{B} | Pumping head at a certain duty point not included in the Q-H curve at nominal frequency | PVIS | Photovoltaic irrigation system |

k_{b0,1}, k_{b1,1}, k_{b2,1} | Parameters that fit the Q-H curve of the individual pump | Q | Water flow |

k_{b0,N}, k_{b1,N}, k_{b2,N} | Parameters that fit the Q-H curve of the equivalent pump of N equal pumps | Q_{1} | Water flow of the individual pump |

k_{p0,1}, k_{p1,1}, k_{p2,1} | Parameters that fit the Q-P_{2} curve of the individual pump | Q_{B} | Water flow at a certain duty point not included in the rated-frequency Q-H curve |

k_{p0,N}, k_{p1,N}, k_{p2,N} | Parameters that fit the Q-P_{2} curve of the equivalent pump of N equal pumps | Q_{min} | Minimum cooling water flow of the pumps |

k_{s0}, k_{s2} | Parameters that fit the system curve | Q_{N} | Water flow of the equivalent pump of N pumps |

${k}_{{z}_{n}0}$$,{k}_{{z}_{n}1}$$,{k}_{{z}_{n}3}$$,{k}_{{z}_{n}4}$ | Parameters that fit the “pseudo system curves” | Q_{s} | Required water flow at the system curve discretized in S samples (0 < Q_{z} < Q_{s}) |

m | Number of active motor pumps in the second group | Q_{s}_{,max} | Maximum required water flow at the system curve discretized in S samples |

M | Number of motor pumps in the second group | S | Number of samples from the system curve |

MP | Motor pump | W_{AC} | AC wiring losses |

n | Number of active motor pumps in the first group | x | Constrained state variable |

N | Number of motor pumps in the first group | Z | Number of equivalent pumps resulting from the possible combinations of N pumps |

P_{1} | Demanded power at the input of the motor | z | Each combination of active pumps |

P_{1B} | P_{1} at point B | η_{FC} | Frequency converter efficiency |

P_{2} | Demanded power at the pump input (1 pump) | η_{PB} | Pump efficiency at point B |

P_{2,N} | Demanded power at the pump input (N pumps) | η_{PC} | Pump efficiency at point C |

P_{2B} | P_{2} at point B | θ | Constant value that depends on the water density and gravity |

P_{AC} | Power at the output of the frequency converter | ω_{max} | Maximum operating frequency of each individual pump |

P_{AC,B} | Power at the output of the frequency converter at point B | ω_{rated} | Rated frequency |

P_{DC} | DC power |

## References

- Palz, W. The French Connection: The rise of the PV water pump. Refocus—Int. Renew. Energy Mag.
**2001**, 2, 46–47. [Google Scholar] - Enochian, R.V. Solar- and Wind-Powered Irrigation Systems; United States Department of Agriculture: Washington, DC, USA, 1982. [Google Scholar]
- Halcrow, W. Small-Scale Solar-Powered Irrigation Pumping Systems—Technical and Economical Review; World Bank: Washington, DC, USA, 1981. [Google Scholar]
- Barlow, R.; McNelis, B.; Derrick, A. Solar Pumping: An Introduction and Update on the Technology, Performance, Costs, and Economics; World Bank Technical Paper Number 168; World Bank: Washington, DC, USA, 1993. [Google Scholar]
- Fedrizzi, M.C.; Sauer, I.L. Bombeamento Solar Fotovoltaico, histórico, características e projetos. In Encontro de Energia no Meio Rural, 4; UNICAMP: Campinas, Brazil, 2002; Available online: http://www.proceedings.scielo.br/scielo.php?script=sci_arttext&pid=MSC0000000022002000100034&lng=en&nrm=iso (accessed on 1 March 2021).
- Van Campen, B.; Guidi, D.; Best, G. Solar Photovoltaics for Sustainable Agriculture and Rural Development; FAO: Rome, Italy, 2000. [Google Scholar]
- Abella, M.A.; Lorenzo, E.; Chenlo, F. PV Water Pumping Systems Based on Standard Frequency Converters. Prog. Photovolt.—Res. Appl.
**2003**, 11, 179–191. [Google Scholar] [CrossRef] - Ghoneim, A.A. Design optimization of photovoltaic powered water pumping systems. Energy Convers. Manag.
**2006**, 47, 1449–1463. [Google Scholar] [CrossRef] - Campana, P.E.; Li, H.; Yan, J. Dynamic modelling of a PV pumping system with special consideration on water demand. Appl. Energy
**2013**, 112, 635–645. [Google Scholar] [CrossRef] [Green Version] - Renu; Bora, B.; Prasad, B.; Sastry, O.; Kumar, K.; Bangar, M. Optimum sizing and performance modeling of Solar Photovoltaic (SPV) water pumps for different climatic conditions. Solar Energy
**2017**, 155, 1326–1338. [Google Scholar] [CrossRef] - Muhsen, D.H.; Ghazali, A.B.; Khatib, T. Multiobjective differential evolution algorithm-based sizing of a standalone photovoltaic water pumping system. Energy Convers. Manag.
**2016**, 118, 32–43. [Google Scholar] [CrossRef] - Bouzidi, B. New sizing method of PV water pumping systems. Sustain. Energy Technol. Assess.
**2013**, 4, 1–10. [Google Scholar] [CrossRef] - Bakelli, Y.; Arab, A.H.; Azouic, B. Optimal sizing of photovoltaic pumping system with water tank storage using LPSP concept. Sol. Energy
**2011**, 85, 288–294. [Google Scholar] [CrossRef] - Muhsen, D.H.; Khatib, T.; Abdulabbas, T.E. Sizing of a standalone photovoltaic water pumping system using hybrid multi-criteria decision making methods. Solar Energy
**2018**, 159, 1003–1015. [Google Scholar] [CrossRef] - Olcan, C. Multi-objective analytical model for optimal sizing of stand-alone photovoltaic water pumping systems. Energy Convers. Manag.
**2015**, 100, 358–369. [Google Scholar] [CrossRef] - Yahyaoui, I.; Atieh, A.; Serna, A.; Tadeo, F. Sensitivity analysis for photovoltaic water pumping systems: Energetic and economic studies. Energy Convers. Manag.
**2017**, 135, 402–415. [Google Scholar] [CrossRef] [Green Version] - Monís, J.I.; López-Luque, R.; Reca, J.; Martínez, J. Multistage Bounded Evolutionary Algorithm to Optimize the Design of Sustainable Photovoltaic (PV) Pumping Irrigation Systems with Storage. Sustainability
**2020**, 12, 1026. [Google Scholar] [CrossRef] [Green Version] - Suehrcke, H.; Appelbaum, J.; Reshef, B. Modelling a permanent magnet DC motor/centrifugal pump assembly in a photovoltaic energy system. Solar Energy
**1997**, 59, 37–42. [Google Scholar] [CrossRef] - Hamidat, A.; Benyoucef, B. Mathematic models of photovoltaic motor-pump systems. Renew. Energy
**2008**, 33, 933–942. [Google Scholar] [CrossRef] - Gherbi, A.D.; Arab, A.H.; Salhi, H. Improvement and validation of PV motor-pump model for PV pumping system performance analysis. Solar Energy
**2017**, 144, 310–320. [Google Scholar] [CrossRef] - Mayer, O.; Baumeister, A.; Festl, T. Design, simulation and diagnosis of photovoltaic pumping systems with DASTPVPS. In Proceedings of the 13th European Photovoltaic Solar Energy Conference, Nice, France, 23–27 October 1995. [Google Scholar]
- Grundfos. Available online: http://de.grundfos.com/grundfos-wincaps.html (accessed on 4 November 2021).
- COMPASS. Available online: https://www.lorentz.de/en/products/submersible-solar-pumps.html (accessed on 4 November 2021).
- SOLARPACK. Available online: http://tools.franklin-electric.com/solar/ (accessed on 4 November 2021).
- Hydraulic Calculation Tool. Available online: http://www.netafim.com/service/hydrocalc-pro (accessed on 4 November 2021).
- GESTAR. SETUP GESTAR. 2016. Available online: http://www.acquanalyst.com/contenido.php?modulo=descargas&cat=2 (accessed on 4 November 2021).
- Estrada, C.; González; Aliod, R.; Paño, J. Improved pressurized pipe network hydraulic solver for applications in irrigation systems. J. Irrig. Drain. Eng.
**2009**, 135, 421–430. [Google Scholar] [CrossRef] - FAO. Land & Water. Available online: http://www.fao.org/nr/water/infores_databases_cropwat.html (accessed on 4 November 2021).
- Langarita, R.; Chóliz, J.S.; Sarasa, C.; Duarte, R.; Jiménez, S. Electricity costs in irrigated agriculture: A case study for an irrigation sheme in Spain. Renew. Sustain. Energy Rev.
**2016**, 68, 1008–1019. [Google Scholar] [CrossRef] - Knecht, R.; Baumgartner, F.P. PV-Battery and Diesel Hybrid System for Irrigation of a Farm in Patagonia. In Proceedings of the 33rd European Photovotaic Solar Energy Conference and Exhibition, Amsterdam, The Netherlands, 25–29 September 2017. [Google Scholar]
- Carrêlo, I.B.; Almeida, R.H.; Narvarte, L.; Martínez-Moreno, F.; Carrasco, L.M. Comparative analysis of the economic feasibility of five large-power. Renew. Energy
**2020**, 145, 2671–2682. [Google Scholar] [CrossRef] - Herraiz, J.I.; Fernández-Ramos, J.; Almeida, R.H.; Báguena, E.; Castillo-Cagigal, M.; Narvarte, L. On the tuning and performance of Stand-Alone Large-Power PV irrigation systems. Energy Convers. Adn Manag. X
**2022**, 13, 100175. [Google Scholar] [CrossRef] - Almeida, R.H.; Carrêlo, I.B.; Lorenzo, E.; Narvarte, L.; Fernández-Ramos, J.; Martinez-Moreno, F.; Carrasco, L.M. Development and Test of Solutions to Enlarge the Power of PV Irrigation and Application to a 140 kW PV-Diesel Representative Case. Energies
**2018**, 11, 3538. [Google Scholar] [CrossRef] [Green Version] - Innovagri. El Riego Solar, una Alternativa para Rentabilizar el Consumo Energético. 2016. Available online: https://www.innovagri.es/comunidad/el-riego-solar-una-alternativa-para-rentabilizar-la-energia.html (accessed on 10 May 2018).
- Energías Renovables. Powen Instala en el Campo de Albacete un Bombeo Solar para Autoconsumo de Casi Doscientos Kilovatios. 2017. Available online: https://www.energias-renovables.com/fotovoltaica/powen-instala-en-el-campo-de-albacete-20171023 (accessed on 10 May 2018).
- EIP Water. European Innovation Partnership Water—Strategic Implementation Plan; European Commission: Brussels, Belgium, 2012. [Google Scholar]
- Ramos, J.F.; Fernandez, L.N.; de Almeida, R.H.T.; Carrêlo, I.B.; Moreno, L.M.C.; Pigueiras, E.L. Method and Control Device for Photovoltaic Pumping Systems. Spain Patent ES 2 607 253 B2, 1 March 2018. [Google Scholar]
- Gasque, M.; González-Alltozano, P.; Gutiérrez-Colomer, R.P.; García-Marí, E. Optimisation of the distribution of power from a photovoltaic generator between two pumps working in parallel. Solar Energy
**2020**, 198, 324–334. [Google Scholar] [CrossRef] - Olszewski, P. Genetic optimization and experimental verification of complex parallel pumping station with centrifugal pumps. Appl. Energy
**2016**, 178, 527–539. [Google Scholar] [CrossRef] - Da Costa Bortoni, E.; de Almeida, R.A.; Carvalho Viana, A.N. Optimization of parallel variable-speed-driven centrifugal pumps operation. Energy Effic.
**2008**, 1, 167–173. [Google Scholar] [CrossRef] - Munoz, J.; Carrillo, J.; Martínez-Moreno, F.; Carrasco, L.; Narvarte, L. Modeling and simulation of large PV pumping systems. In Proceedings of the 31th European Photovoltaic Solar Energy Conference and Exhibition, Hamburg, Germany, 14–18 September 2015. [Google Scholar]
- Bellman, R.E. Dynamic Programming; Princeton University Press: Princeton, NJ, USA, 1957. [Google Scholar]
- Casti, R.L.a.J. Principles of Dynamic Programming, Part I; Dekker: New York, NY, USA, 1979. [Google Scholar]
- Viholainen, J.; Tamminen, J.; Ahonen, T.; Ahola, J.; Vakkilainen, E.; Soukka, R. Energy-efficient control strategy for variable speed-driven parallel pumping systems. Energy Effic.
**2013**, 6, 495–509. [Google Scholar] [CrossRef]

**Figure 1.**Configuration of a PV pumping system: PV generator, frequency converter, centrifugal motor pump, and water pool.

**Figure 3.**Model consisting of two groups of pumps working in parallel and sharing the same PV generator.

**Figure 4.**Characteristic curves of an equivalent pump corresponding to N equal and frequency-synchronized pumps, where N = 1..5.

**Figure 5.**Illustration of the procedure to calculate the P

_{2}required by the equivalent pump working at a certain point (Q

_{B}, H

_{B}). Point A would be the operating point on the system curve if the pump were operating at the rated frequency, ${\omega}_{\mathrm{rated}}$. Point B is the actual operating point at a lower frequency. The pump works at the same efficiency at point B as at point C, therefore allowing P

_{2}at point B to be calculated.

**Figure 6.**P

_{DC}-Q curves of each of the equivalent pumps of a group of five equal and synchronized pumps.

**Figure 7.**For the case N = 2 and M = 1, Q-H curves and the working point when n = 2 and m = 1 active pumps at H

_{s}.

**Figure 8.**Q-H curves and working points for the case N = 2 and M = 1 and the Q-H locus where the second group of variable frequency pumps is required to work for each combination of n active pumps in the first group.

**Figure 10.**For the case N = 2 and M = 2, combination of pumps that maximizes the water flow pumped depending on available P

_{DC}, as well as the frequency at which the two groups of pumps will work; (

**a**) operation view—working frequencies, (

**b**) design view—efficiencies, power balance, not used P

_{DC}.

**Figure 11.**Q-H curves and working points for the case N = 2 and M = 1, and the Q-H “pseudo system area” where the second group of variable frequency pumps is required to work for each combination of n active pumps working at variable synchronized frequency in the first group.

**Figure 12.**For the case N = 2 and M = 2, combination of pumps that maximizes the water flow pumped for a certain available P

_{DC}, as well as the frequency of the two groups of pumps; (

**a**) operation view—working frequencies, (

**b**) design view—efficiencies, power balance, non-useful P

_{DC}.

**Figure 13.**H-Q and P

_{DC}-Q characteristic curves—(

**a**) and (

**b**), respectively—for each of the equivalent pumps for N = 5 and M = 2.

**Figure 14.**(

**a**) A single pump connected to a single PV generator of P

_{nom1}. (

**b**) Group of N pumps sharing a single PV generator of N × P

_{nom1}.

**Figure 15.**Relative gain of the volume pumped in the N case with respect to the base case with and without resizing the hydraulic piping system.

**Figure 16.**Three possible frequency controls under comparison: (

**a**) both pumps work synchronized at a variable frequency, (

**b**) one pump works at a constant (nominal) frequency, and the second pump works at a variable frequency, and (

**c**) one pump works at a variable frequency, and the second pump also works at a variable frequency, which can be different.

**Figure 17.**Combination of pumps that maximizes the water flow pumped for a certain available P

_{DC}, as well as the frequency at which the two groups of pumps will work in control (

**a**)—n pumps at nominal frequency and m pumps at variable frequency; and in control (

**b**)—n pumps at a variable frequency and m pumps at other variable frequency.

**Figure 18.**Cloudy and clear day. Case N = 5 and M = 2. (As there are two groups of different pump types, the best combination (with the chosen criteria) can include 0 or more active pumps in each group. A line disappears (discontinuous effect) in those combinations where there are 0 active pumps in one of the groups).

Frequency Control Type | |||
---|---|---|---|

Control (a) N = 2 (Variable) | Control (b) N = 1 (Nominal) M = 1 (Variable) | Control (c) N = 1 (Variable) M = 1 (Variable) | |

Potential DC energy yield at MPP, E_{DC,MPP} (kWh/kWp) | 2057.9 | 2057.9 | 2057.9 |

DC energy yield, E_{DC} (kWh/kWp) | 1936.3 | 1924.0 | 2036.9 |

E_{DC}/E_{DC,MPP} ratio | 0.950 | 0.935 | 0.990 |

AC energy yield at the motor-pump input, E_{1} (kWh/kWp) | 1859.7 | 1747.7 | 1860.5 |

Mechanical energy yield at the pump input, E_{2} (kWh/kWp) | 1690.0 | 1567.1 | 1690.7 |

Hydraulic energy yield, E_{H} (kWh/kWp) | 1439.5 | 1326.2 | 1440.1 |

Annual water volume yield (m^{3}/kWp) | 9868 | 9141 | 9872 |

Control (a) | Control (b) | |
---|---|---|

Potential DC energy yield (at MPP), E_{DC,MPP} (kWh/kWp) | 2057.9 | 2057.9 |

DC energy yield (MPP), E_{DC} (kWh/kWp) | 1990.3 | 2000.1 |

E_{DC,MPP/}E_{DC} ratio | 0.967 | 0.972 |

AC energy yield at the motor-pump input, E_{1} (kWh/kWp) | 1922.5 | 1936.2 |

Hydraulic energy yield, E_{H} (kWh/kWp) | 1553.1 | 1586.3 |

Annual water volume yield (m^{3}/kWp) | 10,741 | 10,962 |

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**MDPI and ACS Style**

Ledesma, J.R.; Almeida, R.H.; Narvarte, L.
Modeling and Simulation of Multipumping Photovoltaic Irrigation Systems. *Sustainability* **2022**, *14*, 9318.
https://doi.org/10.3390/su14159318

**AMA Style**

Ledesma JR, Almeida RH, Narvarte L.
Modeling and Simulation of Multipumping Photovoltaic Irrigation Systems. *Sustainability*. 2022; 14(15):9318.
https://doi.org/10.3390/su14159318

**Chicago/Turabian Style**

Ledesma, Javier R., Rita H. Almeida, and Luis Narvarte.
2022. "Modeling and Simulation of Multipumping Photovoltaic Irrigation Systems" *Sustainability* 14, no. 15: 9318.
https://doi.org/10.3390/su14159318