Estimation of Centrifugal Pump Efficiency at Variable Frequency for Irrigation Systems

Abstract

The sustainability of the food production achieved with the help of irrigation systems and the sustainability of their energy consumption are major challenges of the current century. Pumping systems currently account for approximately 30% of global electrical energy consumption. As electricity prices rise, there is a growing need for technological advancements to enhance energy efficiency and reduce consumption costs effectively. This study focuses on operating centrifugal pumps at variable frequency as an effective means of achieving this goal. Most centrifugal pump manufacturers/providers traditionally assume that pump efficiency remains constant across various operating frequencies, often equating the efficiency at various frequencies to that at the standard frequency (50/60 Hz). In contrast, this paper introduces a new formula for estimating pump efficiency, crucial to precise power consumption determination, particularly in variable-frequency scenarios. The formula parameters are identified by using experimental data acquired from an existing pumping system. The tests and results presented in this paper demonstrate that the proposed formula outperforms the formulas of the current industry standards in accuracy. Practically, the new relation assures enhanced accuracy in estimating pump efficiency and absorbed power, allowing for the design of a more precise model used for control systems synthesis required for operating centrifugal pumps at variable frequency. This research offers a new way of calculating pump efficiency, which could be very useful for industry practitioners seeking to optimize energy consumption in pumping systems.

1. Introduction

Population growth, global warming, and the energy crisis have led the sustainability of the food production achieved with the help of irrigation systems and the sustainability of their energy consumption to become major challenges of the new millennium. At the same time, the use of renewable energy sources (photovoltaics and wind) to power such systems, which consume a lot of energy, contributes decisively to ensuring energy sustainability, which is so necessary in the context of the continuous increase in world energy consumption.

Pumping systems play a vital role in pressurizing and transporting water from a source to consumers and are extensively used in various industries, such as urban water supply networks, agricultural irrigation, and power plants [1]. Presently, pumping systems consume approximately 30% of the global electrical energy produced [2]. In the European Union, pumping systems represent around 20% of industrial and agricultural electrical energy consumption, while in China, they account for 40% of the generated electrical energy [3,4,5,6]. Rising electricity prices in recent years underscore the need for technological advancements, specifically in enhancing energy efficiency. Operating centrifugal pumps at variable frequency emerges as an effective strategy.

Traditionally, centrifugal pump manufacturers/providers assume that pump efficiency remains constant across various operating frequencies, often equating efficiency at various frequencies to that at the standard one (50/60 Hz) [7,8]. In contrast, the current work introduces a new formula for estimating pump efficiency, crucial to precise power consumption determination, particularly in variable-frequency scenarios. The proposed formula’s parameters are identified by using system identification techniques applied to experimental data acquired from an existing pumping system. The current work commences with a brief description of the considered pumping system and its main components, followed by the presentation of the proposed data acquisition and logging system in Section 2. Section 3 outlines the current and proposed relations used in determining centrifugal pump efficiency at variable frequency. Section 4 provides a comparison between the current and proposed relations with the experimental data of centrifugal pump efficiency. Section 5 details the impact of each formula on the electrical power consumption of the induction motor. Section 5 presents the conclusions.

The main objective and contribution of this paper consist in proposing a new formula, whose parameters are identified by using system identification techniques applied to experimental data collected with the proposed data acquisition and logging system on the centrifugal pump in question. This new formula provides better accuracy in estimating the efficiency of centrifugal pumps and the power absorbed by their induction motors at variable frequency than the ones currently used and provided by most pump manufacturers/providers. It also allows for a more precise model for synthesizing control structures required for operating centrifugal pumps at variable frequency.

2. Experimental Data

Case studies, conducted through simulation in Matlab and Simulink (R2021b), involve an existing industrial high-power pumping system (layout presented in Figure 1). The proposed formula ensures good accuracy and is recommended for modeling, simulating, and synthesizing control structures of pumping system configurations with induction motors driven by Variable-Frequency Drives (VFDs) at various frequencies. In most cases, irrigation systems are installed in regions where they can be connected to the national electricity grid. However, in remote areas where grid access is unavailable, these systems are currently based on diesel or gasoline generators for power. Anticipating that many irrigation system owners in such remote locations would prefer to transition to renewable energy sources (such as wind or photovoltaic generators) due to their sustainability and economic benefits, this research aims to precisely determine the power consumption of irrigation systems when powered by intermittent and variable energy sources, such us the renewable ones, by determining pump efficiency at various frequencies. Electric power supply systems based on renewable energy sources allow for a control strategy by varying the frequency of supply voltage.

The considered pumping system (used for irrigation) is located in the Aragon region in Spain [9]. Figure 1 presents its layout and main components, including a VFD that drives the induction motor of a submersible pump to collect water from a borehole and discharge it into a hydraulic distribution network.

The centrifugal pump [10] in this study is a submersible pump collecting water from a well and delivering it directly to an irrigation hydraulic distribution network at different operating points. Technical specifications and operating points are detailed in

Table A1. Centrifugal pump technical specifications [9,10].

Technical Property	Submersible Pump	Symbol	Unit
Manufacturer	Caprari	-	-
Type	E8P95/7ZC	-	-
Minimum water speed to cool the jacket of the motor	0.5 m/s	-	-
Maximum number of starts in one hour	20	-	-
Minimum immersion depth	507.5 m	-	-
Pump service flow rate	91.85 m3/h	Q	m3/h
Submersible pump service head	78.12 m	H	m
Pump rated efficiency	75.63%	η	%
Pump rated hydraulic power	25.84 kW	Phydraulic	kW
Pump maximum flow rate	169.2 m3/h	Qmax	m3/h
Pump head at theoretical 0 flow rate	97.35 m	Hmax	m
Pump minimum head (at maximum flow rate)	83.29 m	Hmin	m
Pump efficiency at maximum flow rate	81.5%	η	%
Pump hydraulic power at maximum flow rate	30 kW	Ppump	m3/h
Head–flow 1st coefficient	185.2123	A	m
Head–flow 2nd coefficient	0.2608	B	h/m2
Head–flow 3rd coefficient	−0.0076	C	h/m5
Efficiency–flow 1st coefficient	1.8274	D	1/m2
Efficiency–flow 2nd coefficient	−0.104	E	1/m5

in Appendix A [10]. The variable-speed centrifugal pump is installed below the lowest water level at a depth of approximately 40 m (PUMP_DEPTH ≈ 40 m below the ground level in Figure 1), extracting water at a depth of approximately 30.5 m (HG_WATER_SOURCE ≈ 30.5 m, the dynamic level in Figure 1). The pump is capable of discharging flow between 45 m³/h and 95 m³/h.

For further technical specifications and operating points, please refer to Table A1 in Appendix A [10].

A non-intrusive data acquisition and logging system capable of evaluating nominal and transitory regimes while detecting hydraulic shocks was developed and implemented to collect operational data from the pumping system. The system layout is presented in Figure 2. The data logger [11] communicates with the Variable-Frequency Drive (VFD) through Modbus RS485 serial communication. The data acquisition and logging system measures and records every second parameters such as input and output power, voltage, current, pressure, volume flow, and water temperature.

2.1. Pressure Transducer

The pressure transducer [12] measures the relative pressure within a range of −1 to 10 bars, providing an analog signal (4–20 mA) to the data logger [11]. Technical specifications are detailed in

Table 1. Pressure transducer technical specifications.

Property	Value	Unit
Pressure measuring range	−1 to 10	bar
Pressure resistance	25	bar
Bursting pressure	300	bar
Operating voltage temperature	−20 to + 50	°C
Ambient temperature	−40 to 90	°C
Storage temperature	−40 to 100	°C
Pressure cycles (min.) across lifetime	60 million times for 1.2 × nominal pressure
Shock resistance	50 g (DIN EN 60068-2-27 [13]; 11 ms)
Vibration resistance	20 g (DIN EN 60068-2-6 [14]; 10–2000 Hz)

.

2.2. Volume Flow Transducer

The volume flow transducer [15] measures the pumped volume flow, sending frequency signals through impulses to the data logger [11] (each impulse represents 100 L). Technical specifications are outlined in

Table 2. Volume flow transducer technical specifications.

Property	Value	Unit
Nominal flow	60	m3/h
Nominal diameter	100	mm
Maximum flow (short duration)	250	m3/h
Maximum flow (permanent duration)	125	m3/h
Minimum flow	1.5	m3/h
Pressure loss at maximum permanent flow	0.2	bar
Maximum operating temperature	50	°C
Maximum admissible pressure	16	bar

.

2.3. Temperature Sensor

The temperature sensor PT100 [16] was employed for measuring water temperature. It sends an analog signal (4–20 mA) to the data logger [11]. Technical specifications can be found in

Table 3. Temperature sensor technical specifications.

Property	Value	Unit
Temperature measuring range	−50 to +230	°C
Temperature measuring resolution	±0.5	°C
Measuring current	max. 1 mA (no self-heating)

.

2.4. Data Logger

The data logger [11] stores collected data every second from analog (temperature sensor and pressure transducer) and frequency (flow transducer) sensors, as well as the Variable-Frequency Drive (induction motor frequency, angular velocity, voltage, power, and current consumption) through Modbus RTU RS485 communication. Technical specifications are presented in

Table 4. Data logger technical specifications.

Property	Value	Unit
Supply voltage range	5 to 30 V DC	VDC
Frequency channels (pulse counters)	Up to 16 (10 physical plus 6 extra)
Analog channels (voltage)	Up to 23 (15 physical plus 8 extra)
5 V outputs	10 (1 for every physical frequency channel)
Modbus RS485 (data logger as Master)	3	-
Modbus RS485 (data logger as Slave)	1	-
Modbus TCP (data logger as Slave)	1	-
Ethernet port	1	-
Connectivity	Global remote access—2G/3G/4G Modem
Collected data	1 s data storage Averaging each 1, 5, and 10 min

. The experimental measured data are presented in Figure 3.

The data acquisition and logging system was utilized to collect experimental data over one day from an existing pumping system [9] operating at different regimes. Figure 3 depicts the variation in electrical power, induction motor frequency, pumping head, and centrifugal pump discharge during the experiment. Regarding the accuracy of measuring parameter instruments/devices in the experimental setup, multiple measurements were carried out over a long period of time, and the paper presenting the results for only one day (see Figure 3). The paper mentions that it is about an industrial irrigation system, operating under real conditions (in the Aragon region in Spain). By using some industrial sensors with specified guaranteed accuracy (temperature sensor [11], pressure transducer [12], flow transducer, etc.), a Modbus RTU RS485 communication module, and a data logger (their technical parameters are presented in Table 1, Table 2, Table 3 and Table 4), the multiple tests performed guarantee the accuracy of the measurements in the performed experiments. The data logger system (data acquisition and storage) was implemented and tested under real industrial conditions, thus proving its accuracy.

Figure 3a presents the variation in the electrical power that the induction motor absorbed during the experiment. The maximum electrical power that the induction motor absorbed was a bit over 35 kW.

Figure 3b depicts the variation in the pumping system induction motor’s electrical frequency (lying between 0 and 41 Hz). According to this figure, the irrigation system worked from 5 a.m. to 8 p.m.

Figure 3c illustrates the pumping head (pressure) variation during the experiment (varying between 35 and 75 mH₂O).

Figure 3d presents the current centrifugal pump discharge (flow) and the filtered value (lying between 50 and 90 m³/h).

The efficiency of the induction motor (η_IM) is estimated by using system identification techniques in Matlab, considering the customized exponential regression from Equation (1) based on the 5 efficiency points given by the manufacturer (see

Table A2. Induction motor technical specifications [9,10].

Technical Property	Induction Motor	Symbol	Unit
Manufacturer	Caprari	-	-
Type	MAC870-8V	-	-
Nominal power	51	PIM0	kW
Nominal efficiency (51 kW)	85.9%	ηIM0	%
Efficiency at 54.72 kW	86%	ηIM1	%
Efficiency at 48.3 kW	86.2%	ηIM2	%
Efficiency at 31.19 kW	86.5%	ηIM3	%
Efficiency at 27.96 kW	85.5%	ηIM4	%
First coefficient of the efficiency regression	−0.863	aIM	-
Second coefficient of the efficiency regression	0.173	bIM	-
Third coefficient of the efficiency regression	0.863	cIM	-
Nominal frequency	50	fIM	Hz
Nominal voltage	400	VIM	V
Nominal current	101.1	IIM	A
Number of poles	2	Poles	-
Rotor synchronous speed	3000	ωs	rpm
Rotor nominal speed	2910	ω0	rpm
Rotor operating speed	Variable	ω	rpm
Power factor	0.845	cosφ	-

in Appendix A) [10]:

$${\eta }_{IM}=a_{IM}·e^{-b_{IM}·P_{IM}}+c_{IM}$$

(1)

where η_IM represents the induction motor efficiency; P_IM represents the induction motor power; and a_IM = −0.863, b_IM = 0.173, and c_IM = 0.863 are the efficiency exponential regression coefficients.

The shaft power of the induction motor (P_shaft) is estimated by using Equation (2) [17]:

$$P_{shaft}=P_{IM}·{\eta }_{IM}$$

(2)

Additionally, the hydraulic power (P_hydraulic) of the centrifugal pump is estimated by using Equation (3), based on the pumping head (H_s) and discharge (Q_s) of the centrifugal pump [18]:

$$P_{hydraulic}=\rho ·g·H_s·Q_s$$

(3)

where g represents the gravitational acceleration and $\rho$ is the water density.

Figure 4a estimates the induction motor efficiency (η_IM) obtained through system identification techniques in Matlab (R2021b) and by utilizing the customized exponential regression from Equation (1). Figure 4b illustrates the variation in induction motor power, along with the estimated shaft and hydraulic power.

Figure 5 estimates the efficiency of the centrifugal pump (η_pump) by dividing the hydraulic power by the induction motor shaft power, as determined by Equation (4) [19].

$${\eta }_{pump}=P_{hydraulic}/P_{shaft}$$

(4)

3. Centrifugal Pump Efficiency Formulas

The centrifugal pump is modelled based on the A, B, C, D, and E coefficients from

Table 5. The 7 operating points of the variable-speed centrifugal pump used for identifying the coefficients of the pump head–flow and efficiency–flow equations.

50 Hz Curve Fitting Point	Q [m³/h]	H [m]	P [kW]	η [%]	NPSH [m]
1	0	185	0	0	0
2	54	177	36.8	68.1	1.8
3	69.84	166	41.19	76.7	2.37
4	85.68	152	45.11	80	3.22
5	101.52	133.5	48.09	78	4.28
6	117.36	111	50.01	71	5.6
7	133.2	85	50.03	58.35	7.2

. These coefficients are determined through quadratic regression performed on seven operating points found on the pump head–flow (H₀–Q₀) and efficiency–flow (η₀–Q₀) curves at 50 Hz provided by the manufacturer (see Figure 5) by using Equations (5) and (6), where the subscript “1” stands for the nominal values and the subscript “2” stands for the current values.

$$H_1=A+B·Q_1+C·{\left(Q_1\right)}^2$$

(5)

$${\eta }_1=D·Q_1+E·{\left(Q_1\right)}^2$$

(6)

where H₁ represents the nominal pumping head, Q₁ is the nominal pumping flow, and η₁ is the nominal pump efficiency.

In Equation (7), α signifies the ratio between the pump current speed (ω₂) and nominal speed (ω₁), providing a means to account for variations in the pump operating speed:

$$\alpha ={\omega }_2/{\omega }_1$$

(7)

The response of the centrifugal pump to various speeds is modelled by using the pump affinity laws [20]. These laws provide insights into the variations in pump flow (relation (8)), head (relation (9)), and power (relation (10)) as the pump speed changes:

$$\frac{{\mathrm{Q}}_2}{Q_1}=\alpha ·{\left(\frac{G_2}{G_1}\right)}^3$$

(8)

$$\frac{H_2}{H_1}={\alpha }^2·{\left(\frac{G_2}{G_1}\right)}^2$$

(9)

$$\frac{P_2}{P_1}=\left(\frac{{\rho }_2}{{\rho }_1}\right)·{\alpha }^3·{\left(\frac{G_2}{G_1}\right)}^5$$

(10)

where Q₁, H₁, and P₁ represent the nominal pump flow, head, and power, respectively, at nominal speed ω₁. The corresponding parameters Q₁, H₁, and P₁ represent the pump characteristics at the current speed, ω₂. The change in water density (ρ₂) and pump geometry (G₂) during the frequency variation are considered constant (ρ₁ = ρ₂, G₁ = G₂).

By combining the pump head–flow (H₀–Q₀) curve at nominal speed (ω₀), as defined by Equation (5), with the pump affinity laws, Equations (8)–(10), the pump characteristic head–flow curve (H–Q) is expressed as a function of pump speed variation in Equation (11):

$$H_2={\alpha }^2·A+\alpha ·B·Q_2+C·{\left(Q_2\right)}^2$$

(11)

Figure 6 illustrates the pump characteristic head–flow (H–Q) curves, where the x-axis represents the pumping head in meters and the y-axis represents the pump discharge in m³/h. These curves are depicted at different electrical frequencies (ranging from 5 to 50 Hz), equivalent to variations in pump speed.

3.1. Current Efficiency Formula Used by Manufacturers

Most centrifugal pump manufacturers/providers, as noted in reference [7,8], employ Equation (12) for estimating centrifugal pump efficiency at frequencies other than the nominal one, 50/60 Hz:

$${\eta }_2=D·\frac{Q_2}{\alpha }+E·\frac{{\left(Q_2\right)}^2}{{\alpha }^2}$$

(12)

where ${\eta }_2$ represents the current pump efficiency and Q₂ denotes the current pump discharge (pumping flow).

However, according to the pump affinity laws (Equation (13)), the pump flow at a frequency different from the nominal one (50/60 Hz) can be calculated as

$$Q_2=\alpha ·Q_1$$

(13)

By substituting the pump flow at different frequencies into Equation (13), it becomes apparent (Equation (14)) that according to most centrifugal pump manufacturers/providers, pump efficiency is considered constant:

$${\eta }_2=D·\frac{\sout{\alpha ·}{\sout{\lambda }}^{\sout{3}}·Q_1}{\sout{\alpha ·}{\sout{\lambda }}^{\sout{3}}}+E·\frac{{\left(\sout{\alpha ·}{\sout{\lambda }}^{\sout{3}}·Q_1\right)}^2}{{\sout{\alpha }}^{\sout{2}}·{\sout{\lambda }}^{\sout{6}}}=D·Q_1+E·{\left(Q_1\right)}^2={\eta }_1$$

(14)

This observation aligns with statements from centrifugal pump manufacturers/providers, as documented in their reports [8], asserting that pump efficiency remains constant across various operating frequencies.

3.2. Anderson’s Efficiency Formula

Based on thousands of pump tests, Anderson recommends an empirical correlation (Equation (15)) for estimating pump efficiency at different flows [20]:

$$\frac{0.94-{\eta }_2}{0.94-{\eta }_1}={\left(\frac{Q_1}{Q_2}\right)}^{0.32}$$

(15)

In Equation (15), Anderson’s formula acknowledges a practical observation: even in an infinitely large pump, losses exist. Anderson proposes a maximum possible efficiency of 94 percent rather than 100 percent.

By denoting coefficients a₁ = 0.94 and b₁ = 0.32 from Anderson’s empirical correlation and substituting the division between the current (Q₂) and nominal (Q₁) flows with α as defined in Equation (14), Anderson’s empirical correlation can be expressed in the form of Equation (16).

$$\frac{a_1-{\eta }_2}{a_1-{\eta }_1}={\left(\frac{1}{\alpha }\right)}^{b_1}$$

(16)

This formulation considers the observed maximum efficiency, providing a practical model for estimating pump efficiency at various flows.

3.3. Sârbu’s Efficiency Formula

Similar to Anderson, Sârbu proposes a comparable relation (Equation (17)) with a key distinction; Sârbu considers pump efficiency variation in response to pump speed changes [21]:

$${\eta }_2=1-\left(1-{\eta }_1\right){\left(\frac{{\omega }_1}{{\omega }_2}\right)}^{0.1}$$

(17)

Which brought to the form of Anderson’s relation, leads to Equation (18):

$$\frac{1-{\eta }_2}{1-{\eta }_1}={\left(\frac{{\omega }_1}{{\omega }_2}\right)}^{0.1}$$

(18)

By denoting as a₂ = 1 and b₂ = 0.1 the coefficients from Sarbu’s formula and by substituting the division between the current (ω₂) and nominal (ω₁) speed with α as defined in Equation (7), Sarbu’s formula can be expressed in the form of Equation (19):

$$\frac{a_2-{\eta }_2}{a_2-{\eta }_1}={\left(\frac{1}{\alpha }\right)}^{b_2}$$

(19)

Sârbu’s formula considers the influence of pump speed variation on pump efficiency, providing a practical model for estimating efficiency with changing operational speeds.

3.4. Proposed Efficiency Formula

Considering the similarities in the forms of both Anderson’s efficiency formula (Equation (16)) and Sarbu’s efficiency formula (Equation (19)), this work introduces a unified approach for estimating pump efficiency (η₂) at frequencies different from the nominal one. The proposed formula is developed by using system identification techniques in Matlab, taking into account customized regression on experimental data:

$$\frac{a-{\eta }_2}{a-{\eta }_1}={\left(\frac{1}{\alpha }\right)}^b$$

(20)

where η₁ represents the pump efficiency at nominal frequency, α denotes the ratio between the pump current frequency or speed (ω₂) and nominal frequency or speed (ω₁), and the coefficients a = 0.99 and b = 0.8 are the efficiency regression coefficients. These coefficients were identified through system identification techniques in Matlab, ensuring a robust and accurate representation of pump efficiency at various frequencies. The proposed formula (Equation (20)) aims to provide a unified and versatile model for estimating pump efficiency under different operational conditions, offering improved accuracy and applicability.

4. Results

Figure 7 presents a comprehensive comparison between the real (measured) pump efficiency (depicted by blue dots, with each point representing a measured value) and the estimated efficiency derived from various methods, i.e., the approach commonly used by most pump manufacturers/providers (illustrated by the red line), Anderson’s formula (depicted by the yellow line), Sârbu’s formula (shown by the purple line), and the proposed formula (represented by the green line).

Figure 7a showcases the efficiency estimates throughout the day of the experiment, while Figure 7b provides insights into efficiency variations concerning frequency changes.

As can be seen in Figure 7a, when the pumping (irrigation) system is stopped, from midnight to 5 a.m. and after 8 p.m., we cannot speak of pump efficiency, and obviously, this is considered zero. Of all the formulas that estimate pump efficiency, the proposed formula provides both the closest absolute values and the closest dynamic to the real one (estimated based on experimental measurements), followed by Anderson’s formula, then Sârbu’s formula and finally the formula used by manufacturers.

In Figure 7b, depicting the pump efficiency variation according to the frequency variation, it can be seen even better that the values estimated with the proposed formula are much closer to the real values (estimated based on experimental measurements) than those of the other formulas considered in this study.

The comparative analysis reveals that the efficiency estimated with the proposed formula demonstrates the highest accuracy, closely followed by Anderson’s formula and then Sârbu’s formula. In contrast, the least accurate estimation is observed with the method commonly used by most pump manufacturers/providers.

Beyond offering only superior accuracy in estimating centrifugal pump efficiency, the proposed formula stands out by providing a more precise model, which can used to design and optimize modern control structures [22] usable in the most performant manner for operating centrifugal pumps at variable frequency.

Given the recent surge in electricity prices and the substantial share of global electrical energy consumed by pumping systems, precise estimation of pumping system power consumption is imperative for optimizing operational efficiency [2]. In light of this, Figure 8a presents a comparative analysis between the measured induction motor power (represented by blue dots) and the estimated power according to Equation (21). These estimations are based on the centrifugal pump efficiency calculated by different approaches, i.e., the method commonly used by most pump manufacturers/providers (depicted by the red line), Anderson’s formula (illustrated by the yellow line), Sârbu’s formula (shown by the purple line), and the proposed formula (represented by the green line).

$$P_{IM}=P_{hydraulic}/{\eta }_{pump}/{\eta }_{IM}$$

(21)

where η_IM represents the induction motor efficiency, P_IM represents the induction motor power, P_hydraulic represents the hydraulic power, and η_Ipump represents the pump efficiency estimated with each relation approached in this work.

Figure 8a illustrates that the induction motor power estimated with the proposed formula for pump efficiency closely aligns with measured values, especially at operating power close to nominal power and significantly so at power levels much lower than the nominal one.

In Figure 8b, the error analysis reveals that in the most unfavorable case, the error of the induction motor power estimated with the proposed formula is approximately 30%. In contrast, Anderson’s formula exhibits around a 45% error and Sârbu’s formula around 60%, and the least accurate estimation is observed with the method commonly used by most pump manufacturers/providers, with an error of around 75%.

The measurements and tests (not all presented here) were carried out during a limited period, practically a few months (the duration of the irrigation season). Almost surely, the wear of the centrifugal pump for a long-term operation regime affects pump efficiency. The proposed formula does not consider this possible scenario, with measurements being necessary over a period of years. However, the structure (shape) of the proposed formula remains the same, with only the new identification of values of the two parameters a and b being necessary. Compared with the current industry standard formulas [23], the new formula (with a generally valid structure) provides better accuracy in estimating pump efficiency, considering operation in variable-frequency regimes, which is little taken into account in the industrial standards that consider constant frequency. In practice, a more accurate estimation of pump efficiency allows for the design of more efficient control systems, especially in the case of irrigation systems powered by variable energy sources, such as photovoltaic energy.

The main contribution of the proposed formula consists in its structure/shape, which is generally valid for any pump, under the conditions of identifying new values for parameters a and b (based on new experimental measurements). Its generality lies in this feature. For the considered industrial pumping system (implemented in Aragon, Spain), part of the experimentally measured data set was used to identify the two parameters of the proposed formula and another part to validate the formula. The pump efficiency estimation accuracy provided by the new formula proves to be superior to the results provided by standard (conventional) formulas, especially in the case of irrigation systems powered by variable energy sources operating at variable frequency.

5. Conclusions

As stated at the beginning of this research paper, ensuring the energy sustainability of the pumping systems used in agricultural irrigation constitutes a guarantee of sustainable food production for a continuously growing population. In this study, we addressed the critical issue of energy efficiency in pumping systems, which currently account for a substantial 30% of global electrical energy consumption. As electricity prices continue to rise, optimizing pumping system operation becomes imperative, necessitating precise estimations of power consumption.

Our investigation focused on the efficiency of centrifugal pumps when operated at variable frequency, challenging the conventional assumption that pump efficiency remains constant across different operating frequencies. Traditional mathematical relations, as employed by most pump manufacturers/providers, often fall short in accurately estimating efficiency at variable frequency, leading to suboptimal system performance.

Introducing a new formula for estimating pump efficiency, developed through system identification techniques applied to experimental data, proved to be a significant advancement. The proposed formula, with the identified coefficients, demonstrated superior accuracy compared with industry standards. The comparison, encompassing Anderson’s and Sârbu’s formulas, highlighted the effectiveness of our proposed model.

The comparison between measured and estimated induction motor power further validated the efficacy of our approach. The new proposed formula consistently outperformed other methods, exhibiting minimal error even at lower power levels, where accurate estimation is crucial. The main novelty and benefit of this research consist in a more accurate estimation of the useful hydraulic power delivered to the fluid, allowing for the design of more efficient control systems for the irrigation process.

The current work not only contributes to advancing the understanding of pump efficiency in variable-frequency operation but also offers a practical tool for industry practitioners seeking to optimize energy consumption in pumping systems or for determining the new efficiency of a pumping system whose operating point has changed due to the wear and tear caused by intense use. The innovation of the new (proposed) formula consists in estimating with better accuracy the efficiency of a centrifugal pump operating at variable frequency, allowing for the design of more efficient control systems for irrigation.

As we navigate an era of increasing energy demand and environmental consciousness, optimizing the efficiency of pumping systems becomes a crucial step towards sustainable resource management. The insights gained from this study lay the foundation for future advancements in the field, promoting energy-efficient practices for pumping systems worldwide.