Quantifying the Energetic Cost Tradeoffs of Photovoltaic Pumping Systems for Sub-Saharan African Smallholder Farms

As solar technologies have matured, irrigation using photovoltaic pumping systems (PVPSs) has gained popularity in developing markets as an eﬀective means to alleviate poverty and increase food security. Yet there remains a barrier to adoption; the upfront cost of PVPS poses a ﬁnancial burden for many low-income farmers. In a PVPS, the capital cost of the solar array contributes a large portion of upfront system costs. The solar pump is the largest energy consumer in the system, thus its eﬃciency directly impacts the size and cost of the solar array. There is limited quantitative understanding about how solar pump eﬃciency aﬀects the capital cost of the solar array. This study presents a technoeconomic framework to directly quantify the impact of solar pump eﬃciency on the cost of the solar array in a PVPS for a range of hydraulic operating conditions. New empirical eﬃciency scaling laws were created by characterizing the eﬃ-ciencies of 4-inch multistage centrifugal borehole pumps and induction motors. The utility of the technoeconomic framework is demonstrated through a case study comparing solar pump architectures with motors of diﬀerent eﬃciencies. Results indicate that, despite increased motor cost, the use of high eﬃciency motors in solar pumps may lead to an overall cost reduction in a PVPS. Counter to the conventional capital cost-driven process, this work demonstrates that an eﬃciency-driven design process


Introduction
There has been a growing interest in providing low-cost photovoltaic pumping systems (PVPSs) to increase reliable water access to smallholder farmers in Sub-Saharan Africa (SSA) (Schmitter et al., 2018).In SSA alone, there are an estimated 50 million smallholder farmers who collectively produce more than 80% of the food for the region (Lowder et al., 2016;IFAD, 2013).Studies have shown that increasing reliable water access to smallholder farmers is an effective tool to alleviate poverty and strengthen food security (Burney and Naylor, 2012;Giordano and de Fraiture, 2014).With the abundance of groundwater at shallow depths, SSA is suitable for installing electric groundwater pumps to provide reliable water access and improve the livelihood of rural households (MacDonald et al., 2012;Pavelic et al., 2013).However, many of the smallholder farmers are off-grid, forcing them to rely on inefficient diesel-powered pumps with high recurring fuel costs (Closas and Rap, 2017).Solar panel prices have declined rapidly in recent years, falling from 21 USD/W p in 1992 to 0.81 USD/W p as of 2019 (Karekezi and Kithyoma, 2002;Coalition Energy for Access, 2019).The drop in solar panel prices makes solar-electric systems more cost-competitive with diesel-powered systems because the lifetime cost of diesel fuel has started to outweigh the high upfront cost of the solar array (Closas and Rap, 2017).However, many smallholders remain financially hesitant to purchase PVPSs since they are more sensitive to the high upfront cost of PVPS than the high lifetime cost of diesel fuel, creating a barrier to wide-scale adoption of PVPSs in the region (Pavelic et al., 2013).
In a PVPS, the solar array is often the dominating upfront cost, and the size of the solar array is directly related to the overall efficiency of the system (Muhsen et al., 2017).To increase the affordability of PVPSs, numerous efforts have been made in the past decade to provide efficiency improvements to existing PVPSs.These advancements include improving solar cell efficiency, developing more efficient power management algorithms, and designing more efficient mechanical hardware (Abdolzadeh and Ameri, 2009;Corrêa et al., 2012;Karami et al., 2017;Caracas et al., 2014;Sokol et al., 2018;Sashidhar and Fernandes, 2014).However, while these efforts have provided multiple avenues to improve PVPS efficiency and increase affordability, it remains unclear what the quantitative impacts of improving solar pump efficiency are on the overall upfront cost of a PVPS.The solar pump is the primary energy consumer in a PVPS and improving its efficiency can reduce the size of the solar array.
Therefore, it is essential to quantify the relationship between solar pump efficiencies and capital costs of the solar array in order to understand how to effectively reduce the overall cost of a PVPS.
During the design process of a PVPS, the impact of solar pump efficiency on the costs of the solar array is often overlooked since solar pumps are typically designed as stand-alone units by manufacturers.When a new solar pump is designed, pump engineers improve upon existing hydraulic designs, then source a submersible motor from a third-party supplier.Based on interviews with manufacturers such as Xylem, these motors are selected with a proper power rating and to be sufficiently low-cost, but their efficiencies are not necessarily considered a high priority during the design of the pump.When a PVPS is designed, local system engineers such as Davis & Shirtliff do not influence the design of the solar pump, rather they often use off-the-shelf solar pumps in conjunction with solar sizing software provided by pump manufacturers in order to produce a system-level package for the customers.Since the sizing software is formulated with the intent to provide an estimate of the overall panel size based on the manufacturer's existing product portfolio, the system engineers have no control over the solar pump efficiency determined by the pump engineers.This demonstrates a disconnect in design considerations between the system engineers who incorporate the costs of the solar panels, and the pump engineers who dictate the efficiency of the solar pump.This is especially problematic since most solar pumps are designed with an emphasis on capital cost reduction over efficiency improvement due to the capital cost of the product directly impacting sales competitiveness and profit margins.However, for a solar-powered application, the solar pump efficiency has a more significant implication on the capital cost of the overall PVPS than the cost of the solar pump itself due to the relatively high cost of the solar panels.
This work proposes a unique technoeconomic framework that enables solar pump engineers to directly quantify the cost implications arising from the solar pump efficiency on the solar array cost in a PVPS.To implement such a framework, the efficiency of 4-inch borewell pump hydraulics and submersible motors commonly used in SSA were characterized.It was found that the induction motors used in existing pumps suffer from a non-trivial amount of inefficiency.To elucidate the potential usefulness of the proposed technoeconomic framework, a case study is presented to demonstrate how solar pump engineers can utilize this framework to quantify efficiency implications on the overall PVPS costs for a one-hectare farm in SSA.This technoeconomic framework can be valuable to the solar pump industry by allowing solar pump engineers to evaluate the design implications of the decisions they make during the pump's design process, and inform local irrigation system designers about the downstream system-level impacts in PVPSs.

Framework to Quantify the Impacts of Efficiency on System Costs
The technoeconomic framework presented in this section can be used to directly quantify the impact of the efficiency of solar pump components -the pump hydraulics and submersible motor -on the solar power system costs of a PVPS.In the typical power flow structure of a PVPS, the electricity generated from the solar array powers the electric motor of the solar pump.The electric motor then delivers shaft power to the rotor shaft of the pump hydraulics, where the rotating pump hydraulics convert the shaft power into hydraulic power, achieving the flow rate and pressure head needed for the required hydraulic operating conditions.
A PVPS is often designed in a backward power flow manner, as the flow rate and pressure head are given as functional requirements, and the solar array is sized based on the hydraulic power needed and the efficiencies of the pump hydraulics and the electric motor.Our technoeconomic framework (Fig. 1) is presented in this backward power flow structure, where the power flow through each component (solid red) is back-calculated based on the operating conditions (dotted green) and the efficiencies of the pump hydraulics and electric motor (dashed blue).The efficiencies of the pump hydraulics are needed to back-calculate motor power, and the efficiencies of the electric motors are needed to back-calculate panel size (considering geographical factors).Formulating the framework this way allows quantifying the impacts of solar pump efficiency on the size and costs of the associated solar array required to support the power draw of the system while adhering to the customary design flow path in the solar pump industry.

Framework Formulation
When selecting different solar pump hydraulics and motors, components with lower efficiencies will incur a higher energy penalty in the system, attributing to a larger solar array size.The energetic costs associated with the inefficiencies can be quantified by correlating them to the capital costs of the solar array, where the capital cost of the solar array is directly proportional to the power requirement of the system, and therefore the efficiency of the system.
Quantification of the efficiency-related costs in the solar array can be done using the backward power flow structure of Fig. 1.The flow rate (Q) and pressure head (H) define the operating condition of the solar pump hydraulics, and its hydraulic power output (P hyd ) can be calculated using Eq. 1.The shaft power (P shaf t ) required for the solar pump hydraulics can be back-calculated using Eq. 2 given the hydraulic efficiency of the pump (η pump ).The electrical power requirement (P elect ) can be calculated similarly using Eq. 3 with the motor efficiency (η motor ).The electrical power requirement based on the solar pump's hydraulic and motor efficiencies enables the sizing of the solar array power (P array ) using Eq. 4 given the system's daily run time (t irr ) and the Fig. 1 Proposed technoeconomic framework structure to quantify the impact of solar pump efficiencies on the capital cost of the solar array in a photovoltaic pumping system (PVPS).
The framework calculates power flow (solid red) throughout the PVPS based on the component efficiencies (dashed blue), the operating conditions (dotted green) of the system, and the geographical-specific parameters (dot-dashed orange).
location-specific solar irradiance (P V out ).Since this study focuses on solar irrigation as an application of PVPSs, the system runtime will simply be the daily irrigation time of the farm.The P V out parameter has units of kW h/kW p , which indicates the electrical energy generated in kilowatt-hour (kW h) from a given kilowatt-peak (kW p ) of solar array installed, and is based on the daily average PV output potential modeled by Global Solar Atlas (ESMAP, 2019).Lastly, the capital cost of the solar array can be calculated using Eq. 5 which includes the solar panel retail price (C sol ).For this study, a C sol of 810 U SD/kW p is used, which is reported by local NGOs in SSA (Coalition Energy for Access, 2019).
(1) During the motor selection process, solar pump manufacturers can use this technoeconomic framework to quantify design implications on the overall systems when deciding between higher efficiency motors versus lower efficiency motors.In an example scenario, a designer may consider two solar pump architectures: architecture 1 which consists of a more expensive, highly efficient permanent magnet motor, and architecture 2 with a cheaper, lower efficiency AC induction motor.The difference in efficiency between the two motor architectures will lead to a difference in the costs of the solar array in a PVPS (Eq.6), which is the absolute difference in energetic costs (∆C array ).The percentage difference in energetic costs (%∆C array ) can be calculated with Eq. 7. Using these estimates on solar panel costs, designers can compare them to the difference in the capital costs (∆C capital ) between the two motors (Eq.8).In this process, designers can quantitatively determine whether the energetic benefits that arise from a more efficient motor can outweigh its additional capital cost, as a premium efficiency motor tends to be more expensive.
In Equations 6-8, C archi array and C archi capital correspond to the capital cost of the solar array and the capital cost of the hardware (excluding solar panels) in architecture i, respectively.

Efficiency Characterization of 4-Inch Multistage
Centrifugal Borewell Pumps (MSPs) In order to implement the technoeconomic framework and calculate pump shaft power (Eq.2), a means of predicting hydraulic efficiency as a function of the BEP operating flow rate is needed.While the actual efficiency of pump hydraulics depends on a variety of design parameters and manufacturing tolerances Gülich (2014), an empirical scaling approach is more practical and straightforward.To the author's knowledge, a prior empirical centrifugal pump efficiency model was initially published by H. H. Anderson and later modified by I. Karassik, and it can be used to capture the empirical relationship between centrifugal pump hydraulic efficiencies and their BEP operating conditions.The Anderson-Karassik model is formulated using pump efficiencies surveyed in 1979 with a large range of pump design flow rates from 1750 GP M to 254,000 GP M (397.5 m 3 /h to 57,690 m 3 /h) (Anderson, 1979;Karassik et al., 2008).While this model provides a general empirical estimation of efficiencies for a large range of pumps, it may not accurately reflect the low flow rate range that is typical for smallholder farms for SSA (1 m 3 /h to 18 m 3 /h).This is because the pump efficiencies in SSA only correspond to a small subset of the low flow rate pumps from the prior model survey while the overall model is more skewed toward pump efficiencies in the range of larger design flow rates.An efficiency scaling law more specific to the SSA operating conditions can be derived empirically from surveying and characterizing the best-efficiencypoint (BEP) hydraulic efficiencies of 4-inch multistage centrifugal borewell pumps (MSPs).These efficiencies were surveyed from pump designs over a range of operating flow rates and pressures expected in SSA smallholder farms.In SSA, 4-in MSPs are commonly used because the operating conditions correlate to a range of specific speeds that are suitable for the use of MSPs and 4-inch is a standard borewell size used by SSA drillers (Gülich, 2014;Van De Zande et al., 2022).To develop the empirical scaling law, efficiencies of 37 independent impeller designs that are used in 453 4-inch MSPs sold in the SSA market were compiled from prominent manufacturers (Grundfos, 2020e;Xylem, 2020b;Pedrollo, 2020;CNP, 2020).The portfolio made up of these pumps covers an operating flow rate range of up to 18 m 3 /h and a pressure head range up to 250 m, which are sufficient for various sizes of farms and depths of borewells found in SSA (Van De Zande et al., 2022;Lowder et al., 2016).
When comparing the surveyed efficiencies to the Anderson-Karassik model, it was found that the model is limited in predicting efficiencies for the 4-inch MSPs in the current market, leading to an error in efficiency predictions of up to 0.24 in the low flow rate region (Fig. 2. The Anderson-Karassik efficiency scaling model has the form of Eq. 9, where N is in RP M , Q BEP is in GP M , H BEP is in f t for a single impeller according to the Anderson-Karassik formulation1 .The specific speed N s is defined as This model predicts centrifugal hydraulic efficiency as a function of flow rate and specific speed (Eq.10) evaluated at BEP.For multistage pumps, the specific speed is calculated using the head-per-stage of the impeller instead of the total pressure head of the pump (Gülich, 2014).Therefore, in order for the framework to more accurately evaluate 4-inch MSP hydraulic efficiencies at different operating conditions, the empirically fitted coefficients (C1 and C2) in the Anderson-Karassik model were refitted to the surveyed data.As shown in Fig. 2, the refitted model is able to predict hydraulic efficiencies more precisely for the 4-inch borewell pump over a range of flow rates in the SSA market.
The refitted model results in a better RMSE value of 6.4391 as compared to the prior model's 19.5488, as shown in Table 1.The trend of the surveyed 4-inch MSP efficiencies in Fig. 2 conforms to the qualitative descriptions from the literature (Karassik et al., 2008;Gülich, 2014).MSPs experience low efficiency at low specific speeds due to their long and radial impeller geometries, which translates to the low flow rate region for the 4-inch impellers.This geometry generates high secondary losses, such as disk friction losses, as well as a high ratio of leakage flow to total flow.As specific speed and flow rate increase, the impeller design becomes more axial, resulting in a significant reduction of secondary losses and the relative leakage flow, contributing to higher efficiencies.This behavior is apparent in the exponential increase of efficiency with specific speed.The efficiency eventually plateaus to a maximum value of approximately 68% for the surveyed 4-inch MSPs.
Table 1 Statistically fitted parameters used in Eq. 9 for the original Anderson-Karassik model (Karassik et al., 2008) and the refitted model to predict the efficiency of 4-inch MSP hydraulic.

Fitted Parameters
Anderson

Effciency Characterization of 4-Inch Submersible Motors
To characterize motor efficiency, efficiencies of 94 4-inch submersible motors currently sold in the SSA market were compiled, with an output shaft power ranging from 0.37 kW to 7.5 kW (Grundfos, 2020c;Xylem, 2020a;Lorentz, 2020b;Davis & Shirtliff Group, 2020).Since the majority of borewell pumps are originally designed for grid-tied applications, all of the motors surveyed were AC induction motors (IMs), due to their simplicity and plug-and-play capability with the grid.To establish efficiency standards for induction motors, the International Electrotechnical Commission (IEC) published the IEC 60034 specification (IEC, 2014) which rates commercial induction motor efficiencies from IE1 to IE4, in the order of increasing efficiencies.The efficiencies of the 4-inch submersible IMs surveyed from the current solar pump market are compared to these four IE ratings in Fig. 3. Since the efficiency scaling of the IE ratings can be numerically extrapolated as a function of motor shaft power using a 4th-order logarithmic relationship (Eq.11), (11) the efficiency data of the surveyed IMs are also fitted to a scaling law of the same form, resulting in an RMSE of 4.7176.This enables direct motor efficiency scaling in the framework for solar pump motors operating in the various shaft power regimes, calculated based on the hydraulic operating points and efficiencies of the pump hydraulics (Eq. 2. The corresponding interpolation coefficients for scaling efficiencies of the IE ratings and the surveyed IMs are listed in Table 2. As shown in Fig. 3, the 4-inch submersible IMs currently sold on the market significantly underperform even the lowest IE1 motor efficiency rating by an average of 0.07 in terms of efficiency.The low efficiencies of IMs are primarily due to the lack of a permanent magnet field and the induction losses in the coil of the rotor.The lower efficiencies in existing IMs on the market suggest a potential opportunity to improve PVPS system efficiency and achieve potential energetic cost reduction with higher efficiency motors.In practice, BLDC motors are often found to have comparable or even superior efficiency to the IE3 or IE4 efficiency rating as they operate with a permanent magnetic field (De Almeida et al., 2011).For example, the newer BLDC 4-inch submersible motors offered by Lorentz have an efficiency of up to 98% (Lorentz, 2020b).
By simply adopting higher efficiency BLDC motors to replace existing IMs, PVPS designers can effectively reduce the size of the solar array and therefore lower upfront costs of the overall PVPS.Moreover, BLDC motors can also operate directly off the DC current generated from the solar array without the need for a DC-AC boost inverter, resulting in reduced complexity of the electrical system.In fact, many solar pump manufacturers have recognized the benefits of the increased efficiency and reduced electronic complexity in using BLDC motors for solar-powered applications, and the pump industry is slowly transitioning to adopt BLDC motors from conventional IMs (Lorentz, 2020a;Grundfos, 2020g;Xylem, 2014).However, the energetic tradeoffs between the gained efficiency and the additional capital cost of the BLDC motor remain not well understood and are hard to quantify.Therefore, the technoeconomic framework proposed in this paper was developed to help industry address this knowledge gap, enabling manufacturers and designers to balance tradeoffs between efficiencies and capital costs -creating more cost-effective solar pumping systems.

Case Study -Demonstrating the Application of the Framework
The technoeconomic framework was used in combination with the efficiency prediction models formulated for the 4-inch borehole pump hydraulics and motors to conduct a case study analysis for SSA farms.Two analyses were conducted to compare solar pump architectures, primarily from the motor selection perspective.First, the energetic costs between solar pumps with two different motor architectures were compared over a range of operating flow rates and pressures, providing a spatial quantification of capital cost-reduction in the solar array for the SSA operating space.In this analysis, both the absolute energetic cost reduction and the percentage energetic cost reductions between the two solar pump architectures were analyzed to elucidate the cost reduction scaling as a function of the operating conditions and its relative magnitude to the total panel cost.Secondly, the overall capital costs between solar pump architectures were compared for a specific operating point.The goal was to demonstrate how PVPS designers can use this framework to directly compare the quantified energetic costs to the capital costs of the hardware components when designing for a specific customer.This analysis also illustrates why efficiency matters when designing PVPS, as using more expensive but highly efficient hardware can potentially create PVPS with lower upfront costs due to the reduced size of the solar array.The case study demonstrates how solar pump manufacturers and PVPS designers can apply the framework to quantitatively relate solar pump efficiency to upfront cost, enabling them to make informed design decisions during the component selection process.

Case Study Parameters
Two solar pump architectures are compared in this case study: IM-driven MSPs and IE4 motor-driven MSPs.This represents the efficiencies of solar pumps with the surveyed IMs on the current market, and the improved efficiencies when the solar pump industry adopts BLDC motor architectures, respectively.The analysis is first conducted at an operating space level, where the energetic cost tradeoffs between the two solar pump architectures are quantified over a range of operating flow rates and pressures.Furthermore, a specific operating point that represents a typical 1 Ha farm in SSA is picked for more detailed analysis including capital costs.
The operating location for the analyses is in Nairobi, Kenya, where farmers have a high interest in solar irrigation products (Van De Zande et al., 2022).The simulated flow rate ranges from 1 to 18 m 3 /h and the pressure head ranges from 10 to 250 m for the operating space analysis.The range of operating conditions is chosen based on the capable operating range of 4-in MSP designs which are suitable for the SSA market.The specific operating flow rate and

Comparative Analysis of Energetic Cost Reduction
The difference in absolute energetic costs (Eq.6) between the IM-driven MSPs and the IE4-driven MSPs is plotted in Fig. 4(a).This difference represents the CAPEX cost-reduction in the solar array achieved by improving solar pump efficiency using more efficient BLDC motors.Based on the simulated results shown in Fig. 4(a), the absolute energetic cost reductions scale primarily with the solar pump hydraulic power.The largest cost reduction is observed in the high hydraulic power region, up to $1,800 USD.At high hydraulic power, the higher IE4 motor efficiency makes a larger impact on the required electrical power and size of the solar array.These results suggest there may be a large economic incentive for solar pump manufacturers to provide higher efficiency motors in the high power region (e.g.larger farms) because the energetic cost reduction will likely outweigh the additional capital cost of the more efficient motor.Moreover, Fig. 4(a) also provides a guideline on the capital cost premium that a more efficient motor can have before the cost benefits from the efficiency gain break even.
The percentage of energetic cost-reduction can be calculated (Eq.7) to evaluate the magnitude of the cost-reduction in the solar array due to efficiency improvement relative to the total cost of the solar array (Fig. 4(b)).The simulation result shows that the largest percentage of the cost reduction occurs in the low hydraulic power region, which correlates with smaller farms.In the low hydraulic power region, the efficiency difference between an IE4 motor and an IM is larger, and the energetic cost associated with the power losses in the hardware is also more prominent.This directly contrasts the trends in the absolute amount of cost reduction shown in Fig. 4(a).Therefore, although the largest absolute amount of cost reduction is in the high-power region, the economic impact of efficiency gain relative to the total panel cost is most pronounced in the low-power region.It demonstrates the potential need for low-cost, high-efficiency motors for PVPS in the low-power region, which represents the operating space of smallholders, who are more likely to be in poverty.

Capital Cost Tradeoffs About A Specific Operating Point
To further demonstrate how an efficiency-driven design mindset for solarpowered applications can potentially lead to more cost-effective PVPSs, the energetic costs in the solar array, quantified by the presented framework, can be compared to the capital cost difference in the hardware components.In this case, the capital cost difference is the cost premium of a highly efficient but more expensive IE4 motor over the cheaper, less efficient IM.To formulate an explicit example for comparing the tradeoffs between efficiency-related energetic costs and motor capital costs, this analysis focuses on a specific operating point.Within the SSA operating space, an operating point of 3 m 3 /h flow rate and 100 m pressure head was selected to represent a typical 0.25 Ha smallholder farm with a borewell depth of 100 meters (Van De Zande et al., 2022).The capital cost of the pump hydraulics was approximated to be $400 USD and the IM motor was approximated to be $350 USD based on Grundfos SP 3A-25 pricing (Grundfos, 2020f,d).The IE4 efficient BLDC motor was approximated to be $610 USD, which is around 75% more expensive than the conventional IM given the corresponding power requirement.

Breakdown of Capital Costs [USD]
Cost of Solar Array Cost of Pump Hydraulic Cost of Motor Fig. 5 Simulated capital costs comparison of a photovoltaic pumping system (PVPS) when using a conventional induction motor versus an IE4 efficient BLDC motor.The simulated PVPSs were designed for the operating point of a typical 1-Ha farm, with a flow rate of 3 m 3 /h and a borehole depth of 100 m.Capital costs are broken down into solar array costs (blue), pump hydraulic costs (orange), and motor costs (yellow).
The combined costs of the solar array (energetic cost) and the capital costs of the motor for the two solar pump architectures are shown in Fig. 5.The simulation results show that even though the more efficient IE4 motor comes with a more expensive cost premium, the cost reduction in the solar array due to the improved efficiency outweighs the additional motor cost.The use of a more efficient IE4 motor effectively reduces the size of the solar array when compared to the use of conventional IM, leading to lower overall system costs for this specific operating point.The result demonstrates the importance of component efficiency in a solar-powered system, which outweighs the capital costs of hardware components due to the relatively higher cost of the solar panels.This analysis further elucidates how PVPS system designers can leverage the presented technoeconomic framework to quantify tradeoffs between efficiency and capital costs in order to produce a more cost-effective solar pump architecture for an end-user through an efficiency-driven design mindset.

Discussion
In this study, a technoeconomic framework was created to enable solar pump engineers to quantify the cost implications arising from the solar pump efficiency on the solar array cost of the overall system.The application of this framework was demonstrated for a simulated case study representative of smallholder farms in SSA.This case study elucidates the importance of prioritizing efficiency in the design process for solar pumps and PVPSs due to the large efficiency-driven capital costs of the solar arrays.This work has also found that the induction motors currently used in the 4-inch borehole pumps on the market significantly underperform in terms of efficiency when compared to the IE motor efficiency standard.In addition, the adoption of highly efficient motors such as permanent magnet brushless motors in solar pumps can effectively reduce the overall upfront cost of a PVPS.

Generalizing Application of the Framework
The hydraulic operating space considered in the simulated case study was based on the operating ranges of 4-inch MSPs, which represents the operating requirements of smallholder farms ranging from less than 1 Ha up to 5 Ha.These operating conditions are commonly used in the SSA irrigation market, and therefore provide a good representation of the application of the framework.However, the energetic cost framework itself can be applied to other pump types, operating locations, and solar-powered applications.When analyzing a different market with potentially different operating conditions and hardware options, the efficiencies of the corresponding pump hydraulics and motors in that market can be recharacterized using the methods outlined in Section 2. The location-specific parameters used in the framework analysis such as the PV output potential can be adjusted according to the local operating conditions.The solar panel price can also be modified to more accurately reflect the price in the various local markets and the potential price changes in the future.With appropriate modifications, this energetic framework can be adapted for various solar-powered applications outside of solar pumping, and geographical locations with different solar irradiances and solar panel prices.

Efficiency-Based Component Selection to Reduce Cost
When designing solar-powered irrigation systems, an efficiency-driven mindset during the component selection process can be an effective strategy to reduce the capital upfront cost of the system and reduce the financial burdens on smallholder farmers.As shown in the case study, the improved efficiency of a permanent magnet motor (e.g.IE4) has technoeconomic benefits that tend to outweigh the higher motor capital cost compared to a cheaper, lower efficiency IM, making PVPSs more affordable for developing markets.This is because the impact of the efficiencies from hardware components (e.g.motors) on the CAPEX of solar array often outweighs the CAPEX of the components themselves.The solar pump is the primary energy consumer and its efficiency has a direct impact on the size of the solar array which is the dominating cost of the overall system (Van De Zande et al., 2022).
However, a CAPEX-driven design mindset is a common practice amongst solar pump manufacturers during the procurement process, as most of the hardware is designed for grid-tied applications (Grundfos, 2020a;Xylem, 2020d).This is because when designing for grid-tied applications, customers are often more sensitive to the lump sum upfront costs of the hardware than the electricity costs over the hardware's operating lifetime.Yet, the CAPEXdriven design mindset does not fully capture the additional upfront costs in the solar array which arise from the inefficiencies of the system in off-grid solarpowered applications.Therefore, it is important for the industry to rethink the component selection process to prioritize efficiency and be aware of the key difference when designing off-grid solar-powered systems versus grid-tied systems.In a solar-powered system, the primary cost of energy is a CAPEX which is primarily attributed to the upfront cost of the solar array.But in a grid-tied system, the primary cost of energy is an OPEX, deriving from the electricity cost it uses over its lifetime of operation.
The results of this study suggest an efficiency-driven design mindset of utilizing more expensive but efficient hardware in a solar-powered system can potentially drive down the overall system costs, which is counterintuitive to the conventional CAPEX-driven design process in the industry.This technoeconomic implication is especially important when designing systems for the off-grid developing markets where customers are much more sensitive to CAPEX than OPEX.The majority of the CAPEX comes from the costs of the solar array, which is correlated to system efficiencies (Van De Zande et al., 2022).In fact, the shift in design thinking of prioritizing efficiency over component CAPEX aligns with the trend observed in industry, as solar pump manufacturers are starting to pursue higher efficiency permanent magnet BLDC motors specifically for solar-powered applications (Lorentz, 2020a;Grundfos, 2020g;Xylem, 2014).

Implications of Solar Pumping System Design
The case study presented in this paper demonstrates a representative application of how solar pump manufacturers and PVPS designers can use the energetic framework to conduct analytic comparisons between the capital costs and the efficiency-related energetic costs when selecting different components.The framework enables them to make an informed design decision during the design process by quantifying the impacts of component efficiencies on overall system costs.By quantifying the energetic costs over the operating space, the framework allows direct evaluation of the potential energetic cost reduction when comparing two solar pump architectures.Solar pump manufacturers and system designers can repeat this analysis for the different architecture options and relate efficiency performance to the capital cost of the solar array to form a baseline of direct cost comparison to produce more cost-effective future solar pump product lines and better PVPSs.The framework's ability to directly quantify the impact of efficiency to cost during the design process can be valuable to industrial practitioners, as it enables them to provide potentially lower cost, yet more efficient, solar-powered irrigation systems to the smallholder farmers in the developing market of SSA.
While the case study results show that an efficiency-driven design process can be useful in driving down costs of the PVPSs, there's a trend in decreasing solar panel prices over time which may make the affordability of a PVPS less sensitive to solar pump efficiency (Feldman et al., 2020).The importance of efficiency diminishes with lowering solar panel prices because the capital cost associated with the additional solar array required to compensate for the power losses may become less expensive.However, the declining costs of solar panels will continue to make PVPSs more price-competitive than the conventional diesel-powered pumping systems used in many developing communities (Schmitter et al., 2018;Closas and Rap, 2017) Since off-grid, solar-powered pumping systems are the focus of this study, the framework primarily focuses on the impact of efficiency on the cost of the solar array.When considering potential grid-tied, hybrid systems, the electricity cost over the systems' operating lifetime can be aggregated and added to the capital cost of the solar array.A similar efficiency-related energetic cost comparison to the capital cost of the hardware can then be conducted for grid-tied pumping systems using the modified framework.The incorporation of electricity costs will ensure the framework remains useful and relevant to designers for grid-tied applications, as "microgrids" become more popular amongst developing rural communities without reliable grid infrastructure (Murenzi and Ustun, 2015).

Assumptions and Limitations
During the formulation of the presented framework, several assumptions were made and some limitations resulted.The solar array required to support the power demand of a solar pump is sized using the conservation of electrical energy generated from the solar array on a daily basis.This assumes the energy generated can be stored in a sufficiently large energy buffer such as a tank or batteries.By doing so, the average daily PV output potential can be directly used and the intraday variation in solar irradiance is not captured, reducing computational complexity.However, losses can occur during energy transfer in physical systems from pipe loss and electrical resistance.These losses are minor compared to the dominant power losses in the solar pump hydraulic and motor, and therefore were excluded from our model.
Moreover, the costs of the power electronics and energy buffers (e.g.batteries and tanks) are not captured, as their sizing largely depends on the actual design of the system and its operating requirements (e.g.voltage, control scheme, irrigation schedule, etc.).PVPS designers may still want to take into account the costs of these power system components during the design of a specific system.Recently, researchers have created in-depth models to optimize the sizing of system components (e.g.battery and tanks) which designers can leverage in conjunction with the presented framework during the design of a solar-powered irrigation system (Grant et al., 2022).
The efficiencies of the solar pump hydraulics were modeled and characterized at their best efficiency point (BEP) since it is the designed operating Quant.the Energ.Cost Tradeoffs of PVPSs for SSA Smallholder Farms point of any pump hydraulics.In reality, a pump's hydraulics typically have a preferred operating range (POR) from 70% to 120% of the BEP flow rate.Operation away from the BEP may lead to an efficiency penalty, typically up to a drop of 8% depending on the efficiency curve shape of the specific design (Xylem, 2019).This may lead to a potential deviation between the actual operating efficiency of the pump from the model approximation.This deviation is not captured by the model since the efficiency penalty away from BEP is design specific.However, manufacturers should be able to incorporate it from their existing sizing software which already contains the efficiency curves of their specific designs (Xylem, 2020c;Grundfos, 2020b).In addition, the cost reduction calculated based on the efficiency models are generalized approximations of the hardware, and these results do not represent the efficiency performance of any specific manufacturer.This is because the efficiency data from multiple manufacturers are lumped together to formulate the efficiency prediction models to describe the general market.

Conclusions
While the solar array is the primary cost driver in a PVPS and the solar pump is its dominant energy consumer, it is crucial to characterize a quantitative relationship between solar pump efficiencies and upfront costs of PVPSs to enable manufacturers to design more cost-effective PVPS for developing markets.An energetic technoeconomic framework was proposed to quantify the implicit relationship between solar pump efficiencies and the upfront cost implications in the solar array in a solar-powered pumping system (PVPS).Since PVPSs are gaining popularity amongst off-grid, rural farming communities in developing regions such as Sub-Saharan Africa (SSA), solar-powered borehole pumping in SSA was chosen to be the scope of focus in this paper to demonstrate the usefulness of the proposed framework.In order to implement the proposed framework and formulate explicit efficiency scaling laws for the solar pump components, the efficiencies of 4-inch multistage centrifugal hydraulics (MSPs) and motors in the current market were surveyed and characterized.An efficiency scaling law for the 4-in MSP hydraulics was updated based on a previously published model.An efficiency scaling law for the 4-in induction motors used in existing 4-in hydraulics was also formulated and compared to the IE motor efficiency standard.It was found that induction motors in the current market underperform even the lowest IE1 efficiency standard while significantly underperforming permanent magnet motors with typical IE3 to IE4 efficiencies by 15% to 20% in efficiency.
A case study was set forth to compare the upfront cost implications in the solar array when choosing between current induction motors and permanent magnet motors for a PVPS.The simulated results from the case study have shown that using more expensive yet highly efficient PM-BLDC motors in solar pumps can lead to an overall cheaper PVPS design than the lower cost, less efficient conventional induction motors.These results suggested that an efficiency-driven design mindset, which is counterintuitive to the CAPEXdriven mindset in common industry practice, can effectively lower the upfront cost of a PVPS.This is because by using more efficient hardware, the size of the solar array in a PVPS can be effectively reduced, and the cost reduction in the solar array outweighs the additional cost premium of the more efficient hardware.The proposed technoeconomic framework not only identifies a potential pathway for the pump industry to rethink its design process when designing for solar-powered applications, but can also serve as a design tool that enables engineers to directly quantify the upfront cost implications of hardware efficiency during the design process.

Fig. 2
Fig. 2 Surveyed and model-predicted 4-inch multistage centrifugal pump efficiencies for a flow rate range of 1 m 3 /h to 18 m 3 /h.Surveyed efficiencies from the Sub-Saharan Africa (SSA) borehole market are denoted by black dots.Efficiency predictions from the prior Anderson-Karassik model (blue crosses) deviate from surveyed data, especially at low flow rates.Efficiency predictions from the updated model presented in this work (red diamonds) provide a more accurate prediction for the low and refitted flow rate pumps used in the SSA market.
Fig. 4 Absolute capital (energetic) cost reductions (a) and percentage cost reduction (b)in the solar array when using high-efficiency IE4-rated BLDC motors over conventional induction motors, for the operating space in SSA.A larger magnitude of cost reduction (and percentage reduction) is represented by the brighter color (yellow), while the duller color (blue) represents a smaller magnitude.The highest absolute capital cost reduction is in the high power range (high flow and high pressure) while in contrast, the highest percentage cost reduction is in the low power range.
P V out [kW h/kW p], P array [kW p], and C sol [U SD/kW p].
Comparison of International Electrotechnical Commission (IEC) standard efficiency ratings to surveyed 4-inch induction motors (IM) from the Sub-Saharan Africa (SSA) solar pump market.Surveyed efficiencies of 4-inch induction motors (IM) used to drive borehole pumps in the SSA market are denoted by blue dots.The four IE motor efficiency ratings (solid lines) demonstrate a 4th order logarithmic relationship to motor shaft output power.A 4th order logarithmic trendline was fit on the surveyed data for comparison and is represented by the dashed line.The graph indicates the existing induction motors on the market underperform when compared to the IE efficiency ratings.

Table 2
Coefficients for efficiency interpolation of the surveyed IMs and the four IE efficiency classes (2-poles, 3000 rpm) (IEC, 2014).

Table 3
(Van De Zande et al., 2022)he example case studies in SSA.Van De Zande et al., 2022).The irrigation time was chosen to be six hours, which is typical based on interviews with SSA farmers who have a PVPS(Van De Zande et al., 2022).The location-specific PV output potential is 4.1918 kWh/kWp based on modeled solar GIS data for the latitude and longitude of Nairobi(ESMAP, 2019).The retail price of the solar panels is 810 USD/kWp reported locally (Coalition Energy for Access, 2019).These input parameters are listed in Table3.