A model for optimal sizing of photovoltaic irrigation water pumping systems
Zvonimir Glasnovic a,*, Jure Margeta b
a Faculty of Chemical Engineering and Technology, University of Zagreb, 10000 Zagreb, Marulicev trg 19, Croatia b Faculty of Civil and Architectural Engineering, University of Split, 21000 Split, Matice Hrvatske 15, Croatia
Received 11 July 2005; received in revised form 26 September 2006; accepted 27 November 2006 Available online 22 December 2006
Communicated by: Associate Editor Hansjo ¨rg Gabler
Abstract
The previous methods for optimal sizing of photovoltaic (PV) irrigation water pumping systems separately considered the demand for hydraulic energy and possibilities of its production from available solar energy with the PV pumping system. Unlike such methods, this work approaches the subject problem systematically, meaning that all relevant system elements and their characteristics have been ana- lyzed: PV water pumping system, local climate, boreholes, soil, crops and method of irrigation; therefore, the objective function has been defined in an entirely new manner. The result of such approach is the new mathematical hybrid simulation optimization model for opti- mal sizing of PV irrigation water pumping systems, which uses dynamic programming for optimizing, while the constraints were defined by the simulation model. The model was tested on two areas in Croatia, and it has been established that this model successfully takes into consideration all characteristic values and their relations in the integrated system. The optimal nominal electric power of PV generator, obtained in the manner presented, are relatively smaller than when the usual method of sizing is used. The presented method for solving the problem has paved the way towards the general model for optimal sizing of all stand-alone PV systems that have some type of energy storage, as well as optimal sizing of PV power plant that functions together with the storage hydroelectric power plant. 2006 Elsevier Ltd. All rights reserved.
Keywords: Photovoltaic pumping; Irrigation; PV generator; optimal sizing; Dynamic programming
1. Introduction
The previous optimizing of photovoltaic (PV) water pumping systems, which have been the subject of numerous papers, mainly dealt with improvement of effectiveness of various system components, as well as their better mutual adjustment, with the aim of total cost reduction of the PV pumping system (AVICENNE Programme, Papers for distribution, 1997). On the other hand, optimal sizing of the PV pumping system is basically reduced to calculation of the required hydraulic energy at the output of the system and its rela-
tion to monthly average daily solar irradiation. Hydraulic energy for PV pumping systems for irrigation is calculated based on required water quantity data, calculated by agri- cultural experts, and total head of water rise, Kenna and Gillett (1985). The equation for nominal electric power of PV genera- tor Pel expressed in (W), in referential condition (Standard Test Condition STC – intensity of solar irradiation 1000 W/m2, relative air mass AM1.5 and temperature of PV generator 25 C), according to Kenna and Gillett (1985) is as follows: Pel ¼ 1000 fm½1 acðTcell T0Þ gMP EH ES ð1Þ where EH ðkWh=dayÞ is output hydraulic energy, ES ðkWh=dayÞ the solar energy at the PV generator input,
0038-092X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2006.11.003
* Corresponding author. Tel.: +385 1 4597281; fax: +385 1 4597260. E-mail address: zvonglas@fkit.hr (Z. Glasnovic).
www.elsevier.com/locate/solener
Solar Energy 81 (2007) 904–916
fm the load matching factor to characteristics of the PV generator, ac the PV cell temperature coefficient ( C 1), T0 the referential temperature of the cell (25 C), gMP mo- tor–pump unit effectiveness, and Tcell temperature of the cell ( C) which, according to Markvart and Castan ˇer (2003), can be calculated by the equation:
Tcell ¼Ta þ
NOCT 20 800
GS ð2Þ where Ta is air temperature ( C), GS the intensity of solar irradiance (W/m2) and NOCT is the Nominal Operating Cell Temperature ( C). Therefore, the nominal electric power of PV generator is calculated based on the known monthly average daily demand for hydraulic energy EH and available monthly average daily solar irradiation Es in the critical month
and the known efficiency of the motor–pump unit GMP in referential operating conditions, taking into account the effect of outside temperature on the efficiency of the PV generator. Eq. (1) stands for critical month, i.e. the month in which the ratio between hydraulic and radiated solar energy EH=ES is maximum. However, this approach has the fol- lowing flaws:
• Lack of systematic quality – Hydraulic energy value obtained from an agriculture expert is not in any way correlated to the possibilities of its covering with avail- able solar energy in the calculated month. Consequently, the problem is observed particularly and separately, i.e. the agriculture expert first calculates the values of the required hydraulic energy independently from the
Nomenclature
An area of irrigated location (ha) Dr the most active rooting depth (m) EH hydraulic energy (kW h) Es mean daily radiation on horizontal plane (ter- restrial radiation) ðkW h=m2dayÞET 0 potential evapotranspiration (mm) ETr real evapotranspiration (mm) F objective function FC field capacity (mm) fiðxiÞ optimal return function per state variable xi in time stage i fi 1ðxi 1Þ optimal return function per state variable xi 1 in time stage i 1f iðWðiÞÞ optimal return function per state variable WðiÞ in time stage i fði 1ÞðWði 1ÞÞ optimal return function per state variable Wði 1Þ in time stage i 1 fm load matching factor to PV generator character- istics GS intensity of solar irradiance ðW=m2Þ H0T vertical head from the water outlet to the ground (m) HDT dynamic water level in borehole (m) HF head friction loss (local and linear losses) (m) HST static level (groundwater level) (m) HTE total head (m) i time stage (increment) INF infiltration (mm) Kc crop coefficient L total precipitation losses due to superficial drainage (mm) N total number of time stages i nd number of days in time stage i Pel nominal electric power of the PV generator (W) P el optimal nominal electric power of the PV gener- ator (W)
PelðiÞðQdðiÞÞ return function from stage i per decision var- iable QdðiÞQ AP average flow, known also as ‘‘apparent flow’’ rate ðm3=hÞ Qd mean daily water volume at the output of the PV pumping system (decision variable) ðm3=dayÞQ PVðiÞ water from PV pumping system which is,by irri- gation, added to soil in time stage i (m3) Qmax maximum discharge capacity of borehole ðm3=hÞ R total precipitation (mm) Re total effective precipitation (that reach the soil) (mm) ri return from stage i tr mean daily insolation (h) T0 referential temperature of PV cell (generator) (25 C) Ta temperature of the surroundings ( C) Tcell temperature of PV cell (generator) ( C) ui decision variable in stage i WðiÞ soil moisture in time stage i (mm) Wði 1Þ soil moisture in time stage i 1 ðmmÞ xi state variable in stage i xi 1 state variable in stage i 1 ac PV cell (generator) temperature coefficient ( C 1) goc nominal efficiency of PV generator (%) gI inverter efficiency (%) gMP motor–pump unit efficiency (%) gMPI efficiency of motor–pump unit and inverter (%) gN irrigation efficiency (%) si transformation, i.e. relation between output and input state variables for every stage of the sys- tem i # calculation coefficient of average flow
Z. Glasnovic, J. Margeta / Solar Energy 81 (2007) 904–916 905
Designer of the PV system. After that, the designer divides it by available solar energy, and then uses this value as his optimal value. • Static quality – By monitoring only the critical month, it is not possible to observe properly the demands for hydraulic energy and possibilities of their fulfilling in the previous and subsequent months. Furthermore, it is completely ignored that the water static level and water quantity in the borehole can vary from month to month, thus affecting the determining of critical val- ues. The fact that total head in borehole is dependent on the quantity of the pumped water is also disregarded. Ignored is the possibility of water redistribution regard- ing time and quantity, from water abundant periods (days) into dry period, and thus redistribution of critical parameters in system sizing (possibilities of water stor- age). Also, possibilities of fulfilling water requirements from available solar energy with the PV pumping system are not even considered in dynamic sense. Therefore, everything is observed statically.
It is evident that such sizing of the nominal power of the PV generator, whose price is still relatively high, doesn’t yield optimal results. This results in increased investment costs and affects possible economic justification of such systems.
2. System configuration
Unlike the approach where PV pumping systems are observed separately from their surroundings (the work of
Bahaj and Mohammed (1994) is typical in that sense), and in accordance with Glasnovic et al. (1991) (which par- tially unites the main influential elements into one system and properly chooses the optimizing method that is widely used in water resources management, but the objective function is inadequately set and in that sense the optimiz- ing problem is solved), in this work the problem is solved at the level of the system as a technological entirety. This entirety equally encloses all components of the system, including natural processes in the system (climate, hydrol- ogy, boreholes, pumping system, irrigation, agriculture and power supply) during the entire period when the system is in operation (irrigation season), Fig. 1. This means that throughout the entire operation period the system is dynamically analyzed as an integrity, taking into account all changes that occur in re
A model for optimal sizing of photovoltaic irrigation water pumping systems
Zvonimir Glasnovic a,*, Jure Margeta b
a Faculty of Chemical Engineering and Technology, University of Zagreb, 10000 Zagreb, Marulicev trg 19, Croatia b Faculty of Civil and Architectural Engineering, University of Split, 21000 Split, Matice Hrvatske 15, Croatia
Received 11 July 2005; received in revised form 26 September 2006; accepted 27 November 2006 Available online 22 December 2006
Communicated by: Associate Editor Hansjo ¨rg Gabler
Abstract
The previous methods for optimal sizing of photovoltaic (PV) irrigation water pumping systems separately considered the demand for hydraulic energy and possibilities of its production from available solar energy with the PV pumping system. Unlike such methods, this work approaches the subject problem systematically, meaning that all relevant system elements and their characteristics have been ana- lyzed: PV water pumping system, local climate, boreholes, soil, crops and method of irrigation; therefore, the objective function has been defined in an entirely new manner. The result of such approach is the new mathematical hybrid simulation optimization model for opti- mal sizing of PV irrigation water pumping systems, which uses dynamic programming for optimizing, while the constraints were defined by the simulation model. The model was tested on two areas in Croatia, and it has been established that this model successfully takes into consideration all characteristic values and their relations in the integrated system. The optimal nominal electric power of PV generator, obtained in the manner presented, are relatively smaller than when the usual method of sizing is used. The presented method for solving the problem has paved the way towards the general model for optimal sizing of all stand-alone PV systems that have some type of energy storage, as well as optimal sizing of PV power plant that functions together with the storage hydroelectric power plant. 2006 Elsevier Ltd. All rights reserved.
Keywords: Photovoltaic pumping; Irrigation; PV generator; optimal sizing; Dynamic programming
1. Introduction
The previous optimizing of photovoltaic (PV) water pumping systems, which have been the subject of numerous papers, mainly dealt with improvement of effectiveness of various system components, as well as their better mutual adjustment, with the aim of total cost reduction of the PV pumping system (AVICENNE Programme, Papers for distribution, 1997). On the other hand, optimal sizing of the PV pumping system is basically reduced to calculation of the required hydraulic energy at the output of the system and its rela-
tion to monthly average daily solar irradiation. Hydraulic energy for PV pumping systems for irrigation is calculated based on required water quantity data, calculated by agri- cultural experts, and total head of water rise, Kenna and Gillett (1985). The equation for nominal electric power of PV genera- tor Pel expressed in (W), in referential condition (Standard Test Condition STC – intensity of solar irradiation 1000 W/m2, relative air mass AM1.5 and temperature of PV generator 25 C), according to Kenna and Gillett (1985) is as follows: Pel ¼ 1000 fm½1 acðTcell T0Þ gMP EH ES ð1Þ where EH ðkWh=dayÞ is output hydraulic energy, ES ðkWh=dayÞ the solar energy at the PV generator input,
0038-092X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2006.11.003
* Corresponding author. Tel.: +385 1 4597281; fax: +385 1 4597260. E-mail address: zvonglas@fkit.hr (Z. Glasnovic).
www.elsevier.com/locate/solener
Solar Energy 81 (2007) 904–916
fm the load matching factor to characteristics of the PV generator, ac the PV cell temperature coefficient ( C 1), T0 the referential temperature of the cell (25 C), gMP mo- tor–pump unit effectiveness, and Tcell temperature of the cell ( C) which, according to Markvart and Castan ˇer (2003), can be calculated by the equation:
Tcell ¼Ta þ
NOCT 20 800
GS ð2Þ where Ta is air temperature ( C), GS the intensity of solar irradiance (W/m2) and NOCT is the Nominal Operating Cell Temperature ( C). Therefore, the nominal electric power of PV generator is calculated based on the known monthly average daily demand for hydraulic energy EH and available monthly average daily solar irradiation Es in the critical month
and the known efficiency of the motor–pump unit GMP in referential operating conditions, taking into account the effect of outside temperature on the efficiency of the PV generator. Eq. (1) stands for critical month, i.e. the month in which the ratio between hydraulic and radiated solar energy EH=ES is maximum. However, this approach has the fol- lowing flaws:
• Lack of systematic quality – Hydraulic energy value obtained from an agriculture expert is not in any way correlated to the possibilities of its covering with avail- able solar energy in the calculated month. Consequently, the problem is observed particularly and separately, i.e. the agriculture expert first calculates the values of the required hydraulic energy independently from the
Nomenclature
An area of irrigated location (ha) Dr the most active rooting depth (m) EH hydraulic energy (kW h) Es mean daily radiation on horizontal plane (ter- restrial radiation) ðkW h=m2dayÞET 0 potential evapotranspiration (mm) ETr real evapotranspiration (mm) F objective function FC field capacity (mm) fiðxiÞ optimal return function per state variable xi in time stage i fi 1ðxi 1Þ optimal return function per state variable xi 1 in time stage i 1f iðWðiÞÞ optimal return function per state variable WðiÞ in time stage i fði 1ÞðWði 1ÞÞ optimal return function per state variable Wði 1Þ in time stage i 1 fm load matching factor to PV generator character- istics GS intensity of solar irradiance ðW=m2Þ H0T vertical head from the water outlet to the ground (m) HDT dynamic water level in borehole (m) HF head friction loss (local and linear losses) (m) HST static level (groundwater level) (m) HTE total head (m) i time stage (increment) INF infiltration (mm) Kc crop coefficient L total precipitation losses due to superficial drainage (mm) N total number of time stages i nd number of days in time stage i Pel nominal electric power of the PV generator (W) P el optimal nominal electric power of the PV gener- ator (W)
PelðiÞðQdðiÞÞ return function from stage i per decision var- iable QdðiÞQ AP average flow, known also as ‘‘apparent flow’’ rate ðm3=hÞ Qd mean daily water volume at the output of the PV pumping system (decision variable) ðm3=dayÞQ PVðiÞ water from PV pumping system which is,by irri- gation, added to soil in time stage i (m3) Qmax maximum discharge capacity of borehole ðm3=hÞ R total precipitation (mm) Re total effective precipitation (that reach the soil) (mm) ri return from stage i tr mean daily insolation (h) T0 referential temperature of PV cell (generator) (25 C) Ta temperature of the surroundings ( C) Tcell temperature of PV cell (generator) ( C) ui decision variable in stage i WðiÞ soil moisture in time stage i (mm) Wði 1Þ soil moisture in time stage i 1 ðmmÞ xi state variable in stage i xi 1 state variable in stage i 1 ac PV cell (generator) temperature coefficient ( C 1) goc nominal efficiency of PV generator (%) gI inverter efficiency (%) gMP motor–pump unit efficiency (%) gMPI efficiency of motor–pump unit and inverter (%) gN irrigation efficiency (%) si transformation, i.e. relation between output and input state variables for every stage of the sys- tem i # calculation coefficient of average flow
Z. Glasnovic, J. Margeta / Solar Energy 81 (2007) 904–916 905
Designer of the PV system. After that, the designer divides it by available solar energy, and then uses this value as his optimal value. • Static quality – By monitoring only the critical month, it is not possible to observe properly the demands for hydraulic energy and possibilities of their fulfilling in the previous and subsequent months. Furthermore, it is completely ignored that the water static level and water quantity in the borehole can vary from month to month, thus affecting the determining of critical val- ues. The fact that total head in borehole is dependent on the quantity of the pumped water is also disregarded. Ignored is the possibility of water redistribution regard- ing time and quantity, from water abundant periods (days) into dry period, and thus redistribution of critical parameters in system sizing (possibilities of water stor- age). Also, possibilities of fulfilling water requirements from available solar energy with the PV pumping system are not even considered in dynamic sense. Therefore, everything is observed statically.
It is evident that such sizing of the nominal power of the PV generator, whose price is still relatively high, doesn’t yield optimal results. This results in increased investment costs and affects possible economic justification of such systems.
2. System configuration
Unlike the approach where PV pumping systems are observed separately from their surroundings (the work of
Bahaj and Mohammed (1994) is typical in that sense), and in accordance with Glasnovic et al. (1991) (which par- tially unites the main influential elements into one system and properly chooses the optimizing method that is widely used in water resources management, but the objective function is inadequately set and in that sense the optimiz- ing problem is solved), in this work the problem is solved at the level of the system as a technological entirety. This entirety equally encloses all components of the system, including natural processes in the system (climate, hydrol- ogy, boreholes, pumping system, irrigation, agriculture and power supply) during the entire period when the system is in operation (irrigation season), Fig. 1. This means that throughout the entire operation period the system is dynamically analyzed as an integrity, taking into account all changes that occur in re
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