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Dynamic model of a complex system including PV cells


Energy 28 (2003) 1165–1181

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Dynamic model of a complex system including PV cells, electric battery, electrical motor and water pump
Viorel Badescu ?
Candida Oancea Institute of Solar Energy, Faculty of Mechanical Engineering, Polytechnic University of Bucharest, Spl Independentei 313, Bucharest 79590, Romania Received 28 October 2001

Abstract The time dependent operation of several components of a PV pumping system (i.e. the PV cell, the PV array, the battery and the assembly electric motor—centrifugal pump) is modelled in this paper. The system has two main operating modes, which depend on the level of the incident solar global irradiance. The mathematical model consists of systems of seven or four ordinary differential equations, respectively, according to the operating mode. A computer simulation code is developed. Clear days with high incident solar irradiance and high cell temperature Tcell are characterised by lower PV ef?ciency, while the cell ef?ciency is larger on cloudy days, when the temperature Tcell is smaller. The sun-to-user ef?ciency is larger during the winter months. The battery acts as a buffer, as the main part of the electricity supplied by the PV array is used to drive the motor. The value of the PV cell series resistance Rs causes the battery to operate under two different regimes: when Rs is larger, the battery is over-discharged once or twice each month in the cold season. In case Rs is small, the battery is over-charged once per month in the warm season. The electric power used to drive the motor is rather constant during the year. ? 2003 Elsevier Ltd. All rights reserved.

1. Introduction Direct-coupling of the array and the load is certainly the cheapest of all PV systems. A directly coupled PV pumping system is composed of a PV array directly connected to a DC motor driving a centrifugal pump. Thus, such a system is simple, reliable, and low cost because it does not include battery storage or a battery voltage regulator. The system simply stores water instead of storing electrical energy. The advantages of this system led to its widespread use over the world.
?

Tel.: +40-1-410-0400; fax: +40-1-410-4488. E-mail address: badescu@theta.termo.pub.ro (V. Badescu).

0360-5442/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-5442(03)00115-4

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As a result, the system has been thoroughly investigated. Anis and Metwally [1] give a good literature survey on the subject. However, Anis et al. [2] reported that a load composed of a DC motor driving a centrifugal pump represents a non-matched load to the PV array. This is because the motor driving a volumetric pump requires an almost constant current, for a given head, apart from the starting current that tends to be higher. This condition does not match the PV array characteristics where the current varies almost linearly with solar irradiance. On the other hand, the high cost of PV energy dictates that even a small percentage increase in total system ef?ciency will result in a signi?cant economic impact [3]. Batteries are used in most PV systems to perform two essential functions: power buffer between the array and loads, and energy storage bank. In addition, batteries provide a more stable system voltage [3]. In this paper, we focus on a solar water pumping system consisting of four basic units: a PV array, a battery, a DC motor, and a centrifugal pump. Some systems may incorporate a maximum power tracker to match the array maximum power locus. A power conditioning unit may also be included if regulated AC or DC supply is required [3]. Section 2 presents the models used here to describe the time dependent operation of the PV pumping system. These models are similar in accuracy to other theoretical approaches for complex renewable energy systems (see, for example, work on integrated solar heating systems [4], solar-assisted desiccant air conditioners [5] and solar–electric vehicles [6]). The models already developed for meteorological and actinometric data generation are shortly described in Section 3. Discussions and preliminary results are presented in Section 4. 2. Models The four basic units of the PV water pumping system are shown in Fig. 1. The following relation applies if the solar cells are in operation Iarray ? Ib ? Im (2.1)

Fig. 1. The system considered here. B—electric battery; M—electric motor; P—centrifugal pump.

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where Iarray, Ib and Im are the current provided by the PV array, the current through the battery and the current through the electric motor armature, respectively. During the night or during periods with very low solar irradiance, the solar cells are not in operation and the relationships Iarray ? 0 Ib ? Im apply, provided the battery alone is able to drive the electric motor. 2.1. PV cell model A standard solar cell model is used here (see for instance [7, p. 173]). The current Icell provided by the cell is given by (Fig. 2): Icell ? Iph?Ir?Ish (2.2) where Iph, Ir and Ish are the photo-generated current, the reverse current and the shunt current, (2.1a) (2.1b)

Fig. 2. PV cell model. S and D—photo-sensible and diode-type components of solar cell. Rs, Rsh—series and shunt electric resistance, respectively.

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respectively. One denotes by Vcell the voltage across the cell. Then, the voltage V over the shunt resistance Rsh is V = Vcell + RsIcell. Here Rs is PV cell series resistance. The shunt current is Ish = V / Rsh. The reverse current is given by the standard diode theory as: Ir ? I0 exp

? ? ? ?
V ?1 BkTcell

(2.3)

where I0, B, k, and Tcell are the reverse saturation current, a cell (thermal voltage) constant, Boltzmann’s constant and cell temperature, respectively. The above relationships allow the writing of Eq. (2.2) as: Icell ? Iph?I0 exp

? ?

Vcell ? RsIcell q(Vcell ? RsIcell) ?1 ? BkTcell Rsh

? ?

(2.4)

where q is electron’s electric charge. The following relations can be used for silicon solar cells: Iph ? GT·(ta)·sAcell

I0 ? KcellAcellT3 cellexp ?

?

Eg kTcell

?

(2.5) (2.6)

GT, (ta), s, and Acell in Eq. (2.5) are solar global irradiance at the level of solar cell, the effective transmittance–absorptance product of the cell, a constant, and the surface area of a single cell, respectively [6]. Also, Kcell and Eg are a cell constant and cell band gap, respectively [8]. 2.2. PV array model A PV array of M parallel strings, each consisting of N identical solar cells in series is shown in Fig. 3. A simple PV array model is developed here. The array’s series and shunt electric losses are neglected. However, the in?uence of array series resistance could be taken into account simply by increasing the nominal value of each PV cell series resistance. The following relations apply: Vcell ? Icell ? Varray N Iarray M (2.7) (2.8)

Use of Eqs. (2.4), (2.7), and (2.8) leads to: Iarray ? MIph?MI0 exp

? ?

MVarray ? NRsIarray q(MVarray ? NRsIarray) ?1 ? . MNBkTcell NRsh

? ?

(2.9)

The energy balance at the level of the whole PV array is: MNAcellGT·(ta)?MNAcellUcell(Tcell?Ta)?IarrayVarray ? 0 (2.10) where Ucell and Ta are the convection heat loss coef?cient of the PV cell and ambient temperature, respectively. The ?rst term in Eq. (2.10) gives the rate of solar energy absorbed by the PV array

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Fig. 3. PV array model.

while the second term is the heat ?ux transferred by convection from the solar cells to the ambient. The last term is the electric energy leaving the PV array towards battery and electric motor. 2.3. Electric battery model Lead–acid battery storage is commonly used in stand-alone solar photovoltaic energy systems [9]. A common technique to evaluate the performance of a battery is to estimate its state of charge (SOC) in Ah or %. A good literature survey on the subject is given in [9]. However, similar SOC levels correspond to different voltage values depending on the operation. Therefore, the SOC does not introduce a comprehensive and deterministic quantity for accurately identifying the condition of a battery. Moreover, the SOC calculation is meaningless at dynamic operation in which successive charge/discharge currents of a random order ?ow through the battery. Instead of the SOC, the state of voltage, SOV, was introduced in [9] as a measure of a battery operational condition. Emphasis was placed in the referenced paper on successfully predicting the voltage trend of the battery and the magnitude of the voltage in the useful operational range. The model from [9] is used here. For simpli?cation, the model in [9] neglected the reactive element, and an equivalent total internal resistance Rb is employed, considered to be independent of the electrolyte temperature and state of charge. The following equation gives the battery voltage Vb: Vb ? V0 ? Keln 1?

?

Q ? RbIb C(Ib)

?

(2.11)

In Eq. (2.11), Ib ? 0 during charge and Ib ? 0 during discharge. Q is the exchanged electric

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charge, i.e. the charge transferred during one time step, while C(Ib) is the battery capacity as a function of the current. V0 is a constant and represents the battery voltage at an initial condition. Coef?cient Ke is a model parameter. The second term in the right hand side of Eq. (2.11) represents the battery electrolyte concentration changes during charge and discharge processes. The algebraic value of this term is adjusted through the parameter Ke, i.e. Ke ? 0 at charge and Ke ? 0 at discharge. The exchanged charge Q from time point tn to time point tn+1 is given by the integral: Q?

?

tn+1 tn

|Ib|·dt

(2.12)

As de?ned above, Q is not the accumulated charge and, therefore, does not represent the state of charge of the battery. Eq. (2.12) will be used in this work under the form: dQ ? |Ib| dt (2.12a)

which is more convenient during the solving procedure. The relationship between the charge/discharge current Ib and the battery capacity C involves three capacity coef?cients, namely C1, C2 and C3: C(Ib) ? C1 C2·(|Ib|)C3 ? 1 (2.13)

Their values differ for the charge and discharge operations. 2.4. Electric motor and water pump models The DC motor considered here is of the permanent magnet type. For this type of DC motor, no power is required for the ?eld winding excitation. Accordingly, the energy generated by the PV array is partially saved [1]. The PV array (or battery) current and voltage are equal to the motor current and voltage, respectively. DC motor rotational losses are neglected. Motor–pump coupling losses and all other system frictions are also neglected. Then, the equations of the system under transient conditions are [1]: Varray ? Rm·Im ? Km·wm ? Lm· J· dwm ? Km·Im?Kp·(wm)2 dt dIm dt (2.14) (2.15)

Here Rm is motor armature and cable resistance, Lm is the inductance of DC motor armature winding, Km is a motor constant, wm is motor rotational speed, J is the moment of inertia of motor pump system and Kp is a constant of the centrifugal pump. The three terms in the r.h.s. of Eq. (2.14) have obvious meanings. The ?rst and the second terms in the r.h.s. of Eq. (2.15) are motor and pump torque, respectively.

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3. Model for meteorological and actinometric data The diffuse and global solar irradiance on a horizontal surface was simulated in this work by using the model we proposed in [10]. This model uses as entries the point cloudiness, the ambient temperature, the atmospheric pressure, and the air relative humidity. An additional parameter associated to the sky status is the sunshine number xs. The sunshine number xs is 1 or 0 if the sun is not covered, or is covered by clouds, respectively. Using the model of [10] to simulate time-series of solar radiation data depends on the procedure of computing the sequences of sunshine number xs. One expects these sequences to show auto-correlation features, especially in case of short time intervals. These properties of xs are not relevant for the purpose of this article. Comments on xs can be found in [10]. Meteorological data measured by the Romanian Meteorological and Hydrological Institute during 1961 in Bucharest (latitude 44.5°, longitude 26.2° E, altitude 131 m above sea level) were used in this work [11]. The 1961 data was selected and prepared previously as a TMY (typical meteorological year) to be used for simulations of renewable energy systems operation. The climate of Bucharest is temperate—continental with a climatic index of continentality (Ivanov) of 131.9%. The METEORAR database consists of values measured at 1.00, 7.00, 13.00 and 19.00 local standard time (LST) for ambient temperature, air relative humidity and point cloudiness. Also, the database contains daily average values for the atmospheric pressure. The ground albedo was always assumed to be 0.2 [12]. A simple isotropic model was used to compute the direct, diffuse and ground-re?ected solar irradiance on a tilted surface by using as input the ?ux of solar energy incident on a horizontal surface (see, e.g. [13]). Fig. 4 shows the variation of simulated solar global irradiance on a south-oriented surface tilted 22° in July 1961 at Bucharest.

Fig. 4. Simulated solar global irradiance on a south oriented surface tilted 22° in July 1961 at Bucharest. Data were generated on a 15 min time interval.

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4. Results and discussions Simulations were performed during all the months of the year, for a water pumping system operating 24 h a day. The system has two main operating modes, A and B. Switching between operating modes depends on the level of solar global irradiance GT incident on the PV array. 4.1. Operating modes The operating mode A applies when GT exceeds a critical level (Gcr). Then, the PV array supplies electricity to both battery and motor. However, during periods with decreasing solar irradiance, the battery acts as a buffer, supplying electricity to the motor. The following performance indicators could be used during operating mode A: the solar power collected by the PV array, Pcoll, the electric power provided by the PV array, PPV, the electric power stored/supplied by the battery, Pb, and the power used to drive the motor, Pm. They are de?ned as follows: Pcoll ? MNAcellGT·(ta) PPV ? IarrayVarray Pb ? IbVarray Pm ? ImVarray Note that Pb is positive or negative, depending on the sign of Ib. In the ?rst case, the power stored by the battery is denoted Pb, in while the power extracted is denoted Pb, out. Also, three dimensionless indicators are used, namely the PV ef?ciency, hPV, the ratio between the energy stored in the battery and the collected solar energy, ePV?b, and the ratio between the energy delivered to the electric motor and the collected solar energy, ePV?m. They are de?ned as follows: hPV ? PPV Pcoll Pb, in Pcoll Pm Pcoll (4.2) (4.1)

ePV?b ? ePV?m ?

Note that, commonly, by ‘PV ef?ciency’ (or ‘sun-to-user’ ef?ciency), one understands the product between the PV ef?ciency de?ned here and the effective transmittance–absorptance product (ta) of the cell. The operating mode B applies when GT?Gcr. Then, the PV array is not in operation and the battery alone is driving the electric motor. The following performance indicators could be used during operating mode B: the electric power supplied by the battery, Pb, out, and the power used to drive the motor, Pm. They are de?ned as follows: Pb, out ? IbVb( ? Pm ? ImVb) (4.3) The primary function of a charge controller in a PV system is to protect the battery from over-

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charge and over-discharge. The controller may disconnect the PV array or limit the charging current if the battery is fully charged. On the other hand, the controller may disconnect the load if the battery reaches its minimum recommended SOC or SOV [3,14]. The operational voltage range for the particular 12 V / 70 Ah nominal lead acid battery used here was assumed to be as follows [9]: (a) Vmin = 11.8 V minimum (battery empty)—Vmax = 14.1 V maximum (full charge) during charge and (b) Vmax = 13.3 V maximum (full charge)—Vmin = 11.5 V minimum (discharged) during discharge. As recommended in [3], two additional constraints are used during both operating modes. They ensure that: (i) the time during which the array is disconnected because the battery has reached its full state of charge is minimised, and (ii) the time during which the load is disconnected because the battery has reached its minimum state of charge is minimised. In practice, we assumed that a fully charged or empty battery (i.e. V?Vmax or V? Vmin, respectively) is ‘instantaneously’ replaced by a battery at nominal voltage V0 (see Section 4.3 for numerical values). This is in agreement with the procedure used in [15]. 4.2. Solving ODEs systems The LSODI FORTRAN Livermore solver for ordinary differential equations (ODE) [16] was used in this work. LSODI solves the initial value problem for linearly implicit systems of ?rst order ODEs of the form a(t,y)dy / dt = g(t,y) where a(t,y) is a square matrix and y is a vector containing the unknowns. LSODI is a variant version of the LSODE package [17]. In case of operating mode A (PV array in action), there are seven unknowns parameters, namely the array voltage Varray( = Vb), the PV cells temperature Tcell, the currents through the array, battery and motor, Iarray, Ib and Im, respectively, the electric charge received/supplied by the battery Q, and the motor speed wm. The following seven equations are used to describe the system time evolution: (2.1), (2.9) (with the associated equations (2.5) and (2.6)), (2.10), (2.11) (with the associated equation (2.13)), (2.12a), (2.14), and (2.15). To allow usage of LSODI package the ?rst time derivative of the Eqs. (2.1), (2.9)–(2.11) was computed ?rst. In case of the operating mode B (PV array not in action), only four unknowns are considered, namely the battery voltage Vb, the current through the battery Ib( = Im), the electric charge received/supplied by the battery Q, and the motor speed wm. The equations used are (2.11) (with the associated equation (2.13)), (2.12a), (2.14), and (2.15). As required by LSODI package, the time derivative of Eq. (2.11) was ?rst computed. A computer simulation code was developed to evaluate the dynamic performance of the system. The ?rst voltage value was set equal to the battery nominal voltage V0. For each day, the 24 h period is divided into intervals of 15 min. The values of the ambient temperature Ta and of solar global irradiance GT are computed by using the meteorological database for both the beginning and the end of each time interval. Then, the time gradients (transients) of Ta and GT during the given time interval was evaluated. These transients are input for the computer code. The system operating parameters, which represent the response of the system, are computed during each time interval and the values at the end of the interval are stored. Note that previous studies assumed the solar irradiance to be constant for short intervals (6 min) [1]. The authors considered this assumption acceptable because solar irradiance varies at a low rate during the clear days in Cairo city. The procedure we used here ensures a more accurate simulation during cloudy days, when solar irradiance could sometimes have a sudden variation.

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4.3. Sizing the system The PV system designer must carefully select the parameters of the PV array and battery in order to match the load demand [3]. In practice, sizing the systems has been well established based on system performance, component modelling, technical–economical considerations, load matching and system reliability (see [15] and references therein). Here we kept the numerical values of [9] to describe battery operation. Also, the numerical values describing the parameters of the electric motor and of the centrifugal pump were preserved as in the original references. However, the values of certain PV cell and PV array parameters were slightly changed in order to allow matching of system components. The modi?ed parameters were M, N and the PV cell shunt resistance Rsh. The ?nal values were adopted after several trial runs. Silicon solar cells were considered here (Eg = 1.12 eV). They have a recombination ef?ciency around 10?4 and Kcell = 2000 Am?2K?3 [8]. The following common values describe PV cells operation: s = 0.003343 AW?1 [6], B = 2 [7], and (ta) = 0.7. A convection heat loss coef?cient Ucell = 25 Wm?2 K?1 is accepted. It corresponds to a wind average speed of about 5 m/s. Note that for a wind speed of 1 m/s, Ucell lies between 1.2 and 9.6 Wm?2 K?1 [18]. We assume a cell surface area Acell = 0.01 m2. PV cell electric resistances are Rs = 0.3 ? and Rsh = 150 ?. The PV array consists of M = 25 parallel PV cell strings. Each string has N = 4 cells in series. The array is south oriented and tilted 22°, which is the optimum tilt for the extended warm season (April–September) at Bucharest [13]. The particular battery used here is a 12 V ?at plate lead–acid type, manufactured by CELTIC with 70 Ah nominal capacity. This battery has heavy duty, low maintenance characteristics and was designed for use in vehicles [9]. The following values characterise its operation: (i) during charge: Kc = ?0.0995 V, C1 = 68.233 Ah, C2 = 1.323·10?4 A?1, C3 = 3.1903 and Rb = 0.0656 ?, (ii) during discharge: K?c = 0.0992 V, C?1 = 64.712 Ah, C?2 = 1.573·10?4 A?1, C?3 = 3.1903 and R?b = 0.0615 ?. The nominal battery voltage is V0 = 12.8 V while the minimum and maximum allowed voltage are Vmin = 11.5 V, Vmax = 14.1 V, respectively [9]. The DC motor parameters are as follows: the inductance of DC motor armature winding Lm = 2.2 mH, armature and cable resistance Rm = 0.3 ? and motor constant Km = 0.85 V rad?1 s?1. The centrifugal pump parameters are as follows: the pump constant Kp = 0.0052 N m / s2 rad, the moment of inertia J = 0.136 N m / s2 rad. Exact computation of the water discharge necessitates determining the pump ef?ciency at various speeds. However, here the constant value hp = 0.8 is accepted. 4.4. Results Two sorts of PV cells will be considered in the following. The ‘nominal’ cell has a series resistance Rs = 0.3 ?—see Section 4.3. The ‘alternate’ PV cell is characterised by Rs = 0.05 ?. 4.4.1. Battery voltage The time dependence of the battery voltage Vb during July is shown in Fig. 5. Note that Vb = Varray when the PV array operates. Fig. 5a shows the case of the ‘nominal’ PV cell. The voltage generally follows the time variation of global solar irradiance (compare Figs. 5a and 4). The monthly voltage deviation is small, of the order of 0.15 V. Generally Vb is lower than the nominal

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Fig. 5. Time variation of the battery voltage Vb during July 1961 at Bucharest. (a) Nominal PV cell; (b) alternate PV cell. Simulation performed on a 15 min time interval.

voltage V0 = 12.8 V. This makes the battery discharge once or twice per month during the cold season (results not shown here). When the ‘alternate’ PV cell is considered, the voltage time variation is quite large (its monthly deviation is about 1.2 V) and it is slightly dependent on the time variation of GT (Fig. 5b). Most of the time, the battery voltage exceeds the nominal value V0. In fact, the computer code simulated that an over-charged battery was replaced on 24 July. Note that the ‘replacement’ occurred shortly after noon, when the voltage suddenly decreases. 4.4.2. Cell temperature The temperature of the ‘nominal’ PV cell during July is shown in Fig. 6. It is strongly correlated to the incident solar global irradiance, as expected (compare Figs. 6 and 4). This is in agreement

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Fig. 6. Time variation of the cell temperature Tcell during July 1961 at Bucharest. A nominal cell was considered. Simulation performed on a 15 min time interval.

with [18] where, for example, an irradiance drop of 0.4 kW/m2 in 12 min is associated with a decrease of 11 K in cell temperature. During the night, Tcell equates the ambient temperature while during the daytime, Tcell exceeds Ta with as much as 30 K, depending on day type (i.e. cloudy or clear sky). Note that the drop in the capacity of a PV array is approximately 0.5% per unit cell temperature rise [15]. 4.4.3. Electric current through battery and motor The current Ib through the battery is positive during the day (when the battery is generally charging) and negative during the night (when the battery drives the electric motor) (Fig. 7). In case of a PV system consisting of ‘nominal’ cells, the charging current Ib is as much as 1.1 A and follows the time variation of global solar radiation (with damped amplitudes—compare Figs. 7a and 4). The charging current variation is rather constant during cloudy days, when solar irradiance is smaller (see 17 and 18 July on both Figs. 4 and 7a). The discharging current is about 1.3 A and it is almost constant in time. The ‘alternate’ PV cells increase the charging current about four times as compared to the case of the ‘nominal’ cell (compare Fig. 7b and a). The discharging current is almost the same in both cases. One concludes that the charging current makes the difference between a battery working in discharging regime (‘nominal’ PV cells) and a battery working in charging regime (‘alternate’ PV cells). These two regimes were already identi?ed during discussions related to Fig. 5. Note that the above results are in agreement with the experimental results of [9], where the current varied between +0.4 and +8.9 A for the charge test and between ?0.1 and ?8.5 A for the discharging test. Our calculations show that the time variation of motor current Im and motor rotation speed wm is similar in shape to the voltage time variation shown in Fig. 5. In both cases, replacing ‘nominal’ cells by ‘alternate’ cells leads to a difference in the time variation similar to that between Fig. 5a and b. These remarks remain in case of the time variation of motor power Pm.

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Fig. 7. Time variation of the battery current Ib during July 1961 at Bucharest. Solid thick line denotes a null current. (a) Nominal PV cell; (b) alternate PV cell. Simulation performed on a 15 min time interval.

However, the time variation of both PV array power PPV and battery stored/extracted power Pb is similar to that of the battery current Ib depicted in Fig. 7. These comments are in concordance with early results showing that PV current varies almost linearly with solar irradiance [1]. 4.4.4. Performance indicators Fig. 8 shows the time variation during July 1961 of the PV ef?ciency hPV in the case of ‘nominal’ PV cells. In order to obtain the ‘sun-to-user’ ef?ciency, one has to multiply hPV by the effective transmittance–absorptance product (ta), as already noted. Around sunrise and sunset, the rather abrupt variation of solar irradiance (see Fig. 4) is associated to a similar variation of the PV ef?ciency. In the middle of the day, the PV ef?ciency decreases, mainly because cell temperature increases (see Fig. 6). Clear days with high incident solar irradiance and high cell temperature (as 15 and 16 July in Fig. 6 for instance) have lower PV ef?ciency. During cloudy

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Fig. 8. Time variation of the PV ef?ciency hPV during July 1961 at Bucharest. A nominal cell was considered. Simulation performed on a 15 min time interval.

days (as 1st July for instance) when solar radiation is lower (see Fig. 4), the temperature Tcell is smaller (Fig. 6) and the cell ef?ciency is larger (Fig. 8). The time variation of ePV?b is shown in Fig. 9 in the case of ‘nominal’ PV cells. This indicator has a smooth increase around sunrise. There is no obvious difference between clear sky and cloudy days from this point of view. This proves that the battery plays the role of a buffer, as

Fig. 9. Time variation of the ratio ePV?b ? Pb, in / Pcoll during July 1961 at Bucharest. A nominal cell was considered. Simulation performed on a 15 min time interval.

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the main part of the electricity supplied by the PV array is always used to drive the motor (see Table 1). Table 1 summarises several quantities related to a system based on ‘alternate’ PV cells operating during 1961 at Bucharest. The monthly sum of the collected solar energy, ?Pcoll, is obviously larger during the summer, as expected. The relative increase of ?Pcoll between the ‘worst’ month (December) and the ‘best’ month (August) is more than 400%. Also, the power delivered by the PV array, ?PPV, mainly depends on the level of the incident solar irradiance. However, ?PPV increases about 270% between December and August. This is due to increasing solar cell temperature during the warm season, which diminishes the PV power supply. Also, the ‘best’ month from the point of view of ?PPV is July. The power stored by the battery, ?Pb, in, exceeds the power supplied by the battery, ?Pb, out, in April and June–August. However, during the other months the periods when electricity is stored (Ib ? 0) alternate with periods when energy is extracted from the battery (Ib ? 0) (see Fig. 7 for instance). Thus, the battery is never over-discharged. The electric power used to drive the motor, ?Pm, is rather constant during the year. There is no obvious dependence on month and season. The monthly mean of sun-to-user ef?ciency, ? PV, depends on both solar irradiance and ambient temperature. Our results are in reasonable (ta)·h agreement with Khouzam [3], who noted that although sun-to-user ef?ciencies of solar cells are approaching 30%, commercially available PV systems which include storage, are experiencing overall ef?ciencies below 10%. Also, the experimentally derived overall ef?ciency reported in [15] is between 3% and 9% for a PV array of 35 Wp modules (four in series and 25 in parallel). Note that ‘the overall ef?ciency’ in the quoted paper includes the ef?ciencies of PV panels, battery banks, charge controller, inverter and of the distribution network. Other experimental studies report overall ef?ciency for PV arrays based on single crystal silicon p-type cells of 3–12% [19]. ? PV between Winter months have larger sun-to-user ef?ciency and the relative decrease of (ta)·h the ‘best’ month (December) and the ‘worst’ month (August) is about 30%. The monthly mean of the indicator ePV?b is rather constant during the year, with slightly lower values during the
Table 1 Mean values (?) and monthly sums (Σ) for various quantities related to the operation of a PV pumping system based on ‘alternate’ solar cells. The 12 months of 1961 at Bucharest were considered ?Pcoll (kWh) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 42.23 54.42 91.46 95.89 92.38 118.59 126.19 129.77 115.39 71.84 47.01 32.21 ?PPV (kWh) 7.27 8.15 14.09 13.34 12.11 15.45 16.38 15.95 14.76 11.11 8.29 5.87 ?Pb, in (kWh) 3.20 3.58 7.83 7.06 5.42 7.86 8.78 8.86 8.47 5.50 3.61 1.98 ?Pb, out (kWh) 9.09 7.96 8.03 6.65 6.24 6.31 6.59 6.82 7.01 8.28 9.42 9.84 ?Pm (kWh) 13.16 12.52 14.30 12.93 12.93 13.91 14.19 13.92 13.30 13.88 14.11 13.72 ? PV (%) e ? PV?b (%) (ta)h 18.4 16.5 16.5 13.9 13.4 13.6 13.4 12.5 13.0 16.7 19.3 20.1 6.87 6.16 9.70 7.21 5.89 6.85 7.37 7.13 7.86 8.42 7.46 5.75 ? PV?m (%) e 19.4 17.4 13.9 12.7 13.3 12.6 11.7 10.7 10.7 15.4 20.2 22.9

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? PV?m is higher during January–February and October–December winter months. The indicator e ? PV?m always exceeds e ? PV?b, and signi?cantly smaller during the rest of the months. Note that e sometimes being four times larger. This con?rms that the battery plays the role of a buffer. The monthly mean of ePV?m varies during the year between 10.7% and 22.9%. The ‘matching ef?ciency’ of the daily energy is de?ned in [1] as the ratio of energy transferred to the load (obtained by using the operating characteristic curve of the load) to the maximum PV energy (obtained by using the maximum power curve of the array). These authors considered the four extreme days of the year, i.e. vernal and autumnal equinox, and summer and winter solstice days in Cairo city, (30° N). The matching ef?ciency of the system exceeded 90% between 9 a.m. and 3 p.m. Early in the morning or late in the afternoon, where solar irradiance is lower, the matching ef?ciency was poorer, i.e. less than 70%. Our results prove that replacing the hourly mean by monthly mean, on one hand, and clear days by mixed clear and cloudy days, on the other hand, lead to a considerable decrease in the matching ef?ciency. 5. Conclusion The time dependent operation of a solar water pumping system is modelled in this paper. The system consists of four basic units: a PV array, a battery, a DC motor, and a centrifugal pump and has two main operating modes. Switching between operating modes depends on the level of solar global irradiance GT incident on the PV array. The ?rst operating mode applies when GT exceeds a critical level (say Gcr). Then the PV array supplies electricity to both battery and motor. The second operating mode applies when GT?Gcr. Then, the PV array is not in operation and the electric motor is being operated by the battery. Several performance indicators are de?ned as the PV ef?ciency, hPV, the ratio between the energy stored in the battery and the collected solar energy, ePV?b, and the ratio between the energy delivered to the electric motor and the collected solar energy, ePV?m, respectively. Depending on the operating mode, the mathematical model consists of systems of seven or four ordinary differential equations. They are solved by using the LSODI FORTRAN Livermore solver. A computer simulation code is developed to obtain the dynamic performance of the system. The main results of the paper are as follows. The value of the PV cell series resistance is an important design parameter as it makes the battery over-discharge in the cold season or overcharge in the warm season. The PV ef?ciency is larger when the cell temperature is smaller, i.e. around sunrise and sunset, during cloudy days and during the winter months. The battery plays the role of a buffer, as the main part of the electricity supplied by the PV array is used to drive the motor. The electric power used by the motor is rather constant during the year. There is no obvious dependence on month and season. Apart from the technical results, the novelty of this paper consists of the tools it uses. First, one suggests that the operation of such a complex system has to be quanti?ed by using a relatively large number of performance indicators. The three indicators we de?ned here are just examples. Each of them can be used to bring information about a particular aspect of the operation that is often of interest. Another novelty is the graphical tool we used, displaying the detailed time evolution of the performance indicators and other quantities during a larger time period (e.g., a month). This makes it easy to identify, by simple visual inspection, certain aspects that the use of global quantities may overlook. On the other hand, it may suggest correlation between various

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quantities, displayed in different graphics, to be studied subsequently by using a more elaborated apparatus. Third, there are the averaged (or global) quantities, which give an overall picture of the system operation. Despite the fact that these quantities are elliptic by nature, sometimes they allow an easy comparison of different design solution. The combination of these three tools allows a more complete understanding of system behaviour. Acknowledgements The author thanks the referees for valuable comments and suggestions. References
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