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Dynamic performance of a directly coupled PV pumping system
Solar Energy, Vol. 53, No. 4, pp. 369-377, 1994 Copyright © 1994 ElsevierScience Ltd Printed in the USA. All rights reserved 0038-092X/94 $6.00 + .00
Pergamon
0038-092X(94)00051-4
DYNAMIC PERFORMANCE OF A DIRECTLY COUPLED PV PUMPING SYSTEM WAGDY R. ANIS *t a n d HAMID M. B. METWALLY** *Electronics and Communications Dept., Faculty of Engineering, Ain Shams University, 1 Sarayat St., Abbasia, Cairo, Egypt; * *Electrical Eng. Dept., Faculty of Engineering, Zagazig University, Zagazig, Egypt Abstract--Although directly coupled Photovoltaic (PV) pumping systems have been extensively studied under steady state conditions, much less attention has been paid to the study of their transient performance. Actually the system is performing dynamically all the time because the solar irradiance is varying continuously. Thus, the steady state solution for the system performance is only an approximation. In this work both dynamic and steady state models are developed for a directly coupled PV pumping system. The results obtained by a detailed computer simulation program based on the two models are compared. The system is studied under external interruptions caused by sudden appearance or disappearance of clouds. The study showed that there are overshoots of both current and voltage of the DC motor. However, the characteristics of the PV array limit the overshoots during the transient period. The motor speed creeps steadily towards its steady state value without overshoot.
I. I NTRODUCI~ION A directly coupled PV p u m p i n g system is c o m p o s e d of a PV array directly coupled to a DC m o t o r driving a centrifugal p u m p as shown in Fig. 1. Thus, such a system is simple, reliable, a n d of low cost because it does not include battery storage or a battery voltage regulator. T h e system simply stores water instead of storing electrical energy. T h e advantages of this system led to its widespread use over the world. As a result, the system has been thoroughly investigated in previous literature. A p p e l b a u m a n d Bany ( 1 9 7 9 a ) a n d ( 1 9 7 9 b ) analyzed the directly coupled PV p u m p i n g system under the steady state conditions. Roger ( 1 9 7 9 ) proved that a D C m o t o r driving a centrifugal p u m p represents a well-matched load to a PV array because this system utilizes most of the available D C power generated by the array. Anis et al. ( 1985 ), reported that a load composed o f a D C m o t o r driving a volumetric p u m p represents a n o n - m a t c h e d load to a PV array. This is because the m o t o r driving a volumetric p u m p requires an almost c o n s t a n t current, for a given head, apart from the starting current which tends to be higher. This c o n d i t i o n does not m a t c h the PV array characteristics where the current varies almost linearly with solar irradiance. A major advantage of the PV p u m p i n g system is that it is naturally m a t c h e d with solar radiation since, in most cases, water d e m a n d increases during s u m m e r w h e n solar radiation is m a x i m u m . In this work, the transient b e h a v i o r of the directly coupled PV p u m p i n g system is thoroughly investigated. The study includes extreme cases when solar irradiance changes abruptly, from high to low or vice versa, due to sudden appearance or disappearance o f clouds. T h e * Correspondence should be addressed to Dr. Wagdy Anis, Yanbu Industrial College, P.O. Box 30436, Yanbu al Sinaiyah 21477, Kingdom of Saudi Arabia.
developed program enables one to obtain transient a n d / o r steady state response of the system at any instant during the whole year. This study considers the four extreme days of the year as examples, i.e., vernal a n d a u t u m n a l equinox, a n d s u m m e r a n d winter solstice days in Cairo city, 30°N.
2. STEADY STATE MODEL T h e following a s s u m p t i o n s are considered while developing the steady state model: 1. Solar irradiance is assumed to be c o n s t a n t for short intervals (6 minutes). This assumption is practically acceptable because solar irradiance varies at a low rate. 2. DC m o t o r rotational losses are neglected, thus, motor torque is equal to p u m p torque. 3. Solar radiation model of Cairo is considered. The system consists o f three basic units: a PV array, a DC motor, a n d a centrifugal p u m p . T h e relationship between the array current a n d voltage is given by: la = ILoG -- l o [ e x p ( ( V + I a R s ) / V x ) -(V+
-
1]
laRs)/Rsh
(1)
where: la is the PV array current (A), ILo is the light generated current of the PV array at G = 1 KW/m 2, G is the solar irradiance (KW/m2), 1o is the PV array reverse saturation current (A), V is the PV array operating voltage (V), VT is the thermal voltage of the PV array (V), Rs is the series resistance of the PV array (ft), and Rsh is the shunt resistance of the PV array (f~). The D C m o t o r considered is of the p e r m a n e n t m a g n e t type. For this type of D C motor, n o power is 369
W. ANIS and H. M. B. METWALLY
370
DC meier
Neglecting m o t o r - p u m p coupling losses, one may equate both motor and p u m p torques. Also the array current and voltage are equal to the motor current and voltage respectively as seen in Fig. 1. Thus the following equations hold
Water tank
t,~ier distribution netw~)rk Fig. 1. Schematic diagram of a PV powered pumping system.
Tm = G
(5)
Im = I.
(6)
Vm = V
(7)
Combining eqns (2), (6), and (7) then,
V = IaRa + Km~o,.. required for the field winding excitation. Accordingly, the energy generated by the PV array is partially saved. U n d e r steady state conditions the characteristics of the permanent magnet DC motor are determined by the following equations:
Vm = G R . + Km~om
(2)
Tm =Kmlm
(3)
(8)
Using eqns (3), (4), and (5) yields
Kow2m = Kmla
(9)
The amount of pumped water is determined by the following relationship:
Qpgh
where:
=
?'/promT m
(10)
where:
V., is the motor operating voltage (V), lm is the motor current (A), R. is the armature and cable resistance (fl), Km is the motor constant (V/sec per rad), Tm is the motor torque (Nm), and ~0m is the motor speed (rad/sec).
Q g h p %
rate of water discharge (m3/sec), gravitational acceleration (m/sec2), water head (m), water density (Kg/m 3), and pump efficiency.
The three nonlinear algebraic eqns ( 1 ), (8), and (9) enable one to calculate the three unknowns V, la, and w,, with G as a parameter. These equations may be solved at a given instant, where G is considered constant, to obtain the steady state operating parameters of the system. The N e w t o n - R a p h s o n method for solving nonlinear simultaneous equations is used. The choice of N e w t o n - R a p h s o n method is found to be satisfactory where no convergence problem is observed. The computation time is quite short and in most cases
The torque required to drive the centrifugal pump is proportional to the square of its speed as has been shown by Braunstein and Kornfeld (1981). Thus, Tp = Kp~o~
is the is the is the is the is the
(4)
where: Tp is the torque required to drive the pump (Nm), and Kp is a constant for a given centrifugal pump.
20
G=835 15-~ G = 7 2 3
at 12 am atl0am
i \
G = 590 at ' am .........
: : .........
• : .........
°
.........
0
0
,
I
10
20
i :-.-~.
~,~O/L~'I? ~ C~'/
::
~ L . ,t -: ...... a ..........
. : .........
o'I\\
i
.
30 40 Voltage (V)
.
.
.
.
.
.
.
50
.
.
.
.
.
.
.
.
60
70
Fig. 2. The operating characteristics of the system during the vernal equinox (2 1~ of March ). Solar irradiance (G) is in W/m 2.
Directly coupled PV pumping system
371
120 100....
~soC
@ Iii
O
e~~
+
.......
....... i ...............................
i ...... i....... i ........
60-
40200
5
7
9
11
13
15
17
19
21
Day Time (Hours) Fig. 3. Variations of the matching efficiencyduring the vernal equinox.
the solution is reached within 30 iterations. The accuracy of the solution may be improved at the expense of increasing the computation time. As mentioned above, the extreme days of the year are considered. For each day, the period between sunrise and sunset is divided into intervals of 6 minutes. The choice of 6 minute intervals is due to the fact that solar irradiance (G) doesn't change appreciably during that period, hence one may consider G almost constant. The system operating parameters V, la, and ~0m,which represent the steady state response of the system, are computed at each interval. 3. DYNAMIC M O D E L
3. the time during the day, and 4. existence or nonexistence of clouds. All the above factors are considered in the developed model for Cairo. Considering equations (5), (6), and (7), and neglecting the system friction, the equations of the system under transient conditions are:
dla+ Kmo~m V = IaRa + Lm --~ do) m
Tm= Tp + J d---i-
80
L,, is the inductance of the DC motor armature winding (Henry), and J is the moment of inertia of the motor pump system (Nm/ see2).
:
1,000
Solar immdiance
o.iiiiiI
iiiiiii-° i_oi
16
-200
0
0
4
6
(12)
where:
Actually the solar irradiance (G) is continuously varying and its value depends on the following parameters: I. climatic conditions of the location, 2. day and season,
~
(11)
8
10
12
14
16
18
20
Day Time (Hours) Fig. 4. Variations of the solar irradiance (G), voltage, current, and speed of the De motor during the vernal equinox (21 = of March).
372
W. ANISand H. M. B, METWALLY 80
:
1,000
Solar in'adiance
O48-
-600
g
,-,
g 16
-20o
0
¢9
0 4
6
8
10
12
14
16
18
20
Day Time (Hours)
Fig. 5. Variations of the solar irradiance (G), voltage, current, and speed of the DC motor during the summer solistice (21 st of June ).
Substituting Tm and To from equations (3) and (4) into equation (12) and using equation (6) results in: do.) m
(13)
J-'~'f- = Kmla - Kp~2m
Equation ( 1), together with the two first-order differential equations ( 11 ) and (13), represent the dynamic mathematical model of the system. The dynamic performance of the system is obtained by solving these three equations numerically to obtain the instantaneous values of the motor voltage, current, and speed. The two, simultaneous, first-order differential equations ( 11 ) and (13) are solved using the RungeKutta fourth-order method to obtain the motor current and speed. During the numerical integration process, the instantaneous values of the motor voltage are obtained by solving the nonlinear array (eqn 1) numer-
80 !
~,.d
:
.i
" ~ 4 ~ v
......
-
r-
]-- 1,000
Solar irradiance
q
~,~
ically using the Newton-Raphson method. As seen from Fig. 2, for a given value of G, the array current is almost constant for a wide range of voltage. So, small changes in current produce large changes in voltage, specially at low values of solar irradiance (G). This necessitates a very small step length for the RungeKutta method to avoid oscillation in the solution. It is found that a step length shorter than 100 microseconds is necessary to get a reliable solution. A computer simulation program is developed to obtain the steady state and the dynamic performance of the system. The program is general in the sense that it can be used to obtain either the steady state or the dynamic performance of the system at any given instant during the year. The program can also compute the dynamic performance of the system under sudden cloud changes which cause abrupt variations in solar irradianee (G).
;
~
-
:
/......",~ .
.
.
.
: . . . . . :. . . . •
.
]
......
Y
:
,.Iz
......
I
:. . . . . . .
i i 800 i
"~ E
........ i °
ol 4
•
;
i
i
:
i
6
8
10
12
14
16
..... i -..o 18
20
D a y T i m e (Hours) Fig. 6. Variations of the solar irradianee (G), voltage, current, and speed of the DC motor during the
autumnal equinox (21 = of September).
373
Directly coupled PV pumping system
8°1
i
/
="
1,0oo
~olar irradiance
16
...... !.........
Cut ei-it . . . . . . . . . . .
200
i. . . . . .
0
0 4
6
8
10 12 14 Day Time (Hours)
16
18
20
Fig. 7. Variations of the solar irradiance (G), voltage, current, and speed of the DC motor during the winter solistice (21 ,t of December).
4. SYSTEM CONSTANTS
The system investigated hereafter has the following constants: • DC motor: The rated voltage is 48 V, the rated current is 20 A, K m = 0.85 V / s e c per rad, Ra = 0.3 f~ and Lm = 2.2 mH. • Pump: The pump constant Kp = 0.0052 N m / s e c 2 per rad and the m o m e n t of inertia J = 0.136 N m / sec2 per rad. • PV Array: The PV array is tilted at 30 ° facing south and its constants are: ILo = 20 A, VT = 2.808 V, Rs = 0.3 f~ and Rsh = 150 Q. 5. RESULTS
The solar irradiance (G) is computed according to the technique described by Klein ( 1977 ). The detailed simulation program developed has considered the following factors: 1. The effect of temperature variations at each moment according to the climatic data of Cairo city.
2. The effect of clouds. 3. Series and shunt resistances of the PV array. Figure 2 shows the operating points of the system as well as the m a x i m u m power points of the PV array during the vernal equinox day. Figure 3 shows that the matching efficiency of the system exceeds 90% between 9 a.m. and 3 p.m. As a matter of fact, most of the daily solar radiation takes place during this period (over 70% of the daily energy). Hence the matching efficiency of the daily energy, defined as the ratio of energy transferred to the load to the m a x i m u m available PV array energy, is fairly high. Attention must be paid to the fact that each curve of the characteristics shown in Fig. 2 has a different solar irradiance, indicated on the figure, and a different solar cell temperature (not indicated) and they are not drawn at constant temperature as is often done in most of the publications. The solar cell temperature is computed according to the method described by Green (1982). Figures 4, 5, 6, and 7 show the steady state variations of solar irradiance (G), current, voltage, and speed
2.5
2.4
I ......
v
2.3 O
2.2
.............................
Final (G) = 125 Wlm2 [nltlal Current' 15.21Amp. ~ i ~ i CuiT0nF.= 2.~Ai~i~.
2.1 -0,5
0.0
0.5
1,0
1.5 2.0 Time (Sec.)
2.5
3.0
3.5
4.0
Fig. 8. Variations of motor current when a sudden change in clouds takes place at noon during the vernal equinox.
W. ANISand H. M. B. METWALLY
374 60--
50 . . . . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . i . . . . .
.
.
.
.
.
.
.
I\ ! l...~.. L
"--3o ......
.
.
.
.
.
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.
.
.
.
.
.
.
"" ". . . . . . . ":. . . . . . .
.
. . . . .
! !
! !
Rrml (G) ~ 125 W/m2 ,n~l,a vona~e , ~.9 voa
1.0
1.5
100-
-0,5
0.0
0.5
2.0
2.5
3.0
3.5
4.0
Time (See.)
Fig.9. Variationsof motorvoltagewhena suddenchangein cloudstakesplaceat noonduringthe vernal equinox.
during the vernal equinox, the summer solistice, the autumnal equinox and the winter solistice respectively. It is seen that these DC motor parameters follow the solar irradiance. The operating voltage, current, and speed are generally less than their rated values, except at noon when they reach these values. The PV array designer should take into consideration that the DC motor must operate within the rated values of current and voltage to avoid motor damage. The sudden rise and fall of the parameters at sunrise and sunset respectively during winter solistice day (indicated in Fig. 7 ) is due to the selected tilt angle of the PV array which makes the solar irradiance of significant value (greater than zero) when sun rays first touch the PV array surface. Motor current, voltage, and speed rise rapidly, but not abruptly, as will be discussed in the transient analysis given later. Figures 8, 9 and 10 show the transient performance of the motor current, voltage, and speed when solar irradiance (G) drops suddenly from 835 W / m 2 at
noontime to 125 W / m 2 due to sudden coverage of the sky by clouds. This change in solar irradiance is computed according to the worst climatic conditions of Cairo. It can be seen from Fig. 8 that the motor current drops abruptly to a value less than its steady value (a negative overshoot) then it creeps up towards the steady state value. It is seen that the current reaches its steady state value within 4 seconds. Figure 9 depicts the motor voltage drop due to sudden drop of solar irradiance. The voltage is redrawn in Fig. 11 using a semi-log scale to expand the first milliseconds. It can be seen that the voltage drops very fast at first, then it increases, and afterwards it drops slowly to the new steady state value. Figure 10 illustrates the motor speed variations, no overshoot is observed and the steady state speed is obtained in about 4 seconds. Figures 12, 13, 14, and 15 are similar to Figs. 8, 9, 10, and I 1 but show the transient performance of the moter current, voltage, and speed when solar irradiance rises abruptly due to sudden cloud clearance. Figure
60O 500
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l.nltlal (G)
E40 O-
.......
~.300¢/)
....... !:
.\. ...........................
\
.
.
.
.
.
.
.
.
.
.
.
.
.
.
= 835 Wlm2
.FJ.~J.. (..G). . . . . = . ! ~ .
W/.m2.
i
i
i
Inltisl Speed ~ 476,1 rpm
: " :
~i
:i
F'nal i ~ i" "ed=I I 18~:5:" i r'pm
0.5
i 1.0
1.5
200100 -0.5
0.0
2.0
2.5
3.0
3.5
4.0
Time (Sec.) Fig. 10. Variations of motor speed when a sudden change in clouds takes place at noon during the vernal equinox.
Directly coupled PV pumping system
375
60-
50 . . . . . . . . .
:
Initial (G)
: . . . . . . . . :. . . . . . . . . . . . . . . . .
F.inai .(Ca.) = l~.W/m2 nifia:lV-oltag~ = 46.9 V o l t
:
= 835iW/m2
e•40 10 0
0.001
0.01
0.1
1
10
100
1,000 10,000 100,000
Time (m Sec,) Fig. 11. Variations of motor voltage when sudden change in clouds takes place at noon during the vernal equinox.
15 shows that the voltage rises very fast at first, then it decreases, and finally creeps to the steady state value. Speed variation is shown in Fig. 14 where, it gradually approaches the steady state value without overshoot. It should be noted that, in order to draw Figs. 11 and 15, it is assumed that the sudden change in solar irradiance takes place at 0.01 m/sec and not at the zero time. This is because the zero time can not be obtained on the log-scale. It is important here to mention two points, the first concerning the calculation of water discharge and the second related to the system losses. As can be seen from eqn (10), water discharge is proportional to the mechanical power of the motor which, in turn, is proportional to the solar power available. Thus, the general shape of the water discharge curve is expected to follow that of the solar irradiance (G). Exact computation of water discharge necessitates determining the pump efficiency at various speeds. This point will be studied in detail in a future work. Concerning the second point,
motor copper loss and cabling loss are considered in the model. If core loss and motor-pump coupling losses are taken into consideration, the motor torque available to drive the pump will be slightly decreased and hence both motor speed and water discharge will somewhat be decreased.
6. C O N C L U S I O N S
The transient performance of the directly coupled PV pumping system is thoroughly investigated in this work. The detailed model developed shows that the system attains the steady state conditions in a relatively short period and the motor operating conditions always remain within the rated values during both the steady state and the transient periods. The study revealed that the computation of water discharge during sunny days, based on the steady state model, doesn't involve an appreciable error although the solar irradiance is continuously varying. The reasons for this result are:
17,0
16,5 . . . . . . .
~ ' "
Initial (G)
,= 1251Wlm2
Final
!= 835;W/m2:
i .......
('G.)
ez ,hii: 2, :xmP::
.......
v
le.O~
......
:3
o
15.5
.......
15.0 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
I
t
1.2
1.4
.6
Time (Sec.) Fig. 12. Variations of motor current when sudden change in clouds takes place at noon during the vernal equinox.
376
W. ANIS and H. M. B. METWALLY
70 6O
'~
O
O~40-
/ 20-
......
(G) ! = 12s W/m~,
/ ~ .
&Rr.! i.(a)., i . =. 83..sW~.m2.......
:,;::vo:
i :
: i
I
I
I
0.8
1.0
1.2
1.4
10 -0.2
0.0
0.2
0.4
0.6
1.6
T i m e (Sec.) Fig. 13. Variations of motor voltage when a sudden change in clouds takes place at noon during the vernal equinox.
600
500
.
~
.
.
....... i .
.
i 0.0
-0.2
.
i / . : ~ :/ i -J . . . . . . ~ :
200 . . . . . . .
100
.
0.2
.
.
.
.
.
i.. -..~.~..wl..~2 ......
i Final i (G) ! = 8 3 5 W / r n 2 i i Inltla! S p e e d = 188.5 r p m . . . . . . . FJr~l.> S p e e d . - = . 4 7 6 . 1 - r p m . . . . . .
0.4
0.6
i 0.8
t 1.0
~ 1.2
a 1.4
1.6
T i m e (Sec.) Fig. 14. Variations of motor speed when a sudden change in clouds takes place at noon during the vernal equinox.
70 :
60
~
. . . . . . . i . . . . . . . . i. . . . . . . . . . . . . . . . . . . .
.......
.
10 0.001
.
.
.
0.01
.
iiiii .
.
.
Initial (G) . = 125:Wlm2 F'Jnat..(G).. " ~, .835.:Wlm2. Initial Voltage = 17.~i Volt Final Voltagie - 46.9 Volt
•
i
'
~ 0.1
, 100
10
1,000
10,000 100,000
Time (m Sec.) Fig. 15. Variations of motor voltage when a sudden change in clouds takes place at noon during the vernal equinox.
Directly coupled PV pumping system 1. solar irradiance variation is relatively slow, hence one may assume that solar irradiance is almost constant within a few minutes, and 2. the system response is fast so the transient period is relatively short (a few seconds). However, for days with strongly fluctuating clouds, the dynamic model gives more accurate results for water discharge. Hence, the dynamic model must be used if the cloud fluctuation period is comparable with the transient period of the system. Another important conclusion based on this analysis is that the matching efficiency of the directly coupled PV pumping system is high at high solar irradiance. The matching efficiency exceeds 90% during the period from 9 a.m. to 3 p.m. Early in the morning or late in the afternoon, where solar irradiance is low, the matching efficiency is poor, i.e., less than 70%. However the overall matching efficiency is satisfactorily high because the solar energy is concentrated during high solar irradiance period.
377 REFERENCES
W. Anis, R. Mertens, and R. Vas Overstaeten, Coupling of a volumetric pump to a photovoltaic array, Solar Cell 14, 27-42 (1985). J. Appelbaum and J. Bany, Analysis of a direct coupled DC motor and a photovoltaic converter, I s~ Commission of European Community Conf on PhotovoltaicSolar Energy, Luxembourg, Sept. 27-30, Reidel, Dordecht, Netherlands (1979). J. Appelbaum and J. Bany, Performance analysis of DC motor photovoltaic converter system--I, Solar Energy 22, 439445 (1979). A. Braunstein and A. Kornfeld, Analysis of solar powered electric water pumps, Solar Energ.v 27(3), 235-240 (1981). M. Green, Solar cells: Operating principles, technology, and system applications, Prentice-Hall, Englewood Cliffs, NJ (1982). S. Klein, Calculation of monthly average insolation on tilted surfaces, Solar Energy 19, 325-329 (1977 ). J. Roger, Theory of the direct coupling between DC motors and photovoltaic solar arrays, Solar Energy 23, 193-198 (1979).
Pergamon
0038-092X(94)00051-4
DYNAMIC PERFORMANCE OF A DIRECTLY COUPLED PV PUMPING SYSTEM WAGDY R. ANIS *t a n d HAMID M. B. METWALLY** *Electronics and Communications Dept., Faculty of Engineering, Ain Shams University, 1 Sarayat St., Abbasia, Cairo, Egypt; * *Electrical Eng. Dept., Faculty of Engineering, Zagazig University, Zagazig, Egypt Abstract--Although directly coupled Photovoltaic (PV) pumping systems have been extensively studied under steady state conditions, much less attention has been paid to the study of their transient performance. Actually the system is performing dynamically all the time because the solar irradiance is varying continuously. Thus, the steady state solution for the system performance is only an approximation. In this work both dynamic and steady state models are developed for a directly coupled PV pumping system. The results obtained by a detailed computer simulation program based on the two models are compared. The system is studied under external interruptions caused by sudden appearance or disappearance of clouds. The study showed that there are overshoots of both current and voltage of the DC motor. However, the characteristics of the PV array limit the overshoots during the transient period. The motor speed creeps steadily towards its steady state value without overshoot.
I. I NTRODUCI~ION A directly coupled PV p u m p i n g system is c o m p o s e d of a PV array directly coupled to a DC m o t o r driving a centrifugal p u m p as shown in Fig. 1. Thus, such a system is simple, reliable, a n d of low cost because it does not include battery storage or a battery voltage regulator. T h e system simply stores water instead of storing electrical energy. T h e advantages of this system led to its widespread use over the world. As a result, the system has been thoroughly investigated in previous literature. A p p e l b a u m a n d Bany ( 1 9 7 9 a ) a n d ( 1 9 7 9 b ) analyzed the directly coupled PV p u m p i n g system under the steady state conditions. Roger ( 1 9 7 9 ) proved that a D C m o t o r driving a centrifugal p u m p represents a well-matched load to a PV array because this system utilizes most of the available D C power generated by the array. Anis et al. ( 1985 ), reported that a load composed o f a D C m o t o r driving a volumetric p u m p represents a n o n - m a t c h e d load to a PV array. This is because the m o t o r driving a volumetric p u m p requires an almost c o n s t a n t current, for a given head, apart from the starting current which tends to be higher. This c o n d i t i o n does not m a t c h the PV array characteristics where the current varies almost linearly with solar irradiance. A major advantage of the PV p u m p i n g system is that it is naturally m a t c h e d with solar radiation since, in most cases, water d e m a n d increases during s u m m e r w h e n solar radiation is m a x i m u m . In this work, the transient b e h a v i o r of the directly coupled PV p u m p i n g system is thoroughly investigated. The study includes extreme cases when solar irradiance changes abruptly, from high to low or vice versa, due to sudden appearance or disappearance o f clouds. T h e * Correspondence should be addressed to Dr. Wagdy Anis, Yanbu Industrial College, P.O. Box 30436, Yanbu al Sinaiyah 21477, Kingdom of Saudi Arabia.
developed program enables one to obtain transient a n d / o r steady state response of the system at any instant during the whole year. This study considers the four extreme days of the year as examples, i.e., vernal a n d a u t u m n a l equinox, a n d s u m m e r a n d winter solstice days in Cairo city, 30°N.
2. STEADY STATE MODEL T h e following a s s u m p t i o n s are considered while developing the steady state model: 1. Solar irradiance is assumed to be c o n s t a n t for short intervals (6 minutes). This assumption is practically acceptable because solar irradiance varies at a low rate. 2. DC m o t o r rotational losses are neglected, thus, motor torque is equal to p u m p torque. 3. Solar radiation model of Cairo is considered. The system consists o f three basic units: a PV array, a DC motor, a n d a centrifugal p u m p . T h e relationship between the array current a n d voltage is given by: la = ILoG -- l o [ e x p ( ( V + I a R s ) / V x ) -(V+
-
1]
laRs)/Rsh
(1)
where: la is the PV array current (A), ILo is the light generated current of the PV array at G = 1 KW/m 2, G is the solar irradiance (KW/m2), 1o is the PV array reverse saturation current (A), V is the PV array operating voltage (V), VT is the thermal voltage of the PV array (V), Rs is the series resistance of the PV array (ft), and Rsh is the shunt resistance of the PV array (f~). The D C m o t o r considered is of the p e r m a n e n t m a g n e t type. For this type of D C motor, n o power is 369
W. ANIS and H. M. B. METWALLY
370
DC meier
Neglecting m o t o r - p u m p coupling losses, one may equate both motor and p u m p torques. Also the array current and voltage are equal to the motor current and voltage respectively as seen in Fig. 1. Thus the following equations hold
Water tank
t,~ier distribution netw~)rk Fig. 1. Schematic diagram of a PV powered pumping system.
Tm = G
(5)
Im = I.
(6)
Vm = V
(7)
Combining eqns (2), (6), and (7) then,
V = IaRa + Km~o,.. required for the field winding excitation. Accordingly, the energy generated by the PV array is partially saved. U n d e r steady state conditions the characteristics of the permanent magnet DC motor are determined by the following equations:
Vm = G R . + Km~om
(2)
Tm =Kmlm
(3)
(8)
Using eqns (3), (4), and (5) yields
Kow2m = Kmla
(9)
The amount of pumped water is determined by the following relationship:
Qpgh
where:
=
?'/promT m
(10)
where:
V., is the motor operating voltage (V), lm is the motor current (A), R. is the armature and cable resistance (fl), Km is the motor constant (V/sec per rad), Tm is the motor torque (Nm), and ~0m is the motor speed (rad/sec).
Q g h p %
rate of water discharge (m3/sec), gravitational acceleration (m/sec2), water head (m), water density (Kg/m 3), and pump efficiency.
The three nonlinear algebraic eqns ( 1 ), (8), and (9) enable one to calculate the three unknowns V, la, and w,, with G as a parameter. These equations may be solved at a given instant, where G is considered constant, to obtain the steady state operating parameters of the system. The N e w t o n - R a p h s o n method for solving nonlinear simultaneous equations is used. The choice of N e w t o n - R a p h s o n method is found to be satisfactory where no convergence problem is observed. The computation time is quite short and in most cases
The torque required to drive the centrifugal pump is proportional to the square of its speed as has been shown by Braunstein and Kornfeld (1981). Thus, Tp = Kp~o~
is the is the is the is the is the
(4)
where: Tp is the torque required to drive the pump (Nm), and Kp is a constant for a given centrifugal pump.
20
G=835 15-~ G = 7 2 3
at 12 am atl0am
i \
G = 590 at ' am .........
: : .........
• : .........
°
.........
0
0
,
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10
20
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~,~O/L~'I? ~ C~'/
::
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30 40 Voltage (V)
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50
.
.
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.
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.
60
70
Fig. 2. The operating characteristics of the system during the vernal equinox (2 1~ of March ). Solar irradiance (G) is in W/m 2.
Directly coupled PV pumping system
371
120 100....
~soC
@ Iii
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e~~
+
.......
....... i ...............................
i ...... i....... i ........
60-
40200
5
7
9
11
13
15
17
19
21
Day Time (Hours) Fig. 3. Variations of the matching efficiencyduring the vernal equinox.
the solution is reached within 30 iterations. The accuracy of the solution may be improved at the expense of increasing the computation time. As mentioned above, the extreme days of the year are considered. For each day, the period between sunrise and sunset is divided into intervals of 6 minutes. The choice of 6 minute intervals is due to the fact that solar irradiance (G) doesn't change appreciably during that period, hence one may consider G almost constant. The system operating parameters V, la, and ~0m,which represent the steady state response of the system, are computed at each interval. 3. DYNAMIC M O D E L
3. the time during the day, and 4. existence or nonexistence of clouds. All the above factors are considered in the developed model for Cairo. Considering equations (5), (6), and (7), and neglecting the system friction, the equations of the system under transient conditions are:
dla+ Kmo~m V = IaRa + Lm --~ do) m
Tm= Tp + J d---i-
80
L,, is the inductance of the DC motor armature winding (Henry), and J is the moment of inertia of the motor pump system (Nm/ see2).
:
1,000
Solar immdiance
o.iiiiiI
iiiiiii-° i_oi
16
-200
0
0
4
6
(12)
where:
Actually the solar irradiance (G) is continuously varying and its value depends on the following parameters: I. climatic conditions of the location, 2. day and season,
~
(11)
8
10
12
14
16
18
20
Day Time (Hours) Fig. 4. Variations of the solar irradiance (G), voltage, current, and speed of the De motor during the vernal equinox (21 = of March).
372
W. ANISand H. M. B, METWALLY 80
:
1,000
Solar in'adiance
O48-
-600
g
,-,
g 16
-20o
0
¢9
0 4
6
8
10
12
14
16
18
20
Day Time (Hours)
Fig. 5. Variations of the solar irradiance (G), voltage, current, and speed of the DC motor during the summer solistice (21 st of June ).
Substituting Tm and To from equations (3) and (4) into equation (12) and using equation (6) results in: do.) m
(13)
J-'~'f- = Kmla - Kp~2m
Equation ( 1), together with the two first-order differential equations ( 11 ) and (13), represent the dynamic mathematical model of the system. The dynamic performance of the system is obtained by solving these three equations numerically to obtain the instantaneous values of the motor voltage, current, and speed. The two, simultaneous, first-order differential equations ( 11 ) and (13) are solved using the RungeKutta fourth-order method to obtain the motor current and speed. During the numerical integration process, the instantaneous values of the motor voltage are obtained by solving the nonlinear array (eqn 1) numer-
80 !
~,.d
:
.i
" ~ 4 ~ v
......
-
r-
]-- 1,000
Solar irradiance
q
~,~
ically using the Newton-Raphson method. As seen from Fig. 2, for a given value of G, the array current is almost constant for a wide range of voltage. So, small changes in current produce large changes in voltage, specially at low values of solar irradiance (G). This necessitates a very small step length for the RungeKutta method to avoid oscillation in the solution. It is found that a step length shorter than 100 microseconds is necessary to get a reliable solution. A computer simulation program is developed to obtain the steady state and the dynamic performance of the system. The program is general in the sense that it can be used to obtain either the steady state or the dynamic performance of the system at any given instant during the year. The program can also compute the dynamic performance of the system under sudden cloud changes which cause abrupt variations in solar irradianee (G).
;
~
-
:
/......",~ .
.
.
.
: . . . . . :. . . . •
.
]
......
Y
:
,.Iz
......
I
:. . . . . . .
i i 800 i
"~ E
........ i °
ol 4
•
;
i
i
:
i
6
8
10
12
14
16
..... i -..o 18
20
D a y T i m e (Hours) Fig. 6. Variations of the solar irradianee (G), voltage, current, and speed of the DC motor during the
autumnal equinox (21 = of September).
373
Directly coupled PV pumping system
8°1
i
/
="
1,0oo
~olar irradiance
16
...... !.........
Cut ei-it . . . . . . . . . . .
200
i. . . . . .
0
0 4
6
8
10 12 14 Day Time (Hours)
16
18
20
Fig. 7. Variations of the solar irradiance (G), voltage, current, and speed of the DC motor during the winter solistice (21 ,t of December).
4. SYSTEM CONSTANTS
The system investigated hereafter has the following constants: • DC motor: The rated voltage is 48 V, the rated current is 20 A, K m = 0.85 V / s e c per rad, Ra = 0.3 f~ and Lm = 2.2 mH. • Pump: The pump constant Kp = 0.0052 N m / s e c 2 per rad and the m o m e n t of inertia J = 0.136 N m / sec2 per rad. • PV Array: The PV array is tilted at 30 ° facing south and its constants are: ILo = 20 A, VT = 2.808 V, Rs = 0.3 f~ and Rsh = 150 Q. 5. RESULTS
The solar irradiance (G) is computed according to the technique described by Klein ( 1977 ). The detailed simulation program developed has considered the following factors: 1. The effect of temperature variations at each moment according to the climatic data of Cairo city.
2. The effect of clouds. 3. Series and shunt resistances of the PV array. Figure 2 shows the operating points of the system as well as the m a x i m u m power points of the PV array during the vernal equinox day. Figure 3 shows that the matching efficiency of the system exceeds 90% between 9 a.m. and 3 p.m. As a matter of fact, most of the daily solar radiation takes place during this period (over 70% of the daily energy). Hence the matching efficiency of the daily energy, defined as the ratio of energy transferred to the load to the m a x i m u m available PV array energy, is fairly high. Attention must be paid to the fact that each curve of the characteristics shown in Fig. 2 has a different solar irradiance, indicated on the figure, and a different solar cell temperature (not indicated) and they are not drawn at constant temperature as is often done in most of the publications. The solar cell temperature is computed according to the method described by Green (1982). Figures 4, 5, 6, and 7 show the steady state variations of solar irradiance (G), current, voltage, and speed
2.5
2.4
I ......
v
2.3 O
2.2
.............................
Final (G) = 125 Wlm2 [nltlal Current' 15.21Amp. ~ i ~ i CuiT0nF.= 2.~Ai~i~.
2.1 -0,5
0.0
0.5
1,0
1.5 2.0 Time (Sec.)
2.5
3.0
3.5
4.0
Fig. 8. Variations of motor current when a sudden change in clouds takes place at noon during the vernal equinox.
W. ANISand H. M. B. METWALLY
374 60--
50 . . . . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . i . . . . .
.
.
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I\ ! l...~.. L
"--3o ......
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"" ". . . . . . . ":. . . . . . .
.
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! !
! !
Rrml (G) ~ 125 W/m2 ,n~l,a vona~e , ~.9 voa
1.0
1.5
100-
-0,5
0.0
0.5
2.0
2.5
3.0
3.5
4.0
Time (See.)
Fig.9. Variationsof motorvoltagewhena suddenchangein cloudstakesplaceat noonduringthe vernal equinox.
during the vernal equinox, the summer solistice, the autumnal equinox and the winter solistice respectively. It is seen that these DC motor parameters follow the solar irradiance. The operating voltage, current, and speed are generally less than their rated values, except at noon when they reach these values. The PV array designer should take into consideration that the DC motor must operate within the rated values of current and voltage to avoid motor damage. The sudden rise and fall of the parameters at sunrise and sunset respectively during winter solistice day (indicated in Fig. 7 ) is due to the selected tilt angle of the PV array which makes the solar irradiance of significant value (greater than zero) when sun rays first touch the PV array surface. Motor current, voltage, and speed rise rapidly, but not abruptly, as will be discussed in the transient analysis given later. Figures 8, 9 and 10 show the transient performance of the motor current, voltage, and speed when solar irradiance (G) drops suddenly from 835 W / m 2 at
noontime to 125 W / m 2 due to sudden coverage of the sky by clouds. This change in solar irradiance is computed according to the worst climatic conditions of Cairo. It can be seen from Fig. 8 that the motor current drops abruptly to a value less than its steady value (a negative overshoot) then it creeps up towards the steady state value. It is seen that the current reaches its steady state value within 4 seconds. Figure 9 depicts the motor voltage drop due to sudden drop of solar irradiance. The voltage is redrawn in Fig. 11 using a semi-log scale to expand the first milliseconds. It can be seen that the voltage drops very fast at first, then it increases, and afterwards it drops slowly to the new steady state value. Figure 10 illustrates the motor speed variations, no overshoot is observed and the steady state speed is obtained in about 4 seconds. Figures 12, 13, 14, and 15 are similar to Figs. 8, 9, 10, and I 1 but show the transient performance of the moter current, voltage, and speed when solar irradiance rises abruptly due to sudden cloud clearance. Figure
60O 500
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l.nltlal (G)
E40 O-
.......
~.300¢/)
....... !:
.\. ...........................
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.
= 835 Wlm2
.FJ.~J.. (..G). . . . . = . ! ~ .
W/.m2.
i
i
i
Inltisl Speed ~ 476,1 rpm
: " :
~i
:i
F'nal i ~ i" "ed=I I 18~:5:" i r'pm
0.5
i 1.0
1.5
200100 -0.5
0.0
2.0
2.5
3.0
3.5
4.0
Time (Sec.) Fig. 10. Variations of motor speed when a sudden change in clouds takes place at noon during the vernal equinox.
Directly coupled PV pumping system
375
60-
50 . . . . . . . . .
:
Initial (G)
: . . . . . . . . :. . . . . . . . . . . . . . . . .
F.inai .(Ca.) = l~.W/m2 nifia:lV-oltag~ = 46.9 V o l t
:
= 835iW/m2
e•40 10 0
0.001
0.01
0.1
1
10
100
1,000 10,000 100,000
Time (m Sec,) Fig. 11. Variations of motor voltage when sudden change in clouds takes place at noon during the vernal equinox.
15 shows that the voltage rises very fast at first, then it decreases, and finally creeps to the steady state value. Speed variation is shown in Fig. 14 where, it gradually approaches the steady state value without overshoot. It should be noted that, in order to draw Figs. 11 and 15, it is assumed that the sudden change in solar irradiance takes place at 0.01 m/sec and not at the zero time. This is because the zero time can not be obtained on the log-scale. It is important here to mention two points, the first concerning the calculation of water discharge and the second related to the system losses. As can be seen from eqn (10), water discharge is proportional to the mechanical power of the motor which, in turn, is proportional to the solar power available. Thus, the general shape of the water discharge curve is expected to follow that of the solar irradiance (G). Exact computation of water discharge necessitates determining the pump efficiency at various speeds. This point will be studied in detail in a future work. Concerning the second point,
motor copper loss and cabling loss are considered in the model. If core loss and motor-pump coupling losses are taken into consideration, the motor torque available to drive the pump will be slightly decreased and hence both motor speed and water discharge will somewhat be decreased.
6. C O N C L U S I O N S
The transient performance of the directly coupled PV pumping system is thoroughly investigated in this work. The detailed model developed shows that the system attains the steady state conditions in a relatively short period and the motor operating conditions always remain within the rated values during both the steady state and the transient periods. The study revealed that the computation of water discharge during sunny days, based on the steady state model, doesn't involve an appreciable error although the solar irradiance is continuously varying. The reasons for this result are:
17,0
16,5 . . . . . . .
~ ' "
Initial (G)
,= 1251Wlm2
Final
!= 835;W/m2:
i .......
('G.)
ez ,hii: 2, :xmP::
.......
v
le.O~
......
:3
o
15.5
.......
15.0 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
I
t
1.2
1.4
.6
Time (Sec.) Fig. 12. Variations of motor current when sudden change in clouds takes place at noon during the vernal equinox.
376
W. ANIS and H. M. B. METWALLY
70 6O
'~
O
O~40-
/ 20-
......
(G) ! = 12s W/m~,
/ ~ .
&Rr.! i.(a)., i . =. 83..sW~.m2.......
:,;::vo:
i :
: i
I
I
I
0.8
1.0
1.2
1.4
10 -0.2
0.0
0.2
0.4
0.6
1.6
T i m e (Sec.) Fig. 13. Variations of motor voltage when a sudden change in clouds takes place at noon during the vernal equinox.
600
500
.
~
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....... i .
.
i 0.0
-0.2
.
i / . : ~ :/ i -J . . . . . . ~ :
200 . . . . . . .
100
.
0.2
.
.
.
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.
i.. -..~.~..wl..~2 ......
i Final i (G) ! = 8 3 5 W / r n 2 i i Inltla! S p e e d = 188.5 r p m . . . . . . . FJr~l.> S p e e d . - = . 4 7 6 . 1 - r p m . . . . . .
0.4
0.6
i 0.8
t 1.0
~ 1.2
a 1.4
1.6
T i m e (Sec.) Fig. 14. Variations of motor speed when a sudden change in clouds takes place at noon during the vernal equinox.
70 :
60
~
. . . . . . . i . . . . . . . . i. . . . . . . . . . . . . . . . . . . .
.......
.
10 0.001
.
.
.
0.01
.
iiiii .
.
.
Initial (G) . = 125:Wlm2 F'Jnat..(G).. " ~, .835.:Wlm2. Initial Voltage = 17.~i Volt Final Voltagie - 46.9 Volt
•
i
'
~ 0.1
, 100
10
1,000
10,000 100,000
Time (m Sec.) Fig. 15. Variations of motor voltage when a sudden change in clouds takes place at noon during the vernal equinox.
Directly coupled PV pumping system 1. solar irradiance variation is relatively slow, hence one may assume that solar irradiance is almost constant within a few minutes, and 2. the system response is fast so the transient period is relatively short (a few seconds). However, for days with strongly fluctuating clouds, the dynamic model gives more accurate results for water discharge. Hence, the dynamic model must be used if the cloud fluctuation period is comparable with the transient period of the system. Another important conclusion based on this analysis is that the matching efficiency of the directly coupled PV pumping system is high at high solar irradiance. The matching efficiency exceeds 90% during the period from 9 a.m. to 3 p.m. Early in the morning or late in the afternoon, where solar irradiance is low, the matching efficiency is poor, i.e., less than 70%. However the overall matching efficiency is satisfactorily high because the solar energy is concentrated during high solar irradiance period.
377 REFERENCES
W. Anis, R. Mertens, and R. Vas Overstaeten, Coupling of a volumetric pump to a photovoltaic array, Solar Cell 14, 27-42 (1985). J. Appelbaum and J. Bany, Analysis of a direct coupled DC motor and a photovoltaic converter, I s~ Commission of European Community Conf on PhotovoltaicSolar Energy, Luxembourg, Sept. 27-30, Reidel, Dordecht, Netherlands (1979). J. Appelbaum and J. Bany, Performance analysis of DC motor photovoltaic converter system--I, Solar Energy 22, 439445 (1979). A. Braunstein and A. Kornfeld, Analysis of solar powered electric water pumps, Solar Energ.v 27(3), 235-240 (1981). M. Green, Solar cells: Operating principles, technology, and system applications, Prentice-Hall, Englewood Cliffs, NJ (1982). S. Klein, Calculation of monthly average insolation on tilted surfaces, Solar Energy 19, 325-329 (1977 ). J. Roger, Theory of the direct coupling between DC motors and photovoltaic solar arrays, Solar Energy 23, 193-198 (1979).