Performance analysis of d.c.-motor-photovoltaic converter system—I separately excited motor

Solar Energy,Vol. 22, pp. 439-445 © PergamonPress Ltd., 1979. Printed in Great Britain

0038.-092X/79/0501-O439/S02.0010

PERFORMANCE ANALYSIS OF D.C.-MOTOR-PHOTOVOLTAIC CONVERTER SYSTEM--I SEPARATELY

EXCITED MOTOR

J. APPELBAUMand J. BANY School of Engineering, TeI-Aviv University, TeI-Aviv, Israel

(Received 3 March 1978: revision accepted 1 November 1978) Abstract--The performance of a solar-electrical system composed of a solar cell array, d.c. separately exicted motor and a mechanical load was analysed. The system operating points in the mechanical plane, (n, T) were transferred to the photovoltaic converter plane (I, U). The relative position of the load line to the maximum power output line of the photovoltaic converter indicates its utilization. The venitlator type load (centrifugal pump) fits very well the converter in contrast to a constant load. The speed variation of the ventilator type load-motor link remains in reasonable limits during the day. For a centrifugal pump, this characteristic corresponds to an almost constant pumping rate during most of the day.

INTRODUCTION

Part II will analyse the system operation with series and shunt motors.

The decrease in cost of solar cells in the last few years and the prediction of cost reduction in the future extends the terrestrial applications of solar cells to higher powers. Many of these applications are of international importance and economically justified, such as water pumping at remote places[l, 2]. It is an "in phase" rare case where the demand of water (for drinking and irrigation) is accompanied by high solar radiation at these places. Low voltage d.c. motors, at these power levels, are usually considered as prime movers for the mechanical pumps. The photovoltaic converter (solar generator) is an uncontrolled power source that behaves differently from a conventional power supply (of constant voltage). For a conventional power supply, the system designer makes the best adaptation of the electric motor to the mechanical load, since the power supply is considered (in many cases) infinite. The photovoitaic converter designed for the motor-mechanical load link is not "infinite" and is tailored for that purpose only. Due to the high cost of solar cells, it is essential to design the system economically and make its operation as close as possible to the maximum output power of the solar generator. It is the purpose of this study to analyse the performance of a system composed of a photovoltaic converter, a d.c. motor and a mechanical load. The study will assist in understanding the system design and operation. The performance analysis covers the operating points in the (n, T), (n, I) and (I, U) planes. The relative position of the load line to the maximum power output line of the converter in the (I, U) plane is an indication of the efficiency of converter utilization. The variation of the motor input power, its terminal voltage and speed, as well as the utilization of the converter, are discussed. Two mechanical loads are considered: a ventilator type load (centrifugal pump) and a constant load. Part I of this study deals with a separately excited d.c. motor.

PHOTOVOLTMC CONVERTER CHARACTERISTICS

The solar cell is a semiconductor device that converts the solar radiation to electrical energy with efliciencies up to about 20 per cent. The cell is a non-linear device and is described by its I-V characteristics or by the equivalent circuit shown in Figs. 1 and 2[3]. The solar cell volt-ampere equation can be described, with high accuracy, by five parameters resulting in implicit relations of U = f ( l ) and l = f ( U ) . For a single well constructed small area ceil, the shunt resistance, R~h. can be neglected and the U = f(l) relation become explicit and therefore more easily handled mathematically. For large area cells, the shunt resistance might have a considerable effect on the generator output characteristics and the

r

P,

Pmox.

le

14

io

2o

Z2~~ k

0

4

a

a2

m

zo

UlVl~

Fig. 1. I - V and P - V photovoltaic c o n v e n e r characteristics.

439

44O

J. APPELBAUMand J. BANY

[

R$

Up = lpR, 4

A(I.~- :Ip - V,/R~. * Io)]/ l

[1-¢ AR~h(Ip._ BI _ Up/R~h + Io)]" Fig. 2. Equivalent circuit of a solar cell.

Equation (8) is a quadratic equation for lp

following more accurate relation must be used[4]

U=-IR.+-~In

I+ Iph- fllIo

(8)

Alp 2 + Blp + C = 0 (I)

(9)

where A =~Rs

where

D = Iph- (UdR,h) + Io /3 = 1 + RdRsh.

(2)

In the mathematical analysis of the performance of the photovoitaic converter, the current, voltage and power need to be calculated. Since these relations are implicit, iterative procedures must be used and there may be convergence problems. The iterative process for U = [(I, U) leads to rapid convergence if the following approximation is used

B = -~(Up + I/A)- DR.

(I0)

C = UpD + U,,/AR.h whose solution having physical significanceis:

/" =

- B + X./(Bz - 4AC) 2A

(11)

The voltage at the maximum power point is

U'=-IR.+-~In

I+

.

(3)

Up= lpR, + l in (1+ leh-~]lelo The photocurrent l,,his proportional to the radiationflux and is slightlygreater than the short circuitcurrent l.c. The load current is obtained from eqn (I) I

=

l/fl{Iph lo[exp A(U -

+ IR.)- I]- V/Rsh}

(4)

and the short circuitcurrent is

I,~ = lI${lph - lo[exp (AL~R,) - 1]}.

(5)

A first good approximation for I~¢ would be l~c = lph. It is important to operate the photovoltaic converter at its maximum power point, or in its vicinity because of the high cost of solar cells. The power delivered to the load is given by P

=

UL

(6)

The maximum power point is obtained by taking the derivative dP d U - u - =u^ ~-=-~-l+

UelR'h).

(12)

Equations (11) and (12) are simultaneous iterative equations for Iv and Up, respectively, from which the maximum power locus of the photovoltaic converter can be calculated as a function of the photocurrent l,h which in turn corresponds to the solar radiation flux. The solar array used in the present study was a silicon cell array of 300 W peak power output, constructed from 36 parallel strings of 36 series connected cells in a string. The array parameters[4]:t Io=0.0162A, A=0.38~/V, Rs = 0.05 fl R,h = 41ft. Three I-V and P-V curves corresponding to three radiation levels are presented in Fig. I. The maximum power line and two lines of 90 per cent of its value are drawn in the figure by dashed lines. These two lines form a quite wide area (dashed) in which the operating voltage may vary Up-+ 18 per cent and more than 90 per cent of the maximum power will be delivered. c n ~ c r z m s ~ c s or rnz Moron rowraa.o sv rnz PllOTOVOLTAICcoNVgCq'EIi The motor fundamental relations[5] are

E=kdm

(13)

or

d" Ip

Up = -I. ~

(7)

where Up and lp are the voltage and current at the maximum power points p. From eqns (1) and (7), we get

U.=E+IoR,,

(14)

T= k,cMo

(15)

where for separate excitation l.=L.

?lph is a varying parameter corresponding to the solar radiation. For the above array size,/ph = 21.24A at 78 mW/cmz.

(16)

This excitation can come either from a separate source

Performance analysis of a d.c. motor-photovoltaic converter system--I

441

I

Photovoltaic converter

motor

load

Fig. 3. Schematic diagram of motor-photovoltaic converter system.

or from a permanent magnet. By coupling the motor to the photovoltaic onverter, Fig. 3, we get Um= U

and Im = I

(17)

where U is given by eqn (1). The speed characteristics n = f(I) is obtained from eqns (13), (14) and (17) )-IRa n = U ( I k,~

(18)

where U(I) is given by eqn (1) and I is the motor load current. The motor speed in eqn (18), for a specific load current /, is determined by solving the iterative eqn (1) for U and then substituting this voltage into eqn (18). Two specific speeds, the ideal no-toad speed and the no-voltage speed, need to be mentioned. The ideal noload speed is obtained for I = 0 and might be quite high.

no =

U(I = O) kecb

(19)

The no-voltage speed is the motor speed at zero voltage output of the converter (U = 0, I =/~c) and is obtained from eqns (1) and (18) nu~o = - ~

Re/

]sc.

(20)

It is always negative. The motor speed torque characteristic n = f(T) is calculated from eqns (15) and (18). U ClTk,ck ) - TR~/ k,cb

"=

k,~b

(21)

where for a given torque 7", the current I is calculated and the voltage value is determined iteratively from eqn (I); the speed is then obtained from eqn ('21). A computer was used for all the calculations, described above. The data for the motor which was coupled to the array in the previous paragraph are as follows: With an applied voltage of 14 V and a current of 9 A, the motor speed is 2200r.p.m. The speed torque characteristics (eqn (21)) were calculated using the array and motor parameters: lph, Io, A, R,, Rsh and Ro, k,.O, respectively. The installed photovoltaic converter was oriented to the south and tilted at an inclination angle of 45° from the horizontal in the Tel-Aviv area (lat. 32°00'N, long. 34°49~E) in July. The results are plotted in Fig. 4 for different radiation levels. The figure includes also, for comparison, the

Fig. 4. Speed-torque characteristics of separately excited motor.

motor speed torque characteristic when it is supplied from a 14 V constant voltage supply (dashed line). These curves are functions of the radiation level, although the general shape is conserved. The no-load speed increases slowly with radiation and may reach high levels exceeding the motor rated speed. The no-voltage speed (points A,, A2, A3.... corresponding to the short circuit array currents Is¢) increases (becoming more negative) with increasing radiation. These points correspond to the maximum torques produced by the motor. Operation in the fourth quadrant (n<0, I > 0 ) corresponds to motor plugging (braking). The starting torque (n = 0) is lower than the maximum torque and increases with radiation. Generally, the motor characteristics are "softer" than for a constant voltage supply. SYSTEM OPERATION

The operating points of a system composed of an electric motor, mechanical load and a power source are determined by the characteristics, of all three components. Mathematically, the operating point is the intersection of the motor speed-torque relation n = f ( T ) and the load speed-torque n = g(T) for a given voltage level of the power supply. This point is transformed to a current value "viewed" by the power supply. Thus the system operation can be described in two planes, namely, the mechanical plane (n, T) and the electrical plane (I, U). In many cases, another (mechanical-electrical) plane, the (n, I) plane, is used for the description of the motor operation. For a separately excited motor, this is a mixed plane since I is also the photovoltaic converter current. The operation of such a system connected to a conventional power supply (constant voltage) is well known. However, the solar radiation is an uncontrolled time dependent energy source (it changes hourly and daily), and the solar cell is a non-linear current source. This combination produces a special power source for which the performance of the load

442

J. APPELBAUM and J. BANY

must be analysed in terms of the source in the (/, U) plane. The operating points of the mechanical load, together with the motor characteristic, are transformed into operating points (transformed load line) as viewed by the photovoltaic converter. The position of this line is an indication of the efficiency of utilization of the system. The system operation is time dependent, therefore the power input to the motor, the motor voltage and speed, and the array utilization must be determined as a function of time during the day to provide information about system performance. In the work reported in this paper, two types of mechanical loads were examined: a constant and a ventilator load type. The last case represents a mechanical load of a centrifugal pump.

responding values of this point, C', are n = 2450 r.p.m., I = 7.65 A in (n, I) plane, and C", I = 7.65 A, U = 13.7 V in (I, U) plane. Plane (L U) includes also the maximum power output line of the photovoltaic converter, a,//, y, .... The extent to which the operating line of the system A", B", C",... is close to the maximum power line determines the efficiency of utilization of the solar array. The motor starts to rotate when the radiation reaches a certain level (2.5 mW/cm2 in the present example) so that the solar array produces the required starting torque OA. Before starting, the converter sees the load as an ohmic resistance of the armature, Ro, and hence the straight line OA". After starting, the load line is as explained previously A n, B", C", .... Point A" is an inflection point from which the system starts to rotate. The time dependence of the system will now be discussed. The array power delivered to the motor is P~n= UI, where U and I are the array voltage and current at an operating point. The utilization of the array was defined as r/= Pin/Pma~ where both the motor input power Pi° and the maximum array power P~,~ correspond to the radiation level at the same instant. These time dependent functions (using Fig. 5), together with the global radiation G arriving at the array are shown in Fig. 6. Furthermore, the motor speed and voltage as a function of the solar time are shown in Fig. 7. The motor starts to rotate half an hour after the array starts to receive radiation, and stops at 1800hr. As said before, the time elapsed for motor starting depends on the required starting torque.

Ventilator type load Figure 5 shows the operation of the system in three planes: the motor (n, T) and (n, I) planes and the photovoltaic converter plane (I, U) at different radiation levels corresponding to the time (in parentheses) of the day. The ventilator load line ABCD (heavy line) and the motor characteristics for different radiation levels 2.5, 9.5, 28 mW/cm2, etc. are shown in the (n, T) plane. The operating points of the system in this plane are the intersection of the load line and the motor characteristic lines, e.g. A, B, C, etc. These points were transferred to the (n, I) plane, points A', B', C' etc, and to the (L U) plane, points A", B", C", etc. The transformation is illustrated (dashed lines) for point C (n = 2450 r.p.m., T = 0.3 Nm) at radiation level of 42 mW/cm2). The cor-

S9 (9.5)

° 4-

4

II

12

0.3

0.5

7

.AT

Fig. 5. Motor-photovoltaicconverter system operation.

14

UiV)

Performance analysis of a d.c. motor-photovoltaicconvener system--I

443 ~t 90

70

60

I0

|0 f I0 , ~ &O

,:o

i

t

,~

t

t

,.o

i

,f.o

i

s;o

s ,

,,io

m.O t ( ~ }

Fig. 6. Array utilization for a ventilator type load. u

n

(v)

(r.p.m,I

14

3OOO I0 2O00

IO00

ILo

?.0

9.0

n.o

13.0

IILO

11.0

Q.O ll(It(mr)

Fig. 7. Voltageand speed variation for a ventilator type load.

P (w)

G

!.o

gO

70

140 , ? 0

I00 SO

IlO

20 • I0

I0 S.O

7.0

9.0

..0

I&O

m.O

I?.O

m~O t o m ' )

Fig. 8. Array utilization for a constant load.

Figures 6 and 7 reveal interesting system operation features. The array utilization, '7 increases rapidly after motor starting and reaches 100 per cent at the early hours. Then, n decreases to a minimum value of 74 per cent at noon and increases again to 100 per cent late in the afternoon, then drops rapidly before sunset. The array is well utilized more than 75 per cent most of the day, i.e. for about 12 hr. The variation of the motor speed (Fig. 7) is quite small and remains within reasonable limits (2550 +_300 r.p,m.) during most of the day (between

0800 and 1600). Th~ flatness of the speed variation (for a wide range) fits the centrifugal pump operation, since the pumping of water starts at a certain high speed (depending on the head). By knowing the pump characteristics, one can easily translate the above solar-electrical characteristics to quantities of pumped water. Constant load

A constant mechanical load, T = 0.36 Nm is plotted in Fig. 5: the operating points are PQR . . . . in the (n, T)

444

J. APPELBAUMand J. BANY U

n

(v:

[r.p.n9

3O0O

O(3O

zo ;o ' ~o ' ,s'o ' ~o ,7o Fig. 9. Voltage and speed variation for a constant load.

plane. These points were transformed, as for the other load, to the (n, I) and (_1,U) planes, points P', Q', R ' , . . . and P", Q", R', .... respectively. Since for a separately excited motor, the motor torque is directly proportional to the motor current and hence equal to the array current, the load lines in the other planes are constant. Before starting, the converter sees the load as the ohmic resistance of the armature, and hence the straight line OP" where the solar array produces the required starting torque (corresponding to 35 mW/cm 2 or 9.4 A at point P"). If the motor is switched connected to the converter at the radiation level of 78 mW/cm 2, the starting point would be Z, which would move along the I-V curve and the operating point will stab|iiTe at S ". It is obvious that the load line P#Q'R#S" is poorly situated relative to the maximum power output line a,/3, "y, e, and hence, low utilization of the converter is expected. The motor power input, Pi., the global radiation G arriving at the array and the array utilization, are shown in Fig. 8. The motor speed and voltage are described in Fig. 9. The system operation period for this type of load is about 8 hr and the solar array is utilized about 55 per cent of the time. The motor speed remains in reasonable limits during the operation period. A constant mechanical load is not the most suitable for a photovoltaic converter.

CONCLUSIONS The performance of the d.c. motor-solar generator system is radiation dependent and the performance is quite different from that of the motor connected to a conventional power supply. The system operating points are determined by the load, motor and the photovoltaic converter characteristics and are time dependent. The solar array utilization is strongly dependent on the load type. Since the photovoltaic converter has a wide maximum power range, the load line (transferred to the (Jr, U) plane) does not necessarily coincide with the maximum power line for good array utilization. The ventilator type load (centrifugal pump) which is suitable for water pumping fits the photovoltaic converter characteristic better than a constant load.

REFERENCES 1. M. Barland and C. Masselot, Optimisation d'une station de pompage alimentee par generateur photovoltaique. 1977 Photovoltaic Solar Energy Conf., Luxembourg (Sept. 1977). 2. J. A. Rogers et al., Calculations and in situ experimental data on a water pumping system directly connected to a il2 kW photovoltaic converter array. 1977 Photovoltaic Solar Energy Conf., Luxembourg (Sept. 19771, 3. H. J. Hovel, Semi-Conductors and Semi-Metals, Vol. 11. Academic Press, New York (19751. 4. A. Braunstein, J. Bany and J. Appelbaum, Determination of solar cell equation parameters from empirical data. Energy Cony., 17(I), 1.6 (1977). 5. D. S. Langsdorf, Principles of a Direct-Current Machine. McGraw-Hill, New York (19591.

A E G / io

l0 /ph I,~

NOMENCLATURE completion factor electromotive force (e.m.f.) global radiation solar array current armature current reverse saturation current photocurrent motor current

Ip I~ k k,, k, n no n.=0 P Pin P,., q

array current at the maximum output power point solararray short circuitcurrent Bolzmann constant motor constants speed idealno-load speed no-voltage speed solar array output power motor input power maximum array output power electronic charge

Ro Rs R,h t T T Um Up

armature resistance solar array series resistance solar array shunt resistance solar time absolute temperature motor electrpmagnetic torque solar array terminal voltage array voltage at the maximum output power point

Greek symbols I + R,/ R,h solar array utilization ~, magnetic flux

A q/AkT

R~sum&-On (~tudie le fonctionnement d'un syst6me solaire-~lectrique constitu(~ d'un panneau de photopiles, d'un moteur ~, courant continu ~ bobinage induit et d'une charge m~canique. Lcs points de fonctionnement du syst/~me dans I'espace (n, T) m~canique ont (~t~ramen(~sdans I'espace (I,/3) du convertisseur photovoltaique.

Performance analysis of a d.c. motor-photovoltaic converter system--I La position, relativement ++la courbe de puissance maximale du convertisseur photovoltaique, de la courbe de fonctionnement de rensemble moteur-charge, en oriente I'utili~fion. Une charge de type venfilateur (pompe centrifuge) s'adapte tr~s bien au convertisseur, au contraire d'une charge constante. Les variations de la vitesse de rensemble moteur-charge restent clans des limites raisonnables pendant ia journ~. Pour une pompe centrifuge, cette caract~ristique correspond ~, un d~bit de pompage +tpeu pros constant pendant la plus grande pattie de la journ~e.

445