Modelling Photovoltaic Systems Using PSpice - 45271 05

Modelling Photovoltaic Systems Using PSpice - 45271 05

(Parte 1 de 3)

Interfacing PV Modules to Loads and Battery Modelling

Summary

Photovoltaic systems are designed to supply either DC or AC electricity to loads. Depending on the complexity and characteristics of the final loads connected to the photovoltaic system, different elements can be found as a part of the photovoltaic system itself. These different elements may be:

0 DC loads, e.g. lights, electric motors;

0 batteries; 0 AC loads, e.g. lamps, household appliances, electronic equipment;

0 power-conditioning equipment, e.g. protection and control elements, AC-DC converters, charge regulators, DC-AC converters.

Depending on the nature of the loads, their interface to the photovoltaic system implies different complexity grades. The study of the different connection options of the photovoltaic modules to the rest of the photovoltaic system components is the objective of this chapter.

5.1 DC loads Directly Connected to PV Modules

The simplest example of a PV system connection to a load is a direct connection of a DC load to the output of a PV module as shown in Figure 5.1. The simplest load is a resistor R.

If we want to know the coordinates of the operating point, then we have to look for the intersection between the load and the PV module I( V) characteristics. Figure 5.2 shows the Z(V) characteristics of a PV module with a short circuit current of 4 A and an open circuit voltage of 2.1 V. The load is a 3 0 resistor in this example and Figure 5.2 shows a plot of the current as a function of the voltage, hence the load Z(V) characteristic is a straight line with a slope of 1/R.

104 INTERFACING PVMODULES TO LOADS AND %ATTERYMODELLING module

Figure 5.1 PV module directly connected to a DC load 8.0 ov 4v av 12v 16V 20v 24V vbias 0 I (vbias) 0 vbiasl3

Figure 5.2 PV module and load Z(V) characteristics

As can be seen in Figure 5.2, for a given irradiance and temperature there is only one intersection point which sets the values of the voltage and the current. In the example sbwn in Figure 5.2, the current flowing into the load is 3.9 A and the voltage is set to 1.8 V This is the voltage common to the PV module output and the resistor load.

This result means that a resistor load sets the operating point of a PV module, and by extension of a PV system, generally, at a different point than the maximum power point. As a result it can be concluded that the power transfer from the PV generator to the load is nat optimized in the general case in this direct connection, unless interface circuits are inserted between the PV system and the load.

Such interface circuits are required because the PV module I( V) characteristic is subject to time variations due to irradiance and temperature changes, as has been shown in Chapters

3 and 4. This will directly affect the transfer of electrical power to the load. In the simple case of a resistor as a load, analysed here, the result of a direct connection will be that the power efficiency will change as long as irradiance and temperature change.

An important effect of a direct connection to a resistor is that no power can be delivered to the load by night, which in many applications is a key specification.

5.2 Photovoltaic Pump Systems

PV pumping systems are often composed of a PV generator directly connected to a DC motor driving a water pump. Suitable models for DC motors and centrifugal pumps can be derived for PSpice simulation.

PHOTOVOLTAIC PUMP SYSTEMS lfB5 5.2.7 DC series motor PSpice circuit

In a DC motor a magnetic field is established either by a permanent magnet a by kEd winding, which involves a rotor with armature winding when electrical power is qpkd- The ic Bm

D~, and to the armature current i, as follows: result is that a mechanical torque T~ is produced, which is proportional to the where K, is the motor torque constant. Moreover a back electromotive force(em is indud in the armature winding and is related to the magnetic flux and to the angular fre- u, by

As in steady-state, and without losses, the mechanical power (wT,) eqds the ekckzic power (vaia) therefore it follows that,

If the magnetic field is created by field winding, in a series motor, where the kld 4 armature windings are in series, the magnetic flux is proportional to the current 4 in liM winding:

and hence

Ecm = KcmKfifw = Kmifw re = K,Kfifi, = Kmifia

The dynamic equivalent circuit of a DC series motor also has to include other additional elements: resistor losses (R, and Rf), inductors of the windings La and 5.. inertial term J and friction term F. The mechanical load torque is also included TL, as can be seen in Figure 5.3. The differential equations governing the dynamic operation of the DC motor are:

dif V’ = ifRf + Lf- dt dw dt re = J- + FW + TL where the last equation relates the mechanical magnitudes and returns the angular frequency value.

106 INTERFACING PVMODULES TO LOADS AND BATT€RYMODElUNG Ra La

Figure 5.3 Equivalent circuit of a DC series motor with the load of a pump

All the equations shown do not consider that the core may enter into saturation. There are more complicated PSpice models that take into account these effects, however, these are beyond the scope of this book, and the reader is referred to reference [5.1] for more in-depth treatment.

5.2.2 Centrifugal pump PSpice model

When a DC motor is connected to a centrifugal pump, a load torque is applied to the mechanical port of the equivalent circuit in Figure 5.3, which is related to the angular frequency by:

TL = A + Bw' (5.8)

where A and B are constants depending on each particular pump. As a result of this load torque an equilibrium is reached for a value of the angular speed. Moreover, the centrifugal pump has a characteristic curve relating the pumping head, H, the angular frequency and the resulting flow Q. These types of curves are widely available in commercial technical notes or web pages, and are generally given for a constant value of the angular speed. It is assumed that these curves can be approximated by a second-degree polynomial as follows:

H = AISZ + BlSQ + C1@ (5.9) where S is the angular speed in rpm, and Al, B1 and C1 are constants of a given pump.

A schematic circuit showing how the different parameters relate in a DC motor-pump connection is shown in Figure 5.4.

It should be noted that the input power is a function of the output angular speed due to the generator E,, in the armature winding equivalent circuit. This means that the first two blocks in Figure 5.4 are tightly related by the feedback loop closing at the mechanical domain.

5.2.3 Parameter extraction

In order to supply the PSpice simulator with correct data, the main parameters in equations (5.1) to (5.9) have to be extracted. First for the pump, the data of the characteristic curve are taken and summarized in Table 5.1 from manufacturers' data taken from reference r5.1.

PHOTOVOLTAIC PUMP SYSTEMS 107

H=A~s~+B~sQ~c~Q? 0 S Pin_ Motor S=(60/2n) +

0 Pump

Figure 5.4 connection

Schematic circuit showing how Pump the different parameters relate in a DC motor-pump

Table 5.1 Data of the characteristic curve from reference [5.1]

Head H (metre) Flow Q (Us) (W) Pi, Power

13.4 1 480 1.5 1.5 340

By fitting equation (5.9) to the characterisitc curve at 3000 rpm, the following values of the constants Al, B1, C1 have been obtained:

A1 = 1.35 x lop6

B1 = 0.0015 C1 = -3.32

Next, the values of K,,,, A and B are extracted following the procedure as follows.

Step I

Obtain the values of the main operational parameters under nominal conditions. That is the values of Pi, and S.

Step 2

Assume a value for the input voltage if not known, and a value for the armature and field winding resistances.

step 3 Calculate K,,,

108 INTERFACING PVMODULES TO LOADS AND BATTERYMODELLING Write the operational equations of the motor at steady-state conditions:

and (5.10)

(5.1 1)

Since ia = if, we can write the total applied voltage at nominal conditions (w = w,,,,, and i, = if = and finally (5.13)

Step 4

Calculate A from the starting torque. From the mechanical equation at steady-state and at the starting conditions (w = 0) re = rL = A = Kmimin 2 (5.14)

Taking into account that the motor does not start working unless a minimum power is applied Pmin, then

(5.15) and substituting Pfi,, by its value estimated from the technical data sheets and using the nominal voltage for the voltage, we can write

(5.16)

Step 5 Assume a value for F and calculate B.

PHOTOVOLTAIC PUMFplasFMcs, PO9

The value of B is calculated from the mechanical equation at steady-stake cooditkm~at rlae nominal values of angular speed and input current:

re = Fw,,,, +A + Bw;, = Kmi:,,, (5.17j and

Example 5.7 Calculate the values for A, B and K, for the same example as in Table 5.1.

Step

H = 1.5m S = 3000 rpm

Pi, = 340 W w = 3 14 rad/s

Step 2 vi, = 200 v R, $- Rf = 0.3 step 3

Vi, - i,,,(R, + Rf) - 200 - 1.7 x 0.3 K, = - = 0.37 Inom Wnom 1.7 x 3000 x

Step 4

Pmin = 100 w

(5.19) (5..M)

(5.21)

110 INTERFACING PVMODULES TO LOADS AND BATTERY MODELLING Step 5: Ca/cu/ate B

Kmi;,, - Fwnonl -A 0.37 x 1.7' - 8.3 x x 314 - 0.0925 = 7.26 x lop6 - B= - w:,n 3 142

(5.2)

The values of the inertia J and of the inductances Lu and 4 do not enter the steady-state solution. However, some reasonable assumptions can be made if the dynamic response is considered. In order to calculate the small-signal dynamic performance of the motor-pump combination, the variational method to linearize equations can be used and then Laplace transforms taken. We first consider a steady-state operating point (Vi,,, iino, w,) and consider the instantaneous value of these three variables as a superposition of steady-state values and time-dependent increments (with subindex 1) as follows,

(5.23)

Substituting equation (5.23) in equations (5.7) the small-signal equations representing the dynamic operation of the motor-pump system are:

Taking into account the steady-state equations

Neglecting now the second-order terms (wlil, 4, it follows taking Laplace transforms of these linearized equations

(5.25)

(5.26) (5.27)

The mechanical dynamic response of the motor-pump association is msuch slower than the electrical response. We can deduce from equation (5.27) the electrical time constant assmming that the angular speed is constant while the armature current builds up. This means irssmning 01 (s) = 0. Then the relationship between input voltage and armature cumed is where the electrical time constant can be identified: (5.299)

Using now equation (5.27) that can be written as:

where

Equations (5.28) and (5.32) can be used to find suitable values for the parameters L, and 5. and J as shown in the example below.

Example 5.2

Using the results and data in Example 5.1, calculate suitable values for (L, + 4) and J considering that the electrical time constant is 5 ms and the mechanical time constant is 5 s. Simulate the electrical and mechanical response of the motor-pump system when a small step of 1 V amplitude is applied to the input voltage while at nominal operation conditions.

Taking into account equation (5.29) and considering the same value for (& + Rf) = 0.3 R as in Example 5.1 it follows that

La + Lf = 1.16H

And assuming the friction F = 0.0083 equation (5.32) returns J = 0.0608.

112 INTERFACING PVMODULES TO LOADS ANDUATTERYMOOELlfNG 5.2.4 PSpice simulation of a PVarray-series DC motor-centrifugal pump system

Using the parameter values derived in Examples 5.1 and 5.2 the following netlist has been written as a subcircuit of the motor-pump system, according lo the schematics shown in Figure 5.3 for the motor-pump association which we will simulale

.subckt pump 500 501 570PARAMS: RA=l,LA=1.K~=~,A;L.B=1 + F=l, J=l, RF=1, LF=1, Al=l, Bl=l,CI=l, R=I ra 501 502 {RA} la 502 503 {LA} econ 503 504value={{KM}*v(508)*v(507)} rf504505 {RF} If 505 506 {LF} vs 506 0 dc 0 qte O507value={{KM}*v(508)*v(508)} qtl 507 Ovalue={A+B*v(507)*v(507)} rdamping 507 0 {l/{F}} CJ 507 0 {J} d2 0507diode .modeldiode d qifO508value={(v(504)-v(505))/{RF}} rif 508 0 1

.IC ~(507) =O

*** revoluciones rpm=omeqa* (60/2/pi) erpm540Ovalue-{~(507)*60/6.28} eflow550Ovalue={(-B1}*v(540) -sqrt(((B1}"T)*(v(540~'2) -4*{C1}*

+(Al*(V(540)"2) -{H>) ))/(2*{C1>)} eraiz560Ovalue={({B1}"2~*(~(540~~2~-4*~C1}*~Al*~Vd540~~2~-{H}~} eflow2 570 Ovalue={if (~(560) >O, v(550),0)};.endspumnp

Most of the parameters involved can be directly identified fmrn Figure 5.3 and equations on this section. This subcircuit is used to simulate the transient response of the motor and pump, as follows:

*watersump-transient . include pump. lib xpumpO4450pumpparams:RA=0.15,LA=0.58,KM=0.37,A=0.0925,B=7.26e-6 + F=0.00083,RF=0.15,LF=O.58, 5=0.0608,A1=1.35e-6,B1=0.0015.C1=-3.32,H=1.5 vin44OOpulse (0,200,0.10m,10m,50.100)

.tranO.Olu20le-6 .probe

I .end

PVMODULES CONNECTED TO A BATTERYAND LOAD 113 1.5v

1 .ov

0.5V n\i V.

0 V(50)

5s 10s 15s 20s

Time

Figure 5.5 y-axis is the flow (lls)

Transient of flow produced by a centrifugal pump after a voltage step of 200 V. Warning,

The slow mechanical response towards the nominal conditions of operation is shown in

Figure 5.5, where the evolution of the flow after the power supply has been switched ON is shown.

It is seen that the flow remains at zero until approximately 7.5 s and then rises towards the predicted steady-state value of 1.5 Us. The electrical power required reaches the predicted 340 W in Table 5.1 for H = 1.5 m. The model described in this section is used in Chapter 7 to simulate long-term pumping system behaviour.

5.3 PV Modules Connected to a Battery and load

The most commonly used connection of a PV system to a battery and a load is the one depicted in Figure 5.6 where the three components are connected in parallel. Of course, a battery is necessary to extend the load supply when there is no power generated by the PV modules in absence of irradiance, or when the power generated is smaller than required. The battery will also store energy when the load demand is smaller than the power generated by the PV modules.

A battery is an energy storage element and can be interpreted as a capacitive load connected to the PV generator output. As can be seen in Figure 5.6 the voltage Vbat is

Imod Iload

Figure 5.6 Standalone basic PV system

114 INTERFACING PVMODULES TO LOADS AND BATTERYMODELLING common to all the components and by applying Kirchhoff’s current law (KCL), the current flowing through the three elements is related by:

(5.3)

Where Imod is the output current of the PV system, &at is the current flowing to the battery and Iload is the current supplied to the loads. As can be seen in Figure 5.6, Zht has a me sign at the positive terminal and charges the battery, whereas when the ]bat sign is negative, the battery discharges. The sign of Ibat is determined at every time t, by the instantaneous PV system I( V) characteristics, according to the irradiance and temperature values, and by the instantaneous value of the current demanded by the load.

5.3.1 Lead-acid battery characteristics

Lead-acid batteries are the most commonly used energy storage elements for standalw photovoltaic systems. The batteries have acceptable performance characteristics and life- cycle costs in PV systems. In some cases, as in PV low-power applications, nickel-cadmium batteries can be a good alternative to lead-acid batteries despite their higher cost.

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