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# Modelling Photovoltaic Systems Using PSpice - 45271 03

(Parte **1** de 4)

Electrical Characteristics of the Solar Cell

Summary

The basic equations of a solar cell are described in this chapter. The dark and illuminated Z(V) characteristics are analytically described and PSpice models are introduced firstly for the simplest model, composed of a diode and a current source. The fundamental electrical parameters of the solar cell are defined: short circuit current (Isc), open circuit voltage (Voc), maximum power (Pmax) and fill factor (F). This simple model is then generalized to take into account series and shunt resistive losses and recombination losses. Temperature effects are then introduced and the effects of space radiation are also studied with a modification of the PSpice model. A behavioural model is introduced which allows the solar cell simulation for arbitrary time profiles of irradiance and temperature.

3.1 Ideal Equivalent Circuit

As shown in Chapter 2, a solar cell can in a first-order model, be described by the superposition of the responses of the device to two excitations: voltage and light. We start by reproducing here the simplified equation governing the current of the solar cell, that is equation (2.16)

(3.1) J = J,, - JO(& - 1) which gives the current density of a solar cell submitted to a given irradiance and voltage. The value of the current generated by the solar cell is given by

42 ELECTRICAL CHARACTERISTICS OF THE SOLAR CELl where I,, and I. relate to their respective current densities J,, and Jo as follows:

where A is the total area of the device. The metal covered area has been neglected. As can be seen, both the short circuit current and the dark current scale linearly with the solar cell area, and this is an important result which facilitates the scaling-up or down of PV systems according to the requirements of the application. This is the simplest, yet the most used model of a solar cell in photovoltaics and its applications, and can be easily modelled in PSpice code by a current source of value I,, and a diode.

Although I. is a strong function of the temperature, we will consider first that I0 can be given a constant value. Temperature effects are addressed later in this chapter.

3.2 PSpice Model of the Ideal Solar Cell

As has already been described, one way to handle a PSpice circuit is to define subcircuits for the main blocks. This is also the case of a solar cell, where a subcircuit facilitates the task of connecting several solar cells in series or in parallel as will be shown later.

The PSpice model of the subcircuit of an ideal solar cell is shown in Figure 3.l(a) which is the circuit representation of equation (3.2). The case in photovoltaics is that a solar cell receives a given irradiance value and that the short circuit current is proportional to the irradiance. In order to implement that in PSpice the value of the short circuit current, is assigned to a G-device which is a voltage-controlled current source, having a similar syntax to the e-devices:

(302) Subcircuit

CELL-1 .LB , girrad (300)

Node (300) i | : |

Reference

G-device (a,

Figure 3.1 (a) Cell-l.lib subcircuit a solar cell and (b) block diagram

PSPlCE MODEL OF THE IDEAL SOLAR CEU 43 Syntax for G-device g-name node+ node- control-node+ control-node- gain

As can be seen, this is a current source connected to the circuit between nodes ‘node+’ and ‘node-’, with a value given by the product of the gain by the voltage applied between control-node+ and control-node-.

A simplification of this device consists of assigning a value which can be a mathematical expression as follows:

g-name node+ node- value = {expression)

In our case the G-device used is named ‘girrad’ and is given by:

JA girrad = - lo00 (3-5) where G is the value of the irradiance in W/m2. Equation (3.5) considers that the value of J, is given at standard (AMlSG, 1000 W/m2 Tcell = 25 “C) conditions, which are the condi- tions under which measurements are usually made. Solar cell manufacturer’s catalogues provide these standard values for the short circuit current. Equation (3.5) returns the value of the short circuit current at any irradiance value G, provided the proportionality between irradiance and short circuit current holds. This is usually the case provided low injection

conditions are satisfied. The subcircuit netlist follows:

.subckt cell-1 300 301 302params: area=l, jO=l, jsc=l girrad300301value={(jsc/1000)*v(302)*area} dl 301 300 diode .modeldioded(is={jO*area}) .ends cell-1

As can be seen, dummy values, namely unity, are assigned to the parameters at the subcircuit definition; the real values are specified later when the subcircuit is included in a circuit. The diode model includes the definition of the parameter ‘is’ of a PSpice diode as the result of the product of the saturation current density Jo multiplied by the area A. The block diagram is shown in Figure 3.l(b). As can be seen there is a reference node (300), an input node (302) to input a voltage numerically equal to the irradiance value, and an output node (301) which connects the solar cell to the circuit.

The solar cell subcircuit is connected in a measurement circuit in order to obtain the Z(V) characteristic. This is accomplished by the circuit shown in Figure 3.2, where the solar cell area corresponds to a 5” diameter device. A short circuit density current of 34.3 mA/cm2 has

4 ELECTRJCAL CHARACTERISTICS OF THE SOLAR CELL

(31)

(302) XCELLl (301) Subcircuit

(300)

'inad 1 CELL-1.LIB ' (0) Figure 3.2 Measurement circuit of the Z(V) characteristics of a solar cell been assumed and Jo = 1 x lo-" A/cm2 is chosen as an example leading to a PSpice file as follows:

*cell-l.cir .include cell-1. lib xcelll0 31 32 cell-1 params:area=126.6 jO=le-1 jsc=O.O343 vbias 31 0 dc 0 virrad32OdclOOO

.plot dc i(vbias) .probe .dcvbias-0.10.60.01 .end

As can be seen the circuit includes a DC voltage source 'vbias' which is swept from -0.5 V to + 0.6 V. The result of the simulation of the above netlist is shown in Figure 3.3. The intersection of the graph with the y-axis provides the value of the short circuit current of the cell, namely 4.342A, which is, of course, the result of 0.0343 x 126.6=4.342A. Some other important points can be derived from this Z(V) plot as described in the next section.

5.OA1,,::1:,:; :;:: :,:: _,_ _I | i..;...:...:... __:. ..:..:..:. ..... :..;...:...:... .. ... _, I I.,, |

,..........,......,.................,............................. .. ~..,~ ..,... ..~..~ ...,...,.. ,,I, .. .,... ..,.. .... ~..~ ...,. ~~~.. .. | |

, | ,...,... ..,... ... I I ; I ; I . . ...... ...,...,... ..... '...'...,..., ...... ...,... I.. .. ,,,I ,,/I I,,, |

,I, I, ,,,, 2.5A :~i'?,j: :;::

.., | ,... ~.. ...... i..;...; .............. '. ......... ,... '../ ..., ...... |

-1 OOrnV ov lOOrnV 200mV 300mV 400mV 500rnV 600mV

A I(vbias) vbias

Figure 3.3 Z(V) characteristics of the solar cell model in Figure 3.2

OPEN ClRCUlT VOLTAGE 45 3.3 Open Circuit Voltage

Besides the short circuit current, a second important point in the solar cell characteristics can be defined at the crossing of the I(V) curve with the voltage axis. This is called the open circuit point and the value of the voltage is called the open circuit voltage, V,. Applying the open circuit condition, I = 0, to the I(V) equation (3.2) as follows:

the open circuit voltage is given by:

From equation (3.7), it can be seen than the value of the open circuit voltage depends, logarithmically on the ZJZ0 ratio. This means that under constant temperature the value. of the open circuit voltage scales logarithmically with the short circuit current which, in turn scales linearly with the irradiance resulting in a logarithmic dependence of the open circuit voltage with the irradiance. This is also an important result indicating that the effect of the irradiance is much larger in the short circuit current than in the open circuit

value. Substituting equations (3.3) and (3.4) in equation (3.7), results in:

The result shown in equation (3.8) indicates that the open circuit voltage is independent of the cell area, which is an important result because, regardless of the value of the cell area, the open circuit voltage is always the same under the same illumination and temperature conditions.

Example 3.1

Consider a circular solar cell of 6" diameter. Assuming that J,, = 34.3A/cm2 and Jo = 1 x lo-" A/cm2, plot the Z(V) characteristic and calculate the open circuit voltage for several irradiance values, namely G = 200, 400, 600, 800 and 1000 W/m2.

The netlist in this case has to include a instruction to solve the circuit for every value of the irradiance. This is performed by the instructions below:

46 ELECTRICAL CHARACTERISTICS OF THE SOLAR CEll

-1 OOmV ov 100mV 200mV 300mV 400mV 500mV 6oomV o v o I(vbias) vbias

Figure 3.4 I(V) plots of a solar cell under several irradiance values: 200, 400, 600, LUM aind 1000 W/m2

.include cell-1. lib xcelll0 31 32 cell-lparams:area=126.6 jO=le-1 Jsc=O.O343 vbias 31 0 dc 0 .param IR=l virrad32Odc {IR} .stepparamIRlist2004006008001OOO

.plot dc i(vbias)

.probe .dcvbias-0.10.60.01

.end

It can be seen that a new parameter is defined, IR, which is assigned to the value of the voltage source virrad. The statement ‘.step param’ is a dot command which makes PSpice repeat the simulation for all values of the list.

The plots in Figure 3.4 are obtained and, using the cursor utility in Probe, the values of V, and I,, are measured. The results are shown in Table 3.1.

Table 3.1 Short circuit current and open circuit voltages for several irradiance values

Irradiance Short circuit Open circuit G(W/m2) current Zsc(A) voltage I,(V)

1000 4.34 0.567 800 3.47 0.561 600 2.60 0.554 400 1.73 0.543

200 0.86 0.525

MAXIMUM POWER Poirvr 47 3.4 Maximum Power Point

The output power of a solar cell is the product of the output current delivered to the electric load and the voltage across the cell. It is generally considered that a positive sign indicates power being delivered to the load and a negative sign indicates power being consumed by the solar cell. Taking into account the sign definitions in Figure 3.1, the power at any point of the characteristic is given by:

(3.9)

Of course, the value of the power at the short circuit point is zero, because the voltage is zero, and also the power is zero at the open circuit point where the current is zero. There is a positive power generated by the solar cell between these two points. It also happens that there is a maximum of the power generated by a solar cell somewhere in between. This happens at a point called the maximum power point (MPP) with the coordinates V = V,,, and Z = I,. A relationship between V, and I, can be derived, taking into account that at the maximum power point the derivative of the power is zero:

at the MPP, I, = ZL - it follows that, v, = v,, - VTIn (I +?)

(3.10)

(3.1 1)

(3.12) which is a transcendent equation. Solving equation (3.13) V, can be calculated provided V, is known. Using PSpice the coordinates of the MMP can be easily found plotting the I x V product as a function of the applied voltage. We do not need to write a new PSpice '.cir' file but just draw the plot. This is shown in Figure 3.5 for the solution of Example 3.1.

Table 3.2 shows the values obtained using equation (3.12) compared to the values obtained using PSpice. As can be seen the accuracy obtained by PSpice is related to the step in voltage we have used, namely 0.01 V. If more precision is required, a shorter simulation step should be used.

Other models can be found in the literature [3.1] to calculate the voltage V,, which are summarized below.

48 ELECTRICAL CHARACTERISTICS OF THE SOLAR CELL 2.5w

2.0w

1 .ow ow ov 1 OOmV 200mV 300mV 400mV 5mv mv

+ x A Y * V(31) * I (vbias) vbias

Figure 3.5 the irradiance, namely: 200, 400, 600, 800 and 1000 W/m2 Plot of the product of the current by the voltage across the solar cell for several values of

Alternative model number 1:

Alternative model number 2:

1 + Inp ln(1 + lnp) -1- - Vm VOC (2+1n/i) In@

_- (3.13) (3.14) with

(3.15) p=- I,, 10

Table 3.2 PSpice results for several irradiance values

Irradiance V,,, (PSpice) V, from equation Im (PSpice) P- (Wpice) (W/m2) (V) (3.14) (V) (A) (w)

1000 0.495 0.4895 4.07 2.01 800 0.485 0.4825 3.28 1.59 600 0.477 0.476 2.47 1.18 400 0.471 0.466 1.63 0.769 200 0.45 0.4485 0.820 0.37

Flll FACTOR (F) AND POWER CONVERSION EFKIENCY (q) 49 3.5 Fill Factor (F) and Power Conversion Efficiency (q)

A parameter called fill factor (F) is defined as the ratio between the maximum power P,,, and the I,,V,, product:

(3.16) VmIm F = -

The fill factor has no units, indicating how far the product ZscV,, is from the power

The F can also be approximated by an empirical relationship as follows [3.2], delivered by the solar cell.

FFO = u,, - In( u,, + 0.72)

1 + uoc (3.17)

Sub-index 0 indicates that this is the value of the F for the ideal solar cell without resistive effects to distinguish it from the F of a solar cell with arbitrary values of the losses resistances, which will be addressed in the sections below. The parameter v, is the normalized value of the open circuit voltage to the thermal potential V,, as

(3.18)

Equation (3.17) gives reasonable accuracy for u,, values greater than 10.

Example 3.2

From the PSpice simulations in Example 3.1 obtain the values of the F for several values of the irradiance.

Taking the data for Is, and V,, from Table 3.1 and the data for V, and Z, from Table 3.2 the values for F are easily calculated and are shown in Table 3.3. As can be seen, the F is reasonably constant for a wide range of values of the irradiance and close to 0.8.

Table 3.3 Fill factor values for Example 3.1

Irradiance (W/m2) Fill factor

0.816 0.816 0.819 0.818 0.819

50 ELECTRICAL CHARACTERISTICS Of THE SOLAR CELL

Table 3.4 Fill factor values for Example 3.1

Power conversion Irradiance (W/m2) efficiency (%)

800 15.68 600 15.53 400 15.17 200 14.6

The power conversion efficiency q is defined as the ratio between the solar cell output power and the solar power impinging the solar cell surface, Pi,. This input power equals the irradiance multiplied by the cell area:

VocJsc = F- q = - VmIm = F- VOCISC - Voclsc

Pin Pin - F G x Area G (3.19)

As can be seen the power conversion efficiency of a solar cell is proportional to the value of the three main photovoltaic parameters: short circuit current density, open circuit voltage and fill factor, for a given irradiance G.

Most of the time the efficiency is given in %. In Example 3.2 above, the values of the efficiency calculated from the results in Tables 3.2 and 3.3 are given in Table 3.4.

As can be seen the power conversion efficiency is higher at higher irradiances. This can be analytically formulated by considering that the value of the irradiance is scaled by a 'scale factor S' to the standard conditions: 1 Sun, AM1.5 lo00 W/m2, as follows:

G = SG1 (3.20) with G1 = 1000 W/m2. The efficiency is now given by:

SG1 x Area q = F (3.21)

If proportionality between irradiance and short circuit current can be assumed:

( ':I) (';I) e) +VTlnS (3.2) V,,=VTln 1+- ZVTln - =VTln - and finally

GENERALIZED MODEL OFA SOLAR CELL 51 Applying the efficiency definition in equation (3.21) to the one-sun conditions it follows (3.24)

(3.25) provided that F = F1, thereby indicating that the efficiency depends logarithmically on the value of the scale factor S of the irradiance. The assumptions made to derive equation (3.25), namely constant temperature is maintained, that the short circuit current is propor- tional to the irradiance and that the fill factor is independent of the irradiance value, have to be fulfilled. This is usually the case for irradiance values smaller than one sun.

(Parte **1** de 4)