# 598 F15 Lecture 02b PV cell model ```Lecture 2b
Photovoltaic (PV) cell model
•
•
•
•
Single diode based model
I-V and P-V characteristics
Effect of different parameters
PLECS model
Photovoltaic cell
n-type
p-type
contacts
• PV cells consist of p-type and n-type semiconductors (silicon
being the dominant at present) forming a p-n junction
• Metal strips for contact with external circuit
• Anti-reflective coating
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PV cell operation (1/2)
n-type
p-type
contacts
• Absorption of photons (light) generates electron-hole pairs leading to
light-generated current
• Light generated carriers are separated by the electric field at the p-n
junction preventing recombination and enabling current flow through
external circuit
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PV cell operation (2/2)
-
n-type
p-type
+
contacts
• Voltage generated by photovoltaic effect – the light-generated carriers
result in forward bias of the junction
• Current in the external circuit is the difference of the light-generated current
and the forward bias current
Also refer to http://pveducation.org/pvcdrom for more detailed discussions
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Ideal PV cell model
𝐼
𝐼𝑑
𝐼𝑝ℎ
+
𝑉
To
• Ideal PV cell is modeled as a solar-irradiance-dependent current source, 𝐼𝑝ℎ
in parallel with a diode
• The current in the external circuit is the difference between the photon current
and the forward-biased diode current
• Good compromise between model accuracy and model complexity (compared
to, for example, two-diode models) with non-idealities included later
• Useful for understanding V-I characteristics, designing MPPT algorithms,
study of partial shading, design of power converters
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Diode equation
+
𝐼𝑑
𝑉𝑑
-
 qVd

I d  I o  e a k T  1




T  Temperature in K
q  charge of an electron in Coulombs 1.602  10-19


k  Boltzman constant (1.38  10-23 j/K)
I o  dark or reverse saturation current of diode at T
a  diode ideality factor, normally between 1 and 2
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Diode equation- reverse saturation current
+
𝐼𝑑
𝑉𝑑
-
10
 qVd

I d  I o  e a k T  1




5
Id( Vd)
(A)
0
I o  dark or reverse saturation current of diode at T
5
•
𝑘𝑇
𝑞
𝑎𝑘𝑇
𝑜𝑟 𝑞
is 25.84 mV at 300K and a = 1
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1 10
0.4
0.6
8
Id( Vd)
0
(A)
1 10
8
𝐼𝑜 ≈ 15 𝑛𝐴
0.15
is called the ‘thermal voltage, 𝑉𝑡ℎ ’ and
0.2
Vd (V)
• 𝐼𝑜 is an important characteristic of a given diode;
higher material quality relates to smaller 𝐼𝑜
• 𝐼𝑜 increases significantly with temperature;
for silicon cells near room temperature,
𝐼𝑜 doubles for every 100C rise in temperature
0
0.1
0.05
Vd (V)
0
0.05
Ideal PV cell model
𝐼
+
𝐼𝑑
𝐼𝑝ℎ
𝑉
To
-
I  I ph  I o (e
qV
akT
 1)
I ph  Photon current at a given irradiance and given T
V  Voltage across the diode
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Photon current dependencies
I  I ph  I o (e
qV
akT
 1)
I ph  Photon current at a given irradiance and given T
• Magnitude of photon depends on the irradiance level, light spectrum,
and the characteristics of the cell
• Short circuit current which is a direct measure of the photon current is
specified at standard test conditions (STC) which is 1000 W/m2 , module
temperature of 25oC and air mass AM = 1.5
• Magnitude of photon current (short circuit current) is directly proportional
to the irradiance – e.g., 20% lower irradiance results in 20% lower
photon current
• Change in photon current with temperature is not very significant – about
0.05% increase per oC rise for silicon
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Series resistance
• Series resistance RS models the combined resistances of
contacts, metal grids, and p and n layers
𝑅𝑆
𝐼
+
𝐼𝑑
𝐼𝑝ℎ
𝑉
-
I  I ph  I o (e
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q (V  I RS )
akT
 1)
To
Shunt resistance
• RSh models the leakage current of p-n junction
• Typically RSh is a large value, with low values representing defective cells
• Impact (on power loss) more pronounced at low irradiance levels
Complete PV cell model
𝑅𝑆
+
𝐼𝑑
𝐼𝑝ℎ
𝐼
𝑅𝑆ℎ
𝑉
To
I  I ph  I d  I Rsh
I  I ph  I o (e
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q (V  I RS )
akT
 V  I RS 
 1)  

 RSh 
I-V characteristics in dark and light conditions
𝐼
+
𝐼𝑑
𝐼
𝐼𝑑
𝑉𝑑
+
𝑉
No light
 qVd

akT

Id  Io e
 1




I o  dark saturation
current of diode at T
𝑉
𝐼𝑝ℎ
-
-
+
𝐼𝑑
𝐼 = − 𝐼𝑑
𝐼
𝐼 = 𝐼𝑝ℎ − 𝐼𝑑
𝐼
𝐼𝑑
Vd
0
Id ( Vd  300 )
Id ( Vd  300 )
0
Vd
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𝑉𝑑
Vd
𝑉
0
𝑉
Id ( Vd  300 )
L
O
A
D
I-V characteristics of a single PV cell
𝐼 (𝐴)
short circuit
current, 𝐼𝑠𝑐
open circuit
voltage, 𝑉𝑜𝑐
𝑉 (𝑉)
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Short circuit current
𝐼 = 𝐼𝑠𝑐
• With small values of 𝑅𝑠 , 𝐼𝑠𝑐 = 𝐼𝑝ℎ , and it is the maximum possible
current from the PV cell
• 𝐼𝑠𝑐 depends on
• Solar irradiance level and spectrum of light
• Area of solar cell (in commercial silicon cells, 𝐼𝑠𝑐 is roughly 30-35 mA/cm2 )
• Characteristics of material
• For modeling, 𝐼𝑠𝑐 can be considered to vary linearly with irradiance level
• 𝐼𝑠𝑐 variation with temperature can be usually neglected (0.05% increase for every oC)
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Open circuit voltage
•
•
•
•
Maximum possible voltage from PV cell with zero external current
𝑉𝑜𝑐 depends on the quality of the material (low value of 𝐼𝑜 )
Typical values for polycrystalline: 600 mV per cell
𝑉𝑜𝑐 is a strong function of temperature with a
high negative temperature coefficient
• Roughly - 0.35%/oC or about -2.2 mV/ oC
• 𝑉𝑜𝑐 increases slightly with irradiance
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1000 W/m2
𝐼 (𝐴)
750 W/m2
500 W/m2
250 W/m2
open circuit
voltages, 𝑉𝑜𝑐
𝑉 (𝑉)
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𝐼 (𝐴)
Effect of temperature
Higher T
Lower T
(STC)
∆𝑇 ≈ 25℃
𝑉 (𝑉)
• The most dominant effect of temperature
is on the open circuit voltage, 𝑉𝑜𝑐 with a
- 0.35%/oC or about -2.2 mV/ oC
Example commercial PV cell – STP245
• 𝐼𝑠𝑐 increases slightly since the band gap of the material decreases slightly with
temperature and for a given irradiance more electron-hole pairs created(0.05%/oC)
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Effect of series resistance
𝑅𝑅𝑠1
𝑅𝑅𝑠3 𝑅𝑅𝑠2𝑠
𝑠1
𝑅𝑠1
2
𝑠3
𝑅𝑠2
𝑅𝑠4
𝐼 (𝐴)
𝑅𝑠1
𝑅𝑠4 &gt; 𝑅𝑠3 &gt; 𝑅𝑠2 &gt; 𝑅𝑠1
𝑉 (𝑉)
I  I ph  I o (e
q (V  I RS )
akT
 1)
• Terminal voltage at which the current begins to drop rapidly becomes progressively
lower as 𝑅𝑠 increases (e.g., 0.45 V instead of 0.55 V), since 𝑉𝑑 = 𝑉 + 𝐼𝑅𝑠
• Impact at low voltage is not significant for typical values of 𝑅𝑠 since 𝑉 + 𝐼𝑅𝑠 is still low
• 𝑅𝑠 determines the slope near 𝑉𝑜𝑐 which can be used to determine 𝑅𝑠 from datasheets
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Effect of shunt resistance
𝐼 (𝐴)
1000 W/m2
𝑹𝒔𝒉 = 𝟏𝟎𝟎 𝜴
𝑹𝒔𝒉 = 𝟏𝟎 𝜴
𝑹𝒔𝒉 = 𝟑 𝜴
𝑹𝒔𝒉 = 𝟏 𝜴
𝑉 (𝑉)
• Low values of shunt resistance leading to higher power loss indicates
• 𝑅𝑠ℎ determines the slope near 𝑉 = 0 which can be used to
determine 𝑅𝑠ℎ from datasheets
• Impact (on efficiency) more severe at low irradiance
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Effect of shunt resistance: low irradiance
𝐼 (𝐴)
250 W/m2
𝑹𝒔𝒉 = 𝟏𝟎𝟎 𝜴
𝑹𝒔𝒉 = 𝟏𝟎 𝜴
𝑹𝒔𝒉 = 𝟑 𝜴
𝑹𝒔𝒉 = 𝟏 𝜴
𝑉 (𝑉)
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Effect of ideality factor
𝐼 (𝐴)
Smaller 𝑎 −⋙ higher MPP
𝒂 = 𝟏. 𝟎
𝒂 = 𝟏. 𝟐
𝒂 = 𝟏. 𝟓
𝑉 (𝑉)
• Ideality factor ‘a’ is measure of the material quality – lower values reflect better
material, smaller dark saturation current and higher power output
• 𝑎 ranges from 1 to 2
• Impact is mainly near the maximum power point and can be used to estimate
𝑎 from datasheet
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P-V (power-voltage) characteristics
𝑃𝑚𝑝
9
𝐼𝑠𝑐
𝐼𝑚𝑝
4
𝐼 𝑣𝑠. 𝑉
1000
8
3.5
7
3
𝑃 𝑣𝑠. 𝑉
6
𝑰 (𝑨)
I ( Vd )
W/m2
5
2.5
𝑃𝑚𝑝2
500 W/m2
2
𝑷 (𝑾)
P ( Vd )
4
1.5
3
1
2
0.5
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
𝑉 (𝑉)
V ( Vd )  V ( Vd )
Imp  7.95
𝑃𝑚𝑝 : Maximum power
𝑉𝑚𝑝 : Voltage at maximum power
𝐼𝑚𝑝 : Current at maximum power
0.4
0.45
𝑉𝑚𝑝
0.5
0.55
𝑉𝑚𝑝2
0
0.65
0.6
𝑉𝑜𝑐
• MPPT (maximum power point tracking) to ensure that the operation is always at the 𝑃𝑚𝑝
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Fill factor
𝑉𝑜𝑐 , 𝐼𝑠𝑐
𝑉𝑚𝑝 , 𝐼𝑚𝑝
5
8
Curr ent (A )
2.5
4
P ow er (W )
𝑃𝑚𝑝
6
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0
0.65
Voltage (V)
𝐹𝑖𝑙𝑙 𝑓𝑎𝑐𝑡𝑜𝑟, 𝐹𝐹 =
𝑉𝑚𝑝 𝐼𝑚𝑝
𝑉𝑜𝑐 𝐼𝑠𝑐
• Low values of fill factor represent higher losses
in parasitic resistances and other non-idealities
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For the example above,
0.498 &times; 7.95
𝐹𝐹 =
= 0.75
8.33 &times; 0.622
P
0
𝑃(𝑊)
Simulation results: P-V at different irradiance levels
𝑉 (𝑉)
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Effect of temperature
Lower T
Higher T
∆𝑇 ≈ 25℃
• The dominant effect of temperature is the
increase in reverse saturation current , 𝐼𝑜
• Significantly reduces open circuit voltage, 𝑉𝑜𝑐
and maximum power
- 0.35%/oC or about -2.2 mV/ oC in 𝑉𝑜𝑐
and about -0.45 %/oC in 𝑃𝑀𝑃
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(STC)
Example commercial PV cell – STP245
PLECS model of a PV cell
I  I ph  I o (e
q (V  I RS )
akT
 V  I RS 
 1)  

R


Sh
𝐼
+
To