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Tutorial C - PV Module Performance Advanced#

Now that we know how to obtain the plane of array (POA) irradiance and cell temperature, let’s calculate a module’s performance assuming a subset of irradiance and temperature conditions. The objectives for this tutorial are to use pvlib python to do the following:

Retrieve a set of module CEC parameters from the latest NREL System Advisor Model (SAM) library.
Calculate the single diode model (SDM) parameters at standard test conditions (STC) and for a set of PV module test conditions known as IEC61853.
Produce an IV curve for each of the IEC61853 test conditions.
Derive the California Energy Commission (CEC) model parameters based on standard CEC test measurements.

Overview

PV Concepts:#

STC Parameters
IV Curve
PV module energy conversion models
- Point Models
  - PVWatts
  - Sandia Array Performance Model (SAPM)
- Continuous Models
  - CEC
  - PVSyst
  - DeSoto
Low light and temperature module performance
IEC 61853 standard

Python Concepts:#

numpy.meshgrid
try: except clauses to catch errors
Transpose a Pandas dataframe
Pandas series and index string methods: e.g. df.index.str.startswith('Canadian') where df is a dataframe of PV modules.

Standard Test Conditions (STC)#

The most basic condition is called “standard test conditions” or STC, which is considered the reference for most PV modules. For example, all of the PV modules in the SAM CEC module database list their nameplate power at STC.

irradiance: 1000-W/m²
cell temperature: 25°C
angle of incidence (AOI): 0°
spectrum: AM1.5g (ASTM G-173)

Air mass (AM)#

The standard reference AM1.5g (ASTM G-173) is defined as the solar spectrum of global irradiance that passes through 1.5 atmospheres. For more information see NREL Solar Spectra.

IEC 61853 test conditions#

Another common set of test conditions is the IEC 61853 standard which provides a PV module test matrix that covers the expected range of inicident irradiance and cell temperatures for PV modules assuming that the irradiance is normal and the solar spectrum is similar to AM1.5g.

irradiance (W/m²): 100, 200, 400, 600, 800, 1000, 1100
module temperature (°C): 15, 25, 50, 75
angle of incidence: 0°
spectrum: AM1.5g (ASTM G-173)

The figure below shows IEC 61853 test results performed at CFV labs:

Overview

Certain combinations are excluded because they’re unlikely: (1100-W/m², 15°C), (400-W/m², 75°C), (200-W/m², 50°C), (200-W/m², 75°C), (100-W/m², 50°C), and (100-W/m², 75°C). The figure below shows SAM parameters fit to IEC61853 test results for two different cell technologies. The white space shows combinations of irradiance and temperature which were intentionally excluded from testing.

Overview

Attribution: NREL Dobos, Freeman 2019

# if running on google colab, uncomment the next line and execute this cell to install the dependencies and prevent "ModuleNotFoundError" in later cells:
# !pip install -r https://raw.githubusercontent.com/PVSC-Python-Tutorials/PVPMC_2022/main/requirements.txt

# import pvlib and other useful python packages
import pvlib
import numpy as np
from matplotlib import pyplot as plt
import pandas as pd

# set STC reference conditions
E0 = 1000  # W/m^2
T0 = 25  # degC

# set the IEC61853 test matrix
E_IEC61853 = [100, 200, 400, 600, 800, 1000, 1100]  # irradiances [W/m^2]
T_IEC61853 = [15, 25, 50, 75]  # temperatures [degC]

# create a meshgrid of temperatures and irradiances
# for all 28 combinations in the test matrix
IEC61853 = np.meshgrid(T_IEC61853, E_IEC61853)

# meshgrid returns two 2-D arrays in the same order as the input arguments
# so the first item in the output is a 2-D array of temperatures and
# the second item is the mesh of irradiances
# display temperature and irradiance test matrices
IEC61853

[array([[15, 25, 50, 75],
        [15, 25, 50, 75],
        [15, 25, 50, 75],
        [15, 25, 50, 75],
        [15, 25, 50, 75],
        [15, 25, 50, 75],
        [15, 25, 50, 75]]),
 array([[ 100,  100,  100,  100],
        [ 200,  200,  200,  200],
        [ 400,  400,  400,  400],
        [ 600,  600,  600,  600],
        [ 800,  800,  800,  800],
        [1000, 1000, 1000, 1000],
        [1100, 1100, 1100, 1100]])]

Single Diode Model (SDM) & IV Curve#

PV module performance can be modeled using point or continuous IV-curve models.

Point models like PVWatts and The Sandia Array Performance Model (SAPM, aka King model) yield the current (I), voltage (V), and power (P) at a single or discrete set of points. PVWatts only yields the performance at the max power point (MPP) of the module, whereas the SAPM also yields the short circuit current (Isc), open circuit voltage (Voc).

Continuous IV curve models like the CEC, PVsyst, and DeSoto models yield a relation between current and voltage called an IV curve, and therefore yield a continuous set of (V, I) points spanning from Isc to Voc and beyond. The domain of the IV curve is in quadrants 1, 2, and 4 where voltage is on the horizontal and current is on the vertical axis. The figure below from PV Education PVCDROM shows an IV curve of an “ideal” cell.

Attribution: PV Education, UNSW, ASU, et al.

The IV curve relationship is based on an electrical analog called the “single diode model” or SDM which is defined by 5 parameters: the light current (\(I_L\)), shunt resistance (\(R_{sh}\)), series resistance (\(R_s\)), diode saturation current (\(I_o\) or \(I_{sat}\)), and the diode ideality factor (\(n\)). Other symbols for diode ideality factor are (\(\gamma\)) used by PVsyst and (\(a\)) used by SAM, but (\(\gamma\)) is also frequently used for power temperature coefficient. This “ideal” cell is described by the electrical schematic drawing below.

Attribution: Sandia NL PV Performance Modeling Collaborative

Combining the components in the SDM using Ohm’s and Kirchhoff’s laws yields the following equation, which is implicit because current (I) is on both sides of the equation, and cannot be solved explicitly:

\[ I = I_L - I_o \left( \exp \left( \frac{V + I R_s}{n V_T} \right) - 1 \right) - \frac{V + I R_s}{R_{sh}} \]

with the diode voltage (\(V_D = V + I R_s\)), the diode current (\(I_D\)) given by the ideal diode equation:

\[ I_D = I_o \left( \exp \left( \frac{V_D}{n V_T} \right) - 1 \right) \]

the thermal voltage (\(V_T = k_T / q_e\)), elementary charge (\(q_e\)), Boltzmann constant (\(k_T\)), and the shunt current (\(I_{sh} = V_D / R_{sh}\))

CEC Model (aka SAM or 6-parameter model)#

The California Energy Commision (CEC) contracted authorized testing labs to measure at STC the nameplate power (Pmp), Isc, Voc, and the MPP voltage and current (Vmp, Imp), as well as Isc and Voc temperature coefficients, the module dimensions, the number of series cells (Ns), parallel substrings (Np), module area in m² (Ac), and more. Tables of the CEC module parameters are available from the Solar Equipment Lists. These measurements have been fit to the SDM by the NREL System Advisor Model (SAM) and stored in a CSV file that is bundled with SAM. You can access the SAM library on GitHub. This SAM library of module coefficients derived from the CEC measurements are collectively called CEC modules and the SAM model that uses the derived SDM coefficients is called the CEC model. The CEC model used in SAM is sometimes also called the 6-parameter model because of the Adjust additional parameter which differentiates it from the DeSoto model.

pvlib python#

There are several functions we can use in pvlib python:

# use pvlib python to retrieve CEC module parameters from the SAM libraries
# with the "name" argument set to "CECMod"
CECMODS = pvlib.pvsystem.retrieve_sam(name='CECMod')

# the CEC modules are a pandas DataFrame oriented as columns, transpose to arrange
# as indices
CECMODS.T.head()

	Technology	STC	PTC	A_c	Length	Width	N_s	I_sc_ref	V_oc_ref	...	a_ref	I_L_ref	R_s	R_sh_ref	Adjust	gamma_r	BIPV	Version	Date
A10Green_Technology_A10J_S72_175	Mono-c-Si	175.0914	151.2	1.3	1.576	0.825	72	5.17	43.99	...	1.981696	5.175703	0.316688	287.102203	16.057121	-0.5072	N	SAM 2018.11.11 r2	1/3/2019
A10Green_Technology_A10J_S72_180	Mono-c-Si	179.928	155.7	1.3	1.576	0.825	72	5.31	44.06	...	1.988414	5.316148	0.299919	259.047943	16.418983	-0.5072	N	SAM 2018.11.11 r2	1/3/2019
A10Green_Technology_A10J_S72_185	Mono-c-Si	184.7016	160.2	1.3	1.576	0.825	72	5.43	44.14	...	1.984817	5.435676	0.311962	298.424438	15.688233	-0.5072	N	SAM 2018.11.11 r2	1/3/2019
A10Green_Technology_A10J_M60_220	Multi-c-Si	219.876	189.1	1.624	1.632	0.995	60	7.95	36.06	...	1.673094	7.959062	0.140393	123.168404	21.875164	-0.5196	N	SAM 2018.11.11 r2	1/3/2019
A10Green_Technology_A10J_M60_225	Multi-c-Si	224.9856	193.5	1.624	1.632	0.995	60	8.04	36.24	...	1.671782	8.047206	0.14737	164.419479	20.698376	-0.5196	N	SAM 2018.11.11 r2	1/3/2019

5 rows × 25 columns

CEC modules library#

Periodically a static copy of CEC module parameters is copied from the SAM library to pvlib python. The modules are roughly named according the following scheme:

<manufacturer name> <model name>

Whitespace, dashes, and other non-alphanumerical characters are all replaced by underscores in pvlib python.

EG: “Canadian Solar Inc. CS6X-300M” becomes

Canadian_Solar_Inc__CS6X_300M

Let's Explore: Click here to see the panel's Datasheet.

The main CEC module parameters are defined as follows:

parameter	data type	description and units
`Technology`	string	one of “Mono-c-Si”, “Multi-c-Si”, “Thin Film”, “CdTe”, or “CIGS” families of cells
`Bifacial`	boolean	is bifacial?
`STC`	float	nameplate in W at STC
`PTC`	float	nameplate in W at PVUSA test conditions (1-sun, 20° ambient temperature, 1-m/s windspeed)
`A_c`	float	module area in m²
`Length`	float	module length in m;
`Width`	float	module width in m;
`N_s`	int	number of cells in series
`I_sc_ref`	float	short circuit current in A at reference condition
`V_oc_ref`	float	open circuit voltage in V at reference condition
`I_mp_ref`	float	max power current in A at reference condition
`V_mp_ref`	float	max power voltage in V at reference condition
`alpha_sc`	float	short circuit current temperature coefficient in A/Δ°C
`beta_oc`	float	open circuit voltage temperature coefficient in V/Δ°C
`T_NOCT`	float	normal operating cell temperature in °C
`a_ref`	float	diode ideality factor
`I_L_ref`	float	light or photogenerated current at reference condition in A
`I_o_ref`	float	diode saturation current at reference condition in A
`R_s`	float	series resistance in Ω
`R_sh_ref`	float	shunt resistance at reference condition in Ω
`Adjust`	float	adjustment to short circuit temperature coefficient in %
`gamma_r`	float	power temperature coefficient at reference condition in %/Δ°C
`BIPV`	boolean	is building integrated PV?

# One trick to find the modules is to search the indices using string filters
# For example: find all Candian Solar 220-W mono-Si modules
cs_220m_mods = CECMODS.T.index.str.startswith('Canadian_Solar') & CECMODS.T.index.str.contains('220M')
CECMODS.T[cs_220m_mods]

	Technology	Bifacial	STC	PTC	A_c	Length	Width	N_s	I_sc_ref	V_oc_ref	...	a_ref	I_L_ref	I_o_ref	R_s	R_sh_ref	Adjust	gamma_r	BIPV	Version	Date
Canadian_Solar_Inc__CS5P_220M	Mono-c-Si	0	219.961	200.1	1.7	1.602	1.061	96	5.1	59.4	...	2.635926	5.11426	0.0	1.066023	381.254425	8.619516	-0.476	N	SAM 2018.11.11 r2	1/3/2019
Canadian_Solar_Inc__CS6P_220M	Mono-c-Si	0	219.775	198.5	1.549	1.615	0.959	60	7.97	36.9	...	1.515583	7.980784	0.0	0.397651	293.871094	-3.311191	-0.436	N	SAM 2018.11.11 r2	1/3/2019

2 rows × 25 columns

# that was almost too easy, let's use the CS5P-220M
# NOTE: don't transpose CECMODS, get column with desired module 
CS_220M = CECMODS['Canadian_Solar_Inc__CS5P_220M']
CS_220M

Technology            Mono-c-Si
Bifacial                      0
STC                     219.961
PTC                       200.1
A_c                         1.7
Length                    1.602
Width                     1.061
N_s                          96
I_sc_ref                    5.1
V_oc_ref                   59.4
I_mp_ref                   4.69
V_mp_ref                   46.9
alpha_sc               0.004539
beta_oc               -0.222156
T_NOCT                     42.4
a_ref                  2.635926
I_L_ref                 5.11426
I_o_ref                     0.0
R_s                    1.066023
R_sh_ref             381.254425
Adjust                 8.619516
gamma_r                  -0.476
BIPV                          N
Version       SAM 2018.11.11 r2
Date                   1/3/2019
Name: Canadian_Solar_Inc__CS5P_220M, dtype: object

Pop Quiz #1#

Get any CEC module from the CECMODS or pvfree.

Which module did you get?
How did you get it?
What is the module’s nameplate power, Isc, Voc, Imp, and Vmp?
Who is the manufacturer?
What cell technology is it?
How does it differ from CS5P-220M?

# use this cell to search CECMODS.T or pvfree
your_mod = 'your module goes here'
try:
    your_mod = CECMODS[your_mod]
except KeyError:
    print(f"*** Sorry, '{your_mod}' wasn't found in CECMODS. Please try again. ***")
else:
    # display your module
    your_mod

*** Sorry, 'your module goes here' wasn't found in CECMODS. Please try again. ***

Calculate CEC Parameters#

The module parameters are given at the reference condition. Use pvlib.pvsystem.calcparams_cec() to generate the five SDM parameters at your desired irradiance and temperature to use with pvlib.pvsystem.singlediode() to calculate the IV curve information.

`nNsVth`, what’s this?#

The diode ideality factor (n) is combined with the number of cells (Ns) and the thermal voltage (Vth) to create a convenience parameter. This is syntactic sugar.

# finally this is the magic
temp_cell, effective_irradiance = IEC61853
cecparams = pvlib.pvsystem.calcparams_cec(
        effective_irradiance=effective_irradiance,
        temp_cell=temp_cell,
        alpha_sc=CS_220M.alpha_sc,
        a_ref=CS_220M.a_ref,
        I_L_ref=CS_220M.I_L_ref,
        I_o_ref=CS_220M.I_o_ref,
        R_sh_ref=CS_220M.R_sh_ref,
        R_s=CS_220M.R_s,
        Adjust=CS_220M.Adjust,
        EgRef=1.121,
        dEgdT=-0.0002677)
IL, I0, Rs, Rsh, nNsVth = cecparams
# display the photogenerated current
IL

array([[0.50727824, 0.511426  , 0.5217954 , 0.5321648 ],
       [1.01455648, 1.022852  , 1.0435908 , 1.0643296 ],
       [2.02911296, 2.045704  , 2.0871816 , 2.1286592 ],
       [3.04366944, 3.068556  , 3.1307724 , 3.19298881],
       [4.05822592, 4.091408  , 4.1743632 , 4.25731841],
       [5.0727824 , 5.11426   , 5.217954  , 5.32164801],
       [5.58006064, 5.625686  , 5.7397494 , 5.85381281]])

IV Curve Info#

Now that we have the 5 SDM parameters (IL, Io, Rs, Rsh, and nNsVth) corresponding to each of the test conditions in the IEC61853 test matrix, we can calculate the IV curve information for that irradiance and cell temperature.

# flatten the meshgrid to allow single diode to broadcast the output
curve_info = pvlib.pvsystem.singlediode(
    photocurrent=IL.flatten(),
    saturation_current=I0.flatten(),
    resistance_series=Rs,
    resistance_shunt=Rsh.flatten(),
    nNsVth=nNsVth.flatten(),
    ivcurve_pnts=101,
    method='lambertw')
# display the max power points
curve_info['p_mp']

array([ 22.47344592,  21.33362028,  18.40249412,  15.37302335,
        46.09151614,  43.87428661,  38.16846011,  32.26328112,
        93.53042781,  89.2112251 ,  78.09399529,  66.58418353,
       140.31159673, 133.92211364, 117.48149685, 100.47026643,
       186.01692511, 177.5719005 , 155.85443596, 133.40606739,
       230.45399426, 219.96096044, 192.99622092, 165.16045701,
       252.16130617, 240.64592808, 211.06610839, 180.55358287])

# plot the calculated curves:
exclude = [(1100, 15), (400, 75), (200, 50), (200, 75), (100, 50), (100, 75)]
kolor = ['#1f77b4', '#2ca02c', '#8c564b', '#9467bd', '#d62728', '#e377c2', '#ff7f0e']
f, ax = plt.subplots(2, 2, figsize=(16, 10), sharex=True, sharey=True)
for m, irr in enumerate(E_IEC61853):
    for n, tc in enumerate(T_IEC61853):
        if (irr, tc) in exclude:
            continue
        i = n + 4*m
        j = n // 2, n % 2
        label = (
            "$G_{eff}$ " + f"{irr} $W/m^2$"
        )
        ax[j].plot(curve_info['v'][i], curve_info['i'][i], label=label, c=kolor[m])
        v_mp = curve_info['v_mp'][i]
        i_mp = curve_info['i_mp'][i]
        # mark the MPP
        ax[j].plot(v_mp, i_mp, ls='', marker='o', c=kolor[m])
        ax[j].vlines(v_mp, 0, i_mp, linestyle='dashed', color=kolor[m])
        # just repeat this every time doesn't hurt anyone
        ax[j].legend(loc='right')
        if j[0] == 1:
            ax[j].set_xlabel('Module voltage [V]')
        if j[1] == 0:
            ax[j].set_ylabel('Module current [A]')
        ax[j].set_title(f"{CS_220M.name}, " + "$T_{cell}$ " + f"{tc} " + "$^{\circ}C$")
        ax[j].grid(True)
        ax[j].set_xlim([0, 80])

f.tight_layout()

_images/Tutorial C - Modeling Module's Performance Advanced_22_0.png

Conclusions#

So is this what you expected? What do you notice about the relation between voltage and temperature? What about the relation between current and irradiance? What other patterns do you observe?

This work is licensed under a Creative Commons Attribution 4.0 International License.

PVPMC 2022

Tutorial C - PV Module Performance Advanced

Contents

Tutorial C - PV Module Performance Advanced#

PV Concepts:#

Python Concepts:#

Standard Test Conditions (STC)#

Air mass (AM)#

IEC 61853 test conditions#

Single Diode Model (SDM) & IV Curve#

CEC Model (aka SAM or 6-parameter model)#

pvlib python#

CEC modules library#

Pop Quiz #1#

Calculate CEC Parameters#

`nNsVth`, what’s this?#

IV Curve Info#

Conclusions#

PVPMC 2022

Tutorial C - PV Module Performance Advanced

Contents

Tutorial C - PV Module Performance Advanced#

PV Concepts:#

Python Concepts:#

Standard Test Conditions (STC)#

Air mass (AM)#

IEC 61853 test conditions#

Single Diode Model (SDM) & IV Curve#

CEC Model (aka SAM or 6-parameter model)#

pvlib python#

CEC modules library#

Pop Quiz #1#

Calculate CEC Parameters#

nNsVth, what’s this?#

IV Curve Info#

Conclusions#

`nNsVth`, what’s this?#