
Research Article


Sensitivity Analysis of a Standalone Photovoltaic System Model Parameters


Abdul Qayoom Jakhrani,
AlKhalid Othman,
Andrew Ragai Henry Rigit,
Saleem Raza Samo
and
Shakeel Ahmed Kamboh



ABSTRACT

The values of input variables of any model are cause to undergo changes due to influence of environmental conditions. These changes can be investigated by conducting the sensitivity analysis of input variables with respect to output variables. Sensitivity analysis increases the validity, credibility and assurance of model estimates. The purpose of this study was to identify the most important and sensitive input variables and to prioritize the parameters based on their influence on the model outputs of a standalone photovoltaic (SAPV) system. For that, a normalized local sensitivity analysis and sensitivity index of seven input variables of a SAPV system model with reference to three output parameters namely amount of absorbed solar radiation, maximum photovoltaic (PV) module power output and optimum PV array area has been carried out. It was revealed from the analysis that the most important and sensitive input variable was the amount of total solar radiation and the least important variable was solar azimuth angle and the lowest sensitive variable was wind speed.





Received: November 15, 2012;
Accepted: January 16, 2013;
Published: February 21, 2013


INTRODUCTION
The physical examination of highly complex systems and processes are expensive
or sometimes even impossible. Therefore, the investigators turned to use mathematical
or computational models to predict or approximate the behavior of systems (Fellin
et al., 2004; Saltelli et al., 2006).
The cause of uncertainties in the model inputs are not known which results the
ambiguity of model outputs. The role of every input variable in the changing
of model outputs is needed for the evaluation of model suitability and understanding
of the system behavior (Isukapalli and Georgopoulos, 2001).
Various terms could be found in literature for the expression of input parameters
such as sensitive, important, most influential, major contributor, effective
or correlated (Iman and Helton, 1988). However, the
term important is used for those parameters whose uncertainty contributes considerably
to the uncertainty in the output results and the word sensitive referred to
those variables which have a significant influence over output results (Saltelli
et al., 2010). The main parameter is always sensitive because the
parameter changeability will not appear in the results unless the model is sensitive
to the input (Cacuci et al., 2005). A sensitive
parameter is not necessarily important because it may have little contribution
in the output variability (Hamby, 1994). So, the object
of the study was to investigate the most important input variable of a standalone
photovoltaic system.
METHODS OF SENSITIVITY ANALYSIS
Different scholars rather used different sensitivity methods according to the
nature of analysis and required accuracy. In brief, these methods include oneatatime
design, differential analysis, subjective analysis and factorial design (Hamby,
1994). The sensitivity of parameters can also be examined by the construction
of scatter plots, calculation of relative deviation ratios, determination of
rank transformation, rank correlation and partial correlation coefficients and
also by means of regression techniques (Hamby, 1995).
Various statistical tests such as Smirnov statistic, Cramervon Mises, MannWhitney
and the squared rank can also be adopted for sensitivity analysis of model parameters
(Cukier et al., 1978; Iman
and Hora, 1990; Bell and Otto, 1992; Bekele
and Nicklow, 2007). Moreover, another simple method for determining parameter
sensitivity was given by Hoffman and Gardner (1983),
which is based, on the output percent difference by varying one input parameter
from its minimum value to its maximum value. It is very helpful to compare the
performance of many individual indices relative to a composite index (Hamby,
1995). It is given as:
where, SI is the sensitivity index, y_{min} and y_{max} represent the minimum and maximum output values, respectively.
All Sensitivity Analysis (SA) methods have their own characteristic, theories
and range of applications. Therefore, the choice of a sensitivity analysis method
is generally depends on the sensitivity measure employed, the required precision
in the estimates of model predictions and the computational cost involved (Bell
and Otto, 1992; Isukapalli and Georgopoulos, 2000).
However, the Differential Sensitivity Analysis (DSA) method is a major component
of nearly all other sensitivity analysis techniques (Hamby,
1994). DSA method enables a direct examination of the sensitivity of simulation
results to input parameter changes (Purdy and BeausoleilMorrison,
2001). This method requires a base case simulation in which input parameters
are set with the basic estimates of the parameters under consideration. It is
mostly accomplished by computing partial derivatives of the output functions
with respect to the input variables (Saltelli et al.,
1993; Zakayo, 2009).
The generalized form of Differential Sensitivity Analysis (DSA) model contains
several independent variables such as X = (X_{1}.......X_{n})
and one dependent variable Y, where Y = f(X). The sensitivity analysis can be
performed by an explicit arithmetic equation that illustrates the relationship
between the independent variables and the dependent variable (Tolsma
and Barton, 2002). The sensitivity coefficient (S_{i}) for a definite
independent variable can be determined from the partial derivative of the dependent
variable with respect to the independent variable (Saltelli
et al., 2006). Such as:
where, the quotient X_{i}/Y is introduced to normalize the coefficient
by removing the influence of units. It was assumed that the higher order partials
are insignificant and there was no correlation among input parameters due to
the nature of calculations (Christopher and Patil, 2002).
The partial derivative was also approximated as a finite difference for the
large set of equations and the output values was calculated for small changes
in the input parameter (Ostermann, 2005). The nonlinearities
are ignored in the model (Pinjari and Bhat, 2006; Saltelli
et al., 2000). Therefore, the partial derivative can be approximated
as:
The sensitivity coefficient dy/dx was considered to be a linear estimate of
number of units change in the variable y as a result of a unit change in the
parameter x. It means that the sensitivity outcomes depend on physical units
of variables and parameters. Therefore, normalized sensitivity coefficients
()
are used to make the sensitivity results independent of the units of the model
parameters.
The derivative in Eq. 4 can be discretized by using forward finite difference scheme as follows:
where,
is the local normalized sensitivity coefficient. It represents a linear estimate
of the percentage change in the variable y caused by a one percent change in
the parameter x.
Sensitivity of photovoltaic system parameters: El
Shatter and Elhagry (2000) conducted Sensitivity Analysis (SA) on unknown
parameters such as series resistance (R_{s}), shunt resistance (R_{sh}),
light generated current (I_{ph}), reverse diode saturation current (I_{o})
and ideality factor (n) with suggested fuzzy input parameter (h) from 0.2 to
0.8 and output parameter (e) around 10%. They found that the PV module parameters
were severely affected by temperature variation. Kolhe
et al. (2002) conducted an economic feasibility of a standalone
photovoltaic (SAPV) system and a diesel generator. The fuel consumption rate
was compared versus diesel generator rated power capacity at different load
factors. They also analyzed PV/diesel life cycle cost ratio against cost of
the PV array and diesel with energy demand. Ito et al.
(2006) carried out a sensitivity analysis of a very large scale PV system
in deserts. They compared the PV module efficiency with generation cost, energy
payback time and carbon dioxide (CO_{2}) emissions. Loutzenhiser
et al. (2007) used Monte Carlo and fitted effects for Nway factorial
for uncertainty analysis of total solar radiation on a southwest facade building
integrated PV system. Cameron et al. (2008) analyzed
power outputs of different PV models with different PV module technologies at
daily and monthly average yearly basis. Emery (2009)
evaluated uncertainties of measured PV power output with rated PV power output
and measured current and voltage with junction temperature and solar irradiance.
A monthly mean solar radiation with total and beam radiation, PV cell temperature
with ambient temperature and energy output for fixed, optimum and tracking PV
systems was evaluated by Gang and Ming (2009). Ren
et al. (2009) conducted sensitivity analysis of levelized cost of
energy with capital cost, efficiency, interest rate and electrical sale price.
Talavera et al. (2010) conducted SA on Internal
Rate of Return (IRR) of a grid connected PV system with three scenarios on the
parameters of annual yield of PV system, PV module unit price, initial investment
and interest rate. The sensitivity of R_{s} to R_{sh}, R_{sh}
to R_{s} and of current, voltage and power of a single diode PV cell
model was conducted by Zhu et al. (2011).
Kaabeche et al. (2011) carried out a technoeconomical
valuation of a PV system on hourly solar radiation, wind speed and ambient temperature
versus time. The authors compared number of PV modules with storage capacity
of different autonomy days and total annualized cost with different deficiencies
of power supply probabilities and net present cost with various discount rates,
capital cost and project life (Jakhrani et al.,
2012a). The uncertainty analysis of a double diode model was conducted by
Adamo et al. (2011). They compared the amount
of solar irradiance versus temperature, mean relative estimation error on R_{s}
and R_{sh} and standard deviation on R_{s} and R_{sh}.
The sensitivity analysis on levelized cost of electricity versus interest rate
with the inputs of initial installation cost of PV system, energy output and
degradation rate was carried out by Branker et al.
(2011). They compared discount rate versus initial installation cost of
PV system with the inputs of lifetime loan term, energy output, degradation
rate and zero interest loans. They also evaluated lifetime of PV system versus
initial installation cost of PV system with the inputs of discount rate, energy
output, degradation rate and zero interest loan. DufoLopez
et al. (2011) applied Strength Pareto Evolutionary Algorithm to the
multiobjective optimization of standalone PVwinddiesel system with battery
storage. They conducted SA on parameters like inflation of diesel cost, acquisition
cost and emissions of PV panels. Andrews et al.
(2012) presented a methodology for fine resolution modeling of a PV system
using PV module short circuit current (I_{sc}) at 5 min timeintervals.
They identified the pertinent error mechanisms by filtering the data with regressive
analysis. Mbaka et al. (2010) carried out an
economic evaluation among three different power producing systems such as PV
hybrid system, standalone PV system and standalone diesel generator system using
net present value cost. Sensitivity analysis on diesel prices and the unit cost
of PV modules was conducted by Jakhrani et al. (2012b,
c).
It is revealed from the literature review, that the most of mathematical models used in sensitivity analysis are based on systems of algebraic and differential equations. The common problem in the models is that, the role of various parameters is not obvious. Generally, the important parameters, effects of changing parameters and uncertainties of model results due to uncertainty of model inputs are not known. In many applications, this information is exactly needed. Furthermore, the sensitivity analysis methods are mostly used for the analysis of biological, environmental, water quality parameters and chemical kinetics. However, these are rather new in the analysis of PV system parameters. No complete sensitivity analysis of PV system input variables as a function of output parameters has been found in the literature. Most of sensitivity analysis was conducted on the cost analysis of systems and a few on currentvoltage characteristics of PV modules. It was necessary to examine the behavior of most influential input variables of PV model with respect to output parameters. Therefore, this study was conducted using differential sensitivity analysis method along with sensitivity index for the identification and extent of sensitive and important model parameters. These methods are computationally efficient and allow rapid preliminary examination of the model parameters. It also provides the slope of the calculated model output in parameter space at a given set of values. The evaluation of model suitability, identification of most influential and sensitive parameters is identified which are essential for the prediction of system performance. METHODOLOGY
Normalized local sensitivity analysis of PV system model parameters such as
absorbed solar radiation (S_{T}), PV module maximum power output (P_{max})
and optimum PV array area (A_{opt}) has been carried out by differential
sensitivity analysis method. Five input variables namely slope (β), solar
azimuth angle (γ), hour angle (ω), ground reflectance (ρ_{g})
and monthly average daily total solar radiation ()
were used for examination of absorbed solar radiation (S_{T}). The absorbed
solar radiation (S_{T}) with other two input variables namely ambient
temperature (T_{a}) and wind speed (V_{w}) were utilized for
the estimation of PV module maximum power output (P_{max}) and required
optimum PV array area (A_{opt}) as shown in Fig. 1.
In Addition, various empirical equations were adopted for the calculation of
transitional model parameters. Those transitional estimated values were used
as the input of the required models.
In first step of differential sensitivity analysis, the base values, ranges
and distributions were selected for each input variable. Secondly, a Taylor
series approximation to the model output was developed close to the base values
of the model inputs. The first order Taylor series was preferred. Thirdly, the
variance propagation techniques were used for the estimation of the uncertainty
in model output in terms of its projected values and variance, because these
values changes according to the order of approximation. Finally, the first order
Taylor series was used to estimate the magnitude of each input parameter (Helton
and Davis, 2003; Helton et al., 2005). The
sensitivity information by differential analysis methods was carried out by
changing the values of one parameter on a single variable, because of its simplicity.
Furthermore, the results of various output parameters are also examined by conducting
the sensitivity index of input variables.

Fig. 1: 
Model for sensitivity analysis of SAPV system parameters 
RESULTS AND DISCUSSION The values of input parameters were varied around the base values of parameters.
The values of slope (β) were changed with an interval of five degrees from
zero degree to 90° as shown in Fig. 2. The estimated results,
differential change, percentage change and normalized local sensitivity coefficient
values of output parameters such as S_{T}, P_{max} and A_{opt}
are illustrated in Fig. 2ad, respectively.
It was observed that the amount of absorbed solar radiation (S_{T}),
maximum power output (P_{max}) at zero degree were 12. 8 MJ m^{2}
and 93.4 W, at 25° the values were 13.6 MJ/m^{2} and 99.4 W and
at 90° of slope these values were 8.5 MJ m^{2} and 61.2 W, respectively.
The positive change was found up to 20° and then the change became negative
up to 90°. The results of absorbed solar radiation (S_{T}) and maximum
PV module power output (P_{max}) were observed as 2.3% at 0°, 0%
around 25° and 10% at 90° of slope. The estimated optimum PV array area
(A_{opt}) was found to be 16.5 m^{2} at 90° and minimum
area of 10.2 m^{2} was noted around 25° of slope. Negative change
in the optimum PV array area were observed from 0 to 25° of slope and positive
from 25 to 90° of slope. A maximum of 12% change was observed at 90°
of slope and minimum around 25°. It was found that the optimum area of PV
modules is inversely proportional to the absorbed solar radiation and maximum
PV module power output. As power output from PV modules decreases the requirement
of PV array area increases for the system installation.
The input values of the solar azimuth angle (γ) was varied with an interval
of 10° from 90 to +90° as shown in Fig. 3. The estimated
values, differential change, percentage change and normalized local sensitivity
values of output parameters are shown in Fig. 3ad,
respectively. The maximum results of S_{T} and P_{max} was found
at the solar azimuth angle of 0° with 13.3 MJ m^{2} and 97.2 W,
respectively and minimum results were observed at ±90°. Both the
amount of absorbed solar radiation and PV module power output showed positive
change from 90 to 0°and negative change from 0° to +90°. The change
of 10° solar azimuth angle results 0.0043% change in the output values at
0° of solar azimuth angle and 0.06% were observed at +90°. At zero degree
of solar azimuth angle, the normalized coefficients were zero but negative up
to +90° of solar azimuth angle. In contrary to the results of S_{T}
and P_{max}, A_{opt} displayed negative change from 90°
to 0° and positive change from 0° to +90°. The normalized coefficients
of sensitivity analysis for A_{opt} were found positive with almost
same values of S_{T} and P_{max}.

Fig. 2(ad): 
Sensitivity analysis of output parameters with respect to
slope, (a) Values of output parameters, (b) Differential change in parameter
values, (c) Percentage change in parameter values and (d) Normalized local
SA values of parameters 

Fig. 3(ad): 
Sensitivity analysis of output parameters with respect to
solar azimuth angle, (a) Values of output parameters, (b) Differential change
in parameter values, (c) Percentage change in parameter values and (d) Normalized
local SA values of parameters 

Fig. 4(ad): 
Sensitivity analysis of output parameters with respect to
hour angle, (a) Values of output parameters, (b) Differential change in
parameter values, (c) Percentage change in parameter values and (d) Normalized
local SA values of parameters 
The input values of the hour angle (ω) was varied with an interval of
15° from 75 to +75 ° as shown in Fig. 4. The estimated
results, differential change, percentage change and normalized local sensitivity
values of output parameters for the hour angle (ω) are shown in Fig.
4ad, respectively. It was observed that the amount of
solar radiation and maximum PV module power output at zero degree hour angle
was 113.3 MJ m^{2} and 97.0 W and at ±75° hour angle the
values were 11.6 MJ m^{2} and 84.5 W, respectively. Only 2.0 M J m^{2}
and 15.0 W change with 17.7% variation were observed in the results of absorbed
solar radiation and maximum PV module power output, respectively at ±75°
of the hour angle. The normalized sensitivity coefficient was minus 0.88 at
75° and minus 0.60 at +75° of the hour angle. The optimum PV array
area values were approximately 12.0 m^{2} at ±75° of the
hour angle and 10.4 m^{2 }at zero degree of the hour angle. Negative
change was observed in the values from ±45 to ±75° and positive
change from zero degree to ±45° of the hour angle. Fifteen percent
change in the output results was observed at ±75° and no change in
the results was found at zero degree of the hour angle. Normalized sensitivity
results were +0.76 at ±75°, whereas, negative sensitivity coefficients
were seen from zero degree to ±45° of the hour angle values.
The input values of ground reflectance (ρ_{g}) were varied by
an interval of 0.1 from 0.0 to 0.7 as shown in Fig. 5. The
estimated results, differential change, percentage change and normalized local
sensitivity values of output parameters with respect to ground reflectance (ρ_{g})
are illustrated in Fig. 5ad, respectively.
The maximum amount of S_{T} and P_{max} was found at higher
ground reflectance (ρ_{g}) inputs and minimum at low levels of
ground reflectance (ρ_{g}). At 0.0 ρ_{g} the values
of S_{T} and P_{max} were 12.1 MJ m^{2} and 88.6 W,
whereas, at 0.7 ρ_{g}, the values of S_{T} and P_{max}
were 16.3 MJ m^{2} and 118.0 W, respectively. The change of 0.6 MJ
m^{2} and 4.0 W were found with the increment of 0.1 variation of ground
reflectance. The higher percentage changes were observed at lower levels of
ρ_{g} and vice versa. Around 5% changes were found at zero value
of ρ_{g} and 3.6% change was observed at the value of 0.7 ρ_{g}.
The input values of monthly mean daily total solar radiation on horizontal
surface ()
were changed with an interval of 1.0 MJ m^{2} from 5.0 to 25.0 MJ m^{2}
as shown in Fig. 6.

Fig. 5(ad): 
Sensitivity analysis of output parameters with respect to
ground reflectance, (a) Values of output parameters, (b) Differential change
in parameter values, (c) Percentage change in parameter values and (d) Normalized
local SA values of parameters 

Fig. 6(ad): 
Sensitivity analysis of output parameters with respect to
total solar radiation, (a) Values of output parameters, (b) Differential
change in parameter values, (c) Percentage change in parameter values and
(d) Normalized local SA values of parameters 
The estimated results, differential change, percentage change and normalized
local sensitivity values of output parameters are illustrated in Fig.
6ad, respectively. The maximum values of S_{T}
and P_{max} were 26.4 MJ m^{2} and 185.8 W found at the highest
value of .
The change in S_{T} and P_{max} values were 1.03 to 1.07 MJ
m^{2} and 7.8 to 6.7 W at the value of 5 to 25 MJ m^{2} of
total solar radiation (),
respectively. Higher change in the output values of S_{T}, P_{max}
and A_{opt} values were found in the lower level of ,
with 20.4, 22 and 17.8% and minimum percentage change were observed in higher
level of ,
around 4% in all output parameters, respectively. Normalized sensitivity coefficient
were noted at all data points of S_{T} and P_{max}. The maximum
and minimum value of A_{opt} were 28.2 and 5.4 m^{2} found at
5 and 25.0 MJ m^{2} of respectively. Almost minus 5 m^{2} changes
in solar radiation were found at low level of solar radiation values and minus
0.2 m^{2} was found at higher level of solar radiation.
The input values of ambient temperature (T_{a}) were changed with an
interval of 5°C from 15 to 50°C as shown in Fig. 7.
The estimated results, differential change, percentage change and normalized
local sensitivity values of output parameters namely P_{max} and A_{opt}
are illustrated in Fig. 7ad, respectively.
The optimum values of P_{max} were found at low level of ambient temperatures
and the lower results of P_{max} were found at higher temperatures.
At 15°C of ambient temperature, the value of P_{max} was 104 W and
at 50°C its output value was found to be 82 W. The change in output of P_{max}
values per 5°C was found to be around minus 3 W. The percentage change at
low level of ambient temperatures was 3.0 and at higher levels of ambient temperatures
were 3.6. Normalized local sensitivity coefficient was found negative, because,
the increase of ambient temperature was responsible for the decrease in power
output. The values of sensitivity coefficients were minus 0.09 at the lower
level of ambient temperature and minus 0.3 at the higher levels. The A_{opt}
at 15 and 50°C of ambient temperature were 9.2 and 11.6 m^{2}, respectively.
The variation in A_{opt} was 0.3 and 0.4 m^{2} and the change
in percentage of A_{opt} was 3.2 and 3.7 found at 15 and 50°C of
T_{a}, respectively. The normalized sensitivity coefficients were positive
due to increase of A_{opt} with the increase of ambient temperature.
The input values of wind speed (V_{w}) were changed with an interval
of 1 m sec^{1} from zero to 10 m sec^{1} as shown in Fig.
8. The estimated results, differential change, percentage change and normalized
local sensitivity values of output parameters namely P_{max} and A_{opt}
are illustrated in Fig. 8ad, respectively.

Fig. 7(ad): 
Sensitivity analysis of output parameters with respect to
ambient temperature, (a) Values of output parameters, (b) Differential change
in parameter values, (c) Percentage change in parameter values and (d) Normalized
local SA values of parameters 

Fig. 8(ad): 
Sensitivity analysis of output parameters with respect to
wind speed, (a) Values of output parameters, (b) Differential change in
parameter values, (c) Percentage change in parameter values and (d) Normalized
local SA values of parameters 

Fig. 9: 
Sensitivity index of output parameters versus different input
variables 
It was found from the results that the increase of wind speed from zero to
10 m sec^{1} results the increase of P_{max} from 96.6 to 100.0
W and decrease of A_{opt} from 9.88 to 9.56 m^{2}, respectively.
The increase of 1 m sec^{1} wind speed results change in 0.3 W power
increase and 0.03 m^{2} decrease of A_{opt}. The average percentage
change in P_{max} values were 0.34 and in A_{opt} 0.33 per 1
m sec^{1} change of wind speed.
The sensitivity index of all output parameters with respect to input variables
are illustrated in Fig. 9. The highest sensitivity index was
shown by total solar radiation ()
with the sensitivity index of 0.8 in all output parameters. The second and third
most sensitive variables were found to be slope (β) and solar azimuth angle
(γ), both with the sensitivity index of approximately 0.4. The sensitivity
index of ground reflectance (ρ_{g}) was approximately 0.25, the
ambient temperature (T_{a}) with 0.20 and the hour angle (ω) with
0.18. The lowest sensitive variable was found to be wind speed (V_{w})
with the index less than 0.1.
It was established from the sensitivity analysis of input variables and output
parameters that the most important input variable was the amount of total solar
radiation ()
because of its higher contribution in changing the amount of absorbed solar
radiation (S_{T}) level. Consequently, it results the maximum PV module
power output (P_{max}) and the less requirement of optimum PV array
area (A_{opt}). The changes contributed by amount of total solar radiation
()
in the output variables is approximately 2.5 times when its amount was varied
around its typical ranges, followed by slope (β) with 61%, ground reflectance
(ρ_{g}) 33%, ambient temperature (T_{a}) 23% and hour angle
(ω) 20%. The less important variables were found to be wind speed (V_{w})
4% and solar azimuth angle (γ) less than one percent as per oneatatime
(OAT) method. The highest sensitive input variable was found to be total solar
radiation ()
with the index of 0.8, followed by slope (β), solar azimuth angle (γ),
ground reflectance (ρ_{g}), the ambient temperature (T_{a})
and the hour angle (ω). The lowest sensitive variable was found to be wind
speed (V_{w}) with the index less than 0.1.
CONCLUSIONS
It was revealed from the sensitivity analysis of input variables and output
parameters of a standalone photovoltaic system that the most important input
variable was the amount of total solar radiation ()
because of its high contribution in changing the amount of absorbed solar radiation
(S_{T}) level, the maximum PV module power output (P_{max})
and optimum PV array area (A_{opt}).
The changes contributed by amount of total solar radiation ()
in the output variables was approximately 2.5 times when its amount was varied
around its typical ranges, followed by slope (β) with 61%, ground reflectance
(ρ_{g}) 33%, ambient temperature (T_{a}) 23% and hour angle
(ω) 20%. The less important variables were found to be wind speed (V_{w})
with 4% variation in the results and solar azimuth angle (γ) less than
one percent as per oneatatime (OAT) method.
The highest sensitive input variable was found to be total solar radiation
()
with the index of 0.8, followed by slope (β), solar azimuth angle (γ),
ground reflectance (ρ_{g}), the ambient temperature (T_{a})
and the hour angle (ω). The lowest sensitive variable was found to be wind
speed (V_{w}) with the index less than 0.1.
NOMENCLATURE
S_{T} 
= 
Monthly average total absorbed solar radiation on a tilted
surface (MJ m^{2}) 
H_{T} 
= 
Monthly average total solar radiation on a tilted surface (MJ m^{2}) 
H_{b} 
= 
Monthly average daily beam radiation on horizontal surface for a month
(MJ m^{2}) 
H_{d} 
= 
Monthly average daily diffuse radiation on horizontal surface for a month
(MJ m^{2}) 
R_{b} 
= 
Ratio of beam radiation on a tilted surface to the beam radiation on horizontal
surface 
ρ_{g} 
= 
Hemispherical ground reflectance or albedo 
φ 
= 
Latitude of location (°) 
δ 
= 
Declination angle (position of sun in the sky) (°) 
β 
= 
Slope (surface tilt angle from horizon) (°) 
γ 
= 
Surface azimuth angle (°) 
ω 
= 
Hour angle (°) 
θ 
= 
Angle of incidence (°) 
ω_{s} 
= 
Sunset hour angle (°) 
G_{sc} 
= 
Solar constant = 1367 W/m^{2} 
n 
= 
nth day of the year, starting from 1st January (n = 1 on 1st Jan. and
n = 365 on 31st Dec.) 
H_{0} 
= 
Extraterrestrial solar radiation on a horizontal surface for a day (MJ
m^{2}) 
K_{T} 
= 
Clearness index 
K_{θ} 
= 
Incident angle modifier 
M 
= 
Air mass modifier 
T_{a} 
= 
Ambient temperature (°C) 
V_{w} 
= 
Wind speed (m sec^{1}) 
P_{max} 
= 
Maximum power output (W) 
A_{opt} 
= 
Optimum PV array area (m^{2}) 

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