A MODEL DRIVEN OPTIMIZATION APPROACH TO DETERMINE TILT ANGLE OF SOLAR COLLECTOR IN INDIA

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International Journal of Advanced Research in Engineering and Technology (IJARET)
Volume 10, Issue 2, March-April 2019, pp. 431-448, Article ID: IJARET_10_02_042
Available online at http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=02
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
© IAEME Publication
A MODEL DRIVEN OPTIMIZATION
APPROACH TO DETERMINE TILT ANGLE OF
SOLAR COLLECTOR IN INDIA
Sanjaya Kumar Sahu, Nagendra Tripathi and Supriya Tripathi
Department of Electrical Engineering
Bhilai Institute of Technology, Drug, Chhattisgarh, India
ABSTRACT
The solar systems are an intense need to full fill the energy requirement of
developing countries like India. Where, thermal and photovoltaic are the two methods
to utilize the solar energy directly from sun. In these methods solar equipments (e.g. flat
plat collector and Photovoltaic panel) are kept in tilted position for absorbing maximum
solar flux. Hence, finding the optimum tilt angle is the problem of optimization.
Therefore, in this paper model driven optimization approach such as particle swarm
optimization (PSO) estimator has been proposed to find optimum tilt angle and its
results are compared with analytical results. A novel cost function has been applied to
determine periodical optimum tilt angle. To validate the performance of PSO estimator
results, statistical analysis study is carried out. Where, three statistical approaches such
as descriptive method, direct method and Altman-Bland methods are adopted. The PSO
estimator results are found satisfactory to ANA results at 95% confidence interval under
statistical study.
Keywords: Solar panel, PSO, Bland-Altman, Solar Power
Cite this Article: Sanjaya Kumar Sahu, Nagendra Tripathi and Supriya Tripathi, A
Model Driven Optimization Approach to Determine Tilt Angle of Solar Collector in
India, International Journal of Advanced Research in Engineering and Technology,
10(2), 2019, pp. 431-448.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=10&IType=2
1. INTRODUCTION
India is in deep energy crisis owing to rising prices of conventional energy sources as well as
their scarcity. On other hand, the uses of conventional sources of power generations are not
only pollute the environment but also fragile the ecological balance. Thus there is an urgent
need to use environment friendly and renewable energy sources such as solar, wind, geothermal
etc. Where, solar energy is most valuable, clean and safe source of energy gifted by the nature.
Owing to its location between the tropic of Cancer and the Equator, India has an average
annual temperature that ranges from 25O
C to −27.5O
C. On the other hand being a tropical
country, India has huge potential for solar power generation.

The thermal and photovoltaic conversions are the two methods of direct energy utilization
of solar energy. In these two methods most preferable solar equipments are Flat Plat Collector
(FPC) and Photovoltaic Array (PVA) respectively. The solar flux falls on the solar collector
depends on the tilt angle from horizontal surface. In other words the amount of radiation
absorbed by solar collector or panels is affected by their orientation and angle of tilt with the
horizontal surface. It has been seen that often solar data are given at horizontal surface and also
it is not possible to measure the performance of solar equipments at different tilt angle in large
countries like India.
Although, the solar radiation calculation is highly nonlinear, stochastic and site specific,
several complex models areavailable in literature. Further, these models estimate solar radiation
in terms ofclimate parameters such as sunshine hour, relative humidity, cloud layer, maximum
temperature etc. [1-3, 7]. Theproposed model from different researchers helps to derive the
optimum tilt angle at any location around the world from available metrological data on
horizontal surface 0 = o [4-5].
In [1] a number of equations related to mean monthly global solar radiant exposure( )gE
available in literature have been reviewed. And it has been observed that the original Angstrom
type regression equation which relates gE to clear days radiation at site and average possible
sunshine hour ratio are modified by different researchers. These modified model are classified
such as linear, polynomial, other (nonlinear, exponential and logarithm term) models. In [7]
author has proposed an empirical formula to estimate optimum tilt angle( )opt , hour angle ( )
and orientation( ) of solar panel based on generic surface with respect to position of sun. In [8-
11] authors have proposed different empirical formulas to derive optimum tilt angle of solar
panel around the world from available metrological data on horizontal surface.
Recently, the soft-computing techniques such as ANN, GA, ANFIS, and PSO etc. becoming
popular in solar energy estimation problems of optimization.The ANN to determine opt has
been widely used in solar energy estimation problems [12, 13, and 14]. In [12] the author has
taken , , andg LE E  as input to ANN model and estimated the optimum tilt angle. Where,
Mubiru and Banda [13] have developed ANN based model to predict direct solar irradiance.
This proposed ANN model takes sunshine hour, cloud cover, maximum temperature, latitude,
longitude as well as altitude of site as input parameters. In [14] author has proposed ANN model
to predict direct beam radiation on horizontal surface based on five input parameters such as
global solar irradiance, maximum temperature, latitude, longitude and altitude of site. In [15]
author has proposed an ANN estimator to predict periodic optimum tilt angle from available
input data such as ( ) , , andg monthly LE E . Here, ground albedo ( ) is not taken into account
because the variation in  is negligible and mostly considered as constant.
The above data driven approach is suitable only if accurate target data to train the ANN is
available. But for large country like India due to geographical and astrological variations,
accurate and reliable data collection is always not feasible. For reliable data collection it is
utmost importance to measure solar radiation data in different part of country. On other hand,
the network of radiation measuring stations should be modernized as well as monitored in same
data.
In paper [16] GA and Lagrange Multiplier algorithm has been proposed to maximize the
photovoltaic panel output. Where, the GA gives initial population at optimal power and then it
is feed to Lagrange Multiplier algorithm to find the PV module output power. The [17]
presented the opt calculation for fixed flat solar panel based on Genetic Algorithm (GA) and

A Model Driven Optimization Approach to Determine Tilt Angle of Solar Collector in India
Simulated-Annealing (SA) methods. The relation between sun incident angle and its radiation
intensity on solar cell has been presented. The climate data of different location of Taiwan have
been used by GA and SA to obtain the optimum tilt angle opt .
Till date no work has been reported to obtain opt using soft computing (model driven
approach) in India. The objective of this work is to propose a model driven approach to
determine optimum tilt angle ( )opt such that total monthly mean daily solar irradiance (Egs) can
be maximized on panel surface. Although the opt problem can be formulated on daily, weekly,
monthly, quarterly, bi-annual and annual basis. In this paper the quarterly, bi-annual and annual
model of expected irradiance in (MJ/m2
) are presented. The optimum tilt angle of south facing
collectors of entire India is calculated on the basis of mean (monthly) global solar irradiance
data obtained by Metrological center India[22]. We have proposed a PSO estimator to predict
the opt (periodically). A comparative study of the results obtained by Analytical Analysis
(ANA) as well as PSO estimator is carried out. Further, the results of PSO estimators are
analyzed based on statistical error tests [17]. In this paper, the results of selected cities of India
of different latitude and elevation such as Minicoy (MNC), Thiruvananthapuram (TRV), Port
Blair (PBL), Bangalore (BNG), Chennai (CHN) Goa (GOA) Hyderabad
(HYD),Visakhapatnam (VSK), Pune (PNE), Mumbai (MMB), Nagpur (NGP),
Bhavnagar (BHV), Kolkata (KLK), Ahmadabad (AHM), Bhopal (BHP), Ranchi
(RNC),Varanasi (VNS), Shillong (SHL), Patna (PTN), Jodhpur (JDP), Jaipur (JPR), New Delhi
(NDL) are presented.
2. MODELING OF TOTAL SOLAR IRRADIANCE CALCULATOR AT
TILTED SURFACE
In general, sensors are placed in a horizontal position to measure direct solar irradiance at
Metrological centers. But in solar energy applications to maximize capture of total solar
irradiance the panels are normally kept in tilted position. Hence, it becomes essential to derive
total solar radiant exposure from available values of monthly mean of daily/hourly global solar
irradiance and monthly mean of daily/hourly diffuse irradiance on the tilted panel surface.
Sometimes, global solar irradiation data may either not be available or not be desired form. In
such case a typical empirical regression (1) suggested by Gopinathan[3] can be considered
when data on sunshine hours are available.
1 1
max
g
go
E S
a b
E S
 
= +  
  -(1)
Where, the a1, b1=f { , EL, S, Smax} are regression constants can be obtained by (2-3) [7,
22].
1
max
0.309 0.539cos 0.0693 0.290L
S
a E
S

 
= − + − +  
 
-(2)
1
max
1.527 1.027cos 0.0926 0.359L
S
b E
S

 
= + + −  
 
-(3)
The extraterrestrial solar radiation on a horizontal panel for dnth
day is given as [7, 9 and
12]
24 2
1 0.033cos
365
go sc
dn
E G


   
= +   
  
-(4)

Moreover, to simplify the calculation of goE , Klein has suggested that the extraterrestrial
solar radiation in middle of the each month is nearly equal to the monthly mean value [3].The
hour angle corresponding to sunrise or sunset on the horizontal surface is given by (5)
( )1
cos tan tan  − =
 
-(5)
Therefore, the corresponding sunshine hour is
( )12
cos tan tan
15
S  − =
 
-(6)
The angle made by the line joining the centers of the sun and earth with the projection of
this line on the equatorial plane, is known as declination angle (δ). There are four models
(formulas) are proposed in literature to calculated declination angle, all these formulas are
assumption based. All four models are simulated and results are presented in Fig.1.Here could
be seen that all four models have no significant difference as far as declination angle calculation
is concern.
Figure 1 Declination Angle versus day plot
Model-1
( )2 284
23.45sin
365
dn

+ 
=  
 
(7-a)
Model-2
1 2 ( 173)
sin sin(23.45) cos
365.25
dn
 −  −  
=    
  
(7-b)
In equation (7-a) and (7-b) dn=1 at 1st
Jan of every year [3],[6].
Model-3
Rble’s model of inclination angle [5] is given in equation (7-c)
1 2
sin sin(23.27) cos
365.25
D
 −   
= −    
  
(7-c)
Where D- time after winter solstice (i. e D count starts from 22 DEC onwards)
Model-4
This is a data driven model for which data is obtained from Indian Metrological Center Pune
(Ref Table-7) [22]
The downward solar radiant energy scatted by the suspended particle, air molecules and the
clouds. Many researchers have developed empirical for estimating Ed/Eg ratio for various part
50 100 150 200 250 300 350
-25
-20
-15
-10
-5
0
5
10
15
20
25
Days From 1st Jan to 31st Dec
DeclinationAngle(inDegree)
eq(7-b)
eq(7-a)
rabal eq(7-c)
from data

of world [5, 7, 9, and 22]. In this case the prediction of mean monthly daily diffuse irradiance
(Ed) on horizontal surface from (8 and 9) is most suitable than other predictors because in fact
diffuse components are much larger in India [1-3].
max
0.8677 0.7365d
g
E S
E S
 
= −  
 
-(8)
1.411 1.696
gd
g go
EE
or
E E
 
= −  
 
 
-(9)
2.1. Solar Irradiance on Tilted Surface
The total solar irradiance Egs on tilted panel is the summation of direct-beam, diffuse and
ground reflected radiations on the inclined surface. The Egs can be expressed as (10)
gs b b d d g rE E E E=  +  +  (10)
Where,
( )b g dE E E= −
b for sharp south facing 0 = o
can be expressed as
( ) ( )sin sin cos cos cos
sin sin cos cos cos
s
b
s s
       
     
− + −
 =
+
-(11)
(1 )
2d
+ =
-(12)
r : If ground reflection to be considered as isotropic [9] then
(1 )
2
d  − = -(13)
 : It is the reflection coefficient, generally in India its value in between 0.12 to 0.2 [22].
3. OPTIMUM TILT ANGLE FORECASTING OF PV PANEL USING
PSO ESTIMATOR
The Particle Swarm Optimization (PSO) is a stochastic optimizationtechnique based on social
behaviour of bee swarming or fish schooling. This method has been proven effective fornon-
derivable objective function as well as its implementationis very easy. In PSO each particle
adjust its position in searchspace by its own as well as other particle flying experience tofind
the best global solution for each particle. The performanceof each swarm can be improved by
defining initial populationrange of each swarm such that each particle closely boundsto the
expected domain of feasible region [19], [23]. A realcoded PSO estimator is proposed to find
the optimum tiltangle of SOLAR COLLECTOR based on mean monthly global and
diffusesolar irradiance during uncertain climate changes at differentlocations in India. The
mathematical model of optimizationproblem can be described as follow:
 min max , gsor f X E -(17)
Where, X is some equality and Non equality constraints and Egs is the objective function to
be maximize.

3.1. Imposed Parametric Constrains
The imposed parametric constraints in this paper for sharp south facing PV array are as follows:
min max
1 365dn dn dn dn     -(18)
min max
0 90       o o
-(19)
0 ; azimuth angle = o
-(20)
0.2 = -(21)
In this work the performance of swarms are made better by putting parametric constraints
(lower and upper bound constraints) as shown in (18-21), but linear and nonlinear constraints
are not taken into account. In PSOeach particle moves about the cost surface with a velocity.
The velocities of particles are controlled by  inertia weight. By decreasing  global search
converges to local search. The particle updates their velocities and position based on the global
and local best solution by(22-24).Therefore, the new position of particle can be estimated by
following:
( ) ( )1
2 1 1 2n n
j jj j j j
n n n nLB GB
C r P P C r P P +
= + − + − -(22)
( )max min
max
max ter
ter
I
I
 
 
−
= −  -(23)
Therefore, thenewposition of particle can b eestimatedbyfollowing:
1 1j j j
n n np p + +
= + -(24)
The [20] demonstrated that the particle swarm is only stable if the following conditions
are satisfied:
( )
( )
1 2
1 2
0
Condition-1: 0< + 4
+
Condition-2: 1 1
2
C C
C C
C

 
−   + 
 
where, C0,C1 and C2 are inertia, social attraction and cognitive attraction of the particles
respectively. Although, the above conditions ensures the convergence of system but that may
not be global minimum. By selecting inertia weight tactfully optimal global solution can be
achieved in lesser number of iterations. Here,  decreases linearly from 0.9 to 0.4 during
simulation [19]. The boundary constraint of particle can be managed by either ‘Penalize’ or
‘Reflect’ or ‘Soft’ or ‘Absorb’ option [20]. In this paper ‘Soft’ boundary constraint has been
taken which sets fitness scores to infinity if particles leave problem bounds. The ‘soft’ boundary
option saves the simulation time since infeasible points are not evaluated.
3.2. PSO Input Parameters
The selected input parameters and their imposed parametric constraints areas follows:
X= [φ, EL,ρ, S, dn, Eg]
where, dn is a day of the year (dn=1onJan 1st
and dn=365onDec 31st
) and ρ=0.2 [22].

3.3. Problem Formulation
The objective of this work is to maximize the Egs over an appropriate period (seasonal, bi-
annual, yearly) by adjusting tilt angle  of solar collector of cities of India. Hence, optimum
tilt angle opt
can be obtained by maximizing the cost function
( )gs iH 
calculated from (10) for
each city. Therefore, the fitness function to be optimized is obtained by taking harmonic mean
of (10) for each of the latitude. The harmonics mean gives largest weight to smallest item and
smallest weight to largest item.
( )
   ( )
1
1
( ) ( ) ( )
1
, , , { , }
N
M
s
i
iF X
N
M
         
−
−
  
=

 
 + +
 
 =
 
 
  

-(25)
with, ( )C M =  C {1, 2, .......23} number of cities under study, N {12} number of months
and M {1, 2, 4} for annual, bi-annual and seasonal period respectively. Here, , and    
are the arbitrary functions of corresponding variables.
3.4. Pseudo-code of PSO Implementation
The pseudo code to implantation of Particle Swarm Optimization algorithm is as follows:
Begin
• Select the number of variable to be optimized for each city
• Select number of swarm particles (population size) and generations (maximum no of
iterations), as well as set 1 2 min max,C C and and  and terminating criteria.
• Impose the parameter constraints (lower bound, upper bound and other constraints if
any).
• Randomly initialize the swarm position ( j
np ) and velocity( )j
n of each particle.
Here
• Randomly generate population.
Next
• Impose Constraints: Is satisfied?
➢ No
o Remove the population and go to next population set.
o Go to: Next
➢ Yes
o Evaluate fitness function using (25).
o Determine j
LBnP and j
GBnP for each particle of each city.
• Is terminating Criteria reached?
➢ No
o Update the swarm position and velocity using (22-24).
o Update the inertia weight.
o Update j
LBnP and j
GBnP for each particle if its current value is better

o Go to :Here
➢ Yes
o Compute the best value of Egs(max) and corresponding independent variable.
End
4. RESULTS AND DISCUSSION
In this paper the seasonal, bi-annual and annual optimum tile angles are predicted for selected
cities of India using analytical method and PSO estimator. In particular, the solar radiation data
of India are considered by taking average on a monthly basis for period of 11 years (1991-2001)
from Indian Meteorology Department (IMD) Pune, India [22]. The exhaustive study of 23 cities
has been carried out but due to space limitation the results of six selected cities of different
latitude are presented in this paper.
4.1. Comparative study of Analytical and PSO Estimator
Based on analysis presented in section-II several MATLAB program were developed to
calculate the maximum total solar irradiance Egsmax for 0 = o
using (7-10). The optimum tilt
angle was calculated by searching for the value at which maximum Egsmax will be obtained. The
inclination of solar collector may be adjusted to optimum tilt angle hourly, daily and monthly
basis. By shortening the adjustment period the total solar irradiance on solar collector can be
increased but these adjustments spends lot of manpower or electrical power in automatic
tracking system.
Table 1 Optimum Tilt Angle of Solar Collector on Seasonal Basis using Analytical Method and Pso
Estimator
SNL opt WNANA ATANA SRANA WNPSO ATPSO SRPSO
MNC 36.00 11.00 6.00 35.51 10.65 5.70
TRV 35.00 12.00 5.00 34.63 11.86 4.89
PBL 38.00 15.00 9.00 38.33 15.06 8.83
BNG 40.00 17.00 10.00 39.82 17.08 10.23
CHN 36.00 15.00 8.00 35.82 15.03 7.87
GOA 43.00 19.00 13.00 43.41 18.65 13.11
HYD 46.00 21.00 15.00 45.74 20.65 14.59
VSK 44.00 20.00 14.00 43.57 19.61 14.48
PNE 46.00 21.00 17.00 46.35 21.40 16.93
MMB 44.00 20.00 15.00 44.07 20.29 15.06
NGP 48.00 23.00 19.00 47.72 22.73 18.76
BHV 49.00 24.00 21.00 49.45 24.02 21.29
KLK 46.00 22.00 16.00 46.11 21.73 16.49
AHM 50.00 25.00 21.00 50.07 24.96 21.34
BHP 51.00 25.00 22.00 50.55 25.13 22.29
RNC 49.00 25.00 19.00 48.97 24.73 19.08
VNS 49.00 26.00 21.00 48.93 25.74 21.19
SHL 52.00 26.00 19.00 51.85 26.23 18.91
PTN 49.00 26.00 21.00 49.46 26.16 21.43
JDP 53.00 28.00 24.00 53.40 27.78 23.95
JPR 53.00 27.00 24.00 52.60 27.22 24.26
NDL 53.00 28.00 24.00 52.71 27.93 23.79
Note: During summer in all case optimum tilt angle is zero degrees.

4.2. Optimum Tilt Angle on Seasonal Basis
In this section simulation result for seasonal adjustment of tilt angle of expected total solar
irradiance in (MJ/m2
) are presented. The Fig.2 shows the convergence behavior of PSO fitness
value for quarter period of the year on seasonal basis. The parameter setting for implementing
the proposed PSO algorithm is same as analytical approach. The mean of optimized fitness
function reaches to best score 0.0484 at 62 generation. The best individual scores i.e. optimum
tilt angle ( )opt among 62 trails are shown for selected cities of India. The simulation result of
ANA and PSO estimator are presented in Table-I, which represents opt for each season and
selected latitude. The total solar irradiance in (MJ/m2
) for these approaches to predict the
optimum tilt angle is approximately same as shown in Table-2. Form Table-1 and Table-2 it
can be observed that PSO approach has better ability to optimize the tilt angle of solar collector
to maximize the collection solar energy.
Table 2 Maximum Total Solar Irradiance Egsmax(Mj/M2
) Corresponding to Optimum Tilt Angle of
Solar Collector on Seasonal Basis using Analytical Method And Pso Estimator
EgsSNL
max WNANA ATANA SMANA SRANA WNPSO ATPSO SMP SO SRPSO
MNC 19.83641 21.38782 16.89667 18.03976 19.83696 21.38811 16.89665 18.03992
TRV 20.93009 22.62209 18.27667 19.27795 20.93041 22.62214 18.27665 19.27797
PBL 20.87194 21.56291 14.49000 15.48524 20.8722 21.56292 14.49000 15.48528
BNG 22.45604 24.39020 19.10333 17.99929 22.45613 24.39022 19.10332 17.99938
CHN 18.05185 23.34558 20.71000 18.28982 18.05191 23.34559 20.70991 18.28985
GOA 24.70772 24.33953 18.36333 18.56207 24.70823 24.33986 18.36331 18.56209
HYD 24.76370 24.95038 20.83333 19.08179 24.76391 24.95074 20.83333 19.08213
VSK 21.28330 22.55773 18.56333 17.35751 21.28378 22.55811 18.56303 17.35789
PNE 23.00623 24.00730 20.20000 18.39556 23.00657 24.00775 20.19983 18.39557
MMB 20.69352 22.88859 18.10000 16.62857 20.69353 22.88880 18.10000 16.62857
NGP 21.73389 23.03411 19.08333 17.59495 21.73410 23.03430 19.08333 17.59505
BHV 24.92165 25.46497 21.71000 19.95153 24.9223 25.46497 21.70998 19.95170
KLK 17.40086 19.53731 17.63333 15.64165 17.40089 19.53746 17.63333 15.64200
AHM 22.91214 24.20319 20.79000 18.51955 22.91216 24.20319 20.78981 18.51976
BHP 23.62239 23.90539 19.55000 18.10153 23.62300 23.90544 19.54999 18.10168
RNC 21.00614 21.73724 17.48000 15.37815 21.00615 21.73741 17.47995 15.37816
VNS 17.95321 21.99334 19.85667 17.72611 17.95322 21.99351 19.85662 17.72617
SHL 21.50543 20.55688 16.96333 15.20201 21.50549 20.55700 16.96333 15.20202
PTN 18.17297 21.91169 19.55000 17.40627 18.17343 21.91175 19.54998 17.40659
JDP 23.57504 23.44017 22.78333 21.17992 23.57552 23.44030 22.78332 21.17992
JPR 21.74466 23.80239 22.84333 19.67433 21.74510 23.80251 22.84332 19.67447
NDL 19.18953 22.26277 22.01333 18.91019 19.18971 22.26278 22.01333 18.91028
4.3. Optimum Tilt Angle on Bi-annual Basis
The aim of this section is to predict the optimum tilt angle of solar collector such that by
adjusting it twice a year maximum solar irradiance MJ/m2
can be achieved. The bi-annual tilt
angle estimation results of PSO are shown in Fig.3. Presently, the PSO estimator is set for 55
generations but in bi-annual adjustment PSO displayed a premature convergence i.e. mean
fitness value reaches to best score in less time. The convergence of bi-annual fitness function
reaches to best score with in 37 generations. The bi-annual average of daily solar radiant

exposure Egsmax (MJ/m2
) for corresponding opt for different cities is presented in Table-4.
Table-3 presents bi-annual optimum tilt angle of solar collector for south direction( )0 = o .
Figure 2 Seasonal Optimum Tilt angle of Selected Cities of India using PSO
4.4. Optimum Tilt Angle on Annual Basis
The convergence behaviour of the fitness value of PSO method for annual basis is shown in
Fig.4. The best score of PSO fitness function reaches to 0.04344 at 55 generations. The best
individual scores (in degree) among 55 trails are shown for selected cities of India. As shown
in Table-5 and Table-6 the proposed PSO approach has better ability to locate the optimum
solution then other methods.
Figure 3 Bi-annual Optimum Tilt angle of Selected Cities of India using PSO
Figure 4 Yearly Optimum Tilt angle of Selected Cities of India using PSO
10 20 30 40 50 60
0.048
0.05
0.052
0.054
0.056
Generation
Score
Best: 0.0484822 Mean: 0.048502
Mean Score
Best Score
5 10 15 20 25 30 35 40 45 50 55
0.054
0.056
0.058
0.06
0.062
0.064
0.066
0.068
0.07
Generation
Score
Best: 0.0546521 Mean: 0.0546604
Mean Score
Best Score
5 10 15 20 25 30 35 40 45 50 55
0.043
0.044
0.045
0.046
0.047
0.048
0.049
Generation
Score
Best: 0.0434376 Mean: 0.0434599
Mean Score
Best Score

Table 3 Optimum Tilt Angle of Solar Collector on Bi-Annual Basis Using Analytical Method and Pso
Estimator
BYR COLANA HOTANA COLP SO HOTP SO
MNC 23.00 0.00 22.99 0.00
TRV 23.00 0.00 22.99 2.60E-03
PBL 27.00 0.00 26.66 8.60E-02
BNG 28.00 0.00 28.04 6.60E-05
CHN 24.00 0.00 23.86 9.60E-04
GOA 32.00 1.00 31.68 7.40E-01
HYD 34.00 0.00 33.57 9.28E-04
VSK 32.00 2.00 31.65 1.53
PNE 34.00 3.00 34.00 2.56
MMB 32.00 2.00 31.87 2.21
NGP 35.00 5.00 35.37 4.85
BHV 37.00 6.00 37.16 6.47
KLK 34.00 4.00 33.76 4.45
AHM 38.00 7.00 37.71 6.67
BHP 38.00 8.00 38.32 7.67
RNC 37.00 5.00 37.04 5.11
VNS 36.00 8.00 36.48 7.64
SHL 40.00 7.00 40.17 7.03
PTN 37.00 8.00 37.08 7.90
JDP 41.00 10.00 41.36 9.82
JPR 40.00 9.00 39.95 9.39
NDL 40.00 10.00 39.98 10.03
4.5. Statistical Error Analysis
The statistical methods to assess agreement two quantitative methods such as PSO-ANA of
prediction of optimum β have been presented. Where, results obtained by PSO estimator have
been compared with analytical results. Here, there are three statistical methods have been
adopted such as descriptive analysis, direct comparison method and simple graphical (Bland-
Altman) method. In direct comparison method it could be answered that whether one method
of β estimation might replace the other with sufficient accuracy. The merits of this approach
are that the data will always cluster around a regression line and for comparing two methods it
is much more informative. In graphical method difference versus mean of two methods are
scattered and it is recommended that 95% of data points should lie within (±2SD) of the mean
difference. Where, Bland-Altman is a reliably simple graphical parametric approach based on
analysis of variance.
Table 4 Maximum Total Solar Irradiance Egsmax (Mj/M2
Solar Collector on Bi-Annualbasis using Analytical Method and Pso Estimator
Egsmax COLANA HOTANA COLPSO HOTPSO
MNC 20.24228 17.43833 20.25462 17.43825
TRV 21.45145 18.75500 21.46491 18.75496
PBL 20.87102 14.93167 20.88790 14.93166
BNG 23.04798 18.46333 23.06854 18.46333
CHN 20.44831 19.44667 20.45985 19.44666
GOA 24.05584 18.31425 24.08164 18.31419
HYD 24.35346 19.75000 24.38119 19.75000
VSK 21.54058 17.79022 21.56152 17.79020
PNE 23.03781 19.05433 23.06354 19.05415

Egsmax COLANA HOTANA COLPSO HOTPSO
MMB 21.42115 17.20370 21.44067 17.20349
NGP 21.94672 18.07562 21.97135 18.07501
BHV 24.66544 20.45251 24.69699 20.45233
KLK 18.15072 16.46724 18.16775 16.46682
AHM 23.08033 19.31221 23.11013 19.31158
BHP 23.26927 18.46966 23.29900 18.46923
RNC 20.97718 16.20159 21.00206 16.20083
VNS 19.64620 18.48987 19.66862 18.48869
SHL 20.59940 15.90208 20.62798 15.90148
PTN 19.70797 18.17986 19.73142 18.17850
JDP 23.00606 21.52961 23.04057 21.52764
JPR 22.30897 20.82034 22.33835 20.81866
NDL 20.33185 20.09996 20.35808 20.09780
Table 5 Optimum Tilt Angle of Solar Collector on Yearly Basis using Analytical Method and Pso
Estimator
opt ANA PSO opt ANA PSO
MNC 11.00 10.51 BHV 25.00 25.50
TRV 11.00 10.94 KLK 22.00 21.73
PBL 17.00 16.60 AHM 26.00 25.83
BNG 17.00 16.57 BHP 27.00 26.95
CHN 11.00 11.27 RNC 25.00 25.47
GOA 21.00 20.72 VNS 24.00 24.24
HYD 21.00 20.74 SHL 29.00 28.62
VSK 20.00 20.08 PTN 25.00 24.88
PNE 22.00 22.25 JDP 28.00 28.11
MMB 21.00 20.94 JPR 27.00 27.01
NGP 24.00 23.78 NDL 27.00 26.92
4.6. Descriptive Methods
The statistical test results of optimum tilt angle prediction of approaches under study (ANA and
PSO) are presented in Table-VII and Table-VIII for annual, bi-annual and seasonal period.
Where, it can be observed that there is no significance difference in mean, Standard Deviation
(SD) and median of all three methods (ANA and PSO) of opt prediction of solar collector but
this estimation of agreement approach is less effective for larger estimation errors. The
covariance and Pearson test result of correlation for two-sided t-test at 95% confidence interval
for annual, bi-annual and seasonal period are given in Table-IX and X. The null hypothesis (
0 : 0H  = ) is rejected in favor of alternative hypothesis ( 0 : 0H   ) at 5% significance level i.e.
there is a significance correlation in PSO-ANA estimator result. Moreover, the correlation
between ANA and PSO estimator is high [0.999(p < 0.001)]. In this section difference of mean,
standard deviation (SD), median and correlation of multivariate analysis are explored to
estimate the agreement of two methods. The correlation coefficient (ρ) only measure the
relation between methods but it doesn’t grantee the agreement between them. The change in
scale of measurement/estimation does not affect the correlation unlike agreement. On the other
hand some time data shows poor agreement but produces quite high correlation. Therefore, the
correlation coefficient is of no use in practice to compare two methods for their agreement [21].

Table 6 Maximum Total Solar Irradiance Egsmax(Mj/M2
Solar Collector on Bi-Annual Basis using Analytical Method and Pso Estimator
ANA PSO ANA PSO
MNC 18.4607 18.4612 BHV 22.0373 22.0379
TRV 19.7254 19.7254 KLK 16.9679 16.9680
PBL 17.5620 17.5623 AHM 20.6994 20.6995
BNG 20.2919 20.2922 BHP 20.3931 20.3931
CHN 19.5682 19.5683 RNC 18.1462 18.1466
GOA 20.7172 20.7174 VNS 18.6852 18.6853
HYD 21.4251 21.4253 SHL 17.8010 17.8013
VSK 19.2488 19.2488 PTN 18.5548 18.5548
PNE 20.5440 20.5441 JDP 21.7023 21.7024
MMB 18.9284 18.9284 JPR 21.0576 21.0576
NGP 19.5661 19.5662 NDL 19.7709 19.7709
Figure 5 Annual Optimum Tilt angle of ANA versus PSO Estimator Results
4.7. Direct Comparison Method
In direct comparison method it could be answered that whether one method of tilt angle
estimation might replace the other with sufficient accuracy. Fig.5, Fig.6 and Fig.7 show the
scatter plot between ANA-PSO result with line of equality for annual, bi-annual and seasonal
period respectively. The merit of this approach is that the data will always cluster around line
of regression which is more informative for comparing two methods.
Figure 6 Bi-annual Optimum Tilt angle of ANA versus PSO Estimator Results
0 5 10 15 20 25 30
0
5
10
15
20
25
30
PSO Estimator Results
ANAResults
Annual Optimum Tilt Angle of ANA vs PSO
Regression Line
ANA
0 5 10 15 20 25 30 35 40 45
0
10
20
30
40
ANAResults
Bi-Annual Optimum (Cooling Period) Tilt Angle of ANA vs PSO
0 2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
ANAResults
Bi-Annual Optimum (Heating Period) Tilt Angle of ANA vs PSO
Regression Line
ANA
Regression Line
ANA
opt opt

Figure 7 Seasonal Optimum Tilt angle of ANA versus PSO Estimator Results
Table 7 Comparison of Multivariate Analysis Result of Annual opt Based on Annual Results
Parameter
ANNUAL COL PERIOD HOT PERIOD
ANA PSO ANA PSO ANA PSO
Mean 22.0 21.9 33.9 33.9 4.8 4.8
SD 5.40 5.45 5.50 5.68 4.3 4.3
Median 24.0 23.8 35.0 35.4 5.0 4.8
Table 8 Comparison of Multivariate Analysis Result of Seasonal opt
Parameter WINTER AUTUMN SPRING
ANA PSO ANA PSO ANA PSO
Mean 46.0 46.1 22.0 21.9 17.1 17.2
SD 5.7 5.8 5.2 5.2 6.5 6.6
Median 48.0 47.7 23.0 22.7 19.0 18.8
Figure 8 Bland-Altman Plot of Annual Optimum Tilt Angle of ANA versus PSO Estimator Results
0 10 20 30 40 50
0
20
40
ANAResults
Winter Optimum Tilt Angle of ANAvs PSO
0 10 20 30 40
0
10
20
30
ANAResults
Autumn Optimum Tilt Angle of ANAvs PSO
0 10 20 30 40
0
10
20
30
ANAResults
Spring Optimum Tilt Angle of ANAvs PSO
0 0.5 1 1.5 2
0
1
2
PSO Estimator Result
ANAResults
Summer Optimum Tilt Angle of ANAvs PSO
Regression Line
ANA
Regression Line
ANA
Regression Line
ANA
Regression Line
ANA
0 5 10 15 20 25 30 35
-0.5
0
0.5
1
(ANA+PSO)/2
(ANA-PSO)
Bland-Altman Plot of Annual Optimum Tilt Angle for ANAand PSO Estimator Results
Ref Mean Diff 95% Lower LoA 95% Upper LoA Diff vs Avg

Figure 9 Bland-Altman Plot of Bi-Annual Optimum Tilt Angle of ANA versus PSO Estimator Results
5. CONCLUSION
In this study, evolutionary optimization approach such as PSO estimator that maximizes the
total solar irradiance Egsmax on the surface of PV panel has been proposed. The PSO estimator
is a model driven approach hence the selection of appropriate model input becomes extremely
important. Therefore, the cost function to be optimized using PSO and its constraints are also
presented in this work. The PSO estimator predicts the opt off-line and these results can be
used to enhance the energy collection of solar collectors. From the analysis it can be concluded
that the PSO estimator is a model driven approach to predict optimum tilt angler( )opt . This
scheme is more effective than data driven approaches such Neural Network, if the target data
for training are vague, missing not available. For estimation of seasonal optimum tilt angle, the
PSO estimator is more effective, accurate and easy to implement. From statistical study it also
can be concluded that the PSO estimator results do not differ by some considered critical
difference (at 95% difference of interval), the ANA could be replaced by PSO estimator.
Figure 10 Bland-Altman Plot of Bi-Annual Optimum Tilt Angle of ANA versus PSO Estimator
Results
0 5 10 15 20 25 30 35 40 45
-2
-1
0
1
Bland-Altman Plot fo Bi-Annual Optimum(Cooling period)Tilt Angle for ANA vs PSO Results
(ANA+PSO)/2
(ANA-PSO)
0 2 4 6 8 10 12 14 16 18 20
-0.5
0
0.5
1
Bland-Altman Plot fo Bi-Annual Optimum(Heating period)Tilt Angle for ANA vs PSO Results
(ANA+PSO)/2
(ANA-PSO)
Ref Mean Diff 95% Lower LoA 95% Upper LoA Diff vs Avg
Ref Mean Diff 95% Lower LoA 95% Upper LoA Diif vs Avg
0 20 40 60
-1
0
1 Winter
(ANA+PSO)/2
(ANA-PSO)
0 20 40 60
-0.5
0
0.5
1
Autumn
(ANA+PSO)/2
(ANA-PSO)
0 20 40 60
-10
-5
0
5
x 10
-3
Summer
(ANA+PSO)/2
(ANA-PSO)
0 20 40 60
-1
0
1
2
Spring
(ANA+PSO)/2
(ANA-PSO)
Ref Mean Diff 95% Lower LoA 95% upper LoA Data

Nomenclature
EL Elevation of the site in (meter)
Ego Extraterrestrial solar radiation on horizontal surface for dnth
day
Egsmax Maximum solar radiation collection by solar collector on tilted surface
(MJ/m2
/day)
Eg Monthly mean daily global solar irradiance (MJ/m2
/day)
Ed Monthly mean daily diffuse irradiance on horizontal surface (MJ/m2
/day)
Eb Monthly mean daily beam irradiance on horizontal surface (MJ/m2
/day)
S Monthly mean Sunshine hour at horizontal surface
Smax Monthly mean maximum possible Sun shine hour per day at horizontal surface
SCG Solar constant (4.921 MJ/m2
/hr)
MSE Mean square error
SOLAR COLLECTOR Photovoltaic Array
ANA Analytical Analysis/ method
WN Winter
AT Autumn
SM Summer
SR Spring
HOT Heating period of the year
COL Cooling period of the year
SNL Seasonal tilt adjustment
SD Standard Deviation
CI Confidence Interval
LOA Limit of Agreement
Greek Letters
 Slope or tilt angle of panel surface with respect to the horizontal surface (degree)
 Surface azimuth angle (degree)
 Earth declination angle (degree)
 Ground reflectance/ albado
 Latitude of the site in (degree)
 Hour angle (degree)
b Tilt factor for beam radiation
d Tilt factor for diffuse radiation
r
Radiation shape factor for surface w.r.t. surrounding ground
j
np Position of nth
particle at jth
iteration
j
n Velocity of nth
particle in jth
iteration
1 2,C C Acceleration or learning factors

1 2,r r Independent uniform random number, where 1,20 1r 
n
j
LB
P Best local solution /position of nth
until jth
iteration
n
j
GB
P Best global solution/position of nth
until jth
iteration
max
 Initial parameter weight
min
 Final parameter weight
max
terI Maximum iteration count
min
terI Minimum iteration count
terI Current iteration count
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A MODEL DRIVEN OPTIMIZATION APPROACH TO DETERMINE TILT ANGLE OF SOLAR COLLECTOR IN INDIA

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