1. Energy Optimisation ENO 732
1
An Optimal Design for Maximum Power Production from a Solar Field
installed with Stationary Solar Collectors
By
Ambrose Njepu
Department of Electrical, Electronics and Computer Engineering,
Date: 6th
June, 2016

2. Energy Optimisation ENO 732
2
An Optimal Design for Maximum Power Production from a Solar Field installed with
Stationary Solar Collectors
Ambrose Odinaka Njepu,
Abstract- The energy collected from a solar field is a function of the active collector cross-
sectional area, field dimensions, solar irradiation, inter-row spacing, row length and
mutual shading. This paper presents an optimal design for maximising the power output
of a constrained solar field installed with flat stationary solar collectors. The objective is
to maximise the number of solar collectors that can be installed in the solar field while
minimising mutual shading, inter-row spacing.
I. INTRODUCTION
Currently, the use of fossil fuel energy is the leading contributor of global warming. The threat
of global warming and the depleting fossil reserves in the face of the ever-rising energy
demand has led to explosion of research in sustainable and environmentally friendly renewable
energy sources. Solar energy has gained popularity in the last three decades thanks to its natural
abundance and environmental friendliness.
This paper presents a case study of a textile company that wants to use the energy generated
from a solar field to heat steam used for powering purpose. The dimension of the field is given
as length, L=500m and width, W=80m. Also, the dimension of the collector is given as length,
L=1.2m and width, W=0.6m. A model is required for the optimal distribution of collectors in
the solar field in order to maximise power output from the constrained field.
The authors of [1], developed an analytical shading model for rows of non-concentrating
collectors tilted towards the equator, showing than the effects of collector length is negligible
except at sunrise and sunset when the sun’s radiation is low. They also discussed beam
radiation and its shading effects. In [2], a model for total irradiation of shaded collector
assuming an infinite length of collector length in an isotropic model is developed. It discusses
the relationship between the diffused radiation and view factor of the sky as seen by the
collector.

3. Energy Optimisation ENO 732
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Similarly, a model for calculating the optimal tilt angle of a collector is developed in [3], it
shows that the tilt angle, 𝛽, optimises the incidence angle, 𝜃, when 𝑑2
𝑐𝑜𝑠𝜃 𝑑𝛽2
⁄ < 0.
The power produced by a PV is proportional to the solar irradiance (I), the cross-sectional area
of the collector (𝐴 𝑐) and the energy conversion efficiency (𝜂) of the PV collector. The solar
irradiation is made up of beam, diffused and reflected radiation. Its magnitude varies at
different locations because of the relative angle between the sun and the earth. Maximum power
yield from the collector is reached when the incident sun ray is at right angle to the collector
plane. The solar tracking system is designed to track the solar radiations such that the incident
radiation is at right angle to the collector. It has been reported that the effective cross sectional
area has a direct relationship with the quantity of power produced. The lesser the active cross-
sectional area, the lesser the generation capacity of the PV collector [1] [2] [4].
For a large scale system, multiple rows of collectors are employed and this comes with its
advantages and disadvantages. Its major advantage is that it increases the capacity of the field
but it also reduces the active area of the active collectors due to an increase in mutual collector
shading. Mutual shading, inter-row spacing and tilt angle are the control variables for the
objective function [2].There is therefore a need for an optimal solution for the orientation of
the collectors in a given area for optimal energy collection and economic considerations.
The contribution of this paper is an optimal model for maximising the energy collected from
the solar field while minimising the inter-row spacing and mutual shading subject to
constraints.
The rest of this paper is laid out as follows: section II shows the problem formulation, the case
study analysis which includes the field dimensioning, solar angles and geometry, shading
effects and insolation are discussed in section III; modelling and optimisation is presented in
section IV and V respectively; while section V is the conclusion of the paper.
II. PROBLEM FORMULATION
The following assumptions are made for the optimal design of the placement of collectors in a
solar field for maximum power output collected from the solar field:
 All panels are mounted parallel to each other on the ground.
 All panels are tilted at an angle, 𝛽, from the horizontal.

4. Energy Optimisation ENO 732
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 The inclination/tilt angle 𝛽, is the same for all collectors.
 The collectors are mounted to a fixed position, no movement or rotation is allowed.
 All panels must be installed facing the equator and the collector rows are inclined along
the east-west axis [2].
 A clear sky of isotropic model is assumed for modelling the global irradiation [1] [2]
[4].
 All collector panels have the same power capacity.
 The energy conversion efficiency for all collectors is assumed to be the same.
 A continuous row of collector is assumed, no spacing between row collectors.
 No fencing around the field, so the first row of collectors is not shaded.
 A walk-path of x m wide should go round the field for easy movement during
maintenance [4].
 The collectors are mounted along the field length (i.e. south facing orientation).
 An isotropic atmosphere and radiation is assumed
III. CASE STUDY
A. Solar Field Dimensioning
The textile has a field whose dimensions are given as length, L=500m and width, W=80m.
Also the dimension of a single panel is given as length, 𝐿 𝑐=0.6m and height, H=1.2m. The
problem is the optimal placement or positioning of the collectors along the field to yield
maximum power.
A walk-path of x m wide is introduced around the field for easy movement during maintenance.
This reduces the active field length and width on both sides. The active field length becomes
′𝐿1 = 𝐿 − 2𝑥′
and width becomes ′𝑊1 = 𝑊 − 2𝑥′
. The number of collectors per row is
expressed as
𝑛𝑖 =
𝐿 − 2𝑥
𝐿 𝑐
=
500 − 2𝑥
1.2
(1)
W1=80-2x

5. Energy Optimisation ENO 732
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Also, the number of rows (𝑅𝑖) in the field depends on the active field width (W-2x), collector
height (H), tilt angle (𝛽), the horizontal collector component (𝑎) and the inter-row collector
spacing (D). The number of rows is expressed as:
𝑅𝑖 =
𝑊 − 2𝑥
𝑎 + 𝐷
=
80 − 2𝑥
𝐻 sin 𝛽 + 𝐷
=
80 − 2𝑥
0.6 sin 𝛽 + 𝐷
(2)
Where the value of D is defined in equation (11) below
B. Solar Geometry and angles
The relative angles of the sun and the earth determine the variations between the energy
delivered to the earth. Global insolation is a function of the solar angles. Azimuth (𝛾𝑠) and
altitude (𝛼)angles are used to determine the exact location of the sun in space [4]. The azimuth
angles are measured from true south positively in a clockwise direction to the horizontal
component of the sun’s radiation [4]. Other important solar angles are the tilt/inclination angle
(𝛽), incidence angle (𝜃), collector azimuth angle(𝛾𝑐), hour angle (𝜔), latitude (∅) and the
declination angle (𝛿).
The equation (3) below shows the relationship between some solar angles [4]
sin ∝ = sin ∅ sin 𝛿 + cos ∅ cos 𝛿 cos 𝜔 (3)
∝= sin−1(sin ∅ sin 𝛿 + cos ∅ cos 𝛿 cos 𝜔) (4)

6. Energy Optimisation ENO 732
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Altitude angle α =49.6
Tilt angle β =49.6
Due South
PV Module
SUN
Fig. 2: showing the relationship between the tilt and altitude angles.
Fig. 3: showing the solar angles
The relationship between the solar azimuth, declination, hour, latitude and altitude angle is
shown in equations (5) and (6) below:
sin 𝛾𝑠 =
cos 𝛿 𝑠𝑖𝑛 𝜔
cos 𝛼
(5)
cos 𝛾𝑠 =
sin ∅ cos 𝛿 cos 𝜔 − cos ∅ sin 𝛿
cos 𝛼
(6)
Therefore,
𝛾𝑠 = sin−1
(
cos 𝛿 𝑠𝑖𝑛 𝜔
cos 𝛼
) = cos−1
(
sin ∅ cos 𝛿 cos 𝜔 − cos ∅ sin 𝛿
cos 𝛼
) (7)
Equation (8) shows the relationship between incidence, tilt, altitude and azimuth angles [4].
cos 𝜃 = cos 𝛽 sin 𝛼 + sin 𝛽 cos 𝛼 cos 𝛾 (8)
cos 𝜃 = sin(∅ − 𝛽) sin 𝛿 + cos( ∅ − 𝛽) cos 𝛿 cos 𝜔 (9)

7. Energy Optimisation ENO 732
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Fig. 4: showing the inter-row spacing and shading analysis.
The declination angle (𝛿) is the angle between the sun’s north or south-pole and the equator.
The value of declination angle is given as [5] [6] [7]
𝛿 = 23.45 sin [
360
365
(284 + 𝑛)] (10)
Where 𝑛 is the day of the year.
From figure 4, the inter-row spacing can be determined using the solar angles seen in [1]
𝐷 =
𝑊1 − 𝑛𝑖 𝐻 cos 𝛽
𝑅𝑖 − 1
(11)
𝐷 =
(80 − 2𝑧) − 0.6 𝑛𝑖cos 𝛽
𝑅𝑖 − 1
(12)
Where k is the number of rows, 𝑅𝑖=1,2….𝑅 𝑚 and 𝑅 𝑚 is the maximum number of rows.
The vertical component of the collector (h) is important for shading analysis. It is given as
ℎ = 𝐻𝑠𝑖𝑛𝛽 = 𝐻𝑐𝑜𝑠𝛽 𝑡𝑎𝑛𝛽 (13)
Incidence angle is the angle between the collector normal axis and the sun’s radiation; it shows
the relative position of a collector to the point of maximum solar irradiation (i.e. normal axis).
It is expressed as [3]
cos 𝜃 = sin 𝛿 sin ∅ cos 𝛽 − sin 𝛿 cos ∅ sin 𝛽 cos 𝛾 + cos 𝛾 cos ∅ cos 𝛽 cos 𝜔 + cos 𝛿 sin ∅ sin 𝛽 cos 𝛾 cos 𝜔 +
cos 𝛿 sin 𝛽 sin 𝛾 sin 𝜔 (14)
C. Shading analysis
Collectors cast shadows on the preceding collector which reduces the active area of the
collector and also the power production capacity of the collector. The shaded area is dependent
on tilt angle, inter-row distance, collector length, incidence and latitude angle [4].

8. Energy Optimisation ENO 732
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An inclined PV collector has lesser shading effect compared to the upright collector; this is
because some of the shadows are under the inclined collector.
The shadow height is defined interms of the collector height [4]
𝐻𝑠 = 𝐻 (1 −
𝐷 + 𝐻 cos 𝛽
𝑃𝑦
) (15)
𝑃𝑦 = 𝐻 cos 𝛽 +
𝐻 sin 𝛽 cos 𝛾𝑠
tan 𝛼
(16)
Substituting 𝑃𝑦 into the 𝐻𝑠 equation (15), the shadow height becomes:
𝐻𝑠 = 𝐻 (1 −
𝐷 + 𝐻 cos 𝛽
𝐻 cos 𝛽 +
𝐻 sin 𝛽 cos 𝛾𝑠
tan 𝛼
), (17)
where 𝐻𝑠 is the shadow height and 𝑃𝑦 is the vertical component of the shadow.
From fig. 5, the length of the shading, 𝐿 𝑠, is obtained by subtracting the length of the unshaded
portion from the entire length of the collector. This is expressed as [4]
𝐿 𝑠 = 𝐿𝑖 − 𝐿 𝑢𝑛𝑠 (18)
𝐿 𝑠 = 𝐿𝑖 −
(𝐷 + 𝐻 cos 𝛽)𝑃𝑥
𝑃𝑦
(19)
𝑃𝑥 is the horizontal component of the shadow, defined as
Fig. 5: showing the mutual shading effect of collectors in the field

9. Energy Optimisation ENO 732
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𝑃𝑥 =
𝐻 sin 𝛽 sin 𝛾𝑠
tan 𝛼
(20)
The length of the shadow is formulated by substituting the values of 𝑃𝑥 𝑎𝑛𝑑 𝑃𝑦 from equations
(16) and (20) into equation (19) and it becomes:
𝐿 𝑠 = 𝐿 − (𝐷 + 𝐻 cos 𝛽)
sin 𝛽 |sin 𝛾𝑠| tan 𝛼⁄
cos 𝛽 + sin 𝛽 cos 𝛾𝑠 tan 𝛼⁄
(21)
For 𝛾𝑠 > 0, the shadow is eastward, and for 𝛾𝑠 < 0 the shadow is westward [4].
For a more general analysis, the relative quantities are introduced. The area of the shaded
portion is defined by the product of equations (17) and (21). It is expressed as
𝑠 = 𝐿 𝑠 × 𝐻𝑠 (22)
From figure 5, the collector height is given as
𝐻𝑐 = 𝐻 sin 𝛽 (23)
The relative spacing between the rows of collectors is defined as [4]
𝑑 =
𝐷
𝐻𝑐
(24)
The relative collector length
is
𝑙 =
𝐿𝑖
𝐻𝑐
(25)
the relative shadow height is defined as
ℎ 𝑠 =
𝐻𝑠
𝐻
(26)
The relative length of the shadow is
𝑙 𝑠 =
𝐿 𝑠
𝐿𝑖
(27)
The relative shadow height (ℎ 𝑠) is obtained by substituting equations (15) and (19) into
equation (26) as seen below
ℎ 𝑠 = 1 −
𝑑 sin 𝛽 + cos 𝛽
cos 𝛽 + sin 𝛽 cos 𝛾𝑠 tan 𝛼⁄
(28)
And

10. Energy Optimisation ENO 732
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𝑙 𝑠 = 1 − (
|sin 𝛾𝑠|
𝑙 tan ∝
) (
𝑑 sin 𝛽 + cos 𝛽
cos 𝛽 + sin 𝛽 cos 𝛾𝑠 tan 𝛼⁄
) (29)
Therefore, the relative shadow area is
𝑠 =
𝐻𝑠 𝐿 𝑠
𝐻𝐿𝑖
= ℎ 𝑠 𝑙 𝑠 (30)
D. Solar Irradiation
The power produced by a collector is proportional to the intensity of solar irradiation it receives.
The solar irradiation is composed of the beam, diffused and the reflected irradiation. Beam
irradiation is the direct solar radiations coming from the sun while diffused irradiation is the
scattered beam which reaches the earth after it has been scattered by the atmosphere.
1) Beam insolation
The beam irradiation on a tilted collector is expressed as [1] [4]
𝐼 𝐵 = 𝐼 𝑏 cos 𝜃 (31)
The first row is unshaded, so it produces it produces the largest power in the field at any point.
However, it shades other subsequent collectors behind it, thereby reducing the active collector
areas. So the total beam irradiation on the shaded collector seen in [4] as
𝐼 𝑏𝑗
𝑠ℎ
= 𝐻𝐿𝑖(1 − 𝑠)𝐼 𝑏𝑗 cos 𝜃𝑗 (32)
Where 𝐼 𝑏𝑗
𝑠ℎ
is the daily variation of the beam insolation on the shaded collector, H is the
collector height, L is the collector length and 𝑠 is the shaded area of the collector at time j.
The total beam irradiation on a shaded collector is explicitly by substituting equation (9) into
(33)
𝐼 𝑏𝑗
𝑠ℎ
= 𝐻𝐿𝑖(1 − 𝑠)𝐼 𝑏𝑗 sin(∅ − 𝛽) sin 𝛿 + cos( ∅ − 𝛽) cos 𝛿 cos 𝜔 (33)
2) Diffused insolation
The diffused irradiation on a tilted collector is given by [4]
𝐼 𝐷 = 𝐼 𝑑 cos2
𝛽
2
, (34)

11. Energy Optimisation ENO 732
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where 𝐼 𝑑 is the diffused solar insolation on a horizontal surface and the minimum value for
cos2 𝛽
2
applies to an isotropic sky. The daily diffused insolation received by the shaded collector
is [4]
𝐼 𝑑
𝑠ℎ
= 𝐼 𝑑(cos2
𝛽
2
+ cos2
𝜂 𝑎𝑣
2
− 1), (35)
where 𝜂 𝑎𝑣 is the screening angle.
3) Global Insolation
The effect of the reflected insolation on a tilted surface is negligible, so the global insolation
of an inclined collector is the sum of the beam and diffused insolation on the PV collector. This
is expressed as [1] [4]
𝐼 = 𝐼 𝑏 cos 𝜃 + 𝐼 𝑑 cos2
𝛽
2
(36)
The global insolation on a shaded collector is defined by the sum of equations (32) and (35)
[1]:
𝐼 𝑠ℎ
= 𝐻𝐿𝑖(1 − 𝑠)𝐼 𝑏𝑗 cos 𝜃𝑗 + 𝐼 𝑑(cos2
𝛽
2
+ cos2
𝜂 𝑎𝑣
2
− 1) (37)
The power output from a PV collector is dependent on the incident solar insolation upon it.
The total daily solar energy received by the collector, per unit area, in a month is defined as:
𝑄 = ∑ 𝐼𝑗∆
𝑇𝑆𝑆
𝑇𝑆𝑅
𝜏 = ∑(𝐼 𝑏𝑗 cos 𝜃𝑗
𝑇𝑆𝑆
𝑇𝑆𝑅
+ 𝐼 𝑑𝑗 cos2
𝛽
2
)∆𝜏 , (38)
𝑄 = ∑(𝐼 𝑏𝑗 cos 𝜃𝑗
𝑇𝑆𝑆
𝑇𝑆𝑅
∆𝜏 + 𝑄 𝑑 cos2
𝛽
2
) , (39)
where 𝑄 𝑑 is the energy received from the diffused insolation, 𝐼 𝑏𝑗 is the hourly variations of the
beam insolation and ∆𝜏 is the sampling period, which is from sunrise to sunset.
The total energy from the beam insolation on the solar field is the sum of the beam insolation
on the unshaded and the shaded collectors [1] is given as
𝑄 𝑏,𝑇 = 𝐻𝐿𝑖 [∑ 𝐼 𝑏𝑗 cos 𝑠𝜃𝑗
𝑁=1
𝐽
∆𝜏 + (𝑅𝑖 − 1) ∑(1 − 𝑠)𝐼 𝑏𝑗 cos 𝜃𝑗
𝑅−1
𝑗
∆𝜏] , (40)
where 𝑅𝑖=2,3,…. 𝑅 𝑚.

12. Energy Optimisation ENO 732
12
Therefore, the total solar energy delivered to the PV solar collector is the sum of the beam and
diffused radiations on the unshaded and shaded rows of collectors as seen [1] as
𝑄 𝑇 = 𝐻𝐿 [∑ 𝐼𝑏𝑗 cos 𝜃𝑗∆𝜏 +𝑗 𝐼 𝑑𝑗 cos2 𝛽
2
+ (𝑅 − 1) ∑ (1 − 𝑠)𝐼𝑏𝑗 cos 𝜃𝑗∆𝜏𝑗 + (𝑅 − 1) 𝑄 𝑑 (cos2 𝛽
2
+ cos2 𝜂 𝑎𝑣
2
− 1)]
(41)
where 𝑅𝑖=2,3,… 𝑅 𝑚
IV. MODELLING
The total global irradiation on a tilted surface is the sum of the direct beam, diffuse and the
reflected radiation expressed as follows [8] [9]:
𝐼 𝑇 = 𝐼 𝐵 + 𝐼 𝐷 + 𝐼 𝑅, (42)
where 𝐼 𝑇 is the total global irradiation, 𝐼 𝐵, 𝐼 𝐷 𝑎𝑛𝑑 𝐼 𝑅 are the beam, diffused and reflected
component of the solar irradiation on the tilted surface. Equation (42) can be expressed further
as [5]:
𝐼 𝑇 = 𝐼 𝑏 𝑅 𝑏 + 𝐼 𝑑 𝑅 𝑑 + 𝐼𝑟 𝑅 𝑟, (43)
where 𝐼 𝑏, 𝐼 𝑑 𝑎𝑛𝑑 𝐼𝑟 are the beam, diffused and the reflected component of solar insolation on a
horizontal surface. The daily beam irradiation on an inclined surface can be redefined as
𝐼 𝑏 = (𝐼 − 𝐼 𝑑) 𝑅 𝑏 (44)
Where 𝐼 and 𝐼 𝑑 are the global and diffused radiation on a horizontal surface, while 𝑅 𝑏 is the
ratio between the beam radiation on a tilted surface to the horizontal surface. The daily reflected
ray can be redefined as
𝐼 𝑅 =
𝐼𝜌(1 − cos 𝛽)
2
(45)
The values of 𝑅 𝑏 depends on the latitude of the collector. In the northern hemisphere, optimal
collector orientation is achieved by facing the equator (i.e. south facing), 𝑅 𝑏 is given as [8]
𝑅 𝑏,1 =
cos(∅ − 𝛽) cos 𝛿 sin 𝜔𝑠 + 𝜔𝑠 sin(∅ − 𝛽) sin 𝛿
cos ∅ cos 𝛿 sin 𝜔𝑠 + 𝜔𝑠sin ∅ sin 𝛿
, (46)
while, the southern hemisphere, the collector orientation is facing the equation (i.e. north
facing), 𝑅 𝑏 is given as [8]
𝑅 𝑏,2 =
cos(∅ + 𝛽) cos 𝛿 sin 𝜔𝑠 + 𝜔𝑠 sin(∅ + 𝛽) sin 𝛿
cos ∅ cos 𝛿 sin 𝜔𝑠 + 𝜔𝑠sin ∅ sin 𝛿
(47)
Where 𝜔𝑠 is the sunset hour angle for the tilted surface for the mean day of the month.

13. Energy Optimisation ENO 732
13
For an isotropic model, 𝑅 𝑑 = 𝑅 𝑟 expressed as:
𝑅 𝑑 = 𝑅 𝑟 =
1 + cos 𝛽
2
(48)
The total irradiation on a tilted surface is re-expressed by substituting the equations (44) and
(45) into equation (42), given as:
𝐼 𝑇 = (𝐼 − 𝐼 𝑑) 𝑅 𝑏 + 𝐼 𝑑 𝑅 𝑑 +
𝐼𝜌(1 − cos 𝛽)
2
(49)
The effect of reflected radiation is significant on a horizontal surface but negligible on a tilted
surface, so the total insolation on an inclined surface is the sum of the direct beam insolation
and the diffused insolation.
𝐼 𝑇 = (𝐼 − 𝐼 𝑑) 𝑅 𝑏 + 𝐼 𝑑 𝑅 𝑑 (50)
V. OPTIMISATION MODEL
Hourly power produced by the PV is a function of the collector’s efficiency (𝜂), area of the
collector (𝐴 𝑐) and the intensity of the solar irradiation (𝐼 𝑝𝑣). The hourly powered output of a
PV is given as:
𝑃(𝑡) = 𝜂𝐴 𝑐 𝐼 𝑝𝑣 (51)
The power output from an ideal solar field where there is no shading is given as:
𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴 𝑐 𝐼 𝑝𝑣
𝑅 𝑖
𝑖=1
, (52)
where 𝑅 and 𝑅𝑖 are respectively, the number of solar collectors in a row and the total number
of rows in the solar field. Substituting 𝐼 𝑝𝑣 = 𝐼 𝐵 + 𝐼 𝐷 + 𝐼 𝑅 in equation (52) gives equation (53).
𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴 𝑐 (𝐼 𝐵 + 𝐼 𝐷 + 𝐼 𝑅)
𝑅
𝑖=1
(53)
The value of 𝐼 𝑅 on a tilted surface is negligible, thus equation (53) reduces to equation (54):
𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴 𝑐 (𝐼 𝐵 + 𝐼 𝐷)
𝑅
𝑖=1
(54)
Equation (54) is the expression for PV power output from a solar field assuming zero shading
effect. A scenario when there is no shading means that the spacing between the collector rows
is large, the panels are few and the energy produced from the field will be less compared to
when the collectors are more. For the optimal power production from the PV field, the number

14. Energy Optimisation ENO 732
14
of collectors will increase and the inter-row spacing will be reduced. So the design for the
optimal performance of the solar field taking the inter-row spacing and the number of panels
into considerations.
For maximum power from a solar field constrained by the fixed field length 𝐿 and width 𝑊,
𝑅𝑖 number of collector rows, distance, D, between rows, each collector length, 𝑙, collector
height 𝐻 and an tilted angle of 𝛽 to the horizontal. The problem variables are 𝑅𝑖, 𝛽, 𝐷 and 𝐻.
The objective function of this problem is the yearly solar energy produced by the solar field
collector, which is given as follows [10]:
Maximise
𝑄 = 𝜂𝐻𝐿𝑖 [𝑞 𝑏 + 𝑞 𝑑 + (𝑅𝑖 − 1)(𝑞 𝑏
𝑠ℎ
+ 𝑞 𝑑
𝑠ℎ
)] (55)
Where 𝑞 𝑏 is the yearly solar beam irradiation per unit area of an unshaded collector (which is
the first row), 𝑞 𝑑 is the yearly diffused irradiation per unit area of an unshaded collector, 𝑞 𝑏
𝑠ℎ
is
the average yearly beam irradiation per unit area of the shaded collectors and 𝑞 𝑑
𝑠ℎ
is the average
diffused solar irradiation per unit area of a shaded collector. 𝑞 𝑏, 𝑞 𝑑, 𝑞 𝑏
𝑠ℎ
𝑎𝑛𝑑 𝑞 𝑑
𝑠ℎ
are defined
mathematically as:
𝑞 𝑏 = ∑ ∑ 𝐼 𝑏 cos 𝜃 Δ𝑇
𝑇𝑠𝑠
𝑇𝑠𝑟
365
𝑛=1
(56)
𝑞 𝑑 = 𝐹𝑑 ∑ ∑ 𝐼 𝑑Δ𝑇
𝑇𝑠𝑠
𝑇𝑠𝑟
365
𝑛=1
(57)
𝑞 𝑏
𝑠ℎ
= ∑ ∑ 𝐼 𝑏 cos 𝜃(1 − 𝑠) Δ𝑇
𝑇𝑠
𝑇𝑟
365
𝑛=1
(58)
𝑞 𝑏
𝑠ℎ
= 𝐹𝑑
𝑠ℎ
∑ ∑ 𝐼 𝑑ℎΔ𝑇
𝑇𝑠𝑠
𝑇𝑠𝑟
365
𝑛=1
(59)
Where 𝐹𝑑 𝑎𝑛𝑑 𝐹𝑑
𝑠ℎ
are the configuration factors for unshaded and shaded collectors
respectively
𝐹𝑑 = cos2
(
𝛽
2
) (60)
𝐹𝑑
𝑠ℎ
= cos2(𝛽/2) −
1
2
[(𝑑2
+ 1)
1
2 − 𝑑] 𝑠𝑖𝑛𝛽 (61)

15. Energy Optimisation ENO 732
15
Constraints
The objective function in constrained by these control variables are 𝑅𝑖, 𝐷, 𝐻, 𝛽 [10].
The variation of the field and collector parameter is constrained by the width of the field W.
The equation below shows that the sum of the first row and the remaining rows must not greater
than the width of the field.
𝑅𝑖 × 𝐻 × cos 𝛽 + (𝑅𝑖 − 1)𝐷 ≤ (𝑊 − 2𝑥) (62)
The spacing between the rows is a control variable, which the minimum allowable space for
maintenance purposes. The spacing is constrained by the minimum allowable space for
maintenance and for shading reduction.
𝐷 ≥ 𝐷 𝑚𝑖𝑛 (63)
This spacing should be at least the minimum space needed for maintenance purpose.
The collector height is maximum when it is standing at right angle to the horizontal(𝐻 =
𝐴 𝑚𝑎𝑥). However, for a tilted surface the height reduces as the angle decreases, so it’s height
must always be less than 𝐴 𝑚𝑎𝑥. It is given as
𝐻 × sin 𝛽 ≤ 𝐴 𝑚𝑎𝑥 (64)
The collector height is limited by the manufacturer’s design, given as
𝐻 ≤ 𝐻 𝑚𝑎𝑥 (65)
At any time, the tilted collector must be inclined at an angle between 0 and90°
. This expressed
mathematically as
0°
≤ 𝛽 ≤ 90°
(66)
The more the rows, the more the shading effect and the greater the power produced by the solar
field. The number of the rows in the solar field must be greater than 2
𝑅𝑖 ≥ 2 (67)
The PV is constrained by its capacity
0 ≤ 𝑃𝑝𝑣 ≤ 𝑃𝑝𝑣
𝑚𝑎𝑥
(68)
All the variables have non-negative values.
The optimisation problem is maximizing the annual solar energy yield from a solar field,
subject to constraints such as the inter-row spacing (D), tilt angle (𝛽), collector height (H),
solar irradiation (I), and the number of rows(𝑅𝑖).

16. Energy Optimisation ENO 732
16
The optimisation model becomes
Maximize:
𝑄 = 𝜂𝐻𝐿𝑖 [𝑞 𝑏 + 𝑞 𝑑 + (𝑅𝑖 − 1)(𝑞 𝑏
𝑠ℎ
+ 𝑞 𝑑
𝑠ℎ
)] (68)
Where 𝑞 𝑏, 𝑞 𝑑, 𝑞 𝑏
𝑠ℎ
𝑎𝑛𝑑 𝑞 𝑑
𝑠ℎ
are defined in equations [56-59]
Subject to
𝑅𝑖 × 1.2 × cos 𝛽 + (𝑛1 − 1)𝐷 ≤ (80 − 2𝑥) (69)
𝐷 ≥ 𝐷 𝑚𝑖𝑛 (70)
1.2 × sin 𝛽 ≤ 1.2 (71)
𝐻 ≤ 𝐻 𝑚𝑎𝑥 (72)
0°
≤ 𝛽 ≤ 90°
(73)
𝑅𝑖 ≥ 2 (74)
0 ≤ 𝑃𝑝𝑣 ≤ 𝑃𝑝𝑣
𝑚𝑎𝑥
(75)
VI. CONCLUSION
In this paper, it is seen that the optimal orientation for a solar panel should be facing the equator
and the angle should be its latitude angle. A sound knowledge of solar geometry proved to be
very important in determining the optimal position for receiving maximum solar irradiation
and in shadow control. This paper shows that in a fixed solar field, increasing the row spacing
will reduce the number of collector and reduce the power capacity of the field, And reducing
the row spacing will increase the number of collectors and also increase the power capacity of
the solar field even though the shading effect will rise. These quantities are inversely and this
paper presents the optimisation model that is the meeting point of both factors in order to
maximise the annual solar energy generated from the solar field.

17. Energy Optimisation ENO 732
17
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