An Optimal Design for Maximum Power Production from a Solar Field installed with Stationary Solar...

Energy Optimisation ENO 732

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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


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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.


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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 𝑑2𝑐𝑜𝑠𝜃 𝑑𝛽2⁄ < 0.

The power produced by a PV is proportional to the solar irradiance (I), the cross-sectional area

of the collector (𝐴𝑐) 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.


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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, 𝐿𝑐=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

𝐿𝑐=

500 − 2𝑥

1.2 (1)

L1=500-2x

W1=80-2x


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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)


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Altitude angle α =49.6

Tilt angle β =49.6Due 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)


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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 − 𝑛𝑖𝐻 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….𝑅𝑚 and 𝑅𝑚 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].


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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, 𝐿𝑠 , is obtained by subtracting the length of the unshaded

portion from the entire length of the collector. This is expressed as [4]

𝐿𝑠 = 𝐿𝑖 − 𝐿𝑢𝑛𝑠 (18)

𝐿𝑠 = 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


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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:

𝐿𝑠 = 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 = 𝐿𝑠 × 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

ℎ𝑠 =𝐻𝑠

𝐻 (26)

The relative length of the shadow is

𝑙𝑠 =𝐿𝑠

𝐿𝑖 (27)

The relative shadow height (ℎ𝑠) is obtained by substituting equations (15) and (19) into

equation (26) as seen below

ℎ𝑠 = 1 −𝑑 sin 𝛽 + cos 𝛽

cos 𝛽 + sin 𝛽 cos 𝛾𝑠 tan 𝛼⁄ (28)

And


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𝑙𝑠 = 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 =𝐻𝑠𝐿𝑠

𝐻𝐿𝑖= ℎ𝑠𝑙𝑠 (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]

𝐼𝐵 = 𝐼𝑏 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

𝐼𝑏𝑗𝑠ℎ = 𝐻𝐿𝑖(1 − 𝑠)𝐼𝑏𝑗 cos 𝜃𝑗 (32)

Where 𝐼𝑏𝑗𝑠ℎ 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)

𝐼𝑏𝑗𝑠ℎ = 𝐻𝐿𝑖(1 − 𝑠)𝐼𝑏𝑗 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]

𝐼𝐷 = 𝐼𝑑 cos2𝛽

2 , (34)


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where 𝐼𝑑 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]

𝐼𝑑𝑠ℎ = 𝐼𝑑(cos2

𝛽

2+ cos2

𝜂𝑎𝑣

2− 1), (35)

where 𝜂𝑎𝑣 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 = 𝐼𝑏 cos 𝜃 + 𝐼𝑑 cos2𝛽

2 (36)

The global insolation on a shaded collector is defined by the sum of equations (32) and (35)

[1]:

𝐼𝑠ℎ = 𝐻𝐿𝑖(1 − 𝑠)𝐼𝑏𝑗 cos 𝜃𝑗 + 𝐼𝑑(cos2𝛽

2+ cos2

𝜂𝑎𝑣

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 = ∑(𝐼𝑏𝑗 cos 𝜃𝑗

𝑇𝑆𝑆

𝑇𝑆𝑅

+ 𝐼𝑑𝑗 cos2𝛽

2)∆𝜏 , (38)

𝑄 = ∑(𝐼𝑏𝑗 cos 𝜃𝑗

𝑇𝑆𝑆

𝑇𝑆𝑅

∆𝜏 + 𝑄𝑑 cos2𝛽

2) , (39)

where 𝑄𝑑 is the energy received from the diffused insolation, 𝐼𝑏𝑗 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

𝑄𝑏,𝑇 = 𝐻𝐿𝑖 [∑ 𝐼𝑏𝑗 cos 𝑠𝜃𝑗

𝑁=1

𝐽

∆𝜏 + (𝑅𝑖 − 1) ∑(1 − 𝑠)𝐼𝑏𝑗 cos 𝜃𝑗

𝑅−1

𝑗

∆𝜏] , (40)

where 𝑅𝑖=2,3,…. 𝑅𝑚.


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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

𝑄𝑇 = 𝐻𝐿 [∑ 𝐼𝑏𝑗 cos 𝜃𝑗∆𝜏 +𝑗 𝐼𝑑𝑗 cos2 𝛽

2+ (𝑅 − 1) ∑ (1 − 𝑠)𝐼𝑏𝑗 cos 𝜃𝑗∆𝜏𝑗 + (𝑅 − 1) 𝑄𝑑 (cos2 𝛽

2+ cos2 𝜂𝑎𝑣

2− 1)]

(41)

where 𝑅𝑖=2,3,… 𝑅𝑚

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]:

𝐼𝑇 = 𝐼𝐵 + 𝐼𝐷 + 𝐼𝑅 , (42)

where 𝐼𝑇 is the total global irradiation, 𝐼𝐵, 𝐼𝐷 𝑎𝑛𝑑 𝐼𝑅 are the beam, diffused and reflected

component of the solar irradiation on the tilted surface. Equation (42) can be expressed further

as [5]:

𝐼𝑇 = 𝐼𝑏 𝑅𝑏 + 𝐼𝑑 𝑅𝑑 + 𝐼𝑟 𝑅𝑟 , (43)

where 𝐼𝑏 , 𝐼𝑑 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

𝐼𝑏 = (𝐼 − 𝐼𝑑) 𝑅𝑏 (44)

Where 𝐼 and 𝐼𝑑 are the global and diffused radiation on a horizontal surface, while 𝑅𝑏 is the

ratio between the beam radiation on a tilted surface to the horizontal surface. The daily reflected

ray can be redefined as

𝐼𝑅 =𝐼𝜌(1 − cos 𝛽)

2 (45)

The values of 𝑅𝑏 depends on the latitude of the collector. In the northern hemisphere, optimal

collector orientation is achieved by facing the equator (i.e. south facing), 𝑅𝑏 is given as [8]

𝑅𝑏,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), 𝑅𝑏 is given as [8]

𝑅𝑏,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.


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For an isotropic model, 𝑅𝑑 = 𝑅𝑟 expressed as:

𝑅𝑑 = 𝑅𝑟 =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:

𝐼𝑇 = (𝐼 − 𝐼𝑑) 𝑅𝑏 + 𝐼𝑑𝑅𝑑 +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.

𝐼𝑇 = (𝐼 − 𝐼𝑑) 𝑅𝑏 + 𝐼𝑑𝑅𝑑 (50)

V. OPTIMISATION MODEL

Hourly power produced by the PV is a function of the collector’s efficiency (𝜂), area of the

collector (𝐴𝑐) and the intensity of the solar irradiation (𝐼𝑝𝑣). The hourly powered output of a

PV is given as:

𝑃(𝑡) = 𝜂𝐴𝑐𝐼𝑝𝑣 (51)

The power output from an ideal solar field where there is no shading is given as:

𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴𝑐𝐼𝑝𝑣

𝑅𝑖

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 𝐼𝑝𝑣 = 𝐼𝐵 + 𝐼𝐷 + 𝐼𝑅 in equation (52) gives equation (53).

𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴𝑐 (𝐼𝐵 + 𝐼𝐷 + 𝐼𝑅)

𝑅

𝑖=1

(53)

The value of 𝐼𝑅 on a tilted surface is negligible, thus equation (53) reduces to equation (54):

𝑃(𝑡) = 𝑅𝑖 ∑ 𝜂𝐴𝑐 (𝐼𝐵 + 𝐼𝐷)

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


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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 [𝑞𝑏 + 𝑞𝑑 + (𝑅𝑖 − 1)(𝑞𝑏𝑠ℎ + 𝑞𝑑

𝑠ℎ)] (55)

Where 𝑞𝑏 is the yearly solar beam irradiation per unit area of an unshaded collector (which is

the first row), 𝑞𝑑 is the yearly diffused irradiation per unit area of an unshaded collector, 𝑞𝑏𝑠ℎ is

the average yearly beam irradiation per unit area of the shaded collectors and 𝑞𝑑𝑠ℎ is the average

diffused solar irradiation per unit area of a shaded collector. 𝑞𝑏 , 𝑞𝑑 , 𝑞𝑏𝑠ℎ𝑎𝑛𝑑 𝑞𝑑

𝑠ℎ are defined

mathematically as:

𝑞𝑏 = ∑ ∑ 𝐼𝑏 cos 𝜃 Δ𝑇

𝑇𝑠𝑠

𝑇𝑠𝑟

365

𝑛=1

(56)

𝑞𝑑 = 𝐹𝑑 ∑ ∑ 𝐼𝑑Δ𝑇

𝑇𝑠𝑠

𝑇𝑠𝑟

365

𝑛=1

(57)

𝑞𝑏𝑠ℎ = ∑ ∑ 𝐼𝑏 cos 𝜃(1 − 𝑠) Δ𝑇

𝑇𝑠

𝑇𝑟

365

𝑛=1

(58)

𝑞𝑏𝑠ℎ = 𝐹𝑑

𝑠ℎ ∑ ∑ 𝐼𝑑ℎΔ𝑇

𝑇𝑠𝑠

𝑇𝑠𝑟

365

𝑛=1

(59)

Where 𝐹𝑑 𝑎𝑛𝑑 𝐹𝑑𝑠ℎ are the configuration factors for unshaded and shaded collectors

respectively

𝐹𝑑 = cos2 (𝛽

2) (60)

𝐹𝑑𝑠ℎ = cos2(𝛽/2) −

1

2[(𝑑2 + 1)

12 − 𝑑] 𝑠𝑖𝑛𝛽 (61)


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 ≥ 𝐷𝑚𝑖𝑛 (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 =

𝐴𝑚𝑎𝑥). However, for a tilted surface the height reduces as the angle decreases, so it’s height

must always be less than 𝐴𝑚𝑎𝑥. It is given as

𝐻 × sin 𝛽 ≤ 𝐴𝑚𝑎𝑥 (64)

The collector height is limited by the manufacturer’s design, given as

𝐻 ≤ 𝐻𝑚𝑎𝑥 (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 ≤ 𝑃𝑝𝑣𝑚𝑎𝑥 (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

The optimisation model becomes

Maximize:

𝑄 = 𝜂𝐻𝐿𝑖 [𝑞𝑏 + 𝑞𝑑 + (𝑅𝑖 − 1)(𝑞𝑏𝑠ℎ + 𝑞𝑑

𝑠ℎ)] (68)

Where 𝑞𝑏 , 𝑞𝑑 , 𝑞𝑏𝑠ℎ 𝑎𝑛𝑑 𝑞𝑑

𝑠ℎ are defined in equations [56-59]

Subject to

𝑅𝑖 × 1.2 × cos 𝛽 + (𝑛1 − 1)𝐷 ≤ (80 − 2𝑥) (69)

𝐷 ≥ 𝐷𝑚𝑖𝑛 (70)

1.2 × sin 𝛽 ≤ 1.2 (71)

𝐻 ≤ 𝐻𝑚𝑎𝑥 (72)

0° ≤ 𝛽 ≤ 90° (73)

𝑅𝑖 ≥ 2 (74)

0 ≤ 𝑃𝑝𝑣 ≤ 𝑃𝑝𝑣𝑚𝑎𝑥 (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

REFERENCES

[1] J. APPELBAUM and J. BANY, "SHADOW EFFECT OF ADJACENT SOLAR COLLECTORS IN

LARGE SCALE SYSTEMS," Solar Energy, vol. 23, pp. 497-507, 1979.

[2] R. E. JONES and J. F. BURKHART, "Shading effects of collector rows tilted toward the equator," Solar

Energy, vol. 26, pp. 563-565, 1981.

[3] E. A. Handoyo, D.Ichsani, and Prabowo, "The optimal tilt angle of a solar collector," International

Conference on Sustainable Energy Engineering and Application, pp. 166-175, 2013.

[4] J. Bany and J. Appelbaum, "THE EFFECT OF SHADING ON THE DESIGN OF A FIELD OF A FIELD

OF SOLAR COLLECTORS," Solar Ceils, no. 20, pp. 201 - 228, 1987.

[5] F. Mohsen,M. Radzi, M. Amran, F. Mohammad , Z. Mahdi, G. Zohreh, "Optimization and comparison

analysis for application of PV panels in three villages," Energy Science and Engineering, vol. III, no. 2, pp.

142-152, 2015.

[6] M.Kacira, M.Simsek, Y. Babur, and S. Demirkol, "Determining optimum tilt angles and orientations of

photovoltaic panels in Sanliurfa, Turkey," Renewable Energy, no. 29, pp. 1265-1275, 2004.

[7] A. Chandrakar and Y. Tiwari, "Optimization of Solar Power by varying Tilt Angle/Slope," International

Journal of Emerging Technology and Advanced Engineering, vol. III, no. 4, pp. 145-150, 2013.

[8] J. M. Ahmad and N. G. Tiwari, "Optimization of Tilt Angle for Solar Collector to Receive Maximum

radiation," The Open Renewable Energy Journal, no. 2, pp. 19-24, 2009.

[9] H. Danny , W. Li, and Lam N. T. Tony , "Determining the Optimum Tilt Angle and Orientation for Solar

Energy collection based on measured Solar radiance data," International Journal of Photoenergy, p. 9, 2007.

[10] D. Weinstock and J. Appelbaum, "Optimal Solar Field Design of Stationary Collectors," Journal of Solar

Energy Engineering, vol. 126, no. 905, pp. 1-8, 2004.

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