Optimal Solar Field Design of Stationary Collectors Dan Weinstock Ph.D. student e-mail: [email protected]

Joseph Appelbaum Faculty of Engineering, Tel-Aviv University, Tel-Aviv, 69978, Israel, 972-3-6409014

1

The optimal design of stationary photovoltaic and thermal collectors in a solar field, taking into account shading and masking effects, may be based on several criteria: maximum incident energy on collector plane from a given field, minimum field area for given incident energy, minimum cost per unit energy, minimum plant cost, maximum energy per unit collector area or other objectives. These design problems may be formulated as optimization problems with objective functions and sets of constraints (equality and inequality) for which mathematical optimization techniques may be applied. This article deals with obtaining the field design parameters (optimal number of rows, distance between collector rows, collector height and collector inclination angle) that produce maximum annual energy from a given field. A second problem is determination of the minimum field area (length and width) and field design parameters that produce a given required annual energy. The third problem is determination of the optimal field design parameters for obtaining maximum energy per unit collector area from a given field. The results of these optimal designs are compared to a recommended approach of the Israeli Institute of Standards (IIS) in which the solar field design result in negligible shading. An increase in energy of about 20% for a fixed field area and a decrease in field area of about 15% for a given annual incident energy, respectively, may be obtained using the approach formulated in the present article compared to the IIS approach. 关DOI: 10.1115/1.1756137兴

Introduction

The design of stationary photovoltaic and thermal solar collectors in a field involves relationships between the field and collector parameters and solar radiation data. In addition, shading and masking 共expressed by the configuration factor兲 affect the collector deployment by decreasing the incident energy on collector plane of the field. The use of many rows of collectors densely deployed, in a limited field, increases the field incident energy but also increases the shading 共Fig. 1, darkened area兲. Therefore there is an optimal deployment of the collectors yielding maximum field incident energy, minimum required field area, minimum cost per unit energy or other objectives. Field and collector parameters contain field length L 共which is also the collected length兲 and field width W, distance between collector rows D, collector height H, inclination angle ␤ and geometric limitations of these parameters. At a given time, the shaded height Hs and length Ls on the collector, is shown in Fig. 1. The rest of the collector area is unshaded. The optimal design of a solar field may be formulated mathematically as a constrained optimization problem and the solution may be based on applying available optimization algorithms like those described in 关1–3兴. The two articles 关4,5兴 use parametric variation methods but not formal mathematical optimization methods for field design. Several articles deal with shading. An analytical-numerical method was developed in 关6兴 for the evaluation of the effect of shading of concentrating cylindrical parabolic collectors. In the study 关7兴 the shading of a vertical and inclined poles and collectors are investigated. Other authors 关8 –10兴 presented different models for shading calculations. Lighting simulation tool was used in 关11兴 to determine energy losses due to shading. A planning tool for the effect of shading on yield for building-integrated photovoltaic installations was developed in 关12兴. Energy losses for different row distances and inclination angles of a solar field installation were investigated in 关13兴. This article is an extension of the work 关14兴 and its purpose is Contributed by the Solar Energy Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the ASME Solar Division August 2003; final revision April 2004. Associate Editor: A. Kribus.

898 Õ Vol. 126, AUGUST 2004

to formulate the solar collector field design as an optimization problem and to apply constraints optimization techniques to obtain optimal design parameters of the solar field for three cases: 共a兲 obtaining maximum annual incident energy on collector plane from a given field 共b兲 finding minimum field area for a given incident annual energy and 共c兲 obtaining maximum incident energy per unit collector area from a given field. Again, only incident energy on the collector plane is considered in all three cases. The design of optimal solar fields with the objective to obtain minimum cost per unit energy or minimum plant cost may be of more interest than the three cases 共a兲 to 共c兲 dealt with in the present article. As one of the purposes of this article is to introduce a mathematical procedure for solar field design, cases 共a兲 to 共c兲 were chosen for illustration because they contain fewer details of the solar plant. It is the intention of the authors to deal in the future, with economical consideration of the optimal solar field design. The design of optimal solar field pertains to thermal as well as to photovoltaic collectors. For a field of photovoltaic collectors the optimal interconnection of the photovoltaic modules takes into account the shading 共see Sec. 6兲.

2

Formulation of the Optimization Problem

The mathematical formulation of the solar field problem may be described as a ‘‘general programming problem,’’ usually multivariable and nonlinear in both objective and constraint functions, which may be stated in the following form: ¯ 兲 with respect to X ¯ Minimize 共Maximize兲 C共X Subject to: ¯ )⫽0, gj共X ¯ )⭐0, gj共X

j⫽1,2, . . . ,me j⫽me⫹1, . . . ,m

(1)

¯ ⫽ 共 X ,X , . . . ,X 兲 X 1 2 n Xk⭓0, Xk苸R ᭙k⫽1, . . . ,n ¯ C共X兲 is called the objective function of the optimization problem. For case 共a兲 it is the solar energy collected by the solar field. The ¯ are the n design parameters of the solar field problem variables X ¯ ⭓0, non-negative values兲. The design parameters are all free to 共X

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Subject to: ¯ )⫽0, gj共X ¯ )⭐0, gj共X Xk⭓0, Xk⭓0,

Fig. 1 Shading by collectors in a solar field

vary in an allowed range bounded by upper and/or lower physical limits and they may be interrelated to satisfy some conditions. The constraints are expressed by me satisfying conditions called equality constraints and by (m⫺me) bounded conditions called inequality constraints. In the optimal design of many engineering problems 关15兴, including solar fields, the following type of objective function with T terms and n variables is usually encountered: n

T

¯ )⫽ C共X

兺 兿

p⫽1

Xk⬎0,

Xk苸R

Ap

k⫽1

j⫽1,2, . . . ,me j⫽me⫹1, . . . ,m

Xk苸Z⫹ , Xk苸R,

᭙k⫽i⫹1, . . . ,n

where Z⫹ is the natural number set. In our case, one of the optimization variables, e.g. X1 ⫽K, is the number of collector rows, and by definition is an integer. One way of solving an integer optimization problem is by Branch and Bound method 关18兴. This method comprises of the following steps: 1. First, solve the optimization problem with all variables being continuous. This problem is called the relaxation problem and is actually expressed by Eq. 共1兲. Solving this problem gives us a solution in which X1 is not generally an integer, e.g. X1 ⫽6.23. 2. Make a partition 共branching兲 to two problems, each containing a bound on the variable X1 as follows: ¯ 兲 with respect to X ¯ Minimize 共Maximize兲 C共X Subject to: ¯ )⫽0, gj共X ¯ )⭐0, gj共X

j⫽1,2, . . . ,me j⫽me⫹1, . . . ,m

(4a)

X1 ⭓7

m

Xk pk (2)

᭙k⫽1, . . . ,n

where Ap are positive constants and the mpk are real numbers 共positive or negative兲. This type of function is called a posinomial function 关16,17兴 and has the property of having a single minimum in the non-negative region of the n dimensional space Rn, defined by the variables X1 , X2 , . . . ,Xn. The constraints, Eq. 共1兲, generate a ‘‘feasible design region’’ within which 共including its boundary兲 they are satisfied. A feasible design region R for a constrained optimization problem with two variables X1 and X2 is shown in Fig. 2. An objective function of the Eq. 共2兲 type has a single minimum 共maximum兲 which may lie within R, at point b, or outside of R, at point a. In the latter case, the constrained minimum 共maximum兲 is on the boundary of R, at point c. When the optimization problem contains variables that some 共or all兲 of them can have only integer 共binary, natural, etc.兲 values it is called an integer optimization problem. Let us denote the integer variables by X1 , . . . ,Xi and the rest of the variables by Xi⫹1 , . . . ,Xn. The formulation of the optimization problem becomes: ¯ 兲 with respect to X ¯ Minimize 共Maximize兲 C共X

Xk⭓0,

Xk苸R,

᭙k⫽2,3, . . . ,n

and ¯ 兲 with respect to X ¯ Minimize 共Maximize兲 C共X Subject to: ¯ )⫽0, gj共X ¯ )⭐0, gj共X

j⫽1,2, . . . ,me j⫽me⫹1, . . . ,m

Fig. 2 Feasible design region in a constrained optimization problem

(4b)

X1 ⭐6 Xk⭓0,

Xk苸R,

᭙k⫽2,3, . . . ,n

Notice that the region 6⬍X1 ⬍7 is not a feasible one. 3. Solve the problems in Eqs. 共4a兲 and 共4b兲. If both solutions are feasible, meaning that X1 is an integer, choose the solution for ¯ 兲 is minimum 共maximum兲. which C共X 4. If one or both solutions to the problems in Eqs. 共4a兲 and 共4b兲 are not feasible, the process of branch and bound continues until obtaining a feasible solution. 2.1 Maximum Energy. This section outlines the formulation of the optimal solar field problem to obtain maximum incident energy on the collector plane. The field is horizontal and of fixed length L and width W, comprising of K rows of solar collectors with distance D between the rows, each collector is of length L and a height H and inclined with an angle ␤ with respect to the horizontal 共Fig. 1兲. ¯ ⫽K, ␤, D, H where K is a discrete The problem variables are X variable. The objective function in this study is the yearly incident solar energy of the field given by: sh Q⫽H⫻L⫻ 关 qb⫹qb⫹ 共 K⫺1 兲共 qsh b ⫹qd 兲兴

Journal of Solar Energy Engineering

(3)

᭙k⫽1, . . . ,i

(5)

where qb is the yearly beam irradiation per unit area of an un-shaded collector 共first row兲 qd is the yearly diffuse irradiation per unit area of an un-shaded collector 共first row兲 qsh b is the average yearly beam irradiation per unit area of shaded collectors 共共K⫺1兲 rows兲 AUGUST 2004, Vol. 126 Õ 899

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qsh d is the average yearly diffuse irradiation per unit area of shaded collectors 共共K⫺1兲 rows兲 sh The explicit expressions for qb, qd, qsh b and qd are given in Appendix A. The variation of the field and collector parameters values is constrained by the field width W 共Fig. 1兲 and defines the equality constraint of the problem, i.e., K⫻H⫻cos ␤ ⫹ 共 K⫺1 兲 ⫻D⫽W

(6)

Equation 共6兲 may also be introduced as an inequality constraint in the formulation of the optimization problem, i.e., K⫻H⫻cos ␤ ⫹ 共 K⫺1 兲 ⫻D⭐W

(6a)

Because the optimization procedure will always utilize the full field width W to obtain maximum energy from the field or to obtain the smallest field area, the solution will be on the boundary of Eq. 共6a兲, i.e., ⫽W. The advantage of Eq. 共6兲 is that one may reduce the dimension of the problem by one variable by eliminating this variable from the equality constraint. Two spacing approaches between the collector rows may be defined. One approach is based on a required minimum distance for maintenance purpose and the other approach is based on reducing the mutual shading between collector rows 共see Sec. 5兲. The spacing between collector rows appears as an inequality constraint. For the case of required minimum distance for maintenance we write D⭓Dmin

(7)

Similarly, collector installation and maintenance may require that the height of the collector above the ground be limited to Amax , i.e., H sin ␤ ⭐Amax

(8)

The collector height H itself may be limited by the solar field construction, maintenance and by module manufacturer, i.e.: H⭐Hmax

(9)

The collector inclination angle may vary in the range 0 deg ⭐ ␤ ⭐90 deg

(10)

and the number of rows is at least 2 and discrete 2⭐K苸Z⫹

(11)

All variables have physical meaning, therefore having nonnegative values. 2.2 Minimum Field Area. Another possible objective function is to find the smallest area, W⫻L, of a solar field receiving a required yearly incident energy Qmin on the collector plane. One may encounter such problem in places where the ground is expensive or on rooftops where the available area for solar collector installation is limited. The problem variables in this case are ¯ ⫽W, L, K, ␤, D, H. The formulation of the optimization probX lem is as follows: Minimize

W⫻L

(12)

Subject to: K⫻H⫻cos ␤ ⫹ 共 K⫺1 兲 ⫻D⫽W

(13)

共relationship between solar field parameters, Eq. 共6兲兲 Q⭓Qmin

(14)

D⭓Dmin

(15)

H sin ␤ ⭐Amax

(16)

H⭐Hmax

(17)

0 deg ⭐ ␤ ⭐90 deg

(18)

900 Õ Vol. 126, AUGUST 2004

W⭐Wmax

(19)

L⭐Lmax

(20)

a1 ⭐W/L⭐a2

(21)

⫹

2⭐K苸Z

(22)

where Q is given by Eq. 共5兲. The field should receive at least a required amount of yearly incident energy Q⭓Qmin . Equations 共19兲 and 共20兲 are geometrical limits of the solar field. Because the optimization for maximum energy tries to reduce the shading between collectors, the result would lead to two long rows 共see Eq. 共22兲兲. For a solar field to be of a desired shape, Eq. 共21兲 was introduced. 2.3 Maximum Energy Per Unit Area. The third optimization problem we solve in this article is obtaining maximum incident energy per unit collector area for a given incident energy from a given field. Obtaining maximum incident energy per unit collector area from a given field solely would result in a design with two collector rows widely apart from each other because maximum energy is obtained with negligible 共or no兲 shading 共assuming the field width W is large enough兲. The result is rather trivial. Therefore to utilize better the given field we need to add an additional requirement to the problem, which is obtaining also desired incident energy from the field. Higher required energy will result in adding more collector rows and thus decreasing the intercollector distance and shading may be cast on the collectors. This leads to a decrease in energy per unit area of collector. The objective function is given by Eq. 共5兲 divided by K⫻H⫻L 共total collector area兲 and the constraints are given by Eqs. 共7兲–共11兲 and 共14兲. The design parameters are ␤, K, D and H.

3

Optimal Design Results

Results of the optimal deployment of a solar field are shown for solar collectors facing south on a horizontal ground at Tel-Aviv 共latitude 32°N兲. Minimal row distance, for example, Dmin⫽0.8 m was assumed. Hourly radiation data were used and Matlab optimization toolbox command ‘‘fmincon’’ 关1兴 was applied to solve the optimization problems. This command uses a Sequential Quadratic Programming and its implementation consists of three main stages: 1. Updating of the Hessian matrix of the Lagrangian function 2. Quadratic programming problem solution 3. Line search and merit function calculation 3.1 Maximum Energy. The objective is to find the deployment of the collectors K, ␤, D, H that results in maximum incident solar energy on collector plane from a given 共small兲 field of L⫽7.5 m, W⫽12 m and for maximum collector height of Hmax ⫽2 m and maximum of collector height above ground of Amax ⫽2 m. The feasible design region in 共␤-H兲 and 共K-D兲 planes is shown in Figs. 3 and 4, respectively. The light lines are iso-energy of the objective function, Eq. 共5兲, in kWh. The heavy lines represent the equality 共Eq. 共6兲兲 and the inequality constraints 共Eqs. 共7兲–共9兲兲. The arrow points to the feasible design region. The solution of this problem lies on the intersection of the equality constraint and on the boundary of the inequality constraint 共Eqs. 共7兲 and 共9兲, respectively, marked by a circle兲 corresponding to ␤⫽48.19 deg, D⫽0.8 m, K⫽6 and H⫽2 m, and the collected maximum yearly energy is 137,788 kWh. For a small field with few collector rows the incident energy on the un-shaded first row is relatively higher than the other ‘‘shaded’’ rows and the optimization of the field prefers a large inclination angle ␤ permitting to have more collector rows installed in the given field width W. For larger fields, ␤ is smaller, see Sec. 5. 3.2 Minimum Field Area. The objective is to find the solar field area 共length L and width W兲 that collects at least a desired amount of yearly incident energy of Qmin⫽500 MWh and for Transactions of the ASME

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Fig. 3 Feasible design region in „␤-H… plane in optimization for maximum incident energy objective function. Numbers represent iso-energy lines Q in kWh, heavy lines represent the equality constraint „Eq. „6…… and inequality constraints „Eqs. „8… and „9……. The solution is marked by circle.

Hmax⫽2 m and Amax⫽2 m. The field dimensions are limited to Lmax⫽30 m and Wmax⫽15 m. The feasible design region 共hatched兲 in 共L-D兲 plane, for example, is shown in Fig. 5. The light lines represent the iso-field area of the objective function, Eq. 共12兲, in m2 and the heavy lines represent the problem con-

Fig. 4 Feasible design region in „K-D… plane in optimization for maximum incident energy objective function. Numbers represent iso-energy lines Q in kWh, heavy lines represent the equality constraint „Eq. „6…… and inequality constraint „Eq. „7……. The solution is marked by circle.

Fig. 6 Variation of maximum energy per unit of collector area, optimal distance between collector rows and optimal number of collector rows as a function of given amount of incident energy for a given field size of LÄ7.5 m, WÄ12 m

straints. The feasible design region is indicated by the arrows Q ⬎Qmin , D⬎Dmin , W⬍Wmax and L⬍Lmax and the solution lies on the boundaries of the feasible region 共marked by a circle兲 corresponding to D⫽0.8 m and L⫽25.8 m. The other optimal parameters are ␤⫽42.27 deg, K⫽6 and H⫽2 m. The field width is W⫽12.9 m, W/L⫽0.5 and the field area is therefore 333 m2. 3.3 Maximum Energy Per Unit Area. As mentioned in Sec. 2.3, maximum energy per unit collector area for a given field size is obtained for 2 collector rows widely separated assuming the field width is sufficiently large. For the problem to be more practical, the field needs to collect a desired amount of energy. Higher energy is obtained by increasing the number of collector rows 共thus decreasing the collector distance兲 in the given field width. However, the shading will also increase resulting in a decrease of the collected energy per unit area. This is shown in Fig. 6 for a field size of L⫽7.5 m, W⫽12 m, Hmax⫽2 m and Amax⫽2 m. The field dimensions are limited to Lmax⫽30 m and Wmax⫽15 m. The figure shows the variation of the maximum incident energy per unit collector area as a function of the yearly incident energy demand obtained by the optimization procedure. The associated optimal number of collector rows K and the distance between the rows D are also depicted in the figure. It is worthwhile mentioning that the maximum energy this field can collect is 137,788 kWh, the result obtained in section 3.1, corresponding to energy per unit collector area of 1,531 kWh m⫺2. The maximum energy per unit area of 1,906 kWh m⫺2 is obtained for K⫽2 corresponding to a required incident energy of 40,000 kWh from the given field size.

4 The Influence of Row Distance on the Solar Field Design The influence of the row distance on the collector deployment was examined by repeating the optimization process for different values of Dmin . Small 共small W兲 and large 共large W兲 fields were examined. In a small field in which the number of collector rows is small, the effect of the un-shaded first row is more pronounced and affects the optimal results, whereas in a large field with many rows the effect of the first row is much more moderate.

Fig. 5 Feasible design region in „L-D… plane in optimization for minimum field area objective function. Numbers represent isoarea lines in m2 and heavy lines represent the constraint „Eqs. „14… and „15… and „19… and „20……. The solution is marked by circle.

Journal of Solar Energy Engineering

4.1 Maximum Energy. The optimal number of rows and the maximum yearly energy for a large field 共L⫽100 m, W⫽200 m, Hmax⫽2 m and Amax⫽2 m) as a function of the defined distance between collector rows are shown in Fig. 7. The optimal number of rows and the maximum yearly energy decrease as Dmin increases because less rows can be installed in a defined field width W 共see Eq. 共6兲兲. As the number of rows, K, can have only AUGUST 2004, Vol. 126 Õ 901

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Fig. 7 Optimal number of rows and maximum yearly incident energy for different maintenance distance Dmin , for a large field „LÄ100 m, WÄ200 m, HmaxÄ2 m and AmaxÄ2 m…

Fig. 9 Optimal number of rows and minimum field area for different maintenance distance Dmin , for a small field „Qmin Ä500 mWh, HmaxÄ2 m and AmaxÄ2 m…

discrete values, K remains constant in a range of Dmin . The variation of the optimal inclination angle is shown in Fig. 8. 4.2 Minimum Field Area. The optimal number of rows and the minimum field area for low and high required yearly energies as a function of Dmin were calculated. The results are shown in Fig. 9, for example, for a small field (Qmin⫽500 MWh and for Hmax⫽2 m and Amax⫽2 m). As the distance between the collector rows is larger more field area is required for a given Qmin . However, the number of collector rows decreases because less shading is taking place. The increase in field area is mainly due to the increase in W.

5

Recommended Row Distance by Israeli Standards

The Israeli standard 579 Part 4 共Solar water heating systems: Open loop thermosiphonic systems—Design, installation and testing兲 recommends a distance between adjacent collector rows given by

Fig. 8 Optimal inclination angle for different maintenance distance Dmin , for a large field „LÄ100 m, WÄ200 m, HmaxÄ2 m and AmaxÄ2 m…, maximum incident energy objective function Table 1 Optimal design results for DminÄ1.35H sin ␤ „HmaxÄ2 m, AmaxÄ2 m… Field m

Dmin m

(23)

31 deg ⭐ ␤ ⭐50 deg

(24)

There is no a similar Israeli standard for PV collectors and as far as we know also not elsewhere. The formulation of the optimal field design using this recommendation is obtained by replacing Eqs. 共7兲, 共10兲, 共15兲 and 共18兲 with Eqs. 共23兲–共24兲. A comparison of the optimal deployment results 共K, ␤, D and H兲 for D⭓0.8 m and for D⭓1.35H sin ␤ for maximum yearly incident energy is shown in Table 1. Two field sizes are compared: 共a兲 a small field of L⫽7.5 m and W⫽12 m and 共b兲 a large field of L⫽100 m and W⫽200 m, both for Hmax⫽2 m and Amax⫽2 m. The table lists the optimal parameters: number of rows, inclination angle, distance between the rows and the collector height. The optimal solution prefers the distance between the collector rows to be D⫽Dmin⫽0.8 m and the collector height to be H⫽Hmax ⫽2 m. A small field prefers large inclination angles 共48.19 deg兲 and a large field a smaller inclination angle 共31.24 deg兲 for the optimal solution with Dmin⫽0.8 m. For the distance 1.35H sin ␤, the optimal inclination angle is 31.00 deg, i.e., the lower limit 共Eq. 共24兲兲. The increase in yearly collected energy Q for D⭓0.8 m is 23.15% for the small field and 17.31% for the large one as compared to D⭓1.35H sin ␤. The product K⫻H⫻L is the total collector area of the solar field. The increase in yearly energy is achieved, however, by the addition of collectors. The collectors are better utilized giving a higher value of Q/K⫻H⫻L for Dmin ⫽1.35H sin ␤ but the difference decreases for larger fields. The increase in energy and the percentage of additional collector area are shown in Fig. 10 for the large field as a function of the row distance. The economics of this field design 共the increase in yearly energy as opposed to first cost兲 is not evaluated in this article as the objective of this study was to obtain maximum incident energy on collector plane from a given field size. Table 2 compares the energy per unit area of an un-shaded 共first row兲 collector qy and a shaded collector qsh y for the solar fields in Table 1. It should be mentioned that the recommended distance

yearly

collected

K

␤ deg

D m

H m

L⫽7.5 W⫽12

0.8 1.35H sin ␤

6 4

48.19 31.00

0.80 1.71

2.00 2.00

L⫽100 W⫽200

0.8 1.35H sin ␤

80 65

31.24 31.00

0.80 1.39

2.00 2.00

902 Õ Vol. 126, AUGUST 2004

D⭓1.35H sin ␤

energy

Q MWh 137.8 111.9 27,883 23,768

for

DminÄ0.8 m

K⫻H⫻L m2

and

Q/K⫻H⫻L MWh m⫺2

90 60

1.531 1.865

16,000 13,000

1.743 1.832

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Fig. 10 Yearly energy increase and percent of additional collector area for different maintenance distance Dmin , for a large field „LÄ100 m, WÄ200 m, HmaxÄ2 m and AmaxÄ2 m…

D⭓1.35H sin ␤ is based on reducing the mutual shading between the collector rows whereas the distance D⭓0.8 m is based on the required distance for maintenance. This difference in point of view results in the ratio qsh y /qy as shown in Table 2. For a small field and Dmin⫽0.8 m a shaded collector receives 83.28% of the yearly irradiation per unit area as compared to an un-shaded collector, whereas for a large distance 1.35H sin ␤⫽1.71 m, only 2.33% of the yearly energy per unit area is lost due to shading. For a large field the difference of the ratio qsh y /qy for the two spacing approaches between the collectors is smaller than for a small field. The basic difference in the optimal results of a small and a large field, for a small distance 0.8 m between the rows, is in the manner in which the optimum is achieved. To obtain maximum energy, the optimization tries to minimize the shading in the large field 共containing many rows兲 by decreasing the collector height H sin ␤, i.e., by decreasing the inclination angle ␤ 共31.24 deg兲, whereas maximum energy for a small field is obtained by increasing the number of rows resulting in an increase of the angle ␤ 共48.19 deg兲. For a small distance between collectors Dmin⫽0.8 m the shading is more pronounced; 16.72% for a small field as oppose to 8.25% for a large field. Also the effect of an un-shaded 共first兲 collector row on the yearly energy in a small field is larger than for a large one. For a large distance between the rows 共1.35H sin ␤兲 there is a small difference in the shading between a small and a large field. A comparison of the optimal deployment results K, ␤, W, L, D and H for D⭓0.8 m and for D⭓1.35H sin ␤ to obtain the mini-

Table 2 Comparison of yearly incident energy density for shaded and un-shaded collectors in Table 1 Field m L⫽7.5 W⫽12 L⫽100 W⫽200

Dmin m 0.8 1.35H sin ␤ 0.8 1.35H sin ␤

qy MWh m⫺2

qsh y MWh m⫺2

qsh y /qy %

1.7657 1.8981 1.8973 1.8981

1.4704 1.8540 1.7407 1.8312

83.28 97.67 91.75 96.48

mum field area, for a low 共500 MWh兲 and high 共20,000 MWh兲 required yearly incident energy value, Qmin , is shown in Table 3 for Hmax⫽2 m and Amax⫽2 m. The decrease in the field area is 13.6 and 15.2% for low and high amount of incident energy requirement, respectively, for the deployment with Dmin⫽0.8 m as opposed to Dmin⫽1.35H sin ␤. To obtain the required yearly incident energy from a smaller field resulting from the optimization with Dmin⫽0.8 m, more collector area K⫻H⫻L is needed as compared for the distance Dmin ⫽1.35H sin ␤. The economics of this field design 共available field or its cost against the collector cost兲 is not evaluated in this article as the objective of the study was to obtain minimum solar field area. However, it should be pointed out that for larger fields 共higher required yearly energy兲 the additional collector area K⫻H⫻L and Q/K⫻H⫻L decrease as the Qmin increases 共comparing the distance for Dmin⫽0.8 m with Dmin⫽1.35H sin ␤). For Qmin ⫽500 MWh the addition K⫻H⫻L is 14% and for Qmin ⫽20,000 MWh it is only 5.3%. As mentioned before, the recommended rows distance by the Israeli standard emphasize the reduction of mutual shading between collector rows in the solar field design, whereas the proposed optimization approach of this article is not to prevent shading but rather to cope with it to obtain a better solar field design. This is achieved by a smaller row distance between collector rows based on a maintenance requirement. As a result one obtains denser collector fields with higher energies or smaller fields. However, additional collector needs to be used. Economic analysis will show whether the value of energy increase along the system life time surpasses the additional cost of the collectors.

6

Conclusions

A solar field may be divided into two types: 共a兲 a solar field comprising of thermal collectors and 共b兲 a solar field comprising of photovoltaic collectors. The incident energy 共input energy兲 to both field types comes from the same source, i.e., solar radiation. The amount of energy supplied to the user 共output energy兲 of each field type is different. For a thermal collector the output energy depends on collector efficiency and its effective threshold operation. The entire collector area is a one basic unit contributing to the warming of the collector liquid resulting from both the direct beam and diffuse irradiance. A photovoltaic collector comprises of many electrically interconnected modules where the basic unit is the solar cell or module. For a photovoltaic collector the output energy depends on the collector efficiency, its effective threshold operation and on the scheme of the electrically interconnected modules. Series interconnection may have a significant effect on the output energy of the collector in the event of shading. An analysis of the yearly variation of the shading pattern on the photovoltaic modules of the collectors may lead to an optimal interconnection scheme. This scheme may depend on the inclination angle ␤, distance between collector rows D and the collector height H. A functional relation between these parameters may be derived and included in the optimization procedure for objective functions pertaining to the output energy.

Table 3 Optimal design results for minimum field area for DÐ0.8 m and DÐ1.35H sin ␤ „HmaxÄ2 m, AmaxÄ2 m… Qmin MWh 500 20,000

Dmin m 0.8 1.35H sin ␤ 0.8 1.35H sin ␤

K

␤ deg

W m

L m

D m

H m

W⫻L m2

K⫻H⫻L m2

Q/K⫻H⫻L MWh m⫺2

6 5 34 57

42.27 31.00 31.46 31.00

12.9 13.9 84.4 175.6

25.8 27.7 168.9 95.7

0.80 1.36 0.80 1.39

2.00 1.96 2.00 2.00

333 385 14,255 16,805

310 272 11,485 10,910

1.615 1.842 1.741 1.833

Journal of Solar Energy Engineering

AUGUST 2004, Vol. 126 Õ 903

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It should be emphasized that the argument of this article is ‘‘not’’ to avoid collector shading but to take into account the shading and still to obtain an optimal system design. It is possible to obtain a yearly energy increase of about 20% and a decrease of about 15% in field area 共depending on the row distance兲, respectively, using the distance between the collectors based on maintenance as compared to the distance recommended by the Israeli standard. The above two increases are achieved, however, by additional collector area. The economics of the design of these fields where not evaluated in this article.

Acknowledgment

␥ ␥c ␥s ⌬T ␪

⫽ ⫽ ⫽ ⫽ ⫽

difference between the sun and collector azimuth collector azimuth with respect to south sun azimuth with respect to south time interval angle between the solar beam and the normal to the collector

Appendix A The shaded and un-shaded irradiation per unit area in Eq. 共5兲 is given by 关4,7兴. It was assumed an isotropic model for the diffused irradiation, n⫽365 TS

This research was partly supported by The Jokel chair of Electronics.

Nomenclature Amax ⫽ maximum collector height above ground Ap ⫽ positive constants a1 , a2 ⫽ lower and upper limit of the ratio of W/L, respectively as ⫽ relative shaded area ¯ 兲 ⫽ objective function C共X D ⫽ distance between collector rows Dmin ⫽ minimum distance between collector rows d ⫽ normalized distance between collector rows Fd ⫽ configuration factor for un-shaded collectors Fsh d ⫽ configuration factor for shaded collectors Gb ⫽ direct beam irradiance on the collector perpendicular to solar rays Gdh ⫽ horizontal diffuse irradiance ¯ ) ⫽ equality and inequality constraints gj共x H ⫽ collector height Hs ⫽ shadow height hs ⫽ relative shadow height Hmax ⫽ maximum collector height K ⫽ number of solar collector rows L ⫽ solar field length Ls ⫽ shadow length Lus ⫽ un-shadow collector length Lmax ⫽ maximum solar field length ᐉ ⫽ normalized collector length ᐉ s ⫽ relative shadow length m ⫽ number of constraints me ⫽ number of equality constraints mpk ⫽ real numbers 共positive or negative兲 n ⫽ day number Q ⫽ yearly energy Qmin ⫽ required yearly energy qb ⫽ yearly beam irradiation per unit area of an un-shaded collector 共first row兲 qd ⫽ yearly diffuse irradiation per unit area of an unshaded collector 共first row兲 qsh b ⫽ average yearly beam irradiation per unit area of shaded collector 共共K⫺1兲 rows兲 qsh d ⫽ average yearly diffuse irradiation per unit area of shaded collector 共共K⫺1兲 rows兲 Rn ⫽ n dimensional space TR ⫽ sun rise on the collector for the beam irradiance TS ⫽ sun set on the collector for the beam irradiance TSR ⫽ sun rise for the diffuse irradiance TSS ⫽ sun rise for the diffuse irradiance ¯ ⫽ vector of variables X W ⫽ solar field width Wmax ⫽ maximum solar field width Z⫹ ⫽ natural number set ␣ ⫽ sun elevation angle ␤ ⫽ collector inclination angle 904 Õ Vol. 126, AUGUST 2004

兺 兺 G cos ␪ ⌬T

qb⫽

n⫽1

TR

b

(A1)

where qb is the yearly beam irradiation per unit area of an unshaded collector 共first row兲; n⫽365 TSS

qd⫽Fd

兺 兺G

n⫽1

TSR

dh⌬T

(A2)

and qd is the yearly diffuse irradiation per unit area of an unshaded collector 共first row兲; n⫽365 TS

qsh b⫽

兺 兺 G cos ␪ 共 1⫺a 兲 ⌬T

n⫽1

b

TR

s

(A3)

and qsh b is the average yearly beam irradiation per unit area of shaded collectors 共共K⫺1兲 rows兲; n⫽365 TSS sh qsh d ⫽Fd

兺 兺G

n⫽1

TSR

dh⌬T

(A4)

and qsh d is the average yearly diffuse irradiation per unit area of shaded collectors 共共K⫺1兲 rows兲; Gb is the direct beam irradiance on the collector perpendicular to solar rays and Gdh is the horizontal diffuse irradiance. ␪ is the angle between the solar beam and the normal to the collector given by cos ␪ ⫽cos ␤ sin ␣ ⫹sin ␤ cos ␣ cos ␥

(A5)

where ␣ is the sun elevation angle; ␤ is the collector inclination angle and ␥ ⫽ ␥ s⫺ ␥ c is the difference between the sun and collector azimuth with respect to south. ⌬T is the summation time interval from sun rise TR to sunset TS on the collector for the beam irradiance, and from sun rise TSR to sun set TSS for the diffuse irradiance. The other summation is from January 1 共n⫽1兲 to December 31 共n⫽365兲. The configuration factors for un-shaded and shaded collectors, respectively, are given by Fd⫽cos2 共 ␤ /2兲

(A6)

2 2 1/2 Fsh d ⫽cos 共 ␤ /2 兲 ⫺1/2关共 d ⫹1 兲 ⫺d兴 sin ␤

(A7)

where d is the normalized distance between two rows given by d⫽D/H sin ␤

(A8)

as is the relative shaded area given by as⫽ᐉ s⫻hs

(A9)

where ᐉ s⫽1⫺

兩 sin ␥ 兩 d sin ␤ ⫹cos ␤ ⫻ , ᐉ cos ␤ tan ␣ ⫹sin ␤ cos ␥

␣ ⭓0,

兩 ␥ 兩 ⭐90 deg,

0⭐ᐉ s⭐1

(A10)

is the relative shadow length Transactions of the ASME

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hs⫽1⫺

␣ ⭓0,

d sin ␤ ⫹cos ␤ , cos ␤ ⫹ 关 sin ␤ cos ␥ /tan ␣ 兴 兩 ␥ 兩 ⭐90 deg,

0⭐hs⭐1

(A11)

is the relative shadow width and ᐉ⫽L/H sin ␤

(A12)

is the normalized collector length.

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