A method to estimate the energy production of photovoltaic trackers under shading conditions

Energy Conversion and Management 150 (2017) 433–450

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A method to estimate the energy production of photovoltaic trackers under shading conditions Eloy Díaz-Dorado, José Cidrás, Camilo Carrillo

MARK

⁎

University of Vigo, Spain

A R T I C L E I N F O

A B S T R A C T

Keywords: Photovoltaic systems Photovoltaic cells Partial shading Ground cover ratio Planning

Energy produced by a photovoltaic park mainly depends on solar irradiance. However in order to estimate the energy production, it must be taken into account the technology of PV-modules, their layout and the electrical connection between them. Furthermore, the energy losses, especially those related to non-uniform distribution of irradiance must be considered. In this context, in a PV-park it is specially important losses related to shadows between trackers. These must be properly estimated to propose different configurations or to evaluate the efficiency of the installation. In this context, this article presents a methodology to evaluate the energy production of a PV-park where PVtrackers are modeled from their simplest elements to the PV-array. The energy calculation includes losses; therefore, shadows are analyzed and included as irregular distributions of irradiance along the tracker plane. The presented method allows for the analysis of different design criteria: PV-cells and PV-modules arrangement, PVcell electrical connections inside a module and electrical connections between PV-modules, tracker layout on the ground and tracker dimensions. Furthermore, the proposed method allows evaluation of the annual energy generation and the losses due to the trackers’ shadows, accounting for the irradiance and the temperature.

1. Introduction

array with an arbitrary configuration. Usually, the PV-array analysis is done by using different kinds of simplifications. For example, in [13] several simplifications on PV-cell and PV-module models, e.g. the reverse biasing of PV-cells is not considered, have been used to obtain the P-V curve for a partially shaded PV-array. Another kind of simplification consists in restricting the type of shadow to be analyzed [10,11,14–17]. For example, in [15] the MPP values are obtained for long strings and parallel-connected short strings under partial shading conditions. Nevertheless, only complete series of cells with a bypass diode are considered to be shaded. Finally, several authors propose empirical expression to obtain power losses without modeling the PV-array in detail, e.g. in [18]. Regarding field layout of trackers, Refs. [7,19] present the first works dedicated to the minimization of energy losses in PV-tracker parks considering the layout as an input for the optimizations process. In those papers, two examples are used to compare the results of square and hexagonal ground layouts although several simplifications on radiation [19] or granularity of analysis [7] are applied. Most papers related to these ones, [10,16,20–23], put their efforts in modeling shadow geometry for different kind of trackers or field geometries but uses simplifications to obtain the energy losses without considering, for example, the different electrical configurations of PV elements. In [10], a metaheuristic method

One usual matter of concern in the analysis of photovoltaic (PV) installations is the estimation of the energy produced under different working conditions, e.g. those far from the rated ones. In this context, one of the most complex situations that is usually found is the estimation of the energy produced when a PV park has no uniform irradiance on its elements. This is a typical situation, and the matter of this paper, in parks formed by sun trackers where partial shading between trackers is quite common. To accurately analyze the effects of partial shading in PV installations, a detailed model is necessary that takes into account their simplest elements, cells and diodes, and the different electrical connections between them forming panels, strings and, finally, arrays [1,2]. This kind of modeling is usually referred as “cell to array” approach [1] and it is typically implemented in commercially available circuit simulations packages [3–5] or even in specific software for PV systems [6,7]. However, the use of commercial software could limit its use, e.g., for planning purposes and energy assessment. Shading effects of partial shading on the electrical behavior of PV systems have been analyzed by several authors [1,8–12]. Nevertheless, most of them cannot be used to accurately obtain the energy yield for a PV

⁎

Corresponding author. E-mail addresses: [email protected] (E. Díaz-Dorado), [email protected] (J. Cidrás), [email protected] (C. Carrillo).

http://dx.doi.org/10.1016/j.enconman.2017.08.022 Received 7 February 2017; Received in revised form 15 July 2017; Accepted 7 August 2017 0196-8904/ © 2017 Elsevier Ltd. All rights reserved.

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Nomenclature a d D Esunny,αγ Ecloudy,αγ Eannual G Gb Gd Gr h Hsunny,αγ

Hcloudy,αγ

I Isc0 Kb KI KP KV l L Lannual m n ns np nf nc MPP Pcell ,G MPP Parray,G MPP Parray ,xy q rG Rs Rsh u0 V S

s T Tm Tn Varray,xy Vbr Voc0 v0

fraction of ohmic current in avalanche breakdown distance between two trackers width of the module annual energy production of a PV-tracker under sunny weather conditions for sun position (α, γ) annual energy production of a PV-tracker under cloudy weather conditions for sun position (α, γ) annual electrical energy produced by a PV-tracker global irradiance beam irradiance diffuse irradiance reflected irradiance height of the tracker irradiation on a tracker under sunny weather conditions for each sun azimuth α and elevation γ when no shading is considered irradiation on a tracker under cloudy weather conditions for each sun azimuth α and elevation γ when no shading is considered current short circuit current Boltzmann constant temperature coefficient of short circuit current temperature coefficient of MPP power temperature coefficient of open circuit voltage width of the shaded area width of the module annual energy losses avalanche breakdown exponent ideality factor of cell number of PV-modules in each PV-string number of PV-strings in each PV-array number of rows in a PV-string number of columns in a PV-string cell MPP power with irradiance G array MPP power with irradiance G array MPP power with shaded area (x, y) elementary charge relationship between indirect and global irradiance series resistance of PV-cell shunt resistance of PV-cell abscissa of the projection of the center of one tracker on another voltage complete area of the cast shadow on a PV-tracker

w x y

surface area of a PV-cell temperature annual mean temperature nominal temperature array MPP voltage with shaded area (x, y) junction breakdown voltage open circuit voltage ordinate of the projection of the center of one tracker on another width of the tracker horizontal size of the shaded area vertical size of the shaded area

Greek letters α α0 ϕ θ θ0 γ δ Δtsunny,αγ Δtcloudy,αγ ξ ηxy ηαγ ρg σi

azimuth angle of the sun position relative azimuth between two trackers azimuth angle of normal vector of a plane elevation angle of normal vector of a plane fixed elevation angle of normal vector of a single-axis tracker elevation angle of the sun position height of the shaded area annual period of time under sunny weather conditions with sun azimuth α and elevation γ annual period of time under cloudy weather conditions with sun azimuth α and elevation γ rectangular shaded area ratio for a PV-module efficiency coefficients of the tracker with shaded area (x, y) efficiency coefficients of the tracker for sun position (α, γ) albedo or ground reflectance shading coefficient

Subscripts b cloudy d h n r sunny xy αγ γθ

beam or normal under cloudy weather conditions diffuse on horizontal plane on normal plane reflected under sunny weather conditions of sizes x and y azimuth angle α and elevation angle γ of sun position azimuth angle γ and elevation angle θ of tracker

integrate the results in the energy yield calculation, a trigonometric approach that considers one-axis and two-axis trackers has been used in the proposed method [16,20,25]. After analyzing the equations related to shadow geometry, it has been demonstrated in this paper that its shape is rectangular which makes easier the shadows modeling. As a resume, the proposed method allows an accurate estimation of the production of a PV-park by considering the following aspects, which are only partially taken into account by the different methods previously commented: PV technology, a high order model for PV-cell including reverse biasing, cell-to-array modeling, shadow geometry, tracker layout (module layout and electrical connections), field layout and annual irradiance. The proposed method allows the calculation the complete P-V curve in a PV system with rectangular shadows. These values, along with a solar energy chart, allow the energy assessment of a PV-park (see Fig. 1). Finally, it must be considered that the cell to array modeling requires the developing of the entire electric circuit of a PV-tracker so, it

is presented based on evolutionary strategies that are used to obtain the best location of each tracker on a terrain of irregular shape; where it has been taken into account the energy losses caused by shadows from nearby obstacles and between PV-trackers. In [21], results are presented for simulations of the energy yield of flat panels for different locations and tracking strategies as a function of the ground cover ratio, but certain limitations on shadows are applied. In any case, some interesting results for design purposes are shown, such as the optimal position of solar trackers on the ground depending on land availability or the energy gains of each tracking strategy. Similarly, in [22], the energy production for different tracking strategies in a PV park is analyzed, although the layout of PV-modules on the tracker and their electrical connections are not taken into account in the optimization process. One of the aspects to be analyzed in PV-parks is the geometry of the shadows cast between trackers. Although there are several software packages [6,24] that help to obtain those shadows, in order to easily 434

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is quite complex and susceptible to numerical issues [1]. In order to overcome numerical problems, the model has been simulating using the method proposed in [26] where a discrete model is presented to obtain the I-V curve of partially shaded PV-arrays, which is an improvement, generalization and systematization of that introduced in [27]. This paper is organized as follows. In Section 2, the irradiance and irradiation on PV-trackers are analyzed. Then, in Section 3, a model of the irradiance under shadowed conditions and a study of the shadows between trackers are presented. In Section 4, a complete model is presented for a shadowed PV-tracker; furthermore, the method to estimate the energy production is introduced. Next, in Section 5, the application of the proposed method for analyzing different PV-tracker configurations, ground layout and to optimise the annual energy production. In Section 6, the results of the proposed method are compared with results obtained using well-known simplifications. Finally, in Section 7, the main conclusions of the paper are presented.

Gd,αγ,αγ = Gd (ϕ = α,θ = γ,α,γ) = Gd (ϕ = 0°,θ = 90°,α,γ)

Similar values can be obtained for single-axis trackers with fixed elevation angle ϕ0, for each azimuth angle α and elevation angle γ of the sun using:

Gb,αθ0,γα = Gb (ϕ = α,θ0,α,γ) = Gb (ϕ = α,θ = γ,α,γ)cos(γ−θ0) =

Gb (ϕ = 0,θ = 90°,α,γ) sinγ

(5)

1 + sinθ0 2

(6)

Gr ,αθ0,αγ = Gr (ϕ = α,θ0,α,γ) = (Gb (ϕ = 0°,θ = 90°,α,γ) + Gd (ϕ = 0°,θ = 90°,α,γ)) ρg

1−sinθ0 2

(7)

Given that shadows occur only in sunny weather, and for the sake of simplicity, only two weather situations have been taken into account, and, consequently, only two solar charts have been obtained (Fig. 2): sunny weather, where the solar energy is due to beam, diffuse and reflected irradiation, and cloudy weather, where solar energy comes from diffuse and reflected irradiations. As a result, for each sun position, defined by its azimuth angle α and elevation angle γ, the following equations for irradiation on a plane with azimuth ϕ and elevation θ can be obtained:

(1)

where the azimuth and elevation angles, ϕ and θ, of the plane are defined with respect to a normal vector to that plane. In this paper two types of trackers are considered: dual-axis trackers which are always facing the sun position (therefore ϕ = α and θ = γ) and single-axis trackers which are oriented with the same azimuth that the sun (therefore ϕ = α and θ = θ0 constant). Firstly, the irradiance on a dual-axis tracker, where the beam irradiance is normal to the tracker plane, so ϕ = α and θ = γ, can be calculated from solar models [29] (see Appendix A) and measurements of diffuse, reflected and beam irradiance at a weather station. The resulting values depend on azimuth angle α and elevation angle γ of the sun position which can be obtained from horizontal values (ϕ = 0° and θ = 90°) that are typical measurement values in weather stations [30]:

Gb,αγ,αγ = Gb (ϕ = α,θ = γ,α,γ) =

Gb (ϕ = 0°,θ = 90°,α,γ)cos(γ−θ0) sin(γ)

Gd,αθ0,αγ = Gd (ϕ = α,θ0,α,γ) = Gd (ϕ = 0°,θ = 90°,α,γ)

The irradiance on a plane depends on its position and the sun azimuth angle α and elevation angle γ. Thus, the global irradiance G in that plane is composed of beam or direct irradiance Gb, diffuse irradiance Gd and reflected irradiance Gr:

W / m2

(3)

Gr ,αγ,αγ = Gr (ϕ = α,θ = γ,α,γ) = (Gb (ϕ = 0°,θ = 90°,α,γ) + Gd (ϕ = 0°,θ 1−sinγ = 90°,α,γ)) ρg (4) 2

2. Irradiance and irradiation on PV-trackers

Gϕθ,αγ = Gb,ϕθ,αγ + Gd,ϕθ,αγ + Gr ,ϕθ,αγ

1 + sinγ 2

Hsunny,ϕθ,αγ = (Gb,ϕθ,αγ + Gd,ϕθ,αγ + Gr ,ϕθ,αγ )Δtsunny .αγ Hcloudy,ϕθ,αγ = (Gd,ϕθ,αγ + Gr ,ϕθ,αγ )Δt cloudy,αγ

(8)

In this paper, solar charts represent the irradiation incident on the plane of the tracker, and not in the horizontal plane, as is usual. Consequently, the values of Gb, Gd and Gr will be determined depending on the type of tracker, on the relative sun position and on data measured at the meteorological station. In Fig. 2, the irradiation charts calculated using (8) are presented for sunny weather and cloudy weather based on measurements from the weather station at Vigo-Campus [30] (latitude 42°, Spain) for a two-axis

(2)

Fig. 1. Diagram of the proposed process for the evaluation of the energy produced by a PV-park when mutual shadows are considered.

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Fig. 2. Irradiation charts under sunny weather (top) and cloudy weather (bottom) at Vigo (Spain) for a dual-axis PV-tracker.

Fig. 3. Geometry analysis of the mutual shadows between dual-axis PVtrackers.

Fig. 4. Geometry analysis of the mutual shadows between single-axis PVtrackers.

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Fig. 5. Configuration of a PV-array starting from PV-cells and bypass diodes for thin film modules (top) and for crystalline Si modules (bottom).

Fig. 6. Forward and reverse bias I-V curves of CIS and m-Si cells.

Table 1 Parameters of CIS and m-Si PV-cells. CIS 2

I0 (A/cm ) IL0 (A/cm2) Rs (Ω cm2) Rsh (Ω cm2) n Vbr (V) m α s (cm2) KI (A/cm2 K) KV (V/K) V0c0 (V) KP (%/K)

m-Si −10

9.3 · 10 26.8 · 10−3 3.5 1200 1.25 −4 3.8 0.35 125 × 0.8 35 · 10−7 2.38 · 10−3 0.555 −0.6

Fig. 7. PV curves of a CIS cell at different irradiance (G) and temperature (T) values.

5.5 · 10−9 32.7 · 10−3 0.5 1000 1.5 −15 3.8 0.35 10 × 10 17 · 10−8 2.22 · 10−3 0.654 −0.45

As discussed above, the energy received by a tracker under sunny weather conditions depends on many variables: the solar elevation, the precipitable water vapor (PWV), altitude, suspended particles, pollution, etc. There are many theoretical models that attempt to approximate the values of the direct, diffuse and reflected irradiances as a function of these and other parameters [29,31–35]. From these models, it is possible to determine the relationship between the global irradiance and the direct, diffuse and reflected irradiances on a plane for sunny days (Appendix A). Direct irradiance depends mainly on solar elevation, the orientation of the plane, altitude and PWV. The diffuse irradiance is related to the direct irradiance, elevation plane and PWV. Finally, the reflected irradiance depends on global irradiance, elevation plane and albedo. In Appendix A, PWV values are presented for 20 cities in Spain and Portugal [34]. The indirect irradiance coefficient rG (relationship between indirect irradiance Gd + Gr and global irradiance

tracker (grayscale is proportional to the solar energy). The data needed to obtain these charts can either be obtained from weather station measurements or be estimated using celestial mechanics equations in which the weather conditions must be considered [29]. 437

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Fig. 8. I-V curves (left) and P-V curves (right) for a CIS module at shading coefficient values from σ = 0 (fully shaded) to σ = 1 (no shaded).

MPP power in W

y=h

x=w

Shaded area ratio (x×y)/(w×h) for a PV-array Fig. 9. MPP values versus the shaded area ratio ξ on a PV-module with dimensions L × D.

Fig. 11. MPP values versus the shaded area ratio on a PV-array.

3.1. Modeling irradiance conditions under partial shading The partial shading on a PV-array is modeled as a non-uniform distribution of the irradiance on it. In this paper, for modeling purposes, a shadowed model has been considered that relates the shadow area dimensions to the solar irradiance. From (1), the shadow effect over a PV-cell can be modeled by means of the equation:

Gσ = σ·Gb + Gd + Gr

where σ is the shading coefficient (σ = 0 when the shadowed area covers all of the cell and σ = 1 when there is no shadow casted on the cell) which only affects beam irradiance. The shading coefficient σ can be obtained from:

Fig. 10. Simulation of the shadowed area on a PV tracker.

G) is considered to be approximately 10% for dual-axis and single-axis trackers (see details in Appendix A):

rG =

(Gd + Gr ) ≈ 0.1 G

(10)

σ = 1−

(9)

S∩s s

(11)

where s is the area of the PV-cell, S is the complete area of the casted shadow on the PV-tracker (S = x · y) and, finally, S∩s represents the intersection between S and s (see Fig. 5).

Thus, for the sake of simplicity, when numerical results are given, the irradiance for shaded areas is considered to be equal to 10% of that received by sunny areas. In addition, it is assumed that the maximum solar irradiance under sunny conditions is G0 = 1000 W/m2.

3.2. Geometry of the shadow casted by nearby PV-trackers The geometry of the mutual shadows in a PV-park formed by singleaxis and dual-axis trackers is analyzed in this section. Dual-axis trackers are characterized by having two rotation axes to allow the tracker to face the sun at any position. The relative position between two trackers can be defined using the distance between them (see d in Fig. 3) and the relative azimuth of the shading tracker with respect to the shaded one (see α0 in Fig. 3). As an initial step to obtain the geometry

3. Analysis of the mutual shadows between PV-trackers In a solar park, the shadows casted between nearby trackers are the main cause of energy losses. In this section, the irradiance values for shadowed areas are depicted. Furthermore, the geometric characteristics of the shadowed area are presented. 438

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Elevation (º)

Fig. 12. Efficiency coefficient ηαγ for a PV-tracker in cylindrical coordinates.

Azimuth (º)

u 0 = d·sin(α−α 0) proj (O′) ≡ (u 0,v0) → ⎧ ⎨ ⎩ v0 = −d·cos(α−α 0)·sin(γ)

Energy coefficient ȘĮȖ

Considering that the tracker size is WxH, the dimensions of the cast shadow are non-zero when −W ⩽ u 0′ ⩽ W and −H ⩽ v 0′ ⩽ 0 . Those dimensions (see x and y in Fig. 3) can be easily obtained as the intersection of the rectangles of equal dimensions (W × H) whose centers are O and the projected point proj(O′), so:

x = W −abs (u 0) y = H −abs (v0)

Fig. 13. Efficiency coefficients for different irradiances and temperatures of a tracker with m-Si modules.

and dimensions of the shadowed area on a tracker, the projection, along the sun direction, of the shading tracker center O′ on the plane defined by the shaded tracker must be calculated. The resulting projection direction is normal to the plane defined by the tracker because it faces the sun. Therefore, the coordinates of the projected point proj(O′) with respect to the center O of the shaded tracker are (see Fig. 3):

u 0 = d·sin(α−α 0) v0 =

−d·cos(α − α 0)·sin(γ) cos(γ − θ0)

Fig. 14. Annual mean temperature chart from Vigo (Spain).

mean temperature (ºC) 25

80 20

70

Elevation (º)

60 15 50

ºC 40 10 30 20

5

10

50

100

150

200 Azimuth (º)

250

(14)

In these conditions, the projection direction is not normal to the plane of the trackers, although, similar to the case in the previous paragraph, the shaded area is a rectangle whose dimensions (see x and y

90

0

(13)

This result implies that the shadow has a rectangular shape whose dimensions are defined by x and y. A similar analysis can be made for single-axis trackers. The single-axis tracker considered in this paper is supposed to have a fixed elevation angle θ0 and a variable azimuth angle ϕ equal to the sun azimuth ϕ = γ. In this case, the projection of the center O′ of the shading tracker on the plane defined by the shaded tracker (Fig. 4) can be calculated using:

Shaded area ratio for a PV-tracker

0

(12)

0 350

300

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4.3. Partially shaded PV-module characterization

in Fig. 4) can be obtained using (13) and (14).

Partial shading of a PV-module means that some PV-cells may be totally or partially shaded. In this section, the behavior of a PV-module with rectangular shadows is described. For PV-modules it is defined a rectangular shaded area ratio ξ as:

4. Electrical model for a shadowed PV-tracker In this section, it is presented as the electrical model for a shaded PV-tracker, the geometry of the shaded area and the irradiance model have been analyzed in previous sections. The model starts from a PVcell model under shadowed conditions, and ends with an electrical model for the entire tracker. The efficiency coefficient and the annual energy production have also been obtained, and the impact of temperature and irradiance on these values have been analyzed. The I-V curves of partially shaded PV-cells, sets of N PV-cells, PVgroups, PV-modules, PV-strings and the PV-array of a tracker have been obtained using the discrete method proposed in [26].

ξ=

Usually, a PV tracker has only one PV-array installed on it (see Fig. 5). For analysis purposes, this PV-array is assumed to be composed of PV-strings connected in parallel, with each PV-string formed by PVmodules connected in series. Internally, each PV-module is composed of one or more PV-groups connected in series. Finally, a PV-group is formed by PV-cells connected in series and, typically, one parallel bypass diode [39]. As an example, thin film solar modules (CIS, a-Si, CdTe, etc.) are typically formed of one PV-group (Fig. 5, top), while crystalline silicon modules (m-Si and p-Si) are usually formed of two or three PV-groups (Fig. 5, bottom).

4.4. Partially shaded PV-array characterization To explain the proposed methodology, an example is used of a PV dual-axis tracker formed by 54 CIS PV-modules, whose width (w) is 7.2 m and height (h) is 3.6 m. From an electrical point of view, the PVtracker is formed by one PV-array composed of six parallel PV-strings, each one formed by nine PV-modules connected in series. Finally, the PV-modules are installed horizontally, i.e., with their longest sides parallel to the ground; and the PV-strings are arranged in columns, i.e., parallel to the shortest sides of the tracker (see Fig. 10). As discussed in Section 3.1, the shadows between trackers are always rectangular, similar to those shown in Fig. 10. Taking into account that the MPP value depends on the size of the shadow and how the PVcells are affected, Fig. 11 shows the MPP power values that can be obtained for rectangular shadows with x and y dimensions. These values have been obtained by means of the expression:

4.2. Partially shaded PV-cell modeling Generally, the PV-cell behavior is modeled by means of a non-linear relationship between four variables: current I, voltage V, irradiance G and temperature T. In this paper, a non-linear implicit function is used, where T and G are assumed to be known, and V and I are the variables of the I-V model. The PV-cell model that allows for analysis of forward and reverse biasing is [26,27,12]: q (V + IR s ) ⎞− V nkB T −1⎟

⎠

⎛ + IRs a ⎜1 + V + IR Rsh ⎜ 1− V s br ⎝

(

⎞

)

m⎟

⎟ ⎠

(15)

MPP Parray ,xy = max{Varray,xy Iarray,xy }

The dependence on G and T can be expressed [36–38] by:

G IL = (IL0 + KI (T −Tn )),I0 = G0

q (Voc 0 + KV ΔT ) nkB T −1

(16) 4.5. Efficiency coefficient of a PV-tracker

PV-cells have two working zones: the forward bias region (PV-cell voltage V > 0), as generator, and the reverse bias region (V < 0), as load. As an example, the complete I-V curves, including forward and reverse biasing, of a CIS and an m-Si PV-cell (G0 = 1000 W/m2 and T0 = 25 °C) are shown Fig. 6. The parameters of these cells are given in Table 1. In addition, the P-V curves of the CIS cell under different G and T values are presented in Fig. 7. MPP Finally, the maximum power point (MPP) power Pcell ,G,T of a cell at irradiance G and temperature T can be approximated by: MPP MPP Pcell ,G,T ≈ Pcell,G0,T0

G (1−KP (T −T0)) G0

(19)

In this figure, it can be seen that, for a given shaded ratio (x × y )/(w × h) , several MPP values are obtained, similarly to the results obtained for PV-modules.

Isc 0 + KI ΔT e

(18)

where L × D and l × δ, with 0 ≤ l ≤ L and 0 ≤ δ ≤ D, are the dimensions of the PV-module and its shaded area, respectively. As an example, a shaded CIS module composed of 42 PV-cells is analyzed. Firstly, in Fig. 8, it is shown the I-V and P-V curves that result when an arbitrary number of PV-cells are fully shaded, i.e. ξ restricted to l = L. MPP power values are indicated in Fig. 8 with black points. However, for a given rectangular shaded area ratio ξ different MPP values can be achieved, as shown in Fig. 9. The solution space is delimited by the extreme situations that occur when l = L (lowest MPP values) and δ = D (highest MPP values). For example, in Fig. 9, it can be seen that for ξ = 0.2 the values of MPP range from 5 W to 35 W (see examples in Appendix C). Therefore, it can be concluded that the impact of a shadowed area in the behavior of a PV-module strongly depends on its size and proportions.

4.1. PV-tracker configuration

I = IL−I0 ⎛⎜e ⎝

l×δ L×D

MPP Taking into account the MPP power values Parray ,xy , the efficiency coefficients for a shaded tracker ηxy can be defined as a function of the shadow dimensions (x, y) with respect to the tracker in sunny conditions:

ηxy =

MPP Parray ,xy MPP Parray ,G0

MPP Parray ,G0

(20)

is the power at the MPP under sunny conditions (i.e., where x = y = 0) and G0 = 1000 W/m2. Similarly, efficiency coefficient values ηαγ for a dual-axis PV-tracker in a PV-park can be obtained from (12) and (13) for each azimuth angle α and elevation angle γ as:

(17)

MPP where Pcell ,G0,T0 is the MPP power of the cell under irradiance G0 and temperature T0.

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Table 2 Different PV-tracker configurations.

3

Vertical

27s2p_9x3_h

Annual energy losses (%)

Horizontal

27s2p_3x9_v

2.5

2 18s3p-1x18-v 18s3p-3x6-h 18s3p-3x6-v 18s3p-9x2-h 27s2p-3x9-v 27s2p-9x3-h 9s6p-1x9-v 9s6p-3x3-h 9s6p-3x3-v 9s6p-9x1-h

1.5

1

0.5

0

18s3p_9x2_h

18s3p_3x6_v

0

50

100

150 200 Azimuth (º)

250

300

350

2 1.8

18s3p_3x6_h

Annual energy losses (%)

1.6

18s3p_1x18_v

9s6p_9x1_h

9s6p_3x3_v

1.4 1.2 18s3p-1x18-v 18s3p-3x6-h 18s3p-3x6-v 18s3p-9x2-h 27s2p-3x9-v 27s2p-9x3-h 9s6p-1x9-v 9s6p-3x3-h 9s6p-3x3-v 9s6p-9x1-h

1 0.8 0.6 0.4 0.2 0

9s6p_3x3_h

9s6p_1x9_v

0

50

100

150 200 Azimuth (º)

250

300

350

Fig. 16. Annual energy losses caused by a tracker located at 10 m as a function of the configuration and relative azimuth α0 for dual-axis (top) and single-axis (bottom) with an elevation of 35°.

5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14

4.5

Annual energy losses (%)

4 3.5 3 2.5 2

ηαγ

⎧ x = W −|d·sin(α−α 0)| ⎧ (α−α 0) ⩽ Δα ⎪ ηxy ⎧ if: & d·cos(α − α 0)·sin(γ) = ⎨ y = H −| cos(γ − θ ) | ⎨ 0 ⎨ ⎩ ⎩ γ ⩽ Δγ ⎪ otherwise ⎩1

(22)

r= Δα = asin(W / d),Δγ = asin(H . cos(θ0)/r ) where with: d 2·cos(α−α 0)2 + H 2 + d·H ·cos(α−α 0)·sin(θ0) . In Fig. 12, the efficiency coefficients ηαγ are shown in cylindrical coordinates, for shadows casted by a tracker located at 10 m to the south of the shaded tracker (d = 10 m; α0 = 180°).

1.5 1

4.6. Effects of global irradiance and temperature in energy production

0.5 0

0

50

100

150 200 Azimuth (º)

250

300

The efficiency coefficients ηxy were calculated in (20) considering a global irradiance G0 = 1000 W/m2, a fixed indirect irradiance coefficient rG = 0.1 (9) and T0 = 25 °C. However, depending on the solar elevation, the global radiation varies between 0 and G0, and the ambient temperature varies throughout the day and year; therefore, the delivered power also changes. However, if efficiency coefficients ηxy are calculated keeping rG at a constant value but varying the global irradiance G or temperature T, the results are similar to those obtained with G0 and T0. In Fig. 13, they are shown the efficiency coefficients for the example in Fig. 9 (m-Si modules) with irradiance values equal to 1000 W/m2 and 500 W/m2 (rG = 0.1 and T = 25 °C) and with temperature values equal to 25° and 50 °C (G = 1000 W/m2 and rG = 0.1). The resulting values are very close because the MPP power values of PV-cells vary almost linearly with irradiance and temperature (17). That is, the power a given shadowed area varies with T and G and, therefore, its efficiency coefficient does not change. Accordingly, the

350

Fig. 15. Annual energy losses of configuration “9s6p_1 × 9_v” as a function of the distance d (m) between trackers and relative azimuth α0.

ηαγ =

⎧ ⎧ (α−α 0) ⩽ Δα | ·sin(α−α 0)| ⎪ ηxy ⎧ x = W − d if: & ⎨ ⎨ ⎩ y = H −|d·cos(α−α 0)·sin(γ)| ⎨ ⎩ γ ⩽ Δγ ⎪ otherwise ⎩1

(21)

where Δα = asin(W / d),Δγ = asin(H /d·cos(α−α 0)) . For single-axis PV-trackers, the corresponding equations, from (13) and (14), are:

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Table 3 Rank of configurations in terms of energy losses. 1st – Best

2nd

3rd

4th

5th

9s6p_1 × 9_v 6° 18s3p_3 × 6_v

18s3p_1 × 18_v 7° 9s6p_3 × 3_h

27s2p_3 × 9_v 8° 9s6p_3 × 3_v

27s2p_9 × 3_h 9° 18s3p_9 × 2_h

9s6p_9 × 1_h 10° – Worst 18s3p_3 × 6_h

100 99 98 18s3p-1x18-v 18s3p-3x6-h 18s3p-3x6-v 18s3p-9x2-h 27s2p-3x9-v 27s2p-9x3-h 9s6p-1x9-v 9s6p-3x3-h 9s6p-3x3-v 9s6p-9x1-h shaded area shaded module h shaded module v shaded tracker

Efficiency (%)

97 96 95 94 93 92 91

Fig. 17. Top view of PV-tracker farm for different field geometries.

90

2

3

4

5

6 1/GCR

7

8

9

10

Fig. 19. Efficiency versus Ground Cover Ratio for Hexagonal E-W layout, 42° latitude and multiple configurations Fig. A.1. Relations of Gd and Gr with G for different solar altitudes, for dual-axis tracker (left) and single-axis tracker of elevation = 35° (right).

The annual energy production of a PV-tracker shaded by other nearby PV-trackers depends on the PV-cell technology, the module layout and the connections between arrays, modules and cells. According to (1), (8) and (20), the energy produced by a PV tracker in sunny (Esunny,αγ) and cloudy (Ecloudy,αγ) weather for any sun position given by α and γ can be obtained using:

Esunny,αγ =

Ecloudy,αγ = Fig. 18. Efficiency versus Ground Cover Ratio for multiple layouts and latitudes for the configuration “9s6p_1 × 9_v”.

(23)

This result is important when using energy values from solar charts because the energy values are not associated with irradiance values. So, taking into account the linearity of power with temperature (17), the power value can be calculated as a function of temperature T as: MPP MPP Parray ,xy = η xy (1−KP (T −T0 )) Parray,G

(1−KP (Tm,αγ−T0))ηαγ Hsunny,αγ

G0 MPP Parray ,G0

G0

(1−KP (Tm,αγ−T0)) Hcloudy,αγ

(25)

(26)

where the use of the annual mean temperature Tm,αγ of each sun position (Fig. 14) is possible due the linearity between power and temperature shown in (24). The efficiency coefficient ηxy is not taken into account when calculating the annual energy in cloudy weather conditions Eαγ,cloudy because no shadows are produced; therefore, the irradiance is uniform over the tracker. Finally, the annual electrical energy Eannual produced by a PVtracker is:

MPP power Parray ,xy of a shaded tracker can be calculated from the sunny MPP power Parray ,G , without having to know the irradiance G or the temperature T: MPP MPP Parray ,xy = η xy Parray,G

MPP Parray ,G0

Eannual =

∑

(Esunny,αγ + Ecloudy,αγ ) (27)

α,γ

(24)

The annual energy losses Lannual, in p.u., can be written as:

∑

4.7. Evaluation of annual energy production

Lannual = 1− MPP The efficiency of the tracker is defined by Parray ,G0 / G0 and from (17) MPP is also Parray,G / G , and can be approximated using the PV-module efficiency given by the manufacturer multiplied by the number of PVmodules in the tracker. Therefore, the energy generated by a PV-tracker can be obtained from the efficiency of the tracker, the solar energy values (8) and the annual mean temperature.

(ηαγ Hsunny,αγ + Hcloudy,αγ )

α,γ

∑ α,γ

(Hsunny,αγ + Hcloudy,αγ ) (28)

5. Application of the proposed method for planning purposes Now that the simulation methodology has been presented, it can be used 442

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to plan PV-tracker parks, i.e., to evaluate annual energy productions for different PV-tracker configurations. For example, the energy output of a PVpark can be obtained for different installation conditions: electrical configurations, relative position between trackers, arrangement of trackers on the ground, dimension of the trackers, etc. In the following paragraphs, the influence of each of the main relevant installation conditions is presented. For the shake of simplicity, in this section the PV-technology is assumed to be CIS.

5.3. Field geometry of PV-tracker park

5.1. PV-tracker configurations

To compare the efficiency (1 – annual energy losses) of the abovementioned layouts, the Ground Cover Ratio (GCR) has been chosen. GCR is defined as the ratio of tracker area to total ground area [7,21]. As a result, Fig. 18 shows the efficiency of the configuration “9s6p_1 × 9_v” (the highest efficiency) with CIS PV-modules for the four arrangements and three latitudes, versus the GCR. As seen, the efficiency of hexagonal configurations is greater than the efficiency of squared and diagonal configurations. The latitude of the solar park is another key parameter to determine the energy losses in a PV-tracker park. In Fig. 19, the resulting efficiencies are presented for all configurations in Table 2 with the hexagonal E-W arrangement and a latitude of 42°. As an example, for a value of 1/GCR = 4 (103.68 m2 of ground area for each tracker), the efficiency varies from 93.5% to 95.5%. Similarly, for an efficiency of 95%, the value of 1/GCR varies from 3.8 to 4.8. This is equivalent to a ground area of 98.5 m2 to 124.5 m2 per tracker.

The effect of the geometry of PV-tracker layouts is analyzed in this section. The geometries analyzed are as follows (see Fig. 17):

• Squared • Diagonal • Hexagonal E-W • Hexagonal N-S

To test the proposed methodology, it has been applied to 10 configurations for a tracker with 54 CIS modules (see Table 2, where the red lines in the figure have been used to show the modules that form each string). The configurations are defined by the horizontal or vertical module layout, the electrical connection—in six, three and two strings of 9, 18 and 27 modules, respectively—and the different distributions of modules in each string in the tracker. The notation used for the different configurations is (see Table 2):

ns s np pnf x nc o where

• n number of PV-modules of each PV-string • n number of PV-strings in each PV-array • n number of rows of a PV-string • n number of columns of a PV-string • orientation of PV-modules (h: horizontal, v: vertical) s

p f

c

5.4. Planning a solar park The results of planning a solar park in Vigo (Spain) with dual-axis trackers (Fig. 10) of CIS PV-modules are presented, assuming 2450 sunny hours per year, Gh = 1510 kWh/year m2 and a mean daily temperature between 5° and 26 °C. The irradiance on the trackers is 1918 kWh/year m2 (1727 kWh/year m2 for sunny weather and 191 kWh/year m2 for cloudy weather), and the energy generated by each tracker, if there are no shadows, is E = 4297 kWh/year. If 1/GCR = 3 (78 m2 of land per 26 m2tracker) is considered, shadow losses represent between 8.5% and 15% of the energy generated, depending on the layout and configuration selected (Table 4). Accordingly, with the proposed method the best configuration and layout can be selected for a given PV-technology and location.

The resulting MPP power values for different tracker configurations and different PV technologies can be seen in Appendix B, using the configurations shown in Table 2.

5.2. Effect of the relative position between trackers In this section, the performance of PV-trackers based on their relative position is studied. The first case analyzed (Fig. 15) represents the annual energy losses for different distance d values and different relative azimuth α 0 values for PV-trackers with configuration “9s6p_1 × 9_v”. By means of these results, the best configuration and distance between PV-trackers can be selected to minimize losses. The second case presents the annual energy losses for PV-trackers with CIS PV-modules separated by a distance d equal to 10 m with different values of their relative azimuth α 0, for the 10 configurations presented in Table 2 (Fig. 16). In Table 3, the configurations are presented in order from best to worst in terms of energy losses.

6. Comparison with simplified models for tracker shadowing In studying the impact of shading in PV systems, it is usual to make certain simplifications [21]. They can be summarized as:

• Shaded area: the power generated is proportional to the shaded area

Table 4 Best and worst configurations for each layout in terms of energy. Layout

Configuration rank

Configuration

Eannual (kWh/y)

Eannual/E (%)

Lannual (kWh/y)

Squared

Worst Best

18s3p_3 × 6_h 9s6p_1 × 9_v

3683.8 3831.6

85.73 89.17

613.2 465.4

Diagonal

Overall Worst Best

18s3p_3 × 6_h 9s6p_1 × 9_v

3655.5 3813.6

85.07 88.75

641.5 483.4

Hexagonal E-W

Worst Overall Best

18s3p_3 × 6_h 9s6p_1 × 9_v

3812.2 3933.4

88.72 91.54

484.8 363.6

Hexagonal N-S

Worst Best

18s3p_3 × 6_h 9s6p_1 × 9_v

3777.6 3902.9

87.91 90.83

519.4 394.1

The overall worst and best values are marked in bold letters.

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trackers is taken into account. The model presented considers the following aspects: shadow modeling, modeling of entire PV installations that are partially shaded, determination of the energy and temperature charts and the associated losses due to shading. The proposed model allows results to be obtained that can be used in optimization models for planning of parks and to study different scenarios considering the impact of shading. Some crucial aspects have been identified for obtaining the maximum energy: the layout of the PV modules, the electrical configuration, the distribution of the modules of each circuit, the layout of trackers in the field, the size of trackers and the number of axes of the trackers. The results show that energy losses of a shaded tracker can be reduced by more than 40% with an appropriate selection of the configuration of the electrical connection, the arrangement of the modules in each series and the orientation of the modules. Similarly, the layout of the trackers on the ground and the distance between them is crucial to the efficiency of a PV park. For example, the energy losses of the hexagonal layouts of trackers are 10–20% lower than those for the squared or diagonal layout with the same GCR. By addressing these aspects, an increase of approximately 7% in the electric annual energy can be achieved, without major changes to the elements of the PV park.

of the tracker.

• Shaded module h: modules in horizontal arrangement and partially shaded modules are considered to be fully shaded. • Shaded module v: modules in vertical arrangement and partially shaded modules are considered fully shaded. • Shaded tracker: the tracker is considered fully shaded when any module is shaded.

The proposed method is compared with the above-mentioned simplifications; the results of that comparison can be seen in Fig. 19. The “Shaded area” simplification gives the highest efficiency values; therefore, it is the most optimistic method when losses are evaluated. On the other hand, the “Shaded tracker” simplification overestimates the losses; therefore, the efficiency obtained is much lower than those obtained with the proposed method. The “Shaded module h” simplification gives higher efficiency values than those found with the proposed method. The “Shaded module v” simplification behavior is similar to the “Shaded module h” simplification, although it gives values that are not always higher than those given with the proposed method. In conclusion, the simplifications tend to overestimate efficiency, except for the “Shaded tracker” case, whose results are not realistic. This make necessary to use an accurate method, like the one proposed in this paper, in order to obtain “realistic” values. 7. Conclusions

Acknowledgments

In this paper, a new methodology is presented for a complete energy evaluation of photovoltaic-tracker parks where the shading between

This work was supported in part by the Ministry of Science and Innovation (Spain) under contract ENE 2009-13074.

Appendix A. Estimation of irradiance The direct normal irradiance Gbn at the earth’s surface on a clear day applying the The Beer-Bouguer-Lambert law is:

Gbn = ae−b/sinγ

(A.1)

where a is the apparent extraterrestrial irradiance, and b the atmospheric extinction coefficient (a function of the PWV), and they can be estimated using [32] or its revised values [31]. The diffuse irradiance for clear days can be expressed using the dimensionless parameter c [33] or its revised values [31]: (A.2)

Gdh = cGbn

where Gdh is the diffuse irradiance falling on a horizontal plane under a cloudless sky. In Fig. A.1, the values of Gd and Gr with respect to G are shown as functions of γ, with ρ = 0.2 and c = 0.136. The value of (Gd + Gr) represents a reduction of 12% of the global irradiance G. In [33], c values are between 0.57 and 0.136 with PWV values between 8 and 28 mm. Revised values for c between 0.103 and 0.138 are presented in [31] with the same PWV values. In this paper, the c value selected is 0.136 because the daily mean value of PWV from 20 Iberian meteorological stations is between 7 and 32 mm [34] (Fig. A.2).

0.14

0.14

0.12

0.12

0.1

0.1

0.08 0.06

Gd / G

0.08

Gr / G

0.06

Gd / G Gr / G (Gd + Gr) / G

(Gd + Gr) / G 0.04

0.04

0.02

0.02

0

0 0

20

40 Solar altitude (º)

60

80

0

20

60 40 Solar altitude (º)

80

Fig. A.1. Relations of Gd and Gr with G for different solar altitudes, for dual-axis tracker (left) and single-axis tracker of elevation = 35° (right).

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Precipitable Water Vapor (mm)

35 30 25 20 15 10 5

2

4

6 month

8

10

12

Fig. A.2. Daily mean value of PWV for 20 Iberian weather stations [34] Fig. C.1. I-V curves of shaded m-Si cells.

Appendix B. MPP power values for different tracker configurations (See Tables B.1 and B.2).

Table B.1 MPP power values for configurations with horizontal and vertical arrangement of a tracker with 54 CIS PV-modules.

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E. Díaz-Dorado et al. Table B.2 MPP power values for configurations with horizontal and vertical arrangement of a tracker with 54 m-Si PV-modules.

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Appendix C. Behavior of the tracker components In Fig. C.1, they are shown the I-V curves for an m-Si cell for the following xc and yc pairs: (0, 0), (wc, hc), (xc, yc), (xc, hc) and (wc, yc), where xc = 0.6 · wc and yc = 0.3 · hc. Furthermore, MPP power values can also be calculated from the number of shaded cells, equivalent to δ/D, and the length of shadow on the cell defined by l/L, as seen in Fig. C.2. The same analysis performed on an m-Si module gives a very different power result (Fig. C.3). This is because the cells are connected along a zigzag path, with three bypass diodes. MPP In Fig. C.4, the MPP power values Parray ,xy are shown for different dimensions of the shadow (x and y). The resulting plots for several PV-tracker MPP configurations are shown in Appendix B. When m-Si PV-modules (Table 1) are considered, the resulting MPP power values Parray ,xy for different shadow dimensions can be seen in Fig. C.5. The differences between Figs. C.4 and C.5 are due to the internal configuration of each module: power, number, size and geometry of the cells, and the connectivity and the number of bypass diodes. In this case, the tracker is 7.9 m width and 5.9 m height.

Fig. C.1. I-V curves of shaded m-Si cells.

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Fig. C.2. MPP power values for rectangular shadowed areas on a CIS module formed by 42 cells Fig. C.3. Power values of MPP’s for rectangular shadows of a m-Si module of 6 × 12 cells.

4 (0,0) (w ,h )

3.5

c

c

(x ,yc ) c

3

(xc ,hc ) (w ,y ) c

Current (A)

2.5

c

2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4 Voltage (V)

0.5

0.6

0.7

0.8

Fig. C.3. Power values of MPP’s for rectangular shadows of a m-Si module of 6 × 12 cells.

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Fig. C.4. Power values at the MPP point for different shadow dimensions of tracker with CIS modules Fig. C.5. Power values at the MPP point for different shadow dimensions of tracker with m-Si modules.

Fig. C.5. Power values at the MPP point for different shadow dimensions of tracker with m-Si modules.

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