An approach of the spatial planning of a photovoltaic park using the Constructal Theory

Sustainable Energy Technologies and Assessments 11 (2015) 11–16

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Original Research Article

An approach of the spatial planning of a photovoltaic park using the Constructal Theory Pantelis N. Botsaris Democritus University of Thrace, School of Engineering, Department of Production Engineering and Management, 67100 Xanthi, Greece

a r t i c l e

i n f o

Article history: Received 8 October 2014 Revised 29 April 2015 Accepted 21 May 2015

Keywords: Spatial planning Photovoltaic parks Constructal design theory Energy efficiency

a b s t r a c t In this paper, a study performed on two different methods for a spatial planning of a photovoltaic (PV) park. Special attention is given to the Constructal Theory (CT) design method and the criteria required for the PV park design. Based on the aforementioned theory, a spatial planning applied for a PV park, focusing mainly on the wires transition losses of alternative current (AC). An analytical comparison applied on the efficiencies of the two different park design methods, CT and common. In conclusion, the pros and cons of different types of spatial planning design, depending on the park installed power, examined by the results. The economic performance of the two low-power design parks (<200 KW) were similar. Therefore, the economic performance of the CT design for installed power between 200 KW and 1 MW performed slightly better compared to the economic efficiency of the common. Significant difference in the structural design starts to appear for higher power (>1 MW), therefore it makes sense the investigation of the implementation of the structural design in the PV park. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction The Constructal Theory (CT) derived from the constructal law which was first stated by Bejan in 1997 [3]. Since then, the constructal law used by many applications, in different fields of science characterized by flow, i.e. thermodynamics, biology, architecture, and continues to grow [5,6,9]. According to the constructal law: ‘‘For a finite system to persist over time, it must evolve in such a way that it provides easier access to the imposed currents that transverse it’’. The constructal law examines the design evolution (shape, geometry and pattern) as a natural phenomenon applied both on natural and artificial systems. In particular, the physical systems evolves with time to facilitate flow through changes in geometry and design, reaching the optimal design of artificial human systems [13,14,15]. The most common CT problem involves the flow between a point and a surface or a volume. The current of self-organization of a system to minimize the resistance encountered by the flow, is the tree form, a form that occurs very often in nature, such as the arteries or the muscles of the human body, the river estuaries, etc. [4,7]. The aim of the current study is to attempt energy optimization and thus an economic performance of a PV park through the reduction of the AC losses of the electrical cables. The paper presents the

E-mail address: [email protected] URLs: http://medilab.pme.duth.gr http://dx.doi.org/10.1016/j.seta.2015.05.001 2213-1388/Ó 2015 Elsevier Ltd. All rights reserved.

design of a park with 500 KW installed capacity, both in common and under CT design. It also presents the design of a customized park totaling more than 500 KW of power with an AC losses estimation. A comparison of the two designs approach with economic terms (NPV, IRR) presented and ended with the conclusions of the current work. Design methods for photovoltaic (PV) parks Constant basis common spatial planning of a 500 KW photovoltaic (PV) park A common PV park design with 500 KW-installed capacity accomplished in Chrisoupoli, a town very close to Kavala city in Eastern Macedonia & Thrace region in Greece, with latitude: 40.98° and mean elevation: 22 m (Fig. 1). The common design performed in accordance with existing literature by using chain inverters of 27.5 KW nominal power [8]. Fig. 2 shows the layout of the common design. The green colour symbolizes the photovoltaic panels, the red the number of the solar inverters and the blue the transmission lines. Constructal theory based spatial planning of a 500 KW Photovoltaic (PV) park The key point of the application of the CT design is the definition of the system flow. In the current paper, the flow of the system

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P.N. Botsaris / Sustainable Energy Technologies and Assessments 11 (2015) 11–16

Fig. 3. Fundamental element.

Fig. 1. Kavala regional unit (red block). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is the electric current flow through the park’s transmission wires. The next step is the definition of an objective in order to optimize the system performance. In the PV park many targets are available such as the maximization of the output of the energy production, the minimization of the occupied space etc. The goal of the current project, is the minimization of the electrical losses of the transmission wires and more specifically the AC electrical power losses. The minimization of the AC losses leads both to increased energy production output of the solar park and the reduction of the initial cost. In literature, there are researchers working on this problem, improving the design of the photovoltaic park, focusing on minimizing AC losses by using other methods such as genetic algorithms [10]. The design of the current system based on the aforementioned objective. Some of the imposed restrictions must be taken into account, while designing a system. In the case of the current work, the restrictions adopted were that the frames had specific orientation (south), tilt (30°), and the frame tables spaced far enough apart to avoid shading. The next step of the CT application is the definition of the fundamental component of the system (elementary element). In a photovoltaic park, many different fundamental components selected. The investigation of this paper focuses on the efficiency of the AC circuit. The layout of the park as illustrated in Fig. 2, cover a total surface area of 679.2 m2. The fundamental element consists of 136 photovoltaic panels of 31.28 KW total nominal capacity (2 parallel strings of 67 and 69 modules each respectively); and an inverter of 27.5 KW nominal power. The design of the fundamental

element presented in Fig. 3. The green symbolizes the photovoltaic panels and the red the solar inverters. It is assumed that the fundamental elements were in rectangle shape and positioned next to each other in order to develop the complete system. The inverter assumed to be the center of the fundamental components, because the AC current flow considered to start from there. As mentioned before, one of the limitations is the distance between the bases in order to avoid shading. To succeed this better, the distance between the fundamental elements spaced equally on east–west and north–south axis. Based on the CT design, the matching and the linking of the fundamental elements performed next. In fact, the current flow of the system started from the inverters to the substation and the transformer. For convenience and without affecting the result, the flow of charge considered to take place from a central source (substation) to an area where consumers concentrated. The problem is how to deploy the fundamental elements around the source (transformer) to minimize AC losses. This problem simplified on how to deploy the fundamentals around the transformer by minimizing the length of the wiring, a problem that has already been addressed [11]. This simplification does not change the results, since the electrical losses are proportional to the cross section of the cable, the current and the cable length. The current and the cross section of the application however, considered as given; therefore minimizing the length of the cable causes minimum electrical losses. Applying the CT, the solar park can progressively designed. The first step is the connection of the source, i.e. the transformer, with the fundamental element in way Fig. 4 shows. Next, the second level designed. Generally the number of the fundamental elements that make up each level is given by [11]:

Ai ¼ 4i1 ði ¼ 1; 2; . . .Þ ði ¼ 1; 2; . . .Þ:

Fig. 2. Spatial planning and common design of AC wiring with 27.5 KW inverters.

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P.N. Botsaris / Sustainable Energy Technologies and Assessments 11 (2015) 11–16 Table 1 Common vs CT design for a 500 KW PV park.

Fig. 4. A fundamental design element of the CT.

Therefore, the second level formed by four fundamental elements arranged as shown in Fig. 5. Fig. 6 shows the layout of the pv park according to the CT design, and the Table 1 presents the main technical features of the two different design approaches (common and CT). Afterwards, a financial comparison of the two spatial planning performed, based on the data listed in Table 2. In Table 2 observed that the cost differences in DC wiring between two approaches is smaller compared to the AC wiring, with a final profit of 2.611€ for CT approach. Economic indicators estimated with the RETScreen International Clean Energy Project Analysis Software (http://www.retscreen.net) by taking into account the following assumptions: - Inflation rate 2.5%. - Discount rate 5%.

Technical characteristics

Common (with 27.5 KW inverters)

CT (with 27.5 KW inverters)

Frame power (Wp) Number of frames Total power (KWp) Number of inverters Inverter total power (KWp) Energy output (MWh/year) Performance ratio Ground area Ground perimeter Ground cover ratio DC wiring total length DC wiring cross section AC wiring length AC wiring cross section AC total losses Final energy output (MWh/year)

230 2176 500 16 440 735.8 84.40% 11,529 m2 559 m 27.44% 2788 m 4 mm2 1077 m 5  16 mm2 5.75 KW 727

230 2176 500 16 440 735.8 84.40% 11,511 m2 430.8 m 27.48% 3617 m 4 mm2 731 5  16 mm2 3.9 KW 729

Table 2 Apportioned construction cost of a 500 KW PV Park. Cost

Unit price

Common (with 27.5 KW inverters)

CT (with 27.5 KW inverters)

PV panels Inverters Mounts Substations-Panels Lighting-AlarmCommunication DC wiring cost AC wiring cost Pipes and grounding Enclosure Work Total Miscellaneous Final cost

0.65 €/Wp 0.2 €/Wp 0.22 €/Wp 80 €/KWp 20 €/KWp

325.013€ 100.004€ 110.004€ 40.002€ 10.000€

325.013€ 100.004€ 110.004€ 40.002€ 10.000€

0.76 €/m 9.48 €/m 11 €/m 18 €/m 60 €/kWp

2.119€ 10.204€ 11.770€ 10.562€ 30.001€ 649.679€ 64.968€ 714.647€

2.749€ 6.933€ 8.030€ 8.254€ 30.001€ 640.990€ 64.099€ 705.089€

10% of total

- Life of the project equalled to 20 years. - The financing of the project was through own funds, no grants or loans. - Income tax 20%. Fig. 5. Second level of design of the CT.

The main results listed in Table 3. Observing Table 3, the differences of the two designs are insignificant. The design based on the CT seems to be preferable, however with only slightly better results. Therefore, a benchmarking for parks of greater output attempted in order to examine any potential different results.

Spatial planning for parks greater than 500 KW The allocation of the relevant costs to a park of a higher power output was done by applying some assumptions. The most

Table 3 Key financial results.

IRR before tax IRR after tax Payback time (years) NPV Fig. 6. Spatial planning and AC wiring design based on the CT.

Common (with 27.5 KW inverters)

CT (with 27.5 KW inverters)

8.2% 6.3% 11.1 77.804€

8.4% 6.4% 11 86.521

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P.N. Botsaris / Sustainable Energy Technologies and Assessments 11 (2015) 11–16

Table 4 Fundamental element costs.

Frames Inverters Mounts DC wiring AC wiring Enclosure and grounding Work done Miscellaneous electrical equipment AC losses Annual income

Sym

Common design

CT design

Cm0 Ci0 Cb0 Cdc0 CINV0 Cf0 Ce0 Cp0 Closses E0

650 €/KWp * 31.28 = 20.332€ 200 €/KWp * 31.28 = 6.256€ 220 €/KWp * 31.28 = 6.881.6€ 4.23 €/KWp * 31.28 = 132.31€ 435.97€ 3.266.4€ 60 €/KWp * 31.28 = 1.876.8€ 80 €/KWp * 31.28 = 2.502.4€ 326.95 * 0.12€ = 39.234€ 1471.48 €/KWp * 31.28 * 0.12€ = 5.523.35€

650 €/KWp * 31.28 = 20.332€ 200 €/KWp * 31.28 = 6.256€ 220 €/KWp * 31.28 = 6881.6€ 5.49 €/KWp * 31.28 = 1.71.72€ 218.62€ 2.510.4€ 60 €/KWp * 31.28 = 1.876.8€ 80 €/KWp * 31.28 = 2.502.4€ 162.07 * 0.12€ = 19.45€ 1468.74 €/KWp * 31.28 * 0.12€ = 5.513.06€

Table 5 General manufacturing costs of the photovoltaic park. Frames Inverters Mounts DC wiring Enclosure and grounding Work done Miscellaneous electrical equipment

Cm0 Ci0 Cb0 Cdc0 Cf0 Ce0 Cp0

Cm(n) = n * Cm0 Ci(n) = n * Ci0 Cb(n) = n * Cb0 Cdc(n) = n * Cdc0 p Cf(n) = n * Cf0 Ce(n) = n * Ce0 Cp(n) = n * Cp0

important of those had taken, was that the components cost of the park, increased proportionally with the power output. The disadvantage of the present case was that it did not take into account the scale economies. Therefore, the following analysis begins with the cost estimation of the fundamental elements and extends to the greater power outputs. We remind that the fundamental element consists of 136 photovoltaic panels of 31.28 KW total nominal power, and an inverter of 27.5 KW nominal power. In Table 4, the electricity price ratio is taken equal to 12€/100 KWh and in Table 5, the letter n symbolizes the number of the fundamental elements comprising the pv park.

CINV: Cost of procuring and installing the wiring (€) A: Cross section of the AC wiring (mm2) L: Length of AC wiring (m) n: Number of fundamental elements comprising the photovoltaic park k: Coefficient of purchasing cost and wiring installation in €/mm2. Considering the original calculating price as the purchasing cost and wiring installation for the 500 KW park originally examined, the coefficient calculated equal to 1.28 €/mm2. In the current project case, the AC wiring does not go directly from the inverters to the transformer, but it concentrates first in groups of 4 in subpanels and then it moves towards to the transformer. Therefore, Eq. (1) can transform into Eq. (2):

C INV ¼ k

where in this case: m: The number of different AC wiring groups L: Total length of each wiring group At this point, it is important for the analysis to follow the size needs to be defined. This is called load moment, represented by the letter M and given by Eq. (3) [[1,2]]:

M¼

A. The cost of the wiring installation (acquisition cost + installation costs) CINV B. The cost of the AC losses Closses This analysis assumes that the current density J is constant in different designs. This means, that due to the definition of the current density, the cross section of the cable varies with the electric current. On further analysis of the two terms of AC losses [1,2]: n X A: C INV ¼ k Ai Li

ð1Þ

i¼1

ð2Þ

i¼1

Alternative current (AC) losses The costs associated with the AC wiring, divided into two main categories [1,2]:

m X Ai Li

m m X X Mm ¼ Im Lm ðA:mÞ

v ¼1

ð3Þ

v ¼1

where: I: the current load (A) L: the length of line in-group Table 6 presents the load moment as a function of the fundamental elements with I0 and L0, the elemental current and length respectively. If the load moment of the fundamental element regarded as the fundamental load moment M0 then:

M0 ¼ I0  L0

ðA:mÞ

ð4Þ

From Eq. (4) and Table 6 the formula that provides the load moment, depending on the power of the photovoltaic park, ie the number of inverters, calculated as:

where:

Table 6 Load moment as a function of the number of inverters (or fundamental elements). N 4 16 64 256

P M¼ m v ¼1 Iv Lv P M = IiLi = 4LoIo P P P M = IiLi = I1L1 + I2L2 = 16I0L0 + 4I2L2 = 16I0L0 + 4(4I0)(2L0) = 48I0L0 P P P P M = IiLi = I1L1 + I2L2 + I3L3 = 64I0L0 + 16I2L2 + 4I3L3 = 64I0L0 + 16 (4I0)(2L0) + 4(16I0)(4L0) = 448I0L0 P P P P P M= IiLi = I1L1 + I2L2 + I3L3 + I4L4 = 256I0L0 + 64I2L2 + 16I3L3 + 4I4LL4 = 256I0L0 + 64(4I0)(2L0) + 16(16I0)(4L0) + 4(64I0)(8L0) = 3840I0L0

M 4I0L0 48 I0L0 448 I0L0 3840 I0L0

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P.N. Botsaris / Sustainable Energy Technologies and Assessments 11 (2015) 11–16

M ðnÞ

pffiffiffi ¼ ð n  1ÞnM 0

ðA:mÞ

ð5Þ

After defining the size of the load moment, Eq. (3) transformed into Eq. (6).

pffiffiffi C INV ðnÞ ¼ kM ðnÞ ¼ kð n  1ÞnM ð0Þ

ðA:mÞ

ð6Þ

B. The formula for calculating the term Closses is given by Eq. (7) [1,2]:

C losses ¼ Plosses CPW

ð€Þ

ð7Þ

Xn Plosses ¼ 3 i¼1 I2i Ri

ðWÞ

ð8Þ

Ri ¼ q

Li Ai

ðXÞ

CPW ¼ ETNði; v Þ

ð€Þ

ð10Þ

The coefficient N(i, v) calculated by Eq. (11) [12] as: N1

AN ¼ Að 1 þ k Þ

ð11Þ

where N:

P A

¼

1  ð1 þ kÞN  ð1 þ r ÞN rk

ð12Þ

where: P: Present Value. A: Annual sequence of amounts. k: Coefficient of growth of annual sequence of amounts. In the case under consideration k = 0.01, as a reduction of annual energy production of 1% due to the degradation of the quality of the frames considered. r: Discount rate assumed to be 5%. N: Lifetime investment considered equal to 20 years. Combining Eq. (8) with Eq. (9) results in:

Plosses ¼ 3J q

X

pffiffiffi C AC ðnÞ ¼ C INV þ C losses ¼ ðk þ 3J qÞnð n  1ÞIo Lo

ð14Þ

NPVðnÞ ¼ EðnÞ  C m ðnÞ  C i ðnÞ  C b ðnÞ  C dc ðnÞ  C AC ðnÞ  C f ðnÞ  C e ðnÞ  C p ðnÞ  C l ðnÞ  C d

ð15Þ

where: Cl(n): The annual costs for the maintenance and the insurance of the park taken equal to 1% and 0.5% of the total installation cost of the park respectively. Cd(n): Various other costs of the park installation (licensing procedure, design costs, etc.).

ð9Þ

where: Ii: Current load of each AC line (A) Ri: Resistance of each AC line (X) E: Energy output of the solar park (KWh/KWp) T: Electricity price (€/kWh) N(i, v): Cost allocation coefficient of the annual losses in present value

Nði;v Þ ¼

Therefore, the total cost of the AC losses given by the equation:

pffiffiffi pffiffiffi Ii Li ¼ 3J qnð n  1ÞIo Lo ¼ nð n  1ÞP AC0

ð13Þ

Substituting the above values, the final equation for the common design becomes:

pffiffiffi pffiffiffi NPVðnÞ ¼ 14802:5n  4203:86 n  1013:53nð n  1Þ

ð16Þ

And the equation for the design based on the CT:

pffiffiffi pffiffiffi NPVðnÞ ¼ 14633:2n  3230:88 n  505:62nð n  1Þ

ð17Þ

Design evaluation A performance evaluation index for each design is to find the maximum point of the equation calculating the Net Present Value, with the help of Eqs. (16) and (17). Starting from the Eq. (16), the maximum of the function, calculated by zeroing its first derivative. p Setting the term n equals to x and solving the quadratic equation:

pffiffiffi

x ¼ n10:27 ! n ¼ 105:47

ð18Þ

Therefore, since the number of the inverters is an integer, it can be calculated that the maximum value of the Net Present Value of the common design is obtained for n = 106 (Park Power Output = 3.32 MW) and equals to:

NPVð106Þ ¼ 527:115:21 ð€Þ

ð19Þ

By exactly the same procedure as for the design, based on the CT p and starting from Eq. (17), setting x = n and solving the resulting quadratic equation:

pffiffiffi

x ¼ n19:85 ! n ¼ 394:02

Fig. 7. Comparison of Net Present Value for the two designs.

ð20Þ

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P.N. Botsaris / Sustainable Energy Technologies and Assessments 11 (2015) 11–16

As before, the maximum value of the Net Present Value obtained for n = 394 (i.e. power 12.32 MW) is equal to:

NPVð394Þ ¼ 1:946:273:39 ð€Þ The above analysis confirms that the better performance of the design based on the CT is at higher power (>500 KW), as Fig. 7 indicates (CT with blue line, common design with green line). It should noted that the financial model developed for the two designs has practical significance for a certain power range (0–20 MW). For higher power capacity, the assumption made was that the current density remained constant. This leads to a commercially unavailable cable cross section, because even if they produced, they would have much higher cost than of the cost calculated, and so the economic model would not be precise. This expected, since for larger photovoltaic power outputs, practice shows that the parks design with central inverters. Conclusions and recommendations In this paper, there is a comparison between a common design and a design based on Constructal Theory (CT) for a PV park of 500 KW nominal power. The comparison made with economic criteria (Net Present Value/NPV and endogenous rates of return/IRR) and the results indicate a very small advantage over the CT. For safer conclusions on the applicability of spatial planning of CT, the comparison extended to higher power, and the examination of the design performances in different power. For low power (<200 KW) the two designs present similar economic performance. Therefore, for power range between 200 KW and 1 MW, the economic performance of the design based on the CT was little better compared to the economic efficiency of the common design. Significant difference starts to appear in the structural design for high power (>1 MW), therefore it makes sense the investigation of the implementation of the structural design in a PV park for this case. Another point that should be noticed, is the power at which the Net Present Value set to zero and maximizes for each different design. The Net Present Value (NPV) of both designs does not set to zero for the examined interval (0 < n < 150). This shows that for specific power range, the NPV of the designs is positive according to the hypothesis made (discount rate and investment duration time) and therefore the investment considered to be advantageous. The NPV of the common design gives its maximum value for n = 106 (3.32 MW) and n = 394 (12.32 MW) for the design based on the CT. The maximum value expresses the power in which the NPV maximizes and then it starts to decreases to zero. These results confirm the better performance of the constructal design for higher power. Table 7 presents the Net Present Value estimated for both the common design and the design based on the CT for different installed power of PV parks.

Table 7 Net Present Value for various power. Power

NPV(€) common design

NPV(€) constructal theory

1 MW 2 MW 4 MW 5 MW 10 MW 20 MW

298.651.6 459.457 509.335 426.699 812.328 6.380.066

374.311.1 683.610 1.168.210 1.357.206 1.891.833 1.423.208

At this point it should be noted that the Net Present Values taken from the developed financial model, can be diverged from reality for the following reasons: - It does not take into account the economy scale. For higher values of PV parks, the prices of materials per unit are lower. - At the time of the economy model’s configuration it does not take into account other financial parameters such as income taxation and inflation. However the Net Present Values obtained, considered to be satisfactory for the purpose of the current work. References [1] Arion V, Cojocari A, Bejan A. Constructal tree shaped networks for the distribution of electrical power. Energy Convers Manage 2003;44:867–91. [2] Arion V, Cojocari A, Bejan A. Integral measures of electric power distribution networks: load-length curves and line-network multipliers. Energy Convers Manage 2003;44:1039–51. [3] Bejan A. Advanced engineering thermodynamics. New York: Wiley; 1997. [4] Bejan A, Lorente S. Design with constructal theory. Hoboken: Wiley; 2008. [5] Lingen Chen, Shuhuan Wei, Fengrui Sun. Constructal entransy dissipation minimization of an electromagnet. J Appl Phys 2009;105(9):094906. [6] LinGen Chen. Progress in study on constructal theory and its applications. Sci China Technol Sci 2012;55:802–20. [7] Lingen Chen, Shuhuan Wei, Fengrui Sun. The area-point constructal entransy dissipation rate minimization for discrete variable cross-section conducting path. Int J Low-Carbon Technol 2014;9(1):20–8. [8] Deutsche Gesellschaft fur Sonnenenergie (DGS). Planning and Installing Photovoltaic Systems; 2005. [9] Huijun Feng, Lingen Chen, Zhihui Xie, Zeming Ding, Fengrui Sun. Generalized constructal optimization for solidification heat transfer process of slab continuous casting based on heat loss rate. Energy 2014;66:991–8. [10] Gomez-Lorente D, TrigueroI, Gil C, Estrella A. Evolutionary algorithms for the design of grid-connected PV-systems. Expert Syst Appl 2012. [11] Lorente S, Wechsatol W, Bejan A. Tree-shaped flow structures designed by minimizing path lengths. Int J Heat Mass Transf 2002;45:3299–312. [12] Panagiotakopoulos D. Systemic methodology and economic technical: the dimension of sustainability. Zigos 2005. [13] Reis AH. Constructal theory: from engineering to physics, and how flow systems develop shape and structure. Appl Mech Rev 2006;59:269–82. http:// dx.doi.org/10.1115/1.2204075. [14] Reis AH. Design in nature, and the laws of physics. Phys Life Rev 2011;8:255–6. http://dx.doi.org/10.1016/j.plrev.2011.07.001. [15] Tondeur D, Fan Y, Luo L. Constructal optimization of arborescent structures with flow singularities. Chem Eng Sci 2009;64:3968–82.