Energy and economic analysis of a point-focus concentrating photovoltaic system when its installation site varies

Abstract

The concentrating photovoltaic (CPV) systems are a promising technology to obtain clean energy. However, these systems are not equally convenient worldwide due to different climatic conditions. The main aim of this paper is to analyze energy and economic performances of a point-focus CPV system for a residential user when its installation site varies. Three locations, Riyadh, Copenhagen, and Palermo, characterized by very different weather conditions are chosen. A model that links the electrical power of a triple-junction (TJ) cell with its temperature and concentrated radiation incident on it is experimentally developed to evaluate the energy performance of the CPV system. A comparison of the three localities for typical winter and summer sunny days indicates that the higher values of the TJ cell temperature are reached in summer, over 70°C at Riyadh, and its electrical power is reduced compared to a winter day. In winter, a TJ cell in Riyadh supplies an electric power of about 20% higher than that in other two cities, while in summer, the maximum power is observed at Copenhagen. On the contrary, the electrical producibility also depends on the sunlight daily hours number during the year. Hence, considering the real distribution of direct normal irradiance (DNI) and the environmental temperature for each locality, a CPV system composed of modules of 90 cells adopted for a residential user is sized. The electric producibility of the CPV system, by varying its module number, is evaluated for different localities together with the optimal number of the modules which is able to maximize the investment profitability.

Graphical abstract

Keywords

CPV system / point-focus / experimental model / energy and economic analysis

1 Introduction

The growing global demand for primary energy and the increasingly concern of climate changes are undeniable realities [1]. Because of the high dependency on burning of fossil fuels, CO₂ emissions in atmosphere are increased to an alarming level. In the period from 2000 to 2011, CO₂ emissions have been increased from 23.7 Gt to 31.2 Gt and a 43.1 Gt increase is expected in 2035 [2]. As a consequence, there is an urgent need for the development of clean energy sources that are renewable and cost-effective to decrease the dependence on fossil fuel and CO₂ emissions [3]. A good alternative is the use of solar energy that has a high potential thanks to its great availability [4]. The simplest method of solar energy utilization is its direct conversion into electricity by using solar cells [5]. The effective way to reduce the limitations of the traditional photovoltaic (PV) systems such as high cost semiconductor [6] and low conversion efficiency, is to adopt concentrating photovoltaic (CPV) systems [7]. The CPV systems use optical devices such as mirrors or lenses which are able to convey the solar radiation on smaller multi-junction (MJ) cells and to obtain a higher electrical power due to their higher electrical conversion efficiency [8]. Despite the higher costs of MJ cells when compared with the conventional solar cells, a careful selection of the type of concentrator optics and of its concentration factor can determine energy and economic advantages [9]. However, since the optics has to focus sunlight on the cells, these systems can work only with the solar radiation direct component [10] and cannot obtain electricity from diffuse radiations [11]. The rates of diffuse and direct radiation depend on the atmospheric condition of a specific zone [12]. In some regions, the diffuse radiation is a significant part of solar radiation which can affect the electrical producibilty of a CPV system. The electrical performance of a CPV system are also highly affected by the operation temperature of the triple-junction (TJ) cell, which depends both on the levels of concentration and on the environmental temperature [13]. The electrical performances of such systems could be lower when they operate in hotter weather regions (for example Portugal and Algeria) with respect to cold regions (for example Sweden and Norway), other conditions being equal [14].

In recent years the scientific community has proposed several typologies of CPV systems and analyzed their performances in different parts of the world. For example, in Ref. [15], the performances of a two-stage square parabolic CPV receiver dish with an overall geometric concentration ratio of 500 suns have been studied for the city of Chennai (India). In Ref. [2], the long-term performances of a CPV system with Cassegrain reflectors and Fresnel lenses have been analyzed for one year under tropical weather conditions of Singapore. In Ref. [16], the operation of a CPV system for a residential user located in Salerno (Italy) has been studied. In Ref. [17], the authors have monitored the electrical producibility of different CPV power plants at different locations in Castilla La Mancha region (Spain). A high application of CPV systems is also presented in China, as demonstrated by the several studies conducted. At Jilin (China), the energy performances of a novel CPV system based on compact linear Fresnel reflector concentrators have been analyzed [18]. An energy analysis of a hybrid solar CPV/concentrating solar power (CSP) system at Beijing, has been performed in Ref. [19]. Another interesting CPV/CSP system has been studied in Ref. [20], showing its great potential to increase the utilization ratio of solar energy. However, such studies limit the analysis of the performance of CPV systems to a restricted area, without investigating the influence of different climate conditions worldwide from an economical and energy point of view.

The CPV systems cannot be equally convenient worldwide. In fact, the electrical producibility of a given CPV system and its convenience with respect to a traditional PV system are strictly dependent on the climate conditions of its installation site, DNI during the year, environmental temperature, number of hours of sunlight, etc.

As it is known, there is not a standard configuration on the market of CPV systems, but it must be defined according the user characteristics. A point-focus configuration is particularly suitable for small size users, since it allows to achieve higher values of concentration thus requiring small areas for the CPV system installation.

Hence, the aim of this paper is to analyze the energy performances of a point-focus CPV system and its cost effectiveness when its installation site varies and this point-focus CPV system is adopted to satisfy the energy loads of a residential user. Three locations characterized by very different weather conditions have been chosen to evaluate the potential of this system in different parts of the world.

2 Experimental plant

The experimental CPV plant, realized at the Applied Thermodynamics Laboratory of the University of Salerno, is shown in Fig. 1. It is a point-focus CPV system where a Fresnel lens focuses solar radiation on the receiver, consistituted by a TJ solar cell and a finned cooling system. The Fresnel lens, made of acrylic material, has a diameter and a thickness respectively equal to 30 cm and 0.4 cm. The TJ solar cell is constituted by InGaP/GaAs/Ge with an area of 10 mm × 10 mm (Table 1). To unify the concentrated solar radiation coming from the Fresnel lens and to improve the optical efficiency, a kaleidoscope is used as secondary optics. The experimental plant has three degrees of freedom; the first two allow the solar tracking by means of two rotation movements in the horizontal plane in order to follow the sun in the azimuth direction and a rotation in the vertical plane in order to follow the sun in the zenithal direction. The third degree of freedom makes possible to vary the vertical distance among the primary optics, located perpendicularly to the sunrays and the receiver. Hence, the solar radiation incident on the TJ cell can be varied by modifying the concentration factor.

Figure 2 shows the plant scheme with the measurement instruments. A pyrheliometer with an accuracy of 2% measures the DNI; two PT100 thermo-resistances with an accuracy of ±0.2°C are used to measure the cell and outdoor temperatures [21]. A variable electrical load, working as maximum power point tracking (MPPT), is connected to the cell while an acquisition data system is adopted for the experimental measurements of voltage, current, DNI, and temperatures. Moreover, the optical concentration factor (C_opt) has been calculated as the ratio between the solar radiation concentrated on the solar cell (R_cell) and the DNI that represents the power flow incident on the optical system. Hence, C_opt becomes independent of the electrical performance of the TJ cell and it depends only on the optical performance of the system. R_cell has been measured by means of a thermal power sensor with an accuracy of ±3% accurately selected. To make comparable the measurements coming from the two sensors above mentioned, an accurate calibration of the thermal power sensor has been defined.

During the measurement of C_opt, it is necessary that the solar radiation concentrated on the TJ cell and on the power sensor are identical. Hence, the TJ cell and power sensor have been mounted in parallel and with the same kaleidoscope in order to have the same power flux (Fig. 1).

3 Evaluation of energy and economic performances

The aim of this paper is to analyze the energy and economic performances of a point-focus CPV system adopted for a residential user, when its installation site varies. Three locations characterized by different weather conditions have been chosen to evidence the potential of these systems in different localities of the world. The characteristics that differentiate the three localities and influence the performances of the CPV system are environmental temperature (T _env), DNI, and daily hours of sunlight. The values of these variables are available for each locality in Ref. [22].

First of all, typical winter sunny and summer sunny days for each locality, which represent the limit situations of working of a CPV system, have been analyzed. Because DNI is extremely variable with the degree of cloudiness, a clear-sky irradiance has been considered for both the days. In this way, the comparison between the various localities is independent of the particular day considered and from its degree of cloudiness.

Subsequently, an accurate evaluation of the TJ cell electrical producibility of a CPV system taking into account the annual hourly distribution of DNI and T _env for each locality has been performed. This analysis is important to define the necessary number of TJ cells necessary to match the electrical load of a given user.

A correct sizing of a CPV system requires an accurate evaluation of its electrical performances when the operation conditions vary. The electrical power supplied by a TJ cell in a CPV system depends on its operation temperature (T _cell) and on the concentrated solar radiation (R _cell) incident on it. The concentrated solar radiation incident on the cell is given by

(1) {{\displaystyle R}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=\mathrm{D}\mathrm{N}\mathrm{I}\cdot {{\displaystyle C}}_{\mathrm{o}\mathrm{p}\mathrm{t}.}

As for T _cell, an experimental relation that links it to the environmental temperature (T _env) and to R _cell has been found. It has been observed that the increase of T _cell with respect to the environmental temperature increases logarithmically with the concentrated radiation, according to Eq. (2).

(2) {{\displaystyle T}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}-{{\displaystyle T}}_{\mathrm{e}\mathrm{n}\mathrm{v}}=A\cdot \mathrm{l}\mathrm{n}{{\displaystyle R}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}+B,

where the coefficients A and B have been experimentally determined.

The electrical power supplied by the TJ cell has been monitored during a long experimental campaign which has covered both cold winter days and hot summer days with different conditions of solar radiation. The experimental activity conducted has made it possible to collect a large amount of data representing different operation conditions of the CPV system. Adopting a black-box modeling approach, a multivariable regression of the measured data, with a significance level of a = 0.05 has been realized in Matlab [23]. Two different relations that link P _el,cell to R _cell and T _cell have been determined.

(3)(4) {{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}={{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}\left({{\displaystyle R}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}},{{\displaystyle T}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}\right)=\{\begin{array}{l}{{\displaystyle A}}_{\mathrm{1}}\cdot {{\displaystyle R}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}+{{\displaystyle B}}_{\mathrm{1}}\cdot {\displaystyle \frac{\mathrm{1}}{{{\displaystyle T}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}}},\hfill \\ \\ {{\displaystyle A}}_{\mathrm{2}}\cdot {{\displaystyle R}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}+{{\displaystyle B}}_{\mathrm{2}}\cdot {{{\displaystyle T}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}}^{\mathrm{2}},\hfill \end{array}

where the coefficients A ₁, A ₂, B ₁, and B ₂ have been experimentally determined. The validity ranges for the two relations are, respectively expressed in Eqs. (5) and (6).

(5)

(6)

As it can be noted in Eqs. (5) and (6), Eq. (3) can be used for low-middle values of the solar radiation concentrated on the TJ cell. On the contrary, Eq. (4) can be used for higher values of R _cell, when the temperatures of TJ cells also reach about 70°C. Hence, once the values of T _env and DNI for each locality [22] and the value of C _opt are defined for the CPV system, it is possible to calculate T _cell (Eq. (2)) and P _el,cell (Eqs. (3) and (4)).

To easily size a CPV system, a modular configuration can be considered. It allows to match the energy load of a specific user varying both number of cells per module and number of modules. The electric power supplied by a single module is given by [16]

(7) {{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{mod}}={{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}\cdot f\cdot {{\displaystyle n}}_{\mathrm{c},\mathrm{mod}}\cdot \left(\mathrm{1}-{{\displaystyle p}}_{\mathrm{p}\mathrm{a}\mathrm{r}}\right)\cdot {{\displaystyle \eta }}_{\mathrm{mod}},

where P _el,cell can be calculated by Eqs. (3) and (4), n _c,mod is the number of cells per module, p _par is a loss factor taking into account the parasitic current losses generated in the module, and h _mod is the module efficiency which takes into account the coupling in series of the cells along a line. Generally, a cell can operate at an efficiency lower than the nominal one and, considering a non-ideal tracking system, a factor f fequal to 0.9 is considered. Hence, the electric power supplied by a CPV system can be calculated as

(8) {{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}={{\displaystyle P}}_{\mathrm{e}\mathrm{l},\mathrm{mod}}\cdot {{\displaystyle n}}_{\mathrm{mod}}\cdot {{\displaystyle \eta }}_{\mathrm{i}\mathrm{n}\mathrm{v}}

where n _mod is the number of modules that constitutes the plant while h _inv is the inverter efficiency.

Once the number of cells per module is defined, it is possible to calculate the number of modules necessary to match the electric load of the domestic user for each locality chosen. Moreover, the monthly electric load of a given residential user has been defined. It is possible to observe that by increasing the number of modules, the electrical producibility of the CPV system can be increased. However, if the number of modules is excessively high, the electrical producibility exceeds the electrical energy needs of the user, thus leading to an oversized system.

The optimal number of modules for each locality can be calculated with the aim of maximizing the profitability of the investment, expressed in terms of NPV (net present value) in Eq. (9).

(9) \mathrm{N}\mathrm{P}\mathrm{V}=-{{\displaystyle I}}_{\mathrm{0}}+{\displaystyle {\sum }_{i=\mathrm{1}}^{UL}\frac{{{\displaystyle \mathrm{C}\mathrm{F}}}_i}{{\left(\mathrm{1}+r\right)}^i}},

where I ₀ is the initial investment, CF _i is the cash flow for the ith year, r is the discount rate, and UL represents the useful life of the CPV system.

The initial investment is given by the cost of the CPV system, which can be calculated as the product between its peak electrical power ( {{\displaystyle P}}_{{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}_{\max }}) and the cost per unit of power (c _u,P), as expressed in Eq. (10).

(10) {{\displaystyle I}}_{\mathrm{0}}={{\displaystyle c}}_{\mathrm{u},\mathrm{P}}\cdot {{\displaystyle P}}_{{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}_{\max }}.

The cash flow for the ith year has been calculated assuming to sell the monthly energy surplus. It is given by the sum of the cost savings for the purchase of the electricity needed by the user (CS _i) and the gains from the sale of the surplus energy (G _i), as expressed in Eq. (11).

(11) {{\displaystyle \mathrm{C}\mathrm{F}}}_i={{\displaystyle \mathrm{C}\mathrm{S}}}_i+{{\displaystyle G}}_i,

where CS _i is given by the product between the unit purchase cost of electricity (c _e) and the monthly rates of the electrical producibility of the CPV not exceeding the user monthly energy needs, as expressed in Eq. (12).

(12) {{\displaystyle \mathrm{C}\mathrm{S}}}_i={\displaystyle {\sum }_{m=\mathrm{1}}^{\mathrm{12}}{{\displaystyle c}}_{\mathrm{e}}}\cdot \min \left({{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}_m}\mathrm{;}\mathrm{\hspace{0.17em}}{{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{U}}_m}\right),

where {{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}_m} and {{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{U}}_m} indicate, respectively, the electrical energy produced by the CPV system and the user electrical energy need for the month m.

The gains from the sale of the surplus energy (G _i) have been then calculated as the product between the monthly energy surplus produced by the CPV system and the sale price to the energy network (p _e), as expressed in Eq. (13).

(13) {{\displaystyle G}}_i={\displaystyle {\sum }_{m=\mathrm{1}}^{\mathrm{12}}{p}_{\mathrm{e}}\cdot \min \left(\mathrm{0}\mathrm{;}\mathrm{\hspace{0.17em}}{{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{C}\mathrm{P}\mathrm{V}}_m}-{{\displaystyle E}}_{{\mathrm{e}\mathrm{l},\mathrm{U}}_m}\right)}.

The analysis conducted on the NPV also makes it possible to evaluate the discounted pay-back period (DPBP). Another index useful in the investment evaluation is the profit index (PI), defined as the ratio between NPV and I ₀, as expressed in Eq. (14).

(14) \mathrm{P}\mathrm{I}={\displaystyle \frac{\mathrm{N}\mathrm{P}\mathrm{V}}{{{\displaystyle I}}_{\mathrm{0}}}}.

4 Results and discussion

The energy and economic performances of a point-focus CPV system depend on the weather conditions of its installation site. Its operation has been analyzed in three different localities, Riyadh, Palermo, and Copenhagen in order to evidence the potential of this system in different zones of the world. These localities have marked differences in terms of environmental temperature, daily hours of sunlight, and weather conditions, which significantly affect the levels of DNI.

At Riyadh (24°46′27.3540′′ N, 46°44′18.9096′′ E), the winter season is very mild, with an average temperature of about 14°C in January. In summer the rainfall is limited and the temperatures are very high, with an average of 36°C in July. Riyadh has a higher DNI in winter because it is very close to the equator; on the other hand, in summer the high environmental temperatures determine a decrease of the TJ cell electrical efficiency without a proper cooling system [24]. However, the desert climate with a low rainfall and cloud cover even in winter and the consequent high levels of DNI make Riyadh a particularly suitable installation site for CPV systems, with a constant monthly electrical producibility.

Copenhagen (55°40′33′′ N, 12°33′55′′ E) has different climatic conditions compared to Riyadh. Its climate is particularly harsh, such as in the northern Europe, with a high cloud cover that negatively affects the solar radiation. On the contrary, the summer situation is optimal thanks to high daily hours of sunlight and low environmental temperatures, which guarantee higher TJ cell electrical efficiencies.

Palermo (38°6′43′′56 N, 13°20′’11′′76 E) has intermediate climatic characteristics between the two localities previously described. The Mediterranean climate is characterized by a very mild and quite rainy winter and a very hot and sunny summer. Hence, the amount of sunshine is high in summer with clear skies, while in winter the climate is variable with periods of not good weather.

These climatic differences between the three above mentioned localities lead to different CPV system operation conditions, which depend on the values of T_env and DNI [22]. In particular, the experimental analysis conducted in this paper has made it possible to observe that the increase of T_cell with respect to the environmental temperature increases logarithmically with the concentrated solar radiation, as exhibited in Fig. 3.

Moreover, the experimental equations (Eqs. (3) and (4)) that link P_el,cell, T_cell, and R_cell have been obtained. Two relations found, valid respectively for low-middle values (a) and middle-high values (b) of R_cell and T_cell, are displayed in Fig. 4.

The coefficients values A and B in Eq. (2) and the coefficients A₁, B₁, A₂, and B₂ in Eqs. (3) and (4) are reported in Table 2 together with the values of R². The values of such parameters have been obtained by means of a regression based on the experimental data collected by the experimental CPV plant described in Section 2. As shown by the values of R², Eqs. (2)–(4) fit the experimental data distributions accurately.

Hence, once the values of T _env and of DNI for each locality are known [22] and once the value of C _opt of the CPV system is defined, it is possible to calculate the average hourly values of T _cell and P _el,cell for each locality referring to typical winter sunny (a) and summer sunny (b) days, as illustrated in Figs. (5) and (6). These trends are related to a value of C _opt equal to 310, which is the maximum value reached by the experimental CPV system.

As shown in Fig. 5, the temperature of the TJ cell at Riyadh is almost always the highest, reaching a peak of about 70°C in the summer day and 50°C in the winter day. However, the temperatures reached by the TJ cell at Copenhagen are rather low, with a peak of only 30°C in winter and 47°C in summer. At Palermo, the T_cell values are included between 40°C in winter and 55°C in summer.

The negative influence of the temperature of the TJ cell on electrical efficiency is clearly depicted in Fig. 6. It can be seen that in a hot summer day, the values of P_el,cell, despite the higher values of DNI, are lower than those in the winter day with maximum values respectively equal to 9.0 W and 9.6 W. The higher decrease is in Riyadh, where P_el,cell decreases from 9.6 W to 7.6 W, followed by Palermo, where there is a decrease of 0.3 W. At Copenhagen, instead, despite a lower efficiency of the TJ cell due to the higher temperature, the increase of DNI from winter to summer is much high, thus leading to an increase of P_el,cell from 8.6 W to 9.0 W. Moreover, in the central hours of the winter day, a TJ cell in Riyadh supplies an electric power of about 20% higher due to the higher values reached by DNI near the equator. On the contrary, in a hot summer day, the maximum power has been noted at Copenhagen, with an average power increase with respect to Riyadh of about 18%.

However, the daily electrical producibility also depends on the daily hours of sunlight. The results obtained for each locality are reported in Fig. 7, where it can be noted that there is an opposite trend between summer and winter in terms of electrical producibility [25]. The variation of daily hours of sunlight during the year varies between Riyadh, very close to the equator, and Copenhagen. In Riyadh, the electrical efficiency reduction due to the high temperatures prevails over the increase of only two daily hours of sunlight passing from the winter to the summer day. On the contrary, in Copenhagen the electrical producibility of the TJ cell in a summer day is about twice that in the winter day because the number of hours of daily sunlight increases from 8 to 16 and the temperature is always low enough during the year. In Palermo, there is an increase of electrical producibility of about 50%.

This analysis has been conducted in order to underline the influence of climatic conditions on the electrical performances of a TJ cell in a CPV system. In this first level analysis, the cloudiness and the consequent reduction of DNI have not been considered in order to have a comparison between the three localities under same conditions. To have an accurate evaluation of the electrical performances of a CPV system during the year, it is necessary to consider the real annual hourly distribution of DNI and T_env for each locality [22]. Starting from these data, a sizing of a CPV system adopted for a residential user for each locality has been realized. The same monthly electrical load of a residential user with five persons (Fig. 8), has been considered for each locality.

An optical concentration factor equal to 310 and a module made up of 90 cells have been considered referring to the CPV system. This value of C _opt represents the maximum that can be achieved by the experimental CPV plant described in Section 2. It is the upper limit that allows the use of the experimental model defined by Eqs. (3) and (4), thus allowing a more accurate evaluation of the energy performance of the CPV system. It is easily achieved by means of the most common optics, even if more sophisticated optical systems make it possible to obtain different values of concentration up to 1000 suns [26].

Considering the annual hourly values of T_env and DNI, it is possible to calculate by means of Eq. (7), the electrical power supplied by a single module (P_el,mod) and then the monthly and annual electrical producibility of the module for each locality. The electrical power and producibility values of a CPV system depend on the number of modules according to Eq. (8). The difference between the yearly electric load of the user and the producibility of the CPV system, by varying its number of modules, is shown in Fig. 9. If the producibility of the CPV system is higher than the electric load of the user, the surplus energy is sold, otherwise it necessary to purchase it from the energy network.

It is important to analyze monthly the difference between the necessary and produced electrical energy by varying the number of modules in the two extreme cases of Riyadh (Fig. 10) and Copenhagen (Fig. 11). In Riyadh, the electrical producibility of the CPV system is not very variable during the year and four modues are sufficient to match the electrical load. A further increase of the number of modules could lead to an overproduction of energy not required by the user.

On the contrary, at Copenhagen, the producibility of the CPV system is extremely variable [27] during the year, with peaks in the summer season and very low values in the winter season. Hence, it is not possible to match the electrical load in each month, but only in the period between April and September. In this case the optimal number of modules is also equal to 4, and a further increase could not be in any case sufficient to match the electrical load in the period of low producibility.

As said in the previous section, the optimal number of modules for each locality can be calculated in order to maximize the investment profitability in terms of NPV, calculated by means of Eq. (9). The values of the parameters necessary for this evaluation are reported in Table 3 [28].

The calculated values of NPV at the 20th year, which represents the average useful life of a CPV system, are shown in Fig. 12 as a function of the number of modules. It can be noted that, for each locality, the value of NPV increases with the number of modules until it reaches its maximum. A further increase of the CPV system size leads to a surplus energy which can be sold to the energy network at a price lower than the purchase price, thus reducing the cash flows.

The optimal number of modules is equal to 4 for Riyadh and Copenhagen, and 5 for Palermo. As s consequence, the value of NPV is also different, assuming its maximum equal to about 11.6 k€ for a CPV system installed at Riyadh where the annual climatic conditions are most suitable. At Palermo, the NPV is slightly lower, with a value of 9.9 k€.

On the contrary, as for Copenhagen, despite the favorable conditions in terms of the temperature of the TJ cell [29], the high number of cloudy days reduces considerably the DNI and then the electrical producibility of a CPV system. Hence, the NPV assumes a very low value, equal to only 3.7 k€.

The NPV trend over the years corresponding to the optimal number of modules for each locality is plotted in Fig. 13.

Moreover, the DPBP of the investment is equal to about 8 years for Riyadh, 10 years for Palermo, and 13 years for Copenhagen. Finally, the profit index (PI) corresponding to the optimal number of modules has been calculated for each locality. The results are given in Fig. 14 where it can be noted that the most profitable investment is related to Riyadh whose PI is equal to about 150%. The investment is still convenient in Palermo with a PI equal to 84%, but not at Copenhagen, since its PI is equal to only 47%.

5 Conclusions

The main aim of this paper has been to analyze the energy and economic performances of a point-focus CPV system applied to a residential user when its installation site varies. Hence, three localities, Riyadh, Copenhagen, and Palermo, characterized by very different weather conditions have been chosen. Riyadh and Copenhagen have opposite climatic conditions, with a desert climate and a particularly harsh climate, respectively; Palermo has an intermediate situation with a Mediterranean climate. A model that links the electrical power supplied by a TJ cell, with its temperature and concentrated radiation incident on it, has been experimentally determined to evaluate the energy performance of the CPV system. Moreover, an equation for the calculation of T _cell as a function of DNI and R _cell has been experimentally developed. Hence, once the values of T _env and DNI is known for a given locality and the value of C _opt is defined, which is equal to 310 in this paper, it is possible to calculate T _cell and P _el,cell.

First, a comparison between the three above mentioned locality for typical winter sunny and summer sunny days has been presented. It has been noted that the temperature of the TJ cell at Riyadh is higher than those of the other two localities, reaching a peak of about 70°C in the summer day. The electrical power of the TJ cell, because of the higher temperatures, in a hot summer day is lower than that of the winter day with maximum values respectively equal to 9.0 W at Copenhagen and 9.6 W at Riyadh. An opposite trend in terms of electrical producibility between summer and winter, due to different daily hours of sunlight between Riyadh and Copenhagen, has been observed. Moreover, considering the real annual hourly distribution of DNI and T _env for each locality, a modular configuration of the CPV system, with a C _opt equal to 310 and a module of 90 cells, has been applied to a residential user. It has been observed that at Riyadh, the electrical producibility of the CPV system is not very variable during the year and 4 modules are sufficient to match the electrical load. On the contrary, the extreme yearly variability of the electrical producibility of the CPV system at Copenhagen makes it possible to match the electrical load only between April and September. Finally, the optimal number of modules has been defined for each locality in order to maximize the profitability of the investment in terms of NPV at the 20th year which is the average useful life for a CPV system. The most profitable investment is related to Riyadh where, with 4 modules, the NPV assumes its maximum value equal to 11.6 k€, with a DPBP of 8 years and a PI equal to 150%. The investment is quite convenient also in Palermo, with a NPV of 9.9 k€, a DPBP of 10 years, and a PI equal to 84%. Finally, the investment is not profitable in Copenhagen, with a NPV and a PI equal respectively to 3.7 k€ and 47%, with a DPBP of 13 years.

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