*2.3. Inverter*

Inverters are electric and electronic equipment developed to transform DC into AC. The inverters interconnected to the electric grid need to consider the grid electric characteristics such as voltage and frequency.

According to the Mexican standard NOM-001-SEDE-1999, the inverter nominal capacity could be between 75% to 80% of the PVS nominal capacity, because of temperature losses, electric wiring, shadowed and mismatch of the system. The inverter input voltage in PVS interconnected to the grid must be higher than 100 VDC. Additionally, it is recommended to use a maximum interconnection voltage in AC of 13% above the grid's nominal voltage.

The inverter selected is the model Advanced Solar Photonics: PV240-277V and 3 inverters are needed for the PVS. The characteristics of the inverter are presented in Table 3.


**Table 3.** Inverter characteristics.

The inverter efficiency curve is presented in Figure 5.

**Figure 5.** Inverter efficiency.

Figure 5 shows that the selected inverter has 98% efficiency. In this figure, three curves can be seen: the nominal DC voltage (Vdco), or design input voltage; the minimum MPPT DC voltage (MPPT-low), the manufacturer-specified minimum DC operating voltage; and the maximum MPPT DC voltage (MPPT-hi), the manufacturer-specified maximum DC operating voltage.

#### *2.4. Financial Model*

The financial model used calculates financial metrics of the power project based on a project's cash flows over an analysis period studied. The financial model uses the system's electrical output calculated by the performance model to calculate the series of annual cash flows.

According to Short et al. [49], the financial metrics are defined based on the definitions and methods as follows:

The present value (PV) analysis is a measure of today´s value of revenues or costs to be incurred in the future. PV is considered an important financial variable because it shows the cost assumed in moment zero. It is defined by Equation (9).

$$\text{PV} = \frac{1}{\left(1 + \mathbf{d}\right)^{\text{n}}} \tag{9}$$

where d is the annual discount rate, and n is the number of periods studied.

Internal rate return (IRR) is commonly used for many accept or reject decisions because it allows for a comparison with a minimum acceptable rate of return that presents an opportunity cost of capital. It is calculated through iterations until PV cash flow is equal to zero.

A simple payback period (SPB) is the number of years necessary to recover the project cost of investment under consideration and can be worked out by using Equation (10).

$$\sum\_{\mathbf{n}} \Delta I\_{\mathbf{n}} \le \sum\_{\mathbf{n}} \Delta \mathbf{S}\_{\mathbf{n}} \tag{10}$$

where Δ*I* is the non-discounted incremental investment cost, and Δ*S* is the non-discounted sum value of the cash flows net annual costs.

Benefit/cost ratio (B/C) shows whether, and to what extent, the benefits of a project exceed the costs. The B/C ratio is expressed by Equation (11).

$$\text{B}/\text{C} = \left(\frac{\text{PV (all Benzefits)}}{\text{PV (all Costs})}\right) \tag{11}$$

where PV (all Benefits) is the present value of all positive cash flow equivalents, and PV (all Costs) is the present value of all negative cash flow equivalents.

#### *2.5. Sensitivity Analysis*

According to Helton [50] and Saltelli et al., [51] a sensitivity analysis (SA) is a typical measure used to quantify the impact of parameter uncertainty on overall simulation/prediction uncertainty. Evaluating the two indices requires calculating the mean and variance in the parameter space, and this is always done by using Monte Carlo (MC) methods. The quasirandom sampling method is the most computationally-e fficient one among the existing MC methods. Following Saltelli et al. [52], the mean and variance are evaluated with Equations (12) and (13),

$$V\_{\partial\_{\neg i}} \Big( E\_{\partial\_{\neg i}} (\Delta | \theta\_i) \big) = \frac{1}{n} \sum\_{j=1}^{n} f(B\_j) \big( f \big( A\_{B,j}^i \big) - f \big( A\_j \big) \big) \tag{12}$$

and

$$E\_{\partial\_{-i}}\Big(V\_{\partial\_i}(\Delta|\theta\_{\sim i})\Big) = \frac{1}{2\pi} \sum\_{j=1}^{n} \Big(f(A\_j) - f(A\_{B\_j}^i)\Big)^2 \tag{13}$$

where **Δ** = *f*(.) denotes a model execution for its parameters. The calculation requires two independent parameter sample matrices, **A** and **B**, with the same dimension of *n* × *d*, where *n* is the number of samples and *d* is the number of parameters. Matrix *A<sup>i</sup> B* is the same as matrix **A,** except that its *i*th column is from the *i*th column of matrix **B**. Suscript *j* denotes the *j*th row of the corresponding matrix [53].

#### *2.6. Capital Asset Pricing Model (CAPM)*

The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for assets. CAPM has been developed by Sharpe [54], Lintner [55] and Ferreira et al. [56]. This model estimates the cost of equity, which allows for comparison among businesses with an economic rationale for calculations [57]. The CAMP equation is widely used for calculating the expected returns of an asset and can be expressed by Equation (14).

$$ER\_i = R\_f + \beta\_i (ER\_m - R\_f) \tag{14}$$

where *ERi* is the expected return of an asset, *Rf* is the risk-free rate, β*i* is the beta of the investment (according to NASDAQ:FSLR the beta of the solar stock is 1.84), *ERm* is the expected return of the market (Mexican *ERm* is 8.00%), and (*ERm – Rf*) is the market risk premium. The Mexican risk-free rate is taken from the Bank of Mexico and is equal to 7.00%, β*i* is a measure of a stock's risk given by measuring the fluctuation of its price changes relative to the overall market; the market risk premium represents the additional return over and above the risk-free rate.

#### *2.7. Levelized Cost of Electricity (LCOE)*

The levelized cost of electricity (LCOE) has been used to determine the USD per megawatt-hour (\$/MWh) cost of PVS over the life of the capacity [57] and may compare different scenarios. According to Perkins [58], the expression used to calculate LCOE is given by Equation (15).

$$\text{LCOE} = \frac{\sum\_{t=1}^{n} \frac{I\_l + M\_l \quad + F\_t}{(1+r)^t}}{\sum\_{t=1}^{n} \frac{E\_t}{(1+r)^t}} \tag{15}$$

where *It* is the invested capital, *Mt* is the operating and maintenance costs, *Ft* is the solar photovoltaic feedstock, *Et* is the energy delivered to the grid (MWh/yr), *n* is the lifetime of the project, and the discount rate is given by *r*. The Mexican Central Bank shows the value of the discount rate of 7.25% [59].

#### *2.8. Emission Analysis*

Greenhouse gases (GHG) include water vapor, ozone (O3), carbon dioxide (CO2), nitrous oxide (N2O), methane (CH4), nitrous and several classes of halocarbons. GHG allow solar radiation to enter the Earth's atmosphere but prevent the infrared radiation emitted by the Earth's surface from escaping. Instead, this outgoing radiation is absorbed by the GHG and then partially re-emitted as thermal radiation back to Earth, warming the surface [60].

According to the "National Inventory of Emissions of Greenhouse Gases and Compounds (INEGYCEI)" that presents the National Institute of Ecology and Climate Change (INECC) in accordance with Article 74 of the General Law of Climate Change, Mexico emitted 683 million tons of carbon dioxide equivalent (MtCO2e) of GHG in 2018.

The Inventory is an instrument that allows for knowing the emissions of Mexico that originate from human activities throughout the national territory. It is a fundamental exercise in designing emission reduction policies, understanding the main sources and the role that ecosystems play in capturing part of these emissions.

Mexico is conducting an inventory, in accordance with scientific and technical criteria established by the Intergovernmental Panel on Climate Change (IPCC), which is a signatory of the United Nations Framework Convention on Climate Change (UNFCCC).

The most relevant gas emitted by Mexico is carbon dioxide with 71% of emissions, followed by methane with 21%. According to total emissions, 64% corresponded to the consumption of fossil fuels; 10% originated from livestock production systems; 8% came from industrial processes; 7% were issued for waste management; 6% for fugitive emissions from oil, gas and mining extraction, and 5% were generated by agricultural activities. In the inventory 148 MtCO2e absorbed by the vegetation were also counted, mainly in forests and jungles. The net balance between emissions and removals for 2018 was 535 MtCO2e. It was estimated that in 2018, 112,240 tons of this short-lived climatic forcer was generated, which has negative effects on public health.

#### **3. Results and Discussion**

#### *3.1. Photovoltaic Output Generator*

The PV generator is the group of modules connected in parallel before the interconnected boxes. The output generator depends on air temperature; the natural degradation of semiconductors via the photoelectric process; the orientation and solar tilt; dust; and shadows.

The output PV is calculated under the Mexican standard NMX-J-643/1-ANCE-2011 related to photovoltaic power. The PV modules are composed of semiconductors but have some differences because they present some variations in electric parameters which are dependent on the air temperature. In order to define how the temperature impacts these PV electrical parameters, it is necessary to know the thermal coefficients: the thermal coefficient of maximum power (γ, gamma); the thermal coefficient of open-circuit voltage (β, beta), and the thermal coefficient of short circuit current (<sup>α</sup>, alpha). Table 4 shows the typical values of thermal coefficients in different PV technologies.


**Table 4.** The thermal coefficients.

The thermal output is calculated by Equation (16).

$$\text{Thermal}\_{\text{output}} = 1 + (\mathbf{y} \cdot \Delta\_T) \tag{16}$$

With this information, it is possible to forecast the variation of temperature on the module during its use. Table 5 presents the average monthly temperature in Queretaro.

**Table 5.** The average monthly temperature in Queretaro.


In this study, solar panels of the type mono-Si Advance Power API-M330 are evaluated. In Figure 6 is shown the current (Amps) and voltage (Volt) relationship of the module. The PVS general efficiency is calculated considering the inverter and conductors efficiency, resulting in 72.5%.

**Figure 6.** The photovoltaic (PV) panel amperes and voltage relationship.

The module characteristics at the reference conditions are presented in Table 6.


**Table 6.** Module characteristics.

#### *3.2. Solar Resource*

The irradiance recorded each month in Queretaro is shown in Figure 7.

**Figure 7.** Monthly solar radiation in Queretaro.

Once knowing the consumption, the solar resource must be analyzed. In Table 7 is shown the irradiance in Queretaro.


**Table 7.** Average monthly irradiance in Queretaro.

Considering the efficiencies and solar resource, it is possible to calculate the PVS peak power (PVSpp) by using Equation (17).

$$\text{PVS}\_{\text{PP}} = \frac{\text{Daily energy consumption}}{\text{solar resource} \times \text{PVS efficiency}} \tag{17}$$

Then, the PVSpp is 21.08 kW.

The comparison between the electricity delivered to the grid and the electricity load each month is shown in Figure 8.

**Figure 8.** Electricity to the grid and electricity load.

Figure 8 presents both electricity delivered to the electric grid and the electricity load (CNC machine) per month. It can be appreciated that in June, July, and August, the electricity load is higher than the electricity delivered by PVS. This is due to higher temperatures in these months in Queretaro. This variation coincides with Bilcik et al. [14], who found that photovoltaic modules depend on climatic conditions, and as can be seen, in Queretaro these months are the warmest of the year. The electricity delivered to the grid behaves as solar radiation; even if the CNC machine works at night, the period will be short. In Queretaro, the relative humidity is 44%; in this case, this variable does not influence the PVS performance, as exposed by Arikan et al. [21].

The monthly energy production by month is presented in Figure 9. March, April, and May are the most energetic months.

**Figure 9.** Monthly energy production in Queretaro.

#### *3.3. Financial Assessment*

The financial analysis has been done considering the following summary of PVS data: a capacity factor of 20.4%, initial costs 46,575 \$/kW, operation and maintenance (O&M) costs (savings) of 569 \$/kW-year, an electricity export rate-monthly of 0.10 \$/kWh, electricity exported to grid 21.5 MWh, and electricity export revenue of \$2,145, as in the work done by Al-Najideen et al. [20], they study a PVS of 56.7 kW grid connected analyzed under the initial costs and the payback period.

RETScreen has been used in different assessments. Its implementation allows it to determine the viability of PVS. Its financial model allows it to calculate from some input parameters (e.g., discount rate and debt ratio) the output items of financial viability such as IRR, SPB, and NPV. Several authors have assessed solar photovoltaic projects using the RETScreen software obtaining important results, as done by Yendaluru et al., [38] who analyzed the techno-economic feasibility of an integrating grid-tied solar PV plant in a wind farm, or Islam et al., [39] who evaluated an LED system.

In general, given the discount rate, a positive net present value indicates an economically-feasible project, while a negative net present value indicates an economically-infeasible project. It is important to evaluate the NPV along with other metrics, including capacity factor or IRR, and this is expressed as Equation (18). All these parameters allow the project decision-maker to consider various financial parameters.

$$\text{NPV} = \sum\_{n=0}^{N} \frac{\mathbb{C}\_n}{(1 + \text{d}\_{\text{real}})^n} \tag{18}$$

where Cn is the after-tax cash flow in year *n* for the residential and commercial models, *N* is the analysis period in years, and dreal is the real discount rate, because this rate excludes inflation effects.

The financial parameters obtained are presented in Table 8.


**Table 8.** Financial parameters.

As can be seen in Table 8, the parameters indicate that in 1.8 years, the cash flow will be positive. Another very interesting indicator is the cost-benefit ratio, which means that benefits are 6.8 times higher than costs.

Figure 10 shows the cumulative cash flow, which represents the net pre-tax flows accumulated from year 0. It represents the estimated sum of cash that will be paid or received each year during the entire life of the project.

The results of the sensitivity analysis are presented in this section. Tables 9 and 10 show what happens if the electricity price and machine hours parameters, as well as electricity exported to the grid and solar irradiance, vary, respectively.

In Tables 9 and 10, a sensitivity analysis is presented between electricity price (\$)-machine hours (h) and electricity exported to grid (\$)-solar irradiance (kWh/m<sup>2</sup>/day).



Shaded amounts indicate the best scenario if the price increases and bolded are the optimal prices.


**Table 10.** The sensitivity analysis between solar irradiance and electricity price.

Shaded amounts indicate the best scenario if the price increases and bolded are the optimal prices.

A what-if analysis is presented in Tables 9 and 10. In Table 9 it can be observed that the electricity price increases if the hours of the machine increase as well, so that with PVS, money can be saved even if the machine works up to 7 hours per week. In Table 10, the analysis is done between solar irradiance and electricity exported to the grid. As can be seen, if the solar irradiation increases, the price of exported energy does, so that, if a CNC machine works more than 5 h per day and there are more than 6.1 kWh/m<sup>2</sup>/day, the PVS will contribute to saving money.

The CAMP result is 8.84%, which is the expected return of the asset. It is bigger than the Mexican discount rate, which is 7.25%.

LCOE analysis shows that the cost of utility electricity is 15.5 cents/kWh, and with this price the energy produced by PVS will reduce future costs, compared with the cost of an electricity tari ff that is 5 USD per kWh/month [61].
