In the next sections, the SHyFTA modelling formalism will be used in order to illustrate the model of the photovoltaic system in the simulation. The main results will be presented so as to point out the main differences in terms of performance between the two geographic locations. Moreover, an economic assessment comparing the investment linked to the installation of a PV system, with and without battery, will be presented.
4.1. Stochastic Hybrid Fault Tree Automaton
Stochastic Hybrid Fault Tree Automaton (SHyFTA) is a modelling formalism that belongs to the umbrella of Dynamic Reliability [
14,
15], an engineering science that aims to study a system by the use of an holistic model that is able to consider the physics of the system process and its inter-relationships with the system dependability, (i.e., the probability of a system performing its task under some specifications like operative conditions, time of mission, restoration, maintenance resources, and so forth [
32]).
SHyFTA is a formalism based on the separation of concerns [
17] which allows us to break the modelling of a system process into two inter-dependent sub-models, the deterministic and the stochastic, that are coupled by the mean of shared variables. This simplifies the conception of complex models.
Figure 4 shows the coupling between the deterministic and the stochastic processes.
In the SHyFTA formalism, the deterministic model can be described with the mathematical equations of the system process, whereas the stochastic model takes the form of a Dynamic Fault Tree. The graphical representation of a fault tree is a logic diagram constituted by a Top Event (TE), Basic Events (BEs), and Gates. Following a TOP-DOWN approach, the construction of a DFT is realized by identifying the sequence of events that brings into occurrence the TE. The TE is the undesired scenario of the fault tree. On the other hand, BEs are the leaves of the fault tree and represent the elementary events of a process, generally linked with the failure of the system components. Gates are used to logically interconnect the BEs and/or other intermediary events, originated by the triggering of other gates.
The shared variables allow the coupling of these two inter-dependent sub-models so that a variation of the deterministic dynamic can arouse a change on the parameters of the stochastic sub-model and vice-versa. Typical interrelationships between the deterministic to the stochastic model are the variations of the working conditions that can alter the failure behavior of the system. On the other hand, the most common shared variables of the stochastic model affecting the deterministic dynamics are the components status: If a component gets deteriorated (or even broken), its contribution within the deterministic process is nullified.
To implement a SHyFTA model, the modeler must identify the components that participate to the physical process and realize the DFT schema that, on the other hand, describes the system failure logic of the system. For the energy supply system of the household shown in
Figure 2,
Figure 5 and
Figure 6 present the deterministic (or physical) and the stochastic diagram (the DFT) of the SHyFTA model. As it is shown, the Basic Events of the DFT represent the active components of the physical model.
In this paper, the implementation and resolution of the SHyFTA model was achieved exploiting a software toolbox library (SHyFTOO) running under the Matlab® framework.
4.2. SHyFTA Model of the Household Energy Supply
The SHyFTA model hereby presented depicts the process of energy supply for a generic household equipped with a grid-connectected photovoltaic power plant and a storage system. The deterministic schema of the process, in
Figure 5, allows the identification of the main sub-systems: The photovoltaic power plant (PV Generator), the storage system (BAT), and the equipment of the grid connection coupling (GCC) that allow the coupling with the electrical grid.
In detail, they can be decomposed into the following functional blocks:
PV Module (PVM), made up by ten photovoltaic modules (M1–M10);
Direct Current Section (DCS), made up of string protection diodes (SPR), a DC disconnector (DCD), and a surge protection device (SPD);
Alternating Current Section (ACS), made up of an inverter (INV) and an AC circuit breaker (ACB);
Grid Connector Coupling (GCC), made up of an AC disconnector (ACD), a differential circuit breaker (DCB), and a generic sub-system representing the electrical grid (GRD).
Battery (BAT) that is connected in parallel in the AC section.
As already said in
Section 3, for management reasons, a grid-connected power plant must be connected to the national electrical grid. If the electrical grid fails, the power plant must be disconnected, stopping the production and the energy supply of the household. Moreover, in these cases, the battery is also forbidden to supply energy to the household until the grid has recovered. This scenario corresponds to the top event of the Dynamic Fault Tree shown in
Figure 6. This model is constituted by the Top Event AND gate (TE) that takes as input the output of the OR gate GCC (OR (ACD, DCB, GRD)) and the OR gate PV Fault (OR (ACS, DCS)). The former models any type of disconnection of the electrical grid, whereas the latter the unavailability of the photovoltaic power plant that occurs if the electrical circuit of the PV Generator gets open (any failure of the ACS or DCS components). The AND Gate PVM models the failure of the photovoltaic strings; although the modules are connected in series, the by-pass diodes guarantee the electrical isolation of those modules that are not working properly.
As far as it concerns the battery, it must be pointed that its unavailability does not cause a stop of the household energy supply because the grid can fulfill the energy request. On the other hand, likewise the PV power plant operativity, any disconnection from the electrical grid causes the unavailability of the battery. These behaviors can be modelled with the DFT using a FDEP Gate, taking as primary input the output of the GCC and secondary inputs the BAT and the OR Gate of the PV Fault.
As generally assumed in the literature, failure and restoration of the system components follow the exponential probability density function that model a random failure/repair.
Table 5 shows the parameters adopted. Failures are measured in occurrence per year whereas repairs in occurrence per hour. As for repair rates, it was assumed that electrical components such as breakers and disconnectors can be restored as-good-as-new within 12 h, string box and surge protections within 48 h after fault. A failure of the grid is restored within four hours, whereas more critical components like the inverter, the battery and the modules, according to the agreements with the manufacturers, are repaired within three or four weeks, so as to consider the whole process of inspection, ordering, delivery, and replacement.
The SHyFTA model was coded with the SHyFTOO library under the Matlab
® and Simulink framework. The Matlab script used to create the DFT is shown in
Table 6 (parameters have to be defined in number of occurrences per hour).
The physical process can be developed in Simulink, exploiting the built-in blocks available in the Simulink libraries. Thanks to the SHyFTOO library, the coupling between the physical and the stochastic model was easily realized, exploiting the properties of the SHyFTOO components that were used as shared variables. For instance,
Figure 7 shows the PVM section: Ten modules (M1–M10) contributed to the conversion of the solar irradiation. The generic block “Mi_Status” (i = 1, i = 2, …, i = 10) took as input the status of the module from the stochastic process. If the status of the generic module was not good, its contribution was nullified in the “Interpreted MATLAB Fcn” that implemented the physical equations shown in Equations (1)–(4). It was possible to identify the other inputs of these equations: (1) The simulation clock, (2) the historical series of the solar irradiation, and (3) the ambient temperature. In this way, the equations can compute at any instant of the simulation the power produced by the solar string. Clearly, the physical process is not limited to the previous block but, since the Simulink model is not the main object of this paper, the rest of the blocks are not illustrated.
4.3. Simulation Results
In order to evaluate the impact of the battery on the auto-consumption and on the energy withdrawn from the electrical grid, four main scenarios were simulated and compared, as summarized in
Table 7. The simulations were built upon the cases of study described in
Section 3, characterized by two different geographical locations with the same photovoltaic system (peak nominal power and battery). As already mentioned, the historical time series collected data from 2008 to 2017, therefore the results presented in this section are related to ten years of simulation (corresponding to 87,672 h of operation).
The measures of interest for the two locations were (1) the energy produced by the photovoltaic power plant EPV, (2) the energy transferred to the grid ETG, (3) the energy required and withdrawn by the grid EFG, and (4) the loss of energy production EL, due to the energy supply system unavailability (grid and PV system). It is worth reminding that EPV and EL are independent from the battery.
Figure 8 compares the energy produced for both the two locations. It is noticeable that in the Location 2, the expected production E
PV is higher than in the Location 1; if compared to the energy required by the household utilities, it can be seen that the design peak power (3 kW) would be enough to cover the consumption only in the Location 2 (and not in the Location 1).
Figure 9 shows the production loss E
L due to the unavailability of the energy supply system. These value amounts to about the 2% of the energy produced that can be explained by the very high availability (0.9876) of the energy supply system, that can be computed with the stochastic failure process (e.g., the DFT) of the SHyFTA model.
The energy withdrawn from the grid E
FG, needed to satisfy the utilities consumptions, is shown in
Figure 10 and
Figure 11, respectively, for a PV system with and without the battery. In this case, as expected, the energy withdrawn from the grid is higher in Location 1, although this difference is less evident for the plant configuration without battery (
Figure 11). This last behavior can be explained considering that household consumption are mostly concentrated during the second half of the day (late afternoon to night) when the power plants do not produce in both the locations.
The same trends are shown in
Figure 12 and
Figure 13 but grouped with respect to the location. In both the cases, it is possible to notice that the battery increased dramatically the auto-consumption, reducing the energy request from the grid by about 50% in both the locations.
The energy transferred to the grid E
TG because it was not instantaneously consumed by the household utilities is shown in
Figure 14 and
Figure 15.
Beside the fact that, as expected, the power plant of Location 2 injects to the grid more energy than the power plant of Location 1, an interesting fact is observed in
Figure 16 and
Figure 17. In fact, they show the same trends with respect to the same location but highlighting, in the right axis, the percentage of the E
PV instantaneously transferred to the grid, which gives origin to the E
TG.
Although the percentage of unused EPV is drastically reduced, the energy injected to the grid, ETG, is still very high (approximately around the 40% of the energy produced by the power plant in both the locations). In order to reduce the ETG, the most appropriate way would be the increasing of the storage system capacity. But this option has a cost and, as it will be shown in the next section, the economic viewpoint cannot be disregarded.
4.4. Economic Assessment
As expected, the results of the simulations presented in
Section 4.3 have shown that the installation of the battery increases the auto-consumption and, accordingly, decreases the withdrawal of energy from the electrical grid. Now, the main point to evaluate is whether the installation of the battery is convenient not only in terms of self-consumption but, above all, from an economic point of view.
In a similar work [
11], an economic analysis targeting the Italian market showed that the profitability of an investment of a PV power plant equipped with a storage system is questionable, resulting with a payback time higher than 75 years. These results are very discouraging; therefore, in our study, it was decided to extend the simulation in order to consider a “Project Lifetime” of twenty years and to compare the “Net Present Value” [
6] of two design configurations of a photovoltaic power plant with and without a battery. Clearly, some further assumption was needed and the main characteristics of the Italian PV market were considered for estimating the installation costs of a PV system and the costs of the electrical bill. Besides all, it must be observed that in the last decades the Italian market has been one of the most responsive to the photovoltaic offer, thanks to several government-sponsored subsidies that have included, among them, incentives and tax deductions. Therefore, the evaluations hereby summarized can provide valuable and interesting prompts for understanding the market and, in particular, the potential of storage system technology in domestic applications.
Table 8 shows the costs related to the installation and the contract management of a grid-connected photovoltaic power plant subjected to the net-metering regime (known also as SSP) [
11]. They reflect the costs of an ordinary mono-phase low-voltage contract.
The investment costs are related to the installation of the PV power plant and are sustained one time at the beginning of the investment (at the beginning of the first year). The annual costs refer to the net-metering service subscription (also known as SSP). In
Table 9, the main components of the electrical bill are summarized.
The “Fixed Price of Energy”, “Price for the Metering Unit”, and “Power Costs” are annual costs sustained for the energy supply (contract with the energy provider). They are included and spread in the periodic electrical bill (every month) that also includes the costs linked with the amount of energy (kWh) EFG withdrawn from the grid. This latter is a variable whom unitary cost value depends on the “Energy Cost” plus an “Additional Energy Cost” that is applied as follows: As long the yearly EFG is lower than 1800 kWh, the unitary cost is incremented with the value of the “Additional Energy Cost L” per kWh. Afterwards, the unitary cost per kWh is further increased using the “Additional Energy Cost H” (e.g., the auto-consumption limits these costs). The “Excise Duty” and the “VAT” are additional taxes payed on top of the bill.
Table 10 provides further data needed for the economic assessment.
The “Unitary Fixed Exchange Value” is a mean of the variable components paid in the electrical bill, while the “Energy Trade Price” corresponds with the sales value of the energy produced by the power plant and transferred to the grid (e.g., energy not instantaneously used). These values are used to evaluate the net-metering contribution (or SSP). Since, they are slightly volatile (depends on the energy market), they have been fixed considering the current market scenario. Finally, the “Discount Rate” (r = rb + rr + r’) is defined according to the following economic scenario assumptions:
- -
rb = 0.5%, is the borrowing/lending rate of interest. In our case, it was assumed an economic scenario of liquidity (lending rate).
- -
rr = 1% is an interest rate featuring the industrial plant risk assessment. In our case, it was assumed a low value because PV technology is mature.
- -
r’ = 1% is the liquidity risk premium.
Since the “Project Lifetime” has been set to twenty years, the analysis presented in the previous sections was extended accordingly; therefore, the Monte Carlo simulation was modified so as to shuffle, for the missing ten years, the historical time-series available.
The discounted cash flow method for the evaluation of the NPV is based on the cash-flows generated by an investment. Therefore, the NPV (20 year, 2.5%) was computed for the four scenarios of
Table 11 with the formula:
where C
0 are the costs sustained for the installation and the start-up of the photovoltaic system (at the beginning of the investment) and CFD
i is the ith annual “Cash-Flow Discount” (CFD) generated by the investment. As shown in the equation Equation (8), the ith CFD can be obtained using the corresponding “Cash-Flow” (CF) and multiplying it to the “Present Value Single Payment”. The PVSP
i depends on the “Discount Rate” and provides the discount value of an amount (at the time of the economic analysis) expected at the ith year, by the formula:
Table 11 shows the PVSP (20 year, 2.5%) that was used for the proposed economic assessment.
The CF
i is computed considering the annual revenue R
i and the avoided costs AC
i of the ith year due to the investment. In our case of study, the AC
i was computed using the following formula:
where, ΔST
i (“Saving Taxes”) are the savings from the taxes (in Italy, the economic discipline allows a deduction from the taxes of 50% of the investment within the first ten years of the PV installation) and ΔSB
i (“Saved Bill”) are the savings of the electrical bill:
where, BBI stands for “Bill Before the Investment” and BAB stands for “Bill after the Investment”.
The revenue R
i is a credit provided by the Government to the household owner that depends on the energy exchanged with the grid as ruled by the net-metering regulation [
11]. So, the following formula is used:
It makes use of the information presented in
Table 8,
Table 9 and
Table 10 to compute the following variables:
- -
SSP (credit recognized) = min (OE, CEI) + UFEV × ES;
- -
OE (value of energy withdrawn from the grid) = EFG × Energy Cost;
- -
CEI (value of energy injected to the grid) = ETG × Energy Trade Price;
- -
ES (energy exchanged with the grid) = min (EFG, ETG);
- -
* ΔVES (value of the energy surplus) = CEI − OE, only if CEI > OE (otherwise is 0).
Where AFSSP is the “Annual Fee for SSP” (30 €/y), EFG is the energy withdrawn from the grid and ETG is the energy transferred to the grid.
Table 12,
Table 13,
Table 14 and
Table 15 present the results of the cash-flow analysis. It is possible to observe that the NPV (at the 20th year) of a PV system without battery is higher than the same PV system with battery. These results confirm the one shown in [
11] that investigated the economic profitability of a PV system during the first 10 years of lifetime.
Making a pair comparison between
Table 12 and
Table 13 for Location 1 and
Table 14 vs.
Table 15 for Location 2, it is possible to observe that the main economic benefit brought by the installation of a storage solution is the increase of avoided costs. In particular, during the first ten years, the avoided costs depend on the electrical bill reduction and on the tax deduction (Saving Taxes of the 50% of the PV investment). Afterwards, only the electrical bill reduction contributes to the avoided costs. On the other hand, the economic reward (R) of the net-metering service is higher in a PV plant without battery. This result can be expected because, as ruled by Equation (12), in the PV plant equipped with a battery, the amount of energy withdrawn by (and injected to) the electrical grid decreases, and so does the economic value of the net-metering contribution (SSP) and the contribution ΔVES (value of the energy surplus). Unfortunately, the net-metering mechanism, conceived in 2007 to serve PV power plants without storage systems, does not consider the auto-consumption as an added value and does not reward the usage of the stored energy.
Figure 18 shows the “return of investment” trend. This figure allows the identification of the pay-back time for the four investments proposed in
Table 7, considering respectively the PV system with and without battery in both the locations. In particular, it is possible to notice that the PV system without battery in Location 2 returned during the 9th year of activity, whereas the same configuration for Location 1 took two more years. The PV systems with battery required respectively 13 and 11 years to return in Location 1 and Location 2.
The results so far shown have demonstrated that, in Italy the current market scenario is not yet favorable for the installation of a battery in domestic PV applications. In order to understand what conditions can turn the market in favor of the installation of the battery, a sensitivity analysis was performed.
Figure 19 and
Figure 20 show how the payback time and the NPV change with a variation of the battery cost per kWh (from 500 to 100 €/kWh).
As far as it concerns the payback time,
Figure 19 shows that the cost of the battery must decrease to 200 €/kWh and 150 €/kWh, respectively, for the Location 1 and Location 2, in order to equalize the investment of a PV system without battery. Instead,
Figure 20 reveals that the installation of a storage system starts to become more profitable (than the same PV plant without battery) for where the costs of the battery is lower than 300 €/kWh. In this case, the payback time would occur, respectively in Location 1 and Location 2, at the 12th and at the 10th year from the beginning of the investment.