**3. PV System Output Calculation**

In general, there are several ways to calculate the power output of PV systems. We used in this study a simple method for calculating it [31]:

$$P\_{m,i} = \;P\_{rel} \times \frac{I\_{m,i}}{I\_{LTC}} \times \left(1 + \gamma \left(T\_{sen,i} - 25\,^\circ \text{C}\right)\right) \times PLF \tag{1}$$

where *Pm* is power output of the PV system, *Prel* is the rated PV system power (the output power of PV device under standard test conditions), *Im* is the measured solar irradiance, *IUTC* = 1000 W/m2, *Tsen* is the module temperature (in ◦C), γ is power temperature coefficient, and PLF is the power loss factor.

The equation contains the temperature coefficient to take into account the drop of sensor signal because of the temperature and to correct the testing conditions. The losses because of inverter and the degradation mechanisms of the PV sensors (0.5%/a) are included in Equation (1) as a PLF, which is time dependent because of the degradation of sensors.

#### *3.1. Load Profile*

The power generation profiles were calculated by using the Equation (1). A synthesized dataset of actual measured load profiles provided by HTW Berlin [32] is used to simulate a household's consumption pattern of electricity. The data set consists of 74 load profiles of German single-family houses with a temporal resolution of 1 min for every day of the year. The load profile used for the calculations is the average of six selected profiles which have an annual consumption between 3900 kWh and 4055 kWh. The average profile has an annual electricity consumption of 4006 kWh (Figure 2). It can be assumed that the selected profiles represent a four-person household.

**Figure 2.** Six private household profiles which have an annual consumption between 3900 kWh and 4055 kWh [32]. The average profile (black curve) has an annual electricity consumption of 4006 kWh.

#### *3.2. Economic Parameters*

Feed-in tariffs are the most common policy instrument worldwide to support renewable energy. Many PV installations sell their power at local grid, and the majority of feed-in tariff contracts are at a fixed price per kWh for 10–20 years [33]. This results in an optimal orientation that is the same for both maximum economic yield and maximum energy production. The German FIT for solar photovoltaic uses varying rates depending on the size of the project. Countries in which the FIT was eliminated usually replace it by net metering schemes. The net metering is also used in many different countries under different rules, but consists of a system in which the excess electricity injected into the grid can be used at a later time to compensate the consumption when PV generation is not sufficient. The compensation usually covers a specific period (usually 1–3 years) depending on the country's regulations, and any excess energy after this period is not remunerated. So, the main idea is to configure the system settings in a way its annual production does not exceed the annual consumption, minimizing the deviation between them and increasing SC. Examples of countries using net metering schemes are: the United States (with particular conditions depending on the state), Denmark, Greece, Australia, Brazil, Mexico, and Chile [34–36].

The FIT used in the financial model for the calculation is 10.64 Ct/kWh (from July, 2019) and the price is constant for 20 years. The electricity price (30.22 Ct/kWh) considered in the calculations in this study represents the average price level for private households in Germany in 2019, including taxes and levies [37]. The increase of electricity price is expected to slow down to 2% p.a. as an average value during the next 20 years. The levelized cost of PV energy (LC) in northern Germany ranges between 9.89 Ct/kWh and 11.54 Ct/kWh, depending on the annual solar irradiance [38]; a value of 10 Ct/kWh is used in this study.

In the design of PV systems, the self-consumption rate (SC) and the degree of autarky (AD) are two important quantities used to assess the congruence of the PV generation and electricity demand profiles. The self-consumption rate is defined by the ratio of PV directly used (PDU) to the total amount of PV power generated (Pm), according to Equation (2).

$$\text{SC} = \frac{P\_{\text{DU}}}{P\_{\text{m}}} \tag{2}$$

The degree of autarky is defined as a ratio of PV directly used to the total consumption by the household [39], according to Equation (3).

$$\text{AD} = \begin{array}{c} \text{P}\_{\text{DU}}\\ \text{L} \end{array} \tag{3}$$

where L is the energy consumed by the loads.

The electricity price *PE* used to evaluate the economic impact of PV system at specific orientation has been calculated according to Equation (4).

$$P\_E = (P\_G - P\_{Fi}) + LC \tag{4}$$

where *PG* is the grid electricity price, *PFi* is the FIT, and LC is the levelized cost of PV energy.

Figure 3 shows a workflow diagram used in this study to calculate the SC with the feed-in components. The calculations are always dependent on the consumption of electricity, with the primary objective to fulfil the demand from the PV produced energy, before purchasing from the public grid. If the produced electricity exceeds the consumption of the house, the excess is supplied to the public grid. Moreover, the internal rate of return (IRR) for all available orientations has been calculated over the life cycle of the PV system (20 years) in order to enlighten prospective owners/investors of rooftop PV systems. The IRR, defined as a discount rate that makes the net present value from all cash flows from a project equal to zero, is used to evaluate the attractiveness of a project or investment, and it is probably one of the most meaningful metric for investors [40]. The degradation mechanisms of the PV collectors (0.5%/a) and an annual increase of electricity price (2%/a) were taken into account in the IRR calculations.

**Figure 3.** Schematic view of the calculation of system components. The calculations are always dependent on the load demand, with the primary objective to fulfil it from the PV produced energy, before purchasing electricity from the public grid.
