The apparent movement of the sun across the sky is a phenomenon attributed to the Earth’s rotation rather than the sun’s motion. This motion is a result of two primary rotations of the Earth in space: (1) the yearly orbit around the sun, leading to the apparent north-south movement of the sun, and (2) the daily rotation of the Earth about its axis, causing the apparent east-west movement of the sun across the celestial sphere. These apparent changes in the sun’s position allow for the definition of several key angles:

1. Sun Declination ( δ ): This angle represents the sun’s declination at solar noon relative to the equator, with a range of − 23.45 ∘ ≤ δ ≤ 23.45 ∘ . As illustrated in Figure 1, δ completes one cycle over the course of a year, varying from −23.45˚ on December 21 to 0˚ on March 21, then to 23.45˚ on June 21, and back to 0˚ on September 21. The day-to-day change in δ is minimal. It can be estimated using the formula [18] : δ = 23.45 sin ( 360 284 + n 365 ) . where n represents the day of the year ( 1 ≤ n ≤ 365 ).

2. Hour Angle ( ω ): This angle denotes the angular displacement between the sun and the local meridian, with each hour corresponding to a 15˚ increment or decrement. In the morning hours, ω is negative, while in the afternoon, it is positive, with solar noon having an angle of 0˚. Energy generation occurs only

between sunrise and sunset.

The sunrise and sunset angles are calculated using [18] :

ω s r = − cos − 1 [ − tan ϕ tan δ ] , (2)

and

ω s s = cos − 1 [ − tan ϕ tan δ ] , (3)

respectively, where ϕ represents the latitude of the location ( − 90 ∘ ≤ ϕ ≤ 90 ∘ ), with north latitudes being positive and south latitudes negative.

The range of ω for the daytime period is bounded by ω s r and ω s s :

ω s r ≤ ω ≤ ω s s (4)

However, energy generation occurs only when sunlight directly strikes the front face of solar panels, not when striking the rear face. Consequently, the duration of energy generation may be shorter than the time between sunrise and sunset. The calculation of sunrise and sunset times on the solar panel itself will be elucidated following further definitions in subsequent sections.

Figure 2 illustrates the positioning of solar panels along with the definition of various pertinent angles, as summarized below:

l β : Represents the tilt angle of the solar panels relative to a horizontal surface, with a range of 0 ∘ ≤ β ≤ 180 ∘ . Panels facing downward have β > 90 ∘ .

l γ : Denotes the surface azimuth angle, covering − 180 ∘ ≤ γ ≤ 180 ∘ , where zero corresponds to due south, positive azimuth values indicate west-facing panels, and negative azimuth values represent east-facing panels.

l θ : Known as the angle of incidence, it signifies the angle between the incident solar radiation beam and the normal to the panel’s surface.

For a given solar system installed at a specific location, the angles β and γ remain fixed, while ω and θ exhibit significant variation throughout the daily cycle of Earth’s rotation. Meanwhile, δ remains relatively constant within each day, but its gradual change becomes evident over the course of the year.

Given the nearly constant value of solar irradiance at approximately 1367 W/m², the incident solar irradiance on a panel is governed by the angle of incidence, θ , as determined by the following equation:

I = 1367 × cos θ . (5)

The angle θ is influenced by the movement of the sun across the sky, which encompasses both daily and seasonal components. It can be calculated using the following equation [18] :

θ = cos − 1 [ sin δ sin ϕ cos β − sin δ cos ϕ sin β cos γ   + cos δ cos ϕ cos β cos ω + cos δ sin ϕ sin β cos γ cos ω   + cos δ sin β sin γ sin ω ] . (6)

It is imperative that the solar incident angle on the panel remains θ < 90 ∘ for energy generation to occur. Conversely, if θ > 90 ∘ , the sun will strike the rear face of the solar panel, resulting in no energy generation. To determine the effective duration of sunshine on the panel for energy generation purposes, we utilize Equation (6) with θ = 90 ∘ on the left-hand side and solve for ω (assuming all other variables are known for a specific installation geometry and location). This yields two values for ω : one corresponding to the hour angle when the sun rises on the panel, denoted as ω s r p , and the other corresponding to the hour angle when the sun sets on the panel, denoted as ω s s p .

Energy generation occurs only during the interval:

max { ω s r , ω s r p } ≤ ω ≤ min { ω s s , ω s s p } (7)

The mathematical formulations delineated in Sections 2.1 and 2.2 are grounded on an idealized solar irradiance scenario, forming the basis for all simulations conducted in this study.

In the subsequent sections, we propose four distinct system configurations labeled as SYSTEM A, SYSTEM B, SYSTEM C, and SYSTEM D. Each of these configurations shares the common objective of providing a consistent baseload, defined as the daily energy demand that remains constant throughout the entire year. It’s pertinent to clarify that our focus lies on the aggregated daily demand over a twenty-four-hour period. Thus, we deliberately overlook intra-daily power mismatches between supply and demand to streamline the analysis. This approach is adopted partly due to our belief that the inter-seasonal energy variations in solar energy are primarily responsible for impeding higher levels of photovoltaic (PV) energy generation in the electric grid.

For all the systems delineated below, we employ a standardized PV system size of 1 kW as a basis. Subsequently, we optimize the tilt angle, compute the daily supplied baseload energy, and ascertain the requisite energy storage system size accordingly.

To compute the energy storage system size for SYSTEM A and SYSTEM C, we first define the storage state as:

storage   state = ∫ 0 t ( P ( τ ) − P ¯ ) d τ = ∫ 0 t     P ( τ ) d τ − P ¯ t . (12)

Here, P ( t ) represents the time function of the generated power from the panel, and P ¯ denotes the average power generation, which, in this analysis, equals the baseload power. Consequently, the required storage size is calculated as:

storage   size = max { storage   state } − min { storage   state } (13)

In this section, we undertake an economic comparison of SYSTEM A, SYSTEM B, SYSTEM C, and SYSTEM D. The Levelized Cost of Electricity (LCOE) is utilized as the metric for comparison, defined as [20] :

LCOE = system   cost baseload   energy   supplied = PV   system   cost + storage   system   cost baseload   energy   supplied (14)

Here, the PV system cost and the storage system cost are given by:

PV   system   cost = PV   system   size × PV   price   per   kW = 1 × PV   price   per   kW (15)

Storage   system   cost = storage   system   size × storage   price   per   kWhr (16)

Let’s abbreviate the following:

PV price = PV   price   per   kW (17)

storage price = storage   price   per   kWhr (18)

Storage A = storage   system   size   for   SYSTEM   A (19)

Storage C = storage   system   size   for   SYSTEM   C (20)

For SYSTEM A, (15) and (16) become

PV   system   cost = PV price (21)

Storage   system   cost = storage A × storage price (22)

For SYSTEM B, (15) becomes

PV   system   cost = PV price (23)

For SYSTEM C, (15) and (16) become

PV   system   cost = PV price (24)

Storage   system   cost = storage C × storage price (25)

For SYSTEM D, (15) becomes

PV   system   cost = PV price (26)

Additionally, let’s define the following baseload energy ratios:

Energy X/Y = baseload   energy   supplied   by   SYSTEM   X baseload   energy   supplied   by   SYSTEM   Y (27)

where X and Y could be A, B, C, or D.

Figure 3 illustrates the relationship between tilt angle and latitude across the four system configurations. The corresponding tilt angles are detailed in Table 2, alongside additional performance metrics such as baseload supplied, required storage, and average curtailment.

In this section, we will derive mathematical models describing the optimal tilt angles for the four discussed systems. These models consist of piecewise linear functions that closely approximate the optimal tilt angles.

β ^ A = { 1.5 ϕ , 0 ∘ ≤ ϕ ≤ 30 ∘ 2.2 ( ϕ − 30 ) + 45, 30 ∘ ≤ ϕ ≤ 50 ∘ 90 ∘ 50 ∘ ≤ ϕ ≤ 90 ∘ (39)

β ^ B = { 0.85 ϕ , 0 ∘ ≤ ϕ ≤ 45 ∘ 0.44 ( ϕ − 45 ) + 38.25, 45 ∘ ≤ ϕ ≤ 66.6 ∘ Not   applicable 66.6 ∘ ≤ ϕ ≤ 90 ∘ (40)

β ^ C = { 0.85 ϕ , 0 ∘ ≤ ϕ ≤ 45 ∘ 0.44 ( ϕ − 45 ) + 38.25, 45 ∘ ≤ ϕ ≤ 90 ∘ (41)

β ^ D = { 1.56 ϕ , 0 ∘ ≤ ϕ ≤ 45 ∘ 0.92 ( ϕ − 45 ) + 70.1, 45 ∘ ≤ ϕ ≤ 66.6 ∘ Not   applicable 66.6 ∘ ≤ ϕ ≤ 90 ∘ (42)

The findings presented in this section provide insights into the economic viability of implementing the four systems discussed, based on the relative pricing of energy storage compared to photovoltaic (PV) systems.

Both SYSTEMs A and C integrate energy storage, albeit with differing criteria for selecting tilt angles. While SYSTEM C prioritizes maximizing generated energy, SYSTEM A selects tilt angles to minimize the required energy storage. On the other hand, SYSTEM D aims to fulfill baseload energy requirements without relying on energy storage, which necessitates selecting tilt angles maximizing winter generation but may result in substantial curtailment during summer due to the absence of storage facilities.

SYSTEM A represents an optimal configuration benefiting from two key features: efficient energy storage utilization and prudent tilt angle selection to minimize storage size requirements.

In the economic comparison of the four systems, as of 2024, the prevailing scenario sees PV costs per kilowatt relatively low while energy storage costs per kilowatt-hour remain high. Consequently, as depicted in Figure 4, SYSTEM D emerges as the most economically advantageous option (upper region in Figure 4), followed by SYSTEM A, with SYSTEM C being the least favorable. Thus, currently, aligning PV tilt angles with those of SYSTEM D appears optimal for transitioning energy usage away from fossil fuels towards solar-based alternatives.

However, prevailing industry practice often favors SYSTEM B criteria, maximizing immediate energy generation and associated profits. Future shifts in electricity pricing, particularly favoring winter consumption over summer, could prompt installers to prioritize profit maximization over energy generation maximization when selecting tilt angles.

Adopting an electricity pricing scheme favoring winter consumption could accelerate the transition to a fully renewable energy-based grid by incentivizing tilt angles that enhance winter generation. Additionally, rapid declines in storage system prices may alter the economic landscape, potentially favoring SYSTEM A as storage costs decrease.

Ultimately, if storage costs significantly undercut PV costs, a scenario where SYSTEM C becomes economically favorable (lower region in Figure 4) may arise, although this remains speculative given projected future PV and storage prices.

In this section, we compute the LCOE based on projected prices of PV and storage systems sourced from the NREL website [21] . Forecasted PV prices are depicted in Figure 5, while battery storage prices are illustrated in Figure 6. The LCOE is assessed for all systems, as depicted in Figure 7, considering a location at 30˚ N latitude and assuming a 30-year system lifetime.

Consistent with expectations, the lowest electricity prices are achieved using SYSTEM D, followed by SYSTEM B, SYSTEM A, and finally, SYSTEM C, representing the most expensive option. This trend is anticipated to persist throughout the entire forecasted period until 2050.

The energy generated varies with latitude primarily due to changes in the duration of daylight and nighttime throughout the year. Near the equator, both daytime and nighttime are roughly equal, with each lasting around twelve hours. Consequently, these latitudes experience the highest energy generation on Earth. However, as you move towards higher latitudes, there is a greater variation in the duration of daylight and nighttime between the summer and winter seasons.

Although a fixed solar panel’s surface could potentially receive a maximum of twelve hours of sunlight within a twenty-four hour period, during winter, the

duration of daytime is less than twelve hours, resulting in reduced energy generation, particularly in winter. This decrease is not compensated for during the summer because fixed solar panels still receive a maximum of twelve hours of sunlight, even though the daytime exceeds twelve hours. As a result, the peak energy generation during summer remains constant across all latitudes. On the other hand, energy generation in winter decreases as you move towards higher latitudes as can been seen in Figure 8. In extreme polar latitudes, there are consecutive days during winter when the sun does not rise at all, causing energy generation to drop to zero. This overall trend leads to a decrease in generated energy throughout the year as you move towards higher latitudes. This trend is apparent in Table 2 for all systems.

The seasonality of generated energy is closely linked to the latitude of a location on Earth. Let’s explore how latitude influences the seasonality of generated energy:

1. Low-Latitude Regions: Locations near the equator exhibit relatively minimal seasonality in generated energy. Day length remains relatively consistent throughout the year, with slight variations attributed to changes in sun declination across the four seasons.

2. Mid-Latitude Regions: Locations at intermediate latitudes experience more pronounced seasonal variations in generated energy compared to low-latitude regions for SYSTEM A. These regions show distinct differences in energy generation between spring and fall, on one hand, and summer and winter, on the other. Spring and fall exhibit two peaks in energy generation, while summer and winter demonstrate two minima. For instance, at latitude of 40 with a panel tilt angle of 65 for SYSTEM A, the angle of incidence at solar noon is close to zero during winter. However, due to the shorter daytime period, energy generation in winter is minimized. In contrast, during spring and fall, when the angle of incidence is approximately 25 and the daytime is the same for both seasons, we can expect similar energy generation. It’s important to note that the daytime in spring and fall is around twelve hours, whereas in winter, it’s only six hours at latitude of 40. Consequently, even though the angle is optimized for the winter season, energy generation is lower compared to spring and fall due to the shorter day length. Similarly, during summer, with a substantial angle of incidence of 50, the sunshine period on the solar panels cannot exceed twelve hours. This reduces the amount of generated energy compared to spring and fall, resulting in the second minimum in addition to the winter minimum.

3. High-Latitude Regions: Locations near the poles, like the Arctic and Antarctic regions, experience extreme seasonality in generated energy. During summer, when the respective pole is tilted towards the sun, these regions receive 24 hours of daylight. However, the solar panels are exposed to sunlight for only 12 hours, similar to the sun exposure in spring and fall, resulting in a plateau in generated energy observed during these seasons. In contrast, during winter, when the pole is tilted away from the sun, these regions experience polar night, with no sunlight or very limited daylight, leading to low or negligible energy generation.

In summary, the seasonality of generated energy varies based on latitude. Low-latitude regions exhibit minimal variations, mid-latitude regions show more pronounced peaks and minima, and high-latitude regions experience extreme seasonality with distinct plateaus and periods of limited energy generation.

The utilization of energy storage poses considerable economic challenges, particularly in the context of renewable energy reliance. This necessity arises due to the inherent intermittency and seasonal variability of renewable energy sources, which necessitate backup solutions for uninterrupted power supply.

In this study, we delved into a novel approach for selecting the tilt angle of solar panels, with the objective of minimizing the energy storage requirements for standalone PV systems. However, given the current high costs associated with energy storage systems, there is limited incentive to augment PV systems with energy storage.

The absence of energy storage leads to a significant increase in curtailed energy as the adoption of PV energy systems, and renewable energy in general, continues to rise. Curtailment represents a financial loss for energy providers, and in the current economic landscape, avoiding energy storage remains the prevailing paradigm. Nevertheless, as storage technology advances and costs decline, there exists the potential for a future scenario where the incorporation of energy storage becomes economically viable. The timing of this transition is uncertain and hinges on the rate at which storage prices decrease.

It is pertinent to note that energy storage solutions are already present in certain sectors of the energy industry, albeit to a limited extent. These systems are typically employed for short-term energy storage to mitigate the intermittent nature of renewable energy sources. However, if the global aim is to transition entirely to renewable for energy needs, addressing the challenge of long-term energy storage becomes imperative, ideally on a seasonal timescale. The longer duration of storage necessitates larger storage capacities, thereby increasing implementation costs—a significant barrier to the widespread adoption of such systems in reality.