# Generated Value of Electricity Versus Incurred Cost for Solar Arrays under Conditions of High Solar Penetration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Solar Array Geometries

_{AP}between the rows’ bearing and the (east-west) circle of latitude, a tilt angle of τ

_{AP}above the horizontal, a transverse distance of d

_{AP}across each surface, and a spacing of s

_{AP}between the rows. Such a geometry is relatively straightforward to implement. However, its performance over the course of a day can be subject to issues of shadowing, as seen in [24,25], convective cooling, as introduced in [26] and applied in [27,28,29,30], and time-varying electricity prices, as proposed in this work.

_{VG}between the corrugations’ bearing and the (north-south) line of longitude, an interior v-angle of ν

_{VG}between the top surfaces, a transverse distance of d

_{VG}across each surface, and a spacing of s

_{VG}= 2d

_{VG}sin(ν

_{VG}/2) between the rows. This array has been popular for tandem solar cells, with differing bandgaps for opposing sides [31,32], as it yields photogenerated charge carriers with less excess kinetic energy and thus greater conversion efficiency. Nonetheless, in this work, we recognize that the V-shaped grooves can also improve light capture during the early- and late-day hours, when electricity prices are the highest, suggesting that it may generate an enhanced value of electricity.

_{UG}between the walls’ bearing and the (north-south) line of longitude, a height of h

_{UG}for the walls, and a spacing of s

_{UG}between the walls. To the authors’ best knowledge, such a geometry has not been studied for solar arrays, but it has the potential to broaden the duration of light capture over the course of a day beyond that of the V-groove array. This is because the U-groove array’s bottom effectively captures light for high-angled (midday) illumination, while the walls continue light capture for low-angled (early- and late-day) illumination via internal reflections. Our work weighs the increase in generated value of electricity for this array against its increased cost, with a comparison to the angled-panel and V-groove arrays.

## 3. Analyses

#### 3.1. Experimental Analyses (of the Constituent Solar Cells)

^{2}. They had alkaline-etched surface texturing and were capped by an ethylene vinyl acetate-film encapsulant and a 3-mm-thick low-iron-glass protective layer. This was done to mimic the packaging of solar cells within a PV module. (Such capping protects the solar cells but can lead to specular reflections within the array and self-heating of the solar cells). The solar cells were tested in an assortment of array types and orientations on the rooftop of a four-storey building in the locale of 49.939° N, 119.394° W. The elevated setting was chosen to yield characterizations with minimal obstructions over the full course of a day.

_{pv}(θ) is the measured current density at a solar zenith angle of θ, V is the measured voltage, and V

_{T}is the junction thermal voltage. The first term defines the photocurrent density as a product of a responsivity scaling factor, n

_{λ}, the normalized spectral responsivity, R

_{λ}(λ), as a function of the wavelength, λ, and the spectral irradiance distribution, S(λ,AM(θ)), as a function of the wavelength, λ, and air mass (AM) at the given solar zenith angle, AM(θ). The spectral responsivity of the silicon solar cells was estimated from several sources [35,36,37,38] and applied here as a wavelength-dependent quantity. This was done to better characterize the high-value early- and late-day light, for which the long atmospheric path lengths shift the spectrum away from the standard AM1.5 spectrum [39]. The second term contains the first saturation current density, J

_{sat1}, and series resistance, R

_{s}. The third term contains the second saturation current density, J

_{sat2}, and diode ideality factor, n

_{d}, which is assumed to be 2 as a common assumption [40]. The fourth term contains the shunt resistance, R

_{sh}.

_{r}= 0.742. Fitting for the first saturation current density suggests that the temperature of the solar cell is above the ambient. We attributed this to self-heating at a rate of 25 °C/(1000 W/m

^{2}), in accordance with [41]. This J

_{sat1}is the result of bulk and surface recombination in the solar cell, which depends upon the minority charge carrier density and thus scales in proportion to the square of the intrinsic charge-carrier density. The intrinsic charge-carrier varies with temperature, T, according to T

^{3/2}exp(–E

_{g}/2k

_{B}T), where E

_{g}= 1.1 eV is the bandgap energy of silicon and k

_{B}is Boltzmann’s constant, and so J

_{sat1}follows T

^{3}exp(–E

_{g}/k

_{B}T) [42]. Such a relation gives a first saturation current density of J

_{sat1}= 471 fA/cm

^{2}at a solar cell temperature of 25 °C, which is comparable to values in the literature [40]. Fitting for the second saturation current density, J

_{sat2}, suggests that the I-V characteristics have only a weak dependence on it, in that it shows only a slight rise at midday due to increased temperature. This J

_{sat2}is the result of trap-assisted recombination in the junction, with the rate of recombination proportional to the product of intrinsic charge-carrier density and temperature, and so J

_{sat2}follows T

^{5/2}exp(–E

_{g}/2k

_{B}T). The temperature dependence seen here is weaker than that of the first saturation current density, but it is still important [43]. A second saturation current density of J

_{sat2}= 15.4 nA/cm

^{2}is extracted here for a temperature of 25 °C. This is comparable to the values seen in the literature and is attributed to the strong edge recombination of commercial silicon solar cells [40]. Fitting for the series resistance yields large values that are dominated by the peripheral wiring. Thus, a closer to average [39], but still large, value of R

_{s}= 3 Ω·cm

^{2}is used here. Fitting for the shunt resistance yields R

_{sh}= 37.74 kΩ·cm

^{2}, which is in accordance with the literature [40].

#### 3.2. Theoretical Analyses (of the Assembled Solar Arrays)

^{(AM(θ)/1.5)/}S(λ,AM = 0.0)

^{(AM(θ)/1.5−1)}, where S(λ,AM = 1.5) and S(λ,AM = 0.0) are the well-known solar irradiance distributions of the AM1.5 and AM0.0 spectra, respectively [47,48]. The cosθ appears here due to the differing orientations of unit area in the specular solar irradiance and S(λ,AM(θ)). The specular solar irradiance is applied here with wavelength dependence embedded in it, although the analyses can also be carried out using the wavelength-dependent S(λ,AM(θ)), with integration over this distribution at the end. Such an approach enables the use of a wavelength-dependent reflectance. Nonetheless, we have found that both approaches yield similar results, and so the simpler formulation with the specular solar irradiance is presented in this section.

#### 3.2.1. Angled-Panel Array

_{AP}, tilt angle of τ

_{AP}, transverse distance of d

_{AP}, and spacing of s

_{AP}, as shown in Figure 2a.

_{i}, d

_{j}, and d

_{k}, respectively. It can be shown that prior to any reflections, n = 0, the angled-panel array has diffuse incident optical power densities on the front surface, back surface, and ground of

#### 3.2.2. V-Groove Array

_{VG}, an interior v-angle of ν

_{VG}, a transverse distance of d

_{VG}, and a spacing of s

_{VG}= 2d

_{VG}sin(ν

_{VG}/2), as shown in Figure 2b.

_{VG,east}(ϕ,θ;n) and γ

_{VG,west}(ϕ,θ;n), respectively. Likewise, the east- and west-facing sides have distances from the vertex to the lower edges of incident illumination of ℓ

_{VG,east}(ϕ,θ;n) and ℓ

_{VG,west}(ϕ,θ;n), respectively, and distances from the vertex to the upper edges of incident illumination of u

_{VG,east}(ϕ,θ;n) and u

_{VG,west}(ϕ,θ;n), respectively, where n is the number of reflections. By methodically tracking the cascaded incidence, reflection, and projection of light within the V-groove array, we can state

_{VG,east/west}(ϕ,θ;n) > d

_{VG}. When this occurs, the upper edge of the illuminated area is truncated at u

_{VG,east/west}(ϕ,θ;n) = d

_{VG}and the specular incident optical power density is scaled to

_{VG,east/west}(ϕ,θ;n) + 0.5ν

_{VG}> π; the reflected light travels towards but fully misses the opposing side of the V-groove array, i.e., ℓ

_{VG,east/west}(ϕ,θ;n) > d

_{VG}; or the incident optical power density is reduced by nine orders of magnitude. When one of the conditions is met, the power densities on both sides are summed to define the specular captured optical power density of the V-groove array as

_{i}or d

_{j}, respectively.

#### 3.2.3. U-Groove Array

_{UG}, wall height of h

_{UG}, and wall-to-wall spacing of s

_{UG}, as shown in Figure 2c.

_{UG}= s

_{UG}/h

_{UG}, meets the condition tanθ|sinϕ| < N

_{UG}, which we can characterize by θ < θ

_{UG}= arctan(N

_{UG}/|sinϕ|). For this case, the specular captured optical power densities of the directly illuminated and shaded walls are given by

_{UG}= s

_{UG}/h

_{UG}, ascribes to tanθ|sinϕ| ≥ N

_{UG}and thus θ ≥ θ

_{UG}= arctan(N

_{UG}/|sinϕ|). As the number of reflections can become arbitrarily large here, steps are used to track the cascaded incidence, reflection, and projection of illumination within this array. For each east- and west-facing wall, we define the distance above the ground to the lower and upper edges of incident illumination as ℓ

_{UG,east/west}(ϕ,θ ≥ θ

_{UG};n) and u

_{UG,east/west}(ϕ,θ ≥ θ

_{UG};n), respectively. The stepping through n reflections is carried out like that of the V-groove array. However, here we test for a condition where incidence on a wall extends above its upper edge, i.e., u

_{UG,east/west}(ϕ,θ ≥ θ

_{UG};n) > h

_{UG}, and a condition where incidence on a wall reaches the bottom of the U-groove array, i.e., ℓ

_{UG,east/west}(ϕ,θ ≥ θ

_{UG};n = r) < 0, where the integer r denotes the reflection at which this occurs. With such definitions, we can state that the specular optical power density on each wall is given by

_{UG,east/west}(ϕ,θ ≥ θ

_{UG};n) > h

_{UG}, or the incident optical power density has been reduced by nine orders of magnitude. Following either condition, the specular captured optical power density for both walls is computed via

_{i}, a distance above the ground of d

_{j}, or a distance from the east-facing wall of d

_{k}, respectively.

#### 3.3. Economic Analyses (of Generated Value versus Incurred Cost)

- Specular and diffuse solar irradiance is computed via Equations (3) and (4), respectively;
- The captured optical power density is then computed for each of the flat-panel, angled-panel, V-groove, and U-groove arrays via Equations (7), (18), (29) and (51), respectively;
- The generated electrical power is then computed for each array using its captured optical power density, the NOAA temperature data from [53], and the I–V characteristics of Equation (1), assuming that each array has one maximum power point tracking (MPPT) system;
- The generated value of electricity is then computed for each array as an accumulating product of generated electrical power and CAISO OASIS electricity pricing data from [3].

_{G}LA − (c

_{f}+ c

_{a}A + c

_{s}C

_{G}A), where the subscript G identifies the solar array geometry as flat-panel (FP), angled-panel (AP), V-groove (VG), or U-groove (UG). The total generated value is stated here as a product of the solar array’s generated value of electricity per annum and unit area, v

_{G}, lifespan, L, and installation area, A. The total cost is denoted by the terms in parentheses. The first term is the fixed cost, c

_{f}, which incorporates all costs that are roughly independent of the solar array size. In practice, most costs scale with array size to an extent, but costs such as permits and grid interconnection have little dependence on the array size. The second term, c

_{a}A, includes an installation-area-dependent multiplier of c

_{a}for costs that scale in proportion to the installation area, A, such as land cost. Note that the RPD in our work is relative, making the results independent of this (and the first) term. Thus, relative comparisons and generalized conclusions can be obtained without worry over these common costs, which often vary with the locale. The third term, c

_{s}C

_{G}A, scales in proportion to a constant surface-area-dependent multiplier, c

_{s}, and the PV surface area, C

_{G}A. A representative value of c

_{s}= 175 USD/m

^{2}is used in our work, as discussed below, whereas the geometry-dependent cost factor, C

_{G}, is calculated as the ratio of PV surface area to installation area. This gives C

_{G}= 1, d

_{AP}/s

_{AP}, 2d

_{VG}/s

_{VG}, and (2h

_{UG}+ s

_{UG})/s

_{UG}for the flat-panel, angled-panel, V-groove, and U-groove arrays, respectively. Ultimately, this third term shows us that arrays with higher vertical-to-horizontal aspect ratios, or more closely spaced/overlapping PV modules, will have these costs scale at an increased rate with respect to the installation area.

_{f}and c

_{a}. The RPD

_{G}for the solar array geometry G can then be manipulated into

_{FP}, and a geometry-dependent value factor, V

_{G}, that is the ratio of the value generated by the given solar array to that of the flat-panel array. Thus, V

_{G}quantifies the extent to which a given solar array generates value over that of the flat-panel array with the GVD acting as an external scaling factor for this value. The second term characterizes the solar array’s surface-area-dependent costs per annum and unit area with respect to those of the flat-panel array. It includes a geometric cost density (GCD), which is equivalent to the surface-area-dependent multiplier divided by the lifespan, i.e., c

_{s}/L, and a geometry-dependent cost factor, C

_{G}. Thus, C

_{G}quantifies the cost of a solar array over that of a flat-panel array with the GCD as an external scaling factor for this cost.

_{G}, and minimizing the cost factor, C

_{G}. However, this must be done while recognizing that the GVD and GCD magnify the effects of value and cost while being set by prevailing supply-and-demand economics. The GVD is linked to cycles of supply and demand in electricity prices, such that it scales with midday electricity prices. Thus, economies having strong Duck Curve characteristics with reduced electricity prices at midday exhibit low GVD. This can be understood by recalling that the GVD quantifies the flat-panel array’s generated value of electricity per annum and unit area as a nominal value with the greatest sensitivity to diurnal fluctuations in electricity prices. (The other solar arrays have geometries that can better trap light at early- and late-day hours, which increases their value factor, V

_{G}, and offers them some protection from diurnal fluctuations in electricity prices). The GCD is linked to supply and demand in costs of solar infrastructure. Thus, improvements and overall growth in the manufacture of PV modules over the past decade [56] has led to a steady decrease in the GCD. Based on the data of Fu et al. [56], such costs are approximately 7 USD/(a·m

^{2}) at present for a solar efficiency of 17% [57], such that a 1 m

^{2}solar module with a lifespan of 25 years would have an estimated cost (and thus surface-area-dependent multiplier) of c

_{s}= 175 USD/m

^{2}. The decline in GCD suggests that solar arrays with heightened aspect ratios and greater values of C

_{G}may now outperform the industry-standard angled-panel array from the standpoint of value. Such a prospect is explored and discussed in the following section.

## 4. Results and Discussions

#### 4.1. Historical Trends

^{2}), over historical trends [56]. The evolving GCD is in contrast to the GVD, in that it shows a simple monotonic decrease over the displayed 5-year history.

_{G}, and cost factor, C

_{G}. The results show GVD decreasing at a slower rate than GCD, which has the first term dominate with V

_{G}taking on greater importance than C

_{G}. This suggests that the increased costs of complex solar arrays, from greater vertical-to-horizontal aspect ratios, are offset (to an extent) by their increased value generation under Duck Curve characteristics.

_{G}, and minimal cost factor, C

_{G}, then give the greatest profit and maximal RPD. The V

_{G}and C

_{G}for the fully optimized angled-panel, V-groove, and U-groove arrays are shown in Figure 4, with Figure 4a showing the results over a 5-year history plotted against the GCD-to-GVD ratio. The (horizontal) unity line in this subfigure characterizes the flat-panel array. Thus, the separation of V

_{G}and C

_{G}from the unity line portrays the degree to which an optimal structure deviates from a flat-panel array. With this interpretation, we see the V-groove and U-groove arrays degenerate into the flat-panel array via their optimization, with optimal geometric parameters of ν

_{VG}= 180° and s

_{UG}= ∞, respectively, yielding unity V

_{G}and C

_{G}. This trend can be explained by noting that C

_{G}can only be at or above unity for these arrays, and historical values of GCD have been high enough to have the optimization primarily minimize C

_{G}. The trends exhibited by V

_{AP}and C

_{AP}are in contrast to this. They are far below unity, showing that the optimal angled-panel array deviates greatly from the flat-panel array. Thus, the reduced V

_{AP}is deemed to be acceptable (and even optimal) given the associated reduction in C

_{AP}. As the GCD-to-GVD ratio increases along the horizontal axis, the RPD optimization preferentially drives the C

_{AP}down in spite of further reductions in V

_{AP}, causing the optimal angled-panel array to deviate further from the flat-panel array. (An excessively high GCD causes both C

_{AP}and V

_{AP}to drop to zero, characterizing conditions antithetical to solar power generation, whereby a degenerate case of no solar cells is optimal). Conversely, as the GCD-to-GVD ratio decreases, the RPD optimization places greater importance on a heightened V

_{AP}, causing the optimal angled-panel array to approach a flat-panel array. Such trends, of the angled-panel array approaching the flat-panel array and higher C

_{G}values becoming acceptable, suggest that further decreases in GCD could make the angled-panel array non-viable and the V-groove and U-groove arrays viable.

_{AP}and C

_{AP}seen in Figure 4a correspond to larger spacings, s

_{AP}, reducing the array density and thereby both geometry-dependent factors, as well as higher tilt angles, τ

_{AP}, which increase as array self-shadowing decreases. For this value-based optimization, such angles ultimately reach 6° more than the latitude, which is slightly more than the tilt angles generally considered optimal for standalone PV modules under power-based optimization [58]. The rotation angle, ρ

_{AP}, meanwhile, does not closely follow the geometry-dependent factors, but rather generally decreases with time, leading the array to point further west over time. This trend demonstrates the direct relation between the optimal values of ρ

_{AP}and the severity of the Duck Curve characteristics, which has the west-facing solar cells preferentially capture light during the late-day hours when electricity prices are the highest.

_{G}and C

_{G}. Thus, the heightened values for the V-groove and U-groove arrays, even in spite of their higher costs, may allow them to outperform the angled-panel array in the future—a possibility that warrants the investigation in the following subsection.

#### 4.2. Future Trends

^{2}), as defined earlier from the final year of the 5-year history. The electricity prices are also fixed at those observed during this final year. The GCD, in contrast, is easier to define. It has fallen with a rate of decrease that is slowing and is trending to a plateau. Thus, the GCD is treated as an independent variable in this work, with values spanning from the present 7 USD/(a·m

^{2}) down to an extreme of 0 USD/(a·m

^{2}), representing zero infrastructure costs.

^{2}) where the two arrays have equivalent RPDs. If the GCD drops through this crossover point, the best-performing solar array will transition from the angled-panel array to the V-groove array. At very low GCD, the RPDs of the angled-panel and U-groove arrays increase, but neither array experiences enough performance improvement to surpass the V-groove array.

_{G}, and cost factor, C

_{G}, in the future as a function of the GCD (and the corresponding GCD-to-GVD ratio). Given the decreasing GCD over the years, the right side of this subfigure can be interpreted as the present, and progression to the left on this subfigure can be cast as trends into the future. The trends are discussed here in ranges of GCD-to-GVD ratios corresponding to the near, foreseeable, and speculative future. (These ratios are based upon realistic parameters, and thus give reliable predictions of trends, but labels of near, foreseeable, and speculative are used here for timespans in the future, rather than precise years, because uncertainties in solar energy markets can affect the years in which such trends emerge.) The optimal geometric parameters for each array across these ranges are tabulated in Table 2.

_{AP}and C

_{AP}both approaching unity as the GCD decreases. This has the optimal angled-panel array approach the flat-panel array. Over this range, the optimal V-groove array has V

_{VG}and C

_{VG}slightly above unity, representing a small deviation from the flat-panel array, while the optimal U-groove array remains degenerate. These results are reflected in Figure 5, whereby the angled-panel array has a large positive RPD, the V-groove array has a small positive RPD, and the U-groove array has zero RPD over this range. Moreover, the optimal geometric parameters (tabulated in Table 2) show that the optimal V-groove array has a large ν

_{VG}near 180°, and the optimal angled-panel array has initially large s

_{AP}and τ

_{AP}, which decrease towards 76.2 mm and 0°, respectively, as the GCD decreases. Such trends agree with predictions of Awad et al. [55]. Overall, the findings suggest the angled-panel array will continue to show the best performance in the near future, but its relative performance will lessen as the GCD decreases.

_{G}and C

_{G}close to unity, suggesting the optimal arrays are all similar to a flat-panel array. The U-groove array remains degenerate across the entirety of this range. Following previously seen trends, the angled-panel array also becomes degenerate and remains that way across the majority of this range. Both of these are reflected in the zero RPD seen in Figure 5 and the degenerate parameters seen in Table 2. By contrast, while V

_{VG}and C

_{VG}remain close to unity, the V-groove array is not degenerate, but continues to display small positive values for the RPD and large ν

_{VG}near but not equal to 180° over the entirety of this range. Ultimately, it can be said that the V-groove array will show the best performance over the foreseeable future—with stable performance in this range. None of the optimized parameters vary significantly with changing GCD-to-GVD ratios, which suggests that significant swings in the GVD caused by volatile electricity prices would not disrupt the performance of the V-groove array under these conditions.

_{G}and C

_{G}for all three arrays rising significantly above unity, with the increase in C

_{G}being more dramatic than that of V

_{G}. This suggests that increasing C

_{G}above unity yields diminishing returns in increased V

_{G}, which can only be justified if the GCD is very low, making costs unimportant. This scenario will only occur if the cost of silicon solar cells drops dramatically. Figure 5 shows that while the RPDs of all three arrays increase significantly in this range, the V-groove array remains the top-performing array over the full range. The optimal geometric parameters (tabulated in Table 2) show that the increased values and costs of all three arrays can be linked to high vertical-to-horizontal aspect ratios with the key geometric parameters of s

_{AP}, ν

_{VG}, and s

_{UG}all decreasing significantly over this range. These more complex structures have significantly higher costs per unit area, but they yield improved light capture.

_{G}depends upon hourly electricity prices in a manner that is not fully encompassed by the GVD, and the marginal electricity prices used in calculating the V

_{G}and GVD do not fully represent the rates at which electricity is purchased from solar generation systems. Such rates are typically higher and less variable due to power purchase agreements and renewable energy subsidies [60]. Nonetheless, the core conclusions of this work are sound: The trends seen for the supply and demand of solar-generated power and solar cell prices will lead to shallow-angled V-groove arrays outperforming industry-standard angled-panel arrays, in terms of profit, with such arrays having reduced sensitivity to electricity price fluctuations.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- International Energy Agency (IEA). Photovoltaic Power System Programme (PVPS). In 2016 Snapshot of Global Photovoltaic Markets; IEA: Paris, France, 2016. [Google Scholar]
- California Independent System Operator (CAISO). Demand Response and Energy Efficiency Roadmap: Maximizing Preferred Resources; CAISO: Folsom, California, 2013. [Google Scholar]
- California Independent Systems Operator (CAISO) Open-Access Same-Time Information System (OASIS) website. Available online: https://oasis.caiso.com (accessed on 15 September 2018).
- Li, M.; Pan, J.; Su, Q.; Niu, Y. Analyzing sensitivity of power system wind penetration to thermal generation flexibility. In Proceedings of the 2017 13th IEEE Conference on Automation Science and Engineering (CASE), Xi’an, China, 20–23 August 2017; pp. 1628–1632. [Google Scholar]
- Obi, M.; Bass, R. Trends and challenges of grid-connected photovoltaic systems–A review. Renew. Sustain. Energy Rev.
**2016**, 58, 1082–1094. [Google Scholar] [CrossRef] - Janko, S.A.; Arnold, M.R.; Johnson, N.G. Implications of high-penetration renewables for ratepayers and utilities in the residential solar photovoltaic (PV) market. Appl. Energy
**2016**, 180, 37–51. [Google Scholar] [CrossRef] - Hassan, A.S.; Cipcigan, L.; Jenkins, N. Optimal battery storage operation for PV systems with tariff incentives. Appl. Energy
**2017**, 203, 422–441. [Google Scholar] [CrossRef] - Schoenung, S.M.; Keller, J.O. Commercial potential for renewable hydrogen in California. Int. J. Hydrog. Energy
**2017**, 42, 13321–13328. [Google Scholar] [CrossRef] - Zhang, N.; Lu, X.; McElroy, M.B.; Nielsen, C.P.; Chen, X.; Deng, Y.; Kang, C. Reducing curtailment of wind electricity in China by employing electric boilers for heat and pumped hydro for energy storage. Appl. Energy
**2016**, 184, 987–994. [Google Scholar] [CrossRef] [Green Version] - Gür, T.M. Review of electrical energy storage technologies, materials and systems: Challenges and prospects for large-scale grid storage. Energy Environ. Sci.
**2018**, 11, 2696–2767. [Google Scholar] [CrossRef] - Chaudhary, P.; Rizwan, M. Energy management supporting high penetration of solar photovoltaic generation for smart grid using solar forecasts and pumped hydro storage system. Renew. Energy
**2018**, 118, 928–946. [Google Scholar] [CrossRef] - Le Floch, C.; Belletti, F.; Moura, S. Optimal Charging of Electric Vehicles for Load Shaping: A Dual-Splitting Framework With Explicit Convergence Bounds. IEEE Trans. Transp. Electrif.
**2016**, 2, 190–199. [Google Scholar] [CrossRef] - Sanandaji, B.M.; Vincent, T.L.; Poolla, K. Ramping Rate Flexibility of Residential HVAC Loads. IEEE Trans. Sustain. Energy
**2015**, 7, 865–874. [Google Scholar] [CrossRef] - Perez, M.; Perez, R.; Rábago, K.R.; Putnam, M. Overbuilding & curtailment: The cost-effective enablers of firm PV generation. Sol. Energy
**2019**, 180, 412–422. [Google Scholar] [CrossRef] - Perez, R.; Perez, M.; Schlemmer, J.; Dise, J.; Hoff, T.E.; Swierc, A.; Keelin, P.; Pierro, M.; Cornaro, C. From Firm Solar Power Forecasts to Firm Solar Power Generation an Effective Path to Ultra-High Renewable Penetration a New York Case Study. Energies
**2020**, 13, 4489. [Google Scholar] [CrossRef] - Perez, R.; Rábago, K.R.; Trahan, M.; Rawlings, L.; Norris, B.; Hoff, T.; Putnam, M.; Perez, M. Achieving very high PV penetration–The need for an effective electricity remuneration framework and a central role for grid operators. Energy Policy
**2016**, 96, 27–35. [Google Scholar] [CrossRef] [Green Version] - Borenstein, S. The Market Value and Cost of Solar Photovoltaic Energy Production; Center for the Study of Energy Markets: Berkeley, CA, USA, 2008. [Google Scholar]
- Hartner, M.; Ortner, A.; Hiesl, A.; Haas, R. East to west–The optimal tilt angle and orientation of photovoltaic panels from an electricity system perspective. Appl. Energy
**2015**, 160, 94–107. [Google Scholar] [CrossRef] - Rowlands, I.H.; Kemery, B.P.; Beausoleil-Morrison, I. Optimal solar-PV tilt angle and azimuth: An Ontario (Canada) case-study. Energy Policy
**2011**, 39, 1397–1409. [Google Scholar] [CrossRef] - Masrur, H.; Konneh, K.; Ahmadi, M.; Khan, K.; Othman, M.; Senjyu, T. Assessing the Techno-Economic Impact of Derating Factors on Optimally Tilted Grid-Tied Photovoltaic Systems. Energies
**2021**, 14, 1044. [Google Scholar] [CrossRef] - Maleki, S.A.M.; Hizam, H.; Gomes, C. Estimation of Hourly, Daily and Monthly Global Solar Radiation on Inclined Surfaces: Models Re-Visited. Energies
**2017**, 10, 134. [Google Scholar] [CrossRef] [Green Version] - Al-Rousan, N.; Isa, N.A.M.; Desa, M.K.M. Advances in solar photovoltaic tracking systems: A review. Renew. Sustain. Energy Rev.
**2018**, 82, 2548–2569. [Google Scholar] [CrossRef] - Vasantha, P.N.; Dehankar, S.; Raman, M.; Karnataki, K.; Shankar, G. Space optimization and backtracking for dual axis so-lar photovoltaic tracker. In Proceedings of the IEEE International WIE Conference on Electrical and Computer Engineering, Dhaka, Bangladesh, 19–20 December 2015; pp. 451–454. [Google Scholar]
- Brecl, K.; Topic, M. Self-shading losses of fixed free-standing PV arrays. Renew. Energy
**2011**, 36, 3211–3216. [Google Scholar] [CrossRef] - Gordon, J.; Wenger, H.J. Central-station solar photovoltaic systems: Field layout, tracker, and array geometry sensitivity studies. Sol. Energy
**1991**, 46, 211–217. [Google Scholar] [CrossRef] - Vaillon, R.; Dupré, O.; Cal, R.B.; Calaf, M. Pathways for mitigating thermal losses in solar photovoltaics. Sci. Rep.
**2018**, 8, 13163. [Google Scholar] [CrossRef] - Stanislawski, B.; Margairaz, F.; Cal, R.; Calaf, M. Potential of module arrangements to enhance convective cooling in solar photovoltaic arrays. Renew. Energy
**2020**, 157, 851–858. [Google Scholar] [CrossRef] - Glick, A.; Smith, S.E.; Ali, N.; Bossuyt, J.; Recktenwald, G.; Calaf, M.; Cal, R.B. Influence of flow direction and turbulence intensity on heat transfer of utility-scale photovoltaic solar farms. Sol. Energy
**2020**, 207, 173–182. [Google Scholar] [CrossRef] - Glick, A.; Ali, N.; Bossuyt, J.; Calaf, M.; Cal, R.B. Utility-scale solar PV performance enhancements through system-level modifications. Sci. Rep.
**2020**, 10, 1–9. [Google Scholar] [CrossRef] [PubMed] - Glick, A.; Ali, N.; Bossuyt, J.; Recktenwald, G.; Calaf, M.; Cal, R.B. Infinite photovoltaic solar arrays: Considering flux of momentum and heat transfer. Renew. Energy
**2020**, 156, 791–803. [Google Scholar] [CrossRef] - Sista, S.; Hong, Z.; Chen, L.-M.; Yang, Y. Tandem polymer photovoltaic cells—current status, challenges and future outlook. Energy Environ. Sci.
**2011**, 4, 1606–1620. [Google Scholar] [CrossRef] - Andersson, B.V.; Würfel, U.; Inganäs, O. Full day modelling of V-shaped organic solar cell. Sol. Energy
**2011**, 85, 1257–1263. [Google Scholar] [CrossRef] - Ding, K.; Zhang, J.; Bian, X.; Xu, J. A simplified model for photovoltaic modules based on improved translation equations. Sol. Energy
**2014**, 101, 40–52. [Google Scholar] [CrossRef] - Humada, A.M.; Hojabri, M.; Mekhilef, S.; Hamada, H.M. Solar cell parameters extraction based on single and double-diode models: A review. Renew. Sustain. Energy Rev.
**2016**, 56, 494–509. [Google Scholar] [CrossRef] [Green Version] - Aberle, A.G.; Zhang, W.; Hoex, B. Advanced loss analysis method for silicon wafer solar cells. Energy Procedia
**2011**, 8, 244–249. [Google Scholar] [CrossRef] [Green Version] - Du, Y.; Tao, W.; Liu, Y.; Le, Z.; Zhang, M. Cell-to-Module Variation of Optical and Photovoltaic Properties for Monocrystalline Silicon Solar Cells with Different Texturing Approaches. ECS J. Solid State Sci. Technol.
**2017**, 6, P332–P338. [Google Scholar] [CrossRef] - Saynova, D.; Mihailetchi, V.; Geerligs, L.; Weeber, A. Comparison of high efficiency solar cells on large area n-type and p-type silicon wafers with screen-printed Aluminum-alloyed rear junction. In Proceedings of the 2008 33rd IEEE Photovolatic Specialists Conference, San Diego, CA, USA, 11–16 May 2008; pp. 1–5. [Google Scholar]
- Solanki, C.S.; Singh, H.K. Anti-Reflection and Light Trapping in c-Si Solar Cells; Springer Nature: Singapore, 2017. [Google Scholar]
- Riordan, C.; Hulstron, R. What is an air mass 1.5 spectrum? (solar cell performance calculations). In Proceedings of the IEEE Conference on Photovoltaic Specialists, Kissimmee, FL, USA, 21–25 May 1990; pp. 1085–1088. [Google Scholar]
- Breitenstein, O. Understanding the current-voltage characteristics of industrial crystalline silicon solar cells by considering inhomogeneous current distributions. Opto-Electron. Rev.
**2013**, 21, 259–282. [Google Scholar] [CrossRef] - Koehl, M.; Heck, M.; Wiesmeier, S.; Wirth, J. Modeling of the nominal operating cell temperature based on outdoor weathering. Sol. Energy Mater. Sol. Cells
**2011**, 95, 1638–1646. [Google Scholar] [CrossRef] - Singh, P.; Ravindra, N. Temperature dependence of solar cell performance—an analysis. Sol. Energy Mater. Sol. Cells
**2012**, 101, 36–45. [Google Scholar] [CrossRef] - Wolf, M.; Noel, G.T.; Stirn, R.J. Investigation of the double exponential in the current-voltage characteristics of silicon solar cells. IEEE T. Electron. Dev.
**1977**, 24, 419–428. [Google Scholar] [CrossRef] - Boivin, A.B.; Westgate, T.M.; Holzman, J.F. Design and performance analyses of solar arrays towards a metric of energy value. Sustain. Energy Fuels
**2018**, 2, 2090–2099. [Google Scholar] [CrossRef] - Boivin, A.B. Performance and value of geometric solar arrays subject to cyclical electricity prices and high solar penetration. Master’s Thesis, The University of British Columbia, Kelowna, BC, Canada, 2019. [Google Scholar]
- Young, A. Air mass and refraction. Appl. Opt.
**1994**, 33, 1108–1110. [Google Scholar] [CrossRef] - American Society for Testing and Materials (ASTM). G173-03 (2012) Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface; American Society for Testing and Materials: West Conshohocken, PA, USA, 2012. [Google Scholar] [CrossRef]
- American Society for Testing and Materials (ASTM). E490-00a (2014) Standard Solar Constant and Zero Air Mass Solar Spectral Irradiance Tables; American Society for Testing and Materials: West Conshohocken, PA, USA, 2014. [Google Scholar] [CrossRef]
- Becker, S. Calculation of direct solar and diffuse radiation in Israel. Int. J. Clim.
**2001**, 21, 1561–1576. [Google Scholar] [CrossRef] - Koomen, M.J.; Lock, C.; Packer, D.M.; Scolnik, R.; Tousey, R.; Hulburt, E.O. Measurements of the Brightness of the Twilight Sky. J. Opt. Soc. Am.
**1952**, 42, 353. [Google Scholar] [CrossRef] - Coblentz, W. The diffuse reflecting power of various substances. J. Wash. Acad. Sci.
**1912**, 2, 447–451. [Google Scholar] [CrossRef] - Liu, X.; Wu, Y.; Hou, X.; Liu, H. Investigation of the optical performance of a novel planar static PV concentrator with Lambertian rear reflectors. Buildings
**2017**, 7, 88. [Google Scholar] [CrossRef] [Green Version] - National Oceanic and Atmospheric Administration (NOAA) Online Weather Data website. Available online: https://w2.weather.gov/climate (accessed on 15 May 2019).
- Darling, S.B.; You, F.; Veselka, T.; Velosa, A. Assumptions and the levelized cost of energy for photovoltaics. Energy Environ. Sci.
**2011**, 4, 3133–3139. [Google Scholar] [CrossRef] - Awad, H.; Gül, M.; Ritter, C.; Verma, P.; Chen, Y.; Salim, K.M.E.; Al-Hussein, M.; Yu, H.; Kasawski, K. Solar photovoltaic optimization for commercial flat rooftops in cold regions. In Proceedings of the IEEE Conference on Technologies for Sustainability, Phoenix, AZ, USA, 9–11 October 2016; pp. 39–46. [Google Scholar]
- Fu, R.; Margolis, R.; Feldman, D. US Solar Photovoltaic System Cost Benchmark: Q1 2018; National Renewable Energy Laboratory: Golden, CO, USA, 2018. [Google Scholar]
- Barbose, G.; Darghouth, N.; Millstein, D.; Cates, S.; DiSanti, N.; Widiss, R. Tracking the Sun IX: The Installed Price of Residential and Non-Residential Photovoltaic Systems in the United States; Lawrence Berkeley National Laboratory: Berkeley, CA, USA, 2016. [Google Scholar]
- Lave, M.; Kleissl, J. Optimum fixed orientations and benefits of tracking for capturing solar radiation in the continental United States. Renew. Energy
**2011**, 36, 1145–1152. [Google Scholar] [CrossRef] [Green Version] - Mundada, A.S.; Prehoda, E.W.; Pearce, J.M. U.S. market for solar photovoltaic plug-and-play systems. Renew. Energy
**2017**, 103, 255–264. [Google Scholar] [CrossRef] [Green Version] - Timilsina, G.R.; Kurdgelashvili, L.; Narbel, P.A. Solar energy: Markets, economics and policies. Renew. Sustain. Energy Rev.
**2012**, 16, 449–465. [Google Scholar] [CrossRef] - Bouchakour, S.; Valencia-Caballero, D.; Luna, A.; Roman, E.; Boudjelthia, E.; Rodríguez, P. Modelling and Simulation of Bifacial PV Production Using Monofacial Electrical Models. Energies
**2021**, 14, 4224. [Google Scholar] [CrossRef]

**Figure 1.**The Duck Curve profile of electricity price versus time of day showing its characteristic bimodal peaks and midday trough. The data shows hourly electricity prices near Bakersfield, California, on 1 May 2018, compiled from day-head locational marginal pricing on the California Independent System Operator (CAISO) Open Access Same-Time Information System (OASIS) [3].

**Figure 2.**The solar arrays and their geometric parameters. (

**a**) The angled-panel array, with its rotation angle, ρ

_{AP}, tilt angle, τ

_{AP}, transverse distance, d

_{AP}, and spacing, s

_{AP}. (

**b**) The V-groove array, with its rotation angle, ρ

_{VG}, v-angle, ν

_{VG}, transverse distance, d

_{VG}, and spacing, s

_{VG}= 2d

_{VG}sin(ν

_{VG}/2). (

**c**) The U-groove array, with its rotation angle, ρ

_{UG}, spacing, s

_{UG}, and height, h

_{UG}.

**Figure 3.**Historical geometric value density (GVD) and geometric cost density (GCD). The GVD (in red) and GCD (in green) are shown plotted against the left axis, and the average electricity price (in dashed black) is shown plotted against the right axis, for the 5-year span of this study. Each data point characterizes a year of data beginning on 1 July of the stated year and ending on 30 June of the following year. Thus, the data points at 2013, 2014, 2015, 2016, and 2017 characterize data over 2013-14, 2014-15, 2015-16, 2016-17, and 2017-18, respectively.

**Figure 4.**Historical and future value factors (V

_{G}) and cost factors (C

_{G}). (

**a**) Historical V

_{G}(in green) and C

_{G}(in red) as a function of the GCD-to-GVD ratio. The yearly progression is shown, where each data point represents a year of data beginning on 1 July of the labeled calendar year. (

**b**) Future V

_{G}(in green) and C

_{G}(in red) as a function of the GCD-to-GVD ratio on the lower axis and GCD on the upper axis, with the results segmented into ranges denoting the near, foreseeable, and speculative future. The results are shown for the optimized angled-panel array (denoted by slashes), V-groove array (denoted by triangles), and U-groove array (denoted by squares). The best-performing array for each range is displayed with solid lines; the remaining arrays are displayed with dashed lines.

**Figure 5.**Future relative profit densities (RPD). The optimal RPD for the future is shown versus GCD-to-GVD ratio on the lower axis and GCD on the upper axis. The results are shown for the optimal angled-panel array (in green with slashes), V-groove array (in blue with triangles), and U-groove array (in orange with squares).

Year | s_{AP} (mm) | τ_{AP} (°) | ρ_{AP} (°) |
---|---|---|---|

2013–2014 | 98 | 18 | −5 |

2014–2015 | 121 | 27 | −10 |

2015–2016 | ∞ | 41 | −20 |

2016–2017 | 219 | 37 | −20 |

2017–2018 | 104 | 21 | −15 |

**Table 2.**Optimized values of geometric parameters for each array and each geometric cost density. Rotation angles are fixed at zero for optimization. For U-groove arrays at high geometry-dependent cost density values, precise optimization is not possible, as the arrays become arbitrarily large.

GCD (USD/(a·m^{2})) | GCD/GVD | s_{AP} (mm) | τ_{AP} (°) | ν_{VG} (°) | s_{UG} (mm) |
---|---|---|---|---|---|

0 | 0 | 30 | 34 | 31 | 76.2 |

0.5 | 0.062 | 62 | 9 | 68 | 228.6 |

1 | 0.124 | 74 | 1 | 128 | 609.6 |

1.5 | 0.186 | 75 | 1 | 140 | ≥762 |

2 | 0.248 | 76.2 | 0 | 144 | >7620 |

2.5 | 0.31 | 76.2 | 0 | 148 | >7620 |

3 | 0.372 | 76.2 | 0 | 152 | >7620 |

3.5 | 0.434 | 76.2 | 0 | 154 | >7620 |

4 | 0.496 | 76.2 | 0 | 156 | >7620 |

4.5 | 0.558 | 77 | 2 | 156 | >7620 |

5 | 0.62 | 79 | 5 | 160 | >7620 |

5.5 | 0.682 | 82 | 8 | 160 | >7620 |

6 | 0.744 | 86 | 11 | 162 | >7620 |

6.5 | 0.806 | 93 | 15 | 164 | >7620 |

7 | 0.868 | 103 | 19 | 180 | >7620 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boivin, A.B.; Holzman, J.F.
Generated Value of Electricity Versus Incurred Cost for Solar Arrays under Conditions of High Solar Penetration. *Solar* **2021**, *1*, 4-29.
https://doi.org/10.3390/solar1010003

**AMA Style**

Boivin AB, Holzman JF.
Generated Value of Electricity Versus Incurred Cost for Solar Arrays under Conditions of High Solar Penetration. *Solar*. 2021; 1(1):4-29.
https://doi.org/10.3390/solar1010003

**Chicago/Turabian Style**

Boivin, Adrian B., and Jonathan F. Holzman.
2021. "Generated Value of Electricity Versus Incurred Cost for Solar Arrays under Conditions of High Solar Penetration" *Solar* 1, no. 1: 4-29.
https://doi.org/10.3390/solar1010003