^{1}

^{*}

^{2}

The performance of solar panels significantly degrades due to dust accumulation but cleaning too frequently will severely impact the financial benefits of the installation of solar panels. This paper assumes a realistic linear model for accumulation of dust on the solar panels and the resulting hourly average of absolute loss of efficiency in solar panels. This model accurately depicts the fact that energy production by solar panels occurs during sunshine hours only and also accounts for the degradation in the efficiency of solar panels due to dust accumulation throughout the entire day. Based on this, the optimal number of days for maximum financial profit and the critical number of days (above which there is no profit in installing solar panels) have been estimated. Furthermore, we have suggested a formalism to help estimate the finances for self-cleaning technology for PV system and also for calculating the minimum payback period for installing solar panels with the financial cost of the cleaning cycles properly considered. This research could be motivation for companies in developing self-cleaning mechanism for PV system.

Photovoltaic (PV) system is emerging technology that can deliver electricity directly from sunlight. As concerns for clean energy rises, PV technology is becoming more popular. There have been numerous research efforts around the world to make PV technology better in terms of price, durability and efficiency. These efforts include investigating on properties of semiconductor, reducing contact losses, using solar tracker and many more. Recently, the effect of soiling has emerged as an important parameter to consider, as this significantly reduces the overall performance of PV panels.

Soiling losses are losses in efficiency of PV module due to deposition of dust, dirt, snow thereby covering the active area of solar panels. Zaki Ahmad et al. [

Sanaz Ghazi et al. [

While cleaning will definitely help to improve the performance of PV, it certainly comes up with some financial costs. Abu-Naser [

N o p t = 2 P i s α β , #(1)

where α is the average daily degradation in the solar conversion efficiency, i is the capacity of the installed PV system, β is the price of kWh, s is the average sun hours per day. There are basically two things that we would want to upgrade to previous model which were not taken into consideration in the model.

Here we follow a similar methodology as presented in [

Firstly, the author [

This section discusses about implementing the modifications and methodology as stated in previous section. For a solar panel of capacity i, efficiency η 0 , with s sunshine hours per day, β the price of unit electricity in kWh (obtained from sources other than solar) than the financial savings due to installing solar panel is given by,

S = i s β , #(2)

where i = η 0 P i n ; P i n being the incident solar radiation.

The dust accumulation will result in decrease in the efficiency of the solar panel, say η 1 , then the power output of the solar panel would be P ′ o u t = η 1 P i n and the financial savings at this condition is:

S 1 = η 1 P i n s β . #(3)

This would imply that the financial loss due to dust accumulation when the efficiency of the solar panel drops from η 0 to η 1 is

L o s s = ( η 0 − η 1 ) i η 0 s β . #(4)

We have assumed a simple yet a realistic scenario where the solar panel is in operation only during the sunshine hours and assumes it has constant efficiency η 0 when there is no dust accumulation.

If we assume that the dust accumulation is uniform and degradation rate in efficiency is constant per day, we can come up with following scenario for the efficiencies of the solar panels as presented in

The dust accumulation is considered linear, so the financial loss at the end of first s sunshine hours is given as:

L o s s 1 s = ( η 0 − η 01 a v ) i η 0 s β , #(5)

where η 01 a v = ( η 0 + η 1 ) / 2 .

This is also the financial loss for the entire first day since during night no savings is acquired due to installation of solar panels (no sunshine). There is still dust accumulation during the night though and the efficiency drops to η 2 at the start of the second day.

We will introduce a term α ′ which is defined as hourly average of absolute loss of efficiency in solar panels. It is realistic to assume an average hourly loss in solar conversion efficiency as it can most easily be related to the rate of dust accumulation in the solar panels. This would imply, η s = η 0 − α ′ s giving exact efficiency after s hours. Similarly, if we want to define absolute loss in terms of day, lets introduce f as daily average of absolute loss in efficiency so that efficiency after N days will be η N = η 0 − f N as illustrated in the schematic below.

To better understand this in terms of α which is defined as average daily loss in solar efficiency (as used in [^{st} day reduces to 19.998%. So, in this case, f will be 0.002%. With this value of f, we can keep track of exact value of efficiency after every N days.

So, defining average of absolute loss in efficiency preserves the fact of average of daily loss in solar conversion efficiency α is uniform. Similarly, the α ′ in terms of f would simply be α ′ = f / 24 . The relation between f and α follow as f = ( η 0 α ) and the relation between α and α’ will thus be α ′ = ( η 0 α ) / 24 .

If we assume a cleaning cycle of N days, the financial loss for the n^{th} day can be generalized by (see Appendix):

L o s s n = [ 48 ( n − 1 ) + s 2 ] i η 0 s β α ′ . #(6)

The total financial loss per annum is

L 1 = 365 N ∑ n = 1 N [ 48 ( n − 1 ) + s 2 ] i η 0 s β α ′ , #(7)

= 365 2 i η 0 s β α ′ [ 24 ( N − 1 ) + s ] , #(8)

If P is the cost of cleaning the solar array, then the cost of cleaning per annum is

L 2 = 365 N P . #(9)

The total cost of installing solar array is

L = L 1 + L 2 , #(10)

= 365 2 i η 0 s β α ′ [ 24 ( N − 1 ) + s ] + 365 N P . #(11)

The optimal number of days between cleaning cycles, N o p t ; which is also the minimum value of N, calculated by differentiating Equation with respect to N and equating it to zero is

N o p t = η 0 P 12 i s α ′ β . #(12)

If we convert hourly loss in efficiency due to soiling into daily loss in efficiency (i.e. α ′ to α by setting α ′ = ( η 0 α ) / 24 we obtain similar equation in terms of α presented in Equation (1) i.e.

N o p t = 2 P i s α β . #(13)

Similarly, following the same procedures, if we assume γ ′ and γ ″ to be average of absolute loss in efficiency during sunshine hour s and night hours (24 - s) respectively. We assume different rates of soiling during day and night because human activities might induce different soiling rates. For such approximation, if we follow similar procedures as in Appendix 1 and Appendix 2, we get

N o p t = 2 η 0 P i s β [ γ ′ s + γ ″ ( 24 − s ) ] . #(14)

Equation (14) converges to Equation (13) if rates of degradation are same ( γ ′ = γ ″ ) throughout day. Equation (14) is a more general form and accounts for optimal cleaning days if the degradation rates vary during the sunshine and outside it.

In this section, we follow through the different parameters in the formulation section and compare the results with existing models as well as interpret and extend the formalism to yield further outcomes. This includes computing total difference in financial loss per year with this model and the existing model in [

1) Comparison of financial loss: This model and previous model yields same equation for optimal days. However, we see that this model predicts larger overall energy production per annum (accounting the soiling effects, of course) resulting in less financial loss per annum with the same cleaning frequency as compared to model presented in [

G = 365 i s α β 2 ( 2 − s 24 ) . #(15)

2) Payback period and self-cleaning mechanism: Another avenue to explore is to imagine instead of manual cleaning, a self-cleaning mechanism is deployed where the cleaning machine derives power from the solar panel itself. This model implies that the calculated payback period is achieved faster than model presented in [

τ = C + X 365 i s β − l o s s ′ soiling − l o s s cleaning . #(16)

If we compare this with [

τ ′ = C + X 365 i s β − l o s s soiling − l o s s cleaning . #(17)

Obviously, l o s s ′ soiling is less than l o s s soiling by G amount (Equation (15)) giving τ < τ ′ indicating a simple payback period is achieved faster. It is to be noted that if self-cleaning mechanism is deployed, the l o s s cleaning is calculated by equivalently converting power taken to clean the device.

3) Sensible Cleaning Frequency: The other aspect that we want to introduce is Sensible Cleaning Frequency of solar module. It is defined when loss from PV module in a single day is exactly equal to cleaning cost (or equivalent energy) of the module. This is obtained by equating the financial loss at n^{th} day (Equation (6)) with total cost of cleaning i.e.

[ 48 ( n − 1 ) + s 2 ] i η 0 s β α ′ = P . (18)

Upon solving for n, and letting n = N s for sensible cleaning frequency, we get

N s = 1 + P i s β α − s 48 . (19)

After N s days, the loss due to soiling in a single day is greater than cost of cleaning. Therefore, it is more sensible to clean panels.

4) Critical Cleaning Frequency & Limit in Lifetime for Solar Technology: We define critical cleaning frequency as cleaning period above which there is no financial profit in installing solar panels. This is obtained by taking ratio of total cost of installing solar modules with total financial loss in T years such that the ratio is greater or equal to lifetime of solar modules. So, by following from Equations (6)-(9), the total loss in T years in terms of α is:

T 365 2 × 24 i α s β [ 24 ( N − 1 ) + s ] + T 365 N P . #(20)

Let C and X be cost of PV with installation and cost of cleaning machine respectively, and if we consider battery and inverter working until the solar panel lifetime (meaning no replacement and associated cost). Also, we assumed that solar panel works constantly throughout its lifetime at same constant efficiency and only loss is due to soiling. The financial gain from installation of solar panel of capacity i with average s sunshine hour and β price per kWh in T years is 365 T i s β . We want to equate total production including loss with total cost for solar module system in T years as:

T = C + X 365 i s β − 365 48 i α s β [ 24 ( N − 1 ) + s ] − 365 N P . #(21)

which is quadratic equation in N,

N = ( A − s A 24 + B ) ∓ ( A − s A 24 + B ) 2 − 1460 A P 2 A . #(22)

with, A = 365 i s β α 2

B = 365 i s β − C + X T .

For valid solution in N, we must have

( A − s A 24 + B ) 2 − 1460 A P ≥ 0 . #(23)

For minimum solution,

A − s A 24 + B = 1460 A P . #(24)

T min = C + X 365 i s β + A − 1460 A P − s A 24 . #(25)

This T min (Minimum Payback Period) carries two main information. T min is directly proportional to the value of α meaning if we are in environment full of dust, the payback period increases. Also, if there is new solar technology, it must have at least T min life period (based on Equation (25)). Another aspect that above expression provides the manufacturer company who is planning to introduce self-cleaning technology to decide how much value for X to be put to optimize the payback period.

Similarly, in the Equation (22) taking positive roots,

N critical = ( A − s A 24 + B ) + ( A − s A 24 + B { function of T life-cycle } ) 2 − 1460 A P 2 A . (26)

Here, in the B term, if we use T as total life cycle of give solar technology (for example, in case of Silicon-based Solar cells, T = 20 years), we can get critical cleaning frequency suggesting above which there is no pay-back from installing PV system.

We have introduced hourly average loss in efficiency because of which we were able to dive into approximation more closely. We did get better results in terms of total power production in a year or less financial loss in a year. In fact, the difference between the losses is discussed in the discussion part and is plotted (G parameter in Equation (15)) as a function of average daily loss in efficiency in

In discussion part, we have also derived Minimum Payback Period (T_{min} in Equation (25)) that emphasizes the newer solar technology to have minimum of T_{min} in an environment set by α parameter. This model sets limit as we have considered no other losses in efficiency except that PV performance degrades every year by 0.5% [

Place (α) | N_{opt} (days) | N_{s} (days) | N_{critical} (days) |
---|---|---|---|

California (0.051%) | 44.28 | 981.28 | 1679.9 |

Chile (0.14%) | 26.72 | 358.03 | 611.75 |

Qatar (0.55%) | 13.48 | 91.80 | 155.43 |

We have introduced a model with α ′ which is absolute loss in conversion efficiency per hour allowing us to closely correlate the dust accumulation patterns and more accurately calculated the value for optimal, sensible and critical cleaning frequency. In addition to this, the financial savings due to installing a solar panel are achieved during sunshine hours only and this model addresses this, thereby accounting the losses more accurately. We have also calculated the payback period for installing a self-cleaning mechanism and provided a formalism to optimize its pricing for shorter payback period. Also, we have discussed how environmental factors (dust deposition) set the minimum lifetime for newer solar technologies to obtain financial benefit upon installing them. This calculation is useful to set up automatic cleaning systems to certain frequency per year depending on the value of average daily loss in efficiency at given area and to manufacture companies for deciding the cost of automatic cleaning systems (if available).

It should be noted that our model is based on linear degradation which might not be the exact case in real life. More experimental effort in determining the trend of loss due to soiling might help to accurately estimate above mentioned parameters. The only degradation in efficiency on solar panels due to soiling is assumed in this model which opens opportunity for further researches by including loss in efficiency due to other factors. Also, for payback period, more sophisticated models with discount rate and inflation rates can be incorporated better estimation.

The authors declare no conflicts of interest regarding the publication of this paper.

Karkee, R. and Khadka, S. (2019) Closer Approximation to Optimize Solar Panels Performance with Cleaning Cycle: A Follow-Up. Open Journal of Energy Efficiency, 8, 166-178. https://doi.org/10.4236/ojee.2019.84010

If α ′ is the hourly average of absolute loss in solar conversion efficiency loss, then the solar conversion efficiencies can be calculated as:

η 1 = η 0 − s α ′ ,

η 2 = η 0 − 24 α ′ ,

η 3 = η 0 − ( 24 + s ) α ′ ,

η 4 = η 0 − 48 α ′ ,

η 5 = η 0 − ( 48 + s ) α ′ .

and so on.

The dust accumulation is considered linear, so the financial loss at the end of first s sunshine hours is given as:

L o s s 1 s = ( η 0 − η 01 a v ) i η 0 s β ,

where η 01 a v = ( η 0 + η 1 ) / 2 .

Calculation for η 0 − η 1 a v follows as:

η 0 − η 01 a v = η 0 − η 0 + η 1 2 = η 0 − η 0 + η 0 − s α ′ 2 = s α ′ 2 ,

i.e.

L o s s 1 s = s α ′ 2 i η 0 s β .

After the s sunshine hours for the second day, L o s s 2 s = ( η 0 − η 23 a v ) i s β with η 23 a v = ( η 2 + η 3 ) / 2 .

Calculation for ( η 0 − η 23 a v ) follows as:

η 0 − η 23 a v = η 0 − η 2 + η 3 2 = η 0 − η 0 − 24 α ′ + η 0 − ( 24 + s ) α ′ 2 = ( 48 + s ) α ′ 2 ,

i.e.

L o s s 2 s = ( 48 + s ) α ′ 2 i η 0 s β .

Similarly, L o s s 3 s = ( 96 + s ) α ′ 2 i s β which leads us the generalization rule Equation (6).

The total financial loss for N days, L 0 can be calculated by summing over the financial losses incurred during each day.

So,

L 0 = ∑ n = 1 N [ 48 ( n − 1 ) + s 2 ] i η 0 s β α ′ = i η 0 s β α ′ ∑ n ′ = 0 N − 1 [ 48 n ′ + s 2 ] = i η 0 s β α ′ ( s 2 + ∑ n ′ = 1 N − 1 [ 48 n ′ + s 2 ] ) = i s β α ′ 2 η 0 ( s + 48 ( N − 1 ) N 2 + s ( N − 1 ) ) = i s β α ′ 2 η 0 ( s N + 24 N ( N − 1 ) ) .

Finally, the financial loss per annum is calculated as:

L 1 = 365 N L 0 .

The optimal number of days between cleaning cycles, N o p t is also the minimum value of N which calculated by differentiating Equation with respect to N and equating it to zero.

d L d N = 0 .

Or,

d d N ( 365 2 i η 0 s β α ′ [ 24 ( N − 1 ) + s ] + 365 N P ) = 0 ,

365 2 i η 0 s β α ′ × 24 − 365 N 2 P = 0 ,

365 × 12 i η 0 s β α ′ = 365 N 2 P ,

N o p t = η 0 P 12 i s α ′ β .