Estimating the maximum energy that can be generated by solar power plant using unity power curve method.

1. Estimating the maximum energy that can be generated by solar power plant
using unity power curve method
J. Amala Joseph, Graduated Engineering Trainee
Sterling and Wilson Pvt. Ltd
E-mail: josephamal50@gmail.com
Abstract
This paper briefly explains the method called “Unity Power Curve” method to find the maximum
energy of any solar power plant that can be generated for a day. The Unity power curve is named for the
curve which is obtained from the perfect Power – Time data of the solar power plant. That is, the Power
and Time value of a solar plant taken and the power values were converted to unity (i.e. the percentage of
power generated on that day was calculated). From the resultant data, a curve was drawn and the
equation of the curve was found. By using this equation of the curve area of the curve was calculated and
the resultant equation is the Unity power curve formula.
By using the resultant equation or formula, we can easily estimate the maximum energy a solar plant
can be generated when there is no dust on the modules and weather is good and clear sunny.
Keywords: Parabola, SPV- Solar Photo Voltaic, Unity Power Curve,
1. Introduction
We all know that solar power in India is a fast-growing industry. Solar energy is a potential solution to the
environmental problems being caused by fossil fuels. When fossil fuels are burned to generate electricity,
they release harmful greenhouse gases into the atmosphere. The vast majority of scientists believe that
continuing to depend on fossil fuels is going to cause serious environmental problems in the future.
Another important use for solar energy is in satellites. Many satellites are engineered with photovoltaic
panels, which capture sunlight and convert it into electricity that is used to power the satellite. Solar
power is also useful in areas where standard electricity is not available. For example, research facilities in
Antarctica depend on sustainable energy sources, such as the sun and wind turbines, to generate power.
In the solar industries one of the first question they ask us is, How much the solar plant can generate on a
day. We may wonder but there is an idea through which we can predict the energy that can a solar can be
generated for one day by using the AC capacity of the plant (which does not include the losses e.g,
Temperature loss, Dust on the modules, etc,…).
2. Unity Power Curve
The study was done from three solar SPV plants in different locations. From the analysis of power time
data of these plants some similarities were found. The similarity is all the plant have similar data if we
convert those data into percentage or unity.
Kindly refer the following table for your references.

2. 1) Plant 1 (3MW)
Time Pactive Punity = Pactive/2425.508
6:00 0.0 0.00
7:00 105.1 0.04
8:00 631.5 0.26
9:00 1308.0 0.54
10:00 1864.7 0.77
11:00 2227.4 0.92
12:00 2342.2 0.97
12:30 2419.4 1.00
13:00 2425.5 1.00
15:00 1941.1 0.80
16:00 1418.8 0.58
17:00 761.9 0.31
18:00 161.2 0.07
19:00 0.0 0.00
Table 1
2) Plant 2 (3MW)
Time Pactive Punity = Pactive/2421
6:00 0.0 0.00
7:00 115.7 0.05
8:00 657.8 0.28
9:00 1340.2 0.56
10:00 1869.6 0.79
11:00 2209.8 0.93
12:00 2382.7 1.00
12:30 2358.2 0.99
13:00 2358.5 0.99
14:00 2213.7 0.93
15:00 1880.5 0.79
16:00 1335.8 0.56
17:00 745.8 0.31
18:00 173.2 0.07
19:00 0.0 0.00
Table 2

3. 3) Plant 3 (1MW)
Time Pactive Punity = Pactive/780.614
6:00 0.0 0.00
7:00 32.4 0.04
8:00 241.1 0.31
9:00 458.9 0.59
10:00 618.0 0.79
11:00 720.4 0.92
12:00 759.4 0.97
13:00 751.3 0.96
14:00 680.5 0.87
15:00 580.2 0.74
16:00 429.0 0.55
17:00 232.0 0.30
18:00 35.2 0.05
19:00 0.0 0.00
Table 3
Thus by comparing the values of all unity power value and the average of all can be shown as,
Time Plant 1 Plant 2 Plant 3 Average
6:00 0.00 0.00 0.00 0.00
7:00 0.04 0.04 0.05 0.04
8:00 0.26 0.32 0.28 0.28
9:00 0.54 0.59 0.56 0.56
10:00 0.77 0.80 0.79 0.78
11:00 0.92 0.93 0.93 0.93
12:00 0.97 0.99 1.00 0.98
12:30 1.00 1.00 0.99 1.00
13:00 1.00 0.98 0.99 0.99
14:00 0.88 0.89 0.93 0.90
15:00 0.80 0.77 0.79 0.79
16:00 0.58 0.57 0.56 0.57
17:00 0.31 0.31 0.31 0.31
18:00 0.07 0.05 0.07 0.06
19:00 0.00 0.00 0.00 0.00
Table 4

4. The curve can be drawn from the above data,
Graph 1: Plot of the Table 4
3. Finding the equation of the Unity Power Curve
Consider the following unity table.
Time Average
6:00 0.00
7:00 0.04
8:00 0.28
9:00 0.56
10:00 0.78
11:00 0.93
12:00 0.98
12:30 1.00
13:00 0.99
14:00 0.90
15:00 0.79
16:00 0.57
17:00 0.31
18:00 0.06
19:00 0.00
Table 5
0.00
0.20
0.40
0.60
0.80
1.00
1.20

5. The curve shown in Graph 1 & 2 is similar to the parabolic curve. Hence considering the vector form of
the following parabola equation, [1], [2], [3]
( ) (1)
where,
– Focus of parabola
(h, k) – Vertex of the curve
(x, y) – Any point on the curve
Graph 2: Plot of the Table 5
Consider (h, k) = (12.5, 1), i.e. at 12:30 the generation reaches maximum,
Then the equation (1) will be,
( ) (2)
To find the focus ( ,0) of the curve, we have to substitute any point from the curve. Say, (x, y) = (16,
0.57)
Then the equation (2) will be,
( )

6. ( )
Substituting the value of in equation (1) and we get,
( ) (3)
Here =1, because Peak of this curve is 1.
Therefore the above expression is the equation of Unity power curve.
Then we need to find the area of the curve. The formula for the area of any curve for an interval ( , ) can
be given as,
∫ ( ) (4)
Let’s take our equation (3) as ( ),
( )
( ) ( )
Then, the equation (4) can be written as,
∫ ( ( ) ) (5)
Then the maximum units that a solar power plant that can be generated can be given as,
(6)
where,
– Estimated Energy generated
– Energy obtained from unity power curve
– Maximum power reached in that day
4. Proof with an Example
Example 1:
Let us take a real time data from Plant 3 (1MW) dated 29th
March 2017 and the data were listed below,

7. Total Inverter Generation – 5061
Plant Start Time ( ) – 6:40 = 6.67Hrs [10]
Plant Stop Time ( ) – 18:20 = 18.33Hrs [11]
Maximum Power Reach. ( ) – 713.923 kW
Time at Reach. ( ) – 12.20 = 12.33Hrs [12]
Hence the equation (5) can be given as,
∫ ( ( ) )
Solving the above equation,
[14]
Substituting the above value in equation (6)
Therefore,
Example 2:
Let us take a another real time data from Plant 3 (1MW) dated 30th
March 2017 and the data were listed
below,
Total Inverter Generation – 5375
Plant Start Time ( ) – 6:40 = 6.67Hrs [10]
Plant Stop Time ( ) – 18:20 = 18.33Hrs [11]
Maximum Power Reach. ( ) – 747.833 kW
Time at Reach. ( ) – 13.00 = 13Hrs
Hence the equation (5) can be given as,
∫ ( ( ) )
Solving the above equation,
[15]

8. Substituting the above value in equation (6)
Therefore,
Therefore 200 units difference is there. Thus we conclude that tolerance we can assume and
predict the energy generation.
5. Conclusion
Thus by using the above formula, we can predict the energy that can be generated by the solar power
plant, by using the AC capacity of the plant and generation up and down time of the plant.
Even though the values obtain from this formula is approximation we can conclude with the nearby value.
6. References
[1]http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php
[2]https://www.mathsisfun.com/geometry/parabola.html
[3]https://www.mathsisfun.com/data/grapher-equation.html
[4]http://www.mathsisfun.com/data/grapher-equation.html?func2=y%3D-0.0351(x-
12.5)%5E2%2B1&xmin=1.044&xmax=21.58&ymin=-7.338&ymax=8.066
[5]https://www.symbolab.com/solver/derivative-calculator/%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft(-
0.0351%5Cleft(x-h%5Cright)%5E%7B2%7D%2Bk%5Cright)
[6]http://www.derivative-calculator.net
[7]https://www.emathhelp.net/calculators/calculus-2/average-value-of-a-function-calculator/?f=-
0.0351%28x-12.5%29%5E2%2B1&a=7&b=18&steps=on
[8]https://www.emathhelp.net/calculators/calculus-2/definite-integral-calculator/?a=7&b=18&f=-
351%2A%28x%20-%2025/2%29%5E2/110000%20%2B%201/11&steps=on&var=x
[9]https://plot.ly/create/
[10]http://www.calculatorsoup.com/calculators/time/time-to-decimal-
calculator.php?hours=6&minutes=40&seconds=00&action=solve
[11]http://www.calculatorsoup.com/calculators/time/time-to-decimal-
calculator.php?hours=18&minutes=20&seconds=0&action=solve

9. [12]http://www.calculatorsoup.com/calculators/time/time-to-decimal-
calculator.php?hours=12&minutes=20&seconds=0&action=solve
[13]https://www.emathhelp.net/calculators/calculus-2/definite-integral-calculator/?f=-0.0351%28x-
12.33%29%5E2%2B1&var=x&a=6.67&b=18.33&steps=on
[14]https://www.emathhelp.net/calculators/calculus-2/definite-integral-calculator/?f=-0.0351%28x-
12.33%29%5E2%2B1&var=x&a=6.67&b=18.33&steps=on
[15]https://www.emathhelp.net/calculators/calculus-2/definite-integral-calculator/?f=-0.0351%28x-
13%29%5E2%2B1&var=x&a=6.67&b=18.33&steps=on