Elliptical Orbits Mode Application for Approximation of Fuel Volume Change


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 316-331 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: December 19, 2021 Reviewed: January 09, 2022 Accepted: January 10, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i1.14407      

Elliptical Orbits Mode Application for Approximation of Fuel 
Volume Change 

Jovian Dian Pratama*, Ratna Herdiana*, Susilo Hariyanto 

Department of Mathematics, Diponegoro University  
 

*Corresponding Author 
Email: joviandianpratama@yahoo.com*,  

herdiana.math@gmail.com*, sus2_hariyanto@yahoo.com 
 

ABSTRACT 

At a 45.507.21 Candirejo Tuntang gas station, it is difficult to ensure the stock of fuel supplies 
because there is always a difference between calculations using dipsticks and fuel dispensers. 
Because the calculation method used by gas stations throughout Indonesia is linear interpolation 
which is not smooth, then by using the pertalite (pertamina fuel products) measuring book data a 
smooth volume change approximation function will be formed. This article presents the Elliptical 
Orbits Mode (EOM) as a proposed method in approximating the function that describes the 
volume change of fuel with respect to fuel height in Underground Tank (UT). Since the calculation 
by the gas station is not smooth, it is necessary for a smoother data fitting by considering Residual 
Square Error (RSS) and Mean Square Error (MSE). The results of the Elliptical Orbits Mode 
approximation will be compared with the circle orbits mode and least square data fitting. The 
result show that EOM(θ) method with elliptical height control produces smaller RSS and MSE 
compared to using COM, EOM, Least Square degree two and three. In next research, the 
approximation results will be applied to the fuel dispenser data. 

Keywords: Orbits Mode; Data Fitting; Ellipse; Fuel; Approximation 

INTRODUCTION 

Based on the assumptions given in [1] and [2] the previous Orbits Mode Data 
Fitting research which was used to calibrate the Dipstick measuring instrument that 
converts the height to the volume of fuel in the buried tank, it is explained that the 
approximation function of the change in fuel volume in the tank is only based on height. 

In [1] and [2] it is explained that the Orbits Mode Data Fitting-based calibration is 
limited by several assumptions and field conditions, including the following: 
1. The resulting approximation function is the change in the volume of fuel in the UT 

which only depends on the variable height of the fuel in the UT. 
2. Orbits Mode Data Fitting proposed by the author is used only for the distribution of 

data that forms a semicircle or ellipse in the first quadrant. 
3. The data to be approximated is the fuel measurement manual in the UT from the 

Semarang Regency Metrology Agency. 

http://dx.doi.org/10.18860/ca.v7i1.14407
mailto:joviandianpratama@yahoo.com


Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 317 

4. It is assumed that UT is not tilted or flat during measurement, including tank trucks 
that deliver fuel to gas stations or are filling supplies at gas stations. 

5. UT for the right and left have the same shape alias symmetrical, due to the second 
assumption. 

In [3] data fitting is applied to approximate the shape of an island on the map, then in 
[4] the curve fitting which is used to detect enlarged and shrinking eye retina, research in 
[3] and [4] has their additional algorithm to approach the desired result, which will also 
be applied to the Orbits Mode Data Fitting starting from the proposed method and the 
object applied to calibration is something new. 

In [5] Hyper Least Square or HyperLS was also introduced and [6] also calibrated data 
in the form of curves but using an orthogonal matrix where the more data the more 
complicated, so the method will be difficult for large data. 

From [7] there is a design drawing of a buried tank where the tank is in the form of a 
capsule tube with a cross section that is not flat or protruding so that according to [8] also, 
changes in volume in the tank tend to form a semicircle or half an ellipse or a parabola. 

The approximation function used by gas stations throughout Indonesia is linear 
interpolation which is not smooth, then by using the pertalite (pertamina fuel products) 
measuring book data a smooth volume change approximation function with Elliptical 
Orbits Mode will be formed, and then will be any improvement on ellips height control to 
minimize Residual Sum of Square (RSS) and Mean Square Error (MSE), where the data 
used is the change in the volume of fuel in the tank based on changes in the height of the 
fuel in the UT in units (cm) and will be converted to fuel volume (liter). 

Therefore, the author proposes method because the calculation is simpler for small and 
large data and is smoother, although only for data that tends to be semicircular or 
elliptical, to approximate the fuel volume with minimized errors. 

Orbits Mode Data Fitting is a method proposed by the author in approximating the 
function of the data which tends to be in the form of a semi-circle or half an ellipse. In [1] 
and [2] the author introduced the Orbits Mode Data Fitting method only in a circle shape, 
then compared it with Cubic Spline Interpolation and Least Square Data Fitting, but this 
time the authors made the Orbits Mode Data Fitting method in the shape of an ellipse too, 
because in the value approach there is a volume of fuel which has not been detected in the 
function. 
 
Definition 1 (Ellipse Equation) 
In [9] the ellipse equation is presented in equation (1) which (𝑝,𝑞) is the center point of 
the ellipse with the major and minor axes adjusting 𝑎 and 𝑏, 

(𝑥 −𝑝)2

𝑎2
+
(𝑦 −𝑞)2

𝑏2
= 1 (1) 

Definition 2 (𝑨𝒊 set for Ellipse Mode) 
A set 𝐴𝑖 of points formed from two ellipse equations is defined as follows, 

𝐴𝑖 = {(𝑥,𝑦) | 𝑑𝑖1
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑𝑖2
 },  (2) 

with 𝑖 = 1,2,…,𝑘. the set (2) it can be visualized as follows, 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 318 

 
Figure 1. Set Visualization 𝐴𝑖 (2) 

with 𝑥𝑤
2
= (𝑥 −

𝑤

2
) and 𝑤 = height of UT and 𝑙 = a half of maximum fuel change in UT 

Defined 
𝑤

2
(√𝑑𝑖2 −√𝑑𝑖1) =

𝑙

2
(√𝑑𝑖2 −√𝑑𝑖1) = 𝑡𝑖 the thickness of the ellipse from the 

partition interval taken from the maximum and minimum values of the volume change 
[Δ𝑉(ℎ)𝑚𝑖𝑛,Δ𝑉(ℎ)𝑚𝑎𝑥] will be divided by several partitions where the thickness with the 
most points is sought, then 
 

[Δ𝑉(ℎ)𝑚𝑖𝑛,Δ𝑉(ℎ)𝑚𝑎𝑥] = [𝑑11,𝑑12]∪ [𝑑21,𝑑22]∪ [𝑑31,𝑑32] ∪…∪[𝑑𝑘1,𝑑𝑘2] (3) 
 
with 𝑑𝑖2 = 𝑑(𝑖+1)1 and the intersection of the respective sub-blankets of the minimum and 

maximum volume intervals in equation (3) is denoted for each [𝑑𝑖1,𝑑𝑖2]∩
[𝑑(𝑖+1)1,𝑑(𝑖+1)2] is equal to 𝑑𝑖2 or 𝑑(𝑖+1)1, where 𝑖 = 1,2,…,𝑘 with 𝑘 is the number of 
blankets dividing the maximum and minimum intervals of the volume change. 
 
Definition 3 (Partition of 𝑨𝒊 set) 
Partition of 𝐴𝑖 set that divide 𝐴𝑖 set to become partitions or sets of points between 2 
ellipses equations, defined as follows, 

𝐴1 = {(𝑥,𝑦) | 𝑑11
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑12
 } 

𝐴2 = {(𝑥,𝑦) | 𝑑21
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑22
 } 

𝐴3 = {(𝑥,𝑦) | 𝑑31
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑32
 } 

… 

(4) 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 319 

𝐴𝑘 = {(𝑥,𝑦) | 𝑑𝑘1
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑𝑘2
 } 

set of partitions (4) can be described as follows, 

 
Figure 2. The visualization of partitions  set 𝐴1,𝐴2,𝐴3 up to 𝐴𝑘 (4) 

 
Definition 4 (Elliptical Orbits Mode) 
Elliptical Orbits Mode was choose partition has the most points, defined as follows: 

𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑀𝑎𝑥(𝑛(𝐴1),𝑛(𝐴2),𝑛(𝐴3),…,𝑛(𝐴𝑘)) (5) 

with 𝑖 = 1,2,…,𝑘. If there is a condition where 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴𝑘1) = ⋯ = 𝑛(𝐴𝑘𝑚), 

then the average ellipses scale 𝐴𝑘1,𝐴𝑘2,…,𝐴𝑘𝑚 is taken so 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴𝑚
̅̅ ̅̅ ), 

Therefore, the inequality whose ellipse will change is defined 𝐴𝑚̅̅ ̅̅  as follows, 

𝐴𝑚̅̅ ̅̅ = {(𝑥,𝑦) |(
𝑑𝑘11 +⋯+𝑑𝑘𝑚1

𝑚
)

 

< (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< (
𝑑𝑘12 +⋯+𝑑𝑘𝑚2

𝑚
)

 

} 

The next step, because we have obtained 𝐴𝑖 or 𝐴𝑚̅̅ ̅̅  , then we approximate ellipse equation, 
divided which can be devide into two cases: 
 
Case 1, [𝑴𝒂𝒙(𝒏(𝑨𝒊)) = 𝒏(𝑨𝒎)] 

Based on the set with the maximum number of points between 2 ellipses, choose 𝐴𝑚 =

{(𝑥,𝑦) | 𝑑𝑚1
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑𝑚2
 }, so that we get: 

𝑑𝑚1
 < (

𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< 𝑑𝑚2
  

⟹ (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

= (
𝑑𝑚1 +𝑑𝑚2

2
)
 

 

⟹ 𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑥𝑤
2

(
𝑤

2
)
)

2

 

(6) 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 320 

The steps in (6) can be visualized as follows, 

 
Figure 3. Visualization Steps in (6) 

Therefore, from (6) the result of the elliptical orbital mode is a semicircular function by 
substituting 𝑥𝑤

2
= 𝑥𝑑𝑚, as follows: 

𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑥𝑑𝑚

(
𝑤

2
)
)

2

 (7) 

with 𝑥 = the fuel level in UT and 𝑥𝑑𝑚 = 𝑥 −(
𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
. 

 
Theorem 1 (Defined Intervals for Elliptical Orbits Mode Approximation Function) 

If 𝑥𝑤
2
= 𝑥𝑑𝑚 is substituted to 𝑦 = (

𝑙

2
)√(

𝑑𝑚1+𝑑𝑚2

2
)
 

−(
𝑥𝑤
2

(
𝑤

2
)
)

2

 then 𝑦 is defined in ℝ in the 

interval 0 ≤ 𝑥 ≤
𝑤

2
−(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
. 

 
Proof: 

Substitute 𝑥𝑤
2
= 𝑥𝑑𝑚 to 𝑦 = (

𝑙

2
)√(

𝑑𝑚1+𝑑𝑚2

2
)
 

−(
𝑥𝑤
2

(
𝑤

2
)
)

2

 so that it is obtained: 

𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑥𝑑𝑚

(
𝑤

2
)
)

2

⟺ 𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−

(

 
𝑥 −(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2

(
𝑤

2
)

)

 

2

 

⟺ 𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−

(

 
𝑥2 −𝑤𝑥√

𝑑𝑚1+𝑑𝑚2

2
+(

𝑤

2
)
2

(
𝑑𝑚1+𝑑𝑚2

2
)

(
𝑤

2
)
2

)

  



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 321 

⟺ 𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑑𝑚1 +𝑑𝑚2

2
)
 

−

(

 
𝑥2 −𝑤𝑥√

𝑑𝑚1+𝑑𝑚2

2

(
𝑤

2
)
2

)

  

⟺ 𝑦 = (
𝑙

2
)√
𝑥(𝑤√

𝑑𝑚1+𝑑𝑚2

2
−𝑥)

(
𝑤

2
)
2

 

⟺ 𝑦 = (
𝑙

𝑤
)√𝑥(𝑤√

𝑑𝑚1 +𝑑𝑚2
2

−𝑥) 

with 𝑤,𝑑𝑚1,𝑑𝑚2 ∈ ℝ
+. Obviously 𝑦 is defined in ℝ since 𝑥(𝑤√

𝑑𝑚1+𝑑𝑚2

2
−𝑥) ≥ 0, then 

𝑥 must be on both interval 0 ≤ 𝑥 ≤  𝑤√
𝑑𝑚1+𝑑𝑚2

2
 and 0 ≤ 𝑥 ≤

𝑤

2
−(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
,  because 

𝑤√
𝑑𝑚1+𝑑𝑚2

2
>
𝑤

2
−(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
. ∎ 

 

Substitute 𝑥𝑤
2
= 𝑥𝑑𝑚 so that the value 𝑦 is defined in 0 ≤ 𝑥 ≤

𝑤

2
−(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
, so that 

(7) the function of volume change to fuel level in the UT can be visualized as follows, 

  
Figure 4. Visualization of Steps in (6) to (7) 

 
The function is formed from the half ellipse selected for the calibration of the buried tank 
whose cross-sectional area is circular but convex, so that the volume of UT will be 
calculated as a function of the change in volume with respect to height, which then the 
coordinates are taken from the change in units (cm) that will be converted to volume per 
centimeter (liter/cm). 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 322 

 
 
Case 2, [𝑴𝒂𝒙(𝒏(𝑨𝒊)) = 𝒏(𝑨𝒌𝟏) = ⋯ = 𝒏(𝑨𝒌𝒎)] 

Based on the set with the maximum number of points between two ellipses equations, 

𝐴𝑚̅̅ ̅̅ = {(𝑥,𝑦) |(
𝑑𝑘11+⋯+𝑑𝑘𝑚1

𝑚
)
 

< (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< (
𝑑𝑘12+⋯+𝑑𝑘𝑚2

𝑚
)
 

}, then: 

(
𝑑𝑘11 +⋯+𝑑𝑘𝑚1

𝑚
)

 

< (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

< (
𝑑𝑘12 +⋯+𝑑𝑘𝑚2

𝑚
)

 

 

⟹ (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

= (
(
𝑑𝑘11+⋯+𝑑𝑘𝑚1

𝑚
)+(

𝑑𝑘12+⋯+𝑑𝑘𝑚2

𝑚
)

2
)

 

 

⟹ (
𝑥𝑤
2

(
𝑤

2
)
)

2

+(
𝑦

(
𝑙

2
)
)

2

= (
𝑑𝑚1 +𝑑𝑚2

2
)
 

. 

(8) 

The steps in (8) are visualized similarly in Figure 11 with 𝑑𝑚1 = (
𝑑𝑘11+⋯+𝑑𝑘𝑚1

𝑚
) and 𝑑𝑚2 =

(
𝑑𝑘12+⋯+𝑑𝑘𝑚2

𝑚
), then 

⟹ 𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑥𝑤
2

(
𝑤

2
)
)

2

. (9) 

Therefore, the result of the orbital mode ellipse is a half-ellipse function, as follows: 

𝑦 = (
𝑙

2
)√(

𝑑𝑚1 +𝑑𝑚2
2

)
 

−(
𝑥𝑑𝑚

(
𝑤

2
)
)

2

, (10) 

with 𝑥 = is the fuel level in UT and 𝑥𝑑𝑚 = 𝑥 −(
𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
. 

As stated Theorem 1 about defined interval for (10) substituted 𝑥𝑤
2
= 𝑥𝑑𝑚 so that 𝑦 is 

defined values since 0 ≤ 𝑥 ≤
𝑤

2
−(

𝑤

2
)√

𝑑𝑚1+𝑑𝑚2

2
, we get (10) the function of the change in 

volume to the fuel level in the UT which is visualized similarly in Figure 4.  
 

METHODS  

Research Steps 
1. Construction of mathematical model Elliptical Orbits Mode methods with the 

following steps: 

a) Construction of mathematical model Elliptical Orbits Mode. 

 Review for Ellipse Equation 

 Define 𝐴𝑖 set for Ellipse Mode 

 Make Partition for 𝐴𝑖 set (Definition) 

 Choose Partitions of set 𝐴𝑖 with 𝑛(𝐴𝑖) is the maximum value 

 Divide onto two cases singular and plural maximum value 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 323 

 Create function 𝑦 = 𝑓(𝑥) from chosen 𝐴𝑖 sets 

 Translate 𝑦 = 𝑓(𝑥) so that 𝑦 is defined on 0 and so on, and 

 controlling height to find ellipse’s height or vertical axis that minimized 

Residual Sum of Square (RSS) and Mean Square Error (MSE). 

2. Construction of mathematical model Elliptical Orbits Mode will be applied on Data 

from Candirejo Gas Station, measuring book from Government Metrology Agency 

specific for Pertalite (Fuel Product of Pertamina) only and visualize it. 

3. Measuring Performance using RSS [10] and MSE [11,12] and compare with Circle 

Orbits Mode from [1], Least Square with n = 2 and n = 3 from [13], and Elliptical 

with Height Control. 

RESULTS AND DISCUSSION  

Pertalite Measuring Book Data 

A calculation of gas station 45.507.21 Candirejo using a measuring book from 
Government Metrology Agency to determine the volume of fuel in the buried tank, so the 
authors are just obtained the following data which is not proceed by authors, we need this 
data from Metrologi Agency as constructor of Approximation function, as follows: 
 

Table 1. Fuel Volume Measuring Book Data for Pertalite Tanks from Metrology Agency 
Height 

(x) 
Volume Diff (y) 

Height 
(x) 

Volume Diff (y) 
Height 

(x) 
Volume Diff (y) 

 

0 0.0 0.0 75 7085.9 117.7 150 16124.4 111.1  

1 237.1 237.1 76 7203.5 117.6 151 16235.6 111.2  

2 294.3 57.2 77 7321.2 117.7 152 16346.7 111.1  

3 351.4 57.1 78 7438.8 117.6 153 16457.8 111.1  

4 409.4 58.0 79 7556.5 117.7 154 16568.9 111.1  

5 468.2 58.8 80 7674.1 117.6 155 16680.0 111.1  

6 527.1 58.9 81 7791.8 117.7 156 16791.1 111.1  

7 586.1 59.0 82 7909.4 117.6 157 16902.2 111.1  

8 646.7 60.6 83 8027.1 117.7 158 17013.3 111.1  

9 707.3 60.6 84 8144.7 117.6 159 17124.4 111.1  

10 767.9 60.6 85 8262.4 117.7 160 17232.6 108.2  

11 831.6 63.7 86 8380.0 117.6 161 17337.9 105.3  

12 896.1 64.5 87 8497.6 117.6 162 17443.2 105.3  

13 960.6 64.5 88 8615.3 117.7 163 17548.4 105.2  

14 1026.7 66.1 89 8732.9 117.6 164 17653.7 105.3  

15 1093.3 66.6 90 8855.0 122.1 165 17758.9 105.2  

16 1160.0 66.7 91 8980.0 125.0 166 17864.2 105.3  

17 1228.3 68.3 92 9105.0 125.0 167 17969.5 105.3  

18 1297,2 68.9 93 9230.0 125.0 168 18065.7 96.2  

19 1366.2 69.0 94 9355.0 125.0 169 18161.0 95.3  

20 1439.3 73.1 95 9480.0 125.0 170 18256.2 95.2  

21 1513.3 74.0 96 9605.0 125.0 171 18351.4 95.2  

22 1588,0 74.7 97 9730.0 125.0 172 18446.7 95.3  

23 1668.0 80.0 98 9855.0 125.0 173 18541.9 95.2  



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 324 

Height 
(x) 

Volume Diff (y) 
Height 

(x) 
Volume Diff (y) 

Height 
(x) 

Volume Diff (y) 
 

24 1748.0 80.0 99 9980.0 125.0 174 18637.1 95.2  

25 1830.0 82.0 100 10105.0 125.0 175 18732.4 95.3  

26 1913,3 83.3 101 10230.0 125.0 176 18825.5 93.1  

27 1997,4 84.1 102 10355.0 125.0 177 18916.4 90.9  

28 2084,3 86.9 103 10480.0 125.0 178 19007.3 90.9  

29 2171.3 87.0 104 10605.0 125.0 179 19098.2 90.9  

30 2261.8 90.5 105 10730.0 125.0 180 19189.1 90.9  

31 2352.7 90.9 106 10855.0 125.0 181 19280.0 90.9  

32 2446.7 94.0 107 10980.0 125.0 182 19370.9 90.9  

33 2541.9 95.2 108 11105.0 125.0 183 19458.3 87.4  

34 2637.1 95.2 109 11230.0 125.0 184 19545.2 86.9  

35 2732.4 95.3 110 11355.0 125.0 185 19632.2 87.0  

36 2830.0 97.6 111 11480.0 125.0 186 19719.1 86.9  

37 2930.0 100.0 112 11605.0 125.0 187 19804.0 84.9  

38 3030.0 100.0 113 11730.0 125.0 188 19884.0 80.0  

39 3130.0 100.0 114 11855.0 125.0 189 19964.0 80.0  

40 3230.0 100.0 115 11980.0 125.0 190 20044.0 80.0  

41 3330.0 100.0 116 12105.0 125.0 191 20124.0 80.0  

42 3430.0 100.0 117 12230.0 125.0 192 20202.2 78.2  

43 3530.0 100.0 118 12355.0 125.0 193 20276.3 74.1  

44 3630.0 100.0 119 12480.0 125.0 194 20350.4 74.1  

45 3730.0 100.0 120 12605.0 125.0 195 20420.0 69.6  

46 3832.6 102.6 121 12730.0 125.0 196 20486.7 66.7  

47 3937.9 105.3 122 12855.0 125.0 197 20553.3 66.6  

48 4043,2 105.3 123 12980.0 125.0 198 20617.5 64.2  

49 4148.4 105.2 124 13105.0 125.0 199 20680.0 62.5  

50 4257.8 109.4 125 13227.1 122.1 200 20742.5 62.5  

51 4368.9 111.1 126 13344.7 117.6 201 20802.9 60.4  

52 4480.0 111.1 127 13462.4 117.7 202 20860.0 57.1  

53 4591.1 111.1 128 13580.0 117.6 203 20917.1 57.1  

54 4702.2 111.1 129 13697.6 117.6 204 20974.3 57.2  

55 4813.3 111.1 130 13815.3 117.7 205 21028.6 54.3  

56 4924.4 111.1 131 13932.9 117.6 206 21082.7 54.1  

57 5035,6 111.2 132 14050.6 117.7 207 21136.8 54.1  

58 5146.7 111.1 133 14168,2 117.6 208 21187.1 50.3  

59 5257.8 111.1 134 14285.9 117.7 209 21222.9 35.8  

60 5368.9 111.1 135 14403.5 117.6 210 21258.6 35.7  

61 5480.0 111.1 136 14521.2 117.7 211 21294.3 35.7  

62 5591.1 111.1 137 14638.8 117.6 212 21330.0 35.7  

63 5702.2 111.1 138 14756.5 117.7 213 21365,7 35.7  

64 5813.3 111.1 139 14874.1 117.6 214 21387.1 21.4  

65 5924.4 111.1 140 14991.8 117.7 215 21399.1 12.0  

66 6035.6 111.2 141 15109.4 117.6 216 21411.0 11.9  

67 6146.7 111.1 142 15227.1 117.7 217 21422.9 11.9  

68 6262.4 115.7 143 15344.7 117.6 218 21434.8 11.9  



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 325 

69 6380.0 117.6 144 15457.8 113.1 219 21446.7 11.9  

70 6497.6 117.6 145 15568.9 111.1 220 21458.6 11.9  

71 6615.3 117.7 146 15680.0 111.1 221 21470.5 11.9  

72 6732.9 117.6 147 15791.1 111.1 222 21482.4 11.9  

73 6850.6 117.7 148 15902.2 111.1 222.3 21486.0 3.6  

74 6968.2 117.6 149 16013.3 111.1     

 

Approximation using Elliptical Orbits Mode (EOM) 

Table 1 is used as sample data; we derive Elliptical Orbits Mode approach as an 
approximation to the change of fuel volume. To obtain a half-ellipse function from the 
smallest to the largest abscissa, choose, 

𝑤

2
=
maximum height on underground tank

2
=
222.3

2
= 111.15 

𝑙

2
= maximum volume change on undergorund tank = 125 

After that, select the difference 𝑑𝑖1 and 𝑑𝑖2 for the prefix of the 𝐴𝑖 set which is 𝑡𝑖 = 0.2 
defined as follows, 

𝐴𝑖 = {(𝑥,𝑦) |0.8 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 1} 

 

with step-size = 0.02 then the number of partitions is obtained 
𝑡𝑖

𝑠𝑡𝑒𝑝−𝑠𝑖𝑧𝑒
= 10, so that the 

partitions are obtained from the 𝐴𝑖set with 𝑖 = 1,2,…,10, 
 

𝐴1 = {(𝑥,𝑦) | 0.98 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 1} 

𝐴2 = {(𝑥,𝑦) | 0.96 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.98} 

𝐴3 = {(𝑥,𝑦) | 0.94 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.96} 

𝐴4 = {(𝑥,𝑦) | 0.92 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.94} 

𝐴5 = {(𝑥,𝑦) | 0.90 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.92} 

𝐴6 = {(𝑥,𝑦) | 0.88 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.90} 

𝐴7 = {(𝑥,𝑦) | 0.86 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.88} 

𝐴8 = {(𝑥,𝑦) | 0.84 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.86} 

𝐴9 = {(𝑥,𝑦) | 0.82 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.84} 

𝐴10 = {(𝑥,𝑦) | 0.80 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.82} 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 326 

 
Accordingly, the value of 𝑛(𝐴𝑖) for 𝑖 = 1,2,…,10 is provided in Table 2. 
 

Table 2. Calculation of Elliptical Orbits Mode𝑛(𝐴𝑖) 
𝑨𝒊 𝒏(𝑨𝒊) 
𝐴1 11 
𝐴2 15 
𝐴3 16 
𝐴4 26 
𝐴5 25 
𝐴6 19 
𝐴7 9 
𝐴8 3 
𝐴9 0 
𝐴10 0 

 

According to Table 2 it is obtained that 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴4), with 𝑖 = 1,2,…,10 the 

selected 𝐴4 set , after that from the 𝐴4 set the following functions 𝑦 = 𝑓(𝑥) will be formed, 

0.92 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.94 ⇒

(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
= (

0.92+0.94

2
) 

0.92 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.94 ⇒

(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
= 0.93 

0.92 <
(𝑥 −111.15)2

(111.15)2
+

𝑦2

(125)2
< 0.94 ⇒ 𝑦 = 125√0.93−

(𝑥 −111.15)2

(111.15)2
. 

Then the translation 𝑦 = 𝑓(𝑥) to be defined at 0 ≤ 𝑥 ≤ 111.15−111.15√0.93 or around 

0 ≤ 𝑥 ≤ 3,96, substitution 𝑥𝑤
2
= (𝑥 −111.5) with 𝑥𝑑𝑚 = (𝑥 −111.15√0.93), so that we 

get: 

𝑦 = 125√0.93−
(𝑥 −111.15√0.93)

2

(111.15)2
 (11) 

with 𝑦 = change in the volume of fuel with respect to fuel height, x., for visualization of the 
graph of changes in the volume of fuel obtained: 
 

 
Figure 5. Graph of Change in Fuel Volume by EOM 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 327 

Based on Figure 5, the EOM results produce a function that is fit to the pertalite volume 
change data, the volume as function of height (h) is then obtained by the following 
integration: 

𝑉(ℎ) = ∫125√0.93−
(𝑥 −111.15√0.93)

2

(111.15)2

ℎ

0

𝑑𝑥 

Using the help of Maple 2015 the integral results are obtained as follows, 
 

𝑉(ℎ)

=
535945891

800000000000000000
√18792746045928961

+
116250000

17040701
√896879arcsin(

10182971929

93000000000000
√896879√93)

+
1

160000000000
(−8094332975000000000000 ℎ2

+1735249799334822290000000 ℎ +18792746045928961)
1

2 ℎ

−
535945891

800000000000000000
(−8094332975000000000000 ℎ2

+1735249799334822290000000 ℎ +18792746045928961)
1

2

+
116250000

17040701
√896879arcsin(

19

93000000000000
√896879√93(5000000 ℎ

−535945891)); 

(12) 

where 𝑉(ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3√0.93. The EOM version of the fuel 
volume calculation uses (12) with 𝑉𝑚𝑎𝑥 = 20.296.55 liters. 
 

Elliptical Orbits Mode with elliptical height control on Data Pertalite 

Based on Figure 5, it can be seen that the Volume change function according to 
EOM will regress more pertalite data if the ellipse height is higher, so it is necessary to 
adjust the ellipse height. 

 The EOM result at (11) has an elliptical height 125 which represents the equation 
so that it has a volume change function with respect to the fuel level in the UT, with the 
general form of (11): 

𝐸𝑂𝑀(𝜃,𝑥) = 𝜃 ∙√0.93−
(𝑥 −111.15√0.93)

2

(111.15)2
 

(13) 

With 𝜃 is the height of the ellipse, defined on the interval 0 ≤ 𝑥 ≤ 222.3√0.93. Choose 𝜃 
the one that minimizes the residual sum square 

𝐸 = ∑ (𝑦𝑖 −𝐸𝑂𝑀(𝜃,𝑥𝑖))
2

⌊222.3√0.93⌋

𝑖=1

 

𝐸 = ∑(𝑦𝑖
2 −2𝐸𝑂𝑀(𝜃,𝑥𝑖)𝑦𝑖 +𝐸𝑂𝑀

2(𝜃,𝑥𝑖))

214

𝑖=1

 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 328 

𝐸 = ∑𝑦𝑖
2

214

𝑖=1

−2∑𝐸𝑂𝑀(𝜃,𝑥𝑖)𝑦𝑖

214

𝑖=1

+∑(𝐸𝑂𝑀(𝜃,𝑥𝑖))
2

214

𝑖=1

 

𝐸 = ∑𝑦𝑖
2

214

𝑖=1

−2∑𝑦𝑖 ∙ 𝜃
√0.93−

(𝑥 −111.15√0.93)
2

(111.15)2

214

𝑖=1

+∑(𝜃√0.93−
(𝑥 −111.15√0.93)

2

(111.15)2
)

2
214

𝑖=1

 

𝐸 = ∑𝑦𝑖
2

214

𝑖=1

−2∑𝑦𝑖 ∙ 𝜃
√0.93−

(𝑥 −111.15√0.93)
2

(111.15)2

214

𝑖=1

+∑𝜃2 (0.93−
(𝑥 −111.15√0.93)

2

(111.15)2
)

214

𝑖=1

 

Then, to find 𝜃 the minimization of 𝐸, find the solution of the equation 
𝜕𝐸

𝜕𝜃
= 0, we get: 

⟺ −2∑𝑦𝑖
√0.93−

(𝑥 −111.15√0.93)
2

(111.15)2

214

𝑖=1

+2𝜃∑(0.93−
(𝑥 −111.15√0.93)

2

(111.15)2
)

214

𝑖=1

= 0 

⟺

−2∑ 𝑦𝑖√0.93−
(𝑥−111.15√0.93)

2

(111.15)2
214
𝑖=1 +2𝜃∑ (0.93−

(𝑥−111.15√0.93)
2

(111.15)2
)214𝑖=1

2√0.93−
(𝑥−111.15√0.93)

2

(111.15)2

= 0 

⟺ −∑𝑦𝑖

214

𝑖=1

+𝜃∑√0.93−
(𝑥 −111.15√0.93)

2

(111.15)2

214

𝑖=1

= 0 

⟺ 𝜃 =
∑ 𝑦𝑖
214
𝑖=1

∑ √0.93−
(𝑥−111.15√0.93)

2

(111.15)2
214
𝑖=1

 

By using the data in Table 1, it is obtained 𝜃 ≈ 130,37 that from (27) it is obtained: 

𝐸𝑂𝑀(130,37,𝑥) = 130,37 ∙√0.93 −
(𝑥 − 111.15√0.93)

2

(111.15)2
 

and visualized the graph of the function of the change in fuel volume and the actual fuel 
volume change in the reservoir as follows: 

 
Figure 6. Graph of Changes in Fuel Volume by 𝐸𝑂𝑀(𝜃) 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 329 

Based on Figure 6 𝑦 = 𝑓(𝑥) the 𝐸𝑂𝑀(𝜃)results produce a function that is more fit to the 
pertalite data than the EOM results in Figure 5, then we calculate the volume function 
𝑉(ℎ) with integral and with (24) obtained: 
 

𝑉𝜃(ℎ) =
130,37

125
∙𝑉(ℎ) 

 
where .𝑉𝜃(ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3. Calculation of the volume of the 
fuel 𝐸𝑂𝑀(𝜃) version has 𝑉𝑚𝑎𝑥 = 21.166,71 liters. 
 
Comparison of approximate visualization results 

Data visualization results from Circle Orbits Mode, Elliptical Orbits Mode, 𝐸𝑂𝑀(𝜃), 
Least Square Data Fitting 𝑛 = 2, and Least Square Data Fitting 𝑛 = 3, as follows: 

 

Figure 7. Comparison Graph of Approximation Method Results 
 

The results of the calculation of changes in the volume of fuel based on the height of the fuel 

in the UT will be applied to the Pertalite Data to search for RSS and MSE from each result. 

Pertalite Data Approximation RSS and MSE Calculation 

Comparison of the results between the proposed method and other methods can be 
seen from the calculation of RSS and MSE with liter unit in Table 3 below: 

Table 3. Pertalite Data Approximation RSS and MSE Calculation 
Method RSS MSE 
𝐶𝑂𝑀 40.390,49 185,28 
𝐸𝑂𝑀 8.529,37 39,67 

𝐿𝑆(𝑛 = 2) 8.980,63 40,09 
𝐿𝑆(𝑛 = 3) 7.574,51 33,81 
𝐸𝑂𝑀(𝜃) 6.415,32 29,84 

 
Based on Table 3 the calculation of COM which has the largest RSS and MSE, for EOM has 
RSS and MSE which is slightly below 𝐿𝑆(𝑛 = 2), but still above 𝐿𝑆(𝑛 = 3), then by 
controlling the height of the ellipse to find 𝜃 that minimizes the RSS we obtain the 
minimum i.e. 6.415,32 and its MSE 29,84. The smallest value RSS and MSE are obtained 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 330 

when using 𝐸𝑂𝑀(𝜃). Pertalite data approximation is not compared to the calculation of 
gas stations and Cubic Spline Interpolation because the RSS and MSE are definitely 0 and 
have unsmooth approximation. 

Defined domain interval and maximum volume 

In the Orbits Mode Data Fitting construction, there is a reduction in the BBM altitude 
domain in the UT, so that it is only defined at a certain height. The results of the 
comparison of the defined domain height and the maximum volume of each 
approximation method are as follows: 

Table 4. Domain Height Intervals and maximum volume approximation method 

Approximation Method Domain Height (cm) 
𝑽𝒎𝒂𝒙  

maximum volume (liter) 
Gas Station Calculation 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.486.00 liter. 

Circle Orbits Mode 0 ≤ ℎ ≤ 217,1 𝑉𝑚𝑎𝑥 = 18.508,85 liter. 

Elliptical Orbits Mode 0 ≤ ℎ ≤ 222,3√0,93 𝑉𝑚𝑎𝑥 = 20.296,55 liter. 

𝐸𝑂𝑀(𝜃) 0 ≤ ℎ ≤ 222,3√0,93 𝑉𝑚𝑎𝑥 = 21.166,04 liter. 

Least Square 𝑛 = 2 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.248,90 liter. 
Least Square 𝑛 = 3 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.256,66 liter. 

 

The calculation result of Circle Orbits Mode is only defined to altitude 217,1 cm there is a 
reduction of 5,2 cm and Elliptical Orbits Mode is only defined to a height of 222,3√0,93 
cm or about 214,382 cm, there is a reduction of (222,3−222,3√0,93) cm or about 7,92 cm. 
For further research, this has no effect on the application of Daily Sales Data (According 
to Dispenser) if the maximum height of fuel data is below 214,38 cm. So that the value is 
defined for all data as well as each Approximation Method as well. Approximation results 
will be validated by measuring Mean Average Deviation (MAD) based on [14] and then 
Mean Absolute Percentage Error (MAPE) based on [15]. If Aproximation Results has 
MAPE below on 10% then Aproximation Methods is very feasible. 

CONCLUSIONS 

Based on the results and discussion, it can be concluded that the method of approximating 

the pertalite data with the smallest RSS and MSE is 𝐸𝑂𝑀(𝜃) by 𝜃 ≈ 130,37, resulting in RSS 
and MSE respectively are 6.415,32 and 29,81. 𝐸𝑂𝑀(𝜃) also produces a more fit half-ellipse 
function than other approximation methods. The results of the comparison of the approximation 

of the pertalite data are compared with 𝐶𝑂𝑀, 𝐸𝑂𝑀, 𝐿𝑆(𝑛 = 2), and 𝐿𝑆(𝑛 = 3) 
Although 𝐸𝑂𝑀(𝜃) produces RSS and MSE, which are smaller than other methods, there 

is a reduction in the altitude domain and has a different maximum volume compared to the 

calculation of gas stations. According to the Gas Station Metrology Measurement Book, the 

height of the UT is 222,3 cm and has a maximum volume of 21.486 liters, but 𝐸𝑂𝑀(𝜃) only 
detects the volume of fuel up to a height of about 214,1 cm and the maximum volume is below 

the calculation of the gas station. 

The author hopes for the development of this research, applied to different types of 
fuel such as Pertamax and Dexlite. As well as for a more real problem under study, use 
data on changes in the height and volume of BBM based on Daily Sales according to the 
BBM Dispenser which must first be tested for the accuracy of the BBM Dispenser used. As 
well as calculating errors using MAPE, MAD, and other error calculations. 



Elliptical Orbits Mode Application for Approximation of Fuel Volume Change 

Jovian Dian Pratama, Ratna Herdiana, Susilo Hariyanto 331 

REFERENCES 

 

[1]  Pratama, J.D., Herdiana, R., Hariyanto, S., Application of Orbits Mode Data Fitting for 
Dipstick Calibration of Altitude Measuring Instruments into Volume of Fuel Oil in 
the Tank Compared with Cubic Spline Interpolation, SNAST – National Seminar on 
Application of Science and Technology, Akprind – Yogyakarta, 138 - 146, 2021 

[2]  Pratama, J.D., Herdiana, R., Hariyanto, S., Application of Orbits Mode Data Fitting For 
Dipstick Calibration. Journal of Mathematics Education: Judika Education, 4(2), 107-
118., 2021. 

[3]  Urang, Job G., Ebong, Ebong, D., dkk., A new approach for porosity and permeability 
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