Elliptical Orbits Mode Application for Approximation of Fuel Volume Change

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.


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. 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 ( , ) is the center point of the ellipse with the major and minor axes adjusting and ,

Definition 2 ( set for Ellipse Mode)
A set of points formed from two ellipse equations is defined as follows, with 2 = ( +1)1 and the intersection of the respective sub-blankets of the minimum and maximum volume intervals in equation (3) is denoted for each [ 1 , 2 ] ∩ [ ( +1)1 , ( +1)2 ] is equal to 2 or ( +1)1 , where = 1,2, … , with is the number of blankets dividing the maximum and minimum intervals of the volume change.

Definition 3 (Partition of set)
Partition of set that divide set to become partitions or sets of points between 2 ellipses equations, defined as follows,

Theorem 1 (Defined Intervals for Elliptical Orbits Mode Approximation Function)
Proof: ) 2 so that it is obtained: , so that (7) the function of volume change to fuel level in the UT can be visualized as follows, 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).

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:   Accordingly, the value of ( ) for = 1,2, … ,10 is provided in Table 2.   Table 2 it is obtained that ( ( )) = ( 4 ), with = 1,2, … ,10 the selected 4 set , after that from the 4 set the following functions = ( ) will be formed,  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: where (ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3√0.93. The EOM version of the fuel volume calculation uses (12) with = 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):  By using the data in Table 1, it is obtained ≈ 130,37 that from (27) it is obtained: Based on Figure 6 = ( ) the ( )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 (ℎ) with integral and with (24) obtained: where . (ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3. Calculation of the volume of the fuel ( ) version has = 21.166,71 liters.

Comparison of approximate visualization results
Data visualization results from Circle Orbits Mode, Elliptical Orbits Mode, ( ), Least Square Data Fitting = 2, and Least Square Data Fitting = 3, as follows: 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: 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 ( = 2), but still above ( = 3), then by controlling the height of the ellipse to find 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

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: 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 ( ) by ≈ 130,37, resulting in RSS and MSE respectively are 6.415,32 and 29,81.
( ) 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 , , ( = 2), and ( = 3) Although ( ) 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 ( ) 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.