1. Introduction
The International Maritime Organization recently began enforcing the following as regulations to prevent marine pollution: the Energy Efficiency Design Index, an evaluation index in the design stage, and the Energy Efficiency Operational Indicator and Ship Energy Efficiency Management Plan, which are evaluation indices for operation. With the increasing importance of the speedpower performance of ships during sea trials for organizations, ship owners, and shipyards, the ISO established a standard method to analyze speedpower trials (ISO 15016, 2015).
Speedpower trials require a process for correcting the external force of actual sea conditions by replacing the ship’s speedpower measured in actual seas with that obtained in calm waters. ISO 15016:2015 provides guidelines for this process for sea trial data analysis. Therefore, the uncertainty of speedpower corrected from actual sea conditions to that of calm waters may differ significantly depending on the reliability of measurements in actual sea conditions (Insel, 2008).
Ship speedpower trials require a significant number of measurements, the accuracy of which determines the uncertainty of speedpower performance. Although reducing measuring equipment errors and conducting trials under ideal conditions (calm waters) are recommended to minimize the uncertainty of speedpower performance, it is almost impossible to conduct trials under ideal conditions. Therefore, the results of every sea trial include uncertainties from both measuring equipment and environmental conditions.
The International Towing Tank Conference (ITTC) provides guidelines for uncertainty analysis in various hydrodynamic tests in laboratories, such as resistance tests and propelleronly performance tests (ITTC, 2002;2005). However, there is no standard procedure for analyzing the uncertainty of speedpower trials at sea. Because of the absence of raw data from trials for uncertainty analysis, there is a lack of related studies.
This study examines the influence of measuring equipment uncertainty on the essential elements (wind, wave, water temperature) required for speedpower trial analysis. The speedpower uncertainty analysis was performed by considering only TypeB uncertainty owing to difficulties in obtaining TypeA uncertainty through repeated testing.
2. Uncertainty Analysis
This study performed a sea trial uncertainty analysis based on ISO JCGM (2008a). According to this procedure, the evaluation method includes TypeA and TypeB uncertainties, as listed in Table 1. The total uncertainty is the sum of the two uncertainties, as shown in Eq. (1).
where N is the number of variables, and ui and uj are the TypeA and TypeB uncertainties, respectively.
TypeA uncertainty refers to uncertainty components resulting from the randomness of the experimental process, and is evaluated by repeated measurements, TypeB uncertainty is defined as the uncertainty of all components except TypeA uncertainty. Typical elements of TypeB uncertainty include previous measurements, experience, general knowledge, and equipment specifications.
When the input or test value is a function of different variables, it is defined as combined uncertainty. The combined uncertainty is expressed as u_{c}(y) in Eq. (2).
where $\frac{\partial f}{\partial {x}_{i}}$ is the sensitivity coefficient of x_{i}.
The expanded uncertainty used to indicate the confidence interval of the experimental value is defined as Eq. (3).
ISO JCGM (2008a) requires the provision of expanded uncertainty as the final result of uncertainty analysis as well as the kvalue used in the process.
Monte Carlo simulations are used to predict results through repeated sampling based on the concept that increasing the number of samples provides more accurate results (ISO JCGM, 2008b). Increased computing power has systematized the Monte Carlo method, which is now used in various modeling methods to simulate actual situations (Kim, 2011). The implementation of the Monte Carlo process consists of four main steps (Fig. 1).

(1) Define the mathematical model

(2) Assign a probability distribution function to each input variable

(3) Calculate the output variable by entering the input variable in the model

(4) Analyze the results
The Monte Carlo simulation complies with the ISO JCGM (2008b) framework and propagates uncertainty using the system’s mathematical model instead of the Law of Propagation of Uncertainty. This method can make reliable predictions regardless of the model’s linearity, and investigate the effects of the uncertainty of each input variable on the measurement results. It also obtains the most reasonable results from test data in a short period of time.
This study applied the Monte Carlo method and repeated 200,000 calculations using Crystal Ball software to estimate the uncertainty of the correction for the sea trial analysis procedure. TypeB uncertainty was considered by assuming the probability distribution of the input data as normal and uniform distributions, while TypeA uncertainty was not considered owing to the lack of raw data and repeated test data.
3. SpeedPower Uncertainty
3.1 Sea trial results
A 300K DWT very large crude carrier (VLCC) was used for the trial, and roundtrip tests were conducted under three conditions. Table 2 lists the wind direction, wind speed, wave height, wave period, wave direction, speed, shaft speed, and power measured during the trial.
The wind speed measured under all conditions was 3.1 ~ 18.2 m/s. The wave height was 1 ~ 1.7 m, reflecting calm sea conditions during the trial. The ship was a 300K VLCC (LBP: 322 m, B: 60 m, T: 20.5 m). The measured data are the test results of 50 %, 75 %, and 90 % Maximum Continuous Rating (MCR). The data are the average values for each round trip test including the speed, ship direction, measured power, propeller rotation speed, relative wind speed, wind direction, wave, air temperature, and water temperature. As the specifications of each measuring equipment were not provided, they could be estimated through assumptions such as other research papers.
3.2 Speedpower uncertainty factor
The purpose of this study is to examine the influence of measuring equipment on the reliability of trial analysis results, and the factors related to the measuring equipment were examined through various assumptions.
As shown in Fig. 2, the uncertainties that can occur during sea trials are divided into factors related to shaft power measurements and trial analysis. The shaft power is obtained by measuring the torque using a strain gauge installed on the propeller shaft, and then measuring the rotational speed of the shaft using optical sensors. The measured shaft horsepower is also corrected by calculating various added resistances according to the ISO 15016 (2015) trial analysis method to correct the effects of disturbance. Therefore, the uncertainty of each element affects the final speedpower performance.
4. Shaft Power Uncertainty
The uncertainties that may occur while measuring shaft power are divided into TypeA and TypeB. Although there are various instruments for measuring shaft power, existing systems frequently use strain gauges to measure the torque, which is converted into power with respect to the relationship with the shaft speed. This study used the Kyma Shaft Power Meter (KPM). The accuracy of the equipment yields useful data when evaluating TypeB uncertainty.
The TypeB uncertainty (u_{c}(Q_{ms})) factors of the shaft power include the uncertainty of the strain gauge (u(gauge)), the calibration uncertainty (u(ε)), the uncertainty that occurs when installing sensors (u(α)), and torque calculation uncertainty (u(Q)). Each of these can be subdivided as shown in Fig. 3, and the TypeB uncertainty (u_{c}(Q_{ms})) is calculated using Eq. (4).
As a result of examining the specifications of the KPM shaft power measurement system, the total uncertainty including uncertainties that can occur when installing and calibrating the gauge and sensor is 0.29 %. After adding the combined uncertainty (1.2 %) from calculating the shaft torque (shear modulus standard uncertainty, shaft diameter standard uncertainty, relative displacement standard uncertainty), the total combined uncertainty of the shaft power measurement system u_{c}(Q_{ms}) becomes 1.24 %, as shown in Table 3.

u_{c}(P_{Dms}) : Uncertainty of the power measurement system

u_{c}(Q_{ms}) : Combined uncertainty of the total torque

u_{c}(N_{ms}) : Combined uncertainty of the total shaft speed

u_{c}(Q) : Uncertainty due to recalculation of the torque

u_{c}(gauge) : Uncertainty due to the gauge of shaft speed

u_{c}(inst.) : Uncertainty due to installation on a shaft

u_{c}(α) : Uncertainty due to strain gauge installation

u(ε) : Standard uncertainty of the relative strain of the gauge

u(D) : Standard uncertainty of the diameter

u(G): Standard uncertainty of the shear module

u(R_{g}) :Standard uncertainty of the strain gauge bridge

u(Δf) : Standard uncertainty of the transmitter and receiver

u(V_{amp}) : Standard uncertainty of the amplifier plugin module

u(V_{out}) : Standard uncertainty of the digital voltage meter

u(R_{t}) : Standard uncertainty of the strain gauge effective resistance

u(k_{t}) : Standard uncertainty of the gauge factor at 75˚

u(R_{cal}) : Standard uncertainty of the resistance of standard resistor
The total uncertainty of the rotational speed measurement system is ±0.1 %; the total combined uncertainty (u_{c}(N_{ms})) is approximately 0.06 %. Therefore, the total combined uncertainty of the shaft power (u_{c}(P_{Dms})) calculated using Eq. (5) becomes 1.242 %.
Although the shaft power measurement system may also have TypeA uncertainties, quantifying these requires raw data obtained through repeated testing. Therefore, this study excluded TypeA uncertainty because of the absence of such data.
5. Added Resistance Uncertainty Analysis
5.1 Uncertainty of added resistance due to wind
ISO 15016 (2015) provides three methods for analyzing added resistance due to wind. This study used the wind resistance coefficient of tankers provided by the ITTC. The added resistance model due to wind is shown in Eq. (6).
Vane anemometers are currently the most widely used instruments to measure wind speed. The most common types of anemometers used in ships are reviewed, and the accuracy of vane anemometers is listed in Table 4.
With regard to the probability distribution of input data used to calculate the uncertainty of added resistance due to wind each MCR condition, the Monte Carlo simulation was performed by assuming the wind speed and wind direction as a normal distribution and uniform distribution, respectively.
Table 5 shows the results of analyzing the uncertainty of wind resistance. The analysis shows at the 95 % confidence interval and coverage factor (k) of 2, the uncertainty is approximately ±12 ~ 20 % at 50 % MCR, 15 % at 75% MCR, and 16 % at 90 % MCR.
Fig. 4 shows the uncertainty of wind resistance (50 % 1^{st} run) and the frequency distribution and variance contribution graph as a result of the Monte Carlo simulation analysis. The average wind resistance is –24.95 kN. The deviation of wind resistance at 95 % confidence interval is ±2.78 kN, which is approximately 12 % of the average added resistance.
Fig. 5 shows the contribution to a variance chart to examine the magnitude of the factors affecting the uncertainty of added resistance. The contribution to the variance chart is a graph that shows the order correlation coefficients between the data and input elements obtained by Monte Carlo simulation. The values were squared and adjusted to achieve a total of 100 %. As shown in the figure, the largest factor contributing to the variation in wind resistance under the condition of 50 % 1^{st}run was relative wind speed (V_WR_1), which accounts for approximately 57 % of the total. The second factor was relative wind direction (Pis_WR_1) at 36 %; ship direction (Psi_1) accounted for –8 %. This shows that relative wind speed has a greater influence on the calculation of wind resistance than relative wind direction under this condition.
5.2 Uncertainty of added resistance due to waves
ISO 15016 (2015) provides three methods (STAWAVE1, STAWAVE2, NMRI) to correct added resistance due to waves, and STAWAVE2 (Eq. 7) is the most widely used.
RAW is the total amount of added resistance due to waves, which is obtained by the sum of the increased resistance due to ship motion (R_{AWML}) and the increased resistance due to reflected waves (R_{AWRL}). E(ω, α) is the direction spectrum per unit area.
Wave height, wave direction, and wave period are essential to obtain the added resistance due to waves. When a swell occurs in the sea trial conditions, the swell height, swell direction, and swell period are also required.
General sea trials should be performed in calm waters to reduce correction uncertainty due to waves. However, if the sea trial cannot be performed in calm waters, the wave characteristics must be measured accurately. The wave characteristics are usually measured by observation or using buoys, radars, and satellites; thirdparty observation and radars are the most widely used methods. This study assumed the use of radar equipment, and the accuracy of general radar equipment is listed in Table 6.
When assigning the probability distribution of the wave measurement system mentioned above, the wave height and period were assumed and applied as normal and uniform distributions to the model in Eq. (7) to perform the Monte Carlo simulation.
Table 7 shows the results of analyzing the uncertainty of added resistance due to waves using input variables to determine the accuracy of the wave measurement system.
The uncertainty analysis of the added resistance due to waves under all MCR conditions shows that the 95 % confidence interval is approximately ±40 %. The reason is because the standard uncertainty of the instrument (±0.5 m) was significantly greater than the measured wave height (approximately 1–1.7 m).
5.3 Uncertainty of water temperature & density
ISO 15016 (2015) uses Eq. (8) based on 15℃ and 1,026 kg/m^{3} to correct the viscosity effect of the sea to the viscosity of the reference temperature to consider the density change caused by water temperature or the effect on viscosity.
During sea trials, the water temperature is not measured in real time but before the trial starts. The accuracy of the measurement system was assumed to be ±0.1℃, as shown in Table 8.
Although the total resistance (R_{F}) uncertainty of the model ship is necessary for calculating uncertainty when applying the Monte Carlo simulation, this value was considered insignificant and negligible.
Table 9 and Fig. 6 show the results of the uncertainty analysis of the added resistance due to water temperature and density. The expanded uncertainty is approximately 5 % at 95 % confidence interval (k = 2) in all cases.
6. Uncertainty of the Delivered Power
The P_{Did} is calculated by Eq. (9).
where △R is the sum of the added resistance to correct the sea trial conditions to that of calm waters, such as the added resistance due to wind and waves. As previously mentioned, this study only considered TypeB uncertainties for each factor. An additional study that considers TypeA uncertainties will be conducted in the future after acquiring raw data.
Based on the shaft power uncertainty calculated in Section 4 and the uncertainty in added resistance (5.1–5.3), the uncertainty of power (P_{Did}) was analyzed under ideal environmental conditions corrected by sea trial measurements. The results show uncertainties up to 1.5 % at 95 % confidence interval under each MCR condition (Table 10). As shown in Fig. 7, the P_{Did} was 11,381 kW when using the ISO 15016 (2015) sea trial analysis method based on the trial data of 50 % MCR. The uncertainty analysis show a range of ±130 kW (approximately ±1.2 %) at 95 % confidence interval. At 75 % MCR and 90 % MCR, the expanded uncertainty was approximately ±1.5 % and ±1.3 %, respectively, at 95 % confidence interval.
Fig. 8 shows the contribution to variance of the estimated power (P_{Did}) according to the results of roundtrip sea trials at 75 % MCR. The analysis results show that the added resistance due to waves at 75 % MCR condition of the 2^{nd} run had a significant influence on the uncertainty. This is because the wave height was measured to be 1.7 m higher than that of other conditions, which increased the uncertainty. The uncertainty of the shaft power measurement system was approximately 25 %, while the uncertainty of the added resistance due to wind was relatively small. This is because the added resistance due to wind under this condition did not have a significant influence as the average value of the roundtrip trial was small.
7. Conclusion
Based on actual sea trial data, this study applied the ISO 15016:2015 analysis method to analyze the uncertainty in ideal power using the Monte Carlo simulation method. The findings are as follows:

The delivered power was analyzed by considering the TypeB uncertainty of each measuring equipment during sea trials. Under 50 %, 75 %, and 90 % MCR conditions, the expanded uncertainty of delivered power (PDid) at 95 % confidence interval (k = 2) was ±1.2 %, ±1.5 %, and ±1.3 %, respectively.

The speedpower uncertainty due to each measuring equipment was ±1.5 % at 95 % confidence interval, and approximately ±0.13 kts when converted to speed. This means the ship speed may vary by ±0.13 knots owing to the uncertainty of various measuring equipment.

The added resistance due to waves under 75 % MCR condition of the 2^{nd} run had the greatest influence on the uncertainty of ideal power, and the contribution to variance was approximately 46 %.
As previously mentioned, this study considered only TypeB uncertainties for each factor. An additional study that considers TypeA uncertainties will be conducted in the future after acquiring raw data.