Model DevelopmentOne of the main deliverables of the project was a mathematical model that the customer could use in the future. In order to properly develop the mathematical, the Team first identified the model objectives and essential elements of analysis. AssumptionsThe following assumptions^{[1]} used during the course of the project are:
Essential Elements of Analysis (EEAs)
Model DesignThe goal of the model was to predict ship fuel consumption based off of power consumption. Speed and sea state are the major parameters used to calculate power consumption. The model was implemented using Microsoft Access in order to easily store and query the data. There were three major sets of data that were provided by the study customer: Monthly Consumption and Op Hours, Ship Logs, and Speed versus Power data. These data were imported into the database. The pedigree of the data was imperfect due to data being entered by hand by Navy personnel; therefore, there were errors in the data. During the data manipulation phase, the Team spent a great deal of time combing through the data to fix obvious errors using best judgment and outlier analysis. Although, previously mentioning of outliers being removed from the data, all original data was left intact and any changes to data were made in a separate column. This method allows the user to easily write a query to identify any data that was changed by comparing the two columns, and it allows for the original data to be restored easily if necessary. The model development process was implemented using database queries. These queries provided the ability to quickly perform calculations across all the data. The queries were built according to the following model diagram.
Figure 7. Mathematical Model Flowchart Steps The bulk of the mathematical model process is conducted in queries 103 and 104, which calculates Hourly Fuel Consumption and aggregate hourly into Monthly Fuel Consumption, respectively. These monthly fuel consumption figures are referred to as the “predicted” values, since they are modeled values computed from the speed of the ship. In parallel to queries 103 and 104, query 102 calculates Monthly Fuel Consumption. This query calculates the monthly fuel consumption for each vessel based on the recorded monthly data. These monthly fuel consumption figures are referred to as the “actuals,” since the figures are a direct computation based on monthly consumption data. The purpose of this step was to provide baseline data to compare the predicted model output against. Once the modeled and baseline calculations were completed, the Team computed the monthly fuel consumption residuals (queries 105 and 106) and plotted them to identify if fuel consumption trends existed. Trends were found, and they are explained later on, in the Output Analysis section. For additional information on the queries included in the mathematical model, reference Appendix D. The data collected in Task 1, Data Collection, was used for model input: Speed Power Data, Hourly Ship Log Data, and Monthly Fuel Data (identified by the blue boxes in the Model Flowchart figure). The outlier analysis for both the mathematical model and baseline processes was mentioned previously in the Data Manipulation section. Database DesignThe database was designed around the data that was provided by the customer. There were three major sets of data that were provided by the study customer: Monthly Consumption and Op Hours, Ship Logs, and Speed versus Power data. The three tables that were created to house these data are called MonthlyConsumption, ShipLog, and PowerVersusSpeed. Tables
Figure 8. List of Tables within Model The MonthlyConsumption table is a direct import of the MonthlyConsumptionAndOpHours spreadsheet that was provided by the customer. The team found the data in this table had a lot of variability which could render subsequent analysis inconclusive, so the team decided to identify and exclude outliers. The data had to be split into UW and NUW in order to be able to identify UW outliers and NUW outliers separately. The UW data from the MonthlyConsumption table was copied into the MonthlyConsumptionUW_Outlier table and the outlier records were identified and flagged. Then the non-outlier records were copied into the MonthlyConsumptionUW_Updated table. The same was done for the NUW data.
Figure 9. MonthlyConsumption Table The ShipLog data had to be preprocessed in Excel before it was imported into Access. The reason for this was that an [Hours] column needed to be created. Each ShipLog record had a date and time stamp, but the duration between ShipLog entries was not captured, and it is difficult for queries to determine this sort of information on the fly. So, Excel was used to calculate the time between ShipLog records. Once the data was imported into Access, it became apparent that the [Avg Speed] and [Sea State] columns contained errors. The Team’s approach to handling this was to back-up the original data into two columns called [Avg Speed Orig] and [Sea State Orig]. This way, the data in the [Avg Speed] and [Sea State] columns could be modified while maintaining the ability to easily trace or restore the data.
Figure 10. ShipLog Table The PowerVersusSpeed table was a direct import of the PowerVersusSpeed spreadsheet that was provided by the customer. The data in this table was for informational purposes. Analysis was done in Excel to produce a Speed Power regression formula. This formula was built into the queries to perform the calculations necessary to convert speed into power.
Figure 11. PowerVersusSpeed Table The Ship and SeaState tables are definition tables. The Ship table is a unique listing of the six vessels and contains all the identifiers necessary to join other tables together. The SeaState table is a unique listing of the nine sea states as well as some supplemental data that pertains to sea states. The SeaState table can be joined to the ShipLog table to provide more detailed sea state information for each ShipLog record. The ShipLogInfo table was built as a supplement to the ShipLog table. Once the Team noticed how much data was missing from the ShipLog table, a concern arose that this could cause inaccurate data points for months that were missing large amounts of data. Therefore, a query was made to identify the magnitude of missing data for each month and record it in the ShipLogInfo table. This table could then be used to exclude months that were missing too much data. The Month and Year tables were primarily built to support the construction of the ShipLogInfo table. They are merely a list of valid months and years, respectively. QueriesOnce the tables were built, queries were created in order to implement the model. The core set of queries are numbered 101 to 106. These queries build on each other and perform the calculations that drive the model.
Figure 12. List of Queries Contained in Model The qry-101-DSP-AvgSpeed pulls the pertinent data from the ShipLog table, which includes average speed and sea state. The qry-102-DSP-MonthlyFuelUsageUW uses the MonthlyConsumption table to pull the pertinent monthly fuel totals as well as compute monthly fuel consumption in terms of gallons per hour. This data was considered to be the “actuals”. It is the real fuel consumption in the sense that it is based on recorded fuel quantities. The qry-103-DSP-HourlyPredictedFuelConsumptionUW was the first step in producing predicted fuel consumption values. This query computed hourly fuel consumption that are a function of average speed and sea state. For each ship log entry, this query takes the average speed and plugs it into the Speed-Power regression formula in order to convert speed into power. The power, expressed in kW, was then multiplied by the sea state factor in order to account for extra power necessary to overcome the recorded sea state. This calculation was done using the equation: [UW Power] = (2.8837*[Avg Speed]^{3} - 39.889*[Avg Speed]^{2} + 247.63*[Avg Speed] + 800) * [Sea State Factor] The power in kW is then converted to horsepower using the 0.746 kW/hp conversion factor. This computation was done using the equation: [UW HP] = [UW Power] / 0.746 Once the power was expressed in terms of horsepower, it could be plugged into the fuel consumption formula. This formula requires the specific fuel consumption of the engine in lbs/hp/hr. This value was estimated using engine specification documents for marine propulsion engines with similar horsepower to the engines found on the T-AGS 60 class ships. The formula also requires the fuel weight, which is 7.2 lbs/gal for diesel. The fuel consumption formula was: [UW Consumption] = BSFC * [UW HP] / FW Where, BSFC = Brake Specific Fuel Consumption = 0.36 lbs/hp/hr FW = Fuel Weight = 7.2 lbs/gal The qry-104-DSP-MonthlyPredictedFuelUsageUW was the next step in producing predicted fuel consumption values. This query took the hourly data computed from query 103 and aggregated it into months. This was necessary to be able to compare the predicted values to the actuals, which are also in terms of months. The qry-105-DSP-MonthlyResidualsUW was used for output analysis. It computed the residuals between the predicted and actuals for total fuel consumed per month. The qry-106-DSP-MonthlyPropulsionFuelConsumption was also used for output analysis. It computed thee predicted and actual propulsion fuel consumptions per month. This was done by taking the UW fuel consumption and subtracting out the NUW fuel consumption. Query 105 was leveraged to compute the predicted propulsion fuel consumption, and the MonthlyConsumption table was used to compute the actual propulsion fuel consumption. Speed Power CurveOne of the fundamental components of our model was the ability to convert speed into power. In the Ship Log data, the Team was given the average speed of each ship over time periods that spanned several hours. This average speed data was the main input to the model. The average speed data is what eventually gets converted into power, and then converted from power into fuel consumption. But before speed can be converted into power, the Speed Power Curve needed to be developed. The Speed Power Curved expressed the relationship between speed and power for the T-AGS 60 class vessels. To help generate this curve, the customer provided some data points for speed versus power. In order to formulate a curve from the speed versus power data points, regression was used. The regression was performed in Excel by charting the data and using trend lines. Several different types of regression were used including: linear, exponential, power, logarithmic, and polynomial. During the initial efforts, linear regression was a poor fit with an R^{2} value of 0.925. Logarithmic regression curved the wrong way. The curve had a concave increase instead of convex increase and had an R^{2} value of 0.888. The power function was also not great because the fit grew worse at higher speeds. The chart below shows the results of the different types of regression analysis.
Figure 13. Speed Power Curve The Team decided that the exponential and third order polynomial were the best fits based on the fit and the high R^{2} values. The result of the third order polynomial is graphically displayed in the chart below.
Figure 14. Third Order Polynomial Speed Power Curve As the model development and analysis progressed, the Team realized that the curves were not modeling the hotel load, which could significantly impact the accuracy of the model. The Team approximated the average hotel load to be 800 kW (reference section Estimating Hotel Load for additional information). This means that regardless of speed, the predicted power consumption for each vessel should be at least 800 kW. The Speed Power Curve was updated to reflect the hotel load at 0 knots. The addition of this data point caused the polynomial function to improve and the exponential function to worsen. The third order polynomial curve became the clear choice with an R^{2} value of 0.984. This can be seen from the following chart.
Figure 15. Fit Speed Power Curves with Hotel Load Therefore, the function used in the model to convert speed into power was: y = 2.8837x^{3} - 39.889x^{2} + 247.63x + 800 Where, x is speed in knots and y is power in kilowatts. As an aside, it makes sense that the third order polynomial function fits the data well because speed and power have a cubic relationship. This can be derived using physical equations. In physics, the following equation is used to calculate power: Power = Force * Velocity In the case of seagoing vessels, the primary force acting on the vessel is the force of drag through water. The formula for the force of drag is: Force_{d} = (1/2) * Fluid Density * Velocity^{2} When the drag equation is incorporated into the power equation, it results in a cubic function where power is expressed as the cube of velocity: Power = [(1/2) * Fluid Density * Velocity^{2}] * Velocity = (1/2) * Fluid Density * Velocity^{3} Calculating PowerThe Speed Power Curve was used to convert speed into power in kilowatts, but in order to compute fuel consumption the power must be converted into horsepower. The Team was able to convert speed into horsepower through the use of the following conversion: 1 hp = 0.746 kW Therefore, in order to convert k kilowatts into h horsepower: h = k / 0.746 Once the power was expressed in horsepower, the new power value was plugged straight into the fuel consumption formula. The fuel consumption formula used was: FC = BSFC * HP / FW Where, FC = Fuel Consumption (gal/hr) BSFC = Brake Specific Fuel Consumption (lbs/hp/hr) HP = Horsepower (hp) FW = Fuel Weight (lbs/gal) Estimating Hotel LoadHotel load was calculated from the Monthly Consumption and Op Hours data. The data was expressed per ship per month for UW and NUW, and it identifies the number of barrels of fuel consumed over how many hours. For hotel load, only the NUW data was of interest. In order to calculate the hotel load, the formulas from the previous section were used in reverse order. First barrels were converted into gallons. It was estimated that there are 42 gallons in a barrel. Next the NUW gallons were divided by the NUW hours to get NUW fuel consumption in gal/hr. This fuel consumption was then plugged into the fuel consumption formula to calculate the horsepower. By manipulating the fuel consumption formula, the powered consumed was calculated using the following equation: HP = FC * FW / BSFC Where, FC = NUW Fuel Consumption (gal/hr) BSFC = 0.36 (lbs/hp/hr) FW = 7.2 (lbs/gal) for diesel Next, power was converted from horsepower to kilowatts, which provided the hotel load. In order to convert h horsepower into k kilowatts, the following conversion formula was used: k = h * 0.746 The above computations were used to convert NUW fuel consumption into power in kW across the board for every month of every ship. The computations were done using an Access query, and the results were exported to Excel and Minitab to perform the hotel load analysis. The Team developed histograms of the average hotel load experienced for each vessel. The purpose of the histograms was to support the decision that the average hotel load was approximately 800 kW.
Figure 16. Hotel Load Histograms The Team also calculated the mean, median, variance, and standard deviation of each vessels hotel load (see table below). The main reason for this calculation was to support the estimate of average hotel load of 800 kW. Table 1. Hotel Load Statistics
The above histograms and table illustrate that the average vessel hotel load was around 800 kW. The mean estimates the hotel load to be a slightly above 800 kW and the median estimates the hotel load to be slightly below 800 kW, so the Team chose 800 kW as an even approximation. Estimating Brake Specific Fuel ConsumptionBrake Specific Fuel Consumption (BSFC) refers to the engine fuel consumption rate. It is a measure of the pounds of fuel that the engine consumes per horsepower per hour. For example, if an engine has a BSFC of 0.36 and it is producing 1000 HP, then this means that the engine would be consuming 360 lbs of fuel per hour. BSFC is a manufacturer specification that is different for every engine. The T-AGS 60 uses twin 4000 HP General Electric CDF 1944 engines. Since the Team was not able to obtain the manufacturer specifications, the BSFC needed to be estimated in order to properly use the fuel consumption calculations to develop the mathematical model. In order to estimate BSFC for a T-AGS 60 class vessel, a Caterpillar engine was used as a surrogate. The Caterpillar manufactures a comparable marine propulsion engine that produces 3634 HP. The engine model is C280-8 and the engine specifications were posted on their website. The chart and table below show how BSFC changes as engine speed and power increase.
Figure 17. Brake Specific Fuel Consumption for Caterpillar C280-8 Marine Propulsion Engine Table 2. Brake Specific Fuel Consumption for Caterpillar C280-8 Marine Propulsion Engine
The team chose 0.36 as an approximate BSFC to use in the model. The BSFC was not varied in respects to vessel speed changes, and the use of a single BSFC value was approved by the customer. But, this would be a possible area of improvement for the model. Estimating Sea State FactorsSea state data was provided for each entry in the ShipLog table. From discussions with the customer, sea state was thought to be one of the significant factors that would affect ship fuel consumption. In order to model sea state effects in the model, the Team decided to scale the ship’s power consumption proportional to the sea state. Higher sea states would require more power in order to overcome the resistance created by the head waves. During the process of determining how the different sea state levels would affect power and fuel consumption, each sea state level needed to be defined. The following table is based off of the World Meteorological Organization (WMO) sea state definitions. Table 3. Sea State Table
The Team could not find any data on how sea state would impact ship propulsion power. But, a study was found that used a computer simulation to model the effects of head waves on a ship. In that study was a diagram that related wave height to total resistance on the ship. The Team used this diagram to estimate the resistance for each sea state. The diagram is shown below, with guidelines that the Team drew to ascertain the values from the graph.
Figure 18. Sea State vs. Resistance In order to calculate the sea state factors, the Team calculated the proportional increases in resistance for each sea state. For example, the resistance at sea state 0 is 42.5 and the resistance at sea state 4 is 43.2, which is a 1.6% increase in resistance. Power increases proportional to the resistance, so the additional power needed to overcome resistance at that sea state is 1.6%. The factor of 1.016 was assigned to the sea state level of four and this process was used to determine the remaining factors for each sea state level. The sea state factors were used to scale the predicted propulsion power in the model. Below is the general form for this calculation, and the following table shows the complete results of the calculations. Sea State Factor = R_{s} / R_{0} Where, R_{s} = Total Resistance at sea state s R_{0} = Total Resistance at sea state 0 Table 4. Sea State Factors
While reviewing the sea state data, it was found that sea states 0 through 8 were experienced by the vessels. The following table provides a histogram of the recorded sea states. The histograms identify the frequency of occurrences of each sea state for each vessel.
Figure 19. Sea State Histogram
The histograms indicate that sea states three and four appeared the most often in the data. The above tables illustrate that 65 percent of the sea states have values of zero through three, which have a Sea State Factor of one and, therefore, would not have any effect on fuel consumption. In addition, 91 percent of the sea states have values of zero through four, which have Sea State Factors of one or nearly one and, therefore, would have a negligible effect on fuel consumption. This explains why in the model, with the way the sea state factors are currently set, sea state does not have a large impact on fuel consumption. [1] All documented assumptions have previously been discussed and approved by the customer. [2] A hotel load is the amount of power needed to support the everyday operations of the vessel and the needs to keep the vessel a livable area. For example, the hotel load would include the power needed for the air conditioning, lights, medical needs, kitchen/cooking appliances, etc. [3] During the identification of skeg modification dates, other modifications were identified. The Team was able to perform a simple analysis on the other modifications – discussed in Appendix G. |