**4. Mathematical Models**

The ship propulsion and electric generation system model of the benchmark chemical tanker introduced in [41] has been updated in this paper and the model structure is shown in Figure 2. The models of fuel consumption and emissions of diesel engines will be briefly presented in this paper. The models of ship resistance, propeller open water characteristics, wake factor, thrust deduction factor and relative rotative e fficiency can be found in [41] for detail. The onboard electric loads are modelled by a constant value, which is set as 350 kW. The shaft generator/motor and the auxiliary generators are

modelled by constant energy conversion efficiencies (which is consistent with the constant auxiliary power assumption and the practice of switching on/off auxiliary generators to ensure proper loading of the engines). The energy conversion efficiencies of both the generator(s) and motor are set as 95% in the model.

**Figure 2.** Model structure of the integrated ship propulsion and electric generation systems.

### *4.1. Models of Fuel Consumption and Emissions of Diesel Engines*

The fuel consumption is calculated using the engine torque model introduced in [3], in which the engine torque is modelled as a function of the fuel consumption and engine speed in the form of second order Taylor expansion of two variables, including the cross product, as shown in Equation (4). Similarly, the emissions are modelled as functions of the engine torque and engine speed [33] as shown in Equation (5). The constant coefficients in the models can be fitted using engine test data.

$$\begin{aligned} M^\* &= f \big( m\_{f'}^\* N^\* \big) \\ = 1 - a \cdot \left( 1 - m\_f^\* \right) + b \cdot \left( 1 - m\_f^\* \right)^2 - c \cdot \left( 1 - N^\* \right) + d \cdot \left( 1 - N^\* \right)^2 + 2 \cdot c \cdot \left( 1 - m\_f^\* \right) \cdot \left( 1 - N^\* \right) \end{aligned} \tag{4}$$

$$\Phi\_{\rm cm}^\* = f\_2(M^\*, N^\*)$$

$$\varepsilon = 1 - a\_{\rm eff} \cdot \left(1 - N^\*\right) + b\_{\rm eff} \cdot \left(1 - N^\*\right)^2 - c\_{\rm eff} \cdot \left(1 - M^\*\right) + d\_{\rm cm} \cdot \left(1 - M^\*\right)^2 + 2 \cdot e\_{\rm cm} \cdot \left(1 - N^\*\right) \cdot \left(1 - M^\*\right) \tag{5}$$

where *M*<sup>∗</sup>, *N*<sup>∗</sup>, *m*∗*f* and Φ<sup>∗</sup>*em* are the normalised engine torque, engine speed, fuel mass injected per cycle, emission mass flow, which are normalised by dividing the relevant variables using the corresponding nominal value of the variables.

Only the emissions of the carbon dioxide CO2, NOx and HC (hydrocarbons) are investigated in this paper. The carbon dioxide is the direct product of the complete combustion of the fuel. So, the CO2 emission is directly determined by the fuel consumption. Note that, as the test data of the auxiliary engines installed in the benchmark ship is not available, the fuel consumption and emissions models of the auxiliary engines are calibrated using the average data of (a small number of) similar engines that are available in the internal dataset. The calibration of the model of the main engine using the engine test data has been introduced in detail in [41]. More details on the calibration of the NOx and HC emissions models of the main engine and auxiliary engines can be found in Appendix B.

### *4.2. Corrections for Di*ff*erent Fuel Types*

The test results of fuel consumption and emissions for developing and calibrating the models of both the main engine and the auxiliary engines have been corrected at ISO (International Organization for Standardization) standard reference conditions using the standard LHV (Lower Heating Value) of the fuel oil (42,700 kJ/kg), referring to ISO 15550:2016 and ISO 3046-1:2002. However, in this paper, the influence of sailing on different fuel types on the ship fuel consumption and emissions will be investigated. For instance, when the ship is in operation, the fuel type for the main engine can be HFO (heavy fuel oil), MDF (marine diesel fuel) or LNG (liquefied natural gas); while the fuel type for the auxiliary engines can be MDF (marine diesel fuel) or LNG (liquefied natural gas). Therefore, the fuel consumption, NOx emission and HC emission during ship operations need to be corrected accordingly by Equation (6) using the multiplying correcting factors shown in Table 3 (and uncertainty in these factors which is in between brackets).


\* ISO, ISO standard reference conditions, referring to ISO 15550:2016 and ISO 3046-1:2002. HFO, heavy fuel oil; MDF, marine diesel fuel; LNG, liquefied natural gas; LHV, lower heat values.

The underlying assumption for correcting the fuel consumption is that the efficiencies of the engines remain the same when the fuel types are changed. The correcting factors for NOx and HC emissions when using HFO and MDF are set based on the internal dataset. When using LNG as the fuel, the NOx emission will be significantly reduced by approximately 80% compared to diesel fuel [36]; while the HC emission will be much higher than the diesel fuel because of the methane slip during engine operation [34,42,43]. So, for a simple assumption, the correcting factors for NOx and HC emissions when using LNG are set as 0.2 and 10, respectively, based on the information in the available literature and engine specifications. Note that the formation mechanisms and the environmental and human health impacts of HC (hydrocarbons) emissions from diesel fuel and LNG are different although they are all called hydrocarbons in this paper as well as in other literature. The HC emissions from diesel fuel, in general, are the consequence of incomplete combustion and they are hazardous to human health (e.g., carcinogenic). However, the HC emissions from LNG are mainly methane emissions caused by "methane slip" or unburnt methane and it is mainly a greenhouse gas; it has no direct health effects on humans (in modest concentrations) [44], but it may cause suffocation if the concentration of methane in the air is too high [45]. It is assumed that the non-dimensional conversion factor between fuel consumption and CO2 emission is 3.206 kg/kg for diesel fuels and 2.750 kg/kg for LNG [46].

$$
\Phi\_x = \mathbb{C}\_x \cdot \Phi\_{x,ISO} \tag{6}
$$

where Φ*x* is the fuel consumption or the emissions of HFO, MDF and LNG (kg/s); <sup>Φ</sup>*<sup>x</sup>*,*ISO* is the fuel consumption or emissions of fuel at ISO (kg/s); *Cx* is the correcting factors of fuel consumption and emissions for different fuel types represented in Table 3.
