**2. Background**

Efficiency can be defined as the ratio of the useful work performed by a vessel to the total energy expended, but also can be expressed as actions designed to achieve efficiency [15]. Under these definitions, first, we can consider the vessel efficiency as the amount of fuel consumed, as the energy source to be expended, over the transport work performed by the vessel, as the process of carrying cargo, and second, we can consider vessel efficiency as the implementation of technological and operational means to a vessel to achieve higher levels of efficiency. Both definitions can be applied but for this research, the first is going to be applied over the design of new vessels, and the second over existing vessels. Both definitions can be worked together when analyzing the efficiency of a vessel, as to improve its efficiency in the design stage, it is required to know its current operational efficiency, from the EEOI. The current efficiency can be estimated when evaluating the amount of fuel that is consumed by the diesel engines of the main propulsion system and the auxiliary systems

when navigating. This estimation of vessel efficiency provides the baseline from where it is possible to improve it when considering the implementation of technological means, i.e., shaft generators and operational means, i.e., selective control schemes as the described in this research. The amount of fuel saved by the implementation of technological and operational means translated as the improvement into the efficiency of a vessel. Because the consumption of fuel generates emissions, any reduction of fuel consumption leads to lower emission levels. The use of the fuel consumption a as state variable provides the baseline of considering the use of EEDI and EEOI as means of evaluation of vessel design and operational efficiency.

### *2.1. Emissions from the Combustion Process*

Emissions are generated during the process of converting the chemical energy of the fuel into mechanical work, Equation (1) represents the stoichiometric reaction of the fuel and the consequence emissions generation, CO2emissions are the higher amount of all of them.

$$\rm C\_mH\_n + \left(m + \frac{n}{4}\right)O\_2 + pN\_2 \to mCO\_2 + \frac{n}{2}H\_2O + pN\_2\tag{1}$$

### *2.2. Fuel Oil Consumption and Diesel Engine and Shaft Generator Operation*

The stoichiometric air to fuel ratio (*AFRst*) is the minimum amount of air required to burn a kilogram of fuel and, when compared to the actual Air to Fuel Ratio (AFR), the stoichiometric ratio *λ*, presented in Equation (2), can be found [15]. The AFR can be considered at any engine load for the purposes of analysis.

$$
\lambda = \frac{AFR}{AFR\_{st}} \tag{2}
$$

The engine's output power suffers when operating at lower loads condition, because at this condition, less fuel is available and the engine while trying to achieve a higher load demands more fuel to be provided to overcome the demand. The engine trying to maintain the power output at the desired operational condition increases the specific fuel oil consumption until it reaches the desired engine load, which is related to its speed as can be seen in Figure 1. At low engine load *λ* ≈ 4.0, which decreases as the engine load is continuously increased. When the load is within the range of 75 to 80% of the engine maximum continuous rating (MCR), the value of *λ* reaches its minimum. When going above 75–80% load, because higher amounts of air and fuel are required, *λ* increases reaching a value of ≈2.0 at the 100% of the MCR.

**Figure 1.** Power Take Off (PTO)/Power Take In (PTI) operating regions.

Figure 1 shows these *λ* conditions and the optimal fuel oil consumption point or MEOP. The MEOP has been considered in this form because is a representation of the Specific Fuel oil Consumption (SFC), which is one of the factors to evaluate the amount of emissions generated by the engine when using the EEDI and EEOI.

Operationally, Figure 1 differentiate the operating regions of the shaft generator when considering the MEOP over the entire engine load range. Before MEOP the shaft generator operates as PTO and after MEOP operates as PTI. The shaft generator as PTO generates electricity to support the operation of the vessel and reduces the use of the diesel engines of the auxiliary system known as generation set. When operating as PTI, the electrical power to operate the shaft generator as an electric motor is provided by the generator set. Nonetheless, from this assumption, future work will investigate options to improve the efficiency of the generator set operation and the use of electric power sources, i.e., use of batteries and Non-Conventional Renewable Energies (NCRE) sources.

The engine efficiency at the MEOP is the highest and has been used as the evaluation point of the EEDI mandated by the IMO for every new ship constructed. The EEDI started with a minimum value established by 2013 and reduced by a percentage over the next 12 years [10,11], a low EEDI value means a more efficient ship, in terms of its design (hydrodynamics, propulsion system, and auxiliaries).

### *2.3. Energy Efficiency Design Index EEDI*

The EEDI can be defined as a technical measure of CO2 emissions per ship's capacity per nautical mile applied to new ship designs [10]. The EEDI equation is presented in Appendix A. Here, a modified version to be applied for the purposes of this paper is presented in Equation (3). This modified version accounts only for the main engine influence of emissions generation; therefore, it is an approximation that is going to be modified to establish and represent the influence of the shaft generator over the entire main and auxiliary systems of the ship. The auxiliary engines, shaft generator, and WHRS influence over the EEDI equation has been found minimum when compared to the main engine installed power therefore the EEDI value do not ge<sup>t</sup> really affected by them as stated in [11]. The influence of these factors is related to the operation of the ship.

$$EEDI = \frac{P\_B \, SFC \, C\_F}{DWT \, V\_S} \tag{3}$$

The *PB* corresponds to the 75% rated installed brake power in kW, *CF* is the carbon factor in *g* CO2 per *g f uel*, DWT is the capacity of the ship in tonnes, and Vs is the ship's design speed in knots.

### *2.4. Energy Efficiency Operational Indicator EEOI*

The MEOP has been also used to calculate the EEOI. The EEOI is the monitoring tool supporting the Ship Energy Efficiency Management Plan (SEEMP) applied to new and existing ships to measure the amount, in grams, of *CO*2 per tonne cargo transported per nautical mile for a single voyage [11]. The equation to calculate EEOI is presented in Appendix B and here a modified version to be applied for the purposes of this paper is presented in Equation (4). This modified version uses the SFC instead of the total amount of fuel consumed for a single voyage to relate the indicator with the operational performance described to evaluate EEDI and have a comparison point of the ship's design performance and the actual operation at the required MEOP at low and high loads. The *mc* factor accounts for the mass of cargo transported in tonnes and *D* to the distance, in nautical miles, of the cargo transported.

$$EEOI = \frac{SF \gets \mathcal{C}\_F}{m\_{\mathbb{C}} \, D} \tag{4}$$

Having a comparison point between the design and the operational behavior of the ship allows for a better understanding of these tools to evaluate the current efficiency of the ship, also allowing to consider technological and operational options to improve ship's efficiency. The use of a shaft generator is part of these improvements, and its influence into of ship's efficiency is presented when analysing the Shaft Generator/Motors Emissions Factor *fgef* into the EEDI equation, this factor is presented in Equation (5), where the power generated accounts and is related to the power generated by the auxiliary engines or generator set to support the service of the ship.

$$f\_{gef} = \left(f\_{\text{i}}P\_{PTI} - f\_{eff}P\_AE\right) \text{ C}\_{FAE} \text{ SFC}\_{AE} \tag{5}$$

Equation (5) can also be contrasted with the Efficiency Technology Factor (ETF) presented in Equation (6), which relates the reduction in power requirements that any technology generates and is used to improve the ship's efficiency.

$$ETF = f\_{eff} P\_{eff} C\_F \, SFC \,\tag{6}$$

The efficiency technology factor *feff* in Equations (5) and (6) represents the percentage of influence of the power output of the technology and relates to its efficiency. The background presented allows for the introduction of the control scheme selected as the most appropriate to represent the influence of a shaft generator into the propulsion system of a ship, because relates its fuel consumption and its emissions in accordance with known and validated indexes for ship's efficiency evaluation.
