Optimal Selection of the Diesel Generators Supplying a Ship Electric Power System

Featured Application: Evaluation of electric generation system during the ship design or selection process. Abstract: It is very common for ships to have electric power systems comprised of generators of the same type. This uniformity allows for easier and lower-cost maintenance. The classic way to select these generators is primarily by power and secondarily by dimensions and acquisition cost. In this paper, a more comprehensive way to select them, using improved cost indicators, is proposed. These take into account many factors that have a signiﬁcant impact in the life-cycle cost of the equipment. A realistic and detailed proﬁle of the ship’s electric load spanning a full year of her operation is also developed to allow for a solution that is tailor-made to a speciﬁc case. The method used is highly iterative. All combinations of genset quantities and capacities are individually considered to populate a power plant, taking into account the existing redundancy requirements. For each of these and for every time interval in the load proﬁle, the engine consumption is Lagrange-optimized to determine the most efﬁcient combination to run the generators and the resulting cost. The operating cost throughout the year is thus derived. In this way, the method can lead to optimal results as large data sets regarding ship operation and her power system’s technical characteristics can be utilized. This intense calculation process is greatly accelerated using memorization techniques. The reliability cost of the current power plant is also considered along with other cost factors, such as ﬂat annual cost, maintenance, and personnel. The acquisition and installation cost are also included, after being distributed in annuities for various durations and interest rates. The results provide valuable insight into the total cost from every aspect and present the optimum generator selection for minimal expenditure and maximum return of investment. This methodology may be used to enhance the current power-plant design processes and provide investors with more feasible alternatives, as it takes into consideration a multitude of technical and operational characteristics of the examined ship power system.


Introduction
The shipping industry is ever growing and today numbers more than 63,000 commercial ships worldwide. Each year, more than 2000 new ships are built in the world [1], while the global shipbuilding industry market is expected to exceed $195 billion by 2030 [2]. In this context, the cost related to building, acquiring and operating a ship is a major concern to investors, but also has a significant impact in the world economy.
One aspect of ship design is its electric power system. This is usually overshadowed by the propulsion plant and thus overlooked in the decision-making process. However, if properly examined, it can turn out to be a substantial financial concern, especially when the requirements for electric power are increased, such as in large container ships, or even more, as technology moves towards electric propulsion.
For classic power plants with diesel generator sets, the common and easiest way to operate them is by sharing the load proportionally among them and adding or removing generators to the grid when the load reaches certain thresholds. This simplistic management scheme allows for little efficiency improvement.
On the other hand, several techniques have been presented to achieve performance optimization and efficiency increase in a ship energy efficiency management plant (SEEMP) [10,11]. These involve sophisticated load management and distribution [12,13], smart grids and microgrids [10,14], multiagent systems [15,16], distributed power management [17] and other methodologies, even exotic ones using quantum computing [18]. According to a complicated but also efficient approach, the load distribution on the gensets is optimized according to their fuel consumption curves [19], leading to notable fuel savings.
However, little has been discussed on the selection process of the gensets. A classic ship power plant is typically designed using the following steps. First, the number of generators is determined, usually based on reservation or redundancy requirements. Afterwards, the nominal power of the generators is calculated so that the maximum total load and the maximum critical load can be adequately supplied according to the reservation and redundancy requirements. Finally, the manufacturer and the exact type of the generators is determined, based on financial criteria, usually purchase price and average fuel consumption. At this point, if the cost seems too high, the design process is restarted in a spiral fashion and all the parameters are redetermined until an acceptable outcome is eventually reached, as in Figure 1. This may be satisfactory, but optimality is not guaranteed. Furthermore, the whole process is mostly empirical and thus not efficient.
One aspect of ship design is its electric power system. This is usually overshadowed by the propulsion plant and thus overlooked in the decision-making process. However, if properly examined, it can turn out to be a substantial financial concern, especially when the requirements for electric power are increased, such as in large container ships, or even more, as technology moves towards electric propulsion.
For classic power plants with diesel generator sets, the common and easiest way to operate them is by sharing the load proportionally among them and adding or removing generators to the grid when the load reaches certain thresholds. This simplistic management scheme allows for little efficiency improvement.
On the other hand, several techniques have been presented to achieve performance optimization and efficiency increase in a ship energy efficiency management plant (SEEMP) [10,11]. These involve sophisticated load management and distribution [12,13], smart grids and microgrids [10,14], multiagent systems [15,16], distributed power management [17] and other methodologies, even exotic ones using quantum computing [18]. According to a complicated but also efficient approach, the load distribution on the gensets is optimized according to their fuel consumption curves [19], leading to notable fuel savings.
However, little has been discussed on the selection process of the gensets. A classic ship power plant is typically designed using the following steps. First, the number of generators is determined, usually based on reservation or redundancy requirements. Afterwards, the nominal power of the generators is calculated so that the maximum total load and the maximum critical load can be adequately supplied according to the reservation and redundancy requirements. Finally, the manufacturer and the exact type of the generators is determined, based on financial criteria, usually purchase price and average fuel consumption. At this point, if the cost seems too high, the design process is restarted in a spiral fashion and all the parameters are redetermined until an acceptable outcome is eventually reached, as in Figure 1. This may be satisfactory, but optimality is not guaranteed. Furthermore, the whole process is mostly empirical and thus not efficient. The goal of this paper is to present a new method that will definitely produce the optimum result in very little computation time and with little effort. It is noted that there is no best solution fitting all cases. On the contrary, each problem has unique requirements and constraints necessitating particular handling. To this end, the proposed method uses as inputs a detailed load profile of a real passenger ship based on real data, and all upcoming calculations are performed in this realistic context.
Additionally, the cost of gensets is much more than just their acquisition price and their average fuel consumption. Therefore, several parameters are also contemplated, The goal of this paper is to present a new method that will definitely produce the optimum result in very little computation time and with little effort. It is noted that there is no best solution fitting all cases. On the contrary, each problem has unique requirements and constraints necessitating particular handling. To this end, the proposed method uses as inputs a detailed load profile of a real passenger ship based on real data, and all upcoming calculations are performed in this realistic context.
Additionally, the cost of gensets is much more than just their acquisition price and their average fuel consumption. Therefore, several parameters are also contemplated, among them the cost of installation, maintenance and payroll of the crew members assigned to it and detailed and optimized fuel and lubricating oil consumption. The reliability of the installation is also considered and the cost it entails. This way, the true life-cycle cost of the installation is estimated.
Furthermore, insight is provided allowing financiers to preview various interest rates and number of annuities combinations in order to select the most suitable return on investment (ROI) scheme.
In Section 2, the methodology followed is described in detail. In Section 3, the applied computational speed improvement technique is described. In Section 4, a representative case study and the results obtained are provided, while discussion on the presented work and results are given in Section 5.

Overview
An auxiliary graphical overview of the calculation process is shown in Figure 2 and described in detail in the following: Load profile creation: Preliminarily, a detailed load profile of the ship is drafted to become the frame in which all calculations will be based upon. More details are provided in Section 2.2.
Genset pool: A pool of diesel engine generators and their specifications is formed to combine and populate the ship's power plant. More details are provided in Section 2.3.
Algorithm main loop: A loop begins by selecting from the pool one genset type after the other. Genset installed capacity: Their installed capacity is determined so that they are sufficient to supply the maximum load of the ship, taking into account any redundancy requirements. More details are provided in Section 2.4. Note that all generators in the power plant are assumed to be of the same type. If their quantity is excessive (i.e., >18), the current generator type is rejected and the loop continues with the next iteration and type selection.
Operating cost estimation: For every time interval throughout the load profile, the Lagrange optimization method is used to establish the genset combination that will supply this particular load with the smallest fuel and lubricating oil consumption. This produces the lowest operating cost and System Marginal Cost (SMC) for each time period. More details are provided in Section 2.5. All operating costs are summed to produce the total operating cost throughout the year for the particular genset type.
Reliability cost estimation: The Capacity Outage Probability Table (COPT) of the selected power plant is estimated. For every time interval throughout the load profile, this is used to evaluate the expected loss of load energy (LOLE) separately for each type of load and load conditions of each time interval in the profile. Afterwards, these are summed up to produce the total LOLE for each load type throughout the year. More details are provided in Section 2.6.
For every time interval in the load profile, the above SMC and LOLE values are used to calculate the total cost of power loss, for the whole year, for the selected generator type.
Initial cost estimation: The initial cost includes acquisition and installation of the genset and it is broken down to annuities for a range of years and for a range of interest rates. More details are provided in Section 2.7.
Total cost estimation: The flat cost related for maintenance and payroll is estimated. Then, this is added to the aforementioned operating cost and initial cost to form the total annual cost. The reliability cost is also added separately, providing the total annual cost with reliability considerations. These are both calculated for the range of annuities and interest rates mentioned above and for the current generator type. More details are provided in Section 2.9. Afterwards, the loop continues with the next genset type selection.
Optimal genset selection: After the loop completes and all genset types are evaluated, the least expensive is selected and the total annual cost of the plant with and without reliability considerations is displayed, for the given range of annuities and interest rates.
The whole process is illustrated below. Subsequently, each individual aspect is more thoroughly discussed.
The flat cost related for maintenance and payroll is estimated. Then, this is added to the aforementioned operating cost and initial cost to form the total annual cost. The reliability cost is also added separately, providing the total annual cost with reliability considerations. These are both calculated for the range of annuities and interest rates mentioned above and for the current generator type. More details are provided in Section 2.9. Afterwards, the loop continues with the next genset type selection.
Optimal genset selection: After the loop completes and all genset types are evaluated, the least expensive is selected and the total annual cost of the plant with and without reliability considerations is displayed, for the given range of annuities and interest rates.
The whole process is illustrated below. Subsequently, each individual aspect is more thoroughly discussed.

Load Profile
The electric load of a ship varies greatly versus time and is very specific to her condition and performed operations. For example, the load of a ferry is much greater when she is underway filled with passengers than when she is at port with only a skeleton crew. Moreover, as the ship's schedule is usually predetermined, a load profile can be drafted with sufficient accuracy.
On the other hand, the load requirements of each ship are very distinctive and vary greatly, not only among different types and sizes but also among similar ships with different operating schedules. For example, a ship will have a different load signature when she is mostly underway and has only brief port time than when she is on a daily short-cruise routine. Therefore, it makes sense for a generator selection process to be shaped around the specific load requirements of the ship. For the purposes of this study, as well as for further research, a complete profile of the electric load of a passenger ship has been created. It is based on actual data from a real ship and it spans the range of a full operational period with relatively high resolution.
A ship is a complex structure like a small autonomous mobile city, containing a large variety of equipment. These extend from propulsion and energy production to airconditioning, galleys and other hotel facilities. As such, the electric load associated with each of them may be characterized as more or less significant. In general, the total load P load (t j ) for every time t j can be divided into K parts P load−k (t j ), first being the least and K-th being the most significant. In this paper, it is divided into inessential (P load−1 ), essential (P load−2 ), and critical parts (P load−3 ).
Inessential load refers to equipment that may become unavailable for a long time without any significant effect on the ship's operation, the performance of her crew or the living conditions of her passengers. This can be air-conditioning, hot water and lighting in living quarters, etc.
Essential load refers to equipment that when unavailable has a significant impact on the ship's operation, the performance of her crew and the living conditions of her passengers. This can be ventilation and lighting in compartments with running machinery, transfer pumps, air compressors, etc.
Finally, critical load refers to equipment that when unavailable seriously affects the safety of the ship and all those onboard. This can be auxiliaries necessary for running the gensets, propulsion and navigation (when the ship is underway), firefighting, damage control, etc.

Generator Specifications
In order to provide applicable results for this process, more than 30 actual generator sets, from several manufacturers, were studied and used to determine the optimal one (Table 1). The specifications in Table 1 were collected or derived from their datasheets.
The most characteristic information of a generator is its nominal power P nom . This is provided along with its minimum and maximum power P min and P max , respectively. These are the limits of the equipment outside which operation is not permitted.
For the reliability calculations, the probability of a genset not being available, also known as the forced outage rate (FOR) [20], was used.
The fuel type, fuel consumption and lubricating oil consumption were used to estimate the operating cost of the engine, while its physical characteristics and its acquisition price were used to estimate the installation cost.
It is noted that engines have additional restrictions and costs in their operation e.g., minimum running time and a minimum time between shutting down and starting up. There is also a maximum power increase/decrease rate and a starting cost. This information can be considered in future work.

Power Requirements
This paper assumes that all generators used in a single power plant are of the same type. Therefore, the maximum load can be supplied by n* gensets of nominal power P nom each, as shown in (3).
This number is adequate for the ship's needs, if no redundancy is required, or if there is an extra emergency generator to take up all critical loads. However, if no extra emergency generator exists and the power plant is to withstand the failure of a single genset, then n* + 1 generators will be required. Similarly, if the whole compartment may fail, then 2·n* gensets are required in a different location. This is summarized in Table 2.

Redundancy
Generator Quantity n none or emergency generator n* 1 generator n* + 1 full: 1 power compartment 2·n*

Operating Cost
The amount of fuel consumed by an engine is a function of its power output or load. As the power increases, so does the consumption versus time (see the fuel consumption curve in Figure 3). However, the consumption versus power and time (i.e., energy), also called specific consumption, reveals the existence of a point of optimal operation (see the corresponding specific fuel consumption curve in Figure 3).

Operating Cost
The amount of fuel consumed by an engine is a function of its power output or loa As the power increases, so does the consumption versus time (see the fuel consumpti curve in Figure 3). However, the consumption versus power and time (i.e., energy), a called specific consumption, reveals the existence of a point of optimal operation (see t corresponding specific fuel consumption curve in Figure 3). The fuel consumption cost Ffuel may accurately be approximated by a secondthird-degree polynomial function (4) of the electric power Pm produced, with coefficie derived from its fuel consumption curve, or specific fuel consumption curve, provid by the manufacturer or actually measured.

Ffuel(Pm) = a + b·Pm +c·Pm 2 + d·Pm 3
In this paper, the approximation was calculated using a second-degree polynom therefore, Ffuel became: On the other hand, it is specified that the lubricating oil consumption m accurately be approximated as proportional to the electric power Pm produced.

Flub(Pm) = e·Pm
Therefore, the total operational cost became: It is common practice to share the load equally among the running generators. T is efficient when all generators are of the same type and have the exact sa consumption curve. However, this is never reality, since even generators of the same ty will have significant differences in their consumption curves, due to their running hou maintenance history, mechanical ware, etc. These curves can be obtained by taki periodic measurements. It has been proven that taking into account these differences a distributing the load using optimization methods, allows for extra fuel savings [19]. The fuel consumption cost F fuel may accurately be approximated by a second-or third-degree polynomial function (4) of the electric power P m produced, with coefficients derived from its fuel consumption curve, or specific fuel consumption curve, provided by the manufacturer or actually measured.
In this paper, the approximation was calculated using a second-degree polynomial; therefore, F fuel became: F fuel (P m ) = a + b·P m + c·P m 2 On the other hand, it is specified that the lubricating oil consumption may accurately be approximated as proportional to the electric power P m produced.
Therefore, the total operational cost became: It is common practice to share the load equally among the running generators. This is efficient when all generators are of the same type and have the exact same consumption curve. However, this is never reality, since even generators of the same type will have significant differences in their consumption curves, due to their running hours, maintenance history, mechanical ware, etc. These curves can be obtained by taking periodic measurements. It has been proven that taking into account these differences and distributing the load using optimization methods, allows for extra fuel savings [19].
The quantity n of the generators required has been established above. Assuming, for the sake of generality that each one is different, there are 2 n − 1 possible combinations B combination of them running. For every one A operation−v of them and for a particular time period t j , the load requirements P load (t j ) were distributed in each running generator m producing power P m (t j ) with operating cost F operation−m (P m (t j )). This distribution was optimized using the Lagrange method [19,21], because of its suitability to solve optimization problems that are constrained with equalities and/or inequalities. As such, that the total operating cost F operation−A operation-ν for this case became minimal (8) under the constraints (9) and (10).
The system marginal cost SMC operation−v (t j ) was also calculated: Out of all combinations B combination , the most efficient was selected, as in (12), and the total cost due to fuel and lubricating oil consumption throughout the year (i.e., N T time intervals) was calculated, as in (13).
The whole process is illustrated in Figure 4.
power Pm(tj) with operating cost Foperation-m (Pm(tj)). This distribution was optimized usi the Lagrange method [19,21], because of its suitability to solve optimization proble that are constrained with equalities and/or inequalities. As such, that the total operati cost Foperation-Aoperation-ν for this case became minimal (8) under the constraints (9) and (10).
The system marginal cost SMCoperation-v(tj) was also calculated: (1 Out of all combinations Βcombination, the most efficient was selected, as in (12), and t total cost due to fuel and lubricating oil consumption throughout the year (i.e., NT tim intervals) was calculated, as in (13).
The whole process is illustrated in Figure 4.  If all the engines populating the power plant have identical behavior, the optimization process may be simplified using equal distribution. However, the algorithm uses optimization to address different duty cycles of the gensets and any future expansion of this work.

Reliability Cost
The reliability of a system is a factor of paramount importance. However, most of the time, industrial systems use rather simplistic and crude redundancy techniques to achieve the required reliability levels.
A more innovative and detailed way is using the COPT of the power plant. This is formulated, for N P amount of generator combinations each with power-outage probability p i , during an amount of N T time intervals each with duration ∆t i . From this, the expected Loss Of Load Power (LOLP) is derived. This is the amount of time the available power P available_power_i is not sufficient to supply the ship's load P load (t j ), thus leading to a power outage, expressed here using the step function u().
Similarly, the expected LOLE is derived, showing the amount of active energy not supplied to the load for the same time period and is expressed here using the ramp function r().
This is also equal to: Furthermore, the LOLE can be individually expressed for each load category as: One way to calculate the cost of LOLE is by assuming a constant cost per load category Cost loss_energy−k , as seen in: A more innovative way is by assuming a cost proportional to the SMC calculated earlier:

Initial Cost
The first type of cost that comes to mind is the initial cost F initial−total of the generators. This is usually limited to their purchase price A m , provided by the vendors.
However, when building a ship, there is an additional cost resulting from the space allocated for the generators and its impact on the ship's size. This is estimated as a fraction of the total ship cost C, which in turn is approximated using semiempirical relations like the following, where a and b are constants and DWT is the DeadWeight Tonnage [22].
Appl. Sci. 2022, 12, 10463 10 of 20 A more detailed way to approach this is by considering the area E m and the volume V m occupied by the generator and also its mass M m , along with their associated unit costs Cost Area , Cost Volume and Cost Mass , respectively, as shown below: Therefore, the total installation and initial costs become: As the operating period of the ship is set to one year, all costs need to refer to this. In order for the initial cost to be projected to the total annual cost, the investment scheme must be examined. For an interest rate i cap and a number of T per annuities, the Capital Recovery Factor (CRF) [23] becomes: Therefore, the annual cost for the total recovery of the investment, or equivalent initial cost F initial−eq , becomes:

Flat Cost
No machinery may be left running unattended and without adequate maintenance. There is additional cost associated with this: the spare parts and the consumables used. This kind of work also requires specialized crew members, devoting a major portion of their time. As a consequence, their payroll was also included. This flat cost F flat , has been statistically approximated as cost per calendar hour Cost flat−m for the m-th generator and for a whole year became: Cost f lat−m (28)

Total Cost
Taking into account all the above, the equivalent annual cost of the electric power generating equipment is the following: F total i cap , T per = F initial−eq i cap , T per + F f lat + Cost operation (29) If reliability considerations are also taken into account, the equivalent annual cost becomes:

Computational Speed Improvement
Performing the above calculations proved to be a very computationally intensive task, even for modern computers, requiring several hours to complete. The major cause of delay was the Lagrange optimization and its repetition for every combination of running gensets, as well as for every time interval in the load profile, as previously seen in Figure 4.
The classic method of proportional load distribution is trivial and thus much faster. However, it has none of the efficiency benefits provided by the otherwise-rigorous Lagrange optimization. Achieving improved generator efficiency and fuel savings outweighed the convenience and speed of the classic method. Moreover, it is a well-established and documented method [24], especially for load distribution among thermal engines [25,26].
Dynamic programming could also be used, but it seemed more complex and less efficient, as it is a multilayer method that would be better suited to solve time-dependent problems [27].
To alleviate the speed concern, the memorization technique, shown in Figure 5 and described next, was also applied.

Computational Speed Improvement
Performing the above calculations proved to be a very computationally intensive task, even for modern computers, requiring several hours to complete. The major cause of delay was the Lagrange optimization and its repetition for every combination of running gensets, as well as for every time interval in the load profile, as previously seen in Figure 4.
The classic method of proportional load distribution is trivial and thus much faster. However, it has none of the efficiency benefits provided by the otherwise-rigorous Lagrange optimization. Achieving improved generator efficiency and fuel savings outweighed the convenience and speed of the classic method. Moreover, it is a well-established and documented method [24], especially for load distribution among thermal engines [25,26].
Dynamic programming could also be used, but it seemed more complex and less efficient, as it is a multilayer method that would be better suited to solve time-dependent problems [27].
To alleviate the speed concern, the memorization technique, shown in Figure 5 and described next, was also applied.  If the load profile has a duration of m months and a resolution of s samples per hour, then it will contain q intervals, where: Consequently, a profile of one year with a resolution of 30 min contains 365·24·2 = 17,520 intervals. On the other hand, the quantity n of generators populating the power plant, as determined in Section 2.4, can be quite high. Depending on the nominal power of a genset type and the redundancy and load requirements of a certain interval, the combinations of running generators can be as much as 2 n − 1.
Therefore, for an average n and for g different genset types, the Lagrange optimization code is executed on average l times, where: This amount can easily be in the order of several million, hence the large total execution time.
Then again, it is apparent that for the same generator type and the same total load, the optimization outcome is the same. If the calculation of the operating cost is the problem, then the Lagrange optimization section, with all its repetitions, is the subproblem. Due to the uniformity of the load profile, many load conditions are the same; therefore, an overlapping of subproblems exists. This is a strong indication that running time can be reduced [28].
According to the memorization technique, an empty matrix is created for storing all optimization (i.e., subproblem) results. Any time such a calculation is required, the code quickly checks the matrix for an existing solution. If one is found, meaning that this particular optimization was performed before, the results are retrieved and the detailed calculation is bypassed.
This approach achieved a computational time reduction of more than 300 times and the running time of the code was reduced from several hours to less than a minute.

Case Study
As a case study, the above method was applied to a real passenger ship. To populate her power plant and to come up with tangible results, an extended data base comprising the functional parameters from several real diesel generators was used. Of course, many different scenarios can also be tested and numerical data better, may easily be applied.

Load Profile
The ship performs the same routine every year. Its load profile was formed to span this time period with a resolution of 30 min. In detail, she completes an 8-hour cruise every weekday, as shown in Table 3. Weekends are holidays and only maintenance takes place. The crew also has 4 weeks of holidays every year. The total electrical load is therefore drafted as in Figure 6. As mentioned above, the total load is distinguished in critical, essential and inessential load. The critical load was measured and approximated as follows in Table 4.

Status
Critical Load at port 20 kW preparation for departure 100 kW underway 100 kW preparation for arrival 100 kW Noncritical load was divided into essential and inessential, as follows in Table 5. For reliability purposes, the cost of losing essential load was estimated at 100 times more that of losing inessential load. Similarly, the cost of losing critical load was estimated to be 100 times even higher, as shown below in Table 6. Table 6. Relative reliability cost.

Load Type
Relative Cost inessential 1 essential 100 critical 10,000

Generator Data: Electrical
The generators examined [29][30][31][32][33][34][35][36][37] covered an area of nominal power from 30 to 2250 kW. The whole range, along with their respective allowable limits of minimum and maximum power, may be seen in Figure 7. A common FOR equal to 0.0113 was used. As mentioned above, the total load is distinguished in critical, essential and inessential load. The critical load was measured and approximated as follows in Table 4. Table 4. Critical load approximation.

Status
Critical Load at port 20 kW preparation for departure 100 kW underway 100 kW preparation for arrival 100 kW Noncritical load was divided into essential and inessential, as follows in Table 5. For reliability purposes, the cost of losing essential load was estimated at 100 times more that of losing inessential load. Similarly, the cost of losing critical load was estimated to be 100 times even higher, as shown below in Table 6.

Generator Data: Electrical
The generators examined [29][30][31][32][33][34][35][36][37] covered an area of nominal power from 30 to 2250 kW. The whole range, along with their respective allowable limits of minimum and maximum power, may be seen in Figure 7. A common FOR equal to 0.0113 was used.

Generator Data: Mechanical
The equipment runs on light fuel (i.e., marine diesel) with a cost of 0.40 €/kg. Its fue consumption was approximated by a second-degree polynomial (with coefficients a, and c) versus its power output, as seen in Figure 8. The lubricating oil consumption cost was found to be proportional to the outpu power and was approximated in all cases as 0.006 €/kWh. The dimensions and the weigh of the engines are shown in Figure 9.

Generator Data: Mechanical
The equipment runs on light fuel (i.e., marine diesel) with a cost of 0.40 €/kg. Its fuel consumption was approximated by a second-degree polynomial (with coefficients a, b and c) versus its power output, as seen in Figure 8.

Generator Data: Mechanical
The equipment runs on light fuel (i.e., marine diesel) with a cost of 0.40 €/kg. Its f consumption was approximated by a second-degree polynomial (with coefficients a and c) versus its power output, as seen in Figure 8. The lubricating oil consumption cost was found to be proportional to the outp power and was approximated in all cases as 0.006 €/kWh. The dimensions and the weig of the engines are shown in Figure 9. The lubricating oil consumption cost was found to be proportional to the output power and was approximated in all cases as 0.006 €/kWh. The dimensions and the weight of the engines are shown in Figure 9. The lubricating oil consumption cost was found to be proportional to the out power and was approximated in all cases as 0.006 €/kWh. The dimensions and the wei of the engines are shown in Figure 9.

Generator Data: Cost
The acquisition and the maintenance cost of each genset are shown in Figure 10.

Generator Data: Cost
The acquisition and the maintenance cost of each genset are shown in Figure 10. The unit costs of installation due to area, volume and weight used were following, as seen in Table 7.

Installation Cost Type
Unit Cost due to area 0 €/m 2 due to volume 463 €/m 3 due to weight 0 €/kg The complete set of data can be found in Table A1 in the Appendix A.

Results
Assuming that no redundancy (n = n*) is required, the most efficient combinat turned out to be one engine of 1500 kW nominal power when reliability was considered. On the other hand, the most efficient combination turned out to be th engines of 500 kW nominal power each when reliability was considered, as seen in Fig  11. The unit costs of installation due to area, volume and weight used were the following, as seen in Table 7. The complete set of data can be found in Table A1 in the Appendix A.

Results
Assuming that no redundancy (n = n*) is required, the most efficient combination turned out to be one engine of 1500 kW nominal power when reliability was not considered. On the other hand, the most efficient combination turned out to be three engines of 500 kW nominal power each when reliability was considered, as seen in Figure 11.
Assuming that no redundancy (n = n*) is required, the most efficient combination turned out to be one engine of 1500 kW nominal power when reliability was not considered. On the other hand, the most efficient combination turned out to be three engines of 500 kW nominal power each when reliability was considered, as seen in Figure  11. Assuming that redundancy of a whole power compartment (n = 2·n*) is required, the most efficient combination turned out to be two engines of 1500 kW nominal power each when reliability of the ship power system was not considered. On the other hand, the most efficient combination turned out to be four engines of 750 kW nominal power each when reliability of the ship power system was considered, as seen in Figure 12. Assuming that redundancy of a whole power compartment (n = 2·n*) is required, the most efficient combination turned out to be two engines of 1500 kW nominal power each when reliability of the ship power system was not considered. On the other hand, the most efficient combination turned out to be four engines of 750 kW nominal power each when reliability of the ship power system was considered, as seen in Figure 12. The results can be summarized as follows in Table 8.  The results can be summarized as follows in Table 8.

Discussion
As observed in the examined designs, a ship with 1400 kW maximum load requirement can be sufficiently supplied by a single 1500 kW generator, assuming that no redundancy and reliability considerations exist.
When reliability begins to matter, one might expect a solution of two 750 kW engines. However, the proposed combination was three 500 kW engines. Although the total power supply capability remained the same, the larger number of engines is obviously more reliable.
Despite the fact that in the first case, the large engine ran most of the time at a load less than 30% of its nominal value, it was still more economical than the combination of the second case, which probably ran more efficiently per engine.
Next, when full redundancy became a requirement, as was expected, the scheme of the first case (without reliability) doubled, even though again only one generator was running at any certain time.
The same did not occur when both full redundancy and reliability were required, and the scheme of the second case was not doubled like before. Instead, four engines with 750 kW nominal power were selected as more efficient. This configuration is seen in many types of ships. Again, although the total power in both plants with full redundancy was the same, the cost of the reliable one was higher.

Conclusions
In this paper, a novel method was introduced to facilitate the selection process of the generators in a ship power plant. It uses many parameters related to all aspects of the life-cycle cost of the engines and to the actual operating routine of the ship; however, computational time is significantly low. This way, the designers can have a complete idea of the cost involved in their selection and its return on investment.
This method may be used for different operation scenarios simply by changing the numerical data. It can also be used for applications other than shipping, since industrial installations have similar needs. Even more exotic applications may also benefit from this, by calibrating the indicators used here, or simply adding new ones.
An idea for future work could be performing a sensitivity analysis to determine how much each the examined factors affect the outcome.
Another probably useful addition might be the consideration of the minimum running time, the minimum time between shutting down and starting up, the power-increase rate, and the starting cost.
Finally, it might prove advantageous to expand this method by testing combinations of different gensets and possible exploitation of renewable energy onboard. This way, ships with shaft generators and electric propulsion, but also terrestrial power factories, may be examined.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The diesel generator data used are shown in detail below.