1. Introduction
Traditional ship propulsion systems mainly rely on thermal engines, such as diesel engines [
1,
2] or gas turbines [
3], mechanically connected to either fixed or controllable pitch propellers, most of the time through a reduction gear. This propulsion plant layout has several clear advantages, such as being based on simple and well-consolidated technologies [
4], ensuring reliability and safety. Moreover, it relies on a small number of efficient energy transformations, ensuring a relatively high overall propulsion efficiency when operating in design conditions [
5,
6]. The latter makes traditional propulsion the most proficient choice for those marine units characterised by relatively narrow operating profiles, i.e., those ships that steam most of the time at their design speed. Combined propulsion plants [
7,
8,
9,
10] coupled with controllable pitch propellers can match the operating requirements of ships that require more flexible profiles, for instance, ferries that steam at a different speed in winter or summer season or for navy vessels.
In recent decades, diesel–electric propulsion [
11,
12] has grown as a good competitor for ship propulsion, bringing some additional benefits to operating flexibility and reduced footprint emission [
13]. This type of propulsion system has some drawbacks due to additional energy transformations that affect the overall efficiency at maximum speed [
14].
On the other hand, the benefits in terms of layout flexibility are straightforward. No shaft-line neither gearbox needs to be installed, allowing the machines to be allocated more efficiently in the available spaces, reducing the vessel’s acoustic signature and noise irradiation. Moreover, there is no mechanical link between the power generation and the propeller shaft, allowing more flexible control of both engines’ and propellers’ revolution speeds. Eventually, the power demand can be shared between the diesel generators (D/G) with more degrees of freedom, ship safety and availability benefit, and machinery redundancy. These aspects pushed ship designers to consider diesel–electric propulsion for passenger ships, navy ships, and various special units.
The possibility to maintain the D/G in optimal operating conditions makes diesel–electric propulsion an effective solution to meet the strict pollution regulations enforced nowadays by the International Maritime Organisation (IMO) [
15,
16]. In other words, a diesel–electric propulsion architecture is one of the state-of-the-art responses to the design of energy-efficient and environmentally friendly ships [
17].
Diesel–electric propulsion is also installed more and more on yachts and pleasure crafts as well, with a constant increase in new diesel–electric designs [
18,
19]. This is also due to the r improved environmental awareness [
20], the greater comfort that a flexible diesel–electric propulsion system allows in terms of noise and vibrations during navigation [
21], and the potential of saving fuel [
22].
A significant improvement to the efficiency of diesel–electric propulsion systems is due to the recent introduction of variable revolution speed generators [
23], allowing the diesel engines to work in their optimal efficiency conditions. This type of engine control logic is coupled with direct current (DC) distribution in order not to constrain the alternators to produced energy at a fixed distribution frequency, as opposed to alternate current (AC) distribution [
24,
25,
26].
The operating and layout flexibility of diesel–electric propulsion systems allows many degrees of freedom in the design phase compared to traditional propulsion. However, it is not straightforward to take advantage of those degrees of freedom in the design phase; traditional approaches usually reduce the number of design choices to consider, compare and evaluate to a manageable number. The application of more advanced computational approaches can consider and compare unconventional system layouts during the design phase and select the most promising solutions and compare them in a refinement phase.
This paper aims to present a method for the optimal design of diesel–electric ship propulsion systems, based on parametric modelling of the system layout and performance, which is optimised using a genetic algorithm [
27,
28]. Compared to other local minimisation algorithms [
29], a genetic algorithm has interesting features that suit the presented application: it allows one to efficiently deal with categorical or integer variables and non-differentiable cost functions, as it does not require to compute derivatives, and it is a global optimisation algorithm, so it is unlikely to get trapped in local minima of the cost function. For this reason, genetic algorithms find various applications in many industrial areas when it is required to deal with the selection of multiple variables affecting one complex system. Examples are the selection of a diesel engine’s optimal working parameters [
30], the parameter selection of a combined cycle [
31], or the optimal allocation of photovoltaic systems to maximise the performance of an electric microgrid [
32]. The optimisation of a geothermic plant design shown in [
33] is particularly relevant to the present work, as it performs a two-stage optimisation, separating the design phase from the computation of the optimal operating parameters. Moreover, relevant applications to many aspects of ship design can be found in [
34,
35,
36].
In the presented application, the algorithm is used to select the optimal type, number and design working conditions for the diesel generators to minimise the fuel consumption of the propulsion system at design speed. Moreover, the same approach is used to select the optimal plant operating mode and load sharing between the generators in off-design conditions.
The design method is applied to a case study pleasure craft, selecting the optimal propulsion layout using data of different marine diesel engines: some assumptions of the system layout are first made, then the cost function and constraints are formalised based on ship propulsion theory [
37]. The proposed method is used to select optimal layouts in two configurations, characterised by variable speed and constant speed controlled diesel generators, coupled with DC and AC distribution networks, respectively, at the same design speed. Next, the optimal propulsion load sharing in off-design conditions, i.e., at lower speeds, is computed. Results are compared and critically discussed both in design and off-design conditions to show the potential of the proposed approach.
2. Diesel–Electric Propulsion System Schemes
In the proposed approach, a diesel–electric system is considered for power generation and propulsion. The three main aspects to take into account when considering a diesel–electric system as a candidate for ship propulsion are:
Propulsion power demand and the electric load required for auxiliary services are comparable, the efficiency gap to mechanical propulsion might not be an issue;
The operating flexibility might be an advantage for those ship types that have very different operating profiles, characterised by, for example, very different ship speeds;
The layout flexibility might come in handy when considering a ship with limited spaces on-board or when the low noise level is a design criterion.
Ships that match the above-described requirements, and are thus usually powered by diesel–electric systems, are, for instance, passenger or cruise ships and some navy ships or pleasure crafts.
Figure 1 presents two alternative diesel–electric plant layouts considered in this study.
Figure 1a represents a typical diesel–electric propulsion system with an AC power distribution network: the diesel engines produce the alternate current through alternators and are connected to an AC network at constant voltage and frequency. As a consequence, diesel engines need to work at a constant revolution speed to maintain the network frequency.
Figure 1b shows an alternative layout using a DC distribution network (DC-link): this approach requires several DC/AC and AC/DC converters with their associated energy losses, yet it has some advantages. As the frequency is not an issue, the D/G control is only focused on the voltage, and the diesel generators can operate in optimal working conditions at partial loads. In addition, DC distribution is not affected by most of the main typical alternate current issues, such as reactive current losses or harmonic distortions [
24].
The standard layout for diesel–electric generation and propulsion of ships features some diesel engines of the same size, mainly for construction and maintenance convenience, as the same engines share the same spare parts. In the present study, the aim is to remove this constraint, allowing the plant to include engines of different sizes to maximise the plant’s efficiency in design conditions.
3. Plant Optimisation
The purpose of this study is to perform the propulsion system design using an optimisation approach. The design problem is formulated as an optimisation problem, and a genetic algorithm is used to find the solution, i.e., to find the minimum fuel consumption layout at design conditions. As a second step, the same optimisation approach, with slight modifications, is applied to determine the optimal working configuration (load sharing and engine working points) of the obtained layouts in off-design conditions, i.e., partial loads. Two alternative plant types are considered, designed and compared: AC and DC distribution. In the first plant type, represented in
Figure 1a, the revolution speed of the diesel engines is constrained by the network frequency, while in the second (
Figure 1b), diesel engines can be controlled at variable speeds.
In the design phase, the algorithm can select the number and type of diesel engines that are part of the propulsion plant, choosing between a number (four in this study, but the database could be reasonably enlarged) of diesel engines of different sizes and performance features. Moreover, the algorithm selects the optimal power of each engine for AC architecture and optimal power and revolution speed in DC configurations. In the two cases, the ship’s design speed is guaranteed while minimising the fuel consumption.
In the off-design phase, the propulsion system is already selected: the algorithm can select the number of operating engines and their working points (power and, if possible, i.e., in DC configuration, revolution speed) in order to minimise the fuel consumption while providing sufficient power to sustain both the required off-design speed and hotel-load.
In summary:
The algorithm is expected to select the number and type of diesel engines to install on-board;
Moreover, the algorithm is expected to select the power output of each engine if the network distribution is AC, the power output and revolution speed if the distribution is DC;
The selected solution layout should minimise the total fuel mass flow rate;
The selected solution should ensure the ship reaches the expected speed;
To sightly simplify the problem, engines of the same type are assumed to operate in the same conditions (power and revolution speed).
Thus, two alternative problems can be formulated, the first describing the AC power generation plant with constant revolution speed controlled generators, the second describing the DC plant with variable speed controlled generators. The following subsections describe all the aspects of the problem formulation, from the genetic encoding, i.e., the parametrisation of the problem, to the set up of the cost function and constraints, based on the steady-state modelling of the ship’s propulsion system.
3.1. Genetic Encoding
The crucial point when using a genetic approach to solve optimisation problems is the definition of the so-called genetic encoding. Let
be the number of diesel generator models available in the dataset, each one in number
, with power
and revolution speed
, and
identifying the engine model. The encoding in the case of variable speed controlled engines takes the following form:
In a similar way, the genetic encoding of a solution in case the engines are controlled at constant revolution speed with AC distribution is the following:
where
indicates the nominal revolution speed of the
ith engine model.
The total electric power provided if a solution
is selected is expressed by the following relationships, respectively, in case of DC and AC distribution:
where
is the efficiency of the
ith alternate current generator, and
is the efficiency of the
ith AC/DC converter, installed only with DC distribution.
3.2. Cost Function
The solution ranking after each generation in a genetic algorithm is performed using a cost function. In the presented application, the optimisation aims to minimise the total fuel mass flow rate of the power generation plant; thus, the following function is to be minimised:
where
represents the engine load diagram, providing the specific fuel consumption at a given revolution speed and power, implemented in the form of a function, such as using a response surface, and the measurement units in proper accordance.
3.3. Constraints
The definition of the constraints is a crucial passage in the presented approach in order to obtain a reasonable result. First, the bounds of the solutions need to be defined:
The number of engines for each type is required to be non-negative and less than a maximum value. The power and revolution speed boundaries are related to each of the engine models. Note that this framework can be applied both to design and off-design optimisations, setting proper boundaries of the maximum number of running engines, while, in the design phase, the number of engines on-board is to be defined, between zero and a reasonable maximum value, the off-design optimisation aims to determine the number of running engines in a given off-design condition, between zero and the number of engines on-board.
The next step is the formalisation of the required speed in the form of a non-linear constraint. In particular, the generated power
needs to be sufficient to ensure the ship’s speed
. If there are
propellers, the thrust
T required to each propeller is given by the following equation:
where
is the ship’s resistance and
t is the thrust deduction factor.
The power required by the electric propulsion motors is described by the following equations, referring to DC and AC distribution, respectively:
where
w is the wake fraction,
and
are the relative rotational efficiency and the mechanical transmission efficiency, respectively,
is the propeller open water efficiency,
and
are the efficiencies of the electric propulsion motor and the DC/AC converter, respectively. Note that
t,
w,
depend on the ship’s speed, and
depends on the propeller’s working conditions [
21].
Note that:
The selected propulsion layout is such that the propeller’s revolution speed is mechanically independent of the engines’, as there is no gearbox;
The propeller is modelled using the open-water diagrams and is assumed to have a fixed pitch.
The speed constraint is described by the following inequality:
where
the power required to satisfy the auxiliary services. Note that this should be an equality constraint: the power provided by the generation system in its working conditions should instantly match the power load. However, inequality is needed because some of the variables are integer numbers, and the solver cannot deal with integer variables and equality constraints at the same time. Moreover, only the lower bound of the power can be constrained because higher power leads to higher fuel consumption, and the optimisation will naturally lead to the lowest possible installed power that allows satisfying the speed constraint.
3.4. Optimisation Problem
The following optimisation problem, combining Equations (
5), (
6) and (
10), needs to be solved to determine the optimal propulsion plant configuration:
6. Conclusions
In this paper, an optimisation procedure has been presented, oriented to the optimal design of a diesel–electric ship propulsion system. In particular, a genetic algorithm has been used to design the optimal layout of a diesel–electric propulsion plant, including diesel generators of various sizes either with an AC or DC power distribution network. The same approach with slight variations is then applied to find the optimal load sharing strategy in several off-design conditions. The proposed method is applied to a case study vessel; specifically, a pleasure craft is considered. The comparison has been discussed in detail, including the original propulsion plant data as a reference.
DC distribution coupled with variable speed generator control is highly beneficial for vessels that have operating requirements that are very demanding in terms of flexibility. The variable revolution speed control of the diesel engines allows the DC systems to keep more stable SFOC values depending on the vessel’s speed, as the engines’ working point can be optimised further if compared to the constant revolution speed control approach. In particular, diesel–electric propulsion systems allow great flexibility, and optimal design and off-design configurations can be achieved by numeric optimisation, allowing maximisation of propulsive efficiency in the whole vessel’s speed operating range.
Numeric optimisation is an effective way to manage highly under-determined problems such as propulsion system layout design or optimal load sharing determination, and the results obtained are auspicious. The proposed approach comes in handy for propulsion plant designers, allowing them to manage high numbers of alternative options and combinations in a reasonable amount of time. When increasing the complexity of the problem, an exhaustive brute force analysis employing standard methods is not feasible.
It should be noted that the proposed approach is based on two sequential steps: first, the optimal layout to reach the design speed with all the engines running is determined, then the optimal load sharing in off-design conditions is computed, considering the propulsion system obtained in the design phase. In the future development of the proposed approach, these two steps are supposed to be nested to compute the optimal propulsion system design to match a given operating profile with two or more different design speeds.