# Multiple optimal solutions in synthesis, design and operation optimization of a marine energy system with the differential evolution algorithm

Florian Dugast, Stéphane Bénac, Pierre Marty, Pascal Chessé

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Florian Dugast, Stéphane Bénac, Pierre Marty, Pascal Chessé. Multiple optimal solutions in synthesis, design and operation optimization of a marine energy system with the differential evolution algorithm. Ocean Engineering, 2025, 328, pp.121064. 10.1016/j.oceaneng.2025.121064. hal-05018367

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# Multiple optimal solutions in synthesis, design and operation optimization of a marine energy system with the differential evolution algorithm

Florian Dugast<sup>a,*</sup>, Stéphane Bénac<sup>b</sup>, Pierre Marty<sup>a</sup> and Pascal Chessé<sup>a</sup>

## ARTICLE INFO

**Keywords:**

ship architecture design
synthesis optimization
design optimization
evolutionary algorithms

## ABSTRACT

This article deals with the optimization of a ship energy system on multiple levels (synthesis, design and operation). These complex problems often induce many local optima, making it difficult to obtain reliable optimization results. A clustering method based on the absence or presence of components in the architecture coupled with an evolutionary algorithm (i.e Differential Evolution) is proposed to tackle this issue. The method enables multiple optimal solutions to be identified for a real-world optimization problem. The reference ship for this study is a destroyer but the use of the Admiralty coefficient in the model description makes the algorithm easily adaptable for any kind of ships. The specific fuel consumption is pre-calculated for groups of components to alleviate the computational cost of the optimization problem. Two numerical cases are computed representing the ship energy system with or without a heat recovery and a heat creation system to illustrate the capabilities of the method. The objective in both cases is to minimize the weight of the ship (fuel consumption + components weights) for a given mission profile. The influence of the clustering technique, population size and repeatability of the differential algorithm is also investigated to assess the reliability of the method.

## 1. Introduction

From an engineering perspective, optimizing real-world systems is challenging due to factors such as a large design space, nonlinearities, or multiphysics modeling. In the maritime sector, ship design and optimization are crucial for ensuring the sustainability of marine transportation or naval defense while meeting economic and environmental constraints. Various methods can enhance ship efficiency over its lifetime, including route optimization (Krata and Szlapczynska (2018); Ma et al. (2020)) and hull design (Sariöz (2006); Han et al. (2012); Iqbal et al. (2025)).

This research focuses on optimizing the ship energy system during the early stages of the design process. This global system is composed of multiple subsystems (i.e., components), where optimization is typically divided into three levels: synthesis, design, and operation. These levels, discussed in recent reviews (Frangopoulos (2020); Trivyza et al. (2022)), are defined as follows:

* Synthesis: Determines the system architecture, including component selection and their interconnections.

* Design: Specifies component properties such as nominal power, size, and cost.

* Operation: Addresses the real-time state of components (e.g., load, flowrate, voltage) to achieve specific performance targets such as mechanical or electrical output.

In the literature on hybrid vehicle design, these levels are often referred to as topology, component sizing, and control, respectively (Silvas et al. (2016)). For marine energy systems, a fully electric ship design was explored in Shang et al. (2016), where two configurations were analyzed: 4 or 6 diesel generators (DGs) with or without an energy storage system (ESS). Control variables included the state of each DG (on/off), associated power, and speed variations to enable flexible voyage scheduling. The optimization problem was solved using the Non-Dominated Sorting Genetic

<figure><img src="image_2.png" /></figure>

<footnote><sup>a</sup>Nantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, Nantes, F-44000, France</footnote>

<footnote><sup>b</sup>Naval Group, CEMEP, Technocampus Ocean, Bouguenais, France</footnote>

<footnote>* Corresponding author</footnote>

<footnote>florian.dugast@ec-nantes.fr (F. Dugast)</footnote>

<footnote>ORCID(s): 0000-0002-2598-2157 (F. Dugast)</footnote>
Algorithm (NSGA II), capable of handling both integer and real variables. Similarly, design optimization of a hybrid electric drive for submarines was applied in Skinner et al. (2007), testing three topologies: direct-electric drive, geared-electric drive, and hybrid steam-electric drive. The latter, the most complex case, involved optimizing nine parameters, such as propeller diameter, motor size, and steam turbine load. Further, three different vessels were studied in Skjong et al. (2017) and were optimized through energy management and logic constraints, each with configurations including fixed or variable speed generators and optional ESS. By using environmental data, the route, speed and trim were optimized in Li et al. (2024). In addition to these variables, the shaft generator output power was also included in the optimization problem in Liu et al. (2025). Energy management optimization was also investigated in Peng et al. (2023) for a hybrid ship with fuel cells. These studies primarily address optimal control for fixed or limited architecture sets.

In this work, the synthesis level will be incorporated into the optimization process to allow greater flexibility in ship architecture design. Binary variables were used in Wu et al. (2024) to switch between PTO or PTI conditions and to enable or disable emission reduction equipments. More generally, component connections are organized into a superconfiguration or superstructure, encompassing all possible configurations of the energy system using available components (Frangopoulos (2020)). For example, a COGES (COmbined Gas turbine Electric and Steam) architecture was explored in Dimopoulos et al. (2008) using a superconfiguration, while a similar approach is employed in Sakalis and Frangopoulos (2018) with diesel engines, auxiliary boilers, and heat recovery steam generators. However, creating a superconfiguration becomes challenging as the number of component connections increases. To address this, a superconfiguration-free method for heat transfer optimization is proposed in Lazzaretto and Toffolo (2008) and Lazzaretto et al. (2018), where synthesis optimization is split into two steps: generating a basic configuration using elementary Rankine cycles and optimizing thermal power based on heat exchanger flows. This methodology was applied to marine Rankine cycle systems by Kalikatzarakis and Frangopoulos (2017). More generally, topology generation algorithms can derive system architectures based on a component list and predefined rules. For instance, de Vos and Stapersma (2018) combined adjacency matrices with genetic algorithms to design ship energy distribution systems. Similarly, Wijkniet and Hofman (2018) used logic programming in Prolog (Colmerauer (1990)) for vehicle powertrain topology generation. The present authors applied a similar topology generation approach to marine energy systems (Dugast et al. (2023)), which will serve as the foundation for this optimization study.

Dimopoulos et al. (2008) conducted pioneering work in the synthesis, design, and operation (SDO) optimization applied to marine energy systems. The ship architecture studied included two gas turbine models with various possible connections to a heat recovery steam generator (HRSG) system, and three operational modes were considered. The optimization problem was structured into two levels —synthesis-design and operation— and solved using a custom evolutionary algorithm that combined Particle Swarm Optimization (PSO) with the Struggle Genetic Algorithm (SGA) (Dimopoulos (2007)). Additionally, a study was conducted to evaluate the influence of fuel price on the synthesis optimization results. Building on this approach, Sakalis and Frangopoulos (2018) investigated the use of diesel engines as main propulsion units, incorporating more detailed modeling of the HRSG system. Baldasso et al. (2019) compared linear and nonlinear mixed-integer approaches for SDO problems, considering eight diesel engine models across six operational modes. While their cost function was based on fuel consumption and volume, Trivyza et al. (2018) introduced criteria addressing gas emissions. Further advancements were made by Marques et al. (2019), who addressed an SDO optimization problem for a liquid natural gas carrier, integrating additional considerations such as propeller modeling and weather conditions. In most of these studies, the mission profile consisted of a limited number of modes, though time-dependent profiles were explored in Sakalis et al. (2019); Tzortzis and Frangopoulos (2019).

The results of these research efforts are typically presented as a single solution to the SDO optimization problem. However, given a large design space, mission profiles, and available components, it is likely that multiple solutions with nearly identical objective function values exist. At the early stages of a project, when synthesis optimization is applied, a comprehensive understanding of the feasible domain's promising regions becomes invaluable for the designer. In this work, an optimization algorithm is developed to address the SDO problem while providing multiple solutions in a single run. A gradient-free approach is chosen for its ability to avoid premature convergence to local minima and handle non-differentiable functions when necessary. One drawback of gradient-free methods is their high computational cost due to numerous cost function evaluations, but this can be mitigated through parallelization. Among gradient-free algorithms, the Differential Evolution (DE) method (Storn and Price (1997)) was selected for several reasons:
* its ability to handle floating-point variables without requiring binarization, unlike genetic algorithms (GA);

* its simplicity, with only a few parameters to tune: 2 (crossover and mutation rates) compared to 4 in the Particle Swarm Optimization algorithm with adaptive inertia weight presented in Wang et al. (2018);

* its promising mutation strategy, which leverages information from the population (Wong and Dong (2005)): compared to a fixed or a random mutation rate in GA, the mutation is adjusted automatically depending on the convergence progress (high or low magnitude depending on whether the individuals are distant or close respectively).

The latter two reasons are good indications that the differential evolution algorithm may be able to solve various problems with few adjustments to parameters for handling cases beyond the ones presented here. Non-dominated features to track a Pareto front have not been considered in this article as the investigated problem is only composed of a single objective function.

Several studies have explored multi-modal optimization using the DE algorithm. Thomsen (2004) modified the selection process to replace candidates with the most similar individuals in the population, maintaining diversity. Epitropakis et al. (2011) altered mutation strategies by replacing the base vector with the individual nearest to the one being mutated. Duan et al. (2018) introduced clustering-based subpopulations with adaptive sizes to explore a broader search domain than classical DE. Dominico and Parpinelli (2021) combined these approaches, incorporating a local search algorithm and adaptive parameters for crossover rates and mutation scaling factors. However, when using clustering techniques, the creation of groups relies on distance criteria, which can be challenging to fine-tune. Achieving a balance between effective exploration and computational efficiency often varies between problems.

In this article, the synthesis level of the SDO problem is utilized to define clusters, eliminating the need for distance-based criteria and improving algorithm robustness. Specifically, the presence or absence of a particular component in the architecture defines a cluster. The main innovations and contributions of this work compared to the state-of-art are: 1) the obtention of multiple solutions for the synthesis, design and operation of a marine energy system with a clustering technique; 2) a specific determination of the clusters (absence or presence of a component) to avoid a distance criterion in classical clustering methods which can be difficult to adjust; 3) the use of the Admiralty coefficient in the physical modeling so that the algorithm can be adapted very easily to any kind of ship. The synthesis, design and operations levels of the optimization problem are respectively presented in Sections 2.1, 2.2 and 2.3. The classical DE algorithm and its clustering procedure are described in Section 3. Two test cases are presented in Section 4 to demonstrate the adaptability of the proposed method:

* Case 1: Propulsion and electricity generation using diesel engines and/or gas turbines.

* Case 2: Case 1 with the addition of a heating supply using an HSRG and/or a boiler.

The optimization results will be analyzed from a physical perspective to ensure their relevance. Finally, the limitations and potential improvements of the approach are discussed in the conclusion.

## 2. Optimization problem

The system under investigation is responsible for the propulsion, electricity generation and heat supply on a ship. The various components required for this purpose are shown in Figure 1. The connections between component blocks are determined using a constraint-programming approach previously presented in Dugast et al. (2023). Two cases are examined: one considering only propulsion and electricity requirements (Case 1) and another including heat supply as well (Case 2). The component blocks and connections specific to Case 2 are shown in a lighter color in Figure 1. The letters *e* or *m* in diesel engines or gas turbines identification distinguish between electricity production and mechanical propulsion, respectively. Different variables for each component block are shown in Figure 1:

* power outputs $W_X^Y$ for the component block *X* and path *Y*

* component models $m_X$
* efficiencies $\eta_X$

The variables subjected to optimization are written in red. Power outputs $W_X^Y$ are vectors as they contain one value per mode. The three levels of the optimization problem (synthesis, design and operation) are described in the following.

<figure><img src="image_3.png" /><figcaption>Figure 1: Ship energy propulsion system</figcaption></figure>

## 2.1. Synthesis level

Based on Figure 1, the ship architecture structure is composed of 6 different paths for Case 1:

* electric propulsion with diesel engines

* electric propulsion with gas turbine engines

* hotel load with diesel engines

* hotel load with gas turbine engines

* mechanical drive with diesel engines

* mechanical drive with gas turbines

The presence of these paths in the optimal ship architecture defines the synthesis level of the optimization problem. An active path is determined by a non-zero power output for the component associated with it. For example, electric propulsion with diesel engines is active if one element of $W_{DEe}^{prop}$ is superior to zero. The same methodology is applied for Case 2 with additional optimized components (HRSG and AB). More precisely, the variables determined by optimization for the synthesis level are the following ones:

$$ W = [W_{DEe}^{prop}, W_{GTe}^{prop}, W_{DEm}, W_{GTm}, W_{DEe}^{hl}, W_{GTe}^{hl}, W_{HRSG}^{prop1}, W_{AB}^{prop1}, W_{HRSG}^{prop2}, W_{AB}^{prop2}, W_{HRSG}^{hl}, W_{AB}^{hl}, W_{HRSG}^{heat}, W_{AB}^{heat}] \quad (1) $$

where $W_{HRSG}$ and $W_{AB}$ are only required for Case 2. The superscripts prop, prop1, prop2, hl and heat refer to the different paths associated to the function propulsion, hotel load or heat. The subscripts refer to the different component blocks. The same variables $W$ will be used at the operation level to limit the number of optimization variables.
## 2.2. Design level

Different models are available for the diesel engines and gas turbines, as their selection within the ship's architecture significantly impacts performance calculations. The goal of the optimization problem is to minimize fuel consumption and component weights across various modes by selecting the appropriate models for each component block, along with the power outputs that meet the propulsion, hotel load or heating requirements. To make the calculations more efficient, the specific fuel consumption of diesel engines and gas turbines for different power values and groups of models is precomputed before starting the optimization process. To ensure a manageable number of groups for evaluation, a limit of four models per component block is set.

The diesel engine models were chosen from Wartsila products due to the availability of specific fuel consumption data at different loads Wartsila (2025). Three models were selected to cover a large range of output power, as indicated in Table 1. The SFC at 15% and 35% were manually added to improve interpolation at low loads. The gas turbine models are sourced from two manufacturers: Rolls-Royce for the MT30 Herdzik and Cwilewicz (2017) and General Electric for the LM2500 Bonet and Pilidis (2013) and LM500 General Electric (2025). The corresponding data are given in Table 2. For the LM500, SFC at 10%, 25% and 50% were manually added following a similar trend compared to LM2500. A distinction is made between a use for mechanical propulsion (m) or electricity production (e).

<table><caption>**Table 1**</caption><thead><tr><th>Models</th><th>6L46F m</th><th>6L46F e</th><th>12V46F m</th><th>12V46F e</th><th>16V46F m</th><th>16V46F e</th></tr></thead><tbody><tr><td>Max. Power (MW)</td><td>7.2</td><td>7.2</td><td>14.4</td><td>14.4</td><td>19.2</td><td>19.2</td></tr><tr><td>Weight (tons)</td><td>97</td><td>97</td><td>177</td><td>177</td><td>233</td><td>233</td></tr><tr><td>SFC 15% (g/kWh)</td><td>300</td><td>350</td><td>300</td><td>350</td><td>300</td><td>350</td></tr><tr><td>SFC 35% (g/kWh)</td><td>210</td><td>240</td><td>210</td><td>240</td><td>210</td><td>240</td></tr><tr><td>SFC 50% (g/kWh)</td><td>182.2</td><td>197.0</td><td>179.3</td><td>194.1</td><td>188.4</td><td>194.1</td></tr><tr><td>SFC 75% (g/kWh)</td><td>179.2</td><td>186.9</td><td>176.4</td><td>184.0</td><td>182.1</td><td>184.0</td></tr><tr><td>SFC 85% (g/kWh)</td><td>174.9</td><td>176.8</td><td>173.0</td><td>174.9</td><td>173.0</td><td>174.9</td></tr><tr><td>SFC 100% (g/kWh)</td><td>183.4</td><td>182.5</td><td>180.6</td><td>179.6</td><td>179.6</td><td>179.6</td></tr></tbody></table>

List of diesel engine models

<table><caption>**Table 2**</caption><thead><tr><th>Models</th><th>LM500 m</th><th>LM500 e</th><th>LM2500 m</th><th>LM2500 e</th><th>MT30 m</th><th>MT30 e</th></tr></thead><tbody><tr><td>Max. Power (MW)</td><td>4.5</td><td>4.5</td><td>25</td><td>25</td><td>40</td><td>40</td></tr><tr><td>Weight (tons)</td><td>3</td><td>30</td><td>18</td><td>94</td><td>32</td><td>77</td></tr><tr><td>SFC 10% (g/kWh)</td><td>701</td><td>701</td><td>681</td><td>681</td><td>700</td><td>700</td></tr><tr><td>SFC 25% (g/kWh)</td><td>539</td><td>539</td><td>522</td><td>522</td><td>350</td><td>350</td></tr><tr><td>SFC 50% (g/kWh)</td><td>411</td><td>411</td><td>397</td><td>397</td><td>250</td><td>250</td></tr><tr><td>SFC 100% (g/kWh)</td><td>269</td><td>269</td><td>261</td><td>261</td><td>220</td><td>220</td></tr></tbody></table>

List of gas turbine models

Combinations of these models give 34 groups of diesel engines and gas turbines. Details of each group's composition, are provided in Tables 13 and 14 in Appendix A. For instance, the group 18 for diesel engines corresponds to a configuration of 2 × 6L46F engines and 2 × 12V46F engines.

The optimal SFC for each group is calculated under the assumption that the load is evenly distributed among all active components in the group. An example of such a calculation is given in Table 3 for a target power of 5 MW. Although this group includes four diesel engines, not all engines are necessarily active, depending on the required power. As a result, all possible combinations are tested. The first two columns in Table 3 indicate the number of active engines. The percentage load is calculated as the target power divided by the maximum power of the combination. The specific fuel consumption is then computed as the weighted SFC of each engine, derived through linear interpolation of the data in Table 1. Finally, the combination with the lowest SFC is selected and highlighted in bold in Table 3. The corresponding SFC value represents one dot in the scatter plot shown in Figure 2 on the left.
<table><caption>Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm</caption><thead><tr><th>6L46F</th><th>12V46F</th><th>Max. power (MW)</th><th>%load</th><th>SFC (g/kWh)</th></tr></thead><tbody><tr><td>0</td><td>1</td><td>14.4</td><td>35%</td><td>211.25</td></tr><tr><td>0</td><td>2</td><td>28.8</td><td>17%</td><td>289.4</td></tr><tr><td><b>1</b></td><td><b>0</b></td><td><b>7.2</b></td><td><b>69%</b></td><td><b>179.8</b></td></tr><tr><td>1</td><td>1</td><td>21.6</td><td>23%</td><td>263.3</td></tr><tr><td>1</td><td>2</td><td>36</td><td>14%</td><td>300.0</td></tr><tr><td>2</td><td>0</td><td>14.4</td><td>35%</td><td>211.2</td></tr><tr><td>2</td><td>1</td><td>14.4</td><td>17%</td><td>289.4</td></tr><tr><td>2</td><td>2</td><td>14.4</td><td>12%</td><td>300.0</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Figure 2:** Diesel engine (left) and gas turbine (right) specific fuel consumption (g/kWh) for the different groups of models and output powers (MW)</figcaption></figure>

Different models are also available for the heat recovery steam generator and for the auxiliary boiler. Their characteristics are listed in Table 6. The load-dependent efficiency for the HRSG is given in Table 2.2 based on Silva et al. (2020). The points for Load=0.4 and Load=0.6 were manually added as this information is missing in Silva et al. (2020). The load-dependent efficiency for the auxiliary boiler is given in Table 2.2 based on Zhang et al. (2018). The specific fuel consumption for the auxiliary boiler comes directly from its PCI value (42 MJ/kg) and there is no specific fuel consumption associated to the heat recovery steam generator.

<table><caption>**Table 4**

Load-dependent efficiency for the HRSG</caption><thead><tr><th>Load</th><th>Efficiency η<sub>HRSG</sub></th></tr></thead><tbody><tr><td>0.4</td><td>0.71</td></tr><tr><td>0.6</td><td>0.71</td></tr><tr><td>0.8</td><td>0.71</td></tr><tr><td>1.0</td><td>0.86</td></tr></tbody></table>

At the end, the optimization variables *m* associated to the design level are:

$$m = [m_{DEe}, m_{GTe}, m_{DEm}, m_{GTm}, m_{HRSG}, m_{AB}] \quad (2)$$

where the first four components represent groups of models. *m*<sub>HRSG</sub> and *m*<sub>AB</sub> represent models and are only used for Case 2.
<table><caption>**Table 5**

Load-dependent efficiency for the auxiliary boiler</caption><thead><tr><td>Load</td><td>Efficiency η<sub>AB</sub></td></tr></thead><tbody><tr><td>0.25</td><td>0.7</td></tr><tr><td>0.4</td><td>0.7325</td></tr><tr><td>0.6</td><td>0.775</td></tr><tr><td>0.8</td><td>0.8</td></tr><tr><td>1.0</td><td>0.81</td></tr></tbody></table>

<table><caption>**Table 6**

List of HRSG and AB models</caption><thead><tr><td>Models</td><td>Max. Power (MW)</td><td>Weight (tons)</td></tr></thead><tbody><tr><td>HSRG model 1</td><td>0.6</td><td>3</td></tr><tr><td>HSRG model 2</td><td>6.0</td><td>33</td></tr><tr><td>HSRG model 3</td><td>12.0</td><td>67</td></tr><tr><td>AB model 1</td><td>0.6</td><td>5</td></tr><tr><td>AB model 2</td><td>6.0</td><td>43</td></tr><tr><td>AB model 3</td><td>12.0</td><td>74</td></tr></tbody></table>

## 2.3. Operation level

Power outputs are calculated for each component in the ship architecture to fulfill propulsion, hotel load, or heating requirements. The equations are presented for both cases at the same time. The $\delta$ symbol is used to distinguish the two cases ($\delta = 0$ for Case 1 and $\delta = 1$ for Case 2). The simulation is performed here in 0D and in steady-state. The velocity of the ship $V$ in knots is determined by assuming that the propeller thrust $T_{PPE}$ is equal to the ship resistance $R_s$:

$$R_s = a \times V^2 \qquad (3)$$

$$W_{PPE} = T_{PPE} \times V = a \times V^3 \qquad (4)$$

where $W_{PPE}$ is the propeller output power in MW. The ship propeller law coefficient $a$ is calculated with the Admiralty coefficient $C_{am}$ which can be written as:

$$C_{am} = \frac{(D_0 + D_c)^{2/3} V_{kn}^3}{W_{PPE,kW}} \qquad (5)$$

where $D = D_0 + D_c$ is the ship displacement in tons, $V_{kn}$ the ship speed in knots and $W_{PPE,kW}$ the mechanical output of the propeller in kW. The values $C_{am} = 140$ and $D_0 = 7500$ tons are chosen to be representative of a destroyer. Due to its complex mission profile, the destroyer is a good candidate for a challenging optimization problem. $D_c$ represents the weights of the components subject to optimization. Based on (4) the coefficient $a$ is given by:

$$a = 1.94^3 \times 1^{e-3} \times \frac{(D_0 + D_c)^{2/3}}{C_{am}} \qquad (6)$$

where $1.94$ comes from the conversion between $m/s$ and knots for the ship speed and $1^{e-3}$ from the conversion between $MW$ and $kW$ occurring from (4) to (5). The required power for the propellers is then determined by their connected component blocks:

$$W_{PPE} = \eta_{PPE} W_{SHAFT} = \eta_{PPE} (W_{ELM} + W_{DEm} + W_{GTm} + \delta W_{ST2}) \qquad (7)$$

$$W_{ELM} = \eta_{ELM} W_{ELN}^{prop} = \eta_{ELM} (W_{AL1}^{prop} + W_{AL2}^{prop} + \delta W_{AL3}^{prop}) \qquad (8)$$

$$W_{AL1}^{prop} = \eta_{AL} W_{DEe}^{prop} \qquad (9)$$

$$W_{AL2}^{prop} = \eta_{AL} W_{GTe}^{prop} \qquad (10)$$

$$W_{AL3}^{prop} = \eta_{AL} W_{ST1}^{prop} = \eta_{AL} \eta_{ST} (\eta_{HRSG} W_{HRSG}^{prop1} + \eta_{AB} W_{AB}^{prop1}) \qquad (11)$$
Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm

$$W_{ST2} = \eta_{ST} (\eta_{HRSG} W_{HRSG}^{prop2} + \eta_{AB} W_{AB}^{prop2}) \quad (12)$$

Then the hotel load power is given by:

$$W_{HL} = W_{ELN}^{hl} = W_{AL1}^{hl} + W_{AL2}^{hl} + \delta W_{AL3}^{hl} \quad (13)$$

$$W_{AL1}^{hl} = \eta_{AL} W_{DEe}^{hl} \quad (14)$$

$$W_{AL2}^{hl} = \eta_{AL} W_{GTe}^{hl} \quad (15)$$

$$W_{AL3}^{hl} = \eta_{AL} W_{ST1}^{hl} = \eta_{AL} \eta_{ST} (\eta_{HRSG} W_{HRSG}^{hl} + \eta_{AB} W_{ab}^{hl}) \quad (16)$$

Finally the heat power is written as:

$$W_{HEAT} = \eta_{HRSG} W_{HRSG}^{heat} + \eta_{AB} W_{AB}^{heat} \quad (17)$$

The maximum value of the heat recovery steam generator power is:

$$\bar{W}_{HRSG} = \eta_{EXG} (W_{DEe}^{prop} + W_{GTe}^{prop} + W_{DEe}^{hl} + W_{GTe}^{hl} + W_{DEm} + W_{GTm}) \quad \text{with} \quad \bar{W}_{HRSG} = \sum_{j} W_{HRSG}^{j} \quad (18)$$

The different parameters are given in Table 7.

<table><caption>**Table 7**

Parameters for the ship energy system calculation</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td><td>Reference</td></tr></thead><tbody><tr><td>η<sub>PPE</sub></td><td>propeller efficiency</td><td>0.68</td><td>Tadros et al. (2021)</td></tr><tr><td>η<sub>AL</sub></td><td>alternator efficiency</td><td>0.97</td><td>NIDEC (2025)</td></tr><tr><td>η<sub>ELM</sub></td><td>electric motor efficiency</td><td>0.92</td><td>Szénásy and Csikor (2024)</td></tr><tr><td>η<sub>ST</sub></td><td>steam turbine efficiency</td><td>0.4</td><td>Tanuma (2022)</td></tr><tr><td>η<sub>EXG</sub></td><td>exhaust gaz fraction</td><td>0.5</td><td>Marty (2014)</td></tr></tbody></table>

## 2.4. Objective function and constraints

Gathering all information from synthesis, design and operation levels, the optimization problem can be formulated as follows:

min $W,m$ $J$

$$W_{PPE}[k] - aV[k]^3 = 0 \quad (19)$$

$$W_{HL}[k] - HL[k] = 0 \quad (20)$$

$$W_{HEAT}[k] - H[k] = 0 \quad (21)$$

$$\bar{W}_{HRSG}[k] \leq \eta_{EXG} (W_{DEe}^{prop}[k] + W_{GTe}^{prop}[k] + W_{DEe}^{hl}[k] + W_{GTe}^{hl}[k] + W_{DEm}[k] + W_{GTm}[k]) \quad (22)$$

$$\bar{W}_{c[i]}[k] \leq f_{c[i]}(m_{c[i]}) \quad \text{with} \quad \bar{W}_{c[i]}[k] = \sum_{j=1}^{N_{paths,c[i]}} W_{c[i]}^{j}[k] \quad (23)$$

$$i \in [1, N_{comp}]$$

$$j \in [1, N_{paths,c[i]}]$$

$$k \in [1, N_{modes}]$$

$$W \in \mathbb{R}^{+}$$

$$m \in \mathbb{N}^{*}$$
where $J$ is given by:

$$J = \omega_1 J_1 + \omega_2 J_2 \tag{24}$$

$$\text{with } J_1 = \sum_{k=1}^{N_{\text{modes}}} \frac{t[k] \times \sum_{i=1}^{N_{\text{comp}}} 1000 \times \bar{W}_{c[i]}[k] \times g_{c[i]}(m_{c[i]}, \bar{W}_{c[i]}[k])}{10^6} \tag{25}$$

$$\text{and } J_2 = \sum_{i=1}^{N_{\text{comp}}} \left( \max_k \left( \bar{W}_{c[i]}[k] \right) > 0.1 \right) h_{c[i]} \tag{26}$$

$J_1$ stands for the fuel consumption in tons for the considered mission and $J_2$ is the total mass in tons of the optimized components. In (26), the weights $\omega_1$ and $\omega_2$ are chosen equal to one to find a good compromise between a reduction in fuel mass due to an advantageous specific fuel consumption or due to the use of a lighter component. The underlying question behind this setting is: can the lower specific fuel consumption of the diesel engines compensate their larger mass compared to gas turbines? Note that if the objective function had focused on CO2 emissions or on the OPEX of the ship, the choice of $\omega_1$ and $\omega_2$ would have been different.

In these equations, $f_{c[i]}(y)$, $g_{c[i]}(y, x)$ and $h_{c[i]}(y)$ are the functions that give the maximum output power, the SFC, and the weight of each group of models $y$ for the power value $x$. Equations (21) and (22) are only used in Case 2. For the components DEe, GTe, DEm and GTm, the function $g_{c[i]}(y, x)$ is composed of cubic splines based on the scattered data presented in Section 2.2. For the HRSG, $g_{c[i]}(y, x)$ is equal to zero. For the boiler, $g_{c[i]}(y, x)$ has a constant value, following explanations given on Section 2.2. The list of optimized component blocks $c$ is written as:

$$c = [\text{DEe}, \text{GTe}, \text{DEm}, \text{GTm}, \text{HRSG}, \text{AB}] \tag{27}$$

where the last two component blocks are dedicated to Case 2 only. $N_{\text{comp}}$ represents the number of components involved in the optimization; it is equal to the size of $c$. $N_{\text{paths,c[i]}}$ stands for the number of paths for a specific component based on the governing equations defined in Section 2.3 and $N_{\text{modes}}$ is the number of modes based on the ship mission profile; its value will be given later.

As is usually done in gradient-free approaches, both equality and inequality constraints are enforced by using a penalty term in the so-called augmented objective function $J^+$. The contribution for the equality constraints $J^{eq}$ is written as:

$$J^{eq} = \left\| (W_{\text{PPE}}[k] - aV[k]^3) \right\| + \left\| (W_{\text{HL}}[k] - \text{HL}[k]) \right\| + \left\| (W_{\text{HEAT}}[k] - \text{H}[k]) \right\|, \quad k \in [1, N_{\text{modes}}] \tag{28}$$

where the last norm is applied for Case 2 only. Then the contribution for the inequality constraints $J^{ineq}$ is given by:

$$J^{ineq} = \sum_{k=1}^{N_{\text{modes}}} \left[ \max \left( \bar{W}_{\text{HRSG}}[k] - \eta_{\text{EXG}} \left( W_{\text{DEe}}^{\text{prop}}[k] + W_{\text{GTe}}^{\text{prop}}[k] + W_{\text{DEe}}^{\text{hl}}[k] + W_{\text{GTe}}^{\text{hl}}[k] + W_{\text{DEm}}[k] + W_{\text{GTm}}[k] \right), 0 \right) \right] + \\ \sum_{i=1}^{N_{\text{comp}}} \left[ \max_k \left( \bar{W}_{c[i]}[k] - f_{c[i]}(m_{c[i]}, 0) \right) \right] \tag{29}$$

where the first term is applied for Case 2 only. Finally the augmented objective function $J^+$ is written as:

$$J^+ = J + \lambda (J^{eq} + J^{ineq}) \tag{30}$$

with $\lambda$ a user-defined value accounting for the constraints weight.
One challenge in the optimization problem described above is the presence of the categorical variables **m**, representing the different models of optimized component blocks. Although the models are arranged in ascending order based on their maximum power, discontinuities in the objective function values may occur between adjacent models. This introduces difficulties in achieving convergence, as neighboring information cannot be reliably used. Furthermore, optimizing simultaneously **m** and **W** can be problematic, as it becomes unclear whether differences in the objective function values between two individuals arise from **m** or from **W**. To address these issues, **m** is removed from the optimization variables and is instead determined as the models or groups of models that give the lowest objective function for a given **W**, as follows:

$$m_{c[i]} = \arg \min_{l} \left[ \sum_{k=1}^{N_{\text{modes}}} \left[ \frac{t[k] \times 1000 \times W_{c[i]}[k] g_{c[i]}(l, W_{c[i]}[k])}{10^6} \right] + \left( \max_{k} (\bar{W}_{c[i]}[k]) > 0.1 \right) h_{c[i]}(l) \right. \\ \left. + \lambda \max \left( \max_{k} (\bar{W}_{c[i]}[k]) - f_{c[i]}(l, 0) \right) \right] \quad (31)$$

with $i \in [1, N_{\text{comp}}]$ and $l \in [1, N_{\text{groups,c[i]}}]$. $N_{\text{groups,c[i]}}$ corresponds to the number of groups of models or models for the component $c[i]$. By using this approach, only **W** remains as optimization variables in the differential evolution algorithm and each individual is compared with its own optimal models, making the convergence of the algorithm faster. The drawback of this method is an additional computational time required to evaluate the objective function as **m** is not given but calculated inside of the objective function. However from the whole perspective, the number of generations to compute is lower as the convergence is better with this approach so the increase in the global computational time is reasonable and the algorithm is more robust.

## 3. Differential evolution algorithm

The optimization problem is solved using an evolutionary algorithm, which is a gradient-free approach. This method offers advantages such as superior exploration capabilities compared to gradient-based methods and the ability to handle non-differentiable functions. However, its main drawback is the large number of objective function evaluations required. Although various evolutionary algorithms have been proposed in the literature, they all follow similar principles. Different values of $x$ are randomly generated as an initial population where each $x$ vector is called an individual. Then the individuals are modified to search for the optimal $x^*$ that minimizes the function $f$. Each iteration of this process is usually called a generation. The key steps of an evolutionary algorithm are the initialization of the population, the crossover/mutation where new individuals are created, and the selection process that incorporates the new individuals in the population if they perform better than the previous ones. The population is updated repeatedly until the termination criteria are met, which is typically defined as the stagnation of the objective value.

### 3.1. Original approach

Among many evolutionary algorithms, the differential evolution approach (Storn and Price (1997)) was chosen for this study due to its ability to treat floating values without binarization and its minimal parameter tuning requirements. Additionally, the mutation magnitude in DE is dynamically adjusted during the optimization process by considering the differences between two individuals. This makes the algorithm more adaptable to various optimization scenarios Wong and Dong (2005). The classical differential evolution algorithm is outlined in Algorithm 1. For population initialization, a Latin hypercube sampling method is employed to ensure a more uniform exploration of the design space.

### 3.2. Additional feature to mitigate residuals errors

The optimization problem includes equality and inequality constraints for each mode, and certain modes may pose greater challenges in satisfying these constraints. This can occur, for example, when a small constraint violation results in a significant decrease in the objective function value. In addition to the clustering technique discussed later in this section, an enhancement has been introduced to the classical DE algorithm to more effectively address these constraints within the SDO optimization problem. Specifically, when the objective function stagnates for a given individual, a new optimization process is initiated using the DE algorithm, but with fixed models. By excluding model determination from the objective function, this simplified problem runs approximately 50 times faster than the standard problem. To avoid excessive computational effort, this procedure is applied only to the ten most promising individuals and is triggered every 300th generation.
**Algorithm 1** Classical Differential Evolution algorithm

1: **INPUTS**
2:     **npop**: population size
3:     **f**: objective function
4:     **CR**: crossover rate
5:     **F**: mutation factor
6:
7: **INITIALIZATION**
8:     Initialize the population pop with npop random individuals
9:
10: **while** criteria = false **do**
11:
12:     **for** i in 1:npop **do**
13:         Generate 3 random integers: $r_1, r_2$ and $r_3$ in [1,npop] with $r_1 \neq r_2 \neq r_3 \neq i$
14:         Generate a random integer $r_4$ in [1,size(individual)]
15:
16:         **CROSSOVER AND MUTATION**
17:             candidate.pop[i]
18:             **for** j in 1 size(individual) **do**
19:                 **if** rand() < CR or j == $r_4$ **then**
20:                     candidate[j] = pop[$r_3$][j] + F(pop[$r_1$][j] - pop[$r_2$][j])
21:                 **end if**
22:             **end for**
23:
24:         **SELECTION**
25:             **if** f Typing (candidate) ≤ f(pop[i]) **then**
26:                 pop[i] = candidate
27:             **end if**
28:
29:         **end for**
30:     **end while**

3.3. Definition of clusters

In an ideal scenario, the objective function to be minimized has a single global minimum, and the optimal individual is located at this point. However, for complex optimization problems, the objective function is often non-convex, which can cause the algorithm to become trapped in a suboptimal solution. Additionally, in engineering problems with many optimization variables, multiple solutions may exist, and it is valuable for the designer to explore these alternatives. Unfortunately, evolutionary algorithms typically converge to only one or a few solutions. To address these challenges, clustering techniques have been developed for evolutionary algorithms. Typically, individuals are grouped together based on small distances between them (Duan et al. (2018); Dominico and Parpinelli (2021)). This approach aims to increase diversity within the population during optimization, enabling exploration of a larger portion of the feasible solution space. The expected impact of the clustering technique is illustrated in Figure 3 for a two-variable optimization problem with five local optima. One challenge in clustering is determining the threshold distance for forming groups. If the threshold is too large, multiple local optima may fall into the same group, reducing the algorithm's ability to identify distinct solutions. Conversely, if the threshold is too small, the population is divided into too many groups, leaving fewer individuals per group and potentially hindering convergence to the optimal solution.

The differential evolution algorithm is applied in this study to determine the topology, component models, and power values of a marine energy system. This problem is commonly referred to as a synthesis, design, and operation optimization problem. At the synthesis level, the absence or presence of components is identified, resulting in a specific architecture topology that is then used in the design and operation stages. This decomposition is leveraged to define clusters, where individuals belonging to the same architecture topology are grouped into a single cluster. The primary
Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm

<figure><img src="image_5.png" /><figcaption>Figure 3: Differential Evolution with or without clustering</figcaption></figure>

<table><caption>**Table 8**
Architecture topologies for Case 1</caption><thead><tr><th>Topology</th><th>DEe</th><th>GTe</th><th>DEm</th><th>GTm</th><th>Variables per mode</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>2</td></tr><tr><td>2</td><td>0</td><td>1</td><td>0</td><td>1</td><td>3</td></tr><tr><td>3</td><td>0</td><td>1</td><td>1</td><td>0</td><td>3</td></tr><tr><td>4</td><td>0</td><td>1</td><td>1</td><td>1</td><td>4</td></tr><tr><td>5</td><td>1</td><td>0</td><td>0</td><td>0</td><td>2</td></tr><tr><td>6</td><td>1</td><td>0</td><td>0</td><td>1</td><td>3</td></tr><tr><td>7</td><td>1</td><td>0</td><td>1</td><td>0</td><td>3</td></tr><tr><td>8</td><td>1</td><td>0</td><td>1</td><td>1</td><td>4</td></tr><tr><td>9</td><td>1</td><td>1</td><td>0</td><td>0</td><td>4</td></tr><tr><td>10</td><td>1</td><td>1</td><td>0</td><td>1</td><td>5</td></tr><tr><td>11</td><td>1</td><td>1</td><td>1</td><td>0</td><td>5</td></tr><tr><td>12</td><td>1</td><td>1</td><td>1</td><td>1</td><td>6</td></tr></tbody></table>

benefit of this approach is that it eliminates the need to determine a threshold distance, which is a common challenge in clustering techniques. Also in the formulation of the optimization problem, the absence of a component is determined by a zero power value at the operation level. However, such solutions are often difficult for the optimization algorithm to achieve due to the discontinuous behavior near these points. Focusing on a specific architecture topology can improve convergence of the differential evolution approach.

In the optimization problem presented above, 6 component blocks are subject to optimization, namely DEe, GTe, DEm, GTm, HRSG and AB. The different architecture topologies are then based on the absence (C=0) or presence (C=1) of these specific blocks and the architecture topology is defined as $T = [C_{DEe}, C_{GTe}, C_{DEm}, C_{GTm}, C_{HRSG}, C_{AB}]$. Based on the ship energy system presented in Figure 1, the conditions for the architecture topology to fulfill Case 1 or Case 2 requirements are given by :

$$ \text{Case 1 : } C_{DEe} + C_{GTe} \le 1 \qquad (32) $$

$$ \text{Case 2 : } (C_{AB} == 1 \text{ AND } (C_{HRSG} \le C_{DEe} + C_{GTe} + C_{DEm} + C_{GTm})) \qquad (33) \\ \text{OR } (C_{HRSG} == 1 \text{ AND } (C_{DEe} + C_{GTe} + C_{DEm} + C_{GTm} \ge 1)) $$

The logic programming language Prolog was used to get all the possible architecture topologies based on (32) or (33). The list for Case 1 is given in Table 8 as an example.

### 3.4. DE algorithm with clusters

The use of clusters based on architecture topologies requires several modifications to the classical DE algorithm :

* in the initialization process, a specific number of individuals are assigned to each topology. This number is not uniform but instead depends on the number of optimization variables associated with each architecture topology,
Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm

see "Variables per mode" column in Table 8. The idea is that if there are more optimization variables to find, more individuals are generally needed to achieve good convergence.

* in the crossover and mutation process, only the optimization variables related to the active components of the corresponding architecture topology are modified. Note that individual indexes $r_1, r_2$ and $r_3$ are selected from the entire population, regardless of the architecture topology. This ensures that potentially relevant values can be shared across different topologies.

* in the selection process, the architecture topologies of the parent and candidate individuals are compared. If they are the same, the procedure remains unchanged. Otherwise, the candidate is compared with the worst-performing individual among those sharing the same architecture topology.

* in the convergence criteria calculation, the stagnation of the objective function is evaluated at the cluster level. If a cluster meets the convergence criteria, the optimization is terminated for the individuals within that cluster. This saves some computational time if some architecture topologies converge faster than others.

The different steps of the optimization problem are summarized in the flowchart in Figure 4. The rectangles representing input or output data are respectively highlighted in green and red.

<figure><img src="image_6.png" /><figcaption>Figure 4: Optimization problem flowchart</figcaption></figure>

## 4. Numerical applications

Two numerical cases are presented for a ship energy system architecture with or without an HRSG and auxiliary boiler. They will be referred to as Case 1 and Case 2 in the following. The objectives of these numerical tests are:

* get a performance comparison of different choices of propulsion (mechanical or electrical, diesel engines or gas turbines)
* quantify the impact of the HRSG and the auxiliary boiler on the ship energy system

* have more information about the convexity of the optimization problem (presence of several local optima or not)

* check the robustness of this custom version of the differential evolution algorithm from one case to another (computational time, accuracy and parameters choice)

The primary objective in presenting two cases is to demonstrate the adaptability of the algorithm. More than the results only, the article aims to present a methodology that can be applied to various energy systems within the SDO context.

Next, several modes are considered to represent the ship power usage. Table 9 represents a 15-day mission, which corresponds to a full usage of the fuel tank of the considered ship with this power demand. For a design starting point, the duration of the mission is set such that the fuel mass represents approximately 10% of the ship's total mass. Due to the energy system architecture, only speed and hotel load requirements are considered for Case 1. Concerning the mission profile, the navigation route and weather conditions are not explicitly taken into account in the optimization problem. A small number of modes is chosen here as the number of optimization variables is already quite important (70 optimization variables for Case 2) for an evolutionary algorithm. Also, optimizing modes individually is challenging due to the different levels (synthesis, design and operation) considered in the optimization. One can say that each mode can cover intermediate sea states or weather conditions. While a few additional modes could be incorporated to address extreme conditions, such considerations are outside the scope of this study.

<table><caption>**Table 9**
Ship mission profile</caption><thead><tr><th></th><th>Speed (knots)</th><th>Hotel load (MW)</th><th>Heat (MW)</th><th>t (hours)</th></tr></thead><tbody><tr><td>Mode 1</td><td>6.0</td><td>0.5</td><td>0.5</td><td>36.0</td></tr><tr><td>Mode 2</td><td>10.0</td><td>1.0</td><td>0.5</td><td>72.0</td></tr><tr><td>Mode 3</td><td>13.0</td><td>1.0</td><td>0.5</td><td>176.4</td></tr><tr><td>Mode 4</td><td>16.0</td><td>2.0</td><td>0.5</td><td>72.0</td></tr><tr><td>Mode 5</td><td>20.0</td><td>3.0</td><td>0.5</td><td>3.6</td></tr></tbody></table>

The optimization settings for both cases are reported in Table 10. The variable $nb_{params}$ corresponds to the multiplication between the quantity "Variables per mode" column in Table 8 and the number of modes. Because of the randomness included in the evolutionary algorithms, the results obtained from one program execution (run) to another can be different. In order to obtain more reliable results, the program is launched 50 times and only the best solution is shown after completion of the whole process. Some experiments will be presented later for more information about the versatility of the results.

<table><caption>**Table 10**
Optimization settings</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Population size</td><td>180</td></tr><tr><td>Crossover rate</td><td>1/$nb_{params}$</td></tr><tr><td>Mutation factor</td><td>0.6</td></tr><tr><td>Constraints weight $\lambda$</td><td>500.0</td></tr></tbody></table>

## 4.1. Case 1

The optimization results for Case 1 are obtained in approximately 10 min for one run. An example of the convergence process is illustrated on Figure 5, where the objective function and residual errors are plotted along the number of generations. The number plotted on the right axis represents the summation of the residual error for each equality or inequality constraint involved in the optimization problem. A rapid decrease of the objective function occurs from the zeroth to the 500th generation, after which it stagnates. 2000 generations are calculated here because our stopping criterion is quite conservative (relative change of the objective function less than $1e^{-4}$ checked every 300
generations) but a good result is already obtained at the 1000th generation. The final residual error magnitude is 0.2%. Note that Figure 5 represents the best solution among all clusters. For each architecture topology, the optimal installed power and objective functions are represented in Figure 6.

<figure><img src="image_7.png" /><figcaption>**Figure 5:** Convergence of the differential evolution algorithm</figcaption></figure>

The numbering of the architecture topologies is indicated in Table 8. The aim of the optimization is to minimize the fuel consumption and component weights. However, as the violation of the constraints is also contained in the objective function value, a distinction between these three quantities has been made in Figure 6 on the plot on the right for a better analysis.

<figure><img src="image_8.png" /><figcaption>**Figure 6:** Case 1: optimal topologies: installed power (left) and objective function (right)</figcaption></figure>

### 4.1.1. Investigations on constraints violation

It can be seen in Figure 6 on the right that topology 1 exhibits a high residual error of 160. It comes from a constraint violation of the propulsion requirement for the third mode (ship target speed at 13 knots and hotel load target at 1 MW). The output power of the propeller should be 10.9 MW but only 8.04 MW are produced in this configuration despite the fact that enough power is installed in this topology (2 electric GT of 4.5 MW and 1 electric GT of 40 MW). This problem arises because the additional fuel consumption required to respect the constraint is higher than the penalty
associated to the constraint violation, as indicated in Table 11.

<table><caption>**Table 11**</caption><thead><tr><th></th><th>Topology 1 with constraint violation</th><th>Topology 1 without constraint violation</th></tr></thead><tbody><tr><td>Propulsion power (MW) in mode 3</td><td>8.04</td><td>10.9</td></tr><tr><td>Required models</td><td>2 GT 4.5MW (100% load)</td><td>1 GT 40 MW (25% load)</td></tr><tr><td>Fuel consumption (tons)</td><td>423</td><td>654</td></tr><tr><td>Penalty for constraint violation</td><td>178</td><td>0.0</td></tr><tr><td>Total (Fuel consumption + Penalty)</td><td>601</td><td>654</td></tr></tbody></table>

Topology 1 residual error explanations

Although in theory, it is possible to increase the coefficient on constraint violation, in practice it deteriorates the efficiency of the optimization algorithm as the solution is usually found iteratively inside the unfeasible domain. Another possibility would have been to use a different coefficient for each topology, but adjusting the coefficients then becomes cumbersome. The conclusion arising from this high residual error is that this topology is not adapted for this particular mission profile. Then it is up to the designer to either discard this topology or modify the mission requirements. At this design stage, a residual error of less than 5% for most of architecture topologies is considered acceptable.

### 4.1.2. Advantages of the diesel engine and gas turbine combination

Next, the best solution for this problem proved to be topology 8 with a 7.2 MW electrical diesel engine, a 7.2 MW mechanical diesel engine, 2 mechanical gas turbines of 4.5 MW and one of 25 MW. Within a 3% difference, topologies 6, 10 and 12 also exhibit a very good performance with a comparable residual error (less than 0.2%). It should be noted that the power profile for topologies 6 and 10 is quite close because these topologies share similar available components. The same conclusions hold between topologies 8 and 12.

The common point between all these topologies is the combination of a diesel electric engine for low speeds with a mechanical gas turbine for high speeds (16 and 20 knots). These solutions take advantage of the low specific fuel consumption of the diesel engines and the lower weight of mechanical gas turbines compared to diesel engines for a similar power range. The power values and fuel consumption comparison curves are shown in Figure 7 for topologies 6 and 8. For both topologies, the cumulated diesel engine power is 14.4 MW; in topology 8, however, it is divided into two 7.2 MW diesel engines, thus increasing the weight of the components slightly due to the presence of two engines. However the gain in fuel consumption is more interesting because the two 7.2 MW diesel engines are used at a better load compared to the 14 MW version. This is mostly visible for the second and third modes in the fuel consumption comparison in Figure 7 on the right.

<figure><img src="image_9.png" /><figcaption>**Figure 7:** Case 1: power values of the components and fuel consumption comparison for topologies 8 and 6</figcaption></figure>
4.1.3. Compromise between components weights and fuel consumption

Through all the topologies, one can see that the optimal solution lies in a compromise between specific fuel
consumption against component weight. The two extreme topologies illustrating this choice are topologies 2 and 7,
represented in Figure 8 with respectively the lowest total components weight and the lowest fuel consumption. While
a similar objective function is reached in these two cases, it is quite far away from the optimal solution due to either
the fuel consumption or the components weight.

<figure><img src="image_10.png" /><figcaption>**Figure 8:** Case 1 : power values of the components and fuel consumption comparison for topologies 2 and 7</figcaption></figure>

4.2. Case 2

The optimization results for Case 2 are obtained in approximately 12 min for one run. The optimal installed power and objective functions are represented in Figure 9 for each architecture topology. The first difference compared to Case 1 is the number of topologies involved in the optimization problem, 46 for Case 2 instead of 12 for Case 1, due to the addition of the component blocks HRSG and AB in Case 2. The second difference is the presence of multiple topologies with very high residual errors (topologies 1,3,6,9). These topologies are unfeasible given the mission profile and components library. To enhance the clarity of the graph, the maximum value for the objective function has been limited to 1800 but the actual value for topology 1 is about 8000.

4.2.1. *Explanations of unfeasible cases*

The only component in topology 1 is an auxiliary boiler. As this component can fulfill propulsion, hotel load and heating requirements, the architecture topology 1 is considered valid in equation (33) to be included in the optimization process. However, equation (33) does not take into account the mission profile nor the components' capabilities. As the maximum power of the largest auxiliary boiler model is 12 MW whereas the maximum propulsion requirement is about 30 MW, topology 1 exhibits therefore a very high residual error. An architecture topology with a residual error superior to 50% is said to be unfeasible. In topology 3 (see Figure 10), the presence of two mechanical drive gas turbines along with an HRSG system is still not enough for this topology to be feasible. The main residual error comes from the first mode so the analysis is based on this mode. For a speed of 6 knots, the propulsion, hotel load and heat power required are respectively about 1 MW, 0.5 MW and 0.5 MW. The hotel load and heat powers can only be provided through the HRSG and due to the low efficiency of the steam turbine, it would require about 2.5 MW in the HSRG. In the physical problem, the HRSG power corresponds to a maximum of 50% of the mechanical output of the diesel engine or gas turbine, meaning that the gas turbine should produce 5 MW for the hotel load and heating requirements. However in that case, the propulsion power largely exceeds the requirement, thereby inducing an error on the equality between given power and target. The explanations for the infeasibility of the other topologies follow the same analysis, just with different components.

4.2.2. Analysis of the different optimal solutions

Among the feasible solutions, the objective function is much smaller but one can distinguish optimal and non
optimal solutions. Topology 25 is non optimal due to the absence of gas turbines, which leads to a high usage of
<figure><img src="image_11.png" /><figcaption>**Figure 9:** Case 2 : optimal topologies : installed power (top) and objective function (bottom)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10:** Case 2 : power values for typical architecture topologies : unfeasible topology on the left, non optimal topology in the middle and optimal topology on the right</figcaption></figure>

heavy diesel engines, following a similar trend as for Case 1. Then an architecture topology is considered optimal if its objective function lies within a 5% interval compared to the lowest objective function among all while having a residual error less than 5%. This criteria is satisfied by 7 architectures topologies (10,18,21,27,33,39,45). Among them, the lowest objective function comes from the topology 33 (see Figure 10) with a 7.2 MW electrical diesel engine, a 14.4 MW mechanical diesel engine, two mechanical gas turbines (4.5 MW and 25 MW) and a 12 MW HRSG. The weight of the components is greater than for Case 1 but the HRSG, using waste power from the engines, has no direct fuel consumption. As a result, the lowest objective function has a similar value between Case 1 and Case 2 (-1% for Case 1) even with the additional heating supply.

For the three architecture topologies with the lowest objective function values (33, 21 and 10), their residual errors are superior to 1%. A manual adjustment of the optimization variables has been realized for these topologies in order
to check what the impact the residual error would have on the performance. The results are indicated in Table 12. The new objective function is obtained after a decrease of the residual error to 1%. The classification of the topologies remains unchanged after this verification. The change in the objective function is relatively small but one can notice that topologies 33 and 21 are very close for the same residual error. Next the decrease in the residual error has a positive impact on the value of the objective function in that case, meaning that the efficiency of the evolutionary algorithm is not perfect, which is quite common for real-world optimization problems.

<table><caption>**Table 12**
Residual errors analysis</caption><thead><tr><th>Topology</th><th>33</th><th>21</th><th>10</th></tr></thead><tbody><tr><td>Original objective function</td><td>1024.86</td><td>1034.96</td><td>1045.4</td></tr><tr><td>Original error</td><td>2.01%</td><td>3.26%</td><td>1.55%</td></tr><tr><td>New objective function with 1% error</td><td>1021.98</td><td>1023.99</td><td>1042.63</td></tr></tbody></table>

Although topology 27 in Figure 11 has a much lower components weight compared to topology 33 (130 tons difference), the absence of a mechanical diesel engine induces a higher fuel consumption. Topologies 27 and 39 share similar components. Initially, there was an option to incorporate an electrical gas turbine in topology 39, unlike in topology 27. However, this option was not selected by the optimization algorithm. The power repartition and objective value are not the same for these two topologies, illustrating the fact that several local optima can also exist inside the same cluster.

<figure><img src="image_13.png" /><figcaption>**Figure 11:** Case 2: power values and fuel consumption evolution over the different modes for topologies 27 and 39</figcaption></figure>

### 4.2.3. Influence of the HRSG and the boiler on optimization results

Compared to Case 1, the HRSG provides additional heat power without direct fuel consumption, which logically explains its inclusion in all optimal solutions. However, given the relatively low efficiencies of the HRSG and the steam turbine, their impact on the objective function is limited. A similar conclusion can be drawn about the boiler, only present in topologies 10 and 22 among the optimal solutions. For example, in the Sankey diagram for the fourth mode of topology 10 (Figure 13), while the combined input power for the HRSG and the auxiliary boiler represents 10 MW, it gives only 0.8 MW of propulsion, 2 MW of hotel load and 0.5 MW of heat power. Based on this analysis, the low specific fuel consumption of the boiler does not always compensate for its weight. It explains that some topologies perform well even without a boiler; see the comparison between Topology 21 and 10 in Figure 12.

As a general conclusion to Case 2, the following comments can be made:

* similar optimal objective functions are obtained in Case 1 or Case 2, owing that the waste heat recovery compensates for the additional heating requirement;

* all the interesting solutions include an HRSG with a combination of diesel engines and gas turbines, mostly with a mechanical gas turbine;
<figure><img src="image_14.png" /><figcaption>**Figure 12:** Case 2: power values and fuel consumption evolution over the different modes for topologies 21 and 10</figcaption></figure>

* The boiler is less interesting than the HRSG due to its non zero specific fuel consumption.

<figure><img src="image_15.png" /><figcaption>**Figure 13:** Case 2: Sankey diagram for the fourth mode in topology 10</figcaption></figure>

## 4.3. Robustness of the optimization algorithm

In evolutionary algorithms, the use of randomness is crucial to explore the search domain. Consequently, optimal results may vary between runs, even when the core algorithm remains unchanged. The optimization algorithm was launched 50 times for Case 1 and Case 2 with different population sizes to check the robustness of the algorithm. The results are shown in Figures 14 and 15 respectively. Each mark represents the mean objective function value over all runs and the error bars indicate the minimum and maximum value of the objective function over all runs for each topology.

For both cases, similar conclusions can be drawn:
<figure><img src="image_16.png" /><figcaption>**Figure 14:** Case 1: Optimization results depending on population size</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 15:** Case 2: Optimization results depending on population size</figcaption></figure>

* the mean objective function for each topology is quite similar for the different population sizes and the test with 90 individuals provides the best average value for some topologies. However, as the number of optimization variables for Case 2 can be up to 70, a more conservative choice of 180 individuals is made in order to ensure a correspondence between the problem dimensionality *d* and the population size about 3d, as recommended in Piotrowski (2017)

* the error bars are large for almost every topology (about 20% variation for Case 2), meaning that several runs are necessary to obtain a more accurate result. The objective function value is plotted against the number of runs of the optimization algorithm in Figure 16 to evaluate its global search capability. In this study, 50 runs were made, as a compromise between computational time and reliability. Compared to the solution at the 50th run, the average error for one run is equal to 2% and 8% for Case 1 and Case 2 respectively. When using 10 runs, the average error goes down to 1% for Case 1 and 2% for Case 2. Considering the complexity of the optimization
problem—which spans synthesis, design, and operational levels—and the numerous variables involved, these
differences are tolerable at this stage. Figure 16 shows that most of the topologies have stabilized after 30 runs.
For each case, only the five best topologies (lowest objective function) are plotted for more readability.

<figure><img src="image_18.png" /><figcaption>**Figure 16:** Best objective function evolution (Case 1 on the left, Case 2 on the right) for selected topologies depending on the number of runs</figcaption></figure>

The scalability of the algorithm seems reasonable, as the computational time between Case 1 and Case 2 is multiplied by 1.2 with more optimization variables per mode (14 against 6) and more components (HSRG and boiler) in Case 2. An improvement could be made concerning the convergence of the differential evolution algorithm in the vicinity of the optimal solution. For example in Case 2, there is a 10% decrease from generation 100 to 300 but only a 5% decrease from generation 300 to generation 2500. A possible solution might be to use a hybrid algorithm such as the one used in Dimopoulos (2007) with a combination of genetic algorithm and particle swarm optimization. This perspective will be investigated in future work. Another way to improve the efficiency of the optimization algorithm is to reduce the number of objective function evaluations with gradient-based methods. However, the solutions we have found so far with this technique, while obtained faster, are outperformed by the differential evolution algorithm for the objective function values. This technique has therefore not been mentioned in this article.

**4.4. Impact of using clusters in the optimization results**

The optimization results without the use of the clustering method are plotted for Case 1 and Case 2, respectively in Figures 17 and 18. The occurrence of the optimal solution found with clusters is quite small for Case 1 (22% for topology 8). The absence of the clustering technique has more impact on the optimization results for Case 2 because most of the optimal topologies introduced before (33, 21, 27, 45, 39 and 18) are not found by the optimization algorithm even after 50 runs. Only the topology 10 for Case 2 has been found with a low occurrence (4%). Moreover, for both cases, the error bars are higher without clusters, meaning that the convergence is more difficult to obtain. One positive thing for the simulation without clusters is that the best solutions for the available topologies are similar to the ones with clusters. Nevertheless, a major drawback is the lack of diversity in the results based on the different topologies.

Our primary motivation for using a clustering technique in the SDO context was to improve the algorithm robustness and at the same time to obtain multiple optimal solutions. However, another interesting side effect is to gain some feedback about the non optimal solutions in order to perform a much more detailed analysis. In addition to identifying the optimal solution, one can also understand why with questions such as :

* are there unfeasible solutions ?

* how do the gas turbines perform compared to diesel engines ?
<figure><img src="image_19.png" /><figcaption>**Figure 17:** Case 1: Optimization results without using the clustering approach</figcaption></figure>

* what is the influence of the boiler and the HRSG on optimization results ?

Such insights can be very helpful at an early design stage so it is valuable to be able to obtain these answers rapidly.

With respect to the use of clustering technique based on the architecture topology, one perspective for this work concerns the modifications required for the algorithm if the number of potential architecture topologies becomes very large, let us say 1000 or 10000. It may be necessary in this case to group some topologies together at the beginning of the optimization process and then explore the most interesting sub-regions of the design space later after a specific convergence criterion has been met. Such investigations are left for future work.

**5. Conclusion**

A differential evolution algorithm with a clustering technique has been developed to enhance robustness in solving synthesis, design, and operation optimization problems, while also enabling the identification of multiple optimal solutions in an affordable manner. The advantages of the proposed method can be summarized as follows :

* Thorough analysis of the system capabilities by obtaining multiple optimal solutions for a real-world problem: a Differential Evolution algorithm is used with a clustering technique based on the absence or presence of
<figure><img src="image_20.png" /><figcaption>**Figure 18: Case 2: Optimization results without using the clustering approach**</figcaption></figure>

components, eliminating the need for a predefined threshold distance as required by traditional clustering methods.

* Affordable and efficient optimization algorithm: First the computational time is reduced by precomputing response surfaces (specific fuel consumption versus power) for different groups of models. In addition, models are selected based on operational variables, simplifying the treatment of categorical variables.

* General and adaptable methodology: The approach uses constraint programming in Prolog for topology generation and incorporates the Admiralty coefficient for the propeller law in physical modeling. This makes it applicable to a large range of ship sizes or entirely different system architectures.

Two numerical applications were conducted using a ship energy system architecture, with and without heating requirements. In both cases, the parameters of the differential evolution algorithm (crossover rate and mutation factor) and the coefficient for constraint violation remained unchanged along with the population size. This consistency supports the claim that the differential evolution algorithm is relatively easy to tune. Additional information has
been given about the global search capability and convergency of the differential evolution algorithm for real-world problems. For both numerical cases, one run is computed in few minutes with a residual error less than 5% in most cases. The global search is quite effective to determine local minima within a 20% margin but it could be improved in the vicinity of the solution.

Since the physical results are dependent on the component list and mission profile, drawing general conclusions is not appropriate. However, a detailed analysis of the results has been provided to demonstrate that the algorithm suggested interesting solutions. Moreover, the interpretation of the results is made easier with the presence of multiple solutions (optimal and less optimal ones) due to the use of the clustering technique. In Case 1 and Case 2, four and seven optimal solutions have been found (5% interval with the lowest objective function and a residual error less than 5%), demonstrating the need for optimization algorithms capable of finding several local optima. Furthermore, the clustering algorithm based on the absence or presence of components has been very effective, since the majority (6 of 7) of the optimal solutions in Case 2 would not have been found without it.

Future work aims to improve the efficiency of the algorithm (reducing the number of function evaluations) while maintaining high precision. In addition, a general guideline will be developed for handling a very large number of architecture topologies, especially when their magnitude exceeds the population size.

## Acknowledgments

This research work was funded by the Joint Laboratory of Marine Technology (JLMT) from Ecole Centrale de Nantes and Naval Group.

## CRediT authorship contribution statement

Florian Dugast : Methodology, Software, Writing - original draft. Stéphane Bénac : Funding acquisition, Methodology, Writing - review and editing. Pierre Marty : Supervision, Methodology, Writing - review and editing. Pascal Chessé : Funding acquisition, Methodology

## Declaration of generative AI and AI-assisted technologies in the writing process.

During the preparation of this work the authors used ChatGPT in order to improve the English expression and readability of the article. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

## References

* Baldasso, E., Elg, M., Haglind, F., Baldi, F., 2019. Comparative analysis of linear and non-linear programming techniques for the optimization of ship machinery systems. Journal of Marine Science and Engineering 7. doi:10.3390/jmse7110403.

* Bonet, M., Pilidis, P., 2013. Comparative assessment of two thermodynamic cycles of an aero-derivative marine gas turbine. IOSR Journal of Mechanical and Civil Engineering 6, 76–81. doi:10.9790/1684-637681.

* Colmerauer, A., 1990. An introduction to prolog iii. Communications of the ACM 33, 69–90. URL: https://dl.acm.org/doi/10.1145/79204.79210, doi:10.1145/79204.79210.

* Dimopoulos, G.G., 2007. Mixed-variable engineering optimization based on evolutionary and social metaphors. Computer Methods in Applied Mechanics and Engineering 196, 803–817. doi:10.1016/j.cma.2006.06.010.

* Dimopoulos, G.G., Kougioufas, A.V., Frangopoulos, C.A., 2008. Synthesis, design and operation optimization of a marine energy system. Energy 33, 180–188. doi:10.1016/j.energy.2007.09.004.

* Dominico, G., Parpinelli, R.S., 2021. Multiple global optima location using differential evolution, clustering, and local search. Applied Soft Computing 108. doi:10.1016/j.asoc.2021.107448.

* Duan, D., Gong, Y., Huang, T., Zhang, J., 2018. Adaptive clustering-based differential evolution for multimodal optimization, IEEE. pp. 370–376. URL: https://ieeexplore.ieee.org/document/8426184/, doi:10.1109/ICIST.2018.8426184.

* Dugast, F., Bénac, S., Marty, P., Chessé, P., 2023. Topology Generation of Naval Propulsion Architecture, in: 4th International Conference on Modelling and Optimisation of Ship Energy Systems (MOSES 2023), Delft, Netherlands. p. 667. URL: https://hal.science/hal-04492416, doi:10.5949/moses.2023.667.

* Epitropakis, M.G., Plagianakos, V.P., Vrahatis, M.N., 2011. Finding multiple global optima exploiting differential evolution’s niching capability, IEEE. pp. 1–8. URL: http://ieeexplore.ieee.org/document/5952058/, doi:10.1109/SDE.2011.5952058.
Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm

Frangopoulos, C.A., 2020. Developments, trends, and challenges in optimization of ship energy systems. Applied Sciences (Switzerland) 10. doi:10.3390/app10134639.

General Electric, 2025. https://www.geaerospace.com/military-defense/engines/lm500, Accessed: 2025-02-28.

Han, S., Lee, Y.S., Choi, Y.B., 2012. Hydrodynamic hull form optimization using parametric models. Journal of Marine Science and Technology 17, 1-17. doi:10.1007/s00773-011-0148-8.

Herdzik, J., Cwilewicz, R., 2017. Utilization possibility of marine trent 30 gas turbine as prime mover on vessels. Journal of KONES 24, 95-100. doi:10.5604/01.3001.0010.3087.

Iqbal, M., Terziev, M., Tezdogan, T., Incecik, A., 2025. Hull form optimisation to minimise the total resistance and dynamic responses of small fishing vessels. Ocean Engineering 321, 120357. URL: https://www.sciencedirect.com/science/article/pii/S0029801825000721 doi:https://doi.org/10.1016/j.oceaneng.2025.120357.

Kalikatzarakis, M., Frangopoulos, C.A., 2017. Thermo-economic optimization of synthesis, design and operation of a marine organic rankine cycle system. Proceedings of the Institution of Mechanical Engineers Part M: Journal of Engineering for the Maritime Environment 231, 137-152. doi:10.1177/1475090215627179.

Krata, P., Szlapczynska, J., 2018. Ship weather routing optimization with dynamic constraints based on reliable synchronous roll prediction. Ocean Engineering 150, 124-137. doi:10.1016/j.oceaneng.2017.12.049.

Lazzaretto, A., Manente, G., Toffolo, A., 2018. Synthsep: A general methodology for the synthesis of energy system configurations beyond superstructures. Energy 147, 924-949. doi:10.1016/j.energy.2018.01.075.

Lazzaretto, A., Toffolo, A., 2008. A method to separate the problem of heat transfer interactions in the synthesis of thermal systems. Energy 33, 163-170. doi:10.1016/j.energy.2007.07.015.

Li, Z., Wang, K., Hua, Y., Liu, X., Ma, R., Wang, Z., Huang, L., 2024. Ga-Istm and nsga-iii based collaborative optimization of ship energy efficiency for low-carbon shipping. Ocean Engineering 312, 119190. URL: https://www.sciencedirect.com/science/article/pii/S0029801824025289 doi:https://doi.org/10.1016/j.oceaneng.2024.119190.

Liu, X., Wang, K., Guo, X., Li, Z., Wu, J., Ma, R., Huang, L., Li, X., 2025. An integrated energy efficiency optimization method of the wind-assisted hybrid ship for the shipping decarbonization. Marine Pollution Bulletin 212, 117579. URL: https://www.sciencedirect.com/science/article/pii/S0025326X25000542 doi:https://doi.org/10.1016/j.marpolbul.2025.117579.

Ma, D., Ma, W., Jin, S., Ma, X., 2020. Method for simultaneously optimizing ship route and speed with emission control areas. Ocean Engineering 202. doi:10.1016/j.oceaneng.2020.107170.

Marques, C.H., Belchior, C.R.P., Caprace, J.D., 2019. An early-stage approach to optimise a marine energy system for liquefied natural gas carriers: Part a - developed approach. Ocean Engineering 181, 161-172. doi:10.1016/j.oceaneng.2019.04.020.

Marty, P., 2014. Ship energy efficiency study: development and application of an analysis method. Ph.D. thesis.

NIDEC, 2025. https://acim.nidec.com/en/generators/leroy-somer/Downloads/Catalogues, Accessed: 2025-02-28.

Peng, X., Chen, H., Guan, C., 2023. Energy management optimization of fuel cell hybrid ship based on particle swarm optimization algorithm. Energies 16. URL: https://www.mdpi.com/1996-1073/16/3/1373, doi:10.3390/en16031373.

Piotrowski, A.P., 2017. Review of differential evolution population size. Swarm and Evolutionary Computation 32, 1-24. doi:10.1016/j.swevo. 2016.05.003.

Sakalis, G.N., Frangopoulos, C.A., 2018. Intertemporal optimization of synthesis, design and operation of integrated energy systems of ships: General method and application on a system with diesel main engines. Applied Energy 226, 991-1008. doi:10.1016/j.apenergy.2018.06. 061.

Sakalis, G.N., Tzortzis, G.J., Frangopoulos, C.A., 2019. Intertemporal static and dynamic optimization of synthesis, design, and operation of integrated energy systems of ships. Energies 12. doi:10.3390/en12050893.

Sariöz, E., 2006. An optimization approach for fairing of ship hull forms. Ocean Engineering 33, 2105-2118. doi:10.1016/j.oceaneng.2005. 11.014.

Shang, C., Srinivasan, D., Reindl, T., 2016. Economic and environmental generation and voyage scheduling of all-electric ships. IEEE Transactions on Power Systems 31, 4087-4096. doi:10.1109/TPWRS.2015.2498972.

Silva, P., Leiroz, A., Cruz, M., 2020. Evaluation of the efficiency of a heat recovery steam generator via computational simulations of off-design operation. Journal of the Brazilian Society of Mechanical Sciences and Engineering 42, 569.

Silvas, E., Hofman, T., Murgovski, N., Etman, L.F., Steinbuch, M., 2016. Review of optimization strategies for system-level design in hybrid electric vehicles. doi:10.1109/TVT.2016.2547897.

Skinner, B.A., Palmer, P.R., Parks, G.T., 2007. Multi-objective design optimisation of submarine electric drive systems.

Skjong, E., Johansen, T.A., Molinas, M., Sorensen, A.J., 2017. Approaches to economic energy management in diesel-electric marine vessels. IEEE Transactions on Transportation Electrification 3, 22-35. doi:10.1109/TTE.2017.2648178.

Storn, R., Price, K., 1997. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11, 341-359.

Szénásy, I., Csikor, D., 2024. Induction motor energy efficiency investigation. Engineering Proceedings 79. URL: https://www.mdpi.com/ 2673-4591/79/1/75, doi:10.3390/engproc2024079075.

Tadros, M., Ventura, M., Guedes Soares, C., 2021. Design of propeller series optimizing fuel consumption and propeller efficiency. Journal of Marine Science and Engineering 9. URL: https://www.mdpi.com/2077-1312/9/11/1226, doi:10.3390/jmse9111226.

Tanuma, T., 2022. Advances in Steam Turbines for Modern Power Plants. Woodhead Publishing Series in Energy, Elsevier Science. URL: https://books.google.fr/books?id=MgF7EAAAQBAJ.

Thomsen, R., 2004. Multimodal optimization using crowding-based differential evolution, IEEE. pp. 1382-1389. URL: http://ieeexplore.ieee.org/document/1331058/, doi:10.1109/CEC.2004.1331058.

Trivyza, N.L., Rentzelas, A., Theotokatos, G., 2018. A novel multi-objective decision support method for ship energy systems synthesis to enhance sustainability. Energy Conversion and Management 168, 128-149. doi:10.1016/j.enconman.2018.04.020.
Multiple optimal solutions in synthesis design operation optimization of a marine energy system with the differential evolution algorithm

Trivyza, N.L., Rentizelas, A., Theotokatos, G., Boulougouris, E., 2022. Decision support methods for sustainable ship energy systems: A state-of-the-art review. Energy 239. doi:10.1016/j.energy.2021.122288.

Tzortzis, G.J., Frangopoulos, C.A., 2019. Dynamic optimization of synthesis, design and operation of marine energy systems. Proceedings of the Institution of Mechanical Engineers Part M: Journal of Engineering for the Maritime Environment 233, 454–473. doi:10.1177/1475090217749370.

de Vos, P., Stapersma, D., 2018. Automatic topology generation for early design of on-board energy distribution systems. Ocean Engineering 170, 55–73. URL: https://doi.org/10.1016/j.oceaneng.2018.09.023, doi:10.1016/j.oceaneng.2018.09.023.

Wang, K., Yan, X., Yuan, Y., Tang, D., 2018. Optimizing ship energy efficiency: Application of particle swarm optimization algorithm. Proceedings of the Institution of Mechanical Engineers Part M: Journal of Engineering for the Maritime Environment 232, 379–391. doi:10.1177/1475090216638879.

Wartsila, 2025. https://www.wartsila.com/marine/products/engines-and-generating-sets/dual-fuel-engines/wartsila-46f, Accessed: 2025-02-28.

Wijkniet, J., Hofman, T., 2018. Modified computational design synthesis using simulation-based evaluation and constraint consistency for vehicle powertrain systems. IEEE Transactions on Vehicular Technology 67, 8065–8076. doi:10.1109/TVT.2018.2844024.

Wong, K.P., Dong, Z.Y., 2005. Differential evolution, an alternative approach to evolutionary algorithm, in: Proceedings of the 13th International Conference on, Intelligent Systems Application to Power Systems, IEEE. pp. 73–83.

Wu, N., Zhang, F., Zhang, F., Jiang, C., Lin, J., Xie, S., Jing, R., Zhao, Y., 2024. An integrated multi-objective optimization, evaluation, and decision-making method for ship energy system. Applied Energy 373, 123917. URL: https://www.sciencedirect.com/science/article/pii/S030626192401300X, doi:https://doi.org/10.1016/j.apenergy.2024.123917.

Zhang, Q., Yi, H., Yu, Z., Gao, J., Wang, X., Lin, H., Shen, B., 2018. Energy-exergy analysis and energy efficiency improvement of coal-fired industrial boilers based on thermal test data. Applied Thermal Engineering 144, 614–627.
# Appendix A : Relationships between models id and composition of components groups

<table><caption>**Table 13**</caption><thead><tr><th>group of models</th><th>6L46F</th><th>12V46F</th><th>16V46F</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>2</td><td>2</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>1</td><td>0</td></tr><tr><td>4</td><td>0</td><td>0</td><td>1</td></tr><tr><td>5</td><td>3</td><td>0</td><td>0</td></tr><tr><td>6</td><td>1</td><td>1</td><td>0</td></tr><tr><td>7</td><td>1</td><td>0</td><td>1</td></tr><tr><td>8</td><td>4</td><td>0</td><td>0</td></tr><tr><td>9</td><td>2</td><td>1</td><td>0</td></tr><tr><td>10</td><td>0</td><td>2</td><td>0</td></tr><tr><td>11</td><td>2</td><td>0</td><td>1</td></tr><tr><td>12</td><td>0</td><td>1</td><td>1</td></tr><tr><td>13</td><td>3</td><td>1</td><td>0</td></tr><tr><td>14</td><td>1</td><td>2</td><td>0</td></tr><tr><td>15</td><td>0</td><td>0</td><td>2</td></tr><tr><td>16</td><td>3</td><td>0</td><td>1</td></tr><tr><td>17</td><td>1</td><td>1</td><td>1</td></tr><tr><td>18</td><td>2</td><td>2</td><td>0</td></tr><tr><td>19</td><td>0</td><td>3</td><td>0</td></tr><tr><td>20</td><td>1</td><td>0</td><td>2</td></tr><tr><td>21</td><td>2</td><td>1</td><td>1</td></tr><tr><td>22</td><td>0</td><td>2</td><td>1</td></tr><tr><td>23</td><td>1</td><td>3</td><td>0</td></tr><tr><td>24</td><td>2</td><td>0</td><td>2</td></tr><tr><td>25</td><td>0</td><td>1</td><td>2</td></tr><tr><td>26</td><td>1</td><td>2</td><td>1</td></tr><tr><td>27</td><td>0</td><td>0</td><td>3</td></tr><tr><td>28</td><td>0</td><td>4</td><td>0</td></tr><tr><td>29</td><td>1</td><td>1</td><td>2</td></tr><tr><td>30</td><td>0</td><td>3</td><td>1</td></tr><tr><td>31</td><td>1</td><td>0</td><td>3</td></tr><tr><td>32</td><td>0</td><td>2</td><td>2</td></tr><tr><td>33</td><td>0</td><td>1</td><td>3</td></tr><tr><td>34</td><td>0</td><td>0</td><td>4</td></tr></tbody></table>

List of diesel engine models
<table><caption>**Table 14**
List of gas turbine models</caption><thead><tr><th>group of models</th><th>LM500</th><th>LM2500</th><th>MT30</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>2</td><td>2</td><td>0</td><td>0</td></tr><tr><td>3</td><td>3</td><td>0</td><td>0</td></tr><tr><td>4</td><td>4</td><td>0</td><td>0</td></tr><tr><td>5</td><td>0</td><td>1</td><td>0</td></tr><tr><td>6</td><td>1</td><td>1</td><td>0</td></tr><tr><td>7</td><td>2</td><td>1</td><td>0</td></tr><tr><td>8</td><td>3</td><td>1</td><td>0</td></tr><tr><td>9</td><td>0</td><td>0</td><td>1</td></tr><tr><td>10</td><td>1</td><td>0</td><td>1</td></tr><tr><td>11</td><td>2</td><td>0</td><td>1</td></tr><tr><td>12</td><td>0</td><td>2</td><td>0</td></tr><tr><td>13</td><td>3</td><td>0</td><td>1</td></tr><tr><td>14</td><td>1</td><td>2</td><td>0</td></tr><tr><td>15</td><td>2</td><td>2</td><td>0</td></tr><tr><td>16</td><td>0</td><td>1</td><td>1</td></tr><tr><td>17</td><td>1</td><td>1</td><td>1</td></tr><tr><td>18</td><td>2</td><td>1</td><td>1</td></tr><tr><td>19</td><td>0</td><td>3</td><td>0</td></tr><tr><td>20</td><td>1</td><td>3</td><td>0</td></tr><tr><td>21</td><td>0</td><td>0</td><td>2</td></tr><tr><td>22</td><td>1</td><td>0</td><td>2</td></tr><tr><td>23</td><td>2</td><td>0</td><td>2</td></tr><tr><td>24</td><td>0</td><td>2</td><td>1</td></tr><tr><td>25</td><td>1</td><td>2</td><td>1</td></tr><tr><td>26</td><td>0</td><td>4</td><td>0</td></tr><tr><td>27</td><td>0</td><td>1</td><td>2</td></tr><tr><td>28</td><td>1</td><td>1</td><td>2</td></tr><tr><td>29</td><td>0</td><td>3</td><td>1</td></tr><tr><td>30</td><td>0</td><td>0</td><td>3</td></tr><tr><td>31</td><td>1</td><td>0</td><td>3</td></tr><tr><td>32</td><td>0</td><td>2</td><td>2</td></tr><tr><td>33</td><td>0</td><td>1</td><td>3</td></tr><tr><td>34</td><td>0</td><td>0</td><td>4</td></tr></tbody></table>