# Use of Genetic Programming for the Estimation of CODLAG Propulsion System Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature Review

- does the correlation exist, and how strong is the correlation between the parameters of CODLAG propulsion system dataset [9], and
- is it possible to obtain the symbolic expressions using GP algorithm for fuel flow estimation, ship speed estimation, starboard and port propeller torque, and total torque-with and without decay state coefficients.

## 2. Materials and Methods

#### 2.1. Dataset Description

#### 2.2. Correlation Analysis

- $r>0$-the linear correlation between x and y are positive i.e., higher absolute levels of one variable are associated with lower levels of the other,
- $r=0$-indicates the absence of any association between x and y, and
- $r<0$-the linear correlation between x and y is negative i.e., higher absolute levels of one variable are associated with lower levels of the other.

#### 2.3. Genetic Programming

#### 2.4. Evaluation Metrics

## 3. Results and Discussion

#### 3.1. Results

#### 3.1.1. The Symbolic Expressions for Fuel Flow Estimation with and without Decay State Coefficients

#### 3.1.2. The Symbolic Expressions for Ship Speed Estimation with and without Decay State Coefficients

#### 3.1.3. The Symbolic Expressions for Starboard Propeller Torque Estimation with and without Decay State Coefficients

#### 3.1.4. The Symbolic Expressions for Port Propeller Torque Estimation with and without Decay State Coefficients

#### 3.1.5. The Symbolic Expressions for Total Propeller Torque Estimation with and without Decay State Coefficients

#### 3.2. Discussion

## 4. Conclusions

- the Pearson’s and Spearman’s correlation analysis showed that from a total of 18 variables in the dataset 14 of them (without decay state coefficient, T1, and P1) have positive correlation values. The turbo compressor decay state coefficient and turbine decay state coefficient do not correlate with ship speed, have positive Pearsons correlation with starboard and port propeller torque, have positive and negative Spearman’s correlation with starboard and port propeller torque, and negative correlation with fuel flow. The T1 and P1 represent ambient temperature and pressure so they are constant values throughout the entire dataset. Hence there are not any correlation values with other parameters in the dataset.
- the GP algorithm can be used to obtain symbolic expressions for estimation of fuel flow, ship speed, starboard propeller torque, port propeller torque, and total propeller torque with and without decay state coefficients for the observed CODLAG propulsion system,
- the symbolic expressions for estimation of fuel flow, ship speed, starboard propeller, port propeller and total propeller torque with decay state coefficients generally have slightly lower ${R}^{2}$ and slightly higher $MAE$ values when compared to those symbolic expressions obtained without decay state coefficients. However, those symbolic expressions with decay state coefficients are more valuable from the CBM perspective which mean that they could be used to estimate or potentially predict possible degradation system states and schedule the system maintenance,
- the symbolic expressions for estimation of starboard propeller, port propeller, and total propeller torque with and without decay state coefficients showed slightly lower estimation performance for lower ship speeds.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Coefficients in Symbolic Expressions

#### Appendix A.1. Coefficients in Symbolic Expressions for Starboard Propeller Torque Estimation with Decay State Coefficients

#### Appendix A.2. Coefficients in Symbolic Expression for Starboard Propeller Torque Estimation without Decay State

#### Appendix A.3. Coefficients in Symbolic Expressions for Port Propeller Torque Estimation with Decay State Coefficients

#### Appendix A.4. Coefficients in Symbolic Expressions for Port Propeller Torque Estimation without Decay State Coefficients

#### Appendix A.5. Coefficients in Symbolic Expressions for Total Propeller Torque Estimation with Decay State Coefficients

#### Appendix A.6. Coefficients in Symbolic Expressions for Total Propeller Torque Estimation without Decay State Coefficients

## References

- Fernández, I.A.; Gómez, M.R.; Gómez, J.R.; Insua, Á.B. Review of propulsion systems on LNG carriers. Renew. Sustain. Energy Rev.
**2017**, 67, 1395–1411. [Google Scholar] [CrossRef] - Mrzljak, V.; Poljak, I.; Medica-Viola, V. Dual fuel consumption and efficiency of marine steam generators for the propulsion of LNG carrier. Appl. Therm. Eng.
**2017**, 119, 331–346. [Google Scholar] [CrossRef] - Armellini, A.; Daniotti, S.; Pinamonti, P. Gas Turbines for power generation on board of cruise ships: a possible solution to meet the new IMO regulations? Energy Procedia
**2015**, 81, 540–547. [Google Scholar] [CrossRef] [Green Version] - Kothamasu, R.; Huang, S.H. Adaptive Mamdani fuzzy model for condition-based maintenance. Fuzzy Sets Syst.
**2007**, 158, 2715–2733. [Google Scholar] [CrossRef] - Widodo, A.; Yang, B.S. Support vector machine in machine condition monitoring and fault diagnosis. Mech. Syst. Signal Process.
**2007**, 21, 2560–2574. [Google Scholar] [CrossRef] - Altosole, M.; Dubbioso, G.; Figari, M.; Viviani, M.; Michetti, S.; Millerani Trapani, A. Simulation of the dynamic behaviour of a CODLAG propulsion plant. In Proceedings of the WARSHIP 2010 Conference: Advanced Technologies in Naval Design and Construction, Beijing, China, 23–25 November 2010; pp. 9–10. [Google Scholar]
- Altosole, M.; Campora, U.; Martelli, M.; Figari, M. Performance Decay Analysis of a Marine Gas Turbine Propulsion System. J. Ship Res.
**2014**, 58, 117–129. [Google Scholar] [CrossRef] - Martelli, M.; Figari, M. Real-Time model-based design for CODLAG propulsion control strategies. Ocean. Eng.
**2017**, 141, 265–276. [Google Scholar] [CrossRef] - Coraddu, A.; Oneto, L.; Ghio, A.; Savio, S.; Anguita, D.; Figari, M. Machine Learning Approaches for Improving Condition?Based Maintenance of Naval Propulsion Plants. J. Eng. Marit. Environ.
**2014**, 230, 136–153. [Google Scholar] [CrossRef] - Lorencin, I.; Anđelić, N.; Mrzljak, V.; Car, Z. Multilayer perceptron approach to condition-based maintenance of marine CODLAG propulsion system components. Pomorstvo
**2019**, 33, 181–190. [Google Scholar] [CrossRef] - Baressi Šegota, S.; Lorencin, I.; Musulin, J.; Štifanić, D.; Car, Z. Frigate Speed Estimation Using CODLAG Propulsion System Parameters and Multilayer Perceptron. NAŠE MORE Znan. Časopis More Pomor.
**2020**, 67, 117–125. [Google Scholar] [CrossRef] - Anđelić, N.; Baressi Šegota, S.; Lorencin, I.; Car, Z. Estimation of gas turbine shaft torque and fuel flow of a CODLAG propulsion system using genetic programming algorithm. Pomorstvo
**2020**, 34, 323–337. [Google Scholar] [CrossRef] - Cheliotis, M.; Lazakis, I.; Theotokatos, G. Machine learning and data-driven fault detection for ship systems operations. Ocean. Eng.
**2020**, 216, 107968. [Google Scholar] [CrossRef] - Uyanık, T.; Karatuğ, Ç.; Arslanoğlu, Y. Machine learning approach to ship fuel consumption: A case of container vessel. Transp. Res. Part Transp. Environ.
**2020**, 84, 102389. [Google Scholar] [CrossRef] - Berghout, T.; Mouss, L.H.; Bentrcia, T.; Elbouchikhi, E.; Benbouzid, M. A deep supervised learning approach for condition-based maintenance of naval propulsion systems. Ocean. Eng.
**2021**, 221, 108525. [Google Scholar] [CrossRef] - Tsaganos, G.; Nikitakos, N.; Dalaklis, D.; Ölcer, A.; Papachristos, D. Machine learning algorithms in shipping: improving engine fault detection and diagnosis via ensemble methods. WMU J. Marit. Aff.
**2020**, 19, 1–22. [Google Scholar] [CrossRef] - Bachmayer, R.; Kampmann, P.; Pleteit, H.; Busse, M.; Kirchner, F. Intelligent Propulsion. In AI Technology for Underwater Robots; Springer: Berlin/Heidelberg, Germany, 2020; pp. 71–82. [Google Scholar]
- Turing, A.M. Computing machinery and intelligence. In Parsing the Turing Test; Springer: Berlin/Heidelberg, Germany, 2009; pp. 23–65. [Google Scholar]
- Forsyth, R. BEAGLE—A Darwinian approach to pattern recognition. Kybernetes
**1981**, 10, 159–166. [Google Scholar] [CrossRef] [Green Version] - Koza, J.R. Genetic programming for economic modeling. Stat. Comput.
**1994**, 4, 187–197. [Google Scholar] - Loginov, A.; Heywood, M.; Wilson, G. Stock selection heuristics for performing frequent intraday trading with genetic programming. Genet. Program. Evolvable Mach.
**2021**, 22, 35–72. [Google Scholar] [CrossRef] - Anđelić, N.; Baressi Šegota, S.; Lorencin, I.; Jurilj, Z.; Šušteršič, T.; Blagojević, A.; Protić, A.; Ćabov, T.; Filipović, N.; Car, Z. Estimation of covid-19 epidemiology curve of the united states using genetic programming algorithm. Int. J. Environ. Res. Public Health
**2021**, 18, 959. [Google Scholar] [CrossRef] - Salgotra, R.; Gandomi, M.; Gandomi, A.H. Time series analysis and forecast of the COVID-19 pandemic in India using genetic programming. Chaos Solitons Fractals
**2020**, 138, 109945. [Google Scholar] [CrossRef] [PubMed] - Benesty, J.; Chen, J.; Huang, Y.; Cohen, I. Pearson correlation coefficient. In Noise Reduction in Speech Processing; Springer: Berlin/Heidelberg, Germany, 2009; pp. 1–4. [Google Scholar]
- Schirinzi, A.; Cazzolla, A.P.; Lovero, R.; Lo Muzio, L.; Testa, N.F.; Ciavarella, D.; Palmieri, G.; Pozzessere, P.; Procacci, V.; Di Serio, F.; et al. New insights in laboratory testing for COVID-19 patients: looking for the role and predictive value of Human epididymis secretory protein 4 (HE4) and the innate immunity of the oral cavity and respiratory tract. Microorganisms
**2020**, 8, 1718. [Google Scholar] [CrossRef] [PubMed] - Lupton, R. 11. Least Squares Fitting for Linear Models. In Statistics in Theory and Practice; Princeton University Press: Princeton, NJ, USA, 2020; pp. 81–97. [Google Scholar]
- Car, Z.; Baressi Šegota, S.; Anđelić, N.; Lorencin, I.; Mrzljak, V. Modeling the spread of COVID-19 infection using a multilayer perceptron. Comput. Math. Methods Med.
**2020**, 2020, 5714714. [Google Scholar] [CrossRef] - Liu, Y.; Mu, Y.; Chen, K.; Li, Y.; Guo, J. Daily activity feature selection in smart homes based on pearson correlation coefficient. Neural Process. Lett.
**2020**, 51, 1–17. [Google Scholar] [CrossRef] - Sedgwick, P. Spearman’s rank correlation coefficient. BMJ
**2014**, 349, g7327. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Yang, H.; Bao, B.; Guo, H.; Jiang, Y.; Zhang, J. Spearman Correlation Coefficient Abnormal Behavior Monitoring Technology Based on RNN in 5G Network for Smart City. In Proceedings of the 2020 International Wireless Communications and Mobile Computing (IWCMC), Limassol, Cyprus, 15–19 June 2020; pp. 1440–1442. [Google Scholar]
- Okpala, B. A Measure of the Impact of Employee Motivation on Multicultural Team Performance Using the Spearman Rank Correlation Coefficient. SSRN 3702059
**2020**, 2020, 1–12. [Google Scholar] [CrossRef] - Poli, R.; Langdon, W.B.; McPhee, N.F.; Koza, J.R. A Field Guide to Genetic Programming; Lulu. com, Lulupress: Morrisville, NC, USA, 2008. [Google Scholar]
- Strušnik, D.; Avsec, J. Artificial neural networking and fuzzy logic exergy controlling model of combined heat and power system in thermal power plant. Energy
**2015**, 80, 318–330. [Google Scholar] [CrossRef] - Fang, Y.; Li, J. A review of tournament selection in genetic programming. In International Symposium on Intelligence Computation and Applications; Springer: Berlin/Heidelberg, Germany, 2010; pp. 181–192. [Google Scholar]
- Poli, R.; McPhee, N.F. Parsimony pressure made easy. In Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, Atlanta, Georgia, 12–16 July 2008; pp. 1267–1274. [Google Scholar]
- Agrež, M.; Avsec, J.; Strušnik, D. Entropy and exergy analysis of steam passing through an inlet steam turbine control valve assembly using artificial neural networks. Int. J. Heat Mass Transf.
**2020**, 156, 119897. [Google Scholar] [CrossRef]

**Figure 1.**The scheme of CODLAG propulsion system (B-gear box, C-clutch, D-diesel engine, G-electrical generator, GT- gas turbine, M-electrical motor, P-frigate propeller).

**Figure 2.**The scheme of GT component used in CODLAG propulsion system (C-turbo compressor; B-combustion chamber; HP-high pressure turbine; LP-low pressure turbine, O.P.-Operating Point).

**Figure 3.**Thermodynamic process of the gas turbine from the analyzed CODLAG propulsion system in T-s diagram (O.P.-Operating Point).

**Table 1.**The list of physical values in CODLAG dataset with corresponding range of values and units.

Physical Variable | Range | Unit |
---|---|---|

Lever position (${l}_{p}$) | 1.138–9.3 | - |

Ship speed (v) | 3–27 | kn |

Gas turbine shaft torque (GTT) | 253.547–72,784.872 | kNm |

GT rate of revolutions (GTn) | 1307.675–3560.741 | rpm |

Gas generator rate of revolutions (GGn) | 6589.002–9797.103 | rpm |

Starboard propeller torque (Ts) | 5.304–645.249 | kN |

Port propeller torque (Tp) | 5.304–645.249 | kN |

High pressure turbine exit temperature (T48) | 442.364–1115.797 | C |

Turbo compressor inlet air temperature (T1) | 288 | C |

Turbo compressor outlet air temperature (T2) | 540.442–789.094 | C |

HP turbine exit pressure (P48) | 1.093–4.56 | bar |

Turbo compressor inlet air pressure (P1) | 0.998 | bar |

Turbo compressor outlet air pressure (P2) | 5.828–23.14 | bar |

GT exhaust gas pressure (${P}_{exh}$) | 1.019–1.052 | bar |

Turbine injection control (TIC) | 0–92.556 | % |

Fuel flow (${m}_{f}$) | 0.068–1.832 | kg/s |

Turbo compressor decay state coefficient | 0.95–1 | - |

Turbine decay state coefficient | 0.975–1 | - |

**Table 2.**The input and output variables used in the GP algorithm to obtain symbolic expressions for estimation of fuel flow, ship speed, starboard, port, and total propeller torque with and without decay coefficient.

Physical Variable | Representation of Variables in GP | ||||
---|---|---|---|---|---|

Fuel Flow Analysis | Ship Speed Analysis | Starboard Propeller Torque Analysis | Port Propeller Torque Analysis | Total Propeller Torque Analysis | |

Lever position (${l}_{p}$) | ${X}_{0}$ | ${X}_{0}$ | ${X}_{0}$ | ${X}_{0}$ | ${X}_{0}$ |

Ship speed (v) | ${X}_{1}$ | y | ${X}_{1}$ | ${X}_{1}$ | ${X}_{1}$ |

Gas turbine shaft torque (GTT) | ${X}_{2}$ | ${X}_{1}$ | ${X}_{2}$ | ${X}_{2}$ | ${X}_{2}$ |

GT rate of revolutions (GTn) | ${X}_{3}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{3}$ | ${X}_{3}$ |

Gas generator rate of revolutions (GGn) | ${X}_{4}$ | ${X}_{3}$ | ${X}_{4}$ | ${X}_{4}$ | ${X}_{4}$ |

Starboard propeller torque (Ts) | ${X}_{5}$ | ${X}_{4}$ | y | - | - |

Port propeller torque (Tp) | ${X}_{6}$ | ${X}_{5}$ | - | y | - |

High pressure turbine exit temperature (T48) | ${X}_{7}$ | ${X}_{6}$ | ${X}_{5}$ | ${X}_{5}$ | ${X}_{5}$ |

turbo compressor inlet air temperature (T1) | ${X}_{8}$ | ${X}_{7}$ | ${X}_{6}$ | ${X}_{6}$ | ${X}_{6}$ |

turbo compressor outlet air pressure (P2) | ${X}_{9}$ | ${X}_{8}$ | ${X}_{7}$ | ${X}_{7}$ | ${X}_{7}$ |

HP turbine exit pressure (P48) | ${X}_{10}$ | ${X}_{9}$ | ${X}_{8}$ | ${X}_{8}$ | ${X}_{8}$ |

Turbo compressor inlet air pressure (P1) | ${X}_{11}$ | ${X}_{10}$ | ${X}_{9}$ | ${X}_{9}$ | ${X}_{9}$ |

Turbo compressor outlet air pressure (P2) | ${X}_{12}$ | ${X}_{11}$ | ${X}_{10}$ | ${X}_{10}$ | ${X}_{10}$ |

GT exhaust gas pressure (${P}_{exh}$) | ${X}_{13}$ | ${X}_{12}$ | ${X}_{11}$ | ${X}_{11}$ | ${X}_{11}$ |

Turbine injection control (TIC) | ${X}_{14}$ | ${X}_{13}$ | ${X}_{12}$ | ${X}_{12}$ | ${X}_{12}$ |

Fuel flow (${m}_{f}$) | y | ${X}_{14}$ | ${X}_{13}$ | ${X}_{13}$ | ${X}_{13}$ |

Turbo compressor decay state coefficient | ${X}_{15}$ | ${X}_{15}$ | ${X}_{14}$ | ${X}_{14}$ | ${X}_{14}$ |

Trubine decay state coefficient | ${X}_{16}$ | ${X}_{16}$ | ${X}_{15}$ | ${X}_{15}$ | ${X}_{15}$ |

Total Propeller Torque (Ts+Tp) | - | - | - | - | y |

GP Parameter | Lower Bound | Upper Bound |
---|---|---|

Population size | 500 | 1000 |

Number of generations | 100 | 500 |

Tournament selection size | 50 | 100 |

Tree depth | (3–7) | (6–12) |

Crossover coefficient | 0.9 | 1 |

Subtree mutation coefficient | 0.01 | 0.1 |

Hoist mutation coefficient | 0.01 | 0.1 |

Point mutation coefficient | 0.01 | 0.1 |

Stopping criteria value | $1\times {10}^{-6}$ | 0.001 |

Maximum number of samples | 0.9 | 1.0 |

Constant range | −0.1 | 0.1 |

Parsimony coefficient | $1\times {10}^{-4}$ | 0.01 |

**Table 4.**Three best symbolic expressions for fuel flow estimation with decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[930, 243, 81, (3, 11), 0.91, 0.021, 0.015, 0.041, 0.0002, 0.95, (−0.043, 0.021), 0.0003] | ${y}_{mfDF1}=(log(min(\sqrt{sin(log\left(\frac{{X}_{12}}{{X}_{15}{X}_{16}}\right))},$ $tan(sin(tan(sin(log\left(\frac{{X}_{12}}{{X}_{13}{X}_{15}}\right))))){\left)\right))}^{\frac{1}{2}}$ | $0.99398$ | $0.02664$ |

[742, 103, 92, (4, 11), 0.9, 0.026, 0.035, 0.02, 0.0002, 0.91, (−0.071, 0.02), 0.0038] | ${y}_{mfDF2}=log\left({X}_{10}\right)cos(log(cos\left({X}_{16}\right)))$ $cos(log(tan\left({X}_{11}\right)))max({X}_{15},log\left({X}_{10}\right))$ | $0.993$ | $0.03695$ |

[927, 346, 80, (6, 9), 0.9, 0.032, 0.039, 0.019, 0.0002, 0.92, (−0.063, 0.056), 0.0008] | ${y}_{mfDF3}=log\left(\right)open="("\; close=")">{X}_{10}{X}_{15}cos\left(\right)open="("\; close=")">\frac{{X}_{13}+sin({X}_{16}+{X}_{3})}{sin\left({X}_{0}\right)+3.35241}$ | $0.95526$ | $0.08184$ |

**Table 5.**Three best symbolic expressions for fuel flow estimation without decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[962, 289, 52, (6, 8), 0.91, 0.017, 0.035, 0.03, 0.000524, 0.99, (−0.073, 0.0014), 0.0029] | ${y}_{mf1}=\frac{{X}_{10}}{\sqrt{\frac{ln\left({X}_{2}\right)\sqrt{\frac{ln\left({X}_{2}\right)}{ln\left({X}_{10}\right)}}}{{X}_{10}}}}$ | 0.9964 | 0.02276 |

[1000, 141, 83, (5, 9), 0.9, 0.022, 0.012, 0.032, 0.000986, 0.98, (−0.049, 0.0943), 0.0013] | ${y}_{mf2}=\sqrt{tan\left({X}_{1}\right)}sin(\sqrt{tan(max({X}_{1},ln\left({X}_{4}\right)))}$ $sin(sin(sin\left(\right(sin(sin(sin\left(\sqrt{sin\left({X}_{1}\right)}\right)))$ $\sqrt{tan(max({X}_{1},ln\left({X}_{4}\right)))}{)}^{\frac{1}{2}}\left)\right)\left)\right)$ | 0.99591 | 0.02341 |

[582, 365, 85, (4, 7), 0.9, 0.022, 0.027, 0.018, 0.00046, 0.91, (−0.0103, 0.0905), 0.0003] | ${y}_{mf3}=\frac{ln\left({X}_{10}\right)}{tan\left(\right)open="("\; close=")">sin\left(\right)open="("\; close=")">\frac{\frac{ln\left({X}_{10}\right)}{{X}_{13}}}{{X}_{13}}+{X}_{11}}$ | 0.99578 | 0.023027 |

**Table 6.**Three best symbolic expressions for ship speed estimation with decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[548, 311, 87, (3, 8), 0.91, 0.017, 0.017, 0.018, 0.000926, 0.92, (−0.015, 0.044), 0.0013] | ${y}_{ssDF1}=({X}_{15}+{X}_{16})\left(\right)open="("\; close=")">\frac{{X}_{0}}{{X}_{10}+{X}_{12}}+{X}_{0}$ | 0.99843 | 0.2858 |

[784, 458, 77, (4, 7), 0.9, 0.015, 0.015, 0.06, $9.3\times {10}^{-5}$, 0.9, (−0.0083, 0.082), 0.0063] | ${y}_{ssDF2}={X}_{0}{X}_{15}{X}_{16}+{X}_{0}{X}_{15}+{X}_{0}{X}_{16}$ | 0.99788 | 0.32584 |

[585, 286, 69, (3, 12), 0.9, 0.024, 0.025, 0.023, 0.000191, 0.92, (−0.00084, 0.018), 0.0053] | ${y}_{ssDF3}=\left(\right)open="|"\; close="|">\left(\right)open="|"\; close="|">log\left({X}_{14}\right)+\sqrt{{X}_{4}}$ | 0.99593 | 0.41067 |

**Table 7.**Three best symbolic expressions for ship speed estimation without decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[732, 352, 86, (6, 10), 0.92, 0.012, 0.013, 0.023, 0.000231, 0.9, (−0.073, 0.031), 0.003] | ${y}_{sp1}=\frac{{X}_{0}-0.066}{{X}_{12}}+2{X}_{0}-0.279$ | 0.9998925 | 0.06729 |

[945, 479, 70, (6, 7), 0.91, 0.016, 0.016, 0.014, $9.4\times {10}^{-5}$, 0.98, (−0.085, 0.0049), 0.0097] | ${y}_{sp2}=\sqrt{{X}_{0}({X}_{0}-{X}_{14})log\left(\right)open="("\; close=")">{X}_{3}+{X}_{4}\sqrt{{X}_{6}}}$ | 0.999825 | 0.08665 |

[690, 152, 82, (6, 12), 0.9, 0.047, 0.01, 0.018, $3.6\times {10}^{-5}$, 0.94, (−0.023, 0.058), 0.0078] | ${y}_{sp3}={X}_{14}cos({X}_{12}-{X}_{14}cos({X}_{0}-{X}_{10}))+$ $log\left({X}_{0}\right)+\sqrt{{X}_{4}}$ | 0.999541 | 0.11797 |

**Table 8.**Three best symbolic expressions for starboard torque estimation with decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[996, 399, 69, (3, 10), 0.92, 0.013, 0.035, 0.018, $9.06\times {10}^{-7}$, 0.94, (−0.078, 0.077), 0.0061] | ${y}_{stDF1}={X}_{0}+{X}_{1}{X}_{10}+5{X}_{13}+$ ${X}_{8}+{X}_{SPTDF11}+{X}_{SPTDF12}$ | 0.99985 | 1.98477 |

[821, 418, 92, (4, 10), 0.909, 0.044, 0.018, 0.011, $1.46\times {10}^{-7}$, 0.98, (−0.002, 0.07), 0.0022] | ${y}_{stDF2}={X}_{1}{X}_{10}min({X}_{11},{X}_{SPTDF21})$ | 0.99959 | 3.16776 |

[598, 398, 63, (4, 11), 0.9, 0.033, 0.018, 0.032, $1.68\times {10}^{-7}$, 0.96, (−0.0055, 0.014), 0.0016] | ${y}_{stDF3}={X}_{12}{X}_{SPTDF31}$ | 0.99737 | 7.9579 |

**Table 9.**Three best symbolic expressions for starboard torque estimation without decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[554, 233, 81, (5, 11), 0.9, 0.052, 0.025, 0.017, $5.1\times {10}^{-7}$, 0.95, (−0.087, 0.028), 0.0031] | ${y}_{st1}=\sqrt{{X}_{SPT11}{X}_{SPT12}}$ | 0.99994 | 1.0697 |

[792, 144, 63, (5, 8), 0.92, 0.039, 0.013, 0.025, $6.25\times {10}^{-7}$, 0.92, (−0.07, 0.01), 0.0069] | ${y}_{st2}={X}_{0}({X}_{1}+{X}_{SPT21})$ | 0.99989 | 1.3387 |

[824, 297, 57, (6, 7), 0.91, 0.014, 0.032, 0.031, $4.08\times {10}^{-7}$, 0.92, (−0.08, 0.039), 0.0039] | ${y}_{st3}=\frac{{X}_{1}{X}_{10}{X}_{SPT31}}{tan(tan\left({X}_{9}\right))}+{X}_{1}{X}_{10}+$ $log\left({X}_{13}\right)+{X}_{SPT32}$ | 0.99981 | 1.8535 |

**Table 10.**Three best symbolic expressions for port propeller torque estimation with decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[788, 470, 94, (6, 8), 0.93, 0.016, 0.017, 0.032, $6.00\times {10}^{-9}$, 0.93, (−0.072, 0.083), 0.0044] | ${y}_{pptDF1}=(log(log\left({X}_{0}\right))+{X}_{12})$ $log({X}_{9}-{X}_{0})+{X}_{PPTDF11}$ | 0.99964 | 1.9885 |

[979, 263, 77, (6, 8), 0.91, 0.047, 0.012, 0.022, $6.86\times {10}^{-9}$, 0.91, (−0.062, 0.0027), 0.0043] | ${y}_{pptDF2}={X}_{1}{X}_{10}+{X}_{PPTDF21}$ | 0.9996 | 2.61963 |

[986, 394, 53, (3, 12), 0.91, 0.018, 0.051, 0.013, $9.47\times {10}^{-9}$, 0.946, (−0.02, 0.016), 0.0095] | ${y}_{pptDF3}={X}_{0}{X}_{PPTDF31}$ | 0.99427 | 14.0996 |

**Table 11.**Three best symbolic expressions for port propeller torque estimation without decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[709, 445, 70, (5, 11), 0.9, 0.023, 0.029, 0.041, $5.72\times {10}^{-7}$, 0.94, (−0.041, 0.01), 0.0031] | ${y}_{pp{t}_{1}}=\frac{{X}_{0}{X}_{11}{X}_{PPT11}}{{X}_{11}+{X}_{9}}$ | 0.9994 | 3.35254 |

[986, 294, 74, (4, 12), 0.91, 0.042, 0.012, 0.025, $6.99\times {10}^{-7}$, 0.91, (−0.021, 0.081), 0.0061 ] | ${y}_{pp{t}_{2}}={X}_{10}\left(\right)open="("\; close=")">min\left(\right)open="("\; close=")">{X}_{13},log\left(\right|{X}_{PPT21}\left|\right)$ | 0.99922 | 4.06154 |

[769, 415, 69, (3, 11), 0.93, 0.014, 0.011, 0.028, $4.21\times {10}^{-7}$, 0.96, (−0.067, 0.035), 0.0046] | ${y}_{pp{t}_{3}}=({X}_{1}+{X}_{13})(\frac{{X}_{12}sin\left({X}_{0}\right)\sqrt{{sin}^{3}\left(\right)open="("\; close=")">sin\left(\sqrt{{X}_{12}}\right){X}_{PPT31}}}{}{X}_{10}$ $+{X}_{PPT32}{)}^{\frac{1}{2}}$ | 0.99891 | 5.11714 |

**Table 12.**Three best symbolic expressions for total propeller torque estimation with decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[910, 285, 69, (5, 9), 0.91, 0.038, 0.016, 0.015, $8.42\times {10}^{-7}$, 0.94, (−0.029, 0.09), 0.0096] | ${y}_{ttDF1}=\left|{X}_{TTDF11}\right|+{X}_{12}$ $sin(sin(sin(log(tan(sin\left(\sqrt{{X}_{0}}\right))-{X}_{0}))))$ | 0.99848 | 11.697387 |

[664, 116, 75, (3, 10), 0.9, 0.058, 0.015, 0.017, $3.4e\times {10}^{-7}$, 0.93, (−0.048, 0.015), 0.0096] | ${y}_{ttDF2}=min({X}_{13},{X}_{14})max({X}_{5},{X}_{13}{X}_{7})$ $-\sqrt{max\left(\right)open="("\; close=")">{X}_{5},{X}_{13}^{2}{X}_{7}}$ | 0.991606 | 26.33334 |

[790, 112, 79, (3, 12), 0.91, 0.012, 0.031, 0.021, $7.67\times {10}^{-7}$, 0.9, (−0.02, 0.054), 0.007] | ${y}_{ttDF3}={X}_{13}{X}_{15}min({X}_{5},{X}_{7})$ | 0.97971 | 49.89208 |

**Table 13.**Three best symbolic expressions for total propeller torque estimation without decay state coefficients with corresponding ${R}^{2}$ and $MAE$ score.

GP Parameters - Population, Generations, Selection Size, Tree Depth, Crossover Coef., Subtree Mutation Coef., Hoist Mutation Coef., Point Mutation Coef., Stopping Criteria, Samples, Constant Range, Parsimony Coef. | Symbolic Expression | ${\mathit{R}}^{2}$ | $\mathit{MAE}$ |
---|---|---|---|

[682, 172, 56, (4, 7), 0.9, 0.018, 0.025, 0.029, $4.09\times {10}^{-7}$, 0.93, (−0.012, 0.065), 0.0026] | ${y}_{tt1}=|{X}_{12}-{X}_{TT11}|$ $-{X}_{TT12}-\sqrt{\frac{{X}_{3}}{{X}_{8}}}+{X}_{6}{X}_{8}$ | 0.99808 | 9.2407 |

[798, 103, 77, (4, 11), 0.9, 0.01, 0.061, 0.021, $6.14\times {10}^{-7}$, 0.92, (−0.039, 0.046), 0.0069] | ${y}_{tt2}=\frac{{X}_{12}{X}_{8}+\sqrt{{X}_{2}}}{{X}_{TT21}}$ | 0.99806 | 13.25 |

[883, 209, 64, (6, 9), 0.93, 0.013, 0.023, 0.028, $8.75\times {10}^{-7}$, 0.96, (−0.057, 0.057), 0.0099] | ${y}_{tt3}=max(\frac{{X}_{TT31}}{{X}_{TT32}}+\sqrt{{X}_{3}},$ $log\left({X}_{2}\right)-{X}_{12})+{X}_{12}$ | 0.9976 | 13.6284 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Anđelić, N.; Baressi Šegota, S.; Lorencin, I.; Poljak, I.; Mrzljak, V.; Car, Z.
Use of Genetic Programming for the Estimation of CODLAG Propulsion System Parameters. *J. Mar. Sci. Eng.* **2021**, *9*, 612.
https://doi.org/10.3390/jmse9060612

**AMA Style**

Anđelić N, Baressi Šegota S, Lorencin I, Poljak I, Mrzljak V, Car Z.
Use of Genetic Programming for the Estimation of CODLAG Propulsion System Parameters. *Journal of Marine Science and Engineering*. 2021; 9(6):612.
https://doi.org/10.3390/jmse9060612

**Chicago/Turabian Style**

Anđelić, Nikola, Sandi Baressi Šegota, Ivan Lorencin, Igor Poljak, Vedran Mrzljak, and Zlatan Car.
2021. "Use of Genetic Programming for the Estimation of CODLAG Propulsion System Parameters" *Journal of Marine Science and Engineering* 9, no. 6: 612.
https://doi.org/10.3390/jmse9060612