Stochastic Optimization of Onboard Photovoltaic Hybrid Power System Considering Environmental Uncertainties

Abstract

Environmental uncertainties present a significant challenge in the design of onboard photovoltaic hybrid power systems (PV-HPS), a pivotal decarbonization technology garnering widespread attention in the shipping industry. Neglecting environmental uncertainties associated with photovoltaic (PV) output and hull resistance can lead to suboptimal solutions. To address this issue, this paper proposes a stochastic optimization method for PV-HPS, aiming to minimize greenhouse gas (GHG) emissions and lifecycle costs. Copula functions are employed to establish joint distributions of uncertainties in solar irradiance, ambient temperature, significant wave height, and wave period. Monte Carlo simulation, the bi-bin method, and the multi-objective particle swarm optimization (MOPSO) algorithm are utilized for scenario generation, scenario reduction, and design space exploration. The efficacy of the proposed method is demonstrated through a case study involving an unmanned ship. Additionally, deterministic optimization and two partial stochastic optimizations are conducted to underscore the importance of simultaneously considering environmental uncertainties related to power sources and hull resistance. The results affirm the proposed approach’s capability to reduce GHG emissions and lifecycle costs. A sensitivity analysis of bin number is performed to investigate the tradeoff between optimality and computation time.

1. Introduction

1.1. Background

The Fourth International Maritime Organization (IMO) Greenhouse Gas (GHG) study highlights the challenge in achieving a 50% reduction in carbon intensity from ships relative to the 2008 baseline by 2050 with current measures [1]. In response, onboard photovoltaic hybrid power system (PV-HPS), leveraging multiple energy sources, have emerged as a promising technology in the shipping industry [2]. PV-HPS capitalizes on abundant solar energy available on water surfaces through photovoltaic (PV) modules [3]. Energy storage systems (ESS) in the PV-HPS not only enable PV-HPS to optimize diesel engine (DG) operation but also mitigate fluctuations in PV output due to the stochastic nature of solar energy [4]. However, designing such systems remains challenging due to the nonlinear characteristics and complex interactions among components [5].

In contrast to land-based microgrids, the onboard load of PV-HPS, significantly influenced by wave conditions, experiences substantial fluctuations. Moreover, DGs in PV-HPS must provide specific power, often exceeding average load, to ensure vessel safety during extreme maritime conditions. Thus, investigating PV-HPS sizing considering ship characteristics is essential.

1.2. Literature Survey

Researchers have extensively studied PV-HPS sizing using two methods: meta-heuristic and optimization-based. Meta-heuristic methods rely on experience or guidelines to determine the size parameters. For example, Ref. [6] conducted comprehensive studies on the design and control of PV-HPS for a 5000-vehicle space car carrier, demonstrating GHG emission reduction with a 143 kW PV system. Ref. [7] illustrated the financial and environmental benefits of a PV-HPS with an ESS along main navigation routes. Dolatabadi et al. [8] explored the potential benefits of PV systems for bulk carriers. Ref. [9] proposed a rule-based approach to design the PV system in the PV-HPS. The approach consists of three steps, namely, area determination, load calculation and economic evaluation. Ref. [10] retrofitted a PV-HPS for a 57 m long passenger-car ferry, leading to 375 tons of emission reduction per year. Ref. [11] developed a PV-powered ship for environmental monitoring.

In optimization-based methods, sizing problems are transformed into constrained optimization that aim to minimize environmental and economic objectives. For example, Lan et al. [12] proposed a multi-objective method for determining the optimal size parameters of a PV-HPS that takes solar irradiation and temperature along the route from Dalian in China to Aden in Yemen into account. Ref. [13] optimized deck area for solar and wind systems to reduce the GHG emission of a handysize bulk carrier. Ref. [14] developed an optimization model of PV/PEM fuel cell/DG power system for cruise ships, considering a predetermined load profile and average solar irradiance of Stockholm. Ref. [15] employed ε-constraint method to optimize the design and operation of a cruise ship’s power system with multiple energy resources, based on data of energy demands, solar irradiance and temperature in three days as optimization scenario.

As aforementioned, optimal sizing approaches have been successfully applied in studies that focus on specific scenarios. However, most of those studies are deterministic, neglecting environmental uncertainties affecting the power system in two ways: power sources and hull resistance. In terms of power source [16], the output of PV modules has strong randomness because of its high dependence on the uncertainties in solar irradiance and ambient temperature. In fact, the power provided by an on-board PV system varies in different meteorological circumstances [17]. Moreover, the assumption of calm water is widely employed in the sizing of onboard power systems [18], yet waves can add up to 15–30% of the calm water resistance [19]. Neglecting this additional resistance and the uncertainties connected to it, such as significant wave height and wave period, can lead to a system blackout or inefficient operation. Therefore, stochastic optimization is needed to tackle the environmental uncertainties.

To mitigate uncertainties in solar irradiance, Ref. [20] devised a risk-based two-stage stochastic framework for sizing a PV-HPS. Yao et al. [21] proposed a stochastic optimization method to determine the optimal capacity of the ESS in the PV-HPS. Ref. [22] established a data-driven method to estimate uncertainties in onboard PV generation. Zhu and Chen examined the impact of environmental uncertainties on the power of onboard PV systems [23] and sail thrust [24]. Unfortunately, the uncertainties related to hull resistance, namely, the uncertainties in significant wave height and wave period, were not accounted for in the aforementioned studies. Fang et al. developed a robust operation strategy [25] and a multi-objective management [26] to enhance the performance of ship power systems considering the uncertainties in waves and wind. However, their studies focus on robust control rather than optimal sizing. Ref. [27] proposed a two-stage method to optimize hull and propeller under aleatory and epistemic uncertainties. The results suggested that uncertainties resulted in up to 26.91% reductions in the average attainable speed. To address the mixed uncertainty and propagation in hull design, Li et al. presented a unified analysis and propagation method based on evidence theory, which achieves better reliability and robustness than the deterministic method [28]. Nevertheless, the aforementioned studies predominantly focused on optimizing hull parameters under uncertainties. It remains challenging to achieve optimal sizing of the power system while considering uncertainties related to power sources and hull resistance.

The research gap can be identified as follows: (1) the power source and hull resistance uncertainties have not been considered together when sizing PV-HPSs. (2) Scenario generation has been conducted without considering the coordinated variation and synergistic effect of different uncertainty sources, which are closely related. For instance, the ambient temperature is commonly higher during summer when solar energy sources are abundant, since sunlight contributes to temperature rise. Not accounting for the relationship between different uncertainty sources may result in scenarios that do not reflect reality accurately, leading to sub-optimal designs.

1.3. Contributions

To address these challenges, this paper proposes stochastic optimization framework for PV-HPS sizing, with a specific focus on incorporating environmental uncertainties. Our model for PV-HPS incorporates scalable components, and to enhance computational efficiency, we employ a bi-bin method for scenario reduction, addressing the combined impact of uncertainties on power sources and hull resistance. The key contributions of this study include:

(1)

Incorporating four environmental parameters that significantly affect the power system of a ship—namely, solar irradiance, ambient temperature, significant wave height and wave period—the proposed stochastic optimization considers the uncertainties associated with them;

(2)

Joint Cumulative Distribution Functions (JCDFs) of solar irradiance and ambient temperature and of significant wave height and wave period are developed to characterize the synergistic effect of uncertainties on power source and hull resistance.

The rest of this paper is organized as follows: Section 2 depicts the methodology, Section 3 provides the results and discussion. Section 4 draws conclusions.

2. Methods

2.1. System Modeling

The PV-HPS follows a standard configuration as illustrated in Figure 1, which can be segmented into two sections by the power bus. The left part is the power source of the PV-HPS that mainly includes a PV system, a DG and an energy storage system (ESS). The right part is the load of the system, which is directly affected by the hull resistance. The optimization of the PV-HPS cannot be achieved without modeling of the two parts. It is noted that this subsection provides the modeling of ship dynamics, hull resistance, and the PV system, all of which are directly impacted by environmental uncertainties. The modeling of other components of the PV-HPS follows a similar approach to [24] and is given in Appendix A.

2.1.1. Ship Dynamics

The longitudinal dynamics of the PV-HPS is given as below:

$$m{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}t}}=2\left(1-t_1\right)T_{pro}-R_{total}$$

(1)

2.1.2. Hull Resistance

Besides the calm water resistance and aerodynamic resistance, which are commonly considered in existing studies [18], this study takes added resistance in waves into account.

$$R_{total}=R_{cw}+R_{air}+{\overline{R}}_{aw}$$

(2)

where R_total, R_cw, R_air and ${\overline{R}}_{aw}$ are total resistance, calm water resistance, aerodynamic resistance and total added resistance in waves.

The calm water resistance can be calculated neglecting the wave-making resistance [29].

$$R_{cw}=\left(1+k\right)C_f{\displaystyle \frac{1}{2}}{\rho }_{wat}{v}^2{S}_{wet}$$

(3)

$$C_f={\displaystyle \frac{0.075}{{\left({\log }_{10}\mathrm{Re}-2\right)}^2}}$$

(4)

$$\mathrm{Re}={\displaystyle \frac{v{L}_{pp}}{\nu }}$$

(5)

where k is the form factor, C_f is the frictional resistance coefficient, ρ_wat is the water density, S_wet is the wetted area of the bare hull, Re is the Reynolds number, L_pp is the length between perpendiculars, $\nu$ is the kinematic viscosity of water.

The aerodynamic resistance is calculated as:

$$R_{air}=C_{air}{\displaystyle \frac{1}{2}}{\rho }_{air}{v}_{air}^2{S}_{air}$$

(6)

where C_air is the aerodynamic resistance coefficient, ρ_air is the air density, $v_{air}$ is the relative wind speed, S_air is the frontal area of the hull and superstructure.

In order to consider additional resistance in waves and investigate the effect of uncertainties related with wave, an empirical formula developed by Liu et al. is introduced [30]. The additional resistance in waves is obtained based on the added resistance in regular head waves:

$${\overline{R}}_{aw}=2{\displaystyle {\int }_0^{\infty }S\frac{R_{aw}}{{\zeta }_a^2}}\mathrm{d}\omega$$

(7)

where S is the wave spectrum, R_aw is the added resistance in regular head waves, ζ_a is the wave amplitude. The JONSWAP spectrum is expressed as follows [31]:

$$S={\displaystyle \frac{{\alpha }^{\ast }{H}_s^2}{2\pi }}{\displaystyle \frac{{\omega }^{-5}}{{\omega }_{peak}^{-4}}}\exp \left[-{\displaystyle \frac{5}{4}}{\left({\displaystyle \frac{\omega }{{\omega }_{peak}}}\right)}^{-4}\right]{\gamma }^{\exp \left[{\displaystyle \frac{-{\left(\omega -{\omega }_{peak}\right)}^2}{2{\sigma }^2{\omega }_{peak}^2}}\right]}$$

(8)

$${\alpha }^{\ast }={\displaystyle \frac{0.0624}{0.23+0.0336\gamma -0.185/\left(1.9+\gamma \right)}}$$

(9)

$$\sigma =\{\begin{array}{cc}0.09& \omega >{\omega }_p\\ 0.07& \omega \le {\omega }_p\end{array}$$

(10)

$${\omega }_{peak}={\displaystyle \frac{2\pi }{T_{peak}}}$$

(11)

$$T_{peak}=1.1T_h$$

(12)

where the peak enhancement factor γ is considered to be 3.3 [32].

The added resistance in regular head waves contains two parts that are caused by reflection/diffraction effect and motion/radiation effect, respectively [33]:

$$R_{aw}=R_{awr}+R_{awm}$$

(13)

$$R_{awr}={\displaystyle \frac{2.25}{2}}\rho gB{\zeta }_a^2{\alpha }_T{\sin }^{2}E\left(1+5\sqrt{{\displaystyle \frac{L_{pp}}{\lambda }}}\mathrm{Fn}\right){\left({\displaystyle \frac{0.87}{C_B}}\right)}^{1+4\sqrt{\mathrm{Fn}}}$$

(14)

$${\alpha }_T=\{\begin{array}{cc}1-e^{-4\pi \left({\displaystyle \frac{T}{\lambda }}-{\displaystyle \frac{T}{2.5L_{pp}}}\right)}& {\displaystyle \frac{\lambda }{L_{pp}}}\le 2.5\\ 0& {\displaystyle \frac{\lambda }{L_{pp}}}>2.5\end{array}$$

(15)

$$E=\mathrm{arc}\tan \left({\displaystyle \frac{B}{2L_e}}\right)$$

(16)

where g is the gravitational acceleration, B is the ship’s beam, α_T is the draft coefficient, λ is the wave length, Fn is the Froude number, C_B is the block coefficient, T is the draft of the ship. The angle of entrance of the water line E and the length of waterline entrance L_e are defined in Figure 2 [33].

$$R_{awm}=4\rho g{\zeta }_a^2{\displaystyle \frac{B^2}{L_{pp}}}{\overline{\omega }}^{b_1}\exp \left[{\displaystyle \frac{b_1}{{\mathrm{d}}_1}}\left(1-{\overline{\omega }}^{d_1}\right)\right]a_1{a}_2{a}_3$$

(17)

$$\overline{\omega }=\{\begin{array}{cc}2.142\sqrt[3]{k_{yy}}\sqrt{{\displaystyle \frac{L_{pp}}{\lambda }}}\left[1-0.13{\displaystyle \frac{0.85}{C_B}}\left(\ln {\displaystyle \frac{B}{T}}-\ln 2.75\right)\right]\left(\mathrm{Fn}+0.62\right)& \mathrm{Fn}<0.1\\ 2.142\sqrt[3]{k_{yy}}\sqrt{{\displaystyle \frac{L_{pp}}{\lambda }}}\left[1-0.13{\displaystyle \frac{0.85}{C_B}}\left(\ln {\displaystyle \frac{B}{T}}-\ln 2.75\right)\right]{\mathrm{Fn}}^{0.143}& \mathrm{Fn}\ge 0.1\end{array}$$

(18)

The calculation of the factors of the added resistance due to motion effect are given:

$$a_1=60.3C_B^{1.34}{\left(4k_{yy}\right)}^2{\left({\displaystyle \frac{0.87}{C_B}}\right)}^{\left(1+\mathrm{Fn}\right)}{\left(\ln {\displaystyle \frac{B}{T}}\right)}^{-1}$$

(19)

$$a_2=\{\begin{array}{cc}0.0072+0.1676\mathrm{Fn}& \mathrm{Fn}<0.12\\ {\mathrm{Fn}}^{1.5}\exp \left(-3.5\mathrm{Fn}\right)& \mathrm{Fn}\ge 0.12\end{array}$$

(20)

$$a_3=1.0+0.25\arctan {\displaystyle \frac{\delta T}{L_{pp}}}$$

(21)

$$b_1=\{\begin{array}{cc}11.0& \overline{\omega }<1\\ -8.5& \overline{\omega }\ge 1\end{array}$$

(22)

$$d_1=\{\begin{array}{cc}14.0& \overline{\omega }<1,C_B<0.75\\ -566{\left({\displaystyle \frac{L_{pp}}{B}}\right)}^{-2.66}\times 6& \overline{\omega }\ge 1,C_B<0.75\\ 566{\left({\displaystyle \frac{L_{pp}}{B}}\right)}^{-2.66}& \overline{\omega }<1,C_B\ge 0.75\\ -566{\left({\displaystyle \frac{L_{pp}}{B}}\right)}^{-2.66}\times 6& \overline{\omega }\ge 1,C_B\ge 0.75\end{array}$$

(23)

where $\overline{\omega }$ is the resonance position of the added resistance, k_yy is the longitudinal mass radius of gyration, a₁, a₂, a₃, b₁ and d₁ are the factors of the added resistance due to motion/radiation effect.

2.1.3. Photovoltaic System Model

The PV system is placed horizontally on the deck, whose output mainly relies on solar radiation and PV module temperature. To strike a balance between precision and computational efficiency, the Zhou method [34] was selected to establish the PV model. On the assumption that all modules in the PV system are identical, the output power of the PV system P_PVs is calculated as:

$$P_{PVs}=n_{PV}{\eta }_{MPPT}{P}_{PVm}$$

(24)

$$P_{PVm}=I_{PVsc}{V}_{PVoc}{f}_{PV}$$

(25)

where n_PV is the number of PV modules in the PV system, η_MPPT is the efficiency of the maximum power point tracking, P_PVm is the maximum output power of single PV module, I_PVsc, V_PVoc and f_PV are the short-circuit current, open-circuit voltage, and fill factor of the PV module, respectively.

The short-circuit current of PV module is given by [34]:

$$I_{PVsc}=I_{PVsc0}{\left({\displaystyle \frac{G}{G_0}}\right)}^{{\alpha }_{PV}}$$

(26)

The open-circuit voltage is a function of solar irradiance and temperature of PV module.

$$V_{PVoc}={\displaystyle \frac{V_{PVoc0}}{1+{\beta }_{PV}\ln \frac{G_0}{G}}}\cdot {\left({\displaystyle \frac{T_{PV0}}{T_{PV}}}\right)}^{{\gamma }_{PV}}$$

(27)

$$f_{PV}=f_{PV0}\left(1-{\displaystyle \frac{R_{PVs}{I}_{PVsc}}{V_{PVoc}}}\right)$$

(28)

$$f_{PV0}={\displaystyle \frac{v_{PVoc}-\ln \left(v_{PVoc}+0.72\right)}{1+v_{PVoc}}}$$

(29)

$$v_{PVoc}={\displaystyle \frac{V_{PVoc}}{nK{T}_{PV}/q}}$$

(30)

The temperature of PV module is determined by the ambient temperature and solar irradiance [35].

$$T_{PV}=T_a-274.67567+0.01981366G-0.000003451G^2$$

(31)

where T_a indicates ambient temperature.

It is noted that the modeling of the rest of the system is given in Appendix A.

2.2. Energy Management Strategy

A rule-based strategy is established to control the energy distribution of the PV-HPS by virtue of its simplicity and reliability. The proposed strategy is a kind of thermostatic control strategy, which operates the PV-HPS in two modes, namely, the ESS driving mode and the DG driving mode [36]. Figure 3 explains the mechanism of such a strategy. Before the start of the voyage, the ESS is charged to SOC₀. According to the mode switching strategy, the PV-HPS switches to the DG driving mode when the SOC of the ESS is smaller than the lower bound b_low. In such a mode, the DG is operated at its optimal point for fuel efficiency. Here, the optimal point represents the working point with the highest efficiency (lowest specific fuel consumption (SFC)). The excess power is used to charge the ESS. If the SOC of the ESS reaches the upper bound b_up, the PV-HPS switches to the ESS driving mode where the DG is shut down and the ESS supplies all power requirements.

2.3. Uncertainty Characterization

2.3.1. Uncertainty in Power Source

According to the modeling of the PV system, the output of the PV system depends on solar irradiance and ambient temperature. Hence, the uncertainty in power source stems from the uncertainties in solar irradiance and ambient temperature, which are closely related [37]. In this study, the copula-based method is introduced to establish the JCDF of solar irradiance and ambient temperature [38].

$${JF}_{G,Ta}\left(G_n,T_a\right)=C_t\left[F_G\left(G_n\right),F_{Ta}\left(T_a\right)\right]$$

(32)

$$F_G={\displaystyle {\int }_0^G{f}_{\mathrm{G}}\left(G_n\right)\mathrm{d}}{G}_n$$

(33)

$$F_{Ta}={\displaystyle {\int }_0^T{f}_{Ta}\left(T_a\right)d}{T}_a$$

(34)

$$G_n={\displaystyle \frac{G-G_{\min }}{G_{\max }-G_{\min }}}$$

(35)

where JF_G,Ta(G_n,T_a) is the joint cumulative distribution function (JCDF) of wind speed and direction, C_t(⋅,⋅) is the t-copula function, F_G(G_n) and F_Ta(T_a) are the cumulative distribution functions (CDFs) of normalized solar irradiance and ambient temperature, respectively; f_G(G_n) and f_Ta(T_a) are the probability density functions (PDFs) of normalized solar irradiance and ambient temperature, respectively. Normalized solar irradiance 0 ≤ G_n ≤ 1 is calculated based on the minimum G_min and maximum G_max values of solar irradiance.

(1)

Solar irradiance

The literature suggests that the beta distribution function is the most suitable probability density function (PDF) for representing the uncertainty in solar irradiance [39]. After data normalization, the PDF of solar irradiance can be expressed as follows:

$$f_{\mathrm{G}}\left(G_n\right)={\displaystyle \frac{\Gamma \left({\alpha }_G+{\beta }_G\right)}{\Gamma \left({\alpha }_G\right)\Gamma \left({\beta }_G\right)}}\cdot {G_n}^{\left({\alpha }_G-1\right)}{\left(1-G_n\right)}^{\left({\beta }_G-1\right)}$$

(36)

$${\alpha }_G={\displaystyle \frac{{\mu }_G{\beta }_G}{1-{\mu }_G}}$$

(37)

$${\beta }_G=\left(1-{\mu }_G\right)\cdot \left({\displaystyle \frac{{\mu }_G\left(1+{\mu }_G\right)}{{{\sigma }_G}^2}}-1\right)$$

(38)

where α_G and β_G are the shape parameters of the beta PDF, Γ(⋅) is the gamma function, μ_G and σ_G are the mean and standard deviation obtained from the historical data.

(2)

Ambient temperature

Normal distribution is considered the most suitable PDF of ambient temperature [40].

$$f_{Ta}\left(T_a\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{Ta}}}{e}^{\left(-{\displaystyle \frac{{\left(T_a-{\mu }_T\right)}^2}{2{{\sigma }_{Ta}}^2}}\right)}$$

(39)

where μ_T and σ_T are the mean value and standard deviation of the distribution.

2.3.2. Uncertainty in Hull Resistance

Based on the modeling of ship dynamics, the hull resistance is affected by significant wave height and wave period. Therefore, the analysis of system load cannot be achieved without characterizing the uncertainties in significant wave height and wave period. Considering the internal connection between the two parameters, a JCDF of significant wave height and wave period is required for stochastic optimization. To this end, the copula-based method is employed. For the convenience of calculation, significant wave height and wave period are normalized.

$$J{F}_{H_{,}Th}\left({\overline{H}}_s,{\overline{T}}_h\right)=C_t\left[F_H\left({\overline{H}}_s\right),F_{Th}\left({\overline{T}}_h\right)\right]$$

(40)

$$F_H={\displaystyle {\int }_0^{\theta }{f}_H\left({\overline{H}}_s\right)\mathrm{d}}{H}_s$$

(41)

$$F_{Th}={\displaystyle {\int }_0^T{f}_T\left({\overline{T}}_h\right)\mathrm{d}}T$$

(42)

where $J{F}_{H_{,}Th}\left({\overline{H}}_s,{\overline{T}}_h\right)$ is the JCDF of significant wave height and wave period, $F_H\left({\overline{H}}_s\right)$ and $F_{Th}\left({\overline{T}}_h\right)$ are the CDFs of normalized significant wave height and wave period, respectively; ${\overline{T}}_h$ and ${\overline{H}}_s$ are the normalized significant wave height and wave period, respectively; $f_H\left({\overline{H}}_s\right)$ and $f_T\left({\overline{T}}_h\right)$ are the PDFs of significant wave height and wave period, respectively.

(1)

Significant wave height.

The marginal distribution of the normalized significant wave height is assumed to follow a 3-parameter Weibull distribution [41].

$$\begin{array}{cc}{f}_H\left({\overline{H}}_s\right)=\left({\displaystyle \frac{{\beta }_h}{{\alpha }_h}}\right){\left({\displaystyle \frac{{\overline{H}}_s-{\gamma }_h}{{\alpha }_h}}\right)}^{\left({\beta }_h-1\right)}{e}^{-{\left({\displaystyle \frac{H_s-{\gamma }_h}{{\alpha }_h}}\right)}^{{\beta }_h}},& {\overline{H}}_s>{\gamma }_h\end{array}$$

(43)

where α_h > 0 and β_h > 0 and ${\gamma }_h\in R$ are the scale parameter, shape parameter and position parameter of Weibull PDF.

(2)

Wave period

The marginal distribution of normalized wave period is assumed to be log-normally distributed in this study [41].

$$f_{Th}\left({\overline{T}}_h\right)={\displaystyle \frac{1}{{\overline{T}}_h\sqrt{2\pi }{\delta }_t({\overline{H}}_s)}}{e}^{-{\displaystyle \frac{{\left(\ln {\overline{T}}_h-{\mu }_t({\overline{H}}_s)\right)}^2}{2{\delta }_t{({\overline{H}}_s)}^2}}}$$

(44)

$${\mu }_t({\overline{H}}_s)=a_{H1}+a_{H2}{{\overline{H}}_s}^{a_{\mathrm{H}3}}$$

(45)

$${\delta }_t({\overline{H}}_s)=b_{H1}+b_{H2}{e}^{b_{H3}{\overline{H}}_s}$$

(46)

where a_i(i = 1, 2, 3) and b_i(i = 1, 2, 3) are the parameters of the log-normal distribution PDF.

2.4. Optimization Problem Formulation

2.4.1. Optimization Variables

Five design variables are considered in this study, i.e., the displacement of diesel engine in DG V_D, the volume of motor rotor V_M, the number of battery modules in the ESS n_ESS, the number of PV modules in the PV system n_PV and the gear ratio of the gearbox i_GB. Hence, the vector of optimization variables is expressed by:

$$x={\left(V_D,V_M,n_{ESS},n_{PV},i_{GB}\right)}^{\mathrm{T}}$$

(47)

2.4.2. Objective Function

In order to comprehensively assess the environmental and economic benefits of the PV-HPS, the greenhouse (GHG) emission and lifecycle cost are minimized in this study.

$$\min \left\{f_1(x),f_2(x)\right\}$$

(48)

The GHG emission of the PV-HPS mainly derive from the fuel consumption of electricity generation and DG startup.

$$f_1(x)=I_f\left({\displaystyle \sum {\dot{m}}_f}+m_{su}{n}_{su}\right)$$

(49)

where the annual GHG emission is considered to be the product of life cycle GHG intensity of diesel fuel I_f and annual fuel consumption, m_su is fuel consumption per startup, n_su is the number of the DG startup times.

The lifecycle cost of the PV-HEPS includes initial investment C_ini, operation cost C_ope and battery cost C_ESS.

$$f_2(x)=C_{ini}+C_{ope}+C_{ESS}$$

(50)

$$C_{ini}=c_{DG}{P}_{DG-rate}\left(1+\Delta c_{DG}\right)+c_M{P}_{M-rate}+c_{PV}{n}_{PV}{P}_{PVm}$$

(51)

where c_DG and c_M are the unit cost of the DG and motor, Δc_DG is the correction factor of electrical components in DG [42], $P_{DG-rate}$ and $P_{M-rate}$ are the rated power of the DG and motor, c_PV is the unit cost of the PV modules.

The operation cost consists of the fuel cost and the maintenance costs of the DG $C_{DG-m}$ and PV system $C_{PV-m}$.

$$C_{ope}={\displaystyle \sum _{i=1}^Y{c}_f\left({\displaystyle \sum {\dot{m}}_f}+m_{su}{n}_{su}\right){\left(\frac{1+g_f}{1+I_a}\right)}^i}+C_{DG-m}+C_{PV-m}$$

(52)

$$C_{DG-m}=c_{DG-m}{t}_{DG-m}$$

(53)

$$C_{PV-m}=c_{PV-m}{n}_{PV}{P}_{PVm}$$

(54)

where c_f is the unit cost of diesel fuel, g_f is the annual inflation rate of diesel fuel, I_a is the interest rate, $c_{DG-m}$ and $c_{PV-m}$ are the unit maintenance cost of the DG and PV system, $t_{DG-m}$ is the annual operation time of the DG, $C_{PV-m}$ is the maintenance of the PV system.

$$C_{ESS}={\displaystyle \sum _{i=1}^{n_{rep}+1}{c}_{ESS}{n}_{ESS}{Q}_{ESS}{\left(\frac{1+g_{ESS}}{1+I_a}\right)}^{i{Y}_{ESS}}}$$

(55)

$$n_{rep}=\{\begin{array}{cc}{\displaystyle \frac{Y}{Y_{ESS}}}-1& {\displaystyle \frac{Y}{Y_{ESS}}}-\left[{\displaystyle \frac{Y}{Y_{ESS}}}\right]=0\\ {\displaystyle \frac{Y}{Y_{ESS}}}& {\displaystyle \frac{Y}{Y_{ESS}}}-\left[{\displaystyle \frac{Y}{Y_{ESS}}}\right]\ne 0\end{array}$$

(56)

where n_rep is the number of battery replacements during the lifespan of the PV-HEPS Y, c_ESS is the unit cost of the battery module, Q_ESS is the capacity of one battery module, g_ESS is the inflation rate of the acquisition cost of ESS, Y_ESS is the lifetime of the battery module.

2.4.3. Constraints

The design parameters are optimized under the constraints of the following limits:

(1)

Power balance

The DG, PV system, and ESS collectively provide the electricity load that includes motor power requirements and hotel load.

$$P_{DG}+P_{PVs}+P_{ESS}=P_{req}=P_{M-in}+P_H$$

(57)

(2)

Constraints of the DG

The DG is required to have the ability to provide $P_{DG-req}$ for navigational safety.

$$P_{DG-\max }\ge P_{DG-req}$$

(58)

where $P_{DG-\max }$ is the maximum output of the DG.

A specific duration $t_{on/off-\min }$ must be exceeded between two consecutive startup/shutdown operations [43].

$$t_{on/off}(k+1)-t_{on/off}(k)\ge t_{on/off-\min },k\in N+$$

(59)

where t_on/off(k) is the time that a startup/shut-down operation happens.

(3)

Constraints of the ESS

The SOC and current of the battery are constrained for safety.

$$SO{C}_{b-low}<SOC(t)<SO{C}_{b-up}$$

(60)

$$I_{bat-\min }\le I_{bat}\le I_{bat-\max }$$

(61)

where $I_{bat-\min }$ and $I_{bat-\max }$ represents the current limit for charging and discharging.

2.5. Optimization Methodology

Monte Carlo (MC) simulation is a popular tool addressing the uncertainties in power system optimization, and is used in this study to handle the uncertainties in solar irradiance, ambient temperature, significant wave height and wave period. In general, about 1000 scenarios are considered to be sufficient to guarantee the convergence of MC [44]. However, such number of simulation trials leads to huge computational burden. Therefore, a bi-bin method is introduced in the optimization methodology for scenario reduction, due to its efficiency in handling large datasets [45]. By discretizing time-series scenarios into bins, the bi-bin method significantly reduces computational burden. It converts a large number of time-series scenarios of environmental conditions into a smaller set of discrete scenarios of PV module power and hull resistance with corresponding probabilities, thereby alleviating the computational load. Figure 4 depicts the framework of the proposed optimization methodology. Three key steps are listed as follows:

• Step 1: scenario generation.

The Monte Carlo (MC) simulation is performed to generate n_MC = 1000 time-series scenarios for solar irradiance, ambient temperature, significant wave height and wave period considering their coordinated variation based on the two joint distributions given before. Then, time-series scenarios for PV module power and hull resistance are obtained with the help of the PV-HPS model. It’s worth noting that PV module power and hull resistance are cross-correlated and should be matched at each time point.

The scenario generation process is performed as follows. (1) JCDF Sampling: draw random samples from the JCDF, representing the joint distributions of key environmental variables. (2) CDF calculation: use copula functions to calculate the cumulative distribution functions (CDFs) of solar irradiance, ambient temperature, significant wave height, and wave period. (3) Variables calculation: convert these CDF values into numerical values of the environmental variables based on their marginal distribution models.

• Step 2: scenario reduction.

(1) Discretization: Time-series scenarios for PV module power and hull resistance are discretized into one-hour intervals. A joint probability distribution of PV module power and hull resistance is then established;

(2) Interval Division: Divide the ranges of PV module power and hull resistance into k_bin equal intervals, respectively;

(3) Bin Creation: Create a two-dimensional space of $k_{bin}^2$ bins by combining the intervals of PV module power and hull resistance;

(4) Scenario Allocation: Place each discrete scenario (pair of PV module power and hull resistance) into the corresponding bin based on its values;

(5) Scenario Averaging: For each bin, compute the average values of PV module power and hull resistance for the scenarios within the bin. These averages represent the characteristic scenario for that bin. Count the number of scenarios in each bin and calculate the probability of each characteristic scenario by dividing the count by the total number of scenarios.

Up to this point, n_MC time-series scenarios for solar irradiance, ambient temperature, significant wave height and wave period are transformed into $k_{bin}^2$ pairs of module power and hull resistance that will be used as scenarios in the following process. In this way, the data length $k_{bin}^2$ is not related to the number of MC simulations n_MC, but to the number of bins.

In order to guarantee the fairness of comparison, when each one-hour scenarios terminate, the SOC of the ESS is adjusted back to its initial value, SOC₀, using fuel consumption compensation method. That is, based on the efficiency of the diesel generator set at the optimal (highest efficiency) point, the change of the battery SOC is converted into fuel consumption.

• Step 3: Problem solution.

The joint probability distribution of PV module power and hull resistance is sent to the PV-HPS model, which is then employed by the classical multi-objective particle swarm optimization (MOPSO) algorithm [46] to solve the optimization problem. The velocity update is performed based on the following equations.

$$v^{k+1}=w{v}^k+c_1{r}_1\left(p_{best}^k-x^k\right)+c_2{r}_2\left(g_{best}^k-x^k\right)$$

(62)

$$x^{k+1}=x^k+v^{k+1}$$

(63)

where $v^k$ is the speed vector of particle i at k^th iteration, $x_i^k$ is the position of particle i at k^th iteration, $w$ is the inertia, r₁ and r₂ are two random numbers with uniform distribution from [0 1], c₁ is the particle increment, c₂ is the global increment, $p_{best,i}^k$ is the best position experienced by particle i, $g_{best}^k$ is the global best position experienced by all particles.

To prevent velocities from becoming excessively large and causing particles to exceed the search space boundaries, velocity limitations are implemented [47]. This is particularly important when particles are far from their personal and global best positions.

$$\left|v_{i,j}^k\right|\le c_v\left(x_{j,\max }-x_{j,\min }\right)$$

(64)

where $v_{i,j}^k$ is j^th dimension of speed vector $v^k$, $c_v$ is the velocity limit factor, x_j,max and x_j,min are the upper and lower bound of j^th dimension of $x_i^k$.

Note that the duration of the scenarios generated by the joint probability distribution is set to be 1 h. Here, to carry out an overall evaluation for PV-HPS performance, the objective function is modified.

$$\min \left\{{\overline{f}}_1(x),{\overline{f}}_2(x)\right\}$$

(65)

$${\overline{f}}_1(x)={\displaystyle \sum f_{1ij}(x)p_{ij}},(i=1,2,3,\dots ,k,j=1,2,3,\dots ,k)$$

(66)

$${\overline{f}}_2(x)={\displaystyle \sum f_{2ij}(x)p_{ij}},(i=1,2,3,\dots ,k,j=1,2,3,\dots ,k)$$

(67)

where f_1ij(x) and f_2ij(x) are the GHG emission and lifecycle cost of PV-HPS under scenario S_ij(i = 1, 2, …, k, j = 1, 2, …, k), p_ij is the probability of bin B_ij(i = 1, 2, …, k, j = 1, 2, …, k). Here, only non-empty bins are considered in this step to speed up the optimization. The Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method is subsequently employed to rank the Pareto designs and facilitate the selection of the final design [48].

3. Results and Discussion

3.1. Case Study

With the advancement of China’s Marine Power Strategy, the demand for monitoring meteorological and hydrological environments and detecting fishery resources has significantly increased. Unmanned ships have the advantages of low cost and high safety, and can independently carry out data collection and transmission work without relying on drivers. Therefore, the development of green unmanned ships is of great importance for hydrological monitoring, fishery management and combating smuggling. Therefore, the proposed methodology is tested by the design of a PV-HPS for an unmanned ship which usually operates in the Beishuang area (26.7° N 120.3° E) of the East China Sea. As is shown in Figure 5, the PV system is placed horizontally on the deck of the case ship. The main dimensions of the ship are presented in

Table 1. Main dimensions of the case study ship.

Particulars	Value
Length (m)	60.00
Beam (m)	10.50
Draft (m)	3.15
Displacement (m3)	1110.20
Wetted Surface area (m2)	680
Length of waterline entrance (m)	10
Block coefficient	0.61

. Figure 6 illustrates the route of a square with sides spanning 24 nautical miles (about 44.45 km). Every day, the ship leaves the port at 7:00 a.m. and navigates at a speed of 8 kn (4.11 m/s). After returning to the port at 18:00 p.m., the ship is refueled and the ESS is charged to the SOC of SOC₀. The hotel load is assumed to be 10 kW. PV modules are installed on the deck and the top of the superstructure to take the most advantage of solar energy. The following assumptions have been made: (1) the unmanned ship is strictly sailing according to the prescribed route and speed, (2) the situation of being unable to sail due to extreme sea conditions or other low-probability events is ignored, (3) the impact of the energy system sizing on the dynamic parameters of the ship is ignored.

Data of solar irradiance, ambient temperature, significant wave height and wave period in the Beishuang area from 2011 to 2020 were obtained from the China Meteorological Data Service Center and used for uncertainty characterization. Figure 6 presents the hourly data of the four environmental parameters in 2020. Note that only the data during sailing time (7:00 to 18:00) is shown. In Figure 6, significant fluctuations can be found, which shows uncertainties in solar irradiance, ambient temperature, significant wave height and wave period. As aforementioned, the neglect of such uncertainties inevitably leads to misevaluation of the performance of PV-HPS, resulting in sub-optimal designs. To highlight the necessity of considering environmental uncertainties and the effectiveness of the proposed method, a deterministic optimization is performed with the environmental parameters in 2020. In addition, two partial stochastic optimizations that consider two of the four uncertain parameters are also proposed as controls. The uncertainties considered in the optimizations are listed in

Table 2. Uncertainties considered in the optimizations.

Optimization	Name of Pareto Front	Name of Final Solution	Uncertainty
			Solar Irradiance	Ambient Temperature	Significant Wave Height	Wave Period
Stochastic	PFsto	xsto	√	√	√	√
Power source	PFsou	xsou	√	√
Hull resistance	PFres	xres			√	√
Deterministic	PFdet	xdet

. The ranges of optimization variables are given in

Table 3. Results of stochastic optimization and deterministic optimization.

Variable	Range	xsto	xsou	xres	xdet
VD (10−3m3)	[14.00, 40.00]	14.00	17.36	14.00	14.00
VM (10−3m3)	[14.20, 40.00]	14.25	14.48	14.23	25.07
nESS	[40, 300]	55	47	46	61
nPV	[200, 1000]	200	372	200	335
iGB	[3.00,30.00]	30.00	3	30	23.16

. The parameters required in the optimization are given in

Table A1. Parameters used in the optimization.

Parameter	Value
Cair	0.80
Cs ($)	5000.00
cDG ($/kW)	350.00 [42]
$c_{DG-m}$ ($/h)	0.02 [56]
cf ($/t)	520.00 [57]
cM ($/kW)	32.00 [42]
CPV ($/kW)	6500.00 [58]
$c_{PV-m}$ ($/kW)	35.00 [58]
G0 (w/m2)	1000.00 [34]
gESS (%)	3.00 [59]
gf (%)	3.00 [59]
HLHV (J/kg)	4.27 × 107 [60]
Ia (%)	5.00 [59]
$I_{bat-\max }$ (A)	82
$I_{bat-\min }$ (A)	−41.00
IF	3.71 [61]
IPVsc0 (A)	6.50 [58]
iD	0.82
iM	1.92
K (J/K)	1.38 × 10−23
m (kg)	1.14 × 106
nMC	1000
PSh (W)	5000.00
$P_{DG-req}$ (W)	200,000
Qbat (Ah)	41.00 [42]
q (C)	1.6 × 10−19
Sair (m2)	60.50
Swet (m2)	680.00
SOC0 (%)	50.00
$SO{C}_{b-low}$	40.00
$SO{C}_{b-up}$	60.00
T (m)	3.15
TPV0 (K)	298.15
$t_{on/off-\min }$ (s)	60.00
$t_{DG-m}$ (h)	2.99 × 104
tP	0.10 [62]
VPVoc0 (V)	21.00 [58]
Y (year)	25.00
YESS (year)	5.00 [12]
γPV	1.15 [34]
ΔcDG (%)	6.00 [57]
ρair (kg/m3)	1.29
ρw	1.03 × 103 [62]
ηG0	0.97 [54]
ηGB	0.98 [63]
ηMPPT	95% [64]

. All the work is performed on a computer with a 12th Gen Intel(R) Core (TM) i5-12500H CPU of 3.10 GHz and a memory size of 16.00 GB. The parameters of the MOPSO algorithm are given in

Table A2. Parameters of MOPSO algorithm.

Parameters	Value
Maximum iterations	50
Number of particles	40
Inertia	0.4
Global increment	0.9
Particle increment	0.9
Velocity limit factor	0.1

.

3.2. Statistical Characteristic of Stochastic Scenarios V_D (10⁻³m³)

The joint probability distribution, as presented in Figure 7, is employed to investigate the relationship between solar irradiance and ambient temperature. The Pearson correlation coefficient of the two parameters is 0.34, indicating that there is a positive but weak correlation between solar irradiance and ambient temperature. Figure 8 illustrates the joint probability distribution of significant wave height and wave period. The Pearson correlation coefficient is 0.27, indicating a weak positive correlation between the two environmental variables. No distinct linear correlation can be recognized in Figure 7 and Figure 8. Hence, it becomes necessary to employ the copula-based method for uncertainty characterization, as it possesses the ability to effectively describe the nonlinear correlations between uncertainties.

The joint probability distribution of PV module power and hull resistance is shown in Figure 9. It is found that the ship resistance approximately follows a to normal distribution with the expectation of 2.5 × 10⁴ (N). On the other hand, the distribution of PV module power is more closely aligned with an exponential distribution. The PV module power is less than 20 W in over 50% of cases. Two factors are responsible for this. First, the solar irradiance of the Beishuang area is not high enough, especially in spring and winter. Second, only a small amount of solar energy can be used by the PV module in the early morning and at dusk. The Pearson correlation coefficient of PV module power and hull resistance is 0.00. Therefore, the effect of uncertainties on the PV-HPS should be investigated in terms of PV module power and hull resistance, respectively.

3.3. Optimization Results

A probabilistic simulation is conducted over the Pareto front (PF) solutions obtained by the four optimizations (Table 3) to evaluate the average performance of the solutions in real life and assess the effect of uncertainties on optimization. The four groups of PF solutions are simulated with 1000 time-series stochastic scenarios generated in Step 1 of the proposed optimization method. Those scenarios are assumed to be competent to represent various working conditions of the PV-HPS in its lifespan.

The mean performance of the four optimizations is shown in Figure 10 with solid icons. It is observed that the PF of stochastic optimization, PFsto, is located at the upper right of the PF of deterministic optimization, PFdet, as do the PFs of the two partial stochastic optimizations. However, the PFs of the deterministic optimization and two partial stochastic optimizations move to the upper right of PFsto when the uncertainties in solar irradiance, ambient temperature, significant wave height and wave period are introduced. For example, the final solution of the deterministic optimization x_det produces 5.33 × 10⁵ kg of GHG emissions annually and requires $2.02 × 10⁶ of lifecycle cost. After the probabilistic simulation, the mean value of the objectives (represented by μ_det in Figure 10) moves to 5.74 × 10⁵ kg of GHG emissions (increases by 7.69%) and $2.16 × 10⁶ of lifecycle cost (increases by 6.93%), respectively. This indicates that the ignorance of all or parts of uncertainties leads to overvaluation of PV-HPS performance.

In addition, the PF of partial stochastic optimization of hull resistance, PFres, which takes the uncertainties in significant wave height and wave period into consideration, is closer to the PF of stochastic optimization than its counterparts. Analogously, the average Euclidean distance moved by the solutions in PFres after the probabilistic simulation is also the lowest among the three controls. This demonstrates that the uncertainties in significant wave height and wave period, which relate to hull resistance, are the major factors affecting the optimization results. It is worth noting that the PF of the stochastic optimization almost coincides with its mean performance after the probabilistic simulation. Hence, the capability of the proposed stochastic optimization method in uncertainty characterization can be verified.

Table 3 lists the final solutions of the four optimizations found by the TOPSIS method. The optimal solution given by the proposed stochastic optimization method, x_sto, achieves 5.54 × 10⁵ kg of GHG emissions and $1.97 × 10⁶ of lifecycle cost that are 3.48% and 8.84% lower, respectively, compared with the mean performance μ_det of the deterministic solution, x_det, when uncertainties are considered. According to Table 3, small DGs, motors, and ESSs are used in the four solutions because of the low initial investment. However, the constraint of maximum power of the DG prevents V_D, V_M, and n_ESS from further decreasing. It is observed that the deterministic optimization and the partial stochastic optimization considering hull resistance related uncertainties result in a higher number of PV modules compared with the other two optimizations that take uncertainties in solar irradiance and ambient temperature into consideration. The reason behind this is the overestimate of PV module power by ignoring the uncertainties in solar irradiance and ambient temperature. Consequently, more than necessary PV modules are installed to make the most of solar energy.

It is noteworthy that the proposed method in this paper can be applied to other similar problems. For instance, by adjusting the model parameters, various types of ship models can be constructed. With the proposed method, environmental data of different sea areas can be used to generate stochastic scenarios of different routes. Thus, power systems can be designed for ships with different navigation areas and tasks.

3.4. Sensitivity Analysis of Bin Number

The optimal sizing of the PV-HPS performed in the previous sections shows that the stochastic optimization method proposed in this study can deal with the cross-correlated uncertainties of solar irradiance, ambient temperature, significant wave height, and wave period. However, in the mathematical sense, different bin numbers ($k_{bin}^2$) would express different features of the power source (PV module power) and system load (hull resistance), leading to different optimization solutions accordingly. It is easy to imagine that a larger bin number is likely to offer a greater ability in uncertainty characterization at the expense of higher computational complexity. On the contrary, a smaller bin number reduces the computing effort but may end up with a solution with dissatisfactory performance in real life. Hence, a sensitivity analysis is proposed in this section to investigate and determine the appropriate bin number. In the sensitivity analysis, the stochastic optimization is conducted with different k_bin, namely, different bin numbers ($k_{bin}^2$). Then, the probabilistic simulation is also proposed where the final solutions are simulated with 1000 time-series uncertainty scenarios generated in Step 1 to evaluate their mean performance in real life. The settings used in the sensitivity analysis are listed in

Table 4. Settings and results of the sensitivity analysis.

kbin	Bin Number (${\mathit{k}}_{\mathit{b}\mathit{i}\mathit{n}}^2$)	Pareto Front	Length of Bin Sides		Final Solution	Mean Performance	Difference between Final Solution and Mean Performance
			PV Module Power (W)	Hull Resistance (N)			GHG Emission (%)	Lifecycle Cost (%)
5	25	PF5	22	20,000	x5	μ5	17.02	13.76
10	100	PF10	11	10,000	x10	μ10	6.30	6.46
20	400	PF20	5.5	5000	x20	μ20	1.42	2.29
30	900	PF30	3.67	3333.33	x30	μ30	0.02	0.01
40	1600	PF40	2.75	2500	xsto	μ40	0.01	0.01

.

Figure 11 shows the PFs of the optimizations obtained in the sensitivity analysis. The final solutions of the optimizations found by the TOPSIS method are labeled with solid icons. When k_bin = 5, PF5 is located in the lower left of Figure 11. The PFs move to the upper right as k_bin increases. In addition, the distance between two consecutive PFs decreases at the same time. The PF of k_bin = 30, PF30, almost coincides with that of k_bin = 40 (PF40). Figure 12 compares the final solutions obtained with different k_bin. It is observed that all stochastic optimizations result in solutions outperforming the solutions provided by the deterministic optimization. This highlights the importance of considering the uncertainties in solar irradiance, ambient temperature, significant wave height, and wave period. It is also found that the GHG emissions and lifecycle cost increase with k_bin, whereas the mean performance of the solutions decreases with k_bin when the uncertainty scenarios are introduced.

As can be seen in Table 4, the difference between the final solutions and their mean performance under uncertainty scenarios converges to a small neighborhood of zero as long as k_bin reaches 30, i.e., the lengths of the bin sides are 5.5 kW and 5000 N, respectively. As depicted in Figure 13 and Figure 14, the computational time increases along with the number of non-empty bins as k_bin increases. It takes 3000 s to finish the stochastic optimization with k_bin = 20. When k increases to 40, 7250 s is required by the optimization process. Those results prove that more bins contribute to uncertainty characterization within limits, resulting in better solutions, but leading to a higher computational burden. Taken together, k_bin = 30 (bin number = 900) gains a satisfactory tradeoff between solution optimality and computational complexity. Compared with the solution of the deterministic optimization x_det, the solution x₃₀ achieves 3.37% GHG emission and 8.88% cost savings at the expense of 690 s more computation time, which is acceptable in the design stage of the PV-HPS.

The findings of this study, while focused on an unmanned ship, have broader applicability to other types of vessels. The stochastic optimization method proposed for PV-HPS can be adapted to different vessel types by adjusting the input parameters and constraints specific to each vessel’s operational profile and design characteristics. The methodology’s flexibility in handling various environmental uncertainties and its capability to minimize GHG emissions and lifecycle costs make it suitable for a wide range of maritime applications, including commercial ships, passenger vessels, and naval ships.

4. Conclusions

In this study, a stochastic optimization approach, based on MC simulation, bi-bin scenario reduction technique, and MOPSO algorithm, is proposed for the sizing of the PV-HPS, aiming to minimize GHG emissions and lifecycle cost. To examine the stochastic behavior and synergistic effect of uncertainties in solar irradiance, ambient temperature, significant wave height, and wave period, joint distributions of environmental uncertainty sources associated with PV module power and hull resistance are established. The case study indicates that the proposed stochastic model outperforms deterministic optimization and two partial stochastic optimizations.

Moreover, the trade-off between solution optimality and computational complexity is investigated in the sensitivity analysis of bin numbers. It is observed that a large bin number is beneficial to uncertainty characterization and performance enhancement of the PV-HPS, albeit at the cost of higher computational burden. A satisfactory balance between system performance and computational complexity is achieved with k_bin = 30.

In terms of future work, the uncertainties related to other energy conversion equipment, such as the aging of DGs and batteries, as well as uncertainties in component and fuel prices should be considered to improve the performance evaluation of the PV-HPS over its lifespan. The seasonal characteristics of uncertainties should also be investigated, especially for studies focused on optimal operation. Furthermore, in order to further enhance the environmental and economic benefits of ships, different types of renewable energy resources, such as sails and fuel cells, should be incorporated into the system in addition to the PV module.