Joint Optimization of Configuration Design and Energy Management Strategy for a Fuel Cell/Supercapacitor Rubber Tire Gantry Crane

Abstract

A fuel cell (FC)/supercapacitor (SC) hybrid powertrain is proposed for rubber tire gantry (RTG) cranes, aiming to address their characteristics of high peak/low average power demand and huge potential energy recovery. Unlike conventional design methods that neglect the coupling effects of energy management strategies (EMSs), this paper adopts a joint optimization (JO) for the powertrain parameters’ design. Parameters are preliminarily sized based on routine container handling tasks, then refined via a dynamic programming (DP)-based EMS for secondary optimization to minimize the total crane operation costs that cover hydrogen consumption as well as FC degradation. Iterations of the optimization process continue until targets are met. The results indicate that the JO framework achieves dual energy-economic goals, exhibiting a 57.33% enhancement in fuel economy compared to diesel-powered cranes through port validation while concurrently decreasing the SC’s capacity redundancy by 12.7%. These findings aid FC/SC RTG crane configuration design in ports. Additionally, the theoretical optimal operation cost obtained by the DP-based EMS can be used as a benchmark for evaluating other EMSs.

1. Introduction

The growth of maritime logistics, handling over 85% of global cargo, has intensified port emissions. Shipping has become the third largest air pollution source in China [1], motivating vigorous green transition policies under the “dual carbon” goal [2]. Land-side port operations, particularly container handling, contribute substantially to atmospheric pollutant emissions [3]. Notably, diesel-powered rubber tire gantry (DPRTG) cranes are primary emission sources in container operations, with a rated power of more than 400 kW and 14 L/h fuel consumption, yielding the production of 40 kg/h CO₂ [4]. Such conventional cranes can no longer meet modern energy-saving and low-carbon requirements. Moreover, DPRTG cranes suffer from extremely low energy recovery efficiency—merely 15~20% of potential regenerative energy can be recycled during container lowering—resulting in massive energy waste [5].

Various approaches are intended to cut fuel consumption of DPRTG cranes. One cost-efficient solution is to hybridize the powertrain system by utilizing energy storage systems such as supercapacitors (SCs), batteries, or flywheels, either individually or in combination [6,7,8,9]. However, these powertrain systems are typically oversized relative to actual operational power demands: many port cranes operate at only 30~50% of their rated power, increasing energy costs by 20~30% due to excessive energy consumption and idle losses. Given the standardized nature of container handling, in-depth investigation into configuration parameters of RTG cranes is essential to improve resource utilization. Meanwhile, energy management strategies (EMSs) are inherently coupled with powertrain component sizing, a relationship widely recognized in marine propulsion and hybrid electric vehicles [10,11,12], offering a synergistic pathway for the optimal design of RTG systems.

The above statement highlights the following questions needing to be addressed to achieve high efficiency and economic performance of port RTG cranes:

• What powertrain configuration is suitable for RTG cranes requiring efficient energy recovery?
• How to optimize the size of the powertrain components for fixed-operation RTG cranes to balance energy efficiency and cost?
• How to employ EMSs to maximize energy efficiency of RTG cranes and collaboratively optimize configuration parameters?

For RTG cranes, frequent start-stop, substantial regenerative energy, and drastic instantaneous peak power impose stringent performance requirements on the energy storage system. Although lithium-ion batteries are widely used in vehicles and some researchers have investigated their application in RTG cranes [6], they suffer from rapid lifespan degradation under frequent charge–discharge cycles and carry potential risks of thermal runaway after low-temperature cycling [13]. In contrast, SCs feature high power density (>1 kW/kg), long cycle life (>500,000 cycles), good low-temperature performance (−40~+50 °C), and high safety, making them more suitable than lithium-ion batteries to be the energy storage system of RTG cranes [14]. Some researchers have proposed a hybrid powertrain combining a diesel generator (DG) and SC energy storage system [15]. However, it is worth noting that this measure cannot reduce carbon emissions directly. The reduction in port emissions requires a multifaceted approach, including the adoption of cleaner fuel sources and optimization of operational practices. Therefore, this paper proposes a novel fuel cell (FC)/SC hybrid powertrain RTG crane (FCRTG crane) which can achieve near-zero emissions while meeting high-power operation requirements.

Owing to hydrogen’s zero-emission attribute and swift refueling capabilities, fuel cell systems (FCSs) have been applied to heavy-duty equipment such as vehicles and trains [16,17,18]. Since RTG cranes operate along fixed routes and impose low requirements for hydrogen infrastructure, FCs represent a golden solution for the green transition of port cranes. Supported by government subsidies, the capital cost of FCRTG cranes in China has become competitive with that of conventional DPRTG cranes. In 2022, Qingdao Port launched research on hydrogen-powered cranes [19]. As an industry pioneer in hydrogen ports, it also constructed China’s first port hydrogen fueling station for port logistics. Globally, integrated ports in the United States, Japan, and Europe are being transformed into hydrogen ports [20]. However, FCs alone cannot satisfy transient high-power demands during startup and lifting, nor can they recover regenerative potential energy during descending and braking [21]. These limitations necessitate a hybrid powertrain system featuring SCs for auxiliary energy supply to achieve high energy density and rapid dynamic response. Given the stable load profiles and fixed operating patterns of RTG cranes, refined powertrain component sizing is critical to avoiding redundant resource allocation and control costs. The high initial investment remains a major barrier to FC/SC hybrid powertrain applications. Thus, configuration optimization is indispensable.

The configuration optimization of powertrain systems is classified into two categories: single and joint optimization (JO). Single optimization only focuses on component parameters and ignores the coupling effect with EMSs [22]. However, the energy-saving performance of a hybrid powertrain system is a multivariate optimization issue closely related to EMSs, operational conditions, and component capacity. JO, while theoretically comprehensive, faces limitations including high-dimensional search space, slow algorithmic convergence, and compromised solution accuracy. To address these challenges, this paper focuses on the hierarchical JO of configuration parameter design and EMS for the powertrain system of FCRTG cranes, aiming to minimize energy consumption and carbon emissions.

EMSs rationally allocate power from multiple energy sources to adapt to fluctuating power demands, effectively reducing fuel consumption. EMSs have been extensively studied in hybrid vehicles [23,24,25,26], rail coaches [27,28], tractors [29,30,31], and power grids [32,33,34]. Common EMSs include rule-based (deterministic rules, fuzzy rules) and optimization-based strategies (model predictive control (MPC), genetic algorithms (GAs), dynamic programming (DP), and Pontryagin’s minimum principle (PMP)).

Table 1. Research on EMSs and configuration design in the literature.

Equipment	Authors	EMSs	Configuration	JO
Vehicles	Ferrara et al. [10]	DP	FC+battery	√√
	Shen et al. [11]	Fuzzy control	Battery+SC	×
	Chen et al. [12]	DP+GAs	FC+battery	√
	Jia et al. [24]	Twin delayed deterministic EMS	FC+battery	×
	Tang et al. [25]	Rule-DDPG method	FC+battery	√
	Song et al. [35]	MPC	FC+battery	√
	Gharibeh et al. [36]	A bi-level online EMS	FC+battery	√
	Lü et al. [37]	DP+MPC	Gasoline+battery	√
Rail coaches	Sipra et al. [27]	Fuzzy logic	Solar PV–battery	×
	Xu et al. [28]	Electric–thermal collaborative	FC+heat pump	√
Tractors	Varlese et al. [29]	A predictive EMS	FC+battery	×
	Radrizzani et al. [30]	MPC	ICE	×
Power grid	Haffaf and Lakdja [32] Wang et al. [33]	A multi-stage optimization EMS	PV–battery–ICE	√
		Fuzzy neural networks and a modified PSO	PV–battery–turbine	×
RTG cranes	Kusakaka et al. [7]	A rule-based EMS	DG+battery	√
	Takalani and Masisi [8]	Filtering and PMP	Battery+SC+grid	×
	Corral-Vega et al. [15]	Voltage control	DG+SC	√
	Takalani and Masisi [38]	A rule-based EMS	Battery+SC+grid	×
	Chen et al. [39]	Game theory	DG+SC+battery	×
	Alasali et al. [40]	GAs	Grid+storage system	×
	Lin et al. [41]	PSO and tabu search	Electric-powered	×

Note: √√ (Deeper study); √ (Simple study); × (No consideration).

summarizes existing research on EMSs and configuration design of different equipment from the literature, with their characteristics.

As observed in Table 1, while extensive explorations of EMSs target vehicles, rail coaches, and tractors, relevant research on EMSs for RTG cranes integrated with FC and SC remains absent. Notably, RTG cranes differ fundamentally from road vehicles, rail coaches and tractors: they operate under fixed, cyclic working conditions with fewer interfering factors, whereas vehicles are subject to complex and variable road and traffic scenarios [42]. Furthermore, benefiting from their ultra-high load capacity, RTG cranes can store and release considerably more potential energy during lifting operations than conventional transportation equipment, which endows EMSs with outstanding energy-saving potential in such hoisting machinery. Specifically, existing EMS studies on hybrid RTG cranes mainly focus on optimizing diesel engine performance and partially adopting supercapacitors for potential energy recovery. In view of the promising energy-saving prospects of cranes, several scholars have begun to explore EMSs for various hybrid RTG crane configurations. To the authors’ knowledge, however, the overall research on EMSs for hybrid cranes is still very limited. Takalani and Masisi [38] developed a rule-based EMS for distributing output power among the battery, supercapacitor, and grid. Their novel hybrid energy storage system created a decrease in energy consumption and peak power demand for the crane operation. Alasali [43] proposed a deterministic optimal EMS controller and MPC strategy as a feasible solution to minimize the electric expenses linked to dynamic power tariffs and mitigate peak power demand under predefined powertrain parameters and grid constraints.

Nevertheless, most existing EMS studies fail to incorporate JO, as summarized in the last column of Table 1. At the same time, although several studies have discussed the influence of EMSs on crane energy efficiency, the mutual coupling between system configuration design and EMSs has been largely neglected. Accordingly, the research domain of joint optimization focusing on such coupling effects has not been explored so far. In terms of FC/SC hybrid crane systems, preliminary explorations have only been conducted in the broader field of green-energy-driven hoisting equipment. Against this background, this paper proposes a joint optimization framework for FCRTG cranes, with a core focus on the coupling mechanism between configuration design and EMSs.

In the current studies on powertrain system design and EMSs of hybrid RTG cranes, there still exist some gaps requiring urgent attention:

• Although investigations have been conducted on battery-SC hybrid cranes, the application potential of FCSs remains insufficiently explored, which hinders the development of zero-emission port logistics. Most existing FC-related studies mainly focus merely on vehicles, without considering the high-power output and cyclic operating characteristics unique to RTG cranes.
• Integrated research that combines practical operational requirements with EMS optimization for RTG powertrain parameter matching is still lacking. The existing parameter design approaches mostly adopt oversimplified principles or directly transplant methodologies from vehicle research, thereby failing to achieve optimal resource utilization and overall system efficiency.
• The majority of currently developed EMSs for RTG cranes rely on rule-based strategies, which cannot guarantee global optimization performance. Moreover, standardized and benchmark EMS solutions are still absent for RTG cranes operating under fixed-cycle working modes.

As a widely employed global optimization method, DP is adopted to distribute energy among different power sources based on predictive operating cycles. Despite its high demand for computing capacity, DP works well for precomputed analysis including comparing various powertrain architectures, assessing online EMSs, and formulating guidelines for RTG cranes. Under a particular operating cycle, the theoretical minimum value of energy consumption can be derived via DP for port RTG cranes with a determined system structure and steady-state environment. Consequently, DP is employed to address the energy management issues of RTG cranes. The obtained theoretical optimal solutions can serve as benchmarks for other EMSs and be used to guide the optimization of the design parameters of the crane system.

In summary, this paper proposes a JO strategy to address the configuration parameter design of the new powertrain system for RTG cranes. The main contributions of this paper include the following:

• A dedicated FC/SC architecture for hydrogen-powered RTG cranes is presented, which can store regenerated potential energy and deliver instant kinetic power during acceleration.
• A DP-based EMS in FCRTG cranes with fixed-cycle heavy lifting scenarios is explored, which establishes a benchmarks for other EMSs.
• The JO framework achieves dual energy-economic objectives, demonstrating through port validation a 57.33% fuel economy boost while reducing capacity redundancy of the SC by 12.7%.

The subsequent parts of this paper are arranged in the following order. The problem statement and solution framework are detailed in Section 2. Section 3 establishes the powertrain model of the FCRTG crane. The principles and implementation of the DP are explained in Section 4. Four instances are computed in Section 5, along with relevant results comparison and discussion. Finally, the research findings are summarized in Section 6.

2. Problem Statement and Solution Framework

2.1. Problem Statement

In traditional situations, RTG cranes are powered to finish duty cycles by a diesel generator set including a diesel engine coupled with an alternator. The electricity undergoes rectification and is then distributed via a DC network to all motors powering the elevators, carts, and movement systems of the RTG crane. As the container unloads, the electric motor produces electricity and feeds it back into the DC network. Without an energy storage device for recovery, the surplus energy is forced to be dissipated in the resistor bank. However, with adequate energy storage, the excess energy can be retained, retrieved, and utilized during subsequent operations, leading to substantial energy savings, especially considering that RTG cranes often handle hundreds of containers daily. During normal operations, data collected from an RTG crane equipped with suitable instruments can help assess the efficiency of current motor energy consumption and required power under different cycles so that an EMS strategy can be explored.

The notations and their corresponding meanings in this paper are given in

Table 2. The notations and their meanings in this paper.

Notations	Meanings	Notations	Meanings
Pfc	power of the FC	ηfc	the FC efficiency
Psc	power of the SC	ηsc	the SC efficiency
ηfcdc	efficiency of DC/DC converters connected to the FC	ηscdc	efficiency of DC/DC converters connected to the SC
Pt	total power demand of the RTG crane	Csc	capacitance of the SC
$\overline{P}$	average power demand	$m_{H_2}$	hydrogen fuel consumption
Cop	operation cost of the RTG crane	$C_{H_2}$	equivalent hydrogen consumption cost
Cfc	degradation cost of the FCS	∆Vfc	rated voltage drop of a single FCS
$p_{{\mathrm{H}}_2}$	unit price (CNY/kg) of hydrogen	$LH{V}_{H_2}$	hydrogen lower heating value
αon-off	start-stop degradation rate of the FCS	αshift	load-shift degradation rate of the FCS
αlow	low-power degradation rate of the FCS	αhigh	high-power degradation rate of the FCS
Ncycle	number of start-stop cycles of the FCS	Nshift	number of load changes of the FCS
Tlow	duration of low-power operation of the FCS	Thigh	duration of high-power operation of the FCS
SOC	state of charge of the SC	SOC0	initial SOC
Pfcmax	maximum power of the FC	Pfcmin	minimum power of the FC
Pscmax	maximum power of the SC	Pscmin	minimum power of the SC
SOCmax	maximum SOC	SOCmin	minimum SOC
Umax	maximum voltage of the SC	Umin	minimum voltage of the SC
k	discrete moment	Isc	current of the SC
Usc,rate	rated voltage of the SC	Rsc	resistance of the SC
Pup	uplink power	Tup	uplink time
Pdown	downlink power	Tdown	downlink time
Esc	stored energy of the SC	Ereg	regenerative energy
ηreg	percentage of regenerative energy	mrated	container weight at rated load
Echarge	energy absorbed charging from U to Umax	Echarge	energy released discharging from U to Umin

.

A conventional RTG crane completes 20 operating cycles in 1 h. The load characteristics of the RTG crane reveal that the ratio of peak power to average power is close to 10:1, making its powertrain system highly suitable for a hybrid power solution. Therefore, it is necessary to redesign the RTG crane powertrain system by combining carbon emissions and energy-saving goals. This paper aims to minimize the operation costs by a designed powertrain system. Unlike common simple machines, RTG cranes have separate motors responsible for the travel of the trolley and gantry, as well as the lifting and lowering of the containers. When the trolley or gantry motors brake or the hoist motor descends, the SC can store energy. Therefore, the FC and SC jointly supply power to satisfy the power demands of the RTG crane as follows:

$$P_t={\eta }_{fc}{\eta }_{fcdc}{P}_{fc}+{\eta }_{sc}{\eta }_{scdc}{P}_{sc}.$$

(1)

For the preliminary design of the configuration parameters, the capacities of the FC (P_fc) and SC (C_sc) are selected as the decision variables which are to meet the power demands (P_t) of crane operations. The power of the FC P_fc and the capacitance of the SC C_sc satisfy the following constraints:

$$\left\{\begin{array}{l}{P}_{fc}\ge {\displaystyle \frac{\overline{P}}{{\eta }_{fc}}}\\ C_{sc}\ge {\displaystyle \frac{2E_{sc}}{\left(U_{\max }^2-U_{\min }^2\right)}}\end{array}\right..$$

(2)

Simply meeting the aforementioned requirements is insufficient, because the operation cost achieved thereby is not at an optimal level, necessitating secondary optimization via EMSs. In the formulation of the EMS, the main focus is on optimizing the power allocation of the two energy sources to obtain the optimal operation cost. At each time step, the power of the SC P_sc is chosen as the decision variable to achieve power distribution, while the state of charge (SOC) of the SC is selected as the state variable. Additionally, the operation cost C_op is chosen to be the objective function. The output power of the FC and SC are restricted by their inherent capacity limits. Furthermore, the SOC is confined between its initial state and the lower bound. The corresponding constraints are concluded as below:

$$\left(\begin{array}{l}{P}_{fc\min }\le P_{fc}\le P_{fc\max }\\ P_{sc\min }\le P_{sc}\le P_{sc\max }\\ {\mathrm{SOC}}_{\min }\le \mathrm{SOC}\le {\mathrm{SOC}}_{\max }\end{array}\right..$$

(3)

The total operation cost encompasses the hydrogen consumption cost and the FC degradation cost, as formulated below:

$$J=\min C_{op}=\min \left(C_{H_2}+C_{fc}\right)=\min {\displaystyle \sum _{k=1}^{N}L\left(\mathrm{SOC}\left(k\right),P_{sc}\left(k\right)\right)}.$$

(4)

The configuration design of the powertrain system and the specific steps of the DP-based EMS are introduced in detail in Section 3 and Section 4. To achieve the dual goals of the JO, the following assumptions are established:

• The FC operates at an ideal thermal condition and a fixed moisture level, so the model excludes the air cooler and humidifier.
• The FC and SC in the powertrain are actually composed of multiple units. For the convenience of modeling, they are both treated as an independent structure in this paper.
• The design of powertrain system parameters is aimed at RTG cranes operating typical loads of containers. If the load changes, the parameters need to be recalculated.

2.2. Solution Framework

A JO of a powertrain system and EMS for the configuration of parameter designs of RTG cranes is proposed. The designed capacity parameters are supposed to minimize operation cost by the DP-based EMS while meeting the power demands of RTG cranes. Considering the inputs, outputs, operating mechanisms, and constraints of the problem, an IDEF0 chart is created in Figure 1. As shown in Figure 1, the optimal result of an EMS is one highly sensitive to the given configuration parameters. On the other hand, to optimize capacity planning and eliminate resource redundancy, the impact of the given EMS on system configuration design must be carefully evaluated by considering operation cost and power allocation.

First, a configuration of the crane powertrain is designed, and models of key components are established. Then, the preliminary capacity parameters are tailored to meet the power demands of the RTG crane. Next, a DP-based EMS is utilized to tackle the JO problem of the configuration design, accounting for the operation cost. In this stage, operation costs quantify the economic benefit of the EMS. Finally, it is essential to verify whether the designed configuration parameters can fulfill the power demands of the RTG crane in the port. If the power demands of the RTG crane cannot be met, the parameters must be updated, even if this ends up compromising the energy-economic goals to some extent. The power demands of the RTG crane comprise the total power demands along with the instantaneous power demand, especially the highest power demand required during startup or acceleration. In the initial design of powertrain parameters, in addition to considering the power demands of the RTG crane, the FC- and SC-related technical specifications were also taken into account. Of course, these values are relatively large and will be further optimized by the EMS to realize cost reduction and lower carbon emissions. Meanwhile, in the process of EMS secondary optimization, the optimal powertrain size with minimal operational cost is sought to avoid capacity redundancy and reduce powertrain size. The main objective of the secondary optimization is operational cost, but the optimized parameters may not meet the high instantaneous power demands and need to be iteratively optimized again. For example, if the parameters of the powertrain system are too small to meet power demands, which perform well in the EMS, this may prevent the RTG crane from working properly. Although the powertrain system has lower operational cost, because of this, it will still be abandoned during the second optimization process.

3. Powertrain System Design

The schematic layout and powertrain system of the FCRTG crane in this paper are designed as illustrated in Figure 2. Figure 2a provides a lateral view drawing corresponding to the practical structure layout of the RTG crane. The FC and SC are integrated and linked to the DC bus via the DC/DC converters, respectively, as shown in Figure 2b. The DC/DC converters serve to regulate the voltage level of the electrical energy flowing from the FC and SC to match the requirements of the DC bus system. To achieve this control, two PI controllers are employed in a cascade configuration. In the yard of the port, it is imperative that the RTG crane remains operational at all times. Consequently, even when the SC is completely discharged, the RTG crane must be capable of handling the container. Based on this requirement, the FC has been designed to produce sufficient power and charge the SC when conditions permit.

3.1. Fuel Cell

Among various types of FCs, proton exchange membrane FCs (PEMFCs) represent a favorable choice for transport applications owing to their high power density, specific power output, and relatively low operating temperature [44], which collectively enhance efficiency and enable rapid system startup. When applied in port scenarios, the FC delivers higher energy efficiency than traditional internal combustion engines throughout the entire energy conversion process [45]. Since the hydrogen tank of the RTG crane can only hold a limited volume of hydrogen, hydrogen consumption is a crucial assessment metric. The following equation quantifies the transient hydrogen consumption rate [46]:

$${\dot{m}}_{H_2}={\displaystyle \frac{P_{fc}}{{\eta }_{fc}LH{V}_{H_2}}},$$

(5)

where $LH{V}_{H_2}=120\mathrm{MJ}/\mathrm{kg}$ is the lower heating value of hydrogen. Therefore, total hydrogen consumption is determined by integrating its consumption rate over time, calculated as outlined below:

$$m_{H_2}={\displaystyle {\int }_0^T\frac{P_{fc}}{{\eta }_{fc}LH{V}_{H_2}}}dt.$$

(6)

Thus, the hydrogen consumption cost during the operating cycle of the RTG crane can be computed as follows:

$$C_{H_2}=p_{H_2}\cdot m_{H_2}.$$

(7)

Start-stop cycles, variable load transitions, as well as low-power and high-power working states are four typical operating conditions contributing to FC degradation [47]. The established FCS degradation model uses the corresponding degradation rates to faithfully emulate the practical aging progression. The mathematical formula for describing this degradation mechanism is expressed as:

$$\Delta V_{fc}={\alpha }_{on\text{-}off}{N}_{cycle}+{\alpha }_{shift}{N}_{shift}+{\alpha }_{low}{\displaystyle \frac{T_{low}}{3600}}+{\alpha }_{high}{\displaystyle \frac{T_{high}}{3600}},$$

(8)

where ∆V_fc denotes the rated voltage drop of a single FC. α_on-off, α_shift, α_low, and α_high represent the degradation rates corresponding to the above four operating conditions, respectively. N_cycle and N_shift stand for the number of start-stop cycles and load changes, respectively. T_low and T_high represent the durations of low-power and high-power operations, respectively. The relevant degradation rate parameters [48] are summarized in

Table 3. Degradation rates under different operating conditions in the FCS.

Operating Conditions	Decay Rates	Operating Conditions	Decay Rates
αon-off	13.79 μV/cycle	αlow	10 μV/h
αshift	0.0441 μV/kW	αhigh	8.66 μV/h

. The FCS degradation cost can be computed as:

$$C_{fc}={\displaystyle {\int }_0^t\frac{\Delta V_{fc}}{0.1}}\cdot {\displaystyle \frac{P_{fc}}{1000}}\cdot p_{fcs}dt.$$

(9)

3.2. Supercapacitor

As the energy storage component, SC consists of series and parallel connections of single cells. The SC is modeled using an equivalent circuit, similar to that of a battery model, and its key parameters can be expressed as follows [49]:

$$\left\{\begin{array}{l}\mathrm{SOC}={\mathrm{SOC}}_0-{\displaystyle \frac{{\displaystyle \int I_{sc}dt}}{C_{sc}{U}_{sc,rate}}}\\ I_{sc}={\displaystyle \frac{U_{sc}-\sqrt{U_{sc}^2-4R_{sc}{P}_{sc}}}{2R_{sc}}}\end{array}\right..$$

(10)

The fluctuating power demands of RTG cranes lead to continuous changes in the output of the FC and SC. Since the SOC correlates with the real-time power of the SC, its rate is defined as follows:

$$\stackrel{·}{\mathrm{SOC}}=\left\{\begin{array}{l}-{\displaystyle \frac{P_{sc}{\eta }_{sc}}{C_{sc}{U}_{sc}}}, \mathrm{for} \mathrm{charging} (P_{sc}<0)\\ -{\displaystyle \frac{P_{sc}}{{\eta }_{sc}{C}_{sc}{U}_{sc}}}, \mathrm{for} \mathrm{discharging} (P_{sc}\ge 0)\end{array}\right..$$

(11)

In line with the charging and discharging modes of the SC, equivalent hydrogen consumption is discussed in two cases as follows:

$$m_{H_2}=\left\{\begin{array}{l}\left({\displaystyle \frac{P_{fc}}{{\eta }_{fc}}}+P_{sc}{\eta }_{sc}\right){\displaystyle \frac{\Delta t}{LH{V}_{H_2}}}, \mathrm{for} \mathrm{charging} (P_{sc}<0)\\ \left({\displaystyle \frac{P_{fc}}{{\eta }_{fc}}}+{\displaystyle \frac{P_{sc}}{{\eta }_{sc}}}\right){\displaystyle \frac{\Delta t}{LH{V}_{H_2}}}, \mathrm{for} \mathrm{discharging} (P_{sc}\ge 0)\end{array}\right..$$

(12)

3.3. DC/DC Converters

The terminal voltages of the FC and SC are contingent upon the power exchanged between the two energy units and the DC bus voltage. Consequently, DC/DC converters are necessary to adjust the terminal voltages of the two units to match the DC bus and manage their delivered power.

A unidirectional boost DC/DC converter [50], typically composed of a fully controlled device Q, a power diode D, an energy storage inductor L, and a filter capacitor C, is employed for the FC connecting to the DC bus. The internal circuit structure is shown in Figure 3a. Let the switching period of Q be T_s. In each cycle, its on-time is T_on and off-time T_off = T_s − T_on. The duty cycle of drive signal u_Q is T_on/T_s. During a switching cycle, the working process of the converter is divided into two states, and the corresponding equivalent circuit diagrams are shown in Figure A1 of Appendix A. As shown in Figure A1a, while the switch Q is conductive during 0∼T_on, the diode D bears reverse voltage and remains off. At this moment, U_fc, L, and Q form a closed loop, and L stores energy. As shown in Figure A1b, Q turns off during T_on∼T_s. The inductor current i_L freewheels through the conducting diode D to form the loop. A new cycle starts at t = T_s. The voltage gain of the boost DC/DC converter depends on the duty cycle, denoted as T_s/(T_s − T_on). The output voltage can be regulated by controlling the duty cycle of Q. The voltage and current waveforms are presented in Figure A2 (T_on/T_s = 40%) through simulation in Simulink. Under steady-state operation, all physical quantities are closely related to the duty cycle. The waveform of output voltage clearly illustrates the gain characteristic of the boost converter.

On the other hand, a bidirectional full-bridge DC/DC converter is employed for the SC to facilitate power transfer both from the SC to the DC bus and vice versa. The circuit of the converter is shown in Figure 3b, which consists of four MOSFETs and four diodes. Within one switching cycle, the converter undergoes four states: Q1 and Q4 turned on, all MOSFETs turned off, Q2 and Q3 turned on, and all MOSFETs turned off again, as illustrated in Figure A3. The simulated voltage and current waveforms are presented in Figure A4. For the high-frequency transformer, the two terminals of its primary winding are connected to the midpoints of the left bridge leg (Q1, Q2) and the right bridge leg (Q3, Q4), respectively. The output voltage can be regulated by adjusting the duty cycle of the four MOSFETs. State (a), corresponding to the interval t₀~t₁ and shown in Figure A3a: Q1 and Q4 receive drive signals and turn on. The input voltage is applied across the primary winding. Based on the dot polarity of the transformer windings, D1 and D4 are forward-biased and conduct. The inductor current i_L flows through the secondary winding, diodes D1 and D4, output filter capacitor C, and load resistor R, and rises linearly during this period. State (b), corresponding to the interval t₁~t₂ (Figure A3b): All four MOSFETs Q1–Q4 are switched off, and the primary winding current drops to zero. The inductor freewheels through the four diodes D1 and D4, D2 and D3, with each diode carrying half of the inductor current. The inductor current decreases linearly. State (c), corresponding to the interval t₂~t₃ (Figure A3c): Q2 and Q3 are triggered and turned on. The input voltage is applied across the primary winding again. According to the winding dot convention, D2 and D3 become forward-biased and conduct. The inductor current passes through the secondary winding, D2 and D3, filter capacitor C, and load R, rising linearly once more. State (d), corresponding to the interval t₃~t₄ (Figure A3d), exhibits identical operating characteristics to State (b). The bidirectional converter can control the SC’s output and minimize fluctuations in the DC bus voltage, thereby enhancing system stability. Among all isolated DC/DC converters using power devices with identical voltage and current ratings, the full-bridge topology achieves the highest power rating. Therefore, it is widely adopted in medium–high-power industrial power supplies.

3.4. Initial Design Optimization

Given the known cyclic operating conditions of the RTG crane, the power of the FC should meet the demand for average power, which is calculated as follows [51]:

$$\overline{P}=\max \left({\displaystyle \frac{{\displaystyle \int P_{up}dt}}{T_{up}}},{\displaystyle \frac{{\displaystyle \int P_{down}dt}}{T_{down}}}\right).$$

(13)

In the hybrid powertrain system, the output power of the SC must satisfy the maximum power demands during crane acceleration and container lifting, while its energy storage capacity should match the maximum recoverable energy during container lowering. Considering extreme operational scenarios, the SC energy configuration should be engineered to accommodate the maximum regenerative energy recoverable from the hybrid powertrain system during rated load conditions:

$$E_{reg}={\eta }_{reg}{m}_{rated}gh.$$

(14)

During the charge–discharge process of the SC, energy is expressed as follows:

$$\left\{\begin{array}{l}{E}_{ch\arg e}={\displaystyle \frac{1}{2}}{C}_{sc}\left|U^2-U_{\max }^2\right|\\ E_{disch\arg e}={\displaystyle \frac{1}{2}}{C}_{sc}\left|U^2-U_{\min }^2\right|\end{array}\right..$$

(15)

In this paper, U_max = 660 V and U_min = 350 V, since SCs can obtain good charging and discharging effects when operating between 350 V and 660 V, which are determined by [52] and Maxwell technical manuals. A 40-foot container with a lifting height of 18.1 m and a rated load of 30.48 t is considered. In addition to considering the power demands of the RTG crane, the FC- and SC-related technical specifications were also taken into account.

Table 4. The initial values of primary parameters of the FCRTG crane.

Component	Parameter	Value
FC	Pfcmax	80 kW
	Pfcmin	0
	ηfc	0.51~0.57
SC	Pscmax	300 kW
	Pscmin	−90 kW
	Csc	63 F
	ηsc	0.97
	SOCmax	0.9
	SOCmin	0.3
DC/DC converter	ηfcdc/ηscdc	0.93

lists the initial design values of primary parameters of the FCRTG crane. Of course, these values are relatively large and will be further optimized by the EMS to meet the goals of reducing costs and carbon emissions. The hybrid system integrates the commercial brand Maxwell SC with a bidirectional DC/DC converter. During the entire operating cycle (EOC), the container undergoes hoisting, movement, lowering, and unloading at the designated position, culminating in the crane trolley being lifted, shifted, and then lowered back to its idle stance.

4. A DP-Based EMS

In this section, a DP-based EMS for secondary optimization of the RTG crane configuration design is proposed. As a globally recognized optimization algorithm, DP handles multi-source energy distribution efficiently within a specific operating cycle. In this paper, the DP-based EMS is established to find the theoretically best control solution for minimal operation costs and compare the performance of different configuration parameter values for the FCRTG crane. As a comprehensive analysis of DP applications for heavy-duty trucks has been presented in our prior research [3], the principle of DP is not elaborated in this paper.

The operating procedures for the DP-based EMS are as follows:

• Split the overall multi-stage decision task into multiple independent single-stage subtasks.
• The SOC of the SC is selected as the state variable and then discretized. The state transition formula is derived from the SC model and presented below:

$$\mathrm{SoC}(k+1)=\left\{\begin{array}{l}\mathrm{SoC}(k)-{\displaystyle \frac{P_{sc}(k)\Delta t}{{\eta }_{sc}{C}_{sc}{U}_{sc}}}, \mathrm{if} P_{sc}\ge 0\\ \mathrm{SoC}(k)-{\displaystyle \frac{P_{sc}(k){\eta }_{sc}\Delta t}{C_{sc}{U}_{sc}}}, \mathrm{if} P_{sc}<0\end{array}\right..$$

(16)

3.

Select the SC power Psc as the decision variable.

4.

Establish the operation cost as the objective function. All states across each phase come with a certain operation cost, which is computed by Equation (4). Within the interval between time step k to k + 1, C_op takes both equivalent hydrogen consumption and degradation cost of the FCS into account. Here, L stands for the transition cost function:

$$L=C_{op}=\left\{\begin{array}{l}\left({\displaystyle \frac{P_{fc}\left(k\right)}{{\eta }_{fc}}}+P_{sc}(k){\eta }_{sc}\right){\displaystyle \frac{\Delta t}{LH{V}_{H_2}}}\cdot p_{{\mathrm{H}}_2}+{\displaystyle \frac{\Delta V_{fc}}{0.1}}\cdot {\displaystyle \frac{P_{fc}}{1000}}\cdot p_{fcs}, P_{sc}<0\\ \left({\displaystyle \frac{P_{fc}\left(k\right)}{{\eta }_{fc}}}+{\displaystyle \frac{P_{sc}(k)}{{\eta }_{sc}}}\right){\displaystyle \frac{\Delta t}{LH{V}_{H_2}}}\cdot p_{{\mathrm{H}}_2}+{\displaystyle \frac{\Delta V_{fc}}{0.1}}\cdot {\displaystyle \frac{P_{fc}}{1000}}\cdot p_{fcs}, P_{sc}\ge 0\end{array}\right..$$

(17)

5.

Set up the boundary conditions as shown in Equation (3).

6.

Iterate to derive the results.

To sum up, the flowchart of DP is presented in Figure 4. N*S is the total number of nodes, corresponding to time samples and selected states, respectively. In this paper, the time interval is set to 1 s and S is 6001, which means that the state variable is recursively calculated between 0.3 and 0.9 at 0.0001.

5. Computational Study

5.1. Design of Experiments

To comprehensively evaluate parameter design and EMSs, three operationally representative cycles are considered in the experiment design. The three operational cycles are an EOC [52] (this reference discusses design, modeling, and control of hybrid cranes in sequence, while ignoring the impact of EMS on power system design and not considering the JO of configuration design and EMS) and two representative work cycles, loading and unloading. Four instances grounded on these three cycles are executed to showcase the advantages of the FC/SC hybrid powertrain, explore the coupling effect between key parameters and EMSs, and perform the JO on configuration parameters of the powertrain system. The attributes of those four instances are listed in

Table 5. The attributes of the four instances.

Instances	Operating Cycles	Purpose
1	EOC	Demonstrate superiority of DP and FC/SC powertrain
2	EOC	Perform the JO on configuration parameters
3	Loading/unloading	Verify the applicability of the new FC/SC powertrain system
4	EOC	Investigate the operation cost in various working scenarios

. A comparative study is conducted with three optimization methods. DP, GAs, and a rule-based EMS are employed to address parameter design and energy management of the FCRTG crane.

Table 4 presents the primary parameters of the RTG crane for instance 1. In instance 2, the influence of RTG crane configuration parameters on the performance characteristics of different EMSs is systematically investigated. To this end, a series of RTG cranes with different parameters are experimentally evaluated using both DP and GA methods. This instance is grounded on the EOC, and

Table 6. Multiple types of configuration parameter values for FCRTG cranes in instance 2.

Number	Pfcmax/kW	Csc/F	Pscmax/kW
P1	60	55	260
P2	80	55	260
P3	60	63	300
P4	60	63	260
P5	80	63	300
P6	40	43	200

outlines several key RTG crane parameters. The results of this instance can be used for secondary optimization of the configuration parameters. The novel hybrid powertrain system of the FCRTG crane, which has been secondary optimized, is used in instance 3 to verify its applicability. A series of load profiles are configured in instance 4 for analyzing the performance of the FCRTG crane under heavy loads. As stated in Section 2, the load variations are directly discernible through changes in power demands, with elevated loads resulting in proportionally higher power demands. In this instance, increases of 5% and 10% respectively are applied to the original power demands to evaluate the performance of the EMSs under high-load scenarios.

Table 7. Different heavy-load working scenarios in instance 4.

Number	Percentage Increase/%	Ptotal/kW
H1	/	1387.86
H2	5	1457.25
H3	10	1526.65

provides detailed information regarding these adjustments.

5.2. Analysis of Computational Results

In this paper, all algorithms were coded in Matlab R2022a and run on a PC with 16 GB memory and AMD R7 5800H processor with Radeon Graphics 3.20 GHz. Ten runs were conducted for each operation and instance to obtain average values.

The results obtained by DP under the EOC, alongside the findings that employ GAs and rule-based strategy [52] respectively, are presented in

Table 8. Fuel consumption and operational cost of different strategies.

EMSs	DP	GA	Rule-Based	DPRTG
Fuel consumption	0.0775 kg	0.0807 kg	0.085 kg	0.83 L
Equivalent hydrogen cost (CNY)	2.33	2.42	2.55	6.89
Degradation cost of the FCS (CNY)	0.61	0.69	0.81	/
Total operation cost (CNY)	2.94	3.11	3.36	6.89

Note: The best results are highlighted in bold.

, and indicate that DP reduces operational cost by 5.47% and 12.50% compared to GAs and rule-based strategy, respectively. Specifically, DP reduces hydrogen consumption by 3.97% and 8.82% compared to GAs and rule-based strategy, respectively. DP can achieve the theoretical minimum hydrogen consumption and can be used to evaluate the effectiveness of different EMSs. At the same time, since the FCS operates at a relatively high power with fewer load shifts under DP, its degradation cost is lower than that of the other two methods. In addition, the costs of FCRTG and DPRTG cranes are compared. Currently, the hydrogen price stands at CNY 30/kg in China, resulting in a cost of CNY 2.33 for the RTG crane using DP and CNY 2.42 for cranes employing GAs. In contrast, a DPRTG crane requires 0.83 L to complete the EOC, costing CNY 5.98 given the current diesel price of CNY 8.3/L in China. Evidently, FCRTG cranes offer not only zero emissions but also cost savings of over half compared to DPRTG cranes. Although the integration of FCs and SCs leads to a higher initial investment cost, the costs associated with hydrogen production, storage, and transportation are decreasing as FCs gain traction in various countries. An optimized hydrogen energy supply chain can be established through precise planning of the spatial distribution of hydrogen sources and demand sides, and integrating multiple transportation methods including liquid hydrogen tanker trucks, organic hydrogen storage systems, and upgraded pipeline networks [53]. It is particularly important to emphasize that with the implementation of policies and the increase in the penetration rate of the FC market, the overall economy of the hydrogen energy industry chain will experience a qualitative leap in long-distance and large-scale application scenarios. Practical cases from Qingdao Port and Shanghai Port in China demonstrate that the construction costs of hydrogen energy infrastructure are being amortized as hydrogen refueling stations expand in scale [54]. It is anticipated that the retail price of hydrogen will stabilize below CNY 20 per kilogram, further widening the cost advantages of FCRTG cranes. Additionally, hydrogen refueling stations, crucial for the commercialization of hydrogen energy transportation, have undergone rapid development in recent times. China has constructed over 400 hydrogen stations, positioning itself as the global leader. Consequently, when considering the environment, cost, and convenience, the FCRTG cranes proposed in this paper are highly likely to emerge as the ultimate solution for future container handling and loading/unloading in ports. The findings reveal obvious advantages of DP and GAs over the rule-based EMSs. Therefore, subsequent analysis is dedicated to these two algorithms for additional verification of secondary optimization.

Figure 5 depicts the SOC variation in the SC based on DP and GAs during the EOC. Analogously, simulated data further confirm the prominent superiority of DP against GAs. The curve in Figure 5 indicates a 1.84% decline in the SC SOC within 1 cycle by GA. Notably, the SC SOC returns to its initial value of 0.9 after 88 s by DP due to continuous decline and braking. The SC alternates between persistent charge and discharge by GAs, while the SC primarily discharges with sporadic brief charging periods by DP. This suggests that DP could potentially prolong the SC’s lifespan due to reduced charging and discharging cycles. Furthermore, when the container descends (40~52 s), the enormous gravitational potential energy rapidly charges the SC, providing power for subsequent operations and further reducing hydrogen consumption.

The operational cost and variation in the SOC of the FCRTG crane across various parameters are illustrated in Figure 6. The operational cost under P6 when using a DP-based EMS is the lowest among all scenarios, attributed to the smallest values of P_fcmax and C_sc, according to the results in Figure 6a. However, the configuration cannot meet the power demands (as discussed in Section 3.4, the gap from the maximum instantaneous power is approximately 40 kW under EOC conditions), so this optimized design is not feasible. Thus, the values of P1 are chosen for the subsequent instances, resulting in suboptimal operation cost. Additionally, it is evident that variations in P_scmax do not influence operation cost. As the frequency of start-stops increases, a noticeable upward trend in operation cost is observed. Figure 6b depicts the SOC variation curves by DP and GA. In identical conditions, the DP-based SOC fluctuations are comparatively smaller than those observed with GAs. Notably, the SOC under P5 exhibits the highest stability, potentially attributed to the values of P_fcmax and C_sc. Conversely, the curves corresponding to P3 and P4 align consistently. Figure 6b underscores that higher values of P_fcmax and C_sc contribute positively to maintaining SOC stability, while P_scmax has no discernible influence on the SOC. This study prioritizes capacity configuration optimization, with the impact of SOC being deemed negligible. The initial design parameters are sufficient to cover the power demands of the RTG crane, thereby reducing the parameter specifications as P1 to avoid capacity redundancy and resource waste.

The parameter values in P1 are applied in the following instances. The results show that the hybrid powertrain system designed with secondary optimization is suitable for typical loading/unloading operations. From Figure 7, operation costs are 0.0501 and 0.0521 under loading and unloading operations respectively, decreased by 5.11% and 7.25% compared with GAs in one cycle. To a certain degree, merely comparing the performance of these strategies within a single cycle lacks significance. Hence, the method was cycled to encompass a 1 h operational task with 36 cycles. The results showed reductions of 3.84% and 5.50% for loading and unloading operations, respectively. Additionally, it is worth noting that DP performs better in unloading operations compared to loading ones, due to the higher power demands. Consequently, EMSs based on DP are well-suited for the tasks of heavy-duty port RTG cranes, owing to their broader operational range and duration.

A similar upward trend is also evident in Figure 8a, with an increase in power demand. Specifically, the operation cost of the RTG crane using GAs experiences a substantial surge of approximately 5.83% or more, whereas the increase for DP is only 3.99%. These observations highlight that in scenarios involving heavy loads, such as in ports, the advantages of using DP are further emphasized. The variation in the SOC based on DP and GAs is shown in Figure 8b. The results indicate the lower power demands correlate with reduced changes in the SOC, as expected.

Therefore, in order to minimize operation cost under the volume limitations of the hydrogen tank, reducing P_fcmax is feasible while satisfying the power demands. Similarly, the SC capacity configuration can also adopt the same strategy, where P_scmax only needs to satisfy the transient peak power demands during startup or acceleration. This configuration optimization can reduce the volume of the powertrain system and avoid resource waste. However, the prerequisite is that the power demands must be met; for example, the values in P6 cannot make the RTG crane operate normally.

6. Conclusions

This paper introduces a novel FC/SC hybrid powertrain system and a JO design integrated with a DP-based EMS strategy for RTG cranes in ports. The system is designed with the objectives of reducing carbon emission and minimizing operational cost, including equivalent hydrogen consumption and the FCS degradation cost, thereby addressing critical challenges in decarbonizing port equipment. The methodology employs a JO: preliminary parameterization based on standard handling cycles and a DP-based EMS targeting dual energy-economic optimization. An entire operating cycle and two representative operations were considered for verification. This study establishes a pioneering framework for hydrogen-powered port equipment. The core conclusions are summarized below:

• The JO framework achieves dual energy-economic objectives, demonstrating through port validation a 57.33% fuel economy boost while reducing capacity redundancy of the SC by 12.7%.
• As the first implementation of DP in fixed and heavy-load cycles of port cranes, this study provides optimal benchmark solutions for evaluating other EMSs. DP reduces operational costs by 5.47% and 12.50% compared to GAs and rule-based strategies, respectively, in the EOC.
• In the process of the JO design, smaller values of P_fcmax and C_sc can reduce operation cost and prevent resource waste while P_scmax only needs to satisfy the transient peak power demands during startup or acceleration. These findings aid FCRTG crane configuration design in ports.

For further research, the focal points will be centered on the following aspects:

• A comprehensive mission profile and degradation of FCs will be systematically identified to ensure a thorough evaluation of the designed powertrain system for RTG cranes, yielding a solution with pan-scenario adaptability and balanced flexibility and efficiency performance.
• DP is more suitable for offline computing and is limited in real-time EMS applications. Thus, a computationally efficient and feasible EMS should be developed to minimize operation costs for RTG cranes serving port container handling.