# Analysis of Hybrid Ship Machinery System with Proton Exchange Membrane Fuel Cells and Battery Pack
## 4 Developed Numerical Model  
In Figure 2, the final model with all components of the system is shown. The first part consists of the PEMFC stack, which was modelled as a quasi-dimensional model with its numerical background explained in the previous chapter. An equivalent circuit model for the battery pack was used, and the model consists of a controlled voltage source with Ohmic resistance, which is used to describe instantaneous voltage response to current input. Additionally, a thermal model called \"a solid wall\" was implemented to predict the transient thermal behavior of the battery. Moreover, multiple battery elements were connected via electrical pins to construct an electrical network acting as a battery module or battery pack. Open-circuit voltage is defined for classic Li-ion batteries [27], as in Figure 3. Ohmic resistance was taken at the temperature of 25 \({}^{\circ}\)C and defined as in Figure 4.  
Figure 3: Voltage distribution of battery pack for different states of charge. Figure 2: Main components of developed numerical model in AVL CRUISETM M 2023.1. Additionally, the maximal current of charging is defined for a given SoC, which is shown in Figure 5, where the maximal charging current of the battery system is shown with a blue line, and the maximal current of the PEMFC stack is shown with an orange line (500 A). Many combinations of battery packages have been implemented, and the optimal battery package configuration consists of 110 battery packages connected in parallel with 130 batteries connected in series (total 14,300 batteries), with each battery having a voltage range from 2.5 to 4.2 V, where nominal voltage is 3.7 V, and a capacity of 8 Ah. Discharge voltage is defined as in Figure 3. The total calculated combined power of the battery pack is 424 kWh with state of charge (SoC) at 100%. Ohmic losses between each battery for this simulation were neglected, as the main goal was to examine the required power from auxiliary storage to satisfy the required condition. There are two electric motors (EM-s) that are the same, defined as asynchronous engines, with a demanded maximum power of 300 kW each or 600 kW in total. They have a defined torque for rotation speed, which is shown in Figure 6 for a full spectrum of load. For this simulation, the spectrum of load has constant torque, as it is from 0 to 370 rpm. Figure 4: Ohmic resistance distribution of battery pack for different states of charge. Figure 5: Maximal speed of charging for battery system dependent on SoC. Propellers that are connected to EM-s are defined as classic B series propellers by [22], with a diameter of 1.5 m, four blades, a pitch to diameter ratio of 1.2, and a disk area coefficient of 0.7. While on full load advance, the coefficient is 0.73. The marine vessel model, which is powered by a fuel cell and a battery, is a sophisticated system with multiple interlinked components. The vessel was modeled as a longitudinal dynamic model and designed to calculate various key parameters related to a vessel's forward and backward movement. Initially, the model defined parameters such as the ship's mass (80 t), thrust force, external forces, resistance force, initial speed, and thrust deduction factor. With those parameters, the numerical model could calculate ship acceleration with consideration of the net forces acting on it, including both the thrust and resistance forces, then dividing them by the ship's mass. The ship's velocity was then updated by integrating acceleration over time, starting from an initial velocity value. Furthermore, the position of the ship is determined by integrating velocity over time. The ship's resistance was determined from experimentally validated results.