In the first stage, various sets of weight governed by (

31) were assigned to the two objectives in (

10) to establish the solution space.

Table 3 lists the cost values of the objective functions with a various sets of weights. The simulation times are also provided in the same table. As a comparison, the non-linear optimization with the degradation model in (

4) with branch and bound method took more than 3 h to complete for a single set of weight. The simulation time depends on the optimization time-resolution, optimization horizon, number of decision variables, and the operating range of the constraints. By comparing the simulation time, it was concluded that the linearization of the ES degradation function can significantly improve the simulation time. It is also worthwhile to mention that day-ahead operation scheduling of the ship power system were generally carried out just a day ahead before the actual operation. This was because the ship power system needed to estimate the propulsion power requirement of the vessel based on the sea-state and payload with the help of the Equation (

8). Forecast data of the solar irradiation was also required to be carried out in the proposed model. For the solar irradiation forecasting, the longer lead-time from the actual operation resulted in higher uncertainty in the prediction and hence making the Pareto front solution space with the longer lead-time to become insignificant or redundant. From

Table 3, it is observed that the largest change in the objectives occurred between

w_{1} = 0.8,

w_{2} = 0.2 and

w_{1} = 0.6,

w_{2} = 0.4. It is the point of interest in the solution space, as there was high fidelity on the trade off between the objectives. From the simulation time as shown in

Table 3, it also reflected the conflicting nature of the objective functions at these sets of weight as the computation time of the optimization significantly increased. The set of weights

w_{1} = 1,

w_{2} = 0 and

w_{1} = 0,

w_{2} = 1 represent the optimization of the single objective function and the cost values obtained from these optimizations are the optimal point of operation to reduced the specific objective in the multi-objective optimization. For an example, if

w_{1} was assigned a value of 1, the minimum cost of emission, fuel consumption, and operation cost and the maximum value of ES degradation were obtained from the above optimization. These sets of weights could be used as benchmarks to evaluate the performance of the remaining solution. From

Table 3, if only objective 1 was optimized, the cost of operation, fuel consumption and emission improved up to 46.93%. This is proven in many literature that are discussed in detailed in the literature review section. On the other hand, the single objective optimization of objective function 2 improved the ES degradation up to 99.95%. However, in reality, the set points

w_{1} = 0,

w_{2} = 1 were almost equivalent to the system operation with no ES and hence the cost of operation could be drastically reduced for the objective function 2. Given these extrema of the objective functions, it becomes clear that there should be a systematic way to analyse the solution space of the Pareto front and identified an optimal operating point of the ship power system to achieve the overall objective of the conflicting objectives in the model.

Figure 6 illustrates the solution space of the optimization. More data points between

w_{1} = 0.8,

w_{2} = 0.2 and

w_{1} = 0.6,

w_{2} = 0.4 are included to map out the solution space. With the assumption that the solar interval prediction was accurate and had similar output characteristics and shapes for the following day, any possible solutions were bounded by this solution space.

Figure 7 illustrates the normalized effect of the solution space against weight

w_{1}. Users could observe the performance of objective 1 and 2, and decide the sets of weight to be used with (

10) in the second stage of dispatch. From the result, the weight

w_{1} between 0.61 to 0.65 resulted in the effects of the objective 1 and 2 having a similar response from each other whereas the weights above these values gave more emphasis on the objective 1 and values below these provided more emphasis on objective 2. The area bounded by the dashed line and solid line in

Figure 7 was the confident bounds due to the interval forecast data of the solar irradiation. It also illustrated the influence of the penetration of the solar PV on the objective functions. The average change in the SOC and energy with the changes in the weight assignments is illustrated in

Figure 8. With greater emphasis to reduce the degradation, the ES was not utilized effectively. If it was not considered in the objective formulation, the degradation cost was even close to the cost of emission and fuel consumption, indicating the over reliance of the ESS. It should be highlighted again that the degradation cost of the ES was large because it considered degradation, charging, and power conversion cost in the total life cycle cost formulation. Furthermore,

Figure 8 also highlights that with the increase in emphasis of

w_{1}, the rate of change of energy for the ES increased up to 16 kWh/5 min which was equivalent to a ramp rate of 192 kW for the optimization. Although charging and discharging of ES are rated higher than the above ramp rate limits in the simulation, such transient behavior of the charging and discharging characteristic of the ES will significantly affect the life of the ES and hence should be carefully addressed.