A Method for Sizing Shipboard ESSs Based on Generator Output Fluctuation Analysis

Abstract

The International Maritime Organization (IMO) has announced regulations that affect many shipbuilding industries and related companies. They require building companies to demonstrate strict compliance with these regulations in construction activities going forward. In response, shipbuilding companies are testing various electrification methods, with the ultimate aim of making ships more eco-friendly. In large ships, in particular, constructors often take a gradual route by hybridizing the propulsion system. In many large cargo ships, the adoption of energy storage systems (ESSs) is expected as part of this transition. In practice, the most frequently operating units inside the ship are the generator engines (GEs). Therefore, this study targets the fluctuation rate characteristics of GEs, providing a more realistic basis for ESS sizing. By focusing on smoothing the GE output, this study determines the ESS capacity required to maintain system stability using a simple moving average (SMA) method and evaluates the fluctuation rate of the GEs under various load conditions.

1. Introduction

Many countries around the world have declared a goal of achieving carbon neutrality or Net-Zero by 2050 as a countermeasure to the climate crisis. This has had a significant influence on a wide range of fields, such as energy, industry, and transportation, both on land and at sea. On land, the centralized transmission grid system was established and operated based on large-scale power plants using fossil fuels. However, solar and wind power generators, which are inverter-based renewable energy sources, are now replacing large-scale fossil fuel plants, and their connection to transmission and distribution grids is rapidly increasing. The transition of the system toward Net-Zero is causing many problems that were not encountered under the existing centralized operation method. Therefore, many studies are underway to operate the changing system effectively and stably.

Similarly, major changes are also taking place at sea. As of 2024, shipbuilding companies in the International Maritime Organization (IMO) member countries (176 in total) are constructing ships in accordance with IMO regulations. The IMO has also created and announced regulations on carbon dioxide emissions in order to reach Net-Zero [1,2,3]. Recently, the IMO approved mid-term measures to reduce greenhouse gas (GHG) emissions at the 83rd Marine Environment Protection Committee. These regulations require ships of 5000 gross tonnage or more engaged in international voyages to meet strengthened GHG intensity standards from the first half of 2027 [4]. If the standards are not met, then ship operators must pay in proportion to their GHG emissions [5,6]. As such, the IMO intends to gradually strengthen regulations to achieve Net-Zero, and shipbuilding companies are actively conducting research to transform ships, starting with systems-level research to reduce GHG emissions. This regulation may be applied not only to newly built ships but also to existing ships. As a result, modifications may be inevitable, such as installing additional equipment within the limited space of existing ships [7].

The internal systems of large ships typically include a main engine (ME) for propulsion, a generator engine (GE), and an auxiliary engine (AE) for electric power. Figure 1 illustrates a roadmap from two-stroke direct propulsion to electric propulsion. Figure 1a illustrates the traditional two-stroke direct propulsion system, in which the ME directly rotates the propeller. In this case, the electric system is not connected to the propulsion system [8]. Figure 1b illustrates a shaft generator (SG)-linked propulsion system in which the SG is installed between the ME and the propeller. Figure 1c illustrates a hybrid type where an SG and a motor are installed between the ME and the propeller. Additionally, SGs are connected to the internal system through a bidirectional converter with an energy storage system (ESS). In this configuration, the SG is additionally coupled to the propulsion shaft, and its output is utilized in the internal electrical system. Finally, Figure 1d illustrates an electric propulsion system driven by a fuel cell instead of MEs [9,10,11].

The target ship type in this paper is an LNG carrier that undertakes ocean-crossing voyages, and its propulsion system corresponds to Figure 1b. While Figure 1b highlights the expected introduction of SGs and ESS as part of future transitions, in practice, GEs are the most frequently utilized power source in large cargo ships. In reality, hybridization may proceed in various forms; some ships may adopt an ESS without SGs, while others may install SGs and an ESS simultaneously. In both cases, the GE and AE remain the essential generating units, which justifies setting GEs as the main target for ESS sizing. Therefore, this study focuses on analyzing the output fluctuations of GEs and proposes a strategy to calculate the ESS output and capacity based on GE data. This approach reflects a realistic pathway for ESS integration, supporting the transition from Figure 1b toward Figure 1c under IMO regulations [12,13,14]. Furthermore, in the analyzed two-year dataset, GE1 or GE2 exceeded a 100 kW output in 743,785 cases, accounting for 71.45% of all records. This confirms that GEs dominate the shipboard power supply, whereas AEs were not included in this statistic.

The internal power system of the target ship is illustrated in Figure 2. Ships are usually referred to as port (left) and starboard (STBD, right) based on the bow, and the layout is generally symmetrical [15,16]. The internal system consists of a total of five generators. A 2500 kW generator driven by a GE is connected to Port 1 and STBD 1, respectively, and a single 2500 kW generator driven by an AE is also connected to both Port 1 and STBD 1. For the SG, 2500 kW units are connected to Port 1 and STBD 1, respectively. In addition, the ship contains cargo load equipment, which covers facilities such as low-duty and high-duty units, along with other auxiliary systems for LNG handling. These are classified as heavy consumers inside the ship, since they are responsible for demanding processes, including LNG management and cargo transfer. For simplicity, they are collectively described as cargo loads. Furthermore, Port 1, Port 2, and STBD 1 are connected to 440 V transformers, which supply power to other equipment and are essential in the daily life of the crew.

The tightening of IMO mid-term measures indicates that stricter regulations on ship emissions and energy efficiency are inevitable in the near future. As a result, modifications to the internal power systems of ships will be inevitable. Under ocean-going conditions, generators are responsible for covering ship loads. However, the variability of load demand leads to fluctuations in the output of generators. Such fluctuations can accelerate the degradation of generators and shorten their lifespan. In order to mitigate this issue, the integration of an ESS is considered. By analyzing the output fluctuation rates of GEs and sizing the ESS output and capacity according to the characteristics of each ship, it becomes possible to effectively smooth the total power profiles. This not only reduces output variability but also contributes to stabilizing shipboard power systems and extending the lifespan of GEs.

In order to address this issue, a method for determining the required output and capacity of a shipboard ESS is presented. Operational data from LNG carriers were analyzed to capture GE fluctuations over several time intervals, including 1, 5, 10, 15, and 60 min. Sudden transitions such as AE on/off were removed, and the cleaned data were studied through histograms and heat maps to obtain a power conversion system (PCS) size for the maximum ESS output. The ESS capacity was then calculated with a sizing equation that considers the capacity rate (C-rate), state of charge (SOC), operating range, and design margin. In the final step, a simple moving average (SMA) method was applied to smooth the GE power profile [17], reducing short-term variations. Overall, this proposed method provides a data-driven approach to ESS sizing and control that supports compliance with IMO regulations and improves the reliability of shipboard power systems.

2. Methodology for GE Output Smoothing and ESS Sizing

The data used in this study comprise the output of each generator with the speed of the target ship. They are recorded in 1-min intervals and cover about two years in total. Because the original data cannot be shared due to confidentiality agreements, a scaled version was used in this study. The scaling was applied only to adjust the absolute magnitude while preserving the temporal patterns and fluctuation characteristics of the original dataset. Therefore, the methodology and results for ESS sizing remain unaffected by the use of scaled data.

The process of sizing a shipboard ESS begins with the specification of the mathematical tools that will be used in the analysis. In practice, the fluctuation rate of GE outputs is calculated first, and the results are applied to reduce short-term fluctuations in the power profile. Through this procedure, operational data are expressed as quantitative indicators, which then serve as the basis for determining the output and capacity of the ESS. For clarity, the following section introduces the mathematical formulations and computational steps applied in the proposed method.

2.1. Fluctuation Metrics and Smoothing Control Framework

2.1.1. Definition of the Fluctuation Rate

The size of the ESS must be determined based on both its maximum output and power capacity. The load of ships is influenced by many factors, such as the sea salinity, weather conditions, and loading capacity, which makes it difficult to estimate the load demand or power generation with limited data. Therefore, in this study, the fluctuation rates ($F_t^{\left(n\right)}$) of GE1 (connected to Port 1) and GE2 (connected to STBD 1) were analyzed using (1) as follows:

$$F_t^{(n)}=\max \left\{\left|S_{t+1}-S_t\right|,\left|S_{t+2}-S_t\right|,\dots ,\left|S_{t+n}-S_t\right|\right\},\hspace{1em}t=1,2,\dots ,T-n$$

(1)

where (1) provides a unified definition of the fluctuation rate. When n = 1, it reduces to the 1-min fluctuation rate, while larger n values yield the n-minute fluctuation rate. S_t represents the GE output at time (t). T denotes the last index of the time-series data.

2.1.2. Fluctuation Rate Binning and Counting Method

The fluctuation rate obtained from (1) is further classified into consecutive 25 kW bins, denoted as B_m. Each data point $F_t^{\left(n\right)}$ is assigned to its corresponding bin, and the number of occurrences is counted as $C_m^{\left(n\right)}$. Collecting these values yields the vector C⁽ⁿ⁾ $= [C_1^{\left(n\right)}, C_2^{\left(n\right)},\dots , C_M^{\left(n\right)}]$. For example, if several fluctuation values fall between 50 and 75 kW, they are included in bin B₃, and the corresponding frequency $C_3^{\left(n\right)}$ increases accordingly. The overall procedure is illustrated in Figure 3.

2.1.3. Capacity Sizing via C-Rate with SOC Range and Design Margin

The C-rate is a parameter widely used to describe the charge and discharge characteristics of an ESS. It is generally defined as the ratio of power to the rated energy capacity and determines how fast the ESS can deliver or absorb energy relative to its capacity, which also means that the higher the C-rate, the faster the ESS response. However, it may place greater stress on the ESS due to the shorter lifespan.

In this study, the C-rate concept is used to determine the required capacity once the PCS output power of the ESS is obtained from the fluctuation analysis. The energy capacity can be calculated as follows:

$$E_{\mathrm{cap}}={\displaystyle \frac{Power}{\mathrm{C-rate}}}\times {\displaystyle \frac{1}{f}}\times {\displaystyle \frac{1}{\eta }}$$

(2)

where the Power unit is kW and the E_cap unit is kWh. Furthermore, the SOC (f) and the design margin (η) were applied when calculating the capacity.

2.1.4. SMA Target and Constrained ESS Deviation Control [18,19]

In this study, the SMA method was selected because it can be implemented in practical shipboard controllers with minimal computational burden. Note that in a practical shipboard system, more sophisticated predictive or optimization-based controllers require accurate forecasting models and a higher processing capacity, which increases the computational burden.

For each generator GEi (i = 1, 2), let the instantaneous power be denoted by P_GEi(k). An SMA is adopted as the smoothing target; the SMA target $\overline{P_{\mathrm{G}\mathrm{E}i}}\left(k\right)$ is defined in (3), which serves as the smoothing reference, as follows:

$$\overline{P_{\mathrm{GE}i}}\left(k\right)={\displaystyle \frac{1}{n_k}}{\displaystyle \sum _{j=0}^{n_k-1}{P}_{\mathrm{GE}i}}\left(k-j\right),n_k=\min \left\{m,k+1\right\}$$

(3)

where k denotes the current sample index, j denotes the lag index that counts past samples in the summation, m represents the window length (the number of samples in the averaging segment), and n_k stands for the number of samples available at time k. Accordingly, at the beginning of the measurement (where k < m), the average is taken over the available samples, whereas after sufficient progress, the average always spans the most recent m samples. In this paper, the value of m was determined based on various units of fluctuation rates.

ESS control follows a deviation-cancelation principle. At each time k, the ESS attached to GEi is commanded to supply the difference between the measured power P_GEi(k) and its SMA target $\overline{P_{\mathrm{G}\mathrm{E}i}}\left(k\right)$ as follows:

$$v_i^{*}\left(k\right)=P_{\mathrm{GE}i}\left(k\right)-\overline{P_{\mathrm{GE}i}}\left(k\right)$$

(4a)

where the subscript i (in v_i, U_i, L_i, E_i) refers specifically to the ESS connected to GEi.

The control set-point is limited by the power rating and SOC constraints through saturation as follows:

$$v_i\left(k\right)=\mathrm{saturate}\left[v_i^{*}\left(k\right),L_i\left(k\right),U_i\left(k\right)\right]$$

(4b)

where $v_i^{*}\left(k\right)$ is compared with the upper and lower bounds, U_i(k) and L_i(k). This saturation operator ensures that if $v_i^{*}\left(k\right)$ lies within the interval [L_i(k), U_i(k)], then v_i(k) is equal to $v_i^{*}\left(k\right)$. If $v_i^{*}\left(k\right)$ exceeds the upper bound, then v_i(k) becomes equal to U_i(k). Likewise, if $v_i^{*}\left(k\right)$ fall below the lower bound, then v_i(k) becomes equal to L_i(k). In this way, the ESS output is saturated within an allowable range, the bounds of which are defined as follows:

$$U_i\left(k\right)=\min \left[P_{i,\max },{\displaystyle \frac{E_i\left(k\right)-E_{i,\min }}{\Delta t}}\right]$$

(4c)

$$L_i\left(k\right)=-\min \left[P_{i,\max },{\displaystyle \frac{E_{i,\max }-E_i\left(k\right)}{\Delta t}}\right]$$

(4d)

where the upper and lower bounds, U_i(k) and L_i(k), are computed from (4c) and (4d), respectively, using the power rating P_i,max, the stored energy E_i(k), the SOC-derived energy bounds E_i,min and E_i,max, and the sampling interval Δt. In this study, SOC was limited to the 10–90% range as a conservative design margin for safe operation, and SOC management was therefore treated as boundary enforcement only, since the primary focus was on sizing. More advanced SOC control strategies may be combined with this framework in future applications.

The resulting smoothed GE output is given by (5a), and the ESS energy is updated according to (5b), as follows:

$$P_{\mathrm{GE}i,\mathrm{sm}}\left(k\right)=P_{\mathrm{GE}i}\left(k\right)-v_i\left(k\right)$$

(5a)

$$E_i\left(k+1\right)=\min \left\{\max \left[E_i\left(k\right)-v_i\left(k\right)\Delta t,E_{i,\min }\right],E_{i,\max }\right\}$$

(5b)

where P_GEi,sm(k) denotes the smoothed GEi output, and E_i(k + 1) is the updated stored energy at time k + 1. In other words, (5a) represents the smoothed GE output after ESS compensation, while (5b) updates the ESS energy according to the charge or discharge during the sampling interval.

In the absence of saturation, the ESS output v_i(k) cancels the deviation between the raw GE output P_GEi(k) and the SMA reference $\overline{P_{\mathrm{G}\mathrm{E}i}}\left(k\right)$. Consequently, the smoothed GE output P_GEi,sm(k), coincides with the SMA target $\overline{P_{\mathrm{G}\mathrm{E}i}}\left(k\right)$, achieving SMA-level suppression of short-term fluctuations.

2.2. Analyzing and Filtering Data

2.2.1. Operating Conditions and Speed

Using the scaled two-year dataset, the ocean-going conditions for calculating ESS output and capacity were extracted. The operation conditions of the target LNG carrier can be classified into six categories, as summarized in

Table 1. Operating conditions of the target LNG carrier.

	Ocean-Going		Port
	Ballast	Loaded	In/Out	Discharging	Loading	Idle
LNG	NBOG + FBOG	NBOG	NBOG	NBOG	NBOG	NBOG
Speed (knot)	10 +	10 +	0 ~ 10	0	0	0

. First, there is an ocean-going ballast condition in which the LNG tank is not completely empty but partially loaded. During this condition, natural boil-off gas (NBOG) is the evaporated gas naturally generated inside the tank due to pressure and heat, and is used as fuel for the generators. In contrast, forced boil-off gas (FBOG) refers to boil-off gas that is deliberately generated from the tank to be used as fuel. All the engines of the ship are dual-fuel engines capable of operating on either diesel or LNG. Depending on whether NBOG or FBOG is utilized, compressors may be operated to supply or process gas for engines and re-liquefaction systems. The ocean-going loaded condition refers to an ocean-going vessel when the LNG tank is fully loaded. Port in/out refers to port entry and departure, port discharging refers to the unloading of LNG, and port loading refers to the loading of LNG. Lastly, idle refers to the condition in which the ship is stationary and waiting at sea. Among these six conditions of operation, ocean-going conditions are characterized by higher sailing speeds, typically above 10 knots, which are faster than port conditions. During the two-year voyage, the maximum speed recorded was 23.71 knots, and the cumulative time at speeds of 10 knots or higher, representing ocean-going operation, amounted to 843,504 min, accounting for 81.03%. Therefore, this study focused on analyzing the GE output fluctuations under ocean-going conditions, as shown in Table 1.

For the purpose of filtering the data used in sizing the ESS output and capacity, ship speeds were filtered based on the following conditions

(i)

Ocean-going operating conditions, defined as periods where the ship speed was maintained at 10 knots or higher for at least 60 min;

(ii)

The exclusion of nearly flat speed segments, defined as cases where the 10-min average slope of the speed was less than or equal to 0.1% of the rated speed;

(iii)

The retention of segments where both GE1 and GE2 were operating simultaneously.

After this filtering process, a total of 85 periods remained, with a total duration of 322,177 min (223 days, 17 h, and 37 min).

2.2.2. Filtering Data by Operating Condition and Speed

Figure 4 illustrates the longest period under conditions where the speed of the ship exceeds 10 knots and either GE1 or GE2 operates continuously for more than 60 min. The period starts at 05:35 on 21 November 2023, and ends at 03:12 on 2 December 2023, lasting a total of 15,697 min (261 h and 37 min). The blue and cyan solid lines represent the output power of GE1 and GE2, respectively. Since GE1 is plotted first and then overlaid by GE2, the GE1 line is often hidden beneath the GE2 line. However, when the actual output of GE1 becomes larger or smaller than that of GE2, the lines intersect, so that the GE1 line can be observed. This indicates that the two GEs follow a broadly similar trend during ocean-going operation. However, they are not identical, which reflects their independent operation. The brown, yellow, and purple dotted lines represent the output power of SG1, SG2, and AE, respectively. The navy solid line represents the total output power of the ship, which is the summation of GE1, GE2, SG1, SG2, and AE. Note that the unit for output power is MW. Finally, the green solid line represents the speed of the ship, given in knots. In particular, during 25–27 November 2023, the total load shows gradual long-term flow patterns in ocean-going conditions. This observation highlights the necessity of applying SMA for effective smoothing, which is verified in Section 3.3.

From the analysis of Figure 4, it can be seen that from about 23 November to just before 27 November, only GE1 and GE2 supplied all the power for the ship. These periods are marked with a blue background. Around midnight on 23 November, AE was additionally operated along with GE1 and GE2 to supply the total power of the ship, and these periods are marked with a purple background. Around midnight on 27 November, SG1 and AE were additionally operated, together with GE1 and GE2, and these periods are marked with a brown background. This analysis shows that in normal ocean-going conditions, while GE1 and GE2 are operating, SGs may also be operated, depending on the operation of heavy consumers. Moreover, it can be observed that the outputs of GE1 and GE2 change rapidly when SGs or AE are switched on or off. This behavior is regarded as noise, since during SGs or AE on/off switching, the average power level itself shifts, producing step-like changes that cannot be effectively smoothed by the ESS. Therefore, these fluctuations are excluded from the fluctuation rate used in sizing the ESS capacity.

The red circles in Figure 4 illustrate the periods in which the outputs of GE1 and GE2 change rapidly. This can be caused by internal issues, events, or the operation of additional devices. When calculating the fluctuation rate for sizing the ESS output, periods such as those marked with red circles were considered noise and filtered out. In this way, a total of 814 periods were obtained after applying the noise exclusion conditions, with a total duration of 286,097 min (198 days, 16 h, and 17 min). In addition to the three conditions described in Section 2.2.1, the two additional conditions were applied to obtain the stable operating region, as follows:

(iv)

Output power from GE with low variation regions was excluded when the 5-min slope of the GE1 or GE2 output was less than or equal to 50% of the respective rated output.

(v)

The noise from the GE output power was removed when the 10-min output range (maximum minus minimum) exceeded 100 kW.

In particular, Figure 5 illustrates the longest duration under the filtered noise condition. This period lasted a total of 5867 min (97 h and 47 min), starting from 05:57 on 14 November 2024, and ending at 07:44 on 18 November 2024.

3. Results

In this section, the analysis results of GE output fluctuations and the corresponding ESS sizing are presented. Firstly, histograms and heat maps of the fluctuation rate are used to determine the required PCS rating. Next, the ESS capacity is determined based on the derived PCS rating, C-rate, and SOC operating range. Finally, the performance of the proposed SMA-based control strategy is demonstrated by applying the output and capacity derived from actual GE operating data. Through this step analysis, the feasibility and effectiveness of the proposed sizing method are validated.

3.1. Fluctuation Rate Analysis for PCS Sizing

A total of 814 filtered periods through Section 2.2.2. are shown as histograms in Figure 6 and Figure 7, which were calculated using (1) and Section 2.1.2. Figure 6 illustrates the output fluctuation rate of GE1. For the 1-min fluctuation rate, most values were in the range of 1–25 kW, with 244,477 counts (85.70%). For the 5-min fluctuation rate, the range of 1–25 kW contained 144,623 counts, the highest proportion (51.13%). For the 10-min fluctuation rate, the largest frequency was in the 26–50 kW range, with 168,484 counts (60.44%). For the 15-min fluctuation rate, the 26–50 kW range was the most frequent, with 174,689 counts (63.59%). Lastly, the 60-min fluctuation rate histogram shows that the 51–75 kW range had the highest frequency, with 99,344 counts (41.73%).

Figure 7 illustrates the output fluctuation rate of GE2. For the 1-min fluctuation rate, the majority of values were in the range of 1–25 kW, with 243,794 counts (85.46%). For the 5-min fluctuation rate, the range of 1–25 kW contained 142,242 counts, the highest proportion (50.29%). For the 10-min fluctuation rate, the largest frequency was in the 26–50 kW range, with 171,854 counts (61.65%). For the 15-min fluctuation rate, the 26–50 kW range was the most frequent, with 177,005 counts (64.44%). Lastly, the 60-min fluctuation rate histogram shows that the 51–75 kW range had the highest frequency, with 100,103 counts (42.05%). Comparing Figure 6 and Figure 7, we can see that the output fluctuation trends of GE1 and GE2 are very similar. Moreover, the fluctuation rates are concentrated in a relatively low range. Along with these fluctuation statistics, supplementary indicators including the standard deviation (σ), root mean square (RMS), and coefficient of variation (CV) were also computed in Appendix A.

For a more detailed analysis, Figure 8 presents Figure 6 and Figure 7 in the form of a heat map. As shown in the highlighted region in the blue box of the figure, the range in which the 1-min output fluctuation rate of GE1 and GE2 covers 100% is under 100 kW. At the same time, the 5-min and 10-min output fluctuation rates of GE1 and GE2 also fall within the 100% coverage range. In the highlighted region, the proportion of the 15-min output fluctuation rate below 100 kW is 99.74% for GE1 and 99.99% for GE2. Similarly, the 60-min fluctuation rate shows 95.93% and 95.86% below 100 kW, respectively. The values for GE1 and GE2 are nearly identical, confirming that both GEs exhibit similar fluctuation characteristics. For all fluctuation units, the ranges under 75 kW and under 100 kW are reported in

Table 2. Summary of fluctuation rate results under 75 kW and 100 kW.

	GEi	1 min	5 min	10 min	15 min	60 min
Under 75 kW (1~75 kW)	GE1	99.97	99.58	98.33	96.47	82.61
	GE2	99.98	99.63	98.47	96.62	82.50
Under 100 kW (1~100 kW)	GE1	100.00	100.00	100.00	99.74	95.93
	GE2	100.00	100.00	100.00	99.99	95.86

. Therefore, the PCS, which determines the maximum ESS output, is set to 100 kW for both ESSs. This rating covers all short-term fluctuations up to 10 min and nearly 100% of 15-min events, ensuring that the ESS rarely operates beyond its rating in ordinary ocean-going operation.

3.2. ESS Capacity-Sizing Results

When the PCS output power of the ESS is 100 kW, the required capacity of the ESS can be calculated according to the C-rate using (2), which expresses the required energy capacity as the PCS output divided by the C-rate and adjusted according to f and η. The values of f and η may differ in practice depending on the ESS used by shipbuilders and related companies. In this study, the value of f was set to 0.8, while the design margin was set at 15%, so that η = 0.85 [20,21,22]. These coefficients (f and η) were applied as conservative design margins, implicitly covering uncertainties such as efficiency variation, temperature influence, and battery degradation without requiring detailed electrochemical modeling.

Table 3. ESS capacity calculation results based on (2) with PCS = 100 kW.

C-Rate	Capacity Considered with f and η (2)	Full-Capacity Discharge Time
0.50C	294.12 kWh	2.00 h
1.00C	147.06 kWh	1.00 h
1.50C	98.04 kWh	0.67 h (=40 min)
2.00C	73.53 kWh	0.50 h (=30 min)

summarizes the results for a PCS rating of 100 kW across representative C-rates (0.50C, 1.00C, 1.50C, 2.00C), showing that higher C-rates require smaller capacities and yield shorter full-discharge times.

Among these cases, the 1C result corresponds to 147.06 kWh with a full-capacity discharge time of about 1 h, and this value was used in the SMA-based control analysis described in Section 2.1.4 as a representative example. The 1C case was selected because it provides a balanced capacity that is large enough to capture the majority of observed fluctuation events while still representing a practical level for real shipboard application. Reference [6] reports an application case in which a vessel with four 2 MW generators (≈8 MW total generation) was equipped with a 450 kWh ESS. Compared to this case in [6], the proposed size of 294.12 kWh (ESS1 + ESS2) is about 65% of that capacity, which can be advantageous in terms of limited installation space, weight, and cost on board. This confirms that the presented result not only falls within the practical range of existing studies but also offers improved feasibility for real shipboard integration. Nevertheless, Table 3 illustrates that a range of practical C-rates can be considered in practice, since lower C-rates provide larger capacities and longer discharge durations, while higher C-rates reduce both capacity and duration. In (3), the window length m was set to 15 min, as shown in Table 2, where 99.74% for GE1 and 99.99% for GE2 of the fluctuations fall within this interval, indicating their stability among the durations considered.

The sensitivity of the ESS sizing results was first examined with respect to the SMA window length m. As given in

Table 4. Sensitivity of ESS sizing to SMA window.

m [Minute]	Fluctuation Reduction [%]	SOC Limit Contact [%]	PCS Saturation [%]
10	58.12	0.18	0.33
15	59.11	0.23	0.50
20	59.41	0.31	0.68

, fluctuation reduction, SOC-limit contact, and PCS saturation are compared for different values of m. The results show that a longer window achieves greater fluctuation reduction, while slightly increasing the occurrence of SOC-limit contacts and PCS saturation. These findings confirm that the choice of m = 15 min provides a practical compromise between smoothing performance and operational constraints.

The sensitivity of the ESS sizing results was also analyzed with respect to the SOC operating range f. As given in

Table 5. Sensitivity of ESS sizing to SOC range.

f	Required Capacity [kWh] (1C)
0.7 (20–90%)	168.07
0.8 (10–90%)	147.06
0.9 (5–95%)	130.72

, the required capacity [kWh] was recalculated from (2) for f = 0.7 (20–90%), f = 0.8 (10–90%), and f = 0.9 (5–95%). The results demonstrate that narrowing the SOC range increases the required capacity, whereas widening the range reduces it. With f = 0.8, the resulting capacity remains consistent with the PCS sizing derived from the fluctuation statistics, thereby providing a reasonable balance between design margin and practical applicability.

3.3. ESS Smoothing Performance

Figure 9a shows the same period as Figure 4, but with ship speed excluded. When applying the SMA method, m was set to 15. Figure 9b is the smoothed graph, obtained by applying a PCS of 100 kW, a C-rate of 1.00 C, and a capacity of 147.06 kWh for both ESS1 and ESS2. Figure 9c shows the ESS output power and SOC.

Comparing Figure 9a,b, the fluctuation of GE1 and GE2 (solid blue and solid cyan lines) and the total power (solid navy line) is clearly reduced. This indicates that the SMA-based control, combined with the proposed ESS sizing, successfully mitigates the rapid fluctuations that are observed in the raw GE output. These results also indicate that the proposed SMA-based control can be applied to each generator in a modular manner, making the approach scalable for the parallel operation of multiple generators with similar fluctuation characteristics. For the AE (dotted purple line), the fluctuation remains unchanged because the ESS is not applied. In Figure 9c, the output fluctuation of the ESS decreases after the AE is operated around midnight on 27 November. This is because the fluctuation of GE1 and GE2 decreases when the AE is in operation. In Figure 9c, the dynamic behavior of the ESS is illustrated; the dark green line represents the SOC of ESS1, and the light green line represents the SOC of ESS2. The dark gray dotted line represents the output power of ESS1, and the light gray dotted line represents the output of ESS2. In Figure 9b,c, the ESS output decreases after the operation of AE, and at the same time, the fluctuation of the SOC is reduced. It is also confirmed that the ESS outputs always stayed within the 100 kW PCS rating and the SOC range of 10–90%, validating the practical feasibility of the proposed design.

4. Conclusions

In this paper, a method to determine the maximum output and capacity of a shipboard ESS was proposed using generator data from an LNG carrier. The PCS output was derived from the fluctuation rate of the generators, and the ESS capacity was calculated considering the C-rate, SOC, and design margins. For GE1 and GE2, applying the SMA method with 100 kW of PCS and 147.06 kWh capacity effectively stabilized shipboard power by reducing short-term fluctuations during ocean-going conditions. The SMA window was selected as a practical compromise, and the ESS output remained within the PCS and SOC limits, confirming the feasibility of the sizing approach under realistic constraints. Although demonstrated on an LNG carrier, the methodology can also be extended to other ship types and operating conditions.

Consistent with previous studies, the results suggest that ESSs can reduce generator wear, improve fuel efficiency, and contribute to IMO compliance by mitigating rapid load changes. The derived ESS capacity of 294.12 kWh is relatively compact, indicating manageable impacts on space and weight, though cost analysis was beyond the scope of this work. Conservative design margins were also applied, implicitly accounting for efficiency variation, thermal effects, and long-term battery degradation. Overall, the proposed approach provides a practical and adaptable framework for shipboard ESS sizing and integration.

Future work will incorporate the explicit modeling of these factors to further refine the proposed methodology. In particular, a more detailed co-tuning of the window length with actual ESS dynamics and shipboard controller response is left to be explored. The analysis will also be extended to compare the SMA-based approach with alternative smoothing methods and advanced ESS sizing approaches, to verify their relative performance. Future research will further include validation across multiple ship types and the exploration of real-time control implementation to confirm the practical applicability of the methodology.