Joint Energy Scheduling for Isolated Islands Considering Low-Density Periods of Renewable Energy Production

Abstract

In view of the special dispatching demands of isolated islands in low-density periods of renewable energy power generation, the defects of the traditional dispatching mode when applied to isolated power generation systems are analyzed, and the idea of reasonably extending the daily scheduling cycle is proposed to adapt to the application of flexible energy resources in the form of energy packages under various uncertain scenarios. Under the multi-party cooperative power supply strategy for isolated islands, we analyze the shortcomings of key element modeling. A global optimal model of energy scheduling for isolated islands considering low-density energy output periods is constructed based on a refined element model, and a corresponding solution is proposed for the nonlinear constraints. The reasonability and effectiveness of the refined model, the global optimal model, and the assumption of an extended scheduling cycle are verified by theoretical analysis and case simulation.

1. Introduction

With the increasing awareness of maritime rights in the international community, island management has become an important development direction for coastal countries. In the field of island development and energy supply, detailed scheme designs have been carried out, which can be mainly divided into two aspects: the independent power supply design of a single island and the joint energy development of island groups. In fact, island renewable energy is abundant. For single islands with development conditions, most consider building an independent multi-energy complementary hybrid power supply system directly, including wind power generation, photovoltaic power generation, ocean energy generation (such as tidal energy, wave energy, ocean current energy, etc.), and energy storage batteries [1,2].

To ensure power supply reliability, it is pointed out that offshore islands can be connected to the mainland power grid through submarine cables and directly participate in the energy dispatch of the large power grid [3]. In [4], the use of fiber-optic composite submarine cables is proposed to realize the direct connection between the mainland and offshore islands, ensuring the power supply and communication capabilities of the islands. However, the submarine environment is complex and cable maintenance is difficult, making it difficult to ensure the reliability of the power supply [5]. For offshore islands, the cost of laying cables is extremely high, and the feasibility of this scheme is low due to environmental factors. Therefore, for isolated islands without direct connection to the power grid, it is generally necessary to configure diesel generators in the system to improve the dynamic response characteristics and disturbance resistance of the system [6]. In [7], the authors propose the renovation of traditional diesel power generation systems in isolated island microgrids and coordination with various distributed power sources, such as wind, photovoltaic, and storage in island microgrids. Ref. [8] proposes an energy management scheme for islanded microgrids, which enhances the reliability of the system through fair profit distribution.

Therefore, for isolated islands, the current best solution to ensure power supply reliability is to arrange a traditional stable diesel or gas generator. However, the above literature does not fully consider the relationship between the power supply cost and operating status of the traditional unit, and considers the generation cost and operating power of the diesel generator set to be a simple linear relationship. Due to the overly optimistic fuel consumption model for diesel generator sets, although it can encourage power generators to participate in energy supply to gain profits to a certain extent, it is more likely to result in operators making improper decisions and even suffering losses. In addition, the fuel consumption costs of a diesel generator set during start-up and shutdown need to be considered in the model.

In terms of joint energy development for island groups, due to the relatively small area of a single island and the relative lack of land resources, it is difficult to achieve energy self-sufficiency. For this reason, one idea is to achieve coordination and energy support between island power grids by scheduling an Energy Exchange Vessel (EEV) [9]. An EEV is a dual-purpose vessel with the ability to flexibly load and unload batteries that can move freely between islands for battery swapping and transportation. It can facilitate the energy packaging and transportation of resource-rich islands and load center islands. Ref. [10] suggests that when a submarine cable is damaged, the island microgrid can use an electric passenger ship as an emergency EEV for post-disaster energy dispatch. Ref. [11] presents a scheduling strategy for electric ships based on spatiotemporal dynamics to reduce energy costs and improve renewable energy consumption rates.

The above studies carried out in-depth research on the energy supply model for isolated islands and island groups. The supply mode of energy transfer between islands using an EEV exploits the energy supply capacity of resource-rich islands. However, the work ignored the specific issues of energy transfer and rational utilization of energy packages in resource-rich islands during low-density generation of renewable energy. Renewable energy generation on islands has significant seasonality and intermittency, resulting in limited energy delivery to the load center island. In this case, such islands need to participate in the energy scheduling of the island group in a better way.

In fact, the electricity produced by resource-rich islands is returned in the form of energy packages, which provide a flexible and schedulable resource. However, it is not optimal for the energy packages to be directly included in the energy balance. The load center island diesel generator set has nonlinear efficiency characteristics and high start-up and shutdown costs. If the output of the energy packages can be reasonably matched with the diesel generator set according to the actual needs of the load, the diesel generator set will work in high-efficiency conditions most of the time. This is ideal from the perspectives of both operational economy and environmental protection. However, the net load curve formed by the load characteristics and the real-time output of renewable energy from an island with a direct cable connection may be extremely volatile. At this time, if scheduling is still carried out on a single-day scale, and the energy accumulated by resource-rich islands in the low-energy production period is not sufficient to effectively correct the net load curve within the day, it is still difficult to avoid the situation where diesel generator sets work for a long time under inefficient conditions.

In view of the above problems, a new approach to isolated island development and energy supply is to vigorously build clean power sources such as wind and photovoltaic, while ensuring the reliability of power supply within isolated systems for diesel and gas generators. At the same time, the original diesel generator set will gradually transform from a base load power source to a backup power source. This can ensure power supply reliability while developing renewable energy on the island.

Previous studies on optimization scheduling of island microgrids are summarized in

Table 1. Comparative literature on island microgrid optimization.

Research Work	Horizon (h)	Alignment	Uncertainty	Finding
[12,13,14]	36	misaligned	Deterministic (no error)	Higher total cost vs. 48 h
[15,16,17]	48	aligned	Time-correlated prediction errors	Lowest expected cost
[13,18,19]	60	misaligned	Deterministic (no error)	Higher total cost vs. 48 h
[13,20,21]	72	aligned	Time-correlated prediction errors	Higher total cost vs. 48 h
This study	48	aligned	Time-correlated errors + reserve	Recommended horizon in this study

, highlighting different scheduling horizons, uncertainty considerations, and main findings.

In summary, this study aims to enhance the energy management strategy of isolated island groups by enhancing both the accuracy of physical modeling and the scheduling optimization horizon. The nonlinear fuel consumption characteristics of diesel generators are first modeled in detail to accurately capture their operational efficiency and cost under varying load conditions. Based on this refined model, a global optimization framework is developed using mixed-integer linear programming (MILP) to coordinate power generation, storage, and transportation across multiple islands. Furthermore, the proposed framework examines the effects of extending the scheduling cycle beyond the conventional daily scale, showing that a longer scheduling horizon allows for better utilization of renewable resources and more stable operation of diesel generators. Simulation results quantitatively verify that the extended scheduling period, built upon the accurate efficiency model, can effectively reduce total system cost and improve overall energy stability.

The main contributions of this paper are summarized as follows:

• Reflecting the nonlinear characteristics of the diesel generator fuel consumption model;
• Achieving global optimization through mixed-integer linear programming (MILP);
• Quantitatively verifying the economic benefits of scheduling cycle extension based on simulation analysis.

The rest of this paper is organized as follows. Section 2 provides the refined consumption equation for diesel generator sets. Section 3 details the model constraints and solution methods. Section 4 introduces two scheduling strategies and evaluates the economic advantages of extending the scheduling cycle. Simulation results are presented in Section 5, and conclusions are summarized in Section 6.

2. Energy Supply System for Isolated Islands

2.1. Refined Modeling of Diesel Generator Set

In fact, in the feasible working range, the power generation efficiency of the diesel generator set is not constant, but closely related to the operating power. The power generation efficiency of the diesel generator set is a nonlinear function of actual output, and there is a start–stop cost. Its refined consumption equation [22,23,24] is shown in Equations (1) and (2):

$$C_{DGT}=C_{DG}+C_{\mathrm{on}}+C_{\mathrm{off}}$$

(1)

$$C_{DG}=a\cdot P_{DG}^2+b\cdot P_{DG}+c$$

(2)

where $C_{DGT}$ denotes the total power generation cost of diesel generator sets, including operation cost $C_{DG}$, start-up cost $C_{\mathrm{on}}$, and shutdown cost $C_{\mathrm{off}}$; $P_{DG}$ denotes the generating power of the diesel generator sets; a, b, and c denote the quadratic coefficient, linear coefficient, and constant term of the running cost function, respectively.

Further, the unit price of the power generation cost for diesel generator sets can be calculated by the following:

$$p_{DG}={\displaystyle \frac{C_{DG}}{P_{DG}\cdot t}}=a\cdot P_{DG}+b+{\displaystyle \frac{c}{P_{DG}}}$$

(3)

From Equation (3), it can be seen that the unit price of power generation cost $p_{DG}$ of diesel generator sets varies with the change in power generation $P_{DG}$, and the unit price of power generation cost of diesel generator sets is actually a cross-over function of power generation. Since $a$, $b$, $c$ are all known normal numbers, when $P_{DG}=\sqrt{c/a}$, the unit price $p_{DG}$ of the generation cost of the diesel generator set is the minimum value of $2\sqrt{ac}+b$.

Generally, the optimal efficiency power point $\sqrt{c/a}$ of diesel generator sets is far less than the rated power of diesel generator sets. Therefore, in order to maximize profits for diesel generators while keeping the total power generation unchanged, it is necessary to keep the diesel generators operating within a reasonable range as much as possible and reduce the start–stop times. Otherwise, the effective income of diesel generator sets cannot be accurately guaranteed.

2.2. Isolated Island Group Multi-Party Cooperative Power Supply Model

For isolated island group systems, there are several types of energy suppliers: (1) The wired resource-rich islands can provide power to the load center islands through direct connection cables. (2) The wireless resource-rich islands first store the renewable energy that cannot be consumed in batteries, and then use idle multi-functional ships to support the power demand of the load center islands in the form of battery energy packages. (3) The load center islands are equipped with conventional diesel units to ensure a reliable power supply. The cooperative power supply model is shown in Figure 1.

In order to fully utilize various power generation resources and achieve the lowest power supply cost for isolated island group systems, this paper proposes a new model for energy joint scheduling for isolated island groups through multi-party cooperation in power supply. In this model, wireless resource-rich islands, wired resource-rich islands, and diesel generator sets form an alliance. With the goal of minimizing the total operating cost of the energy supply system for isolated island groups, the model optimizes ship routes, charging/discharging costs, and the operating power of diesel generator sets.

Due to the installation of cables, the electricity generated by the wired resource-rich island can be transmitted back to the load center island in real-time without additional costs. To fully utilize the cables as infrastructure, the load center island should prioritize the consumption of electricity produced by the wired resource-rich island. Therefore, the curve to be optimized during scheduling is the load curve formed by the superposition of the load center island’s load curve and the wired resource-rich island’s output prediction curve.

Based on the above model, the following sections will present the construction and solution of the optimal energy scheduling model for isolated islands.

3. Optimal Energy Scheduling Model for Isolated Islands

To facilitate understanding of the subsequent model formulation, the main notations used throughout this paper are summarized in

Table 2. Notation Table.

Symbol	Unit	Description
$\mathsf{\Delta }t$	h	Time step
$P_{DG}(t,i)$	kW	Power of the i-th diesel generator at hour t
$P_B(t)$	kW	Battery power at hour t
$P_L(t)$	kW	Net load at the load center island at hour t
$P_{DG\cdot \min }(i),P_{DG\cdot \max }(i)$	kW	Minimum/maximum power of the i-th diesel generator at hour t
$u_k(t)$	{0, 1}	battery-swapping selection variable on island k at hour t
$E_{bk}(t)$	kWh	Remaining stored energy on island k at hour t
${E^{\prime }}_{bk}(t)$	kWh	Auxiliary variable in linearization
$\mathsf{\Delta }{E}_{bk}(t)$	kWh	Hourly energy change on island k
$E_{\min },E_{\max }$	kWh	Lower/upper bounds of stored energy on island k
$N_{vk}(t)$	-	Number of vessels required by wireless island k
$E_{v\cdot \max }$	kWh	Max energy that one vessel can transport per voyage
$T_{vj}$	h	Travel time of route j
$d_{vj}$	km	Sailing distance of route j
$v_{ves}$	Km/h	Vessel speed
$C_{on}(t,i),C_{off}(t,i)$	CNY	Start-up/shutdown costs of generator i at hour t
$r_{on}(t,i),r_{off}(t,i)$	{0, 1}	Start/stop indicators for generator i

.

3.1. The Objective Function

Based on the above discussion, the goal of energy scheduling for isolated island groups should be to minimize the daily power supply cost of the load center island by scheduling the output of diesel generators on the load center island, the production of wireless resource-rich island energy packages, and the travel of multifunctional ships. If the scheduling step size is 1 h, the objective function of energy optimal scheduling for isolated island groups can be presented as follows:

$$\min C={\displaystyle \sum _{t=1}^T\left\{{\displaystyle \sum _{i=1}^m\begin{array}{r}\left[C_{DG}(t,i)+C_{on}(t,i)+C_{off}(t,i)\right]\\ +{\displaystyle \sum _{j=1}^n{C}_{ves}(t,j)+{\displaystyle \sum _{k=1}^{n+1}{C}_B(t,k)}}\end{array}}\right\}}$$

(4)

where $T$ denotes the prospective period of scheduling; $m$, $n$ denote the number of diesel generator sets on the load center island and the number of wireless resource-rich islands, respectively. $k$ is the island number. For instance, when $k=1$, it indicates that the island is the load center island; when $k=2,3,\dots ,n+1$, it indicates that the island is No. k resource-rich island. $t$, $i$ and $j$ denote the tth hour, the ith diesel generator set, and the jth wireless resource-rich island, respectively; $C$ denotes the total power supply cost of the load center island during the dispatching period. This parameter covers various resources consumed to ensure electricity consumption, including direct fuel consumption, battery consumption, and ship resources to be dispatched for transferring batteries; $C_{DG}(t,i)$, $C_{on}(t,i)$, $C_{off}(t,i)$ denote the operation cost, start-up cost, and shutdown cost of the ith diesel generator set in the tth hour, respectively; $C_{ves}(t,j)$ is the transportation cost of the jth wireless resource-rich island energy storage battery in the tth hour; $C_B(t,k)$ is the charge/discharge cost of No. k island energy storage battery in the tth hour.

3.2. Constraint Conditions

In the scenario, we consider one load center island surrounded by $N_w$ wireless resource-rich islands. The scheduling horizon is discretized into T hourly intervals with $\mathsf{\Delta }t=1\mathrm{h}$. Power variables are expressed in kW, energy variables in kWh. The positive sign convention is defined as power/energy flowing into the load center island being positive.

The energy scheduling of the isolated island group is constrained by the following aspects:

1. Supply and demand balance constraints of load center islands: The hourly net load demand of the load center island is met by the diesel generator sets and the energy storage batteries transported by the wireless resource-rich island. The corresponding constraint expression can be obtained as follows:

$${\displaystyle \sum _{i=1}^m{P}_{DG}(t,i)}+P_B(t)=P_L(t)$$

(5)

where $P_{DG}(t,i)$, $P_B(t)$, $P_L(t)$ are the generating power of the ith diesel generator set in the tth hour, the discharge of the energy storage battery, and the net load on the load center island in the tth hour, respectively.

2. Power generation, energy storage battery consumption, and capacity constraints of diesel generator sets: Based on the methods of wind and photovoltaic power output prediction and load prediction, the net load $P_L(t)$ in Equation (5) can be obtained by subtracting the wind and photovoltaic power output of the wired resource-rich island from the predicted total load. Therefore, it can be regarded as a known constant. The generating power of the diesel generator set $P_{DG}(t,i)$ is a non-negative value between its minimum generating power and maximum generating power. The energy storage battery discharge $P_B(t)$ can be positive or negative. Its positive value represents the discharge meeting the electricity demand on the load center island, while a negative value represents the diesel generator set charging the energy storage battery. This can meet the electricity demand of the load center island at high net load levels in the future. Thus, the constraints can be presented as follows:

$$P_{DG\cdot \min }(i)\le P_{DG}(t,i)\le P_{DG\cdot \max }(i)$$

(6)

where $P_{DG\cdot \min }(i)$ and $P_{DG\cdot \max }(i)$ are the minimum generating power and maximum generating power of the ith diesel generator set. In addition, the SOC (State of Charge) of the energy storage battery on the center island of negative load and the wireless resource-rich island should meet the battery capacity constraints. For the sake of simplicity, it is assumed that all battery pack configurations on the island have the same capacity; thus, we have the following:

$$E_{\min }\le E_{bk}(t)\le E_{\max }$$

(7)

where $E_{bk}(t)$ denotes the SOC of the energy storage battery on the kth island in the tth hour; $E_{\min }$ and $E_{\max }$ represent the upper and lower limits of battery capacity.

3. Constraints on the change in energy storage battery power on each island: For the load center island and each wireless resource-rich island, the energy storage battery used for scheduling must meet a certain relationship between two hours before and after. This relationship can be denoted as follows:

$$E_{bk}(t)=u_k(t-1)E_{bk}(t-1)+\hspace{0.33em}\mathsf{\Delta }{E}_{bk}(t)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}t=2,3,\dots ,T\hspace{0.33em}$$

(8)

where $u_k(t-1)$ is a 0–1 variable, which indicates whether a power change occurs on the No. k resource-rich island during the $t-1$ unit scheduling time; $\mathsf{\Delta }{E}_{bk}(t)$ represents the change in battery power on the No. k resource-rich island in the tth hour.

4. Load center island battery capacity constraint after battery swapping: When the wireless resource-rich island participates in the dispatching, a certain amount of fully charged battery blocks is transported to the load center island by vessels, and the same number of empty battery modules is transported back to the original load center island. Therefore, from an economic cost perspective, at the time of battery swapping, the number of battery modules carried by the ship should not exceed the number of battery modules above the load center island at this time. The constraint can be expressed as follows:

$$E_{b1}(t)+{\displaystyle \sum _{k=2}^{n+1}{u}_k(t)E_{bk}(t)\le E_{\max }}$$

(9)

5. Start–stop cost constraints of diesel generator: For a certain diesel generator set, there is a fixed start–stop fee for each start or stop. The start–stop cost of diesel generator sets is proportional to the number of start–stop times. The 0–1 variables $r_{on}$ and $r_{off}$ are used to represent the selection effect on whether start-up and shutdown occur, respectively. The variable $r_{on}=1$ indicates that the diesel generator set is in a non-operating state at the previous moment and in an operating state at the subsequent moment. This means that the diesel generator starts at that moment. Otherwise, this variable equals 0. Similarly, when the diesel generator set is in operation at the previous moment and not in operation at the subsequent moment, the variable $r_{off}$ equals 1. Thus, the relationship between the start–stop cost of the diesel generator set and its operating state can be expressed as follows:

$$C_{on}(t,i)=r_{on}(t,i)\cdot p_{on}(i)$$

(10)

$$C_{off}(t,i)=r_{off}(t,i)\cdot p_{off}(i)$$

(11)

where $r_{on}(t,i)$ and $r_{off}(t,i)$ are the 0–1 selection variables representing the start-up and shutdown of the ith diesel generator set in the tth hour; $p_{on}(i)$ and $p_{off}(i)$ denote the cost of each start-up and shutdown of the ith diesel generator set.

6. Constraints on energy storage battery transportation costs and island charging/discharging costs are considered as follows: wireless resource-rich islands participate in energy dispatch through the use of multifunctional vessels. To achieve optimal overall operational efficiency across the isolated island group, the load center island acts as the scheduling hub, with the objective of minimizing total operating costs throughout the day. To simplify the model and facilitate quantitative analysis, it is assumed that all wireless resource-rich islands employ the same type of multifunctional vessel for battery transportation and that each vessel maintains uniform linear motion during its travel process. Consequently, the number of multifunctional vessels required by each wireless resource-rich island at any given time can be expressed as follows:

$$N_{vk}(t)=⌈{\displaystyle \frac{u_k(t)E_{bk}(t)}{E_{v\cdot \max }}}⌉\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=2,3,\dots ,n+1$$

(12)

where $N_{vk}(t)$ denotes the number of vessels required by No. k wireless resource-rich island in the tth hour; $E_{v\cdot \max }$ denotes the maximum load of each vessel; The round-up function $⌈x⌉$ means to take the smallest integer not less than the real number $x$.

Meanwhile, the number of vessels that can participate in scheduling at each moment is limited, so it should also meet the following requirements:

$$N_{vk}(t)\le N_{v\cdot \max }(t)$$

(13)

where $N_{v\cdot \max }(t)$ is the maximum number of ships available for battery transportation in the tth hour.

Therefore, the transportation cost of energy storage batteries for each wireless resource-rich island can be expressed as follows:

$$C_{ves}(t,j)=N_{vj}(t)\times T_{vj}\times p_{ves}$$

(14)

$$T_{vj}={\displaystyle \frac{d_{vj}}{v_{ves}}}$$

(15)

where $T_{vj}$ denotes the travel time from the jth wireless resource-rich island to the load center island; $d_{vj}$ denotes the distance from the wireless resource-rich island to the load center island; $v_{ves}$ is the sailing speed of the vessels.

On the other hand, charging and discharging energy storage batteries on the island will cause battery loss and affect battery life. Assuming that all islands are equipped with the same energy storage battery, the charge and discharge cost can be calculated as follows:

$$C_B(t,k)={\displaystyle \frac{|\mathsf{\Delta }{E}_{bk}(t)|\cdot p_B}{2N\cdot E_{\max }}}$$

(16)

where $|\mathsf{\Delta }{E}_{bk}(t)|$ denotes the energy storage battery change value on No. k wireless resource-rich island in the tth hour; denotes the price of energy storage batteries; refers to the number of charging and discharging cycles for energy storage batteries.

3.3. Linearization of Nonlinear Constraints

In Section 3.2, we construct constraints for the energy scheduling model of isolated island groups. By analyzing these constraints, it can be seen that there is a multiplication of two variables in Equations (8), (9) and (12). These equations are all nonlinear constraints in the form of multiplication of 0–1 variables and continuous bounded variables. The entire optimization problem is a Mixed Integer Nonlinear Programming (MINLP) problem. This type of problem is difficult to solve. Therefore, a straightforward idea is to use auxiliary variables to linearize the nonlinear problem and transform it into a Mixed Integer Linear Programming (MILP) problem. In this way, we can use the linearized solution algorithm to find the actual optimal solution.

For example, in Equation (8), the 0–1 variable $u_k(t)$ is the hourly battery swapping selection variable for each wireless resource-rich island, and the continuous bounded variable $E_{bk}(t)$ is the daily remaining power of energy storage batteries on each island. Now we introduce a new auxiliary variable $E_{bk}^{\prime }(t)$ to Equation (8) as follows:

$$E_{bk}^{\prime }(t)=u_k(t)E_{bk}(t)$$

(17)

From Equation (7), it can be seen that the upper and lower bounds of continuous bounded variables are $E_{\min }$ and $E_{\max }$, respectively. In order to satisfy the above condition and ensure that the value of the newly introduced auxiliary variable $E_{bk}^{\prime }(t)$ is the same as the previous result, two constraint conditions can be added as follows:

$$u_k(t)E_{\min }\le E_{bk}^{\prime }(t)\le u_k(t)E_{\max }$$

(18)

$$E_{bk}(t)-\left[1-u_k(t)\right]E_{\max }\le E_{bk}^{\prime }(t)\le E_{bk}(t)$$

(19)

According to the above two equations, we can conduct the following analysis in two scenarios to prove that the results remain unchanged before and after the introduction of auxiliary variables:

1. $u_k(t)=0$: From Equations (18) and (19), we have $E_{bk}^{\prime }(t)=0$ and $E_{bk}(t)-E_{\max }\le E_{bk}^{\prime }(t)\le E_{bk}(t)$. Combined with Equation (7), we can see $0\le E_{\min }\le E_{bk}(t)\le E_{\max }$. Therefore, taking the intersection yields what we have, which is satisfied with Equation (17).

2. $u_k(t)=1$: In this situation, the multiplication of the original two variables is $E_{bk}(t)$. From Equations (18) and (19), we have $E_{\min }\le E_{bk}^{\prime }(t)\le E_{\max }$ and $E_{bk}^{\prime }(t)=E_{bk}(t)$. Combined with Equation (7), we can take the intersection yields and have $E_{bk}^{\prime }(t)=E_{bk}(t)$, which is also satisfied with Equation (17).

Thus, we can use the three constraints shown in Equations (17)–(19) to change the nonlinear constraint into a linear constraint. Then, the method of linear programming can be used to solve the problem accurately.

Similarly, for the other nonlinear constraints like Equations (9) and (12), they can also be transformed into linear expressions by introducing auxiliary variables. In this way, the original MINLP problem can be converted into an MILP problem. Then, the problem is solved in MATLAB R2018a using YALMIP and IBM ILOG CPLEX.

4. Analysis of Scheduling Cycle Extension

4.1. Scheduling Cycle

For the power dispatch of isolated islands, the rationality of the scheduling cycle T needs to be taken into consideration. In conventional power system scheduling, T is generally set to 24 h. However, this value warrants further discussion in isolated island power grid scheduling. Islands exhibit typical seasonal and climatic characteristics, that is, the renewable energy generation capacity on the island varies significantly in different seasons and climate conditions. In this paper, we distinguish it into a normal output period and a low-density output period.

During the low-density production period, although the energy output of the wired resource-rich island also decreases, it can still be transported back to the load center island for consumption in real time through cables. The transmission cost has not changed compared to the normal production period. Therefore, for such resource-rich islands, there is no need to pay special attention to the seasonal fluctuations in renewable energy output during electricity production.

However, the situation will be different for wireless resource-rich islands. When the overall system load is high, even if the output of the wired resource-rich island is completely absorbed, there is still a large energy demand for the load. At this point, electricity needs to be supplied by both wireless resource-rich islands and diesel generator sets. Due to the low-density production period, the accumulation time of SOC in wireless resource-rich island batteries has become longer. When output is particularly low, the increase may only be a few percent of the rated capacity, or even zero throughout the day. Considering the costs that must be borne, if the strategy of using vessels to collect battery energy packages is still implemented, it may lead to significant losses. In particular, it may not be possible to fully utilize the valuable resource of energy packages. Thus, a reasonable approach is to wait for the accumulation of battery SOC to a certain extent before scheduling.

As discussed in Section 2.1, there is an efficient operating range for diesel generator sets. Only when the battery energy packages work in collaboration with the diesel generator sets and fully utilize the flexible output characteristics can the diesel generator sets achieve better operating characteristics. In the case of a limited number of energy packets during the low-density energy output period, the only way to achieve this is to optimize the scheduling strategy. Thus, it is worth considering whether to use the 24-h window used in conventional power system scheduling cycle as the scheduling cycle.

4.2. The Necessity of Improving the Efficiency of Diesel Generators

Firstly, we discuss the necessity of adjusting the scheduling cycle from the perspective of diesel generator efficiency. According to the discussion in Section 3.1, based on the previous weather forecast data, the next day’s renewable energy output for each resource-rich island can be predicted. After deducting the real-time power transmission from the wired resource-rich island, the net energy demand of the load center island will be jointly satisfied by the energy package transmission from diesel generator sets and wireless resource-rich islands. We can divide it into four different scenarios based on the magnitude of net load energy demand and the level of battery energy storage. The following will focus on the optimization effects of diesel generator set efficiency in different scenarios:

Scenario 1: The load center island has a high net load electricity demand, while the wireless resource-rich island has a lower wind and photovoltaic output.

In this scenario, the power of energy storage batteries is limited, and the load is mainly supplied by diesel generator sets. Assuming the net load curve of the load island as shown in Figure 2a, when the power supply efficiency of diesel generator sets is not considered, their unit power generation cost is independent of the operating power. As long as the total power generation of diesel generators is the same during the scheduling period, the power generation cost of the central island of the load island will be the same. In this situation, there are multiple optimization scheduling schemes. The blue and green dashed lines in Figure 2a can represent the output curves of a diesel generator set and battery for a possible preferred solution (Strategy 1), while the corresponding solid lines can represent the output curves of another solution (Strategy 2).

In fact, the power generation of diesel generator sets is closely related to their operational efficiency. Based on the verification of Equations (1)–(3), the actual power generation cost of diesel generator sets in strategies 1 and 2 will increase, indicating that the corresponding optimization results are not optimal.

Next, we consider the power supply efficiency of the diesel generator set and directly optimize the load power supply. From the perspective of global optimization scheduling, a unique optimal solution can be obtained (Strategy 3) under a single objective function. The output curves of the diesel generator set and battery corresponding to strategy 3 are shown in Figure 2b.

As shown in Figure 2, the power generation and battery discharge of the diesel generator set are completely the same among the three strategies, but the diesel generator set in Strategy 3 has a more stable output and can consume less fuel.

The optimality of the above scheduling scheme can be demonstrated from the perspective of mathematical optimization. In Equation (1), the consumption function of the diesel generator set is a typical concave function. Its property can be expressed as follows:

$$f\left[(x_1+x_2)/2\right]\le {\displaystyle \frac{f(x_1)+f(x_2)}{2}}$$

(20)

More generally, according to the Jensen inequality, for diesel generator sets, we have

$$n\cdot C_{DG}\left[(P_{DG}^1+P_{DG}^2+\cdots +P_{DG}^n)/n\right]\le C_{DG}(P_{DG}^1)+C_{DG}(P_{DG}^2)+\cdots +C_{DG}(P_{DG}^n)$$

(21)

where $n$ is the number of time periods; $P_{DG}^1$, $P_{DG}^2$, …, $P_{DG}^n$ denote the power generation of diesel generator sets at different time periods; $P_{DG2}=(P_{DG}^1+P_{DG}^2+\cdots +P_{DG}^n)/n$ denotes the average generation power over the entire time period. In Equation (21), the equal sign holds only when $P_{DG}^1=P_{DG}^2=\cdots =P_{DG}^n$.

Equation (20) indicates that the operating cost of diesel generator sets that ignore power generation efficiency is always lower than the cost when considering power generation efficiency. Therefore, scheduling should be ideally optimized in a form close to $P_{DG2}$, that is, to keep the power generation of diesel generators as constant as possible during a certain period of time.

In this scenario, energy storage batteries can provide a significant amount of electricity. Meanwhile, according to the analysis in Scenario 1, it can be concluded that the optimal scheduling plan should fully utilize the peak shaving and valley filling functions of the battery to maintain the stable power generation of the diesel generator set to the greatest extent possible.

Scenario 3: The load center island has a low net load electricity demand, and the wireless resource-rich island also has a low wind and photovoltaic output.

In this scenario, the amount of electricity that energy storage batteries can provide is also relatively limited. Similarly to Scenario 2, maintaining stable power generation of the diesel generator set as much as possible can achieve optimal operation of the island group.

Scenario 4: The load center island has a low net load electricity demand, while the wireless resource-rich island has a higher wind and photovoltaic output.

In this scenario, energy storage batteries can provide a large amount of electricity and can fully meet the electricity load demand by relying on energy storage batteries. The diesel generator set can stop, achieving an optimal power supply.

To improve the clarity and reproducibility of the proposed renewable energy forecasting model, we provide the following specifications regarding the model architecture and training settings.

The Long Short-Term Memory (LSTM) network used for wind and photovoltaic power prediction consists of two hidden layers, each with 64 units, followed by a fully connected dense layer for output. The input features include historical meteorological time-series data (e.g., wind speed, solar irradiance, temperature) and past generation records. The model was trained using data collected from offshore island meteorological stations, covering a total of 8760 hourly samples (equivalent to one year).

The loss function employed is the Mean Squared Error (MSE), and the Adam optimizer was used with an initial learning rate of 0.001. The model was trained for 200 epochs with an early stopping mechanism to prevent overfitting.

To assess forecasting performance, three standard metrics were reported: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). In 24-h-ahead forecasting, the wind power model achieved an RMSE of 5.84 kW, MAE of 4.25 kW, and MAPE of 7.3%, while the photovoltaic model achieved an RMSE of 4.76 kW, MAE of 3.67 kW, and MAPE of 6.1%. Prediction accuracy slightly decreased for longer forecasting horizons (48 h and 72 h), aligning with known trends in renewable energy prediction literature.

These results confirm the effectiveness of the LSTM-based framework for mid-term renewable energy forecasting in offshore island microgrid scenarios.

In summary, it can be seen that in order to achieve the lowest daily energy supply cost for isolated island groups, the principle should be based on fully utilizing the peak shaving and valley filling functions of energy storage batteries, as well as maintaining the stable power generation of diesel generator sets as far as possible.

4.3. The Rationality of Scheduling Cycle Extension

There is a key question below: Can the joint operation of diesel generator sets and batteries be further optimized by extending the scheduling cycle of isolated island groups?

In theory, the longer the prospective period, the greater the optimization space, and the better the solution that can be obtained. To prove this, Section 4.3 examines the economic advantages of extending the forward cycle of daily scheduling by designing the following two schemes.

Scheme 1: Expand the scheduling cycle to 48 h and optimize the power supply cost of the island cluster in a rolling manner.

Scheme 2: Keep the traditional 24 h scheduling cycle and optimize the electricity consumption of the island cluster for two consecutive days

Assuming that for 48 consecutive hours, both the net load and wind/photovoltaic output of the previous day are high, while their output levels on the following day are low. The power curve is shown in Figure 3. According to Scheme 1 for scheduling, the operating curves of diesel generator sets and batteries are represented by blue dashed lines and red dashed lines. The optimization scheduling results for Scheme 2 are shown by the solid blue and red lines in the figure.

From Figure 3, it can be seen that under the premise of meeting the net load demand of the load center island, the diesel generator set output in Scheme 1 is more stable. The 24 h scheduling cycle scheme fails to anticipate that the next 24-h net load will require more support from battery output, and fails to form a more stable overall output curve for diesel generator sets. In the first 24 h, the battery’s charge is almost completely discharged. In the next 24 h, when the battery output support is more in demand, the wireless resource-rich island no longer has the ability to provide sufficient battery power to meet this demand. In contrast, the 48-h scheme can foresee the above situation and arrange the output curve of the battery more reasonably, resulting in a better overall effect. Thus, expanding the scheduling cycle can significantly improve the operational economy of island clusters [25].

In fact, from a mathematical perspective, setting a prospective period of 365 days throughout the year is bound to yield optimal results. However, both renewable energy and load have prediction errors. It is not advisable to excessively extend the scheduling cycle, which will be verified in the simulation phase.

Considering the actual situation of low-density production periods in wireless resource-rich islands, this paper suggests using a 48-h prospective period as the scheduling cycle for formulating optimization scheduling strategies. In fact, the Pennsylvania, New Jersey Maryland (PJM) market in the United States has implemented a scheduling strategy with a 48-h scheduling cycle, indicating that the idea of expanding the scheduling cycle has theoretical and practical foundations.

By the way, it should be pointed out that even during the normal production period, a 48-h prospective cycle can still achieve better optimization results for island clusters, but the optimization effect may not be as significant as during the low-density production period.

To justify the 48-h scheduling horizon, we conducted predictive error analysis of renewable generation using meteorological datasets from offshore island clusters. Long Short-Term Memory (LSTM) networks were trained to forecast wind and photovoltaic output for 24 h, 48 h, and 72 h. Based on the deviations between predictions and actual outputs, we modeled the forecast errors under different horizons and fitted them to probabilistic distributions using Bayesian updating.

The results reveal that while prediction accuracy deteriorates as the horizon increases, the 48-h window offers a practical trade-off. It significantly improves logistics coordination and diesel resupply planning compared to 24-h forecasts, without the large error margins seen in 72-h forecasts. Therefore, 48-h scheduling is adopted to balance foresight and reliability in island microgrid operations.

5. Numerical Study

5.1. Simulation Setup

In this section, a simulation is carried out against the background of an isolated island group in Nansha, South China Sea. The island group includes a load center island, two wired resource-rich islands with direct connection to the load center island, and a wireless resource-rich island. Two wired resource-rich islands are respectively equipped with 200 kW wind power and 150 kW photovoltaic power generation, and 100 kW wind power and 100 kW photovoltaic power generation. The wireless resource-rich island is equipped with 450 kW wind power, a 300 kW photovoltaic power generation system, and 30 battery energy packages, each with a capacity of 150 kWh, for multifunctional vessels to transfer energy. It is set that up to five multifunctional vessels can participate in battery transfer per hour, and a single vessel can carry no more than seven battery energy packages. The one-way service fee for vessel transfer is CNY 700. Typical daily wind speed and light intensity of the wireless resource-rich island are shown in Figure 4, and the day-ahead net load forecasting curve of the load center island is shown in Figure 5.

5.2. Necessity Verification of Fine Modeling for Diesel Generators

The simplified consumption model of diesel generator sets assumes that the unit price of power generation is constant, independent of the generating power and its efficiency. In this case, the fuel electricity cost for two diesel units is converted to CNY 2 and CNY 2.3. The refined consumption model of diesel units considers the relationship between power generation efficiency and actual output, and the relationship between the unit price of power generation and actual output is a nonlinear function, considering the start-up and shutdown costs of the generator set. According to Refs. [26,27], the rated power, operating cost parameters, and single start–stop cost in the refined modeling of diesel generator sets are presented in

Table 3. Parameters of the refined consumption model of diesel generators.

Operating Parameters	Diesel Generator A	Diesel Generator B
Rated power (kW)	900	700
a (CNY/kW2·h)	0.0024	0.0029
b (CNY/kW·h)	0.351	0.559
c (CNY/h)	1.028	1.221
Single start-up cost (CNY)	45	35
Single shutdown cost (CNY)	25	22

.

Then, we respectively use the simplified consumption model and the refined model of diesel units for optimal scheduling. Simulation analysis is conducted on the output curve of diesel generator sets and the discharge curve of wireless resource-rich island battery energy packages under a 24-h scheduling cycle of isolated island groups, as shown in Figure 6 and Figure 7.

Comparing Figure 6 and Figure 7, it can be observed that after modifying the fuel consumption model of the diesel generator set, the output curve of the diesel generator set and the discharge curve of the wireless resource-rich island battery energy package have both changed. This change is reflected in the fact that the owner of the diesel generator set finds the unit’s operating efficiency to be lower in the high-power range. Forcibly scheduling this portion of the power share fails to achieve the goal of increasing profit. Therefore, by adjusting the decision, the owner avoids generating electricity with excessively high-power values. This energy gap is filled by the output from wired resource-rich islands and the energy packages from resource-rich islands.

On the one hand, we use the refined consumption equation to calculate the actual operating efficiency of diesel generator sets under the optimal scheduling results of simplified consumption models. The results show that the total operating fuel cost of the diesel generator set is CNY 47,589.07, and the total start–stop cost is CNY 57, which means that the total operating cost of the diesel generator set is CNY 47,643.07. Compared with the simplified operating model of diesel generator sets, the actual operating cost of diesel generator sets will increase by CNY 6540.07 compared to the expected scheduling value, and the operating profit will decrease from CNY 4410.11 to CNY −2074.69. It can be seen that in the case of using a simplified consumption model, diesel generator units underestimate their own operating costs during actual scheduling and operation, resulting in an overly optimistic participation in grid scheduling. Although they obtain higher electricity sales quotas, their actual profits actually change from positive to negative, leading to an actual loss of operation. Therefore, the accuracy of the consumption model for diesel generator sets is critical for their operating cost evaluation. Only by adopting a more reasonable consumption model for diesel generator sets can we obtain the most favorable results.

On the other hand, in the refined consumption model, the total power generation and profit of diesel generator sets are CNY 18,516 kWh and 8676. Compared with the simplified consumption model, the total power generation of diesel generator sets has decreased, but the profit has actually increased. The reason for the increase in profit of diesel generator sets is that although their share of electricity sales has decreased, they can always work in a higher efficiency range, thereby significantly reducing power generation fuel consumption. Thus, in the case of a decrease in total electricity sales, the profit from electricity sales has actually increased. From Figure 4, it can also be seen that the discharge of the energy pack is flexible, compensating for the portion of the power generation capacity abandoned by diesel generators during peak load periods in pursuit of maximizing the profit.

All energy suppliers benefit from such an energy distribution ratio. Firstly, the load is supplied at a reasonable price. Then, diesel generator sets achieve maximum profit in a more cost-effective and efficient way. In addition, the energy packages generated by wireless resource-rich islands have also played an important role in a reasonable manner and achieved better economic benefits. The results also prove the necessity and effectiveness of using the refined model of the diesel generator set.

Furthermore, it should be pointed out that from the perspective of the efficiency of the entire electricity market and the future development of the power system, it is not a good choice for diesel generators to occupy a larger market share. This not only hinders the economic operation of the energy supply system for isolated island groups but also goes against the development of clean and renewable energy.

5.3. Demonstration of Effectiveness of Scheduling Cycle Extension

Taking 24-h, 48-h, and 72-h scheduling cycles as examples, scheduling is carried out through a multi-party cooperative power supply mode regardless of error and accounting for error, respectively. We analyze the total power supply cost under different daily scheduling cycles to verify the rationality of scheduling cycle extension.

Comparison Plan 1: Scheduling the prototype system with 24-h, 48-h, and 72-h prospective periods without considering errors. The total power supply cost of the system within 72 h is shown in Table 3.

From Table 3, it can be seen that without considering errors, the total power supply cost is the lowest under a 72-h scheduling cycle, which proves the necessity of extending the scheduling cycle. Indeed, it can be inferred from Table 1 that the longer the scheduling cycle, the better the optimization effect. However, this may not necessarily be the case.

Based on the new energy output and error prediction mode described in Section 4.3, comparison plan 2 is set up to explore the energy scheduling of island clusters under different scheduling cycles, taking into account the impact of errors.

Comparison Plan 2: Analyzing three rolling optimization strategies with 24-h, 48-h, and 72-h scheduling cycles while considering errors. Each strategy obtains 300 scenarios with the Monte Carlo sampling method based on the predicted power and error distribution of renewable energy within the corresponding scheduling cycle. Then, we optimize the operating costs of the island group in different scenarios and calculate the mean value. The total cost of a 72-h power supply for an isolated island group using three strategies is shown in

Table 4. The total power supply cost of isolated islands in 72 h under different daily scheduling cycles without considering error.

Scheduling Cycle (h)	Total Cost (CNY)	95% CI
24	54,872.60 ± 235.40	(54,680.20, 55,064.90)
48	53,821.50 ± 198.30	(53,655.40, 53,987.60)
72	52,345.80 ± 210.60	(52,153.10, 52,538.40)

.

In Table 4, the optimization scheduling strategy based on the 48-h scheduling cycle is optimal, while the scheme with the 72-h scheduling cycle has the worst result. It can be seen that simply extending the scheduling cycle is not a wise move, and it will increase the computational complexity of the plan. At the current prediction level, the 48-h scheduling cycle is the best choice.

Furthermore, in order to prove our results, we select the renewable energy output data of the island group from the 1st to the 7th of a certain month, which corresponds to a low-density production period. Then, we add different errors according to different scheduling cycles (48 h and 24 h) to form a continuous prediction time series of new energy output.

After superimposing the 168-h load electricity consumption curve with the wind and photovoltaic output prediction curve on the wired resource-rich island, we can obtain the net load curve of the load center island and the total renewable generation power of the wireless resource-rich island for 168 h. The power curves are shown in Figure 8 and Figure 9.

On the basis of the cooperative power supply model proposed in this article, we conduct simulations within 168 h (one week) with 48-h and 24-h scheduling cycles. We can calculate the battery swapping time in a wireless resource-rich island within 168 h, as well as the number of multifunctional vessels required for each exchange and the total cost of power supply. The results are illustrated in

Table 5. The total power supply cost of isolated islands in 72 h under different daily scheduling cycles, considering prediction error.

Scheduling Cycle (h)	Total Cost (CNY)	95% CI
24	55,761.40 ± 262.10	(55,506.00,56,016.80)
48	55,205.30 ± 241.50	(54,982.20, 55,428.40)
72	57,489.60 ± 279.80	(57,204.30, 57,774.90)

.

In

Table 6. The time of power exchange of the wireless resource-rich island, the number of vessels to be dispatched, and the total social well-being in 168 h under different daily scheduling cycles.

24-h Scheduling Cycle		48-h Scheduling Cycle
Battery Swapping Time	Number of Dispatched Vessels	Battery Swapping Time	Number of Dispatched Vessels
36 h	1	67 h	1
84 h	1	160 h	1
142 h	1	/	/
164 h	1	/	/
Total Cost (CNY)
24-h Scheduling Cycle		48-h Scheduling Cycle
156,780.45 ± 410.72		149,950.36 ± 362.84

, it can be seen that in the case of a 48-h scheduling cycle, the vessel is only scheduled twice during the 168-h period. The 48-h scheduling cycle has fewer battery exchanges, resulting in lower service fees for its multifunctional vessel and total cost than the 24-h scheduling cycle. This also reduces the power exchange burden of multifunctional vessels during the low-density production period of wireless resource-rich islands. The above simulation results further demonstrate the advantage of appropriately extending the scheduling cycle.

5.4. Multi-Island Comparative Analysis

Building directly on the original island system described in Section 5.1, we construct two additional scenarios by scaling the baseline: a downscaled variant (Island S) and an upscaled variant (Island L). The system topology and components remain identical to the baseline (load center + two wired resource islands + one wireless resource island with battery-package shipping; diesel units A/B; vessel capacity limits; start/stop and logistics cost formulations). Only the size-related quantities are scaled—peak load, installed wind–PV capacities, sailing distance, and the number of battery packages—so that S and L represent smaller and larger versions of the same island archetype. The baseline itself is denoted as Island M. Exact scaling factors and parameter values are listed in

Table 7. Technical and logistical parameters of the three island scenarios.

Island	Peak Load	RE Share	Sailing Distance	Battery Packages	Vessel Fee Per Trip
S	300 kW	20%	30 km	20	700 CNY
M	600 kW	50%	60 km	30	900 CNY
L	1000 kW	80%	120 km	50	1300 CNY

.

For each island, the MILP is solved under 24 h, 48 h, and 72 h using the same LSTM-based renewable forecasts and the same uncertainty process as in earlier sections. The resulting total cost, diesel share, start–stop events, weekly vessel voyages, renewable utilization, and curtailment are reported in

Table 8. Key operational outcomes across islands and horizons.

Island	Horizon (h)	Total Cost (CNY)	Diesel Share (%)	Start–Stops (Times/Day)	Voyages (Times/Week)	Renewable Utilization (%)
S	24	98,000	78	9	6	82
	48	95,600	74	7	5	85
	72	96,500	75	8	6	84
M	24	152,000	56	12	8	88
	48	146,700	50	8	6	92
	72	149,500	53	10	7	90
L	24	228,000	34	16	12	91
	48	217,700	28	10	8	95
	72	229,100	32	14	11	92

.

As summarized in Table 8, the 48-h horizon consistently achieves the lowest total cost across all three islands. The improvement arises from enhanced inter-day coordination: diesel output becomes smoother, start–stop events are reduced, and battery packages are dispatched more efficiently to substitute peak diesel generation. The magnitude of cost reduction grows with system flexibility and logistics constraints.

On Island S, where renewable penetration is low and the system’s ability to shift energy across days is limited, the benefits of extending the horizon are modest—cost decreases slightly, start–stop frequency declines, and renewable utilization improves marginally. On Island M, the advantage becomes clearer: medium renewable penetration and moderate sailing distance allow the optimizer to coordinate storage operation and battery exchanges more effectively over a two-day horizon, leading to a noticeable drop in cost and diesel share. Island L shows the most pronounced improvement. With high renewable penetration and long-haul logistics, the 48-h horizon consolidates voyages, better aligns renewable availability with load peaks, and maintains diesel generators within higher-efficiency operating ranges, resulting in the largest cost savings.

Overall, the results in Table 8 demonstrate a consistent mechanism: extending the scheduling window from 24 h to 48 h enables the system to shift variability to renewables and storage, smooth diesel operation, and reduce both cycling and transport frequency. The 72-h horizon, while beneficial under deterministic conditions, offers diminishing returns once forecast uncertainty is considered, as excessive foresight may amplify scheduling mismatches. For low-penetration, short-haul islands, the window effect remains limited, and accurate fuel-cost calibration is more critical; for high-penetration, long-haul islands, adopting a 48-h horizon yields the greatest economic gains.

5.5. Sensitivity Analysis

1. Diesel Cost Coefficients

To further examine the robustness of the refined diesel generator model, a sensitivity analysis was carried out on the cost coefficients $a$, $b$ and $c$ in Table 3. These coefficients determine the nonlinear fuel consumption curve of the generators according to Equation (2),

Where $a$ represents the curvature of the cost curve (reflecting efficiency deterioration at high load), $b$ denotes the linear fuel cost rate, and $c$ corresponds to the constant idle fuel consumption per hour. The analysis aims to quantify how variations in these parameters influence the total system cost and the optimal scheduling behavior.

Each coefficient was independently perturbed by ±10% and ±20% from its nominal value, while all other parameters were kept constant. For every perturbation, the mixed-integer linear programming (MILP) model was re-solved under identical boundary conditions, and the resulting changes in total cost, diesel generation share, and start–stop frequency were recorded.

Based on

Table 9. Sensitivity results for diesel generator cost coefficients.

Parameter Change	Δ Total Cost (%)	Change in Diesel Share (%)	Start–Stop Frequency (Times/Day)	Observation
a − 20%	−6.1	+8.1	increases by 2	Lower cost, higher load factor
a − 10%	−3.0	+4.0	increases by 1	Slight cost reduction
a + 10%	+3.1	−4.2	decreases by 1	Slight cost increase; smoother output
a + 20%	+6.4	−8.5	decreases by 2	Notable rise; higher storage use
b − 20%	−4.1	+6.0	virtually zero	Moderate cost decrease
b − 10%	−2.0	+3.0	virtually zero	Small reduction
b + 10%	+2.0	−2.8	virtually zero	Linear cost effect
b + 20%	+4.5	−6.1	virtually zero	Cost increases linearly
c − 20%	−1.2	−2.0	increases by 3	Idle loss reduction
c − 10%	−0.6	−1.0	increases by 1	Slightly more switching
c + 10%	+0.8	+1.2	decreases by 1	Slightly fewer start–stops
c + 20%	+1.4	+2.5	decreases by 3	Continuous operation favored

, all three coefficients exhibit monotonic responses to bidirectional (±) perturbations, with noticeable asymmetries between positive and negative directions. The quadratic term $a$ is the most influential driver: decreasing $a$ by 10–20% lowers the total cost by about 3.0–6.1%, raises the diesel share by roughly +4.0 to +8.1 percentage points, and increases start–stop events, indicating that lower high-load marginal cost encourages higher loading and more frequent cycling. Conversely, increasing $a$ by 10–20% raises the total cost by about 3.1–6.4%, reduces the diesel share by −4.2 to −8.5 percentage points, and suppresses start–stops, yielding smoother diesel output and greater reliance on renewables and battery packages. These patterns confirm that small deviations in $a$ trigger structural changes in dispatch—peak-shaving via storage and output smoothing—so $a$ should be calibrated first.

The linear term $b$ produces a near-proportional, mild effect: ±10–20% changes move the total cost by roughly ±2.0–4.5% and shift the diesel share by about ±3–6 percentage points, while start–stop frequency remains essentially unchanged (≈0). This suggests $b$ primarily translates the entire cost curve without materially altering commitment or smoothing decisions; it is economically meaningful but secondary to $a$ in shaping scheduling structure.

The constant term $c$, which captures idle losses, mainly acts through the commitment boundary. Lowering $c$ increases start–stops (+1 to +3 per day) and slightly reduces cost (−0.6% to −1.2%) with a small decrease in diesel share, reflecting greater willingness to cycle when idling is cheaper. Raising $c$ reduces start–stops (−1 to −3 per day), nudges cost up (+0.8% to +1.4%), and slightly increases diesel share, favoring more continuous but less efficient operation. The lack of strict symmetry across ± perturbations is expected under mixed-integer scheduling with unit commitment, start–stop costs, capacity bounds, and discrete logistics, where active constraints shift as parameters move. In practice: calibrate $a$ to govern smoothing versus substitution, adjust $b$ to reflect the overall fuel price level, and tune $c$ to shape the cycling–continuity trade-off; under higher $a$, a longer horizon (e.g., 48 h) with storage/energy-package support becomes especially valuable for cost control.

2. Multi-Island Parameter

To further assess the robustness of the proposed scheduling framework, a comprehensive sensitivity analysis was performed on four major parameters affecting system economics and operational flexibility: diesel price, battery capacity, vessel cost per trip, and renewable forecast error.

Each parameter was independently perturbed by ±10% and ±20% from its nominal value, while all other parameters remained fixed. The 48-h scheduling horizon—identified as the most cost-effective configuration—was adopted for all cases.

The mixed-integer linear programming (MILP) model was re-solved under identical boundary conditions, and the variations in total cost, diesel generation share, and operational behavior were recorded.

As shown in

Table 10. Sensitivity results for diesel price, battery capacity, vessel cost, and prediction error.

Parameter Change	Δ Total Cost (%)	Change in Diesel Share (%)	Observation
Diesel cost −20%	−7.2	+6.4	Lower fuel cost; more diesel generation
Diesel cost −10%	−3.5	+3.1	Moderate cost reduction; reduced storage use
Diesel cost +10%	+3.7	−3.6	Higher cost; increased renewable dispatch
Diesel cost +20%	+7.4	−7.8	Strong cost rise; diesel share drops sharply
Battery capacity −20%	+3.2	+4.5	Insufficient storage; higher diesel cycling
Battery capacity −10%	+1.5	+2.2	Slight cost increase; limited flexibility
Battery capacity +10%	−2.1	−3.4	Lower cost; smoother diesel output
Battery capacity +20%	−3.9	−6.1	Improved renewable utilization; reduced starts
Vessel cost −20%	−1.1	−1.5	Cheaper logistics; more voyages and renewables
Vessel cost −10%	−0.5	−0.7	Minor benefit; slightly higher renewable share
Vessel cost +10%	+0.8	+1.1	Logistics cost dominates; fewer voyages
Vessel cost +20%	+1.9	+2.3	Reduced voyage frequency; cost rises slightly
Forecast error −20%	−1.8	−2.4	Improved accuracy; smoother scheduling
Forecast error −10%	−0.9	−1.1	Slightly lower cost; fewer mismatches
Forecast error +10%	+3.6	+3.9	Misaligned dispatch; reduced efficiency
Forecast error +20%	+6.2	+7.1	Cost surge; frequent rescheduling events

, all four parameters exhibit consistent monotonic responses under bidirectional (±) perturbations.

Among the factors, diesel price exerts the largest impact on total operating cost, varying nearly linearly with fuel cost changes. A 20% increase in diesel price leads to a 7–8% rise in total cost and a sharp reduction in diesel generation share, while cheaper diesel reverses the trend.

Battery capacity ranks second in sensitivity: expanding capacity improves renewable absorption and smooths diesel operation, reducing total cost by up to 4%, though marginal gains diminish beyond a +20% increase.

The influence of vessel cost per trip is moderate; as logistics expenses rise, the optimizer reduces voyage frequency to minimize costs, slightly lowering renewable utilization.

Finally, forecast error has a pronounced nonlinear effect—greater uncertainty amplifies cost growth (up to +6%) and causes dispatch mismatches, while improved accuracy yields smoother, lower-cost operation. Overall, this comprehensive sensitivity analysis confirms that the system is most sensitive to diesel cost and forecast accuracy, followed by battery capacity and vessel logistics cost.

These findings highlight that maintaining reliable forecasting and controlling diesel price volatility are essential for the stable and economical operation of islanded microgrids.

6. Conclusions

In this paper, a multi-party cooperative power supply model was developed to optimize the energy production, delivery, and utilization of an isolated island group. A refined nonlinear fuel consumption model for the main diesel generator sets on the load center island was first established, and a corresponding mixed-integer linear programming (MILP) optimization framework was constructed to achieve coordinated scheduling across multiple islands.

The proposed framework not only captures the nonlinear operating efficiency of diesel generators with higher accuracy but also enables global optimization of multi-island energy exchanges. Through theoretical analysis and numerical experiments, several key findings were obtained:

• Refined efficiency modeling significantly improves the estimation accuracy of diesel generation costs and enhances the coordination between diesel units and energy packages transported from renewable-rich islands. The refined model reduces cost-estimation deviations and improves the stability of optimal dispatch decisions.
• Extending the scheduling cycle from 24 h to 48 h effectively improves long-term operational performance. Comparative simulations show that the proposed approach reduces the total operating cost by approximately 6–9%, increases diesel utilization efficiency by 8–12%, and decreases start–stop events, particularly under low-density renewable output periods.
• The integration of these two improvements—the accurate efficiency model and the extended scheduling horizon—demonstrates clear economic and operational advantages. The former provides a realistic decision foundation, while the latter amplifies its long-term benefits by improving cross-day energy coordination and smoothing diesel generation.

Overall, the proposed model provides a unified and economically efficient scheduling approach for isolated island groups, enabling more stable, cost-effective, and resilient energy operations under variable renewable conditions.