Resilience Improvement of Microgrid Cluster Systems Based on Two-Stage Robust Optimization

Abstract

For microgrids in deep and remote sea areas, ocean currents are widely used as an emerging power generation resource. Implementing ES systems is crucial for smooth power output and grid stability. The stability of power output from sea current energy generation faces challenges due to speed fluctuations. Enhancing resilience requires addressing transmission line failures caused by extreme seabed conditions, and ensuring operational security. An ES configuration method considering line faults based on two-stage robust optimization is presented in this paper. First, in order to simultaneously consider planning and operation, a defender–attacker–defender (DAD) model was established. Additionally, the capacity, rated power, and charging/discharging power of ES during operation were jointly optimized through the column-and-constraint generation (C&CG) algorithm. In addition, the rationality and effectiveness of the proposed method were demonstrated through experiments on modified IEEE six-bus and fifty-seven-bus systems. The results show that a distributed ES configuration increased system resilience by 54.60% and reduced abandoned power rate by 57.06% compared with the situation without ES configuration.

1. Introduction

The development of renewable energy has garnered increasing attention, and the ocean undeniably holds a vast reserve of energy. Nowadays, since wind and solar energies are extensively utilized, the exploration of ocean current energy has been added to the schedule [1].

The power output of current energy generation from sea currents can be affected by fluctuations in their speeds, leading to instability in the power supply. Cyclical variations in current speeds, caused by tidal or seasonal factors, can impact the operation rate of current energy generators and the amount of power produced, leading to an unreliable power supply [2,3]. For submarine microgrid clusters, maintaining a stable power supply is crucial, as it is directly tied to the stability of grid frequency and voltage. Excessive fluctuations in power generation can negatively affect the grid, reducing the quality and stability of power supply. Submarine microgrid clusters are usually operated off-grid due to the difficulty and high economic cost of grid access in deep-sea environments. The conditions in the deep sea are incredibly harsh, featuring extreme water pressures, frigid temperatures, potent corrosive effects, and biofouling, all of which pose significant risks of equipment malfunction [4]. Maintenance operations in such environments are not only costly but also highly complex; when equipment sustains damage, repairs can be both time-consuming and resource-intensive. Therefore, enhancing the resilience of microgrid clusters is crucial to significantly reduce the likelihood of equipment failures and minimize maintenance costs.

The implementation of ES systems is essential to ensure a smooth power output and safeguard the stable operation of the grid. This system can compensate for power supply shortages by releasing previously stored energy when sea currents slow down and power generation capacity decreases. Conversely, it can store excess power when sea currents speed up and power generation capacity exceeds demand. In this manner, ES systems not only enhance the reliability of the sea current energy generation systems but also improve economic efficiency [5,6,7]. ES devices are especially valuable in areas that operate off-grid or have difficulties connecting to the main power grid.

Existing studies have focused on allocating energy storage capacity to achieve economic cost savings. Cheng et al. [8] introduce a multi-stakeholder and ES co-optimized method for distributed generation, balancing interests, and incorporating carbon/green certificate trading. Talib et al. [9] explore solar PV-grid integration with ES to manage power fluctuations, addressing challenges and optimization opportunities. Lei et al. [10] propose an optimal allocation model for distributed generation to enhance revenue, increase renewable consumption, and address subsidy constraints through source–load–storage coordination. Research on enhancing the resilience of submarine microgrid clusters through ES systems is still relatively scarce. Xu et al. [11] investigated the optimal sizing and placement of ES systems for marine renewable energy integration. They developed a mathematical model to optimize the ES configuration considering various factors such as system cost, energy efficiency, and power quality. The results of their study demonstrated the importance of ES in enhancing the resilience and sustainability of marine renewable energy systems.

When enhancing the resilience of submarine microgrid clusters, it is important to consider transmission line failures. Transmission lines can fail easily due to the extreme environment of the seabed, thereby undermining the operational security of the submarine microgrid cluster system. This paper uses the defender–attacker–defender (DAD) model to consider the uncertainty of faulty lines. The two-stage robust optimization method is adopted to improve the robustness of the microgrid cluster system. The main contributions are as follows:

• A DAD model for solving the optimal allocation of ES is proposed, which synergistically optimizes the planning and operation of ES while also considering the uncertainty of damaged lines.
• The column-and-constraint generation (C&CG) algorithm is exploited to solve the problem, with the big-M method for transforming the problem into a mixed integer linear constraint programming problem. The optimal solution can be obtained in a small number of iterations.

The remainder of this paper is organized as follows: In Section 2, the DAD model based on the uncertainty of faulty lines is established, in which the planning and operation of ES are optimized in the first and third levels, respectively. Then, the C&CG algorithm is provided in Section 3. Additionally, extensive numerical experiments are implemented and the findings are discussed in Section 4 to manifest the effectiveness and rationality of the proposed method. We conclude the study and outline promising future research directions in Section 5.

2. Formulation of the Problem

2.1. Ocean Power Generation Model

The method utilized for power generation by microgrids located in deep and remote sea areas involves ocean current energy generation, which drives the rotor to rotate through the seawater of the turbine blades at a certain flow rate, thereby converting mechanical energy into electrical energy. The generation of power is closely linked to the flow rate of seawater at any given moment. The model representing the seawater flow rate follows a sinusoidal pattern with periodic characteristics.

The periodic variation pattern of seawater flow velocity is as follows [12]:

$$vs.=V_m\sin {\displaystyle \frac{2\pi t}{T_{sea}}}=V_m\sin \omega t$$

(1)

$$V_m={\displaystyle \frac{V_s+V_n}{2}}$$

(2)

where $V_m$ is the average flow velocity of seawater, $V_s$ is the maximum flow velocity during high tide, $V_n$ is the maximum flow velocity during low tide, and $T_{sea}$ is the period of seawater flow velocity variation.

Based on the variation curve of seawater flow velocity, the following ocean current energy generation model characterized by a piecewise function can be established [13]:

$$P_{sea}=\left\{\begin{array}{cc}0,\hfill & vs.\le V_c\hfill \\ {\displaystyle \frac{1}{2}}{C}_p\rho S_g{v}^3,\hfill & V_c\le vs.\le V_r\hfill \\ P_g,\hfill & V_r\le vs.\le V_f\hfill \\ 0,\hfill & vs.\ge V_f\hfill \end{array}\right.$$

(3)

where $C_p$ is the power utilization coefficient of the power generation unit, $\rho$ is the average seawater density in the test sea area, $S_g$ is the area swept by the rotor, and $P_g$ is the rated power of the power generation unit. Regarding the speed parameter, $V_c$ is the cutting speed of the power generation unit, $V_r$ is the lowest flow rate to keep the power generation unit at rated power, and $V_f$ is the cutting flow rate.

2.2. The DAD Model

The research object of this paper is a microgrid cluster system, in which a single microgrid is abstracted as a bus with independent power generation and storage capabilities, and all of them form a network topology system through interconnection lines. The DAD model proposed in this paper is based on offensive and defensive games. The upper layer of the model represents the behavior of the defender, that is, the configuration of ES capacity and rated power at each bus. The middle layer is the behavior of the attacker, which is the fault situation of contact lines among microgrids. The lower level denotes the response of the defender, referring to the changes in electrical quantities such as ES charging/discharging power and power flow during system operation. The illustration of the problem formulation and the explanation of the transformation are shown in Figure 1.

The upper-level model is shown in (4)–(8):

$$\underset{W_i^{ESS},P_i^{ESS}}{min}F+\beta =F_{inv}+F_{op}+\beta$$

(4)

$$F_{inv}={\displaystyle \frac{n{(1+n)}^y}{{(1+n)}^y-1}}\sum _{i\in N}{c}_1{W}_i^{ESS}$$

(5)

$$F_{op}=\sum _{t=1}^y\sum _{i\in N}{c}_2{P}_i^{ESS}{({\displaystyle \frac{1+a}{1+b}})}^t$$

(6)

$$\mathrm{s}.\mathrm{t}.W_{min}^{ESS}\le W_i^{ESS}\le W_{max}^{ESS}\forall i\in N$$

(7)

$$0\le P_i^{ESS}\le P_{max}^{ESS}\forall i\in N$$

(8)

Equation (4) is the objective function of the upper layer, where the cost consists of two parts, $F_{inv}$ is the initial investment cost of ES, and $F_{op}$ is the operating cost of ES. The initial investment cost is related to the capacity of ES at each bus, where $c_1$ is the investment cost per unit capacity, n is the annual interest rate, and y is the life cycle of ES. The operating cost is related to the rated power of ES at each bus, where $c_2$ is the annual operating cost per unit of power, a is the inflation rate, and b is the discount rate [14]. Equations (7) and (8) represent the upper and lower bounds of the decision variables, with the former depicting the capacity constraint and the latter signifying the rated power constraint.

The formation of the middle layer is displayed in (9)–(12):

$$\beta =\underset{x_{l,t}}{max}\delta$$

(9)

$$\mathrm{s}.\mathrm{t}.\sum _{l\in L}{x}_{l,T}=n_L-k$$

(10)

$$-x_{l,t}\le x_{l,(t+1)}\le x_{l,t}\forall l\in L,\forall t\in T$$

(11)

$$x_{l,1}=1\forall l\in L$$

(12)

Equation (9) is the objective function, where $\delta$ is obtained from the third layer. Constraint (10) is used to limit the total number of faulty lines, with $x_{l,t}=1$ indicating normal operation of line l and a fault has occurred on it when $x_{l,t}=0$, and $n_l$ represents the total number of system lines. The value constraints of decision variables are represented by (11) and (12). At the beginning, all the lines are running normally, and the value of $x_{l,(t+1)}$ is determined by $x_{l,t}$. If a fault occurs on line l at one moment, it will remain faulty in the following moment without considering the possibility of it being repaired shortly.

The lower model is written as Equations (13)–(23):

$$\delta =\underset{W_{i,t},p_{i,t},p_{l,t},r_{i,t}^{+},r_{i,t}^{-},{\theta }_{i,t}}{min}\sum _{t\in T}\sum _{i\in N}{c}_3{r}_{i,t}^{-}+c_4{r}_{i,t}^{+}$$

(13)

$$\mathrm{s}.\mathrm{t}.p_{i,t}-C_{NL,i}{p}_{l,t}+r_{i,t}^{-}-r_{i,t}^{+}=d_{i,t}-g_{i,t}:{\lambda }_{i,t}^1\forall i\in N,\forall t\in T$$

(14)

$$-M(1-x_{l,t})\le p_{l,t}-B_l({\theta }_{s\left(l\right),t}-{\theta }_{r\left(l\right),t})\le M(1-x_{l,t}):{\lambda }_{l,t}^2,{\lambda }_{l,t}^3\forall l\in L,\forall t\in T$$

(15)

$$\underline{P_l}{x}_{l,t}\le p_{l,t}\le \overline{P_l}{x}_{l,t}:{\lambda }_{l,t}^4,{\lambda }_{l,t}^5\forall l\in L,\forall t\in T$$

(16)

$$0\le r_{i,t}^{+}\le g_{i,t}:{\lambda }_{i,t}^6\forall i\in N,\forall t\in T$$

(17)

$$0\le r_{i,t}^{-}\le d_{i,t}:{\lambda }_{i,t}^7\forall i\in N,\forall t\in T$$

(18)

$$-2\pi \le {\theta }_{i,t}\le 2\pi :{\lambda }_{i,t}^8,{\lambda }_{i,t}^9\forall i\in N,\forall t\in T$$

(19)

$$W_{i,t+1}=W_{i,t}-\eta p_{i,t}:{\lambda }_{i,t}^{10}\forall i\in N,\forall t\in T$$

(20)

$$e_1{W}_i^{ESS}\le W_{i,t}\le e_2{W}_i^{ESS}:{\lambda }_{i,t}^{11},{\lambda }_{i,t}^{12}\forall i\in N,\forall t\in T$$

(21)

$$-P_i^{ESS}\le p_{i,t}\le P_i^{ESS}:{\lambda }_{i,t}^{13},{\lambda }_{i,t}^{14}\forall i\in N,\forall t\in T$$

(22)

$$W_{i,0}=W_{i,T}=0.5W_i^{ESS}:{\lambda }_i^{15},{\lambda }_i^{16}$$

(23)

The decision variables in the third layer are divided into ES operation variables and other operation variables of the microgrid cluster system. The objective function is to minimize the sum of load-shed penalties and ocean current power generation curtailment penalties in the system, where T is the total number of planned time periods, and $c_3$ and $c_4$ are penalty factors for units of power. Constraints (14)–(19) are the operation constraints of the microgrid cluster system in [15], where (14) is the node power balance equation, and $g_{i,t}=n_{g,i}{P}_{sea}$; the generation output of each bus belongs to the parameter category in this model. The charging and discharging power of ES is in the positive direction of discharge, that is, when $p_{i,t}$ is greater than 0, it is in the discharge state. $C_{NL}$ denote the bus–line correlation matrix with dimensions of $n_i\times n_l$, and the position of element 1 represents the sending end of a line, and the position of element −1 represents the receiving end of a line. The DC power flow constraint is shown in (15), and the condition for converting the inequality into an equation is that the line operates normally. Otherwise, the line power is 0, and the difference is limited by M. Equations (16)–(19) represent the boundary constraints of line power, abandoned power, load shed, and power angle, respectively. Equations (20)–(23) represent the operational constraints of ES, where the state of charge (SOC) at each moment of ES is determined by the previous SOC and power of discharging, as shown in (20), where $\eta$ represents the charge/discharge efficiency. The boundary constraints of $W_{i,t}$ and $p_{i,t}$ are shown in (21)–(23), which limit the SOC at the beginning and ending times.

A resilience enhancement strategy for onshore distribution systems incorporating line hardening and ES configuration was proposed in [14], modeling the typhoon paths and solving the problem from the perspective of robust optimization as well. Apart from differences in research subjects and decision variables, the causes of malfunctions are not limited in this paper, and the configured ES units at each bus participate in power regulation under normal operating conditions instead of just discharging during the emergency transition period. Additionally, the investment cost of ES units is added into the objective function, which is different from [14].

3. Solution Method

The common solution method for the proposed model involves dividing the problem into a master problem and a sub-problem, and solving them iteratively for each other. This is represented by the C&CG algorithm [16] and Benders decomposition [17]. The latter may disrupt the original structure of the master problem during the search for the cutting plane, while the C&CG algorithm completely avoids this problem. In addition, the C&CG algorithm shortens the iterations and accelerates the convergence speed by adding variables and constraints in each iteration process. Therefore, the C&CG algorithm is adopted to solve the proposed model.

3.1. The Sub-Problem

Due to the fact that the third layer of the model does not contain binary variables, the strong duality theory can be used to transform the min-problem into a max-problem and incorporate it into the second layer of the model. The dual variables are shown in (14)–(23), where the one with a smaller superscript corresponds to the right-hand inequality for dual variables within the same constraint. For (23), ${\lambda }_i^{15}$ is related to $W_{i,0}$. The merged objective function can be obtained as shown below:

$$\begin{array}{c}\hfill f_s=\underset{x_{l,t},\Lambda }{max}0.5W_i^{ESS}({\lambda }_i^{15}+{\lambda }_i^{16})+\sum _{t\in T}\sum _{i\in N}(d_{i,t}-g_{i,t}){\lambda }_{i,t}^1+2\pi ({\lambda }_{i,t}^8+{\lambda }_{i,t}^9)+\\ \hfill W_i^{ESS}(e_1{\lambda }_{i,t}^{11}-e_2{\lambda }_{i,t}^{12})+P_i^{ESS}({\lambda }_{i,t}^{13}+{\lambda }_{i,t}^{14})+g_{i,t}{\lambda }_{i,t}^6+d_{i,t}{\lambda }_{i,t}^7+\\ \hfill \sum _{t\in T}\sum _{l\in L}M(1-x_{l,t})({\lambda }_{l,t}^2+{\lambda }_{l,t}^3)+{\overline{P}}_l{x}_{l,t}({\lambda }_{l,t}^4+{\lambda }_{l,t}^5)\end{array}$$

(24)

$$\mathrm{s}.\mathrm{t}.\left(11\right)-\left(12\right)$$

(25)

$${\lambda }_{i,t}^1+\eta {\lambda }_{i,t}^{10}+{\lambda }_{i,t}^{13}-{\lambda }_{i,t}^{14}=0$$

(26)

$$-C_{NL}^{l,i}{\lambda }_{i,t}^1+{\lambda }_{l,t}^2-{\lambda }_{l,t}^3+{\lambda }_{l,t}^4-{\lambda }_{l,t}^5=0$$

(27)

$$-C_{NL}^{i,l}{B}_l({\lambda }_{l,t}^2-{\lambda }_{l,t}^3)+{\lambda }_{i,t}^8-{\lambda }_{i,t}^9=0$$

(28)

$${\lambda }_{i,t-1}^{10}-{\lambda }_{i,t}^{10}+{\lambda }_{i,t}^{11}-{\lambda }_{i,t}^{12}\le 0$$

(29)

$$-{\lambda }_{i,t}^1+{\lambda }_{i,t}^6\le c_4$$

(30)

$${\lambda }_{i,t}^1+{\lambda }_{i,t}^7\le c_3$$

(31)

$${\lambda }_{l,t}^2,{\lambda }_{l,t}^3,{\lambda }_{l,t}^4,{\lambda }_{l,t}^5,{\lambda }_{i,t}^6,{\lambda }_{i,t}^7,{\lambda }_{i,t}^8,{\lambda }_{i,t}^9,{\lambda }_{i,t}^{11},{\lambda }_{i,t}^{12},{\lambda }_{i,t}^{13},{\lambda }_{i,t}^{14}\le 0$$

(32)

The constraints of the sub-problem include not only the constraints of the second layer of the model, but also the constraints of the dual variables, i.e., Equations (26)–(32). The objective function of the sub-problem is nonlinear, and the nonlinear term comes from the product of binary and dual variables, i.e., $x_{l,t}({\lambda }_{l,t}^2+{\lambda }_{l,t}^3)$ and $x_{l,t}({\lambda }_{l,t}^4+{\lambda }_{l,t}^5)$. This type of linearization problem is similar to $z=xy$, where $x/y$ are binary/continuous variables, respectively [18]. Therefore, z can be introduced as a new variable to replace the original nonlinear term, and constraints can be added to restrict the value of z, that is, $-m(1-x)\le z-y\le m(1-x)$ and $y_{min}x\le z\le y_{max}x$, which will not be further elaborated in this paper. The result of the sub-problem is the worst-case scenario considering the current ES configuration, and the upper bound of the problem in the j-th iteration resulting in $U{B}^j=\min (U{B}^{j-1},F+f_s)$.

3.2. The Master Problem

The initial form of the main problem is the same as the first layer of the model, but, as the iteration progresses, new variables and constraints are generated based on the faulty line scenario after each sub-problem is solved, corresponding to the decision variables and corresponding operational constraints of the third layer of the model. It is worth noting that the state of faulty line $x_{l,t}$ in the newly added constraints is not a variable, but the current optimal solution of the sub-problem, which can be denoted as $x_{l,t}^{*}$. The lower bound of the problem in the j-th iteration can be expressed as $L{B}^j=F+\beta$, corresponding to the objective of the master problem. The upper and lower bounds of the problem gradually converge during the iteration process. When the lower-bound value is greater than or equal to the upper-bound value, the algorithm converges. The flowchart of the algorithm is depicted in Figure 2. First, basic information and parameters about the studied microgrid are set, and lower and upper bounds are initialized. Second, we solve the master problem and the sub-problem separately, and determine whether the difference between the upper and lower bounds meets the convergence threshold. Third, if convergence has not yet been achieved, we create a new set of variables and constraints for the third layer of the DAD model and add them to the master problem. We repeat steps 2–3 until the algorithm converges.

4. Case Studies

To verify the effectiveness of the proposed model in enhancing system resilience considering line faults, simulations were conducted based on modified IEEE six-bus and fifty-seven-bus systems [19]. Additionally, all the numerical experiments were performed on the Matlab R2021b platform on a PC with Intel Core i9-12900H 2.50 GHZ processor and 16 GB RAM. The master and sub-problems in C&CG computations were solved using Gurobi 10.0.1.

4.1. Six-Bus System

4.1.1. Case Parameters

From Figure 3, the modified six-bus system has its own power generation equipment, ES equipment, and load (i.e., UUV) for each bus, where a bus is abstracted from a microgrid. The number of ocean current energy generation units for each bus is set to 10, with a blade radius of 1 m and a power generation efficiency of 0.95; the total power generation is the sum of the individual power generation units obtained from the model of Section 2.1. The number of ES units at each bus is 1. The system topology is also exhibited in Figure 3. It can be seen that the system has 11 lines, the transmission capacity of each line is set at 20kW, and the degree of each bus is greater than or equal to 3, indicating that the network structure is fully connected.

By reducing the load time series data of a certain region, a typical day was obtained to represent the planning year. Ocean current power generation and typical daily load data are shown in Figure 4.

Although the total power generated by the generators exceeds the total load demand, there are periods, such as the 10th to 15th hours, when the power generation is insufficient to support the load demand. It is necessary to configure ES to alleviate the mismatch between peak power generation and peak load, which is beneficial for improving the resilience of the system.

Regarding the configuration parameters of ES, $n=6\%$, $y=12$, $c_1=120$, $c_2=0.18$, $a=1.5\%$, $b=9\%$, $W_{max}^{ESS}=100$, $W_{min}^{ESS}=1$, and $P_{max}^{ESS}=20$. Additionally, regarding system operating parameters, $k=3$, $c_3=8$, $c_4=10$, $e_1=0.1$, and $e_2=0.9$.

4.1.2. Configuration Scheme for ES

Table 1. Configuration scheme of the IEEE six-bus system.

Bus No.	Capacity/kWh	Rated Power/kW
1	92.2	20
2	1	0.22
3	100	18.10
4	57.44	20
5	94.15	20
6	61.98	13.77

shows the ES capacity and rated power configured at each bus.

This configuration scheme is the most robust solution for any three faulty lines in the system, corresponding to the worst-case scenario where lines 1–4, 2–4, and 4–5 simultaneously fail at $t=2$. Bus 4 will be disconnected from the system and operate independently. The case converges after six iterations.

The charging and discharging power of ES, load shed, and abandoned power are investigated in Figure 5. The charging and discharging of ES in Figure 5a are related to the power generation of ocean current energy, thus exhibiting a similar periodic variation pattern. Due to the system failure at $t=2$, there is a small amount of load shed at the corresponding time in Figure 5b, but it is later balanced by ES, indicating the positive role of ES in improving system resilience in terms of fault recovery. By comparing the power generation and load data of the system, it can be inferred that the load shedding occurring at $t=11-14$ is due to low power generation, and ES cannot bear the entire load demand of the system, resulting in partial load shed. The discarded electricity in Figure 5c comes from bus 4, which is due to the inability of isolated buses to exchange power with other buses. When ES and load cannot consume power generation, excess electricity can only be discarded.

4.1.3. Comparison before and after ES Configuration

Based on the above operational data, the resilience of each bus can be obtained by

$$R_i=1-{\displaystyle \frac{{\displaystyle \sum _{t\in T}}{r}_{i,t}^{-}}{{\displaystyle \sum _{t\in T}}{d}_{i,t}}}\forall i\in N$$

(33)

which is used as a metric to gauge the microgrid’s ability to respond to fluctuations in ocean current generation and line failures. In order to investigate the impact of distributed ES on improving system resilience, this section compares the performance differences among no ES configuration, distributed ES configuration, and centralized ES configuration. The distributed ES configuration scheme follows the previous solution listed in Table 1; other parameters remain unchanged. One among all buses is selected as the access point for centralized ES, and the upper/lower limits of capacity configuration and bound of rated power are set at 600/10 kWh and 200 kW, respectively.

The centralized configuration optimization result specifies configuring ES with a capacity of 277.17 kWh and a rated power of 69.75 kW at bus 3. In addition, the worst fault scenario obtained is the simultaneous tripping of lines 2–3, 2–5, and 3–6 at $t=2$. The resilience and abandoned power rate of each bus under three cases based on this fault scenario is shown in

Table 2. Comparison of before and after ES configuration.

Bus No.	Without ES		With Distributed ES		With Centralized ES
	Resilience	Abandoned Power Rate	Resilience	Abandoned Power Rate	Resilience	Abandoned Power Rate
1	60.08%	57.47%	100%	27.96%	60.74%	42.53%
2	67.31%	52.77%	100%	60.08%	74.59%	70.69%
3	21.52%	51.42%	97.43%	17.24%	100%	12.71%
4	53.86%	43.08%	100%	31.57%	100%	44.36%
5	49.81%	18.15%	95.10%	0	100%	18.15%
6	53.88%	57.08%	100%	0	97.56%	41.26%
Total	64.14%	45.93%	99.16%	19.72%	83.39%	31.27%

.

A numerical comparison can demonstrate the advantages of distributed ES configuration in improving system resilience. Compared to the case without ES configuration, whether it is a centralized or distributed ES configuration, the load satisfaction of buses has been improved to varying degrees, and the abandoned power rate has basically decreased. Therefore, a well-designed ES configuration can improve system resilience and facilitate the integration of renewable energy sources. Furthermore, distributed ES configuration yields better performance in terms of load response ability and renewable energy accommodation from Table 2.

4.2. Fifty-Seven-Bus System

The system has 80 lines; the detailed information and topology are shown in [20]. The number of ocean current energy generation units of each bus is still 10, while the blade radius is 0.8 m. Each bus needs to be configured with a certain scale of ES, setting $W_{min}^{ESS}=1$ kWh, $W_{max}^{ESS}=100$ kWh, $P_{max}^{ESS}=20$ kW, and $\overline{P_l}=50$ kW, the rest parameters remain consistent with the settings of the six-bus system.

Figure 6 shows the results of configuration scheme for the 57-bus system in terms of capacity and rated power. The worst-case scenario solved is if lines 9–12, 12–13, and 18–19 are simultaneously damaged at time $t=2$. The experimental results demonstrate the applicability of the proposed method on large-scale cases. Surprisingly, the proposed method is not only applicable in large-scale systems, but also exhibits exceptional computational efficiency, converging in just four iterations.

5. Conclusions

In order to enhance system resilience, a two-stage robust optimization framework was proposed for configuring distributed ES in microgrid cluster systems. First, a DAD model considering line faults was established, in which the planning and operation of ES have been jointly optimized. Second, the C&CG algorithm was adopted to divide the model into two stages for an iterative solution. It was proven in the six-bus system that the configuration of distributed ES has significant advantages in improving the resilience and promote renewable energy accommodation of the system when dealing with faults compared to not configuring ES or using centralized ES. Specifically, a distributed ES configuration has increased system resilience by 54.60% and reduced the abandoned power rate by 57.06% compared to the situation without ES configuration. Furthermore, the the applicability of the proposed method on large-scale cases is demonstrated in the 57-bus system. In addition to the combination uncertainty of fault lines, the number uncertainty of damaged lines will also be considered in our future work.