1. Introduction
In recent years, hurricanes, earthquakes, and other natural disasters have caused large-scale power interruptions for a long time, which has brought huge economic losses to society. Forexample, in 2008, China’s extreme snow and ice disaster damaged power grid components in 13 provinces, witha total of 36,740 line faults, 2018 substations affected, andpower grids in some areas collapsed and even entered isolated network operation[
1]. Hurricane Sandy in 2012 destroyed the elevated distribution system in New York, resulting in the loss of nearly 1000 power poles and more than 900 transformers, andthe power failure of 8,371,242 customers[
2]. It can be seen from the power outage events in recent years that the power production, transmission, anddistribution modes of centralized power generation and long-distance transmission of large units can not fully meet the power demand. Therefore, research on the resilient power grid that can quickly respond to natural disasters has become a trending topic among scholars and has also become an urgent demand for a stable and reliable energy supply in today’s society. Asa kind of Distributed Generation (DG), Mobile Energy Storage Vehicles (MEVs) play a pivotal role in the construction of resilient microgrids. Withthe characteristics of flexibility and stability, MEVs can quickly move to the damaged areas and supply power to the isolated nodes by forming multi-microgrids after a natural disaster occurs, greatly reducing the blackout time. However, due to the high cost of mobile energy storage vehicles, it is uneconomical to use them in large quantities. Inaddition to the installation of mobile energy storage vehicles, another measure to enhance resilience is the addition of tie lines. When some of the original transmission lines are damaged and failed, thetie lines can be closed remotely to restore those isolated island nodes, which costs less than the installation of mobile vehicles. Thus, bythe usage of tie lines and MEVs, when the fault occurs, MEVs and tie lines can be used to reconstruct the power grid to form a local power supply area, so as to realize the rapid recovery of power lossload.
Most literature focuses on the planning of defense resources before power system damage and the allocation of reconfiguration resources and network topology reconfiguration after power system damage. Leietal.[
3] proposed a two-stage microgrid reconfiguration model which contained pre-positioning and real-time allocation of mobile generators. Furthermore, different from[
3], only critical line switches were considered available in network reconfiguration. In[
4], adistributed multiagent coordination scheme of communication network besides the reconfiguration of electric network was proposed for autonomous communication. Gilanietal.[
5] considered different types of DGs in microgrid formation and used the time series forecasting method to model the loaduncertainty.
A series of models have been proposed to formulate this issue. In[
6], Guoetal. formulated the distribution network reconfiguration as a mixed-integer quadratic programming problem by using electric vehicles and remote-controlled switches to minimize energy loss and switching operation times, withconsidering the battery degradation cost. Maetal.[
7] proposed a two-stage stochastic mixed-integer linear program model by taking measures such as line hardening, installing DGs, and adding switches in transmission lines. Ranjbaretal.[
8] built a two-stage stochastic planning model by using distributed energy resources and considering the different operation modes in normal and emergency conditions and classified the emergency scenarios into medium, serious and complete damage according to the degree of damage. In[
9], Ghasemietal. proposed a three-stage stochastic planning model by hardening lines and placing DGs. They also considered line damage uncertainty compared to the traditional two-stage planning model. Baghbanzadehetal.[
10] built a tri-level Defender–Attacker–Defender (DAD) model by using distribution generation to enhance the resilience of distribution networks. Leietal.[
11] built a multi-period DAD model to realize the defense resources planning and allocation with the optimization objective of minimizing the shed loads. Mousavizadehetal.[
12] firstly defined the index of power system resilience, andthen constructed a two-stage linear model using DGs such as renewable energy which studied the optimal management of energy storage units and DG units and evaluated the resilience of the system. Agrawaetal.[
13] proposed a three-stage self-healing method with distributed energy resources to restore maximum priority loads. To sum up, these models can be roughly classified as either deterministic or stochastic. Since the damaged scenario is uncertain, stochastic methods are better suited for the solutions of the microgrid reconfigurationproblem.
Accordingly, some algorithms were designed to solve these problems. Dingetal.[
14] used a heuristic algorithm to solve the mixed-integer linear programming problem of power grid reconfiguration, which resulted in multiple solutions with different rules and interests. Huangetal.[
15] provided a targeted algorithm based on the nested column-and-constraint generation decomposition to solve a two-stage robust mixed-integer optimization model. Chandaetal.[
16] used a two-stage reconfiguration algorithm to enhance the resiliency of the power system after developing a new method to quantify the resiliency which was based on complex network analysis and network percolation theory. In[
17], thetraditional distribution system was transformed into multiple autonomous microgrids by optimizing the scale and location of DGs to improve the resiliency of the distribution system, andparticle swarm optimization and genetic algorithm were used to solve the problem. In[
18], the key infrastructure nodes in the network were sorted according to their priorities, anda customized PSO algorithm to allocate DGs was designed to maximize resilience and minimize power loss. Khalilietal.[
19] proposed a multi-objective optimization model to construct multi-microgrids, which was solved by an exchange marketalgorithm.
With regard to the use of reconfiguration resources, Taherietal.[
20] not only considered the pre-positioning and scheduling of mobile energy storage vehicles but also crews who could operate switches. Zhouetal.[
21] proposed a distributed fixed-time secondary control scheme that was based on a general directed communication graph. Xuetal.[
22] considered the dispatch of repair crews in addition to mobile power sources in the transportation system to restore critical loads. Zhangetal.[
23] built a three-stage stochastic planning model for mobile emergency generator allocation which added a capacity decision-making procedure in the first stage compared with the two-stage model. Erenougluetal.[
24] proposed a mixed-integer quadratic programming model for dispatching and scheduling multiple types of sources including mobile energy storage systems, mobile emergency generators, and repair crews. Combing through the literature found that mobile energy storage vehicles as a kind of DG were widely studied and applied to the microgrid reconfiguration. However, few studies focused on the application of tie lines, andthey only studied the use of existing tie lines in the power system without considering the addition of tie lines. Shietal.[
25] considered the use of tie lines in addition to original line switches for network reconfiguration, andproposed an algorithm that was based on an incidence matrix to identify radial network topology. In[
26], tie lines were also utilized in network reconfiguration, andthe master–slave control method was deployed to overcome the challenge of voltage loops.
Based on the above discussion, ascenario-based stochastic two-stage mixed-integer linear program model based on hybrid microgrid defense resources is proposed in this paper. This model includes the pre-position of the MEVs and the location selection of tie line addition before damage, andthe dynamic dispatching of MEVs and operation of added tie lines after damage. Amore detailed problem statement will be presented in
Section 2.
The major contributions of this study can be summarized as:
- (1)
The multi-stage application of tie lines including addition and operation is firstly proposed and studied. This novel method effectively enhances power system resilience and the diversity of reconfiguration resources;
- (2)
A two-stage framework for the combination use of microgrid defense resources by dispatching MEVs and employing tie lines is firstly proposed, which can restore the critical loads immediately with the limited use of MEVs in severely damaged scenarios;
- (3)
A scenario-based two-stage mixed-integer linear program model is formulated and explained in detail, which is generalizable and easy to understand.
- (4)
Several experiments have been conducted to compare the performance of the proposed method with other methods, demonstrating that the combined optimization model with MEVs and tie lines is competitive.
The remainder of this paper is organized as follows.
Section 2 describes the action mechanisms of MEVs and tie lines on power grid restoration and proposes a two-stage microgrid reconfiguration framework.
Section 3 formulates the two-stage model. In
Section 4 the progressive hedging algorithm is discussed, followed by
Section 5 which presents numerical case studies. Finally, thepaper is concluded in
Section 6.
4. Progressive Hedging Algorithm
The proposed two-stage microgrid optimization problem is where some of the decision variables are binary variables and the data is uncertain is a multi-stage mixed-integer stochastic programming problem, which is difficult to solve in a short time. To represent the uncertainty in data, we use the common approach by formulating a finite number of scenarios with corresponding probabilities for the values of uncertain parameters in this paper. Since the Progressive Hedging Algorithm (PHA) has proved an effective method for solving multi-stage stochastic programming problems [
27,
28], especially those with discrete decision variables in every stage [
29], we apply this algorithm to solve this scenario-based two-stage stochastic optimization problem. Based on [
3], the implementation of the PHA is described in Algorithm 2. The PHA decomposes the extended form according to the scenario and iteratively solves the penalized versions of the sub-problems to gradually enhance the realizability, thus reducing the computational difficulty related to large problem instances.
Algorithm 2: Progressive Hedging Algorithm. |
|
We consider the stochastic program of the following form:
where
,
,
denote first-stage decision variables, second-stage decision variables, and second-stage objective function, respectively.
is the feasible set of
defined by constraints such as (
2), (
3) and (
10),
is the feasible set of
defined by other constraints. Since the first-stage decisions are independent of specific scenarios, we introduce the non-anticipativity constraints to make use of the block-diagonal structure of the problem. Therefore, the scenario-based two-stage stochastic optimization problem can be reformulated as:
where the equation
is the non-anticipativity constraint enforcing
. By dualizing these non-anticipativity constraints, the form of the Lagrangian duality problem is as follows:
where
is the dual vector;
is the weight of scenario
n. As for the consensus criteria in step2, it is based on the average per-scenario deviation from the average denoted as
:
where
denotes a threshold set as 0.1 in this paper;
denotes the weighted average of
in the previous iteration;
l denotes the
lth element of the corresponding vector;
N denotes the number of scenarios.