1. Introduction
Several major blackouts occurred worldwide in the past few decades, e.g., the 14 August 2003 blackout in the northern USA and Canada [
1], the 4 November 2006 blackout in Western Europe [
2], the 30–31 July 2012 blackout in India [
3] and the 31 March 2015 blackout in Turkey [
4]. A major blackout, caused by natural disasters, cascading failures or human errors, threatens the security and reliability of power systems and has negative impacts on social and economic development [
5,
6]. A more secure and reliable power system considering the severe weather and multiple contingencies is required [
7,
8,
9]. The task of power system restoration is to restore the outage system quickly and safely. Therefore, efficient restoration strategies receive increasing attentions from the power industry [
10,
11].
The primary goal of system restoration is to restore generating units and loads, and recover the transmission network quickly [
12,
13,
14,
15,
16]. Parallel restoration strategies, which sectionalize a power system into several subsystems and restore these subsystems simultaneously, are preferred since it accelerates the system restoration process. The first step for parallel restoration is to determine a sectionalizing scheme. Some utilities and ISOs, including PJM (Pennsylvania–New Jersey–Maryland Interconnection) in United States [
17], National Grid Company in United Kingdom [
18] and Chongqing Power of China State Grid [
19], have adopted parallel restoration plans for their systems. The sectionalizing schemes are designed mainly based on the operators’ knowledge and experience.
In recent years, theoretical investigations on sectionalizing methods for parallel restoration have attracted researchers’ interests. The sectionalizing problem, which is usually determined by solving an optimization problem with constraints on black-start generators and generation-load imbalance, is proved to be a non-deterministic polynomial complete problem (NP-complete) [
20]. There is no polynomial-time algorithm to solve it. Thus, it is difficult and time consuming to solve the sectionalizing problem for large-scale power systems. Wang, C. et al. applied the ordered binary decision diagram-based method (OBDD) to reduce the candidate sectionalizing schemes for determining optimal solution [
20]. Afrakhte et al. applied genetic algorithm (GA) to find the subsystem boundaries and minimize the index of energy not supplied (ENS) [
21]. Liang et al. also used GA to create subsystems with consideration of the start-up sequence of units [
22]. Quiros-Tortos et al. put forward a sectionalizing methodology based on the cut-set matrix theory [
23]. It was simple and useful for parallel restoration. Based on it, Sun et al. developed an optimal network sectionalizing strategy considering cranking generating units [
24]. To minimize the time of units for getting cranking power is one of its objectives. These two methods are based on the recursive bisection and obtain two new subsystems by one iteration. After multiple iterations, multiple subsystems can be generated. Sarmadi et al. proposed a heuristic search algorithm to define an islands’ matrix [
25]. Considering the wide area measurement system (WAMS) information, the partitioning scheme for parallel restoration is determined by modifying the islands’ matrix. A community detection algorithm was employed into the sectionalizing method for parallel restoration by Lin et al. [
26]. The method can also generate the sequence for resynchronization of subsystems. The spectral graph clustering method was used in [
27,
28] to design the optimal sectionalizing schemes. The sectionalized results depended on the eigenvalues of the Laplacian matrix of the system network, which were the impedance characteristics of branches.
The aforementioned methods are focused on the topology of system without consideration of restoration time of the subsystems. A practical and efficient sectionalizing scheme can reduce the system restoration time (SRT) of a power system, which depends on the subsystem with the slowest restoration process, and enhance the efficiency of parallel restoration. Therefore, the restoration time of each subsystem is an important basis for determining sectionalizing schemes and should be estimated. Adibi discussed the estimation of the restoration duration [
29]. The start-up time of generating units and the restoration time of branches are the main time consumption for restoring a power system. The time for the load pick up is ignored [
30,
31].
The constraints, which should be considered in the sectionalizing problem, are summarized in references [
20,
21,
23]. One important constraint is the minimum active power output of generating units, such as thermal units. To meet the minimum output constraint on generating units, researches generally take the total amount of load as the constraint but have not considered the effects of dispatchable loads [
24,
25]. Since dispatchable loads can provide the variable power demand, they are widely used in power system operation, e.g., in demand side management [
32,
33]. For power system restoration, they can be restored flexibly to balance the output of generating units [
30,
34]. Especially, during the start-up process, a generator cannot be controlled effectively since its control system is designed for use between minimum and maximum load [
12]. In this stage, dispatchable loads can be picked up to balance the output of restored units for system security.
This paper proposes a novel sectionalizing method for parallel restoration based on the minimum spanning tree (MST). The objective is to minimize the SRT, considering the start-up time of generating units and the restoration time of branches. A power system can be abstracted as a weighted graph. The weight of each edge, representing a transmission line or a transformer, is specified with its restoration time. For a generating unit, a virtual node and a virtual edge are added. The weight of the virtual edge is set as the start-up time of the unit. Since the MST is a spanning tree with the minimum total weight of edges in a graph, it can determine the skeleton network, which is a minimum adequate network connecting all available buses [
30], with the minimum restoration time of the power system. The effect of dispatchable loads is also considered in this paper. Our method offers some distinct advantages:
- (1)
The start-up time of generating units and the restoration time of branches are considered in the problem formulation, which help minimize the SRT.
- (2)
By employing the dispatchable loads as an important constraint, it ensures that generating units are restored safely.
- (3)
Candidate boundary lines of a system are identified according to the skeleton network and used to generate candidate sectionalizing schemes. Expansion in system size will not significantly increase the complexity of the proposed method.
- (4)
It can be applied to power systems under various conditions, including the conditions when some components are not available, and provide multiple schemes for dispatchers.
The remainder of this paper is as follows.
Section 2 introduces the formulation of the sectionalizing problem.
Section 3 describes the proposed method. Case studies on IEEE 39-bus and 118-bus test systems and discussions are provided in
Section 4. The conclusion is given in
Section 5.
3. The Proposed Method for Sectionalizing
In this section, a sectionalizing method based on the MST for power system parallel restoration is proposed.
Section 3.1 introduces the overall procedure to generate sectionalizing schemes, which consists of four steps.
Section 3.2,
Section 3.3,
Section 3.4 and
Section 3.5 illustrate the steps in detail.
3.1. Procedure of the Proposed Sectionalizing Method
A four-step method is proposed to provide sectionalizing schemes for power system parallel restoration: (1) construct the skeleton network; (2) generate candidate sectionalizing schemes; (3) evaluate constraints; and (4) determine optimal or near-optimal schemes. The flow diagram of the proposed method is shown in
Figure 1.
The first step abstracts a power system as a graph. Faulted devices of the system are removed from the graph in order to meet Equation (5). Based on MST, a skeleton network of the graph is constructed considering the start-up time of generating units and the restoration time of branches.
The second step generates candidate sectionalizing schemes by exhaustive search. For each scheme, the skeleton network is divided into several sub-networks. Each sub-network can be used for estimating the total restoration time of the corresponding subsystem. In a subsystem, sequential restoration strategies, i.e., restoring outage elements one-by-one from a black-start unit, are usually adopted. Thus, there must be at least one black-start unit to provide cranking power in each subsystem. In order to accelerate the restoration process and shorten the outage time, it is preferred to have as many subsystems as possible. In this paper, it is assumed that each subsystem contains only one black-start unit. Therefore, the maximum number of subsystems is achieved and equals the number of black-start units.
The third step is aimed at finding the feasible sectionalizing schemes by examining Equations (2)–(4). These constraints are formulated as three inequalities. Details are provided in
Section 3.4.
The fourth step calculates the SRT of each sectionalizing scheme. In a scheme, the estimation of restoration time depends on the subsystem with the slowest restoration process. The scheme, which consumes the minimum SRT, is selected as the final sectionalizing solution.
3.2. Constructing the Skeleton Network Based on Minimum Spanning Tree
3.2.1. Abstraction of a Power System
A power system can be described as a weighted connected graph . Generating-unit, substation and load buses of a power system are modeled as nodes, i.e., vertices in the graph. For each generating unit, a virtual node, as well as a virtual line connecting the virtual node to the corresponding bus node, are added to the graph. The set of edges represents transmission lines, transformers and the virtual lines associated with generating unites. If there are double-circuit or multi-circuit lines or transformers, they are abstracted as a single edge in the graph. Note that faulted devices should be removed from the graph to satisfy Equation (5).
For each edge
, a weight
is assigned. In this paper, the restoration time (in minutes) of a transmission line or transformer is set as the weight of the corresponding edge while the start-up time (in minutes) of a generating unit is regarded as the weight of the corresponding virtual edge. In general, weather conditions, geographical locations and the operation of dispatchers effect the restoration time of a branch. The “most likely” restoration time of a branch is employed as its restoration time [
24]. The start-up time depends on the inherent characteristics of a generating unit and can be obtained from its design document.
Figure 2 shows a portion of a graph with weights. A transmission line is connected between buses
G1 and
S1. It requires 5 min to restore the transmission line, so the weight of line
is set to be 5. A generating unit locates at bus
G1. A virtual bus
G1’ is used to represent the generating unit. The start-up time of the unit is 30 min. Thus, the weight
of virtual edge
is set to be 30.
3.2.2. The skeleton network Based on Minimum Spanning Tree
Let
denote a spanning tree of
, where
. If the total weight
of
, defined by Equation (6), is the minimum among all spanning trees of
, then
is regarded as a minimum spanning tree (MST) of
[
41]. In this paper, the MST
is used as the skeleton network of the power system. The
represents the set of edges in the skeleton network.
equals the restoration or start-up time for the corresponding component. Therefore, the
equals to the total restoration time of actual lines and the start-up time of generating units in the skeleton network.
Theorem 1. Suppose is a MST of . Let denote a subtree of , where and . is an induced subgraph of , whose vertex set is , the same as , and whose edge set consists of all edges of which have both ends in . Then, is a MST of .
Proof of Theorem 1. Suppose for contradiction that is not a MST of . Let be a MST of , so . Construct a tree , where . It can be verified that is a spanning tree of and , so is not a MST of , and we have a contradiction. □
Based on Theorem 1, the skeleton network is used for sectionalizing a power system and estimating the restoration time of subsystems, considering the start-up time of generating units and the restoration time of branches. The Prim’s algorithm is a common MST method [
41]. It starts from an arbitrary root vertex until the tree spans all the vertices. The minimum weight edge from the growing tree to an isolated vertex is found at each step. It is similar to the greedy algorithm as it adds an edge to the spanning tree, which contributes the minimum amount possible to the tree’s weight at each step. A skeleton network of IEEE 9-bus test system based on the Prim’s algorithm is shown in
Figure 3.
In
Figure 3a, the value between two buses is the edge weight. For example, the weight of edge 2 connecting bus-4 and bus-5 is 5. The algorithm adds edges 1, 2, 3, 5, 6, 8, 7 and 4 to the spanning tree in sequence.
Figure 3b shows the final skeleton network of the test system.
The skeleton network is used for sectionalizing a power system for parallel restoration. The restoration time of each subsystem can be estimated according to each skeleton subnetwork. Therefore, the restoration time
of subsystem
is calculated as
where
is the weight of edge
in the skeleton network;
is a binary variable. If
is equal to 1, line
belongs to subnetwork
.
3.3. Candidate Sectionalizing Schemes Generation
In order to sectionalize a power system for parallel restoration successfully, there must be at least a black-start generating unit in each subsystem. Between any two black-start units, there is only one acyclic path on the skeleton network. Edges on the acyclic paths connecting any two black-start units can be selected as the boundary lines. These edges are named by the candidate edges (CE) for boundary lines in this paper.
Exhaustive search is used to provide all candidate sectionalizing schemes. In a scheme, if the boundary lines include CEs, the skeleton network is sectionalized into subnetworks, namely subsystems. The number of CEs is generally much less than that of all edges. Therefore, the number of candidate sectionalizing schemes for constraints evaluation does not increase largely as the scale of system expands.
3.4. Constraint Evaluation
The proposed objective function is related to the edges in the graph. However, sectionalizing Equations (2)–(4) are related to the nodes. The incidence matrix, representing the relationship between nodes and edges in a graph, is used to formulate sectionalizing constraints.
3.4.1. Constraint on the Number of Subsystems
Equation (2) guarantees that there is at least one available black-start generating unit for supplying the cranking power in each subsystem. This paper defines a black-start incidence matrix
as follows
where
represents the relationship between black-start unit node
and edge
.
is the number of edges and
is the number of nodes of the skeleton network. If an available black-start unit locates at node
which is one terminal of edge
, the
is equal to 1. For example, the black-start units locate at bus-1 and bus-2 in
Figure 3b.
is obtained, with
while other elements equal to 0. It indicates that the black-start unit at bus-1 is related to edge 1, while the black-start unit at bus-2 is related to edge 7. In subsystem
, a black-start indicator index is defined as
In this paper, a black-start judgment matrix is defined as
. The number of available black-start generating units in each subsystem is represented by
. It is equal to the number of non-zero elements in
. To meet Equation (2), all elements of the
must be not less than 1. It is described as
where
.
3.4.2. Minimum Output Constraint on Generating Units
The minimum active power output of a generating unit depends on the unit’s characteristic. It is not the emphasis of this paper. Dispatchable loads are used to balance the minimum output during the starting process of generating units and maintain a necessary amount of power output of each generating unit in each subsystem. This paper defines a minimum output incidence matrix
as follows
If edge
connects a generating unit or a dispatchable load located at bus
, the
is equal to 1. For example, there are generating units at buses 1, 2 and 3, and a dispatchable load at bus-9 in
Figure 3b.
is obtained, with
while other elements equal to 0.
A minimum output judgment matrix is defined as
.
represents the imbalance of generating units’ minimum output and dispatchable loads (IMODL) in subsystem
and is calculated by
where
is the IMODL at bus
. The
is the component-wise logical operator “AND”. To meet Equation (3), all elements of the
must be not larger than 0. It is described as
where
.
3.4.3. Capacity Constraint on Generating Units
Each subsystem should have sufficient capacity to maintain a satisfactory frequency by matching generation capacity and load demand. All critical loads need to be restored before resynchronization of subsystems. Therefore, this paper defines a capacity incidence matrix
:
If edge
connects a generating unit or a critical load located at bus
, the
is equal to 1. For example, there are critical loads at bus-5 and bus-7 in
Figure 3b.
is obtained, with
while other elements are equal to 0.
A capacity judgment matrix, defined as
.
, represents the imbalance of generating units’ output and restored loads (IORL) in subsystem
and is calculated by
where
is the imbalance of units’ output and critical loads (IOCL) at bus
. To meet Equation (4), all elements of the
must be not less than 0. It is described as
The proposed model contains four constraints. Equation (5) is achieved in the abstraction of a power system in
Section 3.2.1. In this section, Equations (2)–(4) are reformulated by Equations (10), (13) and (16), respectively. If Equations (10), (13) and (16) are true, the corresponding constraints are satisfied and feasible solutions are achieved.
3.5. Optimal Sectionalizing Schemes
After examining all kinds of constraints, feasible sectionalizing schemes are obtained. In a scheme, the restoration time of each subsystem is estimated based on its skeleton subnetwork with consideration of the generating units’ start-up time and the branches’ restoration time. Sectionalizing schemes with the minimum SRT are the optimal solutions for parallel restoration.
For a large power system, there can be more than one optimal scheme with the same minimum SRT. One of these schemes can be selected for application according to other factors by power system dispatchers, such as the generation-load imbalance.