1. Introduction
In recent years, due to human operation factors, power equipment failures, and extreme weather disasters, large-scale power outages in domestic and international power systems continue to occur from time to time [
1,
2], such as the “6.16” blackout that affected the whole country in Argentina in 2019 [
3], the “2.15” power outage in the Texas power grid in 2021 [
4], and the “33” island-wide blackout in Taiwan in 2022, all of which brought about serious economic losses. The “2.15” extreme cold wave-induced lasting large-scale power outage in the United States and the “33” island-wide blackout in Taiwan in 2022, for example, each led to serious economic losses and caused bad impacts on residents’ lives and social functioning. These events have caused severe economic losses and impacted the lives of residents and the functioning of society. The complete restoration control process is long and complex, and prioritizing restoration of the underwritten grid can maximize the protection of critical urban loads in the case of extreme disasters and external damages. At the same time, it is difficult to significantly improve the recovery speed after a power outage simply by using a unit-optimal recovery path control strategy; therefore, the reasonable development of a unit recovery path search strategy can improve the grid recovery speed and risk resilience, which has important research significance and engineering value.
Black-start power is the basis of the entire outage restoration process, which refers to generating units with self-starting capability that can provide power support for network restoration without relying on external power conditions in the case of a total system outage [
5]. The mentioned unit start-up refers to the provision of start-up power by the black-start power source to the generating units that are unable to self-start after a major outage to help them start and restore their generation capacity [
6], laying the foundation for network restoration and load recovery. The development of the unit start-up strategy includes the selection of the unit start-up sequence and the recovery path, and the decisions between the two are not fragmented and single, but often internally linked [
7].
A significant amount of research work has been carried out by domestic and international researchers on the optimization of generator starting strategies. The literature [
8] divided the power system restoration decision process into different milestone stages and formulated the unit start-up strategy with the goal of the shortest restoration time at each step; the literature [
9] combined the sequencing method and traversal means to find the unit start-up sequence with the goal of restoring as many non-black-start units as possible in a shorter time; the literature [
10] used the maximum generation capacity restored by the system in a given time period as the goal. In the literature [
11], the influence of the charging time of the recovery path on the start-up phase of the unit is taken into account in the development of the unit start-up sequence; the literature [
12] summarizes various factors that determine whether the unit can be successfully recovered during the black-start period and finds the unit recovery path by combining the former with the K shortest path algorithm. In the literature [
13], the line operation time, remote operation coverage factor, and line recovery probability are considered comprehensively. The line commissioning time expectation function is then constructed, and the unit start-up priority index is defined to decide the start-up order. Meanwhile the recovery path of the unit to be started is optimized based on Dijkstra’s algorithm. In the literature [
14], the unit start sequence decision is abstracted as a multi-constrained backpack problem, and the unit to be started at the next moment is obtained by solving the problem using a data envelopment analysis method and a backtracking algorithm.
The existing research works usually ignore the mutual coupling between the recovery order of the unit to be started and the optimal start path when formulating the unit start strategy, and consider them independently in the solution process, thus easily causing the unit start order and the recovery path to be not optimal after the combination, which affects the speed and reliability of power system recovery. Therefore, this paper will focus on a black-start strategy that takes into account the recovery path and start sequence of the unit, in order to reduce the recovery time of the unit and improve the recovery success rate.
This paper firstly constructs mathematical models for different power units in the black-start stage, including black-start power sources of hydroelectric units, gas turbines and FCB coal-fired units, and non-black-start power sources; secondly, based on the principle of Dijkstra’s shortest path algorithm, it searches for the optimal recovery path of units to be started after a major power outage; it then combines the unit capacity, climbing rate, starting power, and other characteristics with the recovery path. Based on the unit start sequence and optimal recovery path, a complete black-start strategy is obtained by taking into account the unit start constraints, and the effectiveness of this strategy is verified by simulation.
In developing the power system restoration plan, the black-start units in the system are first started to provide initial power for the restoration of the bonded grid, and then for each step of the target generators to be started, a restoration path is developed with the black-start units charging each non-black-start unit.
1.1. Dijkstra-Based Unit Recovery Path Search
Dijkstra’s algorithm is a single-source shortest path algorithm based on the idea of breadth-first search, which gradually expands outward from the origin until the last vertex is covered, and is a classical method for solving the shortest path problem in weighted graphs [
15].
First, the generators and loads in the grid are extracted as undifferentiated nodes, the lines and transformers are considered as branches, and the grid is abstracted as a topology diagram
, where
V represents the set of vertices of the graph, and
E represents the set of branches of the graph. Taking into account the charging time of the line, the operation time of the transformer, and the magnitude of the capacitance value of the line, weights are assigned to the branches as shown in Equation (1) to obtain the weighted topology graph.
where
wij represents the branch weight between node
i and node
j,
tij represents the recovery speed of the branch between node
i and node
j—specifically the normalized recovery time of the branch, including line charging time and transformer recovery time—and
cij represents the recovery success rate of the branch between node
i and node
j, characterized here by the normalized capacitance value.
The connection relationship of the grid topology diagram is inscribed by the adjacency matrix
A, as shown in Equation (2).
Take the topology diagram shown in
Figure 1 as an example, which contains six nodes, seven branches, and branch weights as shown in
Figure 1.
Firstly, construct the adjacency matrix as shown in Equation (4).
Node 1 is the black-start power node, with nodes 2–6 as the nodes of the unit to be started, the set VS = [1] of the nodes that have found the shortest path in the initial state, the set Vo = {2,3,4,5,6} of the other nodes, and the set D = [0,1,2,3, ∞,∞] of the initial nodes and the distances of each node. From set D, it is known that the current nearest point to node 1 is node 2, so node 2 is included in the set VS, which gives VS = [1,2], Vo = [3,4,5,6], and the shortest path of node 2 is recorded. The distance between node 1 and node 2 is 1. Combining this with the second row of the adjacency matrix, it is known that if node 2 is the intermediate node, the distance from node 1 to node 6 is 7, which is smaller than that in the original set D. Accordingly, the distance set D = [0,1,2,3,∞,7] is updated.
From set D, we know that the nearest to node 1 is node 3, except node 2; thus, we update VS = [1,2,3], update Vo = [4,5,6], and record the shortest path of node 3. The distance between node 1 and node 3 is 2, so with node 3 as the intermediate node, the distances from node 1 to node 5 and node 6 are 7 and 6, respectively, and we update the distance set D = [0,1,2,3,7,6].
From the updated D set, the closest point in Vo to node 1 is node 4 (distance is 3), and the shortest path of node 4 is recorded by updating VS = [1,2,3,4] and Vo = [5,6]. With node 4 as the intermediate node, the distance from node 1 to node 5 is 7, and set D remains unchanged.
At this time, the nearest point in Vo to node 1 is node 6, and the distance is 6. Thus, we update VS = [1,2,3,4,6], Vo = [5], and record the shortest path of node 5. Node 1 does not have a path through node 6 to node 5, so the shortest path of node 5 is recorded directly, so the final shortest distance set D = [0,1,2,3,7,6] and the shortest path matrix S are obtained.
The flow chart of the unit recovery path search is shown in
Figure 2.
1.2. Unit Start-Up Sequence Integrated Index
(1) Unit characteristics index
In determining the order of recovery of the units to be started, the following principles should be followed [
16]:
- (1)
In order to maximize the number of units to be hot-started, units with hot-start conditions are arranged first for restoration.
- (2)
When the power generation capacity is insufficient at the initial stage of system recovery, priority is given to units with small starting powers to achieve smooth starting of units to be started.
- (3)
Priority is given to restoring units with larger climbing rates to ensure rapid recovery of the system.
- (4)
Priority is given to units with high recovery capacities to ensure sufficient power generation capacity of the system.
Considering the unit starting power, climbing rate, and unit capacity, the unit characteristic index is defined as shown in Equation (5).
where,
is the characteristic index of the kth unit,
is the normalized value of the starting power of the kth unit,
is the normalized value of the capacity of the kth unit, and
is the normalized value of the climbing rate of the kth unit.
The above normalization method is shown in Equation (6).
where,
represents the normalized value of
x,
represents the minimum value of
x, and
represents the maximum value of
x.
Among them, the power output model of the unit is shown in
Table 1.
In this model, is the start-up time of the unit; is the time when the unit is connected to the grid to deliver power to the outside; is the time when the unit outputs the maximum active power to the outside; represents the climbing rate of the unit; is the average climbing rate of the unit; is the active power of the unit operating with plant power; is the maximum active power of the unit.
(2) Unit distance index
Considering the recovery speed and recovery success rate of the black-start unit to recover the unit to be started, we reasonably set the branch weights, calculate the sum of the branch weights contained in the shortest recovery path of the non-black-start unit obtained based on Dijkstra’s algorithm, and normalize according to Equation (6) to obtain the unit distance index .
Taking into account the unit’s own characteristic index and distance index, a comprehensive index of unit start-up is established as shown in Equation (7).
where
is the composite index of the start-up of the kth unit,
is the characteristic index of the kth unit, and
is the distance index of the kth unit.
1.3. Unit Start-Up Constraints
(1) The start-up time constraint is:
where
is the start time of the non-black-start unit, and
is the maximum hot-start time limit of the non-black-start unit, which is the maximum outage time that the unit can be hot-started.
When the starting moment of the unit is greater than the minimum hot-start time limit, the starting moment of the unit needs to meet the minimum cold-start time limit as shown in Equation (9).
where
is the start time of the non-black-start unit, and
is the minimum cold-start time limit of the non-black-start unit.
(2) The start-up power constraint is defined as follows:
where
p denotes the number of black-start units in the system,
represents the active power output from the ith black-start unit at time
t,
q denotes the number of restored non-black-start units in the system,
represents the active power output from the
jth non-black-start unit at time t, and
represents the start power required for the next unit to be started.
(3) Unit start/stop state constraints
It is assumed that once the unit has been started, it will remain in operation and will not shut down again. Thus,
where,
indicates the start–stop state of unit
k at time
t, and the value is 1 when the unit starts and 0 vice versa.
(4) Power constraints
where,
is the minimum active power allowed to be output by the
th generator set,
is the active power issued by the
th generator set,
is the maximum active power allowed to be output by the
th generator set,
is the reactive power issued by the
th generator set,
is the minimum reactive power allowed to be output by the
th generator set,
is the maximum reactive power allowed to be output by the
th generator set,
is the active power transmitted by the
th line, and
is the upper limit of the power of the
th line.
(5) Voltage constraints
where
is the magnitude of the voltage value of the
th node,
is the lower voltage limit of the
th node, and
is the upper voltage limit of the
th node.
1.4. Unit Start-Up Process
The unit start-up policy development process is shown in
Figure 3.
The specific steps of the unit start-up are as follows.
(1) Read the topology and parameters of the grid, take generators and loads as nodes, take lines and transformers as branches, and select line charging time, transformer operation time, and line capacitance value as branch weights to abstract the grid as a weighted topology diagram, characterized by adjacency matrix A.
(2) Based on Dijkstra’s algorithm to find the shortest path of the unit to be started as the recovery path of the unit, the distance index of each non-black-start unit is calculated.
(3) Combining that with the unit’s own characteristic indexes, calculate the comprehensive indexes of the group starting and rank them, so as to determine the starting order of the unit.
(4) Determine the next non-black-start unit to be started based on the unit start sequence and check its hot-start time limit, start power, power constraint, and voltage constraint. Units that do not meet the start conditions are moved back in the start sequence or included in the cold start until the units to be started that meet the constraints are screened.
(5) Recover the selected units to be started according to the shortest recovery path and update the recovery status of the system at the same time.
(6) Repeat steps (2)–(5) until all non-black-start units have completed starting.