Research on the Unit Black-Start Strategy Considering Recovery Path and Start Sequence

Abstract

Large-scale power outage events bring serious economic losses to national and social development and cause bad impacts, and the reasonable formulation of a unit black-start strategy is the basis of power outage recovery. Firstly, the Dijkstra shortest path algorithm is used to search for the optimal recovery path of the unit to be started after a major blackout occurs, and secondly, a comprehensive index of the unit start sequence considering the unit characteristics and unit distance is defined to guide the unit start sequence formulation by combining the unit capacity, climbing rate, start power, and other characteristics with the recovery time and capacitance value of the recovery path. Finally, based on the unit start sequence and optimal recovery path, and taking into account the unit start constraints, a complete unit black-start strategy is proposed, and the effectiveness of this strategy is verified by IEEE30 node system simulation. The results show that the proposed unit black-start strategy can reduce the unit recovery time and improve the recovery success rate.

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 $G=(V,E)$, 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.

$$w_{ij}=2t_{ij}+c_{ij}$$

(1)

where w_ij represents the branch weight between node i and node j, t_ij 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 c_ij 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).

$$A=\left[\begin{array}{llll}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& \ddots & & \\ ⋮& & \ddots & \\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]$$

(2)

$$a_{ij}=\{\begin{array}{ll}{w}_{ij}& i\ne j\mathrm{and}<i,j>\in E\\ \infty & i\ne j\mathrm{and}<i,j>\notin E\\ 0& i=j\end{array}$$

(3)

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).

$$A=\left[\begin{array}{llllll}0& 1& 2& 3& \infty & \infty \\ 1& 0& \infty & \infty & \infty & 6\\ 2& \infty & 0& \infty & 5& 4\\ 3& \infty & \infty & 0& 4& \infty \\ \infty & \infty & 5& 4& 0& \infty \\ \infty & 6& 4& \infty & \infty & 0\end{array}\right]$$

(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).

$$O(k)=P^{*}(k)-S^{*}(k)-C^{*}(k)$$

(5)

where, $O(k)$ is the characteristic index of the kth unit, $P^{*}(k)$ is the normalized value of the starting power of the kth unit, $S^{*}(k)$ is the normalized value of the capacity of the kth unit, and $C^{*}(k)$ is the normalized value of the climbing rate of the kth unit.

The above normalization method is shown in Equation (6).

$$x^{*}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}$$

(6)

where, $x^{*}$ represents the normalized value of x, $x_{\min }$ represents the minimum value of x, and $x_{\max }$ represents the maximum value of x.

Among them, the power output model of the unit is shown in

Table 1. The black-start mathematical models of the units.

Unit Type	Hydroelectric Units	Gas Turbine Unit
Power output model	$P=\{\begin{array}{ll}0& t\le t_c\\ K(t-t_c)& t_c<t\le t_{\max }\\ P_{\max }& t>t_{\max }\end{array}$	$P=\{\begin{array}{ll}-P_{st}& t\le t_c\\ K(t-t_c)& t_c<t\le t_{\max }\\ P_{\max }& t>t_{\max }\end{array}$ $t_{\max }=\frac{P_{\max }+P_{st}}{K}+t_c$
Unit Type	FCB Coal-Fired Units	Thermal Power Units
Power output model	$P=\{\begin{array}{ll}-P_{st}& t\le t_s\\ -P_{st}+K_N(t-t_s)& t_s<t\le t_{\max }\\ P_{\max }& t>t_{\max }\end{array}$ $t_{\max }=\frac{P_{\max }+P_{st}}{K_N}+t_c$	$P=\{\begin{array}{ll}0& \mathrm{t}\le {\mathrm{t}}_s\\ -P_{st}& \mathrm{t}\le {\mathrm{t}}_{\mathrm{c}}\\ -P_{st}+K_N(t-t_c)& {\mathrm{t}}_{\mathrm{c}}<\mathrm{t}\le t_{\max }\\ P_{\max }& \mathrm{t}>t_{\max }\end{array}$ $t_{\max }=\frac{P_{\max }+P_{st}}{K_N}+t_c$

.

In this model, $t_s$ is the start-up time of the unit; $t_c$ is the time when the unit is connected to the grid to deliver power to the outside; $t_{\max }$ is the time when the unit outputs the maximum active power to the outside; $K$ represents the climbing rate of the unit; $K_N$ is the average climbing rate of the unit; $P_{st}$ is the active power of the unit operating with plant power; $P_{\max }$ 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 $D(k)$.

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).

$$Z(k)=O(k)+D(k)$$

(7)

where $Z(k)$ is the composite index of the start-up of the kth unit, $O(k)$is the characteristic index of the kth unit, and $D(k)$ is the distance index of the kth unit.

1.3. Unit Start-Up Constraints

(1) The start-up time constraint is:

$$0<t_s\le t_s^{\max }$$

(8)

where $t_s$ is the start time of the non-black-start unit, and $t_s^{\max }$ 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).

$$t_s\ge t_s^{\min }$$

(9)

where $t_s$ is the start time of the non-black-start unit, and $t_s^{\min }$ is the minimum cold-start time limit of the non-black-start unit.

(2) The start-up power constraint is defined as follows:

$${\displaystyle \sum _{i=1}^p{P}_{black}^i(t)}+{\displaystyle \sum _{j=1}^q{P}_{nblack}^j(t)}\ge P_{st}$$

(10)

where p denotes the number of black-start units in the system, $P_{black}^i(t)$ 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, $P_{nblack}^j(t)$ represents the active power output from the jth non-black-start unit at time t, and $P_{st}$ 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,

$$\begin{array}{cc}{s}_k(t)\le s_k(t+1)& k=1,2,\dots ,N\end{array}$$

(11)

where, $s_k(t)$ 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

$$\{\begin{array}{ll}{P}_{Gi}^{\min }<P_{Gi}<P_{Gi}^{\max },& i=1,2,3,\mathrm{\dots },n_G\\ Q_{Gi}^{\min }<Q_{Gi}<Q_{Gi}^{\max },& i=1,2,3,\mathrm{\dots },n_G\\ P_{Li}<P_{Li}^{\max },& i=1,2,3,\mathrm{\dots },n_L\end{array}$$

(12)

where, $P_{Gi}^{\min }$ is the minimum active power allowed to be output by the $i$th generator set, $P_{Gi}$ is the active power issued by the $i$th generator set, $P_{Gi}^{\max }$ is the maximum active power allowed to be output by the $i$th generator set, $Q_{Gi}$ is the reactive power issued by the $i$th generator set, $Q_{Gi}^{\min }$ is the minimum reactive power allowed to be output by the $i$th generator set, $Q_{Gi}^{\max }$ is the maximum reactive power allowed to be output by the $i$th generator set, $P_{Li}$ is the active power transmitted by the $i$th line, and $P_{Li}^{\max }$ is the upper limit of the power of the $i$th line.

(5) Voltage constraints

$$U_i^{\min }<U_i<U_i^{\max }$$

(13)

where $U_i$ is the magnitude of the voltage value of the $i$th node, $U_i^{\min }$ is the lower voltage limit of the $i$th node, and $U_i^{\max }$ is the upper voltage limit of the $i$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.

2. Analysis of the Examples

2.1. Analysis of Starting Strategies for Different Black-Start Unit Types

IEEE-30 is used as an example to verify the black-start strategy method of the units proposed in this paper [17], and the system structure is shown in Figure 4. Among them, node 1 is the gas turbine unit, i.e., the black-start power unit, while nodes 2, 5, 8, 11 and 13 are non-black-start units; branches 6–9, 6–10, 4–12, and 28–27 are transformer branches, and the rest of the branches are lines.

The generator set characteristics parameters are shown in

Table 2. Generator set characteristics parameters.

Unit Nodes	Unit Capacity/p.u.	Starting Power/p.u.	Climbing Rate/p.u./h	Grid Connection Time/min
1	3.51	0	1.86	10
2	11.45	0.89	7.43	20
5	7.50	0.32	4.28	15
8	6.62	0.25	3.55	30
11	6.43	0.27	3.34	20
13	9.29	0.42	5.28	15

.

It is assumed that the black-start unit located at node 1 starts at the moment of 0. After 10 min of grid connection time, the starting power is output for the system, and the non-black-start units at nodes 2, 5, 8, 11, and 13 are restored. MATLAB is used to write the unit start-up program for simulation. Firstly, the integrated unit start index of each non-black-start unit is calculated as shown in

Table 3. First start-up sequence of the unit.

Non-Black-Start Unit Node	Comprehensive Index of Unit Start-Up	Start-Up Sequence
2	−1	1
5	−0.02	3
8	0.51	4
11	1.03	5
13	−0.11	2

to obtain the initial unit black-start sequence for the first round.

As can be seen from Table 3, firstly, the black-started unit at node 1 provides starting power to the non-black-started unit located at node 2, and the recovery path is 1–2 based on Dijkstra’s algorithm, which is verified to satisfy the constraint limits, and the unit can start normally. After the unit at node 2 starts, the system operation status is updated, and the non-black-start unit 2 is included in the recovered unit, while the weights of the branches through which the recovery paths 1–2 pass are set to zero. After the branch paths are set to zero, the distance characteristic indexes of the remaining non-black-started units are changed, and the comprehensive indexes of unit start-up for each remaining unit are recalculated as shown in

Table 4. Second start-up sequence of the unit.

Non-Black-Start Unit Node	Comprehensive Index of Unit Start-Up	Start-Up Sequence
5	−0.03	1
8	0.51	3
11	1.02	4
13	0.21	2
5	−0.03	1

. From Table 4, it can be seen that the unit to be started in the next order is changed from the original unit 13 to unit 5, and the above steps are repeated to start the non-black-start unit at node 5 after calibration.

We start each unit in turn until all the non-black-start units of the grid have been traversed, and the start situation of each unit is obtained as shown in

Table 5. Unit start-up for gas turbine black start.

Unit Nodes	Start-Up Moment/min	Recovery Path	Outward Force Moment/min
2	12	1-2	19.15
5	14	2-5	18.64
13	24	1-3-4-12-13	28.72
8	28	5-7-6-8	32.30
11	34	6-9-11	38.85

.

If the black-start power source at node 1 is an FCB coal-fired unit, the remaining characteristic parameters such as the capacity, starting power, and climbing rate of the unit are kept constant. The black-start strategy simulation of the unit is carried out according to the method described in this paper, and the starting situation of each unit is obtained as shown in

Table 6. Unit start-up for FCB black start.

Unit Nodes	Start-Up Moment/min	Recovery Path	Outward Force Moment/min
2	2	1-2	9.15
5	4	2-5	18.64
13	14	1-3-4-12-13	18.72
8	18	5-7-6-8	22.30
11	24	6-9-11	28.85

. Comparing Table 5 and Table 6, it can be seen that compared with the gas turbine unit as the black-start power source, the starting sequence and recovery path of each non-black-start unit remain the same when the FCB coal-fired unit is black-started, while the starting moment and the outgoing power are both 10 min earlier, which can increase the system power generation in the same time period. This is because the coal-fired units keep the boiler running at the lowest load in the fast load-shedding operation state to maintain operating capability for plant power; they can start quickly after a system blackout incident and immediately increase their power output to provide the initial power for power system recovery. Therefore, using FCB coal-fired units as black-start power can restore a blacked-out system more quickly.

2.2. Comparison of Different Unit Start-Up Strategies

The IEEE-39 node system is used to simulate the unit black-start strategy example [17], and the system structure is shown in Figure 5. Node 30 is the black-start power unit, and here the gas turbine unit is considered; nodes 31, 32, 33, 34, 35, 36, 37, 38, and 39 are non-black-start units; branches 11–12, 13–12, 31–6, 32–10, 33–19, 34–20, 35–22, 36–23, 37–25, 30–2, 38–29, and 20–19 are transformer branches, and the rest of the branches are lines.

In order to further verify the effectiveness of the unit start strategy proposed in this paper, the following two unit black-start strategies were used to start each unit in turn until all non-black-start units in the grid were traversed, and the results of the unit start strategies obtained by the two methods were compared and analyzed.

Method 1: black-start strategy considering only the unit’s own characteristic index.

Method 2: black-start strategy using the integrated index of unit start sequence.

The generator set characteristics parameters are shown in

Table 7. Generator set characteristics parameters.

Unit Nodes	Unit Capacity/p.u.	Starting Power/p.u.	Climbing Rate/p.u./h	Grid Connection Time/min
30	3.51	0	1.81	5
31	11.45	0.89	7.43	20
32	7.50	0.32	4.20	15
33	7.71	0.27	5.89	30
34	8.12	0.35	6.99	20
35	6.69	0.49	5.28	15
37	5.13	0.23	6.58	15
38	7.61	0.95	3.50	30
39	10.48	0.39	4.35	20

.

(1) Comparison of recovery effects of different unit start strategies

The unit starts obtained by using method 1 are shown in

Table 8. Method 1 unit start-up situation.

Unit Nodes	Start-Up Moment/ min	Recovery Path	Outward Force Moment/ min	Full Power Operation Time/ min	The Sum of Path Capacitance Weights/10−3 p.u.
34	18	30-2-3-18-17-16-19-20-34	21.03	110.56	3.32
31	28	3-4-5-6-31	35.15	147.57	1.27
33	31	19-33	33.75	142.19	0
39	33	2-1-39	38.313	203.14	1.61
37	37	2-25-37	39.13	100.64	0.46
32	43	6-11-10-32	47.64	169.79	0.67
35	52	16-21-22-35	57.66	148.80	1.63
36	58	22-23-36	67.39	236.50	0.59
38	63	25-26-29-38	79.31	239.60	4.97

. Method 1 only considered the unit’s own characteristic indexes, i.e., the start power, climbing rate, and capacity of the unit, when formulating the unit start strategy. Therefore, when selecting the units to be started, priority was given to start the non-black-start units located at nodes 34 and 31, which require less starting power, a faster climbing rate, and a larger unit capacity for restoration. Also, considering the time required for closing the line and restoring the transformer, the units at node 34 and node 31 were started at 18 min and 28 min, respectively, by the shortest path. The two units started climbing and reached positive output at 21.03 min and 35.15 min, respectively.

The unit start-ups obtained from method 2 are shown in

Table 9. Method 2 unit start-up situation.

Unit Nodes	Start-Up Moment/ min	Recovery Path	Outward Force Moment/ min	Full Power Operation Time/ min	The Sum of Path Capacitance Weights/10−3 p.u.
37	9	30-2-25-37	11.13	72.64	0.46
39	11	2-1-39	16.31	181.14	1.61
31	23	2-3-4-5-6-31	30.15	142.57	2.09
32	29	6-11-10-32	33.64	155.79	0.67
33	42	4-14-15-16-19-33	44.75	153.19	3.12
34	48	19-20-34	51.04	140.57	0
36	54	16-24-23-36	63.40	232.50	1.37
35	57	23-22-35	62.66	153.80	0.59
38	62	25-26-29-38	78.31	238.60	4.97

. Because method 2 introduces the unit distance index when formulating the unit start strategy and considers the location distribution of the units, the unit start-up strategy obtained from the solution gives priority to the non-black-started units that are closer to the black-started units, and the units located at node 37 and node 39 were started at 9 min and the 11 min, respectively. The non-black-started units climbed to positive output at 11.13 min and 16.31 min, respectively, and were able to output power to the outside. The comparison analysis shows that method 2 can start the non-black-start units in a shorter period of time at the early stage of power system restoration, thus providing power generation for the outage system more quickly.

From the perspective of successful unit restoration, method 1 restores the unit at node 34 first, and the shortest restoration path capacitance weight sum of this unit is 3.31 × 10⁻³ p.u., while method 2 takes into account the branch capacitance value in the unit start-up sequence decision and starts the unit at node 37 first, and its shortest restoration path capacitance weight sum is 0.46 × 10⁻³ p.u. The smaller the capacitance value, the smaller the reactive power generated by no-load charging of the line, reducing the possibility of overvoltage at the node, which is conducive to the successful restoration of the unit at the early stage of power outage recovery and the output of reliable generation capacity for the power system.

(2) Unit start-up sequence and recovery path relationship analysis

Taking the non-black-started units located at nodes 35 and 36 as an example, the schematic diagram of the recovery path obtained by method 1 is shown in Figure 6a, and the recovery path of method 2 is shown in Figure 6b. The red line represents the recovered system, the blue line represents the shortest recovery path of the unit to be started at node 35, and the yellow line represents the shortest recovery path of the unit at node 36.

From Table 8 and Table 9, it can be seen that method 1 used the unit characteristic index to decide the unit start order, starting unit node 35 first and then unit node 36; method 2 used the comprehensive unit index to decide the unit start order and started unit node 36, which is closer to the black-start unit, first, before starting node 35.

From Figure 6a,b, it can be seen that the optimal recovery paths of unit nodes 35 and 36 change under both methods due to the change in the unit start order, which further illustrates that the unit start order and recovery paths are interrelated. The method 2 proposed in this paper considers the shortest recovery path of the units when deciding the unit start order, and the start order will affect the recovery path of each unit at the same time.

In summary, the start-up strategy proposed in this paper takes into account the coupled relationship between the start-up sequence and the recovery path of the units, which can increase the recovery speed of the units at the early stage of the system outage recovery and improve the reliability of the units’ recovery.

3. Conclusions

This paper analyzes the mathematical models of different black-start units and non-black-start units, defines a comprehensive index of a unit start sequence combining the unit characteristic index and distance index, and proposes a unit black-start strategy considering recovery path and start sequence. The main conclusions are: (1) After a system blackout, compared with a gas turbine unit as a black-start power source, the start time and outgoing power are reached 10 min earlier when an FCB coal-fired unit is black-started, thus increasing the system power generation in the same time period. Therefore, FCB coal-fired units used as black-start power sources can start quickly and immediately climb to outward power output, and the starting moment and outward power output moment of each unit are advanced, which can restore a blacked-out system more quickly. (2) The proposed unit start strategy considering the recovery path distance index of each unit can start the non-black-start units at node 34 and node 31 9 min and 17 min earlier, respectively, than the unit start strategy that only considers each unit’s own characteristic index, and the recovery efficiency is improved by 50–60%, which can provide power generation to the downed system more quickly and improve the reliability of unit restoration at the same time. (3) The proposed unit start strategy takes into account the coupled relationship between the unit start sequence and recovery path and considers the shortest recovery path of units in the unit start sequence decision, while the start sequence will in turn affect the optimal recovery path of each unit. This paper focuses on the issue of unit start-up strategy from the perspective of safety, while ignoring the fact that the retrofitting and maintenance of gas turbine units, FCB thermal power units, etc., have economic costs. The next research effort will focus on considering both economics and safety in unit start-up decisions, which is more in line with actual grid applications.