# Evaluation of Black-Start Schemes Based on the Evaluation Indicator System

Authors: Zhongming Xiang; Mengke Lu; Chengxiang Ling; Tianyu Li; Ye Wu; Cunzhi Tong; Xiang Ma; Xiaocong Zhang; Chen Zhang
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

The black-start scheme is of great significance for the rapid recovery of the power system after a blackout. Most of the existing black-start scheme evaluations are studied from the perspective of improving the weight of evaluation indicators and evaluation methods, ignoring differences in the importance of evaluation indicators at different stages of recovery. Therefore, this study proposes a black-start scheme evaluation model based on the evaluation indicator system. First, starting from the elements of the black-start scheme, 14 evaluation indicators for the black-start scheme are proposed, concerning the characteristics of the grid nodes and the different stages of system recovery. Based on the analytic hierarchy process, the evaluation indicators are linked to the evaluation criteria of the system recovery stage to construct an evaluation indicator system for the black-start scheme. Second, the weight of the evaluation indicator is calculated by combining the analytic hierarchy process and the entropy weight method to obtain the result of the subjective and objective combination. Third, this study proposes an evaluation method integrating prospect theory and the multicriteria compromise solution ranking method, by combining the psychological perceived value of decision-makers with the characteristics of black-start schemes, to evaluate and rank black-start schemes. Finally, the rationality and accuracy of the evaluation model proposed in this study are verified by an example of the black-start scheme of Gansu power grid, which could provide a new idea for decision-makers to choose an appropriate black-start scheme.


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## Body

## 1. Introduction

In recent years, extreme weather and natural disasters have increased the risk of blackouts in the power system [1, 2]. The black-start scheme formulated for blackouts of the power grid can effectively reduce the time of blackouts and reduce the social impact and economic losses caused by accidents [3, 4]. A reasonable black-start scheme can not only start the corresponding units safely and quickly but also facilitate the recovery of the grid and load in the later period [5, 6]. However, there are often multiple feasible black-start schemes for one power grid. Therefore, evaluating and ranking the black-start schemes are of great significance to ensure the optimum operation of the power grid [7, 8].The evaluation and ranking of black-start schemes are multiattribute decision-making problems. Many scholars have made a lot of useful explorations and achieved fruitful results. References [9–11] introduce methods such as analytic hierarchy process, fuzzy number, and super efficiency. Using these methods to improve the data envelopment analysis method and applying them to the evaluation of black-start schemes can achieve good results, but with few indicators to evaluate, it is difficult to assess the pros and cons of black-start schemes as a whole. References [12, 13] introduce the entropy power decision theory. Using entropy decision theory to evaluate and rank black-start schemes is also feasible. However, the entropy method to determine the weight of the evaluation indicator can easily lead to an excessive difference in the indicator weight, and the phenomenon that a single indicator affects the evaluation and ranking results. References [14–16] introduce the theory of group decision-making. Applying group decision-making theory to the evaluation and ranking of black-start schemes can fully reflect the role of decision-makers and improve the scientific nature of decision-making. However, the subjectivity is relatively large, and the efficiency of group decision-making is low. It is not suitable for the evaluation and ranking of black-start schemes under multiple indicators. References [17] introduce the principal component analysis method. Introducing the principal component analysis method into the evaluation indicator system of the power grid black-start scheme can provide a decision basis for the evaluation of the power grid black-start scheme from an objective perspective of the indicator system. References [18, 19] introduce an evaluation method based on interval numbers. By establishing a linear programming model and combining the risk attitude factors of power decision-makers to evaluate and rank black-start schemes, the decision-making process can be described more realistically. However, this method ignores the influence of the scheme on subsequent recovery. References [20, 21] introduce the vague set theory. By introducing the vague set theory, it is possible to combine the qualitative and quantitative indicators to describe the correlation between the indicators, deal with the fuzzy information in the black-start scheme, and obtain better evaluation results. In addition, some scholars combine different evaluation models to obtain better evaluation results, but they still cannot avoid the shortcomings of the insufficient combination of subjective and objective evaluation methods [22, 23].After comparing and analyzing the relevant literature, there are four main problems in the current evaluation studies of black-start schemes. First, the evaluation indicator system is inadequate and the various stages of the recovery process are not fully considered; therefore, the analytical process for evaluating the schemes is not convincing [3–7, 12, 13]. Second, the weights of the evaluation indicators do not combine subjectivity and objectivity, leading to extreme results [8, 9, 14, 15, 17]. Third, the lack of psychological perceived value of decision-makers in the evaluation scheme makes the evaluation results deviate from the actual [18–22]. To better highlight the differences between the model in this study and existing references and to enable readers to understand the contents of relevant references, some existing literature related to this study are selected for comparison, as listed in Table 1.Table 1
Comparison of this study with other works in the evaluation of black-start schemes.
Type and number of evaluation indicatorsConsidered system recovery phaseMethod for determining the weight of evaluation indicatorsEvaluation methodExampleStart-up powerPath nodeRecovery lineRestarted unitEarlyTransitionalLateThis study4334✔✔✔Analytic hierarchy process and entropy methodProspect theory combined with VIKORGansu power gridLu [6], 20201013✔Gini coefficient and Pearson coefficientProspect theory combined with VIKORGuangdong power gridLeng [13], 20201013✔Collaborative filtering and entropy methodMAE weightTianjin power gridWang [18], 20161013✔Interval valuesA risk attitude factor-based methodSouthern power gridZeng [19], 20121021✔✔Subjective weights and objective weightsThe vague set theoryGuangdong power gridYe [21], 20181012✔✔Impact factor valueConditional value-at-riskHebei power gridZhang [22], 20211031✔Principal component analysis and entropy weight methodOptimal combination weight combined with improved VIKORTianjin power gridTo address the abovementioned issues, in this study, an evaluation model of black-start schemes based on the evaluation indicator system is proposed with the contributions shown below.Aiming at the above problems, this study proposes an evaluation model of the black-start scheme based on the evaluation indicator system, and its contributions are as follows:(1)
The evaluation system of the black-start scheme proposed in this study adopts the analytic hierarchy process (AHP) to establish 14 evaluation indicators, which not only cover the three recovery stages of the power system but also use different evaluation indicators in different recovery stages. Under this system, the evaluation results can be more realistic.(2)
This study realizes the combination of subjective and objective in the process of calculating the indicator weight and ranking scheme by introducing the idea of combined weight. This approach expands the applicability of this assessment system.(3)
This study combines prospect theory and the multicriteria compromise scheme ranking method (VIKOR) to realize the combination of decision-makers’ psychological perceived value and the characteristics of the black-start scheme. This approach successfully takes into account the decision-makers who sort the black-start schemes in practice. The decision-maker is also an important destabilizing factor in the ranking problem and therefore has to be considered.The rest of this study is organized as follows: in Section2, the evaluation indicator system is constructed. In Section 3, the combined weight of each evaluation indicator is calculated. In Section 4, the combination of prospect theory and the multicriteria compromise solution ranking method is introduced. In Section 5, the results for test examples are presented. Finally, in Section 6, conclusions and future work are highlighted.


## 2. Construction of Evaluation Indicator System

### 2.1. Elements of the Black-Start Scheme

#### 2.1.1. Start-Up Power

The pros and cons of the black-start power supply in the black-start scheme are related to the success of the black-start scheme. Therefore, this study takes the black-start power supply unit capacity, unit level, and node importance of the start-up power supply as evaluation indicators to measure the black-start scheme.


#### 2.1.2. Path Node

There is a long electrical distance between the starting power source and the restarted unit in the power grid. Therefore, the process of restoring the restarted units will pass through more grid nodes than in normal working conditions. Most of these nodes carry a certain amount of load. The restoration of important nodes is beneficial to increase the strength of the network skeleton. Therefore, this study takes the average importance of the restoration node, the restored important load, and the total load as the evaluation indicators of the black-start scheme.


#### 2.1.3. Recovery Line

The length of the recovery line and the attributes of the line itself are important factors that cause overvoltage and self-excitation during the black-start process. Therefore, this study takes the attribute indicator and the betweenness indicator of the restoration line as the evaluation indicator of the black-start scheme.


#### 2.1.4. Restarted Unit

In the power system, the restarted units are widely dispersed and have different attributes. At the same time, considering the impact of the restarted units on the subsequent recovery of the system, this study takes the capacity of the restarted units, the importance of the restarted units, the starting power required by the restarted units, and the starting time as indicators for evaluating the black-start scheme.


### 2.2. Evaluation Indicators Based on the Elements of the Black-Start Scheme

The evaluation indicators for determining the black-start scheme should follow the following principles: first, the evaluation indicators can reflect the black-start recovery stage from different aspects. Second, the evaluation indicators can be calculated and obtained easily. Finally, the evaluation indicators should follow the principles of science and practicality. In order to adapt to various actual situations, starting from the elements of the black-start scheme, this study proposes the evaluation indicators of the black-start scheme, as shown in Figure1. The significance of each evaluation indicator is listed in Table 2.Figure 1
Evaluation indicator system for black-start scheme.Table 2
Evaluation indicator description.
IndicatorIndicator meaningI1Start-up power capacityI2Start-up power importanceI3Voltage fluctuations caused by impulse loadI4Frequency fluctuations caused by impulse loadI5Average importance of restoration nodesI6The important load to be restoredI7The total load to be restoredI8Line attribute indicatorI9Recovery line average betweennessI10Line operation timeI11The capacity of the restarted unitI12The importance of the restarted unitI13The required power of the restarted unitI14The starting time of the restarted unitThe combination of the start-up power level and the importance of the node where the starting power is located represents the importance of the start-up power as shown in the following formula:(1)I2=EsGs,In this study, the frequency and the maximum voltage drop caused by the shock load in the black-start scheme are used as evaluation indicators to measure the impact of the shock load on the black-start scheme:(2)I3=maxU′−UU×100%,(3)I4=maxf′−ff×100%,where U is the rated value of the outlet voltage of the starting power supply, U′ is the voltage fluctuation caused by the impact load, f is the rated frequency, and f′ is the frequency fluctuation caused by the impact load.This study takes the average value of the importance of path nodes in the black-start scheme as the evaluation indicator of the black-start scheme, as shown in the following formula (a detailed description is provided in Appendix):(4)Ei=J′i+B′i+CC′i,I5=1Nn∑i=1NnEi.The line attribute indicator is constructed, as shown in the following formula (a detailed description is provided in Appendix):(5)I8=1NlVi∑r=1NlRr+Xr+BrLr.The betweenness indicator of the line reflects the importance of the line in the power grid. The following formula takes the average betweenness of the restored line as one of the evaluation indicators for the black-start scheme (a detailed description is provided in Appendix):(6)I9=1Nl∑r=1NlLr.Formula (7) indicates the importance of the restarted unit. This study adds the ramp rate of the restarted unit based on the definition of the importance of start-up power and defines the importance of the restarted unit:(7)I12=KpEgGg.


### 2.3. Evaluation Indicators for Different Recovery Stages of the System

An excellent black-start scheme can not only start the corresponding unit quickly and stably but also facilitate the subsequent recovery of the black-start. Therefore, this article takes the black-start early stage, the transition stage, and the late stage (impact on subsequent recovery) as three recovery stages of the black-start scheme. Furthermore, it defines the main characteristics that affect each recovery stage as the evaluation criteria and combines the evaluation indicators of the black-start scheme and evaluation criteria to construct the evaluation indicator system of the black-start scheme, as listed in Table3.Table 3
Evaluation indicator system.
Recovery stageEvaluation criteriaEvaluation indicatorsEarlyVoltage fluctuationI1, I3, I8, I13Frequency fluctuationI1, I4, I6, I13Technical indicatorsI3, I4, I8Unit recoveryI11, I12, I14TransitionalTime requiredI10, I14Line attributeI8, I9LatePath nodeI5Unit capacityI1, I11Load recoveryI6, I7Skeleton networkI2, I5, I9, I12The ratio of the influence degree of the three recovery stages isα:β:λ, and the ratio of the influence degree of the criterion included in each stage is α1:α2:α3:α4, β1:β2, λ1:λ2:λ3:λ4, respectively. According to the importance ratio of different stages and the criterion importance ratio, the importance of each indicator can be obtained. For example, for the I1, its importance is α∗α1+ α∗a2+ λ∗λ3, and the importance of other indicators is similar.


## 2.1. Elements of the Black-Start Scheme

### 2.1.1. Start-Up Power

The pros and cons of the black-start power supply in the black-start scheme are related to the success of the black-start scheme. Therefore, this study takes the black-start power supply unit capacity, unit level, and node importance of the start-up power supply as evaluation indicators to measure the black-start scheme.


### 2.1.2. Path Node

There is a long electrical distance between the starting power source and the restarted unit in the power grid. Therefore, the process of restoring the restarted units will pass through more grid nodes than in normal working conditions. Most of these nodes carry a certain amount of load. The restoration of important nodes is beneficial to increase the strength of the network skeleton. Therefore, this study takes the average importance of the restoration node, the restored important load, and the total load as the evaluation indicators of the black-start scheme.


### 2.1.3. Recovery Line

The length of the recovery line and the attributes of the line itself are important factors that cause overvoltage and self-excitation during the black-start process. Therefore, this study takes the attribute indicator and the betweenness indicator of the restoration line as the evaluation indicator of the black-start scheme.


### 2.1.4. Restarted Unit

In the power system, the restarted units are widely dispersed and have different attributes. At the same time, considering the impact of the restarted units on the subsequent recovery of the system, this study takes the capacity of the restarted units, the importance of the restarted units, the starting power required by the restarted units, and the starting time as indicators for evaluating the black-start scheme.


## 2.1.1. Start-Up Power

The pros and cons of the black-start power supply in the black-start scheme are related to the success of the black-start scheme. Therefore, this study takes the black-start power supply unit capacity, unit level, and node importance of the start-up power supply as evaluation indicators to measure the black-start scheme.


## 2.1.2. Path Node

There is a long electrical distance between the starting power source and the restarted unit in the power grid. Therefore, the process of restoring the restarted units will pass through more grid nodes than in normal working conditions. Most of these nodes carry a certain amount of load. The restoration of important nodes is beneficial to increase the strength of the network skeleton. Therefore, this study takes the average importance of the restoration node, the restored important load, and the total load as the evaluation indicators of the black-start scheme.


## 2.1.3. Recovery Line

The length of the recovery line and the attributes of the line itself are important factors that cause overvoltage and self-excitation during the black-start process. Therefore, this study takes the attribute indicator and the betweenness indicator of the restoration line as the evaluation indicator of the black-start scheme.


## 2.1.4. Restarted Unit

In the power system, the restarted units are widely dispersed and have different attributes. At the same time, considering the impact of the restarted units on the subsequent recovery of the system, this study takes the capacity of the restarted units, the importance of the restarted units, the starting power required by the restarted units, and the starting time as indicators for evaluating the black-start scheme.


## 2.2. Evaluation Indicators Based on the Elements of the Black-Start Scheme

The evaluation indicators for determining the black-start scheme should follow the following principles: first, the evaluation indicators can reflect the black-start recovery stage from different aspects. Second, the evaluation indicators can be calculated and obtained easily. Finally, the evaluation indicators should follow the principles of science and practicality. In order to adapt to various actual situations, starting from the elements of the black-start scheme, this study proposes the evaluation indicators of the black-start scheme, as shown in Figure1. The significance of each evaluation indicator is listed in Table 2.Figure 1
Evaluation indicator system for black-start scheme.Table 2
Evaluation indicator description.
IndicatorIndicator meaningI1Start-up power capacityI2Start-up power importanceI3Voltage fluctuations caused by impulse loadI4Frequency fluctuations caused by impulse loadI5Average importance of restoration nodesI6The important load to be restoredI7The total load to be restoredI8Line attribute indicatorI9Recovery line average betweennessI10Line operation timeI11The capacity of the restarted unitI12The importance of the restarted unitI13The required power of the restarted unitI14The starting time of the restarted unitThe combination of the start-up power level and the importance of the node where the starting power is located represents the importance of the start-up power as shown in the following formula:(1)I2=EsGs,In this study, the frequency and the maximum voltage drop caused by the shock load in the black-start scheme are used as evaluation indicators to measure the impact of the shock load on the black-start scheme:(2)I3=maxU′−UU×100%,(3)I4=maxf′−ff×100%,where U is the rated value of the outlet voltage of the starting power supply, U′ is the voltage fluctuation caused by the impact load, f is the rated frequency, and f′ is the frequency fluctuation caused by the impact load.This study takes the average value of the importance of path nodes in the black-start scheme as the evaluation indicator of the black-start scheme, as shown in the following formula (a detailed description is provided in Appendix):(4)Ei=J′i+B′i+CC′i,I5=1Nn∑i=1NnEi.The line attribute indicator is constructed, as shown in the following formula (a detailed description is provided in Appendix):(5)I8=1NlVi∑r=1NlRr+Xr+BrLr.The betweenness indicator of the line reflects the importance of the line in the power grid. The following formula takes the average betweenness of the restored line as one of the evaluation indicators for the black-start scheme (a detailed description is provided in Appendix):(6)I9=1Nl∑r=1NlLr.Formula (7) indicates the importance of the restarted unit. This study adds the ramp rate of the restarted unit based on the definition of the importance of start-up power and defines the importance of the restarted unit:(7)I12=KpEgGg.


## 2.3. Evaluation Indicators for Different Recovery Stages of the System

An excellent black-start scheme can not only start the corresponding unit quickly and stably but also facilitate the subsequent recovery of the black-start. Therefore, this article takes the black-start early stage, the transition stage, and the late stage (impact on subsequent recovery) as three recovery stages of the black-start scheme. Furthermore, it defines the main characteristics that affect each recovery stage as the evaluation criteria and combines the evaluation indicators of the black-start scheme and evaluation criteria to construct the evaluation indicator system of the black-start scheme, as listed in Table3.Table 3
Evaluation indicator system.
Recovery stageEvaluation criteriaEvaluation indicatorsEarlyVoltage fluctuationI1, I3, I8, I13Frequency fluctuationI1, I4, I6, I13Technical indicatorsI3, I4, I8Unit recoveryI11, I12, I14TransitionalTime requiredI10, I14Line attributeI8, I9LatePath nodeI5Unit capacityI1, I11Load recoveryI6, I7Skeleton networkI2, I5, I9, I12The ratio of the influence degree of the three recovery stages isα:β:λ, and the ratio of the influence degree of the criterion included in each stage is α1:α2:α3:α4, β1:β2, λ1:λ2:λ3:λ4, respectively. According to the importance ratio of different stages and the criterion importance ratio, the importance of each indicator can be obtained. For example, for the I1, its importance is α∗α1+ α∗a2+ λ∗λ3, and the importance of other indicators is similar.


## 3. Combination Weight

To avoid the problem of the insufficient subjective and objective combination of evaluation weights and imbalance of evaluation weights, this study uses different subjective and objective weighting methods to calculate evaluation indicator weights from different angles, establishes an optimal weight function based on game theory, and calculates the optimal combination weight. Assuming that there arem black-start schemes and n evaluation indicators, an m∗n evaluation matrix is constructed:(8)X=xijm×n=X1,X2,⋯,Xn.Since the dimensions of the indicators are inconsistent, the indicator data need to be normalized. General evaluation indicators are divided into benefit-type indicators and cost-type indicators. The larger the value, the better the indicator is called the benefit-type indicator, which is normalized by formula (9). The smaller the value, the better the formula:(9)x′ij=xij2∑i=1mxij2.(10)x′ij=1/xij2∑i=1m1/xij2.After normalization, the dimensionless evaluation matrix of the black-start scheme is obtained:(11)X′=x′ijm×n=X1′,X2′,…,Xn′.


### 3.1. Subjective Empowerment

The analytic hierarchy process is a multiobjective decision-making method that combines quantitative and qualitative, with strong subjectivity. Through the established evaluation indicator system, the judgment matrixA′ can be constructed:(12)A′=a11,a12,…,a1n,a21,a22,…,a2n,⋮,⋮,abj,⋮,an1,an2,…,ann.,where a12 = (axa1 + axa2 + λxλ2)/(λxλ4) is the importance of I1 to I2. By normalizing the judgment matrix A′, the subjective weight of the evaluation indicator is calculated by the characteristic root method:(13)H=h1,h2,…,hj,…,hn.


### 3.2. Objective Empowerment

EWM calculates the objective weight of the evaluation indicator from the perspective of the overall confusion degree of the normalized evaluation indicator value:(14)ej=dj∑j=1ndj,dj=θ′−Ej,Ej=−∑i=1mpij1npij1nm,where Rij=x′ij/∑i=1mx′ij, if Rij = 0, then RijlnRij = 0. θ′ is a variable adjustment parameter, and θ′ = 1.5 in this study.


### 3.3. Combination Weighting

To avoid the insufficient combination of subjective and objective evaluation indicator weights, this study combines the weight results of the analytic hierarchy process and the entropy weight method to achieve a combination of subjective and objective evaluation indicator weights:(15)wj=hjej∑j=1nhjej.This study starts with the four components of the black-start scheme: starting power, path nodes, restoration lines, and units to be started. About 14 evaluation indicators are proposed to establish a basic evaluation indicator system based on these elements and combined with the evaluation criteria of different stages of system restoration. In order to realize the combination of subjective and objective in the evaluation process, this study uses the analytic hierarchy process and entropy method to calculate the weight of the evaluation indicator, establishes the optimal weight function based on the idea of game theory, and calculates the optimal combination weight of the evaluation indicator. At the same time, in order to consider the influence of decision-makers in the evaluation process, this study introduces the prospect theory that considers the psychological perceived value of decision-makers and the multicriteria compromise solution ranking method that considers maximizing group benefits and minimizing individual regrets of objections. The two are combined to evaluate and rank the black-start schemes.


## 3.1. Subjective Empowerment

The analytic hierarchy process is a multiobjective decision-making method that combines quantitative and qualitative, with strong subjectivity. Through the established evaluation indicator system, the judgment matrixA′ can be constructed:(12)A′=a11,a12,…,a1n,a21,a22,…,a2n,⋮,⋮,abj,⋮,an1,an2,…,ann.,where a12 = (axa1 + axa2 + λxλ2)/(λxλ4) is the importance of I1 to I2. By normalizing the judgment matrix A′, the subjective weight of the evaluation indicator is calculated by the characteristic root method:(13)H=h1,h2,…,hj,…,hn.


## 3.2. Objective Empowerment

EWM calculates the objective weight of the evaluation indicator from the perspective of the overall confusion degree of the normalized evaluation indicator value:(14)ej=dj∑j=1ndj,dj=θ′−Ej,Ej=−∑i=1mpij1npij1nm,where Rij=x′ij/∑i=1mx′ij, if Rij = 0, then RijlnRij = 0. θ′ is a variable adjustment parameter, and θ′ = 1.5 in this study.


## 3.3. Combination Weighting

To avoid the insufficient combination of subjective and objective evaluation indicator weights, this study combines the weight results of the analytic hierarchy process and the entropy weight method to achieve a combination of subjective and objective evaluation indicator weights:(15)wj=hjej∑j=1nhjej.This study starts with the four components of the black-start scheme: starting power, path nodes, restoration lines, and units to be started. About 14 evaluation indicators are proposed to establish a basic evaluation indicator system based on these elements and combined with the evaluation criteria of different stages of system restoration. In order to realize the combination of subjective and objective in the evaluation process, this study uses the analytic hierarchy process and entropy method to calculate the weight of the evaluation indicator, establishes the optimal weight function based on the idea of game theory, and calculates the optimal combination weight of the evaluation indicator. At the same time, in order to consider the influence of decision-makers in the evaluation process, this study introduces the prospect theory that considers the psychological perceived value of decision-makers and the multicriteria compromise solution ranking method that considers maximizing group benefits and minimizing individual regrets of objections. The two are combined to evaluate and rank the black-start schemes.


## 4. Evaluation Method

### 4.1. Prospect Theory

The prospect theory is composed of value function and weight function. Among them, the value function is shown in equation (16) [6]. It can be seen that the “loss” of the same amount of change has a greater impact on the decision-maker than the “profit,” which is in line with the actual value of the decision-maker. Therefore, for the evaluation and ranking of black-start schemes, the prospect theory can be used to describe the “profit” and “loss” of black-start scheme evaluation indicators. The psychological perceived value of decision-makers can be considered. After the normalized evaluation matrix is processed by the value function, the value matrix V is obtained. Integrating the value function and the optimal combination weight of the evaluation indicator can obtain the prospect matrix U as follows:(16)vx=Δxa,Δx≥0,−c−Δxb,Δx<0.(17)V=vijm×n,(18)U=uijm×n,where Δx = xij′−xrj′ and xrj′ represents the reference value of the jth indicator. In this study, a, b, and c are selected as 0.88, 0.88, and 2.25, respectively. The average value of each index is used as a reference value, xj′=1/m∑r=1mxrj′. In this study, the average value of each indicator is used as a reference point, uij=vijwj.


### 4.2. Multicriteria Compromise Solution Ranking Method

VIKOR is a multiattribute decision-making method. It obtains the degree of closeness to the ideal solution by considering maximizing group benefits and minimizing the individual regrets of opposing opinions. The basic calculation steps are as follows:Ideal Solution Calculation. According to the foreground matrix obtained by equation (18) [6], a positive ideal solution U+ and a negative ideal solution U− are obtained:(19)U+=u1+,…,uj+,…,un+T,U−=u1−,…,uj−,…,un−T,where uj+=maxuij|1≤i≤n and uj−=minuij|1≤i≤n.Eigenvalue Calculation. Calculate the maximum group benefit value S (i), minimum individual regret value Y (i), and profit value Q (i) of the ith black-start scheme:(20)Si=∑j=1nwjuj+−uijuj+−uj−,(21)Yi=maxjwjuj+−uijuj+−uj−,(22)Qi=κSi−S−S+−S=+1−κYi−Y−Y+−Y−,where S−=minSi|1≤i≤n, S+=maxSi|1≤i≤n, Y−=minYi|1≤i≤n, and Y+=maxYi|1≤i≤n.If 0 <κ < <0.5, it indicates that the decision is inclined to regret value. If κ = 0.5, it indicates that the decision-making equilibrium is compromised. If 0.5 < κ < <1, it indicates that the decision is inclined to maximize group benefits, which is in line with most people’s views.Scheme Ranking. Qi is a cost variable, the smaller the value, the better. Therefore, it is possible to rank from small to large according to the Qi value and obtain the ranking result of the scheme.The complete evaluation flowchart of black-start schemes in this study is shown in Figure2.Figure 2
Evaluation flowchart of black-start schemes in this study.


## 4.1. Prospect Theory

The prospect theory is composed of value function and weight function. Among them, the value function is shown in equation (16) [6]. It can be seen that the “loss” of the same amount of change has a greater impact on the decision-maker than the “profit,” which is in line with the actual value of the decision-maker. Therefore, for the evaluation and ranking of black-start schemes, the prospect theory can be used to describe the “profit” and “loss” of black-start scheme evaluation indicators. The psychological perceived value of decision-makers can be considered. After the normalized evaluation matrix is processed by the value function, the value matrix V is obtained. Integrating the value function and the optimal combination weight of the evaluation indicator can obtain the prospect matrix U as follows:(16)vx=Δxa,Δx≥0,−c−Δxb,Δx<0.(17)V=vijm×n,(18)U=uijm×n,where Δx = xij′−xrj′ and xrj′ represents the reference value of the jth indicator. In this study, a, b, and c are selected as 0.88, 0.88, and 2.25, respectively. The average value of each index is used as a reference value, xj′=1/m∑r=1mxrj′. In this study, the average value of each indicator is used as a reference point, uij=vijwj.


## 4.2. Multicriteria Compromise Solution Ranking Method

VIKOR is a multiattribute decision-making method. It obtains the degree of closeness to the ideal solution by considering maximizing group benefits and minimizing the individual regrets of opposing opinions. The basic calculation steps are as follows:Ideal Solution Calculation. According to the foreground matrix obtained by equation (18) [6], a positive ideal solution U+ and a negative ideal solution U− are obtained:(19)U+=u1+,…,uj+,…,un+T,U−=u1−,…,uj−,…,un−T,where uj+=maxuij|1≤i≤n and uj−=minuij|1≤i≤n.Eigenvalue Calculation. Calculate the maximum group benefit value S (i), minimum individual regret value Y (i), and profit value Q (i) of the ith black-start scheme:(20)Si=∑j=1nwjuj+−uijuj+−uj−,(21)Yi=maxjwjuj+−uijuj+−uj−,(22)Qi=κSi−S−S+−S=+1−κYi−Y−Y+−Y−,where S−=minSi|1≤i≤n, S+=maxSi|1≤i≤n, Y−=minYi|1≤i≤n, and Y+=maxYi|1≤i≤n.If 0 <κ < <0.5, it indicates that the decision is inclined to regret value. If κ = 0.5, it indicates that the decision-making equilibrium is compromised. If 0.5 < κ < <1, it indicates that the decision is inclined to maximize group benefits, which is in line with most people’s views.Scheme Ranking. Qi is a cost variable, the smaller the value, the better. Therefore, it is possible to rank from small to large according to the Qi value and obtain the ranking result of the scheme.The complete evaluation flowchart of black-start schemes in this study is shown in Figure2.Figure 2
Evaluation flowchart of black-start schemes in this study.


## 5. Example Studies

### 5.1. Example Description

Taking the eight black-start schemes of a 330 kV power grid in the eastern part of a province as an example study, according to the principles of evaluation indicators mentioned in this study, a total of evaluation indicators corresponding to each black-start scheme are calculated. The parameters of the evaluation example are listed in Table4. Among the evaluation indicators, I1, I2, I5, I6, I7, I9, I11, and I12 are benefit-type indicators, and the others are cost-type indicators. Different types of evaluation indicators are dimensionlessly processed, and the normalized results are listed in Table 5.Table 4
Evaluation example.
I1 (MW)I2I3 (%)I4 (%)I5I6 (MW)I7 (MW)I8I9I10 (min)I11 (MW)I12I13 (MW)I14 (min)S113500.5130.7750.390.492275.281014.192.84772.00206000.91452.836S213500.5130.4850.190.540145.95851.312.24388.75122861.57123.027S313500.5130.4950.190.492275.28967.092.68669.33202861.57123.027S413500.5131.0750.870.516276.841090.601.856100.172013201.890105.660S522000.3990.5000.160.539137.53629.572.14393.50122861.57123.027S622000.3990.6000.160.661162.241269.062.088166.40162861.57123.027S722000.3991.6500.550.504223.70943.181.55087.60166000.91452.836S822000.3992.0000.880.609248.421323.982.151149.33206000.91452.836Table 5
Normalized result.
I1I2I3I4I5I6I7I8I9I10I11I12I13I14S10.2620.3950.3020.2120.3180.4320.3470.2600.2350.2810.3380.2290.2030.309S20.2620.3950.4830.4360.3490.2290.2920.3300.2890.4680.1610.3930.4650.412S30.2620.3950.4730.4360.3180.4320.3310.2750.2260.2810.1610.3930.4650.412S40.2620.3950.2180.0950.3330.4340.3740.3980.3260.2810.7440.4730.1010.185S50.4260.3070.4680.5170.3480.2160.2160.3450.3050.4680.1610.3930.4650.412S60.4260.3070.3900.5170.4270.2550.4350.3540.5420.3510.1610.3930.4650.412S70.4260.3070.1420.1510.3260.3510.3230.4770.2850.3510.3380.2290.2030.309S80.4260.3070.1170.0940.3930.3900.4530.3450.4860.2810.3380.2290.2030.309This study believes that in the overall evaluation of the black-start scheme, the weights of the different recovery stages are 0.45 : 0.2 : 0.35, and the ratios of the evaluation criteria for each stage are 0.25 : 0.25 : 0.18 : 0.32, 0.5 : 0.5, and 0.2 : 0.3 : 0.2 : 0.3, respectively. The subjective, objective weights, and combined weights of the evaluation indicators are listed in Table6. From Table 5, it can be found that there are both links and conflicts between the evaluation indicators. For example, I11, I13, and I14 are positively correlated. The larger the capacity of the restarted unit, the greater the required starting power, and the longer the natural start-up time will naturally be, which conforms to the attributes of general units. I11 has a negative correlation with I3 and I4. When I11 is larger, the values of I3 and I4 are smaller. This is because the larger the capacity of the restarted unit, the greater the impact load it brings, and the greater the system voltage and frequency fluctuations during the start-up process. The capacity of the restarted unit is often used as a benefit indicator, the larger the better, while the corresponding start-up power and start-up time are often cost-based indicators, the smaller the better. Therefore, the evaluation indicators can be correlated with each other. Compared with the literature [13] only using 5 evaluation indicators to evaluate different black-start schemes, the black-start scheme evaluation indicator system proposed in this study is more comprehensive and effective, which can assess the pros and cons of black-start schemes as a whole and can avoid the evaluation error caused by the lack of evaluation indicators. Moreover, this study considers the pros and cons of the black-start scheme as a whole based on the different stages of the system recovery and combines the evaluation criteria of the different stages with the evaluation indicators, which means that the constructed evaluation indicator system is closer to the actual situation of the system recovery.Table 6
Evaluation indicator weights.
I1I2I3I4I5I6I7I8I9I10I11I12I13I14AHP0.11400.03630.06680.06680.06040.06300.02420.10100.07080.06250.08600.08600.07770.0843EWM0.07040.06960.07320.07580.06950.07060.07010.06990.07110.07020.07530.07070.07350.0703w0.11200.03530.06830.07080.05870.06220.02370.09910.07030.06130.09060.08490.07980.0828


### 5.2. Evaluation Result

This study takes the decision coefficientκ=0.8, which tends to maximize the interests of the group, but it also has the smallest individual regret value, which is in line with the choice of most people. The group benefit value, individual regret value, and benefit value of each black-start scheme are shown in Table 7 and Figure 3.Table 7
S, Y, and Q values and ranking of each scheme.
SYQRankS10.6920.11218S20.4320.1120.4563S30.5170.1120.6326S40.5600.1120.7237S50.3460.0910.1962S60.2620.0910.0211S70.5260.0850.5525S80.5200.0850.5394Figure 3
Comparison of evaluation values of different black-start schemes.It can be seen from Table8 that this study believes that S6 is the best overall and S5 is the second best. It can be seen from the 14 evaluation indicators included in S6 that, except for I11, the other evaluation indicators perform well. The S6 has a large starting power supply capacity, and the characteristics of the restarted units are better, with small voltage fluctuations and frequency fluctuations, which largely ensures that the scheme can be successfully put into operation after a major power outage. Moreover, the total recovery load of S6 is the largest. The path nodes and recovery lines passed are the most important hub nodes and lines in the system, and the start-up time of the restarted units is the smallest, indicating that the overall performance of the scheme in the power system recovery process is the best. So, S6 can be used as the first choice for rapid system recovery after a major power outage. S5 can be used as a suboptimal solution. Although the overall performance is not as good as S6, it also has its advantages. The overall performance of voltage fluctuations and frequency fluctuations caused by the impact load of S5 is the best, indicating that the risk of voltage and frequency collapse in the recovery process of this solution is the lowest, and the line operation time of S5 is small, which can save system recovery time. Therefore, S5 can be used as an alternative to S6.Table 8
Comparison of different evaluation methods.
PT-VIKORVIKORTOPSISValueRankValueRankValueRankS108080.0207S20.55630.55730.7033S30.37660.43250.6605S40.28370.30770.6784S50.82120.82720.7822S6111111S70.45850.427608S80.47140.44340.1596


### 5.3. Comparison of Evaluation Method Results

To highlight the superiority of the evaluation method in this study, the evaluation results of VIKOR and the commonly used TOPSIS [24] method are compared. The evaluation result of the VIKOR combination (PT-VIKOR) is compared with the evaluation result of VIKOR. To effectively compare the evaluation results, this study normalizes the evaluation results. The larger the value, the higher the ranking. The evaluation results of the three methods are shown in Table 8 and Figure 4.Figure 4
Ranking results of black-start schemes based on different evaluation methods.It can be seen from the evaluation results of VIKOR and TOPSIS that the difference between the two is mainly in the ranking ofS4. VIKOR believes that S4 ranks seventh, lower than S3’s ranking, while S4 ranks fourth in TOPSIS, higher than S3’s ranking. Through the evaluation indicators included in S4 and S3, S4 has obvious advantages over I11 and I12 compared to S3. This is the benefit brought by the large capacity of the restarted unit. But the capacity of the restarted unit is large, which means it needs to be started with high power and a long start-up time. Moreover, the S4 has a large impact load. The voltage and frequency fluctuations caused by it are also large, indicating that the risk of the system crashing again during the black-start process is not conducive to the recovery of the power failure system. Compared with the shortcomings of S4, S3 only performs poorly on the capacity indicators of units to be started, but other indicators are better than S4. Therefore, the overall ranking of S4 should be lower and should be lower than that of S3. In addition, the TOPSIS method uses the Euclidean distance to be unable to distinguish the black-start scheme on the vertical line in the positive and negative ideal solutions, which may lead to deviations in the evaluation results. Compared with the TOPSIS method, the VIKOR method can not only avoid the shortcomings of the TOPSIS method itself but also bring the evaluation indicator weight into the calculation, which is closer to the actual evaluation requirements.From the evaluation results of PT-VIKOR and VIKOR, it can be seen that the only difference in the ranking of black-start schemes is the ranking ofS3 and S7. S7’s start-up power capacity, the capacity of units to be started, and line indicators perform better than S3’s evaluation indicators. The S3’s voltage fluctuation, frequency fluctuation, and the required power indicator of the restarted unit performed better than the evaluation indicator of S7. Therefore, it is difficult to rank the pros and cons of the two schemes only from the objective point of view of the black-start scheme. However, after introducing the prospect theory to describe the value of the perception of decision-makers, the value of the black-start scheme can be calculated. The return value of S3 is 0.0476, and the return value of S7 is 0.00571. This shows that S7 can bring greater value to the decision-maker. Its ranking should be higher than the S3 ranking. Therefore, the PT-VIKOR method correctly evaluates the black-start scheme. It can describe not only the psychological perceived value of the decision-maker but can also take into account the value of the benefits of the black-start scheme itself.


## 5.1. Example Description

Taking the eight black-start schemes of a 330 kV power grid in the eastern part of a province as an example study, according to the principles of evaluation indicators mentioned in this study, a total of evaluation indicators corresponding to each black-start scheme are calculated. The parameters of the evaluation example are listed in Table4. Among the evaluation indicators, I1, I2, I5, I6, I7, I9, I11, and I12 are benefit-type indicators, and the others are cost-type indicators. Different types of evaluation indicators are dimensionlessly processed, and the normalized results are listed in Table 5.Table 4
Evaluation example.
I1 (MW)I2I3 (%)I4 (%)I5I6 (MW)I7 (MW)I8I9I10 (min)I11 (MW)I12I13 (MW)I14 (min)S113500.5130.7750.390.492275.281014.192.84772.00206000.91452.836S213500.5130.4850.190.540145.95851.312.24388.75122861.57123.027S313500.5130.4950.190.492275.28967.092.68669.33202861.57123.027S413500.5131.0750.870.516276.841090.601.856100.172013201.890105.660S522000.3990.5000.160.539137.53629.572.14393.50122861.57123.027S622000.3990.6000.160.661162.241269.062.088166.40162861.57123.027S722000.3991.6500.550.504223.70943.181.55087.60166000.91452.836S822000.3992.0000.880.609248.421323.982.151149.33206000.91452.836Table 5
Normalized result.
I1I2I3I4I5I6I7I8I9I10I11I12I13I14S10.2620.3950.3020.2120.3180.4320.3470.2600.2350.2810.3380.2290.2030.309S20.2620.3950.4830.4360.3490.2290.2920.3300.2890.4680.1610.3930.4650.412S30.2620.3950.4730.4360.3180.4320.3310.2750.2260.2810.1610.3930.4650.412S40.2620.3950.2180.0950.3330.4340.3740.3980.3260.2810.7440.4730.1010.185S50.4260.3070.4680.5170.3480.2160.2160.3450.3050.4680.1610.3930.4650.412S60.4260.3070.3900.5170.4270.2550.4350.3540.5420.3510.1610.3930.4650.412S70.4260.3070.1420.1510.3260.3510.3230.4770.2850.3510.3380.2290.2030.309S80.4260.3070.1170.0940.3930.3900.4530.3450.4860.2810.3380.2290.2030.309This study believes that in the overall evaluation of the black-start scheme, the weights of the different recovery stages are 0.45 : 0.2 : 0.35, and the ratios of the evaluation criteria for each stage are 0.25 : 0.25 : 0.18 : 0.32, 0.5 : 0.5, and 0.2 : 0.3 : 0.2 : 0.3, respectively. The subjective, objective weights, and combined weights of the evaluation indicators are listed in Table6. From Table 5, it can be found that there are both links and conflicts between the evaluation indicators. For example, I11, I13, and I14 are positively correlated. The larger the capacity of the restarted unit, the greater the required starting power, and the longer the natural start-up time will naturally be, which conforms to the attributes of general units. I11 has a negative correlation with I3 and I4. When I11 is larger, the values of I3 and I4 are smaller. This is because the larger the capacity of the restarted unit, the greater the impact load it brings, and the greater the system voltage and frequency fluctuations during the start-up process. The capacity of the restarted unit is often used as a benefit indicator, the larger the better, while the corresponding start-up power and start-up time are often cost-based indicators, the smaller the better. Therefore, the evaluation indicators can be correlated with each other. Compared with the literature [13] only using 5 evaluation indicators to evaluate different black-start schemes, the black-start scheme evaluation indicator system proposed in this study is more comprehensive and effective, which can assess the pros and cons of black-start schemes as a whole and can avoid the evaluation error caused by the lack of evaluation indicators. Moreover, this study considers the pros and cons of the black-start scheme as a whole based on the different stages of the system recovery and combines the evaluation criteria of the different stages with the evaluation indicators, which means that the constructed evaluation indicator system is closer to the actual situation of the system recovery.Table 6
Evaluation indicator weights.
I1I2I3I4I5I6I7I8I9I10I11I12I13I14AHP0.11400.03630.06680.06680.06040.06300.02420.10100.07080.06250.08600.08600.07770.0843EWM0.07040.06960.07320.07580.06950.07060.07010.06990.07110.07020.07530.07070.07350.0703w0.11200.03530.06830.07080.05870.06220.02370.09910.07030.06130.09060.08490.07980.0828


## 5.2. Evaluation Result

This study takes the decision coefficientκ=0.8, which tends to maximize the interests of the group, but it also has the smallest individual regret value, which is in line with the choice of most people. The group benefit value, individual regret value, and benefit value of each black-start scheme are shown in Table 7 and Figure 3.Table 7
S, Y, and Q values and ranking of each scheme.
SYQRankS10.6920.11218S20.4320.1120.4563S30.5170.1120.6326S40.5600.1120.7237S50.3460.0910.1962S60.2620.0910.0211S70.5260.0850.5525S80.5200.0850.5394Figure 3
Comparison of evaluation values of different black-start schemes.It can be seen from Table8 that this study believes that S6 is the best overall and S5 is the second best. It can be seen from the 14 evaluation indicators included in S6 that, except for I11, the other evaluation indicators perform well. The S6 has a large starting power supply capacity, and the characteristics of the restarted units are better, with small voltage fluctuations and frequency fluctuations, which largely ensures that the scheme can be successfully put into operation after a major power outage. Moreover, the total recovery load of S6 is the largest. The path nodes and recovery lines passed are the most important hub nodes and lines in the system, and the start-up time of the restarted units is the smallest, indicating that the overall performance of the scheme in the power system recovery process is the best. So, S6 can be used as the first choice for rapid system recovery after a major power outage. S5 can be used as a suboptimal solution. Although the overall performance is not as good as S6, it also has its advantages. The overall performance of voltage fluctuations and frequency fluctuations caused by the impact load of S5 is the best, indicating that the risk of voltage and frequency collapse in the recovery process of this solution is the lowest, and the line operation time of S5 is small, which can save system recovery time. Therefore, S5 can be used as an alternative to S6.Table 8
Comparison of different evaluation methods.
PT-VIKORVIKORTOPSISValueRankValueRankValueRankS108080.0207S20.55630.55730.7033S30.37660.43250.6605S40.28370.30770.6784S50.82120.82720.7822S6111111S70.45850.427608S80.47140.44340.1596


## 5.3. Comparison of Evaluation Method Results

To highlight the superiority of the evaluation method in this study, the evaluation results of VIKOR and the commonly used TOPSIS [24] method are compared. The evaluation result of the VIKOR combination (PT-VIKOR) is compared with the evaluation result of VIKOR. To effectively compare the evaluation results, this study normalizes the evaluation results. The larger the value, the higher the ranking. The evaluation results of the three methods are shown in Table 8 and Figure 4.Figure 4
Ranking results of black-start schemes based on different evaluation methods.It can be seen from the evaluation results of VIKOR and TOPSIS that the difference between the two is mainly in the ranking ofS4. VIKOR believes that S4 ranks seventh, lower than S3’s ranking, while S4 ranks fourth in TOPSIS, higher than S3’s ranking. Through the evaluation indicators included in S4 and S3, S4 has obvious advantages over I11 and I12 compared to S3. This is the benefit brought by the large capacity of the restarted unit. But the capacity of the restarted unit is large, which means it needs to be started with high power and a long start-up time. Moreover, the S4 has a large impact load. The voltage and frequency fluctuations caused by it are also large, indicating that the risk of the system crashing again during the black-start process is not conducive to the recovery of the power failure system. Compared with the shortcomings of S4, S3 only performs poorly on the capacity indicators of units to be started, but other indicators are better than S4. Therefore, the overall ranking of S4 should be lower and should be lower than that of S3. In addition, the TOPSIS method uses the Euclidean distance to be unable to distinguish the black-start scheme on the vertical line in the positive and negative ideal solutions, which may lead to deviations in the evaluation results. Compared with the TOPSIS method, the VIKOR method can not only avoid the shortcomings of the TOPSIS method itself but also bring the evaluation indicator weight into the calculation, which is closer to the actual evaluation requirements.From the evaluation results of PT-VIKOR and VIKOR, it can be seen that the only difference in the ranking of black-start schemes is the ranking ofS3 and S7. S7’s start-up power capacity, the capacity of units to be started, and line indicators perform better than S3’s evaluation indicators. The S3’s voltage fluctuation, frequency fluctuation, and the required power indicator of the restarted unit performed better than the evaluation indicator of S7. Therefore, it is difficult to rank the pros and cons of the two schemes only from the objective point of view of the black-start scheme. However, after introducing the prospect theory to describe the value of the perception of decision-makers, the value of the black-start scheme can be calculated. The return value of S3 is 0.0476, and the return value of S7 is 0.00571. This shows that S7 can bring greater value to the decision-maker. Its ranking should be higher than the S3 ranking. Therefore, the PT-VIKOR method correctly evaluates the black-start scheme. It can describe not only the psychological perceived value of the decision-maker but can also take into account the value of the benefits of the black-start scheme itself.


## 6. Conclusion

First, from the perspective of the components of the black-start scheme, this study considers the characteristics of power grid nodes and three stages of system recovery and constructs 14 evaluation indicators for the black-start scheme. Second, the analytic hierarchy process is used to combine the evaluation indicators with the evaluation criteria of different recovery stages, and a reasonable and scientific evaluation indicator system of the black-start scheme is constructed. Third, to better reflect the combination of subjective and objective, combined with the objective results of the entropy weight method and the subjective weight of AHP, this study calculates the combined weight of evaluation indicators to avoid the weight defect caused by the insufficient combination of subjective and objective. Forth, to reflect the role of the decision-maker as the main factor of the black-start scheme, the prospect theory is used to reflect the psychological perceived value of the decision-maker. The prospect matrix with the subjective thoughts of decision-makers is brought into the VIKOR method to evaluate and rank the black-start scheme, and the appropriate black-start scheme can be selected from different interests. Finally, the black-start scheme example of Gansu power grid verifies the rationality and accuracy of the black-start scheme evaluation model based on the evaluation indicator system proposed in this study. By comparing the evaluation results of different evaluation models, the evaluation model in this study is closer to the actual power grid operation requirements and can provide new ideas and methods for decision-makers when they choose the black-start scheme.After a blackout accident, the focus of the black-start scheme is different for power grids with different voltage levels. How to combine the evaluation indicator system with the actual operation of power grid restoration is one of the key work directions. Researchers can supplement and improve the evaluation model proposed in this study and adjust the evaluation indicator system for their actual power grid. At the same time, this study does not make specific improvements to the evaluation weights, and the next step will be to research the evaluation indicator weights based on the evaluation indicator system.



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*Source: 5077812-2022-08-04.xml*