Study on the smart grid vulnerability index based on the moment-generating function and distortion function

3.1 Identification of the vulnerability sources of smart grids

Vulnerable sources refer to a variety of factors that may introduce accident risk to the system, which may be derived from the technical level or natural disasters. Based on summarizing relevant literature on the analysis of many typical large-scale smart grid power outage events and their causes in recent years, the potential vulnerability sources that affect the vulnerability of smart grids are presented as follows:

Structural vulnerability: According to the characteristics of a power grid, the structural vulnerability of a power grid can be considered from the perspectives of the power supply structure and power grid operation structure. Based on a previous analysis of multiple grid faults, the reason for these faults is that the structure of the power grid is unjustifiable. An unjustifiable power grid structure may be attributed to the unjustifiable power supply structure or the unjustifiable layout structure of the power grid. If the distance between the power source location and the power load center is large, the unjustifiable power grid layout structure or the unjustifiable power supply structure may prevent the dispatching from being able to maintain the load changes in a timely manner, and the power grid is likely faulty, which leads to the occurrence of fault events.

3.2 Framework of the indicator system

With the guidance of the selection principles of the smart grid vulnerability evaluation indicator system, the assessment of smart grid vulnerability should start from the perspectives of structure, function, equipment, technology, and environment. Therefore, an evaluation indicator system of smart grid vulnerability is constructed for this smart grid project in this study, as shown in Table 1.

4.1 Distortion function-based index calculation method

For smart grids, events such as crash and collapse are rare but once these low-probability events occur, they will have a significant impact on a power grid. More attention should be paid to these extreme events with small probability and high risk. The essence is to emphasize extreme events with high risk and low probability. By giving high weights to events with high risk, smart grid managers can have a clearer understanding of power grid vulnerability.

In the financial sector, with increased globalization of financial markets and the integration of credit markets, the tail risk of extreme events with small probability and large value also attracts people’s attention. In this section, we apply the functional risk measurement method in the financial sector to calculate the vulnerability index of the power grid based on the distortion function. The distortion function is described as follows:

Definition 1. Assume that the mapping g : [0, 1] → [0, 1] is a nondecreasing function and satisfies g (0) = 0 and g (1) = 1. For the probability measure P over the σ-algebra F, if P^* = g (P) is the distortion probability distribution, g is referred to as the distortion function.

Assume that X is a nonnegative loss random variable with a distribution function of F (x) = P (X ⩽ x), and define its survival function as S (x) = 1 - F (x). Its measure is [22]: E ( x ) = ∫ 0 ∞ xdF ( x ) = ∫ 0 ∞ S ( x ) dx(1)

Definition 2. Wang’s functional risk measure is defined as: ρ g ( X ) = E g ( X ) = ∫ 0 ∞ g [ S ( x ) ] dx(2) where g is the distortion function, which transforms the survival function S into a new function g ∘ S, g ∘ S is a nonincreasing function of x. Define the mean of the corresponding distorted survival function, E _g  (X), as a measure of risk. Obviously, the distribution probability of the distortion function g is changed.

Definition 3. For discrete risk X, the probability distribution is P (X = x _k ) = p _k , k = 1, 2, ⋯ , n, then p₁ ⩾ 0, p₂ ⩾ 0, …… , p _n  ⩾ 0 here p₁ + … + p _n  = 1. Assume that x _k , k = 1, 2, …, n is on the order of x₁ < x₂ < … x _n . The functional risk measure of risk X can be obtained as: ρ g ( X ) = ∑ s = 1 n g ( ∑ k = s n P k ) ( x s - x s - 1 ) , x 0 = 0(3)

Or ρ g ( X ) = ∑ s = 1 n X s [ g ( ∑ k = s n P k ) - g ( ∑ k = s + 1 n P k ) ](4)

The vulnerability index used in this study corresponds to the discrete risk X. We preset five levels of the vulnerability index—slightly vulnerable, mildly vulnerable, moderately vulnerable, strongly vulnerable, and extremely vulnerable—for which we present x = 10, 30, 50, 70, 90, respectively. The discrete probability distribution P (x = x _k ) = P _k satisfies the conditions of Definition 3. Therefore, the distortion function is appropriate to calculate the final vulnerability index.

In the calculation and analysis of the smart grid vulnerability index, we need to determine the weight of each indicator. The subjective-objective weighting method is mainly used to determine the weight of each indicator. Subjective distortion fully utilizes the knowledge and experience of experts to score indicator weights, and based on the scoring results, an objective analysis method is used to determine the weight of each index. However, when experts make judgments, they need to have a certain scoring rule. Here, the index scale method, that is, the different importance of pairwise indicators is represented by different numbers.

When calculating the vulnerability index of a smart grid, due to the complexity of the smart grid itself and the indicator system, assessing the importance of the pairwise indicators based on the scale method shown in Table 1, which causes a relatively large error, is unrealistic. When scaling, the degree of fuzziness is determined using triangular fuzzy numbers combined with the AHP, and then the reliability is calculated. Based on the obtained credibility, obtain the final expected value of the index. All experts are assumed to score the pairwise indicators to determine a set of fuzzy judgment matrices. However, expert scoring would also be influenced by subjective preference. To minimize the influence and bias caused by subjective factors, the following formula can be used to calculate the comprehensive score:

Judgment matrix: the element in A = (a _ij ) _n×m is expressed as a_ij= [l _ij , m _ij , u _ij ]: M ij k = r 1 a ij ( 1 ) + r 2 a ij ( 2 ) + . . . + r t a ij ( T )(5) where n is the number of decision-making factors for pairwise comparison. k is the hierarchical layer of indicators. a ij ( T ) , T = 1, 2, ⋯ , T. T is the triangular fuzzy number given by the T ^th expert, which is the indicator weight assigned by different experts. The comprehensive scoring matrix of all elements of the k ^th layer with respect to the h ^th factor of the (k - 1)  ^th layer can be determined according to this formula, the expected value is solved using y formulas, and eventually the comprehensive judgment matrix M is obtained.

The weighting coefficients of the matrix can be determined using a variety of methods. In this study, the “inverse of column sum” method is applied. Denoting c j = 1 / ∑ i = 1 n k a ij , then the weight coefficient is ω n i j k = c j / ∑ k = 1 n k c k(6)

Thus, the weighting of a single hierarchical sequence ( ω 1 h k , ω 2 h k , . . . , ω n k h k ) T can be obtained, where n _k represents the number of indicators in the k ^th layer. This method mainly distinguishes different objects to be evaluated as the whole; so, methods are needed to distinguish the importance of different indicators. In this study, the AHP is used to assign weight to each indicator, and the weights of different indicators are used to distinguish the importance of different indicators. Via the AHP, the weights of different indicators are obtained. The differences among different indicators can be effectively determined by their assigned weights, that is, x ij * = ω j x ij(7)

Considering the complexity of a smart grid vulnerability assessment, the evaluation indicator is the primary factor in an evaluation. Considering whether the evaluation values of different indicators are balanced, the degree of balance can further supplement and interpret the evaluation results. To more reasonably interpret the values of evaluation indicators, this study introduces the quasi-Gini coefficient in mathematical economy, and on this basis, the AHP is complemented and refined to obtain a credibility-based improved AHP, which is expressed by formula (7) [23]: y i = λ G i ∑ j = 1 n ( ω j x ij * ) + ( i - λ ) ∑ j = 1 n max j { ( ω j x ij * ) }(8) where x _ij is the evaluation value of the evaluation object P _i  (i = 1, 2, …, m, j = 1, 2, …, n) in terms of the indicator O _j  (j = 1, 2, …, n). ω _j is the weight of the k ^th element in this layer; and G _i is a quasi-Gini coefficient, which is expressed as

G i = 1 n + 2 n 1 ∑ j = 1 n X ij ∑ k = 1 n - 1 ( n - k ) X ik , i = ( 1 , 2 , … , n )(9) where n is the number of indicators; 1 and 2 is the ascending order from small to large; and λ is an adjustable parameter. λ = 0 indicates that the relevant indicator that has the largest product of the indicator value and its weight is mainly considered in the computer simulation algorithm, and λ = 1 indicates that, instead of just focusing on the indicator with the largest product of indicator value and weight, all indicators are considered according to their weight and modified by the quasi-Gini coefficient. In the actual calculation, the λ value can be determined appropriately. However, in the actual evaluation process, accurately measuring the importance of different indicators, which causes deviations in determining the importance of indicators, is difficult. With the premise of controlling the influence degree of deviation, testing the deviation and incompatibility of the judgment matrix is necessary. The test can be carried out by calculating the consistency ratio. The specific formula is expressed as follows:

Consistency indicator CI is calculated as

When CR < 0.1, this judgment matrix is considered to have a high level of satisfaction and is acceptable. Although this matrix has high consistency, it needs to be normalized to make the processed result more reasonable, that is: ω ih k = ω ih k ∑ i = 1 n k ω ih k(11)

Therefore, the single ordering ω h k = ( ω 1 h k , ω 2 h k , … , ω n k h k ) T of different elements in the k ^th layer with respect to the h ^th factor in the (k - 1)  ^th layer can be obtained. h is taken throughout 1, 2, …, n_k-1 to obtain the n _k  × n_k-1 weight matrix ω i k . Denoting the ordering weight vector of the (k - 1)  ^th layer with respect to the general objective is expressed as W k - 1 = ( W 1 k - 1 , W 2 k - 1 , . . . , W n k - 1 k - 1 ) T . The weight of the composite ordering W ^k of all elements in the kth layer is expressed as follows: W k = ( W 1 k , W 2 k , . . . , W n k k ) T = ω k W k - 1(12)

The total ordering consistency ratio: CR = ∑ i = 1 n k - 1 ω j CI j ∑ i = 1 n k - 1 ω RI j(13)

If the CR cannot pass the consistency test, then the judgment matrix needs to be adjusted until the adjusted matrix can pass the consistency test before the next step calculation can be performed.

After obtaining the weight for each indicator, normalization can be performed. Each indicator is treated as the discrete distribution of the vulnerability, and the vulnerability index is obtained using formula (2). ρ g ( X ) = ∑ s = 1 n g ( ∑ k = s n P k ) ( x s - x s - 1 )(14) where x _i is an indicator, P _k is the value after normalization of the weight obtained according to the weighting method. In this formula, it can be considered the probability, and g is its distortion function.

5.1 Selection and determination of evaluation indicators

The weights of the secondary and primary indicators are obtained using the credibility-based moment-generating function method described in this study, as shown in Table 3. Similarly, the weights of the tertiary indicators calculated by the above method are correspondingly multiplied by the weights of the second secondary indicators to obtain the final weights of the tertiary indicators.

To more clearly and intuitively calculate and analyze the power grid vulnerability index of smart grids, this study divides the state of the grid vulnerability index evaluation into five levels, namely vulnerable, slightly vulnerable, mildly vulnerable, moderately vulnerable, strongly vulnerable, and extremely vulnerable, and the corresponding scores are set to 10 points, 30 points, 50 points, 70 points, and 90 points, respectively. By referring to the laws and regulations formulated by the state, the rules and standards within the industry, and historical data and expert opinions, the occurrence probabilities of different levels are summarized in Table 4.

When calculating the comprehensive evaluation vector of each primary indicator X _i , the analogue of the primary indicator evaluation vector is made to the discrete risk mentioned in formula (3). For the primary indicator A₁, its secondary indicators X₁₁, X₁₂, X₁₃, X₁₄ are its probability distribution, x_1i ⩾ 0. The sum of its secondary indicators is normalized to ensure that X₁₁ + X₁₂ + X₁₃ + X₁₄ = 1, i.e., (0.053, 0.057, 0.052, 0.046) is transformed to (0.25, 0.27.0.25, 0.23).

Each row in the previous data table is equivalent to the value of its vulnerability. Considering the secondary indicator of the structural vulnerability of the power source as an example, its weight ρ₁₁ = 0.044, and its indicator probability distribution is: P 1 = P ( x = x 1 = 10 ) = 0.25 P 2 = P ( x = x 2 = 30 ) = 0.25 P 3 = P ( x = x 3 = 50 ) = 0.30 P 4 = P ( x = x 4 = 70 ) = 0.15 P 5 = P ( x = x 5 = 90 ) = 0.05

Substitute this distribution into formula 3 and the use of X₁₁ to represent the vulnerability of the power structure calculated by distortion function: X 11 = ∑ s = 1 5 g ( ∑ k = s 5 P k ) ( x s - x s - 1 ) = 40.7

Similarly, the expectations after distortion of all secondary indicators can be calculated. Then, the expectations after distortion of each primary indicator can be obtained by multiplying it with the normalized secondary indicators weight. The results are shown in Table 5.

According to the expectations after distortion of the primary indicators, the final vulnerability index can be calculated as: I vul = ρ 1 × A 1 + ρ 2 × A 2 + ρ 3 × A 3 + ρ 4 × A 4 + ρ 5 × A 5 = 37.44

The calculation results show that this power grid is mildly vulnerable, and the power grid users can be provided with a reliable power supply. The power grid can withstand most disturbances without power outage.

6.1 Analysis of calculation results

By analyzing the previous example and based on the criterion for assessing the vulnerability index level, the vulnerability index of the power grid is between “mildly vulnerable” and “moderately vulnerable” and skewed to being mildly vulnerable. However, a certain risk remains. Based on the specific index of each primary indicator, the foremost factor that causes the vulnerability of this power grid is structural vulnerability, followed by functional vulnerability and equipment vulnerability, while the vulnerabilities of the technology application level and natural environment impact have the lowest degree. Based on the calculation results and the constructed evaluation indicator system, the vulnerability state of the power grid is analyzed from five aspects:

In terms of the structural vulnerability, this power grid project is accessed with wind power, which is a renewable energy that has a substantial influence on the power grid structure due to the instability of wind power. Unstable operation of the power grid is possible, and the ability to control the power grid is weakened. When the main power transmission channels are blocked, the networking effect in local areas decreases, which further increases the structural vulnerability of the power grid system.

6.2 Case comparative analysis

In order to verify the effectiveness of the method proposed in this paper, two additional cases with the same data in this paper are designed for comparative analysis.

Case 1: implementing the credibility-based improved AHP, moment-generating function and distortion function proposed in this paper (the results have been shown in Section 4);

Case 2: replacing the proposed credibility-based improved AHP, and implementing AHP method in Ref. [24] to calculate the indicators weights;

Case 3: replacing the moment-generating function and distortion function proposed in this paper, and implementing the fuzzy Delphi method in Ref. [25] to identify the final evaluation indicators.

The methods in Case 2 and Case 3 are used to evaluate the vulnerability of the smart grid project in Section 4. Correspondingly, the vulnerability indexes of five primary indicators can be obtained, which are shown in Fig. 1.

OPEN IN VIEWER

Fig. 1

The vulnerability indexes of the primary indicators in different cases.

According to the expectations after distortion of the primary indicators, the corresponding variance and coefficient of variation of primary indicators can be obtained, which are shown in Table 6. And the superiorities of the credibility-based improved AHP, moment-generating function and distortion function proposed in this paper is verified by comparing dispersion of the evaluation results of each case.

The calculation results show that the vulnerability indexes of the primary indicators in Case 1 have the smallest variance and coefficient of variation, indicating that the dispersion of the evaluation result in this paper is the smallest. In fact, there are a lot of uncertainties and fuzziness in the process of evaluation. The greater the dispersion of evaluation results, the more absolute the judgment that the evaluation object belongs to a certain level, which can not well reflect the objective situation. Therefore, the credibility-based improved AHP, moment-generating function and distortion function proposed in this paper have better applicability.