Risk group assessment based on the demands of important consumers with standby electric source

Abstract

Consumers equipped with standby electric source will become electricity providers at particular moments in the future smart grid. From the perspective of smart grid risk management, this paper concludes the supply risk factors that may lead to the dissatisfaction of consumer demands for security, quality and benefit, which are further classified to supply risk elements. Then the mapping relations between supply risks and consumer demands is established. For the risk assessment problem involving consumer demands and power supply risk, a risk assessment model based on catastrophe theory and 2-tuple linguistic representation is proposed. Firstly, the experts’ linguistic variables are converted to 2-tuples linguistics, and catastrophe theory expands operations from real numbers to linguistic variables. Secondly, combining the expert subjective weights in the form of linguistic, a compromised weighting method based on similarity and deviation is proposed. Subsequently, the ranking results are obtained according to the total catastrophe 2-tuple linguistic value of each supply risk element. The ranking results are helpful for power supply companies to identify important risks and develop risk aversion strategies. At last, the feasibility and effectiveness of the proposed method is proved by contrastive analysis. Therefore, the study could provide reference for future smart grid risk management.

Keywords

Important consumers with standby electric source supply risk catastrophe theory 2-tuple linguistic

1 Introduction

Important consumers are power users within the power supply range who play important roles in society, politics and economy. Their power outages may cause personal injury or death, serious environmental pollution, great political influence, heavy economic loss, and serious social disorder. Scholars analyzed and investigated the risks in the power supply process from the consumer perspective. Tian et al. [1] proposed the concept of consumer demand risk and built a risk transmission model combining trapezoidal fuzzy number and Mahalanobis distance [1]. Horvat et al. [2] proposed an average risk optimization model to simulate the CHP consumer risk [2]. Li and Gao [3] proposed a risk assessment method based on cloud model and grey interval incidence for the uncertainty risks of important consumers [3]. Chowdhury et al. [4] studied the impact of consumers’ human factor behavior on supply security risk and proposed measures to improve the existing risk aversion schemes [4]. Li et al. [5] considered equipping important consumers with distributed generation as an effective method to reduce power loss and promote social benefits.

Dzobo et al. [6] proposed a multi-dimensional customer segmentation model for the reliability-worth analysis of power systems [6].

The above studies only direct at some aspects of supply risks (such as power supply security and reliability), without considering the impacts of other risk factors of grid companies on important consumers. With the development of smart grid, the potential supply risks as well as consumer demands will increase. In the future, distributed generation will be gradually connected to the consumer side. These important consumers with standby electric source (ICSES) will play dual roles, both the consumers of secondary energy and electrical services, and electricity providers at particular moments [7]. Therefore, as grid stakeholders, ICSES want to reduce their electricity costs and increase their benefits while getting high-quality energy and good electrical services. To protect the interests of ICSES, grid companies need to consider consumers’ benefit risk as a supplement to traditional consumer demand risks, conclude supply risk factors that may lead to the risk, and assess the risk values of these factors. Domestic research idea on electric power risk assessment is studying a smart grid demonstration project, choosing a scientific evaluation index system and then using combined intelligent algorithms [8, 9] (such as fishbone diagram method - Delphi method, ANP, anti-entropy method, principal component clustering analysis method) to calculate risks and draw conclusions. By contrast, study abroad are more specific: Chan et al. [10] examined how residential consumers perceive the smart grid and what factors influence their acceptance of the smart grid through a survey for electricity consumers in Korea, and studied the smart grid management policy risk from the perspective of consumers [10]. Eising et al. [11] identified the risks that arise from the integration of energy and transport supply chains, and proposed a new method as a strategic tool to plan (smart) grid investments [11]. Rice and Anas [12] analyzed the smart grid system risk and proposed that upgrading power grid with networked metrology would introduce new classes of risks [12]. Schuyler (2014) proposed new methods for the normalization and ranking of criteria drawing lessons from a published multi-criterion assessment of power systems [13]. Johansson et al. analyzed the vulnerability of power grid system [14].

Existing studies generally do not involve the situation that consumers participate in power distribution and the interaction between supply risk factors and consumer demands. Therefore, this paper analyzes consumers demands at first, then concludes the supply risk factors that may lead to the dissatisfaction of consumer demands and builds evaluation model. Furthermore, by case analysis and result comparison, the rationality and superiority of model is verified. At last, risk aversion strategies are developed according to the risk values.

2 Demand and supply risk analysis of ICSES

Drawing lessons from [7], consumer demands are divided into three categories: 1) Security demand. When one or more supply risks cause the dissatisfaction of the demand, stakeholders will suffer enormous damages. 2) Quality demand. When one or more supply risks cause the dissatisfaction of the demand, the working condition of stakeholders will be adversely affected, but the consequence is no more serious than the former. 3) Benefit demand. When one or more supply risks cause the dissatisfaction of the demand, stakeholders will suffer economic losses, but the extent of the loss is lower than the former.

Security Demand

From the viewpoint of whether important consumers or grid companies, the security requirements are first. Security risk event may cause casualties and large-scale blackouts, which are great social losses.

Quality Demand

The consequence of dissatisfying quality demand can be divided into three categories in accordance with the severity of the impacts on power users: ➀ Power grid cannot provide consumers with continuous and enough power to meet the load, the unexpected power outages will cause industrial, commercial and residential electricity consumption stagnation, which will cause great social and economic losses. ➁ Power grid provides unqualified power. The unqualified power will shorten electrical equipment service life or even directly scrap electrical equipment, leading to great economic losses. ➂ Grid companies will provide consumers with a range of services in the sale process, which aims to help consumers to utilize power more quickly and easily. If consumers do not enjoy electrical services they deserve, their confidence in grid company will reduce, which will prompt them to consider using other sources of energy, resulting in indirect economic losses.

Benefit Demand

ICSES can access distributed generation for personal use, and they can also sell excess power to obtain economic benefits. This demand includes two aspects: ➀ Cost demand. If consumers’ electricity cost increase for power tariff rise and a series of other reasons, they will consider other forms of energy or increase other power configurations, which will affect the sustainable development of grid companies. ➁ Revenue demand. Consumers earn revenue by participating in power generation through self-contained power supply. If the revenue is not high enough, consumers’ enthusiasm will be dampened, which is not conducive to the interaction between grid companies and consumers.

Risk element refers to the set of uncertain factors which impact on actual results in particular environment and time. Grid companies face many risk factors in actual operations. From the viewpoint of uncertain factors that induce risks, grid companies’ supply risk elements that may cause consumer demand risk in power supply link are grouped into four [1]: system risk element, structural risk element, operational risk element, environmental risk element. ➀ System risk element refers to the risks brought by the change of system operation mode and system equipment failure in the integration of main network and distribution network to important consumers. ➁ Structural risk element is the collection of risk factors in network planning, including power supply layout risk and transmission technology risk. ➂ Operational risk element refers to the risks caused by the change of system device state during the use in power transmission and distribution projects, including primary electrical equipment operating risk, secondary electrical equipment operating risk, other equipment operating risk, load rate risk, duty cycle risk and so on. ➃ Environmental risk element includes natural environment risk and supply management system risk. Natural environment risk refers to that, in long distance power transmission, most power facilities expose to the natural environment, so power supply system faces the environmental impacts of climate, geology, natural disasters and so on, and once these risks occur, the risk in a certain part of the power supply system will be caused, which will lead to the dissatisfaction of consumer demand. Supply management system risk refers to the risks brought by the obvious loopholes in risk management system implemented by grid companies or the loss increase due to inappropriate risk plan, including emergency power supply configuration risk, operational management risk, etc. All above are shown in Table 1.

Table 1
Supply risk elements

Supply risk R System risk element r₁ System operation method change risk r₁₁

System associated equipment failure risk r₁₂

Structural risk element r₂ Power supply layout risk r₂₁

Transmission technology risk r₂₂

Operational risk element r₃ Equipment operating risk r₃₁

Load rate risk r₃₂

Duty cycle risk r₃₃

Environmental risk element r₄ Natural environmental risk r₄₁

Supply management system risk r₄₂

One of the difficulties of risk assessment is to identify the correspondence between supply risk elements and consumer demands [1]. For example, system risk element may cause the dissatisfaction of security or quality or benefit demand, and the influence degree varies according to different consumers. Based on the authors’ previous studies (a real project in Jiangxi, China), this paper considers that one supply risk element may cause the dissatisfaction of multiple demands and one consumer demand can be affected by many risk elements, that is, the mapping relation between supply risk elements and consumer demands is many-to-many, as shown in Fig. 1.

Fig.1

Mapping relation graph between supply risk elements and consumer demands.

When describing the uncertain complex mapping relations between supply risk elements and consumer demands, people tend to use “high”, “low” and other words for evaluation. Methods for processing linguistic information can be roughly divided into two: ➀ Convert the linguistic assessment information into fuzzy numbers or exact numbers, then operate and analyze based on the extension principle [15]. ➁ Directly operate and process based on the order and properties of linguistic assessment set [16]. However, these two methods are flawed. Group assessment information aggregated by individual linguistic assessment information cannot be always expressed by a single semantic phrases previously defined in linguistic assessment set. There must be an approximation, resulting in loss of information and inaccurate results. In this regard, Spain Professor Herrera put forward 2-tuple linguistic method to solve the linguistic information aggregation problem, which overcomes the defects existing in previous researches and applies to multi-granularity linguistic scale multi-attribute decision analysis successfully. Dutta and Guha [17] proposed a partition Bonferroni mean operator in linguistic 2-tuple environment and developed three new linguistic aggregation operators [17]. Tao et al. [18] developed some novel operational laws of 2-tuples linguistic information based on the Archimedean t-norm and s-norm [18]. Truck [19] studied on two models: the 2-tuple semantic model and the 2-tuple symbolic model to represent the information during decision making [19]. Li et al. [20] generalized 2-tuple linguistic induced generalized ordered weighted averaging distance (2LIGOWAD) operator to model a multiple attribute decision making problem [20]. You et al. [21] extended 2-tuple linguistic into interval 2-tuple linguistic to assess the diversity and uncertainty of assessment information, and proposed a VIKOR method for group multi-criterion supplier selection problem [21]. The above studies dig the mechanism of 2-tuple linguistic, and achieve certain theoretical results. However, when it comes to the problem of determining attribute weights, different methods will produce various weights, leading to different results. If the subjective factors and human factors of decision makers weight heavier, it is likely to lead to evaluation result error and decision result deviation. Catastrophe theory proposed by the French scholar R. Thom in 1972 can completely solve this problem, which has been widely used in risk assessment of geology, water, military and other fields [22, 23].

Thus, this paper introduces catastrophe theory in supply risk assessment based on 2-tuple linguistic representation, and proposes a compromised weighting method based on similarity and deviation according to the given experts’ subjective weights in linguistic form. In addition, this paper uses 2-tuple linguistic representation to aggregate information, adopts catastrophe function to get catastrophe values, and then determines the risk ranking results.

3 Basic theory

3.1 Catastrophe theory

In 1972, the French mathematician René Thom proposed catastrophe theory, a theoretical model combining topology, singularity theory and stability mathematical theory for discontinuous phenomena in nature. The model mainly studies the phenomena and laws of a system transiting from one steady state to another steady state in which each state of the system is described by a set of parameters. When the function value of a state is unique, then the system is in a stable state; when the function value is not unique, the parameters must change within a certain range and the system is in an unstable state, which will inevitably change into a stable state with the changes of parameters, thereby leading to the mutation of the system [22]. Therefore, the theory has also been regarded as part of chaos theory.

In catastrophe theory, the critical points of each state in the system are classified by potential function V (x), and each potential function V (x) determines a catastrophe. The set of points which satisfy the first derivative of potential function V (x) is zero is called equilibrium surface M, which can fully describe the whole mutation process of the system. Based on the control variables of potential function, the most basic seven kinds of catastrophe models are fold catastrophe, cusp catastrophe, swallowtail catastrophe, butterfly catastrophe, elliptic umbilical catastrophe, hyperbolic umbilical catastrophe and parabolic umbilical catastrophe. This paper briefly introduces the features of the frequently-used four catastrophe models, as shown in Table 2 [24, 25].

Table 2
The frequently-used four catastrophe models

Catastrophe types Dim Potential function V (x) Equilibrium surface M Normalization formula

Fold catastrophe 1 x³ + ax 3x² + a = 0 $x_{a} = \sqrt{a}$

Cusp catastrophe 2 x⁴ + ax² + bx 4x³ + 2ax + b = 0 $x_{a} = \sqrt{a}$ , $x_{b} = \sqrt[3]{b}$

Swallowtail catastrophe 3 x⁵ + ax³ + bx² + cx 6x⁴ + 3ax² + 2bx + c = 0 $x_{a} = \sqrt{a}$ , $x_{b} = \sqrt[3]{b}$ , $x_{c} = \sqrt[4]{c}$

Butterfly catastrophe 4 x⁶ + ax⁴ + bx³ + cx² + dx 6x⁵ + 4ax³ + 3bx² + 2cx + d = 0 $x_{a} = \sqrt{a}, x_{b} = \sqrt[3]{b}, x_{c} = \sqrt[4]{c}, x_{d} = \sqrt[5]{d}$

Catastrophe types	Dim	Potential function V (x)	Equilibrium surface M	Normalization formula
Fold catastrophe	1	x³ + ax	3x² + a = 0	$x_{a} = \sqrt{a}$
Cusp catastrophe	2	x⁴ + ax² + bx	4x³ + 2ax + b = 0	$x_{a} = \sqrt{a}$ , $x_{b} = \sqrt[3]{b}$
Swallowtail catastrophe	3	x⁵ + ax³ + bx² + cx	6x⁴ + 3ax² + 2bx + c = 0	$x_{a} = \sqrt{a}$ , $x_{b} = \sqrt[3]{b}$ , $x_{c} = \sqrt[4]{c}$
Butterfly catastrophe	4	x⁶ + ax⁴ + bx³ + cx² + dx	6x⁵ + 4ax³ + 3bx² + 2cx + d = 0	$x_{a} = \sqrt{a}, x_{b} = \sqrt[3]{b}, x_{c} = \sqrt[4]{c}, x_{d} = \sqrt[5]{d}$

In Table 2, x is a state variable, indicating the state of the system; coefficient a, b, c, d whose importance is diminishing are the control variables for each state variable, representing various factors that interact with each other within the system; potential function V (x) reflects the relationship between state variables and control variables in the system; normalization formula is used to solve the catastrophe value of each control variable, also known as catastrophe progression. When making multi-objective evaluation analysis by catastrophe theory, we need to consider two principles, namely “non-complementary” principle and “complementary” principle. If the control variables (such as a, b, c, d) in a system cannot substitute each other, that is, one control variable cannot make up the deficiency of another control variable, then state variable x is determined by selecting the smallest one from catastrophe progression x_a, x_b, x_c, x_d corresponding to each control variable calculated by normalization formulas. If there are correlations among the control variables in a system, then state value x is determined by calculating the average value of catastrophe progression x_a, x_b, x_c, x_d corresponding to each control variable calculated by normalization formulas.

3.2 2-tuple linguistic representation

2-tuple linguistic is a representation model proposed by Spain scholar Herrera in 2000 to solve decision-making problems with linguistic assessment information. The basic principle of this model is using 2-tuple linguistic to represent linguistic assessment information based on symbol translation, which can effectively avoid the problem of information loss and distortion during integrating and operating linguistic assessment information and can obtain more accurate results [17]. Its definition and operators are shown below.

Definition 3.1. Let s_i be a linguistic assessment scale in linguistic term set S = {s₀, s₁, ⋯ , s_n}, then call mapping F:

$\begin{matrix} F : S \to S \times [- 0.5, 0.5) \\ F (s_{i}) = (s_{i}, 0) \end{matrix}$ (1) the conversion function of linguistic assessment scale s_i corresponding to 2-tuple linguistic (s_i, 0) [26].

Definition 3.2. [26] Let θ ∈ [0, n] be the result of aggregation operation on linguistic assessment scale set S = {s₀, s₁, ⋯ , s_n}, then call mapping Δ:

$\begin{matrix} Δ : [0, n] \to S \times [- 0.5, 0.5) \\ Δ = {\begin{matrix} s_{i}, i = round (θ) \\ a_{i} = θ - i, a_{i} \in [- 0.5, 0.5) \end{matrix} \end{matrix}$ (2) the conversion function of real number θ corresponding to 2-tuple linguistic (s_i, a_i), where 0 ≤ i ≤ n and round () is the rounding operator.

Definition 3.3. [26] Let S = {s₀, s₁, ⋯ , s_n} be a linguistic assessment scale set and (s_i, a_i) be a 2-tuple linguistic, then call Δ^-1

$\begin{matrix} Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, n] \\ Δ^{- 1} (s_{i}, a_{i}) = i + a_{i} = θ \end{matrix}$ (3) the inverse function of 2-tuple linguistic (s_i, a_i) corresponding to numerical value θ ∈ [0, n].

According to the above three basic definitions, the distance and comparison rules of 2-tuple linguistic are defined as follows.

Definition 3.4. [26] Let (s_i, a_i) and (s_j, a_j) be any two 2-tuple linguistics in linguistic assessment scale set S = {s₀, ⋯ , s_n}, then the distance between (s_i, a_i) and (s_j, a_j) is: $(d, a) = Δ (| Δ^{- 1} (s_{i}, a_{i}) - Δ^{- 1} (s_{j}, a_{j}) |)$ (4) where, d ∈ S and a, a_i, a_j ∈ [-0.5, 0.5).

The ordering of two 2-tuple linguistic (s_i, a_i) and (s_j, a_j) can be determined according to the lexicographic order as follows: (1) If i < j, then (s_i, a_i) < (s_j, a_j). (2) If i = j and a_i = a_j, then (s_i, a_i) = (s_j, a_j); if a_i < a_j, then (s_i, a_i) < (s_j, a_j); if a_i > a_j, then (s_i, a_i) > (s_j, a_j).

4 Weight determination and evaluation procedure

Due to the differences of experts’ subjective experience, using subjective weights as weights could lead to the inconsistency between single expert suggestion and expert group suggestions.

To improve the reliability of expert weights, this paper, combining expert subjective weights and assessment information, based on the similarity of group average information and the deviation of the aggregation result of subjective weights, takes the compromise weights as decision weights. Details are given below.

4.1 Expert weights based on similarity

The determination of expert weights follows the similarity of group average information, that is, the expert will have a larger weight if his assessment information is close to the group average information [23]. Similarity is measured by grey correlation coefficient, and the method to determine expert weights is proposed based on the similarity.

Definition 4.1. [23] Let $x_{ij}^{k}$ be the linguistic assessment information of expert e_k on demand f_i about risk r_j, and $\bar{x_{ij}}$ be the group average information of expert group $\bar{e}$ on f_i about risk r_j, then call $S_{x} (x_{ij}^{k}, \bar{x_{ij}})$ $S_{x} (x_{ij}^{k}, \bar{x_{ij}}) = \frac{ξ (x_{ij}^{k}, \bar{x_{ij}})}{\sum_{k = 1}^{l} ξ (x_{ij}^{k}, \bar{x_{ij}})}$ (5) the linguistic assessment information similarity between expert e_k and expert group $\bar{e}$ on plan f_i about risk r_j, where, 1 ≤ k ≤ l, 1 ≤ i ≤ m, 1 ≤ j ≤ n. $ξ (x_{ij}^{k}, \bar{x_{ij}})$ is grey correlation coefficient, which is calculated by $\begin{matrix} ξ (x_{ij}^{k}, \bar{x_{ij}}) \\ = \frac{min_{i} min_{j} D (x_{ij}^{k}, \bar{x_{ij}}) + ρ max_{i} max_{j} D (x_{ij}^{k}, \bar{x_{ij}})}{D (x_{ij}^{k}, \bar{x_{ij}}) + ρ max_{i} max_{j} D (x_{ij}^{k}, \bar{x_{ij}})}, \end{matrix}$ ρ is discrimination coefficient, the value is typically 0.5. Obviously, the larger the value of $S_{x} (x_{ij}^{k}, \bar{x_{ij}})$ , the higher the consistency of expert e_k and group on f_i about risk r_j.

According to the definition of linguistic assessment information similarity between expert and expert group, the similarity between expert and expert group is defined as follows.

Definition 4.2. [23] Let $S_{x} (x_{ij}^{k}, \bar{x_{ij}})$ be the linguistic assessment information similarity between expert e_k and expert group $\bar{e}$ on demand f_i about risk r_j, then call $S_{e} (e_{k}, \bar{e})$ $S_{e} (e_{k}, \bar{e}) = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} S_{x} (x_{ij}^{k}, \bar{x_{ij}})}{\sum_{k = 1}^{l} \sum_{i = 1}^{m} \sum_{j = 1}^{n} S_{x} (x_{ij}^{k}, \bar{x_{ij}})}$ (6) the similarity between expert e_k and expert group $\bar{e}$ , where 1 ≤ k ≤ l, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Obviously, (6) satisfies: (1) $0 \leq S_{e} (e_{k}, \bar{e}) \leq 1$ , (2) $\sum_{k = 1}^{l} S_{e} (e_{k}, \bar{e}) = 1$ .

According to the definition of the similarity between expert and expert group, the similarity between expert e_k and group $\bar{e}$ can be the weight $w_{e_{k}}^{'}$ of expert e_k, that is $w_{e_{k}}^{'} = S_{e} (e_{k}, \bar{e})$ . Obviously, weight $w_{e_{k}}^{'}$ satisfies these two basic weight restrictions: (1) $0 \leq w_{e_{k}}^{'} \leq 1$ , 1 ≤ k ≤ l, (2) $\sum_{k = 1}^{l} w_{e_{k}}^{'} = 1$ .

4.2 Process of power supply risk group assessment

Security demand, quality demand and benefit demand of consumers equipped with standby electric source are represented by f₁, f₂, f₃. Let R = {r₁, r₂, r₃, r₄} be the set of power supply risk elements, representing system risk element, structural risk element, operational risk element, and environmental risk element respectively. The expert group involved in risk assessment is E = {e₁, e₂, … e_l}, and the decision maker, according to the differences of experts’ knowledge and experience, qualifies expert e_k with weight λ_k in linguistic form, k = 1, ⋯ , l. The impact of each risk element r_j, j = 1, 2, …, n in power supply risks on consumer demands f_i, i = 1, 2, … m is represented by linguistic assessment value $x_{ij}^{k}$ given by expert e_k. Thus, initial matrix of risk assessment decision is $D = (x_{ij}^{k})_{m \times n \times l}$ . The assessment process is as follows [23].

Step 1. Convert the linguistic assessment matrix into 2-tuple linguistic decision matrix.

On the basis of Definition 3.1, Convert the risk assessment decision matrix $D = (x_{ij}^{k})_{m \times n \times l}$ into 2-tuple linguistic decision matrix $D_{2 - T} = (x_{ij}^{k}, 0)_{m \times n \times l}$ .

Step 2. Calculate the weights of experts.

Step 2.1. On the basis of Definition 3.2 and Definition 3.3, calculate the expert group average information $(\bar{x_{ij}}, a_{ij})$ by $(\bar{x_{ij}}, a_{ij}) = Δ^{- 1} (\sum_{k = 1}^{l} Δ (x_{ij}^{k}, 0) / l)$ (7) where l is the number of experts, a_ij ∈ [-0.5, 0.5].

Step 2.2. On the basis of Formula (5) and Definition 3.3, calculate the similarity between expert e_k and expert group $\bar{e}$ on each demand about each risk indicator by

$\begin{matrix} S_{x} ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij})) \\ = \frac{ξ ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij}))}{\sum_{k = 1}^{l} ξ ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij}))} \end{matrix}$ (8) where, the distance formula in grey correlation coefficient $ξ ((x_{ij}^{k}, 0), ({\bar{x}}_{ij}, a_{ij}))$ is $D ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij})) = | Δ^{- 1} (x_{ij}^{k}, 0) - Δ^{- 1} (\bar{x_{ij}}, a_{ij}) |$ .

Step 2.3. On the basis of similarity and formula (6), calculate the weight $w_{e_{k}}^{'}$ based on similarity of expert e_k as $w_{e_{k}}^{'} = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} S_{x} ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij}))}{\sum_{k = 1}^{l} \sum_{i = 1}^{m} \sum_{j = 1}^{n} S_{x} ((x_{ij}^{k}, 0), (\bar{x_{ij}}, a_{ij}))}, 1 \leq k \leq l$ (9)

Step 2.4. According to the 2-tuple linguistic assessment information, experts’ weights based on deviation are calculated. On the basis of Definitions 3.2 and 3.3, calculate the aggregated assessment result of expert group (z_i, c_i) and the assessment result of each expert $(y_{i}^{k}, b_{i})$ by $(y_{i}^{k}, b_{i}) = Δ (\sum_{j = 1}^{n} Δ^{- 1} (x_{ij}^{k}, 0))$ (10) $(z_{i}, c_{i}) = Δ (\sum_{k = 1}^{l} w_{k} \sum_{j = 1}^{n} Δ^{- 1} (x_{ij}^{k}, 0))$ (11) where 1 ≤ k ≤ l, b_i, c_i ∈ [-0.5, 0.5), w_k is the real number converted from expert weight in linguistic form by $w_{k} = Δ^{- 1} (λ_{k}, 0) / \sum_{k = 1}^{l} Δ^{- 1} (λ_{k}, 0)$ .

Step 2.5. Calculate the deviation of expert and expert group H (e_k, e) as $H (e_{k}, e) = \sum_{i = 1}^{m} {(Δ^{- 1} (y_{i}^{k}, b_{i}) - Δ^{- 1} (z_{i}, c_{i}))}^{2}$ (12)

Furthermore, calculate expert weight $w_{e_{k}}^{″}$ by $w_{e_{k}}^{″} = \frac{H (e_{k}, e)}{\sum_{k = 1}^{l} H (e_{k}, e)}, 1 \leq k \leq l$ (13)

Step 2.6. Determine the final weight of expert. On the basis of formula (14), the compromise result between the weight based on similarity and the weight based on deviation is determined to be the final weight w_{ɛ
_k}. $w_{e_{k}} = α w_{e_{k}}^{'} + (1 - α) w_{e_{k}}^{″}$ (14)

α, 1 - α respectively are the eclectic preference coefficients for these two methods with a constraint 0 ≤ α ≤ 1. When α > 0.5, it means the decision maker prefers the mean opinion of the group. When α < 0.5, the individual opinions are more preferred. And α = 0.5 means the same preference to these two.

Step 3. Aggregate expert assessment information.

On the basis of expert weight and 2-tuple linguistic assessment decision matrix in Step 1, aggregate assessment information of each expert on each demand about each risk indicator by $(X_{ij}, A_{ij}) = Δ (\sum_{k = 1}^{l} w_{e_{k}} \times Δ^{- 1} (x_{ij}^{k}, 0))$ (15)

Step 4. Calculate total catastrophe value of each demand.

Step 4.1. Determine the catastrophe type and corresponding normalization formula of primary risk indicator.

Regard the number of secondary indicators in each primary risk indicator as the dimension of control variables to determine the catastrophe type and corresponding normalization formula of primary risk indicator.

For example, when a primary indicator has 4 secondary indicators, butterfly catastrophe model is applicable to the primary indicator, and the corresponding normalization formula is ${x_{a} = \sqrt{a}, x_{b} = \sqrt[3]{b}, x_{c} = \sqrt[4]{c}, x_{d} = \sqrt[5]{d}}$ .

Instead of precise number, 2-tuple linguistic assessment information converted from qualitative language is used in the paper, thus we give the normalization formula of 2-tuple linguistic assessment information $(X_{ij}^{G}, A_{ij}^{G}) = Δ {(Δ^{- 1} (X_{ij}, A_{ij}))}^{1 / n}$ (16) where, $(X_{ij}^{G}, A_{ij}^{G})$ is the 2-tuple linguistic catastrophe value of (X_ij, A_ij) calculated by normalization formula.

Step 4.2. Calculate the total 2-tuple linguistic catastrophe value of each demand by recursive algorithm. When calculating the catastrophe progressions and total catastrophe value of each alternative by recursive algorithm according to normalization formula, we follow “non-complementary” and “complementary” principles. If the control variables in a system can substitute each other or make up for deficiency mutually, then use the “complementary” principle to calculate total catastrophe value in the “average” way according to 2-tuple linguistic information after the normalization of control variables and related definitions of 2-tuple linguistic. If the control variables are mutually independent, then use the “non-complementary” principle. The total catastrophe value is obtained according to the comparison rules of 2-tuple linguistic and “minimum” principle.

Step 5. Sort power supply risk elements.

According to the comparison rules of 2-tuple linguistic and total 2-tuple linguistic catastrophe value of each power supply risk element, following the principle “the bigger the total 2-tuple linguistic catastrophe value, the greater the risk”, sort power supply risk elements and put forward suggestions.

5 Case analysis

This paper takes a project of State Grid as a case. The research object is an ICSES of the province, whose standby electric source is distributed solar power. Three experts of a provincial power supply company are asked to finish a questionnaire about power supply risks based on consumer demands. Experts’ qualitative assessment set in linguistic form is s = {s₀ = VeryLow, s₁ = Low, s₃ = Medium Low, s₄ = MediumHigh, s₅ = High, s₆ = Very High}. The decision maker, according to the differences of experts’ knowledge and experience, subjectively qualifies each expert with weight W_S = {w₁ = VH (s₆) , w₂ = H (s₅) , w₃ = MH (s₄)} in linguistic form. The specific assessment data are shown in Table 3. For example, s₄ in Row 3 and Column 3 means that expert e₁ thinks that system operation mode change risk has medium high influence on security demand.

Table 3
Expert assessment results

System risk Structural risk Operational risk Environmental

element r₁ element r₂ element r₃ risk element r₄

r ₁₁ r ₁₂ r ₂₁ r ₂₂ r ₃₁ r ₃₂ r ₃₃ r ₄₁ r ₄₂

e ₁ f ₁ s ₄ s ₃ s ₄ s ₂ s ₅ s ₂ s ₂ s ₃ s ₂

f ₂ s ₆ s ₃ s ₃ s ₂ s ₀ s ₅ s ₄ s ₄ s ₂

f ₃ s ₃ s ₅ s ₄ s ₁ s ₃ s ₄ s ₄ s ₃ s ₁

e ₂ f ₁ s ₆ s ₂ s ₃ s ₁ s ₂ s ₃ s ₅ s ₂ s ₂

f ₂ s ₄ s ₄ s ₁ s ₁ s ₄ s ₅ s ₆ s ₁ s ₅

f ₃ s ₅ s ₃ s ₀ s ₃ s ₂ s ₃ s ₂ s ₄ s ₄

e ₃ f ₁ s ₃ s ₄ s ₃ s ₁ s ₄ s ₃ s ₂ s ₄ s ₄

f ₂ s ₂ s ₅ s ₁ s ₅ s ₄ s ₅ s ₃ s ₂ s ₃

f ₃ s ₃ s ₁ s ₀ s ₂ s ₃ s ₂ s ₄ s ₁ s ₃

		System risk	Structural risk	Operational risk	Environmental
e ₁	f ₁	s ₄	s ₃	s ₄	s ₂	s ₅	s ₂	s ₂	s ₃	s ₂
	f ₂	s ₆	s ₃	s ₃	s ₂	s ₀	s ₅	s ₄	s ₄	s ₂
	f ₃	s ₃	s ₅	s ₄	s ₁	s ₃	s ₄	s ₄	s ₃	s ₁
e ₂	f ₁	s ₆	s ₂	s ₃	s ₁	s ₂	s ₃	s ₅	s ₂	s ₂
	f ₂	s ₄	s ₄	s ₁	s ₁	s ₄	s ₅	s ₆	s ₁	s ₅
	f ₃	s ₅	s ₃	s ₀	s ₃	s ₂	s ₃	s ₂	s ₄	s ₄
e ₃	f ₁	s ₃	s ₄	s ₃	s ₁	s ₄	s ₃	s ₂	s ₄	s ₄
	f ₂	s ₂	s ₅	s ₁	s ₅	s ₄	s ₅	s ₃	s ₂	s ₃
	f ₃	s ₃	s ₁	s ₀	s ₂	s ₃	s ₂	s ₄	s ₁	s ₃

Step 1. According to Table 3, convert linguistic assessment matrix into 2-tuple linguistic decision matrix.

Step 2. Use formulas (7–9) to calculate the weights based on similarity and use formulas (10–13) to calculate the weights based on deviation. The specific content is shown in Table 4, where the subjective weights given by the decision maker W_S = {s₆, s₅, s₄} are converted into real number weights W_S = {0 . 4, 0 . 333, 0.267} by the inverse function of 2-tuple linguistic. On that basis, to study the effects of different compromise preference coefficients α on experts’ final weights w_{e
_k}, this paper sets α = {0, 0.25, 0.5, 0.75, 1} and calculates the final weights respectively, as shown in Table 5.

Table 4

Experts’ weights based on similarity and deviation

	e ₁	e ₂	e ₃
Similarity	17.01	15.76	17.63
$w_{e_{k}}^{'}$	0.338	0.313	0.349
Deviation	33.20	36.64	39.64
$w_{e_{k}}^{″}$	0.303	0.335	0.362

Table 5

Experts’ final weights

	α	e ₁	e ₂	e ₃
Final Weights w_{e _k}	0	0.303	0.335	0.362
	0.25	0.312	0.329	0.359
	0.5	0.320	0.324	0.356
	0.75	0.329	0.318	0.353
	1	0.338	0.313	0.349

It can be seen from Tables 4 and 5 that expert e₃ has the highest similarity of group average assessment information. Thus, when α = 1, the weight of expert e₃ must be the largest, indeed w_{e
₃} = 0.349. The deviation of the aggregation result of subjective weights between expert e₃ and expert group is the highest. Thus, when α = 0, the weight of expert e₃ must be the largest, indeed w_{e
₃} = 0.362. In order to simultaneously consider the different weights obtained by the two principles, this paper sets α = 0.5 for subsequent operations, that is, the final weights are 0.320, 0.324, 0.356.

Step 3. On the basis of final weights, according to Formula (15), obtain the final assessment information of expert group by aggregation, as shown in Table 6.

Table 6

The assessment information of expert group by aggregation

	System risk		Structural risk		Operational risk			Environmental risk
	element r₁		element r₂		element r₃			element r₄
	r ₁₁	r ₁₂	r ₂₁	r ₂₂	r ₃₁	r ₃₂	r ₃₃	r ₄₁	r ₄₂
f ₁	s₄0.292	s₃0.032	s₃0.32	s₁0.32	s₄–0.328	s₃–0.32	s₃–0.028	s₃0.032	s₃–0.288
f ₂	s₄–0.072	s₄0.036	s₂–0.36	s₃–0.256	s₃–0.28	s₅0	s₄0.292	s₂0.316	s₃0.328
f ₃	s₄–0.352	s₃–0.072	s₁0.28	s₂0.004	s₃–0.334	s₃–0.036	s₃0.352	s₃–0.388	s₃–0.316

Step 4. The relationship among supply risk elements is “complementary”. According to catastrophe theory and “complementary” principle, normalized catastrophe values of secondary indicators are obtained in Table 7 and normalized catastrophe values of primary indicators are obtained in Table 8.

Table 7

Normalized catastrophe values of secondary indicators

	System risk		Structural risk		Operational risk			Environmental risk
	element r₁		element r₂		element r₃			element r₄
	r ₁₁	r ₁₂	r ₂₁	r ₂₂	r ₃₁	r ₃₂	r ₃₃	r ₄₁	r ₄₂
	$x_{a} = \sqrt{a}$	$x_{b} = \sqrt[3]{b}$	$x_{a} = \sqrt{a}$	$x_{b} = \sqrt[3]{b}$	$x_{a} = \sqrt{a}$	$x_{b} = \sqrt[3]{b}$	$x_{c} = \sqrt[4]{c}$	$x_{a} = \sqrt{a}$	$x_{b} = \sqrt[3]{b}$
f ₁	s₂0.072	s₁0.447	s₂–0.178	s₁0.097	s₂–0.084	s₁0.389	s₁0.313	s₂–0.259	s₁0.395
f ₂	s₂–0.018	s₂–0.408	s₁0.281	s₁0.4	s₂–0.351	s₂–0.29	s₁0.439	s₂–0.478	s₁0.493
f ₃	s₂–0.09	s₁0.431	s₁0.131	s₁0.261	s₂–0.364	s₁0.436	s₁0.353	s₂–0.384	s₁0.39

Table 8

Normalized catastrophe values of primary indicators

		System risk	Structural risk	Operational risk	Environmental risk
		element r₁	element r₂	element r₃	element r₄
f ₁	$x_{a} = \sqrt{a}$	s₂–0.251	s₁0.459	s₂–0.461	s₂–0.433
f ₂	$x_{b} = \sqrt[3]{b}$	s₂–0.213	s₁0.340	s₂–0.401	s₂–0.493
f ₃	$x_{c} = \sqrt[4]{c}$	s₂–0.33	s₁0.196	s₁0.475	s₂–0.497

Step 5. Consumer demand analysis shows the three types of consumer demands satisfy f₁ ≻ f₂ ≻ f₃, and the relationship among the three is “non-complementary”. Select swallowtail catastrophe model to calculate the total catastrophe value of each risk element. r₁ = (s₁, 0.137), r₂ = (s₁, 0.046), r₃ = (s₁, 0.102), r₄ = (s₁, 0.105). Ranking result is r₁ ≻ r₄ ≻ r₃ ≻ r₂.

On this basis, in order to study the effects of compromise preference coefficients α on ranking result of risks in power supply company and verify the reliability of the method used herein, this paper sets α = {0, 0.25, 0.5, 0.75, 1} and conducts analysis respectively, as shown in Table 9.

Table 9

The risk ranking of power supply company under the comprise preference coefficient α = {0, 0.25, 0.75, 1}

α	r ₁	r ₂	r ₃	r ₄	Risk ranking
α = 0	(s₁, 0.137)	(s₁, 0.043)	(s₁, 0.101)	(s₁, 0.108)	r₁ > r₄ > r₃ > r₂
α = 0.25	(s₁, 0.137)	(s₁, 0.045)	(s₁, 0.102)	(s₁, 0.106)	r₁ > r₄ > r₃ > r₂
α = 0.75	(s₁, 0.137)	(s₁, 0.047)	(s₁, 0.102)	(s₁, 0.103)	r₁ > r₄ > r₃ > r₂
α = 1	(s₁, 0.137)	(s₁, 0.049)	(s₁, 0.103)	(s₁, 0.101)	r₁ > r₃ > r₄ > r₂

As can be seen from Table 9: (1) The ranking result of risk elements will change with the change of compromise preference coefficient, indicating the value of supply risk is influenced by compromise preference coefficient. If decision maker tends to adopt expert group average assessment to make decisions, namely risk averter, then the bigger α, the better. If decision maker tends to use individual expert assessment with distinctive features to make decisions, namely risk lover, then the smaller α, the better. Therefore, the compromise preference coefficient can be flexibly selected according to the actual situation of decision makers. (2) In various ranking results, the greatest supply risk element has never changed. System risk element is the main risk for consumer demand. Once the system operation mode changes or system equipment breaks down, all consumers are potentially under risk. Thus, power supply companies should focus on system operation control, develop a new risk management method for ICSES, and do their best to protect the interests of important consumers. (3) The risk values of operational risk and environmental risk are almost equal and merely inferior to system risk, which means both risks are the potential threats to ICSES. Therefore, power supply company should not ignore the two risks. For operational risk, advanced technologies such as online monitoring and intelligent scheduling should be introduced to make the system device state normal. For environmental risk, on the one hand, power supply company needs to enhance the self-healing ability of grid to cope with natural environment risk, on the other hand, power supply company ought to improve risk management system to avoid the loss brought by human errors.

6 Conclusions

On the height of the future smart grid risk management, assuming that ICSES can take on power distribution work, this paper analyzes the new demands of ICSES. In addition, we discuss the power supply risk factors that may affect consumer demands in reality and establish the mapping relations between the two.

To increase the validity and reliability of expert group assessment result, combining expert weights in linguistic form, we propose a compromised weighting method based on similarity and deviation. To get rid of restrictions on determining reasonable weights, we combine catastrophe theory and 2-tuple linguistic representation to conduct researches, and verify the validity of the model by case analysis. The study expands the application of catastrophe theory from precise real number to uncertain fuzzy linguistic variable, which provides a new idea and method for the application of catastrophe theory in other fields.

The result of case analysis shows that system risk element is the most minatory power supply risk to ICSES. Once the system operation mode changes or system equipment breaks down, the resulting risk has wide range of effects and serious consequences, thus grid company should strengthen risk management measures on system operation. In addition, risk management on primary, secondary and other equipment is also particularly important, for the cascading failure risk of equipment is a significant contribution to the loss of consumer interests. Furthermore, there are a lot of risks in the management system of grid company, for example, power equipment inspection and maintenance system, emergency risk management system and others, whose threat is higher than expected. Grid company should design more humanized management system from the perspective of consumer demand to ensure the maximization of consumer interests and achieve the overall goal of quality service.

The appetite of different grid companies for risk is different, we can flexibly set compromise preference coefficient α for different risk preferences.

Linguistic variables that describe risk by expert can be updated with intuitionistic fuzzy sets, hesitant fuzzy sets or trapezoidal fuzzy sets [27, 28]. The new linguistic variables could express information with more uncertainty and fuzziness, which are more suitable for complex power system risk analysis. Therefore, the future study will apply fuzzy linguistic variables to complex electric problems.

Footnotes

Acknowledgments

This study is funded by the National Natural Science Foundation of China (Grant Number: 71271084, 71671065) and the technology project of state grid (5204BB1600CN).

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Supply risk R	System risk element r₁	System operation method change risk r₁₁
		System associated equipment failure risk r₁₂
	Structural risk element r₂	Power supply layout risk r₂₁
		Transmission technology risk r₂₂
	Operational risk element r₃	Equipment operating risk r₃₁
		Load rate risk r₃₂
		Duty cycle risk r₃₃
	Environmental risk element r₄	Natural environmental risk r₄₁
		Supply management system risk r₄₂