A Novel Evaluation Model for Hybrid Power System Based on Vague Set and Dempster-Shafer Evidence Theory

Because clean energy and traditional energy have different advantages and disadvantages, it is of great significance to evaluate comprehensive benefits for hybrid power systems. Based on thorough analysis of important characters on hybrid power systems, an index system including security, economic benefit, environmental benefit, and social benefit is established in this paper. Due to advantages of processing abundant uncertain and fuzzy information, vague set is used to determine the decision matrix. Convert vague decision matrix to real one by vague combination ruleand determine uncertain degrees of different indexes by grey incidence analysis, then the mass functions of different comment set in different indexes are obtained. Information can be fused in accordance with Dempster-Shafer D-S combination rule and the evaluation result is got by vague set and D-S evidence theory. A simulation of hybrid power system including thermal power, wind power, and photovoltaic power in China is provided to demonstrate the effectiveness and potential of the proposed design scheme. It can be clearly seen that the uncertainties in decision making can be dramatically decreased comparedwith existingmethods in the literature. The actual implementation results illustrate that the proposed index system and evaluation model based on vague set and D-S evidence theory are effective and practical to evaluate comprehensive benefit of hybrid power system.


Introduction
At present, climate warming and environmental pollution are becoming increasingly prominent, which poses a serious threat to sustainable development of human society.Developing clean energies, such as wind power and photovoltaic power, is widely accepted common choice of the world 1-5 .Clean energy and traditional energy have their own Due to the public service property of electric power, it is necessary to consider social benefit; meanwhile, because of instable clean energy generation, it is needed to consider the security of the power system.This paper establishes comprehensive benefit evaluation index system of hybrid power system from the four aspects of economic benefit, environmental benefit, social benefit, and security.

Economic Benefit
Economic benefit is the most important aspect of comprehensive benefit evaluation.It is mainly composed of the four aspects of profitability capability, solvency capability, operational capability, and development capacity.
Profitability refers to the ability to profit in a certain period.Profitability capability is determined by comprehensive consideration of gross profit margin, net profit margin, ratio of profits to cost, and return on total assets.Solvency capability refers to the ability to repay due debts which be assessed in terms of current ratio, quick ratio, asset-liability ratio, and interest coverage ratio.Operational capacity refers to enterprise's ability to make profits using existing assets.This paper mainly assesses the operational capability through current asset turnover, fixed asset turnover, and total asset turnover.Development capacity refers to the potential ability to expand scale and grow strength in the coming years which can be accessed through profit growth, sale growth, and total asset growth.

Environmental Benefit
Environmental benefit refers to the influence on living environment, which can be divided into positive benefit, negative benefit, direct benefit, and indirect benefit.Fundamentally speaking, environmental benefit is the basis of economic benefit and social benefit.Environmental benefit is a qualitative index.Comprehensively considering circumstances of hybrid power system, this paper evaluates environmental benefit from the two aspects of saving natural resources and reducing scraps.

Social Benefit
Social benefit refers to the good consequences and implications for community and society, also known as external and indirect economic benefit, mainly in forms of public reflection and social assessment system.Social benefit is a qualitative indicator.Comprehensively considering the specific circumstances of hybrid power system, this paper establishes social benefit evaluation index system from the three aspects of pulling gross domestic product, improving people's living standard and promoting social employment.

Security
Security occupies a basic position in comprehensive benefit evaluation of power grid.Combined with exiting researches and characters of hybrid power system, this paper selects four indexes to evaluate security, including integrated voltage qualification rate, grid frequency qualification rate, average loading rate of provincial grid, and power supply reliability rate.Above all, the comprehensive benefit evaluation index system of hybrid power system is shown in Figure 1.

Decision Matrix Based on Vague Set
Vague set proposed by Gau and Buehrer 31 is a generalized form of fuzzy set.Vague set has been successfully applied in the field of fuzzy control, decision analysis and expert systems, and has achieved better results than the traditional fuzzy set theory 32-34 .
Set U the universe of discourse, with a generic element u.A vague set A is characterized by a truth-membership function t A and a false-membership function f A , where t A u is a lower bound on the grade of membership of u, derived from the evidence rising for u; f A u is a lower bound on the negation of u, derived from the evidence against u; 0 ≤ t A u f A u ≤ 1.The grade of membership of u in vague set A is bound to a subinterval t A u , 1 − f A u of 0, 1 .The vague value t A u , 1 − f A u indicates that the exact grade of membership μ A u of u maybe unknown, but it is bound by When the universe of discourse U is discrete, a vague set A can be written as When the universe of discourse U is continuous, a vague set A can be written as The steps of determining the decision matrix is as follows. 1

Construct Evaluation Matrix
Assuming the comment level of index u ij , i 1, 2, . . ., m; j 1, 2, . . ., n is v k k 1, 2, . . ., p , then the relationship matrix of factor set U mapping to comment set V is R i : where r ijk means the comment of factor u ij mapping to comment set, and r ijk can be expressed as For qualitative index, the value can be got from experts' evaluation.For quantitative index, its quantitative value is the index value.Normalizing index value, we can get t Rijk and 1 − f Rijk .
The main calculations are as follows.

Mathematical Problems in Engineering
Finite sum calculation: where B i is a vague subset and b ik k 1, 2, . . ., p is the evaluation value of k to B i .

D-S Evidence Theory
The evidence theory, first proposed by Harvard University mathematician A. P. Dempster in the 1960s, aims to use upper and lower probability to solve the multivalued mapping issues.G. Shafer, Dempster's student, has further developed the evidence theory by introducing the concept of belief function to form a mathematical system to deal with uncertainty reasoning based on "evidence" and "portfolio." , A / φ. 3.9

Design of Comprehensive Evaluation Model
Suppose there is a multiattribute index evaluation problem with comprehensive comment set V {V 1 , V 2 , . . ., V m } and evaluation index set Different mass functions mean different evaluation results.So it is vital to determine the uncertainty degree of evidences.Theoretically, one index which matches average information better than other indexes is more beneficial for decision making, and the information uncertainty degree of the index is lower, vice versa.

3.10
Define the q-order uncertainty degree of index I j as

3.11
where r ij is called gray mean correlation degree with ξ 0.5 in general.
In case of negative number, the normalization method is shown as Mathematical Problems in Engineering Suppose x x 1 , x 2 , . . ., x t is a finite difference information sequence with a length of t t / 0 .An index set exists in the form of sequence x j / 0, j ∈ J, J {1, 2, . . ., t}.The mapping function f is defined as information structure operator of finite sequence x:

3.13
Here y y 1 , y 2 , . . ., y t is called mapping sequence of information structure.The normalized score function matrix can be transformed into mapping sequence of information structure Y y ij m×n .Then the mass function m j i can be constructed: where m j i is a mass function of V m to index I j , m i 1 m j i < 1.So the mass function of overall uncertainty with respect to the index I j is shown as

3.15
According to the proposed combination rule, the final evaluation results can be calculated after the combination of mass functions of comment set with respect to all indexes.
In summary, steps for comprehensive evaluation based on vague set and D-S evidence theory are as follows.
Step 1. Construct the decision-making matrix R based on the vague set theory.
Step 2. Convert R into real matrix G g ij m×n according to vague combination rule.
Step 3. Calculate the uncertainty degree DOI I j j 1, 2, . . ., n combined with real matrix G and 3.6 .
Step 4. Establish the mapping sequence of information structure Y y ij m×n .
Step 5. Build up mass function m j i and mass function of overall uncertainty m j i 1 on the basis of Y y ij m×n and DOI I j .
Step 6. Fuse evidence information using the D-S combination rule.
Step 7. Make evaluation and decision in principle of the maximizing belief function.

Simulation Analysis
Guangdong power system in China, which includes thermal power, wind power, and solar photovoltaic power, is selected to simulation analysis.It is known from Section 2 that the comprehensive benefit evaluation index system of hybrid power system includes the four aspects of economic benefit, environmental benefit, social benefit, and security, which is composed of 13 indexes shown in Table 1.According to data statistics and calculation, the original index value can be got as follows.
The comprehensive comment set is set to be V {v 1 , v 2 , v 3 , v 4 , v 5 } {best, better, good, worse, worst}.For quantitative indexes such as voltage qualification rate, power grid frequency rate, average load rate, and power supply reliability rate, their values can be directly calculated by basic statistical data.For qualitative indicators such as debt paying ability, operation ability, saving natural resources, reducing environmental pollution, their values can be decided by comprehensive experts' opinions with specific situations.

Conclusion
The proportion of clean energy is increasing in recent years.Different power generation models have different advantages and disadvantages.It is of great significance for hybrid power system to evaluate the comprehensive benefits.Analyzing characteristics of hybrid power system, an index system of comprehensive benefit evaluation including economic benefit, environmental benefit, social benefit, and security is established in this paper.Due to advantages of processing abundant uncertain and fuzzy information, vague set is used to determine the decision matrix.Convert vague decision matrix to real one by vague combination rule and determine uncertain degrees of different indexes by grey incidence analysis, then the mass functions of different comment set in different indexes are obtained.Information can be fused in accordance with the D-S combination rule and the evaluation result is got.A simulation of hybrid power system including thermal power, wind power, and photovoltaic power in China is simulated.In order to validate the proposed evaluation model, three other commonly used algorithms including Fuzzy set theory, Vague set theory, and D-S evidence theory are calculated for comparison.The results illustrate that a satisfying conclusion can be obtained and an obvious decrease can be observed in the uncertainty of decision making compared to other commonly used evaluation methods.The actual implementation results prove that the proposed evaluation algorithm based on vague set and D-S evidence theory is effective and practical to evaluate comprehensive benefit of hybrid power system.

Theorem 3 . 6 .
the function m is called the basic probability assignment on Θ; for all A ⊂ Θ, m A is called the basic belief degree.Definition 3.2.Suppose Θ is the frame of discernment, function m : 2 Θ → 0, 1 is the basic probability assignment on Θ, then the belief function is defined as Bel : 2 Θ → 0, 1 , where Bel A B⊂A m B , for all A ⊂ Θ. Definition 3.3.If m A > 0, then A is called the focal element of the belief function Bel.And all focal elements are called its core.Definition 3.4.If function Q : 2 Θ → 0, 1 is defined by Q A A⊂B m B , for all A ⊂ Θ, then Q is called the total belief function of Bel.For all A ⊂ Θ, Q A is called the total belief number of A. Definition 3.5.Suppose Bel : 2 Θ → 0, 1 is a belief function on Θ. Functions Dou : 2 Θ → 0, 1 and pl : 2 Θ → 0, 1 are defined as for all A ⊂ Θ Dou A Bel A and pl A 1 − Bel A , then Dou is called suspicion function of Bel, and pl is called plausibility function.For all A ⊂ Θ, Dou A is called the suspicion degree of A, and pl A is called the plausibility degree.Suppose m 1 and m 2 are two basic probability assignment functions formed based on information obtained from two different information sources, Bel 1 and Bel 2 , in the same frame of discernment Θ.A 1 , A 2 , . . ., A k and B 1 , B 2 , . . ., B l are focal elements of Bel 1 and Bel 2 .If Suppose m 1 , m 2 , . . ., m n are the corresponding basic probability assignment functions formed based on information obtained from different information sources Bel 1 , Bel 2 , . . ., Bel n in the same frame of discernment Θ.If Bel 1 ⊕Bel 2 ⊕• • •⊕Bel n exists and m is its basic probability assignment function, then

Table 1 :
Comprehensive benefit evaluation index system of hybrid power system.