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

Authors: Dongxiao Niu; Yanan Wei; Yan Shi; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

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 compared with existing methods 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.


---

## Body

## 1. 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 different characteristics. As to thermal power, it has advantages of low cost and stable power, but generally causes serious pollution, especially carbon dioxide, whereas clean energy power has the disadvantages of high cost and instability, but is highly environmental friendly [6–9]. A power system usually includes thermal power, wind power, and photovoltaic power. Because of different techniques and economic influences, it is of practical significance to choose scientific, comprehensive, and reasonable indexes, and methods to evaluate comprehensive benefit of hybrid power system.Numerous factors should be considered to evaluate the comprehensive benefit of hybrid power system, such as technique security, influence to environment, and economy [10]. Literatures [11, 12] have analyzed economic aspect of power system; security and technique aspect are considered in [13, 14]; the work in [15, 16] evaluated environmental influence of different energies. However, only a few attempts have been made on social benefit of power system, let alone comprehensive benefit including social and environmental aspects of hybrid power system.The input indexes are critical for comprehensive evaluation and are often complex, which involves production, scheduling, dispatch, marketing, planning, and finance as well as many other aspects [17]. Because of ambiguity of characters of evaluation indexes, it is suitable for information fusion technique to evaluate comprehensive benefit of hybrid power system which allows representation of both imprecision and uncertainty [18, 19]. Information fusion technique includes several kinds of methods, such as Kalman filtering, Bayesian estimation, and D-S evidence theory. Because Kalman filtering is depending on the initial state estimation, and by measuring the noise influence such as large faults, estimate process covariance easily appears morbid, leading to filter positioning results are not stable. Bayesian is a kind of commonly used method for static environment fusion in sensor lower information, suitable for normal distribution measuring result or additive Gaussian noise system. D-S evidence theory tackles the prior probability issue by keeping track of an explicit probabilistic measure of a possible lack of information. It is suitable take into account the disparity between knowledge types [20] because it is able to provide a federative framework [21] and it combines cumulative evidence for changing prior opinions in the light of new evidence [22]. Literature [23] proposed the combined fuzzy logic/D-S evidence theory method. Although Jones et al. [24] noted that it is necessary to ensure the validity of basic probability assignment, they did not give a method to do so. Vague set is proposed to solve the problem in this paper.Due to advantages of vague set processing abundant uncertain and fuzzy information, it is chosen to determine basic decision matrix of evaluation model [25–30]. Combining vague set with D-S evidence theory, a novel evaluation algorithm is established and applied into the comprehensive benefit evaluation of hybrid power system.The paper is organized as follows. Section2 proposes a systematic evaluation index system according to characters of hybrid power system. A comprehensive evaluation model based on vague set and D-S evidence theory is given in Section 3. A simulation of hybrid power system including thermal power, wind power, and photovoltaic power in China is proposed, and comparison between the proposed evaluation model with some existing works is analyzed in Section 4. The paper is conducted in Section 5.


## 2. Evaluation Index System

Comprehensive benefit evaluation of hybrid power generation should not only include economic benefit, but also its impact on the environment, namely, environmental benefit. 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.


### 2.1. 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.


### 2.2. 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.


### 2.3. 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.


### 2.4. 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 Figure1.Figure 1
Comprehensive evaluation index system.


## 2.1. 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.


## 2.2. 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.


## 2.3. 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.


## 2.4. 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 Figure1.Figure 1
Comprehensive evaluation index system.


## 3. Model Establishment

### 3.1. 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].SetU the universe of discourse, with a generic element u. A vague set A is characterized by a truth-membership function tA and a false-membership function fA, where tA(u) is a lower bound on the grade of membership of u, derived from the evidence rising for u; fA(u) is a lower bound on the negation of u, derived from the evidence against u; 0≤tA(u)+fA(u)≤1. The grade of membership of u in vague set A is bound to a subinterval [tA(u),1-fA(u)] of [0,1]. The vague value [tA(u),1-fA(u)] indicates that the exact grade of membership μA(u) of u maybe unknown, but it is bound by tA(u)≤μA(u)≤1-fA(u), where 0≤tA(u)+fA(u)≤1.When the universe of discourseU is discrete, a vague set A can be written as
(3.1)A=∑i=1n[tA(ui),1-fA(ui)]ui,ui∈U.When the universe of discourseU is continuous, a vague set A can be written as
(3.2)A=∫U[tA(ui),1-fA(ui)]uidui,ui∈U.The steps of determining the decision matrix is as follows.(1)
Determination Evaluation Factors and Comment SetThe evaluation factors are expressed asu1,u2,…,um, then the evaluation factor set is U={u1,u2,…,um}.set evaluation indexes of each evaluation factor beuij,(i=1,2,…,m;j=1,2…,n). We can get index set ui={ui1,ui2,…,uij,…,uim}, i=1,2,…,m;j=1,2,…,n.Define the comment set asV={v1,v2,…,vp}.(2)
Construct Evaluation MatrixAssuming the comment level of indexuij, (i=1,2,…,m;j=1,2,…,n) is vk(k=1,2,…,p), then the relationship matrix of factor set U mapping to comment set V is Ri:
(3.3)Ri=[ri11,ri12,…,ri1pri21,ri22,…,ri2p⋮⋮⋮rin1,rin2,…,rinp],
where rijk means the comment of factor uij mapping to comment set, and rijk can be expressed as
(3.4)rijk=[tR~ijk,1-fR~ijk].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 gettR~ijk and 1-fR~ijk.The main calculations are as follows.Multiply calculation:(3.5)k·A~=[k·tA~,k·(1-fA~)]k,k∈[0,1].Finite sum calculation:(3.6)A~⊕B~=[tA~⊕tB~,(1-fA~)⊕(1-fB~)]=[min{1,tA~+tB~},min{1,(1-fA~)+(1-fB~)}],(3.7)bik=[min{1,∑j=1naijtRijk~},min{1,∑j=1naij(1-fRijk~)}],
where Bi~ is a vague subset and bik(k=1,2,…,p) is the evaluation value of k to Bi~.


### 3.2. 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.”Definition 3.1.
SupposeΘ is the frame of discernment. If set function m:2Θ→[0,1] (2Θ is the power set of Θ) is to meet m(Φ)=0, ∑A⊂Θm(A)=1, 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.
SupposeΘ is the frame of discernment, function m:2Θ→[0,1] is the basic probability assignment on Θ, then the belief function is defined as Bel:2Θ→[0,1], where Bel(A)=∑B⊂Am(B), forallA⊂Θ.Definition 3.3.
Ifm(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 functionQ:2Θ→[0,1] is defined by Q(A)=∑A⊂Bm(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.
SupposeBel:2Θ→[0,1] is a belief function on Θ. Functions Dou:2Θ→[0,1] and pl:2Θ→[0,1] are defined as for all A⊂ΘDou(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.Theorem 3.6.
Supposem1 and m2 are two basic probability assignment functions formed based on information obtained from two different information sources, Bel1 and Bel2, in the same frame of discernment Θ. A1,A2,…,Ak and B1,B2,…,Bl are focal elements of Bel1 and Bel2. If ∑Ai∩Bj=ϕm1(Ai)m2(Bj)<1, the combination of m1 and m2 is as follows:
(3.8)m(A)={0,A=ϕ,∑Ai∩Bj=Am1(Ai)m2(Bj)1-∑Ai∩Bj=ϕm1(Ai)m2(Bj),A≠ϕ.Theorem 3.7.
Supposem1,m2,…,mn are the corresponding basic probability assignment functions formed based on information obtained from different information sources Bel1,Bel2,…,Beln in the same frame of discernment Θ. If Bel1⊕Bel2⊕⋯⊕Beln exists and m is its basic probability assignment function, then
(3.9)∀A⊆Θ,A≠ϕ,m(A)=Kn∑A1∩A2∩⋯∩ANm1(A1)m2(A2)⋯mn(An),Kn=1Nn,Nn=∑A1∩A2∩⋯∩AN≠ϕm1(A1)m2(A2)⋯mn(An).


### 3.3. Design of Comprehensive Evaluation Model

Suppose there is a multiattribute (index) evaluation problem with comprehensive comment setV={V1,V2,…,Vm} and evaluation index set I={I1,I1,…,In}. According to vague set, the decision matrix R=(Rij)m×n, Rij=[tij,1-fij] is got. By synthesis law of vague set, we can convert the vague set R into a real set G=(gij)m×n, gij=(tij+(1-fij))/2.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.Suppose the matrixG is
(3.10)G=(gij)m×n,gi¯=1n∑j=1ngij,i=1,2,…,m.Define theq-order uncertainty degree of index Ij as
(3.11)DOI(Ij)=1m[∑i=1m(rij)q]1/q,rij=mini|gij-g¯|+ξmini|gij-g¯||gij-g¯|+ξmini|gij-g¯|,i=1,2,…,m;j=1,2,…,n,
where rij is called gray mean correlation degree with ξ=0.5 in general.In case of negative number, the normalization method is shown as(3.12)gij¯=gij-mini(gij)mini(gij)-mini(gij).Supposex=(x1,x2,…,xt) is a finite difference information sequence with a length of t(t≠0). An index set exists in the form of sequence xj≠0,j∈J, J={1,2,…,t}. The mapping function f is defined as information structure operator of finite sequence x:
(3.13)f:x→yyj=xj∑k=1txk.Herey=(y1,y2,…,yt) is called mapping sequence of information structure.The normalized score function matrix can be transformed into mapping sequence of information structureY=(yij)m×n.Then the mass functionmj(i) can be constructed:
(3.14)mj(i)=(1-DOI(Ij))yij,
where mj(i) is a mass function of Vm to index Ij, ∑i=1mmj(i)<1. So the mass function of overall uncertainty with respect to the index Ij is shown as
(3.15)mj(i+1)=1-∑i=1mmj(i).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 matrixR based on the vague set theory.Step 2.
ConvertR into real matrix G=(gij)m×n according to vague combination rule.Step 3.
Calculate the uncertainty degreeDOI(Ij)j=1,2,…,n combined with real matrix G and (3.6).Step 4.
Establish the mapping sequence of information structureY=(yij)m×n.Step 5.
Build up mass functionmj(i) and mass function of overall uncertainty mj(i+1) on the basis of Y=(yij)m×n and DOI(Ij).Step 6.
Fuse evidence information using the D-S combination rule.Step 7.
Make evaluation and decision in principle of the maximizing belief function.


## 3.1. 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].SetU the universe of discourse, with a generic element u. A vague set A is characterized by a truth-membership function tA and a false-membership function fA, where tA(u) is a lower bound on the grade of membership of u, derived from the evidence rising for u; fA(u) is a lower bound on the negation of u, derived from the evidence against u; 0≤tA(u)+fA(u)≤1. The grade of membership of u in vague set A is bound to a subinterval [tA(u),1-fA(u)] of [0,1]. The vague value [tA(u),1-fA(u)] indicates that the exact grade of membership μA(u) of u maybe unknown, but it is bound by tA(u)≤μA(u)≤1-fA(u), where 0≤tA(u)+fA(u)≤1.When the universe of discourseU is discrete, a vague set A can be written as
(3.1)A=∑i=1n[tA(ui),1-fA(ui)]ui,ui∈U.When the universe of discourseU is continuous, a vague set A can be written as
(3.2)A=∫U[tA(ui),1-fA(ui)]uidui,ui∈U.The steps of determining the decision matrix is as follows.(1)
Determination Evaluation Factors and Comment SetThe evaluation factors are expressed asu1,u2,…,um, then the evaluation factor set is U={u1,u2,…,um}.set evaluation indexes of each evaluation factor beuij,(i=1,2,…,m;j=1,2…,n). We can get index set ui={ui1,ui2,…,uij,…,uim}, i=1,2,…,m;j=1,2,…,n.Define the comment set asV={v1,v2,…,vp}.(2)
Construct Evaluation MatrixAssuming the comment level of indexuij, (i=1,2,…,m;j=1,2,…,n) is vk(k=1,2,…,p), then the relationship matrix of factor set U mapping to comment set V is Ri:
(3.3)Ri=[ri11,ri12,…,ri1pri21,ri22,…,ri2p⋮⋮⋮rin1,rin2,…,rinp],
where rijk means the comment of factor uij mapping to comment set, and rijk can be expressed as
(3.4)rijk=[tR~ijk,1-fR~ijk].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 gettR~ijk and 1-fR~ijk.The main calculations are as follows.Multiply calculation:(3.5)k·A~=[k·tA~,k·(1-fA~)]k,k∈[0,1].Finite sum calculation:(3.6)A~⊕B~=[tA~⊕tB~,(1-fA~)⊕(1-fB~)]=[min{1,tA~+tB~},min{1,(1-fA~)+(1-fB~)}],(3.7)bik=[min{1,∑j=1naijtRijk~},min{1,∑j=1naij(1-fRijk~)}],
where Bi~ is a vague subset and bik(k=1,2,…,p) is the evaluation value of k to Bi~.


## 3.2. 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.”Definition 3.1.
SupposeΘ is the frame of discernment. If set function m:2Θ→[0,1] (2Θ is the power set of Θ) is to meet m(Φ)=0, ∑A⊂Θm(A)=1, 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.
SupposeΘ is the frame of discernment, function m:2Θ→[0,1] is the basic probability assignment on Θ, then the belief function is defined as Bel:2Θ→[0,1], where Bel(A)=∑B⊂Am(B), forallA⊂Θ.Definition 3.3.
Ifm(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 functionQ:2Θ→[0,1] is defined by Q(A)=∑A⊂Bm(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.
SupposeBel:2Θ→[0,1] is a belief function on Θ. Functions Dou:2Θ→[0,1] and pl:2Θ→[0,1] are defined as for all A⊂ΘDou(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.Theorem 3.6.
Supposem1 and m2 are two basic probability assignment functions formed based on information obtained from two different information sources, Bel1 and Bel2, in the same frame of discernment Θ. A1,A2,…,Ak and B1,B2,…,Bl are focal elements of Bel1 and Bel2. If ∑Ai∩Bj=ϕm1(Ai)m2(Bj)<1, the combination of m1 and m2 is as follows:
(3.8)m(A)={0,A=ϕ,∑Ai∩Bj=Am1(Ai)m2(Bj)1-∑Ai∩Bj=ϕm1(Ai)m2(Bj),A≠ϕ.Theorem 3.7.
Supposem1,m2,…,mn are the corresponding basic probability assignment functions formed based on information obtained from different information sources Bel1,Bel2,…,Beln in the same frame of discernment Θ. If Bel1⊕Bel2⊕⋯⊕Beln exists and m is its basic probability assignment function, then
(3.9)∀A⊆Θ,A≠ϕ,m(A)=Kn∑A1∩A2∩⋯∩ANm1(A1)m2(A2)⋯mn(An),Kn=1Nn,Nn=∑A1∩A2∩⋯∩AN≠ϕm1(A1)m2(A2)⋯mn(An).


## 3.3. Design of Comprehensive Evaluation Model

Suppose there is a multiattribute (index) evaluation problem with comprehensive comment setV={V1,V2,…,Vm} and evaluation index set I={I1,I1,…,In}. According to vague set, the decision matrix R=(Rij)m×n, Rij=[tij,1-fij] is got. By synthesis law of vague set, we can convert the vague set R into a real set G=(gij)m×n, gij=(tij+(1-fij))/2.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.Suppose the matrixG is
(3.10)G=(gij)m×n,gi¯=1n∑j=1ngij,i=1,2,…,m.Define theq-order uncertainty degree of index Ij as
(3.11)DOI(Ij)=1m[∑i=1m(rij)q]1/q,rij=mini|gij-g¯|+ξmini|gij-g¯||gij-g¯|+ξmini|gij-g¯|,i=1,2,…,m;j=1,2,…,n,
where rij is called gray mean correlation degree with ξ=0.5 in general.In case of negative number, the normalization method is shown as(3.12)gij¯=gij-mini(gij)mini(gij)-mini(gij).Supposex=(x1,x2,…,xt) is a finite difference information sequence with a length of t(t≠0). An index set exists in the form of sequence xj≠0,j∈J, J={1,2,…,t}. The mapping function f is defined as information structure operator of finite sequence x:
(3.13)f:x→yyj=xj∑k=1txk.Herey=(y1,y2,…,yt) is called mapping sequence of information structure.The normalized score function matrix can be transformed into mapping sequence of information structureY=(yij)m×n.Then the mass functionmj(i) can be constructed:
(3.14)mj(i)=(1-DOI(Ij))yij,
where mj(i) is a mass function of Vm to index Ij, ∑i=1mmj(i)<1. So the mass function of overall uncertainty with respect to the index Ij is shown as
(3.15)mj(i+1)=1-∑i=1mmj(i).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 matrixR based on the vague set theory.Step 2.
ConvertR into real matrix G=(gij)m×n according to vague combination rule.Step 3.
Calculate the uncertainty degreeDOI(Ij)j=1,2,…,n combined with real matrix G and (3.6).Step 4.
Establish the mapping sequence of information structureY=(yij)m×n.Step 5.
Build up mass functionmj(i) and mass function of overall uncertainty mj(i+1) on the basis of Y=(yij)m×n and DOI(Ij).Step 6.
Fuse evidence information using the D-S combination rule.Step 7.
Make evaluation and decision in principle of the maximizing belief function.


## 4. 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 Section2 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.Table 1
Comprehensive benefit evaluation index system of hybrid power system.
Evaluation objective
Influence factors
Evaluation indexes
Original index value
Pulling gross domestic product (I1)
61.24%
Social benefit
Improving people’s living standard (I2)
2.99%
Promoting social employment (I3)
3.79%
Comprehensive benefit
Environmental benefit
Saving natural resources (I4)
17.77%
Reducing environmental pollution (I5)
6.70%
Solvency capacity (I6)
4.47%
Economic benefit
Operational capacity (I7)
4%
Profitability (I8)
4.50%
Development capacity (I9)
5%
Integrated voltage qualification rate (I10)
99.27%
Security
Grid frequency qualification rate (I11)
99.99%
Average loading rate of provincial grid (I12)
80.94%
Power supply reliability rate (I13)
99.81%The comprehensive comment set is set to beV = {v1,v2,v3,v4,v5} = {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.According to vague set, we can get the vague matrixR for hybrid power system. Calculate the uncertainty degree DOI(Ij) and mapping sequence of information structure Y=(yij)m×n. Then we can get mass function mj(i) and mass function of overall uncertainty mj(i+1):
(4.1)R=[v1v2v3v4v5I1[0.72,0.86][0.83,0.89][0.52,0.63][0.24,0.28][0.10,0.16]I2[0.68,0.88][0.76,0.80][0.86,0.93][0.27,0.38][0.34,0.49]I3[0.92,0.87][0.80,0.86][0.43,0.49][0.34,0.40][0.13,0.28]I4[0.67,0.73][0.79,0.87][0.64,0.73][0.41,0.56][0.34,0.39]I5[0.64,0.73][0.79,0.87][0.64,0.73][0.41,0.55][0.34,0.39]I6[0.71,0.84][0.94,0.98][0.74,0.82][0.51,0.64][0.14,0.25]I7[0.75,0.84][0.77,0.86][0.89,0.90][0.23,0.29][0.21,0.31]I8[0.77,0.84][0.74,0.83][0.91,0.96][0.23,0.32][0.14,0.31]I9[0.68,0.76][0.70,0.81][0.84,0.92][0.18,0.22][0.08,0.16]I10[0.82,0.96][0.94,0.97][0.75,0.96][0.34,0.48][0.18,0.33]I11[0.79,0.80][0.94,0.96][0.56,0.63][0.29,0.33][0.13,0.20]I12[0.84,0.90][0.95,0.97][0.34,0.41][0.23,0.26][0.19,0.36]I13[0.75,0.69][0.87,0.84][0.24,0.33][0.17,0.31][0.12,0.24]],M=[0.14,0.12,0.00,0.24,0.11,0.13,0.17,0.20,0.17,0.00,0.24,0.21,0.170.32,0.21,0.37,0.14,0.37,0.29,0.29,0.15,0.15,0.13,0.00,0.18,0.240.14,0.09,0.27,0.28,0.17,0.14,0.00,0.00,0.16,0.21,0.23,0.14,0.000.00,0.00,0.07,0.17,0.19,0.25,0.16,0.23,0.00,0.18,0.15,0.31,0.180.24,0.25,0.13,0.17,0.16,0.19,0.22,0.22,0.39,0.35,0.08,0.16,0.31].Setθ=v1,v2,v3,v4,v5, 2θ={{v1},{v2},{v3},{v4},{v5},{v1,v2,v3,v4,v5}}. According to D-S combination rule, we can get belief functions of {v1},{v2},{v3},{v4},{v5},{v1,v2,v3,v4,v5} which are 0.19, 0.45, 0.13, 0.14, 0.08, 0.01, meaning v2≻v1≻v4≻v3≻v1. The belief functions of v2 is best. According to maximizing principle of belief function, the result of comprehensive benefit evaluation is “better.”In order to validate the proposed evaluation algorithm, three other commonly used algorithms including Fuzzy set theory, Vague set theory, and D-S evidence theory are calculated for comparison. According to Fuzzy set theory and the original index value, the comprehensive evaluation result of hybrid power system is[v1,v2,v3,v4,v5]=[0.2204,0.2060,0.1963,0.1925,0.1849], which means the quality of hybrid power system is “best.” According to Vague set theory and the calculated vague index value, the comprehensive evaluation result is [v1,v2,v3,v4,v5] = [[0.7517,0.8414][0.8587,0.9067][0.6760,0.7513][0.3368,0.4229][0.1843,0.3045]], the corresponding evaluation value is 0.7966, 0.8827, 0.7137, 0.3799, 0.2444, which means the final evaluation result is “better” and the corresponding evaluation value being 0.8827. According to D-S evidence theory, the belief functions of {v1},{v2},{v3},{v4},{v5},{v1,v2,v3,v4,v5} are 0.19, 0.32, 0.12, 0.16, 0.07, 0.14, which means the result of comprehensive benefit evaluation is “better.”Combine the actual conditions and experts’ opinions we know such that the actual comprehensive benefit of hybrid power system in Guangdong province is better, which means that the evaluation results of proposed evaluation model, vague set theory, and D-S evidence theory are all correct but fuzzy set. Therefore we know that fuzzy set has a great limitation in comprehensive evaluation compared to the proposed model. At the same time, the best result of proposed model is 0.45 which is much more than better 0.19 no less than the worst 0.01. Analyze final evaluation result sequence of proposed evaluation model, vague set theory, and D-S evidence theory; it is known that the range of result set of the proposed model is much more apparent than other evaluation algorithms. So the results from different evaluation methods validate that the proposed algorithm based on vague set and D-S evidence theory could obtain satisfying conclusion and decrease uncertainty of decision making.


## 5. 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.



---

*Source: 784389-2012-11-05.xml*