Optimal scheduling strategy for virtual power plants based on credibility theory
 Qian Ai^{1}Email author,
 Songli Fan^{1} and
 Longjian Piao^{1}
https://doi.org/10.1186/s416010160017x
© The Author(s) 2016
Received: 12 May 2016
Accepted: 16 May 2016
Published: 25 June 2016
Abstract
The virtual power plant (VPP) is a new and efficient solution to manage the integration of distributed energy resources (DERs) into the power system. Considering the unpredictable output of stochastic DERs, conventional scheduling strategies always set plenty of reserve aside in order to guarantee the reliability of operation, which is too conservative to gain more benefits. Thus, it is significant to research the scheduling strategies of VPPs, which can coordinate the risks and benefits of VPP operation. This paper presents a fuzzy chanceconstrained scheduling model which utilizes fuzzy variables to describe uncertain features of distributed generators (DGs). Based on credibility theory, the concept of the confidence level is introduced to quantify the feasibility of the conditions, which reflects the risk tolerance of VPP operation. By transforming the fuzzy chance constraints into their equivalent forms, traditional optimization algorithms can be used to solve the optimal scheduling problem. An IEEE 6node system is employed to prove the feasibility of the proposed scheduling model. Case studies demonstrate that the fuzzy chance strategy is superior to conservative scheduling strategies in realizing the right balance between risks and benefits.
Keywords
Introduction
Nowadays, due to the rising prices of fossil fuels and the threat of climate change caused by greenhouse gases, distributed energy resources (DERs) have drawn widespread attention because of their clean and renewable characteristics [1–3]. However, the output of DERs is fluctuating and unpredictable. As the penetration of intermittent renewables in the grids is increasing gradually, more technical challenges need to be addressed in the schedule and control of their operation [4–7]. Meanwhile, the liberalization of the electricity market makes DERs inevitable. However, the small capacity, intermittent output and the lack of appropriate interaction with the system operator are the biggest barriers ahead of DERs for participation in the electricity market. To solve these problems mentioned above, the DERs could be aggregated as an entity which can behave like a conventional generator, naming the virtual power plant (VPP) [8–10].
In [11, 12], a VPP is defined as a coalition of DERs including distributed generations (DGs), storage devices, and interruptible loads. Considering the characteristics of each DER and the impact of network, VPPs generate one unit portfolio which can be utilized to offer services to the system operator, and even to make contracts in the wholesale market. Therefore, by introducing the concept of the VPP, the visibility and controllability of DERs for system operators will be improved considerably, just the same as the conventional transmission–connected power plants.
To realize the concept of the VPP, scholars around the world have done lots of studies on VPPs in many aspects [13–17], including VPP modeling methods, negotiating behaviors in the market, bidding strategies, reliability evaluation, management systems, and so on. Among them, the optimal scheduling strategy of DERs in the VPP is a hot topic. Reference [18] proposed a nonequilibrium model which takes into account the supply and demand balancing constraint and security constraint of VPPs. On this basis, the strategy proposed in [19] considered the effect of reliability and determined the optimum hourly operating strategy of DERs by applying the Monte Carlo simulation method. However, these scheduling strategies are based on the deterministic market prices. Considering the uncertainties in prices, [20] developed a new risk constrained scheduling for VPPs to help the aggregator bid in the dayahead market.
However, the optimal bidding strategy of those researches mainly focused on maximizing the VPP’s profit in different types of markets. With regard to reserve level, these methods spare large capacity as recovery to balance the supply and demand. Although the system stability can be guaranteed in this deterministic way, the obtained scheduling results are always inevitably conservative. Actually, more benefits can be gained via reducing the reserve capacity appropriately.
This paper proposes a fuzzy chanceconstrained scheduling model to provide solutions for the optimal scheduling strategy for VPPs based on credibility theory. In this model, as the prediction errors of renewables are characterized as fuzzy parameters, the related constraints correspondingly contain fuzzy variables which need to be tackled properly. By introducing the concept of the confidence level, the feasibility of the conditions where the fuzzy chance constraints can be satisfied is quantified, so as to characterize the risk tolerance of VPPs. The proposed fuzzy chance model is difficult to solve due to the incorporation of fuzzy variables. By transforming the constraints containing fuzzy variables into their equivalent forms, the chance constraints are converted into deterministic constraints which consider the fuzzy risks (or reliability) as well. Then traditional optimization algorithms can be utilized to solve the optimal scheduling problem, in order to achieve a compromise between risks and benefits.
The rest of the paper is organized as follows: the theoretical basis of this paper is presented in Credibility theory and fuzzy chanceconstrained programming, including credibility theory and fuzzy chance constrained programming; the proposed VPP scheduling strategy based on credibility theory is demonstrated in VPP scheduling strategy based on credibility theory; solution methods used in this paper are stated in Methods; case studies and discussion are given in Case study; and finally, the conclusion is drawn in Conclusions.
Credibility theory and fuzzy chanceconstrained programming
Distinguished from the conventional power plants, the scheduling of VPPs is required to deal with a number of uncertain variables from renewables and loads. The common ways to process those uncertain variables are categorized into determinate and indeterminate methods. Determinate method is to set plenty of reserve aside in order to guarantee the reliability of operation. However, this approach is too conservative to coordinate the risks and benefits of VPP operation. More benefits can be obtained by ignoring the small probability events and making decisions within the scope of risk tolerance [21]. The fuzzy chanceconstrained programming method proposed in this paper is one kind of indeterminate scheduling strategy.
In this approach, the decision result is allowed to violate the constraints which contain fuzzy variables, but the feasibility of satisfying the constraints should be no less than the preset confidence level. The confidence level is a concept which reflects the risk tolerance of the system, or the reliability requirements of the system when facing uncertain variables. In conclusion, the optimal fuzzy chanceconstrained scheduling strategy is a compromise between risks and economic profits. Credibility theory provides theoretical support to solve the confidence level problem of the fuzzy chance strategy decision, which greatly contributes to the development and improvement of the fuzzy chance programming theory.
Credibility theory
The concept of the fuzzy set theory was initiated by Zadeh via the membership function [22]. In order to measure a fuzzy event, the possibility measure is proposed. Since then the possibility theory has been studied by many researchers. Although the possibility measure has been widely used, it does not possess the selfduality property. Thus, if the possibility measure equals one it does not mean the event will happen definitely, while the event may happen even though the possibility measure is zero. So the solution to the fuzzy decision problem has been a mathematical conundrum for a long time.
Credibility theory was propounded by Liu in 2004 as a branch of mathematics for studying the fuzzy behavior [23]. It establishes a complete axiomatic system which is parallel with probability theory in dealing with uncertainty. Based on five axioms mentioned in [23], a credibility measure is defined.
According to Liu [23], the following five axioms should hold to ensure that the number Cr(A) has certain mathematical properties.

Axiom 1: Cr{Θ} = 1;

Axiom 2 :Cr is nondecreasing, i.e., if A ⊆ B, there is always Cr{A} ≤ Cr{B};

Axiom 3: Cr is selfdual: ∀ A ∈ P(Θ), Cr{A} + Cr{A^{c}} = 1;

Axiom 4: ∀ A_{ i } ∈ {A_{ i } Cr{A_{ i }} ≤ 0.5}, there is \( \mathrm{C}\mathrm{r}\left\{\underset{i}{\mathrm{U}}{\mathrm{A}}_i\right\}\wedge 0.5=\underset{i}{ \sup}\mathrm{C}\mathrm{r}\left\{{\mathrm{A}}_i\right\} \).

Axiom 5: Let Θ _{1}, Θ _{2}, ..., Θ _{ n } be the nonempty sets corresponding to that Cr_{1}, Cr_{2}, ..., Cr_{ n } satisfy the axioms as respectively defined above, and let Θ = Θ _{1} × Θ _{2} ×...× Θ _{ n }. Then, we have Cr{(θ _{1}, θ _{2}, ..., θ _{ n })} = Cr_{1}{θ _{1}} ∧ Cr_{2}{θ _{2}} ∧ ... ∧ Cr_{ n }{θ _{ n }} for each (θ _{1}, θ _{2}, ..., θ _{ n }) ∈ Θ _{.}

where Θ is a nonempty set; P(Θ) is the possibility set of Θ _{;} the element of P(Θ) is a fuzzy event; A is the subset of Θ; Cr(A) is a nonnegative number indicating the credibility of Event A which will occur; and ^ represents the minimal operator.
Accordingly, the credibility measure describes the credibility level of a fuzzy event; parallel with the confidence level in probability theory, it satisfies the selfduality. It means that the event whose credibility measure is 1 will definitely occur, and similarly, the event whose credibility measure is 0 will never happen. So credibility theory solves the confusion caused by a subordinate degree, and provides theoretical foundations for the fuzzy decision.
In the fuzzy decision of VPP scheduling, some definitions and theorems of credibility theory are utilized, as listed below:
Fuzzy chanceconstrained programming
where x is the decision vector; ξ is the fuzzy parameter vector; g _{ i }(x, ξ) ≤ 0 is the ith constraint; α is the confidence level of the object function; β is the confidence level of the fuzzy constraint. The decision vector x is feasible only when the credibility of g _{ i }(x, ξ) ≤ 0 is no less than β. Additionally, the optimal solution can maximize the object function f at the confidence level of α.
VPP scheduling strategy based on credibility theory
Traditional VPP scheduling strategy
where J _{ Gi }(t) is the operational cost of unit i at time t; SU _{ i }(t) is the startup cost of the unit i at time t; a _{ i }, b _{ i }, c _{ i } are the coefficients of the operational cost of the unit i;d _{ j } is the coefficient of the cost by curtailing the load j; P _{ Gi } is the output power of the unit i; P _{ DRj } is the virtual generation via curtailing the load; u _{ i }(t) is the binary state variable of the unit i at time t: it equals “1” if the unit i is on at time t, and equals “0” if the unit i is off at time t. Similarly, u _{ j }(t) is the binary state variable of the load j at time t. N _{ G } and N _{ DR } are the numbers of conventional units and loads in the demand response respectively.
 1)Supply–demand balancing constraint$$ {\displaystyle \sum_{i=1}^{N_G}{P}_{Gi}(t)+{\displaystyle \sum_{j=1}^{N_{DR}}{P}_{DRj}(t)+{P}_{RES}(t)={P}_L(t)}} $$(7)
 2)System reserve constraint$$ {\displaystyle \sum_{i=1}^{N_G}{P}_{Gi}^{\max }+{\displaystyle \sum_{j=1}^{N_{DR}}{P}_{DRj}^{\max }+{P}_{RES}(t)\ge {P}_L(t)}}+R(t) $$(8)
 3)DG constraints$$ {u}_i(t){P}_{Gi}^{\min}\le {P}_{Gi}(t)\le {u}_i(t){P}_{Gi}^{\max } $$(9)$$ R{D}_i\le {P}_{Gi}(t){P}_{Gi}\left(t1\right)\le R{U}_i $$(10)
 4)Interruptible load constraint$$ {u}_j(t){P}_{DRj}^{\min}\le {P}_{DRj}(t)\le {u}_j(t){P}_{DRj}^{\max } $$(11)
where P _{ RES }(t) is the power generated by renewables at time t; P _{ L }(t) is the total load at time t; R(t) is the reserve capacity for the fluctuation of load and renewables at time t; \( {P}_{Gi}^{\min } \) and \( {P}_{Gi}^{\max } \) are the minimum and the maximum output limit of the unit i respectively; RU _{ i } and RD _{ i } are the rampup rate and the rampdown rate of the unit i; \( {P}_{DRj}^{\min } \) and \( {P}_{DRj}^{\max } \) are the minimum and maximum allowed curtailed value of the load j respectively.
Fuzzy chanceconstrained VPP scheduling
The fuzzy parameters of VPP scheduling strategy come from the unpredictable output of renewables in VPPs [24]. Accurate mathematical expressions of those uncertain characteristics should be established, which are the foundations for the uncertain scheduling strategy. The main characterizing methods include modeling output of renewables and modeling their predicted errors. In this paper, the modeling of the forecast errors is considered to describe the fuzziness of renewables instead of modeling the output. The modeling of renewables output will make three constraints containing fuzzy variables, which are the supply–demand balance constraint, the spinning reserve constraint, and the static security constraint. If the predicted value is regarded as a deterministic value and the prediction error is utilized to describe the fuzziness of renewables, only the spinning reserve constraint should be considered.
where P _{ RES } and \( {P}_{RES}^F \) are the real value and the forecasted value of the output respectively.
where E _{+} and E _{−} are the average values of positive and negative prediction errors obtained from statistics, respectively; σ is the weighting factor.
where α is the credibility level, which represents the reliability level of the reserve constraint. In the realistic problem, α should be more than 0.5.
It can be proved that K _{ α } is a monotonously increasing function. Equation (18) illustrates the links between the confidence level of the fuzzy constraint and spinning reserve allocation when taking the fuzziness of RES predicted error into account. It shows that to improve the confidence level of the spinning reserve constraint, a greater spinning reserve should be allocated. The form of the formula is similar to the traditional spinning reserve constraint. Thus, the model can be easily solved by correcting the reserve levels using the regular optimization algorithm with the coefficient K _{ α }.
Methods
As introduced above, once chance constraint is transformed into its crisp equivalent, traditional optimization methods can be employed to solve the scheduling problem. Based on the natural selection and genetic manipulation, genetic algorithms (GAs) are heuristic random searching methods, and they possess excellent robustness and commonality. However, there are some shortfalls in the original GA, e.g. the slow convergence speed, falling easily into the locally optimal solution, and so on. A selfadaptive GA method was introduced to improve the performance. The probabilities of crossover and mutation for each generation are adaptively determined, which can overcome the premature convergence and the slow convergence speed in later evolutionary processes.
where P _{ c } and P _{ m } are the crossover and mutation probabilities respectively; f _{ max }, f _{ min }, f _{ avg } are the maximum, minimum, and average fitness; f′ is the greater fitness of the two individuals for genetic operation. Therefore, the superior individual with greater fitness is more likely to be reserved and the inferior individual tends to be transformed into a new one. The improved method could guarantee the population diversity and accelerate its convergence; furthermore, it could avoid the phenomena of premature and slow convergent speed in the later stage of evolution.
Case study
To evaluate the proposed approach, two test systems are studied. At first, a small VPP consisting of three conventional generators, one 50 MW wind farm and loads (conventional and interruptible) is employed. Then a system consisting of 10 generators and one wind farm is implemented to show the effectiveness of the proposed model.
Test systems
Parameters of conventional generators
Node  a  b  c  \( {P}_G^{\max } \)  \( {P}_G^{\min } \)  RU(RD) 

1  0.1  13.5  176.9  220  100  55 
2  0.1  32.6  129.9  100  10  50 
6  0.1  17.6  137.4  20  10  20 
Load data of the VPP in 24 h
Time  Load  Time  Load  Time  Load  Time  Load 

1  175.2  7  173.4  13  242.2  19  246.0 
2  165.2  8  177.6  14  243.6  20  237.4 
3  158.7  9  186.8  15  248.9  21  237.3 
4  154.7  10  206.9  16  255.8  22  232.7 
5  155.1  11  228.6  17  256.0  23  195.9 
6  160.5  12  236.1  18  246.7  24  195.6 
The parameters of the selfadaptive GA are set as follows: the population size Np = 30; maximum iterations N _{ ite } = 300; P _{ c1} = 0.85, P _{ c2} = 0.5, P _{ c3} = 0.2, P _{ m1} = 0.09, P _{ m2} = 0.05, P _{ m3} = 0.01. Considering the stochastic nature of GA, 20 test trials were conducted for each case.
Results and discussion
Optimal solutions compared between deterministic and fuzzy chanceconstrained model
Cost/$  α = 0.6  α = 0.7  α = 0.8  α = 0.9  Deterministic model 

Startup cost of units  400  400  200  161  387 
Operation cost  143,380  144,681  144,280  144,341  144,300 
Cost of curtailing load  2253  934  1644  1620  2333 
Total cost  146,033  146,015  146,124  146,122  147,020 
From Table 3, the total costs vary under different confidence levels; especially when α = 0.6, α = 0.7, α = 0.8, α = 0.9, the total cost are 146,033 $, 146,015 $, 146,124 $, 146,122 $, respectively, which are all less than the cost in deterministic model. It can be inferred that the proper selection of confidence levels will help achieve a tradeoff between economy and reliability. However, as the credibility level is increasing gradually from 0.6 to 0.9, the optimized results (i.e., the total costs) in Table 3 do not show a clear monotonic feature. It should be noted that this does not mean that the results of the fuzzy chance model based on credibility theory are confusing. It is because the results are interfered by the demandside resources.
Optimal solutions without curtailing loads
Cost/$  α = 0.6  α = 0.7  α = 0.8  α = 0.9  Deterministic model 

Startup cost of units  400  400  200  200  200 
Operation cost  145,854  145,865  146,080  146,096  146,287 
Total cost  146,254  146,265  146,280  146,296  146,487 
Wind power and the load of the VPP in 6 h
Hour  1  2  3  4  5  6 

Wind Power/MW  42  63  70  60  58  40 
Load/MW  1036  1110  1258  1406  1480  1628 
Parameters of conventional generators
unit  a  b  c  \( {P}_G^{\max } \)  \( {P}_G^{\min } \)  RU(RD) 

1  0.00043  21.60  958.20  470  150  80 
2  0.00063  21.05  1313.6  460  135  80 
3  0.00039  20.81  604.7  340  73  80 
4  0.00070  23.90  471.60  300  60  50 
5  0.00079  21.62  480.29  243  73  50 
6  0.00056  17.87  601.75  160  57  50 
7  0.00211  16.51  502.71  130  20  50 
8  0.00480  23.23  639.40  120  47  30 
9  0.10908  19.58  455.60  800  20  30 
10  0.00951  22.54  692.40  55  55  30 
Conclusions
VPP is a promising way to integrate the DERs into the power system. As the output of DERs is usually unpredictable, the scheduling of VPPs has to deal with a number of uncertain variables, which is a fuzzy programming problem. To realize the proper balance between benefits and risks, credibility theory is introduced in the optimal scheduling strategy for VPPs in this paper, thereby proposing a fuzzy chance scheduling model. The concept of the confidence level can quantify the possibility of satisfying the fuzzy chance constraints, which represents the risk tolerance of VPPs. Further, through transforming the fuzzy chance constraints into their equivalent forms, the conventional optimization algorithms can be utilized to solve the optimal scheduling problem. The case study proves that the operational cost of VPPs will increase when the confidence level increases. That is to say, unnecessary costs can be reduced when the risk of VPP operation is within tolerance.
Declarations
Acknowledgment
This work is supported by the National Natural Science Foundation of China (No. 51577115). Meanwhile, the authors would like to thank the editor and reviewers for their sincere suggestions on improving the quality of this paper.
Authors’ contribution
QA and SF carried out the study of virtual power plants,and drafted the manuscript. All authors read and approved the final manuscript.
About the authors
Q. Ai received the B.S. degree in electrical engineering and automation from Shanghai Jiao Tong University, Shanghai, China, in 1991 and M.S. degree in electrical engineering from Wuhan University, Wuhan, China, in 1994 and Ph.D. degree in electrical engineering from Tsinghua University, Beijing, China, in 1997. After spending one year at Nanyang Technological University, Singapore and two years at University of Bath, UK, he is currently a professor at Shanghai Jiao Tong University, Shanghai, China. His research interests include load modeling, smart grid, and intelligent algorithms.
S. L Fan received the B.S. degree in electrical engineering and automation from Sichuan University, Chengdu, China, in 2013. She is currently pursuing the Ph.D. degree in electrical engineering at Shanghai Jiao Tong University, Shanghai, China. Her research interest include virtual power plant operation and optimization.
L. J Piao received the B.S. and M.S. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2012 and 2015, respectively. He is currently pursuing the M.S. degree in electrical engineering at Shanghai Jiao Tong University, Shanghai, China. His current research interests include multiagent system in smart grids, and electric vehicle coordinated charging using game theory.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
References
 Bamberger Y, Baptista J, Belmans R. et al. (2006). Vision and Strategy for Europe’s Electricity Networks of the Future: European Technology Platform Smart Grids[R]. Office for Official Publications of the European Communities.Google Scholar
 CollMayor D, Paget M & Lightner E. (2007). Future intelligent power grids: analysis of the vision in the European Union and the United States[J]. Energy Policy, 35(4), 2453–2465.View ArticleGoogle Scholar
 Wang C, & Li P. (2010). Development and challenges of distributed generation, the microgrid and smart distribution system [J]. Automation of Electric Power Systems, 2, 004.Google Scholar
 McDermott TE, Dugan RC. (2002). Distributed generation impact on reliability and power quality indices[C]//Rural Electric Power Conference, 2002. IEEE: D3D3_7.Google Scholar
 Newman DE, Carreras BA, Kirchner M. et al. (2011). The impact of distributed generation on power transmission grid dynamics[C]//System Sciences (HICSS), 2011 44th Hawaii International Conference on. IEEE: 1–8.Google Scholar
 Gomez JC, Vaschetti J, Coyos C. et al. (2013). Distributed generation: impact on protections and power quality[J]. Latin America Transactions, IEEE (Revista IEEE America Latina), 11(1), 460–465.View ArticleGoogle Scholar
 Fahim SR, Helmy W. (2012). Optimal study of distributed generation impact on electrical distribution networks using GA and generalized reduced gradient[C]//Engineering and Technology (ICET), 2012 International Conference on. IEEE: 1–6.Google Scholar
 Braun M, & Strauss P. (2008). A review on aggregation approaches of controllable distributed energy units in electrical power systems[J]. International Journal of Distributed Energy Resources, 4(4), 297–319.Google Scholar
 Mohammadi J, RahimiKian A, Ghazizadeh MS. (2011). Joint operation of wind power and flexible load as virtual power plant[C]//Environment and Electrical Engineering (EEEIC), 2011 10th International Conference on. IEEE: 1–4.Google Scholar
 Pudjianto D, Ramsay C, & Strbac G. (2007). Virtual power plant and system integration of distributed energy resources[J]. Renewable Power Generation, IET, 1(1), 10–16.View ArticleGoogle Scholar
 Pudjianto D, Ramsay C, Strbac G. et al. (2008). The virtual power plant: Enabling integration of distributed generation and demand[J]. FENIX Bulletin, 2, 10–16.Google Scholar
 Saboori H, Mohammadi M, Taghe R. (2011). Virtual power plant (VPP), definition, concept, components and types[C]//Power and Energy Engineering Conference (APPEEC), 2011 AsiaPacific. IEEE: 1–4.Google Scholar
 El Bakari K, Kling WL. (2011). Development and operation of virtual power plant system[C]//Innovative Smart Grid Technologies (ISGT Europe), 2011 2nd IEEE PES International Conference and Exhibition on. IEEE: 1–5.Google Scholar
 Zdrilić M, Pandžić H, Kuzle I. (2011). The mixedinteger linear optimization model of virtual power plant operation[C]//Energy Market (EEM), 2011 8th International Conference on the European. IEEE: 467–471.Google Scholar
 Salmani MA, Tafreshi SMM, Salmani H. (2009). Operation optimization for a virtual power plant[C]//Sustainable Alternative Energy (SAE), 2009 IEEE PES/IAS Conference on. IEEE: 1–6.Google Scholar
 Ruiz N, Cobelo I, & Oyarzabal J. (2009). A direct load control model for virtual power plant management[J]. Power Systems, IEEE Transactions on, 24(2), 959–966.View ArticleGoogle Scholar
 Mashhour E, MoghaddasTafreshi SM. (2009). Trading models for aggregating distributed energy resources into virtual power plant[C]//Adaptive Science & Technology, 2009. ICAST 2009. 2nd International Conference on. IEEE: 418–421.Google Scholar
 Mashhour E, & MoghaddasTafreshi SM. (2011). Bidding strategy of virtual power plant for participating in energy and spinning reserve markets—Part I: Problem formulation[J]. Power Systems, IEEE Transactions on, 26(2), 949–956.View ArticleGoogle Scholar
 Soltani M, Raoofat M, Rostami MA. (2012). Optimal reliable strategy of virtual power plant in energy and frequency control markets[C]//Electrical Power Distribution Networks (EPDC), 2012 Proceedings of 17th Conference on. IEEE: 1–6.Google Scholar
 Taheri H, Ghasemi H, RahimiKian A. et al. (2011). Modeling and optimized scheduling of virtual power plant[C]//Electrical Engineering (ICEE), 2011 19th Iranian Conference on. IEEE: 1–6.Google Scholar
 Xue Z, Li G, Zhou M. (2011). Credibility theory applied for estimating operating reserve considering wind power uncertainty[C]//PowerTech, 2011 IEEE Trondheim. Trondheim, Norway: IEEE, 1–8.Google Scholar
 Zadeh LA. (1965). Fuzzy sets[J]. Information and Control, 8(3), 338–353.MathSciNetView ArticleMATHGoogle Scholar
 Liu B. (2006). A survey of credibility theory[J]. Fuzzy Optimization and Decision Making, 5(4), 387–408.MathSciNetView ArticleMATHGoogle Scholar
 Xin A, & Xiao L. (2011). Dynamic economic dispatch for wind farms integrated power system based on credibility theory[J]. Proceedings of the CSEE, 31(S1), 12–18.Google Scholar
 Kuang H, Jin J, & Su Y. (2006). Improving crossover and mutation for adaptive genetic algorithm[J]. Computer Engineering and Applications, 12, 93–99.Google Scholar