JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 10, Number 4, October 2014
doi:10.3934/jimo.2014.10.1297 pp. 1297–1318
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT BASED ON ELM TRANSFORMATION
Hongming Yang Hunan Province Key Laboratory of Smart Grids Operation and Control School of Electrical and Information Engineering Changsha University of Science and Technology Changsha 410114, China
Dexin Yi School of Electrical and Information Engineering Changsha University of Science and Technology Changsha 410114, China and Jinjiang Electric Power Company Limited Jinjiang 362200, China
Junhua Zhao and Fengji Luo Centre for Intelligent Electricity Networks The University of Newcastle Callaghan, NSW 2308, Australia
Zhaoyang Dong School of Electrical and Information Engineering The University of Sydney Sydney, NSW 2006, Australia
(Communicated by Victor Sreeram) Abstract. To implement the optimal dispatch of distributed energy resources (DER) in the virtual power plant (VPP), a distributed optimal dispatch method based on ELM (Extreme Learning Machine) transformation is proposed. The joint distribution of maximum available outputs of multiple wind turbines in the VPP is firstly modeled with the Gumbel-Copula function. A VPP optimal dispatch model is then formulated to achieve maximum utilization of renewable energy generation, which can take into account the constraints of electric power network and DERs. Based on the Gumbel-Copula joint distribution, the nonlinear functional relationship between the wind power cost and wind turbine output is approximated using ELM. The approximated functional relationship is then transformed into a set of equality constraints, which can be easily integrated with the optimal dispatch model. To solve the optimal dispatch problem, a distributed primal-dual sub-gradient algorithm is proposed to determine the operational strategies of DERs via local decision making and limited communication between neighbors. Finally, case studies based on the 15-node and the 118-node virtual power plant prove that the proposed method is effective and can achieve identical performance as the centralized dispatch approach. 2010 Mathematics Subject Classification. Primary: 90B35, 90B18; Secondary: 90C30. Key words and phrases. Distributed primal-dual sub-gradient algorithm, extreme learning machine, Gumbel-Copula, optimal dispatch, virtual power plant. This work was supported by the National Natural Science Foundation of China (Key Project 71331001, General Projects 71071025 and 51107114).
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1. Introduction. Nowadays, the human society is experiencing a number of severe challenges, such as global warming, energy security and environmental pollution. Developing renewable energies (e.g. wind and solar pow-er) and integrating them into the electric power system as distributed energy resources (DER) is an important solution to these challenges. However, the integration of large-scale intermittent DERs will introduce new difficulties in the secure operation of power systems. How to improve the operation efficiency of the power system subject to security constraints becomes an important subject. The concept of virtual power plant (VPP) is firstly proposed in [15]. A VPP will manage and disptch a large number of DERs, which may include renewable generation units, energy storage (ES) devices and interruptible loads (IL). By integrating these DERs, a VPP can act similarly as an ordinary power plant, and participate in the dispatch process of the electricity market. As a relatively mature renewable energy technology, wind power generation has experienced fast development in the past decade. However, the significant intermittency of wind power will threat the security of the power system (includes the static security, such as no operating limit violations on pow-er flows and nodal power, and the dynamic security, such as post-disturbance stability; in this paper, the static security within a VPP is considered), thus obstruct its large-scale deployment. In [16], the wind power is forecasted with the autoregressive moving average (ARMA) model, based on which the pow-er system optimal dispatch model is developed. Reference [5] models the uncertainty of wind power with the Weibull distribution and proposes an optimal dispatch model, which takes into account optimal power flow (OPF) and line transmission limits. In [26], based on the assumption that the wind speed at each individual wind farm follows the Rayleigh distribution, the analytical relationship between the power output of a single wind farm and its expected penalty cost is derived. Based on the relationship, the power system operation model considering wind farms is formulated. However, the above methods have not considered the correlation between the outputs of multiple wind generators. In practice, the outputs of multiple wind generators can be correlated since they draw energy from an identical wind source. Failure to consider this correlation will degrade the model accuracy. Literatures [4]-[25] adopted the fuzzy integral theory and fuzzy measure to analyze the fuzzy correlation characteristics of the multiple random variables. However, the fuzzy integral theory can only analyze the fuzzy correlation rather than the probability correlation. It thus cannot be used to describe the probabilistic characteristics of multiple wind farms. Therefore, in [14] and [22] the Normal-Copula and Gumbel-Copula functions are pro-posed to model the correlation between multiple wind generator outputs. In existing literature, the VPP adopts a centralized approach to DERs dispatch. As illustrated in Fig. 1(a), the dispatch centre of VPP collects the status information of each DER; it then solves the optimal dispatch model and send the dispatch signals back to all DERs via a communication network. In [10], the VPP containing renewable generators is considered; the control objective is to minimize the overall generation cost. Reference [11] studies the multi-period bidding strategies of VPPs. A genetic algorithm based approach is proposed to determine the power outputs of DERs and maximize the economic benefit of the VPP. Although in theory the centralized approach can ensure finding the optimal solution, it is difficult to be applied in practice. This is because a VPP may contain a large number of DERs and the corresponding optimal dispatch problem therefore can easily suffer from
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT
(a) Centralized Dispatch
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(b) Decentralized Dispatch
Figure 1. The Architecture of Controlling VPP
the curse of dimensionality. Also, the centralized approach requires a reliable communication channel between the dispatch centre and each DER, which indicates that a very large investment on the communication infrastructure is needed [2] and [24]. Furthermore, such centralized dispatch structure is very vulnerable to cyberattackers. A cyber attacker may launch the distributed denial of service (DDoS) attack to disrupt the server in the control center. To overcome these difficulties, a more flexible and robust distributed dispatch approach which relies only on local and limited communication is required, depicted in Fig. 1(b). To this end, we propose a distributed dispatch method based on ELM transformation in this paper. The dispatch objective of the proposed method is to maximize the economic benefit of the VPP. In our method, the joint distribution of maximum available outputs of multiple wind generators is modeled using the Gumbel-Copula function. Since the maximum power outputs of multiple wind farms follow a high dimensional joint distribution, there exists a highly complicated nonlinear relationship between the wind power costs and wind farm outputs. The VPP optimal dispatch model is thus very difficult to be solved. To effectively solve the model, in this paper the nonlinear relationship between the wind power cost and wind generator output is approximated using ELM, and transformed into a set of equality constraints in the optimal dispatch model. A distributed primal-dual sub-gradient algorithm is then proposed to overcome the difficulties of centralized dispatch. The effectiveness and validity of the pro-posed method is finally tested in a 15-node and a 118-node virtual power plant. The main contribution of this paper is three-fold: 1) Model the probabilistic correlation between the maximum outputs of multiple wind farms using Copula function. The output characteristics of multiple wind farms can then be accurately approximated. 2) Employ ELM to estimate the high dimensional nonlinear relationship between wind power cost and wind farm outputs, and transform the relationship into a set of equality constraints in the optimal dispatch model. Incorporating the relationship between wind power cost and wind farms outputs in the optimal dispatch model is a difficult problem that has not been addressed before. Our method therefore has high practical value to wind power research.
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3) Employ the distributed primal-dual sub-gradient algorithm to solve the VPP optimal dispatch model only via limited information exchange between VPPs. The proposed algorithm can be proven to have similar performance as a centralized algorithm. Although the centralized approach and distributed approach can reach an identical solution to the optimal dispatch model, the distributed approach does have some unique advantages: 1) To apply the centralized dispatch method, the information of all DERs must be collected and transmitted to the dispatch center, which will increase the processing time. On the other hand, the pro-posed distributed dispatch method can make decisions locally without transmitting local information to the dispatch center, which will greatly increase the decision efficiency. 2) In a real-world VPP, DERs are usually owned by different entities. These entities may not be willing to disclose their information to the dispatch center. On the other hand, the proposed method does not require the entities to disclose their private information to the dispatch center; it is therefore easier to be accepted. 3) When the proposed distributed algorithm is employed, if a DER is disconnected from the system, other DERs can still adjust their decisions based on local information and limited communication, which significantly improves the robustness of the control system. 4) To implement the centralized dispatch method, a direct communication channel must be established between the dispatch center and each DER. Considering the fact that, there usually exists far distance between the dispatch center and DERs, this requires huge investment on building the communication infrastructure. On the other hand, to implement the proposed distributed algorithm, only the limited communication between neighboring nodes is needed, which can be supported by local area network (LAN) technologies and can greatly save the communication investment. 2. The optimal VPP dispatch model considering uncertain wind power outputs. 2.1. The probability distribution of maximum available outputs of multiple wind generators. The uncertainty of wind speed can usually be described by Weibull, Rayleigh or Lognormal distributions. In this paper, the widely-used Weibull distribution with two parameters is employed, and its probability density function (PDF) takes the following form [3]: κ −1 κ vw,n w,n vw,n w,n κw,n exp − , (1) φwn (vw,n ) = σw,n σw,n σw,n where vw,n represents the actual wind speed of wind farm at node n; κw,n and σw,n denote the shape and scale parameters of the Weibull distribution, and can be estimated with the maximum likelihood estimation (MLE) method. The speed-power relationship of a wind turbine takes the following form [9]: (0 ≤ vw,n < vin,n , vw,n > vout,n ) 0 aw,n vw,n + bw,n (vin,n ≤ vw,n < vrate,n ) Pwn,s (vw,n ) = , (2) max Pw,n (vrate,n ≤ vw,n ≤ vout,n ) where Pwn,s denotes the maximum available output of the wind turbine at node n; vin,n , vrate,n , vout,n denote the cut-in, rated and cut-out speeds of the wind turbine,
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max respectively; Pw,n represents the rated power of the wind turbine at node n; aw,n , bw,n are parameters which can be calculated as:
aw,n =
max pmax Pw,n w,n vin,n , bw,n = . vrate,n − vin,n vin,n − vrate,n
Based on (1) and (2), the cumulative distribution function (CDF) of the maximum available output Pwn,s can be derived as: Fn (Pwn,s ) = R +∞ R vin,n φw,n (vw,n ) dvw,n + vout,n φw,n (vw,n ) dvw,n , Pwn,s = 0 0 R +∞ R vw,n (Pwn ) max φw,n (vw,n ) dvw,n + vout,n φw,n (vw,n ) dvw,n , 0 < Pwn,s < Pw,n 0 max 1, Pwn,s = Pw,n
.
(3) The multiple wind turbines within a distribution network can be correlated since they draw power from an identical wind source. To model this kind of correlation, the Copula function is employed. Assume that the marginal distributions of maximum available outputs of N wind turbines are F1 (Pw1,s ) , · · · , FN (PwN,s ) respectively, then there exists a Copula function C (·) such that the joint distribution < (·) of multiple wind turbine outputs can be expressed as [6]-[23]: < (Pw1,s , · · · , PwN,s ) = C (F1 (Pw1,s ) , · · · , FN (PwN,s )) .
(4)
There exist several different types of Copula functions. Since the Gumbel-Copula function is un-symmetrical and upper fat-tailed, which well match the characteristics of wind power correlation [1]; it is employed to model the joint distribution of maximum available outputs of multiple wind turbines as Eq. (5). n o ς ς 1 < (Pw1,s , · · · , PwN,s ) = exp −[(−InF1 (Pw1,s )) + · · · + (−InFN (PwN,s )) ] ς , (5) where ς is the parameter of the Copula function and can be estimated using the MLE method. 2.2. The optimal dispatch model of VPP. A virtual power plant (VPP) is defined as a virtual electricity generator consisting of a number of DERs, such as micro-gas turbines, renewable units, energy storage (ES) devices and interruptible loads (ILs). By coordinating the behaviours of these DERs, the VPP can act similarly as an ordinary power plant and participate in the power system dispatch. In a VPP, ES devices can absorb energy when the electricity price is low or when the outputs of renewable units cannot be fully consumed. It can then discharge the energy back to the system if necessary. Interruptible loads are the devices which can be interrupted (e.g. air conditioner) when the system has insufficient generation capacity. The objective of VPP optimal dispatch is to maximize its economic benefit by dispatching the out-puts of micro-gas turbines and renewable units, the charging/discharging power of ES devices, and the amount of interruptible load. Therefore, the objective can be formulated as: N X ρd Pdn − Cgn (Pgn ) − Crn (Prn ) max f= − ρ E PS , (6) −Ccn (Pcn ) − E [Cwn (Pwn )] Pwn ,Pgn ,Prn ,Pcn ,PS n=1
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where N denotes the number of nodes in the VPP; ρE , ρd are the prices at which the VPP buys electricity from the market and sells electricity to its customers respectively; PS , Pdn are the power injected into the VPP and the load level (including both interruptible and non-interruptible loads) at node n, respectively. The overall economic benefit of a VPP can be divided into 6 parts. The first part is the income from selling electricity to customers, i.e. the first term in Eq. (6). The left five parts refer to the costs of a VPP include the fuel cost, the charging/discharging costs of ES devices, the cost of interrupting loads, the cost of wind power and the cost of purchasing electricity, which correspond to the second to sixth terms in (6) respectively. It is assumed that the generation cost of micro-gas turbines and the cost of interrupting loads take the following forms: 2 Cgn (Pgn ) = cgn,2 Pgn + cgn,1 Pgn ,
(n = 1, 2, · · · , N ),
(7)
2 Ccn (Pcn ) = ccn,2 Pcn + ccn,1 Pcn ,
(n = 1, 2, · · · , N ),
(8)
where Pgn , Pcn are the output power of the micro-gas turbine, and the interrupted power of ILs at node n, respectively; cgn,1 , cgn,2 are the cost coefficients of the micro-gas turbine; ccn,1 , ccn,2 denote the cost coefficients of ILs. Also, we assume that the charging/discharging cost of energy storage devices takes the following form: 2 Crn (Prn ) = crn,2 Pcn + crn,1 |Prn | ,
(n = 1, 2, · · · , N ),
(9)
where Prn denotes the charging/discharging power of the ES devices at node n. it takes positive values in the charging mode and negative values in the discharging mode. crn,1 , crn,2 represent the cost coefficients. The maximum available output of a wind turbine is random and can be assumed to follow distribution (5). To fully utilize wind power, the cost of wind power includes not only the direct generation cost but also two other parts. The first part is the penalty cost if the dispatched wind turbine output is smaller than the maximum available output (i.e. the available wind power has not been fully utilized). The second part is the cost of providing spinning reserve when the dispatched wind turbine output is greater than the maximum available output (i.e. the available wind power is insufficient). Therefore, the total wind power cost can be expressed as: Cwn (Pwn ) = cwn Pwn + CP,wn (Pwn ) ,
(n = 1, 2, · · · , N ),
(10)
where the first term in (10) is the direct cost of generating wind power; cwn is the cost coefficient; the second term in (10) denotes the penalty cost and reserve cost: CP,wn (Pwn ) =
cun (Pwn,s − Pwn ) Pwn,s > Pwn , con (Pwn − Pwn,s ) Pwn,s < Pwn
(n = 1, 2, · · · , N ),
(11)
When the maximum available output Pwn,s of wind turbine is greater than the dispatched output Pwn , CP,wn (Pwn ) denotes the penalty cost; when the maximum available output of wind turbine is smaller than the dispatched output, CP,wn (Pwn ) is the reserve cost. cun , con are the penalty and reserve cost coefficients, respectively.
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Since the maximum available outputs of wind turbines are random, CP,wn (Pwn ) is a random variable as well. Its mathematical expectation can be expressed as: N R P max P E CP, wn (Pwn ) = 0 w,1 n=1
···
N max P R Pw,N 0
n=1
∂
CP, wn (Pwn )
N
1 P N ς ς exp − (−InF1 (Pwn,s )) n=1 ∂Pw1,s ···∂PwN,s
, dPw1,s · · · dPwN,s (12)
where E (·) denotes the expectation operator. Clearly, the relationship between wind power cost CP,wn (Pwn ) and maximum available output Pwn,s is highly nonlinear and cannot be solved analytically. The operating security of VPP is modelled based on DC power flow, including the power balance constraint, line transmission limits and capacity constraints of DERs. 1) Power balance constraint: PS =
N X
(Pdn − Pwn − Pgn − Prn − Pcn ).
(13)
n=1
2) Line transmission limits: − Tl ≤
N X
ηln (Pwn + Pgn + Prn + Pcn − Pdn ) ≤ Tl (l = 1, · · · , L) ,
(14)
n=1
where ηln denotes the sensitivity of the power injected at node n with respect to the power flow of line l; Tl is the transmission limit of line l; L is the total number of transmission lines. 3) Capacity constraints of DERs: max 0 ≤ Pwn ≤ Pw,n , min max Pgn ≤ Pgn ≤ Pgn , max Pcn ,
(n = 1, 2, · · · , N ), (n = 1, 2, · · · , N ),
0 ≤ Pcn ≤ (n = 1, 2, · · · , N ), dch,max Prn ≤ Prn , Prn ≥ 0 , (n = 1, 2, · · · , N ), ch,max −Prn ≤ Prn , Prn ≤ 0
(15) (16) (17) (18)
min max where Pgn , Pgn are the minimum and maximum outputs of the micro-gas dch,max ch,max turbine at node n respectively; Prn , Prn are the maximum discharging max and charging power of ES devices at node n; Pcn denotes the maximum amount of interruptible load at node n. By substituting Eq. (13) into Eq. (6), the objective function can be simplified as: N [cwn Pwn + E (CP,wn (Pwn )) − ρE Pwn ] + X [Cgn (Pgn ) − ρE Pgn ] + [Crn (Prn ) − ρE Prn ] . (19) min f= Pwn ,Pgn ,Prn ,Pcn n=1 + [Ccn (Pcn ) − ρE Prn ]
The dispatch objective is to minimize the cost of the VPP (the overall cost of all DERs). Since the maximum available outputs of wind turbines are random and correlated, the cost function (12) is highly nonlinear and cannot be solved analytically. In this paper, we employ extreme learning machine to estimate the functional relationship between the wind power cost and wind turbine output, and transform the relationship into a set of equality constraints in the optimal dispatch
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Pw1
ω11
β H1
•
E w1 •
•
•
•
•
PwN
E wN
•
ω NH
• •
β HN
Figure 2. The Topology of the ELM Network
model. Based on this ELM transformation, the transformed model can be easy to be solved. 3. ELM based transformation of VPP optimal dispatch model. 3.1. Estimating the mapping between wind power cost and wind turbine output. Extreme learning machine (ELM) is a novel algorithm for training the single hidden-layer feed-forward neural networks [8]-[7]. ELM has been successfully applied in a number of areas, such as function approximation, evolutionary computation, power system analysis and classification. Compared with the traditional artificial neural networks (ANNs) based on back-propagation algorithm, ELM has the following advantages: 1) ELM randomly assigns the input weights and the thresholds of hidden nodes, without the assumption that the input weights and thresholds are given; 2) it analytically calculates the output weights by using the least square method, rather than the iterative calculation by the gradient descent method. Existing studies show that ELM has superior training speed and generalization ability. The topology of an ELM network can be shown in Fig. 2. Since the maximum available outputs of N wind turbines follow a joint distribution, the wind power cost is a nonlinear function of all wind turbine outputs in Eq. (12). Assuming that the ELM network contains N input nodes where each of which represents the output of a wind turbine, then all the wind turbine outputs T is a vector (Pw1 , Pw2 , · · · , PwN ) . The output layer of the ELM network also has N nodes, which denote the costs of N wind turbines. The number of nodes in the hidden layer is set as H. The relationship between the nth output variable Ewn and input variables (Pw1 , T Pw2 , · · · PwN ) can be expressed as: ! H N X X Ewn = βjn G ωij Pwi + θj , (20) j=1
i=1
where ωij denotes the connection weights between the ith input node and jth hidden node; θj is the threshold of hidden node j; βjn represents the connection weights between the jth hidden node and nth output node; G (·) is the activation function
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in the hidden layer, and we select the Sigmoid function in this paper: G(x) =
1 . 1 + exp(−x)
(21)
The procedure of employing ELM to estimate the relationship between wind power costs and wind turbine outputs can be summarized as follows. T k k k 1) Randomly select the kth set of wind power outputs Pwk = Pw1 , Pw2 , · · · , PwN . k Here Pwn represents the output of the nth turbine; 2) Calculate the joint distribution of maximum available outputs of all N turbines based on Eq.(5). Perform Monte Carlo simulation to draw M sets of m m m maximum available outputs Pw1,s , Pw2,s , · · · , PwN,s (m = 1, 2, · · · , M ) of wind m turbines (Pwn is the maximum available output of the nth turbine in the mth m m set). Calculate corresponding wind power costs CP, m w1 , CP, w2 , · · · , CP, wN (m = 1, 2, · · · , M ) based on Eq.(11), and then obtain the expected wind power T k k k k costs Ew = Ew1 , Ew2 , · · · , EwN represents the expected cost of the nth wind turbine in the kth set, which can be calculated as: M 1 X m C (n = 1, 2, · · · , N ) . (22) M m=1 P,wn k 3) Repeat steps 1) and 2), obtain W sets of Pwk and Ew as the inputs and outputs of the ELM network. 4) Train the ELM network based on the input and output samples. The training of ELM is equivalent to the minimum norm least square solution to the following equations: k Ewn =
Υ0 β = E0 ,
(23)
where N P 1 G ω P + θ ··· n1 wn 1 n=1 ··· Υ0 = ··· N P W G ωn1 Pwn + θ1 ···
G
n=1
β=
1 ωnH Pwn + θH
··· G
n=1
N P
N P n=1
W ωnH Pwn + θH
,
(24)
W ×H
β11 β21 .. .
β12 β22 .. .
··· ··· .. .
β1N β2N .. .
βH1
βH2
···
βHN
,
(25)
H×N
1 2 W T E0 = Ew , Ew , · · · , Ew . W ×N
(26)
It has been proven in [8]-[7] that the by randomly assigning ωij , θj , the weights ˆ β can be calculated analytically as: βˆ = pinv(Υ0 ) · E0 ,
(27)
where pinv(Υ0 ) = ΥT0 Υ0
−1
ΥT0 .
(28)
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3.2. Transforming the optimal dispatch model based on ELM. Based on the relationship between the wind power cost and wind turbine output estimated by ELM, the expected wind power cost in (19) can be approximated as: ! H N X X E (CP,wn (Pwn )) = Ewn = βjn G ωij Pwi + θj . (29) j=1
i=1
This relationship can be transformed into a set of equality constraints and integrated with the dispatch model. The transformed VPP dispatch model takes the following form: min
Pwn ,Pgn ,Prn ,Pcn ,Ewn
f=
N P
{{[cwn Pwn + Ewn − ρE Pwn ] + [Cgn (Pgn ) − ρE Pgn ]
n=1
+ [Crn (Prn ) − ρE Prn ] + [Ccn (Pcn ) − ρE Prn ]} s.t.
N P
− Tl ≤
n=1 H P
Ewn −
j=1
ηln (Pwn + Pgn + Prn + Pcn − Pdn ) ≤ Tl (l = 1, · · · , L) N P βjn G ωij Pwi + θj = 0 (n = 1, 2, · · · , N )
.
1
max min max 0 ≤ Pwn ≤ Pwn , Pgn ≤ Pgn ≤ Pgn dch,max Prn ≤ Prn , Prn ≥ 0 max , 0 ≤ Pcn ≤ Pcn ch,max −Prn ≤ Prn , Prn ≤ 0
(30) In (30), the objective function is the sum of the 4 costs of N DERs. These costs include wind turbine generation costs, micro-gas turbine generation costs, ES charging/discharging costs and the costs of interrupting loads. 4. Optimal dispatch based on distributed primal-dual sub-gradient algorithm. Traditional power systems usually adopt the centralized dispatch approach, which is not suitable for a large number of DERs. This is because the dispatch centre may not be able to access the private information of DERs. Moreover, the centralized dispatch approach relies on reliable communication channels between the dispatch centre and all DERs, which requires very large investments. And the centralized architecture is also vulnerable to cyber attacks. To overcome the difficulties of centralized dispatch, we propose a distributed method to optimally dispatch the DERs within a VPP. In the distributed disptach method, the dispatch centre does not need the private information of DERs. Each DER only communicates with its neighboring DERs and makes decisions independently. Theoretically, distributed dispatch can achieve a performance similar to the centralized dispatch. To simplify the discussion, we denote DERs 1, · · · , N as the wind turbines at nodes 1, · · · , N ; DERs N +1, · · · , 2N represent the micro-micro-gas turbines at nodes 1, · · · , N ; DERs 2N + 1, · · · , 3N represent the ES de-vices at nodes 1, · · · , N ; and DERs 3N + 1, · · · , 4N as the interruptible loads at nodes 1, · · · , N . The topology of the communication network between DERs is represented by an undirected weighted graph Γ. Let Γ = (Λ, Z) be a graph with a nonnegative adjacency matrix A = {aij } ∈ R4N ×4N which characterizes the topology of communication network; if there is an undirected connection between DERs i and j, then 0 < aij = aji ≤ 1(i 6= j), otherwise aij = aji = 0. Let Λi = {i ∈ Λ : 0 < aij ≤ 1} denotes the set of neighbors of DER i, where Λ = {1, · · · , 4N }.
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To simplify the discussion, we define: x = (Pw1 , · · · , PwN , Pg1 , · · · , PgN , Pr1 , · · · , PrN , Pc1 , · · · , PcN , Ew1 , · · · , EwN ) , (31) X=
N Y
max [0, Pwn ]×
n=1
N N Y min max Y ch,max dch,max Pgn , Pwn × −Prn , Prn n=1 N Y
n=1 max [0, Pcn ] × RN ,
(32)
fwn (x) = cwn xn + x4N +n − ρE xn , (n = 1, 2, · · · , N ),
(33)
fgn (x) = Cgn (xN +n ) − ρE xN +n , (n = 1, 2, · · · , N ),
(34)
frn (x) = Crn (x2N +n ) − ρE x2N +n , (n = 1, 2, · · · , N ),
(35)
×
n=1
fcn (x) = Ccn (x3N +n ) − ρE x3N +n , (n = 1, 2, · · · , N ), (36) N P xwn + xN +n + x2N +n ηjn , (j = 1, 2, · · · , L), +x3N +n − Pdn n=1 gj (x) = , N P xwn + xN +n + x2N +n − ηjn − Tj , (j = L + 1, · · · , 2L) +x3N +n − Pdn n=1 (37) ! H N X X hn (x) = x4N +n − βjn G ωij xi + θj , (n = 1, 2, · · · , N ). (38) j=1
i=1
The dual problem of the VPP dispatch problem (30) can be obtained as [13]: max min Ξ =
µj ,γj
s.t.
x
N P
Ξwn +
n=1
N P
N P
Ξgn +
n=1
Ξrn +
n=1
N P
Ξcn
n=1
x∈X µj ≥ 0(j = 1, 2, · · · , 2L) λj ≥ 0(j = 1, 2, · · · , N )
,
(39)
where µj , λj denotes the Lagrange multipliers for constraints (14) and the equality constraints in (30) derived based on ELM transformation. Ξwn , Ξgn , Ξrn , Ξwn are dual functions for wind turbines, micro-gas turbines, ES devices and interruptible loads. They are defined as: Ξwn = fwn (x) +
2L X
+
µj [gj (x)] +
j=1
Ξgn = fgn (x) +
2L X
2L X
+
µj [gj (x)] +
Ξcn = fcn (x) +
j=1 +
(40)
N X
λj |hj (x)|,
(41)
λj |hj (x)|,
(42)
λj |hj (x)|,
(43)
j=1 +
µj [gj (x)] +
j=1 2L X
λj |hj (x)|,
j=1
j=1
Ξrn = frn (x) +
N X
N X j=1
+
µj [gj (x)] +
N X j=1
where [·] denote the projection operator onto the non-negative orthant. To employ the sub-gradient algorithm to solve dual problem (39), firstly the data 2L N of each DER needs to be initialized, i.e. set xi (0) ∈ X, µi (0) ∈ R≥0 , λi (0) ∈ R≥0 ,
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where xi (t), µi (t), λi (t)(t = 0, 1, · · · ) denotes the values of x, µ, λ estimated by DER i at the tth iteration. The sub-gradient algorithm iteratively calculates x, µ, γ based on the following equations: X aij xj (t), υxi (t) = (44) j∈Λi
υµi (t) =
X
aij µj (t),
(45)
aij λj (t),
(46)
j∈Λi
υλi (t) =
X j∈Λi
xi (t + 1) = ψX υxi (t) − α (t) Sxi (t) , µi (t + 1) = υµi (t) + α(t) ·
η · Pinput − Tmax −η · Pinput − Tmax
(47) ,
T xi , xi4N +2 , · · · , xi4N +N 4N +1 i i T λ (t + 1) = vλ (t) + α (t) −βG ω · xi1 , xi2 , · · · , xiN + θ
(48) ,
(49)
where T
Tmax = (T1 , T2 , · · · , TL ) ,
(50)
Pw1 + Pg1 + Pr1 + Pc1 − Pd1 Pw2 + Pg2 + Pr2 + Pc2 − Pd2 .. .
Pinput =
,
(51)
PwN + PgN + PrN + PcN − PdN η=
ω=
η11 η21 .. .
η12 η22 .. .
··· ··· .. .
η1N η2N .. .
ηL1
ηL2
···
ηLN
ω11 ω21 .. .
ω12 ω22 .. .
··· ··· .. .
ω1N ω2N .. .
ωH1
ωH2
···
ωHN
,
(52)
,
(53)
µ = (µ1 , µ2 , · · · , µ2L ) ,
(54)
λ = (λ1 , λ2 , · · · , λN ) ,
(55)
ΨX [·] is the projection operator onto the set X; α(t) is the step-size at time t; Sxi (t) is sub-gradient of the dual function υxi (t) of DER i with respect to its own
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT
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decision variables: 2L N + P P ∇fwn (υxi (t)) + υµi (t)∇ gj (υxi (t)) + υλi (t)∇ |hj | (υxi (t)) j=1 j=1 i = 1, 2, · · · , N, 2L N + P P ∇fgn (υxi (t)) + υµi (t)∇ gj (υxi (t)) + υλi (t)∇ |hj | (υxi (t)) j=1 j=1 i = N + 1, N + 2, · · · , 2N, Sxi (t) = N 2L + P P i υµi (t)∇ gj (υxi (t)) + υλi (t)∇ |hj | (υxi (t)) ∇frn (υx (t)) + j=1 j=1 i = 2N + 1, 2N + 2, · · · , 3N, 2L N + P P υµi (t)∇ gj (υxi (t)) + υλi (t)∇ |hj | (υxi (t)) ∇fcn (υxi (t)) + j=1 j=1 i = 3N + 1, 3N + 2, · · · , 4N,
,
(56) where ∇f (·) is the sub-gradient of function f . In this algorithm, at iteration t, each DER i calculates the estimated υxi , υµi , i υλ based on its own decision xi , µi , λi and neighboring DERs’ decisions xj , µj , λj (j ∈ Λi ) via limited communication. Then, each DER i updates its own decision xi , µi , λi along the sub-gradient direction to minimize the Lagrangian function, and takes a primal and dual projection onto constraints. Without a centralized authority, each DER can determine its respective decisions via limited information exchange between its neighbors to optimize the objective function. It is proved in [13] that for a convex optimization problem, there is a pair of primal and dual optimal solutions to which the distributed sub-gradient algorithm will converge globally. For the non-convex optimization problem in (30), the proposed distributed algorithm converges to a local optimum, which is similar to +∞ +∞ 4N 4N P P P P 2 the centralized approach. If α(t) = +∞, α(t) < +∞, ai,j = ai,j = 1 t=0
t=0
i=1
j=1
and the graph Γ = (Λ, Z) is connected, there exists a primal optimal solution x∗ ∈ X such that
(57) lim xi (t) − x∗ = 0, ∀i = 1, 2, · · · , 4N. t→+∞
Clearly, the distributed primal-dual sub-gradient algorithm can converge to the local optimum based on only limited information sharing between neighboring DERs. 5. Case studies. 5.1. Cast study setting. The proposed method is tested with the 15-node virtual power plant [12] as shown in Fig. 3. The VPP contains 5 renewable units, 2 ES devices and 2 interruptible loads. Two wind turbines (DERs 1 and 2) are connected at nodes 9 and 13, respectively. Three micro-gas turbines are connected at nodes 4, 6 and 8 (DERs 3, 4 and 5). Two ES devices (DERs 6 and 7) are connected at nodes 10 and 13. Loads at nodes 5 and 12 are interruptible loads (DERs 8 and 9). The price at which the VPP purchases (from the market) and sells (to customers) electricity are respectively 0.04$/kWh and 0.045$/kWh. The parameters of wind turbines are given in Table 1. The cost coefficients and output limits of micro-gas turbines, ES devices and interruptible loads are shown in Table 2. The load levels
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PS
Gas Interruptible Generator Load Wind Load Turbine Electrochemical Storage
communication 1
line1
2 line 2
GG
WT
ES
line 4 line 12
5
3
13 line 8
line 3
9
line 5
4
WT line 10
6
line 13
ES
14
WT 11
GG
line 6
GG
line14
line 9 10
7
line 7
15
line 11 12
ES
8
GG
Figure 3. The Structure of the Virtual Power Plant of all nodes in the VPP are listed in Table 3. The parameters of distribution lines are given in Table 4. Table 1. Parameters of Wind Turbines Turbine Cost (DER Coefficient No.) ($/kWh) 1 2
0.0021 0.0023
Penalty Cost Coefficient ($/kWh) 0.013 0.013
Reserve Cost Coefficient ($/kWh) 0.021 0.021
Cut Cut Rated Rated in out Power Speed Speed Speed (kW) (m/s) (m/s) (m/s) 1400 5 45 15 1400 5 45 15
Table 2. The Cost Coefficients and Output Limits of DERs 3 9
DER No. 3 4 5 6 7 8 9
First-order Cost Coefficient 7.2 × 10−6 7.5 × 10−6 7.3 × 10−6 7.6 × 10−6 7.5 × 10−6 2.4 × 10−5 2.2 × 10−5
Second-order Lower Upper Cost Output Output Coefficient Limit(kW) Limit(kW) 0.031 200 800 0.029 250 9000 0.030 150 800 0.027 -250 1000 0.028 -200 1000 0.012 0 750 0.013 0 800
Table 3. Load Data Node 1 2 Load(kW) 0 200 Node 9 10 Load(kW) 250 200
3 300 11 200
4 5 300 450 12 13 600 250
6 350 14 250
7 8 350 350 15 180
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT
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Table 4. Distribution Line Parameters Line 1 2 3 4 5 6 7 no3 tiu bri sti 2 D -l ac 1 iir p m0 E0 no 3 it buri ts 2 i D l-a icir 1 p m0 E0
Reactance (Ω ) 91 96 96 147 96 39 96
1
2
Gumbel-Copula
1
2
Clayton-Copula
Power Flow Limit (kW) 2300 350 300 1300 500 600 400
3
no 3 it buri ts 2 i D l-a icir 1 p m0 E0
3
no 3 it buri ts 2 i D l-a icir 1 p m0 E0
Line 8 9 10 11 12 13 14
1
2
Frank-Copula
1
t-Copula
2
Reactance (Ω ) 96 96 96 103 96 128 13
3
no 3 it buri ts 2 i D l-a icir 1 p m0 E0
3
no 3 it buri ts 2 i D l-a icir 1 p m0 E0
Power Flow Limit (kW) 1200 750 700 700 600 400 320
1
2
3
1
2
3
Normal-Copula
Uncorrelated
Figure 4. QQ Plots for Different Function 5.2. The joint probability density of maximum available power output of two wind farm outputs. The wind speed data at nodes 9 and 13 come from the real-world data of two wind farms (De Bil and Soesterberg wind farms) in Netherlands (see http://www.knmi.nl/samenw/hydra). Denote the wind speeds at nodes 9 and 13 as vw,9 and vw,13 . The Weibull distributions of the wind speeds at nodes 9 and 13 are estimated by using the maximum likelihood estimation (MLE). We use Copula function to analyze the probabilistic characteristics of the maximum available outputs of two wind farms. Fig. 4 shows the Quantile-Quantile (QQ) plots of five Copula functions and the joint distribution without consider-ing the correlation between multiple wind farm outputs. The joint distribution without considering correlation is expressed with Eq. (3); the shape and scale parameters are set as 2.27 and 13.86, and the K-S goodness-of-fit value is 0.8821. The parameters of five Copula functions and the K-S goodness-of-fit values are shown in Table 5. From Fig. 4 and Table 5, it can be seen that the Gumbel-Copula function can better approximate the tail correlation of multiple wind farm outputs; it also has the largest good-ness-of-fit value compared with the five competitors. 5.3. Estimating the mapping between wind power cost and wind turbine output using ELM. By using the Monte Carlo simulation, 5000 maximum available outputs Pw9,s , Pw13,s of wind turbines are drawn randomly from the GumbelCopula joint distribution. Randomly select 300 sets of power outputs (Pw9 , Pw13 )
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Table 5. Copula Parameters Estimation and Goodness Test Copula Upper tail Connection Goodness-of-fit Function correlation parameter value Type coefficient Gumbel 8.02 0.8821 0.9098 Clayton 0.67 0.5318 0.0 Frank 15.93 0.6127 0.0 Normal 0.73 0.5954 0.0 t 0.76 0.7329 0.3520 -3
-6
9.9
x 10
1.8
9.8
x 10
1.7
E S9.7 M R gn9.6 in ia r 9.5 T
E S1.6 M R gn1.5 tis e1.4 T
9.4
1.3
9.3 2
Lower tail correlation coefficient 0.0 0.3560 0.0 0.0 0.3520
4 6 8 10 The number of hidden layer nodes
(a) Training RMSE
1.2 2
4 6 8 10 The number of hidden layer nodes
(b) Testing RM
Figure 5. The Number of Hidden Layer Nodes v.s. RMSE of wind turbines at nodes 9 and 13 (setting Pw9 ∈ [0, 150], Pw13 ∈ [0, 150]) as the inputs of ELM. Based on Eq. (20), calculate 300 sets of expected wind power costs T Ew = (Ew9 , Ew13 ) as the outputs of ELM. Within the 300 samples, 240 of them are used for training, while 60 are test data. The Sigmoid function is selected as the activation function. The relationship be-tween the number of hidden nodes and the root mean square error (RMSE) achieved on the training data set is illustrated in Fig. 5(a); the relationship between the number of hidden nodes and the RMSE on the testing data set is illustrated in Fig. 5(b). As illustrated, when the number of hidden nodes is set as 5, both the training RMSE and testing RMSE will reach their minima. After the training, the number of hidden nodes is determined as 5. The input weights ω, output weights β and thresholds θ are obtained as: T −0.8617 0.0029 0.4763 −0.2446 −0.5623 ω= , −0.8593 −0.0059 0.9116 0.1169 0.3565 T 822.8859 95.5198 198.2061 0.1074 −7.5483 β= , 2590.6 −1167.2 1132.3 −92.3944 −53.2859 T θ = 0.5192 0.9200 0.9057 0.6822 0.3586 . The actual wind power costs and estimated wind power costs are depicted in Fig. 6. As shown clearly, the wind power costs estimated by the ELM are very close to their actual values. On the training data, the RMSE is 9.32 × 10−6 , while on the
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT
1313
16
The samples of DG5 The prediction of DG5 The samples of DG13 The prediction of DG13
)$15 (t ne 14 m hs in13 up fo12 ts oc11 eh T10 9 0
200
400
600
800
Output power(kW)
1000
1200
1400
(a) Performance of ELM on the Training Data
fo sn oit tac ep x E
fo sn oit tac ep x E
12
ts oc 11 tn e 10 m hs in up 9
8 0
Actual value Predictive value
10
20
12
ts oc 11 tn e 10 m hs in up 9
8 0
30 40 The sample of DER1
50
60
Actual value Predictive value
10
20
30 40 The sample of DER2
50
60
(b)Performance of the ELM on Test Data
Figure 6. The Actual and Estimated Wind Power Costs test data the RMSE is 1.25 × 10−3 . Also, we can see that there exists a nonlinear relationship between the wind power cost and wind turbine output. When the outputs of wind turbines are 834.2kW and 715.3kW, their costs reach the minimum values. Table 6 shows the training error, testing error and the training time of the multilayer perceptron (MLP), support vector machine (SVM), least squares support vector machines (LS-SVM), relevance vector regression (RVM) [20]-[17], and ELM. The training errors and testing errors of MLP, SVM, LS-SVM, RVM, and ELM are relatively small. However, the training time of MLP, SVM, LS-SVM and RVM
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H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG
DER7
DER9
DER7
DER9
DER7
DER6
DER2
DER2
DER2 DER1
DER1 DER3
DER3
DER1 DER3
DER5
DER5 DER8
DER9
DER6
DER6
DER8
DER4
(a) Open-loop Connected
DER5 DER8
DER4
(b) Closed-loop Connected
DER4
(c) Irregularly Connected
Figure 7. The Topologies of the Communication Network for DERs is respectively 50, 68, 42 and 43 times higher than ELM, which can greatly affect the dispatch performance. Therefore, in practice, ELM is a preferred method for estimating the relationship between wind power outputs and penalty costs, since it allows faster and better control of the VPP. Table 6. Training Error, Testing Error, and Training Time of Different Algorithms Learning Algorithms Training RMSE Testing RMSE Training time(s)
MLP
SVM
LSSVM
9.63 × 10−6
9.29 × 9.30 × 10.41× 9.32 × 10−6 10−6 10−6 10−6
1.62 × 10−3
1.21 × 1.28 × 1.29 × 1.25 × 10−3 10−3 10−3 10−3
1.211
1.632
1.012
RVM
1.072
ELM
0.024
5.4. The results of distributed optimal dispatch. We firstly employ the centralized approach (the interior point method in this paper) to solve the formulated optimal dispatch model, i.e. the dispatch centre of VPP will collect the status information of each DER; it then solves the optimal dispatch model using the interior point method, so that its results can be compared with the proposed distributed dispatch method. By using the interior point method, the outputs of wind turbines at nodes 9 and 13 are calculated as 967.90kW and 824.34kW; the outputs of microgas turbines at nodes 4, 6, and 8 are 626.3kW, 733.3kW and 684kW; the outputs of ES devices at nodes 10 and 13 are 855.2kW and 800kW; the interrupted load levels at nodes 5 and 12 are 583.3kW and 613.6kW. The maximum economic benefit of the VPP can be obtained as 209.640$. In this study, we assume that the communication network can possibly have three topologies, which are respectively open-loop connected, closed-loop connected and irregularly connected networks, as shown in Fig. 7. To analyze the convergence property of the proposed distributed algorithm, the convergence curves of all DER outputs under the three different network topologies are illustrated in Fig. 8.
DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT
1315
1000 950
) 900 W (kr 850 e 800 wo P 750 s RE 700 D 650
Pw9
Pr10 Pg6 Pg4
Pr13
Pw13
Pg8 Pc5
Pc12
600 550
0
10
20
30
40
50
60
70
80
90 100 110 120
Iteration Number
(a) The Convergence Curves of DER Outputs Under the Open-loop Connected Network
1000 950
) 900 W kr( 850 e 800 wo P 750 s RE 700 D 650
P
P P
P
g6
P g4
P
w9
r10
P
r13
w13
g8
P
P
c12
c5
600 550 0
10
20
30
40
50
60
70
80
90
100 110 120
Iteration Number
(b) The Convergence Curves of DER Outputs Under the Closed-loop Connected Network
1000 950 Pr10
900
) 850 W (kr 800 e w oP750 s700 R E650 D
Pg8
Pg6 Pr13 Pg4
Pw9
Pw13 Pc12
Pc5
600 550
0
10
20
30
40
50
60
70
80
90 100 110 120
Iteration Number
(c) The Convergence Curves of DER Outputs Under the irregularly Connected Network
Figure 8. The Convergence Curves of DER Outputs Under Different Topologies for the 15-node VPP
Fig. 9 further shows the solution error versus the iteration number, where the n
2 o difference of decision variables is defined as ε(k) = max xi (k) − x∗ 2 , the differi P ence of objective functions is defined as i fi (xi (k))−f (x∗ ) (the difference between the sum of the cost for all the DERs in the distributed iteration and the cost of VPP obtained by the centralized interior point algorithm), and x∗ is obtained by
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H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG 10
1
s 10 leb riaa10 V no is ic e10 fD o cen10 er fei D 10
0
-1
Closed-loop Connected Network
-2
-3
Open-loop Connected Network
-4
Irregularly Connected Network 20 40
0
60 80 Decision Number
100
120
10
1
sn10 oti cn Fu10 ev it ecj 10 b fO o ec ne10 re if D 10
0
-1
Closed-loop Connected Network
-2
-3
Open-loop Connected Network
-4
0
(a) The Difference of Decision Variables
20
irregularly Connected Network 40 60 80 100 Decision Number
120
140
160
(b) The Difference of Objective Functions
Figure 9. Solution Error Versus Iteration Number Under Different Topologies for the 15-node VPP
10
10
2
10 se bla 10 ir aV oni 10 si ce fD o 10 cen er 10 ef i D 10
10 sn oti cn10 uF ev tic10 ej b O of10 ec ne re10 fi D 10
1
2
1
0
Closed-loop Connected Network
-1
Open-loop Connected Network
-2
-3
Irregularly Connected Network
-4
0
30
60
90 120 150 180 210 240 270 300 Decision Number
(a) The Difference of Decision Variables
0
-1
-2
Closed-loop Connected Network
Open-loop Connected Network
-3
-4
irregularly Connected Network 0 30 60 90 120 150 180 210 240 270 300 330 360 Decision Number (b) The Difference of Objective Functions
Figure 10. Solution Error Versus Iteration Number Under Different Topologies for the 118-node VPP
using the centralized optimal dispatch with the interior point algorithm. It can be seen that under the three network topologies in Fig. 7, the DERs will communicate with their neighbors for 112, 76 and 63 times, ε(k) < 0.0001; their outputs can then converge to the outputs given by the interior point method. The case study results indicate that, the distributed dispatch method can achieve similar performance to the centralized approach. Moreover, the topology of the communication network does not affect the convergence of the distributed algorithm as long as the network is connected. However, adding more communication lines does help increase the convergence speed. As shown in Figs. 7 and 8, by adding 1 and 8 communication lines, the closed-loop connected and irregularly connected networks can significantly improve the convergence speed of the proposed algorithm. However, this will increase the cost of communication network investment. The proposed method is also tested with a larger VPP system including 118 nodes and 80 DERs. It can be seen from Fig. 10 that under the three network topologies in Fig. 7, the DERs will communicate with their neighbors for 266, 158 and 139 times. Each DER will then reach its optimal decisions; the decision variables and Lagrangian multipliers for all DERs will converge.
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6. Conclusion. This paper proposes a distributed dispatch method for virtual power plant based on ELM transformation. With the proposed method, the VPP can maximize its economic benefit via local decision making and limited communication between distributed energy resources. The main contributions of this paper include: 1) employing ELM to estimate the nonlinear relationship between the wind power costs and wind turbine outputs, and transform the estimated relationship into a set of equality constraints in the optimal dispatch model, so as to ensure the model is computable; 2) proposing a distributed primal-dual sub-gradient algorithm to solve the VPP optimal dispatch problem. The proposed algorithm can determine the optimal outputs of all DERs in the VPP via only limited communication between neigh-boring DERs. REFERENCES [1] N. Br, An Introduction to Copulas, New York: Springer, 2005, 7–48. [2] D. S. Callaway and I. A. Hiskens, Achieving controllability of electric loads, Proceedings of the IEEE , 99 (2011), 184–199. [3] N. Celika, Energy output estimation for small-scale wind power generators using weibullrepresentative wind data, Journal of Wind Engineering and Industrial Aerodynamics, 91 (2003), 693–707. [4] M. Grabisch, The representation of importance and interaction of features by fuzzy measures, Pattern Recognition Letters, 17 (1996), 567–575. [5] J. Hetzer, D. C. Yu and K. Bhattarai, An economic dispatch model incorporating wind power, IEEE Transactions on Energy Conversion, 23 (2008), 603–611. [6] M. Hofert, Sampling archimedean copulas, Computational Statistics and Data Analysis, vol. 52 (2008), 5163–5174. [7] G. B. Huang, Q. Y. Zhu and K. Z. Mao, et al., Can threshold networks be trained directly?, IEEE Transactions on Circuits and Systems II: Express Briefs, 53 (2006), 187–191. [8] G. B. Huang, Q. Y. Zhu and C. K. Siew, Extreme learning machine: Theory and applications, Neurocomputing, 70 (2006), 489–501. [9] N. S. Jens, Wind Energy Systems: Optimising Design and Construction for Safe and Reliable Operation, Woodhead Publishing, 2001. [10] I. Kuzle, M. Zdrilic and H. Pandzic, Virtual power plant dispatch optimization using linear programming, in 10th International Conference on Environment and Electrical Engineering (EEEIC), (2011), 1–4. [11] E. Mashhour and S. M. Moghaddas-Tafreshi, Bidding strategy of virtual power plant for participating in energy and spinning reserve markets-part I: problem formulation, IEEE Transactions on Power Systems, 26 (2011), 949–956. [12] E. Mashhour and S. M. Moghaddas-Tafreshi, Bidding strategy of virtual power plant for participating in energy and spinning reserve markets-part II: numerical analysis, IEEE Transactions on Power Systems, 26 (2011), 957–964. [13] Z. Minghui and S. Martinez, On distributed convex optimization under inequality and equality constraints, IEEE Transactions on Automatic Control, 57 (2012), 151–164. [14] G. Papaefthymiou and D. Kurowicka, Using copulas for modeling stochastic dependence in power system uncertainty analysis, IEEE Transactions on Power Systems, 24 (2009), 40–49. [15] D. Pudjianto, C. Ramsay and G. Strbac, Virtual power plant and system integration of distributed energy resources, IET Renewable Power Generation, 1 (2007), 10–16. [16] B. C. Ummels, M. Gibescu and E. Pelgrum, et al., Impacts of wind power on thermal generation unit commitment and dispatch, IEEE Transactions on Energy Conversion, 22 (2007), 44–51. [17] C. M. Vong, P. K. Wong and L. M. Tam, et al., Ignition pattern analysis for automotive engine trouble diagnosis using wavelet packet transform and support vector machines, Chinese Journal of Mechanical Engineering, 24 (2011), 870–878. [18] X. Z. Wang, A. X. Chen and H. M. Feng, Upper integral network with extreme learning mechanism, Neurocomputing, 74 (2011), 2520–2525. [19] Z. Y. Wang, K. S. Leung and J. K. George, Applying fuzzy measures and nonlinear integrals in data mining, Fuzzy Sets and Systems, 156 (2005), 371–380.
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[20] P. K. Wong, C. M. Vong and L. M. Tam, et al., Data preprocessing and modelling of electronically-controlled automotive engine power performance using kernel principal components analysis and least squares support vector machines, International Journal of Vehicle Systems Modelling and Testing, 3 (2008), 312–330. [21] P. K. Wong, Q. S. Xu and C. M. Vong, et al., Rate-dependent hysteresis modeling and control of a piezostage using online support vector machine and relevance vector machine, IEEE Transactions on Industrial Electronics, 59 (2012), 1988–2001. [22] T. H. Yeh and W. Li, A study on generator capacity for wind turbines under various tower heights and rated wind speeds using weibull distribution, IEEE Transactions on Energy Conversion, 23 (2008), 592–602. [23] L. Yu and E. O. Voit, Construction of bivariate s-distributions with copulas, Computational Statistics and Data Analysis, 51 (2006), 1822–1839. [24] C. Yuen, A. Oudalov and A. Timbus, The provision of frequency control reserves from multiple microgrids, IEEE Transactions on Industrial Electronics, 58 (2011), 173–183. [25] J. H. Zhai, Fuzzy decision tree based on fuzzy-rough technique, Soft Computing, 15 (2011), 1087–1096. [26] J. H. Zhao, F. S. Wen and Z. Y. Dong, et al., Optimal dispatch of electric vehicles and wind power using enhanced particle swarm optimization, IEEE Transactions on Industrial Informatics, 8 889–899.
Received August 2013; 1st revision October 2013; 2nd revision January 2014. E-mail E-mail E-mail E-mail E-mail
address: address: address: address: address:
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