Distributed Optimal Dispatch of Virtual Power Plant Based on Elm Transformation

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JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2014.10.1297 MANAGEMENT OPTIMIZATION Volume 10, Number 4, October 2014 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).

1297 1298 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

1. Introduction. Nowadays, the human society is experiencing a number of se- vere challenges, such as global warming, energy security and environmental pollu- tion. 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 in- termittent 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 intermit- tency 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. Ref- erence [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 analyti- cal 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 de- scribe 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 1299

(a) Centralized Dispatch (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 cyber- attackers. 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. 1300 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

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]:

κw,n−1 κw,n κw,n vw,n vw,n φwn (vw,n) = exp − , (1) 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 (0 ≤ vw,n < vin,n, vw,n > vout,n) Pwn,s (vw,n) = aw,nvw,n + bw,n (vin,n ≤ vw,n < vrate,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, DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1301

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: max max Pw,n pw,n vin,n aw,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 vin,n R +∞ φw,n (vw,n) dvw,n + φw,n (vw,n) dvw,n,Pwn,s = 0  0 vout,n . R vw,n(Pwn) R +∞ max φw,n (vw,n) dvw,n + φw,n (vw,n) dvw,n, 0 < Pwn,s < P 0 vout,n w,n  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 1 o ς ς ς < (Pw1,s, ··· ,PwN,s) = exp −[(−InF1 (Pw1,s)) + ··· + (−InFN (PwN,s)) ] , (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. There- fore, the objective can be formulated as: N X ρdPdn − Cgn (Pgn) − Crn (Prn) max f = − ρEPS, (6) Pwn,Pgn,Prn,Pcn,PS −Ccn (Pcn) − E [Cwn (Pwn)] n=1 1302 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG 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,2Pgn + cgn,1Pgn, (n = 1, 2, ··· ,N), (7)

2 Ccn(Pcn) = ccn,2Pcn + ccn,1Pcn, (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,2Pcn + 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) = cwnPwn + 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:

cun (Pwn,s − Pwn) Pwn,s > Pwn CP,wn (Pwn) = , (n = 1, 2, ··· ,N), (11) con (Pwn − Pwn,s) Pwn,s < Pwn

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. DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1303

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 max P R Pw,1 E CP,wn (Pwn) = 0 n=1  1  N ς N  P ς  , ∂ exp − (−InF1(Pwn,s)) max N  n=1  R Pw,N P ··· CP, (Pwn) dPw1,s ··· dPwN,s 0 wn ∂Pw1,s···∂PwN,s n=1 (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: N X PS = (Pdn − Pwn − Pgn − Prn − Pcn). (13) n=1 2) Line transmission limits: N X − Tl ≤ 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 , (n = 1, 2, ··· ,N), (15) min max Pgn ≤ Pgn ≤ Pgn , (n = 1, 2, ··· ,N), (16) max 0 ≤ Pcn ≤ Pcn , (n = 1, 2, ··· ,N), (17) dch,max Prn ≤ Prn ,Prn ≥ 0 ch,max , (n = 1, 2, ··· ,N), (18) −Prn ≤ Prn ,Prn ≤ 0 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 [c P + E (C (P )) − ρ P ] + X  wn wn P,wn wn E wn  min f = [Cgn (Pgn) − ρEPgn] + [Crn (Prn) − ρEPrn] . (19) Pwn,Pgn,Prn,Pcn n=1  + [Ccn (Pcn) − ρEPrn]  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 1304 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

ω 11 β H1 Pw1 Ew1

• • • • • • P wN EwN • • 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 success- fully 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 de- scent 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 = βjnG ωijPwi + 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 DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1305 in the hidden layer, and we select the Sigmoid function in this paper: 1 G(x) = . (21) 1 + exp(−x) The procedure of employing ELM to estimate the relationship between wind power costs and wind turbine outputs can be summarized as follows. k k k k T 1) Randomly select the kth set of wind power outputs Pw = 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 m set). Calculate corresponding wind power costs CP,w1,CP,w2, ··· ,CP,wN (m = 1, 2, ··· ,M) based on Eq.(11), and then obtain the expected wind power k k k k T 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 Ek = Cm (n = 1, 2, ··· ,N) . (22) wn M P,wn m=1

k k 3) Repeat steps 1) and 2), obtain W sets of Pw 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:

Υ0β = E0, (23) where  N N  P 1 P 1 G ωn1P + θ1 ··· G ωnH P + θH  wn wn   n=1 n=1  Υ0 =  ·········  , (24)  N N   P W P W  G ωn1Pwn + θ1 ··· G ωnH Pwn + θH n=1 n=1 W ×H   β11 β12 ··· β1N  β21 β22 ··· β2N  β =   , (25)  . . .. .   . . . .  β β ··· β H1 H2 HN 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 βˆ can be calculated analytically as: ˆ β = pinv(Υ0) · E0, (27) where T −1 T pinv(Υ0) = Υ0 Υ0 Υ0 . (28) 1306 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

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 = βjnG ωijPwi + 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:

N P min f = {{[cwnPwn + Ewn − ρEPwn] + [Cgn (Pgn) − ρEPgn] Pwn,Pgn,Prn,Pcn,Ewn n=1 + [Crn (Prn) − ρEPrn] + [Ccn (Pcn) − ρEPrn]} N P s.t. − Tl ≤ ηln (Pwn + Pgn + Prn + Pcn − Pdn) ≤ Tl (l = 1, ··· ,L) n=1 H N . P P Ewn − βjnG ωijPwi + θj = 0 (n = 1, 2, ··· ,N) j=1 1 max min max 0 ≤ Pwn ≤ Pwn ,Pgn ≤ Pgn ≤ Pgn dch,max max Prn ≤ Prn ,Prn ≥ 0 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 diﬃcul- ties 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) 4N×4N be a graph with a nonnegative adjacency matrix A = {aij} ∈ R 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}. DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1307

To simplify the discussion, we deﬁne:

x = (Pw1, ··· ,PwN ,Pg1, ··· ,PgN ,Pr1, ··· ,PrN ,Pc1, ··· ,PcN ,Ew1, ··· ,EwN ) , (31) N N N Y max Y min max Y ch,max dch,max X = [0,Pwn ] × Pgn ,Pwn × −Prn ,Prn n=1 n=1 n=1 N Y max N × [0,Pcn ] × R , (32) n=1

fwn(x) = cwnxn + x4N+n − ρExn, (n = 1, 2, ··· ,N), (33)

fgn(x) = Cgn (xN+n) − ρExN+n, (n = 1, 2, ··· ,N), (34)

frn(x) = Crn (x2N+n) − ρEx2N+n, (n = 1, 2, ··· ,N), (35)

fcn(x) = Ccn (x3N+n) − ρEx3N+n, (n = 1, 2, ··· ,N), (36)  N P xwn + xN+n + x2N+n  ηjn , (j = 1, 2, ··· ,L),  n=1 +x3N+n − Pdn gj(x) = N ,  P xwn + xN+n + x2N+n  − ηjn − Tj, (j = L + 1, ··· , 2L) n=1 +x3N+n − Pdn (37) H N ! X X hn(x) = x4N+n − βjnG ωijxi + θj , (n = 1, 2, ··· ,N). (38) j=1 i=1 The dual problem of the VPP dispatch problem (30) can be obtained as [13]: N N N N P P P P max min Ξ = Ξwn + Ξgn + Ξrn + Ξcn µj ,γj x n=1 n=1 n=1 n=1 s.t. x ∈ X , (39) µj ≥ 0(j = 1, 2, ··· , 2L) λj ≥ 0(j = 1, 2, ··· ,N) 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: 2L N X + X Ξwn = fwn(x) + µj[gj(x)] + λj |hj(x)|, (40) j=1 j=1

2L N X + X Ξgn = fgn(x) + µj[gj(x)] + λj |hj(x)|, (41) j=1 j=1 2L N X + X Ξrn = frn(x) + µj[gj(x)] + λj |hj(x)|, (42) j=1 j=1 2L N X + X Ξcn = fcn(x) + µj[gj(x)] + λj |hj(x)|, (43) j=1 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 i i 2L i N of each DER needs to be initialized, i.e. set x (0) ∈ X, µ (0) ∈ R≥0, λ (0) ∈ R≥0, 1308 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG 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:

i X j υx (t) = aijx (t), (44)

j∈Λi

i X j υµ (t) = aijµ (t), (45)

j∈Λi

i X j υλ (t) = aijλ (t), (46)

j∈Λi

i i i x (t + 1) = ψX υx (t) − α (t) Sx(t) , (47)

i i η · Pinput − Tmax µ (t + 1) = υµ(t) + α(t) · , (48) −η · Pinput − Tmax

T xi , xi , ··· , xi λi (t + 1) = vi (t) + α (t) 4N+1 4N+2 4N+N , (49) λ i i i T −βG ω · x1, x2, ··· , xN + θ where T Tmax = (T1,T2, ··· ,TL) , (50)

  Pw1 + Pg1 + Pr1 + Pc1 − Pd1  Pw2 + Pg2 + Pr2 + Pc2 − Pd2  P =   , (51) input  .   .  PwN + PgN + PrN + PcN − PdN

  η11 η12 ··· η1N  η21 η22 ··· η2N  η =   , (52)  . . .. .   . . . .  ηL1 ηL2 ··· ηLN

  ω11 ω12 ··· ω1N  ω21 ω22 ··· ω2N  ω =   , (53)  . . .. .   . . . .  ωH1 ωH2 ··· ωHN

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; i i Sx(t) is sub-gradient of the dual function υx(t) of DER i with respect to its own DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1309 decision variables:

 2L + N  ∇f (υi (t)) + P υi (t)∇g (υi (t)) + P υi (t)∇ |h | (υi (t))  wn x µ j x λ j x  j=1 j=1   i = 1, 2, ··· ,N,  2L N  i P i i + P i i  ∇fgn(υx(t)) + υµ(t)∇ gj(υx(t)) + υλ(t)∇ |hj| (υx(t))  j=1 j=1  i  i = N + 1,N + 2, ··· , 2N, Sx(t) = , 2L + N  ∇f (υi (t)) + P υi (t)∇g (υi (t)) + P υi (t)∇ |h | (υi (t))  rn x µ j x λ j x  j=1 j=1  i = 2N + 1, 2N + 2, ··· , 3N,   2L N  i P i i + P i i  ∇fcn(υ (t)) + υ (t)∇ gj(υ (t)) + υ (t)∇ |hj| (υ (t))  x µ x λ x  j=1 j=1  i = 3N + 1, 3N + 2, ··· , 4N, (56) where ∇f(·) is the sub-gradient of function f. i i In this algorithm, at iteration t, each DER i calculates the estimated υx, υµ, i i i i j j υλ based on its own decision x , µ , λ and neighboring DERs’ decisions x , µ , j λ (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 2 P P 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 i ∗ lim x (t) − x = 0, ∀i = 1, 2, ··· , 4N. (57) 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 Table1. The cost coeﬃcients and output limits of micro-gas turbines, ES devices and interruptible loads are shown in Table2. The load levels 1310 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

Gas Interruptible GG communication Generator P Load S Wind 1 WT Load Turbine 2 line1 Electrochemical Storage ES line 2 line 4 line 12 3 5 13 line 8 line 3 line 5 9 WT line 13 ES 4 line 10 6 WT 14 11 line14 GG GG line 6 line 9 10 7 line 11 15 12 line 7 ES 8

Figure 3. The Structure of the Virtual Power Plant of all nodes in the VPP are listed in Table3. The parameters of distribution lines are given in Table4.

Table 1. Parameters of Wind Turbines

Penalty Reserve Cut Cut Turbine Cost Rated Rated Cost Cost in out (DER Coefficient Power Speed Coefficient Coefficient Speed Speed No.) ($/kWh) (kW) (m/s) ($/kWh) ($/kWh) (m/s) (m/s) 1 0.0021 0.013 0.021 1400 5 45 15 2 0.0023 0.013 0.021 1400 5 45 15

Table 2. The Cost Coeﬃcients and Output Limits of DERs 3 9

First-order Second-order Lower Upper DER No. Cost Cost Output Output Coeﬃcient Coeﬃcient Limit(kW) Limit(kW) 3 7.2 × 10−6 0.031 200 800 4 7.5 × 10−6 0.029 250 9000 5 7.3 × 10−6 0.030 150 800 6 7.6 × 10−6 0.027 -250 1000 7 7.5 × 10−6 0.028 -200 1000 8 2.4 × 10−5 0.012 0 750 9 2.2 × 10−5 0.013 0 800

Table 3. Load Data

Node 1 2 3 4 5 6 7 8 Load(kW) 0 200 300 300 450 350 350 350 Node 9 10 11 12 13 14 15 Load(kW) 250 200 200 600 250 250 180 DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1311

Table 4. Distribution Line Parameters

Reactance Power Flow Reactance Power Flow Line Line (Ω ) Limit (kW) (Ω ) Limit (kW) 1 91 2300 8 96 1200 2 96 350 9 96 750 3 96 300 10 96 700 4 147 1300 11 103 700 5 96 500 12 96 600 6 39 600 13 128 400 7 96 400 14 13 320

n 3 n 3 n 3 tio io io u t t b u u ri ib ib t 2 tr 2 tr 2 is is is D D D l- l- l- a a a ic1 ic1 ic1 ir ir ir p p p m m m E0 E0 E0 0 1 2 3 0 1 2 3 0 1 2 3 Gumbel-Copula Frank-Copula Normal-Copula n 3 n 3 n 3 tio tio tio u u u ib ib ib tr 2 tr 2 tr 2 is is is D D D l- l- l- a a a ic1 ic1 ic1 ir ir ir p p p m m m E0 E0 E0 0 1 2 3 0 1 2 3 0 1 2 3 Clayton-Copula t-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 Table5. From Fig.4 and Table5, 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 Gumbel- Copula joint distribution. Randomly select 300 sets of power outputs (Pw9,Pw13) 1312 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

Table 5. Copula Parameters Estimation and Goodness Test

Copula Upper tail Lower tail Connection Goodness-of-fit Function correlation correlation parameter value Type coefficient coefficient Gumbel 8.02 0.8821 0.9098 0.0 Clayton 0.67 0.5318 0.0 0.3560 Frank 15.93 0.6127 0.0 0.0 Normal 0.73 0.5954 0.0 0.0 t 0.76 0.7329 0.3520 0.3520

-6 -3 x 10 x 10 9.9 1.8

9.8 1.7 E E S9.7 S1.6 M M R R g9.6 g1.5 in in in st ra9.5 e1.4 T T 9.4 1.3

9.3 1.2 2 4 6 8 10 2 4 6 8 10 The number of hidden layer nodes The number of hidden layer nodes

(a) Training RMSE (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

) $15 The samples of DG5 t( n The prediction of DG5 e14 m The samples of DG13 h is The prediction of DG13 n13 u p f o12 t s o c11 e h T10

9 0 200 400 600 800 1000 1200 1400 Output power(kW)

(a) Performance of ELM on the Training Data

12 f t Actual value o s o11 Predictive value s t c n n tio e a m10 ct h e is p n x u 9 E p 8 0 10 20 30 40 50 60 The sample of DER1 12 f st Actual value o o11 s t c Predictive value n n tio e a m10 ct h e is p n x u 9 E p 8 0 10 20 30 40 50 60 The sample of DER2 (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. Table6 shows the training error, testing error and the training time of the multi- layer 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 1314 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

DER9 DER9 DER9 DER7 DER7 DER7

DER6 DER6 DER6

DER2 DER2 DER2

DER1 DER1 DER1

DER3 DER3 DER3

DER5 DER5 DER5

DER8 DER8 DER8 DER4 DER4 DER4

(a) Open-loop Connected (b) Closed-loop Connected (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 MLP SVM LS- RVM ELM Algorithms SVM Training 9.63 × 9.29 × 9.30 × 10.41× 9.32 × RMSE 10−6 10−6 10−6 10−6 10−6 Testing 1.62 × 1.21 × 1.28 × 1.29 × 1.25 × RMSE 10−3 10−3 10−3 10−3 10−3 Training 1.211 1.632 1.012 1.072 0.024 time(s)

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 micro- gas 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 P 900 P w9 ) r10 W k 850 r( e w 800 o P P P P 750 g6 r13 w13 s R P P E 700 g4 g8 D P P 650 c12 c5 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 P 900 P w9 ) r10 W k 850 r( e w 800 o P P P P 750 g6 r13 w13 s P R P g8 E 700 g4 D P P 650 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 P 900 P w9 r10 ) W850 k r( e800 w P P P o750 g6 r13 w13 P P s700 g8 R E650 P P P D g4 c12 c5 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 Dif- ferent Topologies for the 15-node VPP

Fig.9 further shows the solution error versus the iteration number, where the n 2o difference of decision variables is defined as ε(k) = max xi(k) − x∗ , the differ- i 2 P i ∗ ence of objective functions is defined as i fi(x (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 1316 H. YANG, D. YI, J. ZHAO, F. LUO AND Z. DONG

1 1 10 10

0 0 10 s 10 s n le o b ti a c ri -1 n -1 a10 u10 V F Closed-loop e Closed-loop n v io Connected ti Connected s c i -2 Network e -2 Network c j 10 e10 b Open-loop D Open-loop O f Connected f Connected o o e Network e Network c -3 c -3 n10 n10 re re fe fe if if D -4 D -4 10 10 Irregularly irregularly Connected Connected Network Network 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 Decision Number Decision Number

(a) The Difference of Decision Variables (b) The Difference of Objective Functions

Figure 9. Solution Error Versus Iteration Number Under Diﬀer- ent Topologies for the 15-node VPP

2 2 10 10

1 1 10 10 s s le n b 0 io a t 0 ri 10 Closed-loop Connected Network c10 a n V u F Closed-loop n -1 e io10 iv -1 Connected is t 10 c c Network e je Open-loop Open-loop D -2 b f 10 O -2 o Connected f 10 Connected e o c Network Network n ce e -3 n r 10 -3 fe Irregularly re10 if fe D Connected if -4 Network D 10 -4 10 irregularly Connected Network 0 30 60 90 120 150 180 210 240 270 300 0 30 60 90 120 150 180 210 240 270 300 330 360 Decision Number Decision Number

(a) The Difference of Decision Variables (b) The Difference of Objective Functions

Figure 10. Solution Error Versus Iteration Number Under Diﬀer- ent 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 and8, 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. DISTRIBUTED OPTIMAL DISPATCH OF VIRTUAL POWER PLANT 1317

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.

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