Power Generation and Distribution via Distributed Coordination Control

Transcription

1 Power Generation an Distribution via Distribute Coorination Control Byeong-Yeon Kim, Kwang-Kyo Oh, an Hyo-Sung Ahn arxiv: v [math.oc] 8 Jul 204 Abstract This paper presents power coorination, power generation, an power flow control schemes for supply-eman balance in istribute gri networks. Consensus schemes using only local information are employe to generate power coorination, power generation an power flow control signals. For the supply-eman balance, it is require to etermine the amount of power neee at each istribute power noe. Also ue to the ifferent power generation capacities of each power noe, coorination of power flows among istribute power resources is essentially require. Thus, this paper proposes a ecentralize power coorination scheme, a power generation, an a power flow control metho consiering these constraints base on istribute consensus algorithms. Through numerical simulations, the effectiveness of the propose approaches is illustrate. I. INTRODUCTION In recent years, a smart gri has attracte a tremenous amount of research interest ue to its potential benefit to moern civilization. The remarkable evelopment of computer an communication technology has enable a realization of smart power gri. A key feature of a smart power gri is the change of the power istribution characteristics from a centralize power system to a istribute power system. Though centralize power plants still cover the major portion of power eman, the amount of power eman covere by istribute power resources has been increasing steaily []. In istribute power systems, achieving the supply-eman balance, which is one of the funamental requirements, is a key challenging issue ue to its ecentralize characteristics. The School of Mechatronics, Gwangju Institute of Science an Technology, Gwangju, South Korea. hyosung@gist.ac.kr

2 2 supply-eman balance problem has been traitionally consiere as the economic ispatch [2], [3] which minimizes the total cost of operation of generation systems. However, the traitional ispatch problem is a highly nonlinear optimization problem an thus it has been aresse usually in a centralize manner. Distribute control of istribute power systems has been consiere in [4] [6]. In the istribute power system, iniviual power resources are interconnecte with each other through communication an transmission line. Thus, for the coorination of power generation an power flow, sharing information is essential. It is more realistic an economic to use only local information. This kin of problem has been solve in consensus [7] [0]. Distribute resource allocation metho name center-free algorithm has been consiere in []. In the algorithm, since the amount of resource for each noe is etermine by the sum of weighte ifference with its neighbors, we can easily see the equivalence of the algorithm to the consensus. Thus, we can consier that the iea of consensus ha been alreay use in the istribute resource allocation problems. In this paper, power istribution among istribute power resources is stuie. The problem is classifie into three subproblems. In the first problem, which is calle power coorination, the esire net power is etermine. In the secon problem an the thir problem, we consier power generation an power flow control of istribute power resources with an without power coorination respectively. Note that the power flow control without power coorination is significantly important when the net power an the generation capacities of each power resource are limite. Consensus schemes are employe to generate power coorination, power generation an power flow control signals. Subsequently, the main contribution of this paper can be summarize as follows. First, the coorination an control of power generation an power flow are precisely efine an formulate. Secon, two power istribution schemes with an without power coorination are evelope, which may be helpful in coorinating actual power generation an power flow among istribute power plants. The propose approach eals with the esire net power which is not necessarily an average value as in the istribute averaging problem [2]. Thir, it is shown that the istribute The net power at a noe is the sum of generate power at the noe an power flows into it from its neighbor noes. Detaile explanation on net power can be foun in the secon paragraph of the problem formulation section Section 2.

3 Fig. : Interconnecte power systems consensus algorithms ensure power istribution among istribute noes that have net power an generation capacity constraints. The propose approach can be applie even if the esire net power of some power resources oes not satisfy the power generation capacity as oppose to those in [6], [3]. This paper is organize as follows. In Section II, power coorination, power generation an power flow control problems are formulate. Main results of this paper are presente in Section III. Illustrative examples are provie in Section IV an the conclusion is given in Section V, respectively. II. PROBLEM FORMULATION Fig. shows a graph representing interconnecte power systems. Each noe enotes a subsystem, i.e., an area that consists of generators an loas, an it has its power eman etermine by its loas. Thus, the esire net power for each generator in the subsystem is require to be etermine. The power can be transmitte to its neighboring subsystems since they are interconnecte with each other through transmission lines. In this paper, we woul like to control the power generation an power flow to satisfy supply-eman balance uner net power an generation capacity constraints. Let p i be the esire net power at the i-th noe; p Gi be the generate power at the i-th noe; an p Fij be the power flow from the i-th noe to the j-th noe, where p Fij = p Fij an i, j =,, n. The net power at a noe is etermine after the net power exchange of generate powers with its neighbor noes. Thus, the net power at the i-th noe can be efine as p i p Gi + j N i p Fji, where N i is the set of neighbor noes of the i-th noe. The total generate power an total power eman within the power gri are represente by p G = n p Gi an p D, respectively.

4 4 Throughout this paper, we nee the following assumption to represent a reality of power gri network systems. Assumption 2.: Each noe has the net power capacity constraint such as p i p i k p i where k inicates the sampling instant of the physical power layer. The net power capacity is associate with the amount of power that each noe can hanle. Thus, the esire net power for each noe shall be limite by the net power capacity constraint. 2 Each noe has the generation capacity constraint such as p Gi p Gi k p Gi 2 The generation capacity is also assume to be boune by the net power capacity constraint ] i.e., [p Gi, p Gi [ ] p i, p i. 3 The graph which represents interconnections among noes is unirecte an connecte. This assumption comes from the fact that the power an information flows are biirectional. However, this assumption can be relaxe to a irecte graph. Note that ue to the constraints an 2 in the Assumption 2., the power coorination an power flow control problems become non-trivial. Fig. 2 shows a power exchange between pairs of noes. As shown in Fig. 2, the generate power an the net power at each noe can be represente as follows: p Gi k = p Gi k + p Gi k 3 p i k = p Gi k + p Fji k, i =, 2,..., n 4 j N i p i k = p Gi k 5 where p Gi k = p Gi k p Gi k is its own variation in power generation an p Fji is the interaction term representing the power exchange between two noes. To make the paper clear, the following problems are formally state. Problem 2.: Power coorination For a given total power eman p D k within the network, etermine the esire net power p i k of iniviual noe that satisfies the supply-eman balance

5 pg i pi p j p G j p Fji 5 p Fji pg i pi p j p G j p Fji Fig. 2: Power exchange between pairs of noes such as p F p F2 p p2 p G k = p D pk F2 = p F22 p i k 6 uner the constraints of Assumption 2.. Problem 2.2: Power generation an power flow control with power coorination For the esire net power given by the power coorination, esign power generation an power flow control strategies, i.e, p Gi k an p Fij k, to satisfy p i k p i k uner the constraints of Assumption 2.. The Problem 2.2 is consiere in a subsystem i.e., an area which consists of generators an loas. Thus, it is assume that the total power eman is given as usually assume in traitional power ispatch problems. As shown in below, only one noe is require to know the total power eman. Thus, the esire net power for each noe is etermine by power coorination, an power generation is controlle to meet the supply-eman balance. This scenario is formulate uner the name of With power coorination. Problem 2.3: Power generation an power flow control without power coorination For the esire net power which is iniviually given without power coorination, esign power generation an power flow control strategies, i.e, p Gi k an p Fij k, to satisfy p i k p i k uner the constraints of Assumption 2.. The Problem 2.3 is consiere in interconnecte subsystems uner the assumption that each area has only one collective generator an one collective loa. Thus, it is assume that the esire net power is iniviually given to each noe. Thus, power coorination is not necessary, but the power generation an power flow are controlle to meet the supply-eman balance an given-iniviual esire net power for each noe simultaneously. This scenario is formulate uner the name of Without power coorination. Remark 2.: Power system transmission lines have a very high X/R ratio. Thus, the real power changes are primarily epenent on phase angle ifferences among generators while they

6 6 are relatively not affecte by voltages. Further, the phase angle ynamics is generally much faster than the voltage ynamics an thus it is usually assume that the phase angle ynamics an the voltage ynamics are ecouple [4]. In our problem setup, since we eal with the real power, one can see that the phase angle is implicitly consiere. Also, in this paper, we mainly concentrate on power istribution, transmission, an coorination issues in operation level. Thus, the lower level signal characteristics are not consiere. Also, it is assume that power flow information can be calculate from the sequence of voltage an current phasors obtaine by phasor measurement units PMUs [5]. Remark 2.2: The problem formulate by 3, 4, an 5 can cover various attribute istribution problems such as gas, water an oil operation, an supply chain management, traffic control, an renewable energy allocation. This paper just focuses on power istribution problem, which has been recently more researche [4] [6]. III. MAIN RESULTS A. Power coorination In the power coorination, a esire net power is require to be realizable [6], i.e., physical constraints such as generation capacity an power flow constraint are require to be satisfie. If the esire net power oes not satisfy these constraints, we cannot achieve the control goal with any control input since the esire net power is not realizable. From 2 an 6, the following conition for a realization i.e., to make the esire net power realizable can be obtaine: p G p D k p G 7 Let us consier the power coorination issue more systematically. If the esire net power is both upper- an lower-boune, then we can make the following rule for the power coorination. p Gi Lemma 3.: If the esire net power is lower-boune an upper-boune, i.e., p Gi p i k an 7 is satisfie, then the following esire net power p i k = p Gi + p G i p Gi p G p G pd k p G provies a sufficient conition for the supply-eman balance 6. 8

7 7 Proof: If p i k is given by 8, then, uner the assumption 7, we have p Gi p i k = p Gi p Gi p G i p Gi pd k p G p G p G which satisfies the left-sie inequality p Gi = p G i p Gi p G p G pd k p G 0 9 p i k. Similarly, if p i k is given by 8, then, uner the assumption 7, we obtain p i k p Gi = p Gi + p G i p Gi pd k p G pgi p G p G pd pg p G p Gi p Gi + p Gi p Gi k p G = p G p G p Gi p Gi p D k p G = 0 0 p G p G Thus, the right-sie inequality p i k p Gi sies of 8 over i from to n yiels p i k = p Gi + is also satisfie. Furthermore, summing up the both p Gi p Gi p G p G pd k p G = p D k Note that the power coorination 8 requires global information such as p G an p G. The coorination scheme can be, however, achieve by istribute consensus algorithm using only local information as in [7]. For that, we nee the following lemma for a further investigation. Lemma 3.2: [8] If A R n n is nonnegative an primitive, then lim m [ ρa A ] m L > 0 2 where L rl T, Ar = ρ A r, A T l = ρ A l, r > 0 an l > 0 in element-wise, an r T l = In fact, r an l are the right an left eigenvectors corresponing to the eigenvalue ρ A. Now, we state one of the main results of this subsection. Theorem 3.: The power coorination 8 is achieve if the following consensus scheme is use: p pgi p Gi i k = p Gi + y i,ss x i,ss 3

8 8 where x i,ss an y i,ss are steay state solutions of the following equations: x i t + = y i t + = + N i x i t + + N j x j t 4 j N i + N i y i t + + N j y j t 5 j N i with the following initial values p D k p Gi x i 0 = p Gi if i is the leaing noe otherwise 6 y i 0 = p Gi p Gi 7 respectively, where x i t, y i t R, an N i is the egree of the i-th noe an the inex t represents the sampling instant at the consensus algorithm. The consensus algorithm 3-7 is completely ecouple from the physical power layer. Proof: Without loss of generality, it is assume that the first noe is the leaing noe which knows the total power eman p D k. Then, the consensus scheme 4 an 6 can be represente by where xt = [x t,..., x n t] T x t + = Qx t 8 [ ] T x 0 = p D k p G, p G2, p G3,..., p Gn 9 an Q = [q ij ] is nonnegative row stochastic matrix because q ij = / + N j if j N i {i} an 0 otherwise. The matrix Q is irreucible because the associate graph for Q is unirecte an connecte. Also, Q is primitive because it is irreucible an has exactly one eigenvalue of maximum moulus λ = ρ Q = see [8] accoring to Perron-Frobenius theorem [8]. Thus, accoring to Lemma 3.2, the vector x converges to its steay state solution x ss = lim m Qm x 0 if there exists a limit of lim m Qm. Accoring to Lemma 3.2, this limit exists for the primitive matrix Q an the steay state solution is given by x ss = rl T x 0, where Qr = ρ Q r, Q T l = ρ Q l, r > 0, l > 0 in element-wise, an r T l = with l = here, enotes a vector with ones as its element an r = [r, r 2,..., r n ] T satisfies r T =. Thus, this solution is given by which x ss = rl T x 0 = T x 0 r = p D k p G r 20

9 9 In the similar manner, the steay state solution for is given by y t + = Qy t 2 [ ] T y 0 = p G p G, p G2 p G2,..., p Gn p Gn 22 y ss = rl T y 0 = T y 0 r = p G p G r 23 Thus, it follows from 20 an 23 that 3 can be represente by p i k = p Gi + p G i p Gi pg pd k p G ri 24 p G ri which is equivalent to 8. = p Gi + p G i p Gi p G p G pd k p G Since the power coorination law is fully istribute, it can be applie even if some power resources are locally ae to or remove from the power gri network uner the realizability assumption 7. Remark 3.: The convergence of the algorithm 4-7 can be ensure if the sampling of the algorithm is much faster than that of the physical layer. Thus, the power coorination scheme of 3-7 is feasible in practice. In more etail, let T k be the time at the k-th sample instant in the physical power layer. Then, the power coorination 3-7 shoul be complete in k T k + T k. Thus, the time interval t T t + T t for iterations 4-7 shoul be chosen to t k so that the steay state solutions of 4-7 can be obtaine in k. 25 B. Power generation an power flow control As mentione in Problem 2.2 an Problem 2.3, we attempt to esign power generation p Gi k an power flow p Fij k in orer to make net power be equal to the esire net power at each noe with an without power coorination, respectively. From 3-4, the net power of each noe can be escribe as follows: p i k = p Gi k + u i k 26 u i k = p Gi k + p Fji k 27 j N i

10 0 where u i is the control input at the i-th noe. It is remarkable that, consiering the generation capacity, the power generation control input shoul satisfy the following constraint p Gi k p Gi k p Gi k 28 where p Gi k = p Gi p Gi k an p Gi k = p Gi p Gi k. Define the net power flow at the i-th noe as follows p Fi k = p Fji k 29 j N i an efine the coorination error at each noe as p e,i k = p i k p i k 30 In the sequel, we provie power generation an power flow control schemes with an without power coorination taking account of two ifferent scenarios mentione in Section II. With power coorination: With the power coorination 3-7, the esire net power for each noe shoul satisfy the generation capacity of each noe. From 30, the coorination error is given by p e,i k = p Gi k + u i k p i k 3 Our goal is to esign u i k such that the coorination error becomes zero. Theorem 3.2: With the power coorination 3-7, if control input is given by u i k = p Gi k = p i k p Gi k 32 then the constraint 28 an p i k p i k can be ensure. Proof: With the power coorination 3-7, p Gi p i k p Gi can be achieve. Thus, we can have p Gi k p i k p Gi k p Gi k 33 In 27, choosing p Fji k = 0 an by the control law of 32, we have u i k = p Gi k. Therefore, by 33, the constraint 28 is satisfie. Furthermore, by inserting 32 into 26, we can achieve p i k p i k. Fig. 3 shows the overall power flow control with power coorination. Without loss of generality, it is assume that noe i is a leaing noe that knows the total eman p D k. Only the

11 p D k Noe i p Gi p Gi Noe j, pg i k ui k pg i k p k Power Coorination N i N j p G j p G j j N i Unit elay p G k p i i k i Fig. 3: Power generation control with power coorination information of generation capacity an the egree of noe i are exchange between neighboring noes. With this information, the esire net power for each noe is etermine by the power coorination. After the power coorination, power generation control input is given by 32. As a consequence of the control input, p i k p i k is achieve. In this case, p i k = p Gi k because there is no power flow between pairs of noes. 2 Without power coorination: Let us assume that the esire net power for each noe is iniviually given without power coorination. But it is suppose that the esire net power satisfies the realizability conition 7 an net power capacity constraints. In this case, the esire net power for some noes may not satisfy the generation capacity of their noe. Thus, it is not sufficient to provie only the power generation control for each noe. Therefore, the power flow control input can be represente as 27, with p F ji k enable in this case. First, we want to esign the power generation control to achieve the overall supply-eman balance.

12 2 Now, we provie a lemma to erive the main result of this paper. Lemma 3.3: If the power generation control input is esigne by the following law p Gi k= p Gi k+ p G i k p Gi k p D k p G k p Gj k p Gj k p Gj k then, the control input will satisfy the constraint 28 an the supply-eman balance is achieve. Proof: For the left-sie inequality p Gi k p Gi k of 28, if p Gi k is given by 34, then we can obtain the following inequality: p Gi k p Gi k = p G i k p Gi k p Gj k p Gj k p D k p G k = p G i k p Gi k p Gj k p Gj k 34 p Gj k 35 p Gj k p Gj k 0 36 For the right-sie inequality p Gi k p Gi k, if p Gi k is given by 34, with the substitution of ξ n [ p Gi k p Gi k][ p Gj k p Gj k] + [ p Gi k p Gi k][p D k p G k n p Gj k], we can obtain the following inequality: p Gi k p Gi k = p Gi k p Gi k + = p Gi k p Gi k p Gj k p Gj k p D k p G k ξ p Gj k p Gj k = p G i k p Gi k p Gj k p Gj k p Gj k pgj k p Gj k 0 37 Furthermore, summing up the both sies of 34 over i from to n yiels p D k = p G k + p Gi k = p G k 38 Thus, the proof is complete.

13 However, as in 8, 34 requires global information such as n p Gi k, 3 p Gi k, an p D k p G k. For a ecentralize power generation control, we now provie the following theorem: Theorem 3.3: The power generation control input 34 can be achieve if the following consensus scheme is use. pgi k p Gi k p Gi k= p Gi k+ z i,ss 39 w i,ss where w i,ss, z i,ss are steay state solutions of the following equations: z i t + = + N i z i t + j N i + N j z j t 40 z i 0 = p i k p G i k p Gi k 4 w i t + = + N i w i t + j N i + N j w j t 42 w i 0 = p Gi k p Gi k 43 Proof: The consensus scheme 40 an 4 can be represente by z t + = Qz t 44 z 0 = [z 0, z 2 0,..., z n 0] T 45 As in the proof of Theorem 3., the steay state solution is given by z ss = rl T z 0 where Qr = ρ Q r, Q T l = ρ Q l, r > 0, l > 0 in element-wise, an r T l = with l = an r = [r, r 2,..., r n ] T which satisfies r T =. Thus, this solution is given by z ss =rl T z0= p D k p G k p Gi k r 46 In the similar manner, the steay state solution for w t + = Qw t 47 w 0 = [w 0, w 2 0,..., w n 0] T 48 is given by w ss =rl T w0= p Gi k p Gi k r 49

14 4 Thus, 39 can be represente by the following equation: pgi k p Gi k p Gi k = p Gi k + = p Gi k + = p Gi k + w i,ss z i,ss p Gi k p Gi k p D k p G k p Gi k p Gi k p Gi k r i p Gi k p Gi k p D k p G k p Gi k p Gi k p Gi k which is equivalent to 34. Then, from 30, the coorination error after the power generation control input is given by r i 50 p e,i k = p e,i k + j N i p Fji k 5 where p e,i k = p Gi k p i k. Now, to make p e,i k 0, we nee to esign power flow control p Fji k, which is summarize in the following theorem. Theorem 3.4: If the power flow control is esigne by p Fji k = h ij,ss 52 where h ij,ss is the steay state solution of the following equations: h ij t + = h ij t + a ij g j t g i t 53 h ij 0 = 0 54 g i t + = g i t + a ij g j t g i t 55 j N i g i 0 = p e,i k 56 where a ij = +max{ N i, N j } is Metropolis-Hasting weight [9], then we can have p e,i k 0. Proof: From 55 an 56, we can obtain g t + = Sg t 57 g 0 = p e k 58

15 5 where g t = [g t, g 2 t,..., g n t] T an S = [s ij ] is oubly stochastic, where +max{ N i, N j if j N } i s ij = +max{ N i, N j if i = j } j N i 0 otherwise 59 an S T = S. As in the proof of Theorem 3., the steay state solution is given by g ss = rl T g 0 where Sr = ρ S r, S T l = ρ S l, r > 0, l > 0 in element-wise, an r T l = with l = an r which satisfies r T =. Furthermore, is also the right eigenvector with the associateeigenvalue because S is oubly stochastic. Thus, without loss of generality, let r =, which n satisfies r T =. Then, the solution of 57 with 58 converges to the average as follows: g ss = rl T g 0 = n T p e k 60 From p e,i k = p Gi k p i k, we can obtain T p e k = 0 because of p G k = p D k. Thus, we have g ss = 0. Hence Also, from 53-56, it follows that g i,ss g i 0 = p e,i k 6 g i,ss = g i 0 + j N i h ij,ss 62 Thus, from 6 an 62, we have h ij,ss = p e,i k 63 j N i If we choose the interaction control input as 52, it follows from 5 an 63 that p e,i k = p e,i k + h ij,ss = 0 64 j N i Fig. 4 shows the power generation an power flow control scheme without power coorination. As previously mentione, it is assume that the esire net power for each noe p i k is given uner 7. The power generation control input is given by As a consequence of the control input, the supply-eman balance is achieve. Next, the power flow between pairs of noes is etermine by Then, after the power flow control, p i k p i k is achieve. There are iterations for the power generation control input computation an for the power flow

16 6 pi k Noe i Unit elay p k pg i k G i p ei, k pi k p Gi p Gi G i p k G i p k Power generation control input computation G i p k N i Power flow control input computation pf ij k jni p Fji k Noe j, j N i p k pg j k G j p G j k N j p e, j k Fig. 4: Power generation an power flow control without power coorination control input computation. Thus, the time interval for iterations an shoul be chosen to t k so that the steay state solutions of the iterations can be obtaine in k. Remark 3.2: The constraints of the Assumption 2. might be time-varying in renewable power resources such as a win or a solar system. It is possible for the propose approach to account for time-varying bouns if the rate of change is not faster than k an the realization conition 7 is satisfie. This can be easily verifie if the time-varying bouns are substitute into the propose approach instea of the constant bouns. IV. ILLUSTRATIVE EXAMPLES In this section, two illustrative examples are provie. The istribute power resources are interconnecte as epicte in Fig.. The generation capacity an net power capacity of each noe are liste in Table I. A. Power istribution with power coorination This example shows the power istribution with power coorination. In this case, without loss of generality, it is assume that noe is the leaing noe that knows the total power

17 7 TABLE I: Genearation capacity an net power capacity at each noe Noei Generation capacityp G i Net power capacityp i [0, 50] [0, 80] 2 [20, 80] [20, 20] 3 [20, 40] [20, 60] 4 [0, 45] [0, 75] 5 [5, 60] [5, 90] 6 [0, 55] [5, 80] eman p D k of the istribute power system. First, the total power eman p D k satisfying the realization conition 7 is ranomly create an it is epicte in Fig. 5 a. Then, p i is etermine by power coorination 3-7 as shown in Fig. 5 b. Next, the power generation control input given by 32 is shown in Fig. 5 c an the generate power i.e., 3 is epicte in Fig. 5. Consequently, the coorination error given by 30 is zero an the supply-eman balance is also achieve as shown in Fig. 5 b an Fig. 5. B. Power istribution without power coorination This example shows power istribution without power coorination. In this case, the esire net power for each noe is not given by power coorination but they are ranomly create uner the realization conition 7 as epicte in Fig. 6 a. First, the power generation control input are given by as shown in Fig. 6 b an the generate power i.e., 3 is shown in Fig. 6 c. After the power generation control, a coorination of power flows between pairs of noes is necessary to make p i converge to p i. The power flows between pairs of noes are etermine by an it is epicte in Fig. 6. Next, we can obtain the net power flow at each noe from 29 as shown in Fig. 6 e. Then, the net power at each noe is given by 4 as shown in Fig. 6 f. Then, the coorination error given by 30 is zero an the supply-eman balance is also achieve as shown in Fig. 6 a an Fig. 6 f.

18 8 300 p D p D Time sec a Total eman 80 p p 2 p 3 p 4 p 5 p 6 60 p i Delta p Gi Time sec b Power coorination Delta p G Delta p G2 Delta p G3 Delta p G4 Delta p G5 Delta p G Time sec c Power generation control input 80 p G p G2 p G3 p G4 p G5 p G6 60 p Gi Time sec Generate power Fig. 5: Power istribution with power coorination V. CONCLUSION This paper has aresse power istribution problems in istribute power gri with an without power coorination. First, a power coorination using a consensus scheme uner limite net power an power generation capacities was consiere. Secon, power generation an a power flow control laws with an without power coorination were esigne using a consensus

19 9 scheme to achieve supply-eman balance. Since this paper has provie systematic approaches for power istribution among istribute noes on the basis of consensus algorithms, the results of this paper can be nicely utilize in power ispatch or power flow scheuling. The authors believe that consensus algorithm-base power istribution schemes of this paper have several avantages over typical power ispatch approaches. The first key avantage is that the power coorination an control can be realize via ecentralize control scheme without relying upon nonlinear optimization technique. The secon avantage is that the framework propose in this paper can hanle power constraint, generation, an flow in a unifie framework. It is noticeable that this paper has consiere the power coorination, power generation an power flow control in the higher-level moels of gri networks but oes not consier lower-level moels of gri networks such as current, voltage rop, an line impeance. However, in our future research, it woul be meaningful to a links between the higher-level moels an the lower-level moels. Further, it is esirable to investigate various features such as behavior of self-intereste customer, power loss in the transmission line, an structures between the physical an cyber layers such as elays, an mismatches between them. Remark 5.: Though the paper has only focuse on power istribution, we believe that the propose approach can be extene to various attribute istribution problems such as traffic control an supply chain management, because we use a funamental moel escribing flow of attribute between pairs of istribute resources as well as the amount of attribute in each istribute resource. For example, in a traffic control, each freeway section name cell correspons to each istribute power resource, istribution of vehicle ensity correspons to net power of each power resource, esire traffic ensity correspons to esire net power, an onramp traffic flow correspons to power generation control input. Thus, the goal of traffic control which satisfies esire traffic ensity correspons to that of power istribution which satisfies esire net power. VI. ACKNOWLEDGEMENT It is recommene to see Byeong-Yeon Kim, Coorination an control for energy istribution using consensus algorithms in interconnecte gri networks, Ph.D. Dissertation, School of Information an Mechatronics, Gwangju Institute of Science an Technology, 203 for applications

20 20 to various engineering problems of the algorithms evelope in this paper. REFERENCES [] K. Moslehi an R. Kumar, A reliability perspective of the smart gri, Smart Gri, IEEE Transactions on, vol., no., pp , 200. [2] D. Streiffert, Multi-area economic ispatch with tie line constraints, Power Systems, IEEE Transactions on, vol. 0, no. 4, pp , 995. [3] Z. Zhang, X. Ying, an M.Y. Chow, Decentralizing the economic ispatch problem using a two-level incremental cost consensus algorithm in a smart gri environment, in North American Power Symposium NAPS, 20. IEEE, 20, pp. 7. [4] K. Yasua an T. Ishii, The basic concept an ecentralize autonomous control of super istribute energy systems, IEEJ Transactions on Power an Energy, vol. 23, pp , [5] H. Xin, Z. Qu, J. Seuss, an A. Maknounineja, A self-organizing strategy for power flow control of photovoltaic generators in a istribution network, Power Systems, IEEE Transactions on, vol. 26, no. 3, pp , 20. [6] A.D. Dominguez-Garcia an C.N. Hajicostis, Coorination an control of istribute energy resources for provision of ancillary services, in Smart Gri Communications SmartGriComm, 200 First IEEE International Conference on. IEEE, 200, pp [7] A. Jababaie, J. Lin, an A.S. Morse, Coorination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control, IEEE Transactions on, vol. 48, no. 6, pp , [8] R. Olfati-Saber an R.M. Murray, Consensus problems in networks of agents with switching topology an time-elays, Automatic Control, IEEE Transactions on, vol. 49, no. 9, pp , [9] M. Zhu an S. Martínez, Discrete-time ynamic average consensus, Automatica, vol. 46, no. 2, pp , 200. [0] Q. Hui an W.M. Haa, Distribute nonlinear control algorithms for network consensus, Automatica, vol. 44, no. 9, pp , [] L.D. Servi, Electrical networks an resource allocation algorithms, Systems, Man an Cybernetics, IEEE Transactions on, vol. 0, no. 2, pp , 980. [2] M. Baric an F. Borrelli, Distribute averaging with flow constraints, in American Control Conference ACC, 20. IEEE, 20, pp [3] B.A. Robbins, A.D. Domínguez-García, an C.N. Hajicostis, Control of istribute energy resources for reactive power support, in North American Power Symposium NAPS, 20. IEEE, 20, pp. 5. [4] H. Saaat, Power system analysis, WCB/McGraw-Hill, 999. [5] J. De La Ree, V. Centeno, J.S. Thorp, an AG Phake, Synchronize phasor measurement applications in power systems, Smart Gri, IEEE Transactions on, vol., no., pp , 200. [6] H.S. Ahn an K.K. Oh, Comman coorination in multi-agent formation: Eucliean istance matrix approaches, in Control Automation an Systems ICCAS, 200 International Conference on. IEEE, 200, pp [7] S.T. Cay, A.D. Dominguez-Garcia, an C.N. Hajicostis, Robust implementation of istribute algorithms for control of istribute energy resources, in North American Power Symposium NAPS, 20. IEEE, 20, pp. 5. [8] R. A. Horn an C. R. Johnson, Matrix analysis, New York: Cambrige Univ. Press, 985. [9] L. Xiao, S. Boy, an S.J. Kim, Distribute average consensus with least-mean-square eviation, Journal of Parallel an Distribute Computing, vol. 67, no., pp , 2007.

21 2 p p 2 p 3 p 4 p 5 p 6 00 p i Time sec a Desire net power 50 Delta p G Delta p G2 Delta p G3 Delta p G4 Delta p G5 Delta p G6 Delta p Gi Time sec b Power generation control input 00 p G p G2 p G3 p G4 p G5 p G6 p Gi Time sec c Generate power 50 p F2 p F3 p F23 p F25 p F34 p F45 p F46 p F56 p Fij Time sec Power flow between pairs of noes 50 p F p F2 p F3 p F4 p F5 p F6 p Fi Time sec e Net power flow at each noe 00 p p 2 p 3 p 4 p 5 p 6 p i Time sec f Net power at each noe Fig. 6: Power istribution without power coorination