Optimal placement of PMUs and communication links for distributed state estimation in distribution networks

Applied Energy 256 (2019) 113963

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Optimal placement of PMUs and communication links for distributed state estimation in distribution networks

T

⁎

Zhida Zhaoa, Hao Yua, Peng Lia, , Peng Lib, Xiangyu Konga, Jianzhong Wuc, Chengshan Wanga a

Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China Power Research Institute, CSG, Guangzhou 510663, Guangdong Province, China c Institute of Energy, School of Engineering, Cardiff University, Cardiff CF24 3AA, UK b

H I GH L IG H T S

network partitioning method for DSE is developed. • AAndistribution PMUs and communication links placement model for DSE is proposed. • Bothoptimal • the bandwidth cost and length cost of communication links are considered.

A R T I C LE I N FO

A B S T R A C T

Keywords: Communication link Distribution network Distributed state estimation (DSE) Optimal placement Phasor data concentrator (PDC) Phasor measurement unit (PMU)

With the expansion in scale and complexity of distribution networks, distributed state estimation (DSE), a realtime database for other on-line applications, is becoming popular for large-scale active distribution networks (ADN). Measurements from phasor measurement units (PMUs) with the same time stamp can assist DSE to obtain faster and more accurate estimation; however, the configuration of PMUs and communication links should be updated to support data collection and transmission. This paper proposes an optimal PMUs and communication links placement method for DSE in distribution networks. A network partitioning method is presented with the aim of balancing calculation times among subareas. Then, a binary integer linear programming model that simultaneously considers the optimal placement of PMUs, phasor data concentrators (PDCs) and communication links is proposed. The economy of the configuration scheme is guaranteed on the premise that the network is fully observable. Finally, case studies on the IEEE 33-node, PG&E 69-node and IEEE 123-node systems verify the feasibility of the proposed method.

1. Introduction State estimation (SE) is considered a core component of distribution management system (DMS) by providing a complete, consistent and accurate database to all other on-line applications [1]. With the evolution of power distribution systems whose scale and complexity are both challenging, changes are happening with fast-growing distributed generation (DG) access and broad participation of customers [2,3]. Traditional centralized state estimation (CSE) is facing heavy communication burden and high-dimensional state variables [4]. The quadratic relationship between the estimation time needed for most approaches used in conventional state estimation and the size of distribution networks makes CSE unpractical when facing growing system scale. To reduce the communication burden and computation complexity, distributed state estimation (DSE) is gradually becoming an

⁎

effective means to deal with this problem [5,6], replacing CSE to provide real-time and accurate data support for on-line monitoring and control. In addition, DSE can also reduce the risk of privacy exposure. The premise for the convergence of state estimation is that the system is fully observable. However, due to the wide distribution of data collection, incomplete coverage of monitoring points, limited real-time measuring devices and relatively simple measurement types, distribution networks are far from overall observability [7]. The introduction of phasor measurement units (PMUs) has greatly improved the operational monitoring level of distribution networks [8,9]. Compared with traditional measuring devices, PMUs, with a precise time stamp via global position systems (GPS) [10], can obtain real-time synchronous phasor information of node voltage and branch current [11,12]. And the technology improves the performance of various on-line applications, including model parameter calibration

Corresponding author. E-mail address: [email protected] (P. Li).

https://doi.org/10.1016/j.apenergy.2019.113963 Received 25 July 2019; Received in revised form 23 September 2019; Accepted 4 October 2019 Available online 17 October 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Lk Abbreviations

Bk SE CSE DSE DMS DG PMU PDC ICA VNS GA DSO

state estimation centralized state estimation distributed state estimation distribution management system distributed generation phasor measurement unit phasor data concentrator imperialistic competition algorithm variable neighbourhood search genetic algorithm distribution system operator

z i, j wu, l

xi yi

Parameters adjacent matrix of the distribution network and subarea m number of subareas number of nodes in the whole network and in subarea m number of branches degree of node i amount of data that the PMU configured at node i generates per second CCL , CPMU ,CPDC the total cost of communication links, PMUs and PDCs CL , CB the unit cost of the communication link’s length and bandwidth CP1, CP2 the unit cost of a PMU and PDC length of the communication link configured along Sk branchk vector of the communication path between nodes i and j Ei, j

A , Am J N , Nm M di Di

Sets

Πi ΛB Ωm , Ωn ΓZ

set set set set

of of of of

node i and its incident nodes all branches nodes in subareas m and n all subareas

Indices

p k m, n i , j , u ,l

expansion before and after adjustment decision variable of the communication link configured along branchk the bandwidth of the communication link configured along branch k decision variable of data uploading between nodes i and j in the same subarea decision variable of information interaction between nodes u and l in adjacent subareas decision variable of the PMU configured at nodei decision variable of the PDC configured at node i

index of partition matrices index of branches indices of subareas indices of nodes

Variables

Gp' , Gp

partition matrix of the distribution network in the pth

utilize the difference in the phase-angle estimation of the overlapping nodes to achieve consistent angle information for the whole network [32]. That means network partitioning has a direct impact on the configuration scheme of PMUs and communication links. Different partitioning schemes can lead to the changes in PMU locations and the candidate range of PDCs, and then affect the overall configuration scheme. At present, there have been a few studies on the partitioning of distribution networks oriented to DSE. In [32], a network was divided into multiple regional networks according to topological or geographical criteria, which effectively reduced the computational complexity of each subarea. In [33], the partitioning method was based on the switch locations and ensured that the partition scheme was still valid after topology changes. Ref. [34] renumbered branches and nodes by the branch-line layer method and utilized a post-order traversal algorithm to partition a network based on existing measurements. Ref. [35] manually completed the partitioning of a distribution network according to DSE principles. In [36], regional subareas were obtained via graph theory and the K-means method. However, most of the methods above have limitations because the partitioning results are not only influenced by the partitioning principles, but also affected by the positions of switches and existing measurements. Therefore, an optimal model for network partitioning only based on the partitioning principles should be established for DSE. Researches on integrated placement of PMUs, PDCs and communication links are more common in power transmission networks. With network observability as the target, Ref. [37] applied an imperialistic competition algorithm (ICA) to optimize the placement location of PMUs and made use of Dijkstra’s algorithm to obtain the optimal location of PDCs, considering the length of communication links and the

[13,14], line outage detection [15,16], stability assessment [17,18], state estimation [4], system damping control [19], wide-area protection [20], disturbance detection [21], system operation control [22,23] and energy management [24,25]. PMUs have been an important part of smart grid technology [26,27]. When applied to state estimation, phasor measurements collected by PMUs are linear with state variables, and algorithms such as linear state estimation algorithms can be used to obtain the system state, greatly reducing the computation time [28]. Besides, PMUs applied in distribution networks are progressively becoming less expensive and smaller with high accuracy and short latency of 10–30 ms [29]. Using PMU measurements for DSE in distribution networks can not only ensure a rapid state variable solution but also effectively solve the problems of poor quality, low synchronization and long acquisition periods from traditional measurements [30,31], providing a guarantee for real-time analysis and control of distribution networks. Nevertheless, PMUs deployed in distribution networks are not enough to meet the demands of DSE. A communication network composed of phasor data concentrators (PDCs) and communication links must be established to support timely data transmission and very large data storage. Before the optimal placement of PMUs and communication links, network partitioning is essential for two reasons: (1) Partitioning results will influence the performance of DSE. In terms of the number of subareas, too many regions will aggravate the communication burden, while too few cannot effectively reduce the computational complexity. In terms of the scale of subareas, the number of the nodes in each subarea should be as close as possible to balance the calculation times among subareas. (2) The overlapping nodes between adjacent subareas are usually configured with PMUs. They can not only be used to obtain the equivalent power injection of the adjacent subareas but also to

2

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cost of switches and routers at the same time. Ref. [38] first employed a variable neighborhood search (VNS) algorithm to solve the optimal PMU placement problem. Taking the lowest configuration scheme cost as the objective function, the siting of PMUs and communication links were optimized, making the network completely observable and establishing a communication network. On the premise that the PDC location was determined, Ref. [39], taking into account the length and bandwidth cost of communication links, established an integrated model for optimal placement of PMUs and communication links. Ref. [40] first completed the network partitioning and then optimized the placement of PMUs and PDCs for each subarea separately with the assumption that all the subareas and the dispatching center were correlated within a backbone network. There are also some works considering wireless communication between PMUs and PDCs [41], the speed and reliability of which remain to be discussed for on-line applications. At present, it is more reasonable to configure wireless communication as the backup or extension of wired communication. Besides, most of the abovementioned works made use of heuristic algorithms. The algorithms may face the issues of low solving efficiency and falling into locally optimal solutions. This paper aims at implementing PMUs and communication links placement targeted to distribution networks whereas the previous works are mainly on transmission networks. There are some significant differences from the network structure to operation mode which influence the placement of system measurements. (1) Distribution systems are three-phase unbalanced networks because of non-transposed lines, unbalanced loads and asymmetric integration of DG [42]. It is necessary to collect three-phase data to complete three-phase state estimation. (2) Distribution networks have a relatively larger system scale due to widespread massive nodes and they need to be decoupled into subareas to reduce the communication and computational burden. (3) From the operation mode of distribution networks, it usually has a meshed structure but operates radially to increase operating efficiency and ensure power supply in emergency situations [43]. The problem to be solved in distribution networks is the effectiveness of PMU configuration scheme under various operation modes. (4) PMUs currently in transmission networks are mainly used to deal with stability issues whereas there are numerous operational demands in distribution networks. The PMU placement may vary with the different operational requirements. (5) Compared with transmission networks, the communication infrastructure of distribution network still needs to be updated with placement scheme. (6) PMU device used in distribution networks should be with a lower cost and more accurate in phase angle measurement because of the short distance of distribution lines. In this paper, a new method is proposed for simultaneous optimal placement of PMUs and communication links in distribution networks. The main contributions can be summarized as follows:

communication links is presented in Section 4. Case studies are provided in Section 5 to show the feasibility of the proposed method. Section 6 concludes this paper with a discussion.

2. Partitioning of distribution network for DSE 2.1. Preconditions of partitioning (1) The number of subareas. The performance of DSE is directly affected by the number of subareas. In this paper, the number of subareas is decided by the network scale with an empirical value 3 N [44] or given by the distribution system operator (DSO). (2) Adjacent subareas overlapping level. This paper considers that there is only one overlapping node between adjacent subareas, and this node must be configured with a PMU. (3) Adopting a decentralized architecture. Each subarea has its own computing center, which is the PDC node. The computing center gathers measurements and completes state estimation within the subarea and only communicates with adjacent subareas, reducing the communication burden and risk of privacy exposure. In DSE with decentralized architecture, measurements are uploaded to their own control centers firstly, and then state estimation is carried out in each subarea separately. Based on the estimation result, the state information of boundary nodes which can be used as pseudo measurements is exchanged between adjacent subareas. At last, state estimation is performed once again to make the system state more accurate. Therefore, the number of the nodes in each subarea should be balanced as much as possible to ensure that the estimation time of each region is similar, reducing the wait time for communication between adjacent regions.

2.2. Partitioning method For partitioning the distribution network, first, we should select the number of subareas. Then, a central node is allocated to every subarea. Moreover, each subarea is gradually extended outward from the central node until all the nodes are partitioned, which can effectively ensure connectivity within the subareas [40]. Taking the 12-node distribution network shown in Fig. 1 as an example, the procedure is illustrated below. Adjacent matrix A can be built as follows:

⎧1 i = j Ai, j = 1 i ≠ j, j ∈ Πi ⎨ ⎩ 0 i ≠ j, j ∉ Πi

(1) To reduce the computational and communication burden, a network partitioning method for DSE is developed. The partitioning objective is to balance the number of nodes in subareas. By transforming the network partitioning to the problem of allocating a central node of each subarea, the partitioning scheme with balanced scale among subareas is obtained through genetic algorithm (GA). (2) A binary integer linear programming model of optimal PMUs and communication links placement for DSE is established. By introducing the traffic model of PMU and PDC, both the bandwidth cost and length cost of communication link are considered. The economy of the configuration scheme is guaranteed on the premise that the network is fully observable.

(1)

The partition matrix G is a J × N dimensional matrix. Each row in G is correlated with one regional network. If node j is in subarea i , then Gi, j = 1; otherwise, Gi, j = 0 . According to Fig. 1, J = 3 12 ≈ 2. By considering nodes 2 and 4 as the central nodes, the original partition matrix G0 is as follows:

The remainder of this paper is organized as follows. Section 2 builds the partition model for DSE in distribution networks. Section 3 demonstrates the communication model of PMUs, PDCs and communication links. The proposed optimal placement model of PMUs and

Fig. 1. 12-node power distribution system. 3

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0 1 0 0 0 0 0 0 0 0 0 0⎤ G0 = ⎡ ⎣0 0 0 1 0 0 0 0 0 0 0 0⎦

3. Traffic model of PMUs and communication links (2) 3.1. Traffic model of PMUs and PDCs

By multiplying the partition matrix and the adjacent matrix, each subarea is extended step by step. Then, the first expansion of the partition matrix is as follows:

1 1 1 0 0 1 0 0 0 0 0 0⎤ G1 = G0 A = ⎡ ⎣0 0 1 1 1 0 0 0 0 0 0 0⎦

A PMU is used to gather the voltage phasor of the configuration node and the current phasor of the branches directly associated with the node and then upload the measurements to a PDC, which sorts the measurements according to the GPS time stamps. The amount of data produced by a PMU in unit time is directly related to the degree of its configured node. The degree of node i , di , denotes the number of branches incident to node i and can be calculated based on the adjacent matrix as follows:

(3)

In the process of expansion, overlapping nodes will be encountered. When two subareas overlap for the first time, if there is only one overlapping node, the node will be selected as the boundary; however, if there are two overlapping nodes, the boundary node will be selected based on the principle that the difference between the number of the nodes in the two subareas is the least. According to G1, nodes 1, 3 and 6 are added to subarea 1, while nodes 3 and 5 are included in subarea 2. The second extension of the partition matrix is as follows:

2 4 2 1 0 2 1 0 0 1 0 0⎤ G2' = G1 A = ⎡ ⎣0 1 2 3 2 0 0 0 0 1 0 0⎦

N

di =

According to the IEEE synchrophasor data standard [45] and [46], the amount of data that the PMU configured at node i generates per second, Di , can be estimated by:

(4)

Di = ((di + 1)P + a) F

It can be seen that some elements in the partition matrix are greater than 1. The nodes corresponding to these elements are partitioned before this expansion, and the actual meaning of each element is the number of nodes in the subarea connected to the corresponding node, including the node itself, in the partition result of the previous step. The nodes with a value of 1 are newly added into the subareas. For nodes that belong to only one subarea, their corresponding elements are just set to 1. For nodes belonging to multiple subareas at the same time, it is necessary to judge whether there is a boundary between those subareas. If there is a boundary, the partitioned nodes before this extension still belong to the original subareas, while the unpartitioned nodes are added to the subarea with fewer nodes. If there is no boundary, the boundary nodes should be selected based on the principle that the numbers of nodes in the regions have the smallest difference. G2' is adjusted according to the above principles, and the partition matrix G2 after adjustment is as follows:

Di = (di + 1)PF

(10)

A PDC can compress phasor data before transmitting. Suppose that the compression function is H (Di ) ; that is, a PDC changes Di data to H (Di ) , where 0 < H (Di ) ≤ Di [47]. In this paper, there is only one overlapping node between adjacent subareas, and the neighbouring PDCs merely interact with the information of this node. If the compression function is taken as a linear model, then the amount of data transmitted by the PDC is f × PF , where f denotes the linear compression ratio.

(5) 3.2. Communication link cost model In this paper, communication links are configured along lines, and each communication link connects two nodes. Compared with an independent communication system, a dependent communication system can effectively reduce the construction and maintenance costs [48] and is widely used in the simultaneous optimal placement of PMUs and communication links. The cost of communication links is calculated based on their bandwidth and length as shown below [39]:

(6)

That is, subarea 1 is composed of nodes 1, 2, 3, 6, 7, 8, and 9; subarea 2 is composed of nodes 3, 4, 5, 10, 11, and 12; and the boundary node is node 3. After adopting the above partitioning method to ensure connectivity, choosing the appropriate central nodes becomes the key to partitioning. In this paper, the center nodes are selected by an optimization method. According to the principle of DSE, the numbers of nodes in different subareas should be as close as possible to reduce the wait time for communication, so the objective function is to minimize the difference in the scale of all subareas, and the mathematical expression is as follows:

max(min{N1, N2, ⋯, Nm, ⋯, NJ }/max{N1, N2, ⋯, Nm, ⋯, NJ })

(9)

where P is the size of the data portion in a single phasor data frame, a is the size of the frame overhead generated by this PMU, and F is the configured phasor data frame reporting rate. Since the size of the overhead is normally very small compared with the size of the phasor data, Di can be simplified to a linear form as follows:

As shown by G2 , node 7 is added to subarea 1, and subarea 2 includes node 10. Iteration continues according to the above rules until all nodes are partitioned, and the final partition matrix is shown below:

1 1 1 0 0 1 1 1 1 0 0 0⎤ G=⎡ ⎣0 0 1 1 1 0 0 0 0 1 1 1⎦

(8)

j=1

from G2'

1 1 1 0 0 1 1 0 0 0 0 0⎤ G2 = ⎡ ⎣0 0 1 1 1 0 0 0 0 1 0 0⎦

∑ A i, j − 1

CCL =

∑ k ∈ ΛB

Lk (CL Sk + CB Bk ) (11)

where Lk is the decision variable. If branch k is configured with a communication link, Lk is 1; otherwise, it is 0. The bandwidth of a communication link depends on two parts, including the bandwidth required for PMUs to upload data to PDCs and the bandwidth required for information interaction between PDCs. Unlike transmission networks, distribution networks operate in a radial structure, with unique paths between nodes. Thus, the optimal path selection problem is not involved. Whether communication links are configured and their bandwidth only depend on the placement locations of PMUs and PDCs. The communication path vector Ei, j is an M -dimensional column vector. If the communication path between nodes i and j incorporates branch k , the kth element of Ei, j is 1; otherwise, it is 0. Under the precondition that the topology of the distribution network is known and constant, Ei, j is known and unique. If node i is configured with a PMU,

(7)

Genetic algorithm (GA) is used to solve the optimal combination of central nodes. In GA, the length of an individual chromosome is J , and the value set of genes on the chromosome is {1, 2, ⋯, N } . The fitness function is the objective function shown in (7). The combination of central nodes is obtained by individual chromosomes; then, the distribution network is partitioned by the above partitioning method, and the individual fitness is calculated according to the partitioning results. The procedure of the GA is not described in detail. 4

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node j is configured with a PDC, and the PMU of node i communicates with the PDC of node j , then z i, j is 1; otherwise, it is 0. If both nodes u and l are configured with PDCs and the two PDCs communicate with each other, then wu, l is 1; otherwise, it is 0. The bandwidth of communication links is calculated as follows:

⎛ ⎜∑ ⎝ h ∈ ΨA

∑ ∑

∑ ∑ ∑

2fEu, l wu, l+

u ∈ Ωh,1 l ∈ Ωh,2

m ∈ ΓZ i ∈ Ωm j ∈ Ωm

Start Obtain the topology of the distribution network and unit costs of a PMU, PDC and communication link

⎞ (di + 1) Ei, j z i, j PF = B ⎟ ⎠

Construct the adjacent matrix and communication path vectors between nodes

(12) where each element of ΨA is a set that consists of two adjacent subareas, that is, ΨA = {{m , n}|Ωm ∩ Ωn ≠ ∅} ; h is an element in ΨA ; Ωh,1 and Ωh,2 denote the sets composed of the nodes in the first and second subareas of h , respectively; and nodes u and l are nodes in two adjacent subareas. Due to the bidirectional communication between PDCs, the bandwidth is 2 times that of unidirectional communication. Nodes i and j are in the same subarea. Bandwidth vector B is an M -dimensional column vector, and its kth element Bk represents the bandwidth of the communication link configured along branch k . If branch k is not configured with a communication link, Bk = 0 .

Address the partition model and complete the distribution network splitting

Construct the binary integer linear programming model of PMUs and communication links

Solve the model and obtain the optimal placement scheme

4. Integrated optimal placement model of PMUs and communication links 4.1. Objective function

end

The lowest total cost of the configuration scheme for PMUs, PDCs and communication links in a power distribution system is taken as the objective function, and the mathematical expression is as follows:

min(CPMU + CPDC + CCL )

i ∈ Ωm

⎞ x i − NOL ⎟ ⎠

∑

∑

yi

m ∈ ΓZ

i ∈ Ωm

⎛ CPMU = CP1 ⎜ ∑ ⎝ m ∈ ΓZ CPDC = CP2

CCL =

∑

∑

Fig. 2. Flow chart of optimal PMUs and communication links placement for DSE.

(13)

(14)

(15)

Lk (CL Sk + CB Bk )

k ∈ ΛB

(16)

Fig. 3. Network partitioning results of the IEEE 33-node system.

where x i is the decision variable of PMU placement. If node i is configured with a PMU, x i is 1; otherwise, it is 0. NOL represents the number of repeated counts of PMUs caused by overlapping subareas, and yi is the decision variable of PDC placement. yi is 1 if node i is configured with a PDC; otherwise, it is 0.

with a PDC. z i is a row vector composed of the decision variables z i, j which are related to node i in subarea m . ym is a column vector composed of decision variables of PDC placement corresponding to nodes in subarea m . The third constraint guarantees that each subarea has only one PDC.

4.2. Constraints (3) Decision variable constraints on overlapping nodes in adjacent subareas

(1) Observable constraints of subareas

Am xm ≥ 1m , m ∈ ΓZ

(17)

x i = 1, i ∈ Ωm ∩ Ωn , m ∈ ΓZ , n ∈ ΓZ

where xm is a column vector composed of decision variables of PMU placement corresponding to nodes in subarea m . 1m is an Nm -dimensional column vector whose elements are all 1.

where node i is the overlapping node between subarea m and subarea n . (4) Information interaction constraints between PDCs of adjacent subareas

(2) Data uploading constraint of PMUs

∑

∑

z i, j = x i , i ∈ Ωm , m ∈ ΓZ

j ∈ Ωm

(18)

z iT ≤ ym , i ∈ Ωm , m ∈ ΓZ

(19)

∑ i ∈ Ωm

yi = 1, m ∈ ΓZ

(21)

wi, l = yi , i ∈ Ωm , Ωm ∩ Ωn ≠ ∅, m ∈ ΓZ , n ∈ ΓZ

l ∈ Ωn

(22)

wiT ≤ yn , i ∈ Ωm , Ωm ∩ Ωn ≠ ∅, m ∈ ΓZ , n ∈ ΓZ

(23)

The first constraint guarantees that the node in subarea m that communicates with subarea n is configured with a PDC. The second constraint guarantees that the node in subarea n that communicates with subarea m is configured with a PDC. wi is a row vector composed of all information interaction decision variables wi, l related to node i in subarea m . yn is a column vector composed of decision variables of PDC

(20)

The first constraint guarantees that the node transmitting the measurement information is equipped with a PMU, and the second constraint ensures that the node receiving the information is installed 5

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Fig. 4. Network partitioning results of the PG&E 69-node system.

Fig. 5. Network partitioning results of the IEEE 123-node system.

Table 1 Partition scheme of the IEEE 33-node system. Subarea ID

Central node ID

Node IDs in the subarea

Node number in the subarea

1 2 3

1 29 12

1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

12 11 12

Table 2 Partition scheme of the PG&E 69-node system. Subarea ID

Central node ID

Node IDs in the subarea

Node number in the subarea

1 2 3 4

38 49 22 31

1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41 9, 10, 11, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 57, 58 3, 28, 29, 30, 31, 32, 33, 34, 35, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

15 18 19 20

6

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Table 3 Partition scheme of the IEEE 123-node system. Subarea ID

Central node ID

Node IDs in the subarea

Node number in the subarea

1 2

11 45

30 32

3 4

92 111

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 35, 53, 54, 55, 56, 57, 58, 59, 60 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 118 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97 58, 61, 62, 63, 64, 65, 66, 67, 68, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

30 29

Re ≤ r

(28)

R eq e = req

(29)

where w is the coefficient vector; e is a vector composed of binary variables to be optimized, including decision variables of PMU placement, decision variables of PDC placement, decision variables of data uploading, decision variables of information interaction and decision variables of communication link placement; and we is the total cost of the configuration scheme. The first constraint contains all inequality constraints in Section 4.2, including (17), (19), and (23)–(26). The second constraint contains all equality constraints in Section 4.2, including (18) and (20)–(22).

Fig. 6. Configuration scheme of the IEEE 33-node system.

placement corresponding to nodes in subarea n .

4.4. Flow chart

(5) Placement constraints of communication links

KL ≥ B

(24)

L≤B

(25)

0M ≤ L ≤ 1M

(26)

In practical application, firstly the network is partitioned into several areas due to the partitioning principles of DSE. Then the binary linear programming model proposed in this paper is established to determine the locations of PMUs and communication links. The flow chart is shown in Fig. 2.

where K is a large enough positive number and L is an M -dimensional column vector whose kth element Lk indicates whether branch k is configured with a communication link. If branch k is configured with a communication link, Lk is 1; otherwise, it is 0. 0M and 1M represent M -dimensional column vectors whose elements are all 0 and all 1, respectively.

5. Case studies and analysis

4.3. Integrated optimal placement model

The IEEE 33-node system is divided into 3 subareas and the PG&E 69-node system is divided into 4 subareas. According to [32] and [34], the IEEE 123-node system is considered to be divided into 4 subareas after simplifying the switch nodes. The partitioning results are demonstrated in Figs. 3–5, and the details are described in Tables 1–3. In [34], for the IEEE 33-node system, the network is also divided

Case studies and analysis are performed on the IEEE 33-node system [49], PG&E 69-node system [50] and IEEE 123-node system [51]. 5.1. Partitioning of distribution network

The binary integer linear programming model of optimal PMUs, PDCs and communication links placement is as follows:

min we

(27)

Fig. 7. Configuration scheme of the PG&E 69-node system. 7

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Fig. 8. Configuration scheme of the IEEE 123-node system. Table 4 Configuration scheme of PMUs and communication links. IEEE 33-node system

PG&E 69-node system

IEEE 123-node system

PMU locations

2, 5, 7, 10, 11, 14, 17, 21, 24, 26, 29, 32

PDC locations

3 in subarea 1 26 in subarea 2 11 in subarea 3

Communication link locations Total cost (×103 USD)

5 branches without CLs

2, 3, 4, 6, 9, 11, 14, 17, 20, 23, 26, 28, 31, 34, 38, 41, 44, 47, 50, 53, 56, 58, 59, 62, 65, 68 4 in subarea 1 9 in subarea 2 17 in subarea 3 3 in subarea 4 6 branches without CLs

2, 5, 7, 13, 15, 16, 19, 20, 22, 25, 28, 29, 33, 38, 40, 42, 44, 46, 49, 51, 53, 57, 58, 60, 62, 64, 67, 68, 71, 75, 77, 79, 84, 85, 89, 91, 93, 95, 97, 98, 101, 104, 108, 109, 112, 114, 118 14 in subarea 1 19 in subarea 2 77 in subarea 3 98 in subarea 4 19 branches without CLs

445.96

1,596.94

1,744.06

Table 5 Comparison of optimal configuration schemes for CSE and DSE. IEEE 33-node system

Number of PMUs Number of PDCs Length of communication links (km) Bandwidth of communication links (kbps) Difference of bandwidth cost (×103 USD) Total cost (×103 USD)

PG&E 69-node system

IEEE 123-node system

CSE

DSE

CSE

DSE

CSE

DSE

11 1 9.1208 6,375 404.40 830.68

12 3 8.9041 3,005

24 1 20.9387 19,350 1,173.96 2,457.41

26 4 20.8600 9,567

45 1 9.5098 32,500 2,390.04 4,102.26

47 4 9.4031 12,583

445.96

1,596.94

1,744.06

5.2. Integrated placement of PMUs, PDCs and communication links

into 3 subareas and the numbers of the nodes in each subarea are 13, 11 and 11. For the IEEE 123-node system, the number of the nodes in 4 subareas are 36, 34, 26 and 25 respectively. Compared with the results above, it demonstrates that the partitioning scheme in this paper has a more balanced nodes number among subareas. For the PG&E 69-node system, manually completing the distribution network partitioning can obtain the same partition scheme in this paper but there are also many other partition schemes and the partitioning effect may vary greatly. Therefore, the method proposed in this paper can effectively ensure the minimum difference in the number of nodes among subareas, which verifies the feasibility of this method.

After partitioning, PMUs, PDCs and communication links are configured on the IEEE 33-node, PG&E 69-node and IEEE 123-node systems. Set f = 0.56 [47], PF = 25 kbps and the unit costs involved in the configuration process: CPMU = 4, 000.00 USD, CPDC = 8, 000.00 USD [52,53], CB = 120.00 USD/kbps , and CL = 1, 500.00 USD/km [39]. To obtain the approximate distance matrix of each test network, it is assumed that all lines have the same conductors with the same configurations. Thus, the relative distances between system nodes can be extracted from the system impedance matrix [54]. We assume that the total length of lines in the IEEE 33-node and PG&E 69-node systems equal to 10.1092 and 23.6363 km which is proportional to the total line impedance. The distance matrix of IEEE 123-node system can be found 8

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Table 6 Bandwidths of communication links for DSE and CSE in the IEEE 33-node system. Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

825 650 575 575 500 300 300 225 225 225 150 150 150 75 75 75

0 175 103 103 103 103 103 103 103 178 150 150 150 75 75 75

17 2 19 20 21 3 23 24 6 26 27 28 29 30 31 32

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0 75 75 75 0 75 75 0 200 200 125 125 125 50 50 50

0 75 75 75 0 75 75 0 206 150 150 150 75 75 75 0

Table 7 Bandwidths of communication links for DSE and CSE in the PG&E 69-node system. Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 3 28 29 30 31 32 33 34

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

1,675 1,675 1,175 1,050 1,050 975 975 925 525 525 450 375 375 300 300 300 225 225 225 150 150 150 75 75 75 0 225 150 150 150 75 75 75 0

0 75 228 253 253 178 178 128 178 178 128 178 178 253 253 253 225 225 225 150 150 150 75 75 75 0 225 150 150 150 75 75 75 0

3 59 60 61 62 63 64 65 66 67 68 4 36 37 38 8 40 9 42 43 44 45 46 47 48 49 50 51 52 53 11 55 12 57

59 60 61 62 63 64 65 66 67 68 69 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

275 275 200 200 200 125 125 125 50 50 50 125 50 50 50 50 50 300 300 300 225 225 225 150 150 150 75 75 75 0 75 0 75 0

300 225 225 225 150 150 150 75 75 75 0 75 75 75 0 50 50 300 300 300 225 225 225 150 150 150 75 75 75 0 50 50 50 50

demonstrates that the total bandwidth of communication links for DSE is much smaller than that for CSE, which verifies the effectiveness of DSE in reducing the communication burden. However, the number of PMUs for DSE increases compared with CSE. The reason is that the overlapping nodes between adjacent subareas must be configured with PMUs, resulting in measurement redundancy for the whole network. The bandwidths of communication links in the configuration schemes oriented to DSE and CSE are shown in Table 6, Table 7 and Table 8, respectively. For most of the communication links configured with the same branch, the bandwidth required for DSE is less than or equal to that required for CSE. On the main feeder, the closer to the

in [51]. After the integrated binary integer linear programming model of PMUs, PDCs and communication links placement proposed in this paper is established, the mixed integer linear programming function “intlinprog” in MATLAB is used to solve the model. The optimal placement is shown in Figs. 6–8, and Table 4 shows the details. Suppose that only one PDC is configured for each feeder in CSE and that it is configured at the source node, e.g. node 1. Under the same unit costs, the optimal placement model proposed in this paper is employed to obtain the optimal configuration scheme for CSE, in which the whole network is only one zone and the location of PDC is determined. Table 5 shows the comparison of optimal schemes for CSE and DSE. It 9

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Table 8 Bandwidths of communication links for DSE and CSE in the IEEE 123-node system. Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

Beginning of branch

End of branch

Bandwidth for CSE (kbps)

Bandwidth for DSE (kbps)

1 2 2 4 4 6 2 8 9 9 10 15 15 9 14 35 16 16 14 19 20 19 22 22 24 24 26 29 30 31 26 27 32 27 28 19 36 37 37 39 36 41 41 43 43 45 46 45 48 48 50 51 14 53 54 55 56 55 58

2 3 4 5 6 7 8 9 13 10 15 11 12 14 35 16 17 18 19 20 21 22 23 24 25 26 29 30 31 118 27 32 33 28 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

3,050 0 100 50 50 50 2,825 2,825 50 100 100 0 0 2,675 100 100 0 0 875 75 0 350 50 300 50 250 125 50 50 50 125 50 50 75 0 450 100 50 50 50 350 0 250 50 200 75 0 125 50 75 75 0 1,700 1,625 1,625 75 0 1,550 75

0 0 100 50 50 50 225 225 50 100 100 0 0 375 100 100 0 0 228 75 0 400 0 300 50 250 125 50 50 50 125 50 50 75 0 400 100 50 50 50 300 50 250 50 200 75 0 125 50 75 75 0 303 228 228 50 50 178 50

59 58 61 61 63 64 65 66 61 68 69 70 71 68 73 74 75 73 77 78 79 79 81 82 83 82 85 77 87 88 88 90 90 92 92 94 94 96 68 98 99 100 101 98 102 103 104 102 106 107 106 109 109 110 111 111 113 114

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 116 102 103 104 105 106 107 108 109 117 110 111 112 113 114 115

0 1,475 50 125 125 50 50 50 1,300 125 50 50 50 650 75 75 0 575 225 225 0 125 125 50 50 75 0 250 250 50 200 50 150 50 100 50 50 50 525 75 75 75 0 350 75 75 0 275 50 50 225 0 125 125 50 75 75 0

50 128 50 125 125 50 50 50 303 75 75 75 0 228 75 75 0 303 225 225 0 125 125 50 50 75 0 250 250 50 200 50 150 50 100 50 50 50 456 75 75 75 0 350 75 75 0 275 50 50 225 0 125 125 50 75 75 0

substation, the more obvious the bandwidth gap is.

Table 9 Comparison of the four schemes.

Scheme Scheme Scheme Scheme

1 2 3 4

The total length of communication links (km)

The total bandwidth of communication links (kbps)

Number of PMUs

20.8600 20.8600 20.5281 20.5234

9,567 9,567 9,642 9,667

26 26 26 26

5.3. Influence of different unit costs on the placement scheme On the PG&E 69-node system, keep the unit costs of a PMU and PDC unchanged, change the length and bandwidth cost of communication links, and compare the variety of configuration schemes. Suppose that the original scheme is scheme 1. Increase the unit length cost of communication links, and set the scheme with CB = 120.00 USD/kbps and CL = 15, 000.00 USD/km as scheme 2. On this basis, the unit bandwidth cost of communication links is reduced, and the scheme with 10

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CB = 12.00 USD/kbps and CL = 15, 000.00 USD/km is scheme 3. Further reduce the unit bandwidth cost of communication links, and set the scheme with CB = 1.20 USD/kbps and CL = 15, 000.00 USD/km as scheme 4. Table 9 shows the comparison of the four schemes. By comparison, with increasing length cost and decreasing bandwidth cost, the branches configured with communication links in the distribution network are decreasing and the total bandwidth of communication links is increasing, but the gap is not large. The main reason is that the distribution network has a radial structure. To ensure the observability of the end nodes, the nodes or their adjacent nodes must be installed with PMUs. Therefore, the cost of communication links mainly changes whether the terminal branches are configured with communication links and the locations of terminal PMUs.

[9]

[10]

[11]

[12]

[13] [14]

6. Conclusions

[15]

This paper proposes an optimal placement method of PMUs and communication links for DSE in distribution networks. First, a partitioning method targeted to distribution networks is developed only base on the partitioning principle of DSE. Then, a binary integer linear programming model for optimal PMUs and communication links placement is established. Case studies performed on the IEEE 33-node, PG &E 69-node and IEEE 123-node systems support the feasibility of this method. According to the analysis, we can conclude that the partitioning method proposed in this paper can effectively ensure that the numbers of nodes in all subareas are balanced. The optimal placement model can be solved quickly without falling into a locally optimal solution. This method is capable of guaranteeing the economy of the configuration scheme under the premise of fully observable distribution networks, creating conditions for DSE in power distribution systems. In the future, taking into account the topology changes of distribution networks, the communication mode for DSE needs to be further considered.

[16]

[17]

[18]

[19]

[20]

[21]

[22]

Declaration of Competing Interest

[23]

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[24]

[25]

Acknowledgements [26]

This work was supported by the National Key Research and Development Program of China (2017YFB0902900, 2017YFB0902902) and the National Natural Science Foundation of China (U1866207). This work was conducted in cooperation of APPLIED ENERGY UNiLABDEM: Distributed Energy & Microgrid. UNiLAB is an international virtual lab of collective intelligence in Applied Energy.

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