Robust distributed state estimation of active distribution networks considering communication failures

Electrical Power and Energy Systems 118 (2020) 105732

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Robust distributed state estimation of active distribution networks considering communication failures

T

Tingting Zhanga, Peiran Yuanb, Yaxin Dua, Wen Zhanga, , Jian Chena ⁎

a b

Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, School of Electrical Engineering, Shandong University, Jinan 250061, China State Grid Langfang Power Supply Company, Langfang 065000, China

ARTICLE INFO

ABSTRACT

Keywords: Communication failures Consensus-based coordination Network partition PMUs Robust distributed state estimation

Due to the large scale and three-phase unbalance problem of active distribution networks (ADNs), traditional centralized state estimation (CSE) cannot meet the requirements on estimation accuracy and efficiency for realtime analysis and control of distribution systems. This paper presents a novel consensus-based coordination distributed state estimation (CC-DSE) approach considering communication failures in ADNs. A network partition approach coupled with placement of phasor measurement units (PMUs), which is based on topology analysis, is firstly executed to divide the large-scale distribution network into several sub-regions. Then, the local state estimation (SE) of each sub-region is executed in parallel by improved weighted least squares (WLS) method with revising weights adaptive to residuals without changing the initial objective function, which can mitigate the negative effect of inner-region communication failures. The data coordination is embedded in the iteration of improved WLS of each sub-region estimation by consensus algorithm, which is robust to data missing and bad data caused by inter-region communication failures. The CC-DSE has better performance in accuracy, efficiency and robustness compared with CSE and distributed state estimation (DSE), which is demonstrated by the simulation results of the unbalanced IEEE 123 bus distribution network.

1. Introduction The distribution system state estimation (DSSE) is one of the core components of distribution management system (DMS) [1], which can provide accurate data used in system control centers for security-constrained dispatch and control of the system [2]. With increasing penetration of distributed generations (DGs) and controllable loads, the traditional passive distribution networks (PDNs) are moving towards active distribution networks (ADNs) based on the requirement of active monitoring and controlling of the networks [3]. ADNs have some characteristics including flexible topological structure, uncertainty caused by DGs, large-scale and three-phase unbalance problem, which make conventional centralized DSSEs [3] face with challenges in estimation accuracy and computation efficiency. Thus, a distributed state estimation (DSE) algorithm applicable for ADNs is urgently needed, which can provide accurate states in reasonable computation time for ADNs. There are two computational architectures of DSE [4], including hierarchical and decentralized architectures. In the hierarchical architecture, a central coordinator is necessary to coordinate the states uploaded from local estimators. Massive data exchange with the central

⁎

coordinator may bring communication bottlenecks [5–9]. In contrast, the decentralized SE only requires local measurements and coordination among adjacent sub-regions without a central coordinator [10–19]. Therefore, we pay attention to decentralized architecture of DSE in this paper. Network partition is the vital basics for DSE, which can reduce network scale and enhance the estimation efficiency. However, there are few researches providing network partition methods or criteria for DSE, especially in distribution networks. In transmission networks, network partition approaches are executed based on matrix-splitting techniques [10], topology or geography criteria [15], or different voltage level [16] simply but without definite partition criteria. In distribution networks, partition criteria are mentioned concretely, but the network splitting is performed manually in [17–18]. In our previous work [20], a network partition approach is proposed for a given PMU placement. However, there is the contradiction between network partition and PMU placement for DSE. An unsuitable measurement placement will result in unreasonable network partition that even cannot meet the partition criteria, leading to poor estimation performance of DSE, both in accuracy and efficiency. To address this issue, we extend our previous network partition method in [20] by combining network

Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (J. Chen).

https://doi.org/10.1016/j.ijepes.2019.105732 Received 12 April 2019; Received in revised form 20 November 2019; Accepted 22 November 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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partition with PMU placement. In this way, the contradiction between the network partition and PMU placement can be handled to obtain the better performance of DSE while achieving good economy of PMU placement. An accurate state estimation relies on a reliable communication system. DSE has advantages of being scalable for large-scale networks by inner-region and inter-region communication [21,22]. While unexpected communication failures can bring about data missing or bad data problems, which will affect the efficiency and accuracy of DSE negatively. However, there is little research solving these problems, especially inter-region communication failures, for DSE in power systems. In wireless sensor networks, inter-region data missing problem can be solved by the proposed consensus-based DSE in [23]. As for inner-region communication failures, there are two types of solving method distinguished by whether removing the bad data [12,15] or not [24]. In one case, chi-square tests [12] or residual tests [15] are used in bad data analysis and then state estimation is re-solved after removing bad data, which may bring about additional computation time. In addition, WLAV is a widely-used robust state estimation method. In the transmission network with very high measurement redundancy, there are always redundant accurate measurements available to WLAV estimator when the imprecise measurements are rejected as bad data. However, in distribution networks, the number of accurate real-time measurements is far less than that of states to be estimated. What is worse, lots of imprecise pseudo measurements are added to ensure the redundancy. In this situation, it is difficult to meet the precondition for performing robust state estimation via WLAV, which fails to work in distribution networks [25]. In the other case, different objective functions, such as exponential [24] and least absolute value ones, are used in restraining bad data. However, improper objective function may lead to inaccurate estimation. To solve these problems, we propose a novel consensus-based coordination distributed state estimation (CC-DSE) approach which integrates an improved WLS with adaptive weights. In this paper, we focus on the robust DSE considering communication failures. Firstly, network partition approach combined with PMU placement is executed by considering the number of sub-regions, equilibrium among sub-regions, the observability and connectivity of sub-regions as partition criteria. Secondly, the CC-DSE method is proposed with consensus-based coordination embedded into iterations of local SE, with which the exchanged data can be calculated considering the communication topology of sub-regions for ADNs. Thirdly, improved WLS algorithm is proposed with weights adapting to measurement residuals to weaken negative effect caused by inner-region communication failures in local SEs. The main contributions of this paper are as follows.

infrastructure for distributed state estimation in ADNs. The CC-DSE which is robust to communication failures in ADNs is described in Section 4. Section 5 presents test cases and results of the proposed algorithm. Conclusion and discussion of future work are included in Section 6.

(1) A joint model combining network partition with PMU placement considering their tight coupling relation is built by considering the number of sub-regions, the equilibrium among sub-regions, and the observability as well as connectivity of sub-regions as partition criteria. It lays the basis for the implementation of DSE and facilitates the application of SE in large-scale systems. (2) An improved WLS algorithm is proposed with weights adapting to measurement residuals to mitigate negative effects caused by innerregion communication failures in local SEs without destroying the measurement redundancy, which is more applicable for limitedredundancy distribution networks. (3) A CC-DSE method is proposed with consensus-based coordination embedded into iterations of improved WLS in local SEs, with which the exchanged data can be calculated considering the communication topology of sub-regions for ADNs. It has great robustness to inter-region communication failures, which also has shorter computation time and adequate accuracy.

2.2. Network partition approach

2. Network partition approach combined with PMU placement In this section, a joint model of network partition and PMU placement is built considering their tight coupling relation by incorporating network partition in the PMU placement process. Based on the proposed network partition criteria, a partition approach is presented by using the topology analysis approaches [20]. Then, sub-regions are decoupled to allow DSE executed in parallel. To optimize PMU placement and results of network partition, a PMU placement model with multi-objective is established, considering PMU installation cost, partition equalization and DSE calculation performance. 2.1. Network partition criteria (1) Number of sub-regions. The increasing number of sub-regions will cause the reduction of computation time in each local state estimator. However, it will also bring about heavier communication burden and worse global convergence performance, which have negative impact on the efficiency and accuracy of DSE. Therefore, the number of sub-regions is crucial. A reasonable number of subregions can be decided by an empirical value 3 n as recommended by [26], where n represents the total number of node in the distribution network. (2) Equilibrium of each sub-region including number of nodes and measurement redundancy. Better equilibrium of node number and measurement redundancy in each sub-region can bring better efficiency and accuracy of local estimators since they are operated in parallel during DSE. The number of nodes in each sub-region can be given around by the average number calculated by the aforementioned number of sub-regions and the total number node in distribution network, that is n/3 n . (3) Observability and connectivity of sub-regions are the basis for local estimators. Pseudo measurements and virtual measurements are used to help each sub-region observable in DSSE for limited realtime measurements. Numerical method is used in observability analysis by evaluating the non-singularity of Jacobian matrix which is explained in detail in Section 3 as the gradient of measurement equation. Topology analysis approach is used to insure the connectivity of nodes in each sub-region.

For the given PMU placement, network partition approach is established based on topology analysis approaches including branch-line layer method and postorder-traversal algorithm, which has two stages: (1) node numbering and (2) network partition by topology searching. In the first stage, each node is numbered with a specific set representing its positional information. Firstly, branches are layered by their topological locations through branch-line layer method, with the main feeder being the first layer and the feeder connecting with the main feeder as the second layer and so on [27]. For instance, the 15-bus network shown in Fig. 1 is divided into three layers, where the main feeder is treated as the first layer, the three branches connecting with the main feeder as the second layer and the branch connecting with the second layer as the third layer. Then, each node is numbered as (A1, A2, A3), which represents its original bus number, the layer number, the node number in each branch, respectively. For example, the yellow node is numbered as (4, 1, 4), which represents that it is the fourth node of the whole network, at the first layer and the fourth node of this branch.

For the remainder of this paper, the network partition approach is provided in Section 2. Section 3 introduces the communication 2

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of sub-region 1 and sub-region 2 with node 4 as the overlapping node. S3-4, S4-5, and S4-10 represent branch complex power flows of line 3-4, 45, and 4-10, which can be acquired by PMU devices in node 4; while S4 represents injected power of node 4, which is obtained as the pseudomeasurement. It is noticed that the direction of the arrow does not represent the real direction of power flows. Radial networks can be decoupled simply in this way to perform local SE in each sub-region independently.

Sub-region 2 The second layer

(10,2,1) Sub-region 1 PMU

(1,1,1) 1 Root bus

PMU

(3,1,3)

(4,1,4)

The second layer (8,2,1) 82

(12,2,3)

PMU

The first layer (2,1,2)

(11,2,2)

(9,2,2) 922

(5,1,5)

(6,1,6)

(7,1,7)

The third layer

(13,2,1)

The second layer

(15,3,1)

2.4. PMU placement for network partition

Sub-region 3

It can be seen in Section 2.2 that PMU locations have a great impact on network partition. An unsuitable PMU placement may result in a poor partition scheme that fails to meet the criteria, deteriorating the performance of DSE. Hence PMUs should be configured properly to satisfy the partition criteria. In addition, for a given partition scheme, installing new PMUs may cause the disequilibrium of the measurement redundancy in each sub-region. Under this circumstance, the balance of the estimation accuracy of local SEs will be destroyed and further it will have adverse effect on the estimates of DSE. In short, there is the contradiction between network partition and PMU placement for DSE. To address this issue, a joint scheme that focuses on the combination of the network partition approach with PMU placement process is proposed in this paper for better DSE performance and economy. A multi-objective PMU placement model is established containing number of PMUs, equilibrium of partition and calculation performance of DSE and gives several candidate placement schemes, which is shown as follows.

(14,2,2)

Fig. 1. Network Partition of a 15 bus Distribution Network.

In the second stage, postorder-traversal algorithm is used to traverse and partition the distribution network in order to insure the connectivity of the sub-regions with the following process. Firstly, the distribution network is traversed back forward from the end node belonging to the largest layer branch. Then, if the next searching node is a cross node, we first traverse branches connected to this cross-node from the end of them and then the cross-node [28]. Finally, if the searched node deployed with PMU and the number of nodes reach the given amount of nodes, numerical method is used in observability analysis. If it is unobservable, add measurements until it is observable. These nodes compose a sub-region. As shown in Fig. 1, we traverse from the node (15, 3, 1) belonging to the largest layer. After the third layer branch finished searching as A3 = 1, the cross-node (13, 2, 1) is reached. We first traverse node (14, 2, 2) belonging to the second layer branch and then visit the cross-node (13, 2, 1). Likewise, after visiting (13, 2, 1), we turn to (7, 1, 7) to search back forward. The visiting path is (15, 3, 1) (14, 2, 2) - (13, 2, 1) - (7, 1, 7) - (6, 1, 6) - (5, 1, 5) with these nodes being sub-region 3. Then, node (5, 1, 5) acts as the starting node to go on traversing to node (4, 1, 4) with these nodes being sub-region 2. Next, we traverse from node (4, 1, 4) to the root node with these nodes being sub-region 1. It can be seen in Fig. 1 that the 15-bus network is divided into three sub-regions.

min(f1 , f2 , f3 ) f1 = NPMU =

S4 sub-region 2

3

10 4 5

sub-region 1

S4

Decoupling S4 S3-4

3

4

4

RMSEd 3 RMSEc

+

M) M)

Td 4 Tc

(1)

where NPMU is the number of PMU devices; xpii is a binary variable; xpii=1 means node ii deployed with a PMU device; xpii=0 means there is no PMU device in node ii; Ψ1 to ΨM represent measurement redundancy in sub-region 1 to M. f1, f2, and f3 reflect the economics of PMU placement, equilibrium of partition results, and calculation performance of DSE, respectively. Among them, the equilibrium of partition results is formed by the weighted sum of two sub-objectives including the normalized equilibrium of node number and measurement redundancy. Similarly, the calculation performance of DSE is formed by the weighted sum of two sub-objectives including the normalized estimation accuracy and the normalized estimation time. The weights of sub-objectives, β1, β2, β3, and β4, are obtained by the analytic hierarchy process (AHP) [29], which is applicable to all simulation systems. In different distribution systems, the concerns of PMU placement to DSE may be various according to the actual operation needs. Under this circumstances, different weights of β1, β2, β3, and β4 can be derived by adjusting the importance intensity of each sub-objective in the judgment matrix of AHP. The calculation performance f3 of DSE is obtained by simulation method in [20]. For each placement scheme, the network partition is implemented by the partition method described in Section 2.2 to obtain the value of multi objectives f1, f2, and f3. Based on the multi-objective genetic algorithm, an optimal PMU placement and its corresponding network partition result can be determined. The contradiction between network partition and PMU placement can be reconciled considering their tight coupling relation through this combined process, and the effective and balanced network partition for DSE is obtained while achieving good economy of PMU placement.

After partition, the network can be divided into several sub-regions. To ensure the global consistence of DSE, all power flows at branches connected to one overlapping node from adjacent sub-regions are transformed into the injected power of the overlapping node. As shown in Fig. 1, the 15-bus network is divided into three sub-regions with node 4 and node 5 as overlapping nodes. Fig. 2 shows the decoupling process

S3-4

f3 =

n x pii ii=1 nM ) max ( + 2 min ( 1 nM ) 1

s. t. 1 NPMU NPMU,max rank (Hi ) = n xi

2.3. Decoupling of Sub-regions

sub-region 1

f2 =

max (n1 1 min (n1

sub-region 2

10

5

Fig. 2. Decoupling Process of Sub-regions in 15-bus Network. 3

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4.1. Model of CC-DSE

Control Center

Sub-Control Center 1

Sub-region 1

Sub-Control Center 2

Terminals

min J (x ) =

Sub-region 3

Sub-region 2

Terminals

It is assumed that the ADN is divided into M sub-regions. The threephase CC-DSE model is presented as follows

Sub-Control Center 3

s. t .

=

M i=1 (z i

hi (xi ))T Wi (z i

hi (x i ))

a, b, c a, b, c V^O i = V^O j

Terminals

Communication with control center

M i=1 Ji (x i )

^ a, b, c = ^ a, b, c Oi Oj

(2)

where subscript i represents the ith sub-region. Ji(xi) is the sub-function of sub-region i. x is the state vector and z is measurement vector, including real-time measurements, pseudo measurements and virtual measurements. h(x) is the three-phase measurement function, which is detailed in Appendix A. Wi = diag(wil) is the weighted matrix, where wil is the weight factor of the ilth measurement and it is changeable. Wi = Ri−1, where Ri is the covariance matrix of measurement errors. The estimated states on overlapping nodes are required to be consistent among sub-regions in DSE. Equality constraint is the equality of states

Inter-region communication Inner-region communication

Fig. 3. Communication Architecture of ADNs.

3. Distributed communication for DSE A reliable communication system plays a vital role in state estimation, especially in DSE. After network partition, the distribution network is divided into several sub-regions. Accordingly, the communication architecture is changed from a centralized manner to a distributed one with 3level [22] which is shown in Fig. 3. The 3-level communication architecture contains with inner-region communication, inter-region communication, and communication from sub-regions to control center. The inner-region communication collects measurements from distributed terminals to subcontrol center which are used in local SEs in each sub-region. The interregion communication is used in data exchange and information coordination of adjacent sub-regions. The communication from sub-regions to control center is used to upload convergent estimation results to the control center. During the calculation of DSE, inner-region communication and inter-region communication are mainly considered. IEC 61850 and TCP protocol are adopted in the communication architecture. The minimum data transmission interval is 20 ms [21,22]. In this paper, static estimation is performed every 15 min generally, during which only one set of measurements is uploaded to the estimator. Thus, the data transmission interval can meet the requirements of information from adjacent sub-regions. In addition, each communication just takes several tens of milliseconds. It is negligible when compared with the elapsed time of DSE and the management cycle. Thus, the implement of other advanced applications would not be delayed. The DSE can be well performed under circumstance that communication system works in good conditions. However, unexpected failures or mistakes may appear randomly during communication which have a negative impact on DSE. Inner-region communication failure may bring bad data problem, which may increase estimation error and computation time in local state estimation. Meanwhile, inter-region communication failure may bring data missing and bad data problems, which have a great impact on data exchange and coordination between adjacent sub-regions. Data missing existed in adjacent sub-regions may bring data deficiency problem in data coordination and bad data transmission may bring wrong information to adjacent sub-region. All problems above have a negative effect on the accuracy and convergence property of DSE. Thus, we should take communication failures into consideration and give a robust DSE method. In this paper, it is assumed that the probability of data missing and bad data problems are given and the communication failures are inconsecutive.

a, b, c

a, b, c

a, b, c

a, b, c

in overlapping nodes where VOi , VOj , Oi , Oj represent the estimated three-phase values of the voltage magnitudes and angles of the overlapping node in sub-region i and sub-region j, respectively. The equality constraints of states on the overlapping nodes are converted into inequality constraints as shown in (3), which are employed as the global convergence criterion for DSE. When the differences between the local estimated states on overlapping nodes obtained from different sub-regions are less than the setting threshold, the DSE converges and the iteration is concluded.

x ij < xi j =

a, b, c | V a, b, c| = |V^O i

a, b, c V^O j |

a, b, c = | ^0 i

^ a, b, c |

|

a, b, c |

Oj

| (3)

where Δxij is the difference of estimated overlapping states in sub-region i and j. δ is the setting threshold. 4.2. Improved robust CC-DSE After network partition, local SEs of sub-regions are firstly executed by improved WLS in parallel with the objective function being divided into M parts. Then the estimated states of overlapping nodes are exchanged between adjacent sub-regions to help meet the equality constraints. The CC-DSE is considered being converged when the local SEs get converged and overlapping nodes reach equal states among adjacent sub-regions simultaneously. 4.2.1. Local state estimation by improved WLS method Local SE is performed in each sub-region by solving the objective function min Ji(xi) in (2) in parallel. The performance of the classic WLS algorithm can be improved by adjusting the weighted matrix in the objective function. There have been some studies focusing on this. For example, in [30], the adaptive weight assignment is proposed to handle the inaccuracy of real-time state estimation caused by the imprecise interpolated priori information in case of big disturbance. All weights of the interpolated measurements are adapted based on an adaptive negative exponential function according to their distance from the disturbance sources to mitigate the adverse effects on estimation accuracy. In this paper, a similar idea of weight adjustment is applied to deal with the inner-region bad data problems in local SEs which adversely affect the efficiency and accuracy of DSE. The improved WLS algorithm is performed by identifying bad data and revising their weights adaptive to residuals in local SEs, which can weaken the adverse influence of bad data caused by inner-region communication failures. The application of Gauss-Newton method for non-linear optimal condition leads to an iterative solution and updates the state vector by

4. Robust CC-DSE for communication failures Three-phase robust CC-DSE model of ADN is established based on WLS with boundary constraints. Each sub-region performs improved WLS in local SE considering inner-region bad data problems simultaneously with data exchange and consensus-based coordination embedded into the iteration of WLS which is robust to inter-region communication failures. 4

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Δxi in each iteration of sub-region i, which is calculated as follows

xi(k ) = (HiT Wi Hi ) 1·HiT Wi (z i xi(k + 1) Hi =

= xi(k ) + hi (xi(k ) )

/

xi(k) xi(k)

regions and E representing the connected adjacent sub-regions. The neighbor set of sub-region i is defined as

hi (x i(k) ))

Ni = {j

ys

1, j 0, i

lij = li i =

j

Ni j, j i

Ni

li j

(7)

where lij=−1 means that sub-region i and j are neighbors which have the same overlapping node, and the diagonal element lii gives the number of neighbors of sub-region i. Then the states with consensus of overlapping node in sub-region i can be calculated by

wil, yil < ys wi l / yi2l , yi l

(6)

E}

where j represents the jth sub-region. The corresponding Laplacian matrix [33] of the network is formed as L = [lij] based on the interaction topology of networks and its elements are defined as follows

(4)

where k denotes the iteration index and H represents Jacobian matrix defined as the gradient of h(x). For WLS, the measurement weights is calculated from the inverse square of the standard deviation of measurement uncertainty, assuming as Gaussian distribution, by wil = 1/σil2. σil is one third of a given percentage of maximum error about the mean value for a 99.7% coverage factor according to 3σ theorem the same as in [31]. The measurement residual error is defined as ril = zil − hil(x). In the iteration of WLS, adaptive weight changes with residual errors by the following rules

wil,new =

|(i , j )

(5)

xi, tk = x i, tk

where yil = ril/σil is the normalized residual for measurement il, and ys is the boundary setting based on 3σ theorem to judge whether the lth measurement is bad data or not. In (5), σil is a determined value for a certain measurement il and yil is related to its residual. During the iteration process, the residual error ril calculated by the estimated states is unequal to the true measurement error and not strictly Gaussian. Considering this deviation, the residual error of the normal measurement is allowed to slightly exceed the range of 3σ to mitigate the misjudgment of bad data. That is, the boundary ys of the normalized residual error yil can be set a little bit more than 3, which is determined by the parameter sensitivity analysis with detailed explained in Fig. 6 of Section 5. The improved WLS performs in the following steps. First, we determine whether the measurements are bad data or not by their normalized residual errors, which are calculated using all states in the region. Then, if the measurement is a bad data, its yil will be quite large and the weight is reduced by yil2 times. In this way, the negative effects of the detected bad data are mitigated by reducing their weights rather than removing them as shown in (5). Therefore, the improved WLS with adaptive weights can deal with the inner-region bad data without destroying the measurement redundancy, which is more applicable for limited-redundancy distribution networks. The improved WLS estimator can perform robust state estimation with almost no measurement information lost to keep the redundancy, yielding high estimation accuracy and good robustness. Moreover, when judging the inner-region bad data and processing it if necessary, only the measurement itself is used without considering the correlation among measurements. Therefore, the interaction of bad data will not affect the robust performance of the proposed CC-DSE. If the local SEs are executed independently without taking the consistency of estimated states of overlapping nodes into consideration, it may bring mismatch of the overlapping states and result in bad convergence properties. Thus, data coordination among overlapping nodes is very essential to improve the accuracy of DSE.

i, tk

=

1

1

tk

j Ni

1

i, tk 1

tk

(x i, tk (

1

xj, tk 1)

i, tk 1

j, tk 1)

(8)

j Ni

where tk = 1,2,…,TK. TK is the number of nonzero eigenvalue of L; xi,tk and σi,tk are the exchanged states and their deviations in the tk-th iteration of sub-region i; λtk is the tk-th nonzero eigenvalue of L. When tk = TK, the overlapping node states get an average consensus. Consensus-based coordination is used to process exchanged data including estimated voltage states on overlapping nodes and their deviation σ. After calculated by (8), the coordinated data are taken as additional measurements and transferred to adjacent regions to reexecute local state estimations until CC-DSE gets converged. Note that only the voltage states on overlapping nodes are employed in the detection and processing of such inter-region bad data. The additional measurements on the overlapping nodes transmitted from adjacent regions have higher accuracy after the consensus-based coordination, which can help correct the multiple interacting bad data centrally distributed in a sub-region. Moreover, the data exchange is embedded into the iteration of CC-DSE. In this way, each region can make use of the coordinated information in time for correction to mitigate the multiple interacting bad data. Data missing problem changes the communication topology and the corresponding Laplacian matrix. For example, the communication topology of three sub-regions with B and C having a data missing problem is shown in Fig. 4. The corresponding L1 matrix in normal case is changed into L2 with data missing problem shown as follows

L1 =

2

1 1

2

1 1

2

1 1

L2 =

2 -1 -1

-1 1 0

-1 0 1

(9)

In consensus algorithm, the average states calculated with L1 and its eigenvalue are the same as with L2 and its eigenvalue by (9) with the

A

4.2.2. Consensus-based coordination Inter-region communication failures may bring data missing or bad data problems. The proposed CC-DSE uses the consensus algorithm for data coordination and inserts consensus-based coordination into iterations of WLS to solve the communication failure problems. The consensus algorithm based on finite time average consensus protocol [32] is employed to coordinate the states of overlapping nodes among adjacent sub-regions. Data exchange can be described by a communication network, in which one sub-region means a communication node. The interaction of a communication network is represented by a graph G = (Γ, E), with the set of Γ representing sub-

A

B

1. Normal

C

B

C 2. With data missing

Fig. 4. Communication topology in normal case and with inter-region data missing problem. 5

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of simulation accuracy of state estimation algorithm, defined as

START Partition network into M sub-regions

RMSE =

Initialize states; k=0;

k=k+1 ···· ··

Calculate xi and xi of sub-region i

···· ··

Calculate xM and xM of sub-region M

Local SE by improved WLS

= (Tc

N

xi< xij<

(xreal, i

i=1

x^i )2

(10)

Td)/ Tc × 100%

(11)

where Tc and Td are the elapsed time for the CSE and DSE, respectively. The efficiency of distributed state estimator is better when α is closer to 1.

Data exchange between adjacent regions; Calculate xij

Consensus-based coordination

ns

^ is the estimated value of where xreal is the true value of state vector, x state vector; ns is the total number of state vector. Voltage magnitude and angle errors are calculated for different units, respectively. The improved efficiency of the distributed estimator is defined by the following formula [34]

Store updated data

Calculate x1 and x1 of sub-region 1

1 ns

5. Case study

Y

Record the results

The proposed algorithm has been applied to the unbalanced IEEE 123 bus and modified IEEE 8500 bus network. Test data are available in [35]. It is assumed that SCADA measurement devices are installed already in test networks and pseudo measurements are known for all the load nodes. All measurements are randomly extracted according to their probability density functions (PDFs), which is assumed as Gaussian distributions. The standard deviations of measurement noise is equal to one third of a given percentage of maximum error about the mean value for a 99.7% coverage factor. The given percentages are 1% for SCADA measurements, 0.7% for PMU measurements and 50% for pseudo measurements, which are the same as those in [31]. Gaussian mixture model (GMM) is used to approximate the PDF of output power of DGs [36]. By Monte Carlo simulations, the power outputs of DGs are sampled from the PDFs. The power outputs of DGs are used as pseudo measurements and covariance of GMM are used in weight matrix in state estimation, which is solved by WLS method. Real-time measurement placement [37,38] and DG installation in IEEE 123 bus and modified IEEE 8500 bus networks are given in Table 8 and 9 in Appendix B. Some parameters to be used in subsequent simulations are firstly derived. Among them, the parameters involved in network partition and PMU placement model include the weights of sub-objectives, β1, β2, β3 and β4, which are obtained by AHP. First, the relative importance of two sub-objectives under the same objective is determined and quantified based on the 1–9 scale to construct the judgment matrix. Considering the fact that estimation time is slightly more important than accuracy in DSE, the importance intensity of the normalized estimation time to the normalized estimation accuracy is set as 2 in the following judgment matrix Aβ.

END

Fig. 5. Flow chart of CC-DSE.

same data. The coordination integrated into WLS can also help deal with data missing problems. 4.2.3. Convergence of CC-DSE The data coordination is embedded in local SE, thus the convergence of CC-DSE should consider the convergence of local SEs and equation constraints of overlapping node. For local SE, it is converged when Δxi < ε, where ε is the setting value. The global convergence is judged by (3). When the local SEs get converged as Δxi < ε and the estimated states of overlapping nodes among adjacent sub-regions can meet (3) simultaneously, CC-DSE is converged. 4.2.4. Iteration steps The flow chat of CC-DSE is shown in Fig. 5 including the network partition and iteration process of CC-DSE. The entire process of CC-DSE is described in the following steps. (1) The network is divided into M sub-regions. (2) Initialize the states. k = 0. (3) k = k + 1. Local SE of each sub-region is executed in parallel to get Δxi. (4) Data exchange between adjacent sub-regions. Calculate Δxij. (5) Check out the convergence of CC-DSE by Δxi and Δxij. If it is converged, record the results, else turn to (6). (6) Consensus-based data coordination is carried out. The coordinated data are taken as additional measurements for local SE, then turn to (3).

A = 1 1/2 2 1

(12)

Then, the eigenvector β34 = (0.333, 0.667) is derived from the maximum eigenvalue of Aβ, whose elements respectively represent the weight of the estimated accuracy and time under the calculation performance of DSE. That is, β3 = 0.333, β4 = 0.667. The equilibrium of node number mainly influences the estimation time in DSE, while the equilibrium of measurement redundancy is closely related to the estimation accuracy. Therefore, the importance intensity of the equilibrium of node number to that of measurement redundancy is also set as 2. By the above process, we can get β1 = 0.667 and β2 = 0.333. In the improved WLS algorithm with adaptive weights, the judgment boundary ys of bad data is determined by the parameter sensitivity analysis. Given that ys should be greater than 3, we make it continuously change from 3 to 40 and perform SE on IEEE 123 bus networks for each ys respectively. The simulation results of estimation accuracy are shown in Fig. 6, where RMSE is the average value of root-

CC-DSE is proposed to deal with large-scale SE problems of ADNs which provides accurate states in reasonable computation time with robustness to communication failure. Each sub-region performs local SE by improved WLS method in parallel after network partition to decrease computation time which can diminish the negative impact of innerregion communication failure. The data exchange and coordination between sub-regions by consensus algorithm can improve the accuracy of states and it also has better robustness to inter-region communication failures. Moreover, only one iteration loop until convergence is needed. Thus, the computational efficiency is improved. 4.3. Evaluation indexes The root-mean-square error (RMSE) is used as the evaluation index 6

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6.2

RMSE/1e-4

6

Sub-region 4

5.8 5.6 5.4 5.2 5 0

5

10

15

20 ys

25

30

35

40 Sub-region 1

Fig. 6. Estimation accuracy with different ys.

mean-square error of voltage magnitudes and angles. It can be seen that, the estimation error is the smallest when ys is around 5, yielding the best effects on the bad data judgment and mitigation. For simplification, ys is set as the integer 5. The proposed three-phase CC-DSE of distribution network is implemented in a parallel way by using SPMD of MATLAB 2016a on the computer, which has 3.3 GHz processor, 10 GB RAM and 64-bit operating system. Tests have been performed by using Monte Carlo simulations for the consideration of measurement uncertainty, 200 trails for each test, to evaluate the accuracy performance of the DSE algorithm. ε and δ are set as 10 - 4 . Test results are compared with traditional DSE [17]. Detail explanation is given in Appendix C.

Sub-region 3

Fig. 7. PMU placement and network partition result of IEEE 123 bus network.

5.2. Performance comparison of the improved WLS and WLAV The performance of the proposed improved WLS is compared with that of the WLAV, in which both the normal case with no bad data and the cases with different amount of bad data are considered. In particular, when there is no bad data exists in the simulation, the weights of normal measurements will not be adjusted by the improved WLS which performs similarly to the classic one. The distributions of RMSEs of both voltage magnitude and phase angle for different algorithms in the normal case without bad data is shown in Fig. 8. Table 2 shows the estimation accuracy in terms of the RMSEs in each scenario with different bad data case based on the two algorithms. It can be seen that, the estimation accuracy of the improved WLS is better than that of WLAV both in the normal case and the cases where bad data exist, even if the amount of bad data is increasing. The rationality of the reported results can be explained from the analysis of algorithm that the WLAV is not as applicable to the distribution network state estimation as the improved WLS. Because the number of accurate real-time measurements in distribution networks is far less than that of states to be estimated and lots of imprecise pseudo measurement are added to satisfy the required measurement redundancy. Under this condition, it is difficult to meet the precondition for performing precise state estimation via WLAV, yielding lower estimation accuracy and worse robustness compared to the improved WLS. Although the estimates based on the two algorithms differ by nearly three times in numerical, the estimation results on the order of magnitude 10 3 and 10 4 are both accurate enough to meet the accuracy requirement of state estimation. However, we prefer to choose the

5.1. PMU placement and network partition results The PMU placement and partition results are shown in Table 1 for the test networks. Table 1 shows the PMU placement and equilibrium of partition result in test networks. In order to minimize the f1, PMUs are only installed at the overlapping nodes. At the same time, the location of PMUs is optimized to ensure equilibrium among sub-regions. IEEE 123 bus network is divided into 4 sub-regions with 3 overlapping nodes placed with PMUs, which is shown in Fig. 7. Each subregion has similar node number and measurement redundancy. Modified IEEE 8500 bus network is divided into 13 sub-regions with the nodes above being overlapping nodes. The modified IEEE 8500 network is so large that there is no figure to show the partition results in this paper. The process of network partition in IEEE 123 bus network has two stages as illustrated in Section 2.2. First, each node is numbered with a specific set representing its positional information. Then, the distribution network is traversed back forward from the end node in the largest layer branch to the node placed with PMU by using postorder-traversal algorithm. The searching order is detailed in Appendix D.

Table 1 PMU placement and partition results of test networks. PMU placement (node)

IEEE 123 bus network Modified IEEE 8500 bus nerwork

Equilibrium Number of node

Measurement redundancy

19, 61, 73

1.320

1.129

86, 115, 147, 170, 210, 237, 790, 805, 866, 897, 920, 935, 963, 988, 1116, 1161, 1344, 1372, 1422, 1583, 1620, 1692, 1839, 1875, 1977, 2143, 2100, 2262, 2427, 2485

1.321

1.096

Fig. 8. The distributions of RMSE in the normal case without bad data under different algorithms. 7

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Table 2 State estimation results under different algorithms.

Table 4 Simulation results with different types of bad data.

Number of bad data

Improved WLS

WLAV

Number of bad data

RMSE of V (×10−4)

RMSE of θ (×10−4)

RMSE of V (×10−4)

RMSE of θ (×10−4)

0 1 2 3

6.93 7.56 7.35 7.74

3.36 3.46 3.52 3.49

20.7 21.3 20.4 21.1

15.8 15.4 15.0 16.1

Multiple non-interacting bad data Multiple interacting bad data

RMSE of V (×10 p.u.) RMSE of θ (×10−4 p.u.) RMSE of V (×10−4 p.u.) RMSE of θ (×10−4 p.u.)

2

3

4

6.96 4.46 7.22 4.36

7.11 4.49 7.27 4.55

7.37 4.53 7.95 5.01

calculation time has already been shorten compared with CSE since the dimensions of computation can be reduced evidently with 13 sub-regions. Therefore, time reduction of CC-DSE isn’t too far from DSE.

Table 3 State estimation results of distribution networks.

RMSE of V(×10−4) RMSE of θ (×10−4) Simulation time (s) α (%)

−4

IEEE 123 bus network

Modified IEEE 8500 bus network

CSE

DSE

CC-DSE

CSE

DSE

CC-DSE

9.15 4.07 1.308 –

3.97 1.66 0.401 69.4

6.93 3.22 0.233 82.2

4.42 5.65 137 –

2.97 4.71 9.3 93.2

4.46 5.49 6.9 95.0

5.4. Robust CC-DSE results with communication failures The robustness of CC-DSE can be reflected in two aspects, as interregion communication failures and inner-region bad data. First, tests are carried on under situations with multiple non-interacting or interacting bad data to show the robustness of the CC-DSE to various types of bad data. The multiple interacting bad data are resulting from the assumption that the voltage measurement of a node has large errors due to the communication failure and then some the relevant branch measurements related to it is also become bad data relevant. Table 4 shows the estimation accuracy considering different types and amounts of bad data. It can be seen that, for both multiple non-interacting and interacting bad data, the CC-DSE can ensure robust estimation performance and obtain high estimation accuracy. Simulation cases are shown in Table 5. The inner-region bad data are set as the negative of real measurements [24]. The inter-region communication failure and the inner-region bad data are both expressed by the given probability. Selected bad data in this way, in addition to the single and multiple non-interacting bad data, there may be multiple interacting bad data in the simulations. However, regardless of the interaction among bad data, the robust CC-DSE can guarantee the great estimation performance. In Table 5, case 1 is the normal situation for comparison. Inter-region bad data is set only in case 3 for it has similar results with interregion data missing. The simulation results of CC-DSE in eight cases are shown in Tables 6 and 7, which are compared with those of DSE in [17]. For DSE, it cannot get a convergence in case 7–8 in IEEE 123 bus network and in case 6–8 in modified IEEE 8500 bus network. Thus, there are no the simulation results in the table. Intuitive results of IEEE 123 bus and modified IEEE 8500 bus test networks are shown in Figs. 9 and 10.

improved WLS with higher estimation accuracy and better robustness to bad data compared with the WLAV. Therefore, it is reasonable and superior to implement the robust CC-DSE on the basis of the improved WLS algorithm with adaptive weights. 5.3. Results of proposed CC-DSE in normal case The proposed CC-DSE simulation results are given by RMSE and simulation time in comparing with those of CSE and traditional DSE in [17], which is shown in Table 3. It can be seen from Table 3 that CC-DSE method has sufficient accuracy and the shortest simulation time among the three methods in test networks. In terms of accuracy, distributed approaches have better accuracy than the CSE and CC-DSE has lower accuracy than DSE. Explanations are as follows. (1) In distributed approaches, the high-precision PMU measurements are calculated more than once in the overall model since these data are shared among adjacent sub-regions. Therefore, there is a higher measurement redundancy and a better utilization of the highprecision PMU data in DSE, which facilitates enhance SE accuracy. (2) The data interaction among sub-regions also helps improve measurement redundancy. In distributed approaches, each local state estimator uses not only local measurements but also data from neighboring subregions. The estimated states of overlapping nodes from neighboring sub-regions are added as additional measurements into local state estimation. (3) The CC-DSE embeds the coordination into WLS iteration. Thus, compared with DSE, it has only one convergence in each subregion, which bring lower accuracy than DSE. As for elapsed time, CC-DSE executes the shortest computation time for the following reasons. (1) Compared to CSE, the dimensionality of sub-region state estimation is reduced and calculation of each sub-region is performed in parallel. (2) Compared with DSE, the number of iteration times in CC-DSE is less than DSE. It can be seen from the value of α that the efficiency of CC-DSE has been improved by 82.2% in IEEE 123 bus network. CC-DSE is scalable in very large-scale network. The dimensions of computation can be reduced even more evidently with more sub-regions, which can bring even better performance in efficiency. In modified IEEE 8500 bus network, the efficiency of DSE and CC-DSE are improved by 93.2% and 95.0%. The reason is that DSE

Table 5 Simulation cases. Probability of inter-region communication failure

Case Case Case Case Case Case Case Case

8

1 2 3 4 5 6 7 8

Data missing

Bad data

0 10% 0 0 10% 20% 0 20%

0 0 10% 0 0 0 0 0

Probability of inner-region bad data

0 0 0 5% 5% 0 10% 10%

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algorithm has great scalability, which can be used in very large scale distribution networks. They can be explained as the following reasons. (1) The CC-DSE embeds the coordination into WLS iteration, thus, compared with DSE, it has only one convergence in each sub-region, which brings lower accuracy but higher efficiency. (2) DSE has no means to solve communication failure problems. It is unable to get satisfactory results when communication failures exist. (3) Inter-region communication failures have a major impact on global convergence by interfering data interaction between sub-regions, which brings a large Td. Local SEs may not get a convergence when there are too much inner-region bad data, which brings large RMSEs or even affect the global convergence. (4) The proposed CC-DSE can deal with communication failures efficiently by embedding the consensus-based coordination into iteration of local SEs and improving WLS method with adaptive weights. In normal conditions, CC-DSE has a better computational efficiency and adequate accuracy. When simulation conditions get worse, CC-DSE has better robustness in computational precision and simulation time. Therefore, the proposed approach has better calculation performance in state estimation of ADNs.

Table 6 Simulation results of IEEE 123 bus network. Case

1 2 3 4 5 6 7 8

Td (ms)

RMSE of V (×10−4 p.u.)

RMSE of θ (×10−4 p.u.)

DSE

CC-DSE

DSE

CC-DSE

DSE

CC-DSE

400.6 552.7 560.7 460.8 498.0 589.0 \ \

233.1 234.6 235.5 233.8 236.0 237.1 237.0 241.6

3.97 6.40 7.32 9.42 12.80 9.52 \ \

6.93 7.03 7.10 7.88 7.96 6.95 7.17 7.04

1.66 2.81 3.06 5.01 11.57 8.63 \ \

3.22 3.25 3.24 3.34 3.26 3.35 3.31 3.31

Table 7 Simulation results of modified IEEE 8500 bus network. Case

1 2 3 4 5 6 7 8

Td (s)

RMSE of V (×10−4 p.u.)

RMSE of θ (×10−4 p.u.)

DSE

CC-DSE

DSE

CC-DSE

DSE

CC-DSE

9.32 12.4 12.8 10.9 11.3 16.1 \ \

6.93 7.21 7.15 7.32 7.28 7.64 7.47 7.73

2.97 6.52 8.23 12.1 15.7 14.8 \ \

4.46 5.17 5.73 4.92 5.86 6.13 6.11 6.85

4.71 5.99 7.21 10.9 14.1 13.7 \ \

5.49 5.80 5.92 6.01 7.01 6.88 6.54 7.56

6. Conclusion This paper proposes a robust consensus-based coordination distributed state estimation for ADNs to reduce calculation dimension and time for limited real-time measurements in parallel, which is robust to the inter-region data missing and bad data caused by link failures and the existence of inner-region bad data. Network partition approach combined with PMU placement can divide the large-scale networks into several sub-regions with overlapping nodes deployed with PMU devices, which can help the information exchange and take full use of limited real-time measurements. Inter-region communication and consensus-based coordination are embedded into iterations of DSE, which help increase the robustness of dealing with inter-region link failures. The improved WLS method with adaptive weights is robust to innerregion bad data without removing them. Tests on IEEE 123 bus and modified IEEE 8500 bus network show the proposed methodology has better calculation efficiency compared with CSE and traditional DSE. The CC-DSE is more robust to communication failures in accuracy and efficiency than traditional DSE.

Fig. 9. Simulation results of eight cases (a) simulation time, (b) RMSE, by using DSE and CC-DSE in IEEE 123 bus network.

CRediT authorship contribution statement Tingting Zhang: Writing - review & editing, Software, Visualization, Investigation. Peiran Yuan: Conceptualization, Methodology, Writing - original draft, Software. Yaxin Du: Writing review & editing, Software, Investigation. Wen Zhang: Supervision, Project administration, Funding acquisition. Jian Chen: Supervision.

Fig. 10. Simulation results of eight cases (a) simulation time, (b) RMSE, by using DSE and CC-DSE in modified IEEE 8500 bus network.

Declaration of Competing Interest

Several conclusions can be obtained as follows from the simulation results above. (1) In the normal case, CC-DSE has sufficient calculation accuracy and higher efficiency. (2) For DSE, the Td and RMSE increase considerably with the cases getting worse. It cannot even get a convergence in case 7–8 in test networks. (3) Inter-region communication failures have a greater impact on Td and inner-region communication failures have a greater impact on convergence and RMSE. (4) For CCDSE, Td and RMSE can be held steadily with better robustness even when DSE cannot get the convergence. (5) The proposed CC-DSE

The authors declared that there is no conflict of interest. Acknowledgments This work was supported in part by the Natural Science Foundation of Shandong Province under Grant ZR2018MEE038 and in part by the National Key R&D Program of China under Grant 2017YFB0902600.

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Appendix A. Three-phase measurement model The three-phase measurement functions are detailed as Eqs. (13)–(15) 3

[gpq (Vp, cos

pp

Vq, cos

pq )

=1

+bpq (Vp, sin

pp

Vq, sin

pq )]

3

[gpq (Vp, sin

pp

Vq, sin

pq )

=1

bpq (Vp, cos

pp

Vq, cos

pq )]

Ppq, = Vp,

Qpq, = Vp,

(13)

where Ppq,φ and Qpq,φ are the active and reactive branch power flow between node p and node q. φ and ξ represent different phases.

Vp, = Vp, p,

=

(14)

p,

where Vp,φ and θp,φ are the voltage magnitude and angle of node p phase φ.

Pp, = Vp, Qp, = Vp,

ni

3

q= 1 = 1 ni

3

q= 1 = 1

[Vq, (gpq cos

pq

+ bpq sin

pq )]

[Vq, (gpq sin

pq

bpq cos

pq )]

(15)

where Pp,φ and Qp,φ are the active and reactive injected power of node p. Appendix B. Measurement devices and DG placement Since the measurement quantity of SCADA is too large in modified IEEE 8500 bus network, no display is given here. Table 8 Measurement Devices Placement in IEEE 123 bus Network. Type

Node

IEEE 123 bus network SCADA (V)

SCADA (S)

2, 33, 46, 77, 109

24-25, 45-48, 54-55, 77-87, 78-79, 102-106

Table 9 DG placement in test networks. Type

IEEE 123 bus network

Modified IEEE 8500 bus network

Node

11, 34, 50, 64, 79, 105

237, 395, 932, 964, 991, 1116, 1170, 1280, 1372, 1577, 1608, 1692, 1871

Appendix C. Traditional DSE The entire process of traditional DSE is the same as in [17] used as comparison method in this paper, which is depicted in Fig. 11. In traditional DSE, local estimators of sub-regions using WLS method without adaptive weight are performed in parallel. Data exchange without processing is executed when local state estimations get converged. The exchanged data between sub-regions are taken as additional measurements.

Start Store updated state data Local state estimation in a parallel way

Data exchange and coordination

N

Global converge Y End

Fig. 11. Flowchart of traditional DSE. 10

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Appendix D. Partition process in IEEE 123 bus network First, each node is numbered with a specific set representing its positional information. Then, the distribution network is traversed back forward from the end node in the largest layer branch to the node placed with PMU. The searching order is shown in Table 10. Number the sub-regions after partition. Table 10 Searching order in IEEE 123 bus network. Sub-region

Searching order

3 4 2 1

97-96-95-94-93-92-91-90-89-88-87-86-85-84-83-82-81-80-79-78-77-76-75-74-73 112-115-114-113-111-110-117-109-108-107-106-105-104-103-102-116-101-100-99-98-72-71-70-69-73-68-67-66-65-64-64-62-61 119-52-51-50-49-48-47-46-45-44-43-42-41-40-39-38-37-36-33-32-34-28-27-118-31-30-29-26-25-24-23-22-21-20-19 18-17-16-35-60-59-61-58-57-56-55-54-53-19-14-15-12-11-10-13-9-8-7-6-5-4-3-2-1

Appendix E. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijepes.2019.105732.

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