Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Fully Distributed AC State Estimation Method for Power System Based on Information Propagation Algorithm  PDF

  • Qiao Li 1
  • Yu Shen 2
  • Lin Cheng 2
  • David Wenzhong Gao 1
1. University of Denver, 2155 E Wesley Ave, Denver, CO 80210, USA; 2. Tsinghua University, Beijing, China

Updated:2023-07-24

DOI:10.35833/MPCE.2021.000764

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Abstract

This paper proposes a new distributed AC state estimation method. Different from the popular distributed state estimation (DSE) methods based on area partitioning method, the proposed method is a truly distributed method in which the power system is not required to be divided into smaller areas and a centralized state estimator in each area is not needed. In order to achieve fully DSE, the information propagation algorithm is introduced in this paper to help the distributed local state estimators share the measurement data. The information propagation algorithm is developed based on consensus protocol. The proof of the convergence of the information propagation algorithm is provided in this paper. Then, the AC state estimation method is integrated with the information propagation algorithm to realize the proposed method. The proposed method is tested in different standard power system models. The results show that the proposed method reaches the similar accuracy as the traditional centralized state estimation methods and performs faster and more accurate than the existing DSE methods.

I. Introduction

IN power systems, state estimation is a very important technique, which is used to estimate the power system states such as the phase angles and voltage magnitudes on the buses. The traditional state estimation methods are mostly centralized in which the state estimation should be performed in a centralized facility, e.g., centralized state estimator or supervisory control and data acquisition (SCADA) system. The most widely used state estimation method in industry is the Newton’s method with weighted least square (WLS) technique [

1]-[5]. Also, some other methods are based on Kalman filter method [6]-[8]. These methods are suitable for the traditional power systems which have centralized structure. However, with the development of renewable energy, microgrid, and smart grid, the power generation of power systems will become more and more distributed in the future. As a result, the control system and monitor system of the power system will get more decentralized as well. So, the conventional centralized state estimation method will be hard to fit in the future power systems, and the new distributed state estimation (DSE) methods are needed. In addition, the DSE methods are operating in distributed communication network which is more robust than the centralized system. This is because, according to the graph theory, a centralized system is in a star graph which has the lowest connectivity (the minimal number of edges needs to be removed to separate the graph into isolated subgraphs), while distributed network can easily have larger connectivity. Therefore, DSE method will be an important technique for future power systems.

There are some existing researches about DSE methods. The majority of them are based on area partitioning method (or called multi-area method) [

9]-[19]. In these methods, the power system should be partitioned into several smaller areas and the state estimation problem for the entire system is usually decomposed into smaller problems corresponding to the partitioned areas. In each area, there is a centralized area state estimator to perform the state estimation for the buses within the area and communicate with the area state estimator from the nearby areas. Based on this concept, different DSE methods are established. For example, [9] and [10] introduce a DSE method for multi-area power system. The state estimation of each sub-area is performed locally and the border information of the areas are exchanged at a coordination state estimator. In [11], a distributed robust bilinear state estimation (D-RBSE) method for multi-area power systems is proposed. The D-RBSE method is developed based on bilinear state estimation [20] with the area partitioning concept. In this method, the state estimation problem in each area is solved locally and a small amount of data is transmitted between different areas to achieve the globally accurate estimation. In [12], a three-stage DSE method for AC-DC hybrid distribution network is proposed. In the first stage, the alternating direction multiplier method (ADMM) is used to perform the preliminary state estimation over the distributed multi-area system. In the second stage, a centralized nonlinear conversion system which calculates the intermediate variables from the results of the first stage by considering the nonlinearity of the system is used. In the third stage, a centralized state estimation system which computes the final estimated states by the intermediate variables from the second stage is used. Reference [13] proposes a DSE method based on parallelized stream computing on the real-time cloud platform. In the computation, the interconnected power grids with tie lines are decoupled into sub-regions to realize the state estimation. In sum, the latest works on DSE are based on the multi-area method as before, in which they are still not totally decentralized. In [14], a DSE method based on ADMM is proposed. The proposed method is more robust and efficient compared with the conventional least-squared based power system state estimation method. Reference [15] proposes a fully decentralized adaptive re-weighted state estimation (DARSE) scheme. The DARSE is developed based on the gossip-based Gauss-Newton (GGN) algorithm, which enables the agents in sub-areas to solve the global problem collaboratively. Reference [16] develops a secure distributed DSE method for wide-area smart grids. The proposed method is majorly focused on improving the security of the state estimation system by integrating block chain and anomaly detection techniques to protect the DSE system. For the DSE system, it also uses multi-area method where the Kalman filter is performing state estimation in each local center. Reference [17] introduces a distributed moving horizon estimation scheme. The proposed DSE is based on operator splitting techniques. The local estimators are installed in the sub-areas of the system to perform the state estimation. Reference [18] considers the DSE problem in unbalanced active distribution network (ADN). The WLS method is used with the area-partitioning technique to realize the DSE. Finally, [19] proposes a DSE method based on parallelized stream computation and power grid decoupling. The proposed method divides the parallel computing sub-regions based on the regulatory sub-regions of the actual power grid. The sub-region state estimations are corrected by the sensitivity matrix coordination algorithm and regional tie-line state normalization.

In addition to the area partitioning methods, other DSE methods are introduced in [

21]-[23] to address the DSE problem. However, the centralized facilities such as global positioning system (GPS) and SCADA are required in these methods. Therefore, these methods are not totally distributed as well.

Besides, for sensor networks, there is a consensus-based DSE method [

24] which is totally distributed without using area partitioning. For the state estimation in sensor networks, all sensors measure the same parameter of the system and the state estimation algorithm merges all measurements together to improve the reading precision. However, for the state estimation of power systems, all meters measure different parameters (the power flow on different transmission lines) and the state estimation algorithm combines these measurements to figure out the states of the entire power system. Hence, the DSE method for sensor networks cannot be directly used on power systems due to the fundamental difference between the state estimations in sensor networks and in power systems as discussed above. But the consensus-based method is a suitable technique to develop totally distributed algorithms for power systems. For example, the consensus-based method has been used for the development of distributed economic dispatch [25] and distributed optimal power flow [26].

Based on the discussion above and wide literature searching, it can be found that almost all existing DSE methods for power systems are not totally distributed, i.e. the centralized structure is still needed in the system. For example, the multi-area methods are not fully distributed because the centralized structure still exists inside the partitioned areas. Also, some methods need a centralized coordinator to organize the computation between different areas. This drawback makes the distributed power system less robust since the centralized structure still exists in the system. Also, due to the same reason, these methods are not compatible with some other distributed methods in power systems, e.g., distributed economic dispatch methods [

25], so that the development of more advanced distributed techniques in power systems is limited. In addition, some area partitioning methods have strict requirements on the size of sub-areas, the way to divide the system, and the overlapping between the areas, which makes the area partitioning method hard to be utilized in some power systems since not all power systems can be easily divided.

To address these problems, this paper proposes a new distributed AC state estimation method which is totally distributed. The fully DSE method is a DSE method without any centralized structure. Hence, in a fully DSE method, the state estimation system is more robust against network failures. The proposed method is developed based on the information propagation algorithm introduced in [

27]. The information propagation algorithm is a consensus-based method [24] and it is a useful technique for data sharing in a distributed communication system. With the help of information propagation algorithm and AC state estimation method [28], the fully DSE method is established. For this method, no centralized facility or structure is needed. The state estimation can be performed with the DSE nodes in the power system. A DSE node is a meter attached with a distributed local state estimator.

Also, according to the discussion from the literature review, the consensus-based methods have also been used for distributed economic dispatch and distributed optimal power flow. The main idea of the consensus-based methods for these applications is using consensus protocol to synchronize the incremental costs of all generators in the power system so that the cost of power production can be minimized. This is different from the proposed DSE, since the consensus protocol is used to develop the information propagation algorithm to broadcast local sensor data to the entire system.

The major contributions of this paper are listed as follows.

1) A fully distributed AC state estimation method is proposed with fully AC nonlinear power flow equations. Compared with the existing methods, the proposed method can estimate the states of the power system without any centralized facility of structure.

2) The information propagation algorithm is introduced and its convergence is theoretically proven, which is a useful and robust technique for data sharing in distributed communication network.

Note that this paper provides theoretical proof of the convergence of the information propagation algorithm and this method is used with fully AC nonlinear power flow equations, resulting in more precise estimates than the algorithm developed with DC power flow model in [

27].

This rest of the paper is organized as follows. Section II presents the problem statement. Section III presents the proposed method. In Section IV, the simulation results are presented. Finally, Section V concludes this paper.

II. Problem Statement

In AC state estimation, the power system measurement process can be modeled as:

z=hxs+η (1)

where z is the measurement vector of readings from all meters; xs is the state vector of the power system, in which the phase angle θ and voltage magnitude V on the buses are the states, i.e., xs=θ;V; h is the observation model; and η is the noise or error in the measurement. For AC state estimation, the following AC power flow model is used to build the observation model as hθ,V=P;Q.

Pnm= Vn2Gnm-VnVmGnmcos θnm+Bnmsin θnmQnm=-Vn2Bnm-VnVmGnmsin θnm-Bnmcos θnm (2)

where Pnm=Pn-Pm and Qnm=Qn-Qm are the real and reactive power flows between buses n and m, respectively; Vn is the voltage magnitude on bus n; θnm=θn-θm is the phase angle difference between buses n and m; and Gnm and Bnm are the conductance and susceptance between buses n and m, respectively.

For the centralized state estimation, there is a centralized state estimator as an example shown in Fig. 1(a) to collect all measurements from the meters. Then, the centralized state estimator solves the following optimization problem [

28] to obtain the optimal estimation for the states.

minx Jx^s=i=1Nmzi-hix^s2σi2 (3)

Fig. 1  An example of power system state estimation with four buses. (a) Centralized state estimation. (b) Proposed method.

where Jx^s is the sum of measurement residuals; x^s is the estimation of xs; Nm is the total number of measurements (meters) in the power system; zi is the reading on the meter i; hix^s is the estimated measurement on the meter i by inputting the estimated state x^s into (2); and σi2 is the variance for the measurement of the meter i.

However, for the DSE, the measurement system is shown in Fig. 1(b). There is no centralized facility such as the centralized state estimator to collect all measurement data and perform the state estimation. Instead, the DSE nodes are installed on the power lines to perform the measurement and DSE. A DSE node is a combination of a meter and a local state estimator, as shown in Fig. 1(b). The meter detects the real and reactive power in the power line and the local state estimator executes the DSE algorithm. The local state estimator connects to the neighboring local state estimators via the communication lines. Since the local state estimator only connects to its neighbors, it cannot directly obtain the measurement data from other nodes which are not immediate neighbors. As an example shown in Fig. 1(b), node 1 only connects to node 3, so the readings from node 4 and node 5 are not available for node 1 to realize the state estimation. To solve this problem, the information propagation algorithm is proposed to allow the nodes to estimate the readings from the nodes which are not the neighbor. In Fig. 1, Rik is the resistance between node i and node k; Xik is the reactance between node i and node k; and Mik denotes the meters between node i and node k.

III. Proposed Method

A. Information Propagation

The goal of the proposed information propagation algorithm is to allow the DSE nodes to accurately estimate the local information of all other nodes in the system. The local information of a node is its meter reading in this paper. For example, in Fig. 1(b), all local state estimators run the information algorithm. For each node, taking node 1 as an example, its local state estimator calculates an estimation vector z^11;z^12;z^13;z^14 by the information propagation algorithm, where z^ij means the estimation for the reading zj of the node j by the node i. If the estimation vector can converge to the actual measurement Z=z1;z2;...;zn, it means that the measurement data from all other nodes are available for the local state estimator on node 1, and thus the state estimation can be performed.

In this paper, the information propagation algorithm is developed based on the consensus protocol technique [

24]. In this method, the communication network of a power system can be modeled as a graph, in which the DSE nodes, e.g., the four nodes in Fig. 1(b), are treated as vertices and the communication lines between the nodes are the edges of the graph. Suppose that xi is the information state of the ith node in the graph. For the DSE problem in this paper, the estimation vector can be treated as the information state, i.e., xi=z^i1;z^i2;;z^iNm. The information propagation algorithm for the node i is proposed as:

xik+1=xik+τIi0j𝒩iwijxjk-xik (4)

where xi(k) is the information state xi at the time step k (the initial values z^ij0 in xi0 can be arbitrary if ij, but z^ii0=zi since the local information zi is available at the ith node itself); τ is the time interval between two update consecutive time steps; wij is a weight coefficient on the edge between node i and node j; 𝒩i is the set of neighbor nodes of node i; and Ii0 is an Nm×Nm diagonal matrix which is the same as the identity matrix but has a zero at the ith diagonal entry.

Ii0diag1,1,...,1,0,1,...,1 (5)

In order to achieve the goal of the information prorogation algorithm, xi should converge to the actual measurement vector, i.e., limk xik=z1;z2;...;zNm. Here, a proof of this statement is provided below. The proof includes two parts. The first part proves that the estimation vector xi(t) converges to an equilibrium point xilimt xi(t) under the algorithm (4). The second part proves that the equilibrium point xi() equals to the actual measurement vector Z.

1) Proof of Convergency of Information Propagation Algorithm

First, let us define a modified Laplacian matrix L* by the equation L*Ii0Nm LI, where L is the Laplacian matrix of the graph and Ii0NmdiagI10,I20,...,INm0, and I10 is from (5); and the symbol represents the Kronecker product. Then, the consensus protocol of the entire system can be written as the matrix form:

X˙t=-L*Xt=-Ii0NmLIXt (6)

where X˙t is the derivative of the vector Xt; and Xt=x1t;x2t;...;xNmt is an Nm2×1 vector.

The task is to prove the convergency of the consensus-based algorithm (6). Since (6) is a linear differential equation, the Gershgorin circle theorem [

29] can be used to locate the eigenvalues of the matrix L*.

Now, let us look at the Laplacian matrix L. Since L=D-A, the center of each Gershgorin disc is the diagonal entry of the degree matrix D, i.e., the degree dvi of each vertex. And the radius Ri is the sum of the row entries of the adjacency matrix A. By the definition of dvi, it is true that Rid=dvi. Similarly, for the matrix LI=DI-AI, the centers and radiuses are determined by DI and AI, respectively. Now, for the modified Laplacian matrix L*=Ii0NmLI, the matrix Ii0Nm can be considered as an operation to replace Nm rows of LI by zeros. For these replaced Nm rows, the corresponding Gershgorin discs become circles at the origin with zero radius. Therefore, as an example, the Gershgorin discs of the modified Laplacian matrix L* have the form as in Fig. 2, where dVk and dVk' are the degrees of the vertexes k and k', respectively.

Fig. 2  An example of Gershgorin discs for L*.

In Fig. 2, all the smaller Gershgorin discs (including the discs at the origin with zero radius) are contained within the largest Gershgorin discs. Thus, all the eigenvalues of L* are located in the largest Gershgorin discs which are centered at max dvi and has the radius max dvi, where max dvi denotes the largest degree of all vertices. Because all the degrees dvi0, the eigenvalues of L* are either zeros or in the right half plane. Therefore, for the differential equation (6), none of the eigenvalues of the matrix -L* is purely imaginary or has positive real part. Then, the solution of the differential equation (6) converges to a certain value as time goes to infinite.

Xtt=X=C (7)

where C is a constant vector.

2) Proof of xi()=Z

In order to complete the proof, the constant X=C must be found out. Since X˙=0, X is the equilibrium point of the differential equation (6). Therefore, the objective is to find the solution X=X for the function:

L*X=0 (8)

Here, let us define a new matrix Ii0 by the equation Ii0diagI,I,...,I,Ii0,I,...,I, in which the identity matrix I has the same dimension as Ii0, and the matrix Ii0 only appears at the ith diagonal block. Hence, the row (and also the column) Nmi-1+i is all zeros for the particular i. The matrix Ii0Nm can be decomposed as Ii0Nm=INm0INm-10...I10. Then, the matrix L* can be rewritten as:

L*=INm0INm-10...I10(LI) (9)

Besides, define 0i1I-Ii0, and Pi1T1T0i1. Pi1×Nm2 is a vector that only has ones at its (kNm+i)th entries for k=0,1,,Nm-1. According to the mixed-product property of the Kronecker product and the property 1TL=L1=0, we have:

PiLI=1T1T0i1LI=1TL1T0i1I=0(1T0i1I)=0 (10)

where 0 is the zero vector. Then, let PiNm2×Nm2 be a matrix which has only one non-zero row vector Pi at its i-1Nm+ith row, i.e., Pi=0;0;...;0;Pi;0;...;0. Based on (10), PiLI=0 as well. Hence, the right-most two terms of (9) becomes:

I10LI=I10LI+P1LI=I10+P1LI=E1LI (11)

where E1I10+P1. Since right multiplying a matrix by I10 is equivalent to the column operation that eliminates the first column of the matrix, and the first column of P2 is already all zeros, P2I10=P2. Thus,

I20I10(LI)=I20I10LI+P2I10LI=                                     I20+P2I10LI=E2E1LI (12)

where E2I20+P2. By performing the similar process as above for Nm times, (9) can be rewritten as:

L*=Ii0NmLI=ENmENm-1...E1LI (13)

where EiIi0+Pi (for i=1,2,...,Nm) and it has the following form:

Ei=I0Pi0I the Nmi-1+ith row (14)

Since Pi is a vector with ones at its kNm+ith entries, the diagonal of the matrix Ei are all ones. So, each Ei can be decomposed as the product of a sequence of elementary matrices corresponding to row-addition transformations. According to (13), Ii0Nm performs similarly as a series of row-addition transformations on the matrix LI. Because the row operation does not change the null space of the original matrix, L* and LI have the same null space, i.e., NullL*=NullLI.

For a connected network, the rank of Laplacian L is Nm-1. According to the property of Kronecker product, rankLI=rankL rankI=Nm-1Nm. Also, because LI is an Nm2×Nm2 matrix and based on the rank-nullity theorem, the dimension dimNullLI=Nm2-Nm-1Nm=Nm. Thus, dimNullL*=Nm.

Meanwhile, for an arbitrary Nm-dimensional vector x*, we can obtain:

L*1x*=Ii0NmLI1x*=Ii0NmL1Ix*=Ii0Nm0Ix*=0 (15)

Thus, 1x* is in NullL*. Since span1x* NullL* and dimNullL*=dimspan1x*=Nm, NullL*=span1x*.

Therefore, the general solution to (8) is:

X=1x*=x*;x*;...;x* (16)

Since X is one of the solutions and X=x1; x2; ...; xNm,

x1()=x2()= ...=xNm=x* (17)

This result shows that all the states xi will converge to a consensus when the time goes to infinity (in practice, it will converge within an acceptable error in a few reasonable steps). Besides, according to the information propagation algorithm in (4), the term Ii0 sets the ith row of x˙it to be zero, which means that the ith row of xit does not change for the whole time. Hence, it is true that:

xii=xii0=zi (18)

where xiit is the ith element of xit. Finally, all the states xi will converge to the solution as:

xi=x110; x220;  ; xNmNm0=z1; z2;  ; zNm=Z (19)

Now, the statement limk xik=z1;z2;...;zNm is proven.

B. Proposed Information Propagation Algorithm Based DSE Method

Based on the information propagation algorithm, the DSE method can be established. In the centralized state estimation, the centralized state estimator solves the optimization problem (3) to get the best estimation of the states. The Newton’s method [

28] is usually implemented to solve the optimization problem by updating the estimated state vector x^s by the following equation:

Δx^s=HTR-1H-1HTR-1Z-hx^s (20)

where H is the Jacobian matrix of the observation model hx^s; and R is the covariance matrix of measurement errors.

This iteration can be combined with that in the information propagation algorithm to update the states of the power system x^s and the information state xi at the same time. Hence, for the node i between buses n and m, the equations for the proposed method are listed as:

ΔZiPk=τ Ii0j𝒩iwijPZjPk-ZiPk (21)
ΔZiQk=τ Ii0j𝒩iwijQZjQk-ZiQk (22)
ZiPk+1=ZiPk+ΔZiPk (23)
ZiQk+1=ZiQk+ΔZiQk (24)
Zik+1=ZiPk+1;ZiQk+1 (25)
ΔZik+1=Zik+1-hx^sk (26)
Δx^sk+1=HTR-1H-1HTR-1ΔZik+1 (27)
x^sk+1=x^sk+Δx^sk+1 (28)

The procedure of the algorithm can be summarized in Algorithm 1.

Algorithm 1  : distributed AC state estimation

1: Initialize ZiP0 and ZiQ0 with random values

2: Set ZiiP0 as measured real power on node i

3: Set ZiiQ0 as measured reactive power on node i

4: while Δx̂sk>tolerance do

5: Update the estimation vectors ZP and ZQ by information propagation     equations (21)-(24)

6: Combine ZP and ZQ in one vector as in (25)

7: Update estimation of state vector x̂s by equations

8: end while

In Algorithm 1, is the infinity norm which gives the maximal value in the vector. Hence, Δx^sk represents the state estimation error and the algorithm should stop when it is smaller than a given tolerance value.

The variables of the DSE method transmitted between the DSE nodes are shown in Fig. 3.

Fig. 3  Signal flow between DSE nodes.

IV. Simulation Results

In this section, the proposed method and algorithm are tested in MATLAB. First, the information propagation algorithm is simulated. Then, the proposed method is tested in four power system models, i.e., IEEE 14-bus system, IEEE 39-bus system, IEEE 118-bus system, and IEEE 300-bus system.

A. Information Propagation Algorithm in Example Graphs

In order to verify that the proposed information propagation algorithm can share the local value of each vertex to the others, the simulations are conducted in the communication networks with two different graphs shown in Fig. 4, which are the cycle graph and the tree graph, respectively [

30].

Fig. 4  Two different graphs. (a) Cycle graph. (b) Tree graph.

In the two case studies, the vertices V1, V2, ..., V6 have initial local values 8, 2, 25, 12, 13, and 3, respectively. Then, at time t=0.5 s, the local values are turned into 4, 6, 14, 9, 21, and 11. By applying the information propagation algorithm on the graphs, the results are shown in Fig. 5 and Fig. 6, and estimations from nodes 1-6 denote the set of estimations for the values on all nodes in the system computed by nodes 1-6.

Fig. 5  Information propagation algorithm in cycle graph.

Fig. 6  Information propagation algorithm in tree graph.

The results show that the local data can be successfully propagated in the graphs by the information propagation algorithm successfully. According to Figs. 5 and 6, the estimations can converge to the correct values in less than 500 steps.

B. DSE in IEEE Models

1) State Estimation in IEEE 14-bus System

In this case study, the IEEE 14-bus system is employed to verify the proposed method. The IEEE 14-bus system is modeled as in Fig. 7. There are 17 DSE nodes installed on the power lines.

Fig. 7  IEEE 14-bus system.

The trajectory of the estimated states (the phase angles and voltage magnitudes of the 14 buses reflecting through different colored lines) by the proposed method is shown in the Fig. 8. The figure shows that the estimated states reach the actual values in about 200 iterations. For the simulation on a PC with Intel Core i7 2.8 GHz CPU and 24 GB memory, each iteration takes about 0.001 s, so the 200 iterations take about 0.2 s, which is acceptable compared with the 15-min interval between two state estimations in a centralized state estimation method implied in SCADA-based energy management system (EMS).

Fig. 8  Convergence of proposed method in IEEE 14-bus system. (a) Phase angle. (b) Voltage magnitude.

The proposed method is also compared with the traditional centralized method [

28] and the existing ADMM-based DSE method [14]. The estimated phase angles and voltage magnitudes from node 1 by the proposed method are compared with the results from the centralized method and the ADMM-based DSE method, as shown in Fig. 9.

Fig. 9  Results of IEEE 14-bus system. (a) Phase angle. (b) Voltage magnitude.

In Table I, MAEθ and MAEV are the mean absolute error (MAE) of phase angle and voltage magnitude, respectively. Similarly, MSEθ and MSEV are the mean squared error (MSE) of phase angle and voltage magnitude, respectively. These errors are defined as:

MSE=1ni=1nyi-y^is2 (29)
MAE=1ni=1nyi-y^is (30)

where yi is the ith state of the system; and y^is is the estimation of the ith state of the system.

MethodMAEθMAEVMSEθMSEV
Centralized method 0.0017381 0.0027212 4.6002×10-6 1.0118×10-5
Proposed method 0.0017387 0.0027230 4.6090×10-6 1.0133×10-5
ADMM-based method 0.0050613 0.0934330 4.4383×10-5 1.0858×10-2

According to the above results, the accuracy of the proposed method is similar to that of the centralized method and better than the existing ADMM-based DSE method in the IEEE 14-bus system.

In addition, the comparison of errors of different methods at each step is shown in Fig. 10.

Fig. 10  Comparison of errors of different methods at each step.

The figure shows that the error of the centralized method decreases the fastest. This is making sense because all measurement data can be directly sent to the control center in the centralized method. Figure 10 also shows that the proposed method can converge to the accurate result faster than the existing ADMM-based DSE method. The error of the ADMM-based DSE method stays higher than the proposed method and the centralized method. This concludes that, the proposed method is as accurate as the centralized method and faster than the ADMM-based DSE method.

2) State Estimation in IEEE 39-bus System

The IEEE 39-bus system as shown in Fig. 11 is adopted in the simulation. There are 46 DSE nodes in the system. The results of this simulation case are listed in Table II.

Fig. 11  IEEE 39-bus system.

MethodMAEMSE
Phase angle (rad)Voltage magnitude (p.u.)Phase angle (rad)Voltage magnitude (p.u.)
Centralized method 0.00018428 0.00019664 5.5541×10-8 5.0927×10-8
Proposed method 0.00018409 0.00019667 5.5518×10-8 5.0954×10-8

3) State Estimation in IEEE 118-bus System

To verify the performance of the proposed method in large power system, the proposed method is also simulated in the IEEE 118-bus power system model [

31]. There are 179 DSE nodes in the system. The results are shown in Table III.

MethodMAEMSE
Phase angle (rad)Voltage magnitude (p.u.)Phase angle (rad)Voltage magnitude (p.u.)
Centralized method 0.0010439 0.00073950 1.6417×10-6 8.2586×10-7
Proposed method 0.0010585 0.00073666 1.6991×10-6 8.2088×10-7

4) State Estimation in IEEE 300-bus System

Finally, the proposed method is tested in the IEEE 300-bus system [

31] and compared with the centralized method. There are 409 DSE nodes in the ADMM-based system. The results are shown in Table IV. The errors of the DSE method are the same as the centralized method in this case.

MethodMAEMSE
Phase angle (rad)Voltage magnitude (p.u.)Phase angle (rad)Voltage magnitude (p.u.)
Centralized method 0.0032567 0.002517 0.00013898 0.0001109
Proposed method 0.0032567 0.002517 0.00013898 0.0001109

According to the results from the different cases above, the errors of the centralized state estimation method and the proposed method are very close. This result confirms that the proposed method has the accuracy similar to the centralized method.

C. Influence of Communication Network Structure

The performance of the DSE method is usually related to the topology of the communication network. In this case study, the IEEE 118-bus system is used to demonstrate the impact of the communication network structure on the convergence of the estimation. Two different communication network structures are investigated in this paper. One is the chain connection and another is the full connection. The chain connection is that each node connects to its previous node if it is not the first node and connects to the next node if it is not the last node. For example, for the 179 DSE nodes in the IEEE 118-bus system, node 1 connects with node 2, node 2 connects to nodes 1 and 3, node 3 connects to nodes 2 and 4, and goes on until the node 179 only connects to node 178. Besides, the full connection means that each node connects to all other nodes in the system. Obviously, the chain connection structure has the fewest communication lines and the full connection structure has the most communication lines.

The simulation results of the two communication network structures in IEEE 118-bus system are shown in Fig. 12 and Fig. 13, respectively. The figures show the trajectory of the estimated states and the errors of the results. According to the results, the state estimation in the fully connected communication network is much faster than in the chain connected network. The chain connected network spends about 40000 iterations to reach an accurate result, but the fully connected network only takes 30 iterations. Since the chain connected structure and the fully connected structure are the extreme cases, other communication structures will have the speed between these two results. In addition, although the chain connected network takes 40000 iterations to reach the acceptable result, the speed of the algorithm can be improved in real world by properly setting the initial values of the estimated states with the previous estimated results.

Fig. 12  Results of IEEE 118-bus system with fully connected communication network. (a) Phase angle. (b) Voltage magnitude. (c) MAE. (d) MSE.

Fig. 13  Results of IEEE 118-bus system with chain connected communication network. (a) Phase angle. (b) Voltage magnitude. (c) MAE. (d) MSE.

V. Conclusion

This paper proposes a new distributed AC state estimation method. The proposed method does not need any centralized facility like the existing DSE methods. An information propagation algorithm is also introduced as the foundation of the proposed method. The theoretical proof of the information propagation algorithm’s convergence is provided. The proposed method is verified by comparing with the centralized method and the existing ADMM-based DSE method in simulations with several standard power system models. The simulation results show that the proposed method can achieve the same estimation accuracy as the centralized method which is better than the existing ADMM-based DSE method. Also, the proposed method can converge faster than the existing ADMM-based DSE method. In addition, since the proposed method is totally decentralized, it is more robust than the traditional methods during network failures.

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