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.
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 [
There are some existing researches about DSE methods. The majority of them are based on area partitioning method (or called multi-area method) [
In addition to the area partitioning methods, other DSE methods are introduced in [
Besides, for sensor networks, there is a consensus-based DSE method [
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 [
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 [
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 [
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.
In AC state estimation, the power system measurement process can be modeled as:
(1) |
where is the measurement vector of readings from all meters; is the state vector of the power system, in which the phase angle and voltage magnitude on the buses are the states, i.e., ; is the observation model; and 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 .
(2) |
where and are the real and reactive power flows between buses and , respectively; is the voltage magnitude on bus ; is the phase angle difference between buses and ; and and are the conductance and susceptance between buses and , respectively.
For the centralized state estimation, there is a centralized state estimator as an example shown in
(3) |

Fig. 1 An example of power system state estimation with four buses. (a) Centralized state estimation. (b) Proposed method.
where is the sum of measurement residuals; is the estimation of ; is the total number of measurements (meters) in the power system; is the reading on the meter ; is the estimated measurement on the meter by inputting the estimated state into (2); and is the variance for the measurement of the meter i.
However, for the DSE, the measurement system is shown in
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
In this paper, the information propagation algorithm is developed based on the consensus protocol technique [
(4) |
where is the information state at the time step (the initial values in can be arbitrary if , but since the local information is available at the th node itself); is the time interval between two update consecutive time steps; is a weight coefficient on the edge between node and node ; is the set of neighbor nodes of node i; and is an diagonal matrix which is the same as the identity matrix but has a zero at the th diagonal entry.
(5) |
In order to achieve the goal of the information prorogation algorithm, should converge to the actual measurement vector, i.e., . Here, a proof of this statement is provided below. The proof includes two parts. The first part proves that the estimation vector converges to an equilibrium point under the algorithm (4). The second part proves that the equilibrium point equals to the actual measurement vector .
First, let us define a modified Laplacian matrix by the equation , where is the Laplacian matrix of the graph and , and 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:
(6) |
where is the derivative of the vector ; and is an 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 [
Now, let us look at the Laplacian matrix . Since , the center of each Gershgorin disc is the diagonal entry of the degree matrix , i.e., the degree of each vertex. And the radius is the sum of the row entries of the adjacency matrix . By the definition of , it is true that . Similarly, for the matrix , the centers and radiuses are determined by and respectively. Now, for the modified Laplacian matrix , the matrix can be considered as an operation to replace rows of by zeros. For these replaced 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 have the form as in

Fig. 2 An example of Gershgorin discs for .
In
(7) |
where is a constant vector.
In order to complete the proof, the constant must be found out. Since , is the equilibrium point of the differential
(8) |
Here, let us define a new matrix by the equation in which the identity matrix has the same dimension as , and the matrix only appears at the th diagonal block. Hence, the row (and also the column) is all zeros for the particular . The matrix can be decomposed as . Then, the matrix can be rewritten as:
(9) |
Besides, define , and . is a vector that only has ones at its th entries for . According to the mixed-product property of the Kronecker product and the property , we have:
(10) |
where 0 is the zero vector. Then, let be a matrix which has only one non-zero row vector at its th row, i.e., . Based on (10), as well. Hence, the right-most two terms of (9) becomes:
(11) |
where . Since right multiplying a matrix by is equivalent to the column operation that eliminates the first column of the matrix, and the first column of is already all zeros, . Thus,
(12) |
where . By performing the similar process as above for times, (9) can be rewritten as:
(13) |
where (for ) and it has the following form:
(14) |
Since is a vector with ones at its th entries, the diagonal of the matrix are all ones. So, each can be decomposed as the product of a sequence of elementary matrices corresponding to row-addition transformations. According to (13), performs similarly as a series of row-addition transformations on the matrix . Because the row operation does not change the null space of the original matrix, and have the same null space, i.e., .
For a connected network, the rank of Laplacian is . According to the property of Kronecker product, . Also, because is an matrix and based on the rank-nullity theorem, the dimension . Thus, .
Meanwhile, for an arbitrary -dimensional vector , we can obtain:
(15) |
Thus, is in . Since and , .
Therefore, the general solution to (8) is:
(16) |
Since is one of the solutions and ,
(17) |
This result shows that all the states 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 sets the th row of to be zero, which means that the th row of does not change for the whole time. Hence, it is true that:
(18) |
where is the th element of . Finally, all the states will converge to the solution as:
(19) |
Now, the statement is proven.
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 [
(20) |
where is the Jacobian matrix of the observation model ; and 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 and the information state at the same time. Hence, for the node between buses and , the equations for the proposed method are listed as:
(21) |
(22) |
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
The procedure of the algorithm can be summarized in
Algorithm 1 : distributed AC state estimation |
---|
1: Initialize and with random values 2: Set as measured real power on node 3: Set as measured reactive power on node 4: while do 5: Update the estimation vectors and by information propagation equations ( 6: Combine and in one vector as in (25) 7: Update estimation of state vector by equations 8: end while |
In
The variables of the DSE method transmitted between the DSE nodes are shown in

Fig. 3 Signal flow between DSE nodes.
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.
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 Two different graphs. (a) Cycle graph. (b) Tree graph.
In the two case studies, the vertices have initial local values 8, 2, 25, 12, 13, and 3, respectively. Then, at time , 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 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.
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 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 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 [

Fig. 9 Results of IEEE 14-bus system. (a) Phase angle. (b) Voltage magnitude.
In Table I, and are the mean absolute error (MAE) of phase angle and voltage magnitude, respectively. Similarly, and are the mean squared error (MSE) of phase angle and voltage magnitude, respectively. These errors are defined as:
(29) |
(30) |
where is the th state of the system; and is the estimation of the th state of the system.
Method | ||||
---|---|---|---|---|
Centralized method | 0.0017381 | 0.0027212 |
4.6002×1 |
1.0118×1 |
Proposed method | 0.0017387 | 0.0027230 |
4.6090×1 |
1.0133×1 |
ADMM-based method | 0.0050613 | 0.0934330 |
4.4383×1 |
1.0858×1 |
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 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.
The IEEE 39-bus system as shown in

Fig. 11 IEEE 39-bus system.
Method | MAE | MSE | ||
---|---|---|---|---|
Phase angle (rad) | Voltage magnitude (p.u.) | Phase angle (rad) | Voltage magnitude (p.u.) | |
Centralized method | 0.00018428 | 0.00019664 | ||
Proposed method | 0.00018409 | 0.00019667 |
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 [
Method | MAE | MSE | ||
---|---|---|---|---|
Phase angle (rad) | Voltage magnitude (p.u.) | Phase angle (rad) | Voltage magnitude (p.u.) | |
Centralized method | 0.0010439 | 0.00073950 | ||
Proposed method | 0.0010585 | 0.00073666 |
Finally, the proposed method is tested in the IEEE 300-bus system [
Method | MAE | MSE | ||
---|---|---|---|---|
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.
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 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.
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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