Abstract
The volatile and intermittent nature of distributed generators (DGs) in active distribution networks (ADNs) increases the uncertainty of operating states. The introduction of distribution phasor measurement units (D-PMUs) enhances the monitoring level. The trade-offs of computational performance and robustness of state estimation in monitoring the network states are of great significance for ADNs with D-PMUs and DGs. This paper proposes a second-order cone programming (SOCP) based robust state estimation (RSE) method considering multi-source measurements. Firstly, a linearized state estimation model related to the SOCP state variables is formulated. The phase angle measurements of D-PMUs are converted to equivalent power measurements. Then, a revised SOCP-based RSE method with the weighted least absolute value estimator is proposed to enhance the convergence and bad data identification. Multi-time slots of D-PMU measurements are utilized to improve the estimation accuracy of RSE. Finally, the effectiveness of the proposed method is illustrated in the modified IEEE 33-node and IEEE 123-node systems.
Set of all lines
Set of nodes connected to node
Set of upstream nodes connected to node
Set of downstream nodes connected to node
Set of all nodes
Set of nodes with nodal current injection magnitude measurement (NCIMM)
Set of phases,
Index of multi-time slots of distribution phasor measurement units (D-PMUs)
, , f Indices of nodes
Index of lines
Index of supervisory control and data acquisition (SCADA) and advanced metering infrastructure (AMI) measurements
Index of phases
Index of nodes with NCIMM
, Indices of nodes connected to node
Index of D-PMU measurements
Voltage phase angle in phase at node
, Line related state variables
, NCIMM related state variables
, Line related state variables in phase of line
, NCIMM related state variables in phase between nodes and
Estimation time
Voltage magnitude in phase at node
Nodal voltage related state variable
Nodal voltage related state variable in phase at node
Second-order cone programming (SOCP) state variables
, , Calculation measurement values, calculation weights, and residuals of multi-source measurements at estimation time
, Weights of SOCP state variables
Susceptance in phase of line
Shunt susceptance in phase of line
Sum of susceptance and shunt susceptance of connected lines in phase at node
Coefficient of multiple time slots of D-PMU measurements
Conductance in phase of line
Sum of conductance of connected lines in phase at node
Total number of SCADA and AMI measurements
Total number of nodes
Total number of lines
Sampling interval of D-PMU measurements
Total number of D-PMU measurements
, , Number of nodal voltage, line current, and nodal current injection magnitude measurements of D-PMUs
Measurement values of SCADA and AMI
, Measurement values of D-PMUs
, Active and reactive power
, Voltage and current magnitudes
, Standard deviations of voltage magnitude and phase angle measurements of D-PMUs
, Standard deviations of current magnitude and phase angle measurements of D-PMUs
true True values
se State estimation values
WITH the integration of advanced measurement and communication technologies, the informatization of active distribution networks (ADNs) is constantly upgrading [
To realize the real-time monitoring and dispatching of ADNs, state estimation of distribution network is carried out to best approximate the operating states with the available measurements [
Aimed at accelerating the computational process of state estimation, the weighted least squares (WLS) estimator has been widely utilized as the objective function [
Although the performance of D-PMUs is improved, the issues of D-PMUs in practical networks would occur due to the sudden sensing errors, data loss, and time synchronization errors. The typical bad data ratio of PMU in practical networks is reported to range from 10% to 17% [
Recently, with the need for fast and reliable global convergence of state estimation, convex programming methods present excellent performance over Gauss-Newton methods [
Currently, due to the introduction of D-PMUs, the SOCP methods considering the multi-source measurements in ADNs are not fully considered. In addition, bad data frequently exist in the multi-source measurements, and the bad data identification may affect the estimation accuracy and computational efficiency. It is necessary to balance the performance indices including estimation accuracy, computational efficiency, and bad data identification. Oriented to the ADNs with D-PMUs and DGs, this paper proposes an SOCP-based RSE method considering the multi-source measurements in ADNs. The proposed method aims to combine the benefits of the robust estimation method in bad data identification and the SOCP method in estimation performance. The overall framework of the proposed method is shown in

Fig. 1 Overall framework of proposed method.
The main contributions of this paper are summarized as follows.
1) A linearized multi-source measurement model related to the SOCP state variables is formulated. Various types of multi-source measurements in ADNs are thoroughly considered, including the nodal current injection phasor measurements of D-PMUs. To enhance the applicability of the SOCP methods, the phase angle measurements of D-PMUs are converted to equivalent power measurements.
2) With the introduction of nodal current injection phasor measurements, the revised SOCP-based RSE method is proposed to enhance the robustness to bad measurement data. The WLAV estimator is selected as the objective function of RSE. The nodal current injection magnitude measurement (NCIMM) related state variables are added to the SOCP state variables. The second-order conic constraints are introduced into the RSE problem, and the effectiveness is verified by the estimation results. The robustness to the single and multiple bad measurement data of the proposed method is corroborated.
3) Considering the differences in accuracy and time scale of the multi-source measurements, the multiple time slots of D-PMU measurements are utilized to improve the estimation accuracy. The temporal correlation of D-PMU measurements is tackled in response to sudden bad measurement data.
The remainder of this paper is organized as follows. Section II builds the linearized measurement model for the SOCP-based state estimation. In Section III, the revised SOCP-based RSE method considering the multi-source measurements is elaborated. Case studies are conducted in Section IV to verify the effectiveness of the proposed method. Finally, Section V concludes this paper.
The linearized measurement model is a prerequisite for the SOCP-based state estimation. In this section, state variables of the SOCP-based state estimation are introduced. The linearized multi-source measurement model related to the SOCP state variables is formulated.
Distinct from the state variables composed of nodal voltage magnitudes and phase angles which are utilized by the Gauss-Newton method, the SOCP state variables are defined as:
(1) |
where ; ; ; ; and .
For each node in phase , the nodal voltage related state variable is expressed as:
(2) |
For each line connecting nodes and in phase , the line related state variables and are expressed as:
(3) |
(4) |
For node with NCIMM which has more than two connecting nodes, the NCIMM related state variables and are expressed as:
(5) |
(6) |
The considered multi-source measurements in this paper include the fast-rate D-PMU measurements and the slow-rate SCADA and AMI measurements. The differences in measurement error and time scale of the multi-source measurement equipment are reflected in the estimation weight parameter. With the introduced state variables in Section II-A, the linearized multi-source measurement equations are constructed. The original measurement values and standard deviations obtained from the metering instruments are partially converted to the calculation measurement values and calculation weights, respectively. To simplify the description of measurement values, residuals, calculation weights, and the standard deviations of the multi-source measurements in ADNs,
Measurement type | Category | Measurement value at time and | Residual at time | Calculation weight at time | Standard deviation |
---|---|---|---|---|---|
Nodal active and reactive power injection measurements in phase at node | AMI | , | , | , | , |
Line sending-end/receiving-end active and reactive power measurements in phase of line | SCADA |
, , , | , , , | ||
Nodal voltage magnitude measurements in phase at node | SCADA | ||||
D-PMU | |||||
Line sending-end/receiving-end current magnitude measurements in phase of line | SCADA | , | , | , | , |
D-PMU | , | ||||
NCIMMs in phase at node | D-PMU | ||||
Nodal voltage phase angle measurements in phase at node | D-PMU | ||||
Line sending-end/receiving-end current phase angle measurements in phase of line | D-PMU | , | , | ||
Nodal current injection phase angle measurements in phase at node | D-PMU |
1) Nodal Active and Reactive Power Injection Measurements
The nodal power injection measurements can be linearly expressed in terms of the SOCP state variables without extra transformation. The linearized measurement equations are expressed as:
(7) |
(8) |
The calculation weights of the nodal power injection measurements are expressed as:
(9) |
(10) |
2) Line Power Measurements
Since the line related state variables are expressed as the relationship between the sending-end node and the receiving-end node, the measurement equations of the sending-end and receiving-end measurements are different, which will be illustrated, respectively. The measurements of line are shown in

Fig. 2 Measurements of line .
1) Sending-end active and reactive power measurements
Similar to the nodal power injection measurements, the linearized measurement equations of the sending-end power measurements are expressed as:
(11) |
(12) |
The calculation weights of the sending-end power measurements are expressed as:
(13) |
(14) |
2) Receiving-end active and reactive power measurements
The linearized measurement equations of the receiving-end power measurements are described as:
(15) |
(16) |
The calculation weights of the receiving-end power measurements are expressed as:
(17) |
(18) |
3) Nodal Voltage Magnitude Measurements
The nodal voltage magnitude measurements are nonlinearly expressed in terms of the SOCP state variables. To construct the linearized relationship of the nodal voltage magnitude measurements, the squared voltage magnitude measurement value is chosen as the calculation measurement value, which is denoted as:
(19) |
The calculation weight of the calculation measurement value is related to that of the original measurement value based on the error propagation theory [
(20) |
4) Line Current Magnitude Measurements
1) Sending-end current magnitude measurements
Similar to the nodal voltage magnitude measurements, the squared current magnitude measurement values are chosen as the calculation measurement values. The linearized measurement equation and calculation weight of the sending-end current magnitude measurements are expressed as:
(21) |
(22) |
2) Receiving-end current magnitude measurements
(23) |
(24) |
Since the sampling rate of D-PMU measurements is significantly higher than the change in the operating state, multi-time slots of D-PMU measurements are utilized in this paper. The temporal correlation of D-PMU measurements is considered, which is aimed to improve the robustness to sudden bad measurement data in state estimation.
For each D-PMU measurement at the estimation time , the utilized D-PMU measurements are extended from time to time , which is expressed as:
(25) |
(26) |
(27) |
where is an adjustable parameter to verify the validity on the improvement of estimation accuracy.
The installed D-PMU measurement types in ADNs consist of nodal voltage phasor measurements, line current phasor measurements, and nodal current injection phasor measurements. The magnitude measurements of D-PMUs can be linearly related to the SOCP state variables. However, the phase angle measurements of D-PMUs are still nonlinearly related to the SOCP state variables. To tackle the nonlinearity, the phase angle measurements of D-PMUs are converted to the corresponding line power measurements or nodal power injection measurements.
1) Nodal Voltage Magnitude Measurements of D-PMUs
Similar to the nodal voltage magnitude measurements in Section II-B, the linearized measurement equations of nodal voltage magnitude measurements of D-PMUs are extended. The variable in (19) is replaced by , where . The linearized measurement equation and calculation weight of the nodal voltage magnitude measurements of D-PMUs are expressed as:
(28) |
(29) |
where the D-PMU measurement relates to the nodal voltage magnitude measurements in phase at node .
2) Line Current Magnitude Measurements of D-PMUs
Similar to the line current magnitude measurements in Section II-B and the nodal voltage magnitude measurements of D-PMUs, the linearized measurement equation and calculation weight of the line current magnitude measurements of D-PMUs are no longer illustrated, which is similar to (21)-(24).
3) NCIMMs of D-PMUs
With the introduction of the NCIMMs of D-PMUs, the NCIMM related state variables are considered in this subsection. The linearized measurement equation and calculation weight of the nodal current injection magnitude measurements of D-PMUs are expressed as:
(30) |
(31) |
where ; and .
4) Equivalent Active Power Measurements of D-PMUs
The equivalent active power measurements of D-PMUs include the sending-end active power measurements, receiving-end active power measurements, and nodal active power injection measurements. The equivalent active power measurements of D-PMUs are expressed as the correlation of related nodal voltage phasor measurements and current phasor measurements. The measurement value and calculation weight of the equivalent active power measurements of D-PMUs are stated as:
(32) |
(33) |
(34) |
(35) |
(36) |
(37) |
The measurement variables concerning the specific equivalent measurement types of D-PMUs are listed in
Equivalent measurement type of D-PMUs | Variable of D-PMUs | |||||
---|---|---|---|---|---|---|
Equivalent sending-end active and reactive power measurements | ||||||
Equivalent receiving-end active and reactive power measurements | ||||||
Equivalent nodal active and reactive power injection measurements |
The linearized measurement equations of the equivalent active power measurements of D-PMUs are similar to (7), (11), and (15), which are not illustrated here.
5) Equivalent Reactive Power Measurements of D-PMUs
Similar to the equivalent active power measurements of D-PMUs, the measurement value and calculation weight of the equivalent reactive power measurements of D-PMUs are expressed as:
(38) |
(39) |
(40) |
(41) |
(42) |
(43) |
The linearized measurement equations of the equivalent reactive power measurements of D-PMUs are similar to (8), (12), and (16), which are not illustrated here.
To combine the benefits of the WLAV estimation method in bad data identification and the SOCP method in estimation performance, a revised SOCP-based RSE method is proposed in this section.
The WLAV-based state estimation model considering the multi-source measurements in ADNs is formulated as:
(46) |
(47) |
(48) |
(49) |
where is constituted by (7), (8), (11), (12), (15), (16), (19), (21), (23), (28), (30), (32), and (38); is constituted by (9), (10), (13), (14), (17), (18), (20), (22), (24), (29), (31), (33), and (39); is the compact matrix form of linearized measurement equations constituted by (7), (8), (11), (12), (15), (16), (19), (21), and (23); and is the compact matrix form of (44) and (45).
To alleviate the nonconvexity caused by the absolute value variable , the auxiliary variables and are introduced as:
(50) |
The equivalent model of (46) is transformed into:
(51) |
Since the number of the SOCP state variables is larger than that of state variables in the traditional Gauss-Newton method considering D-PMUs, it may lead to the unobservability of RSE. To solve the unobservability problem, the second-order cone constraints are introduced into the WLAV problem.
The equality constraints (52) and (53) are satisfied between the SOCP state variables.
(52) |
(53) |
The above equality constraints are relaxed into the rotating second-order cone constraints:
(54) |
(55) |
To make the second-order cone constraints close to the equality constraints (52) and (53), the term is added to the objective function. Then, the model of the SOCP-based RSE method is transformed into:
(56) |
The number of the multi-source measurements in the Gauss-Newton method is . With the introduced equivalent power measurements, the number of the multi-source measurements in the SOCP method is . Compared with the Gauss-Newton method, the number is increased by . In addition, the second-order cone constraints make the equivalent measurements increase by . Compared with the Gauss-Newton method, the variation of the number of the multi-source measurements is . For radial distribution networks, ; thus, the variation of the number of the multi-source measurements between the Gauss-Newton method and the SOCP method is .
The number of state variables in the Gauss-Newton method is considering D-PMUs. However, with the introduction of auxiliary state variables, the number of state variables in the SOCP method reaches . Compared with the Gauss-Newton method, the number of state variables is increased by .
Since is larger than , the variation of the number of the multi-source measurements is larger than the variation of the number of state variables . It indicates that the introduction of measurement conversion and second-order cone constraints ensures the observability of the SOCP-based RSE.
To solve the above SOCP-based RSE problem, the interior-point algorithm is utilized where the dual problem of the primal problem (56) is formulated [
In this paper, the RSE problem (56) is solved by the optimization software MOSEK [
1) Estimation Values of Nodal Voltage Magnitudes
For any node , the estimation value of nodal voltage magnitude can be obtained from the SOCP state variable as:
(57) |
2) Estimation Values of Nodal Voltage Phase Angles
The estimation values of nodal voltage phase angles at the source node in phases A, B, and C are set equal to , respectively. For any line connecting nodes and , the estimation values of nodal voltage phase angles satisfy the following relationship:
(58) |
Starting from the source node, the voltage phase angles of all other nodes can be obtained by using the breadth-first search algorithm or the depth-first search algorithm.

Fig. 3 Flowchart of proposed SOCP-based RSE method.
Due to the mutual decoupling between phases of the second-order cone constraints (54) and (55), the proposed RSE method is applicable to multi-phase balanced networks and multi-phase decoupled unbalanced networks, and the unbalance of loads can be treated.
In this section, the effectiveness of the proposed SOCP-based RSE method considering multi-source measurements is verified in the modified IEEE 33-node and 123-node systems. The proposed method is programmed using C++ and solved by MOSEK. The numerical experiments are carried out on a computer with an Intel Xeon CPU E5-2650 v2 processor running at 2.60 GHz and 20 GB of RAM.
The topology and multi-source measurement configuration of the modified IEEE 33-node system is presented in

Fig. 4 Topology and multi-source measurement configuration of modified IEEE 33-node system.
To simulate the multi-source measurement data, their true values are obtained by the distribution power flow analysis. Then, the Gaussian distributed measurement noises are added to the true values. The standard deviations of the multi-source measurements are listed in
Measurement type | Standard deviation of measurement errors |
---|---|
Magnitude measurements of D-PMUs | 0.1% |
Phase angle measurements of D-PMUs | 0.01° |
Measurements of SCADA | 1% |
Measurements of AMI | 5% |
For the magnitude measurements of D-PMUs, SCADA, and AMI, the measurement value satisfies:
(59) |
where and are the measurement value and true value, respectively; and is the added measurement noise.
For the phase angle measurements of D-PMUs, the measurement value satisfies:
(60) |
To assess the estimation accuracy, convergence, and computation time of the proposed method, 250 Monte Carlo simulations are performed, and the total time of state estimation is listed. The following four indices are utilized to assess the estimation accuracy.
1) Average relative errors of nodal voltage magnitudes
(61) |
2) Average absolute errors of nodal voltage phase angles
(62) |
3) Average relative errors of nodal power injections
(63) |
4) Average absolute errors of nodal power injections
(64) |
The solution of the proposed SOCP-based RSE method (WLAV-SOCP) is compared with the WLS-based Gauss-Newton (WLS-GN) method and the WLAV-based Gauss-Newton (WLAV-GN) method in [
1) Multi-source Measurements Without D-PMUs
To evaluate the second-order cone relaxations on the estimation performance, the D-PMU measurements in
The results in
Method | %) | (°) | %) | kW) | (s) |
---|---|---|---|---|---|
WLS-GN | 0.5767 | 0.0141 | 6.5740 | 1.5878 | 6.106 |
WLAV-GN | 0.4804 | 0.0143 | 6.4795 | 1.4744 | 15.280 |
WLAV-SOCP | 0.7018 | 0.0165 | 7.0577 | 1.8151 | 11.337 |
Note: total computation time refers to total time of 250 simulations.
The maximum iteration number | Non-converge state estimation number |
---|---|
25 | 75 |
50 | 11 |
100 | 0 |
The results show that when the maximum iteration number is 50, 11 times of the WLAV-GN estimation are not convergent with the termination tolerance of . All the state estimations are convergent with the maximum iteration number 100. In contrast, the proposed WLAV-SOCP method achieves the estimation convergence rate of 100%.
2) Multi-source Measurements Considering D-PMUs
The multi-source measurements considering the D-PMU measurements in
Method | (%) | (°) | (%) | (kW) | (s) |
---|---|---|---|---|---|
WLS-GN | 0.1138 | 0.0241 | 4.7592 | 0.9786 | 7.255 |
WLAV-GN | 0.0356 | 0.0044 | 4.9222 | 0.9918 | 10.289 |
WLAV-SOCP | 0.0482 | 0.0032 | 4.3695 | 1.0682 | 13.155 |
The estimation results in
Different coefficients of multi-time slots of D-PMUs, i.e., , are considered to improve the estimation accuracy of the WLAV-SOCP method. The results in
(%) | (°) | (%) | (kW) | (s) | |
---|---|---|---|---|---|
0 | 0.0482 | 0.0032 | 4.3695 | 1.0682 | 13.155 |
1 | 0.0250 | 0.0030 | 4.1702 | 1.0260 | 23.223 |
2 | 0.0196 | 0.0030 | 4.1419 | 1.0130 | 27.157 |
3 | 0.0164 | 0.0029 | 4.0774 | 1.0098 | 33.121 |

Fig. 5 Estimation errors of nodal voltage magnitudes.
3) More D-PMUs in Multi-source Measurements
Based on the multi-source measurement configurations in

Fig. 6 Structure of modified IEEE 33-node system with more D-PMU and SCADA measurements.
Method | (s) | ||||
---|---|---|---|---|---|
WLS-GN | 0.0761 | 0.0185 | 3.7151 | 0.7167 | 7.276 |
WLAV-GN | 0.0264 | 0.0036 | 4.3458 | 0.8069 | 10.781 |
WLAV-SOCP () | 0.0445 | 0.0023 | 3.0662 | 0.6711 | 15.689 |
WLAV-SOCP () | 0.0235 | 0.0021 | 2.8532 | 0.6352 | 28.526 |
WLAV-SOCP () | 0.0171 | 0.0020 | 2.7498 | 0.6223 | 35.066 |
WLAV-SOCP () | 0.0140 | 0.0020 | 2.6853 | 0.6096 | 44.029 |

Fig. 7 Estimation errors of nodal voltage magnitudes with different methods.
The results show that when the coefficient is 0, the estimation errors of nodal voltage phase angles and nodal power injections outperform the other two methods. When the coefficient increases to 1, the estimation errors of nodal voltage magnitudes are also superior to other methods. With the increase of coefficient , the estimation errors of nodal power injections show a certain reduction.
The bad measurement data are simulated by adding relatively large Gaussian distributed errors to the corresponding true values. The relatively large measurement error is usually set to be larger than 5 times the normal measurement standard deviation. The estimation results of the WLAV-SOCP method are intended to illustrate the robustness to bad measurement data.
1) Single Bad Measurement Data
Single bad measurement data is set in each state estimation, and all the bad measurement data are distributed at different nodes or lines. The multi-source measurement configurations are still based on
The results in
Method | (%) | (°) | (%) | (kW) | (s) |
---|---|---|---|---|---|
WLS-GN | 0.1819 | 0.0367 | 7.3114 | 1.4029 | 7.983 |
WLAV-GN | 0.0284 | 0.0036 | 6.4462 | 1.1113 | 11.208 |
WLAV-SOCP () | 0.0467 | 0.0030 | 4.2436 | 0.8702 | 16.725 |
WLAV-SOCP () | 0.0240 | 0.0024 | 3.0135 | 0.6759 | 30.080 |
WLAV-SOCP () | 0.0175 | 0.0023 | 2.9268 | 0.6640 | 36.001 |
WLAV-SOCP () | 0.0143 | 0.0023 | 2.8468 | 0.6474 | 44.488 |
With the utilization of multi-time slots of D-PMU measurements, the proposed WLAV-SOCP method can identify and correct the bad measurement resulting from the failure of network communication or sudden abnormal measurement. By comparing the results in
2) Multiple Conforming Bad Measurement Data
To describe the effectiveness of the proposed method, the measurement scenarios including multiple conforming bad measurement data are described as follows.
Case 1: the sending-end current magnitude measurement value of D-PMUs in the line connecting nodes 5 and 6 (D-I-5-6) increases from 121.6582 A to 133.6582 A, and the receiving-end active power measurement value in the line connecting nodes 6 and 7 (P-6-7) decreases from -364.4283 kW to -437.6068 kW.
Case 2: the sending-end current magnitude measurement value of D-PMUs in the line connecting nodes 1 and 2 (D-I-1-2) increases from 197.2925 A to 237.0729 A, and the sending-end active power measurement value in the line connecting nodes 1 and 2 (P-1-2) increases from 1219.3742 kW to 1445.9194 kW.
Case 3: the sending-end current magnitude measurement value of D-PMUs in the line connecting nodes 2 and 3 (D-I-2-3) decreases from 179.4144 A to 161.2107 A, and the sending-end current magnitude measurement value of D-PMUs in the line connecting nodes 2 and 19 (D-I-2-19) increases from 18.0837 A to 36.1718 A, and the sending-end active power measurement value in the line connecting nodes 3-23 (P-3-23) decreases from 313.1989 kW to 205.8947 kW.
The coefficient of multi-time slots of D-PMU measurements in the WLAV-SOCP method is set to be 1. The comparison of bad data identification results is shown in
Value | Case 1 | Case 2 | Case 3 | |||||
---|---|---|---|---|---|---|---|---|
D-I-5-6 (A) | P-6-7 (kW) | D-I-1-2 (A) | P-1-2 (kW) | D-I-2-3 (A) | D-I-2-19 (A) | P-3-23 (kW) | ||
Measurement | 121.7011,133.6582,121.6516 | -437.6068 | 197.6049,197.0091,237.0729 | 1445.9194 | 179.3481,161.2107,179.3418 | 18.0776,36.1718,18.0744 | 205.8947 | |
True | 121.6582 | -364.4283 | 197.2925 | 1219.3742 | 179.4144 | 18.0837 | 313.1989 | |
Estimation | WLS-GN | 126.1285 | -382.4433 | 221.8963 | 1404.0266 | 161.9103 | 36.0962 | 207.8455 |
WLAV-GN | 121.6582 | -364.6023 | 200.7107 | 1242.6048 | 161.8424 | 36.1718 | 199.2292 | |
Proposed WLAV-SOCP | 121.6348 | -364.2590 | 197.2085 | 1219.3627 | 179.2867 | 18.0782 | 312.2099 |
It can be observed from the results in Case 1, the WLAV-GN method and the proposed WLAV-SOCP method can identify the conforming bad measurement data. However, in Case 2 and Case 3, the WLAV-GN method cannot identify the conforming bad measurement data. With the multiple time slots of D-PMU measurements, the WLAV-SOCP method identifies the sudden conforming bad measurement data in all three cases. Therefore, the robustness to multiple conforming bad measurement data of the proposed method is verified.
To verify the effectiveness of the proposed method in larger-scale distribution networks, the modified IEEE 123-node system is adopted. The topology of the modified IEEE 123-node system is presented in

Fig. 8 Topology of modified IEEE 123-node system.
The mutual impedances between phases are ignored and the measurement redundancy is 1.651. To assess the estimation accuracy and bad data identification of the proposed method, 100 Monte Carlo simulations are performed. The multi-source measurement scenario considering D-PMU measurements in
Method | (%) | (°) | (%) | (kW) | (s) |
---|---|---|---|---|---|
WLS-GN | 0.0133 | 0.0015 | 0.3876 | 0.0926 | 23.439 |
WLAV-GN | 0.0158 | 0.0019 | 0.3063 | 0.0738 | 27.327 |
WLAV-SOCP () | 0.0255 | 0.0026 | 0.6218 | 0.1556 | 43.345 |
WLAV-SOCP () | 0.0155 | 0.0015 | 0.3765 | 0.0963 | 69.403 |
WLAV-SOCP () | 0.0099 | 0.0009 | 0.3335 | 0.0861 | 87.773 |
WLAV-SOCP () | 0.0076 | 0.0008 | 0.3154 | 0.0817 | 110.373 |
Note: total computation time
The state estimation results in
Similar to the bad measurement data scenarios in Section IV-C, to illustrate the robustness to bad measurement data of the proposed method, three bad measurement data are set in each state estimation and all the bad measurement data are distributed at different nodes or lines. The state estimation results of different methods with three bad measurement data are shown in
Method | (%) | (°) | (%) | (kW) | (s) |
---|---|---|---|---|---|
WLS-GN | 0.0155 | 0.0028 | 0.9268 | 0.2243 | 25.832 |
WLAV-GN | 0.0160 | 0.0019 | 0.5009 | 0.1227 | 27.986 |
WLAV-SOCP () | 0.0264 | 0.0026 | 0.7991 | 0.2000 | 45.544 |
WLAV-SOCP () | 0.0155 | 0.0015 | 0.3783 | 0.0966 | 73.020 |
WLAV-SOCP () | 0.0099 | 0.0009 | 0.3334 | 0.0859 | 87.530 |
WLAV-SOCP () | 0.0075 | 0.0007 | 0.3158 | 0.0818 | 109.238 |
Note: bolded number means that there exists 3 times non-convergence in WLS-GN method.
The above estimation results indicate that with the utilization of multiple time slots of D-PMU measurements, the proposed WLAV-SOCP method can identify the bad measurement data in the multi-source measurements. Although the proposed method takes a relatively long computation time, the computation time of single state estimation is still less than 1 s, which is within the acceptable range.
An SOCP-based RSE method considering the multi-source measurements in ADNs is proposed in this paper. The method incorporates measurement data consisting of D-PMUs, SCADA, and AMI. A linearized estimation model of multi-source measurements related to the SOCP state variables is formulated. The phase angle measurements of D-PMUs are converted to the equivalent power measurements. The revised SOCP-based WLAV method transforms the non-convex problem into the convex problem and is carried out to improve the estimation accuracy and convergence of state estimation. Estimation results in the modified IEEE 33-node and IEEE 123-node systems indicate that with the multiple time slots of D-PMU measurements, the trade-off between the estimation accuracy and computational efficiency of the proposed method is realized. The robustness to the single bad measurement data and the multiple conforming bad measurement data of the proposed method is verified.
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