State Estimation for Active Distribution Networks Considering Bad Data in Measurements and Topology Parameters

Abstract

SE is critical in ADNs integrating renewable DGs. Traditional SE methods face the challenges of increasing SE errors and decreased robustness due to the adverse impact of bad data in measurements and topology parameters. To address these issues, this paper proposes a robust SE method that considers bad data in measurements and topology parameters. First, a bad measurement data processing model is proposed to improve measurement and SE accuracy by generating high-precision pseudo-measurements through adaptive learning from historical data sequences to replace the bad measurement data in measurements. Second, a robust SE model combining network estimation and linear estimation is proposed, which enhances SE accuracy and robustness under bad data generated in measurements and topology parameters in ADNs. In a simulation, the proposed method’s effectiveness is verified on the modified IEEE 33-node system.

1. Introduction

With the increasing penetration of renewable DGs, traditional distribution networks are transitioning towards ADNs [1,2]. The increasingly complex structure and functional requirements of ADNs pose higher demands on SE [3,4,5]. SE involves collecting and processing measurement data to estimate state parameters such as voltage and current at various nodes in ADNs. This process provides essential state information for the operation and control of ADNs. Therefore, ensuring a high-accuracy SE in ADNs is of great significance.

Many studies improve SE accuracy by enhancing the accuracy of measurement data and optimizing SE methods or models. To improve the accuracy of measurement data, high-precision pseudo-measurements can be used to replace the bad data in the measurement, thereby processing the bad data and enhancing the SE accuracy. In the existing research on pseudo-measurement, pseudo-measurement is mainly generated through methods based on linear interpolation [6,7,8] and ANN approaches [9,10], thereby improving the SE accuracy. Ref. [6] proposes an adaptive interpolation strategy to enhance the accuracy of generator SEs. Refs. [7,8] consider the synchronous characteristics of linear interpolation and use average interpolation to supplement pseudo-measurements. Refs. [9,10] use ANNs to obtain pseudo-measurements, thereby improving SE accuracy. To optimize SE methods or models, WLS-based SE [11,12,13,14] is widely applied, with many studies focusing on its improvement. Refs. [15,16] propose enhanced WLS that transform measurement data into equivalent current measurement matrices to improve accuracy. Ref. [17] introduces an improved WLS that dynamically tracks changes in measurement data to reduce SE errors. Ref. [18] optimizes SE by combining branch current and branch power methods, achieving high-precision estimation. Ref. [19] addresses ill-conditioning issues in polar-coordinate SE by representing the Jacobian matrix of the iteration process in Cartesian coordinates. Ref. [20] simplifies measurement equations, avoiding the propagation of measurement errors without increasing redundant measurements. However, when the measurement contains bad data (the measurement deviation is greater than three times the normal measurement standard deviation) [21], since the WLS method does not have SE robustness, the SE accuracy would be reduced due to the bad data. Therefore, to improve the robustness of the SE model to bad data, robust estimations have been proposed. In the existing studies on robust estimation, robust SE is mainly achieved through methods based on WLAV and its related improvements [22,23,24,25,26]. Refs. [22,23] propose a WLAV method to address the residual amplification problem caused by the weighted sum of squares. Ref. [24] proposes a WLAV robust SE method based on the multi-prediction correction inlier approach, reducing the number of iterations. Ref. [25] presents an adaptive WLAV robust SE method to ensure estimation accuracy. Ref. [26] proposes the Huber-M estimation to further improve convergence and robustness through weighting measures. Ref. [27] proposes an exponential target function estimation that applies to measurements with any probability distribution on SE. Ref. [28] proposes a robust SE to address the low SE accuracy caused by insufficient measurement fusion and topology changes.

However, the above methods still face critical issues when used for SE in ADNs with renewable DGs:

(1) The power fluctuation in renewable DGs and complex variations of loads pose challenges for measurement equipment to track and accurately record ADNs’ real-time data, thus impacting the measurement. Also, SE is closely related to measurement accuracy. Bad data in measurements caused by measurement equipment faults or communication noises [29,30] deteriorate measurement accuracy, which may greatly increase SE error and make it difficult to enhance traditional SE methods’ accuracy;

(2) When the ADN topology changes, bad topology parameters may occur due to the delay and loss of topology data [31]. This causes errors in the calculation of the Jacobian matrix, leading to increased SE errors, especially when using mainstream SE methods that rely on accurate topology parameters.

To address these issues, this paper proposes a robust SE method that considers bad data.

Table 1. Comparison of this paper with existing studies.

Models	Strength	[7,8]	[9]	[11]	[22,23]	Ours
Bad data processing	Pseudo-measurement robustness in bad measurement					✓
Robust SE	SE robustness in bad measurement				✓	✓
	SE robustness in bad topology parameters					✓
	SE computational efficiency			✓		✓

provides a comparison with existing studies. The contributions of the paper are as follows.

(1) To tackle the measurement and SE accuracy decline caused by bad measurement data generated due to measurement faults and communication noises, this paper proposes a bad measurement data-processing model. The proposed model utilizes adaptive learning from historical data sequences to generate pseudo-measurements for replacing bad measurements. Compared with the mainstream model, the model still has strong robustness when the measurements contain different proportions of bad data corruption, and it improves the SE accuracy by generating high-precision pseudo-measurements;

(2) The proposed robust estimation model uses a neural network that requires no topology information input compared to mainstream models, and it overcomes the SE accuracy degradation in mainstream models due to wrong topology parameters and bad measurement input. It introduces measurement redundancy in linear estimation, which ensures ADNs’ SE accuracy under conditions involving wrong topology parameters and different bad measurement proportions compared to mainstream methods;

(3) The proposed robust estimation model relies on the parallel computing characteristics of neural networks rather than the Jacobian matrix iteration in the SE model. Thus, it has faster computational efficiency than mainstream methods.

The remainder of the paper is organized as follows: Section 2 introduces the principles of SE; Section 3, Section 4 and Section 5 discuss the overall framework of the proposed method, the bad measurement data-processing model, and the robust SE model, respectively; Section 6 verifies the proposed method in simulation; and finally, Section 7 concludes the paper.

2. State Estimation Principle

SE is the process of obtaining the state variables $x$ through AND’s measurement $z$. The ADNs’ SE [32] is as follows:

$$z=h\left(x\right)+\epsilon$$

(1)

The widely used WLS can be equivalent to solving the optimization problem by the Newton–Raphson method:

$$minJ={[z-h\left(x\right)]}^{T}W[z-h\left(x\right)]$$

(2)

$${\widehat{x}}^{l+1}={\widehat{x}}^l+∆x^l$$

(3)

$$H\left({\widehat{x}}^l\right)={\left[{\displaystyle \frac{\partial h{\displaystyle (}x{\displaystyle )}}{\partial x}}\right]}_{x={\widehat{x}}^l}$$

(4)

$$G\left({\widehat{x}}^l\right)=H^T\left({\widehat{x}}^l\right)WH\left({\widehat{x}}^l\right)$$

(5)

$$∆x^l={G\left({\widehat{x}}^l\right)}^{-1}{H}^T{\displaystyle (}{\widehat{x}}^l{\displaystyle )}W[z-h\left({\widehat{x}}^l\right)]$$

(6)

3. The Overall Framework of State Estimation Considering Bad Data

This section presents the overall framework of the proposed robust state estimation method considering bad data. Based on this framework, the bad measurement data-processing model in Section 4 and the robust estimation model in Section 5 are proposed, respectively. The overall SE framework is shown in Figure 1, which includes bad measurement data processing, robust model design, and robust model estimation.

First, before model estimation, bad measurement data processing is needed to address the issue of increased SE errors caused by decreased measurement accuracy. The bad measurement data processing is as follows. (1) Based on actual load data, obtain the load data by incorporating load fluctuation factors. Then, calculate the true power flow values, and add Gaussian noise to generate the measurement dataset containing bad measurement data. (2) Construct the pseudo-measurement network for replacing bad measurement data, which is composed of the GRU. The network takes historical measurement data as an input and outputs the predicted measurement values. (3) Use the generated pseudo-measurements to replace the bad measurement data.

Second, after bad measurement data processing, robust model design is used to address the decreased SE accuracy issue caused by incorrect topology parameters. The design process is as follows. (1) Design the pre-estimation network. The network input consists of the measurement data at the estimation time, and the outputs are the estimated values of the node voltage magnitude and phase angle. (2) Train the network using the generated dataset.

Finally, after bad measurement data processing and model training, model estimation is used to obtain the final estimation results. The model estimation process is as follows. (1) Feed the measurement data into the DSENN. (2) Based on the input of μPMU measurement and the SE values from DSENN, linear SE is used to obtain the final SE results.

4. Bad Measurement Data Processing Model

This section presents the bad measurement data-processing model, which provides the basis for robust SE of ADNs. The two subsections discuss the dataset generation and the bad measurement data-processing model, respectively.

4.1. Generation of Dataset

The generated dataset is used for bad measurement data processing and robust estimation model training and testing.

First, the load data from Belgium [33] are selected as the baseline data, and the specific daily load curve is shown in Appendix A Figure A1.

Next, the actual power data are calculated based on their respective loads and Matpower software verision 5.0 [34], thus generating large amounts of ADN’s power flow data for simulation.

Finally, based on the power flow data in ADN, the SCADA and μPMU measurements can be acquired by adding noise following the normal distribution. The normal distribution parameters SCADA and μPMU measurement, as shown in [35], are shown in

Table 2. The normal distribution parameters for SCADA and μPMU measurement.

Measurement Equipment	Measurement Type	Mean	Standard Deviation
SCADA	Node injected power	0	0.02
	Branch power flow	0	0.02
μPMU	Voltage magnitude	0	0.002
	Voltage phase angle	0	0.005
	Current magnitude	0	0.002
	Current phase angle	0	0.005

. The SCADA measurements include node power injections and branch power flows. The μPMU measurements include the magnitude and phase angle of voltage and current. The definition of bad data in this article follows the Chinese national standard [21]. When the error is greater than three standard deviations of the normal measurement error, the data can be regarded as bad data. To simulate bad measurement data that may occur in ADNs, the topology is randomly altered before performing the power flow calculation to obtain measurement data under different topologies, and 15% of the selected measurements from each set of generated data are altered by adding normal distribution following ±20% mean and ±20% standard deviation as bad measurement data. Through the above methods, the dataset for training the models is constructed.

4.2. GRU-Based Bad Measurement Data Processing

In the ADNs, considering that the bad measurement data in measurements decreases the SE model’s accuracy, the GRU is used for adaptive learning from historical data to generate high-precision pseudo-measurements, which replace the bad measurement data in measurements for enhancing the measurement and SE accuracy.

The GRU includes an update gate and a reset gate. The update gate controls the degree to which the previous time step’s state information is preserved in the current state, with a larger value indicating more past state information is retained. The reset gate controls the degree to which the current state combines with previous information, with a smaller value indicating more ignored information. Figure 2 shows the structure of the GRU. In Figure 2, the direction of the arrows indicates the direction of data flow, where “$\otimes$” represents matrix multiplication, “$\sigma$” represents the sigmoid activation function, “$\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$” represents the $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ activation function, and $1-s_t$ represents the data propagated forward through this link.

The update gate and reset gate are denoted as $s_t$ and $r_t$, respectively. $z_t$ is the input, and $h_t$ is the output of the hidden layer. The pseudo-measurement $h_t$ is computed based on the following formula:

$$s_t=\sigma {\displaystyle (}{W}^{\left(s\right)}{z}_t+U^{\left(s\right)}{h}_{t-1}{\displaystyle )}$$

(7)

$$r_t=\sigma {\displaystyle (}{W}^{\left(r\right)}{z}_t+U^{\left(r\right)}{h}_{t-1}{\displaystyle )}$$

(8)

$${\stackrel{{\displaystyle ~}}{h}}_t=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}{\displaystyle (}{r}_t\otimes U_t{h}_{t-1}+W_t{z}_t{\displaystyle )}$$

(9)

$$h_t=\left(1-s_t\right)\otimes {\stackrel{{\displaystyle ~}}{h}}_t+s_t\otimes h_{t-1}$$

(10)

5. Robust Estimation Model

Based on Section 2, Section 3 and Section 4 a robust SE method combining pre-estimation and linear estimation models is proposed in this section. The two subsections discuss the neural network-based pre-estimation and linear SE, respectively.

5.1. Neural Network-Based Pre-Estimation

To address the decreased accuracy and computational efficiency of SE caused by incorrect topology parameters, a neural network-based pre-estimation is proposed. It doesn’t require topology parameters and offers high computational efficiency, thereby enhancing the SE accuracy and computational efficiency under incorrect topology parameters in ADN.

The DSENN for pre-estimation is designed. The DSENN design includes its data normalization, input and output, layer structure, and loss function.

(a)

Data normalization

The node injection power and branch power data in the measurement data set are normalized from minimum to maximum, so that their values are between the interval [0, 1], and are used for the training input of the DSENN model.

(b)

Input and output

The DSENN utilizes the normalized measurement data as inputs and directly computes the voltage estimated value through itself. The inputs and outputs of DSENN are as follows:

$$Z={\left[\begin{array}{ccc}{z}_1& \dots & z_m\end{array}\right]}^{\mathrm{T}}$$

(11)

$$X=\left\{\begin{array}{c}{X}^v={\left[\begin{array}{ccc}{x^v}_1& \cdots & {x^v}_n\end{array}\right]}^{\mathrm{T}}\\ X^{\theta }={\left[\begin{array}{ccc}{x^{\theta }}_1& \cdots & {x^{\theta }}_n\end{array}\right]}^{\mathrm{T}}\end{array}\right.$$

(12)

(c)

Layer structure

As shown in Figure 3, the input layer of DSENN is the measurement data of ADN, and the output layer outputs is the voltage amplitude. $X^v$ and voltage phase angle $X^{\theta }$ of ADN. The network structure in the middle layer is composed of FC. The selection of the number of hidden layers in the layer structure, the selection of the number of neurons in a single layer, and the selection of the activation function for each neuron layer are shown in Appendix A 

Table A1. DSENN structure parameters.

Layer Name	The Serial Number of Hidden Layers	Activation Function	Size
Input layer	-	-	$m$ × 512
FC1	1	leaky-relu	512 × 512
FC2	2	leaky-relu	512 × 512
FC3	3	leaky-relu	512 × 512
FC4	4	Tanh	512 × 256
FC5		leaky-relu	512 × 256
Output layer 1	-	-	256 × $n$
Output layer 2	-	-	256 × $n$

.

(d)

Loss function

The DSENN uses the RMSE between the SE and true values as the loss function. The overall loss function for the DSENN is calculated as follows:

$${loss}^v=\sum _{i=1}^n{\left({x^v}_i-{{\widehat{x}}^v}_i\right)}^2$$

(13)

$${loss}^{\theta }=\sum _{i=1}^n{\left({x^{\theta }}_i-{{\widehat{x}}^{\theta }}_i\right)}^2$$

(14)

$${loss}^{all}={loss}^v+{loss}^{\theta }$$

(15)

Based on the above, the DSENN model proposed in this paper is trained to make it have good estimation performance. See Appendix A 

Table A2. DSENN training parameters.

Parameter Name	Value
Learning rate	1 × 10−4
Epoch	60
Gradient decay factor	0.8
Minbatchsize	96

and Figure A2 for detailed training configuration parameters, data set partitioning and loss convergence graphs in training.

5.2. Pre-Estimation Based Linear SE

The DSENN’s performance may decrease because of the significant deviation of input measurement from the dataset. Therefore, the linear SE is adopted. It utilizes the output of DSENN as pseudo-measurements combined with μPMU measurement to further enhance the accuracy of ADN’s SE.

The input of linear SE includes the DSENN’s output voltage vector, μPMU voltage, and current vectors. The outputs are the estimated voltage magnitude and phase angle values. Given ADN’s measurements and topology parameters, the linear SE follows Equations (16) and (17):

$${Z=\left[\left[\begin{array}{c}{X}^v\\ X^{\theta }\end{array}\right]\left[\begin{array}{c}{U}_{\mu PMU}\\ {\theta }_{\mu PMU}\end{array}\right]\left[\begin{array}{c}{I}_{\mu PMU}\\ {\theta }_{I\mu PMU}\end{array}\right]\right]}^{\mathrm{T}}=H\left[\begin{array}{c}U\\ \theta \end{array}\right]+\epsilon$$

(16)

$$\left[\begin{array}{c}U\\ \theta \end{array}\right]={\left[H^{\mathrm{T}}WH\right]}^{-1}{H}^{T}WZ$$

(17)

6. Case Studies

In this section, the proposed method is verified by simulation. Section 6.1 discusses the simulation setup. Section 6.2 and Section 6.3 verify the proposed bad measurement data processing and robust estimation’s effectiveness for improving the estimation accuracy, respectively. Section 6.4 verifies the robust SE model’s effectiveness in enhancing computational efficiency.

6.1. Simulation Setup

In the simulation, the modified IEEE 33-node system in Figure 4 is used to represent the ADN.

In Figure 4, the modified system has node numbers ranging from 1 to 33 and includes a total of 37 branch lines. Additionally, renewable DGs with a rated capacity of 300 kW each are connected at nodes #7, #11, #16, and #33. In terms of measurement equipment configuration, SCADA equipment can measure all node power injections and branch power flows in the system to simulate actual SCADA measurements. Additionally, μPMU measurements that provide synchronous voltage and current measurements are installed at nodes #3, #6, #10, #13, #17, #21, and #25. The difference between the modified IEEE 33-node test system and the original system is that the load data is different, and four nodes are connected to the DG. And the load data can be calculated according to Appendix A Figure A1 and

Table A3. Line parameter and load data of modified IEEE 33-node test system.

Line Parameter				Load Data at Nodes
Branch		Branch Impedance		Node	Active Load/kW	Reactive Load/kVar
Startin node	End node	Resistance/(p.u.)	Reactance/(p.u.)	1	0	0
1	2	0.0922	0.047	2	100	60
2	3	0.493	0.2511	3	90	40
3	4	0.366	0.1864	4	120	80
4	5	0.3811	0.1941	5	60	30
5	6	0.819	0.707	6	60	20
6	7	0.1872	0.6188	7	200	100
7	8	0.7114	0.2351	8	200	100
8	9	1.03	0.74	9	60	20
9	10	1.044	0.74	10	60	20
10	11	0.1966	0.065	11	45	30
11	12	0.3744	0.1238	12	60	35
12	13	1.468	1.155	13	60	35
13	14	0.5416	0.7129	14	120	80
14	15	0.591	0.526	15	60	10
15	16	0.7463	0.545	16	60	20
16	17	1.289	1.721	17	60	20
17	18	0.732	0.574	18	90	40
2	19	0.164	0.1565	19	90	40
19	20	1.5024	1.3554	20	90	40
20	21	0.4095	0.4784	21	90	40
21	22	0.7089	0.9373	22	90	40
3	23	0.4512	0.3083	23	90	50
23	24	0.898	0.7091	24	420	200
24	25	0.896	0.7011	25	420	200
6	26	0.203	0.1034	26	60	25
26	27	0.2842	0.1447	27	60	25
27	28	1.059	0.9337	28	60	20
28	29	0.8042	0.7006	29	120	70
29	30	0.5075	0.2585	30	200	600
30	31	0.9744	0.963	31	150	70
31	32	0.3105	0.3619	32	210	100
32	33	0.341	0.5302	33	60	40

. The specific line parameters and base load data are shown in Appendix A Table A3.

Table 3. Topology configuration of the modified IEEE 33-node system.

Topology Type	Disconnected Switch
Topology 1	7–8, 9–10, 14–15, 32–33, 25–29
Topology 2	7–8, 10–11, 14–15, 30–31, 25–29
Topology 3	7–8, 9–10, 14–15, 28–29, 18–33
Topology 4	7–8, 9–10, 14–15, 31–32, 25–29

shows the four types of topologies configured for the system in this paper.

The proposed method is implemented in the R2024a version of MATLAB. See Appendix A Figure A3 and Figure A4 for detailed algorithm implementation steps. The computer’s processor configuration is a 12th Gen Intel(R) Core(TM) i7-12700, with a RAM of 64 GB.

6.2. Validation of the Bad Measurement Data Processing Model’s Effectiveness

In this section, the RMSE and the MAE are used as evaluation metrics to verify the effectiveness of the proposed method for improving measurement data accuracy and SE accuracy. The evaluation metrics are as follows:

$${RMSE}_m=\sqrt{{\displaystyle \frac{\sum _{i=1}^D{{\displaystyle (}{\widehat{z}}_i-z_i{\displaystyle )}}^2}{D}}}$$

(18)

$${MAE}_m={\displaystyle \frac{\sum _{i=1}^D\left|{\widehat{z}}_i-z_i\right|}{D}}$$

(19)

$${MAE}_v={\displaystyle \frac{\sum _{i=1}^N\left|{\widehat{xx}}_i^v-{xx}_i^v\right|}{N}}$$

(20)

$${MAE}_{\theta }={\displaystyle \frac{\sum _{i=1}^N\left|{\widehat{xx}}_i^{\theta }-{xx}_i^{\theta }\right|}{N}}$$

(21)

The existing studies of linear interpolation [8] and ANN [9] methods are compared with the proposed method for the accuracy of pseudo-measurement and SE. The accuracy of pseudo-measurement is shown in

Table 4. Comparison of the RMSE and MAE of pseudo-measurements.

Method	${\mathit{R}\mathit{M}\mathit{S}\mathit{E}}_{\mathit{m}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{m}}$
Linear interpolation	3.29 × 10−2	1.95 × 10−2
ANN	3.07 × 10−2	8.4 × 10−3
Proposed	2.35 × 10−2	7.7 × 10−3

and

Table 5. Comparison of the MAE of pseudo-measurements in different proportions of bad measurements.

Proportion of Bad Measurement	Method	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{m}}$
10%	Linear interpolation	8.8 × 10−2
	ANN	7.39 × 10−2
	Proposed	2.35 × 10−2
30%	Linear interpolation	1.56 × 10−1
	ANN	1.12 × 10−1
	Proposed	4.87 × 10−2

and Figure 5. The accuracy of SE is shown in

Table 6. Comparison of the MAE of SE based on different pseudo-measurements.

Method	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\theta }}$
Origin data	8.51 × 10−2	1.15 × 10−1
Linear interpolation	5.59 × 10−2	7.85 × 10−2
ANN	4.85 × 10−2	6.57 × 10−2
Proposed	4.68 × 10−2	5.49 × 10−2

and

Table 7. Comparison of the MAE of SE based on pseudo-measurements in different proportions of bad measurements.

Proportion of Bad Measurement	Method	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\theta }}$
10%	Linear interpolation	2.92 × 10−2	1.01 × 10−1
	ANN	2.65 × 10−2	8.63 × 10−2
	Proposed	2.23 × 10−2	6.7 × 10−2
30%	Linear interpolation	3.12 × 10−2	1.56 × 10−1
	ANN	2.72 × 10−2	1.1 × 10−1
	Proposed	2.4 × 10−2	9.58 × 10−2

and Figure 6.

The RMSE and MAE of the proposed method for pseudo-measurement are 2.35 × 10⁻² and 7.7 × 10⁻³ in Table 4, respectively, both of which are smaller than the errors under traditional pseudo-measurement methods. Moreover, from Figure 5, it can be observed that the RMSE fluctuation of the traditional method is relatively large, with a maximum RMSE of 0.09, while the RMSE of the proposed method for pseudo-measurement is more stable compared to the traditional method, with the RMSE fluctuation stable at around 0.02. This is because traditional linear interpolation and ANN methods have difficulty effectively fitting the complex relationship between historical measurement data and pseudo-measurements in ADN, resulting in poor performance in predicting pseudo-measurements. The proposed method adapts to the nonlinear relationship in the historical measurement data through adaptive learning and has better adaptability to fit complex historical measurement data, generating pseudo-measurements with smaller errors. It can also be seen from Table 5 that the pseudo-measurement error of the proposed model increases with the increase in the proportion of bad data, but the increase in the model error is significantly smaller than that of the pseudo-measurement error based on the linear interpolation method and ANN method, indicating that the proposed model has good robustness. In addition, the comparison experiments of the pseudo-measurement accuracy of the proposed pseudo-measurement model under different topologies are given in Appendix A 

Table A4. Comparison of the MAE of pseudo-measurements based on no bad measurement under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{m}}$
Linear interpolation	Topology 1	7.94 × 10−2
	Topology 2	7.51 × 10−2
	Topology 3	4.5 × 10−2
	Topology 4	6.34 × 10−2
ANN	Topology 1	2.43 × 10−2
	Topology 2	2.61 × 10−2
	Topology 3	2.43 × 10−2
	Topology 4	2.58 × 10−2
Proposed	Topology 1	6.3 × 10−3
	Topology 2	6.3 × 10−3
	Topology 3	5.1 × 10−3
	Topology 4	5.7 × 10−3

and

Table A5. Comparison of the MAE of pseudo-measurements based on bad measurement under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{m}}$
Linear interpolation	Topology 1	1.75 × 10−1
	Topology 2	1.63 × 10−1
	Topology 3	1.53 × 10−1
	Topology 4	1.7 × 10−1
ANN	Topology 1	6.26 × 10−2
	Topology 2	6.4 × 10−2
	Topology 3	5.85 × 10−2
	Topology 4	6.19 × 10−2
Proposed	Topology 1	3.27 × 10−2
	Topology 2	3.63 × 10−2
	Topology 3	3.11 × 10−2
	Topology 4	3.29 × 10−2

. The simulation results show that the proposed pseudo-measurement model can still generate high-precision pseudo-measurements under different topologies. Its accuracy is superior to the existing linear interpolation and ANN methods, and it has a strong generalization ability. The proposed method’s effectiveness and robustness in processing bad measurement data are verified by the above simulation results, thereby improving the measurement accuracy by replacing bad measurement data with high-precision pseudo-measurements.

After replacing the bad measurement data in the measurement, the ADN’s SE MAE significantly decreased, which is shown in Table 6 and Table 7. The MAEs using the proposed method are 4.68 × 10⁻² and 5.49 × 10⁻³, respectively, both of which are smaller than the compared pseudo-measurement method. Moreover, with the increase in the proportion of bad data, the SE error of the proposed pseudo-measurement method is significantly lower than the linear interpolation and ANN methods. Additionally, as shown in Figure 6, in terms of voltage magnitude, the MAE at each node under the traditional methods is relatively large, with the MAEs reaching 0.056 and 0.048, respectively. In contrast, the MAE of the proposed method stabilizes around 0.046, which is smaller compared to traditional methods. Regarding voltage phase angles, the proposed method achieves a lower phase angle estimation MAE for most nodes than traditional pseudo-measurement methods. This improvement is attributed to the adaptive learning of the proposed method, which generates high-accuracy pseudo-measurements, replaces bad measurement data, and enhances measurement data accuracy. Consequently, the adverse impact of bad data on SE is reduced, thereby improving SE accuracy. In Appendix A 

Table A6. Comparison of the SE MAE of pseudo-measurements based on no bad measurement under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$
Linear interpolation	Topology 1	1.19 × 10−2	2.29 × 10−1
	Topology 2	7.8 × 10−3	2.35 × 10−1
	Topology 3	1.06 × 10−2	1.97 × 10−1
	Topology 4	1.27 × 10−2	8.03 × 10−2
ANN	Topology 1	1.02 × 10−2	1.74 × 10−1
	Topology 2	7.3 × 10−3	1.84 × 10−1
	Topology 3	1.05 × 10−2	1.85 × 10−1
	Topology 4	1.1 × 10−2	8.33 × 10−2
Proposed	Topology 1	9.4 × 10−3	1.5 × 10−1
	Topology 2	7.2 × 10−3	1.64 × 10−1
	Topology 3	9 × 10−3	1.65 × 10−1
	Topology 4	8.8 × 10−3	6.65 × 10−2

and

Table A7. Comparison of the SE MAE of pseudo-measurements based on bad measurement under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$
Linear interpolation	Topology 1	1.17 × 10−2	3.15 × 10−1
	Topology 2	6.8 × 10−3	3.04 × 10−1
	Topology 3	1.04 × 10−2	2.18 × 10−1
	Topology 4	1.18 × 10−2	1.77 × 10−1
ANN	Topology 1	1 × 10−2	3.62 × 10−1
	Topology 2	6.2 × 10−3	2.39 × 10−1
	Topology 3	1.04 × 10−2	1.88 × 10−1
	Topology 4	9.5 × 10−3	2.27 × 10−1
Proposed	Topology 1	9.4 × 10−3	1.93 × 10−1
	Topology 2	6.2 × 10−3	1.89 × 10−1
	Topology 3	9.6 × 10−3	1.86 × 10−1
	Topology 4	8.7 × 10−3	9.86 × 10−2

, a comparison experiment on the SE accuracy of the proposed pseudo-measurement model under different topologies is presented. The simulation results show that the proposed pseudo-measurement model improves the SE accuracy by generating pseudo-measurements with higher accuracy than the existing linear interpolation and ANN methods. The simulation results validate the proposed bad measurement data processing model’s effectiveness in enhancing SE accuracy.

6.3. Validation of the Robust Estimation Model’s SE Accuracy

In this subsection, the MAE is used as the evaluation metric to verify the proposed robust estimation model under bad topology parameters. The evaluation metric is consistent with Equations (20) and (21).

The existing studied WLS [11] and WLAV [22] SE methods are compared with the proposed model under situations with and without bad measurement data in different topologies. The comparison with the situation without bad measurement data is shown in

Table 8. Comparison of the MAE of SE without bad measurement data under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\theta }}$
WLS	Topology 1	2.94 × 10−2	1.14 × 10−1
	Topology 2	1.14 × 10−2	7.63 × 10−2
	Topology 3	1.17 × 10−2	1.13 × 10−1
	Topology 4	1.42 × 10−2	3.96 × 10−2
WLAV	Topology 1	1.22 × 10−2	1.14 × 10−1
	Topology 2	7.6 × 10−3	5.61 × 10−2
	Topology 3	1.04 × 10−2	1 × 10−1
	Topology 4	6.7 × 10−3	3.52 × 10−2
Proposed	Topology 1	2.2 × 10−3	2.98 × 10−2
	Topology 2	2.8 × 10−3	5.08 × 10−2
	Topology 3	2.8 × 10−3	2.5 × 10−2
	Topology 4	2.7 × 10−3	3.58 × 10−2

and Figure 7; the comparison of the situation with bad measurement data is shown in

Table 9. Comparison of the MAE of SE containing bad measurement data under four topologies.

Method	Topology Type	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\theta }}$
WLS	Topology 1	3.01 × 10−2	1.19 × 10−1
	Topology 2	1.14 × 10−2	8.61 × 10−2
	Topology 3	1.17 × 10−2	1.17 × 10−1
	Topology 4	1.44 × 10−2	4.02 × 10−2
WLAV	Topology 1	1.23 × 10−2	1.23 × 10−1
	Topology 2	7.6 × 10−3	6.81 × 10−2
	Topology 3	1.04 × 10−2	1.02 × 10−1
	Topology 4	6.8 × 10−3	3.28 × 10−2
Proposed	Topology 1	2.21 × 10−3	3 × 10−2
	Topology 2	2.83 × 10−3	5.1 × 10−2
	Topology 3	2.82 × 10−3	2.52 × 10−2
	Topology 4	2.72 × 10−3	3.6 × 10−2

and

Table 10. Comparison of the MAE of SE based on different proportions of bad measurement.

Proportion of Bad Measurement	Method	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{v}}$	${\mathit{M}\mathit{A}\mathit{E}}_{\mathit{\theta }}$
10%	WLS	8.4 × 10−3	8.32 × 10−2
	WLAV	8.7 × 10−3	6.91 × 10−2
	Proposed	3.1 × 10−3	4.97 × 10−2
30%	WLS	8.4 × 10−3	9.62 × 10−2
	WLAV	8.5 × 10−3	9.04 × 10−2
	Proposed	3.1 × 10−3	4.94 × 10−2

and Figure 8. Additionally, in Figure 7 and Figure 8, four types of topologies are selected for simulation experiments, with each topology having 10 estimation sections. The estimation serial numbers 10, 20, 30, and 40 represent the change of topology. The estimation results with bad data in their topology parameters are simulated between serial numbers 1–5, 11–15, 21–25, and 31–35. Topology correction is performed at serial numbers 6, 16, 26, and 36 for each topology to simulate the different topology situations. The horizontal axis represents the estimation section serial number, and the vertical axis represents the MAE of SE.

Based on Table 8, in the situation without bad measurement data, the SE MAE of the proposed method under four different topologies is generally smaller than the WLS and WLAV methods. In terms of voltage magnitude estimation, the MAE of the proposed method under four topologies is nearly an order of magnitude smaller than WLS, and there is a significant reduction compared to WLAV. Regarding the voltage phase angle estimation MAE, the MAE of the proposed method is smaller than that of the WLS and most of the WLAV. From Figure 7, without bad measurement data, the proposed method has a lower MAE in SE under the four topologies compared to WLS and WLAV. When the topology parameter is not corrected, bad data in the topology parameter inputs cause an increase in the MAE of WLS and WLAV under certain topologies. By correcting the topology parameters, the SE accuracy of both WLS and WLAV improves. Regardless of whether the topology parameters are corrected or not, the MAE of the proposed method remains at a low level. The above simulation results fully demonstrate that the proposed method can effectively reduce the adverse bad data in the topology parameters inputs on the SE accuracy without bad measurement data, thereby improving SE accuracy.

From Table 9, it can be seen that, in terms of voltage magnitude estimation, the MAE of the proposed model on four topologies is significantly lower compared to the WLS and WLAV. In terms of voltage angle estimation, the phase angle MAE of the proposed model is smaller than that of the WLS and most results of the WLAV. From Figure 8, it can be observed that, when there is bad measurement data, the traditional WLS lacks robustness, making the bad measurement data have an adverse impact on the SE. Both WLAV and the proposed method exhibit good robustness, effectively reducing the adverse impact of bad measurement data on the accuracy of SE. When the topology parameters are not corrected, the input of bad data into the topology parameters increases the estimation error of the WLS and WLAV under certain topologies. By correcting the topology parameters, the estimation accuracy of both the WLS and WLAV is improved. Regardless of whether the topology parameters are corrected or not, the estimation error of the proposed model remains at a relatively low value. This is because the proposed model does not require the topology parameters to be input during the pre-estimation stage, reducing the impact of bad topology parameter data input on the estimation results. Although there may be bad topology parameter inputs during the linear estimation stage, their adverse effects are still significantly less than WLS and WLAV. In addition, Table 10 shows that, with the increasing proportion of bad measurements, the SE errors of the WLS and WLAV methods rise to varying degrees, while the SE error of the proposed model remains at a low level, demonstrating good robustness. The simulation results fully demonstrate that the proposed method can effectively reduce the impact of bad topology parameters on SE accuracy, contain bad measurement data, and thereby, improve estimation accuracy.

6.4. Validation of the Computational Efficiency of the Robust Estimation Model

In this subsection, the average estimation time is used as the evaluation metric. The results of the average estimation time for 500 estimation times are used to verify the effectiveness of the proposed method in improving model computational efficiency. The evaluation metric is as follows:

$$T_s={\displaystyle \frac{\sum _{i=1}^{Num}{t}_i}{Num}}$$

(22)

From

Table 11. Comparison of estimation times for different methods.

Method	${\mathit{T}}_{\mathit{s}}$/(ms)
WLS	2.437
WLAV	603.99
Proposed	0.14

, it can be seen that the computation time of the proposed robust estimation model is significantly lower than WLS and WLAV.

The proposed SE model takes advantage of the fast computation of neural networks to obtain the SE results. In contrast, the traditional estimation requires repeated iterations, with each iteration needing to recompute the Jacobian matrix. Therefore, the proposed method demonstrates a significant advantage in estimation efficiency. Although the linear SE in the estimation process may slightly increase computation time, the overall estimation time remains much shorter than the WLS and WLAV methods. These results verify the proposed robust estimation model’s effectiveness in improving computational efficiency.

7. Conclusions

Aiming to effectively address the decreased SE accuracy and long computation time caused by bad data, this paper proposes a robust SE method that considers bad data in measurements and topology parameters. The following conclusions are drawn from the proposed method:

(1) Reducing the impact of bad data on measurements and SE accuracy: The bad measurement data-processing model adaptively learns historical data sequences and has good robustness. It can generate high-precision pseudo-measurements under different proportions of bad data and different topologies, thereby improving the measurement and SE accuracy of ADNs;

(2) Strong estimation robustness and computational efficiency: The proposed robust SE model exhibits higher estimation accuracy and computational efficiency compared to traditional methods. The proposed model enhances estimation accuracy through pre-estimation and linear estimation, ensuring SE robustness in situations with bad data in measurements and topology parameters.

Future research directions include considering the interpretability and measurement correlation in the SE model, which can further enhance SE accuracy.