Abstract
As the phasor measurement unit (PMU) placement problem involves a cost-benefit trade-off, more PMUs get placed on higher-voltage buses. However, this leads to the fact that many lower-voltage levels of the bulk power system cannot be observed by PMUs. This lack of visibility then makes time-synchronized state estimation of the full system a challenging problem. In this paper, a deep neural network-based state estimator (DeNSE) is proposed to solve this problem. The DeNSE employs a Bayesian framework to indirectly combine the inferences drawn from slow-timescale but widespread supervisory control and data acquisition (SCADA) data with fast-timescale but selected PMU data, to attain sub-second situational awareness of the full system. The practical utility of the DeNSE is demonstrated by considering topology change, non-Gaussian measurement noise, and detection and correction of bad data. The results obtained using the IEEE 118-bus system demonstrate the superiority of the DeNSE over a purely SCADA state estimator and a PMU-only linear state estimator from a techno-economic viability perspective. Lastly, the scalability of the DeNSE is proven by estimating the states of a large and realistic 2000-bus synthetic Texas system.
POWER utilities attain situational awareness of their transmission system through the process of state estimation. Particularly, state estimation provides the inputs for performing real-time contingency analysis, optimal power flow, and even network expansion planning [
Due to the asynchronous nature of their inputs, purely SCADA state estimators suffer from problems such as non-linearity, divergence, and low accuracy [
We have investigated the reality of the PMU unobservability problem by collecting data from two U.S. power utilities.
Voltage level (kV) | Number of buses | Number of PMU-equipped buses | Percentage of observed buses (%) |
---|---|---|---|
500 | 52 | 28 | 79 |
230 | 15 | 5 | 53 |
161 | 1185 | 92 | 27 |
115 | 42 | 2 | 10 |
69 | 144 | 2 | 3 |
Voltage level (kV) | Number of buses | Number of PMU devices | Percentage of observed buses (%) |
---|---|---|---|
500 | 18 | 53 | 90 |
230 | 47 | 89 | 80 |
115 | 30 | 23 | 30 |
69 | 258 | 207 | 50 |
To counteract the impact of unobservability on state estimation, pseudo-measurements obtained by interpolated observations or forecasts obtained using historical data can be used. However, as demonstrated in [
Motivated by the knowledge gaps outlined above, we propose a deep neural network-based state estimator (DeNSE) that estimates all the transmission system voltages in a time-synchronized manner from PMUs that are only placed at the highest-voltage buses. By performing TSSE using very few PMUs, the DeNSE also circumvents the need for a massive supporting communication infrastructure [
The first is the scalability of the state estimator. Classical LSE formulation involves a matrix inversion step, whose computational complexity is [
In summary, this paper advances the state-of-the-art for time-synchronized state estimation in transmission systems by making the following salient contributions.
1) A high-speed time-synchronized state estimator, i.e., DeNSE, is developed for the BPS that satisfies the need to observe the full system by PMUs.
2) A robust BDDC algorithm is created that ensures the performance of DeNSE under diverse types of bad data and loading conditions.
3) The ability of the DeNSE to tackle topology changes and non-Gaussian measurement noise is demonstrated.
We also provide a logical explanation along with a numerical example in Appendix A to illustrate how DeNSE can perform state estimation for unobservable power systems.
PMU-only LSE solves a variant of the MLE problem, with the most common being the least squares formulation. However, the least squares solution requires the system equations to have full rank, which translates to the constraint of full system observability by PMUs. One way to circumvent this constraint is to reformulate the TSSE problem within a Bayesian framework, where the states and the PMU measurements are treated as random variables. Then, the following minimum mean squared error (MMSE) estimator can be formulated:
(1) |
where is the estimated value of the states; is the optimal estimate; and is the expectation operator.
However, there are two challenges in computing the expected conditional mean of (1). First, the conditional expectation, defined by , requires the knowledge of the joint probability distribution function (PDF) between and , denoted by . When the number of PMUs is scarce, is unknown or impossible to specify, making the direct computation of intractable. Second, even if the under-lying joint PDF is known, it can be difficult to find a closed-form solution for (1). The DNN used in DeNSE overcomes these difficulties by providing an approximation of the conditional expectation of the MMSE estimator.
The DNN has a feed-forward architecture with inputs and outputs, where is the number of measurements coming from PMUs and is the total number of states to be estimated (i.e., and ). Due to incomplete observability of the system by PMUs, . The DNN has hidden layers, in which the input vector entering the layer is expressed in terms of the inputs from the layer as:
(2) |
where is the input vector entering the layer; is the weight between the and the layer; is the output of the layer; and is the bias value of the layer. Next, is passed through an activation function to yield :
(3) |
This propagation continues through all the hidden layers and the resulting value is obtained at the output layer. The loss function compares the estimated output and corresponding true output. The error between them is represented by:
(4) |
where is the error; is the true value of the output; is the estimated value of the output by the DNN in the current epoch; and is an appropriate loss function that indicates how well the DNN has been trained. To improve the training accuracy, is minimized by optimally tuning the weights and biases through a process called backpropagation. The process is repeated until the loss becomes acceptable.
A unique feature of the DeNSE that sets it apart from other ML-based state estimators (such as [
Training the DNN by using the above-mentioned process of indirectly combining inferences from SCADA and PMU data has two advantages: ① the problem of temporal differences and synchronization issues are completely circumvented, and ② any reasonable errors in the SCADA data do not impinge on the performance of the DeNSE. The DeNSE can be impacted by noisy as well as bad PMU data since these data are input to the trained DNN during online operation. The effects of the quality of input data are investigated analytically in Section III-B, and experimentally in Sections IV-B, IV-E, and IV-F.
A DNN trained using the framework proposed in Section II will perform fast and accurate time-synchronized state estimation for PMU-unobservable BPS during real-time operation as long as the topology does not change. However, if the topology used for training and testing changes, the joint PDF between the measurements and the states will change; this can deteriorate the performance of the DeNSE. A possible alternative is to train the DNN from scratch for the new topology. However, it will take a very long time to do so. Instead, we use transfer learning to update the DNN of the DeNSE when topology changes. Transfer learning refers to utilizing models learned from an old problem and leveraging them for a new problem, in order to maintain the learning performance and accuracy. In the context of TSSE, transfer learning is particularly useful because when a topology changes, the mapping between measurements and states of only a small portion of the system gets altered. This implies that the re-learning will be localized.
We employ inductive transfer learning [
To determine when transfer learning via fine-tuning should be implemented, we make use of the topology processor of the BPS. After updating the DNN, the new topology is designated as the base topology to make it consistent with the DeNSE. The overall implementation of transfer learning to handle topology changes is shown in

Fig. 1 Implementation of transfer learning to handle topology changes.
During online implementation, streaming PMU data will be fed as inputs to the proposed DeNSE framework. However, PMU data obtained from the field often suffer from bad data in the form of data dropouts and outliers [
A technique to detect bad data before it enters an ML-based state estimator is proposed in [
(5) |
where is the tail of the distribution, , and is a tunable parameter that specifies the false positive limit. Essentially, the Wald test makes use of the fact that DNN training is done using good quality data. Hence, once the limits of good quality data become known during training, any testing data that lie outside that limit can be termed as bad. This bad data detection method based on Wald test developed in [
Since the Wald test is applied independently and simultaneously to all the input features of a given sample of the testing dataset, it is unlikely that all the features will be bad at the same time. For a given testing dataset sample , the set of indices that correspond to features flagged as bad by the Wald test are called . Then, if denotes the set of indices corresponding to all the features of , the difference of these two sets gives the set of indices corresponding to the good features of , which is denoted by . Now, can be used to find that operating condition (OC) in the training database that most closely resembles the OC captured by . Once that OC (called the nearest OC (NOC)) is found, its entries corresponding to should replace the flagged features of . The overall procedure is depicted in
The Wald test is very sensitive to the choice of . A very small value of may result in bad data being treated as good data, while a large value may result in an extreme scenario data being treated as bad data. This can happen because by definition, extreme scenarios are those OCs that are unlikely to occur normally. In the worst case, data corresponding to an extreme scenario will get flagged as bad data and be replaced by normal data from the training database, making the DeNSE produce an incorrect picture of the operating state of the system. We combine our knowledge of how PMUs are placed in a power system with how extreme OCs actually manifest to design an extreme scenario filter that prevents this problem.
Algorithm 1 : bad data correction using NOC in training dataset |
---|
Input: , Output: the corrected testing dataset sample 1: Create array of indices from , and set 2: Conduct Wald test on and flag the indices of bad data to create 3: 4: 5: |
Furthermore, if PMUs are placed only at the highest voltage buses (which is the premise of this paper), they will be automatically (electrically) close to each other even for PMU-unobservable BPS. This is because the highest voltage buses are connected to each other by the highest voltage lines. Thus, when an extreme scenario manifests, measurements of multiple PMUs will be simultaneously impacted. Conversely, bad data occur randomly in both space and time. This realization leads to the proposal of the following logic for designing the extreme scenario filter: if one or more features of the testing data sample are simultaneously identified as bad by the Wald test for different PMUs, each of which is within hops of each other, then the data sample corresponds to an extreme OC and should not be treated as bad data. This logic is implemented in the manner shown in
Note that in
Algorithm 2 : implementation of extreme scenario filter |
---|
Input: features flagged as bad by Wald test, Output: features passing extreme scenario filter, 1: locations corresponding to 2: 3: = 4: =List of subsets of with elements 5: For : If (every element of is within hops of each other):=List of all features corresponding to =– End if 6: End for 7: 8: If : 9: End 10: Else go to Step 3 |

Fig. 2 Bayesian framework for proposed DeNSE.
The effectiveness of the DeNSE is first illustrated using the IEEE 118-bus system. Each bus of this system is mapped to a bus in the 2000-bus synthetic Texas system [
It is assumed that PMUs are only placed on the highest voltage buses of this system, namely 8, 9, 10, 26, 30, 38, 63, 64, 65, 68, and 81. PMUs located at these 11 buses measure the voltage of the corresponding bus as well as the currents flowing in the lines emanating from that bus. The 41 PMU measurements (11 bus voltage phasors and 30 branch current phasors) are the inputs to the DNN. The outputs of the DNN are the 118 voltage magnitudes and angles of this system.
The training and testing of the DNN is carried out using Keras with TensorFlow as the backend library in Python [
Type | Name | Value |
---|---|---|
Hyperparameter | Number of hidden layers | 4 |
Number of neurons per hidden layer | 500 | |
Activation functions | ReLU (hidden layers), linear (output layer) | |
Loss function | MSE | |
Optimizer | Adam | |
Batch size | 128 | |
Learning rate | 0.0207 | |
Number of epochs | 2,000 | |
Early stopping | ||
Dropout | 30% | |
Dataset size | Training | 7500 |
Validation | 2500 | |
Testing | 4000 | |
Total | 14000 |

Fig. 3 Performance evaluation of DeNSE for IEEE 118-bus system as a function of distance from buses where PMUs are placed. (a) MAPE of voltage magnitude. (b) MAE of voltage angle.
The subplots shown in
Noise type | MAPE of voltage magnitude (%) | MAE of voltage angle (rad) |
---|---|---|
Gaussian | 0.1676 | 0.0042 |
GMM | 0.1667 | 0.0047 |
Laplacian | 0.1678 | 0.0049 |
The performance of the DeNSE is now compared with two other state estimators, namely a purely SCADA state estimator and a PMU-only linear state estimator. For fairness of comparison, 1% TVE Gaussian noise is added to all the PMU measurements. The SCADA measurements comprise all sending-end active power flows and voltage magnitudes [
Type | Number of PMU locations | Average MAPE (%) | Average MAE (rad) |
---|---|---|---|
Purely SCADA state estimator | 0.9816 | 0.0079 | |
PMU-only linear state estimator |
3 | 0.2709 | 0.0026 |
DeNSE | 11 | 0.1676 | 0.0042 |
Note: * means that PMUs are optimally placed to ensure complete system observability.
We have also compared the performance of DeNSE with the NN-based state estimator developed in [
Noise amplitude (standard deviation of noise) | RMSE of [ | RMSE of DeNSE with PMUs at 11 buses |
---|---|---|
0.000 | ||
0.001 | ||
0.010 | ||
0.030 | ||
0.050 |
Note: * means that PMUs are not optimally placed (five buses left unobserved).
Next, we investigate the ability of transfer learning in updating the DNN of DeNSE after a topology change takes place. A set of likely topologies is identified for the IEEE 118-bus system by removing one line at a time between any two buses of the system such that an island is not formed. 177 such topologies have been identified. The training data for these likely topologies are saved in the database. When a topology change is detected by the topology processor in real-time, transfer learning via fine-tuning is activated as described in
Let the base topology be denoted by T1. By opening different lines, three new topologies are created from T1. T2 is created by opening the line between buses 75 and 77, neither of which has a PMU. T3 is obtained when the line between buses 38 and 37 is removed; note that bus 38 has a PMU on it. T4 is realized by opening the line between buses 26 and 30, both of which have a PMU on them. The changes in topology and their influences on TSSE with and without transfer learning are studied, as shown in Figs.

Fig. 4 Efficacy of transfer learning in terms of average MAPE of voltage magnitudes.

Fig. 5 Efficacy of transfer learning in terms of average MAE of voltage angles.
When transfer learning is used to update the DNN, fine-tuning only takes 30 s of re-training time to give similar results for the new topologies, as obtained for the base topology (the heights of the green and blue bars are similar). Note that if we had trained the DNN from scratch for every new topology, it would have taken three hours for every topology change, making the DeNSE inconsistent with the current state of the system for a much longer time period. The reason why fine-tuning is so fast is that it only needs 2000 samples and 90 epochs compared with 10000 samples and 2000 epochs that are needed to train the DNN from scratch (see Table III). Conversely, if the DNN trained for T1 is used throughout, the performance of DeNSE degrades significantly (shown by the heights of blue and orange bars in Figs.
It can also be observed from Figs.
To investigate the performance of the proposed NOC-based BDDC algorithm, we simulate two different scenarios. In the first scenario, we increase the amount of testing samples that are bad, while fixing the severity of the bad data. To do this, the probability of bad data is randomly varied from to in steps of 10%, while the severity is kept at , where the standard deviation of good quality data is computed from the training dataset. The value of is set to be 0.05 to ensure that the false alarm (false positive) probability does not exceed 5%. The results are shown in

Fig. 6 Bad data replacement with increasing amount of bad data. (a) Average MAPE of voltage magnitude. (b) Average MAE of voltage angle.
In the second scenario, we increase the severity of the bad data while fixing the amount of testing samples that are bad. To do this, the severity is increased from to , while setting . The results are shown in

Fig. 7 Bad data replacement with increasing severity of bad data. (a) Average MAPE of voltage magnitude. (b) Average MAE of voltage angle.
Considering the high speed at which DeNSE is expected to operate during its online implementation (30 samples per second), it must be ensured that the Wald test and data preprocessing are performed within that time frame. The most time-consuming portion in this regard is the proposed bad data correction module, which must compare the current testing sample with all the samples in the training database to find the optimal replacement(s). It is observed that with 10000 training samples and 41 phasor measurements as inputs, the bad data replacement for the IEEE 118-bus system could be carried out in ()ms. As this is much less than the speed at which a PMU produces an output (≈33 ms), the proposed algorithm meets the high speed and high accuracy expectations of purely PMU-based state estimation.
In Section IV-E, the superiority of the BDDC based on the Wald test and NOC is demonstrated. In this sub-section, the need and impact of the extreme scenario filter are discussed. 1000 extreme scenarios are created for the IEEE 118-bus system by significantly increasing the loading of buses 8 and 10. Due to the physics of the power system, PMUs located at buses 8 and 10 as well as the ones located in the vicinity of the two buses are impacted in these scenarios. Consequently, one or more measurements coming from the impacted PMUs (i.e., input features of the DeNSE) are flagged as bad data by the Wald test. At the same time, bad data are also added to the PMUs placed at buses 68 and 81, which are far away from the stressed region of the system. The extreme scenario filter identifies the set of features for which the BDDC should be suppressed, using the logic described in Section III-B. Three different outcomes are analyzed, as shown in
Method | Average MAPE of voltage magnitudes (%) | Average MAE of voltage angles (rad) | ||
---|---|---|---|---|
Mean | Standard deviation | Mean | Standard deviation | |
DeNSE without BDDC | 0.3337 | 0.0254 | 0.0267 | 0.0023 |
DeNSE with BDDC but without extreme scenario filter | 0.1853 | 0.0035 | 0.0059 | 0.0002 |
DeNSE with BDDC and extreme scenario filter | 0.1812 | 0.0037 | 0.0053 | 0.0002 |
The first row of
In the proposed DeNSE, it is necessary to solve a variety of power flows under different operating conditions to create a comprehensive database for DNN training. The determination of the requisite number of samples is contingent upon the accuracy of the DNN relative to the number of samples utilized. In general, augmenting the training samples can further diminish DNN error until a point of performance saturation is reached. This is realized for the IEEE 118-bus system by progressively training the DNN with an increasing number of samples. It is observed that beyond the threshold of 10000 samples, no discernible improvement occurs, as shown in

Fig. 8 Impact of database sizes on DNN performance.
To demonstrate the applicability of the DeNSE to large transmission systems, we use the publicly available 2000-bus synthetic Texas system [
The error estimates obtained with PMUs placed at 120 buses and under different noise types are shown in
Method (noise type) | Average MAPE of voltage magnitudes (%) | Average MAE of voltage angles (rad) | Number of buses with PMUs |
---|---|---|---|
LSE (Gaussian) | 0.2809 | 0.0026 |
51 |
DeNSE (Gaussian) | 0.2800 | 0.0024 | 120 |
DeNSE (GMM) | 0.2714 | 0.0024 | 120 |
DeNSE (Laplacian) | 0.2890 | 0.0027 | 120 |
Note: * means that PMUs are optimally placed to ensure complete system observability.

Fig. 9 Performance evaluation of DeNSE for 2000-bus synthetic Texas system as a function of distance from buses where PMUs are placed. (a) Average MAPE of voltage magnitudes. (b) Average MAE of voltage angles.
Type | Name | Value |
---|---|---|
Hyperparameter | Number of hidden layers | 4 |
Number of neurons per hidden layer | 500 | |
Activation functions | ReLU (hidden layers), linear (output layer) | |
Loss function | MSE | |
Optimizer | ADAM | |
Batch size | 256 | |
Learning rate | 0.001 | |
Number of epochs | 3000 | |
Early stopping | Patience=10 | |
Dropout | 30% | |
Dataset size | Training | 7500 |
Validation | 2500 | |
Testing | 4000 | |
Total | 14000 |
Remark 1: note that for the test systems analyzed in this paper, the DeNSE performs state estimation using (1) in real-time based on a limited set of PMU measurements and without requiring knowledge of the system model and parameters. However, in the offline training phase, essential information is derived from power flow computations, which requires knowledge of the system model and parameters. In other words, the proposed DeNSE remains model-agnostic during online operation but depends on system model and parameters during offline training. One way to avoid this dependency for an actual power system implementation is by directly utilizing historical SCADA state estimator results for creating the requisite training database of the DNN.
Remark 2: when making additions to any existing system, a variety of factors must be considered. Therefore, it is not surprising that the final locations where PMUs would be placed are often decided based on negotiations with the grid operators, rather than through a purely mathematical optimization procedure (such as solving an OPP problem) [
In this paper, a Bayesian framework for high-speed time-synchronized TSSE is proposed, which does not require complete observability of the system by PMUs for its successful execution. The proposed state estimator, i.e., the DeNSE, overcame unobservability by indirectly combining inferences drawn from slow-timescale SCADA data with fast-timescale PMU measurements. The robustness of the DeNSE is demonstrated by its ability to successfully tackle practical challenges such as topology changes, non-Gaussian measurement noise, and different types of bad data under diverse operating conditions.
The IEEE 118-bus system and the 2000-bus synthetic Texas system are used as the test systems for the analysis conducted here. In comparison to conventional methods, the proposed DeNSE is able to bring the estimation errors of all the buses to reasonable levels, which requires less than half the number of PMUs required for full observability for the IEEE 118-bus system and less than one-quarter for the 2000-bus Synthetic Texas system. The future scope of this study will involve developing strategies to further improve accuracy of the DeNSE by determining locations for adding new PMUs, extending the proposed framework to handle events such as faults and load/generation losses, and providing provable performance guarantees [
Appendix
The DeNSE is an MMSE estimator, in which the DNN approximates the conditional expectation . For the state , the conditional expectation can be written in terms of the probability distributions as shown below:
(A1) |
where and are the conditional probability and the joint probability between and , respectively; and is the probability distribution of . Now, it can be inferred from (1) and (A1) that can be obtained for any value of (where ), as long as is known. Moreover, increasing can improve the estimation quality only if the new measurements are not correlated with the existing measurements, or are constant.
To better understand these inferences in the context of TSSE, consider the 3-bus system shown in Fig. A1. The reference bus (bus 1) has an angle of , but its magnitude is an unknown variable. Bus 2 has both load and generation, while bus 3 has only load. The system has three sensors (depicted by blue boxes) that are measuring the magnitude of the current flowing in lines 1-2 and 2-1, and the magnitude of the current injection at bus 3.

Fig. A1 3-bus system.
Let the goal be to estimate the voltage magnitude of bus 3, i.e., . The system is unobservable because cannot be estimated from the given measurements in the conventional least squares sense. Note that this example simply illustrates how the Bayesian framework of DeNSE can be used to estimate states that cannot be estimated using conventional methods due to limited observability. In an actual system, the DeNSE will estimate all bus voltage magnitudes and angles without differentiating among unobserved buses, directly observed buses, and indirectly observed buses as it only relies on the joint PDF between the PMU measurements and the states.
To generate and for this system, power flows are solved. The simulation parameters used for solving the power flows of the 3-bus system are provided in Table AI.
Parameter | Value (p.u.) | Parameter | Value (p.u.) |
---|---|---|---|
Series Imp_1-2 | 2+ | ||
Series Imp_2-3 | 0.5+ | ||
Series Imp_3-1 | 0.1+ | ||
Shunt Imp_1 | -j100 | 2+ | |
Shunt Imp_2 | Inf | 0.5+ | |
Shunt Imp_3 | -j40 | 1+ |
Due to the reasons mentioned in Section II-A, it is usually not possible to analytically compute for all and , which is why its approximation by a DNN is needed in the first place. However, for this 3-bus system, it is observed that the probability distributions of the relevant random variables , , , and could be well-approximated by multivariate normal distributions. In such a scenario, the conditional probability of given is written as [
(A2) |
where and are obtained from power flow solutions, with comprising all variables in and , and comprising variables in only; and are the mean and covariance, respectively, and is determinant of the covariance. Now, using (A1) and (A2), we compare with the actual value of for five MMSE estimator cases: ① Case 1: ; ② Case 2: ; ③ Case 3: ; ④ Case 4: ; ⑤ Case 5: . Note that , , , and are dependent variables as they correspond to converged power flow solutions, while is a constant. The estimation results are shown in Table AII. In Case 1, is a constant, and so , which is the mean value of across all power flows. As this case is not able to track the variations in OCs across different power flows, its estimate is the worst. Cases 2 and 3 give similar results as they separately track the variations in and to estimate . Despite having two measurements, the results of Case 4 are worse than those in Cases 2 and 3 because and are highly correlated. The expected values of Case 5 are closest to the ground-truth values as this estimator is able to use both and to estimate . This analysis confirms that the knowledge of and a non-large value of are the basis for the DeNSE to overcome unobservability. It is also worth mentioning that the estimation quality of the DeNSE improves if is increased, because with more training samples, the DNN will be able to better approximate the probability distributions, and in turn, .
Case | ||
---|---|---|
1 | 0.00100 | |
2 | 0.00014 | |
3 | 0.00021 | |
4 | 0.00094 | |
5 | 0.00005 |
All the Python codes required for implementing the DeNSE method developed in this paper can be accessed through the following GitHub repository: https://github.com/Anamitra-Pal-Lab/DeNSE. The Read Me file provided in this repository contains all the information that is needed to run the files and obtain the results.
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