Power flow (PF) analysis aims at obtaining complete voltage angle and magnitude information for each bus in a power system, given specified loads, generator real power and voltage conditions. The obtained information can be used by system operators for a variety of tasks such as energy loss analysis, voltage control, reactive power planning, security analysis, and best operation determination. Conventionally, PF analysis is achieved by first describing a power system as its state-space representation using a set of differential algebraic equations (DAEs) derived from prior knowledge such as the system topology and network parameters. These DAEs, which are nonlinear and have no-closed form solutions, are then solved using iterative numerical methods such as Newton-Raphson method to obtain voltage angles and magnitudes.
However, as modern power systems get more dynamic and uncertain, which are too complex to describe timely and precisely with DAEs, conventional PF analysis approaches become ineffective. For example, line profiles, corresponding to the parameters of DAEs, can be inaccurate due to aging and weather conditions, or even unavailable in some secondary distribution grids. Distributed energy resources (DERs), corresponding to control rules in power system modeling, can be difficult to describe with DAEs due to missing information from vendors and individual owners[27, 13]. Such deficiencies in power system modeling and analysis have posed not only safety threats to power systems such as blackouts[17, 10], but also challenges for adoption of DERs.
To address these concerns, researchers leverage the availability of massive measurements from phasor measurement units and smart meters, and have proposed various data-driven approaches to solve the PF problem. We classify them into two categories:model rebuilding[31, 30, 7, 6] and solver learning[29, 19, 15, 23, 4]. Model rebuilding focus on calibrating or rebuilding PF models by rediscovering model parameters (e.g., line parameters, bus admittance matrix, PF Jacobian matrix) from historical operation data. The rebuilt models are then solved using conventional numerical approaches. Although model rebuilding is intuitive and can potentially reflect changes of PF models, it suffers from drawbacks including high computational overhead, error accumulation and non-convergence[24, 4]. In contrast, solver learning focuses on modeling mapping rules between inputs and outputs of a power system based on historical measurement data[29, 19, 15, 23, 4], thus sidestepping drawbacks in model rebuilding. Inspired by the structure of DC and AC power flow equations, several solver learning19]
and support vector regression (SVR) with polynomial kernels, to uncover the forward and inverse mapping rules between power injections and bus voltages. However, LR overly simplifies PF models and SVR suffers from high computational overhead and scalability. Other solver learning approaches[15, 23, 4]
view the outputs of the PF problem as implicit functions of the operating conditions, and employ neural networks such as multi-layer perceptron (MLP) and radial basis function neural network (RBFNN), to map the power-flow inputs to the power-flow outputs. Since pure RBFNN is subject to overfitting, subsequent efforts[23, 4] try to incorporate physical knowledge to avoid overfitting. However, they make impractical assumptions such as the availability of accurate system parameters and PF models.
In this paper, we propose physics-guided deep neural networks to solve the PF problem under the assumption that system parameters and PF models can be inaccurate or unavailable. The key idea is to utilize the physical knowledge of power systems to regularize neural networks and alleviate the over-parameterization issue. Unlike existing physics-related neural network approaches[23, 4] which make impractical assumptions, our method utilizes generic physical knowledge of power systems. We design a neural network architecture with the capability of multi-task learning and set the training goal to minimize both voltage prediction errors and power reconstruction errors (i.e., power mismatches). Our architecture is inspired by supervised auto-encoders[11, 16] and models the PF solver by minimizing voltage prediction errors, with an auxiliary task to rebuild PF models from historical measurement data by minimizing power reconstruction errors. The auxiliary task encodes the basic physical knowledge of reducing power mismatches and acts as a regularizer to alleviate overfitting and improve generalization performance. In addition, we propose variants of the regularizer, to facilitate the integration of other types of physical knowledge such as the structural property of AC power flow equations and the topology of power systems.
We evaluate our physics-guided neural networks on the standard IEEE 57 and 118 bus systems. The simulation results show that our PF solvers achieve an order of magnitude higher accuracy, compared to previous data-driven methods. Meanwhile, our solvers have better performance on test samples that deviate from training data. All these results show that by incorporating generic physical knowledge of power systems, our neural networks are regularized and guided to focus on physically consistent solutions, thus have better performance and generalizability. Furthermore, we demonstrate that the weight matrices of our regularizer variants, which integrates the structural property of PF equations and the topology of power systems, share similar patterns with bus admittance matrices. This shows not only the effectiveness of physical knowledge integration but also the potential of using our auxiliary task for parameter estimation and PF model rebuilding.
The contributions of this paper can be summarized as follows:
We propose the first physics-guided neural network for PF analysis under the assumption that system parameters and PF models can be inaccurate or unavailable.
We quantitatively evaluate and compare all data-driven PF solvers and show that our methods consistently achieve better accuracy and generalizability.
We demonstrate that our two regularizer variants embody physical knowledge in their weight matrices and elaborate corresponding use cases.
Ii Power Flow Analysis Problem Formulation
Ii-a Model-based Methods for Power Flow Analysis
The basic active power flow (ACPF) problem can be formulated as a set of nonlinear trigonometric equations in polar coordinate, which represent Kirchhoff’s laws :
where and are the real and reactive power injections at bus , and are the real and imaginary parts of the th element in the bus admittance matrix respectively, is the voltage magnitude, is the voltage phase angle difference between bus and bus , and is the total bus number in a power system.
In the basic formulation of the PF problem, three types of buses are defined. PQ buses are load buses, the real and reactive power injections of which are specified, but the voltage magnitudes and angles are unknown. PV buses are generation buses, the real power injections and voltage magnitudes of which are specified, but the reactive power injections and voltage angles are unknown. V bus is called reference bus, where the voltage magnitude and angle are specified, but the real and reactive power injections are unknown. According to the three bus types, a set of PF equations with the same number of equations and unknowns are obtained:
where and are ACPF equations defined by Equation 1 and 2, and are known real and reactive power injections into PQ and PV buses. To fully specify the voltages of all buses, numerical methods are conventionally used to iteratively solve Equation 3.
Another formulation of the PF problem is to represent ACPF equations in rectangular coordinates. Let and be the real and imaginary components of a voltage phasor, ACPF equations become,
The power injection and are second-order polynomial functions of the voltage vector .
Ii-B From Model-based Methods to Data-driven Methods
Numerical methods typically require accurate PF equations. Therefore, numerical PF solvers are subject to the inaccuracy and unavailability of PF models. The issues of modeling the PF problem usually come from the missing knowledge of system parameters and control rules. Especially, in distribution grids, it is hard for substation engineers to construct admittance matrices due to the inaccurate or missing information of line parameters . Modeling DERs is also challenging since DERs are usually independently owned, and manufacturers do not readily provide model data and control policies .
Intuitively, numerical PF solvers are expected to be feasible again by employing data-driven approaches [31, 30, 7, 6] to rediscover system parameters to calibrate or rebuild PF models. However, errors are accumulated in the parameter rediscovery and numerical approximation processes
. Numerical solutions are sensitive to differences in system parameters. A small variance in system parameters can induce big disturbance on the system operation setting, potentially causing errors to “blow-up”. In addition to error accumulation, the convergence issue of numerical methods is beyond the capabilities of rebuilding PF models. Especially, when DERs are incorporated into power grids, power electronic interface provides some degrees of freedom in the models of DER units[24, 3], thus new power-flow solver is required to solve a set of non-linear equations with an unequal number of equations and unknowns.
Ii-C Data-Driven Power Flow Solvers
Data-driven power flow solvers produce PF solutions by inferring the mapping rules between the observed system inputs and outputs based on historical system operation data. To obtain voltage values of all buses, data-driven methods use machine learning algorithms to solve Equation 3 by learning the mapping rules from specified variables to bus voltages:
where, according to the bus type definition, the specified variables or power-flow inputs are power injections , on load buses, real power injections and voltage magnitudes , on generation buses, and voltages , on the reference bus.
The challenges of data-driven PF solvers lay in two main aspects: to avoid overfitting and to improve generalizability. Since the PF solutions, i.e., Equation 6, do not have closed-form expressions, there is no guide to select a proper machine learning model family. Besides, as measurement data may not be various to reflect all aspects of power systems, data-driven methods may have generalization issues. Incorporating physics knowledge of power systems into data-driven models is a good way to reduce overfitting and improve generalizability . Some neural-network PF solvers have explored to incorporate physical knowledge. The MLP method proposed in  incorporates physics by additionally inputting system parameters (e.g., bus admittance matrix). The work in  assumes the availability of accurate PF models, and considers physics by repeatedly retraining RBFNN until the power mismatch of the RBFNN outputs satisfies a predefined tolerance. Unfortunately, both of these methods cannot consider physics if system parameters and the PF models are inaccurate or unavailable. The open question is how to incorporate more generic physics into data-driven PF solvers.
Iii Physics-Guided Deep Neural Network for Power Flow Analysis
In this section, we describe how we add regularization terms to a “black-box” multilayer perceptron neural network (MLPNN) to solve the PF problem under the constraints of generic physical knowledge of power systems.
Iii-a Data-Driven Power Flow Solver with Regularizer
In spite of the universal function approximation theorem of MLPNN, multiple relations in the hypothesis space of MLPNN can map power-flow inputs to bus voltages. To guide the hypothesis search procedure, we introduce regularizers through multi-task learning. The architecture of our physics-guided data-driven PF solver with regularization is illustrated in Fig. 1
. It is inspired by supervised AutoEncoder and consists of two parts respectively focusing on the inverse and forward problems: power flow solving and power flow modeling.
The “encoder” part is a black-box MLPNN focusing on solving the PF problem by learning the mapping from power-flow inputs to bus voltages . Assume the MLPNN “encoder” has 3 layers, its outputs are
is a point-wise activation function at the-th layer, and are respectively the weight matrix and bias of the -th layer.
The “decoder” part is a neural network carrying out the auxiliary task, which is rebuilding PF models mapping bus voltages to power injections. We will discuss the structure of the “decoder” neural network later. Here let’s denote the outputs of the “decoder” as
The loss function is a weighted sum of voltage prediction losses produced by the “encoder” and power injection reconstruction losses produced by the “decoder”. Let, , the loss for a single training sample is
where typically is the squared error function.
A variety of works [21, 20, 18, 16] have theoretically and empirically demonstrated that auxiliary task or multi-task learning serves as a regularizer to constrain the solution and improve generalization performance. When training our neural network, the errors at the voltage prediction layer is
where is similar to the PF Jacobian matrix . The first term directs learning the MLPNN “encoder” towards parameters that are effective for minimizing bus voltage prediction errors. This is desired and what we target to. However, solely training an MLPNN according to the supervised losses is an under-constrained problem, and will find parameters that over-fit the data and do not generalize well. With the guide of the second term , among similarly effective parameters that produce good voltage estimation, the learning algorithm will prefer the one that generates small power mismatches. In this way, the combination of the two types of losses is supposed to generate a more robust and physically consistent data-driven PF solver.
Iii-B Encode Physical Knowledge in Regularizer
In this section, we describe how we design different “decoders” to rebuild PF models according to different prior physical knowledge of power systems.
To calculate power reconstruction errors without PF models, the intuitive method is to employ MLPNNs to learn any functions to approximate PF equations. Since active and reactive power injections are different nonlinear functions of bus voltages, we decouple their mapping rule learning process and utilize two MLPNNs with a shared input layer. Given the same inputs, i.e. bus voltages estimation of the “encoder”, the output layers of these two MLPNNs predict active and reactive power injections respectively:
where we assume both MLPNNs have 3 layers. We call the neural network with the MLP ”encoder” and the MLP ”decoder” as ”MLP+MLP” PF solver.
Different from the MLPNN regularizer, we design a “decoder” by utilizing the structural properties of ACPF equations to approximate PF models and incorporate power reconstruction errors. As shown in Equation 4 and 5, a power injection basically is a linear combination of the 2-th Kronecker power of bus voltages in the rectangular coordinate. This bilinear function formulation is indicated by Kirchhoff’s laws, except that the function parameters, i.e. bus admittance matrix and , may be unavailable or inaccurate due to system dynamics, uncertainties, and erroneous measurements. To utilize this prior knowledge of Kirchhoff’s laws which indicate the structure of PF models, we design a Bilinear Neural Network (BNN), depicted in Fig. 2.
To learn the bilinear mapping rules from bus voltages to power injections, BNN first builds bilinear maps based on the outer products of the given voltage vectors and . Then, these bilinear maps are weighted summed by weight matrices and , which are the weight matrices of BNN and are related to bus admittance matrices.
By multiplying an all-one vector to the bilinear maps and , we flatten the bilinear maps and further linearly combine the bilinear terms to predict power injections:
where bias terms are added to absorb noise in the input voltage vectors and enhance the regression capability. We call the neural network with the MLP ”encoder” and the BNN ”decoder” as ”MLP+BNN” PF solver.
Built upon the BNN architecture, we can further encode topology information into the “decoder” module if system topology is available. When bus and bus are not directly connected, their corresponding system parameters , , , and are zero. Based on this fact, the weights of monomials , , , and in BNN can be zeroed out. Therefore, in Equation 13 and Equation 14, some parameters in the weight matrices and are redundant.
To reduce the redundancy in the parameter space and improve generalization, when the topology of a power grid is available, we prune the parameters of BNN by employing the adjacent matrix as a hard attention mask. Let be the adjacent matrix of a power system. The Hadamard product of bus admittance matrix and adjacent matrix is bus admittance matrix itself, i.e. , . As the role of weight matrices and is equivalent to the role of bus admittance matrices, we expect that , . Therefore, by augmenting topology knowledge to BNN, the power injections are calculated by
In the process of backpropagation, compared to the corresponding BNN, the gradients of the-th element of the weight matrices and are
When the bus and bus are not neighbors, the corresponding is zero and the gradients of and are zeros. By initializing and as zeros, the weight matrices and will keep the same sparsity pattern as admittance matrices. In this way, we encode the topology information into the BNN by entry-wisely multiplying the adjacent matrix to the weight matrices. We call the BNN of topology information as topology-pruned BNN (TPBNN), and the neural network with the MLP ”encoder” and the TPBNN ”decoder” as ”MLP+TPBNN” PF solver.
Iv Experimental Results
Iv-a Experiment Setup
We test our physics-guided NNs for PF analysis on the IEEE 57 and 118 bus systems. To simulate the power system operation measurement data, we use the real-world load data from GEFCom 2012  as load consumption, where load series were subsampled and scaled to match and balance the system generation capacity. Then, the MATPOWER 
is employed to obtain the associated voltage magnitudes and phase angles at each bus. Unless otherwise stated, we use the first 60% of samples as training data, the next 10% of samples as validation data, and the last 30% of samples as testing data. Besides, Gaussian noise with a 1% relative standard deviation is added to the training data because measurement noise generally exists in practical settings. While we evaluate our methods, we assume that the accurate line parameters are unavailable. We fit our physics-guided NNs on the training dataset to rebuild and solve PF models with the awareness of physical laws. To alleviate randomness in the learning process of neural networks, we train and test all NNs for 10 times and average the results. The source code is available online111Source code will be made available once published..
Iv-B Power Flow Solution Results
|Root Mean Squared Error (p.u.)|
|Case||V||LR ||SVR ||MLP ||MLP+MLP||MLP+BNN||MLP+TPBNN|
In this section, we evaluate the effectiveness of our three kinds of physics-guided neural network PF solvers. To compare our methods with existing data-driven power-flow solvers, we modify the LR , SVR , and MLPNN  methods, such that they have the same inputs (i.e. power-flow inputs) and outputs (i.e. bus voltages) with our methods. We compare with the MLPNN method and skip the RBFNN method  because both MLPNN and RBFNN methods are based on the universal function approximation theorem and have similar performance. Since our goal is to investigate the impact of the physical constraints, the MLPNN method  and our MLP “encoder” have the same architecture.
Iv-B1 Basic Results
Table I shows the average results of different data-driven PF solvers in terms of RMSE. The average voltage errors of the LR, SVR, and MLPNN methods on the two test cases are , and . We can observe that among these unconstrained data-driven PF solvers, nonlinear PF solvers (i.e. the SVR and MLPNN methods) outperform the LR-based one, and the MLPNN method has better scalability than the SVR method. Compare with the counterpart MLPNN method, our three physics-guided neural networks have more superior performance. The average error of voltage solutions produced by our physics-guided neural networks is and about 4 times smaller than the MLPNN method. This indicates that the reconstruction errors of “decoders” included as an auxiliary task (i.e. PF modeling) clearly regularize the MLPNN “encoder” and help it improve performance. Besides, we can see the more prior physical laws the auxiliary PF modeling task uses, the more improvement the primary PF solving task achieves. Fig. 3 is the distributions of MAPE of bus voltages produced by the data-driven PF solvers on all the testing samples. 90% of testing errors of the LR, SVR, MLP, and our physics-guided NN methods are less than 32.39%, 20.03%, 19.16%, and 5.23%. It is clear that the unconstrained data-driven PF solvers have very bad solutions at some testing points, and our physics-guided data-driven PF solvers have a more stable performance on all the testing points.
Iv-B2 Interpolation and Extrapolation Capability
Power systems are dynamic, the load demands change significantly over time, and historical measurements may not observe the whole system dynamics. Therefore, it is important to evaluate the generalization performance of data-driven PF solvers when given new power-flow inputs deviating a lot from training measurement data. In this case, the measurement data values are normalized into , then these values are divided equally into 20 portions. We split the whole measurement data such that the input variables’ values of training samples and testing samples are within totally different ranges. Then, we evaluate the interpolation and extrapolation capabilities of the above-mentioned data-driven PF solvers.
The RMSEs of interpolation and extrapolation results in each portion are illustrated in Fig. 4
. The points in the shade areas are validation errors, and the points outside the shade areas are testing errors. As for the interpolation, we can see that it is difficult for the LR method to find a suitable hyperplane to fit the training data as well as to generate well on the testing data. Although other methods fit the training data better than LR, the average differences between the testing errors and validation errors of the SVR, MLPNN and our physics-guided NN methods are 0.0185, 0.0145, and 0.00712 respectively. The interpolation capabilities of our physics-guided PF solvers are about 2 times better than the SVR and MLPNN methods. As for the extrapolation, the LR method over-fits the training data and has the biggest testing errors. The average differences between the testing errors and validation errors of the SVR, MLPNN and our three physics-guided NN methods are 0.0125, 0.0122, and 0.00685 respectively. The extrapolation capabilities of our physics-guided PF solvers are about 2 times better than the SVR and MLPNN methods. All these results show that with the constraints of the physical regularization, the generalization capabilities of our physics-guided PF solvers are at least 2 times better than those of other unconstrained PF solvers.
Iv-B3 Robustness to Data Outliers
It is common that measurement data is contaminated by noisy signals. Therefore, we test the robustness of the data-driven PF solvers to noise and outliers in training data on the IEEE 118 bus system. In this test, different amounts of training data are corrupted with Gaussian noise, the standard deviation of which is (Interquartile Range) of the training samples. The percentage of outliers resulting from this data corruption ranges from 0 to 10%.
The performance of estimation accuracy under different levels of outliers is illustrated in Fig. 5. The MAE (Mean Absolute Error) of the linear regression method increases fast when the percentage of outliers in the training data increases. This shows that linear regression is sensitive to the presence of outliers. On the other hand, the performances of the SVR and neural network methods are relatively stable when the level of outliers increases. Especially, our physics-guided neural networks continuously perform best and are robust enough even for 10% outliers in training data.
Iv-C Physical Consistency Evaluation
While in the process of learning mapping rules to specify bus voltages, our physics-guided NNs are also learning to approximate system parameters and rebuild PF models. In this section, we evaluate the capabilities of our “decoder” parts to integrate physical knowledge and rebuild PF models.
First, the weight matrices of our BNN and TPBNN “decoders” are related to the bus admittance matrices and . Fig. 6 visualizes the bus admittance matrix and the corresponding weight matrices of BNN and TPBNN under the IEEE 118 bus system. We observe similar results of the admittance matrix approximation, but we do not show them due to page limitation. Both BNN and TPBNN learn the sparsity of bus admittance matrices. Although BNN learns the diagonal non-zero parameters not as well as TPBNN, it roughly captures the power system topology through measurement data. This means network topology is latently encoded in system operation measurement data [30, 28], and the 2nd-polynomial structure can help infer the topology pattern through data. With the prior knowledge of topology, TPBNN provides a very good approximation of bus admittance matrices.
Regarding the performance of PF modeling, we compare our BNN, TPBNN, and MLPNN “decoders” with the LR  and SVR  methods which are also used to learn the mapping rules from bus voltages to power injections. Given voltage measurements as inputs, the power injection modeling results of all these data-driven PF models are measured in terms of RMSE, shown in Table II. The average errors of the LR, SVR, and our MLP, BNN, and TPBNN “decoder” methods on the two test systems are , , , , and . The LR method has the worst performance, which indicates that linear regression has limited capabilities to represent nonlinear and complicated systems. The SVR method and our MLPNN “decoder” have comparable accuracy performance, but MLPNN scales better than SVR when system complexity increases. Explicitly utilizing the good scalability of neural networks and the bilinear structural knowledge indicated by Kirchhoff’s laws, our well-designed BNN is better at capturing nonlinear dynamics of power systems. Since ACPF equations are sparse polynomials, when the system topology is available, topology augmented BNN (i.e. TPBNN) prunes parameters, pays attention to a subset of its features and parameters, and consistently achieves the best accuracy performance than other methods.
The proposed physics-guided neural networks enable the PF analysis in situations where the exact system parameters and control logic are difficult to obtain. While our three kinds of physics-guided data-driven PF solvers achieve comparable calculation accuracy, they work in different situations according to the different prior physical knowledge of power systems. The “MLP+MLP” model is a general method and especially suitable for power grids where some control policies are unknown due to the various active controllers introduced by the increasing penetration of DERs. The “MLP+BNN” and “MLP+TPBNN” models are suitable for power grids where control policies indicate that a basic PF model is quadratic. Additionally, by utilizing the quadratic structural information, the BNN and TPBNN modules provide a new way to approximate system parameters and rebuild PF equations. However, TPBNN is a restricted method requiring accurate topology information. In addition to PF analysis, since the calculation speed of a trained neural network is very fast, our physics-guided neural networks can apply to other power system problems requiring a large amount of PF calculations or real-time operations, such as probabilistic load flow analysis , security constraints , etc.
Although our physics-guided neural network methods solve the PF problem when PF models are inaccurate or unavailable, our methods cannot settle this matter once and for all. Power grids are varying and evolving over time. The well-trained neural network PF solvers need to adapt to future system changes. One intuitive method is to periodically train the neural networks with new measurement data in an off-line manner and update the power-flow solvers with the newest neural network models. Some more advanced methods are to learn neural networks on the fly , and to utilize time-series patterns of power systems , etc. Another future work is to incorporate other generic physical knowledge and constraints that are helpful for the PF analysis, such as the symmetric structure of admittance matrix, inequality constraints (e.g., reactive power limits), etc.
To account for the increasing dynamics and uncertainties of modern power systems, which make conventional PF analysis approaches ineffective, researchers have proposed various data-driven methods to rebuild and solve power flow models through historical measurements. However, existing works make invalid assumptions such as availability of power flow models and suffer from poor performance and generalizability. We, therefore, propose physics-guided neural networks for PF analysis which incorporate physical constraints with practical assumptions. Simulation results show that the proposed data-driven power flow solvers are regularized by corresponding physical laws and achieve higher accuracy and better physical consistency. Besides, the results show that algorithms (BNN and TPBNN), which incorporate the physical laws of the quadratic structure of ACPF equations and system topology, are promising alternatives to approximate system parameters and rebuild power flow equations.
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