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Fast Calculation of Probabilistic Power Flow: A Model-based Deep Learning Approach

Yan Yang, Zhifang Yang, Juan Yu, Baosen Zhang

arXiv:1906.06017v2eess.SPcs.LGcs.SYeess.SY

TL;DR

Sample-based probabilistic power flow is computationally burdensome because it repeatedly solves power flow for many injection samples. This paper trains a physics-informed DNN to approximate those calculations, reporting at least three orders of magnitude faster PPF calculation with similar accuracy to standard iterative solvers.

  • Problem

    Sample-based PPF requires power-flow calculations for many random injection samples, making its cumulative computational burden considerable.

    Source: §I · p. 1

  • Method

    A DNN approximates power-flow calculations, trained with a branch-flow-based composite loss, physically simplified gradients, and an improved initialization method.

    Source: §I · p. 2, §VI · p. 9

  • Results

    At least three orders of magnitude faster PPF calculation is reported while maintaining similar accuracy compared with standard iterative power flow solvers.

    Source: §I · p. 2

  • Takeaways & Limitations

    The proposed approach is presented as a way to accelerate PPF calculation while retaining similar accuracy to standard iterative solvers.

    Source: §I · p. 2

    A trained DNN may not apply when grid topology or renewable-resource distributions change; transfer learning is proposed as future work.

    Source: §VI · p. 9

Abstract

from arXiv · show

Probabilistic power flow (PPF) plays a critical role in power system analysis. However, the high computational burden makes it challenging for the practical implementation of PPF. This paper proposes a model-based deep learning approach to overcome the computational challenge. A deep neural network (DNN) is used to approximate the power flow calculation and is trained according to the physical power flow equations to improve its learning ability. The training process consists of several steps: 1) the branch flows are added into the objective function of the DNN as a penalty term, which improves the approximation accuracy of the DNN; 2) the gradients used in the back propagation process are simplified according to the physical characteristics of the transmission grid, which accelerates the training speed while maintaining effective guidance of the physical model; and 3) an improved initialization method for the DNN parameters is proposed to improve the convergence speed. The simulation results demonstrate the accuracy and efficiency of the proposed method in standard IEEE and utility benchmark systems.

I. INTRODUCTION

The paper targets the repeated power-flow solves that make sample-based PPF computationally burdensome, using a physics-informed DNN to approximate power flow and reporting substantially faster PPF calculation with similar accuracy.

I. INTRODUCTION

  • Analytical methods rely on assumptions that may not hold in practical systems, while sampling, parallel, and DC alternatives retain sample-count, infrastructure, or accuracy constraints.Source: §I · pp. 1–2
  • Sample-based PPF repeatedly solves power flow for many injection samples, creating a cumulative computational burden as renewable uncertainty grows.Source: §I · p. 1
  • Prior RBF-based power-flow approximations use shallow networks and lack guidance from the physical power-flow model.Source: §I · p. 2
  • The proposed DNN approximates power-flow calculations and uses a composite objective based on branch-flow equations to guide learning beyond output square loss.Source: §I · p. 2
  • Training gradients are simplified by removing specified voltage-magnitude and phase-angle dependencies, accelerating training while maintaining similar accuracy.Source: §I · p. 2
  • The method combines rectifier linear and linear activations with a new parameter initialization designed to improve DNN training speed.Source: §I · p. 2
  • At least three orders of magnitude faster PPF calculation is reported while maintaining similar accuracy compared with standard iterative power flow solvers.Source: §I · p. 2

II. BASIC STRUCTURE OF THE PROPOSED METHOD

  • For PPF, the DNN maps sampled active and reactive injections to complex bus voltages, after which statistical indexes are computed across samples.Source: §II · pp. 2–3

III. MODEL-BASED DEEP LEARNING TECHNIQUES

The method maps uncertain injections to bus voltages and embeds power-flow physics in DNN training. Branch-flow penalties supplement voltage fitting, while modified gradients guide parameter updates.

A. Feature vector selection

  • Inputs are renewable-generation and load-demand injections, and outputs are complex bus voltages from which branch flows can be computed.The paper uses bus-voltage magnitudes and phase angles as the output features.Source: A. Feature vector selection · p. 3

B. Loss Function Design Based on the Power Flow Equations

  • Branch-flow equations add active- and reactive-flow penalties to voltage-output fitting, supplying side information because flows can be computed from predicted voltages.The flow terms are normalized so their components are comparable and weighted equally.Source: B. Loss Function Design Based on the… · p. 3
  • The modified loss changes the output-loss derivative, increasing weight updates when voltage and flow losses decrease together and potentially reducing voltage overfitting when their gradients conflict.Numerical studies are said to demonstrate the method's effectiveness in promoting training convergence.Source: B. Loss Function Design Based on the… · p. 4

C. Model-based Simplification of the Learning Process

The training simplification removes selected physical-model gradients to reduce computation while retaining guidance judged most useful for learning phase angles.

C. Model-based Simplification of the Learning Process

  • Voltage-magnitude guidance is removed because magnitudes usually vary within ±5% and are easier to learn, while its guidance costs approximately twice as much computation as angle guidance.The paper also argues that model guidance has a smaller impact on magnitude than on phase angle.Source: C. Model-based Simplification of the… · p. 4
  • The training process drops voltage-magnitude guidance and reactive-branch-power guidance for phase angles, simplifying physical-model gradient calculations.The paper describes these omissions as retaining the more influential guidance while reducing computational complexity.Source: C. Model-based Simplification of the… · pp. 4–5
  • Reactive branch-power guidance for phase angles is omitted because its estimated training impact is much lower than that of active power flow.The stated reasoning includes easier voltage-magnitude learning and generally smaller reactive-power deviations than active-flow deviations.Source: C. Model-based Simplification of the… · p. 5

A. Designing Activation Function

The DNN uses ReLU in hidden layers and a linear output activation, with z-score normalization for input and output samples.

A. Designing Activation Function

  • ReLU is selected for the hidden activation, while the final layer is linear because ReLU cannot output negative values.The linear output provides a wider range for the network's predictions.Source: A. Designing Activation Function · p. 5
  • Z-score normalization uses the sample mean and standard deviation, handles outliers, and preserves input-distribution shape better than min-max preprocessing.The paper applies this normalization to improve training efficiency and mitigate numerical and outlier effects.Source: A. Designing Activation Function · p. 5

B. Initialization Method

The initialization method is designed for a DNN with ReLU hidden layers and a linear output, balancing signal behavior during forward and backward propagation.

B. Initialization Method

  • The method derives a new initialization for the paper's ReLU and linear activations, using variance analysis of layer activations and arguments.The derivation assumes independent elements and zero-mean symmetric weights in the stated analysis.Source: B. Initialization Method · p. 5
  • The initialization design seeks to prevent input-signal magnitudes from shrinking or growing exponentially and also constrains gradient magnitudes across layers.The derivation balances forward- and backward-propagation requirements.Source: B. Initialization Method · pp. 5–6
  • The resulting weights use a zero-mean Gaussian initialization with zero biases, and experiments use equal numbers of hidden neurons in each layer.The paper states equal hidden-layer widths meet the initialization criteria it derives.Source: B. Initialization Method · p. 6

V. SIMULATION TESTS

The effectiveness of the proposed methods is tested on modified IEEE 30-bus and 118-bus systems and a 661-bus utility system.

  • The proposed methods are evaluated on modified IEEE 30-bus and 118-bus systems and a 661-bus utility system.Source: §V · p. 6

A. Test Information and Methods for Comparison

The experiments compare DNN variants against Monte Carlo sampling with Newton-Raphson, using 10,000 validation and test samples. Fixed-epoch and threshold-stopped comparisons show M4 improves accuracy and convergence.

MEETING THE ACCURACY REQUIREMENTS

Table VI is titled as a comparison of M1, M5, and M6 when meeting the accuracy requirements.

MEETING THE ACCURACY REQUIREMENTS

  • Table VI presents performance comparisons among M1, M5, and M6, but the supplied passage does not specify the conditions under which those comparisons were made.Source: Table VI · p. 8

C. Validation of the Model-based simplification

The simplified methods maintain comparable performance while reducing training time, with the time savings most evident in the large system. The tested simplifications remove selected gradient guidance for voltage magnitudes and phase angles.

C. Validation of the Model-based simplification

  • In Case 661, M6 further reduces Nepoch versus M5, and M1's Nepoch can be reduced by 74.1%.In Case 30, M5's epoch count rises from 507 to 576 versus M4, which the passage describes as tolerable for the small-scale case.Source: C. Validation of the Model-based… · p. 8
  • 6.64 seconds per epoch are saved by M5 versus M4, and M6 saves a further 8.35 seconds by removing reactive-branch-power guidance for phase angles.The time advantage grows with the number of epochs; differences were less significant in Cases 30 and 118 than in the large practical system.Source: C. Validation of the Model-based… · p. 8, Table VII · p. 8
  • The simplified methods reduce computational pressure versus M4 while maintaining comparable performance.Ignoring voltage-magnitude guidance and reactive-branch-power guidance for phase angles yields similar or better performance than full information.Source: C. Validation of the Model-based… · p. 8

D. Performance of the Proposed Method for the PPF

The proposed DNN accelerates PPF calculation relative to the benchmark, with training performed once offline and excluded from the reported computation-time comparison. The conclusion also identifies changing grid topology or renewable-resource distributions as a boundary for applying a trained DNN.

D. Performance of the Proposed Method for the PPF

VI. CONCLUSION AND DISCUSSION

  • Three orders of magnitude faster power-flow calculation could allow operators to consider 1000 times more samples under the same time constraints.The simulations use IEEE and utility test benchmarks.Source: §VI · p. 9
  • Changes in grid topology or renewable-resource distributions may make the trained DNN inapplicable in those scenarios.Transfer learning is identified as a future direction for updating parameters and reducing the effort to rebuild the DNN.Source: §VI · p. 9
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