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Data-Driven Power Flow Linearization: A Regression Approach
Yuxiao Liu, Ning Zhang, Yi Wang, Jingwei Yang, Chongqing Kang
TL;DR
The paper addresses the challenge of simplifying nonlinear power-flow calculations when model-based linearizations may be limited by inaccurate or unavailable system information. It proposes forward and inverse data-driven regressions with PLS and BLR, and reports higher calculation accuracy than model-based approaches while preserving power-system-related parameter patterns.
Problem
Existing model-based PF linearizations rely on analytical approximations, while changing grid conditions and available measurements motivate data-driven alternatives that can support accurate, flexible calculation.
Method
The paper learns forward and inverse linearized PF mappings from historical data using PLS- and BLR-based regression algorithms.
Results
The proposed data-driven approaches are more accurate than or at least as accurate as model-based DCPF and DLPF methods across the examined cases.
Takeaways & Limitations
Regression parameter matrices retain physical significance through patterns similar to model-based power-system matrices such as the admittance matrix.
Abstract
from arXiv · showhide
The linearization of a power flow (PF) model is an important approach for simplifying and accelerating the calculation of a power system's control, operation, and optimization. Traditional model-based methods derive linearized PF models by making approximations in the analytical PF model according to the physical characteristics of the power system. Today, more measurements of the power system are available and thus facilitate data-driven approaches beyond model-driven approaches. This work studies a linearized PF model through a data-driven approach. Both a forward regression model ((P, Q) as a function of (theta, V)) and an inverse regression model ((theta, V) as a function of (P, Q)) are proposed. Partial least square (PLS)- and Bayesian linear regression (BLR)-based algorithms are designed to address data collinearity and avoid overfitting. The proposed approach is tested on a series of IEEE standard cases, which include both meshed transmission grids and radial distribution grids, with both Monte Carlo simulated data and public testing data. The results show that the proposed approach can realize a higher calculation accuracy than model-based approaches can. The results also demonstrate that the obtained regression parameter matrices of data-driven models reflect power system physics by demonstrating similar patterns with some power system matrices (e.g., the admittance matrix).
I. INTRODUCTION
The paper proposes data-driven linearized power-flow models that use historical measurements instead of requiring exact grid models. Forward and inverse regressions, supported by PLS and BLR, target accurate and flexible PF calculation across operating settings.
- Motivation: PF linearization simplifies power-system control, scheduling, market clearing, and optimization by reducing nonlinear calculation difficulty and computational burden.The paper motivates linearization as a way to address non-convergence and high computation costs in PF-based algorithms.
- Motivation: Data-driven PF linearization uses historical measurements to rebuild model relationships without requiring complete knowledge of system topology or parameters.This is especially relevant in distribution grids where reconfiguration, distributed resources, and active-device controls complicate accurate modeling.
- Contributions: Compared with prior data-driven approaches, the method directly maps operating variables, avoiding intermediate model identification and reducing computational burden in subsequent PF applications.The paper also emphasizes flexibility for different bus types and direct data-to-data calculation.
- Contributions: The proposed framework provides both forward and inverse regression models, supporting PF calculations under different bus-type settings.The approach maps between power injections and voltage states in both directions.
- Method: PLS- and BLR-based algorithms address data collinearity and reduce overfitting when estimating regression parameters from operational data.The regression formulation treats PF-equation identification as multiparameter fitting of a high-dimensional operating surface.
- Method: The approach approximates the nonlinear AC power-flow surface with a hyperplane, accepting approximation error in exchange for simpler fitting and model extension.The paper argues that the high degree of linearity in PF relationships makes this approximation practical.
B. PF Linearization Visualization
A two-bus example visualizes AC power flow alongside DCPF and DLPF approximations. The AC surface is sufficiently linear for regression, while both model-based approximations retain visible errors and DLPF is closer to AC power flow than DCPF.
- Two-bus comparison: The two-bus example compares the nonlinear ACPF surface with traditional DCPF and decoupled DLPF approximations.The system uses active and reactive power injections as independent variables and voltage magnitude and angle as dependent variables, with bus #1 as reference.
- Observed geometry: The ACPF surface has a high degree of linearity, supporting its approximation with linear regression.The visualization is used to motivate data-driven linearization of PF equations.
- Observed geometry: DLPF approximates ACPF more closely than DCPF does in the two-bus visualization.The comparison concerns the relative closeness of the two model-based linear approximations to the nonlinear ACPF surface.
- Observed geometry: Both DCPF and DLPF still produce clear errors relative to ACPF, leaving accuracy room for data-driven linearization to improve.The example therefore motivates replacing fixed model-based approximations with regression-based relationships learned from operating data.
C. PF Mapping Directions
The paper considers both directions for linearized PF mappings: forward regression maps voltage magnitude and angle to active and reactive power, while inverse regression maps power injections to voltage variables. The framework combines these models with regression methods and multiple data types to address collinearity and validate the approach.
- Forward regression maps (V, θ) to (P, Q), following the explicit direction of the analytical PF equations.
- Inverse regression maps (P, Q) to (V, θ), matching the PF-calculation procedure in which powers are known and voltage variables are calculated.
- Collinearity among voltage angles and magnitudes can make regression ill-conditioned and increase PF calculation errors.
- The proposed framework has three parts: linearization models, regression methods, and data types.
- PLS and BLR methods are applied to both regression directions, while Monte Carlo and public collinear data are used for validation.
III. POWER FLOW LINEARIZATION MODELS
The paper formulates forward and inverse data-driven PF linearization models for different calculation settings. The inverse formulation supports changing bus types, while its structure addresses rank and collinearity issues that affect regression feasibility.
- Forward regression: Forward regression includes constant terms to improve capability when some independent variables remain unchanged.The constants absorb the influence of independent variables whose regression coefficients cannot be estimated.
- Inverse regression: Inverse regression is designed to map known variables to unknown variables across PQ, PV, and Vθ bus settings.
- Inverse regression: Buses are reordered as PQ, PV, and Vθ so the inverse model can accommodate bus-type transformations by reordering its parameter matrix.
- Inverse regression: The inverse mapping requires a full-ranked matrix, whereas forward-regression matrices cannot provide the same mapping when zero-power-injected PQ buses prevent full rank.
- Inverse regression: The inverse formulation removes R_P from the independent variables to reduce collinearity caused by its near-determination from other active injections.The formulation also represents the power-balance relationship and indirectly accounts for network losses through reactive injections and constants.
C. Relationship with Physical Parameter Matrices
The regression parameter matrices have theoretical and empirical connections to physical power-system matrices. Forward-regression parameters correspond to a constant approximation of the PF Jacobian, and these relationships can help indicate overfitting.
- The paper states that regression parameter matrices are numerically similar to several power-system matrices, with the relationship derived theoretically in Fig. 4.
- The forward-regression parameter matrix can be interpreted as a constant approximation of the PF Jacobian across operating points.The regression matrix is constant, whereas the Jacobian varies with the operating point.
- Inverse-regression parameter relationships require implicit differentiation because voltage variables are difficult to express explicitly as functions of active and reactive power.
- These relationships between regression parameters and physical matrices can serve as an indicator of overfitting.
D. Mapping of Branch Power Flow
The paper extends data-driven linearization to branch power-flow mappings. Historical active and reactive branch-flow data can support mappings either from power injections to branch flows or from voltage variables to branch flows.
- Branch power-flow mappings can be regressed from historical active and reactive branch-flow data.
- The mapping direction may be from (P, Q) to (PF, QF) or from (V, θ) to (PF, QF).The formulation shown takes the power-injection-to-branch-flow direction and removes R_P from the independent variables.
IV. REGRESSION ALGORITHMS
The paper formulates forward and inverse PF relationships as generalized matrix regressions, with datasets represented by variable rows and operating snapshots as columns. PLS- and BLR-based algorithms are then introduced for these regressions.
- Forward and inverse PF models are expressed as linear regressions using parameter matrix A, independent-variable matrix X, and dependent-variable matrix Y.The formulation applies to forward, inverse, and branch PF regression models.
- In the data-driven formulation, columns represent datasets or operating snapshots, while rows represent different system variables.The last row of X contains constant terms in the forward-regression example.
- The regression matrices are also written row-wise as collections of variable vectors for theoretical derivation.
- Both PLS- and BLR-based algorithms are proposed to estimate the regression relationships.
A. PLS-Based Algorithm
PLS extracts correlated low-dimensional components from the independent and dependent data blocks, while BLR uses Bayesian priors and MAP estimation to control coefficients and promote sparsity. These designs target collinearity, limited observations, and overfitting in PF regression.
- A. PLS-Based Algorithm: PLS projects X and Y onto extracted components that capture their shared variation and correlation.The method combines ideas from PCA and CCA to address collinearity and limited observations.
- A. PLS-Based Algorithm: PLS predicts the dependent data from updated matrices using a regression relation based on the extracted component matrices.
- B. BLR-Based Algorithm: BLR models each dependent-variable vector with a linear regression containing additive noise and coefficient vector a_i.The observations and dependent vector are centered before regression.
- B. BLR-Based Algorithm: BLR assigns coefficient-specific Gaussian priors whose standard deviations adapt to observed data, with hierarchical hyperparameters encouraging sparsity.The prior is introduced to avoid overfitting, while the hierarchical formulation promotes a sparser prior than a flat Gaussian hierarchy.
- B. BLR-Based Algorithm: MAP optimization can set coefficients to zero below a threshold, providing adjustable sparsity that suits the sparse structure of PF regression matrices.The regression parameter matrix is described as having patterns similar to the sparse constant Jacobian matrix.
V. EXPERIMENTAL RESULTS
The experiments train the proposed regressions on simulated and public power-system data, then evaluate forward, inverse, and branch PF calculations. Across transmission and distribution cases, the data-driven methods generally match or outperform DCPF and DLPF, with especially strong gains in selected quantities.
- Training and testing use Monte Carlo and public measurement data processed with MATLAB and MATPOWER 6.0.Parameters are fitted on training data and evaluated on newly generated testing data.
- The Monte Carlo study covers meshed IEEE 5-, 30-, 57-, and 118-bus systems plus radial IEEE 33- and modified 123-bus systems.
- The public-data study uses hourly synthetic NREL-118 load data with added Gaussian noise and scaling to balance the system.
- The proposed PLS- and BLR-based approaches are more accurate than or at least as accurate as model-based DCPF and DLPF methods across PF calculations.Table I reports forward, inverse, and branch-calculation errors for the compared methods.
- Errors for reactive-power injection in forward calculations and active-power branch flow are more than one order lower than those of model-based methods.
- PLS is more accurate in most cases, while BLR performs better for IEEE 33-bus P and Q calculations and NREL-118 voltage-angle calculations.
C. Calculation Results under Data Collinearity
The collinearity-focused evaluation uses worst-case inverse-calculation errors from 300 NREL-118 test groups and compares the proposed regressions with least-squares and model-based structures. The results support distinct strengths for PLS and BLR and show that learned matrices retain physical patterns.
- C. Calculation Results under Data Collinearity: The NREL-118 inverse-calculation figure reports the largest error among 300 testing groups to assess algorithm robustness.Only PQ buses are shown for voltage-magnitude results.
- C. Calculation Results under Data Collinearity: The proposed algorithms outperform ordinary least-squares regression, particularly for voltage-angle calculation under data collinearity.
- C. Calculation Results under Data Collinearity: PLS is more accurate for voltage magnitude, whereas BLR is more accurate for voltage angle in the NREL-118 comparison.
- D. Regression Parameters: Forward regression matrices for the IEEE 5-bus system are extremely similar to the constant Jacobian matrix for both PLS and BLR.
- D. Regression Parameters: Regression matrices approximate the inverse of B in DCPF, while zero columns correspond to zero-power-injected buses and do not affect accuracy when those injections remain zero.
- D. Regression Parameters: The study proposes forward, inverse, and branch PF mappings with PLS and BLR across meshed, radial, and public-data cases.The paper presents the approach as a bridge between model-based PF linearization and data-driven power-system analysis.
Appendix-Data-Driven Power Flow Linearization:
The appendix compares regression parameter matrices with several power system matrices for IEEE 5-bus and IEEE 57-bus systems.
- The listed authors are Yuxiao Liu, Ning Zhang, Yi Wang, Jingwei Yang, and Chongqing Kang.
- IEEE 5-bus results compare regression parameter matrices with several power system matrices.
- IEEE 57-bus results compare regression parameter matrices with several power system matrices.