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Load-Flow in Multiphase Distribution Networks: Existence, Uniqueness, Non-Singularity and Linear Models
Andrey Bernstein, Cong Wang, Emiliano Dall'Anese, Jean-Yves Le Boudec, Changhong Zhao
TL;DR
Nonlinear AC load flow in multiphase distribution networks may lack globally guaranteed existence or uniqueness, especially with mixed connection models. The paper extends a fixed-point Z-bus approach to these settings, deriving conditions that guarantee a unique solution and convergence, plus Jacobian non-singularity and linear approximations validated on IEEE feeders.
Problem
Nonlinear AC load-flow equations may have multiple solutions, motivating explicit existence and uniqueness conditions for multiphase distribution networks with varied connection models.
Method
The paper extends classical Z-bus load flow through a fixed-point formulation applicable to generic topologies and wye, delta, mixed, and transformer-secondary connection models.
Results
The derived conditions guarantee existence, uniqueness within an analytically characterized domain, convergence of the iterative algorithm, and non-singularity of the load-flow Jacobian.
Takeaways & Limitations
The methodology provides a unified analytical and computational framework for mixed-connection multiphase load flow, with linear models whose accuracy is analyzed on IEEE test feeders.
Abstract
from arXiv · showhide
This paper considers unbalanced multiphase distribution systems with generic topology and different load models, and extends the Z-bus iterative load-flow algorithm based on a fixed-point interpretation of the AC load-flow equations. Explicit conditions for existence and uniqueness of load-flow solutions are presented. These conditions also guarantee convergence of the load-flow algorithm to the unique solution. The proposed methodology is applicable to generic systems featuring (i) wye connections; (ii) ungrounded delta connections; (iii) a combination of wye-connected and delta-connected sources/loads; and, (iv) a combination of line-to-line and line-to-grounded-neutral devices at the secondary of distribution transformers. Further, a sufficient condition for the non-singularity of the load-flow Jacobian is proposed. Finally, linear load-flow models are derived, and their approximation accuracy is analyzed. Theoretical results are corroborated through experiments on IEEE test feeders.
I. INTRODUCTION
The paper addresses solvability and computation of multiphase distribution-network load flow under generic topologies and mixed connection models. It extends fixed-point Z-bus analysis with conditions for solution guarantees, Jacobian non-singularity, and linear approximations.
- Problem setting: The problem concerns multiphase networks with radial or meshed topology, one slack bus, and grounded-wye, ungrounded-delta, mixed, or transformer-secondary connection models.The mixed models cover both aggregated primary-side injections and line-to-line plus line-to-grounded-neutral devices at transformer secondaries.
- Problem setting: Because nonlinear AC load-flow equations can have multiple solutions, the paper studies explicit existence and uniqueness conditions for a solution within a characterized domain.The paper focuses on the high-voltage solution in a specified domain rather than claiming global uniqueness.
- Method and scope: The fixed-point interpretation extends classical Z-bus methodologies and yields conditions that guarantee convergence to the unique load-flow solution.The iteration is obtained from the nonlinear AC equations and is analyzed using Banach fixed-point theory.
- Method and scope: The proposed unified method covers load models that existing analytical methods do not handle, including combined wye-delta and transformer-secondary line-to-line or line-to-neutral connections.The paper identifies OpenDSS as the only freely available solver cited as supporting all four model classes.
- Additional contributions: A sufficient condition for load-flow Jacobian non-singularity is provided, and the guaranteed solutions satisfy that condition.The paper links Jacobian non-singularity to the operating point's static voltage-stability characterization.
- Additional contributions: Two approximate linear load-flow models relate voltages and complex power injections, using respectively a first-order Taylor approximation and the fixed-point formulation.The fixed-point-based model provides a non-local approximation intended for computationally affordable optimization and control applications.
III. PROBLEM FORMULATION
The formulation represents multiphase distribution networks with wye and delta power injections, transformer-secondary connections, and a fixed slack voltage. It converts the network equations into a fixed-point problem while defining the real-coordinate Jacobian used for non-singularity analysis.
- Network and variables: The model considers a generic three-phase network with one slack bus and N three-phase PQ buses, with electrical quantities collected in vectors for voltages and injections.The framework is stated for three phases but extends to mixed three-, two-, and single-phase buses.
- Network and variables: Wye and delta source or load injections are represented separately, with delta quantities also covering phase-to-phase connections at transformer secondaries.The notation uses sY for grounded-wye or line-ground quantities and s∆ for delta or line-line quantities.
- Network equations: The admittance submatrices are formed from network topology, transmission-line π-models, and passive devices, while H encodes phase-to-phase connections.The slack-bus voltage v0 is fixed and known, and YLL is the load-bus admittance block.
- Network equations: Kirchhoff’s current law, delta power-current relations, and Ohm’s law define the coupled load-flow equations for voltages, currents, and powers.These relations are initially written as a set of equations involving the delta current i∆.
- Fixed-point formulation: Eliminating i∆ yields a fixed-point equation for v, whose zero-load voltage profile supports analysis through Banach fixed-point theory.The formulation remains valid without phase-to-phase connections after removing the corresponding H and s∆ term.
- Jacobian analysis: The load-flow equations are also expressed in real coordinates, where the Jacobian maps state variables to power injections and non-singularity means invertibility of that Jacobian.The paper identifies this property as a sufficient condition for static voltage stability of the operating point.
IV. EXISTENCE, UNIQUENESS, AND NON-SINGULARITY
The paper uses a fixed-point formulation to establish explicit conditions for existence, uniqueness, convergence, and non-singularity in general multiphase load-flow problems.
- The fixed-point equation yields an iterative voltage-update procedure extending the classical Z-bus method to the paper’s general setting.Convergence is analyzed for this iteration.
- Theorem 1 guarantees a unique load-flow solution within an analytically characterized voltage region under explicit sufficient conditions.The region is defined using a radius ρ constrained by γ of the reference voltage.
- Theorem 2 guarantees uniqueness, iterative reachability, solution localization, and non-singularity under explicit conditions around a known operating point.The iteration converges from any initialization in the specified larger region, while another region localizes the solution.
- The conditions can be applied using the zero-load voltage profile when no known solution is available, or improved around a measured operating point.The latter setting is identified as relevant to real-time power-network control.
- The explicit conditions can serve as convex constraints in optimal-power-flow settings and support successive construction of non-singular load-flow solutions.For networks without phase-to-phase connections, the theory reduces by removing terms involving H, L, and delta variables.
V. LINEAR MODELS
The paper derives two approximate linear representations of the AC load-flow equations: a local first-order Taylor model and a fixed-point linearization.
- The first-order Taylor model linearizes the load-flow solution around a given operating point and is the best local linear approximator.
- The fixed-point linearization uses a single iteration of the fixed-point method to approximate the relationship between injected powers and voltages.
- The models represent complex voltages and voltage magnitudes as linear functions of active and reactive wye and delta power injections.The injection vectors stack active and reactive components for both connection types.
A. First-Order Taylor (FOT) Method
The FOT method is constructed by differentiating the AC load-flow equations at a specified operating point and solving the resulting Jacobian system.
- The FOT model is obtained by substituting the power-balance equations and differentiating with respect to wye and delta injection coordinates.
- At the operating point, the unknown derivative matrices are solved from the resulting linear equations after substituting the corresponding voltage and delta-current values.
- The derivative system has equal numbers of real equations and variables, and its unique solution is equivalent to invertibility of the load-flow Jacobian.A sufficient condition for invertibility is supplied by Theorem 2’s condition (12).
- The voltage-magnitude model is obtained using a separate derivation rule to convert the operating-point derivatives into matrices K_Y and K_Δ.
B. Fixed-Point Linearization (FPL) Method
The FPL method derives a linear load-flow model from the fixed-point formulation, providing a non-local approximation alongside explicit error bounds and practical application pathways.
- FPL derivation: The FPL model is obtained by initializing the fixed-point iteration at a known solution and yields an explicit linear relationship between injections and voltages.Its derivation uses the first fixed-point iteration around (bv, bs).
- Error bound: The FPL approximation error is upper bounded when the reference solution satisfies the existence conditions and the target solution lies in the guaranteed domain.The bound uses q = ξY(s)(α(bv) − ρ†(bv,bs,s))^2 + ξ∆(s)(β(bv) − ρ†(bv,bs,s))^2 < 1.
- FPL versus FOT: Unlike FOT, which is a tangent-plane approximation, FPL interpolates between the zero-load and reference load-flow solutions and can provide a better global approximation.The paper presents FOT as locally strongest, while FPL is non-local in its construction.
- Potential applications: The results support OPF and real-time-control applications by supplying linear models and convex constraints that ensure existence and non-singularity of the exact high-voltage solution.The proposed methodology is evaluated on IEEE 37-Bus, 123-Bus, and 8500-Node feeders.
- Illustrative example: The illustrative voltage-space region contains the guaranteed solution and can cover feasible voltage magnitudes with angles between ±35.78° and strong reverse-power-flow magnitudes.Accounting for the power injections localizes the guaranteed solution more accurately through Dρ(bv) with ρ = ρ†(bv,bs,s).
- Illustrative example: The iterative update gradually converges, and empirical evidence indicates that precision of 10^-6 is generally reached in fewer than ten iterations when the conditions hold.The observed convergence rate is usually less than one-third of the contraction modulus.
B. IEEE 37-Bus Feeder
On the IEEE 37-Bus feeder, the proposed conditions certify solutions over a wider injection range than the comparison method, while both linear models remain accurate over the tested range.
- Experimental setup: The IEEE 37-Bus experiment uses purely delta-connected sources and loads, converts constant-current and constant-impedance elements to constant-power sources, and fixes voltage regulators at default values.The target injections are scaled as s = κsref.
- Existence and uniqueness: Intervals 1, 2, and 3 are numerically identical, while the proposed method certifies existence and uniqueness for additional injections in Intervals 4 and 5.The proposed conditions also require substantially less computation than verifying the comparison conditions.
- Linear models: Both linear models have relative errors below 1% for κ ∈ [−1.5, 1.5], with FOT more accurate locally and FPL providing a better global approximation.Voltage-magnitude approximation errors are reported at a similar level.
C. IEEE 123-Bus Feeder
The IEEE 123-Bus experiments evaluate the proposed conditions and linear models on an unbalanced multiphase network with mixed connections, including large-scale secondary-transformer modeling. Results agree with OpenDSS, while the FPL approximation remains accurate and computationally scalable.
- Conditions evaluation: The IEEE 123-Bus evaluation uses an unbalanced multiphase network with one-, two-, and three-phase sources and loads.The experiment also considers mixed delta-wye connections by adding sources and loads listed in Table III.
- Conditions evaluation: The mixed-connection results match those obtained in OpenDSS, the only freely available solver identified for such connections.
- Large-scale applicability: The 8500-node feeder analysis evaluates applicability using voltage-feasibility constraints defined relative to the zero-load voltage profile.The feasibility condition is |(v)j| ≥0.9|(w)j| for all j.
- Large-scale applicability: For the 8500-node feeder, FPL relative errors remain below 1.4% for κ ∈[-1, 2].The result covers both phasor and corresponding voltage-magnitude approximations.
- Complexity evaluation: Condition verification has worst-case complexity O(N^2) and is not real-time for the 8500-bus network, while one FPL iteration scales as O(N) in radial networks.The paper reports fewer than 10 iterations for accuracy 10^-6 in almost all experiments.
APPENDIX
The appendix establishes the fixed-point properties needed for convergence by reparametrizing voltages and proving self-mapping on a bounded domain.
- The proof reparametrizes the voltage variables as u := W^-1v, preserving solution properties through an invertible transformation.
- The Banach fixed-point theorem reduces the argument to showing that the fixed-point mapping is self-mapping and contractive on a chosen domain.
- For ρ ∈(0, γ(bv)) satisfying condition (9), membership in the domain is preserved from one iteration to the next.
- The proof bounds the update using triangle inequalities, induced matrix norms, and the voltage quantities α(·) and β(·).
2) Proof of Contraction:
The contraction proof shows that successive fixed-point updates become strictly closer when the radius additionally satisfies condition (10).
- Under the radius condition in (10), each successive update difference is strictly smaller than the preceding one.The proof derives this through bounds involving ξY(s) and coordinatewise voltage denominators.
- The contraction argument reuses rearrangements analogous to the self-mapping proof to establish the decrease in iteration differences.
C. Proof of Theorem 2
The proof of Theorem 2 establishes Jacobian invertibility under the stated conditions and then shows that those conditions imply the self-mapping and contraction requirements.
- Jacobian non-singularity: Jacobian invertibility is analyzed through the equivalent linear system obtained from the Jacobian relation.Because the system is square, uniqueness of the linear solution is equivalent to triviality of the corresponding homogeneous system.
- Jacobian non-singularity: Assuming a nonzero homogeneous solution leads to a contradiction by constructing perturbed networks and invoking the theorem’s guaranteed solution properties.
- Fixed-point conditions: The proof shows that conditions (12) and (13) imply the self-mapping condition (9) over the interval [ρ†(bv,bs,s), ρ‡(bv,bs)].
- Fixed-point conditions: Using that ξ(·) is a norm, the same conditions also imply the contraction condition over the stated radius interval.
- Theorem conclusion: The remaining bounds complete the theorem’s proof, including the implication from guaranteed solutions to Jacobian non-singularity.
D. Proof of Theorem 3
The proof identifies the stated update as one fixed-point iteration and bounds its error using contraction and the Banach fixed-point theorem.
- Equation (18) is a single fixed-point iteration initialized at b_v, with v(0) = b_v and v(1) = e_v.
- The error bound uses the contraction coefficient q < 1 established in Theorem 2.
- The first inequality follows from the Banach fixed-point theorem, while subsequent inequalities use v = Wu and v ∈ D_ρ†(b_v).