⚡ electrical engineering

Three-phase model of unbalanced distribution networks with DERs

This paper introduces Dist3Flow, a rigorous non-approximated three-phase branch flow model that extends classical DistFlow equations to accurately analyze unbalanced radial and closed-ring distribution networks with distributed energy resources, utilizing a backward/forward sweep algorithm validated against OpenDSS.

Original authors: S. Perna, C. Lillo, A. R. Di Fazio, M. Russo, G. M. Casolino, P. Varilone, P. Verde

Published 2026-06-17

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CC BY 4.0

Original authors: S. Perna, C. Lillo, A. R. Di Fazio, M. Russo, G. M. Casolino, P. Varilone, P. Verde

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a city's electrical grid as a complex plumbing system. In the old days, engineers treated the pipes as if they were all identical, carrying water (electricity) smoothly and evenly in three separate lanes. They used a simple rulebook called "DistFlow" to predict how much water would be in each pipe and how much pressure (voltage) would be left at the end of the line.

However, real-world neighborhoods aren't perfect. Some houses use a lot of power, some use little; some power lines are old and crooked; and now, many homes have their own solar panels or batteries (called DERs) pushing power back into the pipes. This creates a "three-phase" system that is messy, unbalanced, and full of twists. The old rulebook fails here because it assumes everything is perfectly symmetrical, like a calm lake, when in reality, the water is churning like a stormy sea.

The New Solution: "Dist3Flow"
The authors of this paper created a new, super-detailed rulebook called Dist3Flow. Think of it as upgrading from a simple map to a high-definition, 3D simulation.

Instead of guessing, this new model looks at the electrical grid with a magnifying glass. It doesn't just look at the total power; it tracks the "real" and "imaginary" parts of the electricity (like tracking both the speed and the direction of a spinning top) for every single phase (Lane A, Lane B, and Lane C).

How It Works: The "Sweep" Analogy
To solve the puzzle of how electricity flows through this messy grid, the authors use a method called the Backward/Forward Sweep (BFS). Imagine you are trying to figure out the water pressure in a long, winding garden hose with several sprinklers attached:

The Backward Sweep (Looking Back): You start at the very end of the hose (the farthest sprinkler) and walk backward toward the faucet. You calculate how much water each sprinkler is taking and how much is left in the hose behind it.
The Forward Sweep (Looking Forward): Once you reach the faucet, you walk forward again. You adjust your calculations based on the pressure you know exists at the source and see how the water flows out to each sprinkler.
The Loop: You keep walking back and forth, adjusting your numbers each time, until your calculations stop changing. At that point, you know exactly what the pressure and flow are at every single point in the hose.

Handling the "Ring" and the "Solar Panels"
The paper also tackles two tricky scenarios:

The Closed Ring: Sometimes, the hose is connected back to the faucet to form a circle (a closed ring). This makes the water flow in two directions at once. The new model handles this by adding a special "correction step" to ensure the pressure matches up perfectly where the circle closes.
The Solar Panels (DERs): When homes push power back into the grid, it reverses the flow. The old models struggled with this, but Dist3Flow treats these reverse flows naturally, just like water flowing backward in a pipe.

The Results
The authors tested their new model against a famous, highly accurate simulation software called OpenDSS (think of OpenDSS as the "gold standard" referee). They ran tests on a fake neighborhood with unbalanced loads and solar panels, both in a straight line and in a ring.

The result? Their new "Dist3Flow" model matched the referee's results almost perfectly. It was accurate enough to catch tiny details, like how the magnetic fields between the three phases of the wire can sometimes actually reduce energy loss in one lane (a phenomenon that sounds weird but is physically real).

In Summary
This paper presents a more accurate, non-simplified way to calculate how electricity moves through modern, messy, unbalanced neighborhoods. By using a "back-and-forth" walking algorithm, it can handle solar panels, uneven loads, and ring-shaped grids without needing to make lazy approximations. It proves that you can get a precise, high-definition picture of the grid's health without needing to throw away the old math entirely—you just need to make it three-dimensional and a bit more flexible.

Technical Summary: Three-Phase Model of Unbalanced Distribution Networks with DERs

Problem Statement
Classical DistFlow equations, traditionally used for steady-state distribution network analysis, are insufficient for modern distribution grids. These classical methods fail to capture the inherent imbalances of three-phase systems caused by asymmetrical line parameters, unbalanced loads, and the integration of Distributed Energy Resources (DERs). Furthermore, standard Power Flow (PF) algorithms designed for symmetric transmission networks often prove inadequate for radial or weakly-meshed distribution topologies characterized by high resistance-to-reactance (r/x) ratios. Existing three-phase extensions of DistFlow in the literature frequently rely on simplifying assumptions—such as constant phase-voltage amplitude ratios, neglect of mutual impedances, or linearization of load models—that compromise accuracy.

Methodology
The authors propose Dist3Flow, a rigorous, non-approximated three-phase formulation that extends the classical DistFlow equations. The model utilizes a Branch Flow Model (BFM) where the state variables consist of the real and imaginary components of nodal voltages and the active and reactive power flows in each branch.

Key modeling components include:

Line Modeling: Distribution lines are represented by 3×3 impedance matrices ( $\dot{Z}_{n-1,n}$ ) that account for self and mutual coupling between phases. The model employs nonlinear forward and backward equations to relate variables at the sending and receiving nodes without neglecting squared current terms or mutual impedances.
Load and DER Modeling: Uncontrolled loads are modeled using the standard ZIP (Constant Impedance, Constant Current, Constant Power) formulation. DERs are represented assuming P–Q control, injecting assigned active and reactive power values.
Solution Algorithm: The system is solved using a Backward/Forward Sweep (BFS) algorithm.
- Initialization: Variables are initialized based on the slack bus conditions and sequential forward sweeps.
- Boundary Conditions: For radial networks, the end node enforces zero power outflow. For closed-ring (meshed) topologies, the algorithm enforces voltage equality at the reconnection point using a compensation method with an equivalent impedance matrix and a relaxation parameter ( $\rho$ ) to ensure convergence.
- Iteration: The algorithm alternates between backward sweeps (aggregating currents/adjusting powers) and forward sweeps (updating voltages) until convergence criteria are met.

Key Contributions

Exact Formulation: The paper presents a complete Dist3Flow formulation that avoids the simplifying assumptions common in previous three-phase extensions, thereby ensuring high accuracy in capturing self- and mutual-coupling effects.
Topology Versatility: The formulation generalizes PF analysis to both radial and closed-ring (looped) topologies by incorporating specific boundary conditions at terminal nodes.
Algorithmic Extension: The work adapts the BFS algorithm, traditionally used for radial systems, to handle closed-loop configurations in unbalanced networks with DERs.
State Variable Definition: The model explicitly uses the real and imaginary components of nodal voltages and branch power flows, allowing for a detailed analysis of phase-specific imbalances.

Results
The proposed model was validated using a modified IEEE 13-node test feeder configured as both an open-ring and a closed-ring network, with and without balanced three-phase DERs. The results were benchmarked against OpenDSS.

Accuracy: The Dist3Flow results showed maximum absolute voltage magnitude errors of $5 \times 10^{-5}$ p.u. and zero phase angle error compared to OpenDSS. Relative errors for active and reactive power flows remained within acceptable engineering tolerances.
Convergence: Both the proposed method and OpenDSS achieved convergence in fewer than ten iterations when an appropriate relaxation parameter ( $\rho = 4$ ) was used.
Physical Phenomena: The model successfully captured complex phenomena, such as negative active power losses in specific phases (e.g., phase b) due to mutual coupling effects, while maintaining strictly positive total system losses. It also demonstrated how DER injection and ring closure alter voltage profiles, flattening them and reducing total active power losses (e.g., reducing losses from 69.96 kW in a radial case to 9.27 kW in a closed-ring case with DERs).

Significance
The paper claims that Dist3Flow provides a precise tool for analyzing unbalanced distribution networks without resorting to the approximations that limit the fidelity of existing methods. By validating the approach against OpenDSS across various configurations, the authors demonstrate that their exact formulation is suitable for steady-state assessment in systems with high DER penetration and complex topologies. The work establishes a foundation for future research aimed at extending these equations to more complex systems and adapting them for use within linearized frameworks.

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