DistFlow: AC Branch-Flow Model in Distribution Networks

• DistFlow is a branch-flow formulation for AC power flow in radial distribution networks that relates branch active/reactive power, squared voltage magnitudes, and squared current magnitudes through nonlinear equations.
• Linearized and convex relaxations of DistFlow, such as LinDistFlow and SOC formulations, enable computationally efficient approximations with quantifiable voltage and current errors on test systems like the IEEE 33-bus feeder.
• Extensions of DistFlow include three-phase unbalanced models, continuum ODE formulations, and data-driven state estimation, which enhance optimization, control, and sensor placement in modern active distribution networks.

DistFlow usually denotes the branch-flow formulation of AC power flow on radial distribution networks, in which branch active and reactive power flows, squared voltage magnitudes, and squared current magnitudes are related through power-balance, voltage-drop, and current-flow equations. In contemporary usage, the term covers the classical nonlinear Baran–Wu equations, linearized and second-order-cone relaxations, data-driven formulations built from historical trajectories, and three-phase generalizations for unbalanced feeders; the same name has also been adopted independently in an unrelated reinforcement-learning context (Molodchyk et al., 9 Apr 2026, Coffrin et al., 2015, Perna et al., 16 Jun 2026, Wang et al., 18 Jul 2025).

1. Classical branch-flow formulation

In its standard form, DistFlow is posed on a balanced, radial network $G=(\mathcal N,\mathcal E)$ with a slack bus and directed branches. The canonical state variables are branch active and reactive power flows $P_{ij},Q_{ij}$, squared line-current magnitude $\ell_{ij}=|I_{ij}|^2$, and squared bus-voltage magnitude $v_i=|V_i|^2$. Net nodal injections are denoted $p_j,q_j$, and line parameters by $r_{ij},x_{ij}$. A common statement of the model is

$$P_{ij}=\sum_{k:(j,k)\in\mathcal E}P_{jk}+p_j+r_{ij}\ell_{ij},$$

$$Q_{ij}=\sum_{k:(j,k)\in\mathcal E}Q_{jk}+q_j+x_{ij}\ell_{ij},$$

$$v_j=v_i-2\bigl(r_{ij}P_{ij}+x_{ij}Q_{ij}\bigr)+(r_{ij}^2+x_{ij}^2)\ell_{ij},$$

$P_{ij}^2+Q_{ij}^2=v_i\ell_{ij}.$

These equations encode branch-flow balance, Ohm’s law in voltage-drop form, and the exact current–power–voltage relation on each line (Molodchyk et al., 9 Apr 2026, Hoosh et al., 2023).

The model is exact for steady-state AC power flow on a radial feeder, but it is nonconvex because of the quadratic equality $P_{ij},Q_{ij}$0 (Molodchyk et al., 9 Apr 2026). Relative to bus-injection formulations, DistFlow eliminates explicit voltage-angle variables and works directly with branch quantities, which makes it structurally natural for feeders whose topology is a tree (Ageeva et al., 2020). In a numerical comparison on the IEEE 33-bus radial distribution system, AC-OPF and exact DistFlow were reported to coincide exactly for the flexibility-domain problem considered there (Ageeva et al., 2020).

2. Linearizations and convex relaxations

A large part of the DistFlow literature concerns tractable approximations of the nonlinear model. The classical simplified DistFlow, often written sDistFlow or LinDistFlow, neglects the loss terms $P_{ij},Q_{ij}$1, $P_{ij},Q_{ij}$2, and $P_{ij},Q_{ij}$3, yielding

$P_{ij},Q_{ij}$4

$P_{ij},Q_{ij}$5

The resulting model is linear, and in matrix form it becomes $P_{ij},Q_{ij}$6 after appropriate elimination of branch variables (Marković et al., 2022, Saha et al., 2020). This approximation is accurate when voltage deviations are small, angle differences are small, and losses are modest, but it omits the nonlinear coupling that represents ohmic losses and current magnitudes (Saha et al., 2020, Gupta et al., 12 Jan 2025).

A second line of development replaces the nonconvex equality by the conic inequality

$P_{ij},Q_{ij}$7

which yields a second-order-cone program once the remaining DistFlow constraints are written in $P_{ij},Q_{ij}$8-space (Zholbaryssov et al., 2019, Hoosh et al., 2023). This relaxation preserves the exact branch balances and voltage-drop equations while convexifying the current relation. In the Convex DistFlow formulation, the same idea is often written as $P_{ij},Q_{ij}$9, where $\ell_{ij}=|I_{ij}|^2$0. For transmission systems, DistFlow has been extended to include bus shunts, line charging, and transformers, and the extended Convex DistFlow relaxation was shown to be equivalent to the corresponding extended SOC relaxation (Coffrin et al., 2015).

Several papers modify the linearization itself rather than only relaxing the nonlinear constraint. A parameterized linear power-flow model introduces a branch-specific parameter $\ell_{ij}=|I_{ij}|^2$1 through

$\ell_{ij}=|I_{ij}|^2$2

and sDistFlow is recovered exactly by setting $\ell_{ij}=|I_{ij}|^2$3 (Marković et al., 2022). A different “modified DistFlow” formulation replaces $\ell_{ij}=|I_{ij}|^2$4 and $\ell_{ij}=|I_{ij}|^2$5 by their ratios to voltage magnitude, using these ratios as state variables so that conventional cold-start linearization errors are reduced (Yang et al., 2020).

3. Extensions beyond balanced radial feeders

Although DistFlow originated as a balanced radial feeder model, later work generalizes it in several directions. The most direct is the three-phase unbalanced extension “Dist3Flow,” which uses the real and imaginary components of nodal voltages together with active and reactive power flows as state variables. Lines are represented by nonlinear forward and backward equations, loads are modeled with ZIP behavior, and DERs are modeled by P–Q control. The same formulation accommodates both radial and closed-ring topologies by changing the boundary conditions, and it is solved by a backward/forward sweep algorithm (Perna et al., 16 Jun 2026). In validation against OpenDSS on a six-node unbalanced feeder derived from the IEEE 13-node test feeder, the reported maximum voltage magnitude error was $\ell_{ij}=|I_{ij}|^2$6 p.u., the maximum phase-angle error was $\ell_{ij}=|I_{ij}|^2$7, and convergence occurred in fewer than 10 iterations for all tested cases (Perna et al., 16 Jun 2026).

A different extension takes a continuum limit along a long feeder. In “DistFlow ODE,” the discrete branch equations are homogenized into ordinary differential equations for $\ell_{ij}=|I_{ij}|^2$8, $\ell_{ij}=|I_{ij}|^2$9, and $v_i=|V_i|^2$0 along the feeder length, with boundary conditions at the substation and end of line (Wang et al., 2012). This continuous model supports feeder-level analyses of voltage drop, total import, power losses, and critical feeder length, and it can include inverter-based, voltage-dependent reactive-power control. One notable result is that sufficiently long PV-rich feeders may exhibit multiple stable solutions, with the associated risk that a short fault near the head of the line can return the system to a low-voltage branch rather than the high-voltage operating point (Wang et al., 2012).

DistFlow has also been extended from distribution to transmission settings. The transmission-oriented extension retains the branch-flow viewpoint while adding bus shunts, line charging, and off-nominal transformer taps, thereby enlarging the original applicability of Convex DistFlow without abandoning its SOC structure (Coffrin et al., 2015). Taken together, these developments show that DistFlow is not restricted to the single-phase, lossless, or strictly radial simplifications that are often associated with introductory presentations.

4. Data-driven inference, state estimation, and sensing

Recent work uses DistFlow as a structural prior in data-driven estimation. A notable example is the data-driven power flow framework for balanced radial distribution networks with sparse real-time data. There, the inputs are $v_i=|V_i|^2$1, the outputs are $v_i=|V_i|^2$2, and block-Hankel matrices built from historical trajectories provide a behavioral representation. Under the rank condition

$v_i=|V_i|^2$3

a candidate pair $v_i=|V_i|^2$4 satisfies the affine DistFlow equations if and only if there exists a coefficient vector $v_i=|V_i|^2$5 with $v_i=|V_i|^2$6 and $v_i=|V_i|^2$7, while the nonlinear current relation is imposed explicitly or through an SOCP relaxation (Molodchyk et al., 9 Apr 2026).

The same framework couples behavioral DistFlow with a sensor-placement problem based on Kron reduction and a mixed-integer assignment model. The goal is to choose a measured subset of buses under a sensor budget while minimizing worst-case voltage-reconstruction error and respecting radial-network structure (Molodchyk et al., 9 Apr 2026). On 47-, 85-, and 141-node feeders, the reported maximum voltage-magnitude error was at most $v_i=|V_i|^2$8 p.u., with a sensor budget as low as $v_i=|V_i|^2$9 of nodes and a typical per-step solution time below $p_j,q_j$0 s; the reduced real-time SOCP was reported to solve in less than $p_j,q_j$1 ms (Molodchyk et al., 9 Apr 2026).

State-estimation work extends DistFlow in another practical direction by adding shunt elements such as cable capacitances. In an enhanced DistFlow weighted least-squares estimator, shunt terms are inserted into the reactive-power balance, and measurement-device placement is determined by a greedy procedure that adds sensors until voltage-magnitude and line-current accuracy limits are satisfied (Paruta et al., 2020). DistFlow-derived bounds have also been used for safety certification rather than direct estimation: in resilient software-update scheduling for intelligent electronic devices, nonlinear DistFlow equations are transformed into affine worst-case voltage and current bounds, and the resulting rollout problem is reformulated as a vector bin-packing problem solvable in real time, including on a benchmark with 10,476 buses (Sou et al., 2023).

5. Optimization and control applications

DistFlow is deeply embedded in distribution-system optimization. In storage planning, it is used in a mixed-integer quadratically constrained program for the optimal sizing, siting, and operation of energy storage systems, where the AC network constraints are written in DistFlow form and the nonconvex current relation is relaxed to an SOC inequality (Hoosh et al., 2023). In microgrid resilience under wildfire uncertainty, SOC-relaxed DistFlow serves as the network model in a two-stage stochastic optimal power flow, with scenario-dependent line outages and PV reductions encoded in the second stage (Chowdhury et al., 2024). In service-restoration and hardening problems, linearized DistFlow underlies stochastic multi-period MILP models for undergrounding lines and coordinating mobile generators while preserving radiality (Taheri et al., 2022).

The same modeling pattern appears in more recent multi-energy and transportation-coupled problems. A thermal-electrical co-optimization framework for active distribution grids with EVs and heat pumps uses convex DistFlow to couple DER schedules, building thermal dynamics, voltage limits, and line ampacity limits. On a realistic 61-node LV network, the paper reports reductions of $p_j,q_j$2 in transformer aging, $p_j,q_j$3 in losses, and complete elimination of voltage violations, while convex DistFlow maintained sub-second runtimes across the tested DER penetrations (Panagi et al., 29 Jan 2026). In the open-source platform V2Sim, an optimized DistFlow model is used to represent the power distribution network side of microscopic vehicle-to-grid simulations, with EV charging loads entering as time-varying nodal demands and slow-charging-station V2G injections treated as dispatchable generation (Qian et al., 2024).

DistFlow also supports optimization under discrete structural decisions. A mixed-integer linear programming approach to phase allocation in unbalanced networks uses linearized DistFlow together with binary phase-consistency constraints so that downstream phase configurations cannot introduce phases absent upstream (Gupta et al., 12 Jan 2025). In distributed control and coordination, the SOCP relaxation of a DistFlow-based OPF has been solved by distributed algorithms over time-varying communication networks using primal–dual iterations, consensus, and gradient tracking (Zholbaryssov et al., 2019).

6. Accuracy, exactness, and computational trade-offs

A recurring issue in the DistFlow literature is the trade-off between fidelity and tractability. Exact DistFlow retains losses and branch-current physics, but it remains nonconvex. Linearized DistFlow is far cheaper computationally, yet its omissions are systematic: sDistFlow always overestimates voltages and its error grows as loading departs from the zero-load linearization point (Marković et al., 2022). On the IEEE 33-bus system, the reported maximum voltage-magnitude error of sDistFlow can exceed $p_j,q_j$4 p.u. at base loading and reach $p_j,q_j$5 p.u. at $p_j,q_j$6 nominal loading, while parameterized corrections remain more accurate across the same range (Marković et al., 2022). Offline parameter tuning can push the linear approximation substantially closer to nonlinear DistFlow: optimized LinDistFlow was reported to improve $p_j,q_j$7-norm and $p_j,q_j$8-norm voltage losses by up to $p_j,q_j$9 and $r_{ij},x_{ij}$0, respectively, relative to traditional LinDistFlow (Taheri et al., 2024).

The status of SOCP relaxations is more nuanced than the common presentation that “radiality implies exactness.” Several papers state sufficient conditions under which the relaxation is exact, including radial topology and mild loading assumptions, and some recent case studies found numerical exactness even when theoretical conditions were violated (Molodchyk et al., 9 Apr 2026, Panagi et al., 29 Jan 2026). In the 61-node active-distribution-grid co-optimization study, the maximum gap

$r_{ij},x_{ij}$1

at the SOCP solution was reported to be on the order of $r_{ij},x_{ij}$2 or smaller (Panagi et al., 29 Jan 2026). At the same time, the wildfire microgrid study notes that heavy loading or extreme conditions can make the relaxation inexact and yield nonphysical solutions, requiring feasibility checks or adjustment steps (Chowdhury et al., 2024).

A sharper controversy appears in TSO–DSO flexibility-domain identification. In that comparison, exact DistFlow and AC-OPF coincided on the IEEE 33-bus feeder, whereas DistFlow-SOCP had the worst accuracy among AC-OPF, DistFlow, DistFlow-SOCP, and LinDistFlow (Ageeva et al., 2020). The reported 24-hour computational costs for approximately 4,800 OPF solves were $r_{ij},x_{ij}$3 s for AC-OPF, $r_{ij},x_{ij}$4 s for exact DistFlow, $r_{ij},x_{ij}$5 s for DistFlow-SOCP, and $r_{ij},x_{ij}$6 s for LinDistFlow (Ageeva et al., 2020). The significance of that result is not that the conic relaxation is generally unusable, but that its approximation error is application-dependent and can be more consequential than its popularity suggests.

7. Other uses of the term

Outside power systems, “DistFlow” has been used as the name of an unrelated fully distributed reinforcement-learning framework for large-language-model post-training (Wang et al., 18 Jul 2025). In that context, DistFlow denotes a multi-controller architecture in which each worker carries a DAG worker, distributed dataloader, and distributed databuffer, eliminating the centralized node used by hybrid-controller designs. The framework was reported to achieve near-linear scalability up to thousands of GPUs and up to a $r_{ij},x_{ij}$7 end-to-end throughput improvement over state-of-the-art frameworks (Wang et al., 18 Jul 2025).

In the power-systems literature, however, DistFlow remains identified primarily with the branch-flow modeling paradigm. Its continuing importance lies in the fact that the same core equations support exact radial AC analysis, linear approximations, SOC relaxations, three-phase and transmission-oriented generalizations, sparse-sensor state estimation, and a wide spectrum of optimization models for modern active distribution networks.