LinDistFlow: Linearized Radial Feeder Model

• LinDistFlow is a linearized, lossless approximation of AC power flow equations tailored for radial distribution networks using squared voltage magnitudes.
• The model simplifies power flow analysis by omitting quadratic loss terms, facilitating convex OPF formulations and enabling sensitivity analysis.
• Its limitations include underestimating losses and reduced accuracy away from linearization points, spurring research on calibration and improved generalizations.

LinDistFlow is a linearized, lossless approximation of the AC power flow equations tailored for radial distribution systems. It originates from the nonlinear DistFlow model, exploits the typical radial (tree) topology of distribution systems, and uses squared voltage magnitudes directly in the formulation, which makes voltage drops explicit and preserves linear dependence on active and reactive power variables. In the contemporary literature, it serves both as a foundational linear power flow model for feeder analysis and optimization and as a reference point for multiphase generalizations, uncertainty-aware formulations, loss-compensated variants, and parameter-optimized surrogates (Huang et al., 2021, Taheri et al., 2024).

1. Canonical formulation

In radial distribution networks, LinDistFlow is commonly written as a simplified branch-flow model in which active and reactive power balances are enforced along the feeder and squared-voltage drops are expressed linearly in line flows. One standard statement is

$$\begin{align} (d_i^p - g_i^p) + \sum_{j \in \mathcal{C}_i} f_j^p &= f_{i}^p, \quad p \in \{P,Q\}, \quad \forall i \in \mathcal{N}^+ \ u_{A_i} - 2(f_i^P R_i + f_i^Q X_i) &= u_i, \quad \forall i \in \mathcal{N}^+ , \end{align}$$

with $u_i = v_i^2$, line resistances $R_i$, reactances $X_i$, and parent/child relations defined by the feeder tree (Mieth et al., 2018). Equivalent branch-oriented forms appear throughout the literature, for example

$$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$

where $w_i=v_i^2$ and $(i,j)$ is a radial line (Ageeva et al., 2020).

A compact sensitivity form is also standard: $$\mathbf{v} = R \mathbf{p} + X \mathbf{q} + \tilde{\mathbf{v}},$$ where $\mathbf{v}$ collects squared voltage magnitudes, $\mathbf{p}$ and $u_i = v_i^2$0 collect active and reactive injections, and the entries of $u_i = v_i^2$1 and $u_i = v_i^2$2 are sums of line resistances and reactances over the common path from the slack bus to the buses of interest. For single-phase radial networks,

$u_i = v_i^2$3

with $u_i = v_i^2$4 denoting the unique path from the substation to node $u_i = v_i^2$5 (Zhou et al., 2019).

These forms encode the same modeling idea: branch losses and current-squared terms are omitted, while voltage drops are retained through squared-voltage variables. That representation is what makes LinDistFlow especially useful in OPF, sensitivity analysis, and convex relaxations built for radial feeders (Bose et al., 2023).

2. Assumptions and modeling regime

The key assumption underpinning LinDistFlow is the neglect of quadratic loss terms, specifically terms proportional to the square of line current. In the nonlinear DistFlow model, terms such as $u_i = v_i^2$6, $u_i = v_i^2$7, and $u_i = v_i^2$8 appear in branch power balance and voltage equations; LinDistFlow drops these terms to obtain a linear approximation (Jiang et al., 6 Aug 2025, Taheri et al., 2024).

The model therefore assumes radial topology, modest system losses, and explicit use of squared voltage magnitudes. This is why it differs structurally from classic DC power flow. Unlike the DC power flow model, LinDistFlow is tailored to feeders with higher $u_i = v_i^2$9 ratios, unbalanced conditions, and significant voltage deviations, and it explicitly incorporates squared voltage magnitudes and their evolution along branches (Jiang et al., 6 Aug 2025). In many formulations it is also described as a lossless approximation of the AC power flow equations for radial networks, with computational tractability as a primary motivation (Mieth et al., 2018).

These assumptions delimit its scope. LinDistFlow is most naturally associated with radial, single-phase or balanced equivalent feeders, although later work extends it to unbalanced three-phase systems. It remains attractive because, once losses are ignored, voltage, flow, and sometimes uncertainty propagation constraints become linear or convex. A plausible implication is that LinDistFlow is less a single equation than a modeling family organized around three recurring choices: radial topology, squared-voltage variables, and omission of quadratic loss terms.

3. Multiphase and generalized variants

Although the classical model is tied to radial single-phase feeders, later work extends its structure in two directions: multiphase unbalanced modeling and topological generalization. For multi-phase radial networks, the linearization becomes

$R_i$0

with phase-indexed voltage and injection vectors and sensitivity entries

$R_i$1

where $R_i$2 and $R_i$3 aggregates mutual impedances over common paths (Zhou et al., 2019).

A more explicit generalization is the Generalized LinDistFlow (GLDF) model: $R_i$4 with open-circuit voltage $R_i$5, $R_i$6, and

$R_i$7

The offset $R_i$8 enforces exactness at the linearization point $R_i$9 (Huang et al., 2021).

This generalized model handles multiphase systems, applies to generic topologies including radial and meshed networks, allows linearization at arbitrary operating points, and reduces to the classic or multiphase LinDistFlow model for a radial network at zero injection (Huang et al., 2021). When both GLDF and fixed-point linearization are linearized at zero injection, GLDF always yields lower errors in squared voltage magnitude than FPL, for any loading, in radial networks; the paper states this as

$X_i$0

A related formulation appears in uncertainty analysis, where LinDistFlow is treated as equivalent to the Linear Coupled Power Flow model under radial network conditions: $X_i$1 with a closed-form inverse on trees involving effective resistances and reactances (Talkington et al., 20 Oct 2025).

4. Role in optimization, control, and market design

LinDistFlow is central to OPF formulations because it converts AC-feeder constraints into linear or convex constraints that remain sensitive to voltage and reactive power. In data-driven distributionally robust OPF for radial systems, the model enables explicit propagation of nodal injection uncertainty into random voltages and flows, so that chance constraints can be recast as second-order conic constraints; the resulting approach solved systems up to the IEEE 8500-bus network in about $X_i$2 seconds (Mieth et al., 2018). In chance-constrained flexibility-request design for local flexibility markets, LinDistFlow likewise supports SOCP reformulations and yields suboptimality gaps of $X_i$3 on a 15-bus case and $X_i$4 on an 81-bus real network relative to a stochastic market-clearing benchmark (Prat et al., 2021).

The model also underpins distributionally robust formulations that account for renewable dependence. In a LinDistFlow-based radial OPF with a copula-based ambiguity set, a conic reformulation preserves tractability while embedding wind-farm dependencies, and reported cost savings reach up to $X_i$5 relative to traditional DRO (Arrigo et al., 2021). In local electricity market design, LinDistFlow equations are combined with intrusive generalized Polynomial Chaos to propagate uncertainty through the grid model and produce probabilistic locational marginal prices in a two-stage convex formulation (Austnes et al., 14 Oct 2025).

In control and distributed computation, LinDistFlow’s sensitivity structure is used to decompose large OPF problems hierarchically. For multi-phase radial networks, the block structure of $X_i$6 and $X_i$7 enables a hierarchical distributed primal-dual algorithm in which regional coordinators and a central coordinator exchange aggregated quantities rather than full-network sensitivities. On a 4,521-node feeder, the algorithm achieves more than 10-fold acceleration in the speed of convergence compared to a centrally coordinated implementation (Zhou et al., 2019). A related hierarchical formulation based on autonomous-grid partitions reports more than tenfold improvement in convergence speed on the same scale of feeder (Zhou et al., 2018).

LinDistFlow is also used in interface flexibility and topology optimization. For TSO-DSO flexibility domain identification on the IEEE 33-bus radial system, the LinDistFlow formulation is an LP solved with Gurobi and required $X_i$8 seconds for 200 boundary points, compared with $X_i$9 seconds for AC-OPF and $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$0 seconds for DistFlow (Ageeva et al., 2020). In daily topology reconfiguration for improving fairness in photovoltaic curtailment, day-ahead grid constraints are modeled using LinDistFlow, while the real-time stage uses a first-order Taylor linearization of AC power flow (Gupta et al., 2024). In optimization-based control of distributed battery storage, a LinDistFlow-based convex multi-period formulation yields reductions of $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$1 per cent in losses and $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$2 per cent in peak substation power compared to other state-of-the-art algorithms (Carvalho et al., 2024).

A further strand of work uses LinDistFlow for economic interpretation. In single-phase radial LDF-OPF, closed-form marginal values of real and reactive demand are derived from dual variables, and upper bounds are obtained for the change in marginal demand prices when apparent power flow limits become binding (Bose et al., 2023).

5. Accuracy, limitations, and failure modes

The principal limitation of LinDistFlow is the omission of losses. Because the model ignores quadratic loss terms, all flows are treated as lossless; the only effect of line parameters is on voltage drops (Jiang et al., 6 Aug 2025). This makes the approximation most accurate when system losses are modest, but it also creates systematic distortions in operating regions where losses materially affect interface exchange, feasible domains, or voltage profiles.

One documented consequence is that LinDistFlow overestimates capability at the point of common coupling. In flexibility aggregation, the lossless model omits the accumulated line-loss terms $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$3 and $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$4, so a flexibility region computed via LinDistFlow overestimates the feasible set at the PCC because, in reality, more power needs to be injected to cover losses (Jiang et al., 3 May 2025). In TSO-DSO flexibility domain identification, the main discrepancies appear in the negative orthant, because LinDistFlow does not consider losses and therefore overestimates the capability of energy that could be generated from the active distribution network (Ageeva et al., 2020).

A second failure mode is temporal error accumulation. Simulations on the KIT Campus Nord network with real demand and solar data show that line losses are generally underestimated by linear models, and line-loss errors tend to accumulate both at the point of common coupling and over extended time horizons (Jiang et al., 6 Aug 2025). In day-ahead scheduling, this can manifest as scheduled ESS state of charge drifting below its intended value because of accumulated undercounting of system losses (Jiang et al., 6 Aug 2025).

A third issue concerns robustness away from the linearization regime. This is one reason later work compares LinDistFlow with generalized linearizations around nonzero operating points. GLDF and FPL are both exact at the linearization point, but GLDF is reported to be more robust and to exhibit more consistent performance as operating points depart from that point (Huang et al., 2021).

Recent work also studies LinDistFlow under uncertain network parameters. Using matrix concentration inequalities, the expected operator norm error of the LinDistFlow or LCPF matrix under independent, bounded uncertainties scales as $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$5, and the probability of large deviation decays exponentially in $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$6 (Talkington et al., 20 Oct 2025). This suggests that the model’s approximation error can be bounded probabilistically under bounded and independent line uncertainty, rather than treated only as a deterministic modeling residual.

6. Compensation, calibration, and current directions

Several recent directions aim not to replace LinDistFlow, but to preserve its structure while correcting its dominant errors. One approach is explicit loss compensation. In flexibility aggregation for integrated transmission-distribution systems, a system loss compensation method first computes the LinDistFlow-based flexibility set and then corrects each candidate interface point by adding estimated active and reactive losses written as quadratic functions of the PCC exchange: $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$7 The compensated set

$$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$8

is reported to closely approximate the actual feasible region while preserving data privacy (Jiang et al., 3 May 2025).

Another approach is parameter optimization. Rather than fixing LinDistFlow coefficients at physical line parameters, the Optimized LinDistFlow formulation introduces trainable coefficient and bias parameters: $$\begin{aligned} p_{ij} &= p_{j} + \sum_{k:(j,k) \in \mathcal{L}}p_{jk}, \ q_{ij} &= q_{j}+\sum_{k:(j,k)\in \mathcal{L}}q_{jk}, \ w_i - w_j &= 2(r_{ij} p_{ij} + x_{ij} q_{ij}), \end{aligned}$$9 The parameters are trained offline with sensitivity information using the Truncated Newton Conjugate-Gradient method to minimize discrepancies relative to the nonlinear DistFlow model (Taheri et al., 2024). Reported improvements reach up to $w_i=v_i^2$0 in $w_i=v_i^2$1-norm loss and $w_i=v_i^2$2 in $w_i=v_i^2$3-norm loss relative to traditional LinDistFlow, and the optimized approximation is also assessed under topology changes and in hosting-capacity optimization (Taheri et al., 2024).

LinDistFlow has also been adapted to settings where the load model itself is voltage-dependent. For load-altering attacks with ZIP loads, the combination of LinDistFlow with a ZP approximation yields a matrix-based closed-form expression for squared voltages,

$w_i=v_i^2$4

which makes analytical attack-impact calculations possible even though the original ZIP model is nonlinear in bus voltage (Maleki et al., 2023).

Taken together, these developments indicate that LinDistFlow remains a reference model not because it is exact, but because its linear, squared-voltage structure is unusually adaptable. This suggests that the most active line of research is no longer the use of an unmodified lossless approximation alone, but the systematic refinement of LinDistFlow through generalized operating-point selection, loss compensation, uncertainty quantification, and parameter calibration (Huang et al., 2021, Taheri et al., 2024).