DistFlow Equations

DistFlow models power flow as a set of recursive equations that propagate from the substation outward to feeder endpoints. Unlike the full AC power flow, DistFlow leverages the radial topology to calculate branch flows and bus voltages sequentially.

• Real power flow: P_{k+1} = P_k - r_k · (P_k² + Q_k²)/V_k² - p_{k+1}
• Reactive power flow: Q_{k+1} = Q_k - x_k · (P_k² + Q_k²)/V_k² - q_{k+1}
• Voltage magnitude: V_{k+1}² = V_k² - 2(r_k·P_k + x_k·Q_k) + (r_k² + x_k²)(P_k² + Q_k²)/V_k²

This formulation eliminates the need for iterative Jacobian matrix inversions, making it computationally efficient for real-time applications.

For many optimization problems, the full DistFlow equations are linearized by neglecting the quadratic loss terms. This simplification, known as LinDistFlow, assumes losses are small relative to line flows.

• Linearized voltage drop: V_{k+1} ≈ V_k - (r_k·P_k + x_k·Q_k)/V_0
• Lossless branch flow: P_{k+1} = P_k - p_{k+1}, Q_{k+1} = Q_k - q_{k+1}

This approximation transforms the power flow into a linear set of equations, enabling formulation as a convex optimization problem suitable for mixed-integer linear programming solvers. The error introduced is typically less than 1% for well-designed distribution feeders.

The DistFlow equations are solved computationally using the Backward/Forward Sweep (BFS) method, which exploits the radial structure for guaranteed convergence.

• Backward sweep: Starting from the terminal nodes, calculate branch currents by summing downstream loads and losses, propagating toward the substation
• Forward sweep: Starting from the substation with a known voltage, calculate voltage drops along each branch using the currents computed in the backward pass
• Iteration: Repeat until voltage magnitudes converge within a specified tolerance

BFS converges in 3-5 iterations for most distribution networks, compared to 10-20 iterations for Newton-Raphson on the same system.

When embedded in optimization problems like network reconfiguration or Volt-VAR control, the DistFlow equations are relaxed using Second-Order Cone Programming (SOCP) to guarantee global optimality.

• Define auxiliary variables: l_k = (P_k² + Q_k²)/V_k² (squared current), v_k = V_k² (squared voltage)
• The non-convex equality l_k = (P_k² + Q_k²)/v_k is relaxed to the convex inequality: l_k ≥ (P_k² + Q_k²)/v_k
• This SOCP relaxation is exact for radial networks under mild conditions, meaning the relaxed solution satisfies the original non-convex constraints

This technique enables provably optimal solutions for problems that would otherwise be NP-hard.

DistFlow equations serve as the measurement model in distribution system state estimation (DSSE), relating sensor measurements to unknown bus voltages.

• Pseudo-measurements: Historical load profiles are treated as virtual measurements at unmonitored nodes, weighted by their variance
• Weighted Least Squares (WLS): The estimator minimizes the sum of squared residuals between measured and calculated values using DistFlow constraints
• Observability analysis: DistFlow enables assessment of whether the available measurement set is sufficient to uniquely determine all bus voltages

The recursive structure allows hierarchical state estimation, where feeder segments can be solved independently before aggregating results.

Standard DistFlow assumes balanced three-phase operation, but distribution systems are inherently unbalanced due to single-phase loads and asymmetric line configurations. The three-phase extension models each phase independently with mutual coupling.

• Phase impedance matrix: Z_{abc} ∈ ℂ^{3×3} captures self and mutual impedances between phases
• Coupled power flow: P_{a,k+1} depends on currents in all three phases through the full impedance matrix
• Neutral and ground modeling: The return path is explicitly modeled using Carson's equations for earth return impedance

This extension is critical for accurate voltage estimation in networks with high penetrations of single-phase rooftop solar.

The iterative load flow algorithm that directly implements the DistFlow logic. It calculates branch currents from the load end backward toward the substation, then updates voltage magnitudes from the source forward to the endpoints.

• Converges reliably in 3-5 iterations for radial networks
• Avoids the Jacobian matrix inversions required by Newton-Raphson
• Handles high R/X ratios that cause traditional methods to diverge
• Forms the computational core of most distribution automation platforms

A fundamental operational rule requiring the distribution network to maintain a tree structure with no closed loops. DistFlow equations are only valid under this topology.

• Enables simple protection coordination with unidirectional fault currents
• Each load node has exactly one parent source node
• Reconfiguration algorithms must verify radiality after every switching operation
• Violations create circulating currents that invalidate the DistFlow voltage drop assumptions

Computational logic that uses DistFlow equations as its power flow engine to evaluate candidate topologies. The algorithm searches for switch combinations that minimize losses while respecting voltage and thermal constraints.

• Branch exchange method: iteratively closes a tie switch and opens a sectionalizing switch
• MILP formulations: encode DistFlow as linear constraints with binary switch variables
• Heuristic approaches: use genetic algorithms or particle swarm optimization guided by DistFlow evaluations
• Real-time implementations re-run DistFlow every few seconds as load conditions change

An efficiency technique that lowers service voltage to the lower ANSI C84.1 bound (typically 114V on a 120V base). DistFlow equations provide the voltage profile visibility needed to implement CVR without violating minimum service levels.

• DistFlow calculates the voltage at every node under reduced setpoints
• Typical energy savings of 2-4% without customer impact
• Requires accurate DistFlow models to identify the limiting node on each feeder
• Combined with Volt-VAR optimization for coordinated reactive power control

The algorithmic process of inferring voltage magnitudes and phase angles across the entire feeder using limited real-time sensor data. DistFlow equations serve as the measurement model that relates sensor readings to system states.

• Weighted least squares estimators use DistFlow as the process model
• Pseudo-measurements from historical load profiles fill gaps where sensors are absent
• Enables bad data detection when measurements violate DistFlow relationships
• Provides the situational awareness layer for advanced distribution automation

A power electronic device replacing normally open tie switches with back-to-back converters. Unlike mechanical switches modeled by binary DistFlow variables, SOPs enable continuous active and reactive power flow control between feeders.

• DistFlow equations are extended with converter power injection terms
• Provides voltage support and loss minimization without topology changes
• Enables controlled loop operation while maintaining radial protection schemes
• Typically rated 1-5 MVA for medium-voltage distribution applications