Three-Phase Infeasibility Analysis (TPIA)

• TPIA is an optimization-based framework that augments three-phase power flow with slack variables to diagnose infeasible conditions in unbalanced networks.
• It employs physically interpretable slack sources like current injections, reactive power, and susceptance adjustments to quantify and localize power deficits.
• Different formulations (L1, L2, exact-global) and voltage-bound constraints enable scalable, actionable diagnostics in both distribution and combined transmission–distribution systems.

Three-Phase Infeasibility Analysis (TPIA) is an optimization-based framework for diagnosing infeasible or non-convergent three-phase AC power-flow conditions in unbalanced distribution networks, and, in later generalizations, in combined transmission–distribution systems. In its canonical form, TPIA augments the network equations with physically interpretable slack variables—most commonly per-phase current injections—and minimizes their norm subject to the full three-phase network constraints. If the original network equations are feasible, the optimal slack variables are zero and the TPIA solution coincides with standard three-phase power flow; if they are nonzero, they quantify and localize the amount and location of “missing power,” thereby identifying weak or power-deficient areas of the grid (Foster et al., 2021, Foster et al., 2022).

1. Historical emergence and conceptual scope

The foundational formulation of TPIA in distribution-grid studies was introduced as an equivalent-circuit, current-injection optimization for unbalanced three-phase feeders. The core motivation was the increasing difficulty of planning and operating feeders with high penetrations of distributed energy resources and electric vehicles, under conditions where conventional three-phase power flow either diverges or yields no actionable explanation of the underlying deficiency. In this setting, TPIA adds slack current sources at non-slack nodes and minimizes their norm while enforcing Kirchhoff’s Current Law (KCL) in a three-phase current–voltage formulation (Foster et al., 2021).

A subsequent development made the framework explicitly “actionable” by adding operational bounds, especially voltage magnitude limits, and by expanding the admissible slack sources beyond current injections to reactive power and susceptance sources. In that formulation, nonzero slack variables both certify infeasibility and indicate where deployable assets such as capacitor banks, STATCOMs, or additional DERs would be effective. The term “actionable” is used specifically to mean that the returned operating point respects utility voltage limits and uses slack-source models that correspond to realizable equipment with realistic ratings (Foster et al., 2022).

The concept has since broadened. One line of work extends infeasibility analysis from standalone feeders to combined positive-sequence transmission and three-phase distribution networks, using a unified equivalent-circuit model and distributed primal–dual interior-point optimization to localize weak nodes across the transmission–distribution interface (Ali et al., 2024). Another line reformulates the original non-convex TPIA into an exact non-convex bilinear program and applies spatial branch-and-bound to drive the optimality gap to below $10^{-4}$ on feeders with more than 5k nodes, addressing the risk that local minima or saddle points can distort upgrade recommendations (Panthee et al., 21 Aug 2025).

Several adjacent methodologies are closely related but not identical. The phase-switching device study in low-voltage distribution networks does not explicitly use the term TPIA, but its three-phase workflow—detect infeasibility in nonlinear three-phase unbalanced power flow, solve a corrective optimization in a linearized model, and validate restored feasibility in the original nonlinear model—maps naturally to a three-phase infeasibility analysis procedure (Liu et al., 2019). Likewise, the convex-inner-approximation methodology for hosting capacity provides a feasibility-assessment and mitigation workflow for unbalanced three-phase feeders; its relation to TPIA is best understood as a feasibility-screening variant rather than a slack-injection formulation in the narrow sense (Mavalizadeh et al., 2023). This suggests that, in current usage, TPIA denotes both a specific equivalent-circuit optimization family and a broader class of three-phase infeasibility diagnostics.

2. Mathematical formulation in the equivalent-circuit framework

The standard TPIA formulation models the feeder as a general unbalanced three-phase AC system with phases indexed by $\phi \in \{A,B,C\}$ or $\phi \in \{a,b,c\}$ and bus variables expressed in rectangular coordinates. The current–voltage formulation uses the block admittance matrix $Y_{abc} = G + jB$, assembled from equivalent-circuit models for lines, transformers, regulators, shunts, switches, fuses, triplex loads, inverters, and related devices. KCL is enforced at every bus-phase rather than using bus power-balance equations (Foster et al., 2022).

In the actionable voltage-bounded formulation, the nodal equations are

$f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$

$f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$

with nonlinear current injections

$$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$

The corresponding optimization introduces slack variables $s$ and solves

$$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$

where the inequalities enforce operational realism, most notably voltage magnitude bounds (Foster et al., 2022).

The voltage bounds are written as

$$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$

$\phi \in \{A,B,C\}$0

In the experiments of the actionable formulation, the limits are set to $\phi \in \{A,B,C\}$1 p.u. $\phi \in \{A,B,C\}$2 p.u. (Foster et al., 2022). Earlier TPIA formulations did not enforce such operational bounds, focusing instead on feasibility restoration and localization under KCL constraints (Foster et al., 2021).

The original formulation of TPIA can also be written compactly as

$\phi \in \{A,B,C\}$3

where $\phi \in \{A,B,C\}$4 is the vector of per-phase slack current injections. The L2 variant minimizes

$\phi \in \{A,B,C\}$5

while the L1 variant minimizes the absolute values of the slack-current components using a split-variable construction. The L1 form is used specifically because its inherent sparsity sharpens localization of weak nodes and phases (Foster et al., 2021).

A different mathematical route appears in the exact-global-optimization work, which reformulates the original non-convex nonlinear program as an exact non-convex bilinear program. There, the reciprocal load relations are lifted through

$\phi \in \{A,B,C\}$6

with exact bilinear equalities

$\phi \in \{A,B,C\}$7

thereby exposing the non-convexity in a form amenable to McCormick relaxations and spatial branch-and-bound (Panthee et al., 21 Aug 2025).

3. Slack-source models, localization, and physical interpretation

The central diagnostic mechanism of TPIA is the introduction of slack sources whose optimal nonzero values reveal where the AC constraints cannot be satisfied without corrective action. In the earliest three-phase formulation, these are per-phase slack current sources. In later actionable variants, three source types are used: current slack sources, reactive power slack sources, and susceptance slack sources (Foster et al., 2021, Foster et al., 2022).

Formulation	Slack variable	Physical mapping
TPIA-I	Per bus-phase real and imaginary current injections $\phi \in \{A,B,C\}$8, $\phi \in \{A,B,C\}$9	Generic power injections or withdrawals, mobile generators, batteries, flexible loads, aggregated DERs
TPIA-Q	Per bus-phase reactive power injections $\phi \in \{a,b,c\}$0	Inverter-based VAR support, D-STATCOMs, synchronous condensers
TPIA-B	Per bus-phase shunt susceptances $\phi \in \{a,b,c\}$1	Fixed or switched capacitor banks, D-STATCOMs in admittance form

For TPIA-I, the slack variables enter KCL directly:

$\phi \in \{a,b,c\}$2

and the actionable objective uses the squared complex power contributed by the slack current at the prevailing voltage,

$\phi \in \{a,b,c\}$3

This objective is emphasized as “voltage-invariant” in the sense that it avoids preferring injections at lower-voltage buses merely because a fixed current yields less power there (Foster et al., 2022).

For TPIA-Q, the slack variable is reactive power, with objective

$\phi \in \{a,b,c\}$4

and the slack enters through the same nonlinear current expressions used for PQ injections, but with reactive support supplied by $\phi \in \{a,b,c\}$5. For TPIA-B, the decision variable is shunt susceptance, entering linearly in voltage as

$\phi \in \{a,b,c\}$6

with objective

$\phi \in \{a,b,c\}$7

Its reactive-power interpretation is

$\phi \in \{a,b,c\}$8

Under the sum-of-squares complex-power objective, TPIA-Q and TPIA-B become mathematically equivalent for optimizing reactive compensation, but they support different planning workflows because one optimizes reactive injections directly and the other optimizes shunt susceptance directly (Foster et al., 2022).

Localization is obtained from the spatial pattern of nonzero slack variables. Infeasible networks yield nonzero slack only where additional power is needed to satisfy KCL and the operational bounds; clusters of nonzero slack near the ends of radial feeders identify weak sections with insufficient voltage support or supply capacity. The actionable paper reports $\phi \in \{a,b,c\}$9 and $Y_{abc} = G + jB$0 as sizing metrics, while per bus-phase slack values guide exact siting (Foster et al., 2022). In the original formulation, the local complex power deficit is computed from the slack current through

$Y_{abc} = G + jB$1

with

$Y_{abc} = G + jB$2

providing a direct translation from infeasibility current to remedial power injection (Foster et al., 2021).

The choice of norm changes the diagnostic character of the solution. L2 slack minimization is smoother and computationally easier, but it tends to distribute corrections across many nodes. L1 minimization is sparsity-promoting and therefore sharper for weak-location identification. In the large infeasible combined transmission–distribution case study, the same distinction appears explicitly: with $Y_{abc} = G + jB$3, $Y_{abc} = G + jB$4 spread across approximately $Y_{abc} = G + jB$5 of nodes, whereas with $Y_{abc} = G + jB$6, the nonzeros occurred at $Y_{abc} = G + jB$7 nodes, enabling targeted interventions (Ali et al., 2024). This suggests that sparsity is not merely a numerical preference but a substantive interpretive choice in TPIA.

4. Algorithms and computational structure

The original TPIA problems are non-convex because the current–voltage relations for constant-power devices are nonlinear and because phase couplings in the three-phase admittance matrix introduce additional structure. In the earliest formulation, Newton’s method is used within an equivalent-circuit framework, with the nonlinear elements linearized by Taylor approximation and sparse linear solves dominating runtime. The L2 solver assembles a KKT system from stationarity and primal feasibility, while the L1 solver uses split nonnegative variables and a primal–dual interior-point method with perturbed complementarity and a “diode limiting” heuristic to maintain feasibility of inequality constraints (Foster et al., 2021).

The actionable voltage-bounded framework adopts a primal–dual interior-point (PDIP) method that solves perturbed KKT conditions. Circuit-simulation heuristics are used for robustness, including a “diode limiting” technique that damps Newton updates for primal and dual variables associated with inequality constraints so that iterates do not step into infeasible regions. Additional equivalent-circuit heuristics from prior work are reported to help avoid saddle points and improve scalability (Foster et al., 2022).

The distributed combined transmission–distribution extension preserves the PDIP structure but embeds it within a Gauss–Jacobi–Newton decomposition. The unified equivalent-circuit model yields a bordered block-diagonal KKT system, and each transmission or distribution subproblem solves its internal PDIP Newton system in parallel while boundary primal and dual variables are exchanged at the points of interconnection using symmetrical-component mappings. The approach is specifically motivated by the need to analyze infeasible combined networks with $Y_{abc} = G + jB$8k+ transmission and $Y_{abc} = G + jB$9k+ distribution nodes (Ali et al., 2024).

A different algorithmic agenda appears in the exact-global TPIA work. There, the non-convex nonlinear program is reformulated exactly as a non-convex bilinear program, and Gurobi’s spatial branch-and-bound is applied with McCormick envelopes to generate global lower bounds and feasible incumbents. To strengthen the relaxations, the paper introduces bound tightening with variable filtering and decomposition, followed by sequential bound tightening (SBT). The reported result is that SBT reduces the runtime of spatial branch-and-bound by up to $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$0 and the number of explored nodes by up to $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$1, while achieving optimality gaps below $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$2 on all but one large instance under the time limit (Panthee et al., 21 Aug 2025).

A separate feasibility-assessment route is provided by the vectorized bus-injection SDP formulation with exact-penalty dual reformulation and a three-cut proximal bundle method. That method targets feasibility assessment in unbalanced three-phase distribution networks through a convex dual problem in which the PSD constraint is absorbed into the objective via a spectral exact penalty. The implementation is reported as numerically over $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$3 times faster than MOSEK with less than $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$4 of its memory on the monolithic BIM-SDP, and approximately $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$5 times faster with $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$6 less memory on the decomposed BIM-SDP (Fang et al., 25 May 2026). Although this is not a slack-current TPIA in the narrowest sense, it functions as an infeasibility-detection and localization mechanism for three-phase networks.

5. Empirical results and benchmark evidence

The original three-phase TPIA study evaluated realistic unbalanced test cases up to approximately $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$7 nodes. On the baseline feeder R4-12.47-1, standard three-phase power flow converged and both TPIA formulations returned zero slack currents, reproducing the same solution. On overloaded cases such as R1-12.47-3_OV, R2-25.00-1_OV, R3-12.47-3_OV, and the $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$8node_OV case, standard power flow failed to converge, while both L2 and L1 TPIA converged with nonzero slack currents. The localization contrast was pronounced: for R1-12.47-3_OV, L2 produced nonzero slack at $f_r(x) = G V_R - B V_I + I^{NL}_R(V_R, V_I) = 0,$9 of $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$0 nodes, whereas L1 localized the issue to $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$1 nodes; for R2-25.00-1_OV, L1 identified $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$2 node; for R3-12.47-3_OV, $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$3 nodes; and for $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$4node_OV, $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$5 nodes (Foster et al., 2021).

The actionable TPIA study reported results on six PNNL taxonomy feeders ranging from $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$6 to $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$7 nodes. Two were deliberately overloaded so that standard power flow did not converge: R1-12.47-3_OV with $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$8 nodes and R2-25.00-1_OV with $f_i(x) = B V_R + G V_I + I^{NL}_I(V_R, V_I) = 0,$9 nodes. Voltage-bounded TPIA-I converged on both, enforcing the $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$0–$$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$1 p.u. limits, with $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$2 equal to $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$3 kW/$$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$4 kVAR and $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$5 kW/$$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$6 kVAR, respectively; the reported iteration counts were $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$7 and $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$8, and the runtimes were $$I^{NL}_R(V_R, V_I) = \frac{P \odot V_R + Q \odot V_I}{V_R^2 + V_I^2}, \qquad I^{NL}_I(V_R, V_I) = \frac{P \odot V_I - Q \odot V_R}{V_R^2 + V_I^2}.$$9 s and $s$0 s (Foster et al., 2022).

On feeders where standard power flow converged but violated voltage limits, the voltage-bounded TPIA formulation removed all violations. For example, on R3-12.47-3 with $s$1 nodes, standard power flow had $s$2 buses below $s$3 p.u., whereas voltage-bounded TPIA-I reduced the number of violations to $s$4 by injecting slack corresponding to $s$5 kW and $s$6 kVAR, in $s$7 iterations and $s$8 s. Across the taxonomy feeders, voltage-bounded TPIA converged in $s$9–$$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$0 iterations with runtimes from $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$1 s to approximately $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$2 s, and the study states that scaling was approximately linear with system size (Foster et al., 2022).

The reactive-compensation study using TPIA-B on R1-12.47-3_OV restricted slack to nodes at or above $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$3 V, reflecting practical siting on medium-voltage buses. The optimizer found that about $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$4 MVAR of reactive power was required to enforce the $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$5–$$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$6 p.u. voltage bounds. After capacitors were added according to the susceptance prescription, standard three-phase power flow converged within $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$7 iterations and all bus voltages lay within $$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$8–$$\begin{aligned} \min_{x,s}\quad & \sum_{m \in \mathcal{N}} \sum_{\phi \in \{A,B,C\}} \alpha \, g_m^{\phi}(s) \ \text{s.t.}\quad & f_r(x) - h_r(s) = 0, \ & f_i(x) - h_i(s) = 0, \ & \overline{l}_i^{\phi}(x) \le 0,\quad \underline{l}_i^{\phi}(x) \ge 0, \end{aligned}$$9 p.u. Across all feeders, restricting compensation to reactive sources resulted in zero real-power slack and runtime increases of at most $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$0 relative to TPIA-I; on R3-12.47-3, the reactive-only formulation needed approximately $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$1 kVAR in $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$2 iterations and $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$3 s (Foster et al., 2022).

In the EV scenario of the original TPIA paper, both GridLAB-D and SUGAR-D failed on R4-12.47-1_EV. L1 TPIA identified a single power-deficient node; the corresponding per-phase deficit $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$4 was used to size a battery, after which standard power flow and both TPIA formulations returned zero slack currents (Foster et al., 2021). A plausible implication is that TPIA can serve not only as a diagnostic after failure but also as a direct siting-and-sizing aid for localized corrective assets.

6. Related three-phase infeasibility methodologies

One related methodology addresses infeasible low-voltage distribution operation through phase-switching devices. In that framework, infeasibility is defined operationally as failure of the traditional iteration-based nonlinear three-phase unbalanced power-flow solver to converge for a given snapshot with fixed customer phase assignments. The corrective step is a mixed-integer linearized three-phase unbalanced power-flow model with binary phase-selection variables for flexible customers, exact mixed-integer linear reformulations of binary–continuous products, and a least-squares linearization of $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$5 over prescribed voltage-magnitude and voltage-angle windows. The algorithm proceeds in three phases: infeasibility detection in nonlinear TUPF, corrective MILP with PSDs, and validation in nonlinear TUPF. In the Australian $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$6-customer case with $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$7 PSDs, the stressed scenario became feasible after rephasing customers $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$8 $$\overline{l}_i^{\phi}(x) = V_{R,i,\phi}^2 + V_{I,i,\phi}^2 - (\overline{V}_{i,\phi})^2 \le 0,$$9, $\phi \in \{A,B,C\}$00 $\phi \in \{A,B,C\}$01, $\phi \in \{A,B,C\}$02 $\phi \in \{A,B,C\}$03, and $\phi \in \{A,B,C\}$04 $\phi \in \{A,B,C\}$05 (Liu et al., 2019). The procedure is not slack-injection TPIA, but it is explicitly described as naturally mapping to a three-phase infeasibility analysis procedure.

Another related methodology uses convex inner approximations to compute nodal positive and negative DER injection bounds that guarantee voltage and current feasibility in radial three-phase feeders. The per-phase decomposition solves a convex program on each phase separately, then validates the resulting injections in the full three-phase model. Under Assumption 1, $\phi \in \{A,B,C\}$06, and Assumption 2, transposed three-phase lines with identical mutual impedances, the paper proves that modified per-phase impedances $\phi \in \{A,B,C\}$07 guarantee that per-phase feasibility implies three-phase feasibility. When those assumptions do not hold, the paper introduces “Mod-Z” and an iterative relaxation of per-phase voltage bounds (Mavalizadeh et al., 2023).

The empirical hosting-capacity results show how this inner-approximation route functions as a three-phase feasibility diagnostic. On the IEEE 37-node feeder, baseline per-phase CIA produced $\phi \in \{A,B,C\}$08 MW and $\phi \in \{A,B,C\}$09 MW with no three-phase voltage violations. Iterative bounds increased these to $\phi \in \{A,B,C\}$10 MW and $\phi \in \{A,B,C\}$11 MW, again with no violations at termination. On the 534-node feeder, iterative bounds increased hosting capacity from $\phi \in \{A,B,C\}$12 MW and $\phi \in \{A,B,C\}$13 MW to $\phi \in \{A,B,C\}$14 MW and $\phi \in \{A,B,C\}$15 MW (Mavalizadeh et al., 2023). This suggests that three-phase infeasibility analysis, broadly construed, includes not only localization after failure but also conservative feasible-region construction for planning.

The combined transmission–distribution PDIP framework also extends the domain of TPIA beyond standalone feeders. It embeds current, power, or impedance infeasibility sources at positive-sequence transmission buses and three-phase distribution nodes, minimizes their norm, and uses the surviving nonzero sources to identify weak nodes and areas. In its case studies, D-PDIP converged in $\phi \in \{A,B,C\}$16 iterations and $\phi \in \{A,B,C\}$17 s on PEGASE2869 plus GC-12.47.1, where ADMM failed to converge within $\phi \in \{A,B,C\}$18 s, and solved the largest network—ACTIVsg70k transmission plus $\phi \in \{A,B,C\}$19 three-phase nodes—within $\phi \in \{A,B,C\}$20 minutes (Ali et al., 2024). In that formulation, TPIA becomes a cross-domain localization tool rather than a feeder-only diagnostic.

7. Limitations, assumptions, and open technical issues

The major limitation emphasized across the nonlinear TPIA literature is non-convexity. Both the original L2/L1 formulations and the actionable voltage-bounded formulations inherit non-convexity from the nonlinear current-injection relations and the phase-coupled admittance structure. As a result, solutions are generally local optima, and initialization and circuit heuristics materially affect robustness (Foster et al., 2021, Foster et al., 2022). The exact-global reformulation is motivated precisely by the concern that local minima and saddle points can produce suboptimal upgrade recommendations (Panthee et al., 21 Aug 2025).

Modeling assumptions also matter. The actionable study uses constant PQ loads in the reported experiments, while noting that the framework supports other models; it further states that model uncertainty and measurement errors can affect localization accuracy (Foster et al., 2022). The original three-phase TPIA equations omit additional PV-bus constraints for brevity, though the underlying equivalent-circuit solver supports PV buses (Foster et al., 2021). In the combined transmission–distribution extension, the transmission system is modeled in positive sequence and the distribution system in full three-phase form, so the approach inherits the balanced-transmission assumption (Ali et al., 2024).

Another technical issue concerns what counts as “actionable.” Earlier TPIA and related equivalent-circuit infeasibility analyses used slack current sources only and did not enforce operational bounds, which could yield non-actionable voltage excursions. The actionable formulation addresses this by imposing voltage magnitude limits and by modeling slack types that correspond directly to realizable assets (Foster et al., 2022). A related misconception is that any infeasibility current pattern directly prescribes equipment placement. The papers are more precise: slack patterns expose deficient areas and quantify missing power, but asset mapping depends on the slack-source model, deployment constraints, and any bounds imposed on candidate locations.

There is also a methodological trade-off between interpretability and computational convenience. L1 objectives sharpen localization but can be slower and more sensitive to initialization; L2 objectives are smoother and often faster but may spread injections too broadly for crisp diagnosis (Foster et al., 2021). In the exact-global study, the worst-case complexity of spatial branch-and-bound remains high, and its tractability depends strongly on tight variable bounds and good warm starts (Panthee et al., 21 Aug 2025). In the SDP-based feasibility route, rank-one recovery is exact on radial networks under conditions cited in prior work, but not guaranteed in meshed networks (Fang et al., 25 May 2026).

The open directions listed in the literature are correspondingly practical. They include explicit thermal limits on branches and transformers, time-varying or location-specific voltage limits, discrete control variables such as switched capacitor steps, mixed-integer enhancements, hybrid objectives that combine sparsity and net-complex-power minimization, and integration with corrective optimization for reactive planning, DER curtailment, or topology control (Foster et al., 2022, Ali et al., 2024, Panthee et al., 21 Aug 2025). Across these variants, the enduring role of TPIA is diagnostic: it converts solver divergence or violated operating envelopes into structured information about where the network is deficient, by how much, and under what modeling assumptions.