Less-Conservative Sufficient Conditions for Exactness of the augmented relaxed optimal power flow

This work provides less-conservative sufficient conditions for the exactness of the augmented relaxed optimal power flow (AR-OPF) formulation.

Author

Wenyu Liu

Published

September 24, 2026

Problem Statement

Notations and Definitions

The network is radial. Index 0 is for the slack bus and its voltage is fixed (\(v_0\)). Without loss of generality, we can assume that only bus 1 is connected to the slack bus (otherwise the problem separates into several independent problems). Buses other than the slack bus are denoted with \(1,\dots, L\); \(\mathcal{L}\) denotes the set \(\mathcal{L} = \{1,2,…,L\}\) and \(\mathrm{up}(l)\) is the label of the bus that is upstream of bus \(l\). We also label with \(l\) the line whose downstream bus is bus \(l\); its upstream bus is, therefore, \(\mathrm{up}(l)\). For a set of lines \(\mathcal{M} = \{1,2,…,M\}\), we denote the lines for which \(\mathbf{H}_{l,m}= 1, \forall m \in \mathcal{M}\) upstream lines/buses, (\(\mathcal{L}^M\)) [\(\mathbf{H}\) is defined in (1)]. Finally, \(\mathcal{L}\) denotes the set of the buses that are the leaves of the graph that maps the grid topology.

Let \(S^t_l=P^t_l +jQ^t_l\) be the complex power flow entering line \(l\) from the top, i.e., from bus \(\mathrm{up}(l)\) ; \(S^c_l=P^t_l +jQ^c_l\) be the complex power flow entering the central element of line \(l\) (it is equal to Stl minus the reactive power associated with the shunt admittance connected to bus \(\mathrm{up}(l)\) , see Fig. 1); \(S^b_l=P^b_l +jQ^b_l\) be the complex power flow entering bus \(l\) from the bottom part of line \(l\); and \(f_l\) be the square of the current in the central element of line \(l\) (see Fig. 1). Let \(z_l=r_l +jx_l\) and \(2b_l\) be the longitudinal impedance and shunt capacitance of line \(l\). We denote with \(z^∗l\) the complex conjugate of \(z_l\). We assume that \(r_l\), \(x_l\), and \(b_l\) of the all lines are positive.

Let \(v_l\) be the square of voltage magnitude and \(s_l = p_l + jq_l\) be the power absorption at bus \(l\). \(p_l \ge 0\) and \(q_l \ge 0\) denote power consumptions, \(p_l \le 0\) and \(q_l \le 0\) denote powers injections. Let \(2B_l\) denote the sum of the susceptances of the lines connected to bus \(l\).

\(v^{\max}\) and \(v^{\min}\) are the square of maximum and minimum magnitude of nodal voltages. \(I_l^{\max}\) is the square of maximum current limit of line \(l\) (\(l \in \mathcal{L}\)). \(\Re(.)\) and \(\Im(.)\) denote the real and imaginary parts of complex numbers, and \(j := \sqrt{-1}\) is the imaginary unit; \(\max\{a, b\}\) returns the maximum of \(a\) and \(b\) and \(\min\{a, b\}\) returns the minimum of \(a\) and \(b\).

A notation without subscript, such as \(v\), denotes a column vector with \(L\) rows as \[ v = \begin{pmatrix} v_1 \\ \vdots \\ v_L \end{pmatrix}, S = \begin{pmatrix} S_1^t \\ \vdots \\ S_L^t \end{pmatrix}, P = \begin{pmatrix} P_1^t \\ \vdots \\ P_L^t \end{pmatrix}, I^{\max} = \begin{pmatrix} I_1^{\max} \\ \vdots \\ I_L^{\max} \end{pmatrix}, \text{etc.} \]

Note that for \(S\), \(P\), \(Q\) and their related auxiliary variables (\(\bar{S}\), \(\hat{S}, \ldots\)) the vectors \(S\), \(P\), \(Q\) represent the relevant values at the upper side of line \(l\) (\(S_l^t\)). The notation \(|P|\) represents the column vector with \(L\) rows whose \(l\text{th}\) element is \(|P_l|\). The comparison of vectors is entry-wise, i.e., \(P \le P'\) means \(P_l \le P_l'\) for every \(l \in \mathcal{L}\). The transposed of \(P\) is denoted with \(P^T\).

Matrices are shown with bold nonitalic capital letters such as \(\mathbf{A}\). We use the Euclidean (Frobenius) norm for vectors (\(v = \sqrt[2]{\sum_{k=1}^L (v_k^2)}\)) and also the Frobenius norm \(\mathbf{A}\) for matrices (\(\mathbf{A} = \sqrt[2]{\sum_{i=1}^L (\sum_{j=1}^L (a_{ij}^2))}\)). For two matrices \(\mathbf{A}\), \(\mathbf{B}\) of equal dimensions, the notation \(\mathbf{A} \circ \mathbf{B}\) denotes their Hadamard product, defined by \((\mathbf{A} \circ \mathbf{B})_{k,l} = \mathbf{A}_{k,l} \, \mathbf{B}_{k,l}\) for all \(k, l\).

For the reader’s convenience, the matrices defined in the paper are listed below.

\(\mathbf{I}\) is the \(L \times L\) identity matrix.
For a vector such as \(r\), \(\operatorname{diag}(r)\) denotes the diagonal matrix whose \(l\text{th}\) element is \(r_l\).
\(\mathbf{G}\) is the adjacency matrix of the oriented graph of the network, i.e., \(\mathbf{G}_{k,l}\) is defined for \(k, l \in \mathcal{L}\), and \(\mathbf{G}_{k,l} = 1\) if \(k = \operatorname{up}(l)\) and \(0\) otherwise. Diagonal elements are zero.
\(\mathbf{H}\) is the closure of \(\mathbf{G}\), i.e., \(\mathbf{H}_{k,l} = 1\) if bus \(k\) is on the path from the slack bus to bus \(l\) or \(k = l\), and \(\mathbf{H}_{k,l} = 0\) otherwise. Because the network is radial, \(\mathbf{G}^L = 0\) and \[ \mathbf{H} = \mathbf{I} + \mathbf{G} + \mathbf{G}^2 + \ldots + \mathbf{G}^{L-1} = (\mathbf{I} - \mathbf{G})^{-1}. \tag{1} \]
\(\mathbf{M} = 2\mathrm{diag}(x)\mathbf{H} + \mathrm{diag}(b)\).
\(\mathbf{C} = (I-G^\top-M)^{-1}\).
\(\mathbf{D}\) is the entry-wise non-negative matrix defined by \[ \mathbf{D} = \mathbf{C} \left[2\mathrm{diag}(r)((\mathbf{H}-\mathbf{I})\mathrm{diag}(r)) + 2\mathrm{diag}(x)((\mathbf{H}-\mathbf{I})\mathrm{diag}(x)) + \mathrm{diag}(|z|^2) \right] \]
\(\pi\), \(\varrho\), and \(\vartheta\) are the vectors defined by

\[ \pi_{l}=\frac{\max \left\{P_{l}^{\max },\left|\mathbf{H} p^{\min }\right|_{l}\right\}}{v_{l}^{\min }} \] \[ \varrho_{l} = \frac{\max \left\{Q_{l}^{\max }+b_{l} v^{\max },\left|\mathbf{H} q^{\min }-\mathbf{H} \text{diag}\left(b\right)\left(\mathbf{I}+\mathbf{G}^{\mathrm{T}}\right)\left(v^{\max }\right)\right|_{l}\right\}}{v_{l}^{\min }} \] \[ \vartheta_{l}=\left(\pi_{l}\right)^{2}+\left(\varrho_{l}\right)^{2} \] where \(p^{\min}\) and \(q^{\min}\) are the vectors of minimum absorptions level on the buses of the system. \(P_{l}^{\max}\) and \(Q_{l}^{\max}\) are the maximum allowed active and reactive power flows of line \(l\). The values of \(P_{l}^{\max}\) and \(Q_{l}^{\max}\) can be chosen so that these constraints do not affect the feasible solution space of A-OPF (by performing a load flow with maximum consumption/injection level of the system and obtain the maximum possible values).

\(\mathbf{E}\) and \(\mathbf{F}\) are the entry-wise non-negative matrices defined by \[ \begin{align} \mathbf{F} &= (\mathbf{H}\text{diag}\,(x) + \mathbf{H}\text{diag}\,(B)\,\mathbf{D}) \\ \mathbf{E} &= 2\text{diag}\,(\pi)\,\mathbf{H}\text{diag}\,(r) + 2\text{diag}\,(\varrho)\,\mathbf{F} + \text{diag}\,(\vartheta)\,\mathbf{D} \end{align} \]

Power Flow Equations with Transverse Parameters

In this section, we introduce the power-flow equations inferred from the transmission line two-port \(\Pi\) model written with the notation above. For the sake of clarity, the transmission line two-port \(\Pi\) model is shown in Fig.~1.

For a given radial power network, the power-flow equations are given as

\[ \begin{align} S_l^t &= s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \, S_m^t + z_l f_l - j \left( v_{\operatorname{up}(l)} + v_l \right) b_l \quad \forall\, l \in \mathcal{L} \tag{8} \\ v_l &= v_{\operatorname{up}(l)} - 2 \Re \left( z_l^* \left( S_l^t + j v_{\operatorname{up}(l)} b_l \right) \right) + |z_l|^2 f_l \quad \forall\, l \in \mathcal{L} \tag{9} \\ f_l &= \frac{\left| S_l^t + j v_{\operatorname{up}(l)} b_l \right|^2}{v_{\operatorname{up}(l)}} = \frac{\left| S_l^b - j v_l b_l \right|^2}{v_l} \quad \forall\, l \in \mathcal{L} \tag{10} \\ S_l^b &= s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \, S_m^t \quad \forall\, l \in \mathcal{L}. \tag{11} \end{align} \]

Equations~(8), (9), and (10) are directly derived by applying the Kirchhoff’s law to Fig.~1. Equation~(11) represents the complex power-flow of line \(l\) at its bottom side (see Fig.~1). \(S_l^b\) is a derived variable, which is introduced here to simplify the notation.

Note that (10) represents the square of current flow in the central part of the two-port \(\Pi\) model of the line. It is worth noting that the term \(f_l\) does not represent the square of the current that we can measure at the line terminals; it is indeed an internal state variable of the two-port \(\Pi\) model.

Original optimal power flow (O-OPF)

\[ \underset{s, S, v, f}{\operatorname{minimize}} \sum_{l \in \mathcal{L}} \left( \mathcal{C} \left( \Re\left(s_l\right), \Im\left(s_l\right) \right) \right) + \mathcal{C}^e\left(P_1^t\right) \tag{12} \]

Subject to:

\[ \begin{align} \text{Set of Equations } (8-11) \tag{13} \\ v_l \le v^{\max} &\quad \forall\, l \in \mathcal{L} \tag{14} \\ v^{\min} \le v_l &\quad \forall\, l \in \mathcal{L} \tag{15} \\ \left| S_l^b \right|^2 \le I_l^{\max} v_l &\quad \forall\, l \in \mathcal{L} \tag{16} \\ \left| S_l^t \right|^2 \le I_l^{\max} v_{\operatorname{up}(l)} &\quad \forall\, l \in \mathcal{L} \tag{17} \\ p_l^{\min} \le \Re\left(s_l\right) \le p_l^{\max} &\quad \forall\, l \in \mathcal{L} \tag{18} \\ q_l^{\min} \le \Im\left(s_l\right) \le q_l^{\max} &\quad \forall\, l \in \mathcal{L} \tag{19} \end{align} \]

where \(\mathcal{C}(.)\) is the cost function of nodal absorption (injection), \(\mathcal{C}^e(P_1^t)\) is the cost function related to energy import from the grid. Both \(\mathcal{C}(.)\) and \(\mathcal{C}^e(.)\) are assumed to be convex, and as mentioned above, \(\mathcal{C}^e(.)\) is strictly increasing. \(I_l^{\max}\) and \(v^{\max}/v^{\min}\) represent the square of the current limit of the lines and the maximum/minimum of square of nodal voltage-magnitudes.

In order to account for the voltage and current operational constraints, (14)–(19) are added to the optimization problem. It is worth noting that the lines’ ampacity limits must not be applied to \(f_l\) as it does not represent the exact value of the current at its terminals. Additionally, a line-ampacity limit has to be applied to both ends of the line. The constraints (18) and (19) represent the upper and lower limits of nodal absorption. Note that the power injection \(s_l\) of a bus (\(l \in \mathcal{L}\)) is normally constrained to be in a prespecified set \(\mathcal{S}_l\) that is usually convex.

Relaxed optimal power flow (R-OPF)

O-OPF is non-convex due to (10), it becomes convex if we replace (10) by the following inequality (20):

\[ {f_l} \geq \frac{{{{\left| {S_l^t + j{v_{{\text{up}}\left(l \right)}}{b_l}} \right|}^2}}}{{{v_{{\text{up}}\left(l\right)}}}},\ \forall l \in {\mathcal L} \tag{20} \]

Augmented relaxed optimal power flow (AR-OPF)

The main idea for modifying the OPF problem is to put the security constraints on a set of variables that 1) are upper bound for nodal voltage-magnitudes and current magnitude of the lines and 2) do not depend on \(f\). These are achieved by introducing an upper bound (\(\bar{f}\)) and a lower bound (a vector of zeroes) for \(f\). The upper and lower bounds of \(f\) are used to define the above-mentioned set of constraints. Note that the case of lower bound of \(f\) (a vector of zeroes) is known in the literature as the linear DistFlow formulation. In this respect, we first introduce the following sets of auxiliary variables \(\bar{f}\), \(\hat{S}\), \(\bar{S}\) for the lines of the grid and \(\bar{v}\) for the buses of the network, defined as \[ \begin{align} \hat{S}_l^t &= s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \hat{S}_m^t - j \left(\bar{v}_{\operatorname{up}(l)} + \bar{v}_l\right) b_l \quad \forall\, l \in \mathcal{L} \tag{21} \\ \bar{v}_l &= \bar{v}_{\operatorname{up}(l)} - 2\Re \left( z_l^* \left(\hat{S}_l^t + j \bar{v}_{\operatorname{up}(l)} b_l\right) \right) \quad \forall\, l \in \mathcal{L} \tag{22} \end{align} \]

\[ \begin{align} \bar{S}_l^t &= s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \bar{S}_m^t + z_l \bar{f}_l - j \left(v_{\operatorname{up}(l)} + v_l\right) b_l \quad \forall\, l \in \mathcal{L} \tag{23} \\ \bar{f}_l v_l &\ge \max \left\{ \left|\hat{P}_l^b\right|^2, \left|\bar{P}_l^b\right|^2 \right\} + \max \left\{ \left|\hat{Q}_l^b - \bar{v}_l b_l\right|^2, \left|\bar{Q}_l^b - v_l b_l\right|^2 \right\} \quad \forall\, l \in \mathcal{L} \tag{24} \\ \bar{f}_l v_{\operatorname{up}(l)} &\ge \max \left\{ \left|\hat{P}_l^t\right|^2, \left|\bar{P}_l^t\right|^2 \right\} + \max \left\{ \left|\hat{Q}_l^t + \bar{v}_{\operatorname{up}(l)} b_l\right|^2, \left|\bar{Q}_l^t + v_{\operatorname{up}(l)} b_l\right|^2 \right\} \quad \forall\, l \in \mathcal{L} \tag{25} \\ & \bar{S}_l^b = s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \bar{S}_m^t \quad \forall\, l \in \mathcal{L} \tag{26} \\ & \hat{S}_l^b = s_l + \sum_{m \in \mathcal{L}} \mathbf{G}_{l,m} \hat{S}_m^t \quad \forall\, l \in \mathcal{L}. \tag{27} \end{align} \]

Augmented relaxed optimal power flow (AR-OPF) :

\[ \underset{s, S, v, f, \hat{S}, \hat{v}, \bar{S}, \bar{f}}{\operatorname{minimize}} \sum_{l \in \mathcal{L}} \left( \mathcal{C} \left( \Re\left(s_l\right), \Im\left(s_l\right) \right) \right) + \mathcal{C}^e\left(P_1^t\right) \tag{28} \]

\[ \begin{align} &\text{Set of Equations } (8, 9, 11), (18-19), (20), (21-27) \tag{29} \\ &{v^{{\min}}} \leq {v_l}\ \ \forall \ l \in \ {\mathcal L} \tag{30}\\ &{\bar{v}_l} \leq {v^{{\max}}}\ \ \ \forall \ l \in \ {\mathcal L} \tag{31}\\ &{\left| {\left| {\max \left\{ {\left| {\hat{P}_l^b} \right|,\left| {\bar{P}_l^b} \right|} \right\}} \right| + j{\max}\left\{ {\left| {\hat{Q}_l^b} \right|,\left| {\bar{Q}_l^b} \right|} \right\}} \right|^2} \leq {v_l}I_l^{{\max}}\ \forall l \in {\mathcal L} \tag{32}\\ &\left| \max \left\{ {\left| {\hat{P}_l^t} \right|,\left| {\bar{P}_l^t} \right|} \right\} + j({\max}\left\{ {\left| {\hat{Q}_l^t} \right|,\left| {\bar{Q}_l^t} \right|} \right\} \right|^2 \leq {v_{{\text{up}}\left(l \right)}}I_l^{{\max}}\ \forall l \in {\mathcal L} \tag{33}\\ &P_l^t \leq \bar{P}_l^t,\ Q_l^t \leq \bar{Q}_l^t\quad \forall l \in {\mathcal L}. \tag{34} \end{align} \]

Note

We need to prove that the optimal solution of AR-OPF is also a solution of O-OPF, i.e., the relaxation is exact. In other words, we need to prove that at the optimal solution of AR-OPF, (20) holds with equality for all \(l \in \mathcal{L}\).

Exactness of AR-OPF

In this section, we provide conditions under which the relaxation (20) in AR-OPF is guaranteed to be exact. They can easily be verified ex ante from the static parameters of the grid.

Statement of the Conditions

The five conditions are as follows:

Condition C1: \[ \|\mathbf{H}^T\mathbf{M}\| < 1. \]
Condition C2: \[ \|\mathbf{E}\| < 1. \]
Condition C3: There exists an \(\eta_1 < 0.5\) such that \[ \mathbf{D}\mathbf{E} \le \eta_1 \mathbf{D}. \]
Condition C4: There exists an \(\eta_2 < 0.5\) such that \[ \left(\mathbf{H} \text{diag}\,(r)\,\mathbf{E}\right) \circ \mathbf{H} \le \eta_2 \mathbf{H} \text{diag}\,(r). \]
Condition C5: There exists an \(\eta_3 < 0.5\) such that \[ \left(\mathbf{H} \text{diag}\,(r)\,\mathbf{E}\mathbf{E}\right) \le \eta_3 \mathbf{H} \text{diag}\,(r)\,\mathbf{E}. \]

Concerning the interpretation of the above conditions, C1 implies that \(\mathbf{I} - \mathbf{G}^T - \mathbf{M}\) is invertible and has non-negative entries. C2 ensures the convergence of the proposed iterative power-flow solution process. Condition C3 implies that the voltage-magnitude of all buses increases when one or more than one entry of \(f\) decreases. Finally, C3 and C4 ensure that if \(f\) (specifically the losses on line \(l\)) decreases, the direct active power flow of all the lines upstream of line \(l\) decreases.

Exactness

Theorem I:

Under conditions C1–C3: For every feasible solution \((s, S, v, f, \hat{S}, \bar{v}, \bar{f}, \bar{S})\) of AR-OPF there exists a feasible solution \((s, S^*, v^*, f^*, \hat{S}, \bar{v}, \bar{f}^*, \bar{S}^*)\) of A-OPF and also O-OPF with the same power-injection vector \(s\).
Under conditions C1–C5: Every optimal solution \((s, S, v, f, \hat{S}, \bar{v}, \bar{f}, \bar{S})\) of AR-OPF satisfies (8.c) and is, thus, an optimal solution of A-OPF.

Part 1 of Theorem I implies that the vector of absorptions (\(s\)) of any feasible solution of the proposed OPF formulation belongs to a region where the upper and lower limits of nodal voltage-magnitudes and the lines’ ampacity limits are satisfied. This is where we use C2–C3 (C1 is related to the existence of Matrix C). Part 2 of Theorem I is the exactness of the relaxation. Here we use C4 and C5.

The main idea of the proof of Theorem I is as follows. If \((s, S, v, f, \hat{S}, \bar{v}, \bar{f}, \bar{S})\) is feasible for AR-OPF, then \((S, v, f)\) is in general not a load-flow solution for the power injections \(s\) [as (20) replaces (10)] but it is always possible to replace \((S, v, f)\) by \((S^*, v^*, f^*)\) obtained by performing a load-flow on \(s\). The technical difficulties are to show existence of such a load-flow solution and to find the good one (as there are multiple solutions), specifically the one that satisfies the voltage and ampacity constraints. This ``good’’ load-flow solution is obtained using an ad hoc iterative scheme shown in algorithm I. Furthermore, we show that an optimal solution of AR-OPF is also a load-flow solution. This is where conditions C4 to C5 are required. The Theorem I is proved using Lemma III introduced in the following.

Lemma III: Under conditions C1–C5, let \(\eta = \max(\eta_1, \eta_2, \eta_3) < 0.5\). For \(n \ge 1\) \[ \begin{align} \left|\Delta f^{(n)}\right| &\le \mathbf{E}^{n-1}\left|\Delta f^{(1)}\right| \tag{35} \\ \left|\Delta v^{(n)}\right| &\le \eta^{n-1}\left|\Delta v^{(1)}\right| \tag{36} \\ v &\le v^{(n)} \tag{37} \\ \left|\Delta P_l^{t(n)}\right| &\le \eta^{n-1}\left|\Delta P_l^{t(1)}\right| \quad \forall\, l \in \mathcal{L}^M \tag{38} \\ P_l^{t(n)} &\le \bar{P}_l^t \quad \forall\, l \in \mathcal{L}^M \tag{39} \\ f_l^{(n)} &\le \bar{f}_l \quad \forall\, l \in \mathcal{L} \tag{40} \\ P_l^{t(n)} &\le \bar{P}_l^t \quad \forall\, l \in \mathcal{L} \tag{41} \\ Q_l^{c(n)} &\le \bar{Q}_l^c + b_l v_{up(l)} \quad \forall\, l \in \mathcal{L} \tag{42} \end{align} \] where \(\Delta f^{(n)} = f^{(n)} - f^{(n-1)}\) for \(n \ge 1\) and similarly with \(P\), \(Q^c\), and \(v\).

The proof of Lemma III is provided in Appendix V.

Proof of Theorem I

Item 1: Let \(\omega = (s, P+jQ, v, f, \hat{P}+j\hat{Q}, \bar{v}, \bar{f}, \bar{S})\) be a feasible solution of AR-OPF. Let \(\mathcal{L}^{\ne}\) be the set of lines where (20) holds with strict inequality. If \(\mathcal{L}^{\ne}\) is empty, \(\omega\) is a load flow solution and Theorem I is trivially true. Assume now that \(\mathcal{L}^{\ne}\) is not empty and \(M\) lines have strict inequality in (20). Denote the set of the lines with strict inequality \(\mathcal{M} = \{1, 2, \dots, M\}\).

We denote the lines for which \(\mathbf{H}_{l,m} = 1\), \(\forall m \in \mathcal{M}\) upstream lines/buses (\(\mathcal{L}^M\)). We now create a load flow solution for \(s\). Using Lemma III, first we will show that, under conditions C1–C3, the created load flow solution is feasible (satisfies the constrains of A-OPF); then we show that, under conditions C1–C5, it has a lower objective function.

Consider the sequence \((P^{(n)}, Q^{c(n)}, v^{(n)}, f^{(n)})\) defined for \(n \ge 0\) by means of Algorithm 1. We now show that this sequence converges.

For \(n \ge 1\) let \(\Delta f^{(n)} \triangleq f^{(n)} - f^{(n-1)}\). Using item 1 of Lemma III we have \[ \left\|\Delta f^{(n)}\right\| \le \|\mathbf{E}\|^{(n-1)}\left\|\Delta f^{(1)}\right\| \tag{43} \] when C2 holds we have \[ \|\mathbf{E}\| < 1. \]

This implies that \(\|\Delta f^{(n)}\| \to 0\) as \(n \to \infty\), which implies that the sequence \(f^{(n)}\) converges. It follows that \((P^{(n)}, Q^{c(n)}, v^{(n)}, f^{(n)})\) converges to some limit, say \((P^*, Q^{c*}, v^*, f^*)\).

Since \(f^{(n)} \ge 0\) by construction, it follows also that \(f^* \ge 0\), and since \(\mathbf{D}\) is non-negative and from item 3 of Lemma III (C3 is used here. See proof of Lemma III). \[ v^{\min} \le v \le v^* \le \bar{v} \le v^{\max}. \tag{44} \]

Furthermore, by step 1 of Algorithm 1 \[ f_l^* = \frac{(P_l^{t*})^2 + (Q_l^{c*})^2}{v_{up(l)}^*} \quad \text{for all } l \in \mathcal{L}. \tag{45} \]

Let \(Q^* = Q^{c*} - \text{diag}(b)\mathbf{G}^T v^*\). It follows that \((s, S^* = P^* + jQ^*, v^*, f^*)\) satisfies (20) with equality, i.e., it is a load flow solution and satisfies (8). Furthermore, item (2) of Lemma I guarantees that there exist \(\bar{P}^*\) and \(\bar{Q}^*\) such that \(\bar{P}^* \le \bar{P} \le P^{\max}\) and \(\bar{Q}^* \le \bar{Q} \le Q^{\max}\). Using \((s, S^*, v^*, \hat{S}, \bar{v}, f^*)\) and (11.d) and (11.e) we can create \(\bar{f}^*\). Let \(\omega^* = (s, S^*, v^*, f^*, \hat{S}, \bar{v}, \bar{f}^*, \bar{S}^*)\). From (30), we can observe that the voltage security constraints are satisfied. Furthermore, from item (1) of Lemma I, (16), and (17) we have \(\hat{P} \le P^* \le \bar{P}^* \le \bar{P} \le P^{\max}\) and \(\hat{Q} \le Q^* \le \bar{Q}^* \le \bar{Q} \le Q^{\max}\) which show that constraints are also satisfied. Thus, \(\omega^*\) is a feasible solution of AR-OPF and of A-OPF.

Furthermore, from Lemma II, we have that every feasible solution of A-OPF is also feasible for O-OPF.

This proves the first item of Theorem I. \(\blacksquare\)

Item 2: Assume that \(\omega\) is an optimal solution of AR-OPF but not a feasible solution of A-OPF, i.e., \(\mathcal{L}^{\ne}\) is non-empty and \(M\) lines have strict inequality in (20). First note that at the first line (\(l=1\)) we have (\(\mathbf{H}_{l,1} \ne 0\ \forall\, l \in \mathcal{L}\)), thus, it is always in the set of upstream lines and we have \[ \begin{align} P_1^{t(1)} - P_1^t &= \sum_{l \in \mathcal{L}^{\ne}} \mathbf{H}_{1,l}\, r_l \left(f_l^{(1)} - f_l\right) \nonumber \\ &= \sum_{l \in \mathcal{L}^{\ne}} r_l \left(f_l^{(1)} - f_l\right) < 0 \tag{46} \end{align} \] thus, \(P_1^{t(1)} < P_1^t\). Furthermore, by item 4 of Lemma III we have (C4 and C5 are used here. See proof of Lemma III): \[ \begin{align*} \left|P_1^{t(n)} - P_1^{t(1)}\right| &\le \left|\Delta P_1^{t(n)}\right| + \ldots + \left|\Delta P_1^{t(2)}\right| \\ &\le \left(\eta^{n-1} + \ldots + \eta\right) \left|\Delta P_1^{t(1)}\right| \le \frac{\eta}{1-\eta} \left|\Delta P_1^{t(1)}\right| \\ &= \frac{\eta}{1-\eta} \left(P_1^t - P_1^{t(1)}\right). \end{align*} \] Since \(0 < \eta < 0.5\) and \(P_1^{t(1)} < P_1^t\), thus, \[ \left|P_1^{t(n)} - P_1^{t(1)}\right| \le \frac{\eta}{1-\eta}\left(P_1^t - P_1^{t(1)}\right) < \left(P_1^t - P_1^{t(1)}\right) \tag{47} \] therefore, \((P_1^t)^* < P_1^t\). Now \(P_1^t\) [resp. \((P_1^t)^*\)] is the net active power import from the external grid for the solution \(\omega\), [resp. \(\omega^*\)]. Since the power injections \(s\) are identical for \(\omega\) and \(\omega^*\), it follows that the objective function of \(\omega^*\) is strictly less than that of \(\omega\), which contradicts the optimality of \(\omega\).

This proves the second item of Theorem I. \(\blacksquare\)

To summarize, the AR-OPF is a combination of the original load-flow equations and the linear DistFlow [8] models with the inclusion of transverse parameters of the lines. Under the sufficient conditions provided above, the feasible solution-space of the AR-OPF is a subset of the one of the O-OPF, whereas the solution of R-OPF could lie outside the feasible solution space of O-OPF. These concepts are schematically represented in Fig.~2 (see Lemma II).

Less-Conservative Sufficient Conditions for Exactness of AR-OPF

The original conditions C1–C5 provide sufficient conditions for exactness by enforcing several geometric contraction properties along the iterative load-flow construction. A closer examination of the proof of Theorem I shows, however, that these properties are stronger than what is ultimately required.

The iterative construction serves two logically different purposes. First, starting from an arbitrary feasible AR-OPF point, it must converge to a physical load-flow solution with the same nodal power injections \(s\). For this purpose, it is sufficient to establish convergence of the sequence and to preserve the operational constraints while the physical solution is recovered. In particular, convergence is essential because the update of \(f\) in Algorithm 1 becomes the exact branch-current equality at the fixed point. Preservation of the lower-voltage limit is also required because the initial AR-OPF point satisfies \(v\ge v^{\min}\).

Second, when the initial AR-OPF point is assumed optimal but inexact, the recovered physical solution must have a strictly smaller objective value in order to obtain a contradiction. Since the nodal injections \(s\) are unchanged throughout the construction, it is sufficient to guarantee a strict reduction of the active power \(P_1^t\) imported from the slack bus. Therefore, geometric decay of every voltage correction and every upstream active-power correction is not required; it is sufficient to control their cumulative effects.

The line-ampacity constraints constitute a separate issue. As in the original proof of Theorem I, their preservation is established through the augmented variables and Lemmas I–II, after the physical load-flow solution has been recovered. The conditions introduced below replace only the contraction arguments associated with C1–C5.

Let

\[ \mathbf A:=\mathbf H\operatorname{diag}(r), \]

and let

\[ e_1:= \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}^{\top}\in\mathbb R^L \]

denote the first canonical basis vector. Hence,

\[ \mathbf a_1^\top := e_1^\top\mathbf A \]

is the first row of \(\mathbf A\), corresponding to the unique line connected to the slack bus. Since line \(1\) is upstream of every non-slack bus,

\[ \mathbf a_1^\top=r^\top. \]

Whenever \(\rho(\mathbf E)<1\), define

\[ \mathbf R_E := \sum_{k=1}^{\infty}\mathbf E^k = \mathbf E(\mathbf I-\mathbf E)^{-1}. \]

The following conditions are introduced.

Condition C1’ \[ \rho(\mathbf H^\top\mathbf M)<1. \]
Condition C2’ \[ \rho(\mathbf E)<1. \]
Condition C3’ \[ \mathbf D\mathbf R_E\le \mathbf D. \]
Condition C4’ \[ \mathbf a_1^\top\mathbf R_E < \mathbf a_1^\top, \]

where the inequality is componentwise.

Conditions C3’ and C4’ admit less conservative solution-dependent versions. For a given feasible AR-OPF point, let

\[ d:=f^{(0)}-f^{(1)}\ge0. \]

It is sufficient to require

\[ \mathbf D\mathbf R_Ed\le\mathbf Dd \tag{C3'-local} \]

and, whenever \(d\neq0\),

\[ \mathbf a_1^\top\mathbf R_Ed < \mathbf a_1^\top d. \tag{C4'-local} \]

The global conditions C3’–C4’ imply the corresponding local conditions for every \(d\ge0\).

Interpretation of C1’

Condition C1’ preserves the structural property provided by the original C1. Recall that

\[ \mathbf H^\top = (\mathbf I-\mathbf G^\top)^{-1}. \]

Using this identity,

\[ \begin{aligned} \mathbf I-\mathbf G^\top-\mathbf M &= (\mathbf I-\mathbf G^\top) \left( \mathbf I-\mathbf H^\top\mathbf M \right). \end{aligned} \]

Let

\[ \mathbf K:=\mathbf H^\top\mathbf M. \]

Since \(\mathbf K\ge0\) entrywise, C1’,

\[ \rho(\mathbf K)<1, \]

implies that \(\mathbf I-\mathbf K\) is nonsingular and

\[ (\mathbf I-\mathbf K)^{-1} = \sum_{k=0}^{\infty}\mathbf K^k \ge0. \]

Moreover, because the network is radial, \(\mathbf G\) is nilpotent and

\[ (\mathbf I-\mathbf G^\top)^{-1} = \mathbf H^\top \ge0. \]

Consequently,

\[ \begin{aligned} \mathbf C &:= (\mathbf I-\mathbf G^\top-\mathbf M)^{-1}\\ &= (\mathbf I-\mathbf H^\top\mathbf M)^{-1} \mathbf H^\top \ge0. \end{aligned} \]

Thus C1’ guarantees exactly the property of C1 required in the construction: the matrix \(\mathbf C\) exists and is entrywise nonnegative. In turn, the matrices \(\mathbf D\) and \(\mathbf F\) defined from \(\mathbf C\) retain their entrywise nonnegative structure.

Since

\[ \rho(\mathbf H^\top\mathbf M) \le \|\mathbf H^\top\mathbf M\|, \]

the original C1 implies C1’, whereas the converse does not generally hold. Hence C1’ is less conservative.

Revised Lemma III

Let

\[ \omega^{\mathrm{AR}} = (s,S^{(0)},v^{(0)},f^{(0)}, \hat S,\bar v,\bar f,\bar S) \]

be a feasible solution of AR-OPF, and let

\[ (P^{(n)},Q^{c(n)},v^{(n)},f^{(n)}) \]

be the sequence generated by Algorithm 1.

Define

\[ \Delta f^{(n)} := f^{(n)}-f^{(n-1)} \]

and

\[ d:=-\Delta f^{(1)} = f^{(0)}-f^{(1)}. \]

Because \(f^{(0)}\) satisfies the relaxed current constraint,

\[ d\ge0. \]

Furthermore,

\[ d\neq0 \]

if and only if the relaxation is strict for at least one branch.

Under C1’–C2’,

\[ \boxed{ |\Delta f^{(n)}| \le \mathbf E^{\,n-1}d, \qquad n\ge1, } \tag{35'} \]

and the sequence generated by Algorithm 1 converges.

If C3’-local additionally holds,

\[ \boxed{ v^{(n)}\ge v^{(0)}, \qquad n\ge1. } \tag{36'} \]

If C4’-local additionally holds and \(d\neq0\),

\[ \boxed{ P_1^{t(n)} < P_1^{t(0)}, \qquad n\ge1. } \tag{37'} \]

Proof

The Jacobian estimate used in the original proof gives

\[ |\Delta f^{(n+1)}| \le \mathbf E|\Delta f^{(n)}|. \]

Since

\[ |\Delta f^{(1)}|=d, \]

induction yields

\[ |\Delta f^{(n)}| \le \mathbf E^{n-1}d. \]

Importantly, this induction does not require a norm bound on \(\mathbf E\).

Under C2’,

\[ \rho(\mathbf E)<1, \]

and therefore, the Neumann series

\[ \sum_{k=0}^{\infty}\mathbf E^k = (\mathbf I-\mathbf E)^{-1} \]

exists and is finite. Hence,

\[ \begin{aligned} \sum_{n=1}^{\infty} |\Delta f^{(n)}| &\le \sum_{n=1}^{\infty} \mathbf E^{n-1}d\\ &= (\mathbf I-\mathbf E)^{-1}d. \end{aligned} \]

Thus \(f^{(n)}\) is absolutely Cauchy componentwise and converges to some \(f^*\). Since

\[ P^{(n)} = \hat P+\mathbf A f^{(n)}, \]

\[ Q^{c(n)} = \hat Q^c+\mathbf Ff^{(n)}, \]

and

\[ v^{(n)} = \bar v-\mathbf Df^{(n)}, \]

the sequences \(P^{(n)},Q^{c(n)}\), and \(v^{(n)}\) converge as well.

We next establish the voltage property. From

\[ v^{(n)}=\bar v-\mathbf Df^{(n)}, \]

we obtain

\[ \begin{aligned} v^{(n)}-v^{(0)} &= -\mathbf D \sum_{k=1}^{n}\Delta f^{(k)}\\ &= \mathbf Dd - \mathbf D \sum_{k=2}^{n}\Delta f^{(k)}. \end{aligned} \]

Using \(\mathbf D\ge0\),

\[ \begin{aligned} v^{(n)}-v^{(0)} &\ge \mathbf Dd - \mathbf D \sum_{k=2}^{n} |\Delta f^{(k)}|\\ &\ge \mathbf Dd - \mathbf D \sum_{j=1}^{n-1}\mathbf E^j d\\ &\ge \mathbf Dd-\mathbf D\mathbf R_Ed. \end{aligned} \]

Under C3’-local,

\[ \mathbf D\mathbf R_Ed \le \mathbf Dd, \]

and therefore

\[ v^{(n)}\ge v^{(0)} \]

for every \(n\ge1\).

Finally,

\[ P^{(n)} = \hat P+\mathbf A f^{(n)} \]

implies

\[ \begin{aligned} P_1^{t(n)}-P_1^{t(0)} &= \mathbf a_1^\top \sum_{k=1}^{n}\Delta f^{(k)}\\ &= -\mathbf a_1^\top d + \mathbf a_1^\top \sum_{k=2}^{n}\Delta f^{(k)}\\ &\le -\mathbf a_1^\top d + \mathbf a_1^\top \sum_{k=2}^{n}|\Delta f^{(k)}|\\ &\le -\mathbf a_1^\top d + \mathbf a_1^\top\mathbf R_Ed. \end{aligned} \]

Under C4’-local,

\[ \mathbf a_1^\top\mathbf R_Ed < \mathbf a_1^\top d, \]

which gives

\[ P_1^{t(n)} < P_1^{t(0)} \]

for every \(n\ge1\). Taking the limit yields

\[ P_1^{t*}<P_1^{t(0)}. \]

\(\blacksquare\)

Revised Exactness Theorem

Theorem. Suppose C1’–C3’ hold. For every feasible AR-OPF solution, Algorithm 1 converges to a physical load-flow solution with the same nodal power-injection vector \(s\). The recovered solution satisfies the original branch-current equality on every line and the nodal voltage limits. Together with the unchanged augmented-flow results of Lemmas I–II, it is feasible for A-OPF and O-OPF.

If C4’ additionally holds, every optimal solution of AR-OPF satisfies the branch-current equality on every line and is therefore an optimal solution of A-OPF and O-OPF.

Proof: recovery of a physical load-flow solution

Let

\[ \omega^{\mathrm{AR}} = (s,S^{(0)},v^{(0)},f^{(0)}, \hat S,\bar v,\bar f,\bar S) \]

be an arbitrary feasible solution of AR-OPF.

By the revised Lemma III, Algorithm 1 converges:

\[ (P^{(n)},Q^{c(n)},v^{(n)},f^{(n)}) \longrightarrow (P^*,Q^{c*},v^*,f^*). \]

We first show that the limiting point satisfies the physical branch-current relation on every line.

Step 1 of Algorithm 1 imposes, for each \(l\in\mathcal L\),

\[ f_l^{(n)} = \frac{ \left(P_l^{t(n-1)}\right)^2+ \left(Q_l^{c(n-1)}\right)^2 }{ v_{\operatorname{up}(l)}^{(n-1)} }. \]

Taking \(n\to\infty\) and using convergence and continuity gives

\[ \boxed{ f_l^* = \frac{ (P_l^{t*})^2+(Q_l^{c*})^2 }{ v_{\operatorname{up}(l)}^* }, \qquad \forall l\in\mathcal L. } \tag{38'} \]

Thus the relaxed inequality is recovered as an equality for every branch. This conclusion follows directly from convergence of Algorithm 1; it does not depend on C4’ and does not concern only the slack line.

The voltage constraints follow separately. Since C3’ gives

\[ v^{(n)}\ge v^{(0)} \]

for all \(n\),

\[ v^* \ge v^{(0)} \ge v^{\min}. \]

Moreover, Step 1 gives \(f^{(n)}\ge0\), and therefore

\[ v^{(n)} = \bar v-\mathbf Df^{(n)} \le \bar v. \]

Since the AR-OPF constraints impose

\[ \bar v\le v^{\max}, \]

we have

\[ \boxed{ v^{\min} \le v^* \le \bar v \le v^{\max}. } \tag{39'} \]

Hence the recovered physical load-flow solution satisfies both voltage limits.

It is important to distinguish (38’) from the line-ampacity constraints. Equation (38’) establishes the exact physical relation between branch power, voltage, and current for every line. It does not by itself establish the terminal ampacity constraints.

The latter are handled exactly as in the original AR-OPF proof. Once the physical load-flow solution \((S^*,v^*,f^*)\) has been obtained, Lemma I is used to construct the corresponding augmented variables and to bound the physical branch flows by the augmented flow envelopes. The augmented ampacity constraints of AR-OPF then imply the original terminal-current constraints. Lemma II subsequently establishes feasibility for O-OPF.

Therefore, provided the unchanged conclusions of Lemmas I–II hold under C1’—which preserves the existence and nonnegativity of \(\mathbf C\) required by their construction—the recovered point is feasible for A-OPF and O-OPF.

This proves the feasibility part of the theorem.

Proof: exactness at an optimum

We now distinguish clearly between two points.

Let

\[ \omega^{\mathrm{AR}} \]

denote an assumed optimal solution of AR-OPF.

Suppose, for contradiction, that \(\omega^{\mathrm{AR}}\) is inexact; namely, that the relaxed branch-current inequality is strict for at least one line. Then

\[ d = f^{(0)}-f^{(1)} \neq0. \]

Starting from \(\omega^{\mathrm{AR}}\), Algorithm 1 generates a second point, denoted by

\[ \omega^{\mathrm{rec}}, \]

which is the limiting physical load-flow solution established above. The two points have exactly the same nodal injections \(s\):

\[ s^{\mathrm{rec}}=s^{\mathrm{AR}}. \]

By C4’ and the revised Lemma III,

\[ \boxed{ P_{1,\mathrm{rec}}^t < P_{1,\mathrm{AR}}^t. } \tag{40'} \]

The AR-OPF objective has the form

\[ J(\omega) = \sum_{l\in\mathcal L} \mathcal C(p_l,q_l) + \mathcal C^e(P_1^t). \]

Because

\[ s^{\mathrm{rec}}=s^{\mathrm{AR}}, \]

all nodal-cost terms are identical:

\[ \sum_l \mathcal C( p_l^{\mathrm{rec}}, q_l^{\mathrm{rec}} ) = \sum_l \mathcal C( p_l^{\mathrm{AR}}, q_l^{\mathrm{AR}} ). \]

On the other hand, \(\mathcal C^e\) is strictly increasing and (40’) gives

\[ \mathcal C^e(P_{1,\mathrm{rec}}^t) < \mathcal C^e(P_{1,\mathrm{AR}}^t). \]

Consequently,

\[ \boxed{ J(\omega^{\mathrm{rec}}) < J(\omega^{\mathrm{AR}}). } \]

However, the first part of the theorem establishes that \(\omega^{\mathrm{rec}}\) is feasible for AR-OPF as well as for A-OPF and O-OPF. We have therefore constructed an AR-OPF-feasible point with a strictly smaller objective value than the assumed optimal solution \(\omega^{\mathrm{AR}}\), which is impossible.

The assumption that \(\omega^{\mathrm{AR}}\) is inexact must therefore be false. Hence

\[ d=0, \]

and

\[ \boxed{ f_l^{\mathrm{AR}} = \frac{ (P_{l,\mathrm{AR}}^t)^2+ (Q_{l,\mathrm{AR}}^c)^2 }{ v_{\operatorname{up}(l),\mathrm{AR}} }, \qquad \forall l\in\mathcal L. } \]

Thus every optimal AR-OPF solution satisfies the original branch-current equality for every line, and the relaxation is exact.

\(\blacksquare\)

Relation to the Original Conditions

The revised conditions target directly the three cumulative properties required in the proof.

The original C2,

\[ \|\mathbf E\|<1, \]

is used to prove convergence of Algorithm 1. Since convergence of the matrix powers and of the Neumann series requires only

\[ \rho(\mathbf E)<1, \]

C2’ is sufficient and less conservative.

The original C3 imposes

\[ \mathbf D\mathbf E \le \eta_1\mathbf D, \qquad \eta_1<\frac12. \]

This leads to the geometric estimate

\[ |\Delta v^{(n)}| \le \eta_1^{n-1} |\Delta v^{(1)}|. \]

However, the proof of feasibility ultimately requires only

\[ v^{(n)}\ge v^{(0)}. \]

Condition C3’ imposes this cumulative property directly. Indeed,

\[ \mathbf D\mathbf R_E = \mathbf D \sum_{k=1}^{\infty}\mathbf E^k \]

bounds the total voltage effect of all corrections after the first iteration. Thus no common scalar contraction ratio smaller than \(1/2\) is required.

Similarly, the original C4–C5 establish geometric decay of the active-power corrections on upstream lines. In the proof of exactness, this property is subsequently used only to show that all corrections after the first iteration cannot cancel the initial strict decrease of \(P_1^t\). Condition C4’ imposes this cumulative requirement directly through

\[ \mathbf a_1^\top\mathbf R_E < \mathbf a_1^\top. \]

Accordingly, the constants \(\eta_i<0.5\) are sufficient bounds arising from scalar geometric-series arguments rather than intrinsic thresholds for exactness.

Impact of Slack-Bus Reactive Power on Exactness

The exactness result above relies not only on the properties of the network and of the iterative load-flow construction, but also on the monotonicity of the objective function with respect to the power exchanged with the slack bus. In the original AR-OPF formulation, the objective is

\[ J = \sum_{l\in\mathcal L} \mathcal C(p_l,q_l) + \mathcal C^e(P_1^t), \]

where \(\mathcal C^e(P_1^t)\) is assumed to be strictly increasing in the active power imported from the external grid. Accordingly, the original exactness proof obtains a contradiction by constructing, from an assumed inexact optimal solution, a physical load-flow solution having the same nodal power injections but a strictly smaller value of \(P_1^t\).

This section examines how this argument changes when the reactive power exchanged with the slack bus, \(Q_1^t\), also enters the objective function.

A. Variation of the slack-bus active and reactive powers

For a fixed nodal power-injection vector \(s\), the active branch flows generated by Algorithm 1 satisfy

\[ P^{(n)} = \hat P+\mathbf A f^{(n)}, \qquad \mathbf A := \mathbf H\operatorname{diag}(r). \]

Let

\[ \mathbf a_P^\top := e_1^\top\mathbf A. \]

Because line \(1\) is upstream of all non-slack buses,

\[ \mathbf a_P^\top=r^\top, \]

and therefore

\[ P_1^{t(n)} = \hat P_1^t+\mathbf a_P^\top f^{(n)}. \tag{41} \]

Similarly, Algorithm 1 gives

\[ Q^{c(n)} = \hat Q^c+\mathbf Ff^{(n)}. \]

For the first line,

\[ Q_1^c = Q_1^t+b_1v_0. \]

Since the slack-bus voltage \(v_0\) is fixed, variations of \(Q_1^t\) and \(Q_1^c\) coincide. Defining

\[ \mathbf a_Q^\top := e_1^\top\mathbf F, \]

we obtain

\[ Q_1^{t(n)} = \hat Q_1^t+\mathbf a_Q^\top f^{(n)}, \tag{42} \]

where the constant term \(\hat Q_1^t\) includes the fixed shunt contribution. Since \(\mathbf F\ge0\),

\[ \mathbf a_Q\ge0. \]

Consider an inexact feasible AR-OPF point and define, as before,

\[ d := f^{(0)}-f^{(1)} \ge0, \qquad d\neq0. \]

The first iteration therefore gives

\[ P_1^{t(1)}-P_1^{t(0)} = -\mathbf a_P^\top d <0, \tag{43} \]

because \(r_l>0\), and

\[ Q_1^{t(1)}-Q_1^{t(0)} = -\mathbf a_Q^\top d \le0. \tag{44} \]

Hence the first correction from an inexact relaxed point toward a physical load-flow point decreases both the active and reactive powers supplied by the slack bus, except that the decrease in reactive power may be zero in a degenerate case.

Under C2’, define

\[ \mathbf R_E = \mathbf E(\mathbf I-\mathbf E)^{-1}. \]

From

\[ |\Delta f^{(n)}| \le \mathbf E^{n-1}d, \]

the cumulative corrections following the first iteration satisfy

\[ \left| \sum_{n=2}^{\infty}\Delta f^{(n)} \right| \le \mathbf R_Ed. \tag{45} \]

Consequently,

\[ P_{1,\mathrm{rec}}^t-P_{1,\mathrm{AR}}^t \le -\mathbf a_P^\top d + \mathbf a_P^\top\mathbf R_Ed, \tag{46} \]

and

\[ Q_{1,\mathrm{rec}}^t-Q_{1,\mathrm{AR}}^t \le -\mathbf a_Q^\top d + \mathbf a_Q^\top\mathbf R_Ed. \tag{47} \]

Here \(\omega^{\mathrm{AR}}\) denotes the initial relaxed point and \(\omega^{\mathrm{rec}}\) the physical load-flow point obtained as the limit of Algorithm 1.

Therefore, the conditions

\[ \mathbf a_P^\top\mathbf R_Ed < \mathbf a_P^\top d \tag{48} \]

and

\[ \mathbf a_Q^\top\mathbf R_Ed < \mathbf a_Q^\top d \tag{49} \]

guarantee

\[ P_{1,\mathrm{rec}}^t < P_{1,\mathrm{AR}}^t, \qquad Q_{1,\mathrm{rec}}^t < Q_{1,\mathrm{AR}}^t. \tag{50} \]

Condition (48) corresponds to C4’-local introduced above. Equation (49) is its reactive-power counterpart.

B. General objective depending on \(P_1^t\) and \(Q_1^t\)

Consider the generalized objective

\[ J(\omega) = \sum_{l\in\mathcal L} \mathcal C(p_l,q_l) + \Psi(P_1^t,Q_1^t), \tag{51} \]

where \(\Psi:\mathbb R^2\rightarrow\mathbb R\).

The exactness argument can be generalized as follows.

Proposition 1. Assume that C1’–C3’ hold and that the physical point \(\omega^{\mathrm{rec}}\) recovered from every inexact feasible AR-OPF point is feasible for AR-OPF, A-OPF, and O-OPF. Suppose further that

\[ P_{1,\mathrm{rec}}^t \le P_{1,\mathrm{AR}}^t, \qquad Q_{1,\mathrm{rec}}^t \le Q_{1,\mathrm{AR}}^t, \tag{52} \]

with at least one inequality strict whenever the initial point is inexact.

If \(\Psi(P,Q)\) is componentwise nondecreasing in \(P\) and \(Q\), and is strictly increasing with respect to at least one component that decreases strictly in (52), then every optimal AR-OPF solution is exact.

Proof. Assume, by contradiction, that an optimal AR-OPF solution \(\omega^{\mathrm{AR}}\) is inexact. Algorithm 1 produces a physical feasible point \(\omega^{\mathrm{rec}}\) having the same nodal injections:

\[ s^{\mathrm{rec}}=s^{\mathrm{AR}}. \]

Hence

\[ \sum_l \mathcal C( p_l^{\mathrm{rec}}, q_l^{\mathrm{rec}} ) = \sum_l \mathcal C( p_l^{\mathrm{AR}}, q_l^{\mathrm{AR}} ). \]

By (52) and the assumed monotonicity of \(\Psi\),

\[ \Psi( P_{1,\mathrm{rec}}^t, Q_{1,\mathrm{rec}}^t) < \Psi( P_{1,\mathrm{AR}}^t, Q_{1,\mathrm{AR}}^t). \]

Therefore,

\[ J(\omega^{\mathrm{rec}}) < J(\omega^{\mathrm{AR}}), \]

which contradicts optimality of \(\omega^{\mathrm{AR}}\). Thus an optimal AR-OPF point cannot be inexact. \(\blacksquare\)

This proposition shows that introducing \(Q_1^t\) into the objective does not by itself invalidate exactness. What matters is whether the objective improves along the physical-recovery direction.

For a differentiable function \(\Psi\), a convenient sufficient condition is

\[ \frac{\partial\Psi}{\partial P} \ge0, \qquad \frac{\partial\Psi}{\partial Q} \ge0 \tag{53} \]

over the relevant range of slack-bus powers, with at least one derivative strictly positive. Under (48)–(49), the recovered physical point then has a strictly smaller objective value.

C. Linear \(P\)-\(Q\) objectives

A particularly relevant case is

\[ \min\; \sum_l\mathcal C(p_l,q_l) + \alpha P_1^t+\beta Q_1^t. \tag{54} \]

\[ \alpha\ge0, \qquad \beta\ge0, \]

and at least one coefficient is strictly positive, then decreasing the corresponding slack-bus powers is favorable to the objective. In particular, using (46)–(47),

\[ \begin{aligned} J(\omega^{\mathrm{rec}}) - J(\omega^{\mathrm{AR}}) \le{}& -\left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right)d \\ & + \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_Ed. \end{aligned} \tag{55} \]

Hence the solution-dependent condition

\[ \boxed{ \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_Ed < \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right)d } \tag{56} \]

is sufficient for the exactness contradiction.

A solution-independent sufficient condition is

\[ \boxed{ \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_E < \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right). } \tag{57} \]

Thus, for a given objective, it is not necessary to impose separate monotonicity conditions on both \(P_1^t\) and \(Q_1^t\). It is sufficient to guarantee a strict decrease of the particular linear combination appearing in the objective.

D. Reactive-power maximization

The situation changes fundamentally when the objective rewards a larger value of \(Q_1^t\). For example,

\[ \max Q_1^t \]

is equivalent to the minimization problem

\[ \min -Q_1^t. \tag{58} \]

The physical-recovery procedure typically decreases \(Q_1^t\), as shown in (44) and (47). Therefore, while the recovered point is physically feasible, its objective value in (58) may be larger rather than smaller. The contradiction used in the exactness proof is then lost.

More importantly, this is not only a limitation of the proof. Reactive-power maximization can make the relaxation genuinely inexact.

Proposition 2. There exists a one-line radial network for which AR-OPF with objective

\[ \max Q_1^t \]

has a strictly inexact optimal solution.

Proof. Consider a two-bus radial network consisting of the slack bus \(0\) and one downstream bus \(1\). Let

\[ z=r+jx, \qquad r>0,\quad x>0, \]

and set

\[ b=0, \qquad s_1=0. \]

Assume

\[ 0<v^{\min}<v_0, \]

and choose the remaining upper-voltage and ampacity limits sufficiently large so that they are inactive.

Since bus \(1\) has no downstream branches, the branch-flow equations give

\[ P_1^t=rf, \tag{59} \]

and

\[ Q_1^t=xf. \tag{60} \]

The voltage equation becomes

\[ \begin{aligned} v_1 &= v_0 - 2\left(rP_1^t+xQ_1^t\right) + (r^2+x^2)f \\ &= v_0-(r^2+x^2)f. \end{aligned} \]

Defining

\[ |z|^2=r^2+x^2, \]

we have

\[ v_1=v_0-|z|^2f. \tag{61} \]

The relaxed current constraint is

\[ f \ge \frac{(P_1^t)^2+(Q_1^t)^2}{v_0}. \]

Using (59)–(60),

\[ f \ge \frac{|z|^2f^2}{v_0}. \tag{62} \]

For \(f\ge0\), (62) is equivalent to

\[ f \le \frac{v_0}{|z|^2}. \tag{63} \]

The lower-voltage constraint gives, from (61),

\[ v_0-|z|^2f \ge v^{\min}, \]

\[ f \le \frac{v_0-v^{\min}}{|z|^2}. \tag{64} \]

Since \(v^{\min}>0\), the bound (64) is strictly tighter than (63).

The relaxed optimization problem is

\[ \max Q_1^t = \max xf. \]

Because \(x>0\), the optimum is attained at

\[ \boxed{ f_{\mathrm{AR}}^\star = \frac{v_0-v^{\min}}{|z|^2}. } \tag{65} \]

Consequently,

\[ Q_{1,\mathrm{AR}}^{t,\star} = \frac{x(v_0-v^{\min})}{|z|^2} >0. \tag{66} \]

We now examine the physical current equality. A physical load-flow solution must satisfy

\[ f = \frac{|z|^2f^2}{v_0}, \]

or equivalently

\[ f \left( 1-\frac{|z|^2f}{v_0} \right)=0. \]

Hence the only possible physical solutions are

\[ f=0 \]

and

\[ f=\frac{v_0}{|z|^2}. \]

The second solution gives

\[ v_1 = v_0 - |z|^2 \frac{v_0}{|z|^2} = 0, \]

which violates

\[ v_1\ge v^{\min}>0. \]

Therefore the only feasible physical solution is

\[ f_{\mathrm{AC}}^\star=0, \]

with

\[ Q_{1,\mathrm{AC}}^{t,\star}=0. \]

On the other hand,

\[ 0 < f_{\mathrm{AR}}^\star < \frac{v_0}{|z|^2}, \]

and therefore

\[ f_{\mathrm{AR}}^\star > \frac{ |z|^2(f_{\mathrm{AR}}^\star)^2 }{ v_0 }. \]

The relaxed current inequality is thus strict at the AR-OPF optimum. Hence the relaxation is inexact. \(\blacksquare\)

The mechanism in Proposition 2 is direct. From (60),

\[ Q_1^t=xf. \]

Since \(x>0\), maximizing \(Q_1^t\) rewards an increase of \(f\). However, the relaxed current constraint only imposes a lower physical consistency requirement,

\[ f \ge \frac{|S_1^t|^2}{v_0}, \]

and therefore allows \(f\) to exceed its physical value. The optimizer can exploit this additional degree of freedom to create artificial reactive losses \(xf\) and thereby increase the reactive power supplied by the slack bus.

The monotonicity assumption on the slack-bus objective is therefore not merely a technical feature of the exactness proof. An objective that rewards additional branch losses can fundamentally change the exactness properties of the relaxation.

E. Competing active- and reactive-power objectives

An intermediate situation arises when active-power import is penalized while reactive-power import is rewarded. Consider

\[ \min\; \sum_l\mathcal C(p_l,q_l) + \alpha P_1^t - \beta Q_1^t, \qquad \alpha>0,\quad\beta>0. \tag{67} \]

Let

\[ \delta_P := P_{1,\mathrm{AR}}^t - P_{1,\mathrm{rec}}^t \]

and

\[ \delta_Q := Q_{1,\mathrm{AR}}^t - Q_{1,\mathrm{rec}}^t. \]

If both are positive, the variation of the interface-dependent part of the objective is

\[ \Delta J = -\alpha\delta_P+\beta\delta_Q. \tag{68} \]

Therefore the physical-recovery argument proves exactness only when

\[ \boxed{ \alpha\delta_P > \beta\delta_Q. } \tag{69} \]

The reduction of active import must dominate the loss of the rewarded reactive import.

Using the cumulative bounds, one has

\[ \delta_P \ge \mathbf a_P^\top (\mathbf I-\mathbf R_E)d, \tag{70} \]

while

\[ \delta_Q \le \mathbf a_Q^\top (\mathbf I+\mathbf R_E)d. \tag{71} \]

A sufficient condition for (69) is consequently

\[ \boxed{ \alpha\, \mathbf a_P^\top (\mathbf I-\mathbf R_E)d > \beta\, \mathbf a_Q^\top (\mathbf I+\mathbf R_E)d. } \tag{72} \]

This condition explicitly quantifies the competition between the loss-reducing incentive associated with \(P_1^t\) and the loss-increasing incentive associated with maximizing \(Q_1^t\).

F. Implication for flexibility-region boundary tracing

The preceding results are particularly relevant when AR-OPF is used to trace the active/reactive flexibility region at the interface with the external grid. A typical support-function problem has the form

\[ \max\; \lambda_P P_1^t + \lambda_Q Q_1^t. \tag{73} \]

Different values of \((\lambda_P,\lambda_Q)\) correspond to different boundary directions. The exactness properties of the relaxation may therefore depend on the direction being traced.

In directions for which the objective penalizes the artificial increase of line losses, the physical-recovery construction is compatible with the objective and an exactness proof may be obtained using the conditions above. In contrast, directions that reward larger slack-bus reactive power can favor an artificial increase of \(f\), and exactness is no longer guaranteed. Proposition 2 shows that this difficulty already arises in the simplest one-line radial network.

Consequently, exactness of AR-OPF for a conventional loss-minimizing or import-minimizing OPF objective does not by itself imply exactness for arbitrary \(P\)-\(Q\) boundary-tracing objectives. The objective direction must be considered explicitly when AR-OPF is used to construct a flexibility region.

Impact of Slack-Bus Reactive Power on Exactness

The exactness result above assumes that the cost associated with the external grid is a strictly increasing function of the active power imported from the slack bus. The argument can be extended to an objective depending on both active and reactive powers exchanged with the slack bus.

Consider

\[ J(\omega) = \sum_{l\in\mathcal L} \mathcal C(p_l,q_l) + \Psi(P_1^t,Q_1^t), \tag{41} \]

where \(\Psi:\mathbb R^2\rightarrow\mathbb R\) denotes the cost associated with the slack-bus power exchange.

For fixed nodal injections \(s\), the slack-bus active and reactive powers can be expressed as

\[ P_1^t = \hat P_1^t+\mathbf a_P^\top f, \qquad \mathbf a_P^\top := e_1^\top\mathbf H\operatorname{diag}(r), \tag{42} \]

and

\[ Q_1^t = \hat Q_1^t+\mathbf a_Q^\top f, \qquad \mathbf a_Q^\top := e_1^\top\mathbf F. \tag{43} \]

Let \(\omega^{\mathrm{AR}}\) be an inexact feasible AR-OPF point and define

\[ d := f^{(0)}-f^{(1)} \ge0, \qquad d\neq0. \]

Under C2’, the cumulative correction after the first iteration satisfies

\[ \left| \sum_{n=2}^{\infty}\Delta f^{(n)} \right| \le \mathbf R_Ed, \qquad \mathbf R_E = \mathbf E(\mathbf I-\mathbf E)^{-1}. \tag{44} \]

Consequently, for the recovered physical solution \(\omega^{\mathrm{rec}}\),

\[ P_{1,\mathrm{rec}}^t-P_{1,\mathrm{AR}}^t \le -\mathbf a_P^\top d + \mathbf a_P^\top\mathbf R_Ed, \tag{45} \]

and

\[ Q_{1,\mathrm{rec}}^t-Q_{1,\mathrm{AR}}^t \le -\mathbf a_Q^\top d + \mathbf a_Q^\top\mathbf R_Ed. \tag{46} \]

Therefore, if

\[ \mathbf a_P^\top\mathbf R_Ed < \mathbf a_P^\top d \tag{47} \]

and

\[ \mathbf a_Q^\top\mathbf R_Ed < \mathbf a_Q^\top d, \tag{48} \]

then

\[ P_{1,\mathrm{rec}}^t<P_{1,\mathrm{AR}}^t, \qquad Q_{1,\mathrm{rec}}^t<Q_{1,\mathrm{AR}}^t. \tag{49} \]

This leads to the following extension of the exactness result.

Corollary 1.
Suppose C1’–C3’ hold and that the recovered load-flow solution is feasible as established in the preceding theorem. Let the objective be given by (41), where \(\Psi(P,Q)\) is componentwise nondecreasing in \(P\) and \(Q\). If, for every inexact optimal AR-OPF point, the physical-recovery procedure produces

\[ \Psi( P_{1,\mathrm{rec}}^t, Q_{1,\mathrm{rec}}^t) < \Psi( P_{1,\mathrm{AR}}^t, Q_{1,\mathrm{AR}}^t), \tag{50} \]

then the AR-OPF relaxation is exact.

In particular, conditions (47)–(48), together with strict monotonicity of \(\Psi\) in at least one of the corresponding arguments, are sufficient for (50).

Proof.
The recovered point \(\omega^{\mathrm{rec}}\) has the same nodal power injections as the assumed optimal relaxed point \(\omega^{\mathrm{AR}}\). Hence,

\[ \sum_l \mathcal C( p_l^{\mathrm{rec}}, q_l^{\mathrm{rec}} ) = \sum_l \mathcal C( p_l^{\mathrm{AR}}, q_l^{\mathrm{AR}} ). \]

If (50) holds, then

\[ J(\omega^{\mathrm{rec}}) < J(\omega^{\mathrm{AR}}), \]

which contradicts the optimality of the inexact point \(\omega^{\mathrm{AR}}\). Therefore every optimal AR-OPF solution must satisfy the branch-current equality on all lines. \(\blacksquare\)

The essential requirement is therefore not that the objective depend exclusively on \(P_1^t\), but that the physical-recovery direction be a strict descent direction for the slack-bus cost \(\Psi\).

Linear Active-Reactive Power Objective

An important special case is

\[ J(\omega) = \sum_l\mathcal C(p_l,q_l) + \alpha P_1^t + \beta Q_1^t, \tag{51} \]

where \(\alpha\) and \(\beta\) are constant coefficients. Combining (45)–(46) gives

\[ \begin{aligned} J(\omega^{\mathrm{rec}}) - J(\omega^{\mathrm{AR}}) \le{}& - \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right)d \\ & + \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_Ed, \end{aligned} \tag{52} \]

provided \(\alpha,\beta\ge0\). Hence the following condition directly guarantees exactness:

\[ \boxed{ \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_Ed < \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right)d. } \tag{53} \]

This gives the following special case of Corollary 1.

Corollary 2.
For the linear slack-bus objective (51) with

\[ \alpha\ge0, \qquad \beta\ge0, \qquad (\alpha,\beta)\neq(0,0), \]

condition (53), together with C1’–C3’, is sufficient for exactness of AR-OPF.

A solution-independent version is

\[ \boxed{ \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right) \mathbf R_E < \left( \alpha\mathbf a_P^\top + \beta\mathbf a_Q^\top \right). } \tag{54} \]

The original AR-OPF objective corresponds to \(\beta=0\), in which case (54) reduces to C4’. If only reactive power is penalized, i.e., \(\alpha=0\) and \(\beta>0\), the corresponding condition becomes

\[ \mathbf a_Q^\top\mathbf R_E < \mathbf a_Q^\top. \tag{55} \]

Thus the condition required for exactness depends naturally on the particular slack-bus quantity appearing in the objective.

Remark.
The sign of the objective coefficients is important. If \(\beta<0\), the objective rewards an increase of \(Q_1^t\), as occurs when reactive-power import is maximized. In this case the physical-recovery procedure, which tends to reduce \(Q_1^t\), is not necessarily a descent direction of the objective, and the above contradiction no longer follows. Indeed, the relaxed current variable \(f\) contributes positively to reactive line losses through \(\mathbf a_Q^\top f\); an objective rewarding larger \(Q_1^t\) may therefore favor artificial relaxation slack. Consequently, exactness for the conventional import-minimizing objective does not imply exactness for arbitrary active-reactive power objectives.

Interior Feasibility by Continuous Boundary Contraction

For a fixed regularization coefficient \(\varepsilon\), define the continuously traced interface curve

\[ \Gamma_\varepsilon := \left\{ y_{\rm AR,\varepsilon}^{\star}(\gamma) : \gamma\in[0,2\pi) \right\}, \qquad y:= \begin{bmatrix} P_1^t & Q_1^t \end{bmatrix}^{\top}. \tag{58} \]

The finite set \(\mathcal S_{\rm AR,\varepsilon}\) obtained from a discrete set of directions is regarded as a numerical approximation of \(\Gamma_\varepsilon\). Let \(\mathcal K_{\bar\varepsilon}\) denote the bounded region enclosed by \(\Gamma_{\bar\varepsilon}\).

We consider the following assumptions.

For every finite \(\varepsilon\ge\bar\varepsilon\), there exists a continuous selection \[ (\gamma,\varepsilon) \longmapsto y_{\rm AR,\varepsilon}^{\star}(\gamma). \tag{59} \]
The initial curve \(\Gamma_{\bar\varepsilon}\) is a simple closed curve.
There exists a point \(y^\infty\in\mathbb R^2\) such that \[ \lim_{\varepsilon\rightarrow\infty} \sup_{\gamma\in[0,2\pi)} \left\| y_{\rm AR,\varepsilon}^{\star}(\gamma) - y^\infty \right\| =0. \tag{60} \]

That is, the complete directional boundary contracts uniformly to the same limiting point.

Proposition.
Under (59)–(60), every point

\[ y\in\operatorname{int}(\mathcal K_{\bar\varepsilon}), \qquad y\neq y^\infty, \]

is attained by at least one intermediate boundary. More precisely, there exist

\[ \varepsilon_y>\bar\varepsilon, \qquad \gamma_y\in[0,2\pi) \]

such that

\[ \boxed{ y = y_{\rm AR,\varepsilon_y}^{\star}(\gamma_y). } \tag{61} \]

Proof

Let

\[ y\in\operatorname{int}(\mathcal K_{\bar\varepsilon}), \qquad y\neq y^\infty. \]

Since \(\Gamma_{\bar\varepsilon}\) is a simple closed curve and \(y\) lies in its interior, the winding number of \(\Gamma_{\bar\varepsilon}\) around \(y\) is nonzero; choosing a consistent orientation,

\[ \operatorname{wind}(\Gamma_{\bar\varepsilon},y)=1. \tag{62} \]

By (60), for sufficiently large \(\varepsilon_1\),

\[ \Gamma_{\varepsilon_1} \subset B_\delta(y^\infty), \]

where \(\delta\) can be chosen such that

\[ 0<\delta<\|y-y^\infty\|. \]

The disk \(B_\delta(y^\infty)\) does not contain \(y\). Hence \(\Gamma_{\varepsilon_1}\) is contractible in \(\mathbb R^2\setminus\{y\}\), and therefore

\[ \operatorname{wind}(\Gamma_{\varepsilon_1},y)=0. \tag{63} \]

Suppose, by contradiction, that

\[ y \notin \Gamma_\varepsilon \qquad \forall\, \varepsilon\in[\bar\varepsilon,\varepsilon_1]. \tag{64} \]

Because of the continuity assumption (59), the family \(\Gamma_\varepsilon\), \(\varepsilon\in[\bar\varepsilon,\varepsilon_1]\), then defines a continuous homotopy in \(\mathbb R^2\setminus\{y\}\). The winding number around \(y\) is invariant under such a homotopy. Consequently,

\[ \operatorname{wind}(\Gamma_{\bar\varepsilon},y) = \operatorname{wind}(\Gamma_{\varepsilon_1},y), \]

which contradicts (62)–(63).

Thus (64) is impossible. There must exist some

\[ \varepsilon_y \in (\bar\varepsilon,\varepsilon_1] \]

and some direction \(\gamma_y\) for which

\[ y = y_{\rm AR,\varepsilon_y}^{\star}(\gamma_y). \]

This proves (61). \(\blacksquare\)

The proposition does not require the regions \(\mathcal K_\varepsilon\) to be nested. Continuity of the boundary deformation and contraction to a point are sufficient.

Consequence for Interior Feasibility

Let \(\mathcal Y_{\rm AC}\) denote the projection of the O-OPF feasible set onto the interface variables \((P_1^t,Q_1^t)\).

Corollary.
Suppose that, in addition to the assumptions of the proposition, the losses-regularized AR-OPF is exact for every

\[ \varepsilon>\bar\varepsilon \]

and every \(\gamma\in[0,2\pi)\). Then

\[ \boxed{ \operatorname{int} (\mathcal K_{\bar\varepsilon}) \setminus\{y^\infty\} \subseteq \mathcal Y_{\rm AC}. } \tag{65} \]

If \(\mathcal Y_{\rm AC}\) is closed, the limiting point \(y^\infty\) is also feasible and therefore

\[ \boxed{ \operatorname{int} (\mathcal K_{\bar\varepsilon}) \subseteq \mathcal Y_{\rm AC}. } \tag{66} \]

Proof.
For any

\[ y\in \operatorname{int}(\mathcal K_{\bar\varepsilon}), \qquad y\neq y^\infty, \]

the proposition guarantees the existence of \(\varepsilon_y>\bar\varepsilon\) and \(\gamma_y\) such that

\[ y = y_{\rm AR,\varepsilon_y}^{\star}(\gamma_y). \]

By assumption, AR-OPF is exact at this value of \((\varepsilon_y,\gamma_y)\). Hence the corresponding optimizer is feasible for O-OPF and

\[ y\in\mathcal Y_{\rm AC}. \]

This proves (65).

Furthermore, by (60) one can choose a sequence \(\varepsilon_k\rightarrow\infty\) and corresponding exact boundary points satisfying

\[ y_{\rm AR,\varepsilon_k}^{\star}(\gamma) \rightarrow y^\infty. \]

If \(\mathcal Y_{\rm AC}\) is closed, it follows that

\[ y^\infty\in\mathcal Y_{\rm AC}. \]

Thus (66) follows. \(\blacksquare\)

The above result relies on three regularity assumptions. First, the mapping \((\gamma,\varepsilon)\mapsto y_{\mathrm{AR},\varepsilon}^{\star}(\gamma)\) is assumed to admit a continuous selection. This is reasonable because \(\gamma\) and \(\varepsilon\) enter only the objective, while the feasible set is independent of these parameters. Under compactness of the feasible set and continuity of the objective, the optimizer correspondence is well behaved in the sense of parametric optimization; continuity of \(y^\star\) follows, in particular, when its optimal projection onto \((P_1^t,Q_1^t)\) is unique.

Second, \(\Gamma_{\bar\varepsilon}\) is assumed to be a simple closed curve. Equivalently, the continuous map \(\gamma\mapsto y_{\mathrm{AR},\bar\varepsilon}^{\star}(\gamma)\) has no self-intersections, apart from the identification of \(\gamma=0\) and \(2\pi\). This assumption can be directly verified from the traced boundary and is required only to unambiguously define its enclosed region.

Finally, the contraction \[ \Gamma_\varepsilon\rightarrow\{y^\infty\} \] as \(\varepsilon\rightarrow\infty\) follows naturally when the loss term dominates the directional term. More precisely, if the feasible set is compact, the loss function \(L\) is continuous, and all minimizers of \(L\) have the same interface projection \(y^\infty\), then optimality of \[ h_\gamma(y)+\varepsilon L \] implies that the loss suboptimality is \(O(1/\varepsilon)\), uniformly in \(\gamma\). Consequently, all directional optimal interface points converge to the common minimum-loss projection \(y^\infty\).

Monotonicity with Respect to the Loss-Regularization Parameter

Consider the losses-regularized directional objective

\[ \begin{aligned} J_\varepsilon &= \cos\gamma\,P_1^t + \sin\gamma\,Q_1^t + \varepsilon(P_{\rm loss}+Q_{\rm loss}) \\ &= (\varepsilon+\cos\gamma)P_1^t + (\varepsilon+\sin\gamma)Q_1^t - \varepsilon\sum_{l\in\mathcal L}(p_l+q_l). \end{aligned} \tag{58} \]

Since the load-flow recovery procedure preserves the non-slack nodal injections \(s\), the last term in (58) is unchanged between an AR-OPF point and its recovered physical point. Hence, for the exactness argument, the relevant slack-bus coefficients are

\[ \alpha_{\varepsilon,\gamma} = \varepsilon+\cos\gamma, \qquad \beta_{\varepsilon,\gamma} = \varepsilon+\sin\gamma. \tag{59} \]

For \(\varepsilon\ge1\),

\[ \alpha_{\varepsilon,\gamma}\ge0, \qquad \beta_{\varepsilon,\gamma}\ge0, \]

for every \(\gamma\), and they cannot vanish simultaneously.

Define

\[ \mathbf c_{\varepsilon,\gamma}^{\top} := (\varepsilon+\cos\gamma)\mathbf a_P^\top + (\varepsilon+\sin\gamma)\mathbf a_Q^\top. \tag{60} \]

According to Corollary X, the solution-independent objective condition for exactness is

\[ \mathbf c_{\varepsilon,\gamma}^{\top}\mathbf R_E < \mathbf c_{\varepsilon,\gamma}^{\top}, \tag{61} \]

or equivalently,

\[ \mathbf c_{\varepsilon,\gamma}^{\top} (\mathbf I-\mathbf R_E) > 0, \tag{62} \]

where all inequalities are componentwise.

Proposition.
Suppose C1’–C3’ hold and let \(\varepsilon_1\ge1\). If

\[ \mathbf c_{\varepsilon_1,\gamma}^{\top}\mathbf R_E < \mathbf c_{\varepsilon_1,\gamma}^{\top} \qquad \forall\gamma\in[0,2\pi), \tag{63} \]

then, for every \(\varepsilon_2>\varepsilon_1\),

\[ \mathbf c_{\varepsilon_2,\gamma}^{\top}\mathbf R_E < \mathbf c_{\varepsilon_2,\gamma}^{\top} \qquad \forall\gamma\in[0,2\pi). \tag{64} \]

Consequently, once the solution-independent exactness conditions are satisfied for all directions at \(\varepsilon_1\), they remain satisfied for all directions at every larger loss-regularization coefficient.

Proof.
Let

\[ \mathbf B:=\mathbf I-\mathbf R_E \]

and define

\[ \mathbf u^\top := \mathbf a_P^\top\mathbf B, \qquad \mathbf v^\top := \mathbf a_Q^\top\mathbf B. \tag{65} \]

Condition (63) is equivalent to

\[ \varepsilon_1(\mathbf u^\top+\mathbf v^\top) + \cos\gamma\,\mathbf u^\top + \sin\gamma\,\mathbf v^\top > 0 \qquad \forall\gamma. \tag{66} \]

Consider an arbitrary component \(j\). Equation (66) implies

\[ \varepsilon_1(u_j+v_j) + u_j\cos\gamma + v_j\sin\gamma > 0 \qquad \forall\gamma. \tag{67} \]

The minimum of the directional term over \(\gamma\) is

\[ \min_{\gamma} \left( u_j\cos\gamma+v_j\sin\gamma \right) = -\sqrt{u_j^2+v_j^2}. \]

Therefore (67) implies

\[ \varepsilon_1(u_j+v_j) - \sqrt{u_j^2+v_j^2} > 0. \tag{68} \]

In particular,

\[ u_j+v_j>0. \]

Since this holds for every component,

\[ \boxed{ (\mathbf a_P^\top+\mathbf a_Q^\top) (\mathbf I-\mathbf R_E) > 0. } \tag{69} \]

Now let

\[ \varepsilon_2 = \varepsilon_1+\Delta\varepsilon, \qquad \Delta\varepsilon>0. \]

Using (60),

\[ \mathbf c_{\varepsilon_2,\gamma}^{\top} = \mathbf c_{\varepsilon_1,\gamma}^{\top} + \Delta\varepsilon (\mathbf a_P^\top+\mathbf a_Q^\top). \]

Multiplication by \(\mathbf I-\mathbf R_E\) gives

\[ \begin{aligned} \mathbf c_{\varepsilon_2,\gamma}^{\top} (\mathbf I-\mathbf R_E) ={}& \mathbf c_{\varepsilon_1,\gamma}^{\top} (\mathbf I-\mathbf R_E) \\ &+ \Delta\varepsilon (\mathbf a_P^\top+\mathbf a_Q^\top) (\mathbf I-\mathbf R_E). \end{aligned} \tag{70} \]

The first term on the right-hand side is strictly positive by (63), while the second is strictly positive by (69) and \(\Delta\varepsilon>0\). Hence

\[ \mathbf c_{\varepsilon_2,\gamma}^{\top} (\mathbf I-\mathbf R_E)>0 \]

for every \(\gamma\), which is equivalent to (64).

\(\blacksquare\)

Corollary.
Under C1’–C3’, if the solution-independent exactness condition is satisfied for every direction at some \(\bar\varepsilon\ge1\), then AR-OPF is certified exact for every

\[ \varepsilon\ge\bar\varepsilon \]

and every

\[ \gamma\in[0,2\pi). \]

Hence the set of regularization coefficients certified by the proposed sufficient conditions is an upper interval.

Suppose that Assumptions 1–3 hold and that \(\bar{\varepsilon}\ge1\). Assume C1’–C3’ and that the solution-independent exactness condition \[\begin{equation} \mathbf c_{\bar{\varepsilon},\gamma}^{\top}\mathbf R_E < \mathbf c_{\bar{\varepsilon},\gamma}^{\top}, \qquad \forall\gamma\in[0,2\pi), \label{eq-exactness-eps-bar} \end{equation}\] holds, where \[ \mathbf c_{\varepsilon,\gamma}^{\top} = (\varepsilon+\cos\gamma)\mathbf a_P^\top + (\varepsilon+\sin\gamma)\mathbf a_Q^\top. \] Then \[ \boxed{ \mathcal K_{\bar{\varepsilon}} \subseteq \mathcal Y_{\rm AC}. } \]

By the monotonicity result with respect to the loss-regularization parameter, \(\eqref{eq-exactness-eps-bar}\) implies \[ \mathbf c_{\varepsilon,\gamma}^{\top}\mathbf R_E < \mathbf c_{\varepsilon,\gamma}^{\top}, \qquad \forall\varepsilon\ge\bar{\varepsilon}, \quad \forall\gamma\in[0,2\pi). \] Hence, under C1’–C3’, the losses-regularized AR-OPF is exact for every \(\varepsilon\ge\bar{\varepsilon}\) and every direction \(\gamma\). In particular, \[ \Gamma_\varepsilon \subseteq \mathcal Y_{\rm AC}, \qquad \forall\varepsilon\ge\bar{\varepsilon}. \tag{A} \]

Consider an arbitrary point \[ y\in\mathcal K_{\bar{\varepsilon}}. \]

If \[ y\in\Gamma_{\bar{\varepsilon}}, \] then \(y\) is AC feasible directly from (A).

If \[ y=y^\infty, \] AC feasibility follows from Assumption 3.

It remains to consider \[ y\in \operatorname{int}(\mathcal K_{\bar{\varepsilon}}), \qquad y\neq y^\infty. \]

Since \(\Gamma_{\bar{\varepsilon}}\) is a simple closed curve, its winding number around \(y\) is nonzero; with a consistent orientation, \[ \operatorname{wind} (\Gamma_{\bar{\varepsilon}},y)=1. \tag{B} \]

Suppose, by contradiction, that \[ y\notin\Gamma_\varepsilon, \qquad \forall\varepsilon>\bar{\varepsilon}. \tag{C} \]

By Assumption 1, the family \(\Gamma_\varepsilon\) then defines a continuous deformation in \[ \mathbb R^2\setminus\{y\}, \] and therefore its winding number around \(y\) is invariant with respect to \(\varepsilon\). Hence, \[ \operatorname{wind}(\Gamma_\varepsilon,y)=1, \qquad \forall\varepsilon>\bar{\varepsilon}. \tag{D} \]

On the other hand, Assumption 3 gives \[ \Gamma_\varepsilon \rightarrow \{y^\infty\} \qquad \text{as }\varepsilon\rightarrow\infty. \] Since \(y\neq y^\infty\), for sufficiently large \(\varepsilon\) the whole curve \(\Gamma_\varepsilon\) is contained in a neighborhood of \(y^\infty\) that does not contain \(y\). It is therefore contractible in \[ \mathbb R^2\setminus\{y\}, \] which implies \[ \operatorname{wind}(\Gamma_\varepsilon,y)=0, \] contradicting (D).

Thus, there must exist \[ \varepsilon_y>\bar{\varepsilon}, \qquad \gamma_y\in[0,2\pi) \] such that \[ y = y_{\rm AR,\varepsilon_y}^{\star}(\gamma_y). \]

By (A), this optimal interface point is AC feasible, and therefore \[ y\in\mathcal Y_{\rm AC}. \]

Since \(y\in\mathcal K_{\bar{\varepsilon}}\) was arbitrary, \[ \mathcal K_{\bar{\varepsilon}} \subseteq \mathcal Y_{\rm AC}. \]