#  Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear
Convex Envelopes

Mohammad Rasoul Narimani,∗ Daniel K. Molzahn†, Katherine R. Davis∗∗, and
Mariesa L. Crow‡ ${\ast}$: Department of Electrical and Computer Engineering,
California State University Northridge (CSUN<EMAIL_ADDRESS>Support from NSF contract #2308498.†: School of Electrical and Computer
Engineering, Georgia Institute of Technology<EMAIL_ADDRESS>Support from
NSF contract #2023140.${**}$: Electrical and Computer Engineering Department,
Texas A&M University<EMAIL_ADDRESS>Electrical and Computer
Engineering Department, Missouri University of Science and Technology.
<EMAIL_ADDRESS>
###### Abstract

AC optimal power flow (AC OPF) is a fundamental problem in power system
operations. Accurately modeling the network physics via the AC power flow
equations makes AC OPF a challenging nonconvex problem. To search for global
optima, recent research has developed a variety of convex relaxations that
bound the optimal objective values of AC OPF problems. The well-known QC
relaxation convexifies the AC OPF problem by enclosing the non-convex terms
(trigonometric functions and products) within convex envelopes. The accuracy
of this method strongly depends on the tightness of these envelopes. This
paper proposes two improvements for tightening QC relaxations of OPF problems.
We first consider a particular nonlinear function whose projections are the
nonlinear expressions appearing in the polar representation of the power flow
equations. We construct a convex envelope around this nonlinear function that
takes the form of a polytope and then use projections of this envelope to
obtain convex expressions for the nonlinear terms. Second, we use certain
characteristics of the sine and cosine expressions along with the changes in
their curvature to tighten this convex envelope. We also propose a coordinate
transformation that rotates the power flow equations by an angle specific to
each bus in order to obtain a tighter envelope. We demonstrate these
improvements relative to a state-of-the-art QC relaxation implementation using
the PGLib-OPF test cases. The results show improved optimality gaps in 68% of
these cases.

###### Index Terms:

Optimal power flow, Convex relaxation

##  I Introduction

The optimal power flow (OPF) problem seeks an operating point that optimizes a
specified objective function (often generation cost minimization) subject to
constraints from the network physics and engineering limits. Using the
nonlinear AC power flow model to accurately represent the power flow physics
results in the AC OPF problem, which is non-convex, may have multiple local
optima [1], and is generally NP-Hard [2]. Since it was first formulated by
Carpentier in 1962 [3], a wide variety of algorithms have been applied to the
OPF problem [4].

A plethora of research has focused on algorithms for obtaining locally optimal
or approximate OPF solutions [5]. Recently, many convex relaxation techniques
have been applied to OPF problems to obtain bounds on the optimal objective
values, certify infeasibility, and in some cases, achieve globally optimal
solutions. The objective value bounds are useful for determining how close a
local solution is to being globally optimal. Thus, local algorithms and
relaxations can be used together in spatial branch-and-bound methods [6].
Power flow relaxations are also valuable for a range of other applications,
including solving robust OPF problems [7, 8], calculating voltage stability
margins [9], exploring feasible operating ranges [10, 11], computing optimal
switching decisions [12], etc. Convex relaxation methods are under active
research with ongoing efforts targeting to improve their tightness. See [13]
for a survey of power flow relaxations.

The well-known “Quadratic Convex” (QC) relaxation encloses the trigonometric,
squared, and product terms in a polar representation of power flow equations
within convex envelopes [14]. Since the quality of these envelopes determines
the tightness of the QC relaxation, a number of research efforts have focused
on improving these envelopes. These include tighter trigonometric envelopes
that leverage sign-definite angle difference bounds [15, 16]; Lifted Nonlinear
Cuts that exploit voltage magnitude and angle difference bounds [15, 17]; cuts
based on the voltage magnitude differences between connected buses [18];
tighter envelopes for the product terms [19, 6]; and other valid inequalities,
convex envelopes, and cutting planes [20, 21]. Most recently, we developed a
“rotated QC” relaxation [22] which applies a coordinate transformation via a
complex per unit base power normalization to tighten envelopes for the
trigonometric terms.

This paper proposes two additional improvements for tightening the QC
relaxation. The first improvement considers a particular nonlinear function
which has projections that are the nonlinear expressions appearing in a polar
representation of the power flow equations. We construct a convex envelope
around this nonlinear function that takes the form of a polytope and then use
projections of this envelope to obtain convex expressions for the nonlinear
terms in the OPF problem. The second improvement uses certain characteristics
of the sine and cosine expressions along with the changes in their curvature
to tighten the first improvement’s convex envelope. We also extend our
previous work on the coordinate transformation [22, 23] via rotating the power
flow equations by an angle specific to each bus in order to obtain a tighter
envelope. A heuristic approach is proposed for choosing reasonable values for
these rotation angles. The proposed relaxation improves the optimality gaps
for 68% of the PGLib-OPF test cases cases compared to a state-of-the-art QC
relaxation [24].

This paper is organized as follows. Sections II and III review the OPF
formulation and the previous QC relaxation, respectively. Section IV presents
a rotated OPF problem and associated QC relaxation with multiple rotation
angles (one per bus). Section V describes a nonlinear function which has
projections that are the nonlinear expressions appearing in the polar
representation of the power flow equations. This section also presents a
convex envelope that encloses this function. Section VI exploits
characteristics of the trigonometric terms to tighten this envelope. Section
VIII presents a method for selecting the rotation angles at each bus to
tighten the relaxation. Bringing this all together, Section VII formulates our
proposed tightened QC relaxation. Section IX provides an empirical evaluation,
and Section X concludes the paper.

##  II Overview of the Optimal Power Flow Problem

This section formulates the OPF problem using a polar voltage phasor
representation. The sets of buses, generators, and lines are $\mathcal{N}$,
$\mathcal{G}$, and $\mathcal{L}$, respectively. The set $\mathcal{R}$ contains
the index of the bus that sets the angle reference. Let
$S_{i}^{d}={P}_{i}^{d}+j{Q}_{i}^{d}$ and $S_{i}^{g}={P}_{i}^{g}+j{Q}_{i}^{g}$
represent the complex load demand and generation, respectively, at bus
$i\in\mathcal{N}$, where $j=\sqrt{-1}$. Let $V_{i}$ and $\theta_{i}$ represent
the voltage magnitude and angle at bus $i\in\mathcal{N}$. Let
$g_{sh,i}+jb_{sh,i}$ denote the shunt admittance at bus $i\in\mathcal{N}$. For
each generator, define a quadratic cost function with coefficients
$c_{2,i}\geq 0$, $c_{1,i}$, and $c_{0,i}$. Upper and lower bounds for all
variables are indicated by $\left(\overline{\,\cdot\,}\right)$ and
$\left(\underline{\,\cdot\,}\right)$, respectively. For ease of exposition,
each line $\left(l,m\right)\in\mathcal{L}$ is modeled as a $\Pi$ circuit with
mutual admittance $g_{lm}+jb_{lm}$ and shunt admittance $jb_{c,lm}$. The
voltage angle difference between buses $l$ and $m$ for $(l,m)\in\mathcal{L}$
is denoted as $\theta_{lm}=\theta_{l}-\theta_{m}$. The complex power flow into
each line terminal $(l,m)\in\mathcal{L}$ is denoted by ${P}_{lm}+j{Q}_{lm}$,
and the apparent power flow limit is $\overline{S}_{lm}$. The OPF problem is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\min\quad\sum_{{i}\in\mathcal{G}}c_{2,i}\left(P_{i}^{g}\right)^{2}+c_{1,i}\,P_{i}^{g}+c_{0,i}$
(1a) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\text{subject to}\quad\left(\forall
i\in\mathcal{N},\;\forall\left(l,m\right)\in\mathcal{L}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!P_{i}^{g}-P_{i}^{d}=g_{sh,i}\,V_{i}^{2}+\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}\\!P_{lm}+\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}\\!\\!P_{ml},$ (1b)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!Q_{i}^{g}-Q_{i}^{d}=-b_{sh,i}\,V_{i}^{2}+\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}\\!\\!Q_{lm}+\\!\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}\\!\\!\\!\\!Q_{ml},$ (1c)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\theta_{r}=0,\quad r\in\mathcal{R},$ (1d)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\underline{P}_{i}^{g}\leq
P_{i}^{g}\leq\overline{P}_{i}^{g},\quad\underline{Q}_{i}^{g}\leq
Q_{i}^{g}\leq\overline{Q}_{i}^{g},$ (1e)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\underline{V}_{i}\leq
V_{i}\leq\overline{V}_{i},$ (1f)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\underline{\theta}_{lm}\leq\theta_{lm}\leq\overline{\theta}_{lm},$
(1g)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!P_{lm}\\!=\\!g_{lm}V_{l}^{2}\\!-\\!g_{lm}V_{l}V_{m}\cos\left(\theta_{lm}\right)\\!-\\!b_{lm}V_{l}V_{m}\sin\left(\theta_{lm}\right),$
(1h)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!Q_{lm}=-\left(b_{lm}+b_{c,lm}/2\right)V_{l}^{2}+b_{lm}V_{l}V_{m}\cos\left(\theta_{lm}\right)$
$\displaystyle\qquad\qquad-g_{lm}V_{l}V_{m}\sin\left(\theta_{lm}\right),$ (1i)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!P_{ml}\\!=\\!g_{lm}V_{m}^{2}\\!-\\!g_{lm}V_{l}V_{m}\cos\left(\theta_{lm}\right)\\!+\\!b_{lm}V_{l}V_{m}\sin\left(\theta_{lm}\right),$
(1j)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!Q_{ml}=-\left(b_{lm}+b_{c,lm}/2\right)V_{m}^{2}+b_{lm}V_{l}V_{m}\cos\left(\theta_{lm}\right)$
$\displaystyle\qquad\qquad+g_{lm}V_{l}V_{m}\sin\left(\theta_{lm}\right),$ (1k)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\left(P_{lm}\right)^{2}+\left(Q_{lm}\right)^{2}\leq\left(\overline{S}_{lm}\right)^{2},\quad\left(P_{ml}\right)^{2}+\left(Q_{ml}\right)^{2}\leq\left(\overline{S}_{lm}\right)^{2}.$
(1l)

The objective (1a) minimizes the generation cost. Constraints (1b) and (1c)
enforce power balance at each bus. Constraint (1d) sets the reference bus
angle. The constraints in (1e) bound the active and reactive power generation
at each bus. Constraints (1f)–(1g), respectively, bound the voltage magnitudes
and voltage angle differences. Constraints (1h)–(1) relate the active and
reactive power flows with the voltage phasors at the terminal buses. The
constraints in (1l) limit the apparent power flows into both terminals of each
line.

##  III Traditional QC Relaxation

As typically formulated, the QC relaxation convexifies the OPF problem (1) by
enclosing the nonconvex expressions ($V_{i}^{2}$, $\forall i\in\mathcal{N}$,
$V_{l}V_{m}\cos(\theta_{lm})$ and $V_{l}V_{m}\sin(\theta_{lm})$,
$\forall(l,m)\in\mathcal{L}$) in convex envelopes [14, 24]. The envelope for
the generic squared function $x^{2}$ is $\langle x^{2}\rangle^{T}$:

$\displaystyle\langle
x^{2}\rangle^{T}=\begin{cases}\widecheck{x}:\begin{cases}\check{x}\geq
x^{2},\\\
\widecheck{x}\leq\left({\overline{x}+\underline{x}}\right)x-{\overline{x}\underline{x}},\\\
\end{cases}\end{cases}$ (2)

where $\widecheck{x}$ is a lifted variable representing the squared term.
Envelopes for the generic trigonometric functions $\sin(x)$ and $\cos(x)$ are
$\left\langle\sin(x)\right\rangle^{S}$ and
$\left\langle\cos(x)\right\rangle^{C}$:

$\displaystyle\left\langle\sin(x)\right\rangle^{S}\\!\\!\\!=\\!\begin{cases}\\!\widecheck{S}\\!:\\!\begin{cases}\widecheck{S}\\!\leq\\!\cos\left(\frac{x^{m}}{2}\right)\\!\\!\left(x\\!-\\!\frac{x^{m}}{2}\right)\\!\\!+\\!\sin\left(\frac{x^{m}}{2}\right)\text{if\leavevmode\nobreak\
}\underline{x}\\!\leq 0\\!\leq\overline{x},\\\
\widecheck{S}\\!\geq\\!\cos\left(\frac{x^{m}}{2}\right)\\!\\!\left(x\\!+\\!\frac{x^{m}}{2}\right)\\!\\!-\\!\sin\left(\frac{x^{m}}{2}\right)\text{if\leavevmode\nobreak\
}\underline{x}\\!\leq 0\\!\leq\overline{x},\\\
\widecheck{S}\geq\frac{\sin\left({\underline{x}}\right)-\sin\left(\overline{x}\right)}{{\underline{x}-\overline{x}}}\left(x-{\underline{x}}\right)+\sin\left({\underline{x}}\right)\text{if\leavevmode\nobreak\
}\underline{x}\geq 0,\\\
\widecheck{S}\leq\frac{\sin\left({\underline{x}}\right)-\sin\left({\overline{x}}\right)}{{\underline{x}-\overline{x}}}\left(x-{\underline{x}}\right)+\sin\left({\underline{x}}\right)\text{if\leavevmode\nobreak\
}{\overline{x}}\leq 0,\end{cases}\end{cases}$ (3)
$\displaystyle\left\langle\cos(x)\right\rangle^{C}=\begin{cases}\widecheck{C}:\\!\begin{cases}\widecheck{C}\leq
1-\frac{1-\cos\left({x^{m}}\right)}{\left(x^{m}\right)^{2}}x^{2},\\\
\widecheck{C}\geq\frac{\cos\left(\underline{x}\right)-\cos\left({\overline{x}}\right)}{{\underline{x}-\overline{x}}}\left(x-{\underline{x}}\right)+\cos\left({\underline{x}}\right),\end{cases}\end{cases}$
(4)

where $x^{m}=\max(\left|\underline{x}\right|,\left|\overline{x}\right|)$. The
envelopes $\left\langle\sin(x)\right\rangle^{S}$ and
$\left\langle\cos(x)\right\rangle^{C}$ in (3) and (4) are valid for
$-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}$.

The lifted variables $\widecheck{S}$ and $\widecheck{C}$ are associated with
the envelopes for the functions $\sin(\theta_{lm})$ and $\cos(\theta_{lm})$.
The QC relaxation of the OPF problem in (1) is:

$\displaystyle\min\quad\sum_{\footnotesize{{i}\in\mathcal{N}}}c_{2,i}\left(P_{i}^{g}\right)^{2}+c_{1,i}\,P_{i}^{g}+c_{0,i}$
(5a) $\displaystyle\text{subject to}\quad\left(\forall
i\in\mathcal{N},\;\forall\left(l,m\right)\in\mathcal{L}\right)$ $\displaystyle
P_{i}^{g}-P_{i}^{d}=g_{sh,i}\,w_{ii}+\\!\\!\\!\\!\\!\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}}\\!\\!\\!P_{lm}+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}}\\!\\!P_{ml},$ (5b) $\displaystyle
Q_{i}^{g}-Q_{i}^{d}=-b_{sh,i}\,w_{ii}+\\!\\!\\!\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}}\\!\\!Q_{lm}+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}}\\!\\!Q_{ml},$ (5c)
$\displaystyle(\underline{V}_{i})^{2}\leq
w_{ii}\leq(\overline{V}_{i})^{2},\qquad w_{ii}\in\left\langle
V_{i}^{2}\right\rangle^{T},$ (5d) $\displaystyle
P_{lm}=g_{lm}w_{ll}-g_{lm}c_{lm}-b_{lm}s_{lm},$ (5e) $\displaystyle
Q_{lm}=-\left(b_{lm}+b_{c,lm}/2\right)w_{ll}+b_{lm}c_{lm}-g_{lm}s_{lm},$ (5f)
$\displaystyle P_{ml}=g_{lm}w_{mm}-g_{lm}c_{lm}+b_{lm}s_{lm},$ (5g)
$\displaystyle
Q_{ml}=-\left(b_{lm}+b_{c,lm}/2\right)w_{mm}+b_{lm}c_{lm}+g_{lm}s_{lm},$ (5h)
$\displaystyle\left(P_{lm}\right)^{2}+\left(Q_{lm}\right)^{2}\leq\left(\overline{S}_{lm}\right)^{2},\quad\left(P_{ml}\right)^{2}+\left(Q_{ml}\right)^{2}\leq\left(\overline{S}_{lm}\right)^{2},$
(5i)
$\displaystyle{c}_{lm}=\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\mu_{lm,k}\,\rho^{(k)}_{lm,1}\rho^{(k)}_{lm,2}\rho^{(k)}_{lm,3},\quad\widecheck{C}_{lm}\in\left\langle\cos(\theta_{lm})\right\rangle^{C},$
$\displaystyle
V_{l}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\mu_{lm,k}\rho^{(k)}_{lm,1},\;V_{m}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\mu_{lm,k}\rho^{(k)}_{lm,2},$
$\displaystyle\widecheck{C}_{lm}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\mu_{lm,k}\rho^{(k)}_{lm,3},$
$\displaystyle\sum_{k=1,\ldots,8}\mu_{lm,k}=1,\quad\mu_{lm,k}\geqslant 0,\quad
k=1,\ldots,8,$ (5j)
$\displaystyle{s}_{lm}=\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\\!\gamma_{lm,k}\,\zeta^{(k)}_{lm,1}\zeta^{(k)}_{lm,2}\zeta^{(k)}_{lm,3},\quad\widecheck{S}_{lm}\in\left\langle\sin(\theta_{lm})\right\rangle^{S},$
$\displaystyle
V_{l}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\gamma_{lm,k}\zeta^{(k)}_{lm,1},\;V_{m}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\gamma_{lm,k}\zeta^{(k)}_{lm,2},$
$\displaystyle\widecheck{S}_{lm}=\\!\\!\\!\\!\sum_{\footnotesize{k=1,\ldots,8}}\\!\\!\
\gamma_{lm,k}\zeta^{(k)}_{lm,3},$
$\displaystyle\sum_{\footnotesize{k=1,\ldots,8}}\gamma_{lm,k}=1,\quad\gamma_{lm,k}\geqslant
0,\quad k=1,\ldots,8,$ (5k) $\displaystyle P_{lm}^{2}+Q_{lm}^{2}\leq
w_{ll}\,\ell_{lm},$ (5l)
$\displaystyle\ell_{lm}\\!=\\!\left(Y_{lm}^{2}-\frac{b_{c,lm}^{2}}{4}\right)\\!\
w_{ll}+Y_{lm}^{2}w_{mm}-2Y_{lm}^{2}c_{lm}-b_{c,lm}Q_{lm},$ (5m)
$\displaystyle\begin{bmatrix}\mu_{lm,1}+\mu_{lm,2}-\gamma_{lm,1}-\gamma_{lm,2}\\\
\mu_{lm,3}+\mu_{lm,4}-\gamma_{lm,3}-\gamma_{lm,4}\\\
\mu_{lm,5}+\mu_{lm,6}-\gamma_{lm,5}-\gamma_{lm,6}\\\
\mu_{lm,7}+\mu_{lm,8}-\gamma_{lm,7}-\gamma_{lm,8}\end{bmatrix}^{\intercal}\begin{bmatrix}\underline{V}_{l}\underline{V}_{m}\\\
\underline{V}_{l}\overline{V}_{m}\\\ \overline{V}_{l}\underline{V}_{m}\\\
\overline{V}_{l}\overline{V}_{m}\end{bmatrix}=0,$ (5n)
$\displaystyle\text{Equations\leavevmode\nobreak\
}\eqref{eq:angref}\text{--}\eqref{eq:bus_angle_limits},\eqref{eq: apparent
power limit sending},$ (5o)

where $\ell_{lm}$ represents the squared magnitude of the current flow into
terminal $l$ of line $(l,m)\in\mathcal{L}$ and
$\left(\cdot\right)^{\intercal}$ is the transpose operator. The relationship
between $\ell_{lm}$ and the power flows $P_{lm}$ and $Q_{lm}$ in (5l) tightens
the QC relaxation [25, 14]. Also, as shown in (5d), $w_{ii}$ is associated
with the squared voltage magnitude at bus $i$. Note that (5i) and (5l) are
convex quadratic constraints, while all other constraints are linear.

The lifted variables $c_{lm}$ and $s_{lm}$ represent relaxations of the
product terms $V_{l}V_{m}\cos(\theta_{lm})$ and $V_{l}V_{m}\sin(\theta_{lm})$,
respectively, with (5) and (5) formulating an “extreme point” representation
of the convex hulls for the product terms $V_{l}V_{m}\widecheck{C}_{lm}$ and
$V_{l}V_{m}\widecheck{S}_{lm}$[26, 6, 19].111An extreme point representation
formulates a polytope as a convex combination of its vertices [26]. The
auxiliary variables $\mu_{lm,k},\;\gamma_{lm,k}\in[0,1]$, $k=1,\ldots,8$,
$(l,m)\in\mathcal{L}$, are used in the formulations of these convex hulls. The
extreme points of $V_{l}V_{m}\widecheck{C}_{lm}$ are
$\rho^{(k)}_{lm}\in[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]\times[\underline{\widecheck{C}}_{lm},\overline{\widecheck{C}}_{lm}]$,
$k=1,\ldots,8$, and the extreme points of $V_{l}V_{m}\widecheck{S}_{lm}$ are
$\zeta^{(k)}\in[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]\times[\underline{\widecheck{S}}_{lm},\overline{\widecheck{S}}_{lm}]$,
$k=1,\ldots,8$. Since sine and cosine are odd and even functions,
respectively, $c_{lm}=c_{ml}$ and $s_{lm}=-s_{ml}$.

“Linking constraints” (5n) associated with the $V_{l}V_{m}$ terms that are
shared in $V_{l}V_{m}\cos(\theta_{lm})$ and $V_{l}V_{m}\sin(\theta_{lm})$ are
also enforced to tighten the QC relaxation [24].

##  IV Exploiting Rotational Degrees of Freedom

To provide tighter envelopes for the nonlinear terms in the OPF problem, our
previous work in [22] considered a polar representation of the branch
admittances, $Y_{lm}\mathrm{e}^{j\delta_{lm}}$, as opposed to the rectangular
admittance representation $g_{lm}+jb_{lm}$ used in (5). We also used a per
unit normalization with a complex base power, i.e., $S_{base}e^{j\psi}$, to
improve the QC relaxation’s trigonometric envelopes. The angle of the base
power, $\psi$, affects the arguments of the trigonometric functions [22]:

$\displaystyle\tilde{S}_{lm}=S_{lm}/e^{j\psi}=\left(Y_{lm}e^{-j(\delta_{lm}+\psi)}+(b_{c,lm}/2)e^{-j(\frac{\pi}{2}+\psi)}\right)V_{l}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad-
Y_{lm}V_{l}V_{m}e^{j(-\delta_{lm}+\theta_{lm}-\psi)},$ (6a)
$\displaystyle\tilde{S}_{ml}=S_{ml}/e^{j\psi}=\left(Y_{lm}e^{-j(\delta_{lm}+\psi)}+(b_{c,lm}/2)e^{-j(\frac{\pi}{2}+\psi)}\right)V_{m}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad-
Y_{lm}V_{m}V_{l}e^{-j(\delta_{lm}+\theta_{lm}+\psi)}.$ (6b)

The angle of the complex base power, $\psi$, linearly enters the arguments of
the trigonometric terms, thus providing a rotational degree of freedom in the
power flow equations [22]. In [22], we exploited this rotational degree of
freedom to improve the QC relaxation’s envelopes. In this section, we extend
this prior work by considering multiple rotation angles (one per bus) as
opposed to the single rotation angle in [22]. We first describe the new
rotated OPF formulation and then formulate its convex relaxation.

###  IV-A Rotated OPF Formulation

Permitting each bus to have a different rotation angle extends our previous
work [22]. We define an angle $\psi_{l}$ for each bus $l$ via a unit-length
complex parameter $e^{j\psi_{l}}$. To ensure that the power balance
constraints at each bus consider quantities that have been rotated
consistently, the power flow equations for each line connected to bus $l$ must
use the same angle $\psi_{l}$. Thus, when formulating the power balance
equations for a specific bus, e.g., bus $l$, the power flow equations for
every line connected to bus $l$ are rotated by a consistent angle, denoted as
$\phi_{l}$. To achieve this, we form rotated versions of the line flow
equations for all lines connected to bus $l$ as follows:

$\tilde{S}_{lm}=\frac{S_{lm}}{e^{j\psi_{l}}},\quad\tilde{S}_{ml}=\frac{S_{ml}}{e^{j\psi_{l}}}.$

Rotated quantities are accented with a tilde,
$\left(\tilde{\,\cdot\,}\right)$. The power generation and load demands are
adapted by the rotation angles as formulated in (7) and (8):

$\displaystyle\begin{bmatrix}\tilde{P}_{l}^{g}\\\
\tilde{Q}_{l}^{g}\end{bmatrix}=\begin{bmatrix}\cos(\psi_{l})&\sin(\psi_{l})\\\
-\sin(\psi_{l})&\cos(\psi_{l})\end{bmatrix}\begin{bmatrix}P_{l}^{g}\\\
Q_{l}^{g}\end{bmatrix},$ (7) $\displaystyle\begin{bmatrix}\tilde{P}_{l}^{d}\\\
\tilde{Q}_{l}^{d}\end{bmatrix}=\begin{bmatrix}\cos(\psi_{l})&\sin(\psi_{l})\\\
-\sin(\psi_{l})&\cos(\psi_{l})\end{bmatrix}\begin{bmatrix}P_{l}^{d}\\\
Q_{l}^{d}\end{bmatrix}.$ (8)

The rotation angles, $\psi_{l}$, linearly enter the arguments of the
trigonometric terms in the power flow equations in the rotated OPF problem, as
shown in (9), where $\Re(\,\cdot\,)$ and $\Im(\,\cdot\,)$ are the real and
imaginary parts of a quantity:

$\displaystyle\tilde{P}_{lm}\\!=\\!\Re(\tilde{S}_{lm})\\!=\\!\left(Y_{lm}\cos(\delta_{lm}+\psi_{l})-(b_{c,lm}/2)\sin(\psi_{l})\right)V_{l}^{2}$
$\displaystyle\qquad\quad\quad\qquad\qquad-
Y_{lm}V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l}),$ (9a)
$\displaystyle\tilde{Q}_{lm}\\!=\\!\Im(\tilde{S}_{lm})\\!=\\!-\left(Y_{lm}\sin(\delta_{lm}+\psi_{l})\\!+\\!(b_{c,lm}/2)\cos(\psi_{l})\right)\\!V_{l}^{2}$
$\displaystyle\qquad\quad\quad\qquad\qquad-
Y_{lm}V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l}),$ (9b)
$\displaystyle\tilde{P}_{ml}\\!=\\!\Re(\tilde{S}_{ml})\\!=\\!\left(Y_{lm}\cos(\delta_{lm}+\psi_{l})\\!-\\!(b_{c,lm}/2)\sin(\psi_{l})\right)\\!V_{m}^{2}$
$\displaystyle\qquad\quad\quad\qquad\qquad-
Y_{lm}V_{m}V_{l}\cos(\theta_{lm}+\delta_{lm}+\psi_{l}),$ (9c)
$\displaystyle\tilde{Q}_{ml}\\!=\\!\Im(\tilde{S}_{ml})\\!=\\!-\left(Y_{lm}\sin(\delta_{lm}+\psi_{l})\\!+\\!(b_{c,lm}/2)\cos(\psi_{l})\right)\\!V_{m}^{2}$
$\displaystyle\qquad\quad\quad\qquad\qquad+Y_{lm}V_{m}V_{l}\sin(\theta_{lm}+\delta_{lm}+\psi_{l}).$
(9d)

Applying (7)–(9) to (1) yields a “rotated” OPF problem. The rotation angles,
$\psi_{l}$, add degrees of freedom to the arguments of the trigonometric terms
in (9). As we will discuss in Section VIII, appropriately chosen values for
$\psi_{l}$ can yield tighter envelopes for these terms.

###  IV-B Rotated QC Relaxation

Enclosing the product and trigonometric terms in the rotated OPF problem yield
a “Rotated QC” (RQC) relaxation:

$\displaystyle\min\quad\textstyle\sum_{\footnotesize{{k}\in\mathcal{G}}}c_{2,k}\left({\tilde{P}_{k}^{g}\cos(\psi_{l})-\tilde{Q}_{k}^{g}\sin(\psi_{l})}\right)^{2}$
$\displaystyle\quad\qquad\qquad+c_{1,k}\left({\tilde{P}_{k}^{g}\cos(\psi_{l})-\tilde{Q}_{k}^{g}\sin(\psi_{l})}\right)+c_{0,k}$
(10a) $\displaystyle\text{subject to}\quad\left(\forall
i\in\mathcal{N},\forall\left(l,m\right)\in\mathcal{L}\right)$
$\displaystyle\tilde{P}_{i}^{g}-\tilde{P}_{i}^{d}=\left(g_{sh,i}\cos(\psi_{l})-b_{sh,i}\sin(\psi_{l})\right)w_{ii}$
$\displaystyle\quad\quad\quad\quad+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}}\tilde{P}_{lm}+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}}\tilde{P}_{ml},$ (10b)
$\displaystyle\tilde{Q}_{i}^{g}-\tilde{Q}_{i}^{d}=-\left(g_{sh,i}\sin(\psi_{l})+b_{sh,i}\cos(\psi_{l})\right)w_{ii}$
$\displaystyle\quad\quad\quad\quad+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}}\tilde{Q}_{lm}+\sum_{\footnotesize{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip 3.0ptm=i\end{subarray}}}\tilde{Q}_{ml},$ (10c)
$\displaystyle(\underline{V}_{i})^{2}\leq
w_{ii}\leq(\overline{V}_{i})^{2},\qquad w_{ii}\in\left\langle
V_{i}^{2}\right\rangle^{T},$ (10d) $\displaystyle\theta_{ref}=0,$ (10e)
$\displaystyle\underline{P}_{i}^{g}\leq\tilde{P}_{i}^{g}\cos(\psi_{l})-\tilde{Q}_{i}^{g}\sin(\psi_{l})\leq\overline{P}_{i}^{g},$
(10f)
$\displaystyle\underline{Q}_{i}^{g}\leq\tilde{Q}_{i}^{g}\cos(\psi_{l})+\tilde{P}_{i}^{g}\sin(\psi_{l})\leq\overline{Q}_{i}^{g},$
(10g) $\displaystyle\underline{V}_{i}\leq
V_{i}\leq\overline{V}_{i},\quad\qquad\underline{\theta}_{lm}\leq\theta_{lm}\leq\overline{\theta}_{lm},$
(10h)
$\displaystyle(\tilde{P}_{lm})^{2}+(\tilde{Q}_{lm})^{2}\leq(\overline{S}_{lm})^{2},\quad(\tilde{P}_{ml})^{2}+(\tilde{Q}_{ml})^{2}\leq(\overline{S}_{lm})^{2},$
(10i)
$\displaystyle\tilde{P}_{lm}\\!=\\!\left(Y_{lm}\cos(\delta_{lm}\\!+\\!\psi_{l})-b_{c,lm}/2\sin(\psi_{l})\right)w_{ll}$
$\displaystyle\qquad-Y_{lm}\tilde{c}_{lm},$ (10j)
$\displaystyle\tilde{Q}_{lm}\\!=\\!-\left(Y_{lm}\sin(\delta_{lm}\\!+\\!\psi_{l})+b_{c,lm}/2\cos(\psi_{l})\right)w_{ll}$
$\displaystyle\qquad-Y_{lm}\tilde{s}_{lm},$ (10k)
$\displaystyle\tilde{P}_{ml}\\!=\\!-Y_{lm}\tilde{c}_{lm}\\!+\\!\left(Y_{lm}\cos(\delta_{lm}\\!+\\!\psi_{l})-b_{c,lm}/2\sin(\psi_{l})\right)w_{mm},$
(10l)
$\displaystyle\tilde{Q}_{ml}\\!=\\!Y_{lm}\tilde{s}_{lm}-\left(Y_{lm}\sin(\delta_{lm}\\!+\\!\psi_{l})+b_{c,lm}/2\cos(\psi_{l})\right)w_{mm},$
(10m) $\displaystyle\tilde{P}_{lm}^{2}+\tilde{Q}_{lm}^{2}\leq
w_{ll}\,\tilde{\ell}_{lm},$ (10n)
$\displaystyle\tilde{\ell}_{lm}=\bigg{(}b_{c,lm}^{2}/4+Y_{lm}^{2}-Y_{lm}b_{c,lm}\cos(\delta_{lm}+\psi_{l})\sin(\psi_{l})$
$\displaystyle\quad\quad+Y_{lm}b_{c,lm}\sin(\delta_{lm}+\psi_{l})\cos(\psi_{l})\bigg{)}V_{l}^{2}+Y_{lm}^{2}V_{m}^{2}$
$\displaystyle\quad\quad+\left(-2Y_{lm}^{2}\cos(\delta_{lm}+\psi_{l})+Y_{lm}b_{c,lm}\sin(\psi_{l})\right)\tilde{c}_{lm}$
$\displaystyle\quad\quad+\left(2Y_{lm}^{2}\sin(\delta_{lm}+\psi_{l})+Y_{lm}b_{c,lm}\cos(\psi_{l})\right)\tilde{s}_{lm},$
(10o)
$\displaystyle\tilde{c}_{lm}=\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\lambda_{k}\,\eta^{(k)}_{1}\eta^{(k)}_{2}\eta^{(k)}_{3}\\!\\!,\;\;\;\;\tilde{s}_{lm}=\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\lambda_{k}\,\eta^{(k)}_{1}\eta^{(k)}_{2}\eta^{(k)}_{4}\\!\\!\\!\\!,$
$\displaystyle
V_{l}=\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\\!\\!\lambda_{k}\eta^{(k)}_{1},\;\;V_{m}=\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\\!\\!\lambda_{k}\eta^{(k)}_{2}\\!\\!,\;\;\tilde{S}_{lm}=\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\\!\\!\lambda_{k}\eta^{(k)}_{4}\\!\\!,$
$\displaystyle\tilde{C}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\\!\\!\\!\\!\lambda_{k}\eta^{(k)}_{3}\,\\!\\!\\!,\\!\\!\\!\sum_{k=1,\ldots,4\tilde{N}}\\!\\!\\!\\!\lambda_{k}=1,\qquad\\!\\!\\!\\!\\!\\!\lambda_{k}\geqslant
0,\quad\\!\\!\\!k=1,\ldots,4\tilde{N},$
$\displaystyle\tilde{C}_{lm}\in\left\langle\cos(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{C}\\!\\!,\;\tilde{S}_{lm}\in\left\langle\sin(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{S}\\!\\!$
(10p) $\displaystyle\text{Equation\leavevmode\nobreak\
}\text{\eqref{eq:qc_linking}}.$ (10q)

##  V Convexifications and Projections of an Alternative Nonlinear Function

In this section, we propose tighter convex envelopes for the nonlinear terms
in the power flow equations. Most previous research convexifies the
$V_{l}V_{m}\cos(\theta_{lm})$ and $V_{l}V_{m}\sin(\theta_{lm})$ terms in
(1h)–(1) independently. Instead of individually convexifying these terms, we
focus on a different function,
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$.
Fig. 1 shows a projection of this function. As we will describe, projections
of a convexified form of this function provide tighter envelopes for the
product terms in the rotated power flow equations (9). We first summarize
prior QC formulations for comparison purposes and then discuss our proposed
formulation.

$V_{l}V_{m}\cos(x)$$V_{l}V_{m}\sin(x)$$V_{l}V_{m}\sin(x)\cos(x)$ Figure 1: A
projection of $V_{l}V_{m}\cos(x)\sin(x)$. The argument $x$ corresponds to
$\theta_{lm}-\delta_{lm}-\psi_{l}$.

|
---

$\sin(x)$$\cos(x)$$V_{l}$ (a) The original QC formulation from [14].

$\sin(x)$$\cos(x)$$V_{l}$ (b) The rotated QC relaxation from [22].

$\sin(x)$$\cos(x)$$V_{l}$ (c) The proposed QC relaxation from Section V-B. The
envelope (light green region) appears nearly coincident with the function
itself (orange region).

Figure 2: Projections of various envelopes for the function
$V_{l}V_{m}\cos(x)\sin(x)$ in terms of $V_{l}$, $\cos(x)$, and $\sin(x)$. The
argument $x$ indicates the angle difference $\theta_{lm}$ for the original QC
relaxation in Fig. 2a and the rotated argument from the polar admittance
representation, $\theta_{lm}-\delta_{lm}-\psi_{l}$, for the rotated QC
relaxations in Figs. 2b and 2c. The pink region common to Figs. 2a–2c is the
function $V_{l}V_{m}\cos(x)\sin(x)$ that we seek to enclose in a convex
envelope. The light green regions in Figs. 2a, 2b, and 2c are the surfaces of
the convex envelopes proposed in the original QC relaxation [14], the rotated
QC relaxation from [22], and our proposed formulation from Section V-B,
respectively. $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
---
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
---
Figure 3: Projection of the function
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
in terms of $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$. The black line is the function
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
that we seek to enclose in a convex envelope. The light green lines are
tangent to the function at the equally spaced black points. The convex region
enclosed by these lines is depicted in light green, encompassing the nonconvex
trigonometric function shown in orange. The extreme points of the convex
envelope for this function are shown by the red squares and are at the
intersections of the green lines. See Algorithm 1 for details.

###  V-A Previous Envelopes for Product Terms

By independently convexifying the terms in the products
$V_{l}V_{m}\cos(\theta_{lm})$ and $V_{l}V_{m}\sin(\theta_{lm})$, the original
QC relaxation proposed in [14] effectively encloses these terms in a rectangle
defined by the bounds on $\cos(\theta_{lm})$, $\sin(\theta_{lm})$, $V_{l}$,
and $V_{m}$. Fig. 2a shows a projection of this envelope.

The approach in [22] also uses the bounds on
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$,
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$, $V_{l}$, and $V_{m}$ to create a
rectangle enclosing the expressions in (9a) and (9). Another rectangle is
similarly constructed using the bounds on
$\cos(\theta_{lm}+\delta_{lm}+\psi_{l})$,
$\sin(\theta_{lm}+\delta_{lm}+\psi_{l})$, $V_{l}$, and $V_{m}$ in (9) and
(9d). Considering the intersection of these rectangles yields a convex
envelope in the form of a polytope. As shown in Fig. 2b, the envelopes from
the rotated QC relaxation [22] can be tighter than those from the original QC
relaxation [14].

###  V-B Proposed Envelopes for Product Terms

This paper tightens the QC relaxation by constructing an envelope tailored to
the function
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$.
To accomplish this, we consider the projection of this function in terms of
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$, as shown by the solid black line in
Fig. 3. In this projection, the function is an arc of the unit circle defined
using the angle difference bounds $\underline{\theta}_{lm}$ and
$\overline{\theta}_{lm}$, i.e., $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$
and $\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$.

To construct the convex envelope in Fig. 3, we first compute the green lines
that are tangent at the equally spaced black dots. The extreme points defining
the polytope that forms the convex envelope are then obtained from the
intersections of neighboring tangent lines, denoted by the red squares in Fig.
3. Finally, the polytope is extended using the bounds on $V_{l}$ and $V_{m}$
in the same manner as in both the original and rotated QC relaxations [14,
22].

Formally, let
$\mathcal{T}_{lm}=\\{(C_{lm}^{int,1},S_{lm}^{int,1}),\,(C_{lm}^{int,2},S_{lm}^{int,2}),\,\ldots,\,\\\
(C_{lm}^{int,N_{seg}},S_{lm}^{int,N_{seg}})\\}$ denote the coordinates of the
extreme points (red squares) in Fig. 3, where $N_{seg}$ is a user-selected
parameter for the number of extreme points. Extend these extreme points using
the bounds on the voltage magnitudes to obtain the extreme points for a convex
envelope enclosing the function
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$,
denoted as
$\eta^{(k)}_{lm}\in[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]\times\mathcal{T}_{lm}$,
$k=1,\ldots,4N_{seg}$. Algorithm 1 describes how to compute these extreme
points.

1 function
Extreme_Point($\overline{\theta}_{lm}$,$\underline{\theta}_{lm}$,$\delta_{lm}$,$\psi_{l}$,$N_{seg}$)

2 $U\leftarrow\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$,
$L\leftarrow\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$.

3 Divide the arc between $U$ and $L$ into $N_{seg}$ equal segments.
for $i=1,\ldots,N_{seg}$ do

4
$E=\left[(\tilde{C}_{lm,1},\tilde{S}_{lm,1}),\ldots,(\tilde{C}_{lm,N_{seg}},\tilde{S}_{lm,N_{seg}})\right]$
$\leftarrow$ Intersection of the tangent lines corresponding to both ends of
the $i$-th segment.

5 end for

6
$EXT=\left[E,(\tilde{C}_{lm,L},\tilde{S}_{lm,L}),(\tilde{C}_{lm,U},\tilde{S}_{lm,U})\right]$
$\leftarrow$ Add the points on the closest tangent line to the arc at the
endpoints $U$ and $L$ to $E$; see Fig. 3.

7 Extend the resulting points by the upper and lower bounds on voltage
magnitudes,
$[EXT]\times[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]$.

8 return
$[EXT]\times[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]$.

9

Algorithm 1 Compute Extreme Points

By introducing auxiliary variables denoted as $\lambda_{lm,k}\in[0,1]$,
$k=1,\ldots,4N_{seg}$, we next form a convex envelope for the function
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
as the convex combination of the extreme points $\eta^{(k)}$. Finally, we take
projections of this convex envelope to obtain envelopes enclosing the products
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$.

Using this procedure, we obtain the following constraints that link the lifted
variables $c_{lm}$ and $s_{lm}$ corresponding to the expressions
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ with the remainder of the
variables in the problem (i.e., the lifted variables $\widecheck{C}_{lm}$ and
$\widecheck{S}_{lm}$ for the cosine and sine terms,
$\widecheck{C}_{lm}\in\left\langle\cos(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{C}\\!\\!$
and
$\widecheck{S}_{lm}\in\left\langle\sin(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{S}$,
and the variables $\theta_{lm}$, $V_{l}$, and $V_{m}$):

$\displaystyle\tilde{c}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\lambda_{lm,k}\,\eta^{(k)}_{lm,1}\eta^{(k)}_{lm,2}\eta^{(k)}_{lm,4}\\!\\!,\;\;\;\;V_{l}=\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,1},\;\;$
$\displaystyle\tilde{s}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\,\eta^{(k)}_{lm,1}\eta^{(k)}_{lm,2}\eta^{(k)}_{lm,5},\leavevmode\nobreak\
V_{m}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,2},$
$\displaystyle\tilde{C}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,4},\quad\tilde{S}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,5}\,,\;\;$
$\displaystyle\theta_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,3},$
$\displaystyle\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\lambda_{lm,k}=1,\qquad\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\geqslant
0,\quad\\!\\!\\!k=1,\ldots,4N_{seg}.$ (11)

Fig. 2c visualizes a projection of the convex envelope obtained using this
approach. Comparing Fig. 2c with Figs. 2a and 2b demonstrates the superiority
of the proposed approach in providing tighter envelopes compared to those in
[14, 22]. Note that (V-B) precludes the need for the linking constraint (5n)
that relates the common term $V_{l}V_{m}$ in the products
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$.

##  VI Tighter Trigonometric Envelopes

Having addressed the product terms, we next turn our attention to the
trigonometric functions $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$. This section leverages certain
characteristics of the sine and cosine functions along with the changes in
their curvature to provide tighter convex envelopes derived using multiple
tangent lines to these functions. Figs. 5a and 5b illustrate these tangent
lines for the sine and cosine functions, respectively. The remainder of this
section focuses solely on the cosine envelopes since the sine envelopes can be
constructed as rotated versions of the cosine envelopes. In this section, we
present an overview of the key ideas without delving into extensive
mathematical details. The complete mathematical derivations related to the
concepts discussed in this section can be found in the appendix.

$\theta_{lm}-\delta_{lm}-\psi_{l}$$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
$\underline{\mathcal{R}}_{lm}$
---
$\overline{\mathcal{R}}_{lm}$
---
$L$
---
$U$
---
Figure 4: Envelope for $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$,
$\theta_{lm}\in\text{[}-60^{\circ},60^{\circ}\text{]}$,
$\delta_{lm}=-5^{\circ}$, and $\psi_{l}=0^{\circ}$. The curvature of the sine
function changes within the interval considered here.
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$ are shown by
red and yellow stars, respectively. Tangent lines to these points from the
endpoints of the interval are plotted. The endpoints of the interval, i.e.,
$L=\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$U=\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$, are shown by black circles.
The trigonometric function’s curvature does not change sign within the
intervals $[L,\underline{\mathcal{R}}_{lm}]$ and
$[\overline{\mathcal{R}}_{lm},U]$. Thus, tangent lines to points in these
ranges can be selected to form an enclosing envelope for the trigonometric
function.

We note that the method proposed in this section is a specific form of an
approach recently developed in [27] that uses a sequence of linear programming
relaxations which converge towards the convex hull of a univariate function.
Our proposed method is an explicit form for a sequence of polyhedral
relaxations that convexify the trigonometric terms in the power flow
equations. In contrast to the approach in [27], which requires solving a
series of linear programs to identify the convex hull of a univariate
function, our proposed method does not necessitate solving any optimization
problems to construct convex envelopes for the trigonometric function.

$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$

$\theta_{lm}-\delta_{lm}-\psi_{l}$ (a) Envelope for
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l}),\theta_{lm}\in{[}{-}75^{\circ},75^{\circ}{]}$
and $\psi_{l}+\delta_{lm}=5^{\circ}$

$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$

$\theta_{lm}-\delta_{lm}-\psi_{l}$ (b) Envelope for
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$,
$\theta_{lm}\in{[}{-}75^{\circ},75^{\circ}{]}$ and
$\psi_{l}+\delta_{lm}=5^{\circ}$

Figure 5: Convex regions (light green) constructed using tangents (solid green
lines) to the sine and cosine functions,
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$, as described by Algorithm 2 in the
appendix.

Convex envelopes constructed using tangent lines were also previously used to
convexify the cosine function in the Linear Programming AC (LPAC)
approximation proposed in [28]. However, those envelopes are specific to
arguments ranging from ${-}90^{\circ}$ and $90^{\circ}$. Since the arguments
for the trigonometric functions in our formulation change with the values of
$\delta_{lm}$ and $\psi_{l}$, we must consider ranges that admit any possible
argument, including ranges for which the curvature changes. This is
challenging since a tangent line to the trigonometric function at one point
may intersect the function in another point, with the resulting envelope
failing to enclose the function.

This section addresses this issue by finding the largest ranges of values for
which tangent lines to the trigonometric function form an enclosing envelope.
These ranges are defined by the lower bound of the argument,
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$, to a point denoted as
$\underline{\mathcal{R}}_{lm}$ as well as from a point denoted as
$\overline{\mathcal{R}}_{lm}$ to the upper bound of the argument,
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. Fig. 4 shows
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$ via red and
yellow stars, respectively. More specifically, to assist in finding
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$, we define a
function $F(\theta_{lm})$ which represents the difference between the
trigonometric function $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ itself and the
line which connects the endpoints of $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
at $\underline{\theta}_{lm}$ and $\overline{\theta}_{lm}$:

$\displaystyle
F(\theta_{lm})=\cos(\theta_{lm}-\delta_{lm}-\psi_{l})-\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})$
$\displaystyle\qquad\quad-\frac{\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})-\cos(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l})}{\overline{\theta}_{lm}-\underline{\theta}_{lm}}$
$\displaystyle\qquad\quad\times\left(\theta_{lm}-\overline{\theta}_{lm}-\psi_{l}\right).$
(12)

The set of zeros of the first derivative of $F(\theta_{lm})$, i.e., the set of
solutions to $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$, is a key quantity to
determine if the curvature of $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ changes
between $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$.

If $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$ has no solutions, then the
curvature of the trigonometric function does not change between
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. Accordingly, any tangent lines
can be selected to form an enclosing envelope for the trigonometric function.
We select equally spaced tangent lines within the range
$[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l},\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}]$
as illustrated in Fig. 5a.

Conversely, if $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$ has one or more
solutions, then the trigonometric function’s curvature changes. This
necessitates special consideration, i.e., finding
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$, to select
appropriate tangent lines to $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$.

To compute $\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$
for the cosine function, we first identify the tangent line to the cosine
function that also passes through the endpoint
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. We then define another
auxiliary function representing the difference between this tangent line and
the cosine function. The value of $\underline{\mathcal{R}}_{lm}$ is given by
the root of the first derivative of this auxiliary function that is between
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. Note that the voltage angle
difference restriction, i.e., ${-}90^{\circ}\leq\theta_{lm}\leq 90^{\circ}$,
ensures that the sine and cosine functions have at most one curvature sign
change in any given interval. $\overline{\mathcal{R}}_{lm}$ is computed
similarly by formulating the tangent line to the cosine function that also
pass through the endpoint $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
following the steps above. A comprehensive explanation of how to compute
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$ is available
in the appendix.

By construction, the trigonometric function’s curvature does not change sign
within the intervals
$[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l},\underline{\mathcal{R}}_{lm}]$
and
$[\overline{\mathcal{R}}_{lm},\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}]$.
Accordingly, tangent lines to points in these ranges can be selected to form
an enclosing envelope for the trigonometric function. As shown in Fig. 5b, we
choose equally spaced tangent lines within each of these ranges.

Our proposed QC relaxation uses envelopes
$\left\langle\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{S^{\prime}}$
and
$\left\langle\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{C^{\prime}}$
based on the tangent lines described above. The formulations of the upper and
lower bounds of these envelopes depend on the curvature’s sign and the number
of solutions for $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$. For brevity, we
present a summary of the envelopes for the cosine function. Future details for
the cosine function along with expressions for the sine envelopes are given in
the appendix.

If $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$ has one or more solutions:
$\displaystyle\left\langle\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{C^{\prime}}=\begin{cases}\\!\widecheck{C}^{\prime}\\!:\begin{cases}\widecheck{C}^{\prime}\\!\leq\overline{L}_{\cos,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\\\
\widecheck{C}^{\prime}\\!\geq\underline{L}_{\cos,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\end{cases}\end{cases}$ (13a) If
$\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$ has no solutions & curvature sign is
negative:
$\displaystyle\left\langle\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{C^{\prime}}=\begin{cases}\\!\widecheck{C}^{\prime}\\!:\begin{cases}\widecheck{C}^{\prime}\\!\leq\overline{L}_{\cos,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\\\
\widecheck{C}^{\prime}\\!\geq\underline{L}_{\cos,i},\leavevmode\nobreak\
i=1\end{cases}\end{cases}$ (13b) If $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$
has no solutions & curvature sign is positive:
$\displaystyle\left\langle\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{C^{\prime}}=\begin{cases}\\!\widecheck{C}^{\prime}\\!:\begin{cases}\widecheck{C}^{\prime}\\!\leq\overline{L}_{\cos,i},\leavevmode\nobreak\
i=1\\\
\widecheck{C}^{\prime}\\!\geq\underline{L}_{\cos,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\end{cases}\end{cases}$ (13c)

where $\overline{L}_{\sin,i}$, $\overline{L}_{\cos,i}$,
$\underline{L}_{\sin,i}$, and $\underline{L}_{\cos,i}$ are the $i^{th}$
tangent lines which upper and lower bound the sine and cosine functions,
respectively. When the sign of the trigonometric function’s curvature does not
change within an interval, either the upper or lower boundary of the envelope
(depending on the sign of the curvature) is defined via the line connecting
the endpoints of the trigonometric function, as defined in (13) and (13).

The envelopes in (13) are valid for any argument
$\theta_{lm}-\delta_{lm}-\psi_{l}$. The lifted variables
$\widecheck{S}^{\prime}$ and $\widecheck{C}^{\prime}$ are associated with the
envelopes for the functions $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$. Detailed expressions for
$\overline{L}_{\sin,i}$, $\overline{L}_{\cos,i}$, $\underline{L}_{\sin,i}$,
and $\underline{L}_{\cos,i}$ are available in the appendix.

##  VII The Linear Rotated QC Relaxation

This section brings together each improvement from this paper (multiple angle
rotations associated with each bus in Section IV, new envelopes for the
product terms in Section V, and tighter trigonometric envelopes in Section VI)
to formulate our proposed QC relaxation in (14) below. We subsequently call
this formulation the “Linear Rotated QC” (LRQC) relaxation since the polytopes
for the convex envelopes are constructed with linear inequalities and the
power flow equations are rotated versions of the original expressions.

$\displaystyle\min\quad\eqref{eq:RQC obj}$ (14a) $\displaystyle\text{subject
to}\quad\left(\forall
i\in\mathcal{N},\forall\left(l,m\right)\in\mathcal{L}\right)$
$\displaystyle\text{Equations\leavevmode\nobreak\ }\eqref{eq:RQC active power
injection}\text{--}\eqref{eq:shifted OPF power flow limit lm},$ (14b)
$\displaystyle\tilde{P}_{lm}\\!=\\!\left(Y_{lm}\cos(\delta_{lm}\\!+\\!\psi_{l})-b_{c,lm}/2\sin(\psi_{l})\right)w_{ll}$
$\displaystyle\qquad-Y_{lm}\tilde{c}^{\prime}_{lm},$ (14c)
$\displaystyle\tilde{Q}_{lm}\\!=\\!-\left(Y_{lm}\sin(\delta_{lm}\\!+\\!\psi_{l})+b_{c,lm}/2\cos(\psi_{l})\right)w_{ll}$
$\displaystyle\qquad-Y_{lm}\tilde{s}^{\prime}_{lm},$ (14d)
$\displaystyle\tilde{P}_{ml}\\!=\\!-Y_{lm}\tilde{c}^{\prime}_{lm}\\!+\\!\left(Y_{lm}\cos(\delta_{lm}\\!+\\!\psi_{l})-b_{c,lm}/2\sin(\psi_{l})\right)w_{mm},$
(14e)
$\displaystyle\tilde{Q}_{ml}\\!=\\!Y_{lm}\tilde{s}^{\prime}_{lm}-\left(Y_{lm}\sin(\delta_{lm}\\!+\\!\psi_{l})+b_{c,lm}/2\cos(\psi_{l})\right)w_{mm},$
(14f) $\displaystyle\tilde{P}_{lm}^{2}+\tilde{Q}_{lm}^{2}\leq
w_{ll}\,\tilde{\ell}^{\prime}_{lm},$ (14g)
$\displaystyle\tilde{\ell}^{\prime}_{lm}=\bigg{(}b_{c,lm}^{2}/4+Y_{lm}^{2}-Y_{lm}b_{c,lm}\cos(\delta_{lm}+\psi_{l})\sin(\psi_{l})$
$\displaystyle\quad\quad+Y_{lm}b_{c,lm}\sin(\delta_{lm}+\psi_{l})\cos(\psi_{l})\bigg{)}V_{l}^{2}+Y_{lm}^{2}V_{m}^{2}$
$\displaystyle\quad\quad+\left(-2Y_{lm}^{2}\cos(\delta_{lm}+\psi_{l})+Y_{lm}b_{c,lm}\sin(\psi_{l})\right)\tilde{c}^{\prime}_{lm}$
$\displaystyle\quad\quad+\left(2Y_{lm}^{2}\sin(\delta_{lm}+\psi_{l})+Y_{lm}b_{c,lm}\cos(\psi_{l})\right)\tilde{s}^{\prime}_{lm},$
(14h)
$\displaystyle\tilde{c}^{\prime}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\lambda_{lm,k}\,\xi^{(k)}_{lm,1}\xi^{(k)}_{lm,2}\xi^{(k)}_{lm,4},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
V_{l}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\xi^{(k)}_{lm,1},$
$\displaystyle\tilde{s}^{\prime}_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\lambda_{lm,k}\,\xi^{(k)}_{lm,1}\xi^{(k)}_{lm,2}\xi^{(k)}_{lm,5},$
$\displaystyle\theta_{lm}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,N_{seg}}\\!\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\eta^{(k)}_{lm,3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
V_{m}=\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\lambda_{lm,k}\xi^{(k)}_{lm,2},$
$\displaystyle\tilde{S}_{lm}^{\prime}=\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\\!\lambda_{lm,k}\xi^{(k)}_{lm,5},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{C}_{lm}^{\prime}=\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\lambda_{lm,k}\xi^{(k)}_{lm,4},$
$\displaystyle\\!\\!\\!\sum_{k=1,\ldots,4N_{seg}}\\!\\!\\!\\!\lambda_{lm,k}=1,\qquad\\!\\!\\!\\!\\!\\!\lambda_{lm,k}\geqslant
0,\quad\\!\\!\\!k=1,\ldots,4N_{seg},$
$\displaystyle\tilde{C}_{lm}^{\prime}\in\left\langle\cos(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{C^{\prime}}\\!\\!,\;\tilde{S}_{lm}^{\prime}\in\left\langle\sin(\theta_{lm}\\!-\\!\delta_{lm}\\!-\\!\psi_{l})\right\rangle^{S^{\prime}}\\!\\!.$
(14i)

The lifted variables $\tilde{c}^{\prime}_{lm}$ and $\tilde{s}^{\prime}_{lm}$
represent relaxations of the product terms
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$, respectively, with (14i)
formulating an “extreme point” representation of the convex hulls for the
product terms
$V_{l}V_{m}\widecheck{C}^{\prime}_{lm}\widecheck{S}^{\prime}_{lm}$. The
extreme points of
$V_{l}V_{m}\widecheck{C}^{\prime}_{lm}\widecheck{S}^{\prime}_{lm}$ are
$\xi^{(k)}\in[\underline{V_{l}},\overline{V_{l}}]\times[\underline{V_{m}},\overline{V_{m}}]\times\mathcal{T}_{lm}$,
$k=1,\ldots,4N_{seg}$. $\mathcal{T}_{lm}$ denotes the coordinates of the
extreme points (red squares) in Fig. 3

|  |
---|---
|
|

$V_{l}V_{m}\sin(x)$

$V_{l}V_{m}\cos(x)$

$V_{l}V_{m}\cos(x)\sin(x)$

(a) Envelope from the original QC relaxation in [24].

$V_{l}V_{m}\sin(x)$

$V_{l}V_{m}\cos(x)$

$V_{l}V_{m}\cos(x)\sin(x)$

(b) Envelope from the original QC relaxation in [24] (alternate view).

$V_{l}V_{m}\sin(x)$

$V_{l}V_{m}\cos(x)$

$V_{l}V_{m}\cos(x)\sin(x)$

(c) Envelope from the proposed LRQC relaxation (14).

$V_{l}V_{m}\sin(x)$

$V_{l}V_{m}\cos(x)$

$V_{l}V_{m}\cos(x)\sin(x)$

(d) Envelope from the proposed LRQC relaxation (14) (alternate view).

$V_{l}V_{m}\sin(x)$

$V_{l}V_{m}\cos(x)$

$V_{l}V_{m}\cos(x)\sin(x)$

(e) Envelope from the proposed LRQC relaxation (14). The black and white
region shows the entire function $V_{l}V_{m}\cos(x)\sin(x)$ while the orange
region is the portion of this function within the voltage magnitude and phase
angle difference bounds as in Figs. 6a–6d.

Figure 6: Projections of the function $V_{l}V_{m}\cos(x)\sin(x)$ in terms of
$V_{l}V_{m}\cos(x)$ and $V_{l}V_{m}\sin(x)$. The argument $x$ indicates the
angle difference $\theta_{lm}$ for the original QC relaxation in Figs. 6a and
6b and the rotated argument from the polar admittance representation,
$\theta_{lm}-\delta_{lm}-\psi_{l}$, for the LRQC relaxation from (14) in Figs.
6c–6e. The orange region common to Figs. 6a–6e is the function
$V_{l}V_{m}\cos(x)\sin(x)$ that we seek to enclose in a convex envelope. The
light green regions correspond to the surfaces of the convex envelopes
proposed in the original QC relaxation [14] for Figs. 6a and 6b and the
proposed LRQC relaxation (14) from Section VII for Figs. 6c–6e. Note that
Figs. 6b and 6d on the right side show rotated views of the same projections
as Figs. 6a and 6c, respectively, on the left side.

To illustrate the tightness of the envelopes in the LRQC relaxation, Figs.
6a–6e show a projection of the function
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
along with the convex envelopes from both the approach in [24] and our
proposed method. The orange region common to each figure corresponds to
different views of the function itself and the light green polytopes are the
convex envelopes. Figs. 6a–6b show envelopes from the original QC relaxation
and Figs. 6c–6d show our proposed envelopes. Observe that our proposed
envelopes can be significantly tighter than those in the original QC
relaxation. Fig. 6e shows these same envelopes with the full function
$V_{l}V_{m}\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
where regions outside of the voltage magnitude and angle difference bounds are
transparent rather than orange.

##  VIII Choosing the Rotation Angles

The rotation angles $\psi_{l}$ play an important role in the performance of
the proposed LRQC relaxation (14). Since the admittance angles $\delta_{lm}$
vary between branches, different rotation angles $\psi_{l}$ may yield tighter
envelopes for the trigonometric terms $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
and $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$. To illustrate the impact of the
rotation angle $\psi_{l}$ on the convex envelopes, Fig. 7 shows two envelopes
associated with different choices of $\psi_{l}$. This section proposes and
analyzes a heuristic approach for choosing the rotation angle $\psi_{l}$ for
each bus. This heuristic is based on minimizing the convex envelopes’ volumes
using the intuition that smaller volumes correspond to tighter envelopes. The
results in Section IX show this heuristic’s merits via improved optimality
gaps for various test cases.

For each bus, we determine the best $\psi_{l}$ by calculating the summation of
volumes associated with the convex envelopes that enclose
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
terms for all the lines connected to bus $l$. To this end, we begin by
sweeping the value of $\psi_{l}$ from $-90^{\circ}$ to $90^{\circ}$ in
$1^{\circ}$ increments. We then compute the volume of the polytope depicted in
Fig. 7 for all lines connected to bus $l$. Finally, we choose the rotation
angle for each bus based on the minimum sum of the volumes.

Finding the volume-minimizing rotation angle for each bus is a time consuming
process especially for larger test systems due to the need to perform many
volume computations. Since this volume-minimization heuristic only requires
knowledge of the line admittances connected to each bus, the volume
computations can be performed once offline and can be reused for multiple OPF
problems with the same system so long as the topology remains unchanged.
Furthermore, the evaluation of this heuristic can be performed in parallel for
each line. If the topology does change, only the values of $\psi_{l}$
associated with buses $l$ that are directly associated with the modified
topology need to be updated. Thus, while potentially time consuming in its
first evaluation, we anticipate this heuristic would nevertheless be
practically relevant. However, if one wishes to avoid time-consuming offline
computations, we observed that most of the resulting rotation angles
$\psi_{l}$ for the PGLib-OPF test cases are in the intervals
$[-90^{\circ},-85^{\circ}]$ and $[85^{\circ},90^{\circ}]$. The numerical
results indicate that selecting a value of $\psi^{\dagger}_{l}=-85^{\circ}$
for all buses $l$ yields small optimality gaps for most test cases, suggesting
that this value could be used directly with limited impacts on the
relaxation’s tightness.

$V_{l}V_{m}\sin(x)$$V_{l}V_{m}\cos(x)$$V_{l}V_{m}\sin(x)\cos(x)$
$\psi_{l}=-67.8^{\circ}$
---
$\delta_{lm}=83^{\circ}$
$\psi_{l}=-174^{\circ}$
---
$\delta_{lm}=83^{\circ}$
Figure 7: Projection of $V_{l}V_{m}\cos(x)\sin(x)$ (orange regions), where $x$
indicates the $\theta_{lm}-\delta_{lm}-\psi_{l}$, for different values of
$\psi_{l}$. The associated envelopes in light orange and light green show how
choosing $\psi_{l}$ affects the convex envelopes.

##  IX Numerical Results

This section demonstrates the proposed improvements using selected test cases
from the PGLib-OPF v18.08 benchmark library [29]. With large optimality gaps
between the objective values from the best known local optima and the lower
bounds from various relaxations, these test cases challenge a variety of
solution algorithms and are therefore suitable for our purposes. Our
implementations use Julia 0.6.4, JuMP v0.18 [30], PowerModels.jl [31], and
Gurobi 8.0 as modeling tools and the solver. The results are computed using a
laptop with an Intel i7 1.80 GHz processor and 16 GB of RAM.

Table I summarizes the results from applying the QC (5), RQC (10), and the
proposed LRQC (14) relaxations to selected test cases. To get illustrative
results for the LRQC relaxation, we set $N_{seg}=5$. The first column lists
the test cases. The next group of columns represents optimality gaps as
defined in (15). The optimality gaps are computed using the local solutions to
the non-convex problem (1) from PowerModels.jl.

$\displaystyle\text{\emph{Optimality\leavevmode\nobreak\
Gap}}=\left(\dfrac{\text{\emph{Local\leavevmode\nobreak\
Solution}}-\text{\emph{QC Bound}}}{\text{\emph{Local\leavevmode\nobreak\
Solution}}}\right).$ (15)

Upon comparing the second and third columns of Table I, it is evident that the
RQC relaxation from our previous work in [22] outperforms the original QC
relaxation for all test cases by converging to tighter lower bounds. The best
rotation angle $\psi^{*}$ for the RQC relaxation in the fourth column of Table
I is obtained by sweeping $\psi$ from $-90^{\circ}$ to $90^{\circ}$ in steps
of $0.5^{\circ}$. The RQC relaxation in [22] with $\psi^{*}$ (the best value
of $\psi$ for each case) provides optimality gaps that are at least as good as
those obtained by the original QC relaxation (5) for all test cases, resulting
in an improvement of 1.36% on average compared to the original QC relaxation.

By comparing the fifth and third columns of Table I, it can be seen that the
proposed LRQC relaxation is superior to the RQC relaxation as it converges to
tighter lower bounds for all test cases. The fifth column in Table I lists the
results for the LRQC relaxation with rotation angles computed by minimizing
the volume of the envelope enclosing the function
$V_{l}V_{m}\cos(\theta_{lm}-\delta_{lm}-\psi_{l})\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$.
Comparing the fifth column with the second and third columns demonstrates that
the LRQC relaxation improves the optimality gaps for all the test cases in
Table I compared to the QC and RQC relaxations, with some cases exhibiting
substantial improvements. For instance, RQC and LRQC relaxations have 0.63%
and 0.27% optimality gaps for “case3$\\_$lmbd” test case, respectively. Note
that the semidefinite programming (SDP) relaxation has a 0.39% optimality gap
for the “case3$\\_$lmbd” test case [14]. The proposed LRQC relaxation improves
the optimality gaps for both small and large systems. Notably, it closes the
optimality gap for the “case2868$\\_$rte$\\_\\_$api” test case, where the
previous RQC relaxation had an optimality gap of 0.16%. Moreover, the LRQCs
relaxation with both the volume-minimizing $\psi_{l}$ and suggested
$\psi^{\dagger}_{l}=85^{\circ}$ outperform the QC and RQC relaxations for all
test cases. As expected from the analysis in Section VIII, applying the
suggested $\psi^{\dagger}_{l}=85^{\circ}$ results in good performance across a
variety of test cases. As shown in Table I, the LRQC relaxation’s improved
tightness comes at the cost of slower (but still tractable) computational
times for some test cases. When analyzing the last two columns of Table I, it
becomes evident that the impact of adding the proposed envelopes on execution
time is quite diverse. For instance, in cases like “case300$\\_$ieee,”
implementing these envelopes leads to a reduction in execution time by over
$57$%. On the contrary, for other test cases, there is a considerable increase
in execution time, reaching up to $300$% in some instances. However, on
average across all the test cases, enforcing the proposed envelopes results in
a moderate increase of less than $38$% in the time required to solve the RQC
relaxation from [22].

Table I: Results from Applying the QC and RQC Relaxations with Various Options
to Selected PGLib Test Cases

Test Cases |  QC Gap (%) |  RQC $(\psi^{*})$ |  LRQC |  LRQC Gap (%) $(\psi^{\dagger}_{l}=85^{\circ})$ |  QC Time (sec) |  RQC Time (sec) |  LRQC Time (sec)
---|---|---|---|---|---|---|---
Gap (%) |  $\psi^{\ast}$ |  Gap (%)
case3_lmbd | 0.97 | 0.63 | 11.00 | 0.26 | 0.26 | 0.26 | 0.01 | 0.02
case14_ieee | 0.11 | 0.10 | -23.00 | 0.09 | 0.09 | 0.37 | 0.02 | 0.05
case30_ieee | 18.67 | 11.82 | -25.00 | 9.08 | 11.24 | 0.33 | 0.03 | 0.12
case39_epri | 0.54 | 0.51 | 0.00 | 0.50 | 0.50 | 0.38 | 0.06 | 0.12
case89_pegase | 0.75 | 0.74 | 77.00 | 0.73 | 0.74 | 0.87 | 0.44 | 1.10
case118_ieee | 0.77 | 0.62 | 70.00 | 0.55 | 0.54 | 0.55 | 0.23 | 0.37
case240_pserc | 2.72 | 2.54 | 8.00 | 2.39 | 2.39 | 1.13 | 0.92 | 1.42
case300_ieee | 2.56 | 2.24 | -13.00 | 2.18 | 2.08 | 1.54 | 3.15 | 1.33
case1951_rte | 0.13 | 0.11 | -10.00 | 0.11 | 0.11 | 7.70 | 8.97 | 12.65
case2316_sdet | 1.79 | 1.78 | -9.00 | 1.76 | 1.77 | 6.11 | 7.37 | 7.36
case2848_rte | 0.12 | 0.12 | -48.00 | 0.11 | 0.11 | 10.56 | 10.63 | 18.04
case2868_rte | 0.09 | 0.09 | 80.00 | 0.09 | 0.09 | 13.79 | 13.15 | 23.47
case2869_pegase | 1.00 | 0.98 | -10.00 | 0.98 | 0.98 | 15.54 | 13.17 | 19.88
case3_lmbd__api | 4.57 | 3.93 | -71.00 | 3.65 | 3.68 | 0.51 | 0.01 | 0.02
case14_ieee__api | 5.13 | 5.13 | 63.00 | 5.13 | 5.13 | 0.37 | 0.03 | 0.05
case24_ieee_rts__api | 11.02 | 6.98 | -11.00 | 4.47 | 5.66 | 0.71 | 0.04 | 0.04
case30_ieee__api | 5.45 | 5.29 | -23.00 | 4.50 | 5.18 | 0.31 | 0.06 | 0.08
case30_fsr__api | 2.75 | 2.69 | 78.00 | 2.58 | 0.37 | 0.05 | 0.14 | 0.12
case73_ieee_rts__api | 9.54 | 7.24 | -10.00 | 6.21 | 6.70 | 1.00 | 0.37 | 0.43
case118_ieee__api | 28.67 | 26.38 | -8.00 | 24.17 | 25.72 | 0.53 | 0.97 | 0.41
case162_ieee_dtc__api | 4.32 | 4.27 | -9.00 | 4.24 | 4.25 | 0.68 | 0.44 | 0.83
case179_goc__api | 5.86 | 4.06 | -78.00 | 3.16 | 3.11 | 0.82 | 0.64 | 1.51
case300_ieee__api | 0.83 | 0.70 | -15.00 | 0.64 | 0.59 | 1.22 | 2.36 | 1.56
case2848_rte__api | 0.22 | 0.21 | 79.00 | 0.18 | 0.20 | 12.60 | 12.56 | 18.52
Case2868_rte__api | 0.17 | 0.16 | -10.00 | 0.00 | 0.13 | 12.52 | 13.29 | 18.63
case2869_pegase__api | 1.32 | 1.30 | -10.00 | 1.11 | 1.28 | 10.72 | 11.00 | 17.26
case3_lmbd__sad | 1.38 | 1.02 | 68.00 | 0.92 | 0.92 | 0.44 | 0.01 | 0.02
case14_ieee__sad | 19.16 | 15.39 | -12.00 | 12.70 | 14.20 | 0.35 | 0.03 | 0.05
case24_ieee_rts__sad | 2.74 | 2.12 | -12.00 | 1.82 | 1.82 | 0.40 | 0.06 | 0.06
case30_ieee__sad | 5.66 | 4.45 | 66.00 | 3.94 | 4.10 | 0.32 | 0.06 | 0.13
case39_epri__sad | 0.20 | 0.17 | 82.00 | 0.16 | 0.16 | 0.36 | 0.11 | 0.12
case57_ieee__sad | 0.32 | 0.31 | 84.00 | 0.29 | 0.29 | 0.38 | 0.11 | 0.16
case73_ieee_rts__sad | 2.37 | 1.82 | 78.00 | 1.55 | 1.55 | 0.41 | 0.41 | 0.44
case118_ieee__sad | 6.67 | 5.07 | 69.00 | 4.00 | 4.43 | 0.58 | 0.39 | 0.68
case162_ieee_dtc__sad | 6.22 | 5.54 | 76.00 | 4.51 | 4.96 | 0.86 | 0.84 | 0.91
case300_ieee__sad | 2.34 | 1.59 | 83.00 | 1.32 | 1.32 | 1.94 | 2.06 | 1.94
case1951_rte__sad | 0.43 | 0.42 | -10.00 | 0.40 | 0.40 | 7.89 | 8.45 | 17.25
case2746wop_k__sad | 1.99 | 1.84 | 80.00 | 1.77 | 1.76 | 6.90 | 7.91 | 10.47

_QC Gap_ : Optimality gap for the QC relaxation from (5), _RQC Gap_ :
Optimality gap for the Relaxation from [22, Eq. (16)], _LRQC Gap_ : Optimality
gap for the Relaxation from (14), $\psi^{*}$: Use of the best $\psi_{l}$ for
this case.

##  X Conclusion

This paper has proposed tighter envelopes for the product and trigonometric
terms in the power flow equations to improve the tightness of the QC
relaxation. These envelopes are developed by considering a particular
nonlinear function whose projections are the expressions appearing in the
power flow equations. Additionally, we exploit characteristics of the sine and
cosine expressions along with the changes in their curvature to tighten convex
envelopes associated with the trigonometric terms. Comparison to a state-of-
the-art RQC relaxation implementation demonstrates the value of these
improvements via reduced optimality gaps on challenging test cases while
maintaining computational tractability.

$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ | $F(\theta_{lm},\delta_{lm},\psi_{l})$
---|---
|
(a) $-165^{\circ}\leq\theta_{lm}-\delta_{lm}-\psi_{l}\leq 15^{\circ}$

$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ | $F(\theta_{lm},\delta_{lm},\psi_{l})$
---|---
|
(b) $-105^{\circ}\leq\theta_{lm}-\delta_{lm}-\psi_{l}\leq 75^{\circ}$

Figure 8: The figures in the left column show visualizations of the function
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ (black curve) and the line connecting
the endpoints of this function at $\underline{\theta}_{lm}$ and
$\overline{\theta}_{lm}$ (dashed magenta line) for different values of
$\delta_{lm}$, $\underline{\theta}_{lm}$, and $\overline{\theta}_{lm}$. The
figures in the right column show the corresponding function $F(\theta_{lm})$
and its roots between $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$.

This appendix details the process of selecting appropriate tangent lines to
the sine and cosine functions for arguments that take a general range of
values in order to construct valid convex envelopes. While finding tangent
lines for sine and cosine functions at any point is relatively straightforward
— one can simply take the derivative to obtain the slope and align the line
with the tangent point — the real challenge lies in identifying tangent lines
that do not intersect the sine or cosine function at any other points. This
occurs due to the possible variation in curvature of the sine and cosine
functions for arguments within a general range of values. If the curvature’s
sign does not change, tangent lines will not intersect the sine and cosine
function at any other points.

The case that poses a challenge is when the curvature’s sign changes, leading
to the possibility of a tangent line intersecting the function at another
point. Hence, the initial step in selecting appropriate tangent lines is to
examine whether the curvature’s sign changes within the specified range for
the trigonometric function’s argument. To tackle this issue, we introduce an
auxiliary function, denoted as $F(\theta_{lm})$ in (X), which helps us
ascertain whether the curvature of the trigonometric function changes within
the range $[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$,
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}]$. The function $F(\theta_{lm})$
captures the difference between the trigonometric function
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ and the straight line connecting its
endpoints at $\underline{\theta}_{lm}$ and $\overline{\theta}_{lm}$:

$\displaystyle
F(\theta_{lm})=\cos(\theta_{lm}-\delta_{lm}-\psi_{l})-\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})$
$\displaystyle\qquad\quad-\frac{\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})-\cos(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l})}{\overline{\theta}_{lm}-\underline{\theta}_{lm}}$
$\displaystyle\qquad\quad\times\left(\theta_{lm}-\overline{\theta}_{lm}-\psi_{l}\right).$
(16)

The set of zeros of the derivative of $F(\theta_{lm})$, i.e., the set of
solutions to $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$, is a key quantity when
determining if the curvature of $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$
changes between $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. In Fig. 8, on the left side, we
present illustrative examples of the function
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ (depicted by the black curve) along
with the dashed magenta line representing the connection between its endpoints
at $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. The right side visualizes the
function $F(\theta_{lm})$ itself, with its roots indicated by yellow circles.

The derivative of $F(\theta_{lm})$ is

$\displaystyle\frac{dF(\theta_{lm})}{d\theta_{lm}}=-\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$
$\displaystyle\quad-\frac{\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})-\cos(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l})}{\overline{\theta}_{lm}-\underline{\theta}_{lm}}.$
(17)

We denote the set of zeroes for $\frac{dF(\theta_{lm})}{d\theta_{lm}}=0$ by
$\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}$,
where the subscripts indicate that the set is parameterized by
$\underline{\theta}_{lm}$, $\overline{\theta}_{lm}$, $\delta_{lm}$, and
$\psi_{l}$:

$\displaystyle\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}=$
$\displaystyle\Big{\\{}(-1)^{\kappa}\arcsin\left(\frac{\cos(\underline{\theta}_{lm}-\delta_{lm})-\cos(\overline{\theta}_{lm}-\delta_{lm})}{\left(\overline{\theta}_{lm}-\underline{\theta}_{lm}\right)}\right)+\pi\kappa,$
$\displaystyle\qquad\kappa=\ldots,-3,-2,-1,0,1,2,3,\ldots\Big{\\}}.$ (18)

Let $\left|\,\cdot\,\right|$ denote the cardinality of a set. The cardinality
of
$\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}$
determines whether the tangent lines to the trigonometric function at
different points within $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ have multiple intersections with
the trigonometric function. If
$\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}$
is empty, it indicates that the curvature of the trigonometric function does
not change. Thus, no tangent lines will have another intersection with the
function. In this case, the slopes of the $k^{th}$ tangent line (referred to
as $m_{k}$) at an arbitrary point within the interval,
$(\phi_{lm},\cos(\phi_{lm}-\delta_{lm}-\psi_{l})$, is the derivative of the
trigonometric function at this point
$\left(m_{k}=\frac{d(\cos(\theta_{lm}-\delta_{lm}-\psi_{l}))}{d\theta_{lm}}|_{\phi_{lm}}=-\sin(\phi_{lm}-\delta_{lm}-\psi_{l})\right)$.
The coordinate of the point itself gives us the offset, represented as
$b_{int}$, of the tangent line which is equal to
$\cos(\phi_{lm}-\delta_{lm}-\psi_{l})$.

Conversely, when
$\left|\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}\right|\geq
1$, the tangent line to the cosine function at some point may intersect the
cosine function again at a different point. To address this concern, we
identify points, denoted as $\underline{\mathcal{R}}_{lm}$ and
$\overline{\mathcal{R}}_{lm}$, at the boundaries of ranges for which tangent
lines do not intersect the cosine function. $\underline{\mathcal{R}}_{lm}$ and
$\overline{\mathcal{R}}_{lm}$ are shown by red and yellow stars, respectively,
in Fig. 4. Note that the voltage angle difference restriction, i.e.,
${-}90^{\circ}\leq\theta_{lm}\leq 90^{\circ}$, guarantees that the sine and
cosine functions experience at most one curvature sign change. Consequently,
selecting tangent lines to the cosine function in the ranges
$[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l},\underline{\mathcal{R}}_{lm}]$
and
$[\overline{\mathcal{R}}_{lm},\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$] is
a straightforward process since the curvature of the cosine function remains
consistent, with the same sign, within these ranges. This method yields linear
envelopes that provide a close outer approximation to the cosine function. The
technique is applicable to any angle difference range, irrespective of the
trigonometric function’s curvature. Next, we will elaborate on the computation
process for $\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$.

To determine $\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$
for the cosine function, we start by formulating the tangent lines to the
cosine function that pass through the endpoints
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$, respectively. The tangent line
to the cosine function that pass through the point
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ is given by
$\underline{F}_{tang}(\theta_{lm})$:

$\displaystyle\underline{F}_{tang}(\theta_{lm})=-\sin(\theta_{lm}-\delta_{lm}-\psi_{l})(\theta_{lm}-\overline{\theta}_{lm})$
$\displaystyle\qquad\qquad\qquad+\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}).$
(19)

We subsequently define an auxiliary function, denoted as
$\underline{G}(\theta_{lm})$, which represents the difference between
$\underline{F}_{tang}(\theta_{lm})$ and
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$:

$\displaystyle\underline{G}(\theta_{lm})=\sin(\theta_{lm}-\delta_{lm}-\psi_{l})(\theta_{lm}-\overline{\theta}_{lm})$
$\displaystyle\qquad-\cos(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})+\cos(\theta_{lm}-\delta_{lm}-\psi_{l}).$
(20)

The root of the derivative of $\underline{G}(\theta_{lm})$, i.e., the solution
to $\frac{d\underline{G}(\theta_{lm})}{d\theta_{lm}}=0$, within the interval
between $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ corresponds to the value of
$\underline{\mathcal{R}}_{lm}$.

Similarly, to find $\overline{\mathcal{R}}_{lm}$, we first formulate the
tangent line to the cosine function that pass through the point
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$:

$\displaystyle\overline{F}_{tang}(\theta_{lm})=-\sin(\theta_{lm}-\delta_{lm}-\psi_{l})(\theta_{lm}-\underline{\theta}_{lm})$
$\displaystyle\qquad\qquad\qquad+\cos(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}).$
(21)

Next, we define an auxiliary function, denoted as $\overline{G}(\theta_{lm})$,
which represents the difference between $\overline{F}_{tang}(\theta_{lm})$ and
$\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$:

$\displaystyle\overline{G}(\theta_{lm})=\sin(\theta_{lm}-\delta_{lm}-\psi_{l})(\theta_{lm}-\underline{\theta}_{lm})$
$\displaystyle\qquad-\cos(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l})+\cos(\theta_{lm}-\delta_{lm}-\psi_{l}).$
(22)

Analogously to the discussion above, the root of the derivative of
$\overline{G}(\theta_{lm})$, i.e., the solution to
$\frac{d\overline{G}(\theta_{lm})}{d\theta_{lm}}=0$, within the interval
between $\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$, corresponds to the value of
$\overline{\mathcal{R}}_{lm}$.

To locate the roots $\frac{d\underline{G}(\theta_{lm})}{d\theta_{lm}}=0$ and
$\frac{d\overline{G}(\theta_{lm})}{d\theta_{lm}}=0$ within the interval
$\left[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l},\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}\right]$,
we start by applying a bisection method to obtain a close initialization for a
locally convergent Newton method to determine the precise values of the roots.
The reason for employing the bisection method initially is that the Newton
method may converge to solutions beyond the interval of interest for periodic
functions like $\frac{d\underline{G}(\theta_{lm})}{d\theta_{lm}}$ and
$\frac{d\overline{G}(\theta_{lm})}{d\theta_{lm}}$. The bisection method finds
approximate solutions within the interval of interest that are refined with a
Newton method. The slope of tangent line to the cosine function at
$\underline{\mathcal{R}}_{lm}$ (referred to as
$m_{k,\underline{\mathcal{R}}_{lm}}$), is the derivative of the cosine
functions at the corresponding argument for $\underline{\mathcal{R}}_{lm}$,
i.e.,
$\left(m_{k,\underline{\mathcal{R}}_{lm}}=-\sin(\phi_{lm,\underline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})\right)$.
The coordinate of the point itself gives us the offset, represented as
$b_{int,\underline{\mathcal{R}}_{lm}}$, of the tangent line which is equal to
$\cos(\phi_{lm,\underline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})$. The slope
and offset for $\overline{\mathcal{R}}_{lm}$ is computed similarly.

1 function
Find_Tangent($\overline{\theta}_{lm}$,$\underline{\theta}_{lm}$,$\delta_{lm}$,$\psi_{l}$,$N_{tan}$)

2 $U\leftarrow\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$,
$L\leftarrow\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$.

3 Define $F(\theta_{lm})$ and $\frac{dF(\theta_{lm})}{d\theta_{lm}}$.

4
$\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}\leftarrow$
Find the roots of $\frac{dF(\theta_{lm})}{d\theta_{lm}}$ within $[L,U]$.

5 if
$\left|\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}\right|<1$
then

6 At $N_{tan}$ equally spaced points within $[L,U]$:

7 for $i=1:N_{tan}$ do

8 $m_{k,i}=-\sin(\phi_{lm,i}-\delta_{lm}-\psi_{l})$
$b_{int,i}=\cos(\phi_{lm,i}-\delta_{lm}-\psi_{l})$

9 end for

10

11 else if
$\left|\mathcal{Z}_{\underline{\theta}_{lm},\overline{\theta}_{lm},\delta_{lm},\psi_{l}}\right|\geq
1$ then

12
$\underline{\mathcal{R}}_{lm}\leftarrow\frac{dG(\theta_{lm},U)}{d\theta_{lm}}=0$

13
$\overline{\mathcal{R}}_{lm}\leftarrow\frac{dG(\theta_{lm},L)}{d\theta_{lm}}=0$

14 Find the tangent lines to $\cos(\theta_{lm}-\delta_{lm}-\psi_{l})$ at
$\underline{\mathcal{R}}_{lm}$ and $\overline{\mathcal{R}}_{lm}$:

15
$m_{k,\underline{\mathcal{R}}_{lm}}=-\sin(\phi_{lm,\underline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})$

16
$b_{int,\underline{\mathcal{R}}_{lm}}=\cos(\phi_{lm,\underline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})$

17

18
$m_{k,\overline{\mathcal{R}}_{lm}}=-\sin(\phi_{lm,\overline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})$

19
$b_{int,\overline{\mathcal{R}}_{lm}}=\cos(\phi_{lm,\overline{\mathcal{R}}_{lm}}-\delta_{lm}-\psi_{l})$

20

21 Equally divide the ranges $[\overline{\mathcal{R}}_{lm},U]$ and
$[L,\underline{\mathcal{R}}_{lm}]$ into $N_{tan}$ segments

22 for $i=1:N_{tan}$ do

23 $m_{k,i}=-\sin(\phi_{lm,i}-\delta_{lm}-\psi_{l})$
$b_{int,i}=\cos(\phi_{lm,i}-\delta_{lm}-\psi_{l})$

24 end for

25

26 end if

27 return $m_{k}$ and $b_{int}$

28

Algorithm 2 Finding tangent lines

After computing $\underline{\mathcal{R}}_{lm}$ and
$\overline{\mathcal{R}}_{lm}$, we select equally spaced tangent lines to the
cosine function within the intervals
$\left[\underline{\theta}_{lm}-\delta_{lm}-\psi_{l},\underline{\mathcal{R}}_{lm}\right]$
and
$\left[\overline{\mathcal{R}}_{lm},\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}\right]$.

Algorithm 2 outlines the procedure for computing these convex envelopes using
carefully selected tangent lines. For notational convenience, define
$L=\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$U=\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. Equations (23) and (24)
define the tangent lines which form the lower and upper bounds, respectively,
of the cosine envelope:

If curvature changes within $[L,U]$ from positive to negative:
$\displaystyle\underline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[L,\underline{\mathcal{R}}_{lm}]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[\underline{\mathcal{R}}_{lm},U]\end{cases}$
(23a) If curvature changes within $[L,U]$ from negative to positive:
$\displaystyle\underline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[\overline{\mathcal{R}}_{lm},U]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[L,\overline{\mathcal{R}}_{lm}]\end{cases}$
(23b) If curvature does not change within $[L,U]$ and it is negative:
$\displaystyle\underline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}\frac{\cos(U)-\cos(L)}{U-L}(x-L)+\cos(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U],\end{cases}$ (23c) If curvature does
not change within $[L,U]$ and is positive:
$\displaystyle\underline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}\frac{\cos(U)-\cos(L)}{U-L}(x-L)+\cos(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (23d)

Similarly, the upper bound for the cosine envelope is:

If curvature changes within $[L,U]$ from positive to negative:
$\displaystyle\overline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i\\!=\\!1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[\overline{\mathcal{R}}_{lm},U]\\\
m_{k,\overline{\mathcal{R}}_{lm}}(x-x_{0,\overline{\mathcal{R}}_{lm}})+b_{int,\overline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[L,\overline{\mathcal{R}}_{lm}]\end{cases}$
(24a) If curvature changes within $[L,U]$ from negative to positive:
$\displaystyle\overline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i\\!=\\!1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[L,\underline{\mathcal{R}}_{lm}]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[\underline{\mathcal{R}}_{lm},U]\end{cases}$
(24b) If curvature does not change within $[L,U]$ and is negative:
$\displaystyle\overline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}\frac{\cos(U)-\cos(L)}{U-L}(x-L)+\cos(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (24c) If curvature does not
change within $[L,U]$ and is positive:
$\displaystyle\overline{L}_{\cos,i}\\!\\!=\\!\\!\begin{cases}\frac{\cos(U)-\cos(L)}{U-L}(x-L)+\cos(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (24d)

where $\overline{L}_{\cos,i}$ and $\underline{L}_{\cos,i}$ are the $i^{th}$
tangent lines which upper and lower bound, respectively, the cosine function;
$x$ equals $\theta_{lm}-\delta_{lm}-\psi_{l}$; $x_{0,i}$,
$x_{0,\underline{\mathcal{R}}_{lm}}$, and $x_{0,\overline{\mathcal{R}}_{lm}}$
represent the horizontal coordinates of the corresponding points within their
respective intervals; $b_{int,i}$, $b_{int,\underline{\mathcal{R}}_{lm}}$, and
$b_{int,\overline{\mathcal{R}}_{lm}}$ denote the vertical coordinates of these
same points, which determines the offset for the tangent lines; and $m_{k,i}$,
$m_{k,\underline{\mathcal{R}}_{lm}}$, and $m_{k,\overline{\mathcal{R}}_{lm}}$
signify the slopes of the tangent lines at these points. Note that when the
curvature of the trigonometric function does not change within an interval,
either the lower or upper envelope can be defined as a line connecting both
ends of the trigonometric function. For instance, consider a specific angle
$\left(\phi_{lm}-\delta_{lm}-\psi_{l}\right)$ within the interval
$[L,\underline{\mathcal{R}}_{lm}]$. In this case, $m_{k,i}$ is equal to
${-}\sin\left(\phi_{lm}-\delta_{lm}-\psi_{l}\right)$, which represents the
first derivative of the cosine function at
$x_{0,i}=\phi_{lm}-\delta_{lm}-\psi_{l}$. The corresponding offset for this
point is $b_{int,i}=\cos\left(\phi_{lm}-\delta_{lm}-\psi_{l}\right)$. Tangent
lines for the lower and upper bounds of the sine function are formulated
similarly.

Equation (13) formulates the cosine function. To complete the full exposition,
we similarly formulate the lower and upper envelopes for the sine function by
defining a function $H(\theta_{lm})$. Here, the function $H(\theta_{lm})$
represents the difference between the trigonometric function
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ itself and the line which connects
the endpoints of $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ at
$\underline{\theta}_{lm}$ and $\overline{\theta}_{lm}$:

$\displaystyle
H(\theta_{lm})=\sin(\theta_{lm}-\delta_{lm}-\psi_{l})-\sin(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})$
$\displaystyle\qquad\quad-\frac{\sin(\overline{\theta}_{lm}-\delta_{lm}-\psi_{l})-\sin(\underline{\theta}_{lm}-\delta_{lm}-\psi_{l})}{\overline{\theta}_{lm}-\underline{\theta}_{lm}}$
$\displaystyle\qquad\quad\times\left(\theta_{lm}-\overline{\theta}_{lm}-\psi_{l}\right).$
(25)

The number of zeros of the first derivative of $H(\theta_{lm})$, i.e., the
number of solutions for $\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$, is a key
quantity to determine if the curvature of
$\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ changes between
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. If
$\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$ has no solutions, then the curvature
of the $\sin(\theta_{lm}-\delta_{lm}-\psi_{l})$ does not change between
$\underline{\theta}_{lm}-\delta_{lm}-\psi_{l}$ and
$\overline{\theta}_{lm}-\delta_{lm}-\psi_{l}$. Our proposed QC relaxation uses
envelopes
$\left\langle\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{S^{\prime}}$
formed by combining the tangent lines. Depending upon the sign of the sine
function’s curvature and the number of solutions to
$\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$, there are different upper and lower
bounds on the sine function’s envelopes:

If $\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$ has one or more solutions:
$\displaystyle\left\langle\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{S^{\prime}}=\begin{cases}\\!\widecheck{S}^{\prime}\\!:\begin{cases}\widecheck{S}^{\prime}\\!\leq\overline{L}_{\sin,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\\\
\widecheck{S}^{\prime}\\!\geq\underline{L}_{\sin,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\end{cases}\end{cases}$ (26a) If
$\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$ has no solutions & curvature sign is
negative:
$\displaystyle\left\langle\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{S^{\prime}}=\begin{cases}\\!\widecheck{S}^{\prime}\\!:\begin{cases}\widecheck{S}^{\prime}\\!\leq\overline{L}_{\sin,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\\\
\widecheck{S}^{\prime}\\!\geq\underline{L}_{\sin,i},\leavevmode\nobreak\
i=1\end{cases}\end{cases}$ (26b) If $\frac{dH(\theta_{lm})}{d\theta_{lm}}=0$
has no solutions & curvature sign is positive:
$\displaystyle\left\langle\sin(\theta_{lm}-\delta_{lm}-\psi_{l})\right\rangle^{S^{\prime}}=\begin{cases}\\!\widecheck{S}^{\prime}\\!:\begin{cases}\widecheck{S}^{\prime}\\!\leq\overline{L}_{\sin,i},\leavevmode\nobreak\
i=1\\\
\widecheck{S}^{\prime}\\!\geq\underline{L}_{\sin,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan}\end{cases}\end{cases}$ (26c)

where $\overline{L}_{\sin,i}$ and $\underline{L}_{\sin,i}$ are the $i^{th}$
tangent lines which upper and lower bound, respectively, the sine function.

Equations (27) and (28) mathematically represent the tangent lines for the
lower and upper bounds of the sine function, respectively:

If curvature changes within $[L,U]$ from positive to negative:
$\displaystyle\underline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[L,\underline{\mathcal{R}}_{lm}]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[\underline{\mathcal{R}}_{lm},U]\end{cases}$
(27a) If curvature changes within $[L,U]$ from negative to positive:
$\displaystyle\underline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[\overline{\mathcal{R}}_{lm},U]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[L,\overline{\mathcal{R}}_{lm}]\end{cases}$
(27b) If curvature does not change within $[L,U]$ and is negative:
$\displaystyle\underline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}\frac{\sin(U)-\sin(L)}{U-L}(x-L)+\sin(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U],\end{cases}$ (27c) If curvature does
not change within $[L,U]$ and is positive:
$\displaystyle\underline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}\frac{\sin(U)-\sin(L)}{U-L}(x-L)+\sin(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (27d)

Similarly, the upper bound for the sine function can be represented as
follows:

If curvature changes within $[L,U]$ from positive to negative:
$\displaystyle\overline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[\overline{\mathcal{R}}_{lm},U]\\\
m_{k,\overline{\mathcal{R}}_{lm}}(x-x_{0,\overline{\mathcal{R}}_{lm}})+b_{int,\overline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[L,\overline{\mathcal{R}}_{lm}]\end{cases}$
(28a) If curvature changes within $[L,U]$ from negative to positive:
$\displaystyle\overline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}m_{k,i}(x-\\!x_{0,i})+b_{int,i},\leavevmode\nobreak\
i=1,\ldots,N_{tan},\leavevmode\nobreak\ \text{if\leavevmode\nobreak\
}x\in[L,\underline{\mathcal{R}}_{lm}]\\\
m_{k,\underline{\mathcal{R}}_{lm}}(x-x_{0,\underline{\mathcal{R}}_{lm}})+b_{int,\underline{\mathcal{R}}_{lm}}\leavevmode\nobreak\
\text{if\leavevmode\nobreak\ }x\in[\underline{\mathcal{R}}_{lm},U]\end{cases}$
(28b) If curvature does not change within $[L,U]$ and is negative:
$\displaystyle\overline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}\frac{\sin(U)-\sin(L)}{U-L}(x-L)+\sin(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (28c) If curvature does not
change within $[L,U]$ and is positive:
$\displaystyle\overline{L}_{\sin,i}\\!\\!=\\!\\!\begin{cases}\frac{\sin(U)-\sin(L)}{U-L}(x-L)+\sin(L)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in[L,U]\end{cases}$ (28d)

where $x$ equals $\theta_{lm}-\delta_{lm}-\psi_{l}$; $x_{0,i}$,
$x_{0,\underline{\mathcal{R}}_{lm}}$, and $x_{0,\overline{\mathcal{R}}_{lm}}$
represent the horizontal coordinates of the corresponding points within their
respective intervals; $b_{int,i}$, $b_{int,\underline{\mathcal{R}}_{lm}}$, and
$b_{int,\overline{\mathcal{R}}_{lm}}$ denote the vertical coordinates of these
points within their corresponding intervals, determining the offset for the
tangent lines; and $m_{k,i}$, $m_{k,\underline{\mathcal{R}}_{lm}}$, and
$m_{k,\overline{\mathcal{R}}_{lm}}$ signify the slopes of the tangent lines at
specific points within their respective intervals.

## References

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