# Robust Short-term Operation of AC Power Network with Injection Uncertainties

Anamika Tiwari, Abheejeet Mohapatra, and Soumya Ranjan Sahoo The authors are
with the Department of Electrical Engineering, Indian Institute of Technology
Kanpur, Kanpur, Uttar Pradesh, India 208016. (email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

With uncertain injections from Renewable Energy Sources (RESs) and loads,
deterministic AC Optimal Power Flow (OPF) often fails to provide optimal
setpoints of conventional generators. A computationally time-efficient,
economical, and robust solution is essential for ACOPF with short-term
injection uncertainties. Usually, applying Robust Optimization (RO) for
conventional non-linear ACOPF results in computationally intractable Robust
Counterpart (RC), which is undesirable as ACOPF is an operational problem.
Hence, this paper proposes a single-stage non-integer non-recursive RC of
ACOPF, using a dual transformation, for short-term injection uncertainties.
The proposed RC is convex, tractable, and provides base-point active power
generations and terminal voltage magnitudes (setpoints) of conventional
generators that satisfy all constraints for all realizations of defined
injection uncertainties. The non-linear impact of uncertainties on other
variables is inherently modeled without using any affine policy. The proposed
approach also includes the budget of uncertainty constraints for low
conservatism of the obtained setpoints. Monte-Carlo Simulation (MCS) based
participation factored AC power flows validate the robustness of the obtained
setpoints on NESTA and case9241pegase systems for different injection
uncertainties. Comparison with previous approaches indicates the efficacy of
the proposed approach in terms of low operational cost and computation time.

_Keywords:_ Convexity, Optimal Power Flow, Participation Factor, Robust,
Short-Term Operation, Uncertainty.

##  1 Introduction

###  1.1 Motivation

Optimal setpoints of conventional generators are obtained every 5-15 min from
OPF (or economic dispatch) for secure and reliable AC power network operation
[1]. Given the forecasted loads for the upcoming time slot, ACOPF finds
setpoints with minimum generation cost to meet the load and satisfy network
constraints. The integration of RESs, such as solar and wind generation, has
brought about a paradigm shift in the operation of AC power network. Unlike
conventional generators, the output of RES is uncertain due to dependence on
weather conditions. Also, demand response enablers have increased the
unpredictability of loads. Often, the deterministic ACOPF solution cannot
ensure economic and secure operation of power network in the presence of
uncertain loads and RESs injections [2, 3]. _This necessitates designing a
robust, time-efficient, optimal dispatch policy to address short-term
injection uncertainties so that the operational generation cost in AC network
is minimum without any constraint violations_.

###  1.2 Literature review

In literature, various methods like Stochastic Programming [4, 5, 6], Chance-
Constrained Programming (CCP) [3, 7], Distributionally RO (DRO) [8, 9], and RO
have been proposed to handle uncertainties in OPF. RO only requires the
information of possible range of uncertain parameters [10], and assumes hard
constraints111Soft constraints (whose violations are tolerated if violations‘
magnitude and duration of violation are not high) in power networks can be
considered via CCP [7] and DRO [8]. However, modeling such constraints is
beyond the scope of this paper and shall be explored in future., which
obviates any violation during the AC network operation with uncertainties
[11]. However, due to the non-convex power flow equations, direct application
of RO is strenuous. Hence, the initial robust OPF formulations [12, 13, 14,
15] use the linear DCOPF model instead of the actual ACOPF model. In DCOPF,
voltage magnitudes are all unity, and thus, the obtained solution is not
always feasible for the actual AC network [16]. Further, the presence of
uncertain terms in usual non-linear equality constraints of OPF poses a
difficulty in directly obtaining a tractable robust ACOPF model [17]. In
previous works, this challenge is addressed by either converting the infinite-
dimensional non-deterministic problem into a finite-dimensional deterministic
problem by obtaining its RC [18, 19, 20], or by decomposition into multi-stage
optimization problem in which the uncertainty is realized in one stage and the
other variables are adjusted accordingly in other stages [21, 22, 23, 24]. The
associated multi-stage problem is solved iteratively using Cutting Plane (CP)
methods.

The RC formulations are computationally efficient but obtaining RC of a
problem with uncertain equality constraints is not a straightforward exercise.
To exclude uncertain terms in equality constraints, [18] considers average
wind generation instead of actual uncertain generation. In [19], the equality
constraints are transformed into inequality constraints by assuming adjustable
generators to be present at each bus, which is not always true and practically
infeasible. The recursive inner-approximated robust ACOPF in [20] gives
conservative solutions with possibility of being infeasible at times due to
convex restriction. The initialization of convex restriction is by actual
deterministic ACOPF solution, and in each recursion (multi-stage formulation),
the states are obtained from AC power flow equations, which increase the
computation time. In [21], a tri-stage robust operational problem is posed,
where uncertainty is realized in the middle stage and other variables are
adjusted via primal and dual cuts. It also approximates actual non-linear
power flow equations via linearization for the worst-case uncertainty
realization. The formulation in [23] also uses linearized AC equations, and is
solved iteratively by Benders Decomposition (BD). In [22, 24], the multi-
period formulation is solved by column and constraint generation and BD,
respectively. Effectively, the formulations of [21, 23, 22, 24] can handle the
uncertain equality constraints but with relatively high computation time and
convergence issues.

###  1.3 Research gap

A qualitative comparison of the attributes of formulations in [18, 19, 20, 21,
22, 23, 24] is given in Table 1. It is evident from Table 1 and Section 1.2
that the majority of previous formulations, except [18], are multi-stage
formulations that need to be solved recursively (repetitively solving the same
formulation) or iteratively (CP methods, such as BD). The formulation in [18],
although being non-recursive, does not consider the impact of uncertainty on
all variables, except active power generation, needs integer variable
representation, and gives highly conservative solutions as the budget of
uncertainty constraints are not considered. Unfortunately, such formulations
with recursive solution process or integer variable representation are
computationally burdensome and may not be suitable for the short-term (5-15
min) operational problems, such as OPF and optimal transmission switching, of
large-scale practical AC power networks [25].

Table 1: Attributes of a few previous robust AC OPF formulations and proposed
formulation (SSNR - Single Stage, Non-Recursive)

Ref. | Robust | SSNR | Optimization | Non- | Uncertainty
---|---|---|---|---|---
Formulation | Model | integer | Budget
[18] | RC | $\checkmark$ | Convex | $\times$ | $\times$
[19] | RC | $\times$ | Convex | $\checkmark$ | $\times$
[20] | RC | $\times$ | Convex | $\checkmark$ | $\times$
[21]∗ | CP | $\times$ | LP1,2 ,NLP3 | $\times$ | $\checkmark$
[22] | CP | $\times$ | Convex | $\checkmark$ | $\checkmark$
[23] | CP | $\times$ | LP | $\checkmark$ | $\checkmark$
[24] | CP | $\times$ | Convex | $\times$ | $\checkmark$
Prop. | RC | $\checkmark$ | Convex | $\checkmark$ | $\checkmark$

∗ In [21], a tri-stage formulation is proposed, where some stages are linear
while other is non-linear. The superscripts indicate the stages of
formulation.

###  1.4 Contribution and paper organization

Hence, _a single-stage, non-recursive, time-efficient, dual, robust OPF
formulation is proposed in this paper for the short-term operation of AC power
network with injection uncertainties. The proposed formulation does not
require integer variables to consider budget of uncertainty constraints_. The
formulation provides conventional generators’ dispatch setpoints, i.e., base-
point active power generations and terminal voltage magnitudes, which are
immune to short-term injection uncertainties. _Also, in contrast to previous
single-stage, non-recursive approach in[18], the non-linear impact of
uncertainties on variables (such as active loss, reactive generations, and
load bus voltages), except the variables related to conventional generators’
dispatch setpoints, is inherently modeled without using any affine policy_. As
stated earlier, the worst-case realization of uncertainties is an arduous task
for actual non-convex ACOPF. This issue is addressed by means of Second-Order
Cone (SOC) relaxations that are further tightened by a set of linearized
arctangent constraints. Using the property of dual of convexified ACOPF, a
novel linear constraint is introduced that helps in the worst-case uncertainty
realization. SOC relaxation for mesh network is exact for assumptions stated
in [26, 27]. Hence, MCS based participation factored AC power flows [28]
numerically verify the robustness and feasibility of obtained setpoints. Thus,
the key attributes of the proposed formulation are

  * •

A novel single-stage, non-recursive, RC of ACOPF, which utilizes the
properties of dual transformation, is proposed to handle short-term
uncertainties in loads and RESs, and provides time-efficient robust
conventional generators’s setpoints, i.e., associated base-point active power
generations and terminal voltage magnitudes, immune to all realizations of
injection uncertainties.

  * •

The non-linear impact of uncertainties on variables, except conventional
generators’ setpoints, is inherently modeled without any affine policy by
considering ACOPF constraints for base-case (no uncertainties) and worst-case
uncertainties, along with ramp constraints on active power generations of
conventional generators.

  * •

The budget of uncertainty constraints are incorporated for a trade-off between
conservatism and robustness of the solution, while avoiding integer variables.

Mathematical proofs, extensive numeric results, and comparison with
deterministic ACOPF, [18, 19] reveal that the proposed formulation can be
efficiently solved to obtain optimal, robust and feasible conventional
generators’ setpoints for all uncertainty realizations. Deterministic ACOPF
with associated convex formulation are discussed in Section 2. Section 3
discusses the proposed RC formulation of ACOPF. Associated numeric results are
discussed in Section 4 while Section 5 concludes the paper and gives
directions for future work.

##  2 Deterministic ACOPF and Associated Convex Formulation

Let $\mathbb{R}$ be the set of real numbers. $\mathcal{N}$, $\mathcal{B}$,
$\mathcal{G}$, $\mathcal{L}$ and
$\mathcal{R}(\subseteq{\mathcal{G}\cup\mathcal{L}})$ denote the set of buses,
branches, generators, load and RES connected buses, respectively.
$\mathcal{M}(i)$ is the set of buses connected to bus $i$. $P_{gi}$ ($Q_{gi}$)
is the active (reactive) power output of generator $i\in\mathcal{G}$.
$V_{i}\angle\theta_{i}$ is voltage phasor of bus $i\in\mathcal{N}$.
$\theta_{ij}=\theta_{i}-\theta_{j}$ and $P_{ij}$ is the active power flow on
line $(i,j)\in\mathcal{B}$ from bus $i$ to bus $j$. $P_{di}$ ($Q_{di}$),
$P_{ri}$ ($Q_{ri}$) are the nominal active (reactive) load and RES generation
at bus $i$, respectively. $\overline{(.)}$ and $\underline{(.)}$ denote the
maximum and minimum bounds. $P_{gi}$ and $Q_{gi}$ are undefined for
$i\notin\mathcal{G}$. $P_{di}$ and $Q_{di}$ ($P_{ri}$ and $Q_{ri}$) are
undefined for $i\notin\mathcal{L}$ ($i\notin\mathcal{R}$). Similar definitions
of injections are true for case with uncertainties.

The base-case (deterministic) ACOPF can be stated as

$\displaystyle\min\sum_{i\in\mathcal{G}}\left[a_{i}P_{gi}+b_{i}\right]$ (1a)
$\displaystyle\text{s.t. }P_{gi}-P_{i}=P_{di}-P_{ri},\forall i\in\mathcal{N}$
(1b) $\displaystyle Q_{gi}-Q_{i}=Q_{di}-Q_{ri},\forall i\in\mathcal{N}$ (1c)
$\displaystyle G_{ii}c_{ii}+\sum_{j\in
M(i)}\left[G_{ij}c_{ij}+B_{ij}s_{ij}\right]-P_{i}=0\,\forall i\in\mathcal{N}$
(1d) $\displaystyle-$ $\displaystyle B_{ii}c_{ii}+\sum_{j\in
M(i)}\left[G_{ij}s_{ij}-B_{ij}c_{ij}\right]-Q_{i}=0,\forall i\in\mathcal{N}$
(1e) $\displaystyle
c_{ii}g_{ij}/|t_{ij}|^{2}+c_{ij}G_{ij}+s_{ij}B_{ij}-P_{ij}=0,\forall(i,j)\in\mathcal{B}$
(1f) $\displaystyle
c_{jj}g_{ij}+c_{ij}G_{ji}-s_{ij}B_{ji}-P_{ji}=0,\forall(i,j)\in\mathcal{B}$
(1g) $\displaystyle
c_{ij}^{2}+s_{ij}^{2}=c_{ii}c_{jj},\forall(i,j)\in\mathcal{B}$ (1h)
$\displaystyle\tan\theta_{ij}=s_{ij}/c_{ij},\forall(i,j)\in\mathcal{B}$ (1i)
$\displaystyle\underline{P}_{ij}\leq
P_{ij}\leq\overline{P}_{ij},\underline{P}_{ij}\leq
P_{ji}\leq\overline{P}_{ij},\forall(i,j)\in\mathcal{B}$ (1j)
$\displaystyle\underline{Q}_{ri}\leq Q_{ri}\leq\overline{Q}_{ri},\forall
i\in\mathcal{R}$ (1k) $\displaystyle\underline{P}_{gi}\leq
P_{gi}\leq\overline{P}_{gi},\underline{Q}_{gi}\leq
Q_{gi}\leq\overline{Q}_{gi},\forall i\in\mathcal{G}$ (1l)
$\displaystyle\underline{V}_{i}^{2}\leq
c_{ii}\leq\overline{V}_{i}^{2},\underline{\theta}_{i}\leq\theta_{i}\leq\overline{\theta}_{i},\forall
i\in\mathcal{N},\theta_{ref}=0$ (1m)
$\displaystyle\underline{\theta}_{ij}\leq\theta_{ij}\leq\overline{\theta}_{ij},\forall(i,j)\in\mathcal{B}$
(1n)

where $a_{i}$ and $b_{i}$ are positive cost coefficients of $i^{th}$
generator. Linear cost characteristics are considered as the focus is on the
short-term operation of AC power network [25, 19]. The nodal power balance
equations are (1b), (1c). The active and reactive injections ($P_{i}$ and
$Q_{i}$) at each bus are, respectively, (1d) and (1e), where
$c_{ii}=V_{i}^{2},\forall i\in\mathcal{N}$, and
$c_{ij}=V_{i}V_{j}\cos\theta_{ij}$,
$s_{ij}=V_{i}V_{j}\sin\theta_{ij},\forall(i,j)\in\mathcal{B}$. $G_{ij}$ and
$B_{ij}$ are real and imaginary parts of $(i,j)^{th}$ element of the bus
admittance matrix, respectively. (1f) - (1g) are the active power flows
($P_{ij}$ and $P_{ji}$) on each branch $(i,j)\in\mathcal{B}$, where $g_{ij}$
is the real part of the associated branch impedance and $t_{ij}$ is the
complex transformer tap. (1h) and (1i) are valid as per the definitions of
$c_{ii}$, $c_{ij}$, and $s_{ij}$. $P_{ij},P_{ji},\forall(i,j)\in\mathcal{B}$
are bounded in (1j), where $\underline{P}_{ij}=-\overline{P}_{ij}$. The
$i^{th}$ RES reactive power output is bounded in (1k), where
$\overline{Q_{ri}}=-\underline{Q_{ri}}=\sqrt{(S_{ri}^{max})^{2}-P_{ri}^{2}}$
and $S_{ri}^{max}$ is the maximum rating of $i^{th}$ RES generator equipped
with an inverter [19]. The conventional generators power outputs are bounded
in (1l). Bus voltage magnitudes and associated phase angles are bounded in
(1m), where $\theta_{ref}$ is the reference bus voltage phase angle, and
$\overline{\theta}_{i}=\pi/2=-\underline{\theta}_{i},\forall i\in\mathcal{N}$.
The branch power flows are further constrained by bounds on
$\theta_{ij},\forall(i,j)\in\mathcal{B}$ in (1n) (details on page 17 of [29]),
where $\underline{\theta}_{ij}=-\overline{\theta}_{ij}$. Limits on branch
apparent flow, instead of active power flow, can be considered to limit the
branch current magnitude. However, this is beyond the scope of present paper,
and for simplicity, constraints in (1j) are considered.

The formulation in (1) is non-convex due to the non-convexity of (1h) and
(1i). With injection uncertainties, the non-convexity of ACOPF poses
difficulty in quick realization of the associated worst-case, and thus,
obtaining a tractable robust solution is arduous [17]. Hence, in the
associated SOC relaxation, (1h) is convexified as rotated cones [30] as

$\displaystyle C_{ij}=2c_{ij},S_{ij}=2s_{ij},\forall(i,j)\in\mathcal{B}$ (2a)
$\displaystyle
E_{ij}=c_{ii}-c_{jj},D_{ij}=c_{ii}+c_{jj},\forall(i,j)\in\mathcal{B}$ (2b)
$\displaystyle C_{ij}^{2}+S_{ij}^{2}+E_{ij}^{2}-D_{ij}^{2}\leq
0,\forall(i,j)\in\mathcal{B}$ (2c) (1i) can be represented through appropriate
constraints using the properties of AC power network. Usually, for typical
operating condition of the power network, bus voltage magnitudes are close to
$1$ pu and voltage phase angle differences across lines rarely exceed
$45^{\circ}$ in (1n), as [31] reports these differences to not exceed
$30^{\circ}$, with most differences being less than $10^{\circ}$. Hence, under
such conditions, $c_{ij}\simeq 1,\forall(i,j)\in\mathcal{B}$, and
$s_{ij}\simeq(1+\Delta V_{i})(1+\Delta
V_{j})\theta_{ij}\simeq\theta_{ij}\pm\epsilon_{\theta},\forall(i,j)\in\mathcal{B}$
(2d) Consequently, from (2d), (1i) can be expressed as [30]
$-\epsilon_{\theta}\leq\theta_{ij}-s_{ij}\leq\epsilon_{\theta}$ (2e)

where $\epsilon_{\theta}$ is the allowable error, chosen as per [30]. For the
same SOC relaxation, i.e., (2a) - (2c) and (2e), the results for Case 2 in
Section IV.A of [30] strongly indicate the existence of zero optimality gap
with same solution between actual non-linear, non-convex ACOPF and associated
convex relaxed ACOPF with fixed line plans. Thus, inspired by results of [30],
the convexified compact form of (1) is (elements in $<>$ are associated
Lagrange multipliers)

$\displaystyle\min_{\mathbf{x},\mathbf{y},\mathbf{y}_{c}}\mathbf{c}^{T}\mathbf{x}+d$
(3a) s.t. $\displaystyle\mathbf{P}\mathbf{x}+\mathbf{Q}\mathbf{y}=\mathbf{f}$
$\displaystyle<\mathbf{z}_{2}>$ (3b)
$\displaystyle\mathbf{M}\mathbf{x}+\mathbf{N}\mathbf{y}\leq\mathbf{e}$
$\displaystyle<\mathbf{z}_{1}>$ (3c)
$\displaystyle\mathbf{y}_{c}+\mathbf{V}\mathbf{x}+\mathbf{W}\mathbf{y}=\mathbf{0}$
$\displaystyle<\mathbf{z}_{3}>$ (3d)
$\displaystyle\mathbf{y}_{cij}^{T}\mathbf{H}\mathbf{y}_{cij}\leq\mathbf{0},\forall(i,j)\in\mathcal{B}$
$\displaystyle<z_{cij}>$ (3e)

where $\mathbf{x}=\\{P_{gi},c_{ii},\forall i\in\mathcal{G}\\}$.
$\mathbf{y}_{c}=\\{\mathbf{y}_{cij},\forall(i,j)\in\mathcal{B}\\}$ is the
vector of variables used to represent the rotated cones in (2c) while
$\mathbf{y}$ is the vector of other variables. Hence, (3a) refers to (1a) with
$d=\sum_{i\in\mathcal{G}}b_{i}$. (3b) denotes to (1b) - (1g) while (3c)
denotes (1j) - (1n) and (2e). (3d) represents (2a) and (2b) while (3e) refers
to (2c), where $\mathbf{y}_{cij}=\\{C_{ij},S_{ij},E_{ij},D_{ij}\\}$.
$\mathbf{H}$ in (3e) is a $4\times 4$ diagonal matrix with the first three
diagonal elements as $1$ and last diagonal element as $-1$. $\mathbf{0}$ is a
zero matrix of appropriate dimension. Based on (3), the proposed robust ACOPF,
discussed next, gives robust $\mathbf{x}$ for short-term operation of AC power
network, whose feasibility is discussed in Section 3.3. The results in Section
4.3 also indicate this feasibility for short-term injection uncertainties,
which is further validated by MCS based participation factored AC power flows.

##  3 Proposed Methodology

The formulation in (1) can provide base-point active power generations and
terminal voltage magnitudes of conventional generators for ACOPF with no
uncertainties. However, with injection uncertainties, the same settings are
not optimal and result in line overloads with potential of cascaded outages
[3]. This phenomenon is illustrated in Fig. 1 for the NESTA $118$ bus system.
Hence, short-term injection uncertainties are taken into account for the
secure operation of the AC network.

Figure 1: Line flow violation due to solution from (1) in NESTA $118$ bus
system for all RESs and loads injection uncertainties. Conventional generator,
RES and load buses are marked by brown, green and blue dots, respectively. RES
penetration is $30\%$ of available generation capacities. Deviations in RESs
(loads) are $15\%$ ($5\%$) of respective nominal values. Lines in red exceed
their flow limits by $3\%-48.6\%$ of respective maximum capacity.

###  3.1 Modeling uncertainties and associated response

####  3.1.1 Uncertainty model

The uncertain injections of active power load and RES generation at bus $i$
are modeled as [7]

$\displaystyle\hat{P}_{di}$ $\displaystyle=P_{di}+\mu_{di},\forall
i\in\mathcal{L}$ (4a) $\displaystyle\hat{P}_{ri}$
$\displaystyle=P_{ri}+\mu_{ri},\forall i\in\mathcal{R}$ (4b) where $P_{di}$
($P_{ri}$) is the forecasted nominal real power load (RES generation).
$\mu_{di}$ ($\mu_{ri}$) is the associated deviation in the real power load
(RES generation) with zero mean around the respective forecasted value [7].
Further, $\bm{\mu}\in\mathcal{U}$,
$\mathcal{U}:=\\{\bm{\mu}\in\mathbb{R}^{|\mathcal{L}|+|\mathcal{R}|}:\bm{\mu}\in[-\overline{\bm{\mu}},\overline{\bm{\mu}}]\\}$,
where $\overline{\bm{\mu}}$ is the maximum deviation from the respective
nominal load and RES generation (RO requires range of uncertain parameters
[10]). Assuming constant power factor operation of loads, the uncertainty in
$Q_{di}$ can be stated as $\hat{Q}_{di}=Q_{di}+LR_{i}\mu_{di},\forall
i\in\mathcal{L}$ (4c) where $LR_{i}=\tan\phi_{i},\forall i\in\mathcal{L}$, and
$\cos\phi_{i}$ is the associated constant power factor of $i^{th}$ load.

####  3.1.2 RES generator model

RES generators, equipped with power electronics inverters can inject both
active and reactive powers. As per (4b) and defined range of uncertainty, the
uncertain active power output of $i^{th}$ RES generator, i.e.,
$\hat{P}_{ri},\forall i\in\mathcal{R}$ is in range of
$P_{ri}+[-\overline{\mu}_{ri},\overline{\mu}_{ri}]$. Consequently, instead of
(1k), the uncertain reactive power output of $i^{th}$ RES generator should
satisfy

$\underline{\hat{Q}_{ri}}\leq\hat{Q}_{ri}\leq\overline{\hat{Q}_{ri}},\forall
i\in\mathcal{R}$ (4d)

where
$\overline{\hat{Q}_{ri}}=-\underline{\hat{Q}_{ri}}=\sqrt{(S_{ri}^{max})^{2}-(P_{ri}+\overline{\mu}_{ri})^{2}}$
[19].

####  3.1.3 Modeling system response

The deviations in nominal loads and RESs generations in (4a) - (4d) due to
injection uncertainties change the network power flows even for the short-term
operation. To accommodate such changes, the controllable (conventional)
generators economically adjust their setpoints to ensure power balance and
maintain the desired voltage profile in the network at $\bm{\mu}\neq 0$ [7].
Hence, the active power output affinely adjusts as per the respective
Automatic Generation Control (AGC) action [28], i.e.,

$\hat{P}_{gi}=P_{gi}+\rho_{i}\psi,\forall i\in\mathcal{G}$ (4e)

where $\hat{P}_{gi}$ is the active power output of $i^{th}$ generator at
$\bm{\mu}\neq 0$. $P_{gi}$ and $\rho_{i}$ are the associated base-point output
at $\bm{\mu}=0$ and participation factor, respectively, with
$\sum_{i\in\mathcal{G}}\rho_{i}=1$ [7]. $\psi$ is the unknown real-time active
power mismatch, due to non-zero uncertain injections and resultant change in
network’s active power loss with respect to $\bm{\mu}=0$. Hence, as per (4a),
(4b), and (4e), (1b) for $\bm{\mu}\neq 0$ is

$P_{gi}+\rho_{i}\psi-\mu_{di}+\mu_{ri}-\hat{P}_{i}=P_{di}-P_{ri},\forall
i\in\mathcal{N}$ (4f)

where $\hat{P}_{i}$ is the active power injection at bus $i$ for $\bm{\mu}\neq
0$.

Similarly, (1c) for $\bm{\mu}\neq 0$, as per (4c) and (4d), is

$\hat{Q}_{gi}-LR_{i}\mu_{di}+\hat{Q}_{ri}-\hat{Q}_{i}=Q_{di},\forall
i\in\mathcal{N}$ (4g)

where $\hat{Q}_{gi},\forall i\in\mathcal{G}$ and $\hat{Q}_{i},\forall
i\in\mathcal{N}$ are the respective reactive generation and injection for
$\bm{\mu}\neq 0$. Also, (1l) for $\bm{\mu}\neq 0$ is

$\underline{P}_{gi}\leq
P_{gi}+\rho_{i}\psi\leq\overline{P}_{gi},~{}\underline{Q}_{gi}\leq\hat{Q}_{gi}\leq\overline{Q}_{gi},\forall
i\in\mathcal{G}$ (4h)

As per [32], the conventional generators regulate their reactive power outputs
so that $c_{ii},\forall i\in\mathcal{G}$ remain unchanged for $\bm{\mu}\neq 0$
and $\bm{\mu}=0$, due to the infrequent action of automatic voltage control as
compared to AGC in conventional generators [33]. Hence, keeping
$c_{ii},\forall i\in\mathcal{G}$ unchanged is desirable for the short-term
operation with injection uncertainties. It can be noted that $P_{i}$, $Q_{i}$,
$Q_{gi}$, $Q_{ri}$, defined for $\bm{\mu}=0$, respectively, change to
$\hat{P}_{i}$, $\hat{Q}_{i}$, $\hat{Q}_{gi}$, $\hat{Q}_{ri}$ for $\bm{\mu}\neq
0$ in (4d), (4f) - (4h). Similarly, all variables in (3), except $\mathbf{x}$,
i.e., $\mathbf{y}$ and $\mathbf{y}_{c}$, respectively, non-linearly vary to
$\mathbf{\hat{y}}$ and $\mathbf{\hat{y}}_{c}$ for $\bm{\mu}\neq 0$.
Additionally, the change in generators’ active power in (4e), i.e.,
$\rho_{i}\psi,\forall i\in\mathcal{G}$ is bounded by active power ramp limits
as

$-\overline{r}_{i}\leq\rho_{i}\psi\leq\overline{r}_{i},\forall
i\in\mathcal{G}$ (4i)

where $\overline{r}_{i}\geq 0$ is the respective limit.

###  3.2 Proposed robust ACOPF formulation

Thus, as per the above discussion, (3b) - (3e), respectively, transform to
(5b) - (5e) for $\bm{\mu}\neq 0$. Hence, the proposed primal robust ACOPF,
which minimizes the total generation cost for the worst-case uncertainty and
forecasted operating point while satisfying network constraints for base-case
(no uncertainty) and all non-zero realizations of uncertainty, is

$\displaystyle\max_{\bm{\mu}}\min_{\mathbf{x},\mathbf{y},\mathbf{y}_{c},\mathbf{\hat{y}},\mathbf{\hat{y}}_{c},\psi}\mathbf{c}^{T}\mathbf{x}+d\text{
s.t. }\eqref{m2eq}-\eqref{m2cone2}$ (5a)
$\displaystyle\mathbf{P}\mathbf{x}+\mathbf{Q}\mathbf{\hat{y}}+\mathbf{h}_{E}\psi+\mathbf{K}_{E}\bm{\mu}=\mathbf{f}$
$\displaystyle<\mathbf{\hat{z}}_{2}>$ (5b)
$\displaystyle\mathbf{M}\mathbf{x}+\mathbf{N}\mathbf{\hat{y}}+\mathbf{h}_{I}\psi\leq\mathbf{\hat{e}}$
$\displaystyle<\mathbf{\hat{z}}_{1}>$ (5c)
$\displaystyle\mathbf{\hat{y}}_{c}+\mathbf{V}\mathbf{x}+\mathbf{W}\mathbf{\hat{y}}=\mathbf{0}$
$\displaystyle<\mathbf{\hat{z}}_{3}>$ (5d)
$\displaystyle\mathbf{\hat{y}}_{cij}^{T}\mathbf{H}\mathbf{\hat{y}}_{cij}\leq\mathbf{0},\forall(i,j)\in\mathcal{B}$
$\displaystyle<\hat{z}_{cij}>$ (5e)
$\displaystyle\mathbf{h}_{r}\psi\leq\mathbf{r}$
$\displaystyle<\mathbf{\hat{z}}_{r}>$ (5f)

where $\mathbf{c}$, $d$, $\mathbf{P}$, $\mathbf{Q}$, $\mathbf{f}$,
$\mathbf{M}$, $\mathbf{N}$, $\mathbf{V}$, $\mathbf{W}$, and $\mathbf{H}$ are
as per (3). $\mathbf{e}$ in (3c) modifies to a known $\mathbf{\hat{e}}$ in
(5c) due to (4d) replacing (1k) for $\bm{\mu}\neq 0$ while bounds of other
constraints in (3c) remain unchanged, similar to the case in (1l) and (4h).
$\mathbf{h}_{E}$ and $\mathbf{h}_{I}$ are non-zero column vectors
corresponding to (4f) and (4h), respectively. Similarly, $\mathbf{K}_{E}$ is a
coefficient matrix corresponding to $\bm{\mu}\neq 0$ in (4f), (4g). (5f)
corresponds to (4i). The following remarks present three critical aspects of
(5).

######  Remark 1.

For a known $\bm{\mu}\neq 0$, it can be proven that without (3b) - (3e), there
exists a generator $i\in\mathcal{G}$ for which (5) gives
$\rho_{i}\psi=\overline{r}_{i}$ and
$P_{gi}\in[\underline{P}_{gi},\overline{P}_{gi}]-\overline{r}_{i}$, where it
is possible that $\underline{P}_{gi}\leq\overline{r}_{i}$ as
$\underline{P}_{gi}\geq 0$ and $0\leq\overline{r}_{i}\leq\overline{P}_{gi}$
(non-negative conventional active generation and ramp limits). Consequently,
(5) gives $P_{gi}\leq 0$ as optimal solution for minimum cost, which is
impractical. With (3b) - (3e) part of (5), this does not exist, due to (5f)
and
$P_{gi}\in[\max(\underline{P}_{gi},\underline{P}_{gi}-\rho_{i}\psi),\min(\overline{P}_{gi},\overline{P}_{gi}-\rho_{i}\psi)]\forall
i\in\mathcal{G}$.

######  Remark 2.

In (5), $\mathbf{y}$ and $\mathbf{y}_{c}$ non-linearly adjust to
$\mathbf{\hat{y}}$ and $\mathbf{\hat{y}}_{c}$ for $\bm{\mu}\neq 0$,
respectively, without any affine control policy.

######  Remark 3.

For $\bm{\mu}=0$, the resultant change in network’s active power loss with
respect to base-case (no uncertainty) is $0$. Hence, $\psi=0$ for
$\bm{\mu}=0$. Consequently, $\hat{P}_{gi}={P}_{gi}$ and $P_{i}=\hat{P}_{i}$ in
(4e), (4f), respectively. In fact, (5b) - (5e) are, respectively, equivalent
to (3b) - (3e) for $\bm{\mu}=0$. Nonetheless, (5f) couples the conditions
$\bm{\mu}=0$ and $\bm{\mu}\neq 0$ in (5). Since (5f) regulates the real-time
change in active generation of conventional generators, the level of short-
term injection uncertainties that can be handled depends on ramp limits.

It is evident from (5b) that uncertain injections appear in equality
constraints. Uncertain terms in equality constraints restrict the primal
feasible region [11]. However, in the dual form, the uncertain terms appear
only in the dual objective with dual constraints free of uncertain injections.
Hence, a novel single-stage non-recursive convex RC of (5) is proposed next,
which provides computationally time-efficient robust setpoints of conventional
generators for the short-term operation of AC power network with injection
uncertainties.

###  3.3 Novel single-stage non-recursive dual RC

The dual of (5) aids in the worst-case realization of
$\bm{\mu}\in\mathcal{U}$, as $\bm{\mu}$ appears only in dual objective to be
maximized. Further, the dual of (5) from conic dual theory [34] does not
guarantee strong duality. Hence, a novel dual transformation is utilized using
the _Karush-Kuhn-Tucker_ conditions of the associated Lagrange that guarantees
strong duality between dual of (5) and (5). This novel dual transformation of
(5) for $\bm{\mu}\in\mathcal{U}$ is

$\displaystyle\max
d-\mathbf{f}^{T}(\mathbf{z}_{2}+\mathbf{\hat{z}}_{2})-\mathbf{e}^{T}\mathbf{z}_{1}-\mathbf{\hat{e}}^{T}\mathbf{\hat{z}}_{1}-\mathbf{r}^{T}\mathbf{\hat{z}}_{r}+\bm{\mu}^{T}\mathbf{K}_{E}^{T}\mathbf{\hat{z}}_{2}$
(6a) $\displaystyle\text{s.t.
}\mathbf{Q}^{T}\mathbf{z}_{2}+\mathbf{N}^{T}\mathbf{z}_{1}+\mathbf{W}^{T}\mathbf{z}_{3}=\mathbf{0}\;<\mathbf{y}>$
(6b)
$\displaystyle\mathbf{Q}^{T}\mathbf{\hat{z}}_{2}+\mathbf{N}^{T}\mathbf{\hat{z}}_{1}+\mathbf{W}^{T}\mathbf{\hat{z}}_{3}=\mathbf{0}\;<\mathbf{\hat{y}}>$
(6c)
$\displaystyle\mathbf{P}^{T}(\mathbf{z}_{2}+\mathbf{\hat{z}}_{2})+\mathbf{M}^{T}(\mathbf{z}_{1}+\mathbf{\hat{z}}_{1})+\mathbf{V}^{T}(\mathbf{z}_{3}+\mathbf{\hat{z}}_{3})=-\mathbf{c}\;<\mathbf{x}>$
(6d)
$\displaystyle\mathbf{h}_{E}^{T}\mathbf{\hat{z}}_{2}+\mathbf{h}_{I}^{T}\mathbf{\hat{z}}_{1}+\mathbf{h}_{r}^{T}\mathbf{\hat{z}}_{r}=0\;<\psi>$
(6e)
$\displaystyle\mathbf{z}_{3ij}^{T}\mathbf{H}\mathbf{z}_{3ij}\leq\mathbf{0},\mathbf{\hat{z}}_{3ij}^{T}\mathbf{H}\mathbf{\hat{z}}_{3ij}\leq\mathbf{0},\forall(i,j)\in\mathcal{B}$
(6f)
$\displaystyle\mathbf{z}_{3ij}+2z_{cij}\mathbf{H}\mathbf{y}_{cij}=0,\forall(i,j)\in\mathcal{B}\;<\mathbf{y}_{cij}>$
(6g)
$\displaystyle\mathbf{\hat{z}}_{3ij}+2\hat{z}_{cij}\mathbf{H}\mathbf{\hat{y}}_{cij}=0,\forall(i,j)\in\mathcal{B}\;<\mathbf{\hat{y}}_{cij}>$
(6h)
$\displaystyle\mathbf{z}_{1},\mathbf{z}_{c},\mathbf{\hat{z}}_{1},\mathbf{\hat{z}}_{c},\mathbf{\hat{z}}_{r}\geq\mathbf{0},\mathbf{z}_{2},\mathbf{z}_{3},\mathbf{\hat{z}}_{2},\mathbf{\hat{z}}_{3}\text{
unrestricted}$ (6i)

where $\mathbf{z}_{3}=\\{\mathbf{z}_{3ij},\forall(i,j)\in\mathcal{B}\\}$,
$\mathbf{\hat{z}}_{3}=\\{\mathbf{\hat{z}}_{3ij},\forall(i,j)\in\mathcal{B}\\}$,
$\mathbf{z}_{c}=\\{z_{cij},\forall(i,j)\in\mathcal{B}\\}$ and
$\mathbf{\hat{z}}_{c}=\\{\hat{z}_{cij},\forall(i,j)\in\mathcal{B}\\}$. (6a) -
(6f) are obtained from conic duality theory [34], whereas (6g) - (6h) are
normalization constraints to ensure strong duality, similar to extended self
dual model in [35, 36]. More details on derivation of (6) and proof of strong
duality are in Appendix A of [30]. _Since strong duality exists, the Lagrange
multipliers in ( 6) are same as the primal variables of (5)_. Further,
$\bm{\mu}^{T}K_{E}^{T}\mathbf{\hat{z}_{2}}$ in (6a) can be re-written as

$\sum_{j=1}^{|\mathcal{U}|}\\{\max(R_{j},0)\overline{\mu}_{j}+\min(R_{j},0)\underline{\mu}_{j}\\}$
(7)

where $\overline{\mu}_{j}$ and $\underline{\mu}_{j}$ are upper and lower
bounds of $\mu_{j}$ in Section 3.1.1. $|\mathcal{U}|$ is cardinality of
$\mathcal{U}$ and $\mathbf{R}=\mathbf{K}_{E}^{T}\mathbf{\hat{z}_{2}}$.

For the worst-case realization of short-term injection uncertainties, it is
evident that a positive (non-positive) element of $\mathbf{R}$ in (6a) should
correspond to the associated element of $\bm{\mu}$ being at its maximum
(minimum). This is inherently ensured in (7). Therefore, the proposed novel
single-stage non-recursive RC of (6) can be stated by its epigraph [11] as

$\displaystyle\max
d-\mathbf{f}^{T}(\mathbf{z}_{2}+\mathbf{\hat{z}}_{2})-\mathbf{e}^{T}\mathbf{z}_{1}-\mathbf{\hat{e}}^{T}\mathbf{\hat{z}}_{1}-\mathbf{r}^{T}\mathbf{\hat{z}}_{r}+\mathbf{1}^{T}\mathbf{t}$
(8a) $\displaystyle\text{s.t. }R_{j}^{+}-R_{j}^{-}R_{j}=0,\forall
j=1,\dots,|\mathcal{U}|,\eqref{uncs}-\eqref{unce1}$ (8b) $\displaystyle
R_{j}^{+}\underline{\mu}_{j}-R_{j}^{-}\overline{\mu}_{j}\leq t_{j}\leq
R_{j}^{+}\overline{\mu}_{j}-R_{j}^{-}\underline{\mu}_{j},\forall
j=1,\dots,|\mathcal{U}|$ (8c) $\displaystyle 0\leq R_{j}^{+},R_{j}^{-}\leq
T,\forall j=1,\dots,|\mathcal{U}|$ (8d)

where $\mathbf{1}$ and $\mathbf{t}$ are column vectors of length
$|\mathcal{U}|$. $R_{j}$ is the $j^{th}$ element of $\mathbf{R}$, and is the
coefficient of $\mu_{j}$, the $j^{th}$ element of $\bm{\mu}$ in (6a). $T$ is a
large Big-M number. $R_{j}^{+}$ and $R_{j}^{-}$ take appropriate values as per
(8b) and (8d), based on the sign of $R_{j}$. Depending on $R_{j}^{+}$ and
$R_{j}^{-}$, $t_{j}$ in (8c) chooses an appropriate value so that the
objective in (8a) is maximized. The proposed RC (8) is convex and provides
robust conventional generators’ setpoints as per the following discussion for
all realizations of short-term injection uncertainties.

######  Remark 4.

To the best of authors’ knowledge, this is the first single-stage, non-
recursive RC formulation of robust ACOPF using duality theory, and is the main
contribution of this paper. (5) and (6) are primal-dual pairs, between which
strong duality exists [30], and (8) is the RC of (6). Hence, $\mathbf{x}$,
obtained as Lagrange multipliers of (6d), are actually the desired robust
primal variables in (5). Since, the desired robust $\mathbf{x}$ is directly
obtained from (8), there is no need of a recursive solution procedure. Hence,
(8) efficiently provides the robust values of $P_{gi}$, $c_{ii},\forall
i\in\mathcal{G}$ in a single-stage non-recursive manner, suitable for the
short-term operation of AC power network.

######  Lemma 3.1.

The feasible set of RC in (8) is a closed convex set, so that the RC of an
uncertain convex program is a convex program. If the RC is feasible, then all
instances of uncertain program are feasible, and the robust optimal value is
greater or equal to optimal values of all instances.

###### Proof.

Please see the proof of Proposition 2.2 in [10]. ∎

######  Lemma 3.2.

As per Theorem 1 of [26], with availability of flexible generation in (4e),
and $\underline{Q}_{gi}=-\infty,\forall i\in\mathcal{G}$, (8) is exact for
robust $\mathbf{x}$ in a cyclic network with cycle of size 3.

###### Proof.

Please see proof in Appendix A. ∎

######  Remark 5.

In Lemma 3.2, by exact, it is meant that the optimal value of (8) is same as
the exact actual robust ACOPF model, and also, for any admissible value of
$\bm{\mu}$, there exists $c_{ii},c_{ij},s_{ij},\theta_{i}$ for which (1h) and
(1i) are satisfied. It is to be noted that the robustness and exactness of
proposed formulation is ensured only for robust $\mathbf{x}$. Further, even if
the assumptions in Lemma 3.2 are not true, the feasibility of robust setpoints
($P_{gi}$, $c_{ii},\forall i\in\mathcal{G}$) obtained from (8) is numerically
ensured via MCS based participation factored AC power flows in Section 4.3.

The paper focuses on the short-term operation of AC power network, where the
deviations in short-term injection uncertainties of loads and RES generations
are not high [25]. Hence, no infeasibility issues are encountered in the
proposed formulation for the considered test cases. However, if infeasibility
occurs for large injection uncertainties (please see Remark 3), load
curtailment can be used as additional emergency corrective action [37], which
shall require solution of an exclusive corrective security constrained ACOPF
for the specific case. Although, it may be noted that solving corrective
security constrained ACOPF for specific infeasibility may be computationally
burdensome for short-term operation of power network. The budget of
uncertainty constraints to reduce conservatism of the robust setpoints from
(8) are as follows.

###  3.4 Budget of uncertainty constraints

With respect to (8), the budget of uncertainty constraints are

$\mathcal{U}=\\{\acute{\mu}_{j}:\acute{\mu}_{j}=\alpha_{j}\mu_{j},\mu_{j}\in[\underline{\mu}_{j},\overline{\mu}_{j}],\alpha_{j}\in\mathcal{A};\forall
j=1,\dots,|\mathcal{U}|\\}$ (9a)
$\mathcal{A}=\\{\alpha_{j}:\sum_{j=1}^{|\mathcal{U}|}\alpha_{j}=\Gamma,\alpha_{j}\in\\{0,1\\},\forall
j=1,\dots,|\mathcal{U}|\\}$ (9b) where $\Gamma$ is the budget parameter.
$\Gamma$ takes integer values in $[0,|\mathcal{U}|]$ and $\alpha_{j}$
indicates the status of $\mu_{j}$. The incorporation of (9a), (9b) in (8)
ensures that only a subset of the uncertain parameters affect the optimal
robust solution. (9a) can be considered in (8) by modifying (8c) as
$\alpha_{j}(R_{j}^{+}\underline{\mu}_{j}-R_{j}^{-}\overline{\mu}_{j})\leq
t_{j}\leq\alpha_{j}(R_{j}^{+}\overline{\mu}_{j}-R_{j}^{-}\underline{\mu}_{j})$
(9c) $\forall j=1,\dots,|\mathcal{U}|$. The constraint (9c) consists of
bilinear terms. With the symmetric uncertainty, i.e.,
$\underline{\mu_{j}}=-\overline{\mu}_{j}$, considered, (9c) can be written as
$\alpha_{j}(R_{j}^{+}+R_{j}^{-})\underline{\mu}_{j}\leq
t_{j}\leq\alpha_{j}(R_{j}^{+}+R_{j}^{-})\overline{\mu}_{j}$, or
$|{t_{j}}|\leq\alpha_{j}(R_{j}^{+}+R_{j}^{-})\overline{\mu}_{j}$ which is a
convex constraint. Further, to avoid consideration of the integer variables
$\alpha_{j}$ in (9b) so that the computational burden of solving the proposed
formulation for short-term operation of AC power network is minimized, (9b)
can be restated as
$\sum_{j=1}^{|\mathcal{U}|}\alpha_{j}=\Gamma;\alpha_{j}\in[0,1];\alpha_{j}(\alpha_{j}-1)\leq
0$ (9d)

$\forall j=1,\dots,|\mathcal{U}|$. In (9d), $\alpha_{j}$ need not be an
integer variable. Nevertheless (9b) is automatically satisfied, when (9d) is
satisfied. Hence, the proposed single-stage non-recursive non-integer dual RC
with budget of uncertainty constraints and with respect to (8) is

$\displaystyle\max
d-\mathbf{f}^{T}(\mathbf{z}_{2}+\mathbf{\hat{z}}_{2})-\mathbf{e}^{T}\mathbf{z}_{1}-\mathbf{\hat{e}}^{T}\mathbf{\hat{z}}_{1}-\mathbf{r}^{T}\mathbf{\hat{z}}_{r}+\mathbf{1}^{T}\mathbf{t}$
$\displaystyle\text{s.t. }\eqref{16b},\eqref{16d},\eqref{bud18},\eqref{bud19}$
(10)

######  Remark 6.

(10) is the dual RC of (5) with budget of uncertainty constraints, which
allows a trade-off between conservatism and robustness of optimal solution.

Depending on the specified $\Gamma\in\mathbb{Z}$, the following are true

  * •

If $\Gamma=0$, then no parameters are uncertain, and (10) is same as (3) (due
to strong duality stated in Section 3.3).

  * •

If $\Gamma=|\mathcal{U}|$, then all parameters are uncertain, and (10) is same
as (8).

  * •

$0<\Gamma<|\mathcal{U}|$, then (10) automatically chooses the short-term
injections that are supposed to be uncertain, and consequently, a trade off is
ensured between robustness and conservatism of the obtained $\mathbf{x}$.

##  4 Numeric Results

The efficacy of the proposed novel dual, convex RC of ACOPF with short-term
injection uncertainties and associated formulation with budget of uncertainty
constraints are verified on four NESTA [29] and the case9241pegase [38]
systems in terms of generation cost and computation time. Robustness analysis
for in-range (interval uncertainty) and out-of-range samples on independent
and identically distributed scenarios and feasibility verification of the
obtained robust solution is done through MCS based participation factored AC
power flow [28]. The robust solution from the proposed formulation is further
compared with the solution from the deterministic ACOPF and the solutions of
formulations reported in [18, 19]. RESs are assumed to be present in the
systems with a maximum penetration capacity of $30\%$. The ramp rate of a
conventional generator in (4i) is $\pm 0.75$ times the associated base-point
output. Since, the emphasis is on ACOPF with short-term injection
uncertainties, uncertainties beyond $15\%$ are not considered. All numeric
results are obtained in MATLAB using SeDuMi [36] on an Intel i7-2600 CPU with
3.40 GHz processor and 8 GB RAM.

###  4.1 Performance of (8) and impact of variable participation factors

Table 2: Optimal objective and computation time for $5\%$ load uncertainty and
varying uncertainties in RES generations

$\%$ RES | $\Gamma=|\mathcal{U}|$ | $\Gamma=1$ | $\Gamma=3$ | $\Gamma=5$ | Violation Probability
---|---|---|---|---|---
uncertainty | Obj. value | Time | Obj. value | Time | Obj. value | Time | Obj. value | Time | in- | out-of-
| ($\$/h$) | (s) | ($\$/h$) | (s) | ($\$/h$) | (s) | ($\$/h$) | (s) | range (%) | range (%)
NESTA 14
0 | 162.630 | 0.563 | 158.994 | 0.569 | 160.700 | 0.571 | 161.490 | 0.629 | 0 | 5.96
5 | 164.379 | 0.635 | 158.994 | 0.563 | 160.925 | 0.592 | 162.442 | 0.545 | 0 | 12.1
10 | 166.138 | 0.565 | 158.994 | 0.594 | 162.472 | 0.543 | 164.193 | 0.577 | 0 | 45.0
15 | 167.906 | 0.561 | 159.500 | 0.568 | 164.225 | 0.552 | 165.954 | 0.566 | 0 | 53.9
NESTA 57
0 | 778.488 | 0.839 | 747.790 | 0.780 | 757.769 | 0.716 | 762.010 | 0.7552 | 0 | 34.7
5 | 789.461 | 0.853 | 747.790 | 0.780 | 758.609 | 0.725 | 768.603 | 0.724 | 0 | 48.3
10 | 800.599 | 0.767 | 747.790 | 0.741 | 769.592 | 0.704 | 779.599 | 0.690 | 0 | 67.6
15 | 811.900 | 0.789 | 752.904 | 0.722 | 780.739 | 0.723 | 790.760 | 0.718 | 0 | 86.1
NESTA 118
0 | 2547.294 | 1.913 | 2398.124 | 1.307 | 2410.392 | 1.311 | 2420.288 | 1.284 | 0 | 19.1
5 | 2592.993 | 1.825 | 2398.124 | 1.332 | 2410.392 | 1.398 | 2419.372 | 1.326 | 0 | 27.4
10 | 2635.046 | 1.825 | 2398.124 | 1.387 | 2415.020 | 1.421 | 2431.605 | 1.238 | 0 | 54.2
15 | 2656.603 | 1.843 | 2398.890 | 1.871 | 2424.595 | 1.678 | 2449.642 | 1.236 | 0 | 73.1
NESTA 300
0 | 14321.926 | 29.224 | 12605.007 | 13.024 | 12700.644 | 14.180 | 12829.027 | 12.692 | 0 | 49.2
5 | 14402.808 | 21.697 | 12605.007 | 13.024 | 12700.644 | 14.868 | 12900.267 | 13.780 | 0 | 84.6
PEGASE 9241
0 | 284496.88 | 126.741 | 268621.05 | 115.691 | 268691.84 | 128.132 | 268756.49 | 121.285 | 0 | 68.2
5 | 286849.86 | 128.423 | 268635.39 | 119.671 | 268716.11 | 114.568 | 268786.58 | 118.172 | 0 | 94.6

The performance of the formulation in (8) is tested in the presence of all
injection uncertainties from loads and RESs, i.e., $\Gamma=|\mathcal{U}|$.
$5\%$ uncertainty is assumed in all system loads. With fixed participation
factors, i.e., $\rho_{i}=a_{i}^{-1}/(\sum_{i\in\mathcal{G}}a_{i}^{-1}),\forall
i\in\mathcal{G}$, the optimal objective and computation time for different
systems are shown in the second and third columns of Table 2 for different
levels of RES injection uncertainties. It can be observed that the optimal
objective in second column of Table 2 only increases by a maximum of $5\%$ of
the associated objective costs with no uncertainty in the RES generations. The
reported computation times also indicate that the proposed formulation may be
used for short-term ACOPF with short-term injection uncertainties.

With generation participation factors as additional variables, the optimal
objective can be further reduced. However, with this, (4f) becomes a bilinear
constraint. Hence, (8) is solved separately to analyze the impact of variable
participation factors with $\psi$ and $\bm{\mu}$ fixed for the worst-case
uncertainty realization (obtained with fixed $\rho_{i}$ in (8)) and
considering $\rho_{i},\forall i\in\mathcal{G}$ as new additional optimization
variables. With this formulation, for $5\%$ load, the reduction in optimal
objective is $0.139\%$ for $5\%$ RES generation uncertainty, and $0.328\%$ for
$15\%$ RES generation uncertainty, in NESTA $118$ bus system.

###  4.2 Impact of budget of uncertainty in (10)

A trade off between the robustness and conservatism of the obtained setpoints
can be ensured by solving the proposed formulation in (10). $\Gamma$ with (9d)
automatically decide the number of uncertain parameters that are allowed to
deviate from their respective nominal values, as discussed in Section 3.4. For
different $\Gamma$, the optimal cost and associated computation time for
different NESTA systems and case9241pegase system are given in second to ninth
columns of Table 2. The maximum reduction in optimal cost is $5.00\%$,
$7.26\%$ and $9.70\%$, for the NESTA $14$ bus, $57$ bus and $118$ bus systems,
respectively, corresponding to $\Gamma=1$ and $15\%$ RES uncertainty. While
for $\Gamma=1$ and $5\%$ RES uncertainty, the same reduction is $12\%$ and
$6\%$ for NESTA $300$ bus system and case9241pegase system, respectively.
Further, the computation time of solving the proposed formulation in (10),
generally, reduces with reduced $\Gamma$ due to lesser number of uncertain
injections.

Also, it can be noted from Table 2 that certain objectives are same for
$\Gamma=1$ and $\Gamma=3$, even for different RES uncertainty, as load
uncertainties are more dominant. With reduced $\Gamma$, the AC power network’s
security margin also tends to reduce, as shown in Fig. 2, because
incorporating the budget of uncertainty constraints causes critical lines to
carry more power, thus reducing the overload margin. For $5\%$ load and $10\%$
RES uncertainty, the brown line in Fig. 2 is the maximum active power limit of
each transmission line in the NESTA $57$ bus system, while the blue and orange
lines indicate the active power flows on lines for $\Gamma=5$ and
$\Gamma=|\mathcal{U}|$, respectively. These active power line flows are the
maximum possible values for each line, obtained from MCS ($N_{s}=10000$) based
participation factored AC power flow with robust generator setpoints from (10)
in Algorithm 1 for associated realizations of uncertainties. None of the
$N_{s}$ scenarios have power flow divergence or constraint violation, which
also indicates the feasibility of obtained robust setpoints for actual ACOPF.

Figure 2: Active power flows for NESTA $57$ bus system for different $\Gamma$
at $5\%$ load and $10\%$ RES injections’ uncertainties

Initialization: Obtain robust setpoints $\mathbf{x}$ from (10). For the chosen
$\mathcal{U}=[\underline{\bm{\mu}},\overline{\bm{\mu}}]$, create $N_{s}$
random scenarios of uncertain injections. Set $count=1$;

while  _$count\neq N_{s}$_ do

run participation factored AC power flow [28] using $\mathbf{x}$ for each
scenario if  _converged_ then

Evaluate the inequality constraints’ values in exact ACOPF if  _all inequality
constraints are satisfied_ then

go to next sample, $count\leftarrow count+1$

else

$\mathbf{x}$ is not robust

end if

else

$\mathbf{x}$ is not robust

end if

end while

$\mathbf{x}$ is robust

Algorithm 1 MCS based robustness check

###  4.3 Validating robustness and feasibility of obtained setpoints

The conventional generators’ setpoints obtained from (10) or (8) are robust
against the injection uncertainties, when for all realizations of
uncertainties as defined by the associated interval uncertainty sets, the AC
power flow converges and provides a solution that satisfies all the physical
and operational limits of the AC power network. Also, the proposed
formulations use SOC relaxation, which are exact for mesh networks with
assumptions in [26, 27]. Hence, the feasibility of the obtained robust
solution also needs to be validated for exact ACOPF. Further, as the paper
focuses on short-term operation of AC power network, validating the robustness
and feasibility of the obtained setpoints (for actual non-determistic ACOPF)
by running deterministic ACOPFs is not a viable option. Additionally,
validating robustness for short-term operation by solving exclusive actual
non-linear non-convex ACOPFs may defeat the purpose of obtaining time-
efficient robust setpoints. Therefore, the feasibility and robustness of the
setpoints from (8) and (10) are verified with MCS based participation factored
AC power flow [28] in Algorithm 1 for all cases in Table 2. For each scenario
of the $N_{s}=10000$ independent and identically distributed scenarios, the
participation factored AC power flow converges and all inequality constraints
of exact ACOPF are satisfied at the converged solution for the considered
uncertainties. Hence, the tenth column of Table 2 reports $0\%$ violation for
all the in-range scenarios robustness analysis.

Further, the maximum possible values of inequality constraints in exact ACOPF
after robustness check with $N_{s}=10000$ scenarios in Algorithm 1 for $5\%$
uncertainty in all system loads and $15\%$ uncertainty in all RES injections
in NESTA $118$ bus system are shown in Fig. 3 and Fig. 4. In Fig. 3(a), the
normalized active power generations and associated ramp are shown. Generators
$14$ and $46$ in Fig. 3(a) refer to buses $31$ and $103$ in Fig. 4(a),
respectively. Actual $\hat{P}_{g}$, $P_{g}$ and ramp for bus $31$ are
$0.23$pu, $0.217$pu and $0.0131$pu, respectively. For bus $103$, these values
are $0.98$pu, $0.9646$pu and $0.0154$pu, respectively. It is clear from Fig. 2
\- Fig. 4 that inequality constraints of exact ACOPF are within their limits.
Thus, the robust setpoints are feasible for actual ACOPF (as stated in
Appendix A) and immune to uncertainties.

Figure 3: Normalized maximum limit, obtained maximum values, and minimum limit
of (a) active power generations and generation ramp in (4i), (b) active power
flows, and (c) reactive power generations for $N_{s}=10000$ scenarios in
Algorithm 1 for NESTA $118$ bus system with $5\%$ uncertainty in all loads and
$15\%$ uncertainty in all RES injections Figure 4: Maximum limit, obtained
maximum values, and minimum limit of (a) bus voltage magnitudes (pu), (b) bus
voltage phase angles (radians), and (c) phase angle differences (radians)
across lines for condition in Fig. 3. The blue line in Fig. 4(a) shows the
obtained minimum bus voltage magnitudes

The robustness of obtained setpoints for unseen realization of uncertainty
using out-of-range analysis is also tested. For $\Gamma=|\mathcal{U}|$ and a
given $\overline{\bm{\mu}}$ (in Section 3.1.1), $10000$ random scenarios for
out-of-range analysis are generated as
$\bm{\mu}\in(-\overline{\bm{\mu}}-0.05,-\overline{\bm{\mu}})\cup(\overline{\bm{\mu}},\overline{\bm{\mu}}+0.05)$.
The violation probability in the last column of Table 2 indicates the %
scenarios in which at least one of the constraints in exact ACOPF is
unsatisfied, which is expected as the obtained setpoints are robust only for
the given interval of uncertainty, i.e.,
$\bm{\mu}\in[-\overline{\bm{\mu}},\overline{\bm{\mu}]})$[10]. Also, with
increase in uncertainty level, the violation probability increases due to
limited resources in respective AC networks.

Table 3: Maximum absolute difference ($\eta$) between robust setpoints from (8) and actual ACOPF with uncertain injections set at worst-case for $5\%$ uncertainty in all load and RES injections Test | NESTA | NESTA | NESTA | NESTA | PEGASE
---|---|---|---|---|---
system | 14 | 57 | 118 | 300 | 9241
$\eta$ ($\times 10^{-5}$) | 0.00492 | 8.50 | 6.56 | 11.3 | 39.5

The feasibility of the obtained robust setpoints from (8) is further verified
by comparing the same with the conventional generator setpoints obtained from
the actual non-linear and non-convex ACOPF with uncertain injections set at
the realized worst-case in (8). The associated maximum absolute differences
($\eta$) for the considered systems with $5\%$ uncertainty in all loads and
RES generations is given in Table 3, which indicate that the obtained robust
conventional generator setpoints from proposed formulation are also feasible
to the actual non-linear and non-convex ACOPF. Since, the setpoints are
feasible, the robustness verification via MCS based participation factored AC
power flow also yields $0\%$ violation for all in-range scenarios. Similar low
values of $\eta$ have also been observed for other cases in Table 2, which
further support the tightness of (2c) and (2e).

###  4.4 Comparison with previous formulations

####  4.4.1 Comparison with deterministic ACOPF

The solution of deterministic ACOPF in (1) is analyzed for injection
uncertainties in the NESTA $118$ bus system. Considering nominal values of
loads and RES generations, (1) is solved using MATPOWER to obtain the
conventional generator settings. The optimal objective cost from (1) and the
associated computation time are given in Table 4.

Table 4: Proposed formulation in (8) versus deterministic (Det.) ACOPF for NESTA $118$ bus system with $5\%$ load and $15\%$ RES injections uncertainty in Fig. 1 Attributes | Det. | Proposed
---|---|---
Objective Cost ($\$/hr$) | 2382.38 | 2656.603
Computation time ($s$) | 0.43 | 1.843
% violation in active power flow | 100 | 0
% violation in voltage magnitude | 56.9 | 0

For the obtained settings, all constraints of the AC power network are
satisfied, if all loads and RES generations are at their respective nominal
values. However, for non-zero injection uncertainties, the same generator
settings fail to be feasible. This is verified by testing the obtained
generator settings (at nominal load and RES generation) against $10,000$
independent and identically distributed scenarios (using Algorithm 1) for
$5\%$ uncertainty in all loads and $15\%$ uncertainty in all RES generations.
Similar results are shown in Fig. 1, where 6 lines (in red color) exceed their
limits by $3\%$ to $48.6\%$ from their maximum capacity for all generated
$10,000$ scenarios. Also, bus voltage magnitude constraints are violated for
$5690$ scenarios for similar conditions. For the same load and RES injection
uncertainties, the optimal objective from (8) is $11.51\%$ higher than the
optimal cost from (1). However, no constraints’ violations are observed for
the robust solution obtained from (8).

####  4.4.2 Comparison with [18]

The robust ACOPF formulation in [18] is also a single-stage formulation.
Hence, the solution from proposed formulation in (8) is compared with [18] in
terms of objective cost and feasibility for AC power network. Table 5 shows
the comparison between the objective cost of proposed formulation with respect
to the objective cost obtained by [18], where, for same uncertainty level,
[18] provides more than $5\%$ higher cost than the proposed formulation. [18]
checks for robustness of solution via real-time OPF, which is impractical from
the short-term operation perspective. Also, if the validation is to be done by
OPF, then the purpose of solving robust ACOPF is lost. Hence, using Algorithm
1 with $N_{s}=10,000$, the robustness of setpoints from [18] (assuming unity
voltage magnitudes) for considered uncertainties in Table 5 is less than
$1.06\%$, whereas from (8), $0\%$ violations for in-range robust analysis is
observed in Table 2.

Table 5: Proposed formulation in (8) versus [18] for NESTA $14$ bus system with $5\%$ in all loads and different levels of RES injections uncertainty $\%$ RES Unc. | $5\%$ | $10\%$ | $15\%$ | $20\%$
---|---|---|---|---
$\%$ change in objective cost | 5.5912 | 5.4003 | 5.2083 | 5.0165

####  4.4.3 Comparison with [19]

The solution from (8) is also compared with the solution obtained from the
formulation of [19] in terms of optimal objective, computation time, and load
curtailment for the IEEE 14 bus system data given in [19]. For the same
network condition, the robust solution from (8) has low cost as compared to
the cost reported in Fig. 1(b) on page 10 of [19], without any infeasibility
until $\sigma=4.5$ ($30\%$ uncertainty in RES) in Fig. 5. For situation just
beyond $\sigma=4.5$, exclusive corrective security constrained ACOPF with load
curtailment option is solved for the latest feasible trend of the worst-case
injection scenario identified from the proposed formulation. The load
curtailment penalty is $\$4,000$/MWh, same as in [19]. The maximum load
curtailment from our formulation is $0.0109\%$ of total active power load at
non-generator buses, while the same in [19] is $0.16\%$. Thus, the proposed
formulation provides $14.91\%$ load saving. Also, the reported average
computation time of formulation in [19] is $130.4$s, while the same for (8)
for feasible cases in only $1.367$s (approximately $98.95\%$ less).

Figure 5: Optimal operation cost (black) and load curtailment (blue) from (8)
for different $\sigma$ tested on IEEE 14 bus system of [19]

##  5 Conclusion

In this paper, a novel single-stage non-recursive non-integer dual RC of ACOPF
is proposed that provides robust setpoints (base-point active power
generations and terminal voltage magnitudes) of conventional generators, which
are immune to the short-term injection uncertainties in loads and RES
generations. By considering constraints for both base-case (no uncertainties)
and worst-case uncertainties, and coupling them with ramp constraints on
active power generations of conventional generators, the non-linear impact of
uncertainties on optimization variables (except conventional generators’
setpoints), including change in total active power loss, is inherently ensured
without resorting to any affine policies. Using the strong duality property of
the novel dual form of convexified ACOPF, novel linear constraints are
proposed for the worst-case realization of short-term injection uncertainties.
The strong duality of novel dual aids in directly retrieving the desired
robust variables in a non-recursive single-stage manner. As compared to
previous formulations, the proposed formulation does not rely on cutting
plane, mixed-integer or multi-stage approaches. Instead, a novel convexified
robust ACOPF model is proposed, which can be solved by existing commercial
solvers, such as SeDuMi. The efficiency of proposed formulation is
demonstrated through numeric simulations on NESTA systems and the
case9241pegase system. For an uncertainty range of $10-15\%$, the obtained
worst-case optimal generation cost is $10-25\%$ higher as compared to the
nominal generation cost. To reduce this over conservatism, budget of
uncertainty constraints are included, which reduce the increase in the
generation cost to the range of $5\%$ \- $12\%$ of the nominal cost. The
obtained robust setpoints are feasible for all realizations of short-term
injection uncertainties, which is verified through proofs and MCS based
participation factored AC power flows. This verification also justifies the
feasibility of obtained setpoints for actual non-linear, non-convex ACOPF with
injection uncertainties. In summary, the proposed dual formulation provides
time-efficient realization of worst-case uncertainty in the equality
constraints of ACOPF. The formulation in [18] gives higher cost (more than
$5\%$) than proposed formulation, with more than $99\%$ probability of
violation for in-range robust analysis. Also, compared to [19], the proposed
formulation has low cost ($14.91\%$ load saving) and computation time
($\approx 98.95\%$ less). It will be of interest to extend the proposed idea
to solve robust multi-period security constrained OPF and transmission
switching problems.

## Acknowledgement

We are grateful to Science and Engineering Research Board, New Delhi, India
for providing financial support to carry out this research work via project
no. CRG/2020/001306.

##  Appendix A Proof of Lemma 3.2

Let $\mathbf{x}^{*}$ be the Lagrange multiplier of (6d) obtained from the
optimal solution of (8). Since (5) and (6) are strong primal-dual pairs [30],
thus, $\mathbf{x}^{*}$ along with other Lagrange multipliers are also the
optimal solution of (5). Thus, (5), (6), or (8) is exact, if for
$\mathbf{x}=\mathbf{x}^{*}$, the following feasibility problem

$\displaystyle\max_{\mathbf{y},\mathbf{y}_{c}}\sum_{i\in\mathcal{G}}Q_{gi}\text{
s.t. }\eqref{m2eq}-\eqref{m2cone2}$ (A.1)

has an optimal solution such that (1h) and (1i) are satisfied. A problem
similar to (A.1) can also be defined in terms of $\mathbf{\hat{y}}$,
$\mathbf{\hat{y}}_{c}$, $\psi$ with $\mathbf{x}=\mathbf{x}^{*}$, worst-case
$\bm{\mu}$ in (8), (5b) - (5f) as constraints, and objective in terms of
$\hat{Q}_{gi}\forall i\in\mathcal{G}$. As per Theorem 3 of [27], for a network
with cycles of size 3, semi-definite programming formulation is equivalent to
relaxed version of cyclic constraints combined with SOC constraints.
Therefore, under the assumption of presence of flexible generator in a
lossless weakly cyclic network, (A.1) is same as the one used for proof of
Theorem 1 in [26]. Hence, the same reasoning can be used here to show the
exactness of (5), (6), or (8).

## References

  * [1] S. R. Khuntia, B. W. Tuinema, J. L. Rueda, and M. A. M. M. van der Meijden, “Time-horizons in the planning and operation of transmission networks: an overview,” _IET Gener. Transm. Distrib._ , vol. 10, no. 4, pp. 841 – 848, 2016.
  * [2] A. J. Lamadrid, D. Muñoz-Alvarez, C. E. Murillo-Sánchez, R. D. Zimmerman, H. Shin, and R. J. Thomas, “Using the matpower optimal scheduling tool to test power system operation methodologies under uncertainty,” _IEEE Trans. Sustainable Energy_ , vol. 10, no. 3, pp. 1280–1289, 2019.
  * [3] D. Bienstock, M. Chertkov, and S. Harnett, “Chance-constrained optimal power flow: Risk-aware network control under uncertainty,” _Siam Review_ , vol. 56, no. 3, pp. 461 – 495, 2014.
  * [4] J. Liang, D. D. Molina, G. K. Venayagamoorthy, and R. G. Harley, “Two-level dynamic stochastic optimal power flow control for power systems with intermittent renewable generation,” _IEEE Trans. Power Syst._ , vol. 28, no. 3, pp. 2670 – 2678, 2013.
  * [5] Y. Guo, K. Baker, E. Dall’Anese, Z. Hu, and T. H. Summers, “Data-based distributionally robust stochastic optimal power flow—part i: Methodologies,” _IEEE Trans. Power Syst._ , vol. 34, no. 2, pp. 1483 – 1492, 2019.
  * [6] R. Lu, T. Ding, B. Qin, J. Ma, X. Fang, and Z. Dong, “Multi-stage stochastic programming to joint economic dispatch for energy and reserve with uncertain renewable energy,” _IEEE Trans. Sustainable Energy_ , vol. 11, no. 3, pp. 1140–1151, 2020.
  * [7] L. Roald and G. Andersson, “Chance-constrained ac optimal power flow: Reformulations and efficient algorithms,” _IEEE Trans. Power Syst._ , vol. 33, no. 3, pp. 2906 – 2918, 2018.
  * [8] P. Xiong, P. Jirutitijaroen, and C. Singh, “A distributionally robust optimization model for unit commitment considering uncertain wind power generation,” _IEEE Trans. Power Syst._ , vol. 32, no. 1, pp. 39–49, 2017\.
  * [9] C. Duan, W. Fang, L. Jiang, L. Yao, and J. Liu, “Distributionally robust chance-constrained approximate ac-opf with wasserstein metric,” _IEEE Trans. Power Syst._ , vol. 33, no. 5, pp. 4924–4936, 2018.
  * [10] A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” _Math. Oper. Res._ , vol. 23, no. 4, pp. 769 – 805, 1998.
  * [11] B. L. Gorissen, İ. Yanıkoğlu, and D. den Hertog, “A practical guide to robust optimization,” _Omega_ , vol. 53, pp. 124 – 137, 2015.
  * [12] H. Yu and W. D. Rosehart, “An optimal power flow algorithm to achieve robust operation considering load and renewable generation uncertainties,” _IEEE Trans. Power Syst._ , vol. 27, no. 4, pp. 1808 – 1817, 2012.
  * [13] D. Bertsimas, E. Litvinov, X. A. Sun, J. Zhao, and T. Zheng, “Adaptive robust optimization for the security constrained unit commitment problem,” _IEEE Trans. Power Syst._ , vol. 28, no. 1, pp. 52 – 63, 2013\.
  * [14] R. A. Jabr, “Adjustable robust opf with renewable energy sources,” _IEEE Trans. Power Syst._ , vol. 28, no. 4, pp. 4742 – 4751, 2013.
  * [15] T. Ding, Z. Bie, L. Bai, and F. Li, “Adjustable robust optimal power flow with the price of robustness for large-scale power systems,” _IET Gener. Transm, Distrib._ , vol. 10, no. 1, pp. 164 – 174, 2016.
  * [16] P. N. Yasasvi, A. Mohapatra, and S. C. Srivastava, “Efficient sensitivity based approach for ac transmission expansion planning,” _IET Gener. Transm. Distrib._ , vol. 14, no. 12, pp. 2378 – 2388, 2020.
  * [17] A. Ben-Tal and A. Nemirovski, “Robust optimization–methodology and applications,” _Math. Program._ , vol. 92, no. 3, pp. 453 – 480, 2002.
  * [18] X. Bai, L. Qu, and W. Qiao, “Robust ac optimal power flow for power networks with wind power generation,” _IEEE Trans. Power Syst._ , vol. 31, no. 5, pp. 4163 – 4164, 2016.
  * [19] R. Louca and E. Bitar, “Robust ac optimal power flow,” _IEEE Trans. Power Syst._ , vol. 34, no. 3, pp. 1669 – 1681, 2019.
  * [20] D. Lee, K. Turitsyn, D. K. Molzahn, and L. A. Roald, “Robust ac optimal power flow with robust convex restriction,” _IEEE Trans. Power Syst._ , vol. 36, no. 6, pp. 4953–4966, 2021.
  * [21] A. Attarha, N. Amjady, and A. J. Conejo, “Adaptive robust ac optimal power flow considering load and wind power uncertainties,” _Int. J. Electr. Power Energy Syst._ , vol. 96, pp. 132 – 142, 2018.
  * [22] A. Lorca and X. A. Sun, “The adaptive robust multi-period alternating current optimal power flow problem,” _IEEE Trans. Power Syst._ , vol. 33, no. 2, pp. 1993 – 2003, 2018.
  * [23] H. Heidarabadi, F. Pourahmadi, and S. H. Hossieni, “Robust linear model for multi-period ac optimal power flow,” _IET Gener. Transm, Distrib._ , vol. 14, no. 13, pp. 2535 – 2548, 2020.
  * [24] H. Yang, D. P. Morton, C. Bandi, and K. Dvijotham, “Robust optimization for electricity generation,” _INFORMS J. Computing_ , vol. 33, no. 1, pp. 336 – 351, 2021.
  * [25] Security constrained economic despatch of inter-state generating stations pan-india. [Online]. Available: https://tinyurl.com/4jabrdnu
  * [26] M. Chamanbaz, F. Dabbene, and C. M. Lagoa, “Probabilistically robust ac optimal power flow,” _IEEE Control Netw. Syst._ , vol. 6, no. 3, pp. 1135–1147, 2019.
  * [27] H. Hijazi, C. Coffrin, and P. Van Hentenryck, “Polynomial sdp cuts for optimal power flow,” in _Proc. Power Syst. Comput. Conf. (PSCC)_ , 2016, pp. 1–7.
  * [28] J. Meisel, “System incremental cost calculations using the participation factor load-flow formulation,” _IEEE Trans. Power Syst._ , vol. 8, no. 1, pp. 357 – 363, 1993.
  * [29] (2014) NESTA, The NICTA Energy System Test Case Archive. [Online]. Available: http://arxiv.org/abs/1411.0359
  * [30] P. N. Yasasvi, A. Mohapatra, and S. C. Srivastava, “Robust ac transmission expansion planning using a novel dual-based bi-level approach,” _IEEE Trans. Power Syst._ , vol. 37, no. 4, pp. 2881 – 2893, 2022.
  * [31] S. M. Fatemi, S. Abedi, G. B. Gharehpetian, S. H. Hosseinian, and M. Abedi, “Introducing a novel dc power flow method with reactive power considerations,” _IEEE Trans. Power Syst._ , vol. 30, no. 6, pp. 3012 – 3023, 2015.
  * [32] L. Halilbašić, P. Pinson, and S. Chatzivasileiadis, “Convex relaxations and approximations of chance-constrained ac-opf problems,” _IEEE Trans. Power Syst._ , vol. 34, no. 2, pp. 1459–1470, 2019.
  * [33] M. A. El-Kady, B. D. Bell, V. F. Carvalho, R. C. Burchett, H. H. Happ, and D. R. Vierath, “Assessment of real-time optimal voltage control,” _IEEE Trans. Power Syst._ , vol. 1, no. 2, pp. 98 – 105, 1986.
  * [34] S. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * [35] Z. Q. Luo, J. F. Sturm, and S. Zhang, “Conic convex programming and self-dual embedding,” _Optim. Methods and Software_ , vol. 14, no. 3, pp. 169 – 218, 2000.
  * [36] J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” _Optim. Methods and Software_ , vol. 11, no. 1-4, pp. 625 – 653, 1999.
  * [37] A. Conceicao and C. Castro, “Corrective control strategies in case of infeasible operating situations,” in _2001 Power Engg. Society Summer Meeting. Conf. Proc. (Cat. No.01CH37262)_ , vol. 2, 2001, pp. 962 – 967.
  * [38] Matpower case9241pegase test system data. [Online]. Available: https://matpower.org/docs/ref/matpower6.0/case9241pegase.html