# Chance Constrained Economic Dispatch Considering the Capability of Network
Flexibility Against Renewable Uncertainties

Yue Song, David J. Hill, and Tao Liu This work was supported by the HKU Seed
Fund for Basic Research for New Staff under Project No. 202009185007.Y. Song
and T. Liu are with the Department of Electrical and Electronic Engineering,
The University of Hong Kong, Hong Kong (e-mail<EMAIL_ADDRESS>taoliu@eee.hku.hk).D. J. Hill is with the Department of Electrical and
Electronic Engineering, The University of Hong Kong, Hong Kong, and also with
the School of Electrical Engineering and Telecommunications, The University of
New South Wales, Kensington, NSW 2052, Australia (e-mail: dhill@eee.hku.hk).

###### Abstract

This paper introduces network flexibility into the chance constrained economic
dispatch (CCED). In the proposed model, both power generations and line
susceptances become variables to minimize the expected generation cost and
guarantee a low probability of constraint violation in terms of generations
and line flows under renewable uncertainties. We figure out the mechanism of
network flexibility against uncertainties from the analytical form of CCED. On
one hand, renewable uncertainties shrink the usable line capacities in the
line flow constraints and aggravate transmission congestion. On the other
hand, network flexibility significantly mitigates congestion by regulating the
base-case line flows and reducing the line capacity shrinkage caused by
uncertainties. Further, we propose an alternate iteration solver for this
problem, which is efficient. With duality theory, we propose two convex
subproblems with respect to generation-related variables and network-related
variables, respectively. A satisfactory solution can be obtained by
alternately solving these two subproblems. The case studies on the IEEE 14-bus
system and IEEE 118-bus system suggest that network flexibility contributes
much to operational economy under renewable uncertainties.

###### Index Terms:

economic dispatch, network flexibility, chance constraint, transmission
congestion

##  I Introduction

Economic dispatch (ED) is a representative class of optimal power flow
problems, which aims to find the most economical generation scheme that meets
load consumption and satisfies operational constraints regarding generations
and line flows. The normal ED problem, which does not consider uncertain power
injections and takes a deterministic formulation, has been extensively studied
and leads to many classic results [1]. For instance, when the generation and
line flow limits are ignored, the optimal dispatch scheme is determined by the
so-called equal incremental cost criterion. When the generation and line flow
limits are included, transmission congestion may occur and the optimal
dispatch scheme induces the locational marginal price at each bus. These
results are of fundamental importance in power system operation.

With the growing penetration of renewable energy, nowadays system operators
are asking for more from ED to tackle the challenge posed by the uncertainty
nature of renewables. Under this background, the chance constrained economic
dispatch (CCED) is a notable extension that receives popularity. Note that the
system states become random under renewable uncertainties. The CCED replaces
the deterministic constraints by chance constraints to guarantee a low
probability of constraint violation in case that the renewable generations
follow a certain probability distribution [2]. The CCED solution is much less
conservative than the solution given by robust optimization and achieves a
sufficiently low risk of insecurity. The DC power flow model is commonly
adopted in CCED and the problem is equivalent to a second-order cone
programming (SOCP) [2, 3]. The AC power flow-based CCED formulations are
emerging and solved by more sophisticated methods such as convex relaxation
[4], sequential linearization [5] and polynomial chaos expansion [6]. In
addition, the CCED is being extended to the distributionally robust version
[7, 8, 9] to make the solution work under a set of probability distributions
for renewables.

Traditionally, ED problems rely on the flexibility of generation side. On the
other hand, the power network, which is another component of power flow (in
terms of admittance matrix), used to be treated as constant in ED. Thanks to
the advances in power electronics and communication technologies, this
viewpoint becomes outdated as network flexibility is enabled by the discrete
and continuous adjustments of line parameters via remote controlled switches
and series FACTS, respectively [10]. Network flexibility is now adding a new
dimension of flexibility to system dispatch. The ED problem with discrete
change of line parameters (line switching) is usually known as the optimal
transmission switching (OTS) problem [11, 12, 13]. The continuous change of
line parameters is “softer” than OTS and attracts attention recently. It has
been shown that the inclusion of flexible line parameters greatly improves the
system performance in terms of economy and security under normal operating
scenarios or contingencies [14, 15, 16].

Despite the respective success of chance constraints and network flexibility
in ED problems, there are no studies to combine these two thoughts and better
address the impact of renewable uncertainties. Some existing works managed to
include a set of operating scenarios to represent the uncertain renewable
outputs (e.g., [17, 18]), where the line parameters are just treated as extra
decision variables in the numerical optimization. Although the obtained
solution improves the operational economy of high-renewable systems, it does
not explain why and how the improvement is made. Overall, there is a lack of
explicit mechanisms of how the network parameters affect system operation
under renewable uncertainties, which prevents a further benefit from network
flexibility.

In this paper, we establish a CCED formulation considering continuously
adjustable line susceptances, which finds the optimal generation and line
susceptance scheme with the minimal expected generation cost while satisfying
the chance constraints regarding generations and line flows. The main
contributions are twofold. First, we derive an analytical form of the CCED
problem that reveals the role of network flexibility. It turns out that
renewable uncertainties take up some line capacities and shrink the feasible
region for base-case flows, while network flexibility tunes the base-case line
flows and reduces the line capacities taken up by uncertainties. With the help
of network flexibility, transmission congestion is greatly mitigated so that
the cost-effective generations can be better utilized. Second, we design an
alternate iteration method to efficiently solve the CCED problem with network
flexibility. The problem formulation is highly complex; however, by the
property of its analytical form together with duality theory, we establish two
subproblems with respect to generation variables and network variables. These
two subproblems are both convex and easy to solve. A satisfactory solution to
the original problem can be obtained by alternately solving these two
subproblems. This work exploits the capability of network flexibility in
handling the impact of uncertainties on operational economy.

The remainder of the paper is organized as follows. Section II formulates the
CCED model considering network flexibility and explores the mechanism of
network flexibility against renewable uncertainties. The solution methodology
is elaborated in Section III, and the case studies on two IEEE test systems
are given in Section IV. Section V concludes the paper.

##  II Formulating CCED with network flexibility

###  II-A The original problem formulation

We first introduce some notations that will be used throughout the paper.
Consider a transmission system with the set of buses $\mathcal{V}$ and set of
lines $\mathcal{E}$. The cardinalities of $\mathcal{V}$ and $\mathcal{E}$ are
$n$ and $l$, respectively. The buses may connect traditional dispatchable
generators, renewable generators or loads. The active power generation from
dispatchable generators, renewable generation and load at bus
$i\in\mathcal{V}$ are respectively denoted as $P_{gi},P_{wi},P_{di}$. The set
of buses with dispatchable generators is denoted as
$\mathcal{V}_{g}\subseteq\mathcal{V}$. The line $k$ connecting bus $i$ and $j$
is denoted as $e_{k}=(i,j)\in\mathcal{E}$. The susceptance of line $(i,j)$ is
denoted as $b_{ij}$, and the line flow from bus $i$ to $j$ is denoted as
$f_{ij}$. Assume there is a subset of lines that install series FACTS devices
and hence have adjustable susceptances, denoted as
$\mathcal{E}_{\textup{flex}}\subseteq\mathcal{E}$. For convenience, we define
the vectors $\bm{P}_{g},\bm{P}_{w},\bm{P}_{d}\in\mathbb{R}^{n}$ and
$\bm{b},\bm{f}\in\mathbb{R}^{l}$ that stack the quantities
$P_{gi},P_{wi},P_{di},b_{ij},f_{ij}$, respectively. In addition, the incidence
matrix $\bm{E}\in\mathbb{R}^{n\times l}$ is defined as follows. Suppose each
line is assigned an orientation, i.e., $e_{k}=(i,j)$ originates at bus $i$ and
terminates at bus $j$, then $\forall e_{k}=(i,j)\in\mathcal{E}$, $E_{ik}=1$,
$E_{jk}=-1$ and $E_{mk}=0$ if $m\neq i,j$. The admittance matrix is then given
by $\bm{B}(\bm{b})=\bm{E}\widehat{\bm{b}}\bm{E}^{T}\in\mathbb{R}^{n\times n}$,
where $\widehat{\bm{b}}\in\mathbb{R}^{l\times l}$ is a diagonal matrix with
the main diagonal being $\bm{b}$.

The renewable generations are modelled as a combination of forecast values and
forecast errors

$\begin{split}\bm{P}_{w}=\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{w}+\bm{u}\end{split}$ (1)

where $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{w}\in\mathbb{R}^{n}$ denotes the forecast values and
$\bm{u}=[u_{wi}]\in\mathbb{R}^{n}$ denotes the renewable uncertainties caused
by forecast errors. We assume $\bm{u}$ follows a multivariate probability
distribution with a zero mean vector and covariance matrix
$\bm{\Sigma}\in\mathbb{R}^{n\times n}$. If bus $i$ does not connect a
renewable generator, then $\mkern 3.5mu\overline{\mkern-3.5muP\mkern
0.0mu}\mkern 0.0mu_{wi}$, $u_{wi}$ and $i$-th row and column of $\bm{\Sigma}$
are set to zero.

The traditional generators adopt the common affine control to balance the
renewable fluctuation, so that $\bm{P}_{g}$ also consists of two parts

$\begin{split}\bm{P}_{g}=\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}-\bm{\alpha}_{g}\bm{1}_{n}^{T}\bm{u}\end{split}$ (2)

where $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}\in\mathbb{R}^{n}$ denotes the base-case generations for the forecast
scenario; $\bm{1}_{n}\in\mathbb{R}^{n}$ denotes a vector with all entries
being unity; and $\bm{\alpha}_{g}=[\alpha_{gi}]\in\mathbb{R}^{n}$ denotes the
vector of participation factors that determines power sharing under renewable
fluctuation. We have $\bm{1}_{n}^{T}\bm{\alpha}_{g}=1$, and a greater
$\alpha_{gi}$ means the dispatchable generator at bus $i$ undertakes more
renewable fluctuation in the balancing. We set $\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ and $\alpha_{gi}$
to be zero for bus $i\notin\mathcal{V}_{g}$.

Since we focus on the impact of renewable uncertainties, for simplicity we
assume the forecasted loads $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{d}\in\mathbb{R}^{n}$ is accurate. Nevertheless, the load
uncertainties can be handled in a similar way to the method proposed in this
paper. With the above notations, we express the DC power flow equation as

$\displaystyle\bm{1}_{n}^{T}(\bm{P}_{g}+\bm{P}_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d})=0$ (3a)
$\displaystyle\bm{f}=\bm{T}_{f}(\bm{b})(\bm{P}_{g}+\bm{P}_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d})$ (3b)

where

$\begin{split}\bm{T}_{f}(\bm{b})=\widehat{\bm{b}}\bm{E}^{T}\bm{B}^{{\dagger}}\in\mathbb{R}^{l\times
n}\end{split}$ (4)

is the power transfer distribution factor matrix and $\bm{B}^{{\dagger}}$
denotes the Moore-Penrose inverse of admittance matrix $\bm{B}$. We write
$\bm{T}_{f}$ as a function of $\bm{b}$ as $\bm{b}$ is a variable in this
paper. (3a) describes the balance between generations and loads, and (3b)
gives the line flows under the power injection profile. Note that (3b) is
derived from $\bm{P}_{g}+\bm{P}_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{d}=\bm{B}\bm{\theta}$ and
$\bm{f}=\widehat{\bm{b}}\bm{E}^{T}\bm{\theta}$, where $\bm{\theta}$ is the
vector of voltage phase angles. It can be seen from (1)-(3) that $f_{ij}$ is
dependent on $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g},\bm{b},\bm{u}$, and hence we write it into the
compact form $f_{ij}(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g},\bm{b},\bm{u})$.

Then, the CCED problem considering network flexibility, which takes $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g},\bm{b}$ as decision variables, is formulated as

$\displaystyle\min_{\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g},\bm{b}}\leavevmode\nobreak\
\mathbb{E}\\{\bm{P}_{g}^{T}\bm{a}_{2}\bm{P}_{g}+\bm{a}_{1}^{T}\bm{P}_{g}\\}$
(5a) $\displaystyle s.t.\leavevmode\nobreak\
\eqref{Pw},\eqref{Pg},\eqref{DCpower}$ (5b) $\displaystyle b_{ij}^{\min}\leq
b_{ij}\leq b_{ij}^{\max},\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}_{\textup{flex}}$ (5c)
$\displaystyle\bm{1}_{n}^{T}\bm{\alpha}_{g}=1$ (5d)
$\displaystyle\alpha_{gi}=0,\leavevmode\nobreak\ \forall
i\notin\mathcal{V}_{g}$ (5e) $\displaystyle\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}=0,\leavevmode\nobreak\ \forall i\notin\mathcal{V}_{g}$ (5f)
$\displaystyle\Pr\\{P_{gi}\leq P_{gi}^{\max}\\}\geq
1-\varepsilon_{i},\leavevmode\nobreak\ \forall i\in\mathcal{V}_{g}$ (5g)
$\displaystyle\Pr\\{P_{gi}\geq P_{gi}^{\min}\\}\geq
1-\varepsilon_{i},\leavevmode\nobreak\ \forall i\in\mathcal{V}_{g}$ (5h)
$\displaystyle\Pr\\{f_{ij}(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g},\bm{b},\bm{u})\leq f_{ij}^{\max}\\}\geq
1-\varepsilon_{ij},\leavevmode\nobreak\ \forall(i,j)\in\mathcal{E}$ (5i)
$\displaystyle\Pr\\{f_{ij}(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g},\bm{b},\bm{u})\geq-f_{ij}^{\max}\\}\geq
1-\varepsilon_{ij},\leavevmode\nobreak\ \forall(i,j)\in\mathcal{E}$ (5j)

where $\bm{a}_{1}=[a_{1,i}]\in\mathbb{R}^{n}$ and
$\bm{a}_{2}=\textup{diag}\\{a_{2,i}\\}\in\mathbb{R}^{n\times n}$ denote the
linear and quadratic coefficients for generation cost; $\mathbb{E}\\{\cdot\\}$
and $\Pr\\{\cdot\\}$ denote the mathematical expectation and probability;
$\varepsilon_{i}$ and $\varepsilon_{ij}$ are predefined small positive numbers
for regulating the risk of constraint violation; $b_{ij}^{\min},b_{ij}^{\max}$
denote the minimum and maximum susceptance of line $(i,j)$;
$P_{gi}^{\min},P_{gi}^{\max}$ denote the minimum and maximum output of the
dispatchable generator at bus $i$ that are determined by generation capacity
and ramp rate; and $f_{ij}^{\max}$ denotes the transmission capacity of line
$(i,j)$ that is possibly regulated by thermal, voltage, or stability
considerations. Constraints (5e) and (5f) are consistent with the previous
discussion on those buses without dispatchable generators. The probabilities
in chance constraints (5g)-(5j) are evaluated under the renewable
uncertainties $\bm{u}$ and a certain fixed profile of base-case generations
$\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}$,
participation factors $\bm{\alpha}_{g}$ and line susceptances $\bm{b}$.

The optimal solution to problem (5) has two features. First, it achieves the
minimal expectation of generation cost under renewable uncertainties. Second,
the probabilities of violating the generation limits and line flow limits are
lower than the predefined values, which guarantees a highly secure operating
status under renewable uncertainties. It will be seen later that the
introduction of variable line susceptances $\bm{b}$ into (5) significantly
reduces the transmission congestion under uncertainties and enhances
operational economy.

###  II-B Analytical form of CCED

Since the original problem formulation in (5) is intractable, we transform it
into an analytical form that helps to reveal the role of decision variables in
CCED.

According to (1), (2) and (3b), the dispatchable generation $P_{gi}$ and line
flow $f_{ij}$ are random variables. Let us first derive the expression of
their mean values and standard deviations. By (2) and properties of random
variables [19], we have

$\begin{split}\textup{mean}\\{P_{gi}\\}&=\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}\\\
\textup{std}\\{P_{gi}\\}&=\alpha_{gi}s_{\Sigma}\end{split}$ (6)

where $s_{\Sigma}=\sqrt{\bm{1}_{n}^{T}\bm{\Sigma}\bm{1}_{n}}$ is a constant;
and $\textup{mean}\\{\cdot\\}$ and $\textup{std}\\{\cdot\\}$ denote the mean
value and standard deviation, respectively. For line flows, substituting (1)
and (2) into (3b) gives

$\begin{split}\bm{f}=\mkern 3.5mu\overline{\mkern-3.5mu\bm{f}\mkern
0.0mu}\mkern 0.0mu(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g},\bm{b})+\bm{T}_{f}(\bm{b})\bm{T}_{g}(\bm{\alpha}_{g})\bm{u}.\end{split}$
(7)

where $\mkern 3.5mu\overline{\mkern-3.5mu\bm{f}\mkern 0.0mu}\mkern
0.0mu=[\mkern 3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern
0.0mu_{ij}]\in\mathbb{R}^{l}$ denotes the base-case line flows that is a
function of $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g},\bm{b}$

$\begin{split}\mkern 3.5mu\overline{\mkern-3.5mu\bm{f}\mkern 0.0mu}\mkern
0.0mu(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{b})=\bm{T}_{f}(\bm{b})(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}+\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d})\end{split}$
(8)

and $\bm{T}_{g}\in\mathbb{R}^{n\times n}$ is a function of $\bm{\alpha}_{g}$

$\begin{split}\bm{T}_{g}(\bm{\alpha}_{g})=\bm{I}_{n}-\bm{\alpha}_{g}\bm{1}_{n}^{T}.\end{split}$
(9)

where $\bm{I}_{n}\in\mathbb{R}^{n\times n}$ denotes the identity matrix. Then,
it follows

$\begin{split}\textup{mean}\\{f_{ij}\\}&=\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern
0.0mu_{ij}=\bm{T}_{f_{ij}}(\bm{b})(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}+\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d})\\\
\textup{std}\\{f_{ij}\\}&=\sqrt{\bm{T}_{f_{ij}}(\bm{b})\bm{T}_{g}(\bm{\alpha}_{g})\bm{\Sigma}\bm{T}_{g}^{T}(\bm{\alpha}_{g})\bm{T}^{T}_{f_{ij}}(\bm{b})}.\end{split}$
(10)

where $\bm{T}_{f_{ij}}\in\mathbb{R}^{1\times n}$ denotes the row of
$\bm{T}_{f}(\bm{b})$ indexed by line $(i,j)$ that takes the expression

$\begin{split}\bm{T}_{f_{ij}}=b_{ij}\bm{E}_{ij}^{T}\bm{B}^{{\dagger}}\end{split}$
(11)

with $\bm{E}_{ij}\in\mathbb{R}^{n}$ being the column of $\bm{E}$ indexed by
line $(i,j)$.

Assume $\bm{u}$ follows a multivariate Gaussian distribution, then chance
constraints (5g)-(5j) are equivalent to

$\displaystyle\textup{mean}\\{P_{gi}\\}+c_{i}\cdot\textup{std}\\{P_{gi}\\}$
$\displaystyle\leq P_{gi}^{\max}$ (12a)
$\displaystyle\textup{mean}\\{P_{gi}\\}-c_{i}\cdot\textup{std}\\{P_{gi}\\}$
$\displaystyle\geq P_{gi}^{\min}$ (12b)
$\displaystyle\textup{mean}\\{f_{ij}\\}+c_{ij}\cdot\textup{std}\\{f_{ij}\\}$
$\displaystyle\leq f_{ij}^{\max}$ (12c)
$\displaystyle\textup{mean}\\{f_{ij}\\}-c_{ij}\cdot\textup{std}\\{f_{ij}\\}$
$\displaystyle\geq-f_{ij}^{\max}$ (12d)

where $c_{i}=\Phi^{-1}(1-\varepsilon_{i})$ and
$c_{ij}=\Phi^{-1}(1-\varepsilon_{ij})$; $\Phi^{-1}(\cdot)$ is the inverse
cumulative distribution function of the standard Gaussian distribution. Note
that this type of equivalence also applies many other types of probability
distributions, such as truncated Gaussian distribution, uniform distribution
on ellipsoidal supports or even distributionally robust optimization, where
$c_{i},c_{ij}$ will be determined by the corresponding distribution-dependent
functions (see a discussion in [20, 21]).

In addition, according to the derivation in [2], the objective function (5a)
is equivalent to

$\begin{split}h(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g})=\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{T}\bm{a}_{2}\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}+s_{\Sigma}^{2}\bm{\alpha}_{g}^{T}\bm{a}_{2}\bm{\alpha}_{g}+\bm{a}_{1}^{T}\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}.\end{split}$
(13)

Thus, by (6)-(13) we obtain the following analytical form of the CCED problem
(5)

$\displaystyle\min_{\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g},\bm{b}}\leavevmode\nobreak\ h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g})$ (14a) $\displaystyle s.t.\leavevmode\nobreak\
\bm{1}_{n}^{T}(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}+\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{w}-\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{d})=0$ (14b) $\displaystyle b_{ij}^{\min}\leq b_{ij}\leq
b_{ij}^{\max},\leavevmode\nobreak\ \forall(i,j)\in\mathcal{E}_{\textup{flex}}$
(14c) $\displaystyle\bm{1}_{n}^{T}\bm{\alpha}_{g}=1$ (14d)
$\displaystyle\alpha_{gi}=0,\leavevmode\nobreak\ \forall
i\notin\mathcal{V}_{g}$ (14e) $\displaystyle\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}=0,\leavevmode\nobreak\ \forall i\notin\mathcal{V}_{g}$ (14f)
$\displaystyle\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}\leq P_{gi}^{\max}-c_{i}s_{\Sigma}\alpha_{gi},\leavevmode\nobreak\
\forall i\in\mathcal{V}_{g}$ (14g) $\displaystyle-\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}\leq-
P_{gi}^{\min}-c_{i}s_{\Sigma}\alpha_{gi},\leavevmode\nobreak\ \forall
i\in\mathcal{V}_{g}$ (14h) $\displaystyle\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g},\bm{b})\leq
f_{ij}^{\max}-c_{ij}\big{\|}\bm{T}_{f_{ij}}\bm{T}_{g}\bm{\Sigma}^{\frac{1}{2}}\big{\|}_{2},\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}$ (14i) $\displaystyle-\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g},\bm{b})\leq
f_{ij}^{\max}-c_{ij}\big{\|}\bm{T}_{f_{ij}}\bm{T}_{g}\bm{\Sigma}^{\frac{1}{2}}\big{\|}_{2},\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}$ (14j)

where the notation $\big{\|}\cdot\big{\|}_{2}$ in (14i) and (14j) denotes the
2-norm. Note that (14b) is equivalent to (3a) since the amount of renewable
fluctuation is fully balanced by the dispatchable generators.

###  II-C Role of network flexibility in CCED

Problem (14) takes a similar form to a normal ED problem that does not
consider renewable uncertainties. The right-hand side of (14g)-(14j) can be
regarded as the equivalent generation limits and equivalent line capacities,
say

$\begin{split}P_{gi}^{\textup{emax}}(\bm{\alpha}_{g})&=P_{gi}^{\max}-c_{i}s_{\Sigma}\alpha_{gi}\\\
P_{gi}^{\textup{emin}}(\bm{\alpha}_{g})&=P_{gi}^{\min}+c_{i}s_{\Sigma}\alpha_{gi}\\\
f_{ij}^{\textup{emax}}(\bm{\alpha}_{g},\bm{b})&=f_{ij}^{\max}-c_{ij}\big{\|}\bm{T}_{f_{ij}}\bm{T}_{g}\bm{\Sigma}^{\frac{1}{2}}\big{\|}_{2}\end{split}$
(15)

which consist of the physical limit ($P_{gi}^{\max},P_{gi}^{\min}$ or
$f_{ij}^{\max}$) and an uncertainty margin ($c_{i}s_{\Sigma}\alpha_{gi}$ or
$c_{ij}\big{\|}\bm{T}_{f_{ij}}\bm{T}_{g}\bm{\Sigma}^{\frac{1}{2}}\big{\|}_{2}$).
The major difference between (14) and a normal ED problem is that the limit
values in (15) are dependent on $\bm{\Sigma}$, $\bm{\alpha}_{g}$ and $\bm{b}$
rather than constants. From the right-hand-side of (15), it can be seen that
the physical limits are partly taken up by renewable uncertainties and the
actually usable capacities shrink. This leads to the following discussion that
reveals the role of decision variables against the uncertainties in different
constraints.

1) Generation constraints (14g)-(14h). Greater renewable uncertainties (i.e.,
a greater $s_{\Sigma}$) induce a greater uncertainty margin
$c_{i}s_{\Sigma}\alpha_{gi}$, which makes the generation constraints easier to
be binding (i.e., the constraint holds with equality at the optimal solution).
Suppose the generation constraint with respect to bus $i$ is binding, it
implies that $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}$ is cost effective and should take a greater value; however, the
renewable uncertainties limit a further increase of $\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$. In contrast, the
participation factors $\bm{\alpha}_{g}$ tend to reduce the impact of renewable
uncertainties and expand the dispatch interval for $\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$.

2) Line flow constraints (14i)-(14j). Transmission congestion occurs when some
line flow constraints are binding. In this case, the capacities of those
congested lines, rather than the generation dispatchability, become the major
bottleneck for further utilizing the cost-effective generators, which may
drastically impact operational economy [22]. This issue becomes more
significant in the CCED as $f_{ij}^{\textup{emax}}$ is even less than the
physical capacity due to the presence of uncertainty margin. It is
straightforward to see the role of participation factors $\bm{\alpha}_{g}$ in
the line flow constraints—they help to increase $f_{ij}^{\textup{emax}}$ of
those congested lines by tuning $\bm{T}_{g}$ and hence improve operational
economy. The role of line susceptances $\bm{b}$ is composite. On one hand,
$\bm{b}$ unleashes the line capacity by tuning $\bm{T}_{f_{ij}}$. On the other
hand, $\bm{b}$ re-routes the power injections $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}+\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d}$ to improve the
base-case line flows $\mkern 3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern
0.0mu_{ij}$ to better utilize the line capacity saved by $\bm{\alpha}_{g}$ and
$\bm{b}$ (see the two sides of (14i)-(14j)).

According to the above discussion, the contributions of the decision variables
are summarized in Table I. We conclude that network flexibility introduces a
new mechanism against renewable uncertainties. The base-case generations
$\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}$ appear
in the left-hand side of (14g)-(14j) and regulate the base-case power flow.
The participation factors $\bm{\alpha}_{g}$ appear in the right-hand side of
(14g)-(14j) and help to expand the equivalent generation limits and equivalent
line capacities. By comparison, the flexible line susceptances $\bm{b}$ appear
in both sides of (14i)-(14j) and specialize in exploiting the capacities of
congested lines. Although $\bm{\alpha}_{g}$ and $\bm{b}$ both contribute to
mitigating transmission congestion, it will be seen in the case study that
$\bm{b}$ is much more effective than $\bm{\alpha}_{g}$. In the next section, a
novel solution methodology for problem (14) will be designed based on the
features of $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g},\bm{b}$ in the problem.

Table I: Contribution of decision variables to CCED

Constraint component | $\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}$ | $\bm{\alpha}_{g}$ | $\bm{b}$
---|---|---|---
$\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ in gen. constraint | Y1 | N | N
$P_{gi}^{\textup{emax}},P_{gi}^{\textup{emin}}$ in gen. constraint | N | Y | N
$\mkern 3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}$ in line flow constraint | Y | N | Y
$f_{ij}^{\textup{emax}}$ in line flow constraint | N | Y | Y

  * 1

Y (or N) means the decision variable contributes to the corresponding
constraint component (or not).

##  III Solution methodology

Problem (14) is hard to solve due to the non-convexity of (14i) and (14j). On
the other hand, the CCED without network flexibility (i.e., $\bm{b}$ is fixed)
is an SOCP problem (constraints (14g)-(14h) are second-order cones if $\bm{b}$
is fixed), where convex solvers apply. In addition, it is shown in Section
II-C that $\bm{b}$ plays a different role from $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}$ and
$\bm{\alpha}_{g}$ in the problem. Therefore, we separate the decision
variables into two groups, say $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g})$ and $\bm{b}$, which correspond to generation
flexibility and network flexibility, respectively. An alternate iteration
framework is then developed to iteratively solve the subproblems with respect
to the two groups of decision variables.

###  III-A Subproblem w.r.t. generations and participation factors

Let $\bm{b}^{*}=[b_{ij}^{*}]\in\mathbb{R}^{l}$ denote the current profile of
line susceptances, then $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$ is obtained by solving the
following subproblem

$\displaystyle\min_{\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g},\bm{\alpha}_{g}}\leavevmode\nobreak\ $ $\displaystyle
h(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g},\bm{\alpha}_{g})$ (16a) $\displaystyle s.t.\leavevmode\nobreak\ $
$\displaystyle\bm{1}_{n}^{T}(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}+\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{w}-\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{d})=0$ (16b)
$\displaystyle\bm{1}_{n}^{T}\bm{\alpha}_{g}=1$ (16c)
$\displaystyle\alpha_{gi}=0,\leavevmode\nobreak\ \forall
i\notin\mathcal{V}_{g}$ (16d) $\displaystyle\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}=0,\leavevmode\nobreak\ \forall i\notin\mathcal{V}_{g}$ (16e)
$\displaystyle\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern
0.0mu_{gi}\leq P_{gi}^{\textup{emax}}(\bm{\alpha}_{g}),\leavevmode\nobreak\
\forall i\in\mathcal{V}_{g}$ (16f) $\displaystyle-\mkern
3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}\leq-
P_{gi}^{\textup{emin}}(\bm{\alpha}_{g}),\leavevmode\nobreak\ \forall
i\in\mathcal{V}_{g}$ (16g) $\displaystyle\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g},\bm{b}^{*})\leq
f_{ij}^{\textup{emax}}(\bm{\alpha}_{g},\bm{b}^{*}),\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}$ (16h) $\displaystyle-\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g},\bm{b}^{*})\leq
f_{ij}^{\textup{emax}}(\bm{\alpha}_{g},\bm{b}^{*}),\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}$ (16i)

where the line susceptances are fixed as $\bm{b}^{*}$, i.e., the traditional
CCED problem. As mentioned before, subproblem (16) is an SOCP problem that can
be efficiently solved by commercial convex solvers such as CVX. Under the
current network structure $\bm{b}^{*}$, $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}^{*}$ and
$\bm{\alpha}_{g}^{*}$ lead to the optimal generation and line flow profiles
that incur the lowest operational cost.

If there is no transmission congestion at $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, it means that the current network
structure is satisfactory and changing line susceptances will not further
reduce the operational cost. This can also be seen from the later analysis
showing that the objective function has a zero sensitivity to line
susceptances in case of no congestion (i.e., (22) with zero dual variables).
Thus, $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*},\bm{b}^{*})$ provides an optimal solution
for CCED. If some lines are congested at $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, it means that the current network
structure is inadequate and needs an adjustment, which will be addressed in
the following subproblem.

###  III-B Subproblem w.r.t. line susceptances

Based on the obtained $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, we now formulate the
subproblem with respect to line susceptances to find an effective adjustment
for the power network structure to mitigate transmission congestion and reduce
operational cost. Since the line flow constraints (16h)-(16i) are highly
nonlinear with line susceptances, we consider the influence after applying a
small change to line susceptances so that the sensitivity analysis is
applicable.

Let $\mathcal{E}_{\textup{con}}^{+},\mathcal{E}_{\textup{con}}^{-}$
respectively denote the set of lines with binding constraints (16h) and
binding constraints (16i) (i.e., the congested lines consist of
$\mathcal{E}_{\textup{con}}^{+},\mathcal{E}_{\textup{con}}^{-}$), which are of
interest here. We first derive the sensitivity of $f_{ij}^{\textup{emax}}$,
$(i,j)\in\mathcal{E}_{\textup{con}}^{+}\cup\mathcal{E}_{\textup{con}}^{-}$ to
the susceptance of line $(k,m)\in\mathcal{E}_{\textup{flex}}$, say $b_{km}$.
By (15) we have

$\begin{split}\frac{\partial f_{ij}^{\textup{emax}}}{\partial
b_{km}}=-\frac{\partial\bm{T}_{f_{ij}}}{\partial
b_{km}}\frac{c_{ij}\bm{T}_{g}(\bm{\alpha}_{g}^{*})\bm{\Sigma}\bm{T}_{g}^{T}(\bm{\alpha}_{g}^{*})\bm{T}_{f_{ij}}^{T}(\bm{b}^{*})}{\big{\|}\bm{T}_{f_{ij}}(\bm{b}^{*})\bm{T}_{g}(\bm{\alpha}_{g}^{*})\bm{\Sigma}^{\frac{1}{2}}\big{\|}_{2}}.\end{split}$
(17)

According to (11), the partial derivative in (17) is given by

$\begin{split}\frac{\partial\bm{T}_{f_{ij}}}{\partial
b_{km}}=\left\\{\begin{array}[]{ll}b_{ij}\bm{E}_{ij}^{T}\frac{\partial\bm{B}^{{\dagger}}}{\partial
b_{km}}+\bm{E}_{ij}^{T}\bm{B}^{{\dagger}},\leavevmode\nobreak\ (k,m)=(i,j)\\\
b_{ij}\bm{E}_{ij}^{T}\frac{\partial\bm{B}^{{\dagger}}}{\partial
b_{km}},\leavevmode\nobreak\ \textup{otherwise}\end{array}\right.\end{split}$
(18)

where it follows from [23] that

$\begin{split}\frac{\partial\bm{B}^{{\dagger}}}{\partial
b_{km}}=-\bm{B}^{{\dagger}}\frac{\partial\bm{B}}{\partial
b_{km}}\bm{B}^{{\dagger}}=-\bm{B}^{{\dagger}}(\bm{b}^{*})\bm{E}_{km}\bm{E}_{km}^{T}\bm{B}^{{\dagger}}(\bm{b}^{*}).\end{split}$
(19)

Thus, the sensitivity can be obtained by substituting (18) and (19) into (17).

Next, by duality theory, we derive the sensitivity of the optimal objective
value of (16) to $b_{km}$. Let $\lambda_{ij}^{+},\lambda_{ij}^{-}$,
$\forall(i,j)\in\mathcal{E}$ be the optimal dual variables associated with
line flow constraints (16h) and (16i), respectively. These dual variables are
byproduct of solving (16), which are obtained without additional computation.
According to the KKT condition [24], it follows that

$\begin{split}&\lambda_{ij}^{+}\geq 0,\leavevmode\nobreak\
\lambda_{ij}^{-}=0,\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}_{\textup{con}}^{+}\\\
&\lambda_{ij}^{+}=0,\leavevmode\nobreak\ \lambda_{ij}^{-}\geq
0,\leavevmode\nobreak\ \forall(i,j)\in\mathcal{E}_{\textup{con}}^{-}\\\
&\lambda_{ij}^{+}=0,\leavevmode\nobreak\
\lambda_{ij}^{-}=0,\leavevmode\nobreak\
\forall(i,j)\in\mathcal{E}\backslash(\mathcal{E}_{\textup{con}}^{+}\cup\mathcal{E}_{\textup{con}}^{-}).\end{split}$
(20)

Then, by Theorem 8.2 in [25], the sensitivity of the optimal objective value
to $b_{km}$ is give by

$\begin{split}\frac{\partial h}{\partial
b_{km}}=&\sum_{(i,j)\in\mathcal{E}}\lambda_{ij}^{+}(\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial
b_{km}}-\frac{\partial f_{ij}^{\textup{emax}}}{\partial b_{km}})\\\
&-\sum_{(i,j)\in\mathcal{E}}\lambda_{ij}^{-}(\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial
b_{km}}+\frac{\partial f_{ij}^{\textup{emax}}}{\partial b_{km}})\end{split}$
(21)

which includes the effect of $\bm{b}$ in both sides of the line flow
constraints. Further, (21) can be simplified into the following form since a
majority of dual variables are zero (see (20))

$\begin{split}\frac{\partial h}{\partial
b_{km}}=&\sum_{(i,j)\in\mathcal{E}_{\textup{con}}^{+}}\lambda_{ij}^{+}(\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial
b_{km}}-\frac{\partial f_{ij}^{\textup{emax}}}{\partial b_{km}})\\\
&-\sum_{(i,j)\in\mathcal{E}_{\textup{con}}^{-}}\lambda_{ij}^{-}(\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial
b_{km}}+\frac{\partial f_{ij}^{\textup{emax}}}{\partial b_{km}}).\end{split}$
(22)

In (22), the formula for $\frac{\partial f_{ij}^{\textup{emax}}}{\partial
b_{km}}$ has been presented in (17)-(19), and $\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial b_{km}}$
can be calculated as follows. By (10), we have

$\begin{split}\frac{\partial\mkern 3.5mu\overline{\mkern-3.5muf\mkern
0.0mu}\mkern 0.0mu_{ij}}{\partial
b_{km}}=\frac{\partial\bm{T}_{f_{ij}}}{\partial b_{km}}(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}^{*}+\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{w}-\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d})\end{split}$
(23)

and substituting (18) into (23) gives the formula for $\frac{\partial\mkern
3.5mu\overline{\mkern-3.5muf\mkern 0.0mu}\mkern 0.0mu_{ij}}{\partial b_{km}}$.

With the above sensitivities, we propose the following subproblem that aims to
improve the operational economy by adjusting line susceptances

$\displaystyle\min_{\Delta b_{km}}\leavevmode\nobreak\
\sum_{(k,m)\in\mathcal{E}_{\textup{flex}}}\frac{\partial h}{\partial
b_{km}}\Delta b_{km}$ (24a) $\displaystyle s.t.\leavevmode\nobreak\
b_{km}^{\min}\leq b_{km}^{*}+\Delta b_{km}\leq
b_{km}^{\max},\leavevmode\nobreak\ \forall(k,m)\in\mathcal{E}_{\textup{flex}}$
(24b) $\displaystyle-\Delta b_{km}^{\max}\leq\Delta b_{km}\leq\Delta
b_{km}^{\max},\leavevmode\nobreak\ \forall(k,m)\in\mathcal{E}_{\textup{flex}}$
(24c)

where the partial derivative terms have been given in (17)-(23); $\Delta
b_{km}^{\max}$ is a predefined small positive number; and (24c) is the trust-
region constraint that enforces the line susceptance adjustment to be small so
that the sensitivity analysis is valid. Subproblem (24) is a simple linear
programming problem, which can be solved efficiently.

Let $\Delta\bm{b}=[\Delta b_{km}]\in\mathbb{R}^{l}$ be the susceptance
adjustment, where $\Delta b_{km}=0$,
$\forall(k,m)\notin\mathcal{E}_{\textup{flex}}$ and $\Delta b_{km}$,
$\forall(k,m)\in\mathcal{E}_{\textup{flex}}$ are given by (24). When solving
subproblem (16) again by using the updated line susceptances
$\bm{b}^{*}+\Delta\bm{b}$, the generations and participation factors are
expected to exploit the line capacity saved by $\Delta\bm{b}$ and achieve a
more economic dispatch scheme.

###  III-C Alternate iteration algorithm

Based on the proposed subproblems (16) and (24), we design an alternate
iteration algorithm for the original problem (14). The solution procedure is
presented below. The algorithm flow chart is also depicted in Fig. 1.

  1. Step 1:

Parameter setting. Set input parameters $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{d},\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{w},\bm{\Sigma}$;
$a_{1,i},a_{2,i},\varepsilon_{i},P_{gi}^{\max},P_{gi}^{\min}$ for bus
$i\in\mathcal{V}_{g}$; $b_{ij}^{\max},b_{ij}^{\min}$ for line
$(i,j)\in\mathcal{E}_{\textup{flex}}$; $b_{ij}$ for line
$(i,j)\in\mathcal{E}\backslash\mathcal{E}_{\textup{flex}}$ and
$\varepsilon_{ij},f_{ij}^{\max}$ for line $(i,j)\in\mathcal{E}$. Set algorithm
parameters $\delta>0$ (convergence criterion), $\Delta b_{ij}^{\max,0}>0$
(benchmark for trust-region size), $0<\beta<1$ (reduction factor for trust
region). Let $\Delta b_{ij}^{\max}=\Delta b_{ij}^{\max,0}$ be the initial
trust-region size, and $\bm{b}^{*}$ be the initial line susceptances.

  2. Step 2:

Initial solution. Solve subproblem (16) under the fixed line susceptances
$\bm{b}^{*}$, and obtain the solution $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$. If all the constraints (16h)-(16i) are
not binding at $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, stop the algorithm and output $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*},\bm{b}^{*})$ as the optimal solution.

  3. Step 3:

Solve subproblem (24) under the profile $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$ and obtain the line susceptance adjustment
$\Delta\bm{b}$.

  4. Step 4:

Solve subproblem (16) under the fixed line susceptances
$\bm{b}^{*}+\Delta\bm{b}$, and obtain the tentative solution, say $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})$.

  5. Step 5:

If all the constraints (16h)-(16i) are not binding at $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})$, update the solution
$\bm{b}^{*}\leftarrow\bm{b}^{*}+\Delta\bm{b}$, $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*}\leftarrow\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{\prime}$,
$\bm{\alpha}_{g}^{*}\leftarrow\bm{\alpha}_{g}^{\prime}$, stop the algorithm
and output $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*},\bm{b}^{*})$ as the optimal solution.

  6. Step 6:

If $h(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})>h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, the tentative solution is aborted, reduce
the trust-region size by $\Delta b_{ij}^{\max}\leftarrow\beta\Delta
b_{ij}^{\max}$. Go back to Step 3.
If $h(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})\leq h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$, the susceptance adjustment $\Delta\bm{b}$
is accepted, update the solution
$\bm{b}^{*}\leftarrow\bm{b}^{*}+\Delta\bm{b}$, $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*}\leftarrow\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{\prime}$,
$\bm{\alpha}_{g}^{*}\leftarrow\bm{\alpha}_{g}^{\prime}$ and recover the trust-
region size $\Delta b_{ij}^{\max}\leftarrow\Delta b_{ij}^{\max,0}$. Further,
if $|\Delta b_{ij}|<\delta$, $\forall(i,j)\in\mathcal{E}_{\textup{flex}}$,
stop the algorithm and output $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{*},\bm{\alpha}_{g}^{*},\bm{b}^{*})$ as the optimal
solution; otherwise go back to Step 3.

We further explain the classification in Step 6. The inequality $h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})>h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$ implies that subproblem (24) does not
provide a correct susceptance adjustment, which is possibly due to that the
trust-region size is too large to guarantee the validity of sensitivity
analysis. In this case, we need to solve subproblem (24) again with a reduced
trust-region size. The inequality $h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\prime},\bm{\alpha}_{g}^{\prime})\leq h(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*})$ implies that the operational cost is
reduced as expected after applying the susceptance adjustment $\Delta\bm{b}$,
and hence $\bm{b}^{*}$ is updated to $\bm{b}^{*}+\Delta\bm{b}$.

According to the stop criteria in the algorithm, the final output of this
algorithm, say $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\textup{opt}},\bm{\alpha}_{g}^{\textup{opt}},\bm{b}^{\textup{opt}})$,
have two types of physical meanings. If the algorithm is stopped in Step 2 or
Step 5, then $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\textup{opt}},\bm{\alpha}_{g}^{\textup{opt}},\bm{b}^{\textup{opt}})$
fully eliminates transmission congestion, which is desirable. If the algorithm
is stopped in Step 6, then $(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern
0.0mu}\mkern 0.0mu_{g}^{\textup{opt}},\bm{\alpha}_{g}^{\textup{opt}})$ and
$\bm{b}^{\textup{opt}}$ reach the following matching state though transmission
congestion is not fully eliminated. On one hand, under the power network
structure $\bm{b}^{\textup{opt}}$, any change on $(\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{\textup{opt}},\bm{\alpha}_{g}^{\textup{opt}})$ cannot decrease the
operational cost. On the other hand, under the generations $\mkern
3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern 0.0mu_{g}^{\textup{opt}}$
and participation factors $\bm{\alpha}_{g}^{\textup{opt}}$, the congestion
cannot be further mitigated by changing $\bm{b}^{\textup{opt}}$. Therefore,
the capabilities of generations, participation factors and line susceptances
in CCED have been fully exploited.

Another merit of the proposed algorithm is that each accepted solution
$(\mkern 3.5mu\overline{\mkern-3.5mu\bm{P}\mkern 0.0mu}\mkern
0.0mu_{g}^{*},\bm{\alpha}_{g}^{*},\bm{b}^{*})$ generated during the iteration
is a feasible solution to problem (14) and has a lower objective value than
the previous one. Thus, even when the algorithm has not yet converged, we can
still apply the latest accepted solution for system dispatch to improve the
operational economy. This feature makes the algorithm more friendly to the
real-time implementation where a satisfactory dispatch scheme needs to be
obtained within a rather limited period of time.

Figure 1: The algorithm flow chart.

##  IV Case study

We take the IEEE 14-bus system and IEEE 118-bus system to test the proposed
method, where it will be observed that network flexibility can help to fully
eliminate transmission congestion in small systems and significantly reduce
congestion in large systems. We refer to the MATPOWER package [26] for the
parameters of these two systems.

###  IV-A IEEE 14: congestion fully eliminated after optimization

Based on the original parameter profile of IEEE 14-bus system, we further
adopt the following settings for our tests.

  * •

Loads and dispatchable generations. There are five dispatchable generators
$\mathcal{V}_{g}=\\{1,2,3,6,8\\}$. To highlight the control effect, we double
the values of loads and generation capacities in [26].

  * •

Renewable generations. Assume that buses 1, 3, 6, 9 have renewable generators
and the load data represents the net loads, i.e., the consumptions deducted by
renewable generations. The covariance matrix for renewable uncertainties takes
value as $\bm{\Sigma}=\textup{diag}\\{\Sigma_{i}\\}\in\mathbb{R}^{14\times
14}$, where $\Sigma_{i}=0.05$ p.u. for $i=1,3,6,9$ and otherwise
$\Sigma_{i}=0$.

  * •

Lines. Assume $\mathcal{E}_{\textup{flex}}=\\{(1,5),(2,3),(6,11)\\}$. The
flexible susceptance is realized by a constant susceptance
$b_{ij}^{\textup{rated}}$ in series with an adjustable susceptance
$b_{ij}^{c}$ such that $|b_{ij}^{\textup{rated}}/b_{ij}^{c}|\leq d_{ij}$,
where $d_{ij}$ is called the degree of flexibility. Thus, for any
$(i,j)\in\mathcal{E}_{\textup{flex}}$ we have

$\begin{split}b_{ij}^{\min}=\frac{b_{ij}^{\textup{rated}}}{1+d_{ij}},\leavevmode\nobreak\
b_{ij}^{\max}=\frac{b_{ij}^{\textup{rated}}}{1-d_{ij}}.\end{split}$ (25)

Here we take $d_{ij}=0.7$, $\forall(i,j)\in\mathcal{E}_{\textup{flex}}$. For
line capacity, we set $f_{ij}^{\max}=140$MW for line (1,2),
$f_{ij}^{\max}=100$MW for line (7,9), and $f_{ij}^{\max}=200$MW for all the
other lines.

  * •

CCED and algorithm parameters. In subproblem (16), we set $c_{i}=c_{ij}=2.326$
(corresponding to $\varepsilon_{i}=\varepsilon_{ij}=0.01$ for the original
problem (5)). In subproblem (24), we set $\Delta
b_{ij}^{\max,0}=0.3b_{ij}^{\textup{rated}}$,
$\forall(i,j)\in\mathcal{E}_{\textup{flex}}$. In the algorithm, we take the
rated line susceptances to be the initial values, and
$\delta=10^{-4},\beta=0.1$.

With the above settings, we obtain the following four solutions for
comparison.

  * •

S1: CCED with network flexibility, which is obtained by solving (14) using the
proposed algorithm.

  * •

S2: CCED without network flexibility, which is the first solution during the
iteration for obtaining S1.

  * •

S3: Normal ED with network flexibility, which is obtained by solving (14)
using the proposed algorithm with $\bm{\Sigma}=\bm{0}$.

  * •

S4: Normal ED without network flexibility, which is the first solution during
the iteration for obtaining S3.

Before looking into these four solutions, let us first detail the process of
finding S1 to verify the proposed algorithm. During the iteration, the
generation constraints are not binding, while two lines (1,2) and (7,9) are
congested at the first solution. Then the susceptances of lines
$\mathcal{E}_{\textup{flex}}$ start to adjust to address the congestion. The
blue and red curves in Fig. 2 respectively shows the trajectories of dual
variables of the binding flow constraints with respect to lines (1,2) and
(7,9). We observe that the two dual variables are decreasing with the
iteration, which implies that the congestion is being gradually mitigated.
After eight iterations, these two dual variables become zero, and hence the
congestion is fully cleared and the algorithm stops. In the meantime, the
generation cost (i.e., objective value), which is denoted by the black curve
in Fig. 2, decreases from 18578.8$/h to 18186.4$/h, which achieves 2.1% cost
reduction.

We now check the performances of the four solutions to show the merits of
network flexibility. Since the size of IEEE 14-bus system is rather small, it
is convenient to present the comprehensive information of the solutions in
Table II, Table III and Table IV. We have some interesting results from the
following comparisons.

1) $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S1
and S2. As network flexibility helps to mitigate transmission congestion under
renewable uncertainties, the dispatchable generator at bus 1, which are more
cost-effective, are better utilized to output more power at S1 than S2.

2) $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S1
and S3. S1 and S3 both consider network flexibility. Additionally, S1
considers renewable uncertainties that shrink the usable line capacities (see
(15)). However, the values of $\mkern 3.5mu\overline{\mkern-3.5muP\mkern
0.0mu}\mkern 0.0mu_{gi}$ at S1 and S3 are nearly identical, which implies that
the network flexibility eliminates the impact of renewable uncertainties. With
network flexibility, the generation cost keeps almost unchanged after
including renewable uncertainties, except for the additional term
$s_{\Sigma}^{2}\bm{\alpha}_{g}^{T}\bm{a}_{2}\bm{\alpha}_{g}$ caused by
participation factors which is slight.

3) $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S2
and S4. S2 and S4 both exclude network flexibility. Additionally, S2 considers
renewable uncertainties. Compared to the generation profile at S4, S2 cannot
resort to network flexibility and has to sacrifice those cost-effective
generations in order to satisfy the line flow chance constraints.

Table V further shows the generation costs of these four solutions. The cost
difference between S1 and S3 (6.1$/h) and the cost difference between S2 and
S4 (290.9$/h) can be interpreted as the cost of uncertainty. The cost
difference between S1 and S2 (392.4$/h) and the cost difference between S3 and
S4 (107.6$/h) can be interpreted as the cost of network non-flexibility. We
make some important observations from these cost data. According to the fourth
column of Table V, network flexibility helps to greatly save generation cost
by congestion mitigation no matter renewable uncertainties are included or
not. According to the third column of Table V, renewable uncertainties cause
much more additional generation cost when network flexibility is unavailable.
Overall, network flexibility has been shown to significantly improve
operational economy specially under renewable uncertainties.

Figure 2: IEEE 14: Generation costs and dual variables of flow constraints during the iteration. Table II: IEEE 14: Generation info of CCED with/without network flexibility (in MW) Gen. $i$ | $a_{2i}$ | $a_{1i}$ |  $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S1 |  $\alpha_{gi}$ at S1 |  $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S2 |  $\alpha_{gi}$ at S2
---|---|---|---|---|---|---
1 | 4.3 | 20 | 249.84 | 0.07 | 161.76 | 0.23
2 | 25 | 20 | 43.00 | 0.00 | 47.98 | 0.00
3 | 1 | 40 | 75.05 | 0.31 | 144.36 | 0.20
6 | 1 | 40 | 75.05 | 0.31 | 76.41 | 0.39
8 | 1 | 40 | 75.06 | 0.31 | 87.49 | 0.18
Table III: IEEE 14: Line susceptance info of CCED with/without network flexibility (in p.u.) Line $(i,j)$ | $b_{ij}^{\textup{rated}}$ | $b_{ij}^{\min}$ | $b_{ij}^{\max}$ |  $b_{ij}$ at S1 |  $b_{ij}$ at S3
---|---|---|---|---|---
(1,5) | 4.48 | 2.64 | 14.95 | 13.90 | 8.52
(2,3) | 5.05 | 2.97 | 16.84 | 2.97 | 2.97
(6,11) | 5.03 | 2.96 | 16.76 | 15.59 | 9.55
Table IV: IEEE 14: Generation info of normal ED with/without network flexibility (in MW) Gen. $i$ | $a_{2i}$ | $a_{1i}$ |  $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S3 |  $\mkern 3.5mu\overline{\mkern-3.5muP\mkern 0.0mu}\mkern 0.0mu_{gi}$ at S4
---|---|---|---|---
1 | 4.3 | 20 | 249.84 | 203.57
2 | 25 | 20 | 43.00 | 45.60
3 | 1 | 40 | 75.05 | 111.24
6 | 1 | 40 | 75.05 | 74.48
8 | 1 | 40 | 75.05 | 83.11
Table V: IEEE 14: Comparison of generation costs (in $/h) with/without network
flexibility or renewable uncertainty

Solution | Gen. cost | Cost of uncer.1 | Cost of no NF1
---|---|---|---
ED with NF (S3) | 18180.3 | 0 | 0
CCED with NF (S1) | 18186.4 | 6.1 (S1-S3) | 0
ED without NF (S4) | 18287.9 | 0 | 107.6 (S4-S3)
CCED without NF (S2) | 18578.8 | 290.9 (S2-S4) | 392.4 (S2-S1)

  * 1

Uncer. is short for renewable uncertainty.

  * 2

NF is short for network flexibility.

###  IV-B IEEE 118: congestion mitigated after optimization

We turn to IEEE 118-bus system to show that the proposed method also works
well in large systems. Based on the original parameter profile of the system,
we further adopt the following settings for our tests.

  * •

Loads and dispatchable generations. There are 54 dispatchable generators. To
highlight the control effect, we double the values of loads and generation
capacities in [26].

  * •

Renewable generations. Assume that buses 3, 8, 11, 20, 24, 26, 31, 38, 43, 49,
53 have renewable generators and the load data represents the net loads. The
covariance matrix for renewable uncertainties takes value as
$\bm{\Sigma}=\textup{diag}\\{\Sigma_{i}\\}\in\mathbb{R}^{14\times 14}$, where
$\Sigma_{i}=0.05$ p.u. if bus $i$ has a renewable generator and otherwise
$\Sigma_{i}=0$.

  * •

Lines. Assume nine lines have flexible susceptances
$\mathcal{E}_{\textup{flex}}=\\{(13,15),(26,30),(46,48),(49,54),(54,59),$
$(59,61),(64,65),(47,69),(69,77)\\}$. Similar to the test on the IEEE 14-bus
system, these susceptances have the degree of flexibility $d_{ij}=0.7$,
$\forall(i,j)\in\mathcal{E}_{\textup{flex}}$. For line capacity, we set
$f_{ij}^{\max}=100$MW for lines (8,9), (8,5), (60,61), (63,64), and
$f_{ij}^{\max}=200$MW for all the other lines.

  * •

CCED and algorithm parameters. The setting is identical to the test on the
IEEE 14-bus system.

Again, we obtain the following four solutions for analysis.

  * •

S1: CCED with network flexibility.

  * •

S2: CCED without network flexibility.

  * •

S3: Normal ED with network flexibility.

  * •

S4: Normal ED without network flexibility.

We first check the iteration process of finding S1. Fig. 3 shows the
trajectories of dual variables of the binding flow constraints with respect to
some lines. Note that there are quite a few congested lines during the
iteration. For simplicity, here we just plot those lines with rather severe
congestion, i.e., the values of dual variables have ever been greater than 20
during the iteration. We observe that some dual variables have oscillation and
some dual variables are even increasing during the iteration. However, the
generation cost monotonically decreases (see the black curve in Fig. 3), which
implies that the overall congestion is mitigated after each iteration. After
ten iterations, the adjustment of line susceptances becomes sufficiently small
and the algorithm stops. The generation cost finally decreases from
321570.9$/h to 310209.8$/h, i.e., 3.5% cost reduction. Although the congestion
is not fully eliminated, the cost reduction is significant and hence S1 is
satisfactory. Also note that the system has totally 186 lines and the
reduction is achieved by assuming only nine lines to have flexible
susceptances (less than 5% of lines).

We then compare the performances of the four solutions in Table VI. We have a
similar observation to the test on the IEEE 14-bus system, i.e., the presence
of network flexibility makes a great contribution to cost reduction especially
when the system has high penetration of uncertain renewables. This again
highlights the capability of network flexibility against the impact of
renewable uncertainties on operational economy.

Moreover, the degree of network flexibility $d_{ij}$ also plays a role in the
generation cost reduction and we have fixed it to be 0.7 so far. Fig. 4 shows
how the generation cost of S1 changes with the degree of network flexibility.
We observe that the cost decreases nearly linearly with the degree of
flexibility. For this system, the degree of flexibility should be greater than
0.5 in order to achieve more than 2% cost reduction.

Figure 3: IEEE 118: Generation costs and dual variables of flow constraints during the iteration. Table VI: IEEE 118: Comparison of generation costs (in $/h) with/without network flexibility or renewable uncertainty Solution | Gen. cost | Cost of uncer. | Cost of no NF
---|---|---|---
ED with NF (S3) | 309044.4 | 0 | 0
CCED with NF (S1) | 310209.8 | 1165.4 (S1-S3) | 0
ED without NF (S4) | 317738.6 | 0 | 8694.2 (S4-S3)
CCED without NF (S2) | 321570.9 | 3832.3 (S2-S4) | 11361.1 (S2-S1)
Figure 4: IEEE 118: Relationship between the generation cost of CCED and
degree of network flexibility.

##  V Conclusion

Network flexibility has been incorporated into CCED problems to cope with the
impact caused by renewable uncertainties. From the analytical form of the CCED
problem, we have figured out how network flexibility play a role in the ED
under uncertainties. The flexible line susceptances tune the base-case line
flows and reduce the line capacities shrunk by uncertainties. Thus, network
flexibility greatly contributes to transmission congestion mitigation and
generation cost saving. Furthermore, we have proposed an efficient solver the
CCED problem with network flexibility. Two convex subproblems with respect to
generation variables and network variables are established using duality
theory. Then alternately solving these two subproblems gives the solution to
the original problem. Case studies have shown that the operational economy
under uncertainties is much improved with the help of network flexibility.

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