# Short Circuit Current Constrained UC in High IBG-Penetrated Power Systems

Zhongda Chu, Fei Teng,

###### Abstract

Inverter Based Generators (IBGs) have been increasing significantly in power
systems. Due to the demanding thermal rating of Power Electronics (PE), their
contribution to the system Short Circuit Current (SCC) is much less than that
from the conventional Synchronous Generators (SGs) thus reducing the system
strength and posing challenges to system protection and stability. This paper
proposes a Unit Commitment (UC) model with SCC constraint in high IBG-
penetrated systems to ensure minimum operation cost while maintaining the SCC
level at each bus in the system. The SCC from synchronous generators as well
as the IBGs are explicitly modeled in the formulation leading to an SCC
constraint involving decision-dependent matrix inverse. This highly nonlinear
constraint is further reformulated into linear form conservatively. The
influence of the SCC constraint on the system operation and its interaction
with the frequency regulation are demonstrated through simulations on IEEE 30-
and 118-bus systems.

###### Index Terms:

unit commitment, inverter based generators, short circuit current, frequency
regulation

##  I Introduction

Concerns on power system operation and stability have been raised with the
increasing penetration of Renewable Energy Sources (RESs), which are connected
to the grid through power electronic inverters. One of the most significant
impact of these IGBs is the decrease of system inertia, which would lead to
larger frequency RoCoF and deviation during system disturbances, such as the 9
Aug 2019 London blackout [1]. Extensive research has been carried out to
investigate the negative impact on system inertia reduction with various
solutions being proposed.

However, fewer attention has been paid on another influence associated with
the high IBG penetration in the system, i.e., shortage of Short Circuit
Current (SCC). Unlike the conventional synchronous generators which are able
to provide a short circuit current of 5-10 p.u.[2], the IBGs only supply fault
current of approximate 1-2 p.u. [3] due to the limited overloading capability
of the semiconductors. Therefore, as the IBGs in the system continue to
increase, the SCC is about to reduce. It has been predicted by the National
Grid that a decline of 15% on the average GB short circuit level will be seen
by the year 2025 with the largest declined area being identified as North-East
and Midlands regions [4]. Similarly, the SCC in Germany power system is
expected to decrease by up to 50% at certain buses [5].

The insufficient SCC in the system would have negative impact on various
aspects. Protection devices may take longer or even be not able to detect
faults in the system, thus posing risks to system operation and stability [6].
In addition, the inadequacy in SCC may lead to unsettled voltage oscillations
in the network and would further cause lose of synchronization of the Phase-
Locked Loop (PLL) [7].

Attempts in various respects have been proposed to enhance the SCC and address
the issues it brings in a high IBG-penetrated system. A method to improve the
semiconductor’s transient thermal capacity and over-current capacity is
discussed in [8] and the experiments show that an SCC of 3 p.u. for 3 s or 1.5
p.u. for 30 s is feasible. This is achieved by inserting phase-change material
right under the chips and setting the phase-change temperature slightly below
the maximum permissible chip temperature. However, additional manufacture cost
is unavoidable and the effectiveness may be limited without wide applications.
On the other hand, it is demonstrated in [9] that distance protection will be
compromised in terms of successful detection and response time as the IBGs’
penetration approaches 100%. It is also shown that the deployment and control
of synchronous condensers can help to mitigate the decline of the short
circuit level and its impact on the protection system. An optimal synchronous
condenser allocation approach is presented in [10] to maintain the system
minimum SCC. However, the mixed-integer nonlinear based problem formulation
and the considerable investment cost (up to $1\mathrm{M\$}$ fixed cost plus
$3\mathrm{M\$}$ per $100\mathrm{MVar}$) may limit its large scale deployment.
In the operational timescale, [11] proposes a system strength and inertia
constrained optimal generator dispatch model, where explicit constraints are
developed to achieve the optimal system scheduling while satisfying the SCC
requirement. However, since the nonlinear SCC constraint is linearized by
neglecting the interaction between different generators, an iterative process
is necessary to determine the optimal scale factor for SCC calibration and the
contribution from IBGs is not considered.

In fact, due to the significantly distinguished fault characteristic from the
conventional synchronous generators, the quantification of the fault current
from the IBGs is complicated. A method considering the impact of reinforcement
on voltage dip required by the grid code is proposed in [12] where the IBGs
are modeled as voltage sources behind an impedance.This equivalent impedance
is iteratively updated to match the terminal voltage. The IEC 60909-0-2016
standard [13] models the IBGs as constant current source and approximates the
SCC by adding the contribution from IBGs to the classic SCC expression yet
neglecting the pre-fault operating condition, which may lead to overestimation
of the SCC in the system. [11] proposes a fault current iterative solver,
which properly models the current limitation and voltage control logic of
IBGs. However, all these methods either require iterations or are too complex
to obtain an analytical expression, thus unable to be directly incorporated
into a UC problem.

In order to alleviate the influence of the SCC reduction in a high IBG-
penetrated system, an SCC constrained UC model is proposed in this paper. The
total number of online generators and their locations are optimized such that
the system operation cost is minimized while maintaining the SCC at each bus
within a predefined range specified by system operators. The contribution of
this paper is identified as:

  1. a.

A novel SCC constraint is proposed by applying the superposition principle to
calculate the SCC in a high IBG-penetrated system, which is then effectively
linearized through data-driven method while maintaining both conservativeness
and accuracy.

  2. b.

A Stochastic UC (SUC) model with SCC and frequency constraints is proposed and
formulated as Mixed-Integer Linear Programming (MILP). The SUC model, for the
first time, allows to optimize the system operation with simultaneous
consideration of variability, uncertainty, low inertia capability and low SCC
contribution from IBGs.

  3. c.

The effectiveness and the scalability of the model are assessed through IEEE
30- and 118-bus systems. The results illustrate the interactions between SCC
and frequency constraints and highlight the necessity to consider SCC limit in
the system scheduling once frequency support from IGBs becomes available. In
addition, the case study demonstrate the potential benefit of over-sizing or
expanding short-term PE overloading capability of IBGs.

The rest of the paper is organized as follows. Section II introduces the
classic SCC calculation method from SGs first based on which a modified
approach is proposed to include the contribution from IBGs. A SCC constrained
SUC model is formulated in Section III with the highly nonlinear SCC
constraint being effectively linearized. Simulation results are provided in
Section IV demonstrating the effectiveness of the proposed method. Finally,
Section V concludes the paper.

##  II Short Circuit Analysis

This section first revises the conventional SCC calculation method where the
synchronous generators are considered as the only sources of the SCC in the
system. An approach incorporating the SCC contribution from the IBGs is then
proposed based upon this. Note that only the three-phase nodal short circuit
faults are considered in this paper.

###  II-A Classic Short Circuit Current Calculation

Figure 1: Synchronous generator model for SCC computation: (i) pre-fault
equivalent circuit; (ii) circuit for $I_{sc_{F}}^{{}^{\prime\prime}}$
calculation.

In a conventional power system with low IBG penetration, the SCC at the fault
bus mostly comes from the synchronous machines, which can be modeled as an
internal voltage in series with a reactance [14]. Fig. 1 shows the equivalent
circuit of a SG, $g\in\mathcal{G}$ with a balanced three-phase load $z$.
$X_{1}$ is the external reactance between the SG and the fault location F.
During normal operation as in Fig. 1-(i), the load current $I_{Lg}$ is
supplied by the SG internal voltage $E_{g}$ in series with synchronous
reactance $X_{dg}$ leading to a terminal voltage $V_{t}$. If a three-phase
fault occurs at point $F$, Fig. 1-(ii) is used to calculate the subtransient
SCC $I_{sc_{F}}^{{}^{\prime\prime}}$, which is supplied by the subtransient
internal voltage $E_{g}^{{}^{\prime\prime}}$ in series with subtransient
reactance $X_{dg}^{{}^{\prime\prime}}$. $E_{g}^{{}^{\prime\prime}}$ is
determined by the pre-fault terminal voltage and the load current as follows
[15]:

$E_{g}^{{}^{\prime\prime}}=V_{t}+\mathrm{j}X_{dg}^{{}^{\prime\prime}}I_{Lg}.$
(1)

The voltage source $E_{g}^{{}^{\prime\prime}}$ in series with a reactance
$X_{dg}^{{}^{\prime\prime}}$ is further converted to a current source $I_{g}$
in parallel with a susceptance $Y_{g}$ according to the Norton’s theorem:

$\displaystyle I_{g}$
$\displaystyle=\frac{E_{g}^{{}^{\prime\prime}}}{X_{dg}^{{}^{\prime\prime}}}$
(2) $\displaystyle Y_{g}$
$\displaystyle=\frac{1}{\mathrm{j}X_{dg}^{{}^{\prime\prime}}}.$ (3)

The short circuit current at different buses in the system can be then
computed based on the classic superposition method [15]. The pre-fault
operating conditions are calculated first through power flow analysis to
determine the pre-fault voltage $V_{F}(0)$ at the fault bus $F$. The sources
$I_{g}$ of SGs are the only current injections in the system whereas the load
currents are ignored at this stage. The voltage $V_{F}(0)$ is then applied in
an inverse direction at the short circuit bus while all the other sources in
the network are disregarded (open circuit for the current sources) to
represent the pure-fault situation. Due to the linear modeling of system
elements, the post-fault system can be represented by a linear combination of
the pre- and pure-fault conditions. As a result, the initial subtransient
short circuit current of the F-th bus $I_{SC_{F}}^{{}^{\prime\prime}}$ can be
determined by:

$I_{SC_{F}}^{{}^{\prime\prime}}=\frac{-V_{F}(0)}{Z_{FF}}$ (4)

where $Z_{FF}$ is the F-th element of the main diagonal in the system
impedance matrix $Z$. It should be noted that the subtransient reactance of
the SGs are also included in the $Z$ matrix formulation.

###  II-B Modeling of SCC with IBGs

With the increasing penetration of IBGs, most of the grid codes require the
provision of reactive current from those units to the grid during fault
conditions to support the system. However, due to the limited SCC capacity of
Voltage Source Converters (VSCs) and the manufacturer-specific control
strategies, it is difficult to formulate an accurate method to quantify the
SCC contribution from the IBGs on a system level especially with a high IBG
penetration.

Figure 2: Equivalent circuit to calculate the SCC from IBGs: (i) post-fault
condition; (ii) pre-fault condition; (iii) ‘pure’-fault condition.

An analogous superposition method is adopted based on the classic SCC
calculation method and the standard IEC 60909-0-2016 to quantify the SCC in a
high IBG-penetrated system. The IBGs are viewed as constant current source
during a system short circuit fault as demonstrated in [13] and [16, 17, 18].
The equivalent circuit of a loaded IBG during a short circuit fault at point F
is shown in Fig. 2-(i) where $I_{f}$ is a pre-defined parameter representing
the short circuit injection from the VSC, e.g., 1.0 p.u. and the other
quantities are the same as in Fig. 1. Applying the superposition method, the
post-fault system in Fig. 2-(i) can be viewed as the combination of pre- and
‘pure’- fault conditions as shown in Fig. 2-(ii) and (iii) respectively. In
the pre-fault condition, $I_{L}$ is the load current from the VSC and
$V_{F}(0)$ is the bus voltage at the fault location before the fault. A
voltage source with the same magnitude is applied at this point without
changing the system. Different from the calculation of SCC from SGs where all
the SGs in the pure-fault condition are disregarded, the inverter-based units
in Fig. 2-(iii) has to be considered as a current source with the output
current being $I_{f}-I_{L}$ so that the superposition theorem holds. In order
to validate the method to calculate the SCC from VSC unit, the SCC is derived
from the post-fault system and the two sub-system respectively. The SCC from
the IBG $c\in\mathcal{C}$ can be derived from Fig. 2-(i) as:

$I_{sc_{F}}^{{}^{\prime\prime}}=I_{fc}.$ (5)

It can also be derived as the linear combination of
$I_{sc_{F1}}^{{}^{\prime\prime}}$ and $I_{sc_{F2}}^{{}^{\prime\prime}}$, i.e.:

$I_{sc_{F}}^{{}^{\prime\prime}}=I_{sc_{F1}}^{{}^{\prime\prime}}+I_{sc_{F2}}^{{}^{\prime\prime}}$
(6)

where $I_{sc_{F1}}^{{}^{\prime\prime}}$ is zero and
$I_{sc_{F2}}^{{}^{\prime\prime}}$ is computed by applying KCL in Fig. 2-(ii):

$I_{sc_{F2}}^{{}^{\prime\prime}}=(I_{fc}-I_{L})-I_{z}.$ (7)

The load current in the ‘pure’-fault condition $I_{z}$ is derived from the
pre-fault condition as follows:

$I_{z}=\frac{-V_{F}(0)}{z}=-I_{L}.$ (8)

Combining (6), (7) and (8) gives (5), which justifies the proposed
superposition method to calculate the SCC from IBGs.

Combining the classic SCC superposition approach with the proposed model of
IBG enables the SCC calculation in a general power system with both SGs and
IBGs. The SCC at the fault bus $F$ can be computed through KCL in the pure-
fault system where the only sources are the current at the VSC buses and the
fault bus:

$-V_{F}(0)=\sum_{c\in\mathcal{C}}Z_{F\Phi(c)}(I_{fc}-I_{Lc})+Z_{FF}I_{sc_{F}}^{{}^{\prime\prime}}$
(9)

where $\Phi(c)$ maps the IBG $c\in\mathcal{C}$ to the corresponding bus index.
Rearranging (9) yields the expression of the SCC at bus $F$ as follows:

$I_{sc_{F}}^{{}^{\prime\prime}}=\frac{-V_{F}(0)-\sum_{c\in\mathcal{C}}Z_{F\Phi(c)}(I_{fc}-I_{Lc})}{Z_{FF}}.$
(10)

Furthermore, the voltage $V_{F}(0)$ can be derived according to the pre-fault
system condition:

$V_{F}(0)=\sum_{g\in\mathcal{G}}Z_{F\Psi(g)}I_{g}+\sum_{c\in\mathcal{C}}Z_{F\Phi(c)}I_{Lc}$
(11)

where $\Psi$ maps the SG $g\in\mathcal{G}$ to the index of the bus it located
at. Substituting (11) into (10) leads to:

$I_{sc_{F}}^{{}^{\prime\prime}}=\frac{-\sum_{g\in\mathcal{G}}Z_{F\Psi(g)}I_{g}-\sum_{c\in\mathcal{C}}Z_{F\Phi(c)}I_{fc}}{Z_{FF}}.$
(12)

One can notice from (12) that pre-fault load current of the IBGs appears only
in the derivation but has no impact to the final expression of the nodal SCC
in the system. Based on this derivation, the SCC constraint is formulated and
then linearized in the next section. It should be noted that, although a
single IBG is plotted in Fig. 2, it can actually represent the aggregation of
all the IBGs that are connected at this bus since the SCC contribution from
IBGs is linear on $I_{fc}$.

##  III SCC Constrained UC Problem Formulation

In this section, the frequency-constrained SUC model proposed in [19] which
simultaneously optimizes the system PFR and the Synthetic Inertia (SI) from
the Wind Turbines (WTs) is applied and adapted to incorporate the SCC
constraint. With the linearization of the SCC constraint discussed in Section
III-B, the whole problem is formulated as an MILP. Note that only the short
circuit fault at each bus in the system is considered, which is a common
practice in fault analysis. Although different types of contingencies would
lead to different SCC level in the system, we do not focus on the detailed
contingency analysis in this paper. Instead, we take its results (minimum
permissible SCC levels) as the input to the proposed model. In addition, it is
possible that the SCC of some specific faults are critical given some prior
knowledge about the grid, in this case, the proposed model can be extended by
Norton’s theorem and sequence components analysis to cope with this situation
[14].

###  III-A Objective Function

The objective of the SUC problem is to minimize the expected cost over all
nodes in the given scenario tree:

$\min\sum_{n\in\mathcal{N}}\pi(n)\left(\sum_{g\in\mathcal{G}}C_{g}(n)+\Delta
t(n)c^{s}P^{s}(n)\right)$ (13)

where $\pi(n)$ is the probability of scenario $n\in\mathcal{N}$ and $C_{g}(n)$
is the operation cost of unit $g\in\mathcal{G}$ in scenario n including
startup, no-load and marginal cost; $\Delta t(n)c^{s}P^{s}(n)$ represents the
cost of the load shedding in scenario n with the three terms being the time
step of scenario n, load shedding cost and shed load.

The scenario tree is built based on user-defined quantiles of the forecasting
error distribution to capture the uncertainty associated with the demand the
wind generation. A rolling plan approach is implemented in the SUC model. At
each time step, a 24-hour horizon SUC problem is performed with only the
decisions of the current node being applied and all the future decisions
discarded. At next time step, the scenario tree is updated according to the
realizations of the uncertain variables and the process repeats. Due to this
rolling planning method, different realizations of nodes in the scenario tree
are consistently generated at every hourly timestep according to the forecast
eror quantile based probability. Therefore, the out-of-sample scheduling
performances can be well captured in the simulation process rather than
through Monte-Carlo simulations afterwords.

The objective function (13) is subjected to a number of constraints. While SCC
and frequency constraints are discussed below, all other conventional UC
constraints such as those of power balance, thermal units and transmission
system are not listed in the paper. The readers can refer [20] for details.

###  III-B Formulation and Linearization of Short Circuit Current Constraint

This subsection formulates the SCC constraint in the unit commitment problem.
The associated quantities that influence the SCC in the system are the
commitment status of the SGs and IBGs. The former can be viewed as binary
decision variables i.e., $x_{g},\,\forall g\in\mathcal{G}$. However, for the
IBGs, their operating conditions are determined by the current available
wind/solar resources rather than the system operator. Therefore, a node-
dependent parameter $\alpha_{c}\in[0,1],\,\forall c\in\mathcal{C}$ is
introduced to represent the percentage of IBGs’ online capacity. The approach
proposed in [21] and [22] is used in this paper where the online wind capacity
is estimated given the current available power based on historical data.

By definition, the system admittance matrix $Y$ is formulated as follows:

$Y=Y^{0}+Y^{g}$ (14)

where $Y^{0}$ is the admittance matrix of the transmission lines only; $Y^{g}$
denotes the additional $Y$ matrix increment due to SGs’ subtransient
reactance. Depending on state of the SGs, the elements in $Y^{g}$ can be
expressed as:

$Y_{ij}^{g}=\begin{cases}\frac{1}{X_{dg}^{{}^{\prime\prime}}}x_{g}\;\;&\mathrm{if}\,i=j\land\exists\,g\in\mathcal{G},\,\mathrm{s.t.}\,i=\Psi(g)\\\
0\;\;&\mathrm{otherwise}.\end{cases}$ (15)

The SCC constraint $I_{sc_{F}}^{{}^{\prime\prime}}\geq
I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}$ can be derived by modifying (12):

$\displaystyle-\sum_{g\in\mathcal{G}}Z_{F\Psi(g)}I_{g}x_{g}$
$\displaystyle-\sum_{c\in\mathcal{C}}Z_{F\Phi(c)}I_{fc}\alpha_{c}$
$\displaystyle\geq I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}Z_{FF},\,\,\forall
F\in\mathcal{N}$ (16a) $\displaystyle ZY$ $\displaystyle=\mathbb{I}_{N}$ (16b)

where $\mathbb{I}_{N}$ is $N$-dimensional identity matrix. $I_{g}$ is computed
according to (2) where $E_{g}^{{}^{\prime\prime}}$ is assumed to be a
constant:

$E_{g}^{{}^{\prime\prime}}=\beta V_{n}$ (17)

with $V_{n}$ being the nominal voltage of the system and $\beta$ a parameter
varying between $[0.95,1.1]$. Since the minimum initial SCC is concerned,
$0.95$ is chosen in this paper.

In order to incorporate the SCC constraint into the MILP-based UC model, the
nonlinearity in (16) has to be linearized. However, (16b) is a matrix inverse
constraint with binary decision variables which is difficult to deal with in
an optimization problem in general. Different linearization methods are
considered and compared in order to achieve a better trade-off between
accuracy and computational efficiency.

####  III-B1 Analytical Method

$N\times N$ linear equality constraints are used to enforce the relationship
in (16b):

$\sum_{k=1}^{N}Z_{ik}(Y_{kj}^{0}+Y_{kj}^{g})=\begin{cases}1\;\;&\mathrm{if}\,i=j\\\
0&\mathrm{otherwise}\end{cases}\;\;\;\;\;\forall i,j\in\mathcal{N}$ (18)

where each element in matrix $Z$ is viewed as a decision variable. Matrix
$Y^{g}$ contains binary variables $x_{g}$ as in (15), therefore the only
nonlinearity in (18) is the product of a continuous variable and a binary one.
Take the term $\sum_{k}Z_{ik}Y_{k\Psi(g)}^{g}$ for example:

$\displaystyle\sum_{k}Z_{ik}Y_{k\Psi(g)}^{g}$
$\displaystyle=\frac{1}{X_{dg}^{{}^{\prime\prime}}}\mu_{g}$ (19a)
$\displaystyle\mu_{g}$ $\displaystyle=Z_{i\Psi(g)}x_{g}.$ (19b)

The nonlinear constraint (19b) can be effectively linearized using McCormick
Relaxation as follows:

$\displaystyle\mu_{g}$ $\displaystyle\leq x_{g}Z_{\mathrm{max}}$ (20a)
$\displaystyle\mu_{g}$ $\displaystyle\geq-x_{g}Z_{\mathrm{max}}$ (20b)
$\displaystyle\mu_{g}$ $\displaystyle\leq
Z_{i\Psi(g)}+(1-x_{g})Z_{\mathrm{max}}$ (20c) $\displaystyle\mu_{g}$
$\displaystyle\geq Z_{i\Psi(g)}-(1-x_{g})Z_{\mathrm{max}}$ (20d)

where $Z_{\mathrm{max}}$ is the upper bound of the magnitude of the elements
in $Z$ matrix. All the other nonlinear terms in (16) and (18) have the same
form as (19). Therefore, they can be linearized using a similar approach.

####  III-B2 Data-driven Methods

Since (16) contains $|\mathcal{G}|$ decision variables ($x_{g}$) and
$|\mathcal{C}|$ parameters ($\alpha_{c}$), the linearization of the SCC
constraint can be formulated as follows:

$\displaystyle I_{F_{L}}^{{}^{\prime\prime}}$
$\displaystyle=\sum_{g\in\mathcal{G}}k_{Fg}x_{g}+\sum_{c\in\mathcal{C}}k_{Fc}\alpha_{c}+\sum_{m\in\mathcal{M}}k_{Fm}\eta_{m}$
$\displaystyle\geq I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}},\,\,\forall
F\in\mathcal{N}$ (21)

where $I_{F_{L}}^{{}^{\prime\prime}}$ is the linearized SCC at Bus $F$. In
order to account for the nonlinearity in (16), the term $k_{Fm}\eta_{m}$ are
included representing the interactions of every two SGs, which is defined as
follows:

$\displaystyle\eta_{m}$
$\displaystyle=x_{g_{1}}x_{g_{2}},\quad\mathrm{s.t.}\\{g_{1},\,g_{2}\\}=m$
(22a) $\displaystyle m\in\mathcal{M}$ $\displaystyle=\\{g_{1},\,g_{2}\mid
g_{1},\,g_{2}\in\mathcal{G}\\}.$ (22b)

Note that since the 2nd order terms lead to accurate enough results as
illustrated in Section IV-A2, the higher order terms are neglected in order to
achieve a balance between the accuracy and computational time. The nonlinear
constraint (22a) can be then linearized using a similar way as (20). The
coefficients $\mathcal{K}=\\{k_{Fg},\,k_{Fm},\,k_{Fc}\\},\,\forall F,g,m,c$
are determined by solving the minimization problem $\forall F\in\mathcal{N}$:

$\displaystyle\min_{\mathcal{K}}\sum_{\omega\in\Omega}\left(I_{F_{L}}^{{}^{\prime\prime}(\omega)}-I_{sc_{F}}^{{}^{\prime\prime}(\omega)}\right)^{2}$
(23a)
$\displaystyle\left.I_{F_{L}}^{{}^{\prime\prime}(\omega)}=I_{F_{L}}^{{}^{\prime\prime}}\right\rvert_{x_{g}^{(\omega)},\,\eta_{m}^{(\omega)},\,\alpha_{c}^{(\omega)}}$
(23b)

with
$\omega=\\{x_{g}^{(\omega)},\,\alpha_{c}^{(\omega)},\,I_{sc_{F}}^{{}^{\prime\prime}(\omega)}\\}\in\Omega$
denoting the data set. It is generated by evaluating the SCC at each bus in
representative system conditions. For the SGs, all the possible generator
combinations are considered, whereas for the continuous parameters
$\alpha_{c}$, there are infinite possible conditions in theory. However,
according to (16), $\alpha_{c}$ only influences the SCC in a linear manner as
it does not appear in (16b). Hence, two values are sufficient to capture the
linear behavior, i.e., $\alpha_{c}\in\\{0,\,1\\}$, which leads the size of
$\Omega$ to $2^{|\mathcal{G}|+|\mathcal{C}|}$ in total.

Since the optimization problem (23a) aims to minimize the total squared error
between the linearized SCC and its actual value, the conservativeness cannot
be guaranteed. In order to force the relationship
$I_{F_{L}}^{{}^{\prime\prime}(\omega)}\leq
I_{sc_{F}}^{{}^{\prime\prime}(\omega)},\,\forall\omega$, (23a) is modified as
follows:

$\displaystyle\min_{\mathcal{K}}\quad$
$\displaystyle\sum_{\omega\in\Omega}\left(I_{sc_{F}}^{{}^{\prime\prime}(\omega)}-I_{F_{L}}^{{}^{\prime\prime}(\omega)}\right)$
(24a) $\displaystyle\mathrm{s.t.}\quad$ $\displaystyle
I_{F_{L}}^{{}^{\prime\prime}(\omega)}\leq
I_{sc_{F}}^{{}^{\prime\prime}(\omega)},\,\,\forall\omega\in\Omega.$ (24b)

With the constraint (24b), the linearized function is always below the real
one. It is understandable that the conservativeness is achieved at a cost due
to the nonlinearity. At some data points, larger errors which could have been
compensated with each other in the least square optimization (23a) now become
inevitable. As a result, there exists situation where the system actual SCC is
above its permissible limit but not according to the linearized value, thus
increasing the operational cost.

Another reason for the over-conservativeness caused by (24) is that its
objective function trys to minimize the errors of all the data points. Since
the SCC limit is a hard constraint in the SUC problem, it is not essential to
force the linearized SCC as close to its real value as possible everywhere.
Instead, the idea of “classification” can be utilized to train the linear
function. That is as long as the linearized SCC is on the same side of the
limit as its real value, the error is of no concern. However, since the SCC is
a continuous variable, which is different from the classic classification
problem, some modifications need to be made in the formulation:

$\displaystyle\min_{\mathcal{K}}\quad$
$\displaystyle\sum_{\omega\in\Omega_{2}}\left(I_{sc_{F}}^{{}^{\prime\prime}(\omega)}-I_{F_{L}}^{{}^{\prime\prime}(\omega)}\right)^{2}$
(25a) $\displaystyle\mathrm{s.t.}\quad$ $\displaystyle
I_{F_{L}}^{{}^{\prime\prime}(\omega)}<I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}},\,\,\forall\omega\in\Omega_{1}$
(25b) $\displaystyle I_{F_{L}}^{{}^{\prime\prime}(\omega)}\geq
I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}},\,\,\forall\omega\in\Omega_{3}$ (25c)

where $\Omega_{1},\,\Omega_{2}$ and $\Omega_{3}$ are the subsets of $\Omega$,
being defined as follows.

$\displaystyle\Omega$
$\displaystyle=\Omega_{1}\bigcup\Omega_{2}\bigcup\Omega_{3}$ (26a)
$\displaystyle\Omega_{1}$ $\displaystyle=\left\\{\omega\in\Omega\mid
I_{sc_{F}}^{{}^{\prime\prime}(\omega)}<I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}\right\\}$
(26b) $\displaystyle\Omega_{2}$ $\displaystyle=\left\\{\omega\in\Omega\mid
I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}\leq
I_{sc_{F}}^{{}^{\prime\prime}(\omega)}<I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}+\nu\right\\}$
(26c) $\displaystyle\Omega_{3}$ $\displaystyle=\left\\{\omega\in\Omega\mid
I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}+\nu\leq
I_{sc_{F}}^{{}^{\prime\prime}(\omega)}\right\\}.$ (26d)

Given (25b) and(26b), all the below-limit SCC can be classified correctly by
the linearized function. Ideally, it is also desired to identified all the
above-limit SCC. However, this may cause infeasibility due to the restricted
linear structure. Therefore, a parameter $\nu\in\mathbb{R}^{+}$ is introduced
to define $\Omega_{2}$ and $\Omega_{3}$ as in (26c) and (26d). In this way,
all the data points in $\Omega_{3}$ will be classified correctly and
misclassification can only occur in $\Omega_{2}$.

Furthermore, $\nu$ should be chosen as small as possible while ensuring the
feasibility. Since it is only the data points in $\Omega_{2}$ whose errors are
penalized in the objective function (25a), a good behavior could be trained,
thus avoiding being over-conservative. The overall linearization performance
of the three data-driven methods as well as the analytical method are compared
in Section IV-A2.

###  III-C Frequency Constraints

The frequency security constraints are elaborated here. A novel wind turbine
SI control scheme is proposed in [19], which eliminates the secondary
frequency dip due to WT over production and allows the WT dynamics to be
analytically integrated into the system frequency dynamics. As a result, the
frequency nadir constraint can be formulated as:

$HR\geq\frac{\Delta P_{L}^{2}T_{d}}{4\Delta f_{\mathrm{lim}}}-\frac{\Delta
P_{L}T_{d}}{4}\left(D-\sum_{j\in\mathcal{F}}\gamma_{j}H_{s_{j}}^{2}\right)$
(27)

where $H$ is the sum of conventional inertia $H_{c}$ and SI of all the wind
farms $\sum_{j}H_{s_{j}}$ with $j\in\mathcal{F}$ being the set of wind farms
and $R,\,T_{d}$ represent the system Primary Frequency Response (PFR) and its
delivery time; $\Delta P,\,\Delta f_{\mathrm{lim}}$ and $D$ denote system
disturbance, limit of frequency deviation and system damping. The last term
can be interpreted as: SI provision from each wind farm introduces a negative
damping to the system proportional to $H_{s_{j}}$ with the coefficient being
$\gamma_{j}$. Furthermore, the multi-dimensional hyperboloid constraint (27)
can be effectively piece-wise linearized as follows:

$a_{i}H+b_{i}R+\tilde{c}_{i}\tilde{H}_{s}+d_{i}\leq 0,\;\forall
i\in\mathcal{P},$ (28)

with $a_{i},\,b_{i}$,
$\tilde{c}_{i}=[c_{i,1},c_{i,2},...,c_{i,|\mathcal{F}|}]$ and $d_{i}$ being
the linearized coefficients, $\mathcal{P}$ the set of hyperplanes and
$\tilde{H}_{s}=[H_{s_{1}},H_{s_{2}},...,H_{s_{|\mathcal{F}|}}]^{\mathsf{T}}$.

Since both the frequency and SCC constraints are influenced by the SGs and
IBGs, it is of interest to investigate their combined effects on system
operation.

##  IV Results

Case studies on the IEEE 30- and 118-bus systems are carried out to illustrate
the validity of the proposed model and the impacts of the SCC constraint on
system operation as well as its interaction with system frequency constraints.
Although it may be helpful to illustrate some results based on deterministic
formulation first, it turns out that the deterministic results do not present
significant differences in terms of system SCC thus not being included in the
paper. The network data of the systems can be obtained in [23]. The MILP-base
UC problem is solved by FICO Xpress on a PC with Intel(R) Core(TM) i7-7820X
CPU @ 3.60GHz and RAM of 64 GB.

###  IV-A IEEE 30-Bus System

The system is modified by adding wind generation at Bus 1, 23 and 26 to model
the IBG-penetrated system as shown in Fig. 3. The system parameters are set as
follows: load demand $P_{D}\in[160,410]\,\mathrm{MW}$, damping
$D=0.5\%P_{D}/1\,\mathrm{Hz}$, FR delivery time $T_{d}=10\,\mathrm{s}$ and
maximum power loss $\Delta P_{L}=50\,\mathrm{MW}$. The frequency limits of
nadir, steady state and RoCoF set by National Grid are: $\Delta
f_{\mathrm{lim}}=0.8\,\mathrm{Hz}$, $\Delta
f_{\mathrm{lim}}^{\mathrm{ss}}=0.5\,\mathrm{Hz}$ and
$\Delta\dot{f}_{\mathrm{lim}}=0.5\,\mathrm{Hz/s}$. The SCC from the IBGs are
assumed to be $1\,\mathrm{p.u.}$ for conservativeness.

Figure 3: Modified IEEE-30 bus system.

####  IV-A1 Impact of wind penetration on system SCC

######  Remark.

In this section, the SCC constraint is not included in the SUC model in order
to study the impact of wind penetration on SCC in an unconstrained system.

The SUC problem of one year is solved with total installed wind generation
capacity being $800\,\mathrm{MW}$. Hours with different instantaneous wind
penetration in the optimal solution are selected. The SCC at each bus is
further calculated and depicted in Fig. 4 and Fig. 5 where $\rho_{w}$ is the
instantaneous wind penetration rate defined by:

$\rho_{w}=\frac{P_{w}-P_{w}^{sh}}{P_{D}}$ (29)

with $P_{w}/P_{w}^{sh}$ and $P_{D}$ being total available/shed wind power and
system demand. Note that only the cases with $\rho_{w}=0,\,0.4$ and $0.8$
representing zero, medium and high wind penetration are considered for
clarity.

The situation where no SI is provided from WTs (Fig. 4) is studied first. The
SCC is calculated in p.u. based on $S_{B}=100\,\mathrm{MVA}$. For the zero
wind penetration case, the SCCs in the system represented by the blue bars
vary at different buses depending on their electric distance to the online
conventional generators. The magnitude of the SCC in this case is the largest
with an average of $12.53\,\mathrm{p.u.}$. In the medium wind penetrated
system, the SCCs (red bars) at all buses decrease to some extent due to the
disconnection of the conventional generators located at Bus 26 and 30. Larger
reductions are therefore spotted at the nearby buses. However, as the wind
penetration doubles to 80 %, the resulted SCC even increases at each bus. This
is because only one more SG at Bus 5 is disconnected whereas significant
amount of SGs are forced to be online for the sole purpose of system inertia
and PFR provision to maintain the frequency constraints. The resulted SCC is,
in fact, dominated by the SCC boost contributed from the increased IBGs. It is
clear that in the system without SI, the reduction of SCC is limited and hence
it is not necessary to include such constraints.

In the case where SI from WTs is available in the system, the SCC at each bus
presents a different trend against the wind penetration. As $\rho_{w}$
increases to $0.4$, a remarkable SCC reduction in the range of $[21\%,\,45\%]$
is identified at all the buses. This much more significantly decreased SCC
compared to that in Fig. 4 is due to the fact that with the SI from WTs, much
fewer SGs are needed in the system to maintain the frequency constraints. For
the same reason, in the high wind penetration system, the SCC at every bus
further declines for another $33\%$ on average. Actually, the only synchronous
generators that are online in this hour locate at Bus 2 thus resulting in the
highest SCC there. The SCCs at the rest buses in the system reduce gradually
as their electric distance to Bus 2 increases. Therefore, the two buses with
minimum SCC are identified: Bus 26 ($4.13\,\mathrm{p.u.}$) and Bus 30
($4.23\,\mathrm{p.u.}$). Under these conditions, it becomes critical to
incorporate SCC constraints into the SUC model in order to maintain reliable
system operation.

Figure 4: SCC at different wind penetration without SI.

Figure 5: SCC at different wind penetration with SI.

Figure 6: SCC violation percentage at different wind penetration.

Furthermore, the SCC constraint violation in the grid during the transition
towards a high wind penetration system is also investigated. The results are
shown in Fig. 6 with the minimum permissible SCC,
$I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}$ being $5\,\mathrm{p.u.}$. Without
SI from WTs, no SCC constraint violation can be found regardless of installed
wind capacity. On the contrary, once the SI from WTs becomes available, an
increasing proportion of operating hours will face the SCC shortage along with
the higher penetration of wind generation. About $28\%$ of the total hours
exhibit SCC violation at different buses at $720\,\mathrm{MW}$ wind capacity.

####  IV-A2 Linearized SCC constraint validation

TABLE I: Comparison of SCC Constraint Linearization

Method | Type I Error | Type II Error |  Cost $[\mathrm{k\pounds}/h]$ |  Time $[\mathrm{s/step}]$
---|---|---|---|---
$N_{e}$ | $err$ | $N_{e}$ | $err$
AM | $0$ | $0$ | $0$ | $0$ | $28.60$ | $>1e3$
DM-1 | $18$ | $1.18\%$ | $213$ | $-2.60\%$ | $28.92$ | $1.51$
DM-2 | $0$ | $0$ | $7379$ | $-16.27\%$ | $32.25$ | $3.41$
DM-3 | $0$ | $0$ | $13$ | $-0.45\%$ | $28.60$ | $3.78$

Before incorporating the SCC constraint into the SUC problem, the performance
of the linearized SCC constraints proposed in Section III-B is examined first.
The results are shown in Table IV-A2 where AM and DM-1/2/3 denote the
analytical method based on matrix inverse (18) and the three data-driven
methods according to (23) - (25) respectively. Two types of error are analyzed
with Type I being the SCC which satisfies the constraint according to the
linearized value but actually violates and Type II being the opposite. $N_{e}$
and $err$ are the total number and the averaged value of the errors defined as
follows:

$\displaystyle N_{e}$ $\displaystyle=|\mathcal{E}|$ (30a) $\displaystyle err$
$\displaystyle=\frac{1}{N_{e}}\sum_{\mathcal{E}}\frac{I_{F_{L}}^{{}^{\prime\prime}(\mathcal{E})}-I_{sc_{F}}^{{}^{\prime\prime}(\mathcal{E})}}{I_{sc_{F}}^{{}^{\prime\prime}(\mathcal{E})}}$
(30b)

with $\mathcal{E}$ being the set of errors. For the analytical method, no
error is observed since there is no approximation in the formulation. However,
the computational time is extremely long due to the presence of the matrix
inverse with decision variables (18) - (20), which limits its practical
application in large scale systems. This challenge is resolved by the data-
driven methods with much shorter computation time. The least square
minimization is used in DM-1. It can be seen that both types of the error
occur with relatively small $err$. As Type-I error leads to insecure operation
point, (24b) is added in DM-2 to eliminate such error. The effectiveness is
manifested by zero Type-I error in the table. However, an averaged Type-II
error of $-16.27\%$ and an operational cost increase of $13\%$ demonstrate its
over-conservativeness in some operating conditions. DM-3 adapts the idea of
classification in the optimization, which only minimizes the error around the
limit. The results are promising with few Type II errors and negligible $err$.
These errors almost have no impact on the system operation cost compared with
the analytical method. Although the time is slightly longer than DM-2, it is
still acceptable and represents significant reduction over AM. As a result,
DM-3 is used for the rest of the study.

####  IV-A3 SCC constrained UC problem

Figure 7: (Un)constrained SCC in the system. TABLE II: Assessment of SCC
Constraint Level

$I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}$ $[\mathrm{p.u.}]$ | min SCC | Online SG Number |  Cost $[\mathrm{k\pounds}/h]$
---|---|---|---
p.u. | Bus | $N_{2}$ | $N_{3}$ | $N_{4,5}$ | $N_{26}$ | $N_{30}$
$0.00$ | $4.13$ | 26 | $2$ | $0$ | $0$ | $0$ | $0$ | $10.60$
$5.00$ | $5.49$ | 26 | $3$ | $1$ | $0$ | $0$ | $0$ | $21.02$
$6.00$ | $6.19$ | 26 | $3$ | $0$ | $0$ | $1$ | $1$ | $26.41$
$7.00$ | $7.14$ | 29 | $3$ | $1$ | $0$ | $2$ | $1$ | $37.74$

In this section, the SCC constraint is incorporated into the SUC model to
evaluate its effectiveness as well as the influence on system operation. The
SCC at each bus in the system with and without the SCC constraint is shown in
Fig. 7. The instantaneous wind penetration rate in the system is $0.8$ and the
minimum permissible SCC in this case is set to be 5 $\mathrm{p.u.}$, i.e.,
$I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}=5,\,\,\forall F\in\mathcal{N}$ such
that a short circuit ratio of 3 (strong grid) can be maintained. It should be
noted that depending on the type of bus that is of concern, different
strategies shall be considered to determine this minimum permissible SCC
level. Specifically, for the point of common coupling of IBGs, a stiff voltage
signal is desired to ensure the synchronization of PLLs and the stability of
current grid-following converters. This is often indicated by a system
strength (short circuit ratio) of 3 or more [10], which would lead to an
appropriate SCC constraint level at this point given the knowledge of IBGs’
capacity. For other buses, the SCC constraint level may be decided by the
settings of critical protection devices, the requirement of voltage stability
margin. In general, the determination of the minimum SCC level can be very
complex [24] and should be conducted before the system scheduling process thus
being out of the scope of the problem at hand.

If not being constrained, the SCC at some buses in the system is below 5 p.u.
as illustrated by the red bars in Fig. 7. Once the SCC constraint is included,
the SCC at all buses is above the limit, which demonstrates the effectiveness
of the proposed method. This is achieved by increasing the online SGs to
provide the SCC during faults.

Different levels of the SCC limit are also considered to better review their
impacts on generator dispatch and system operation cost. The results are shown
in Table IV-A3 where the base case
($I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}=0$) corresponds to the red bars in
Fig. 7 and $N_{i}$ represents the online SG number at bus $i$. As the level of
the SCC limit increases from $5\,\mathrm{p.u.}$ (Case A) to $7\,\mathrm{p.u.}$
(Case C), the actual minimum SCC in the system is always higher than the limit
and the bus at which it locates also changes. It is understandable that higher
SCC in the system would lead to larger system operation cost. For example, the
cost in Case A is about two times that of the base case since the number of
online SGs doubles. It is also worth noting that in Base B, the SGs at Bus 26
and 30 are connected to the grid to replace the SG at Bus 3. Although the SG
at Bus 3 has lower operating cost, the SGs at Bus 26 and 30 are more effective
in supplying SCC for the limiting bus, demonstrating the spacial phenomena of
the SCC constraints. Moreover, the minimum SCC in the system is changed from
Bus 26 to 19 in Case C because of the specific SG combination, which
illustrates that only confining certain bus with minimum SCC may not be
sufficient during some operating conditions.

####  IV-A4 Assessment of IBGs’ PE overcapacity

Instead of dispatching more SGs, the SCC in the system can be improved by
increasing the PE capacity of the IBGs as well, through either increasing the
capacity of the semiconductor devices or utilizing advanced technologies [8].
In this section, the value of IBGs’ short-term PE overcapacity is
investigated. The results are depicted in Fig. 8 where the horizontal axis
represents the ratio to the base PE capacity $P^{C}$. The averaged system
operation cost is reduced by increasing the PE capacity due to the additional
SCC capability from IGBs. Moreover, as the SCC limit becomes higher, the IBGs’
overcapacity is more valuable. However, the marginal value of overcapacity
quickly reduces as it approaches to the unconstrained case. This results and
further studies can be used to inform the optimal sizing of IBGs’ PE capacity.

Figure 8: System operation cost against PE capacity.

###  IV-B IEEE 118-bus system

The scalability of the proposed method is demonstrated through IEEE 118-bus
system with the demand $P_{D}\in[2000,5600]\,\mathrm{MW}$, largest loss of
generation $\Delta P_{L}=500\,\mathrm{MW}$ and $S_{B}=1000\,\mathrm{MVA}$.
Wind generation is added at Bus 3, 41, 72 and 87 and the parameters for the
frequency regulation remains unchanged.

The SUC problem is performed in different conditions to compare the effects of
the SCC constraint and the IBGs’ overcapacity. The results are listed in
Table. IV-B with the base size of PE capacity $P^{C}=8000\,\mathrm{MW}$.
Nearly two thirds of the total hours present shortage of SCC without SCC
constraint. This number reduces significantly to $18.97\%$ if the total PE
capacity of IBGs is increased by $50\%$. Once the SCC is constrained in the
SUC model, the violation disappears with an increased averaged cost of
$11.6\%$. The overcapacity in this case decreases the system operation cost
since more SCC is provided by the IBGs thus reducing the committed SGs during
high wind penetration period. In addition, there is a clear increase in the
computational time when SCC constraints are added but it is still in a
manageable range. In fact, it can be also noted when overloading capability of
IBGs increases, the computational time decreases as fewer SCC constraints are
active. Moreover, an example of SCC variation during 48 hours at Bus 87 is
plotted in Fig. 9-(i) with the PE capacity being $P^{C}$ and
$I_{F_{\mathrm{lim}}}^{{}^{\prime\prime}}=6\,\mathrm{p.u.}$. Without the
constraint, SCC violation is observed most of the time (blue curve), which is
successfully eliminated by the SCC constraint (red curve). This is achieved by
increasing the online SGs as in Fig. 9-(ii) where more SGs are dispatched
during the hours with SCC constraint violation. In details, the SGs that will
be switched on depend on the combination effects of the SG capacity, marginal
cost and its electric distance to the critical bus. If a large SCC shortage
occurs, e.g., Hour 24, the closest SG at Bus 87 is dispatched regardless of
its higher marginal cost; On the contrary, if the SCC shortage is small (Hour
36), farther SGs located at Bus 69 may be switched on to reduce the overall
operation cost. Furthermore, because of these additionally dispatched SGs,
less wind power is utilized during the time $\mathrm{[20,48]\,h}$ when there
is more than enough wind resource in the system as illustrated in Fig.
9-(iii). Unsurprisingly, larger SCC shortage is likely to lead to more SGs
being online thus more wind power being curtailed (red curve), which also
justifies the cost increase in Table III.

TABLE III: SCC comparison in IEEE 118-bus system

SCC Constraint |  IBGs’ PE Capacity |  SCC Violation |  Cost $[\mathrm{k\pounds/h}]$ |  Time $[\mathrm{s/step}]$
---|---|---|---|---
w.o. | $P^{C}$ |  $67.93$% | $56.71$ | $19.55$
$1.5P^{C}$ | $18.97\%$ | $-$ | $-$
with | $P^{C}$ | $0$ | $63.31$ | $32.32$
$1.5P^{C}$ | $0$ | $58.78$ | $23.04$

Figure 9: SUC results within two days: (i) SCC; (ii) online SG capacity; (iii)
Wind generation.

##  V Conclusion

A short circuit current constrained SUC model is proposed in this paper in
order to deal with the SCC shortfalls due to the increasing penetration of
IBGs. The SCC from SGs as well as IBGs is modeled in the formulation leading
to a highly nonlinear constraint, which is effectively linearized based on the
data obtained offline. It is validated that the linearized constraint provides
almost the same accuracy as the nonlinear version but consumes much less
computational time.

The influence of the SCC constraint on the system operation and its
interaction with the frequency constraints are thoroughly investigated via
IEEE 30-bus system. The results demonstrate that the SCC in the system
declines dramatically as the wind penetration increases if SI is available
from WTs. Overcapacity of the IBGs helps to improve system SCC but needs more
investment cost. Employed with the SCC constraint, the system SCC limit can be
maintained with higher operational cost depending on the wind penetration. The
scalability of the proposed model is assessed through IEEE 118-bus system.

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