This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2018.2866099, IEEE
Transactions on Power Systems

Adaptive Formation of Microgrids with Mobile

Emergency Resources for Critical Service
Restoration in Extreme Conditions
Liang Che, Member, IEEE, Mohammad Shahidehpour, Fellow, IEEE
Abstract—The existing microgrid-based distribution system 𝑃,𝑄 Generation capacities of MER g (kW and kVar)
restoration methods follow traditional restoration criteria which prs Probability of scenario s
achieve a maximum coverage of distribution systems. In extreme R Vectors of line resistances (Ohm)
conditions (e.g., natural disasters and physical and cyber attacks), V0 Reference voltage in the system (volt)
the traditional restoration plans could pose additional financial VR Vector of rated voltage at nodes (kV)
and security risks to local utility customers. That is, restored 𝑥̅ Upper limit for x, i.e., the maximum number of energized
services could be further interrupted by subsequent outages in the lines connected to a single microgrid node
extended event. To address such resilience challenges, this paper α, β Vectors of weights for ∆D and v, respectively
proposes the formation of adaptive multi-microgrids as part of the ηD, ηλ Penalty factors in the objective function of the MF model
critical service restoration (CSR) strategy. The proposed strategy ϵG, ϵL Ratios for MER kW capacity reserve and branch flow
comprises microgrid formation (MF) and load switching sequence security margin (0<{ϵG, ϵL}≤1 and are small)
(LSS) steps. The MF forms multiple microgrids with minimum Ρ Ratio for allowed p.u. nodal voltage variation
overall scale and radial or looped topologies and includes Γ Ratio for reactive component of load shedding
optimally positioned mobile emergency resources (MERs) to
Variables
address potential risks in extreme conditions. The LSS determines ∆D Load shedding for ensuring feasible MF Stage 1 (kW)
a proper load switching sequence for ensuring the system dynamic LSs Critical load loss in scenario s in MF Stage 2 (kW)
performance in the restoration process. The effectiveness of the F, FQ Vectors of line real /reactive flow (kW/kVar)
proposed strategy is verified by a case study on the IEEE 132- Lmax, Lsw The maximum and the actually-switched critical load in a
node test feeder system and the time-domain simulation in single step of LSS (kW)
MATLAB Simulink. P, Q Vectors of real /reactive power output of MER
Index Terms—Microgrids, mobile emergency resources, (kW/kVar)
distribution system, critical service restoration, resilience. V Vector of nodal voltages (kV)
Λ Coefficient for controlling topology of ℳ in MF Stage 1
xi Number of energized lines connected to node i
NOMENCLATURE (xi≥1↔ wi=1; xi=0↔ wi=0)
Indices and Sets 𝑢 ,𝑢 Binary, =1 if MER g is connected at node i in MF Stages
d∈𝒟 Index of critical loads 1 and 2, respectively
g∈G Index of distributed generation resources 𝑣,𝑣 Binary, =1 if branch l is energized (included in a
i, j, n∈𝒩 Index of nodes microgrid) in MF Stages 1 and 2, respectively
l∈ℒ Index for distribution lines wi Binary, =1 if node i is energized (included in microgrid)
m∈ℳ Index of microgrids
K Index of load picked up in the LSS
S Index of scenarios in MF Stage 2 I. INTRODUCTION
Gm Set of resources allocated to microgrid m in MF Stage 1
ODAY’s power grids are facing an increasing frequency
OL, OI
Os
Set of nodes on outage (cannot be energized)
Set of nodes failed in scenario s T and a higher intensity of outages caused by natural disasters
like Hurricane Sandy, and the threat of terrorism (both physical
Parameters and cyber) [1]-[2]. The impacts of extreme events on power
D, DQ Vector of critical loads (kW) and (kVar) grids underscore the need for enhancing the resilience, which is
𝑭 Vectors of line flow limit (kW) defined as “the ability to prepare for and adapt to changing
GRg, cg Governor primary response reserve (kW) and ramp rate
(kW/s) conditions and withstand and recover rapidly from extreme
H System inertia (kW∙s/Hz) outages” [3]. In this context, microgrid has been recognized as
KL, KD Branch-bus and load-bus incidence matrices a promising solution to resilience enhancement and service
Ni Maximum number of MERs connected at node i restorations after extreme events.
The use of microgrid for distribution service restoration
(DSR) [4]-[7] and critical service restoration (CSR) [8]-[11]
Manuscript received March 20, 2018; revised June 29, 2018; accepted were studied in the literature and will be reviewed further in
August 14, 2018. Paper no. TPWRS-00407-2018. L. Che is with the Section II. DSR methods partition a distribution system (DS)
Midcontinent Independent System Operator (MISO), Carmel, IN 46032 into islands to form microgrids with the objective to maximize
USA (e-mail: lche@misoenergy.org). M. Shahidehpour is with the Galvin the sum of restored loads, and CSR methods typically focus on
Center for Electricity Innovation, Illinois Institute of Technology, Chicago, restoring critical loads (CLs) that deliver vital society services,
IL 60616 USA, and also with the Renewable Energy Research Group, King e.g., hospitals, police and fire stations [8]. These methods
Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: ms@iit.edu). provided insight into the microgrid-based service restoration
problem; however, their applications can be limited considering
the following challenges.

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First, the existing methods basically construct a maximum To undertake the above challenges, this paper proposes a
DS coverage by forming microgrids in DS, which is referred to microgrids-based CSR strategy to ensure the highest service to
as the maximum coverage criterion. The application of such CLs when DS encounters outages and damages under extended
criterion still follows the traditional restoration principles [12], extreme events such as natural disaster and physical and cyber
which do not pertain to the flexibility of microgrids or the DS incidents. The contributions of the paper are listed as follows:
vulnerability following extended extreme events (discussed 1. The CSR strategy (consisting of MF and LSS steps) is
below), thereby risking the continuity of services to CLs. proposed for serving CLs while taking into account the
This paper focuses on CSR for extended extreme events, microgrid survivability in extreme events.
which are extreme events that can impose subsequent damages 2. MF addresses the aforementioned challenges and enhances
to DS and subject the initially-formed microgrids to additional the microgrid survivability in extended events by forming
risks of damages. For example, flooding in Hurricane Katrina multiple minimum-scale microgrids with radial and looped
left behind 3×1013 gallons of water in 2017 which caused topologies and properly positioning MERs.
subsequent damages to electricity infrastructures in Texas [13]. 3. A proper LSS is developed by considering the relationship
In addition, earthquake aftershocks can impose physical risks between switching actions and frequency deviations which
that would last for several weeks [14], and physical or cyber includes a necessary condition for satisfying the dynamic
incidents can launch coordinated multi-stage attacks to impose frequency nadir limit.
extended damages to multiple infrastructures [15]. The Ukraine The remainder of this paper is organized as follows. Section
grid attack (which caused outages by manipulating distribution II provides a review of related works. Section III presents the
switches) demonstrated the vulnerability of distribution basics of CSR strategy. The two building blocks of CSR,
systems to malicious attacks [16]. including the MF model and the LSS tool, are presented in
When microgrids are formed by the maximum-coverage Sections IV and V, respectively. The strategy is validated by
criterion, they will involve a maximum number of energized detailed simulations of the IEEE 123-node system in Section
components in DS, and are likely to be affected by the VI. Finally, Section VII concludes the paper.
extended extreme events. Specifically, this could risk the
service restorations of CLs in the following ways. 1) Loss of II. RELATED WORK
lines which link large non-CLs will trigger large power The microgrid-base DSR methods typically form
imbalance in microgrids which could cause CL equipment microgrids to maximize the sum of restored loads in DS, e.g.,
damages (due to frequency/voltage deviations), service novel switch placement method aiming at maximizing the
interruptions (as protection services would function), or a restoration capability [4], distributed multi-agent coordination
microgrid collapse (due to voltage or frequency instability). 2) strategy for the global information discovery in restoration [5],
Mobile resources (discussed later) being positioned away from sequential restoration [6], and dynamically adjusting microgrid
CLs might result in higher risk of inflicting additional CL boundaries [7]. For comparison, CSR typically focuses on
outages when the energized paths encounter new damages in restoring critical loads [8]-[11]. The key difference is that, DSR
the extended event. 3) The use of maximum-coverage criterion identifies microgrid boundaries and forms microgrids by
may require the operation of a large number of switches which opening switches on these boundaries, while CSR will
would lead to a more complicated and delayed restoration. optimally choose a set of paths and restore CLs by energizing
Besides, the limited DER capacity and fuel availability could the selected paths [8]. Notably, ref. [8] put forth a novel CSR
serve CLs more comprehensively, as opposed to achieving the strategy that defines restoration trees and load groups and
maximum coverage when DS is facing such extreme conditions. effectively determines the maximum coverage and restorative
Such a feature was exercised for supplying several CLs after actions of CLs. Other works on CSR include the look-ahead
the Hurricane Maria hit Puerto Rico [17]. Based on the above restoration involving feeder selection and operational dispatch
considerations, this study takes into account the DS [9], and the investigations of renewable energy and demand
vulnerability and the microgrid survivability, and proposes an uncertainties in CSR [10]-[11]. Other works also investigated
optimal critical service restoration (CSR) strategy for serving microgrid formations [20] and [21]; however, they essentially
CLs in extended extreme events. studied long-term planning problems rather than service
Second, the previous research on resource mobility and the restoration issues.
provision of flexibility of forming microgrids might be deemed This paper considers the CSR, but differs from previous
inadequate, which would need to be expanded. The mobile studies in the following respects. 1) The key problem
emergency resource (MER), typically a truck-mounted considered in this paper is not to identify restoration trees/paths
generator or battery storage, can be quickly dispatched after an [8], but is the formation of microgrid topology and the
event [18], e.g., 400 truck-mounted generators delivered positioning of MER. 2) A node is not a load zone [8] but is a
emergency services in the aftermath of Hurricane Sandy [19]. connection of lines; a microgrid is a DS sub-network which is
Ref. [18] investigated the MER pre-positioning and allocation, formed by connecting MERs and CLs with energized branches.
which is essentially another application of the maximum- 3) The restorative action in this paper focuses on an efficient
coverage criterion under the traditional restoration principle. In load switching process which satisfies the frequency nadir
practice, the flexible MERs can enhance the microgrid requirement.
survivability when staged properly and dispatched from secure The proposed CSR strategy includes LSS which considers
locations. Our proposed CSR will form microgrids with MERs frequency nadir requirement in microgrids. Microgrid
for managing the supply continuity to CLs. frequency dynamics and recovery were investigated in the
Moreover, previous studies did not fully investigate an literature, including the microgrid dynamic performance
appropriate load switching sequence (LSS) for restoring CLs. assessment (under hypothetical disturbances) [22], microgrid
The LSS dynamics involve frequency deviations when picking frequency recovery by load shedding (when the microgrid is
up large loads. A proper LSS will consider the microgrid islanding) [23], and microgrid frequency control under normal
dynamic performance in CSR. condition [2],[24]. Different from such studies, the LSS in this

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paper considers the frequency dynamics in the load restorative Step-2. LSS Strategy: For each microgrid m, determine the
process in which the goal is to realize an efficient restoration appropriate LSS to pick up the CLs in 𝒟m. The LSS solution is
by determining the maximum amount of CLs that can be presented in Section V.
picked up at a single load switching step while maintaining the C. Microgrid Topology in CSR
frequency above its nadir.
We use x and w (|𝒩|×1 vectors) to express the properties of
DS nodes, where xi (integer) is the number of energized lines
III. BASICS OF CSR connected at node i, and wi (binary) is the node energizing
This section provides the CSR basics, including the CSR status. We have xi≥1↔wi=1 and xi=0↔wi=0.When serving a
definition and overview, analyses of microgrid topology, and CL, a looped microgrid can have a higher survivability as
the derivation of power-voltage characteristics used in the MF. demonstrated at the IIT Microgrid1 which consists of seven
loops [21],[26]. So, the MF model considers both radial and
A. CSR Definition in This Paper looped microgrids. Loop is defined as follows.
Our previous work [21] conducted a graph theory-based Definition: A loop is defined as a series of nodes {ndi} where
analysis for developing the microgrid topology. Similarly, i={1,2,…N}⊂𝒩m, in which each pair of adjacent nodes ndi and
topology analyses are performed here for defining CSR. The ndi+1 are connected by a branch, nd1 is connected to ndN, and
DS is represented by a graph DS={𝒩,ℒ}, where 𝒩 is the set of there is no connection between any two nodes ndi and ndj with
nodes and ℒ is the set of branches. In CSR, microgrids are |i-j|≠1 and |i-j|≠N-1. Examples are given in Figs. 2c-2e.
formed in the set ℳ. Each microgrid m∈ℳ includes nodes For topology control when forming microgrids, MF places
𝒩m⊂𝒩 and branches ℒm⊂ℒ. The MER set allocated to the following limit on xi for any node i:
microgrid m is Gm⊂G, which supply CLs in 𝒟m⊂𝒟. Any two
𝒙 𝐊 𝐋 ∙ 𝒗 𝑥̅ ∙ 𝟏 (1.1)
microgrids m1 and m2 satisfy 𝒩m1∩𝒩m2=∅ and ℒm1∩ℒm2=∅.
Once DS loses the grid supply in an extreme event (which where 𝑥̅ (user-defined) is the maximum number of lines
can be initiated from either an outage in upstream networks or connected to a node (i.e., upper limit of xi), |KL| is a matrix
a fault in distribution substation/feeder), the distribution whose elements are the absolute values of those in KL (the
supervisory control and data acquisition (SCADA) system will nbus×nbranch bus-branch incidence matrix), and 1=[1,1…1 𝑻|𝒩| 1 .
collect the DS network information including the lines on Based on x, node i will only be energized when it is connected
outage. Then, the proposed CSR strategy flexibly forms to at least one energized branch. This is expressed as
multiple microgrids ℳ to serve CLs in 𝒟 by determining the xi=0↔wi=0 and xi≥1↔wi=1, which is equivalently stated as:
status of key switches in 𝒩 and MER positions. Moreover, if a 𝒙/𝜋 𝒘 𝒙 (1.2)
microgrid encounters subsequent topology changes, the where π is sufficiently large to ensure 0<xi/π<1 (∀i).
proposed method will be used to adaptively change the Furthermore, the MF model will not form energized islands
microgrid formation, i.e., determine a new ℳ by changing without CLs, which is due to the minimization of the overall
certain switches and repositioning MERs [4],[5],[25]. scale of microgrids. This will be discussed later.
B. Principles of CSR When 𝑥̅ =2, a microgrid can take the topologies depicted in
Fig. 1 illustrates the CSR strategy consisting of MF and LSS.
Figs. 2a or 2c; while if 𝑥̅ =3, all topologies in Fig. 2 are possible.
Thus, (1.1) essentially ensures that each formed microgrid is
either radial or looped, i.e., takes one of the topologies in Fig. 2.

(a) (b) (c) (d) (e)

Fig. 2. Microgrid topologies under constraint (1.1): (a) and (c) can occur with
𝑥̅ =2, and (a)-(e) can occur with 𝑥̅ =3.
Based on x and w, the following corollary provides an
Fig. 1. Illustration of steps in the proposed CSR strategy. analysis on the topology of ℳ and defines λ by (1.3) which
The proposed steps are discussed as follows: will be included in the MF model later. λ is defined for ℳ in
Step 1. MF Strategy: After any DS outages resulting from the MF Stage 1 (not for a certain microgrid). Based on
extreme event damages, the microgrids (ℳ) will be formed by Corollary 1, a smaller λ indicates more loops and/or more inter-
a two-stage MF framework, in which the first stage determines microgrid connections, e.g., Figs. 2a-2e have λ=1, 1, 0, 0 and -1,
the microgrid topology and the second stage performs a as they have nmg=1 while nlp=0, 0, 1, 1 and 2, respectively.
network reconfiguration if necessary. The MF is presented in Corollary 1: The λ defined below gives the value of nmg–nlp for
Section IV. Once determined by MF, microgrids will be ℳ (nmg and nlp are the numbers of microgrids and loops in ℳ):
established as follows. For each microgrid m∈ℳ, switches 𝜆 𝟏𝑻 𝒘 𝒙/2 (1.3)
with precisely one end in m (i.e., lines ij with i∈m and j∉m) are Proof: The numbers of nodes and branches in ℳ are nd=1Tw
opened to isolate m from the rest of DS, while lines within and nb=(1Tx)/2, respectively. A tree graph has nd–nb=1 [27]. So,
microgrids (i.e., ij with i∈m and j∈m) are closed. All load ℳ containing a single radial microgrid (i.e., nmg=1, nlp=0) has
switches are also opened. MERs in Gm are allocated to m and nmg–nlp=1, and λ=nd–nb=1 based on (1.3). Accordingly, λ=nmg–
connected at nodes determined by MF. Each microgrid m will nlp. Next, for each additional radial microgrid formed, ℳ will
be black started by its MERs (each MER has a black start
capability, while the detail of black start is out of the scope of 1
IIT Microgrid refers to the campus microgrid project at the Illinois Institute of
this paper). Technology [26]. It is the world’s first fully functional campus microgrid.

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advance (nmg–nlp) by 1 because nmg=nmg+1, and increase λ by 1 IV. MICROGRID FORMATION PROCESS
because of an additional tree graph. For each loop formed, ℳ A. Two-Stage Framework of MF
will lower (nmg–nlp) by 1 because nlp=nlp+1, and decrease λ by 1 The MF is the first CSR step which properly forms multiple
as the increase in 1Tx is 2 more than that in 1Tw. Therefore, ℳ microgrids (ℳ) to achieve the following goals:
will have λ=nmg–nlp regardless of the number of microgrids or 1) Microgrids ensure the CSR of CLs.
loops. ■
2) Microgrids maintain a high survivability in extreme
D. Power Flow and Voltage Characteristics in the MF events.
In the literature, there are two types of models for microgrid These goals are fulfilled by a two-stage MF framework given
operations. One uses a DC flow model to ensure the real power in Fig. 3, which links the two stages and the goals stated above:
balance, which typically does not consider the line ampere  The MF Stage 1 determines the microgrid topologies in
rating [11],[28]. The other uses the linearized DisFlow model ℳ and the resource allocations where the key decision
to handle reactive power and nodal voltages [5],[18]: variable is the individual line switching status.
Nodal inflow: ∑ ∈ 𝐹 𝐹 𝐷 ; ∑∈ 𝐹 𝐹 𝐷 (2.1)  The MF Stage 2 reconfigures the microgrids, including
Voltage drop: 𝑉 𝑉 𝑟𝐹 𝑥 𝐹 /𝑉 (2.2) re-positioning MERs and further shrinking the microgrids,
while the line ampere rating is either ignored [5] or applied by if necessary, that will minimize the loss-of-CL in the
limiting apparent power flows [18]. In (2.1)-(2.2), V0 is the extended extreme event.
system voltage reference, 𝐹 / 𝐹 are the net real/reactive in-
flows of node i [5], 𝐷 /𝐷 and Vi are real/reactive loads and
voltage at node i.
In this paper, the MF in CSR has three features: 1) A
microgrid will cover a small network (discussed later). 2) A
microgrid will have loops. 3) The MF goal is to form feasible
microgrids in which the reactive power and voltages are
regulated by the microgrid control. Considering these features,
MF will handle the reactive power and nodal voltages by a Fig. 3. Two-stage MF framework including the goals and the models.
simplified method presented below. In the following, the two MF Stages are presented in
A small-scale microgrid will have large R/X ratio [29]: Subsections B and C, respectively.
𝑟 ≫ 𝑥 , ∀𝑙 ∈ ℒ𝑚 (3.1)
B. MF Stage 1 (Forming Microgrid Topology)
Introducing (3.1) into (2.2), we have:
The first MF goal is represented in (5.1) where ηD as a large
𝑉 𝑉 ∆𝑉 ∙𝑟 ∙𝐹 (3.2) penalty factor reduces the not-served CLs (∆D) weighted by α:
which shows the relationship between real power and voltage min: 𝜂 ∙ 𝜶𝑻 ∆𝑫 (5.1)
magnitudes in small-scale microgrids. In such cases, the real In Fig. 3, the MF Stage 1 realizes the second goal by
power-frequency characteristic commonly used for the minimizing two terms. One term is stated as:
microgrid droop control is maintained by adding a virtual min: 𝜷𝑻 𝒗 (5.2)
impedance in the control module to obtain an overall inductive
where v (binary) is the branch energizing status, weighted by β.
impedance for the resource [30]-[31].
If β=1, then (5.2) is the number of lines in ℳ. Eq. (5.2)
Based on (3.1)-(3.2), the MF model (given later) will
include (4.1)-(4.4) for handling reactive power and voltages: minimizes the network scale of ℳ, which has two implications.
One is that a smaller microgrid will energize fewer components
1) Nodal voltage: Eq. (3.2) is re-written as: and thus can have a lower exposure to potential risks of new
𝑉 ∙ 𝒖𝑻 𝑽 𝒓 ∗ 𝑭 (4.1) outages in the extended event. The other is that, β can be
where * is the element-wise multiplication operator. In addition, practically set to manage risks. For example, in a physical
the nodal voltage limits are stated as [5]: incident, a line with a higher exposure to outages can have a
1 𝜌 𝑽𝑹 𝑽 1 𝜌 𝑽𝑹 (4.2) larger βl. In this context, (5.2) basically minimizes the overall
2) Reactive power balance and limit: The nodal balance for risk in the extended extreme event. In addition, (5.2) will
reactive power is satisfied in (4.3), in which 𝛾∆D is the reactive eliminate the unnecessary switching of a branch, which is
load shedding based on the assumption that if the load is reflected in Corollary 2 below.
reduced by ∆D (kW), its reactive component will be reduced by Corollary 2: In the optimal MF solution, a radially-connected
𝛾 ∆D (kVar), where ratio 𝛾 is determined based on the power node (connecting only one line) must be either a source (i.e.,
factor. Eq. (4.4) is the capacity limit which is discussed later. connecting to an MER) or a sink (i.e., connecting a CL).
𝐊 𝐋 ∙ 𝑭𝑸 𝒖 ∙ 𝑸 𝐊 𝐃 ∙ 𝑫𝑸 𝛾∆𝑫 (4.3) Proof: Assume that in an optimal solution, node i is radially
connected to node i1 which is neither a source nor a sink. In this
0 𝑄 1 𝜖 ∑∀ 𝑢 𝑄 , ∀𝑔 (4.4)
case, branch i-i1 will have zero flow and its exclusion (vi-i1=0)
3) Branch rating: The cable rating given in ampere is will not affect any CLs. So, vi-i1=0 is another feasible solution,
converted to kVA and then to kW rating (based on a typical which is actually more appealing due to (5.2). This contradicts
power factor) [32]. So, the distribution line rating can be stated with the above assumption so it proves the corollary. ■
by the kW flow limit (𝑭). Besides, as discussed later, a small The other term at Stage 1 which contributes to the second
ratio ϵL is used to set a security margin for line flows. The goal (see Fig. 3) is expressed as:
impact of large reactive flows on network security can be min: 𝜂 ∙ 𝜆 (5.3)
accommodated by a slightly larger ϵL. mg lp
where λ provides the value of (n –n ) in ℳ as defined in (1.3).
The minimization of (5.3) tends to form additional loops (i.e., a

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larger nlp) and/or interconnecting adjacent microgrids (i.e., a Constraint (6.8) collectively represents (4.1)-(4.4). It
smaller nmg) which would otherwise be operated as separated establishes the relationship between F and V, enforces the
microgrids. Thus, the MF Stage 1 will first form ℳ with a voltage limits, and sets the reactive power balance. Constraint
minimum overall scale by minimizing (5.2), and then considers (6.9) represents (1.1)-(1.3). It configures the radial or looped
more loops and interconnections by minimizing (5.3). It, microgrid topologies and gets the value of λ based on (1.3).
therefore, contributes to the reduction of the likelihood of CL Finally, the adaptiveness of the model is enhanced by adding
service interruptions under new outages during the extended (6.10), which restricts the use of any branches in set OL and
event, which consequently enhances the microgrid survivability any nodes con in set OI. Basically, OL is the set of branches on
(the second goal in Fig. 3). outage. Additional branches and nodes can be added into OL
The complete MF objective function is given in (6.1), which and OI, respectively, based on a practical consideration that
reflects (5.1)-(5.3). ηD and ηλ are set to offer the highest priority energizing these branches/nodes might incur additional risks.
to the first term while offering the lowest priority to the third For example, in a physical or cyber incident, if branch l is
term. Thus, (6.1) states the following strategy: first ensure highly vulnerable to subsequent attacks and is difficult to be
services to CLs, and then enhance the microgrid survivability safeguarded, then it can be added to OL. In another example,
by maintaining the minimum microgrid scales and taking node n whose capability for integrating MERs is impaired by
advantage of loops if possible. the event can be added to OI.
MF‐1: min: 𝜂 ∙ 𝜶𝑻 ∆𝑫 𝜷𝑻 𝒗 𝜂 ∙𝜆 (6.1) C. MF Stage 2 (Reconfiguration)
MF Stage 2 will further enhance the microgrid survivability
∑∀ 𝑢 1, ∀𝑔 (6.2) and contribute to realizing the second goal (see Fig. 3) by
implementing two reconfigurations in ℳ:
∑∀ 𝑢 N , ∀𝑖 (6.3)
1) repositioning MERs within microgrids and
𝐊𝐋 ∙ 𝑭 𝒖∙𝑷 𝐊𝐃 ∙ 𝑫 ∆𝑫 (6.4) 2) further shrinking the microgrid coverage,
where the survivability enhancement is quantified by
0 𝑃 1 𝜖 ∑∀ 𝑢 𝑃 , ∀𝑔 (6.5) minimizing the loss-of-CL when microgrids encounter new
𝟎 ∆𝑫 𝑫 (6.6) outage scenarios in the extended event as expressed by the first
term in (7.1) which has ηD as a large penalty (the same as in
1 𝜖 𝒗∗𝑭 𝑭 1 𝜖 𝒗∗𝑭 (6.7) (6.1)).
𝑉 ∙𝒖 𝑽 𝒓∗𝑭 𝑻 The MF stage 2 includes binary variables 𝑢 and 𝑣 , which
⎧ are only assigned to node i and branch l with 𝑢 =1 and 𝑣 =1
⎪ 1 𝜌 𝑽𝑹 𝑽 1 𝜌 𝑽𝑹
𝐊 𝐋
∙ 𝑭 𝑸
𝒖 ∙ 𝑸 𝐊 𝐃
∙ 𝑫𝑸 𝛾∆𝑫
(6.8) solved by (6), respectively; KL/KD are re-defined based on the
⎨ topology form by (6), which are not explicitly expressed in (7)
⎪0 𝑄 1 𝜖 ∑∀ 𝑢 𝑄 , ∀𝑔
⎩ for brevity. A small number of variables 𝑢 and 𝑣 is needed in
𝒙 𝐊 𝐋 ∙ 𝒗 𝑥̅ ∙ 𝟏 (7) since only few nodes and branches are in scope after the
𝒙/𝜋 𝒘 𝒙 (6.9) network scale of ℳ is minimized in the MF stage 1. In (7),
𝜆 𝟏𝑻 𝒘 𝒙/2 each scenario s refers to the failures in set Os, defined as the
failure of a branch, which is expressed in (7.2). In scenario s,
𝑣 0, ∀𝑙 ∈ OL LSs is the loss-of-CLs, which is different from ∆D used in the
∑∀ 𝑢 0, 𝑤 0, ∀𝑖 ∈ OI
(6.10)
MF Stage 1 for ensuring the feasibility of (6), and prs is the
scenario’s probability.
Constraints (6.2)-(6.3) are the MER allocation constraints, The first reconfiguration mentioned above is implemented
where the binary variable 𝑢 =1 denotes MER g is connected at by determining 𝑢 (=1 means MER g is re-connected to node i)
node i, while ∑∀ 𝑢 =0 means it is not allocated. An MER in (7.3), which ensures MERs allocated in Stage 1 will be
cannot be allocated to more than one node as specified in (6.2). selected in Stage 2 and acknowledges the resource-connecting
In (6.3), Ni (which is system dependent) represents the physical capabilities of nodes. The second reconfiguration is carried out
capability of node i for connecting MERs: Ni=0 means node i by determining the branch status 𝑣 through (7.4), which means
does not have such capability, and Ni=1 and Ni>1 mean that it that the branches opened in Stage 1 are kept open while those
can integrate one and multiple MERs, respectively. closed in Stage 1 will have their status (𝑣 ) re-determined in
Constraints (6.4)-(6.7) model the CSR of CLs by ensuring Stage 2, which may further shrink the microgrids when solving
the nodal real power balance. Constraint (6.4) balances the (7). Note that these two reconfigurations are determined by
nodal flows, in which the MER output (P), CL shedding (∆D) minimizing (7.1) which will contribute to minimizing the loss-
and line flows (F) are obtained from (6.5)-(6.7), respectively. of-CL.
Constraint (6.5) ensures that MER g has zero output if it is not Constraints (7.5)-(7.6) perform a generation re-dispatch to
allocated (∑∀ 𝑢 =0). Since the proposed model is to ensure the minimize (7.1) when the lines in scenario s are opened in (7.2).
feasibility of the microgrid formation, a small ratio ϵG is used in The re-dispatch implements the power balance in (7.5), places
(6.5) to define a capacity reserve for each MER. Constraint (6.6) physical limits on generation and loss-of-load in (7.6), and
represents the critical-load shedding (∆D), which will be driven includes the real power - voltage characteristic and reactive
to zero by the large ηD in (6.1). Such load shedding is power / voltage constraints in (7.7) which is modified from
implemented in the formulation to ensure the feasibility of the (6.8).
solution. Constraint (6.7) models branch flows, in which vl=1
means branch l is energized and included in a microgrid (its MF‐2: min: 𝜂 ∙ ∑∀ 𝑝𝑟 𝜶𝑻 𝑳𝑺𝒔 𝜷𝑻 𝒗 (7.1)
switches are closed), and a small ratio ϵL is prescribed to set a 𝐹 0, ∀𝑙 ∈ O
𝑠
(7.2)
network security margin.

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∑∀ 𝑢 1, ∀𝑔 which means that Lmax for the LSS in microgrid m is:

∑∀ 𝑢 N, ∀𝑖 (7.3) 𝐿 2𝑀 𝑓 𝑓 ∙∑ 𝑐 (8.3)
∈
∑∀ 𝑢 0, 𝑖 ∈ OI
Eq. (8.3) is the necessary condition for the microgrid frequency
1 𝜖 𝒗 ∗𝑭 𝑭 1 𝜖 𝒗 ∗𝑭 (7.4) satisfying the fmin requirement in LSS. It indicates that a
𝐊 𝐋 ∙ 𝑭𝒔 𝒖 ∙ 𝑷𝒔 𝐊𝐃 ∙ 𝑫 𝑳𝑺𝒔 (7.5) microgrid with larger inertia (MH) and/or quicker primary
response (cg) can pick up a larger size load at each LSS step. In
𝟎 𝑷𝒔 𝑷 (7.6) practical applications, fmin can be defined based on the
𝟎 𝑳𝑺𝒔 𝑫 composition of load and the performance of generations and
𝑻
𝑉 ∙ 𝒖 𝑽𝒔 𝒓 ∗ 𝑭𝒔 protection relays [36]. Also, governor ramp-rate (cg) can be
⎧ obtained from a stress test which simulates large frequency
⎪ 1 𝜌 𝑽𝑹 𝑽𝒔 1 𝜌 𝑽𝑹
𝐋 𝑸𝒔 (7.7) deviations to register the ramp-rate of governor response [37],
⎨ 𝐊 ∙ 𝑭 𝒖 ∙ 𝑸𝒔 𝐊 𝐃 ∙ 𝑫𝑸 𝛾∆𝑫𝒔
⎪ and inertia (MH) can be estimated either from empirical data [38]
⎩ 0 𝑄 1 𝜖 ∑∀ 𝑢 𝑄 , ∀𝑔 or online testing [39].
The two-stage MF are formulated as mixed-integer linear B. LSS Procedure
programming (MILP) problems (6)-(7), which can be solved by Based on (8.3), LLS is implemented in each microgrid
commercial MILP software packages like CPLEX. The very m∈ℳ as follow. Considering that a CL can represent multiple
large-scale computation can be shortened by acceleration loading facilities, we use k to index individual switchable loads
techniques, e.g., forcing the binary variables to zeros in areas in CLs located in m, in which N is the size of k in m.
that have no CLs and are away from the nearest CL. 0) Initialize the index of the first picked-up load as k=1.
1) Based on Lmax from (8.3), identify loads Dk through Dk+∆k
V. LOAD SWITCHING SEQUENCE
with a total amount of Lsw, to be switched at this step:
This section presents the LSS process which is used as the
second step of CSR. Since MERs are typically diesel 𝐿 ∑ ∆ 𝐷 𝐿 ∑ ∆ 𝐷 ; 𝑘 ∆𝑘 N (8.4)
generators, which are considered as synchronous generator- 2) Since the last switching action, the microgrid has
interfaced DERs [22], LSS will be investigated based on DER experienced a frequency drop which is kept above fmin by
characteristics. However, similar analyses will apply to (8.3). Then, the primary and secondary controls will
inverter-interfaced DERs. eventually restore the frequency [2]. The next load
Picking up a large load can cause frequency variations, switching action (i.e., picking up loads Dk through Dk+∆K) is
which is essentially a primary frequency control problem. Ref. allowed once the frequency is stabilized. A stabilized
[33] was the first to develop a detailed model of the governor frequency will be realized if the frequency oscillation is
primary response for generation dispatch. Then, governor rate damped [23] or a pre-defined time delay is met [8].
constrained optimal power flow and frequency dynamics- 3) LSS and thus CSR are completed if k+∆k=N , i.e., all the
constrained unit commitment were examined in [34]-[35]. CLs in microgrid m are restored. Otherwise, assign
Governor ramp-rate (cg) represents the prime mover k←k+∆k+1 and go back to Step 1.
ramping restrictions, i.e., the fastest rate of change of
On the one hand, such an optimal restoration sequence (as
mechanical power output in response to sudden frequency
opposed to a single switching action) is necessary considering
deviations [34]. Ref. [34] proved that the necessary (but
the microgrid dynamic performance. On the other hand, LSS
insufficient) condition for ensuring the frequency nadir limit is:
contributes to an efficient restoration by determining the
𝐺𝑅 𝑃 ∙𝑐 optimal Lmax and Lsw, and properly strikes a balance between a
(8.1) fast restoration (i.e., larger load picked up at a step and smaller
∑𝑔 𝐺𝑅𝑔 𝐿
number of switching actions) and a required dynamic
which means that when a system with inertia of MH (MW∙s/Hz) performance (i.e., satisfactory frequency nadir).
and individual governor ramp-rate of cg (MW/s) encounters a
sudden power loss of size L (MW), its frequency can be kept VI. CASE STUDY
above the nadir, fmin, if its total governor primary response
reserve ΣgGRg (MW), determined by (8.1), is enough to cover L. A. Simulation Setup
In this section, the simulation is conducted on the modified
A. Condition for Satisfying Frequency Nadir Limit in LSS IEEE 123-node test feeder system [40]. The optimization
Since the frequency nadir limit is the most key concern in problems are solved by CPLEX 12.4 and the time-domain
system operations, this paper focuses on satisfying this limit. simulation is performed in MATLAB Simulink. The test
The key process in the proposed LSS is to determine the system is shown in Fig. 4, in which the dashed lines are added.
maximum load size which can be picked up at a single step The double-slashed lines are the normally-opened lines in [40];
(Lmax) for each microgrid such that the frequency can always be in this study, they form the set OL, and represent the lines that
kept above fmin. We develop the condition for satisfying the fmin are outaged by the first strike of the event.
requirement as follows. L as a parameter in (8.1) is replaced by In the two-stage MF (6)-(7), system-dependent parameters
Lmax which is the variable to be determined by LSS, and the are obtained as follows. Line resistance (rl=0.4576 Ω/mile),
inequality in (8.1) is changed to an equality in LSS since GRg line length in feet (lenl) and load power factors (γ) are obtained
can reach its maximum of 𝑃 if the unit has not reached its from [40], and the length of each added line is set to be the
capacity limit 𝑃 [34], which is already ensured by (6.4)-(6.5) mean value of lenl, i.e., len=333 feet. As discussed earlier, Ni
in the MF. Thus, (8.1) is transformed to: (maximum number of MERs connected at node i) is system
dependent. In the case study, without losing generality, we set
𝐿 ∙∑ ∈ 𝑐 (8.2) Ni=2 (∀i), as the values of Ni do not change the nature of the

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model. Set G includes ten MERs: six with Pmax=100kW, three shows the final solution of ℳ, in which the combinations of
with Pmax=50kW and one Pmax=20kW. Also, the outage set the shaded and hatched areas represent the microgrids formed
OI=∅, each line and CL are switchable, and distribution line in Stage 1, while the hatched areas are excluded (𝑣 =0) in the
loss is ignored. In (6)-(7), the power is stated in kW/kVar and reconfiguration in Stage 2. For example, microgrids 3,6,8 are
voltage in kV, except for V0 (set as 1000V) in (6.8) to match formed as a single microgrid in Stage 1, which is because
the unit of its right-hand side. Since [40] does not provide line energizing the gray-hatched lines leads to smaller λ and larger
ratings, we set 𝐹 =245 kW (∀l) which is the largest kW load in βTv and finally a smaller, i.e., more optimal, (6.1) than the case
the original system based on [40]. of excluding them in Stage 1. Next, these three microgrids will
User-defined parameters in MF are set as follows. MF be separated in Stage 2 since the minimization of (7.1)
Stage 1 has 𝑥̅ =3, which means that any topologies in Fig. 2 are determines 𝑣 =0 for those gray-hatched lines and thus excludes
allowed. In the objective function, the penalty factors are them from ℳ.
ηD=106 and ηλ=len×4, which imply that an additional loop or By minimizing the overall scale of ℳ, i.e., βTv in (6.1), the
microgrid interconnection will be formed if it does not incur an solution of MF Stage 1 has λ=2 (calculated by (1.3)). On the
increase in βTv above len×4; and the weights are αd=1 (∀d) and other hand, Fig. 5 shows that ℳ in Stage 1 has seven
βl=lenl. The ratios for generation, flow, and voltage limits in (6) microgrids and five loops (nmg–nlp=7–5=2) which verifies the
are ϵG=0.2, ϵL=0, and ρ=1%. In MF Stage 2, each scenario s Corollary 1. Fig. 5 also shows that each radially linked node
considers the loss of a line (l) in DS, with a probability of (e.g., nodes 2,5,12) is connected to either a CL or an MER,
𝑝𝑟 = len / ∑ len . This assumption is reasonable considering which verifies the Corollary 2.
that a longer line has a higher risk of new outages in the
extended event (disaster/attack). One may consider other
scenario definitions, which will not affect the nature of the
model.
For LSS, a dynamics simulation is conducted in MATLAB
Simulink with a modified diesel governor model (DEGOV) [41]
in which primary and secondary microgrid controls are
considered as shown in Appendix A. For frequency
stabilization, following the criterion mentioned in Section V,
the next switching action will take place after a 10-sec time
delay. Parameters fmin, cg and MH will be defined later.

Fig. 5. Final solution of the two-stage MF.

Microgrids 2,5,8 demonstrate that a loop will be formed if it

results in an increment of βTv that is less than ηλ= len ×4.
Additional tests show that, microgrid 2 will only have the gray-
shaded loop in Stage 1 (Fig. 5) when ηλ is reduced to len×3,
Fig. 4. One-line diagram of the modified IEEE 123-node test feeder. and will become radial (including nodes 31-26-27-29) when
ηλ= len ×2. The reason is that forming the left loop adds
B. Simulation Results
len×2.56 to βTv, and forming the right loop with the left one in
Case 1. Demonstration of the proposed CSR strategy existence will further add len ×3.56. On the other hand,
During an extreme event, DS encounters outages as shown microgrid 4 is formed as radial in Stage 1 since the loop of 54-
in Fig. 4, in which the proposed CSR strategy is implemented. 57-60-61 would increase the scale by len×5.96>ηλ.
The two-stage MF and LSS results are shown as follows. The two MF goals (see Fig. 3) are reflected in the formation
1) MF Stage 1. As the first CSR step, the two-stage MF, (6)- of Fig. 5. The first goal is realized by zero CL shedding (∆D=0)
(7), is solved to determine the microgrid formation ℳ. Fig. 5 determined by the solution of the MF Stage 1 model (6). The

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two terms in Stage 1 aiming at the second goal (survivability) as fmin=59.3 Hz [42], and the microgrid with 40kW load has
are reflected as follows. Σgcg=10 kW/s and an inertia MH=9.4 kW∙s/Hz which is
First, the minimum scale of ℳ is obtained in Fig. 5 by converted from the empirical data provided in [38] using the
minimizing βTv, which can reduce the impact of the loss of same inertia-load ratio. The realization of cg and MH in the
microgrid components. For example, if microgrid 3 covered simulation model is discussed in Appendix A.
nodes 93,95,96 and MER was connected at node 96, then the The key LSS problem is to determine Lmax for each
loss of any line between nodes 96 and 91 would interrupt the switching step by (8.3), which is determined for Microgrid 9 as:
supply to the two CLs. Also, if non-CLs were connected, then
𝐿 2𝑀 𝑓 𝑓 ∙∑ 𝑐 √2 ∙ 9.4 ∙ 0.7 ∙ 10 = 11.5 kW (9)
any loss of lines connecting non-CLs would trigger an
imbalance issue (discussed earlier). which means that the frequency can be maintained above
Second, loops enhance the microgrid survivability by fmin=59.3 Hz when picking up a load that is less than 11.5 kW.
reducing the likelihood of any load supply interruptions. The Next, assuming that CLs are comprised of multiple 1-kW
three microgrids with loops (2,5,8) are re-drawn in Fig. 6 in switchable load blocks, we use (8.4) to determine the switched
which the loss of any lines in the loop of microgrid 2 will not load at each step as Lsw=11kW (which satisfies Lsw≤Lmax<Lsw+1
interrupt the CL service at node 29. In Fig. 6, arrows show line kW). Consequently, following the procedure stated in Section
flows (kW) and numbers in parentheses represent p.u. node V, the LSS in Microgrid 9 is executed in four switching steps
voltage drops (∆Vp.u.) where the p.u. values are based on V0. It with Lsw={11,11,11,7} kW, respectively.
can be seen that the voltage drop is very small in such small- Fig. 7a shows the microgrid frequency dynamics based on
scale microgrids. ∆Vp.u. will be even smaller in a DS with a the proposed LSS design. The results show that the microgrid
higher voltage level (V0), because the kV drop (∆VkV) will be frequency drops when the load is switched on. However, as the
smaller due to (3.2) and ∆Vp.u.=∆VkV/V0 will be further reduced rate-of-change-of-frequency (ROCOF) is limited by the
at a higher V0. microgrid inertia, the microgrid primary control ramps up the
MER to pick up the load as the frequency drops, followed by
the secondary control (in longer time horizon) to restore the
system rated conditions. Specifically, lowest frequency after
each LSS switching action is effectively maintained above
fmin=59.3 Hz. The first three actions have very similar
dynamics as they all pick up Lsw=11 kW, while higher
frequency is observed in the last action (Lsw=7 kW). The
dynamic performance is compared to that in the benchmark
case which switches on all CLs simultaneously in place of LSS
(see Fig. 7b), in which the frequency drops to approximately
Fig. 6. The kW flows and p.u. voltage drops in the three microgrids with loops. 52.5 Hz. This drop will be an issue for microgrid operations,
which will be worse when inverter-interfaced DERs with a
2) MF Stage 2. In Stage 2, solving (7) re-positions MERs and lower inertia are considered [43]. In another comparative case
shrinks microgrids. In Fig. 5, triangles show the final MER (see Fig. 7c), which manually divides the process into 10
connections, and gray-hatched areas are excluded by opening switching steps (4 kW for each step), the lowest frequencies are
branches ( 𝑣 =0) based on (7)’s solution mentioned earlier. higher those in Fig. 7a which is due to a lower load picked up
When solving (7) in Stage 2, the exclusion of the gray-hatched at each step. However, such a manually switching process is
lines reduces the second term in (7.1) which is the scale of ℳ. inefficient and will extend the restoration time (Fig. 7a vs. Fig.
On the other hand, opening gray-hatched lines in Stage 2 can 7c).
also reduce the impact of power balancing when microgrids Such an inefficiency will be further aggravated by longer
3,6,8 are split by the new outages of gray-hatched lines. The time intervals between adjacent switching actions. Here it can
contribution of MER’s optimal position is considered in the be seen that the proposed LSS determines the optimal Lsw and
following examples. In microgrid 1, the first term in (7.1) is thus properly strikes a balance between a required dynamic
1.31ηD with MER positions shown in Fig. 5. The corresponding performance and a fast restoration (i.e., larger load picked up at
value will increase to 1.83ηD if the 100kW/20kW MERs are a step and smaller number of switching steps). Furthermore, the
placed at nodes 16/12, and 2.27ηD if both are placed at node 16. recovery of frequency after each switching action will be
The above results show a general trend that these mobile affected by the governor ramp and microgrid
resources tend to be positioned at nodes where they can serve primary/secondary control functions [22]-[23].
CLs by energizing adjacent microgrids. In this way, additional
damages to the network under extended events are less likely to 60.1
60.0
interrupt restored CL services. On the other hand, serving a CL 59.9
Freq. (Hz)

59.8
through a long path (which implies larger microgrids and 59.7
MERs positioned farther from CLs) can be more risky under 59.6
59.5
such events. Note that such a trend is determined by the 59.4
59.3
optimal solution of the two-stage MF model whose objective 5 10 15 20 25 30 35 40 45 50
minimizes the loss-of-CL and considers the minimal network- Time (s)
scale for microgrids. The merit of the proposed strategy will be (a)
further demonstrated in Case 2.
3) LSS Process. After the microgrids are formed and black
started, the second CSR step is to determine LSS to pick up
CLs. Due to limited space, Microgrid 9 with a total CL of
40kW is selected for the LSS simulation. Frequency nadir is set

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61.0
60.0
Table I. Loss-of-CL indices in Case 1 vs. in the comparative case
59.0 Microgrids in Fig. 5 CL nodes Loss-of-CL indices (ΣsprsαTLSs)
58.0
Freq. (Hz)

57.0 (m) (i) Case 1 Comparative case

56.0
55.0 MG1 2 0.50 0.47
54.0
53.0 5 0.41 0.66
52.0 12 0.40 0.25
51.0
16 - 0.60
5 10 15 20 25 30 35 40 45 50
MG2 29 - 1.07
Time (s) 31 0.11 0.86
(b) MG3 88 - 1.65
92 0.93 2.24
60.10
MG4 53 0.32 1.28
60.05
56 - 0.63
Freq. (Hz)

60.00 MG5 37 - 1.24

59.95 45 0.31 0.27
59.90 MG6 80 - 0.55
59.85 MG7 85 - 2.10
59.80 MG8 74 - 1.61
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110 98 - 1.34
Time (s) MG9 111 0.16 0.79
114 - 0.95
(c) Total 3.14 18.54
Fig. 7. Simulated microgrid frequency dynamics with CLs picked up by: (a) the
proposed LSS, (b) a single-step switching, and (c) a manual 10-step switching. The enhancement of microgrid survivability is
demonstrated here which indicates that Case 1 has a lower loss-
Case 2. A Comparative Study of MF of-CL for all CLs except those in nodes 2,12,45, with a total
This case further demonstrates the microgrid survivability value of 3.14 vs. 18.54 in the comparative case. The two
enhancement by the proposed strategy (Fig. 3) using a small examples indicate that the CL at node 31 will be entirely
comparative case. Considering that [5] also simulated the interrupted when any branch between nodes 18 and 31 is on
microgrid formation on the IEEE 123-node test feeder, our outage in Fig. 8. However, it will only be lost when line 26-31
comparative case is constructed by resolving (6)-(7) using the is on outage in Fig. 5. The CL at node 74 is lost when any line
same resource-connecting nodes as in [5]. In another word, if between nodes 77 and 74 is on outage (Fig. 8). However, it will
node i connected resources in [5], then it is enforced with be served even under any line outages in the loop of microgrid
“Σg 𝑢 ≥1” in (6) and “Σg 𝑢 ≥1” in (7), meaning that it will 8 (Fig. 5). The comparison demonstrates the contributions of
connect at least one MER as depicted by a triangle in Fig. 8. minimal-scale microgrids, looped topologies, and optimal MER
Otherwise, Σg𝑢 =0 and Σg𝑢 =0 which points out that node i positions, to the lower loss-of-CL (i.e., enhanced microgrid
will have no MER connected. Also, ηλ=0 in (6.1) since the survivability), which are realized by the proposed two-stage
comparative case does not consider the merit of loops. Fig. 8 MF in extended extreme events.
depicts the final microgrid formation in the comparative case,
in which its loss-of-CL indices are compared with those in VII. CONCLUSIONS
Case 1 in Table I, where “Total” provides the value of The CSR strategy proposed in this paper integrates MF and
∑∀ 𝑝𝑟 𝜶𝑻 𝑳𝑺𝒔 in (7.1). The corresponding values for each CL LSS, and exploits the mobility, flexibility and resilience of
node are also listed in the table. microgrids to address the risk of service restoration in extended
extreme events. The MF enhances the microgrid survivability
by minimizing microgrid scales, applying radial or looped
topologies, and properly positioning MERs. The MF can
partition DS into multiple microgrids or cluster adjacent CLs to
form a microgrid. LSS further addresses the microgrid dynamic
performance. In CSR, adjacent CLs are likely to be clustered
into the same microgrid, while a CL located farther from other
CLs is more likely to be directly served by a local MER. In the
meantime, multiple loops are implicitly designed based on the
optimal MF solution. These measures enhance the microgrid
resilience and survivability, which is essential for serving the
critical loads in risky events.

APPENDIX A
This appendix presents the model for simulating the LSS
frequency dynamics in Case 1. The simulation is performed
using the modified governor (DEGOV) model [41] integrated
with microgrid primary and secondary control functions
depicted in Fig. 9. The default DEGOV parameters used in the
industry are provided in [41]. Considering that microgrids can
have smaller time constants, the parameters are modified to:
T1=1s, T2=T5=T6=0.2s, T3=T4=1s, Td=0.01s, K=1,
Tmax/Tmin=1/0 p.u., R=0.01， and m=1. The detailed microgrid
primary and secondary control functions are given in [2].
The governor ramp rate (cg=10 kW/s) and microgrid inertia
Fig. 8. Final MF solution in the comparative case. (MH=9.4 kW∙s/Hz) mentioned earlier are realized in the model

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Transactions on Power Systems

as follows. In practical applications, cg can be estimated from www.nerc.com/pa/CI/ESISAC/Documents/E-

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2018.2866099, IEEE
Transactions on Power Systems

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Liang Che (M’15) received the B.S. degree from Shanghai Jiaotong University,
China, in 2006 and the Ph.D. degree from Illinois Institute of Technology,
Chicago, IL, in 2015, both in Electrical Engineering. He was a power system
planning consultant at Siemens PTI, Minnetonka, MN, in 2015-2016. Presently,
he works as an EMS engineer at the Midcontinent Independent System
Operator (MISO), Carmel, IN. His research interests include microgrid
operations and planning, and power system security analysis.

Mohammad Shahidehpour (F’01) received an Honorary Doctorate from the

Polytechnic University of Bucharest, Bucharest, Romania. He is a University
Distinguished Professor and Bodine Chair Professor and serves as Director of
the Robert W. Galvin Center for Electricity Innovation at Illinois Institute of
Technology. He is a member of the US National Academy of Engineering, a
Fellow of the American Association for the Advancement of Science (AAAS),
and a Fellow of the National Academy of Inventors (NAI).

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