Applied Energy 199 (2017) 205–216

Contents lists available at ScienceDirect

Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

A resilient microgrid formation strategy for load restoration considering

master-slave distributed generators and topology reconfiguration
Tao Ding a,⇑, Yanling Lin a, Zhaohong Bie a, Chen Chen b
a
State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an 710049, China
b
Energy Systems Division, Argonne National Laboratory, Lemont, IL 60439, USA

h i g h l i g h t s

A resilient microgrid-forming model is set up considering master-slave DG operation.

The topology reconfiguration and microgrid-forming are coordinated in the model.

A mixed-integer second-order cone programming is employed to solve the model.

a r t i c l e i n f o a b s t r a c t

Article history: Recent severe power outages caused by extreme weather hazards have highlighted the importance and
Received 6 February 2017 urgency of improving the resilience of electric distribution grids. Microgrids with various types of dis-
Received in revised form 10 April 2017 tributed generators (DGs) have the potential to enhance the electricity supply continuity and thus facil-
Accepted 2 May 2017
itate resilient distribution grids under natural disasters. In this paper, a novel load restoration
optimization model is proposed to coordinate topology reconfiguration and microgrid formation while
satisfying a variety of operational constraints. The proposed method exploits benefits of operational flex-
Keywords:
ibility provided by grid modernization to enable more critical load pickup. Specifically, a mixed-integer
Resilient distribution network
Topology reconfiguration
second order cone programming is employed to reduce the computational complexity of the proposed
Microgrid optimization with optimality guaranteed. Finally, the effectiveness of the proposed method has been ver-
Master-slave control ified on an IEEE 33-bus test case and a modified 615-bus test system.
Mixed-integer second-order cone Ó 2017 Elsevier Ltd. All rights reserved.
programming

1. Introduction optimal operation. Reconfiguration has become a common feature

in urban distribution systems, and has contributed greatly to
Electrical power production is the large-scale conversion pro- enhancing power system reliability by redirecting faulted areas
cess of transforming different types of primary energy into easily to alternative supply sources through sectionalizing switches
transportable electrical energy. In particular, the distribution sys- within a feeder or tie switches between feeders [3–13].
tem is at the end of the whole power system, and directly affects In recent years, the world has witnessed several natural disas-
the power supply quality and reliability seen by customers. Statis- ters and resulting severe power outages and blackout. For example,
tics from electric power companies show that more than 80% of a hurricane hit Zhanjiang, China in October 2015, causing a loss of
power outages are caused by faults in the distribution network 4.24 million kW h of energy. In this scenario, the substations may
level [1]. Hence, load restoration from outages for resilient opera- be at fault so that the distribution system cannot be supplied by
tion is the core function of the distribution network, and it has the main grids, or the damages to the distribution facilities may
great significance for serving customer demand and improving lead to several isolated areas without power. Thus, the traditional
the reliability of the power supply [2]. With the development of load restoration approaches (e.g., [3–13]), which rely entirely on
information technology and distribution automation, remote con- reconfiguration, may not guarantee supply continuity of energy
trol enables fast switch operations, allowing the distribution sys- after natural disasters, and thus customers may experience
tem topology to adapt quickly to isolate faults and achieve extended outages [14].
Today’s power system is transiting from a centralized bulk sys-
tem to smart grid, with the distribution system being smarter and
⇑ Corresponding author. more active by the integration of distributed generators (DGs) [15].
E-mail address: tding15@mail.xjtu.edu.cn (T. Ding).

http://dx.doi.org/10.1016/j.apenergy.2017.05.012
0306-2619/Ó 2017 Elsevier Ltd. All rights reserved.
206 T. Ding et al. / Applied Energy 199 (2017) 205–216

Nomenclature
Sijmax the maximum capacity of the branch ij
Indices and sets Iijmax the maximum branch current of the branch ij
i, j, s, k index of buses N the number of substations
ij index of branch from bus i to bus j R the number of load islands
E set of lines in the network w weight of load
P set of DGs M a big number
Eo set of lines in the open state aj the power factor angle of the j-th load demand
V set of nodes in the network
H set of master DGs Decision variables
d(j) set of all children of bus j yij binary variable for line ij. If the line is open, yij = 0;
p(j) set of all parents of bus j otherwise, yij = 1
|E| the numbers of lines Hij the active power flow on distribution line ij
|V| the numbers of nodes Gij the reactive power flow on distribution line ij
|P| the number of DGs PDG,j the active power output of DG j
|H | the number of master DGs QDG,j the reactive power output of DG j
rij the resistance of branch ij PL the active load demand under faulted condition
xij the reactance of branch ij QL the reactive load demand under faulted condition
bs,j the charging capacitance connected to bus j lij the square of branch current on line ij
P0L the active load demand under normal condition ui the voltage magnitude square on bus i
Q0L the reactive load demand under normal condition Fij the fictional flow on the distribution line ij
P0DG,j the j-th DG output under normal condition Wj the power supplied by the ‘‘source” buses in the ficti-
U0j the given voltage magnitude on master DG bus j tious network
SDG,jmax the maximum capacity of the DG j
hj the maximum power factor angle of the j-th DG

DGs can make the distribution system more diverse, flexible, and (i) Different types of control strategies for DGs may lead to dif-
secure. DGs are increasingly integrated in the distribution network ferent operation rules for system restoration. Originally,
due to their benefits such as loss reduction, voltage stability, sys- droop-control-based methods were widely adopted for
tem reliability enhancement and lowered global warming [16– DGs in microgrid, which does not require communication
19]. In addition to their ability to satisfy the increasing energy among DGs for effective grid control. However, droop con-
demand, DG intentional islanding is gradually recognized as an trol faces the problem of circulating current among DGs
essential capability in providing the load in contingency, which is because it uses a voltage loop at each DG node [36]. Subse-
further validated by IEEE 1547.4 [20]. It is found in many studies quently, the master-slave control technique was deployed
DGs can be used to enhance load restoration, providing an alterna- to solve the above problem; in this technique, the voltage
tive way to improve distribution network resilience [14]. The effi- and frequency of the system are controlled by only one gen-
cient way to manage a power system with significant level of DGs eration unit, which serves as the master unit, and the rest of
is to break the distribution system into small clusters or microgrids the DGs work in current control mode and serve as the slave
[21–23]. Thanks to the microgrids powered by the DGs, the supply units. The master unit can be a diesel generator, storage
to the customers can still be guaranteed, even for isolated areas device or DG with large capacity, etc. In contrast, DG units
[24–30]. based on renewable energy, such as solar and wind, are usu-
On the basis of this idea, the concept of ‘‘resilience” in distribu- ally chosen as the slave units. However, the existing studies
tion network was proposed in [31–35] to restore the system after on service restoration haven’t considered the DG control
natural disasters by use of microgrids. Ref. [31] built a model suit- strategies.
able for re-configuration of a distribution system with microgrids. (ii) The network reconfiguration and the control strategy of the
Once a fault occurs in a distribution system, some DG-based microgrids haven’t been coordinated in the previous works.
islands will be formed to guarantee the power supply of important The existing studies are based on either reconfiguration [1–
customers. Ref. [32] studied multi-agent coordination for dis- 8] or microgrids [31–35]. To our best knowledge, how to
tributed information discovery, but the restoration process did coordinate the reconfiguration and microgrids should be
not consider the topology of the distribution network. Ref. [33] dis- investigated.
cussed a spanning tree method for distribution network restora-
tion with embedded microgrids to enhance the self-healing To address the above two shortcomings identified in the previ-
capability. However, this method only considered a single fault in ous research, this paper proposes a new resilient microgrid forma-
the distribution network. When natural disasters occur, multiple tion strategy for load restoration with both topology
faults could lead to several unsupplied, isolated islands. An opera- reconfiguration and master-slave DG control framework. The main
tional approach to restore loads after natural disasters was given in contributions can be summarized as follows:
[14], where multiple microgrids were dynamically formed to con-
tinue supplying critical loads. Ref. [34] reviewed the contribution (i) A resilient microgrid-forming model is formulated consider-
of reconfiguration in reducing load shedding. Ref. [35] presented ing master-slave DG operation, where there is only one mas-
the reconfiguration scheme for minimal load shedding considering ter DG in each island to guarantee a self-adequate system.
soft-open points. (ii) The topology of the whole system can be reconfigured by
The above references presented sound results and investigated sectionalizing and using tie switches, such that the load at
the basic framework of resilience in distribution networks, but one feeder can be transferred to another feeder in the
there are still two points that haven’t been addressed: microgrid-forming model to pick up more loads.
 T. Ding et al. / Applied Energy 199 (2017) 205–216 207

(iii) A mixed-integer second-order cone programming (MISOCP) of microgrids is equal to the number of master control units. That
relaxation is employed to solve the proposed model. means, |H| microgrids will be formed for restoration.

The rest of the paper is organized as follows: Section 2 intro- 2.2. Modeling of the radiality with the network topology
duces the modeling of the restoration by microgrids with consider- reconfiguration constraints
ation of the master-slave control framework and network topology
reconfiguration. Section 3 presents the simulation results of the Traditionally, to guarantee radiality in the distribution reconfig-
proposed method on 33-bus and 615-bus test systems. Finally, uration, graph-theory-based methods are proposed to eliminate
conclusions are drawn in Section 4. unconnected buses and loops. For example, the branch exchange
method was used in [37] to simultaneously open and close a pair
of switches within one loop to maintain radiality; all switches were
2. Modeling of restoration by microgrids
closed and then opened them one by one to form a radial network
in [38]; and a spanning tree search algorithm was proposed in
When multiple faults from natural disasters isolate parts of the
[39,40] to find the optimal radial topology.
distribution system into unsupplied islands, traditional distribu-
Theoretically, a necessary and sufficient condition for radiality
tion system restoration approaches that only change system topol-
was proposed in [41], i.e., the graph is radial if and only if the fol-
ogy cannot guarantee restoration of the energy supply. However, a
lowing two conditions are satisfied: (a) the number of closed
promising approach is to intentionally divide the distribution sys-
branches equals the number of buses minus the number of sub-
tem into several microgrids by means of sectionalizing switches
graphs, and (b) the connectivity of each sub-graph is guaranteed.
and DGs to continue supplying critical loads, while ensuring vari-
To achieve the first condition, the number of sub-graphs should
ous constraints in each island. A DG-based microgrid is a localized
be acquired at first. After a natural disaster, the faulted lines are
grouping of small DG units and loads to guarantee a self-sufficient
opened which result in three isolated areas, as shown in Fig. 1.
system. Moreover, it should be noted that for the master-slave con-
However, as shown in Figs. 2 and 3, with the consideration of the
trol framework, only one DG acts as the master unit that sets the
network reconfiguration and microgrids, the sub-graphs could be
voltage and frequency of the microgrid, while other DG units are
of three types—microgrids supplied by DGs, buses supplied by sub-
slave units that follow the set voltage and frequency.
stations, and unsupplied load islands—so condition (a) above will
To meet the above requirements, the following constraints
be represented by the following equality constraint:
should be satisfied for forming microgrids in case of severe natural X
disasters: yij ¼ jVj  jHj  N  R; ð2Þ
ij2E
(a) Distribution System Condition Constraints: The state of
It can be observed from Figs. 2 and 3 that N is actually a prede-
switches and the microgrid control mode should be consid-
termined parameter, but R is related to the locations of faulted
ered for each microgrid.
lines, tie switches and DGs.
(b) Radiality with Network Topology Reconfiguration Constraints:
After a natural disaster, the system can pick up load through
The distribution system is operated radially, and the micro-
network reconfiguration and microgrids powered by DGs. As a
grids should also adopt a radial topology. Moreover, the
result, the number of isolated load islands can be determined by
topology of microgrids can be further reconfigured by open-
closing all the tie lines to form a new mesh network where substa-
ing and closing sectionalizing switches
tions and DGs are considered. For example, in Fig. 3, close all the tie
(c) Microgrid-forming Constraints: The distribution system
lines and a new mesh network can be constructed, in which Area 1
should be split into several microgrids. For each microgrid,
is a load island and R = 1. Similarly, in Fig. 2, close all the tie lines
the power flow constraints should be satisfied.
and a new tree network will be constructed, in which there is no
(d) Power System Physical Constraints: DG generation limits, line
load island and R = 0.
capacity limits, and voltage magnitude restrictions should
Generally, finding the number of load islands in a graph (i.e.,
be met for each microgrid.
determining R) is equivalent to finding the number of connected
components that only contain load buses. A connected component
2.1. Modeling of the distribution system condition constraints of an undirected graph is a connected sub-graph of the graph [42].
If an undirected graph is a connected graph, there is only one con-
Let G = (V, E) be a connected and undirected graph with set of nected component. If there are multiple connected components, a
vertices and edges denoted by V and E respectively. After a natural traversal algorithm can be employed, either depth-first or breadth-
disaster, some lines will be at fault and a substantial amount of first, to find connected components of an undirected graph. After a
time will be needed for repairs. Therefore, these lines should be traversal starting from a vertex, all the vertices that can be reached
in the open state. Let these lines be defined by the set EO , where from this vertex will be visited. If there are other connected com-
EO  E. Let the binary decision variable yij represent the status of ponents, there will still be unvisited vertices after the traversal is
distribution line ij. If the line is open, yij = 0; otherwise, yij = 1. complete. Starting from one of those unvisited vertices, another
Therefore the constraints regarding the binary decision variables connected component can be found. If continuing this procedure
based on the actual conditions of the distribution system lines until all vertices are visited, all the connected components can be
can be written as follows: identified. Table 1 shows the flowchart of the algorithm finding
yij ¼ 0; 8ði; jÞ 2 EO : ð1Þ all connected components of a graph.
According to Table 1, R can be determined and condition (a) can
It is assumed that all lines that are in service (i.e., 8ði; jÞ 2 E n EO ) be satisfied by constraint (6) below. However, considering only
are equipped with a switch. Both the microgrid formation and condition (a) cannot sufficiently guarantee the radiality. As shown
topology reconfiguration will employ open/close operation of the in Fig. 4, the microgrid after reconfiguration may be disconnected
switches. or meshed, or may contain several master units in one microgrid,
In addition, the master-slave control mode in the microgrid so condition (b) to guarantee the network connectivity and the
suggests that only one DG is serving as a master control unit and constraint of only one mater unit being in one microgrid should
the other DGs are serving as slave control units. Thus, the number be satisfied, too.
208 T. Ding et al. / Applied Energy 199 (2017) 205–216

Area 1

Area 2

Master Control DG

Substation

Area 3

Fig. 1. Topology after natural disasters.

Area 1

Area 2

Master Control DG

Substation

Area 3

Fig. 2. Network reconfiguration without the load island.

Area 1

Area 2

Master Control DG

Substation

Area 3

Fig. 3. Network reconfiguration with the load island.

Table 1
Algorithm for finding connected components.
In order to enforce the subgraph connectivity via mathematical
programming formulations, the single commodity flow method Step 1 for i = 1: |V|
[43] is employed in this paper. In essence, a fictitious network is Step 2 visited[i] = false; compNum = 0;
constructed, where each microgrid has only one ‘‘source,” and all Step 3 for v = 1: |V|
other buses are ‘‘sink” buses that have unit load demands (1.0 p. Step 4 if (!visit[V])
Step 5 compNum + + ; q = [];
u.). Since each microgrid is only allowed to contain one master
Step 6 q.enqueue(v);
control unit, the master control unit should be chosen as the Step 7 visited[v] = true;
‘‘source” bus. Then, the connectivity can be expressed by the fol- Step 8 while (! q.isempty())
lowing constraints: Step 9 w = q.dequeue();
X X Step 10 for each edge from w to vertex k
F js  F ij ¼ 1; j2 VnP ð3Þ Step 11 if (! visited [k])
s2dðjÞ i2pðjÞ Step 12 visited [k] = true;
Step 13 q.enqueue(k);
X X Step 14 end
F js  F ij ¼ W j ; j2P ð4Þ Step 15 end
s2dðjÞ i2pðjÞ Step 16 end
Step 17 end
Myij 6 F ij 6 Myij ; 8ij 2 E ð5Þ Step 18 end
Step 19 end
q.enqueue() adds an item to the queue.
Mð2  yij Þ 6 F ij 6 Mð2  yij Þ; 8ij 2 E ð6Þ q.dequeue() removes an item from the queue and returns the value.
 T. Ding et al. / Applied Energy 199 (2017) 205–216 209

W j P 1; j2P ð7Þ Intuitively, if branch (i, j) is opened, the branch current and
branch flow should be zero, such that Hij = 0, Gij = 0 and lij = 0;
Since the fictitious network has the same topology structure as
meanwhile, the voltage magnitudes between the two buses need
the original distribution network, they have the same connectivity.
not be restricted by the third equation in (8). The formation of
Thus, if the power balance constraint is guaranteed in the fictitious
microgrid primarily needs to find a cut set of the network. The
network, it suggests that for any sink bus, there exists at least one
edges belonging to the cut set should be open, and the other edges
path from this sink to a source bus, so that each microgrid should
remain closed. Thus, the splitting constraints for the power flow
be connected; otherwise, the microgrid will remain disconnected.
equations of each microgrid can be given as:
The connectivity constraints can be illustrated via an example in X X
8
Fig. 5. The power in microgrid 1 can be balanced by satisfying those >
> PDG;j  PL;j ¼ Hjk  ðHij  r ij lij Þ; 8j 2 V
constraints, so microgrid 1 is connected. The load demand at buses < k2dðjÞ i2pðjÞ
X X ð9Þ
#7–#10 in microgrid 2 cannot be satisfied, implying that there is no > ðGij  xij lij Þ þ bs;j uj 8j 2 V
: Q DG;j  Q L;j ¼
> Gjk 
path between these buses and the source denoted by Source2. k2dðjÞ i2pðjÞ

(
2.3. Modeling of the microgrid-forming constraints 2ðr ij Hij þ xij Gij Þ þ ðr 2ij þ x2ij Þlij if ði; jÞ 2 E \ yij ¼ 1
uj  ui ¼
The microgrid-forming problem can be viewed as a graph parti- arbitrary otherwise
tion problem [44]. A split in G is a partition of V with several split ð10Þ
sets. For instance, splitting G into Np sub-graphs Gi ¼ ðVi ; Ei Þ can (
be represented by G ¼ G1 [ G2 . . . [ GNp , and Gi \ Gj ¼ £ for 8i–j, 2 ½Smax max
ij ; þSij  if ði; jÞ 2 E \ yij ¼ 1
which can also be expressed as V ¼ V1 [ V2 . . . [ VNp , Vi – £, Gij or Hij : ð11Þ
¼0 otherwise
and Vi \ Vj ¼ £ for 8i – j. Here, the cut set ðV1 ; V2 ; . . . ; VNp Þ is
defined as a set of edges Eb # E with one end vertex in Vi and (
max
2 ½0; lij  if ði; jÞ 2 E \ yij ¼ 1
another end vertex in Vj , i – j [43]. lij : ð12Þ
For each microgrid, the power flow constraint should be satis- ¼0 otherwise
fied. It has been well recognized that a distribution network is often However, constraints (10)–(12) are ‘‘if-else” type constraints,
a radial network, and the power flow can be formed by a set of implying that only one constraint can hold. However, these con-
recursive equations, called branch flow formulation, which yields: straints can be equivalently transformed into a set of affine con-
8 X X
> PDG;j  PL;j ¼ Hjk  ðHij  rij lij Þ; 8j 2 V straints using the big-M approach [43,45] as follows:
>
> 8
>
> i2pðjÞ Mð1  yij Þ 6 uj  ui þ 2ðr ij Hij þ xij Gij Þ  ðr 2ij þ x2ij Þlij 6 Mð1  yij Þ
>
>
k2dðjÞ
X X >
>
<
>
<Q
DG;j  Q L;j ¼ Gjk  ðGij  xij lij Þ þ bs;j uj 8j 2 V Smax yij 6 Hij 6 Smax yij ;Smax yij 6 Gij 6 Smax yij ; 8ði; jÞ 2 E
i2pðjÞ ð8Þ >
>
:
ij ij ij ij
>
>
k2dðjÞ max 2
0 6 lij 6 ðIij Þ yij
>
>
>
> uj ¼ ui  2ðr ij Hij þ xij Gij Þ þ ðr2ij þ x2ij Þlij ; 8ði; jÞ 2 E
>
> ð13Þ
: 2
Hij þ G2ij ¼ lij ui ; 8ði; jÞ 2 E

Master Control Unit

Fig. 4. Notional system topology.

v3
v8 v9
v1 v2
v4 v10
Microgrid1 v7
Source1 Microgrid2
Source2
v5
v11
v6 v13

Fig. 5. Connectivity constraints of the fictitious network.

210 T. Ding et al. / Applied Energy 199 (2017) 205–216

1 1
x S1 X
S18 2 S18 2
S2 S2
19 3 S22 19 3
S19 \\S3 \\ S19 S3
C 20 4 C 20 20 4
S20 S4 C 23 S20 S4 C 23
21 5 S23 21 21 5 S23
S21 S5 24 \\ S5 24
22 6 S24
22 6 \\
\\S6 S25 25 \\ 25
7 S33 7
S7 S37 S7 26 S37
26 8
8 S26 S26
S8 27 S8 27
9 S27 S35 9 S27
S9 28 S9 28
10 S28 10 S28
S10 S10 29
29 11
11 S29 S29
S11 C 30 \\S11 C 30
12 S30 12 \\
S12 31 S12 31
C C 13 S31
13 S31 S13
S13 32 32
14 14 S32
S32 S14
33 15 33
15 S36 S36
S15 S15
16 16
S16
S16 17
17
S17
S17
18
18

(a) non-reconfigurable (b) reconfigurable

1 1
x S1 X S1
S18 2 S18 2
S2 S2
19 3 S22 19 3
S19 S3 \\ S19 S3
C 20 20 4 C 20 20 4
S20 S4 C 23 S20 S4 C 23
21 21 5 S23 21 21 5 S23
S21 x S5 24 \\ X S5 24
22 6 S24 22 6 \\
S25 S25
S6 x 25 S6 X 25
7 S33 7
S7 26 S7 26
8 S26 8 S26
S8 27 S8 27
9 S27 S35 9 S37
S27
S9 28 S9 28
10 S28 10 S28
S10 29 S10 29
11 S29 11 S29
S11 C 30 \\ C 30
12 xS30 12 XS30
S12 31 S12 31
C 13 S31 C 13 S31
S13 32 S13 32
14 S32 14 S32
S14 S14
15 33 15 33
S36 S36
S15 S15
16 16
S16 S16
17 17
S17 S17
18 18

(c) non-reconfigurable (d) reconfigurable

Fig. 6. Comparison of different strategies (see Table 2) applied to IEEE 33-bus system.
 T. Ding et al. / Applied Energy 199 (2017) 205–216 211

Table 2
Loads restored by different strategies depicted in Fig. 6.

Scenario Computational time (s) Gap (MW) Restored load (MW) Ratio (%)
5
(a) 0.0862 2.1681  10 2.3418 63.03
(b) 36.7805 2.4676  105 2.3692 63.78
(c) 0.0134 2.9157  105 2.1061 56.69
(d) 0.1989 2.4676  105 2.3692 63.78

H2ij þ G2ij ¼ lij ui ; 8ði; jÞ 2 E; ð14Þ mate optimal solution, which can be obtained by reformulating the
problem in terms of its continuous-variables convex second-order
If yij = 1, the first constraints of (10)–(12) hold; otherwise, the last cone programming [46–50]. Thus, the whole problem can be for-
constraints hold. mulated as a MISOCP.
Specifically, for the quadratic equalities (14), conic relaxation is
2.4. Modeling of distribution system physical constraints performed by relaxing the quadratic equalities into inequalities.
Thus, the relaxation yields
Distribution system physical constraints guarantee that the
power generation of DGs, voltage magnitudes, and load shedding H2ij þ G2ij 6 lij ui ; 8ði; jÞ 2 E ð20Þ
should be restricted within their physical limits: Furthermore, (14) can be reformulated as a standard second-
8  9
>  jQ DG;j j 6 PDG;j tanðcos1 hj Þ > order cone formulation, such that
>
<  >
=  
  2Hij 
ðPDG;j ; Q DG;j Þ P2DG;j þ Q DG;j 6 ðSDG;j Þ
max 2
; 8j 2 P
2
ð15Þ  
>
>  >
>  
:  0 6 PDG;j 6 P0 ;  2Gij  6 lij þ ui ; 8ði; jÞ 2 E ð21Þ
DG;j  
 lij  ui 
2
2 2
ðU min
j Þ 6 uj 6 ðU max
j Þ ; 8j 2 V n H ð16Þ Then, the model can finally be reformulated as
X
2
max wi PL;i ð22:aÞ
uj ¼ ðU 0j Þ ; 8j 2 H ð17Þ j2V

s:t: ð2Þ—ð7Þ; ð9Þ; ð13Þ; ð15Þ  ð18Þ; and ð21Þ ð22:bÞ

(
0 6 PL;j 6 P0L;j
; 8j 2 V ð18Þ
Q L;j ¼ PL;j tanðcos1 aj Þ 3. Numerical results
where constraint (15) describes the feasible region of each DG; con-
straints (16) and (17) specify the voltage magnitude limit and con- In this section, the proposed method is tested on a modified
straint (18) implies that the active and reactive power should be IEEE 33-bus system and a 615-bus test system composed of five
curtailed at the same time and the power factor of each load should IEEE 123-bus systems. The test is carried out on a personal com-
be maintained. puter with an IntelÒ CoreTM i5 Duo Processor T420 (2.50 GHz) and
Finally, let wj denote the priority weight associated with the 4 GB of RAM (32-bit system) using CPLEX 12.5. In the simulation,
load at bus j. Then, the proposed model can be explicitly formu- a reasonable voltage magnitude is assumed to stay within the
lated as (19) to maximize the total priority-weighted loads picked range [0.90, 1.10] p.u. The voltages of the master generators in
up, such that island operation are kept at 1.0 p.u. and the priority weight of all
X load is chosen the same value.
max wi PL;i ð19:aÞ
j2V 3.1. IEEE 33-bus test system
s:t: ð2Þ—ð7Þ; ð9Þ; ð13Þ—ð18Þ ð19:bÞ
The 12.66-kV 33-bus radial distribution test system is adopted
from a standard IEEE 33-bus test system. Four master generators
2.5. Convex relaxation for solution methodology are installed at buses 13, 20, 23, and 29, respectively. The capacity
of each generator is 0.5 MVA; 5 photovoltaic (PV) panels are
The proposed model (19) is actually a mixed-integer nonlinear installed at buses 9, 15, 18, 27, and 33, respectively. The individual
nonconvex programming (MINNP) problem, owing to the fact that capacity is 0.15 MVA. Two disaster scenarios are deployed for this
the equality constraints (14) are nonlinear. Unlike the mixed- test system, and each fault location is indicated with a red cross in
integer convex programming model, which can be generally solved Fig. 6: (1) a single fault occurs at the main transformer, and the
by the branch-and-bound method, cutting plane, etc., the MINNP is optimal microgrid-forming solutions with/without reconfiguration
very hard to handle, since the global optimality of the sub-problem are shown in Fig. 6(a) and (b); (2) multiple faults occur within the
obtained by relaxing the integrality constraints is challenged. To system, and the optimal solutions with/without reconfiguration
deal with this problem, convex relaxation approaches are widely are shown in Fig. 6(c) and (d). In this paper, non-reconfigurable
utilized in optimal power flow problems to find a ‘‘good” approxi- system indicates that the system has no tie switches among feed-
Table 3 ers, while a configurable system retains the additional tie switches
Loads restored under different PV capacity. from the test system.
Scenario PV capacity Restored Load (MW) Ratio (%)
(A) Effect of reconfiguration on load restoration
(a) 1/2 of the original 2.1236 57.16
(b) 2 times of the original 2.8498 76.71 In all the scenarios, the system is cut off from the main grid, and
(c) 3 times of the original 3.2349 87.08 microgrids are formed by opening/closing line switches, each pow-
(d) 4 times of the original 3.5269 94.94
ered by one master generator. The total DG installation capacity,
212 T. Ding et al. / Applied Energy 199 (2017) 205–216

1
X S1 1
S18 2 X S1
S18 2
S2
19 3 S2
S22 3
S19 S3 // 19
//
S22
C 20 S19 S3
20 4 C 20 20
// S20 S4 C 23 4
21 21 // S20 S4 C 23
5 S23 21 21 5 S23
S21 S5 24 S5 24
S21
22 6 //S24 22 6 //S24
//S6 S25 S25
25 //S6 25
S33 7 S33 7
S7 26 S37 S7 S37
8 26
S26 8 //S26
S8 27 S8 27
S35 9 S27 S35 9 S27
S9 28 S9 28
10 S28 10 S28
S10 29 S10 29
// 11 //S29 // 11 S29
S11 C 30 S11 C 30
12 S30 12 S30
S12 31 //S12 31
C 13 // S31 C 13 S31
S13 32 S13 32
14 14 S32
S32 S14
S14 33
15 33 15 S36
S36
// S15
16 16
S16 //S16
17 17
S17 S17
18 18

(a) PV capacity: 1/2 of the original (b) PV capacity: 2 times of the original

1 1
X S1 X S1
S18 2 2
S18
S2 // S2
19 3 S22 3
// 19 S22
S19 S3 S19 S3
C 20 20 4 C 20
20 4
// S20 S4 C 23 S20 S4 C 23
21 21 5 S23 21
// S21 // S5 24
21 5 //S23
// S21 // S5 24
22 6 //S24 22 6
//S6 S25 25 S25 S24
S33 7 S6 25
S33 7
S7 26 S37 S37
8 // S7 26
S8
S26 8 //S26
27 S8 27
S35 9 S27 S35 9 S27
S9 28 S9 28
10 S28 S28
S10
10
29 S10 29
11 S29 11
//S11 C S29
30 //S11 C 30
12 //S30 12
S12 S30
31 S12 31
C 13 S31 C 13 S31
S13 32 S13
14 32
S32 14 S32
S14
15 33 //S14
S36 15 33
S15 S36
S15
16 16
S16 S16
17 17
S17 S17
18 18

(c) PV capacity: 3 times of the original (d) PV capacity: 4 times of the original
Fig. 7. Comparison of DG islanding under different PV capacity in IEEE 33-bus system.

including PVs, in the 33-bus system is 2.75 MVA, and the maxi- extreme disasters break the system into different parts, and those
mum possible restored load is 74.02% of the total active load. A without a controllable DG will not be supplied. For example, a fault
comparison of loads restored by different strategies is given in at line #30–#31 (i.e., S30) isolates loads without a controllable DG,
Table 2. For a single fault that occurs at the root bus, the microgrids and the load must be lost without reconfiguration, as seen in
can pick up 63.08% of the load, and with the help of topology Fig. 6c. However, with a reconfigurable tie switch between bus
reconfiguration, an additional 0.7% of the load will be restored. In 18 and 33 (Fig. 6d), isolated load buses can be reconnected to
contrast, for multiple faults, an additional 12.5% load is restored another island and supplied by the DG at bus 13.
by coordinating topology reconfiguration with microgrid forma- On the other hand, it can be found from Table 2 that the pro-
tion. That is because the multiple faults that could result from posed method that integrates topology reconfiguration and micro-
 T. Ding et al. / Applied Energy 199 (2017) 205–216 213

Table 4
Resilience strategies for distribution system.
Substation 1 Substation 2
Strategy Sectionalizing Switches Tie switches
① – –
p
② –
p
③ –
p p
④

Table 5
Restoration of loads by different strategies.

Faults ① ② ③ ④
F1 69.63% 78.80% 81.81% 82.09%
F1/F2 69.63% 78.37% 79.00% 81.02%
F1/F2/F5 69.63% 77.94% 78.98% 80.65%
F1 + internal 63.32% 73.07% 75.21% 81.23%
Fig. 8. 615-bus test system. F1/F2 + internal 62.46% 71.68% 72.79% 77.94%
F1/F2/F5 + internal 61.89% 69.48% 71.79% 75.28%

grid formation takes more computational time than the method

without topology reconfiguration. This is because the feasible
region will become larger when topology reconfiguration is
(B) Effect of DG capacity on load restoration
considered.
To evaluate the accuracy of the SOCP relaxation, a gap (relax-
In this section, the DG capacity of the system is changed to
ation error) is defined as
investigate the effect of PV capacity on the final reconfiguration
Gap ¼ max jH2ij þ G2ij  lij ui j ð23Þ result. As indicated by Table 3, the restored load is increasing stea-
8ði;jÞ2E dily as the capacity of the PV installation increases. In addition, the
The maximum gap of conic relaxation on the test system is DG islanding schemes under scenarios of different PV capacity are
listed in the Gap column in Table 2; it is obvious that the gap on sketched in the following figure. The fault is assumed to take place
any test system is smaller than 104, which implies that the conic in line #1. The effect of the PV capacity is evident in the result.
relaxation is actually an exact match to the original nonconvex From (a) to (d) in Fig. 7, the most significant change is about the
model to guarantee the optimality. DG island supplied by DG at bus 2. It can be seen that the DG island

30 122
29
32 111
48 47 49 50 51 116 115
28 109 110
33 31 108
x

26 25 44 45 46
27 106 107 112
43 105
24 23 42 66 65 64 113
39 102 103 104
22 21 41 101
114
40 38
63
20 19 117
18
x 62 98 99 100 118
9 121 35
11 14 37 97
2 59 58 57 60 119 67 68 69 70 71
x
1 7 8
x123 13 54 55 56
72 73 74 75
120 52 53
3 34 77
x

96 94 92 90 88 78 79
5 6 15 17 76
4
85
80
16
95 93 91 89 87 86 84
81

82
83

Fig. 9. IEEE 123-bus test system.

214 T. Ding et al. / Applied Energy 199 (2017) 205–216

formed by this DG is becoming smaller as PV capacity increases. In and moreover, some parts of the lost load can be picked up by
(a), this DG island supplies load at bus 18–19, 2–6, and 26–28. Due nearby feeders through the tie switches.
to limited capacity of the PV in this scenario, this island is formed To compare the traditional intentional-microgrids method
to supply more loads, even though the long distance within the without reconfiguration, four resilience strategies, designated ①
island may cause voltage problem. As the PV capacity increases, through ④, are described in Table 4. The strategies involving topol-
the load at bus 26–28 are then supplied by the controllable DG ogy reconfiguration include both tie switches among feeders and
nearby. As can be seen, the increased PV capacity increase the flex- sectionalizing switches within feeders. In Strategy ①, topology
ibility of the DG islanding scheme. reconfiguration is not considered; in Strategy ②, only the section-
alizing switches within feeders are considered; in Strategy ③, only
3.2. 615-bus test system the tie switches among feeders are considered; in Strategy ④,
which is the method proposed in the present paper, both tie
The 615-bus test system (Fig. 8) is composed of five IEEE 123- switches and sectionalizing switches are considered.
bus systems, designated F1 through F5, each of which is described The ratios of restored load to the total lost load for all fault sce-
by the single line diagram shown in Fig. 9. The load of each feeder narios under the different resilience strategies are summarized in
is 3.49 MW + 1.92 MVar. The capacities of substations 1 and 2 are Table 5. It can be observed that the picked-up loads are limited
set to be 15 MVA and 10 MVA, respectively. There are four addi- by the capacity of DGs, so the lost load cannot be 100% restored.
tional switches among the feeders, which could pick up loads after However, in addition to microgrid, reconfiguration by both tie
failures, as shown in Fig. 9. The total capacity of the installed DGs, switches and sectionalizing switches can improve the performance
including PVs, is 3.15 MVA, or 79% of the total active load. Therein, of load restoration. Comparing ② with ①, the reconfiguration of
8 DGs are set as the master control units and the others are PV sectionalizing switches increases the restored load from 11.93%
units which are slave control units. to 15.40%, while comparing ③ with ①, the reconfiguration of tie
Six fault scenarios are considered where the faults occur either switches increases the restored load from 13.43% to 18.78%.
at the feeder bus only, or within internal feeders. For the faults at These results show that reconfiguration by tie switches yields
the feeder buses only, three scenarios are studied, with the faults 3–4% more restored loads than reconfiguration by sectionalizing
occurring at Feeder 1, Feeders 1 & 2, and Feeder 1, 2, & 5, respec- switches in the test cases. Moreover, the comparison between
tively. For the faults both at the feeder buses and within internal strategy ③ and ④ shows that for the faults at the feeder buses,
buses, another three scenarios are set up with faults occurring at the restored loads mainly come from the tie switches and micro-
the above feeder buses and within feeders (see Fig. 9). grids, and the effect of sectionalizing switches is slight (no more
With the help of topology reconfiguration, the strategies of DG than 3%). This is because the lost loads are supplied mainly from
formulation for 615-bus systems to deal with faults are manifold: the microgrids, and additional loads that cannot be supplied by
when faults occur, the restoration can be implemented by inten- the microgrid will be picked up by the nearby feeders from the
tional microgrid formation with master-slave control frameworks, tie switches.

F1
F2 F1
1.00 F2
F3 1.00
F3
0.98 F4
Voltage(p.u.)

F4
Voltage (p.u.)

0.98
F5
0.96 F5
0.96
0.94
0.94

0.92 0.92

0.90 0.90

0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

Bus no. Bus no.
(a) voltage profile for strategy (b) voltage profile for strategy
F1
F1
1.00 F2
F2
F3 1.00
F3
F4
Voltage (p.u.)

0.98
F4
Voltage (p.u.)

0.98
F5 F5
0.96
0.96
0.94 0.94

0.92 0.92

0.90 0.90

0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

Bus no. Bus no.
(c) voltage profile for strategy (d) voltage profile for strategy
Fig. 10. voltage profiles with faults at root buses of feeder 1 & feeder 2.
 T. Ding et al. / Applied Energy 199 (2017) 205–216 215

However, for the faults within the feeder buses, the improve- [5] Gomes FV, Carneiro S, Pereira JLR, Vinagre MP, Garcia PAN, De Araujo LR. A new
distribution system reconfiguration approach using optimum power flow and
ment due to sectionalizing switches becomes significant, at about
sensitivity analysis for loss reduction. IEEE Trans Power Syst Nov. 2006;21
5% to 8%. This is because there will be several load islands when (4):1616–23.
faults occur within the feeders. The capacity of the DGs is enough [6] Schmidt HP, Ida N, Kagan N, Guaraldo JC. Fast reconfiguration of distribution
in some parts, but can be insufficient in other parts. If the section- systems considering loss minimization. IEEE Trans Power Syst 2005;20
(3):1311–9.
alizing switches within the feeders are not taken into account, the [7] Baran ME, Wu FF. Network reconfiguration in distribution systems for loss
load will be curtailed in the insufficient load islands to satisfy the reduction and load balancing. IEEE Trans Power Deliv 1989;4(2):1401–7.
power balance constraints. Therefore, the reconfiguration by sec- [8] Civanlar S, Grainger JJ, Yin H, Lee SSH. Distribution feeder reconfiguration for
loss reduction. IEEE Trans Power Deliv 1988;3(3):1217–23.
tionalizing switches can further increase the value of load picked [9] Enacheanu B, Raison B, Caire R, Devaux O, Bienia W, HadjSaid N. Radial
up. Finally, it can be concluded that the restored load with micro- network reconfiguration using genetic algorithm based on the matroid theory.
grid formulations by the proposed method ④ can pick up more IEEE Trans Power Syst 2008;23(1):186–95.
[10] Zhu JZ. Optimal reconfiguration of electrical distribution network using the
critical load than the traditional method ③, i.e., more than 15%. refined genetic algorithm. Electric Power Syst Res 2002;62(1):37–42.
In Fig. 10, the voltage profiles of the 615-bus system corre- [11] Parada V, Ferland JA, Arias M, Daniels K. Optimization of electrical distribution
sponding to the four strategies are sketched. The buses in the five feeders using simulated annealing. IEEE Trans Power Deliv 2004;19
(3):1135–41.
feeders are denoted as F1, F2, F3, F4, and F5. When faults occur at [12] Srinivasa Rao R, Narasimham SVL, Ramalinga Raju M, Srinivasa Rao A. Optimal
F1 and F2, the DG forms microgrids and the voltage at the master network reconfiguration of large-scale distribution system using harmony
DG bus is kept at 1.0 p.u. For strategies ① and ②, there are no tie search algorithm. IEEE Trans Power Syst 2011;26(3):1080–8.
[13] Jabr RA, Singh R, Pal BC. Minimum loss network reconfiguration using mixed-
switches among feeders, so F1 and F2 are independent from F3, F4,
integer convex programming. IEEE Trans Power Syst 2012;27(2):1106–15.
and F5 and the voltage magnitudes in F1 and F2 are close to 1.0 p.u. [14] Chen C, Wang J, Qiu F, Zhao D. Resilient distribution system by microgrids
In contrast, a noticeable voltage drop exists at bus #80 in both F1 formation after natural disasters. IEEE Trans Smart Grid 2016;7(2):958–66.
and F2 under strategy ③ and ④, because DG microgrid formation [15] Mazidi M, Monsef H, Siano P, et al. Robust day-ahead scheduling of smart
distribution networks considering demand response programs. Appl Energy
and tie switches among feeders are applied as restoration strate- 2016:929–42.
gies are coordinated. Meanwhile, the isolated buses are supplied [16] Hung DQ, Mithulananthan N. Loss reduction and loadability enhancement
by nearby feeders through tie switches instead of being supplied with DG: a dual-index analytical approach. Appl Energy 2014;115:
233–41.
by DG, which lie at the end of the substations and have low voltage [17] Marzband Mousa et al. Distributed generation for economic benefit
magnitude. However, owing to the voltage limit constraints, the maximization through coalition formation–based game theory concept. Int
voltage magnitudes always stay within the allowable feasible Trans Elect Energy Syst 2017.
[18] Marzband Mousa et al. A real-time evaluation of energy management systems
region. for smart hybrid home Microgrids. Elect Power Syst Res 2017;143:624–33.
[19] Marzband Mousa et al. Non-cooperative game theory based energy
management systems for energy district in the retail market considering
4. Conclusions DER uncertainties. IET Gener Transm Distrib 2016;10(12):2999–3009.
[20] Mohamad H, Mokhlis HH, Ping HW. A review on islanding operation and
In this paper, a new load restoration method is proposed to control for distribution network connected with small hydro power plant.
Renew Sustain Energy Rev 2011;15(8):3952–62.
facilitate resilient distribution grids after natural disasters, Both
[21] Lasseter RH. Smart distribution: Coupled microgrids. Proc IEEE 2011;99
the microgrid formulation and reconfiguration are considered (6):1074–82.
and coordinated by sectionalizing the switches so that operational [22] Marzband M, Moghaddam MM, Akorede MF, et al. Adaptive load shedding
scheme for frequency stability enhancement in microgrids. Elect Power Syst
flexibility can be better exploited to enhance electricity supply
Res 2016;140:78–86.
continuity. The master-slave control technique is integrated in [23] Marzband Mousa et al. Distributed smart decision-making for a
the optimization to coordinate multiple DGs in one microgrid. Fur- multimicrogrid system based on a hierarchical interactive architecture. IEEE
thermore, a mixed-integer second-order cone programming is Trans Energy Convers 2016;31(2):637–48.
[24] Liu G, Starke M, Xiao B, et al. Microgrid optimal scheduling with chance-
employed to efficiently reduce the computational complexity of constrained islanding capability. Elect Power Syst Res 2017:197–206.
the proposed optimization model with optimality guaranteed. [25] Marzband M, Azarinejadian F, Savaghebi M, et al. An optimal energy
Two tests on an IEEE 33-bus test system and a modified 615-bus management system for islanded microgrids based on multiperiod artificial
bee colony combined with Markov chain. IEEE Syst J 2015;100(99):1–11.
test system show that the proposed method can pick up more crit- [26] Marzband M, Yousefnejad E, Sumper A, et al. Real time experimental
ical load than the existing methods which do not coordinate the implementation of optimum energy management system in standalone
microgrid formulation and reconfiguration. microgrid by using multi-layer ant colony optimization. Int J Electr Power
Energy Syst 2016;75:265–74.
[27] Marzband M, Ghadimi M, Sumper A, et al. Experimental validation of a real-
Acknowledgments time energy management system using multi-period gravitational search
algorithm for microgrids in islanded mode. Appl Energy 2014;128:
164–74.
This work was supported by National Natural Science Founda- [28] Marzband M, Parhizi N, Adabi J. Optimal energy management for stand-alone
tion of China (Grant 51607137), National Key Basic Research Pro- microgrids based on multi-period imperialist competition algorithm
gram of China (2016YFB0901904), China Postdoctoral Science considering uncertainties: experimental validation. Int Trans Elect Energy
Syst; 2015.
Foundation (2015M580847), Natural Science Basis Research Plan [29] Marzband M, Sumper A, Ruiz-Álvarez A, et al. Experimental evaluation of a real
in Shaanxi Province of China (2016JQ5015), and the project of State time energy management system for stand-alone microgrids in day-ahead
Key Laboratory of Electrical Insulation and Power Equipment in markets. Appl Energy 2013;106:365–76.
[30] Marzband M, Sumper A, Domínguez-García JL, et al. Experimental validation of
Xi’an Jiaotong University (EIPE16301).
a real time energy management system for microgrids in islanded mode using
a local day-ahead electricity market and MINLP. Energy Convers Manage
References 2013;76:314–22.
[31] Xiaodan Y, Hongjie J, Chengshan W, Wei W, Yuan Z, Jinli Z. Network
reconfiguration for distribution system with micro-grs. In: 2009
[1] Billinton R, Billinton J. Distribution system reliability indices. IEEE Trans Power
International conference on sustainable power generation and supply,
Deliv 1989;4(1):561–8.
Nanjing; 2009. p. 1–4.
[2] Song IK, Jung WW, Kim JY, Yun SY, Choi JH, Ahn SJ. Operation schemes of smart
[32] Xu Y, Liu W. Novel multiagent based load restoration algorithm for microgrids.
distribution networks with distributed energy resources for loss reduction and
IEEE Trans Smart Grid 2011;2(1):152–61.
service restoration. IEEE Trans Smart Grid March 2013;4(1):367–74.
[33] Li J, Ma XY, Liu CC, Schneider KP. Distribution system restoration with
[3] Lee C, Liu C, Mehrotra S, Bie Z. Robust distribution network reconfiguration.
microgrids using spanning tree search. IEEE Trans Power Syst 2014;29
IEEE Trans Smart Grid 2015;6(2):836–42.
(6):3021–9.
[4] Sultana B, Mustafa MW, Sultana U, et al. Review on reliability improvement
[34] Vaskantiras G, Shi Y. Value assessment of distribution network
and power loss reduction in distribution system via network reconfiguration.
reconfiguration: a Danish case study. Energy Proc 2016;100:336–41.
Renew Sustain Energy Rev 2016;66:297–310.
216 T. Ding et al. / Applied Energy 199 (2017) 205–216

[35] Qi Qi, Wu Jianzhong, Zhang Lu, et al. Multi-objective optimization of electrical [43] Ding T, Sun K, Huang C, et al. Mixed-integer linear program based splitting
distribution network operation considering reconfiguration and soft open strategies for power system islanding operation considering network
points. Energy Proc 2016:141–6. connectivity. In: IEEE systems journal, vol. 99; November 2015. p. 1–10.
[36] Shoeiby B, Davoodnezhad R, Holmes DG, McGrath P. Voltage-frequency [44] Bondy JA, Murty USR. Graph theory with applications, vol.
control of an islanded microgrid using the intrinsic droop characteristics of 6. London: Macmillan; 1976.
resonant current regulators. In: 2014 IEEE energy conversion congress and [45] Ding T, Bo R, Gu W, Sun H. Big-M based MIQP method for economic dispatch
exposition, Pittsburgh, PA; 2014: p. 68–75. with disjoint prohibited zones. IEEE Trans Power Syst March 2014;29
[37] Baran ME, Wu FF. Network reconfiguration in distribution systems for loss (2):976–7.
reduction and load balancing. IEEE Trans Power Deliv 1989;4(2):1401–7. [46] Jabr RA, Singh R, Pal BC. Minimum loss network reconfiguration using mixed-
[38] Shirmohammadi D, Hong HW. Reconfiguration of electric distribution integer convex programming. IEEE Trans Power Syst 2012;27:1106–15.
networks for resistive line losses reduction. IEEE Trans Power Deliv 1989;4 [47] Farivar M, Low SH. Branch flow model: relaxations and convexification-Part I.
(2):1492–8. IEEE Trans Power Syst 2013;28(3).
[39] Li J. Distribution system restoration with microgrids using spanning tree [48] Low SH. Convex relaxation of optimal power flow-Part II: Exactness. IEEE Trans
search. IEEE Trans Power Syst 2014;29(6):3021–9. Control Network Syst 2014;1(2):177–89.
[40] Enacheanu B. Radial network reconfiguration using genetic algorithm based on [49] Low SH. Convex relaxation of optimal power flow-Part I: formulations and
the matroid theory. IEEE Trans Power Syst 2008;23(1):186–95. equivalence. IEEE Trans Control Network Syst 2014;1(2):15–27.
[41] Balakrishnan R, Ranganathan K. A textbook of graph theory. Springer Science & [50] Madani R, Sojoudi S, Lavaei J. Convex relaxation for optimal power flow
Business Media; 2012. problem: mesh networks. IEEE Trans Power Syst 2014;30(1):199–211.
[42] Skiena SS. The algorithm design manual. Springer; 2008.