> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <
1
Abstract—Extreme weather events, many of which are climate
change-related, are occurring with increasing frequency and
intensity and causing catastrophic outages, reminding the need to
enhance the resilience of power systems. This paper proposes a
proactive operation strategy to enhance system resilience during
an unfolding extreme event. The uncertain sequential transition of
system states driven by the evolution of extreme events is modeled
as a Markov process. At each decision epoch, the system topology
is used to construct a Markov state. Transition probabilities are
evaluated according to failure rates caused by extreme events. For
each state, a recursive value function, including a current cost and
a future cost, is established with operation constraints and
inter-temporal constraints. An optimal strategy is established by
optimizing the recursive model, which is transformed into a mixed
integer linear programming by using the linear scalarization
method, with the probability of each state as the weight of each
objective. The IEEE 30-bus system, the IEEE 118-bus system, and
a realistic provincial power grid are used to validate the proposed
method. The results demonstrate that the proposed proactive
operation strategies can reduce the loss of load due to the
development of extreme events.
Index Terms—Extreme weather events, generation redispatch,
Markov model, power system resilience, sequentially proactive
strategy
NOMENCLATURE
Indices and Sets
,,i i i
Index of Markov states
,tt
Index of time periods
,aa
Index of actions
l
Index of lines
k
Index of electrical devices
This work was supported by the National Natural Science Foundation of
China under Grant 51677160, the Theme-based Research Scheme through
Project No. T23-701/14-N, the Research Grant Council of Hong Kong SRA
under Grants ECS739713 and GRF17202714, Basic Research
Program-Shenzhen Fund (JCYJ20150629151046877), and research funding
from China Southern Power Grid (ZD2014-2-0004).
C. Wang, Y. Hou, and S. Lei are with the Department of Electrical &
Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong,
and HKU Shenzhen Institute of Research and Innovation, Shenzhen, China
(wangc@eee.hku.hk, yhhou@eee.hku.hk, leishunbo@eee.hku.hk).
F. Qiu is with the Energy Systems Division, Argonne National Laboratory,
Argonne, IL 60439 USA (e-mail: fqiu@anl.gov ).
K. Liu is with China Southern Power Grid Co., Ltd., Guangzhou, China
(liukai@csg.cn)
j
Index of generators
,nn
Index of nodes
r
Index of paths between states
,Ct
Set of components that might be in failure at t
,St
Set of Markov states at t
A
Set of actions
T
Set of time periods
N
Set of nodes
G
Set of generators
,1
S
it
Set of states at t+1 following the state S
i,t
at t
,
S
it
Set of states
G
n
Set of generators connected with node n
D
n
Set of loads connected with node n
N
n
Set of nodes connected with node n
Path
i
Set of paths from initial state to state S
i,t
Notation for Failure Rate
w
t
Failure rate due to typhoon, windstorms etc
w
A given wind speed
t
w
Wind speed at t
A given parameter
Failure rate under normal weather conditions
I
t
Failure rate due to ice storm
,tL
M
Total load in kN on line
L
M
Maximum load in kN on line
Li
m
Ice load on lines
Lw
m
Wind load on lines
()
L
f
Joint probability density function of ice load and
wind load on lines
Notation for Optimization Model
,it
S
Markov state at t
,1it
S
Markov state at t+1
,it
S
Markov state at
t
,kt
s
Status of component k at t and t+1. ‘0’ and ‘1’ denote
a failure state and normal operating status
,1kt
Failure rate of component k at t+1
()
t
R
Immediate cost at t ($)
Resilience Enhancement with Sequentially
Proactive Operation Strategies
Chong Wang, Student Member, IEEE, Yunhe Hou, Senior Member, IEEE, Feng Qiu, Member, IEEE,
Shunbo Lei, Student Member, IEEE, and Kai Liu
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <
2
Pr( )
Transition probability
,Lt
Penalty due to loss of load ($/MWh)
()
t
v
Value function at t
1
()
t
v
Value function at t+1
()
t
v
Optimal value function
,,n t i
L
Load demand of node n in state S
i,t
at t (MW)
min
j
P
Lower generation of generator j (MW)
max
j
P
Upper generation of generator j (MW)
UP
j
R
Ramp-up rate limit of generator j (MW/h)
DN
j
R
Ramp-down rate limit of generator j (MW/h)
,max
,
L
nn
P
Max capacity of line
nn
(MW)
,min
,
L
nn
P
Min capacity of line
nn
(MW)
max
n
Max limit of Phase angle of node n
min
n
Min limit of Phase angle of node n
,at
A
Action a at t
,nn
B
Electrical susceptance of line n-
n
ON
j
D
Min on time of generator j (Decision Epoch)
OFF
j
D
Min off time of generator j (Decision Epoch)
T
Duration of each period (h)
,,
G
j t i
P
Generation of generator j in state S
i,t
at t (MW)
, 1,
G
j t i
P
Generation of generator j in state
,1it
S
at t (MW)
, , ,
L
n n t i
P
Power from node
n
to
n
in state S
i,t
at t (MW)
,,n t i
L
Load shedding of node n in state S
i,t
at t (MW)
,,n t i
Phase angle of node n in state S
i,t
at t
,,j t i
o
Binary variable to indicate status of generator j in
state S
i,t
at t
, 1,j t i
o
Binary variable to indicate status of generator j in
state
,1it
S
at t+1
,,j t i
o
Binary variable to indicate status of generator j in
state
,it
S
at
t
, , ,n n t i
u
Binary outage indicator to indicate status of line
nn
in state
,it
S
at t
T
N
Number of time periods
N
A large number
()PDF
i
P
Probability of state S
i,t
,,r i t
p
Probability from initial state to state S
i,t
via path
r
I. INTRODUCTION
XTREME weather events, e.g., wind storms, typhoons, and
hurricanes, are occurring with increasing intensity [1][2]
and causing complete or partial power outages. These outages
suggest the vulnerability of current power systems. Since
power systems are critical infrastructures for society and
economic development [3], an outage might cause severe
consequences. In the United States, weather-related outages
cause estimated $25 billion economic losses each year [4]. In
China, Typhoon Rammasun, which struck the Guangdong
province on July 2014, took several 220 kV transmission lines
out of service. The severe consequences of such extreme
weather events in power systems have brought power system
resilience to the attention of organizations and governments in
the world. A Policy Framework for the 21
st
Century Grid [5],
which was released by the U.S. government in June 2011,
emphasized the significance of resilient grids in countering the
effects of increasingly intense weather events. The United
States National Research Council (NRC) [3] and the House of
Lords in the United Kingdom [6] have also emphasized the
importance of a resilient energy infrastructure. The North
American Electric Reliability Corporation (NERC) [7]-[9] and
the United States Electric Power Research Institute (EPRI) [10]
have both recognized the functionalities of system resilience.
Based on the requirements of power system resilience [3],
some conceptual frameworks have been proposed [11]. To
ensure resilience against extreme weather events, the strategies
in the three stages of a severe event [8] (i.e., prior to the event,
during the event and after the event) should be considered.
Prior to an extreme weather-related event, an accurate outage
prediction contributes to manage preparedness and restoration
efforts. To improve accuracy of predictions, a negative
binomial regression model is proposed [12]. Since this model is
based on data regarding outages caused by three hurricanes,
i.e., Fran (1996), Bonnie (1998), and Floyd (1999), it is only
suitable for a specific service area. To overcome this limit, a
generic model for the full U.S. coastline is proposed in [13]. To
estimate power outage durations in face of hurricanes, a
statistical model is proposed in [14]. With outages and the
duration predictions, some preventive strategies prior to an
extreme weather-related event can be performed to increase
power system resilience. Considering the stochastic and
sequential characteristics of events, the events’ potential
impacts on the resilience of power systems are analyzed by
using sequential Monte Carlo simulations [15]. To minimize
negative impacts, the response before a hurricane is modeled as
a mixed-integer programming problem [16]. In addition,
preparation of sufficient blackstart generating units and
emergency generators also plays an important role in
improving power system resilience before an extreme
weather-related event. To assess blackstart capacities, a
GRM-based algorithm is developed [17]. To provide enough
blackstart resources at right locations, [18] proposes a model to
establish a procurement plan with a minimal cost while
guarantee sufficient blackstart capacities. [19] focuses on
dispatch strategies of mobile emergency generators to
minimize the loss of load. Furthermore, some strategies, e.g.,
maintenance planning [20] and wide-area controls in response
to communication failures [21], [22], can also be performed to
enhance power system resilience before an weather-related
event.
During the event, hardening, which refers to physically
changing power systems, is a measure to make systems less
susceptible to weather-related events. In [23], a resilient
E
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <
3
distribution network planning problem is formulated as a
two-stage robust optimization model. In addition to hardening,
islanding schemes can also be used to improve power system
resilience. In [24], a unified resilience evaluation and an
enhancement method, including a novel defensive islanding
algorithm, are proposed. The proposed islanding scheme can
mitigate potential cascading effects during weather-related
events. Considering non-dispatchable and dispatchable
distributed generators, [25] proposes a novel comprehensive
operation and self-healing scheme, which sectionalizes a
distribution system into several micro-grids, to improve
distribution system resilience.
After an extreme weather-related event, it is necessary for
system operators to implement restoration strategies [26]-[28]
to restore loads as quickly as possible. A generic conventional
power system restoration can be divided into three stages, i.e.,
preparation, system restoration and load restoration [29]. In the
preparation stage, the system status, i.e., blackstart units,
non-blackstart units and critical loads, should be evaluated. In
the system restoration stage, the main goal is to establish a
strong bulk power network by restarting appropriate blackstart
and non-blackstart units associated with appropriate
transmission lines and some critical loads [30]. In the load
restoration stage, the critical objective is to restore all loads as
quickly as possible. Many approaches, e.g., expert systems [31]
and heuristic approaches [32], have been proposed to deal with
load restoration. However, outages caused by weather-related
events usually have their own unique characteristics, which
might result in inapplicability of the existing recovery schemes.
Therefore, new techniques should be proposed to deal with
restoration after weather-related events. [28] proposes a novel
operational approach for distribution systems by establishing
multiple microgrids energized distributed generators to restore
critical load from power outages. In [33], the impact of
microgrids as blackstart resources after a natural disaster is
evaluated. Furthermore, decentralized restoration schemes [34]
can be employed.
Most research studies have focused on assessment/strategies
prior to an weather-related event and restoration strategies after
an weather-related event. However, strategies during an event
are still in their infancy. In this work, we focus on operational
strategies during an event to enhance power system resilience
against extreme weather-related events. During extreme
weather events, operating strategies should be established
subject to both current system/equipment statuses and potential
future statuses as the weather-related events unfolding. Due to
the essentially sequential characteristics during an event
unfolding, the operation strategies should be a sequence of
actions associated with uncertainties caused by development of
the event and faults of components.
The main contributions of this paper are two-fold. 1) A
Markovian method for sequentially proactive generation
redispatch is proposed. At each decision epoch, the system
topology, which may change due to the failure of some
components (such as transformers or transmission lines) due to
extreme event, constitutes a Markov state. Transition
probabilities between different states, i.e., different topologies,
are determined by component failure rates and development of
the event. In each state, a recursive value function that includes
a current cost and a future cost is established subject to
operation constraints (such as ramping rates of generators). 2)
The optimal strategy for each state is obtained by optimizing
the proposed recursive model. The recursive model is
transformed into a mixed integer linear programming by using
the linear scalarization method, with the probability of each
state used as the weight of each objective. The linear
programming is solved with the CPLEX solver. Two IEEE test
systems and a modified realistic system are used to validate the
proposed model, with the results showing that the proposed
model provides insight for proactive generation redispatch
under extreme weather events.
This paper is organized as follows. Section II describes the
impacts of extreme weather events on system states. Section III
introduces sequentially proactive operation strategies, while
Section IV shows the solution. Section V presents the case
studies, and the work is concluded in Section VI.
II. INFLUENCES OF EXTREME WEATHER EVENTS ON SYSTEM
STATES
This section introduces the influences of extreme weather
events on system states. First, several component failure rate
models are introduced. Second, system states on the trajectory
of extreme weather events are presented. Third, transition
probabilities between different system states are modeled.
A. Component Failure Rate
In certain extreme weather events, such as hurricanes,
tornados, typhoons, windstorms, floods, lightning storms [35],
the intensity of the hazardous forces will change both
temporally and geographically as the trajectories of the weather
events move passing a region. The component failures are
correlated with hazardous forces. Usually, a generic fragility
curve, in Fig. 1, can be used to relate failure probabilities of a
component to the weather intensity [24].
0
0.2
0.4
0.6
0.8
1
Weather Intensity
Failure Probability
Fig. 1 Generic fragility curve
Typically, several models for calculating component failure
probabilities under different extreme weather events may be
employed.
1) Failure rate caused by hurricanes, tornados, typhoon and
windstorms
For hurricanes, tornados, typhoons and windstorms, the
pressure of high wind is a key reason of component failures. It
is deemed that the pressure exerted on components is
approximately proportional to the square of wind speeds [36].
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <
4
The failure rates of components during these events can be
expressed as
22
11
w
tt
ww
(1)
2) Failure rate caused by ice storm
Many research studies have focused on transmission
reliability assessment in consideration of ice storms. Take
transmission lines for example, their failure rates under ice
storms can be expressed as
,
( , )
tL
L
M
L
t L Li Lw Li Lw
M
f m m dm dm
(2)
The failure rates of transmission towers under ice storms
have the similar expression.
B. System States on Trajectories of Extreme Weather Events
Because of the time sequence in which components
experience extreme weather events, different components, e.g.,
lines, transformers, loads and generators, might be in failure at
sequential time intervals. For example, there are two
components A and B on the trajectory of a typhoon, as shown in
Fig. 2. Since the failure rates depend on the wind speed, the
failure rates of each component at different sequential time
intervals are usually different, as shown in Fig. 3(a).
A
B
t
1
t
2
t
4
t
3
t
5
Trajectory of Typhoon
Fig. 2 Two components on the trajectory of a typhoon.
Failure Rates
Time
A
B
P
A,1
P
B,2
P
A,1
P
B,3
(a)
Markov States
Time
(b)
S
0
S
0
S
A
t
1
t
2
t
4
t
3
t
5
t
1
t
2
t
4
t
3
t
5
S
0
S
A
S
B
S
AB
S
0
S
A
S
B
S
AB
S
0
S
A
S
B
S
AB
Fig. 3 (a) Failure rates at different time intervals. (b) Markov states at different
time intervals.
At each decision epoch, the system topology may be changed
due to component failures in the face of an event. As for the
system operators, they should make decisions based on the
current system topology at each decision epoch in the face of
the event, as well as consider possible topology scenarios
caused by the extreme weather event in the subsequent time
intervals. In this paper, the system topology is defined as a
Markov state. The number of Markov states depends on the
number of components that might be in failure due to the
extreme weather event. Let
be the set of components that
might be in failure at time t. Sets at different time intervals
satisfy the following equations.
, , 1
, {1, 2, }
C t C t
t
(3)
where (3) denotes that possible failed components at one
interval should include the possibly failed components from the
previous time intervals. For example, the typhoon only directly
influences the component B at t
3
, but
at t
3
in Fig. 2 includes
the component A as well as the component B. The Markov
states include S
0
, S
A
, S
B
and S
AB
, as shown in Fig. 3(b). The
subscript ‘0’ denotes that no components are in failure. In
general, the number of Markov states at time t is
, and
is the number of components that might be in failure at time t.
Usually, a realistic trajectory is uncertain and the forecast of
the trajectory cannot be entirely accurate. When considering the
uncertainty of a trajectory, its influences on component failures
can be included in component failure probabilities.
C. Transition Probabilities between Different System States
Let
be the set of states at time t. Let
and
be
states at time t and t+1, respectively. The transition probability
from
to
can be expressed as
,1
, , 1 , , 1 ,
Pr , Pr , ,
Ct
i t i t k t k t S t
k
S S s s i
(4)
, , 1
, , 1
, , 1
, 1 , , 1
, 1 , , 1
1 0, 0
0 0, 1
Pr ,
1 1, 1
1, 0
k t k t
k t k t
k t k t
k t k t k t
k t k t k t
ss
ss
ss
ss
ss
(5)
where is the transition probability,
is the failure
probability of the component k at time t+1,
and
are
statuses of the component k at time t and t+1, respectively. The
lowercase s denotes a status of an electrical device on the
trajectory of a weather event. It can be a failure status or a
non-failure status. ‘0’ and ‘1’ denote a failure status and a
normal operating status, respectively. The uppercase S denote a
system state, which should consider all statuses of electrical
devices on the trajectory of a weather event.
III. SEQUENTIALLY PROACTIVE OPERATION STRATEGY
This section introduces the optimization model to establish
sequentially proactive operation strategies. First, a recursive
model for each system state, including current and future
influences, is established. Second, constraints for sequentially
proactive operation strategies are presented.
A. Sequential Decision Processes
Since the failure of a component on the trajectory of an
extreme weather event is uncertain, the best decision making
for system operators is to adjust strategies, according to
real-time states of the system, to optimize their objective.
Meanwhile, when making decisions based on a state, the
system operators should consider not only the current
influences, but also any future influences caused by the
decisions. This decision process is a Markov decision process.
For a realistic system, the system operators make decisions
continuously. To simplify the model, we assume that the
decisions are made at discrete decision epochs. In this context,
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <
5
we model the whole decision process as a discrete-time Markov
decision process.
Since the system operators should consider both current and
future influences, the value function for each state can be
expressed as a recursive formula.
,1
, , , ,
, , 1 1 , 1 , 1
,
( , ) ( , )
Pr( , ) ( , )
, , , , 1
S
it
t i t a t t i t a t
i t i t t i t a t
i
S t A A T T
v S A R S A
S S v S A
i a a t t
(6)
where
denotes actions, i.e., redispatching the system. In
this paper, “redispatching the system” means “generation
redispatch”, i.e., adjusting outputs of available generators.
is the expected cost, from time t to the terminal
time, with state
under action
. Before an event hits a
power grid, the system topology can also be considered as a
Markov state, which can be included in the model.
is the immediate cost at time t for state
under action
.
This immediate cost is the cost of loss of load. Usually, when a
system is under a severe weather event, the reliability,
represented as expected loss of load in the paper, has a higher
priority. Therefore, the expected loss of load works as the
objective of the system operators.
Based on (6), the optimal strategy with state
at t can be
obtained by using the following formula.
, , , ,
( ) min ( , ), , ,
t i t t i t a t A S t T
v S v S A a i t
(7)
where
is the minimal expected cost of state
at time
t. The immediate cost
is the cost of loss of load,
which can be expressed as
, , , , ,
( , )
N
t i t a t L t n t i
n
R S A L T
(8)
B. Constraints for Sequentially Proactive Operation Strategy
During generation redispatch in state
at time t, the
constraints of power balance, upper and lower limits of
generators, upper and lower limits of voltage, power flows
through lines and load limits should be satisfied. Typically, the
ramping rates between possible states should be satisfied.
1) Power balance
The power balance constraint in state S
i,t
at time t can be
expressed as
, , , , , , , , ,
0,
GN
nn
GL
j t i n t i n t i n n t i
jn
P L L P n
(9)
where (9) denotes power balance at each node in state S
i,t
at
time t.
2) Ramping rates of generators
The ramping rates of each generator in state S
i,t
at time t
should be satisfied during the implementation of any proactive
generation redispatch such that
min
, 1, , , , , , 1,
, , , 1, , 1
(2 )
(1 ) ,
GG
j t i j t i j t i j t i j
UP S
j t i j t i j i t
P P o o P
o o R i j
(10)
min
, , , 1, , , , 1,
, , , 1, , 1
(2 )
(1 ) ,
GG
j t i j t i j t i j t i j
DN S
j t i j t i j i t
P P o o P
o o R i j
(11)
where the term
ensures that ramping rates should be
satisfied between the state
at t and its possible following
states at t+1. Considering potential online/offline statuses of
generators, binary variables regarding generators’ statuses are
included in constraints (10) and (11).
3) Minimum up time and down time constraints of generators
When performing generation redispatch, minimum up time
and down time constraints of generators should be satisfied.
, , , 1, , ,
, 1 ,
0 , 1
,,
ON
j t i j t i j t i j
SS
i t i t
o o o t t D
i i j
(12)
, , , 1, , ,
, 1 ,
1, 1
,,
OFF
j t i j t i j t i j
SS
i t i t
o o o t t D
i i j
(13)
where the terms
and
ensure that
minimum up time and down time constraints should be satisfied
between possible transition states.
4) Power flows of lines
The limits for power flows through online lines in state S
i,t
at
time t should be satisfied.
, , , , , , , , , , ,
(1 ) 0
,,
L
n n n t i n t i n n t i n n t i
l
B P u N
n n Line l
(14)
, , , , , , , , , , ,
(1 ) 0
,,
L
n n n t i n t i n n t i n n t i
l
B P u N
n n Line l
(15)
,min ,max
, , , , , , , , , , ,
,,
L L L
n n n n t i n n t i n n n n t i
l
P u P P u
n n Line l
(16)
where (14) and (15) represent the physical relations between
voltage angles and power flows through transmission lines.
u
n,n’,t,i
is a binary outage indicator. If the line l is in outage in
state S
i,t
at time t, u
n,n’,t,i
=0; otherwise u
n,n’,t,i
=1. N is a
disjunctive parameter. With a sufficiently large N, (14) and (15)
are redundant when lines are outages. (16) shows the limits of
transmission lines. The models of power flows through
transformers are similar to (14)-(16).
5) Upper and lower limits of the outputs of generators
During the implementation of generation redispatch in state
S
i,t
at time t, the upper and lower limits of generators should be
satisfied by
min max
, , , , , ,
G
j j t i j t i j j t i
P o P P o j
(17)
6) Load limits
When performing generation redispatch in state S
i,t
at time t,
load shedding might be conducted to ensure power balance
when considering the ramping rates of the generators, power
flows through the lines and so on. When conducting load
shedding , the following constraints should be involved
, , , ,
0
n t i n t i
L L n
(18)
where L
n,t,i
is the forecasted load of the bus n at t. Its value can
be predicted based on existing load forecasting methods.
7) Voltage limits
The following constraint regarding voltage in state S
i,t
at t
should be satisfied.
min max
,,n n t i n
n
(19)
