1

Power-Traffic Coordinated Operation for Bi-Peak Shaving and Bi-Ramp Smoothing
–A Hierarchical Data-Driven Approach
Huaiguang Jiang, Member, IEEE, Yingchen Zhang, Senior Member, IEEE, Yuche Chen, Changhong
Zhao, Member, IEEE, Jin Tan, Member, IEEE

arXiv:1712.00211v1 [] 1 Dec 2017

National Renewable Energy Laboratory, Golden, CO 80401

Abstract—With the rapid adoption of distributed photovoltaics
(PVs) in certain regions, issues such as lower net load valley during the day and more steep ramping of the demand after sunset
start to challenge normal operations at utility companies. Urban
transportation systems also have high peak congestion periods
and steep ramping because of traffic patterns. We propose using
the emerging electric vehicles (EVs) and the charing/discharging
stations (CDSs) to coordinate the operation between power
distribution system (PDS) and the urban transportation system
(UTS), therefore, the operation challenges in each system can be
mitigated by utilizing the flexibility of the other system.
The proposed operation approach is designed hierarchically
consists of a higher and a lower level. In the higher level, we
assume integrated operation of both the PDS and UTS, and target
of the operation is to minimize the social cost. Meanwhile, the
target for the EVs and the CDSs as customers is to minimize their
own expenditures. Then, there exists an equilibrium between
two targets to determine the optimal charging/discharging price.
In the lower level, the temporal & spatial models of PDS
and UTS are developed to provide a detailed analysis of the
power-traffic system. Specifically, the PDS is built with a threephase unbalanced AC power flow model, the optimal power flow
(OPF) problem is relaxed with the semidefinite relaxation programming (SDP), and solved with alternating direction method
of multiplier (ADMM). A dynamic user equilibrium (DUE)
problem is formulated for the UTS, which is based on the static
user equilibrium (SUE) with additional constraints to ensure
a temporally continuous path of flow. The EVs and the CDSs
function as reserves for both the PDS and UTS, and the state of
charge (SOC) is considered to optimize the charging/discharging
schedule and reduce the impacts to the PDS. We conducted the
simulation and numerical analysis using the IEEE 8,500-bus for
the PDS and the Sioux Falls system with about 10,000 cars for
the UTS. Two systems are simulated jointly to demonstrate the
feasibility and effectiveness of the proposed approach.

Index terms— Power distribution system, duck curve, urban
transportation system, electrical vehicle, charging/discharging
station, state of charge, dynamic user equilibrium, traffic
congestion, traffic pattern, optimal power flow, distributed
computation
N OMENCLATURE

Functions
t
FUT
Y
t
FCSO
t
FCDS

fPt S
t
fUT
S
t
fW
T
QtP DS
QtDP DS
QtDUT S
FPt DS
fka (xtka )

Parameters
G UT S

Pru su

Cka
G P DS

Υ, Υ
γ1
γ2

Abbreviations
PDS
UTS
CDS
SUE
DUE
OPF
SDP
ADMM
EV
SOC

Power distribution system
Urban transportation system
Charging/discharging station
Static user equilibrium
Dynamic user equilibrium
Optimal power flow
Semidefinite relaxation programming
Alternating direction method of multiplier
Electrical vehicle
State of charge

Objective function of the utility and customer in time interval t, t ∈ Dt
Objective function of the customer in time
interval t, t ∈ Dt
Objective function of the smart charging/discharging in a CDS
Cost function of PDS in time interval t
Cost function of UTS in time interval t
Cost function of charging/discharging waiting time
Netload of PDS in time interval t
Total charging/discharging power of all
CDS
Total EVs for charging/discharging
Objective function of OPF in PDS
Travel time function on link ka in time
interval t, where traffic flow is xtka

̺1 , ̺2
̺3 , ̺4
χi1

The graph for UTS with a node set and
a link set: G UT S = [V UT S , E UT S ], where
V UT S = {1, 2, · · · , nV U T S }, and E UT S =
{1, 2, · · · , mE U T S }
A set of path used to connect the OD
S
and
pair, which is defined as ru ∈ VrUT
u
UT S
UT S
UT S
su ∈ Vsu , Vru and su ∈ Vsu are the
original node set and destination node set,
respectively
The traffic flow capacity on link ka
The graph for PDS with a node set and a
link set: G P DS = [V P DS , E P DS ], where
V P DS = {1, 2, · · · , nV P DS }, and E P DS =
{1, 2, · · · , mE P DS }
The upper bound and lower bound of SOC
Weight factor between PDS and UTS
Iteration step coefficient for gradient descent
The electrical power price and the congestion fee
The EV parking ratios to the CDSs
The capacity of CDS i1

Variables
θkt a
π ∗t

The traffic flow on link ka in time interval
t
The optimal charging/discharging price in
time interval t

qrdtu su
hdpr1u su
δpdr1ut su ka

Viφt
Iiφt
sφt
i
ziφ
Siφt

viφt , iφt
i
xt,k , y t,k , λt,k
t1
Cev,i
3

Number of vehicles from ru to su departing
in time interval d via any path, d ∈ Dt , Dt
is the time interval set
Number of vehicles assigned to path pru su
with departing time interval d1
A 0-1 variable to indicate in time interval t,
whether the trip from ru to su is assigned to
path pru su ka via link ka departing in time
interval d1
Complex voltage on bus i with phase φ, φ
⊆ {a, b, c}
Complex current on bus i with phase φ, φ
⊆ {a, b, c}
φt
Power injection on bus i, where sφt
i = pi
φt
+ iqi
The complex impedance matrix ziφ = riφ +
ixφi
The complex power from bus i to bus i1 ,
where bus i1 is the ancestor of bus i, Siφt
= Viφt (Iiφt )H , H indicates the hermitian
transpose
viφt = Viφt (Viφt )H , liφt = Iiφt (Iiφt )H
The kth iterative variables of ADMM for
distributed OPF computation
The discharging (positive) and charging
(negative) speed of EV i3 at time t1

and transportation information. In [18], based on locational
marginal pricing (LMP), an optimal deployment of charging
stations is proposed for EVs. In [19], an optimal traffic-power
flow is proposed with the wireless charging technology and
congestion toll manipulation.
In this paper, a two-step power-traffic coordinated operation approach is proposed to address the bi-peak and biramp problems for both PDS and UTS. It also provides a
multifunction platform for the researches such as emission
reduction, multisystem integration, and future city planning.
The main contributions of this paper are:
1) Considering the complexity of the power-traffic system,
a two-step coordinate approach is designed from the
higher level to lower level to operate the system in
spatial and temporal domain hierarchically. In the higher
level, both the PDS and the UTS are regulated and
treated together as an utility to minimize the social
cost. The EVs and CDSs are treated as customers to
minimize the expenditure. Then, an equilibrium exists
between the utility and the customers to determine the
operation variables such as optimal charging/discharging
price, total electrical demand, and total available EVs. In
the lower level, the detailed models of PDS, UTS, EVs
& CDS are considered to specifically determine these
variables in spatial and temporal domains.
2) In the lower level, the PDS is built with the branch
flow model, which is equivalent to the classic businjection model and more suitable for the PDS with
radial topology [20]. There are several developed approaches for solving an optimization problem associated
with a PDS, usually called OPF such as DC OPF [21],
genetic algorithm [22], [23], and OPF relaxation with the
second-order cone program (SOCP) [24]. Based on [25],
[26], we relax an OPF with the SDP for the threephase unbalanced PDS. Meanwhile, several distributed
algorithm are designed to solve the OPF such as dual
decomposition [27] and predictor corrector proximal
multiplier (PCPM) [28]. Based on [25], [29], the ADMM
is applied to decompose the OPF problem and reduce
computation time.
3) In the lower level, for the UTS, the DUE is widely
used to estimate the traffic volume, congestion time,
and travel cost for each link [30]. Based on the assumption that all travelers can choose the best route
to minimize the cost of transportation, in the Wardrop
user equilibrium (UE), the travel times of all paths are
equal for each original-destination (OD) pair and less
than any unused paths [31], [32]. In [19], a static user
equilibrium assignment is used to integrate with the
PDS. Considering that the EV behaviors can be impacted
by the charging/discharging prices, a DUE is applied to
keep the temporal domain continuous for the UTS.
4) In this paper, the EVs and the CDSs are designed as
the “reserves” for both PDS and UTS, respectively.
Considering the SOC [33], [34], an optimal charging/discharging schedule is designed to meet the requirements of PDS and UTS and reduce the impacts to

I. I NTRODUCTION
Rooftop photovaltaic (PV) gained a foothold in many power
systems because of its continuously declining cost [1]–[3].
It brings many benefit to costumers such as reduced energy
cost; however, it also presents unprecedented challenges to
power systems operation and control. The upper right-hand
plot in Fig. 1 shows the so-called “duck curve”, which has
been recorded in some high-distribution PV regions such as
California [4]–[6]. The peak solar generation in the middle
of the day sinks the netload to lower valley and then when
the peak load occurs right after sunset, a huge volume of
energy demand ramps up in a short time frame. This creates
the artificial peak and ramping that are costly to balance using
the current power system assets.
It is not surprising that other man-made complex systems
also suffer similar issues. As shown in the lower right-hand
plot in Fig. 1, the transportation system also has high peaks
and steep ramps due to fairly constrained traffic patterns at the
rush hours in urban areas [7]–[9]. Thanks to the spawn of EVs
and the widely located CDSs, the two originally independent
systems: the PDS and the UTS can be coupled together.
Specifically, the increasing number of EVs can be seen
as a significant amount of distributed and highly manipulatable small reserves that can be used to provide demand
response, enhance the system reliability, and provide other
services for the power systems [10]–[16]. In these studies,
the geographical information and transportation information
are ignored, because the EVs are treated as the aggregated
loads/reserves. In [17], the optimal placement of the charging stations is studied with the geographical information
2

Fig. 1. The main idea and the bi-peak and bi-ramp problems in a power distribution system and a transportation system in a urban area.

tion VIII.

PDS. The test bench consists of real systems, the PDS is
the IEEE 8,500-bus distribution system, and the UTS is
the Sioux Falls transportation system with about 10,000
EVs, to demonstrate the proposed approach.
Based on Fig. 1, we acknowledge that cybersecurity is
a critical aspect to consider in the proposed power-traffic
system operation. Also, the large volume of data generated
by the proposed power-traffic system requires a high-speed,
flexible and reliable communication network. Because the
proposed power-traffic system contains two complex systems,
many industrial communication network infrastructures for
PDS and UTS operations need to be attached, such as 3G,
4G, and WiFi [35]. The multi-network integration brings a
lot of challenge for the 1). real-time monitoring & anomaly
detection [36], [37], 2). data transmission, storage, & security,
3). attack analysis & mitigation [38]–[41]. In this paper, we
assume that all the messages such as electrical power price,
traffic congestion information, and various control signals can
be transmitted correctly in real-time without any cybersecurity
issues.
The paper is organized as follows. In Section II, the
flowchart of the proposed approach is introduced. In Section III, the equilibrium between the utility and the customers
is designed to determine the optimal system variables. In
Section IV, based on SUE, a DUE is applied to model the
UTS and compute the dynamic traffic flow. In Section V, based
on the branch flow model, the OPF problem is relaxed with
SDP and solved with ADMM. In Section VI, considering the
SOC and the EVs behaviors, an optimal charging/discharging
schedule is designed to reduce the impacts to the PDS. In
Section VII, the numerical results are presented to validate
the proposed approach. The conclusion is presented in Sec-

II. T HE F LOWCHART OF THE P ROPOSED A PPROACH
The proposed two-step power-traffic operation approach is
shown in Fig. 2. In the higher level, the power-traffic system
consists of two major parts: the utility part (taking the PDS
and the UTS as a whole) shown in green, and customer
part (taking the CDSs and related EVs) shown in yellow.
At time interval t, the duck curve of the netload is recorded
with the renewable energy generation data and end user load
data in PDS. Meanwhile, the travel data is collected with the
traffic monitoring data and congestion model in UTS. Then,
as shown in the two green blocks, the PDS and UTS are
treated as a regulated utility to minimize the social cost. In the
yellow block, the EVs and CDSs are treated as the customers
to minimize their expenditures. There exists an equilibrium
to determine the system operation variables such as optimal
charging/discharging prices, total demand electrical power, and
total required EVs. These variables are fitted into the lower
level as inputs.
In the lower level, the power-traffic system consists of three
major parts: the PDS part, the EVs & CDSs part, and the UTS
part, which determine the system operation variables in spatial
and temporal domains with detailed models. First, in the UTS,
to accurately simulate the dynamic traffic system in spatial
and temporal domain, the DUE is built based on SUE with
temporal continuous constraints. The proposed DUE problem
can be transfered into a convex problem and solved with
optimal results. Second, in the PDS, the objective function
FP DS is designed to minimize the cost of the PDS, which
is built with a three-phase unbalanced branch flow model.
Based on the SDP relaxation, FP DS can be minimized in an
3

Synchronized time and unified GIS information

Load
data
Net load
(Duck curve)

EV

EV

CDS

CDS

CDS

Utility PDS

Traffic Flow
Monitoring

Congestion

EV

Traffic flow curve

Upper Level

Renewable
energy

Utility UTS

Customers

An equilibrium between the utility and customers

SCADA
AMI

Demand electrical
power

Required power,
charging/discharging
price

PDS
information

SDP
relaxation

Three-phase unbalanced power
distribution system
(Branch flow model)

Power distribution system

Road

Battery
SOC
Dynamic traffic
assginment
(DUE Model)

Model of
EVs

Charging/
discharging power
Optimal Power Flow
(ADMM)

GIS

Charging/
discharging price
and schedule

Charging/
discharging
speed

GIS

Total available EVs

Bi-direction charging/
discharging and schedule

Congestion
model

Congestion, available
EVs

Temporal continuosly
constraints

User
Equilibrium

EVs & charging/discharging stations

Lower Level

Optimal charging/
discharging price

Urban transportation system

Fig. 2. Flowchart of the proposed power-traffic coordinate operation approach.

number of available EVs on CDS i1 . The constraints of (1)
are illustrated as follows:

OPF problem, which is solved with ADMM. Third, with the
specified electrical power demands and available EVs for each
CDS, an objective function FCDS is designed to minimize the
charging/discharging impacts to the PDS while considering the
SOC.
In addition, in this paper, it is assumed that all the drivers
know the traffic information very well, and can get the optimal
charing/discharging price through the wireless communication
immediately. The user-equilibrium in UTS network can be
reached in a very short time.

Pit1 ≤ Pit1 ≤ Pit1 ,

(2a)

0 ≤ γ1 ≤ γ1

(2b)

Nit1

(2c)

0≤

Nit1

≤

fPt S is defined as a convex function [42], which determine
the PDS cost in time interval t, QtP DS is the netload of PDS,
and QtDP DS (Pit1 ) is the total charging/discharging power of
all CDSs, which can be defined as

III. E QUILIBRIUM D ESIGN B ETWEEN U TILITIES AND
C USTOMERS (H IGHER L EVEL )

QtDP DS (Pit1 ) =

X

Pit1

(3)

i1 ∈NCDS
t
where NCDS is a set of CDS. fUT
S is defined as a convex
function, which determines the UTS cost in time interval
t, QtUT S is the traffic load, and QtDUT S (Nit1 ) is the total
t
EVs for charging/discharging. fW
T (τ1 ) is the time cost with
charging/discharging time τ1 . Considering the number of EVs
is very large, QtDUT S (Nit1 ) can be estimated as

A. Equilibrium Design
In this paper, we focus on a small city with tens of thousands
of cars, which means any single user behavior has a very weak
impact on the equilibrium design in the higher level. The utility
consists of PDS and UTS. A customer consists of a CDS and
related EVs.
Utility Objective (Minimize Social Cost)
t
As shown in (1), FUT
Y is the objective function of the
utility, which is designed to minimize the social cost with the
variables Pit1 and Nit1 , t is a time interval t ∈ Dt . Pit1 is the
electrical power charging (negative) or discharging (positive)
at CDS i1 . γ1 is the weight coefficient of UTS. Nit1 is the

QtDUT S (Nit1 ) =

X

i1 ∈NCDS

Nit1 =

X

i1 ∈NCDS

Pit1
C1t

(4)

where C1t is the average charging/discharging speed of all
t
the EVs. fW
T is the waiting cost when an EV is parking
for charging/discharging, which is also defined as a convex
4

t
FUT
Y = min
t
t
Pi ,Ni
1

1



t
t
t
t
t
t
t
))
))
+
f
(τ
Q
(N
(Q
−
Q
(N
fPt S (QtP DS − QtDP DS (Pit1 )) + γ1 fUT
1
WT
DUT S
i1
S
UT S
DUT S
i1

function. In summary, the proposed utility objective function
t
FUT
Y is convex, which indicate an optimal point existed.
Customer Objective (Minimize Own Expenditure) Based
on (4), the objective function of a customer (a CDS and related
EVs) can be designed as follows:
t
t
FCSO
= min
fW
T (τ1
t
Pi

X Pit

1

i1

1

C1t

) − π t Pit1

IV. DYNAMIC U SER E QUILIBRIUM A SSIGNMENT
(L OWER L EVEL )

Compared with the SUE, the DUE can accurately describe
the traffic dynamics in a set of successive time intervals. In
this paper, the DUE is based on the SUE with a temporal
generalization [32], [46].
A. Basic Model of UTS

(5b)

The UTS can be represented in a graph with a node
set and a link set: G UT S = [V UT S , E UT S ], where V UT S
= {1, 2, · · · , nV U T S }, and E UT S = {1, 2, · · · , mE U T S }. The
S
origin and destination (OD) pair can be defined as ru ∈ VrUT
u
UT S
and su ∈ Vsu , a set of paths Pru su is used to connect the
OD pair, and the traffic flow between the OD pair can be
represented as qrdu su with departing time dt ∈ Dt , where Dt
is a set of all time intervals. Then, the congestion model can
be presented as follows [47]:


θt
(8)
fka (θkt a ) = fk0a 1 + 0.15( ka )4
Cka

where the objective function of the customer consists of two
parts: the time consumption in the CDSs and the benefit/cost
in charging/discharging. π t is the electrical power price, and
π t Pit1 is the benefit (by discharging, positive) or cost (by
charging, negative) at CDS i1 . As discuss above, the objective
t
function of the customer FCSO
is also convex.
B. Solutions
exists, and the optimal price
An equilibrium π ∗t and Pi∗t
1
π can be computed as [43], [44]
∗t

where fka (θkt a ) is the travel time (travel impedance) when the
traffic flow at link ka ∈ KA is θkt a in time interval t ∈ Dt , fk0
is the initial travel time, and Cka is the traffic flow capacity.

X Pi∗t
γ1 ′t
t
1
+ t fUT
−
π =
S (QUT S −
t )
C
C
1
1
i1
i1
(6)
where the optimal price depends on the PDS part fP′tS and the
′t
UTS part fUT
S . Based on the gradient descent, a distributed
algorithm is proposed to optimize the utility and the customers
jointly the equilibrium. Specifically, the objective function of
each customer can be computed independently to the optimal
price π ∗t . At k1 -th iteration:
X

fP′tS (QtP DS

)
Pi∗t
1

B. Dynamic User Equilibrium
Based on the SUE, the DUE can be built as follows [46]:
t

JDUE = mint

ka ,θka

π k1 ,t =


X k ,t  γ 1
X Pik1 ,t 
1 ′t
′t
t
t
1
1
+ t fUT S QUT S −
Pi1
fP S QP DS −
t
C
C
1
1
i
i
1

1

s.t. θkt a =

(π

k1 ,t

=

Pik11 ,t

, Pik11 +1,t )

+ γ2
=



X Pit
τ1 ′t
k1 ,t
1
f
(τ
1
W
T
t )−π
C1t
C
1
i

[π k1 ,t , Pik11 +1,t ]̟

1



qrdu1 su =

fkta (w)dw

X

(9a)
(9b)

hdpr1u su

(9c)

fkta (θkt a )δpdr1ut su ka

(9d)

pru su ∈Pru su

bdpr1u su =

(7b)

θka
X X Z

ka ∈KA t∈Dt 0
X
X
hdpr1u su δpdr1ut su ka
pru su ∈Pru su d1 ∈Dt

(7a)

Pik11 +1,t

UTS

(5a)

s.t. (2a), π t ≤ π t ≤ π t

∗t

FOR

(1)

X X

t∈Dt ka ∈Ka

where hdpr1u su is the number of vehicles in path pru su with
departing time interval d1 , bdpr1u su is the travel time from ru
to su via path pru su , and δpdr1ut su ka is a 0-1 variable and can
be defined as

(7c)

where the optimization stepsize can be set as 0 ≤ γ2 , and the
algorithm can converge when γ2 is small enough [45]. [·]̟
means that the results are projected onto the set ̟ defined by
(5b).
In summary, in this section, the optimal price π ∗t can be
gives
computed in the higher level, and the optimal results Pi∗t
1
the upper bound of power injection in (20). In the following
sections, the models of the UTS, PDS, and CDS are built
to provide detailed information in both temporal and spatial
domains about how to operate the power-traffic system in the
lower level.

δpdr1ut su ka =


 1


0

If in time interval t, link ka is in
path pru su with departing time d1
otherwise

(10)

In time interval t, the traffic flow θkt a on link ka is given as
in (9b), which equals to the sum of the traffic flows hdpr1u su ≥ 0
via link ka for any departing time interval t ∈ Dt and any path
pru su ∈ Pru su . The number of vehicles from ru to su is given
5

Algorithm 1 DUE solution
Step 1: Initialization. Set the iteration number n1 = 1, the
.
threshold ε1 . Update the new demand as qrdtu su = qrdt−1
u su
Then using all-or-nothing assignment [32], [48] to generate
θkt a .

as in (9c), which equals to the sum of traffic flows hdpr1u su on
any assigned path pru su ∈ Pru su . The total time consumption
of path pru su from ru to su is given as in (9d), which keeps the
temporal continuous traffic flow for all time intervals t ∈ Dt
with a temporal unique constraint:
X

δpdr1ut su ka = 1

1
1
),
= fkta (θkt,n
Step 2: Update the travel time with θkt a : fkt,n
a
a
which is the beginning of the loop (Step 2 to Step 6).

(11)

t∈Dt

which ensures that a vehicle cannot be assigned on two links
in a certain time interval t.

Step 3: Determine the descend direction. Find the shortest
1
using all-or-nothing assignment to generate
path with fkt,n
a
t,n1
φ , which is the shortest route pattern.

C. Solutions
Step 4: Determine the iteration step ζ n1 , which can be
designed as:

Considering the temporal continuous traffic flow, in time
interval t, take the initial state fk0a in (8) as fkt−1
. In this
a
paper, the test bench is based on Sioux Falls network, and the
departing time is the same during each time interval t. Then,
in time interval t, the DUE problem can be transfered as a
SUE problem. The SUE is based on (9a), (9b), and (9c), and
the Lagrangian can be generated as (12).
The Hessian matrix of (9a) is positive definite, which
indicate the convexity. The optimal travel flow θk∗ta can be
computed with the KKT conditions [32], [43]. Therefore, the
solution of the UTS in time interval t can be designed as in
Algorithm 1.

η

minζ n1

1
1
)
+ ζ n1 (φt,n1 − θkt,n
s.t. η = θkt,n
a
a

0≤ζ

=

viφt

−

j∈Ci
φ φt
2(ri Pi +

φ 2 φt
xφi Qφt
i ) + |zi | li ,

Step 6: Compare the result to the threshold ε1 as:
qP
t,n1 +1
1 2
)
− θkt,n
ka (θka
a
≤ ε1
P
t,n1
ka θka

(13b)
(13c)

(14)

(15)

where if (15) is fulfilled, the result of traffic flow in time
1 +1
; otherwise, go back to step 2, with
interval t is θkt,n
a
n1 = n1 + 1.

B. Objective Function and SDP Relaxation of OPF in PDS
The objective function contains two major parts: the generation cost and system line loss, which can be designed as
follows:

(16a)
(16b)

FPt DS =

φt
φt
where sφt
i = pi + iqi indicates the power injection on bus
φt
φt φt H
i, Si = Vi (Ii ) is the complex power from bus i to bus
i1 , viφt = Viφt (Viφt )H , liφt = Iiφt (Iiφt )H , ziφ = riφ + ixφi , Ci is
the set of buses that have a common ancestor bus i, and H
indicates the Hermitian transpose.
In addition, according to [50], the matrix Mt is defined as
" φt
#
vi
Siφt
t
M =
(17)
(Siφt )H liφt
t

≤1

1
1
1 +1
)
+ ζ n1 (φt,n1 − θkt,n
= θkt,n
θkt,n
a
a
a

The PDS can be represented in a radial graph as: G P DS =
[V
, E P DS ], where V P DS = {1, 2, · · · , nV P DS }, and E P DS
= {1, 2, · · · , mE P DS }. Similar as UTS, the PDS also contains
the temporal characteristics, the superscript (·)t , t ∈ Dt .
Considering the three-phase unbalanced system, the phase configuration is defined as φ ⊆ {a, b, c}. For a radial distribution
system, there is an unique i1 as the ancestor bus for bus i,
and the branch between bus i1 and i can be named as i [25],
[49]. The branch flow model in time interval t can be defined
as:

viφt
1

n1

1 +1
with
Step 5: Update θkt,n
a

P DS

(Sjφt − zjφt ljφt )),

(13a)

The Golden-section search is employed to determine ζ n1 in
a short time.

A. Branch Flow Model

X

fkta (ω)dω

ka 0

V. O PTIMAL P OWER F LOW OF P OWER D ISTRIBUTION
S YSTEM (L OWER L EVEL )

φt
sφt
i = diag(Si −

XZ

X X

φt
2
[α1 (pφt
i ) + β1 (pi )]

(18)

i∈V PDS φi

where α1 and β1 are two weight coefficients for the generation
cost and system line loss, respectively.
The constraints of the voltage viφt and current liφt are
illustrated as follows:

t

where in (17), M  0 ( M is positive semidefinite) and
Rank(Mt ) = 1.
6

viφ ≤ viφt ≤ viφ ,

(19a)

liφ ≤ liφt ≤ liφ ,

(19b)

t

L1 (θ, λ1 , µ1 ) =

θka
X X Z

ka ∈KA t∈Dt 0





X
X X
hdpr1u su (12)
hdpr1u su δpdr1ut su ka + µd1,ru su qrdu1 su −
fkta (w)dw + λt1,ka θkt a −

In this paper, the injection power sφt
i plays an important
role in active control and reactive control for the OPF.
sφi

≤

sφt
i

≤

Pi∗t ,

min

Sit ,sti ,vit ,lti

(20)

min

FPt DS

s.t. (16), (19), and Mt  0

FPt DS =

min

Sit ,sti ,vit ,lti

s.t. (16), (19), and M  0

where Pi∗t is the optimal power injection in (7b). Considering
the relationship with the UTS, the optimal injection power s∗t
i1
decides the number of EVs in CDS i1 (Here for CDS, i1 = i
for the PDS and UTS, which can be explained as Fig. 3 with
yellow circles).
According to the SDP relaxation introduced in [25], [50],
the constraint Rank(Mt ) = 1 of (17) can be removed, and
the relaxed objective problem is shown as follows:

φt φt
φt
sφt
i ,vi ,li ,Si

Pru su

Pru su d1 ∈Dt

φt,(x)
φt,(y) φt,(x)
, li,i1
Si,i1 = Si
φt,(y) φt,(x)
φt,(x)
, li
= Si
Si
φt,(y)
φt,(x)
= si
si

=
=

X

FPt DS,i

(24a)

i∈V PDS

φt,(y) φt,(x)
, vi,i1
li
φt,(y) φt,(x)
, vi
li

(24b)
=
=

φt,(y)
vi
φt,(y)
vi

(21a)
(21b)

(26a)

y t,k+1 ∈ arg min Lρ (xt,k+1 , y t , λt,k )

(26b)

λt,k+1 = arg min λt,k + ρ(xt,k+1 − y t,k+1 )

(26c)

y∈Ky

C. OPF Solution with ADMM
In the power-traffic system, the OPF computation of the
three-phase unbalanced PDS is the bottleneck of the whole
system operation. Firstly, because the real PDS usually contains a lot of feeders, which brings high computation loads
for the system. Secondly, the PDS is a highly dynamic
system, which requires a short computation time. Therefore,
the ADMM, a distributed algorithm is proposed to solve this
problem in a short time, which is based on Lagrange multiplier
with an additional penalty term (augmentation term).
The standard form of the ADMM can be formulated as
follows:

s.t. x ∈ Kx , y ∈ Ky , and x = y

(24e)

xt,k+1 ∈ arg min Lρ (xt , y t,k , λt,k )
x∈Kx

x,y

(24d)

where the variables (Sit , sti , vit , lit ) is maintained in bus i,
which can be seemed as an local agent. In the ADMM, the
x-update and y-update indicate as (·)(x) and (·)(y) .
Then, as in [25], [29] the augmented Lagrangian of (24)
can be derived as (25).
The detail formulation of fPt DS,i,y and fPt DS,i,x can be
found in [25], and at iteration k, the variables x, y, and λ
are illustrated as follows:

where (21) is a convex problem and can be solved with
ADMM.

min fA (x) + gA (y)

(24c)

x∈Kx

Considering the time intervals of the UTS, the ADMMbased distributed OPF computation architecture is built to
dramatically reduce the time consumption to the proposed
power-traffic coordinated operation approach.
Therefore, in addition to the designed user equilibrium in
the higher level, the detail models of UTS and PDS are build
to provide the spatial and temporal information in the lower
level. In the next section, a smart charging/discharging strategy
is proposed for the EVs and bidirectional CDSs to further
reduce the impact to PDS.

(22a)

VI. S MART EV C HARGING / DISCHARGING (L OWER
L EVEL )

(22b)

In this paper, the EVs and CDSs act as the reserves for both
PDS and UTS. In this section, we take CDS i1 as an example,
other CDSs contain the same performances. In time interval
t, nti1 EVs are parked in CDS i1 with parking time τ2 , which
is defined in (1). In real world, nti1 depends on many factors,
for example, different habits of the drivers, which are beyond
the scope of this paper. Here, nti1 is determined as follows:

where fA and gA are convex, and Kx and Ky are convex sets.
With an additional penalty term ρ/2||x − y||22 , the efficient
augmented Lagrangian function is shown as follows [29]:
ρ
Lρ (x, y, λ) = fA (x) + gA (y)+ < λ, x − y > + ||x − y||22 ,
2
(23)
where 0 ≤ ρ is a coefficient for convergence, and ρ2 ||x − y||22
is the quadratic penalty term to increase the converge speed.
Therefore, the relaxed objective function (21) can be formulated as
7

nti1 =

t t
∗t
≥ ̺1 + ̺2
 ̺3 ∗ min{s∗t
i1 /C1 , θka } If π
t t
∗t
̺4 ∗ min{si1 /C1 , θka } If ̺1 ≤ π ∗t ≤ ̺1 + ̺2

0
If ̺1 ≥ π ∗t

(27)

X

Ltρ (xt , y t , λt ) =

FPt DS,i + < λt , xti − yit > + < µti , xti,i1 − yit > +

X

FPt DS,i + < λt , xti − yit > +

X

where s∗t
i1 is the optimal power injection, which is determined in the OPF solution (26). C1t is the average discharging/charging speed of EVs. ̺1 is the electrical power price,
̺2 is the congestion fee, ̺3 is the ratio for EV parking to the
CDSs. nti1 < χi1 , χi1 is the capacity of CDS i1 .
During the parking time τ2 , the stochastic EVs charging/discharging can impact the stability of the PDS. Considering the SOC, a smart EVs charging/discharging approach is
proposed to reduce the impact to PDS as follows:
t
FCDS
= min
t

1
Cev,i
3 t1 ∈τ2

nt

t1
Cev,i
3

(28c)

i3

X

t1
Cev,i
, 0 ≤ i3 ≤ nti1
3

(28d)

t1 ∈τ2

Υ ≤ SOC ≤ Υ,

(28e)

t1
Cev,i
3

where
is the discharging (positive) and charging (negt1
t1
ative) speed of EV i3 at time t1 . In (28c), PCDS
(Cev,i
) is
3
the netload at CDS i1 , which equals to original load of PDS
QtP1DS minus the total charging/discharging power. In (28d),
2
Qτev,i
is the total demand of charging/discharging of EV i3 ,
3
which contains a SOC constraint as (28e), Υ is the ratio of
SOC.
VII. N UMERICAL S IMULATION

AND

(25b)

The simulation flowchart is shown in Fig. 4, and the
corresponding description is shown in Algorithm 2. In Step 2,
a distributed algorithm is used to reduce the time consumption
of the equilibrium computation. In Step 3, for each time
interval t, the DUE problem is transferred as a SUE problem.
The the Golden-section search is employed to determine the
iteration step ζ n1 in a short time for the UTS with about 10,000
EVs and 11 CDSs. In Step 4, because the test bench is based
on the IEEE 8,500-bus PDS, the ADMM is implemented to
computed the OPF in distributed manner with the constraints
from Step 3. In Step 5, for each CDS, the charging/discharging
are independent, the smart EVs charging/discharging can be
computed in parallel, which also helps to reduce the computation time. The computation time analysis is shown in Fig 5.
The two curves indicate the average computation times of
the proposed approach are less than 50 s with 100 scenarios.
Compared with the time interval 15 minutes, the proposed
approach is quick enough to support continuously operations
for the power-traffic systems.

(28b)
i1
X

(y)t

(25a)

A. Flowchart of the Simulation

(28a)

t1
t1
PCDS
(Cev,i
) = QtP1DS −
3

(x)t

ρ
> + fPt DS,i,y
2
ρ
> + fPt DS,i,x
2

̺1 = 45 $/M W h, and the congestion fee ̺2 = 2 $/h [54],
[55]. The EVs parking ratio ̺3 = 0.8, ̺4 = 0.3. The parking
capacity for each CDS is χ1 = 100. The simulations are
executed using a server with 3.60 GHz Intel Xeon CPU and 32
GB RAM, and the software are Python, MATLAB, MATLAB
global optimization toolbox, and parallel computing toolbox.
The communication network is good enough to transmit all
the information in real-time without any cybersecurity issues.

2
t1 +1
t1
t1
t1
PCDS (Cev,i3 ) − PCDS (Cev,i3 )

t1
≤ Cev
s.t. Cev ≤ Cev,i
3

2
Qτev,i
=
3

(y)t

< µtj , xtj,i − yjt > + < γit , vi1 ,i − vi1

j∈Ci

i∈V PDS

X 

(x)t

< γjt , vi,j − vi

j∈Ci

i∈V PDS

=

X

B. Bi-Peak Shaving and Bi-Ramp Smoothing

R ESULTS

As shown in Fig. 6(a), the blue curve is the netload peak of
PDS from 17:00 to 22:00, and the orange curve is the result of
proposed approach. The peak load decreases from 78 MW to
72 MW with 15% EVs discharging. The yellow curve is the
traffic delay peak in UTS, and the purple curve is the reduced
traffic delay with the proposed approach. The total peak traffic
delay time of UTS decreases from about 4,100 hours to 2,900
hours. In Fig. 6(b), it is also clear that from 8:00 to 15:00,
the netload of PDS increases from 48 MW to 51 MW, and the
total traffic delay of UTS decreases from 3,500 hours to 2,800
hours. From Fig. 6, it is clear that the proposed approach can
benefit both PDS and UTS to reduce the bi-peak and smooth
the bi-ramp.
In Fig. 7, the detailed traffic congestion information of
a congestion scenario is shown with the traffic maps. The
OD pair of this scenario is shown in Table I. The red and
pink arrows indicate the traffic congestions in the roads. The
traffic congestion information without the proposed approach

As shown in Fig. 3, the test bench is based on the IEEE
8,500-bus PDS and the Sioux Falls UTS [51], [52], which
cover the similar geographical areas. It is assumed that a small
city with 10,000 EVs has the bi-peak and bi-ramp problem,
the ”duck curve” data is based on [53] and the traffic behavior
data is based on [7]. 11 CDSs are located in node 2, 3, 5, 8,
10, 11, 17, 18, 20, 22, 23, which are shown with yellow circles
and illustrated in both PDS and UTS, respectively. The time
interval t is 15 minutes for the power-traffic systems. The
PDS cost function fPt DS is defined as quadratic function as
t
in [42], and the UTS cost function fUT
S is defined similarly.
t
t
Both fP DS and fUT S are convex, and the quadratic form
indicates the utility cost increase fast with the increasing
of PDS load and UTS load. As discussed above, the utility
t
objective function FUT
Y is convex. The weight factor γ1
equals to 1, which indicates the same importance of PDS
t
and UTS. Similarly, the customer objective function FCSO
is also set as a convex function. The electrical power price
8

7

18

6

8

16

5

9

10

15

22

21

4

11

14

23

24

3

12

18
2

2

17

19

20

17
8
20
22

10

5

23
11

3
1

13

Fig. 3. The test bench consists of a PDS and a UTS.

Algorithm 2 Simulation Process
Step 1: Initialization and data collection. Collect the data
and parameters of the PDS, UTS, CDSs, and EVs, such
as the topology information, electrical power price ̺1 ,
congestion fee ̺2 , set the simulation time interval t = 15
min, etc.

Objective
function of Utility
(PDS and UTS)

Objective
function of
Customers (EVs)

Optimal Charging/
EV Information
Discharging Price
∗
∗

Step 2: Higher level: equilibrium computation for the
defined utility part (PDS & UTS) and customer part (CDSs
& EVs). Build the utility and customer objective function
as (1) and (5). Then, compute the equilibrium π ∗t and Pi∗t
1
with (6) and (7).

1

UTS (DUE)

Parking EVs
in CDSs

Step 3: Lower level: DUE in UTS. Receive the information
, build the DUE objective
such as π ∗t , Nit1 , qrdu1 tsu and Pi∗t
1
function with constraints as in (9), solve it with Algorithm
1, and generate the traffic information such as θkt a and bdru1 su .

1

Traffic
Data
Information Collection
(OD Pair)

Traffic Information
1

1

∗

CDSs (Smart Charging/discharging)
CDS 3

CDS 5

Optimal
Power Flow

Step 4: Lower level: OPF in PDS. Receive the information
such as π ∗t , θkt a , and Pi∗t
, build the OPF objective function
1
as in (18) with the constraints (19) and (20), relax it as (21),
solve it as (25) and (26). Then, generate the s∗t
i1 to CDS.

∗

Upper Bound of
Power Injection
∗

1

Step 5: Lower level: smart EVs charging/discharging. Re∗t
t
ceive the information such as s∗t
i1 , π , and Ni1 , compute
the parking EVs for each CDS as (27), build the smart
charging/discharging as (28), then generate the feedback
Pnt t1
for PDS.
information nti1 for UTS and i3i1 Cev,i
3

1

CDS 23

Optimal Charging/
Discharging Price

Optimal
Power
Injection
1

,3

PDS (OPF)
Demand
Load

Data
Collection
(Load)

Fig. 4. The test bench consists of a PDS and a UTS.

is shown in Fig. 7(a) in 18:30. The red arrows indicate that
the traffic delays on these roads are larger than 15 minutes.
As shown in Fig. 7(b), with the proposed approach, the pink
arrows indicate that the traffic delays on these roads are less
than 15 minutes. The number of pink arrows (6 pink arrows)

Step 6: The power-traffic system update with the new UTS
and PDS information, then back to Step 2 with t = t + 1.

9

6
50

6.5

7

7.5

EV (thousand)
8

8.5

9

9.5

10

7

18

6

8

16

5

9

10

15

22

21

4

11

14

23

24

3

12

Computation time (Traffic)
Computation time (Power)

48
46

2

17

19

20

Computation Time (S)

44
42
40
38
36
34
32
30
0.9

0.95

1

1.05

1.1
Power (P.U.)

1.15

1.2

1.25

1.3
1

Fig. 5.

Computation time comparison with different traffic and load factor.
78

13

(a)
7

18

6

8

16

5

9

10

15

22

21

4

11

14

23

24

3

12

4.2
Original Netload
Proposed Approach
Original Traffic Hours
Proposed Approach

66

1.4

Power (MW)

2.8

60
17

18

19

Time (h)

20

Total Traffic Delay (Kh)

2
72

(a)

20

1

13

3.5

(b)
2.8

57

2.1

54

1.4

51

0.7

8

9

10

11

12

13

14

Fig. 7. (a) The traffic congestion scenario in 18:30, (b) The traffic congestion
scenario with proposed approach in 18:30.
Total Traffic Delay (Kh)

Original Netload
Proposed Approach
Original Traffic Hours
Proposed Approach

60

Power (MW)

19

0
22

21

63

48

17

Nodes
1
2
3
5
6
10

0
15

Time (h)

(b)
Fig. 6. (a) The bi-peak shaving and bi-ramp smoothing for PDS and UTS
from 17:00 to 22:00, (b) The peak-shaving and ramp-smoothing for UTS and
over-generation compensation for PDS from 8:00 to 15:00.

TABLE I
T HE OD PAIR OF UTS
13
14
18
20
200
350
240
300
270
210
360
200
200
300
220
345
250
320
260
450
300
210
270
250
200
200
345
345

21
256
200
270
230
300
200

23
345
310
345
345
300
300

C. Social Cost Analysis

Fig. 8, the cost of the proposed approach is lower, and it is
clear that the proposed approach can benefit more for the social
cost with the increasing number of EVs. In Fig. 8, the red
curve and cyan curve indicate the social cost with different
power factors. The EV number is 8000 in the UTS. From
Fig. 8, it is also clear that the proposed approach can benefit
more for the social cost with the increasing power factor. In
summary, the prosed approach can benefit both PDS and UTS
with different traffic scenarios and load factors. In addition, the
social cost increases more with the increasing of PDS load or
EV number, which indicates the quadratic objective function
of PDS and UTS. This design benefits the power-traffic system
to reduce the total social cost.

In Fig. 8, the blue curve with circle and green curve with
square are in one group, which indicates the social costs with
different number of EVs on the road. The load of PDS is 1.0
P.U., which means the total load of PDS is 60 MW. From

D. Smart EVs Charging/Discharging
To test the robustness of the proposed approach, we select
two continuous intervals load data, which contains the biggest

is much less than the number of red arrows (16 red arrows),
which also indicates the alleviation of traffic congestion with
the proposed approach. The total congestion delay reduces
from 4100 hours to 2900 hours.
In summary, from Fig. 6 and Fig. 7, the bi-peak and biramp problems can be reduced and smoothed by the proposed
approach, which benefits both PDS and UTS.

10

6
15000

6.5

7

EV (thousand)
8

7.5

8.5

9

9.5

discharging speeds are controlled to meet the constraint (28b).

10

Original Cost 1 (Traffic)
Proposed Approach (Traffic)
Original Cost 2 (Power)
Proposed Approach (Power)

VIII. C ONCLUSION
In this paper, a hierarchical approach is proposed to shave
the bi-peak and smooth the bi-ramp problem in the powertraffic system from higher level to lower level. The higher
level gives an “overview” for the whole system, and the lower
level gives the detailed description for each part. The EVs
and CDSs function as reserves for both the PDS and UTS
to utilize the flexibility and optimize the operations of the
power-traffic system. In the higher level, the PDS and UTS are
treated together to minimize the social cost, and the EVs and
CDSs are treated as customers to minimize their expenditure.
Then, an equilibrium is designed to determine the optimal
charging/discharging prices, total demand electrical power, and
total required EVs. In the lower level, considering the spatial
and time domain, the detail models of PDS and UTS are
built to specifically determine the power injection and EVs
behaviors. In each CDS, a smart EVs charging/discharging
approach is proposed to further reduce the impacts to the PDS.
In the numerical results, the test bench consists of the IEEE
8,500-bus PDS and Sioux Falls UTS with about 10,000 EVs
and 11 CDSs, which are used to demonstrate the feasibility
and effectiveness of the proposed approach.
In real-world implementation, the weather, human behavior,
and social issues bring the stochastic impacts to the powertraffic system, which increase the uncertainties and bring
more challenges for system operation. In addition, the other
systems such as the natural gas delivery system and the water
system can also impact the proposed system and result in
nonnegligible consequences. In the next step, other factors
such as stochastic of renewable energies, the departure time of
EVs, cybersecurity, multi-energy system, and human behaviors
will be taken into consideration.

Social Cost ($)

10000

5000

0
0.9

0.95

1

1.05

1.1
Power (P.U.)

1.15

1.2

1.25

1.3

Fig. 8. The different social costs comparison with different traffic and load
factors.

1000

180
Original Netload
Original Netload+EV Discharging

Power (KW)

120

500

60

EV Discharging Power (KW)

750

250

0
0

5

10

15

20

0

25

30

Time (Minutes)

(a)
1000

180
Original Netload
Original Netload+EV Discharging

Power (KW)

120

500

60

EV Discharging Power (KW)

750

R EFERENCES
[1] C.-M. Cheng, S.-L. Tsao, and P.-Y. Lin, “Seeds: A solar-based energyefficient distributed server farm,” IEEE Transactions on Systems, Man,
and Cybernetics: Systems, vol. 45, no. 1, pp. 143–156, 2015.
[2] M. Cui, J. Zhang, A. Florita, B.-M. Hodge, D. Ke, and Y. Sun,
“Solar power ramp events detection using an optimized swinging door
algorithm,” in Proc. ASME Int. Design Eng. Tech. Conf. Comput. Inf.
Eng. Conf, 2015.
[3] Y. Gu, H. Jiang, Y. Zhang, J. J. Zhang, T. Gao, and E. Muljadi,
“Knowledge discovery for smart grid operation, control, and situation
awarenessła big data visualization platform,” in North American Power
Symposium (NAPS), 2016. IEEE, 2016, pp. 1–6.
[4] Y. A. Katsigiannis, P. S. Georgilakis, and G. J. Tsinarakis, “A novel colored fluid stochastic petri net simulation model for reliability evaluation
of wind/pv/diesel small isolated power systems,” IEEE Transactions on
Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 40,
no. 6, pp. 1296–1309, 2010.
[5] Y. Tian and C.-Y. Zhao, “A review of solar collectors and thermal energy
storage in solar thermal applications,” Applied energy, vol. 104, pp. 538–
553, 2013.
[6] D. Connolly, H. Lund, B. V. Mathiesen, and M. Leahy, “A review of
computer tools for analysing the integration of renewable energy into
various energy systems,” Applied Energy, vol. 87, no. 4, pp. 1059–1082,
2010.
[7] E. J. Gonzales and C. F. Daganzo, “The evening commute with cars and
transit: Duality results and user equilibrium for the combined morning
and evening peaks,” Procedia-Social and Behavioral Sciences, vol. 80,
pp. 249–265, 2013.

250

0
0

5

10

15
Time (Minutes)

20

25

30

0

(b)
Fig. 9. (a) The normal EV discharging in CDS 3 without considering the
impacts to the PDS, (b) smart EV discharging in CDS 3 considering the
impacts to the PDS.

deviations within the 30 minutes. In Fig. 9, EV discharging
is taken as an example and 50 EVs are employed to test
the proposed approach. As shown in Fig. 9(a), without the
proposed approach, the stochastic EV discharging behaviors
increase the deviations of the netload. In Fig. 9(b), with the
proposed smart charging/discharging approach, the deviations
of the netload are smoothed, which decreases the impacts to
the PDS during the EV discharging. At the same time, the
11

[8] Y.-H. Cheng, Y.-H. Chang, and I. Lu, “Urban transportation energy and
carbon dioxide emission reduction strategies,” Applied Energy, vol. 157,
pp. 953–973, 2015.
[9] Z. Huang, X. Xu, H. He, J. Tan, and Z. Sun, “Parameterized batch
reinforcement learning for longitudinal control of autonomous land vehicles,” IEEE Transactions on Systems, Man, and Cybernetics: Systems,
2017.
[10] A. Foley, B. Tyther, P. Calnan, and B. Ó. Gallachóir, “Impacts of electric
vehicle charging under electricity market operations,” Applied Energy,
vol. 101, pp. 93–102, 2013.
[11] C. Liu, J. Wang, A. Botterud, Y. Zhou, and A. Vyas, “Assessment
of impacts of PHEV charging patterns on wind-thermal scheduling by
stochastic unit commitment,” IEEE Transactions on Smart Grid, vol. 3,
no. 2, pp. 675–683, 2012.
[12] D. Wu, D. C. Aliprantis, and L. Ying, “Load scheduling and dispatch for
aggregators of plug-in electric vehicles,” IEEE Transactions on Smart
Grid, vol. 3, no. 1, pp. 368–376, 2012.
[13] E. Sortomme and M. A. El-Sharkawi, “Optimal scheduling of vehicle-togrid energy and ancillary services,” IEEE Transactions on Smart Grid,
vol. 3, no. 1, pp. 351–359, 2012.
[14] H. Jiang, Y. Zhang, J. J. Zhang, D. W. Gao, and E. Muljadi,
“Synchrophasor-based auxiliary controller to enhance the voltage stability of a distribution system with high renewable energy penetration,”
IEEE Transactions on Smart Grid, vol. 6, pp. 2107–2115, 2015.
[15] C. Heymans, S. B. Walker, S. B. Young, and M. Fowler, “Economic
analysis of second use electric vehicle batteries for residential energy
storage and load-levelling,” Energy Policy, vol. 71, pp. 22–30, 2014.
[16] X. Geng and L. Xie, “Learning the lmp-load coupling from data: A
support vector machine based approach,” IEEE Transactions on Power
Systems, vol. 32, no. 2, pp. 1127–1138, 2017.
[17] P. Sadeghi-Barzani, A. Rajabi-Ghahnavieh, and H. Kazemi-Karegar,
“Optimal fast charging station placing and sizing,” Applied Energy, vol.
125, pp. 289–299, 2014.
[18] F. He, D. Wu, Y. Yin, and Y. Guan, “Optimal deployment of public
charging stations for plug-in hybrid electric vehicles,” Transportation
Research Part B: Methodological, vol. 47, pp. 87–101, 2013.
[19] W. Wei, S. Mei, L. Wu, M. Shahidehpour, and Y. Fang, “Optimal
traffic-power flow in urban electrified transportation networks,” IEEE
Transactions on Smart Grid, vol. 8, no. 1, pp. 84–95, 2017.
[20] B. Subhonmesh, S. H. Low, and K. M. Chandy, “Equivalence of
branch flow and bus injection models,” in Communication, Control, and
Computing (Allerton), 2012 50th Annual Allerton Conference on. IEEE,
2012, pp. 1893–1899.
[21] B. Stott, J. Jardim, and O. Alsaç, “Dc power flow revisited,” IEEE
Transactions on Power Systems, vol. 24, no. 3, pp. 1290–1300, 2009.
[22] A. G. Bakirtzis, P. N. Biskas, C. E. Zoumas, and V. Petridis, “Optimal
power flow by enhanced genetic algorithm,” IEEE Transactions on
power Systems, vol. 17, no. 2, pp. 229–236, 2002.
[23] Y. Gu, H. Jiang, Y. Zhang, and D. W. Gao, “Statistical scheduling of
economic dispatch and energy reserves of hybrid power systems with
high renewable energy penetration,” in 2014 48th Asilomar Conference
on Signals, Systems and Computers, 2014, pp. 530–534.
[24] Q. Peng and S. H. Low, “Distributed algorithm for optimal power flow
on a radial network,” in 2014 IEEE 53rd Annual Conference on Decision
and Control (CDC). IEEE, 2014, pp. 167–172.
[25] ——, “Distributed algorithm for optimal power flow on an unbalanced
radial network,” in Decision and Control (CDC), 2015 IEEE 54th Annual
Conference on. IEEE, 2015, pp. 6915–6920.
[26] E. Dall’Anese, H. Zhu, and G. B. Giannakis, “Distributed optimal power
flow for smart microgrids,” IEEE Transactions on Smart Grid, vol. 4,
no. 3, pp. 1464–1475, 2013.
[27] A. Y. Lam, B. Zhang, and N. T. David, “Distributed algorithms for
optimal power flow problem,” in Decision and Control (CDC), 2012
IEEE 51st Annual Conference on. IEEE, 2012, pp. 430–437.
[28] N. Li, L. Chen, and S. H. Low, “Demand response in radial distribution
networks: Distributed algorithm,” in Signals, Systems and Computers
(ASILOMAR), 2012 Conference Record of the Forty Sixth Asilomar
Conference on. IEEE, 2012, pp. 1549–1553.
[29] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed
optimization and statistical learning via the alternating direction method
of multipliers,” Foundations and Trends R in Machine Learning, vol. 3,
no. 1, pp. 1–122, 2011.
[30] M. Patriksson, The traffic assignment problem: models and methods.
Courier Dover Publications, 2015.
[31] N. Jiang, C. Xie, J. C. Duthie, and S. T. Waller, “A network equilibrium
analysis on destination, route and parking choices with mixed gasoline

[32]
[33]
[34]

[35]

[36]
[37]

[38]

[39]

[40]
[41]

[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]

[53]
[54]
[55]

12

and electric vehicular flows,” EURO Journal on Transportation and
Logistics, vol. 3, no. 1, pp. 55–92, 2014.
Y. Sheffy, “Urban transportation networks: equilibrium analysis with
mathematical programming methods,” Traffic engineering control.
Prentice-Hall, ISBN 0-13-93-972, 1985.
Y. He, X. Liu, C. Zhang, and Z. Chen, “A new model for state-of-charge
(soc) estimation for high-power li-ion batteries,” Applied Energy, vol.
101, pp. 808–814, 2013.
B. Pattipati, C. Sankavaram, and K. Pattipati, “System identification and
estimation framework for pivotal automotive battery management system
characteristics,” IEEE Transactions on Systems, Man, and Cybernetics,
Part C (Applications and Reviews), vol. 41, no. 6, pp. 869–884, 2011.
H. Sedjelmaci, S. M. Senouci, and N. Ansari, “A hierarchical detection
and response system to enhance security against lethal cyber-attacks in
uav networks,” IEEE Transactions on Systems, Man, and Cybernetics:
Systems, 2017.
H. Jiang, J. J. Zhang, W. Gao, and Z. Wu, “Fault detection, identification,
and location in smart grid based on data-driven computational methods,”
IEEE Transactions on Smart Grid, vol. 5, pp. 2947 – 2956, 2014.
H. Jiang, X. Dai, W. Gao, J. Zhang, Y. Zhang, and E. Muljadi, “Spatialtemporal synchrophasor data characterization and analytics in smart
grid fault detection, identification and impact causal analysis,” IEEE
Transactions on Smart Grid, vol. 7, no. 5, pp. 2525–2536, 2016.
C.-W. Ten, G. Manimaran, and C.-C. Liu, “Cybersecurity for critical
infrastructures: Attack and defense modeling,” IEEE Transactions on
Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 40,
no. 4, pp. 853–865, 2010.
K. Islam, W. Shen, and X. Wang, “Wireless sensor network reliability
and security in factory automation: A survey,” IEEE Transactions on
Systems, Man, and Cybernetics, Part C (Applications and Reviews),
vol. 42, no. 6, pp. 1243–1256, 2012.
X. Geng and L. Xie, “A data-driven approach to identifying system
pattern regions in market operations,” in Power & Energy Society
General Meeting, 2015 IEEE. IEEE, 2015, pp. 1–5.
H. Jiang, Y. Li, Y. Zhang, J. J. Zhang, D. W. Gao, E. Muljadi, and Y. Gu,
“Big data-based approach to detect, locate, and enhance the stability of
an unplanned microgrid islanding,” Journal of Energy Engineering, vol.
143, no. 5, p. 04017045, 2017.
J.-B. Park, K.-S. Lee, J.-R. Shin, and K. Y. Lee, “A particle swarm
optimization for economic dispatch with nonsmooth cost functions,”
IEEE Transactions on Power systems, vol. 20, no. 1, pp. 34–42, 2005.
S. Boyd and L. Vandenberghe, Convex optimization.
Cambridge
university press, 2004.
N. Li, L. Chen, and S. H. Low, “Optimal demand response based on
utility maximization in power networks,” in Power and Energy Society
General Meeting, 2011 IEEE. IEEE, 2011, pp. 1–8.
D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation:
numerical methods. Prentice hall Englewood Cliffs, NJ, 1989, vol. 23.
B. N. Janson, “Dynamic traffic assignment for urban road networks,”
Transportation Research Part B: Methodological, vol. 25, no. 2, pp.
143–161, 1991.
T. A. Manual, “Bureau of public roads,” US Department of Commerce,
1964.
C. Chekuri, S. Khanna, and F. B. Shepherd, “The all-or-nothing multicommodity flow problem,” in Proceedings of the thirty-sixth annual
ACM symposium on Theory of computing. ACM, 2004, pp. 156–165.
S. H. Low, “Convex relaxation of optimal power flowłpart i: Formulations and equivalence,” IEEE Transactions on Control of Network
Systems, vol. 1, no. 1, pp. 15–27, 2014.
L. Gan and S. H. Low, “Convex relaxations and linear approximation for
optimal power flow in multiphase radial networks,” in Power Systems
Computation Conference (PSCC), 2014. IEEE, 2014, pp. 1–9.
Distribution
test
feeder.
[Online].
Available:
http://ewh.ieee.org/soc/pes/dsacom/testfeeders/index.html
W. H. Lam, H. Shao, and A. Sumalee, “Modeling impacts of adverse
weather conditions on a road network with uncertainties in demand and
supply,” Transportation research part B: methodological, vol. 42, no. 10,
pp. 890–910, 2008.
California ISO today’s outlook details. [Online]. Available:
http://www.caiso.com/Pages/Today’s-Outlook-Details.aspx
Eia
us
energy
information
administration
regional
wholesale
markets.
[Online].
Available:
https://www.eia.gov/electricity/monthly/update/wholesale markets.php
A. de Palma and R. Lindsey, “Traffic congestion pricing methodologies
and technologies,” Transportation Research Part C: Emerging Technologies, vol. 19, no. 6, pp. 1377–1399, 2011.

