JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 1
Distributionally Robust Day-ahead Scheduling for Power-traffic
Network Considering Multiple
Uncertainties under a Potential Game Framework Haoran Deng, Bo
Yang, Jiaxin Cao, Chao Ning, Cailian Chen and Xinping Guan
Abstract—Widespread utilization of electric vehicles (EVs) incurs
more uncertainties and impacts on the scheduling of the
power-transportation coupled network. This paper investigates
optimal power scheduling for a power-transportation coupled network
in the day-ahead energy market considering multiple uncertainties
related to photovoltaic (PV) generation and the traffic demand of
vehicles. Specifically, the EV drivers choose the lowest-cost
routes in response to electricity prices and the power operators
maximize their operating profits. Furthermore, we show the
interactions between the power system and EV users from a potential
game-theoretic perspective, and the scheduling problem of the
equilibrium of power-transportation coupled network can be
interpreted by a two-stage distributionally robust optimization
(DRO) model. In addition, uncertainties of PV generation and
traffic demand are established as scenario- based ambiguity sets
combined with the historical distribution information,
respectively. A combination of the duality theory and the Benders
decomposition is developed to solve the DRO model. Simulation
results demonstrate the effectiveness and applicability of the
proposed approach.
Index Terms—Power-traffic coupled network, day-ahead power
scheduling, potential game, ambiguity set, distributionally robust
optimization (DRO).
I. INTRODUCTION
IN the past decade, transportation electrification grows rapidly as
a result of the increasing electric vehicles
(EVs) charging demand with a positive role in alleviating
environmental pollution of EVs for metropolis [1]. Public fast
charging stations (FCSs) become the main source of energy for EVs
gradually. The spatial-temporal distribution characteristics of EVs
flow can lead to the uncertainty of charging load in the power
network (PN), which will result in a direct impact on the
scheduling strategy [2]. On the other hand, with the large-scale
application of renewable energy, the stability of the power network
is impacted seriously by the uncertainty of generated energy. As a
result, the day-ahead energy dispatch of the power-transportation
coupled network gains more and more research interest. Under this
context, an urgent need for uncertainty modeling tools of this
coupled network is required.
In the traffic engineering literature, the modeling of traffic flow
patterns considering the charging demand more prac- tically has
been focused on by numerous researchers. In
Corresponding author: Bo Yang. The authors are with the Department
of Automation, Shanghai Jiao Tong
University, Shanghai 200240, China, the Key Laboratory of System
Con- trol and Information Processing, Ministry of Education of
China, Shanghai 200240, China, and also with Shanghai Engineering
Research Center of Intelligent Control and Management, Shanghai
200240, China.
this connection, He et al. proposed an equilibrium modeling
framework that can capture the destination and route selection of
EV travelers to maximize social welfare [3]. On the other hand, the
travel plans of EVs can also be affected by queuing time and
electricity prices, which further impacts the flow distribution of
transportation networks (TN) [4], [5]. Alizadeh et al. proposed a
scheme in which independent operators can collaborate to manage
each network towards a socially optimum operating point taking into
account traffic congestion and the variations of electricity prices
for battery charging in spatial space [6]. Considering a locational
marginal pricing (LMP) scheme of the coupled networks, Wei et al.
proposed a traffic assignment problem (TAP) to derive the network
equilibrium, which can alleviate traffic congestion effectively
[7], [8].
EV route choices and traffic distribution, which could be
influenced by electricity price strategies, can redistribute the
charging load of PN. To this end, the authors in [9] studied the
interdependency of the coupled networks by combining with TAP. On
the other hand, a game relation exists between the two networks.
Game theory provides an effective method to investigate the
profit-maximizing strategies among players. Moreover, extensive
game-theoretic approaches about the traf- fic assignment [10], [11]
and EV charging problems [12]- [14] have been developed, which
belong to the cooperative game. Meanwhile, there are some
researches about the non- cooperative game combined with the
coupled network through the charging behavior. The interconnections
of the coupled networks are reflected in the matching of the supply
and demand, which can be described by the power balance con-
straints considering the charging demand of EVs. However, few
studies concern the day-ahead scheduling optimization of the
coupled network based on the potential game-theoretic. In [15], the
authors formulated the charging problem as a potential game to
achieve the co-optimization of the payoff for utility companies and
customers, while the model of the transportation network was
simplified massively through some assumptions proposed. The authors
in [17] analyzed the interdependency between the EV travelers and
power operators by establishing the potential game-theoretic model,
due to the complexity of solution methodology for the ambiguity set
of uncertain variables, the uncertainty of the coupled network was
not discussed and the temporal demand of EV loads was neglected,
which will reduce the practicability of the proposed method. In our
paper, we obtain the interdependence between the EVs and PNs
considering the uncertainties aiming
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JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 2
to investigate the day-ahead power scheduling with the Nash
equilibrium.
For the power-transportation coupled network, researchers usually
describe the PN by optimal power flow models to formulate a
coordinated operation framework, e.g. [3], [16], [18]–[20], while
the power scheduling of PN with a macro- scopic perspective is
investigated in our research. On the other hand, in some literature
[19], [20], deterministic optimization scheduling models are
formulated with the specified value on system parameters. However,
the uncertainties due to the variations of traffic demand and
charging demand are hardly considered in the deterministic models,
let alone the renewable power generation in the
power-transportation coupled systems. Authors in [21] established a
two-stage robust optimization (RO) model considering multiple
uncertainties for the coupled network of PN and TN, which was
solved by a delayed con- straint generation algorithm. Similarly,
in [22], the model with two-stage robust optimization considering
the uncertainties of wind power generation and electric demand was
developed, and the column-and-constraint generation (C&CG)
method was developed to solve it. In [23], the authors introduced
an adaptive three-stage robust model considering the uncertain load
and power production with bounded intervals. However, the RO model
in the above literature considering the up- per/lower boundary
values of uncertain variables can result in over-conservative
decision-making [24], due to the probability distribution
information of uncertain variables and the scenario representation
are not incorporated in the optimization prob- lem.
In this regard, a robust optimization framework, called
distributionally robust optimization (DRO), has gained the
attention of numerous researchers for analyzing uncertainty
problems. The risk aversion caused by the unknown prob- ability
distribution of uncertainty can be charactered more adequately by
DRO models. A distributionally robust chance- constrained model was
established in [25] with uncertainties of wind power generation and
load demand for power distribu- tion system planning. The authors
in [26] developed a DRO model with an ambiguity set based on
Wasserstein distance (DROW) to handle the uncertainties from
electricity price, wind power, and PV generation effectively. In
Ref. [27] and [28], a scenario-based DRO model considering long and
short term uncertainties of the total net demand was applied.
Authors in [29] proposed a data-driven RO method, and the norm-1
and norm-inf were adopted to construct the confidence set of wind
power probability distribution, and then the availability of this
method was testified.
Based on the existing literature, the studies on a distri-
butionally robust day-ahead scheduling model of the power-
transportation coupled network considering source-load un-
certainties are rarely reported. How to efficiently model the
optimization problem with multiple uncertainties considering the
game relations of the EV charging behavior to attain the optimum
operating point for the two networks respectively and solve the
two-stage DRO model with scenario-based ambiguity set are
challenging. The major contributions of this paper are as
follows:
1) The spatial-temporal interaction of TN and MGs coupled
via FCSs is investigated. In the TN model, a TAP considering the
charging cost of EVs is proposed and incorporated with the Wardrop
user equilibrium (UE) to capture the selfish behaviors of EV
travelers. Meanwhile, some nonlinear constraints about latency
travel time and UE complementary condition are proposed and handled
by linearization techniques.
2) A game-theoretic model is established to represent the charging
behavior of EVs and derive the revenue-maximizing microgrids (MGs)
and cost-minimizing EV travelers with multiple uncertainties. We
show that the game is an ordinal potential game and propose a
concave potential function that can be transformed into an
equivalent mathematical problem to obtain the optimal strategies of
two networks simultaneously.
3) The uncertainties of the coupled network are transformed into
source-load uncertainties of the power network and the uncertainty
sets can be built simultaneously. Furthermore, a day-ahead power
scheduling model based on the potential function is proposed
considering intra-day charging game and traffic flow state. Then
the model can be transformed into a two-stage DRO model on account
of the existence of the PV generation and traffic demand
uncertainties. For reducing the conservativeness of this model, an
ambiguity set incorporating the probability distribution of the
uncertain parameters is established. In this way, the
master-subproblem framework can be obtained by the DRO model
through the duality theory, which can be solved by the Benders
decomposition approach effectively.
The rest of the paper is organized as follows. In Section II, the
basic structure and mathematical model of the coupled network are
introduced. In Section III, we propose a game- theoretic model with
uncertainties for the coupled network and present the equilibrium
properties. Section IV and V define the ambiguity set and present a
DRO two-stage optimization model, and then the corresponding
solution methodology is introduced. Numerical results and
conclusions are analyzed in Sections VI and VII,
respectively.
II. PROBLEM FORMULATION AND MODEL
In this paper, daily 24-hour profiles are employed to rep- resent
the dynamic characteristics of active network manage- ment.
According to Ref. [2], the characteristics varying with the
situation in hours are regarded as scenarios, which are represented
by the subscript s. The hour set can be defined as T , ∀s ∈ T , and
T = {1, 2, · · · , 24}.
A. Transportation System Modeling
Assumptions: Herein, we make some important modeling assumptions
for the transportation system. 1) It is a non- atomic measure for
each traveler to control a negligible traffic flow and the
influence of a single vehicle is infinitesimal. 2) The monetary
value of travel time for EV travelers is homogeneous, which is
represented by ω. 3) The heteroge- neous information of vehicles is
neglected, such as the state of charging, unit energy consumption,
and capacity of the battery. 4) The gasoline vehicles (GVs) and EVs
without plenty of electricity to destinations are neglected. 5) We
assume that EV travelers can acquire the traffic congestion
information
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 3
for each path and the electricity price of each FCS, and then each
traveler chooses the minimum cost route spontaneously to travel and
charge. 6) Assume that the charging demand of each EV in scenario s
is uniform, in other words, the power demand for each link is
proportional to the link flow, and the charging service fee is
neglected.
A TN can be represented by GTN = (N ,L), where N and L ⊆ N × N
denote the set of nodes and links, respectively. A pair (a, b) ∈ L
denotes the link from node a to node b.
1) Traffic demand In the transportation network, each vehicle has a
pair of
origin and destination and travels between them, which are named
O-D pairs. And the set of O-D pairs is denoted by R, specifically,
(o, d) ∈ R. The traffic demand (also called trip rate) of O-D pairs
in scenario s can be described by qods , denoted in (1), which
defines the number of vehicles in each scenario intending to travel
from o to d.
qods = [ qod1 , · · · , qod24
]T (1)
2) O-D flow Each O-D pair is connected by several paths, which
consists
of certain links [30]. The set of feasible routes p is denoted by p
∈ Pod,Pod = {1, · · · , Nod}, where Nod denotes the number of the
routes for each O-D pair. Let fodp,s be the traffic flow of path p,
and the traffic demand balance equation is described by (2), and
Eq. (3) denotes the non-negativity of path flows.
qods =
fodp,s ≥ 0 (3)
3) Link flow In the transportation system, the set of all links
with FCS
is represented by LC ⊆ L. The traffic link flow is equal to the
total number of vehicles of the paths through it, which is
described by (4).
xl,s(f od p,s) =
fodp,sδ od l,p (4)
where an indicator variable δodl,p is noted as reflecting the link-
path relation. The value of δodl,p is set to 1, if link l ∈ Lp,
where Lp is a set of links belonging to path p; otherwise, δodl,p
is set to 0.
4) Traffic expense EV travelers choose their driving routes mainly
based on
the total travel time, which is affected by the congestion level.
The travel time can be described by a latency function tl,s(xl,s),
which is also called the Bureau of Public Roads (BPR) function [31]
and can reflect the delayed travel time in links accurately.
tl,s(xl,s) = t0l [ 1 + 0.15(xl.s/Cl)
(l ∈ L) (5)
xl,s ≤ Cl (6)
where t0l is free-speed time and Cl is the link flow when tl,s =
1.15t0l , i.e., the link capacity. For constraint (6), if xl,s >
Cl, the travel time will be penalized by a quick growth.
Since only EVs are considered in this study, the traffic expense
should include charging expenses for vehicles. λl,s is defined as
the electricity price of FCS on link l in scenario s, and then the
charging energy cost of a single vehicle is λl,ses, where es is the
charging demand of one vehicle.
In summary, the total expense by a single EV on path p from o to d
in scenario s can be calculated as
costods,p = ∑ l
[ ωtl,s(xl,s) + λl,ses
] δodl,p (l ∈ L) (7)
where ω is the monetary value of travel time and λl,s is equal to 0
in link l without an FCS.
5) Wardropian traffic assignment The balanced distribution of
traffic flow is analyzed by the
Wardrop UE principle, which can achieve a stable situation of
traffic network that no vehicle can decrease the total cost by
changing its driving route unilaterally. In other words, a stable
situation occurs when total travel costs on all active routes are
equal. The UE principle, which is also called Wardrop’s first
principle, is closer to the real situation than the second
principle [32].
Then a traffic assignment problem considering the charging cost is
described as follows, in which each traveler wishes to minimize his
travel expense.
TAP : min fod p,s
(8)
According to classic transportation theory, the UE traffic state
can be characterized as (9).{
fodp,s(cost od s,p − uods ) = 0,
costods,p − uods ≥ 0 (9)
where uods is the minimal travel cost for one vehicle between an
O-D pair.
Proposition 1: The solution to problem (8) is an UE flow pattern
satisfying (9).
Proof: The proof of this equivalency can be acquired by
generalizing the method in [33]. In detail, the Lagrangian function
of TAP for the equality constraint (2) and inequality constraint
(3) can be formulated as
LTN(fodp,s, u od s ,v
od s,p) =
]
where uods and vods,p are the vectors of Lagrangian multipliers.
The optimal solution of (8) satisfies the Karush-Kuhn-Tucher
conditions as follows:
Stationarity: ∂LTN
∂fod p,s
Complementary slackness: vods,pf od p,s = 0.
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 4
Further considering that
s − vods,p
= 0
Then vods,p = costods,p − uods ≥ 0 and vods,pf od p,s = 0 can
be
obtained, which is equivalent to (9). It is observed that
satisfying condition (9) is equivalent to
reaching the traffic UE state, and thus it can be considered as a
complementary constraint for the optimization problem (8), whose UE
state of the optimal solution can be ensured [34]. Meanwhile, the
constraint (9) can be linearized as follows by the big-M method
[35].{
0 ≤ fodp,s ≤M(1− wod p,s),
0 ≤ costods,p − uods ≤Mwod p,s
(10)
where M is a large enough constant and wod p,s is the
introduced
binary variable, i.e., wod p,s ∈ {0, 1}.
Furthermore, since the bivariate terms exist in Eq. (5), a
piecewise linearization method [2] is adopted to transform it as
(11).
tl,s = t0l +
J∑ j
(gl,jxl,j,s)
(11)
where J is the number of linear segments; gl,j is the linear
segment slope; and xl,j,s is the link flow of segment j.
Consequently, the linear optimization problem for TN can be
constructed as follows:
min FTN(fodp,s, λl,s) , ∑ s
∀l ∈ L, s ∈ T , od ∈ R, p ∈ Pod
(12) The objective of the traffic network optimization
problem
(12) is to find the optimal traffic flow distribution at a certain
electricity price.
B. Power System Modeling
An urban power network consists of multiple independently- owned
MGs, which is depicted in Fig. 1. MGs are constructed by a
decentralized structure with a flexible energy allocation and
managed by an MG operator [36]. In this study, an MG is equipped
with a PV power generator and a dispatchable generator (DG) to
provide energy to the traditional load and the charging load of EV
users. Note that we assume MGs can sell their redundant energy to
the main grid. Generally, we can assume that it is sufficient for
the transmission line
capacity, and the line loss and charging loss can be neglected. In
the MG operation model of this paper, we omit the energy storage
due to the limited space of paper, but it can be readily introduced
into this paper to make it more practical. The index of MGs is
denoted by i, and i ∈ B = {1, 2, · · · , Nb}, where Nb is the
number of buses or MGs.
Fig. 1. Illustration of MG electrical network.
The objective is to maximize the operation revenue of MG i, which
consists of five parts as shown in (13a). And the cost of PV
generation is ignored.
max pi
0 ≤ pi0,s, p0i,s ≤ P 0i (13c)
|p0i,s − p0i,(s−1)| ≤ PR 0i , s ∈ T \[1] (13d)
pGi,s − pi0,s + p0i,s + pPV i,s = pLti +
∑ l
∀l ∈ L, s ∈ T , i ∈ B
where pi , {pi0,p0i,pGi} is the decision vector of MG i. pi0, p0i
and pGi denote the energy transfer from MG i to the main grid, the
energy import from the main grid to MG i of the day-ahead energy
scheduling, and the DG generation power. The first item in (13a) is
the revenue for providing energy to the traditional load, and λC is
the contact price of the main grid. The second item is the income
of EVs charging, where δl,i = 1 when the link of MG i serving for
is l, otherwise δl,i = 0. The third and fourth items are energy
trading revenue between MGs and the main grid, in which λB is the
buyback price determined by the main grid. The operation cost of DG
generation in MG i can be calculated by Eq. (13f), where a, b and c
denote the cost parameters.
The constraints (13b) and (13c) specify the feasible range of DG
power output and the energy trading between MGs and the main grid,
respectively. The constraint (13d) indicates that the ramping rate
limits PR
0i of the energy imported from the main grid to MG i should be
satisfied at two adjacent hours. Eq. (13e) denotes the energy
conservation constraint within each MG, where pPV
i,s is the PV power output. In addition, the electricity demands of
FCS depend on the traffic flow on the pertaining link.
In this power-transportation coupled network, each EV traveler
intends to minimize its total travel cost by route selection
autonomously under the uncertainty of traffic de- mand, and the
traffic flow distribution can be analyzed by the
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 5
convex optimization model (8). Meanwhile, each MG operator
determines the day-ahead power schedule according to the charging
load caused by traffic flows under the uncertainty of PV
generation. As a result, the supply and demand of the power network
are matched. In addition, the uncertainties of this network will be
discussed in Section IV.
III. GAME-THEORETIC ANALYSIS
In this section, we construct the EV charging performance as a
potential game to optimize the costs of MGs and the payoffs of
drivers simultaneously. In addition, obtaining a Nash equilibrium
of this game is equivalent to solving a centralized optimization
problem, which is transformed from Section II [37], [38].
A. Game-theory with uncertainty
In the power network, MGs determine the power schedule and maximize
their operation revenues by providing energy at the electricity
prices. And in the transportation network, selecting the
lowest-cost routes and charging in FCSs for EV drivers is
determined by the electricity prices and the degree of traffic
congestion. Meanwhile, the uncertain variables in the two networks
are pPV
i,s and qods , respectively. There are interconnections between the
strategies and revenues of the EV travelers and MGs, and thus it
can be interpreted as a noncooperative game with
uncertainties.
Definition 1: The game can be defined by a triplet Ξ ={ {Q∪B},
{{Pod}od∈R, {Pi}i∈B}, {(−FTNv)v∈Q, (Ri)i∈B}
} , and uncertain variables of the coupled network exist in the
strategy sets. The components of game Ξ are described as
follows:
1) Customer Side (EVs) The players noted as Q in the traffic
network are the
EV travelers corresponding to the travel demand in dif- ferent O-D
pairs, and player v belongs to Q , [0, qods ], which is an interval
for the number of players. The strategy set consists of all routes
connecting the O-D pair, i.e., Pod = {p1, p2, · · · , pNod
}. Each EV traveler aims to selfishly choose its route and FCS to
maxi- mize its traveling utility with uncertain travel demand,
i.e., max fv∈Fv
(−FTNv) = max fv∈Fv
0 costods,p(θ)dθ], where
FTNv is the total traffic cost of traveler v, fv is the path flow
on the route selected by traveler v, and Fv is the set of fv
.
2) Supplier Side (MGs) The players noted as B in the power network
are MG
operators, and player i belongs to B , {1, 2, · · · , Nb} , where
Nb is the number of MGs. The power decision vector for MG i is pi,
and the set of Pi = {p1,p2, · · · ,pNi
} is the strategy set, where Ni is the number of MG i’s power
generation strategies. And the revenue maximization problem for MG
i with uncertain PV generation can be described by max pi∈Pi
Ri(pi,s, λl,s).
The above game-theoretic perspective of this coupled net- work is
shown in Fig. 2, where qods and pPV
i,s denote the uncertain parameters of traffic demand and PV power
output.
The game is designed to derive the global optimums for EVs and MGs
with the constraints of the optimization problem (12) and (13),
respectively. It is noted that the number of player v is uncertain
due to the uncertainty of traffic demand, which leads to the
existence of an uncertain vector in the strategies of the
transportation network. Meanwhile, when qods is determined, fodp,s
is variational with the changing of the route selection and can be
the decision variable of EV travelers. Note that the power network
revenue function does not contain the uncertain variable pPV
i , which is only reflected in constraints.
Fig. 2. The game-theoretic perspective of the power-transportation
coupled network.
B. Potential function construction
Under the above mathematical framework, the whole prob- lem turns
out to be seeking the Nash equilibrium of the game. The
corresponding Nash equilibrium is a strategy profile on which no
traveler can improve its utility by unilaterally chang- ing its
route at a certain electricity price, i.e., the Wardrop UE, and no
MG can benefit more by switching to another strategy except the
optimal power strategy.
Proposition 2: The Nash equilibrium can be obtained by solving an
equivalent centralized optimization problem, and the game is a
potential game with potential function as follows:
Φ(fodp,s,pi,s) = − ∑ s
[−c(pGi,s)− λCp0i,s + λBpi0,s] (14)
where the variables satisfy the constraints of the optimization
problems (12) and (13).
Proof: 1) Variable Declaration There are two kinds of decision
vectors as follows. In
transportation network, f i = (fi,1, fi,2, · · · , fi,24) is the
traffic flow in the route selected by EV i, and f−i is the vector
of the traffic flow in the route selected by all EVs except EV i,
defined as f−i = (f1, · · · ,f i−1,f i+1, · · · ,f qods
). (f−i,f i)
and (f−i, f i) are the arbitrary two strategies of EV i. In the
same way, in the power network, pi is the power profile of MG i,
and p−i is the power profiles of all MGs except MG i defined as p−i
= (p1, · · · ,pi−1,pi+1, · · · ,pNb
). (p−i,pi) and (p−i, pi) are the arbitrary two strategies of MG
i.
2) Transportation Network Side Let f = (f1,1, · · · , fi,24, · · ·
, fqods ,1, · · · , fqods ,24) denote
the travel flow profile vector of all EVs in the O-D pair.
Considering the definition of ordinal potential game [38], for two
strategy profiles f = (f−i,f i) and f = (f−i, f i),
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if [−FTNi (f−i,f i)] − [−FTNi
(f−i, f i)] ≥ 0, since that the function FTN is increasing with
respect to fodp,s, and the value of FTN is nonnegative, then
Φ(f−i,f i,pi) − Φ(f−i, f i,pi) ≥0 can be obtained. Thus Φ is a
potential function for EV travelers.
3) Power network Side In the power network side, for two strategy
profiles p =
(p−i,pi) and p = (p−i, pi), if Ri(p−i,pi)−Ri(p−i, pi) ≥ 0, it can
be proofed readily that Φ(p−i,pi,f i) − Φ(p−i, pi,f i) ≥0. Thus Φ
is a potential function for MGs. It is noted that as the
traditional load under consideration is non-responsive in the power
system, this part is a constant and does not affect the optimal
solution, and thus it can be ignored in potential function.
In the potential game Ξ, the utility and revenue function for each
player can be mapped to the potential function Φ(fodp,s,pi,s). The
Nash equilibrium of game Ξ is equivalent to the set of optimal
solutions of the potential function (14), and the feasible set of
decision variables can be defined by constraints of the
optimization problems (12) and (13). Note that the potential
function is concave with respect to decision variables, which
guarantees the existence and uniqueness of the Nash equilibrium.
Therefore, we can determine the optimal strategy of the coupled
network by locating the local optima of the above potential
function [37].
IV. AMBIGUITY SET AND REFORMULATION OF DRO PROBLEM
The exogenous uncertainties of the power-transportation coupled
network include the PV power output and charging demand of FCS
which is led by the uncertainty of the traffic demand. Furthermore,
these uncertainties can be transformed into the source-load
uncertainties of the power network, which are built as
scenario-based ambiguity sets incorporating the probability
distribution information to reduce the conservative- ness by RO in
this study. In this framework, the uncertainties in discrete
scenarios of the coupled network can be captured by a two-stage DRO
model, in which all game players seek to optimize their profits.
Generally, the decision variables in the DRO model include
two-stage strategies. The first-stage strategies are mainly the
design strategies, such as price setting and investment in the
day-ahead, which must be made “here- and-now” before the
realization of uncertainties. The second- stage strategies are the
operational decisions and can also be called a “wait-and-see”
pattern, which is determined after the uncertainty realization of
the coupled network.
A. Uncertainty Handling
The uncertainty of traffic demand for each O-D pair in scenario s
can be characterized as a box uncertainty set [16] in (15).
qods ∈ BOX(qod s , qods ) , {qod
s ≤ qods ≤ qods ,∀od ∈ R} (15)
The traffic flow space-time distribution led by the uncer- tainty
of traffic demand can be transformed into uncertainties of the
charging demand of FCSs connected to the buses, which is
illustrated in Fig. 3.
Fig. 3. Illustration of uncertain variables conversion in a
transportation system.
According to the Wardrop UE Principle, there is a unique solution
for the convex optimization problem (8), and the box uncertainty
set fodp,s ∈ BOX(fod
p,s , f
f od
(16)
The link flow is monotonically increasing for the path flow
according to Eq. (4). And then the box set of the link flow can be
derived as
xl,s ∈ BOX(xl,s(f od
p,s)) (17)
Due to the charging demand for each FCS being regarded as a linear
relation with the link flow, the uncertainty box set of charging
load in MG i can be represented as
ei,s ∈ BOX(xl,s(f od
p,s )es, xl,s(f
p,s)es) (18)
On the other hand, it is assumed that the available energy of PV
generation fluctuates with the interval between pPV
i,s and
pPV i,s , and then the uncertainty set of PV power output can
also be characterized by a box set as follows:
pPV i,s ∈ BOX(pPV
B. Scenario-Based Ambiguity Set Construction
There are two common methods to establish an ambi- guity set,
including moment-based [27] and distance metric approach [39]. In
this study, the moment-based method is adopted to construct a
scenario-based ambiguity set consid- ering the mean and support
information.
According to Ref. [40], the uncertain variable ei,s and pPV
i,s
can be written as{ ei,s = ei,s,pr + ei,s,deαi,s
pPV i,s = pPV
i,s,pr + pPV i,s,deβi,s
where ei,s,pr and pPV i,s,pr are the predicted charging de-
mand and predicted PV generation. Parameters ei,s,de and pPV i,s,de
denote the maximum deviations relative to the pre-
dicted values, satisfying Eq. (21). Random variables αi,s
and βi,s take values within [−1, 1] indicating the degree of
fluctuation relative to the predicted values. We assume that αs =
[α1,s, α2,s, · · · , αi,s, · · · , αNb,s] ∈ RNb and βs = [β1,s,
β2,s, · · · , βi,s, · · · , βNb,s] ∈ RNb are variable
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 7
vectors, and then the uncertain vector σs is defined as σs =
{αT s β
od
} (21)
In this way, the ambiguity set Ps is defined as [41]
Ps = { Ps ∈ RNb × RNb : EPs [σs] = 0,
Ps [σs ∈ Cm,s] = Pm,s,∀m ∈M} (22)
where Ps denotes that the joint probability distribution on RNb ×
RNb of uncertain vector αs and βs is Ps, which is obtained by the
historical data of the PV power output and the traffic demand. The
first item in (22) indicates that the expectation of the uncertain
vector in scenario s is equal to 0. The second item implies that
the probability of confidence set Cm,s occurring is Pm,s with M =
{1, · · · ,M0}, and M0 is the number of confidence sets. Meanwhile,
the value of Pm,s
belongs to [0, 1] for all m ∈M and PM0,s = 1 is assumed. According
to the Ref. [41], the confidence set Cm,s is
defined as
∀s ∈ T ,∀m ∈M} (23)
where the first item denotes that all the elements in σs take
values between -1 and 1, and the second item implies that the sum
of absolute values for all components in σs take values lower than
the budget of uncertainty defined as Γm,s, which is added to avoid
that σs always achieves boundary values to reduce the
conservativeness of this model. For obtaining tractable DRO, we
assume that Γm,s is strictly increasing with the increase of m and
Cm,s⊂Cm+1,s. Fur- thermore, the probabilities in (22) satisfy
Pm,s≤Pm+1,s for ∀m ∈ {1, · · · ,M0−1}. It is noted that if Γm,s is
given, Pm,s is the probability for the uncertain vector σs
occurring in Cm,s.
C. Reformulation of the DRO Problem
Due to the existence of uncertain variables, the Nash equi- librium
state of potential function (14) can be derived by an equivalent
optimization problem as follows:
min f ,p
EPs [ ∑ l
(ωtl,s + λl,ses)xl,s
+ c (pGi,s) + λCp0i,s − λBpi0,s] s.t. (2)− (4), (6), (10), (11),
(13b)− (13f)
(24)
where p(s) represents the occurrence probability of the sce- nario
s. In the following, a two-stage distributionally robust
optimization model is proposed with the underlying compact matrix
form:
min x cTx+ min
(25)
where A, Bs, Cs, and Ds denote constant matrices; h and ds are
constant vectors. Vectors x and ys represent decision variables and
are listed as follows:
x = {p0i,s, λl,s, δodl,p} ys = {pGi,s, pi0,s, xl,s, f
od p,s, tl,s,xl,j,s}
(26)
The variable x is a vector of the first-stage decision consisting
of the variables determined in the day-ahead and the indicator
variable. The variable ys is a vector of the second-stage decision,
which includes link flow, EV travel time, path flow, DG output, and
other related variables.
From the objective of (25), the second item differentiates with
classical robust approaches, and the physical meaning of this item
is minimizing the worst situation depending on the distribution Ps.
Combined with (22), the second item in the objective of (25) can be
converted as follows:
min ys
(min bTys)dP (σs)
(27) Similarly, constraints in the ambiguity set (22) about the
probability of σs can be transformed as∫
CM0,s
As a result, we can reformulate (25) as follows [42]:
min x cTx+ min
Ax ≤ h
where ηs, νs and γms denote the dual vectors for the corresponding
constraints.
V. SOLUTION METHODOLOGY
Generally, the two-stage optimization problem (25) can be derived
as a master problem (MP) and a subproblem (SP). It should be noted
that the term including the uncertainties in (25) has a max-min
form, sup
Ps∈Ps
EPs(min bTys), which is
classified as a bi-level programming problem. For solving the above
problem, duality theory can be used to transform the bi-level
problem into a single-level model with corresponding dual
variables.
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 8
Proposition 3: Concerning the ambiguity set Ps, the bilevel
programming problem sup
Ps∈Ps
reformulated as the following optimization problem:
min
] b = (Cs)
T νs
Proof: Concerning the confidence set Cm,s⊂Cm+1,s for all m ∈M, the
objective (27) can be discretized into [40]
max
∫ CM0,s
C1,s
∫ Cj+1,s\Cj,s
dP (σs) + m−1∑ j=1
∫ Cj+1,s,Cj,s
) (30)
where ηs and γms represent the dual vector and variable
corresponding constraints, and ηs ≥ 0, γms ≥ 0.
Based on the dual theory, the maximization problem (29) and (30)
can be written as an equivalent minimization problem as
follows:
min
(σs) T ηs +
∀m ∈M\[1] (31)
M0∑ j=m−1
γjs ≥ max σs∈Cm,s
[min bTys − (σs) T ηs],∀m ∈M\[1]
(32) Combining with (25) and (32), the second stage of the
bi-
level programming problem can be reformulated as:
max σs∈Cm,s
s.t. Bsx+Csys ≤ ds +Dsσs : νs
(33)
where ∀m ∈ M\[1], and νs is the dual vector of the second stage
constraints, νs ≥ 0.
According to the duality theory, (33) can be converted as a
single-level optimization model as follows:
max σs,νs
(34)
Assuming two uncertain vectors σ1,s and σ2,s being defined as 0 ≤
σ1,s,σ2,s ≤ 1, and σ1,s,σ2,s ∈ RNb × RNb , the constraints of
confidence set Cm,s can be converted as [43]{
σs = σ1,s − σ2,s
1′ · (σ1,s + σ2,s) ≤ Γm,s (35)
Then the bilinear term in the constraint of Proposition 3 can be
written as
max σs∈Cm,s
(36)
where τ 1,s, τ 2,s and ρs represent the dual vectors and variable
corresponding to the constraints, respectively.
Based on the dual theory, the single-level optimization model in
the constraint of Proposition 3 can be converted as follows:
maxνT s (Bsx− ds) + 1T · (τ 1,s + τ 2,s) + ρsΓm,s
s.t. b = (Cs) T νs
νT sDs + ηT
− ηT s − νT
τ 1,s, τ 2,s ≥ 0, ρs ≥ 0,νs ≥ 0
(37)
The complementary slackness conditions converted by the big-M
method can be written as
0 ≤ τ 1,s ≤Mε1,s 0 ≤ τ 2,s ≤Mε2,s 0 ≤ νT
sDs + ηT s + τ 1,s + 1 · ρs ≤Mι1,s
0 ≤ −ηT s − νT
sDs + τ 2,s + 1 · ρs ≤Mι2,s M (ε1,s − 1) ≤ σ1,s ≤M (1− ι1,s) M
(ε2,s − 1) ≤ σ2,s ≤M (1− ι2,s)
(38)
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 9
Then the subproblem of the optimal problem (25) can be expressed as
the following MILP problem:
SP : maxνT s (Bsx− ds) + 1T · (τ 1,s + τ 2,s) + ρsΓm,s
s.t. b = (Cs) T νs
0 ≤ τ 1,s ≤Mε1,s 0 ≤ τ 2,s ≤Mε2,s 0 ≤ νT
sDs + ηT s + τ 1,s + 1 · ρs ≤Mι1,s
0 ≤ −ηT s − νT
sDs + τ 2,s + 1 · ρs ≤Mι2,s M (ε1,s − 1) ≤ σ1,s ≤M (1− ι1,s) M
(ε2,s − 1) ≤ σ2,s ≤M (1− ι2,s) 1′ · (σ1,s + σ2,s) ≤ Γm,s
(39) where ε1,s, ε2,s, ι1,s and ι2,s denote binary variables.
Next, we implement the Benders decomposition algorithm to solve the
DRO model (25) [42]. According to Ref. [44], the Benders cut is
expressed as
M0∑ j=m−1
T (Bsx− ds −Dsσ
∗ ms)
(40) where σ∗s and ν∗s are the optimal solution of the SP. And the
MP can be written as
MP : min
) (41)
where σ∗kms and ν∗kms are the optimal solution of the SP at
iteration k.
On this basis, the solving procedure of the DRO model (25) with a
master-subproblem framework relying on the Benders decomposition
algorithm is presented in Algorithm 1, and the parameter setting of
the ambiguity set can affect the precision of optimal
solution.
VI. CASE STUDY
In this section, numerical analysis is performed for the 24-h
scheduling in an 8-MG power system coupled with a bench- mark urban
TN, as depicted in Fig. 4 and Fig. 5. The arrows of each link
represent the direction of vehicles permitted to drive. This urban
TN is extensively utilized in the research related to power and
transportation coupled networks (e.g., [8], [21], [45]). Relevant
parameters of traffic links are listed in Table I. It is noted that
the proposed approach is extensible, and it can be applied to the
urban TN with bidirectional-link and multiple O-D pairs. Owing to
the spatial limitation of this paper, only three O-D pairs in this
transportation system are considered, and the parameters are listed
in Table II, in which the traffic demand is the average value from
the historical data [21] and the basic value (p.u.) is 100 vehicles
per hour. The simulations are implemented on a laptop with an Intel
Core
Algorithm 1 Benders decomposition algorithm on DRO model.
Result: Obtain the optimal day-ahead power scheduling strategy 1.
Initialization. Let the lower bound LB = −∞, upper bound UB = +∞
and iteration number k = 0. Choose a convergence tolerance level ξ
> 0. Fix a feasible solution (x∗k, γ∗kms,η
∗k s ) of MP.
2. Solve the SP related to (x∗k, γ∗kms,η ∗k s ), to get the
ob-
jective value M0∑
j=m−1 γ∗kjs and optimal solution (σ∗kms,ν
∗k ms).
3. Obtain the objective value OB = cTx∗k +∑ s∈T
p(s) M0∑ m=0
M0∑ j=m−1
γ∗kjs Pm,s of MP with respect to x∗k
and M0∑
j=m−1 γ∗kjs , and let UB=min{OB,UB}.
4. Update the Benders cut related to (σ∗kms,ν ∗k ms) and
solve
MP. Then update the optimal solution and objective value to
(x∗(k+1), γ
∗(k+1) ms ,η
∗(k+1) s ) and z∗(k+1), respectively. Let
LB = z∗(k+1). 5. If (UB − LB)/UB ≤ ξ, terminate the procedure and
return x∗k as the optimal solution. Otherwise, let k = k+1 and go
to Step 2.
i9-10885H CPU 2.40 GHz using MATLAB with YALMIP and CPLEX 12.9.0
solver.
Fig. 4. Topology of TN with FCS.
Fig. 5. Topology of MGs.
In the power network, MG1-MG8 serves FCSs of C1-C8, respectively,
and the redundant power can be sold to the main grid. We assume
that the charging power of each EV is a constant 0.015MWh and the
monetary value of travel time ω is 10$/h. The relevant parameters
of DGs are a = $0.1/(MWh)2, b = $90/MWh, c = 0/h, PGi,s =
0MWh,
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 10
TABLE I PARAMETERS FOR THE 20-LINKS TRANSPORTATION SYSTEM
link Ca(p.u.) t0l (min) link Ca(p.u.) t0l (min) 1 18 6 11 13.8 12 2
8.5 6 12 17.5 6 3 9.8 5 13 8.9 5 4 20 10 14 9.76 5 5 13.5 12 15 7.9
5 6 19 10 16 17 6.5 7 14 11 17 8.2 6.5 8 20 9 18 9.15 5.9 9 13.2 11
19 8.97 5.8 10 20 10 20 18.2 6.1
TABLE II PARAMETERS FOR O-D PAIRS
O-D pair From node To node Average traffic demand (p.u.)
1-6 N1 N6 15 3-11 N3 N11 12 4-12 N4 N12 15
and PGi,s = 30MWh. The contract price of main grid energy selling
is set as λC = $140/MWh.
In this paper, there are three kinds of cases being adopted,
including the DRO model proposed by this paper, the RO model solved
by the column-and-constraint generation (C&CG) method [46], and
the deterministic model (DM) without uncertain variables, to
research the influence on the day-ahead power scheduling, cost and
other variables of the coupled network. In the DRO model case, the
number of the confidence sets is set as 6, i.e., M0 = 5. It should
be noted that the model complexity and the computing time will
increase with the rising of M0. While if a small value is assigned
to M0, the statistical features of uncertain parameters cannot be
represented adequately. Hence, the value of M0, the corresponding
uncertainty budgets Γm,s and probabilities Pm,s are set by the
distribution information of historical data. The traffic demand and
PV generation box sets are set as {0.7qods,av ≤ qods ≤
1.3qods,av,∀od ∈ R} and {0.75pPV
i,s,av ≤ pPV i,s ≤ 1.25pPV
i,s,av,∀i ∈ B}, respectively. Then the charging demand of FCSs can
be obtained by (15)-(17). In the RO case, the uncertainty set
intervals are the same as the DRO model. In the DM case, the
uncertainties of this coupled network are not considered. The
average value in all time slots of PV generation is adopted, and
the UE traffic assignment is determined as a certain variable by
solving (8) with the average traffic demand computed from
historical data. And the UE link flow pattern obtained by (8) is
shown in Fig.6, in which the flow of 10th, 13th, 15th, 18th, and
20th links is zero. As shown in Fig.6, the numerous vehicles travel
through links 5, 7, 14, and 19.
TABLE III SIMULATION RESULTS IN DIFFERENT CASES
Model RO DRO DM P0i(MWh) 21.4211 20.2155 19.4345
Totalcost(104$) 57.5800 54.3390 52.9370
Table III provides the total cost and the day-ahead average power
import from the main grid results of the three cases. As expected,
the costs of the DRO and RO model are more than
Fig. 6. The link flow with the average traffic demand in 24h.
those of DM. The reason is that the robust approach considers the
worst situation of the higher-level charging load and the
lower-level PV generation, and then the power reservation is not
considered. Meanwhile, the uncertain factors can change the flow
patterns and the route choices of vehicles, and the results of the
game between EVs and MG operators are changed, which is neglected
in DM. Hence the total cost will increase when uncertainties in
this coupled network are taken into account. It can also be
observed that the DRO method can obtain an excellent scheduling
strategy with more economical performance compared with RO, which
indicates that the DRO model proposed by this paper can reduce the
conservativeness of strategies and total cost.
In addition, taking MG1 and MG3 as examples, the solu- tions of
uncertain variables by DRO and RO for the coupled network compared
with the average value computing from historical data are shown in
Fig. 7, including PV generation and EV charging demand in all time
slots. The DG power output and the situation of selling power to
the main grid with the DRO and RO model of MG1 and MG3 are shown in
Fig. 8. In Fig. 7, the uncertainty of traffic demand influences the
UE state, so further causes the uncertainty of charging demand to
MG and generates the reservation of charging power for EVs in the
DRO and RO methods. In Fig. 8, there is only energy selling to the
main grid in the early morning. As shown in Fig. 7 and Fig. 8, the
power outputs of PV generation by DRO model are more than the ones
by RO overall, on the other hand, the charging demand, DG power
output, and the power selling to the main grid by DRO are all less
than the ones by RO. In the DRO model proposed by this paper, the
probability distribution for uncertain parameters can be captured
based on the historical data. In contrast, the RO model adopts the
boundary information of uncertainty box sets to obtain the optimal
solution in the worst case increasing the conservativeness of
strategies. Meanwhile, the DRO model can avoid the dilemma of
selling more surplus electricity back to the main grid.
Furthermore, about the performance of the above three cases, the
total computing times of the DRO, RO, and DM are 196s, 76s, and
16s, respectively. As expected, the DM has a minimal calculation
time. It should be noted that the DRO requires a longer computing
time than RO, due to there being more auxiliary variables in the
computing process of DRO. However, it can attain more practical
scheduling and can adapt
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 11
Fig. 7. The solutions of PV generation and charging demand of MG1
and MG3 by DRO and RO.
to multiple uncertainties.
Fig. 8. The solutions of DG power output and selling power to the
main grid of MG1 and MG3 by DRO and RO.
VII. CONCLUSION
This paper proposes a two-stage DRO model considering the
uncertainties of PV generation and the traffic demand of vehicles
under a potential game framework to obtain the optimal day-ahead
power scheduling strategy for PN-TN sys- tems. This DRO model
considers the probability distribution for uncertain parameters and
designs to derive the equilibrium of the potential game in the
worst situation over ambiguity sets. Taking an MGs system coupled
with a double-ring trans- portation network as an instance,
simulation results confirm that the capability of DRO with
scenario-based ambiguity sets in this paper is superior to the
traditional robust optimization approach. The limitation of this
study is that there are some assumptions in the PN and TN, which
restrict the practical application of this method. As a result, the
future extension of this investigation is to take the endogenous
uncertainty for the travelers’ charging demand into account to
enhance the application value in practice.
REFERENCES
[1] J. A. P. Lopes, F. J. Soares, and P. M. R. Almeida,
“Integration of electric vehicles in the electric power system,”
Proc. IEEE, vol. 99, no. 1, pp. 168–183, Jan. 2011.
[2] S. Xie, Z. Hu, and J. Wang, “Two-stage robust optimization for
expansion planning of active distribution systems coupled with
urban transportation networks,” Appl. Energy, vol. 261, pp. 114412,
2020.
[3] F. He, D. Wu, Y. Yin, and Y. Guan, “Optimal deployment of
public charging stations for plug-in hybrid electric vehicles,”
Transp. Res. Part B Methodol., vol. 47, pp. 87-101, Jan.
2013.
[4] X. Dong, Y. Mu, X. Xu, H. Jia, et al, “A charging pricing
strategy of electric vehicle fast charging stations for the voltage
control of electricity distribution networks,” Appl. Energy, vol.
225, pp. 857–868, 2018.
[5] K. Mahmud, and G. E. Town, “A review of computer tools for mod-
eling electric vehicle energy requirements and their impact on
power distribution networks,” Appl. Energy, vol.172, no. 3, pp.
37–59, 2016.
[6] M. Alizadeh, H. Wai, M. Chowdhury, A. Goldsmith, et al,
“Optimal Pricing to Manage Electric Vehicles in Coupled Power and
Transporta- tion Networks,” IEEE Trans. Control Netw. Syst., vol.
4, no. 4, pp. 863 – 875, Dec. 2017.
[7] W. Wei, L. Wu, J. Wang, and S. Mei, “Network equilibrium of
coupled transportation and power distribution systems,” IEEE Trans.
Smart Grid, vol. 9, no. 6, pp. 6764–6779, Nov. 2018.
[8] W. Gan, M. Shahidehpour, M. Yan, J. Guo, et al, “Coordinated
Planning of Transportation and Electric Power Networks with the
Proliferation of Electric Vehicles,” IEEE Trans. Smart Grid, vol.
11, no. 5, pp. 4005–4015, Sep. 2020.
[9] F. He, Y. Yin, J. Wang, and Y. Yang, “Sustainability SI:
Optimal prices of electricity at public charging stations for
plug-in electric vehicles,” Netw. Spatial Econ., vol. 16, no. 1,
pp. 131–154, Mar. 2016.
[10] J.R. Correa, and N.E. Stier-Moses, “Wardrop equilibria,” In
Cochran J, editor. Wiley Encyclopedia of operations research and
management science, Feb. 2011.
[11] D. Acemoglu, and A. Ozdaglar, “Competition in parallel-serial
net- works,” IEEE J. Sel. Areas Commun., vol. 25, no. 6, pp.
1180–1192, Aug. 2007.
[12] W. Lee, L. Xiang, R. Schober, and V. W. S. Wong, “Electric
vehicle charging stations with renewable power generators: a game
theoretical analysis,” IEEE Trans. Smart Grid, vol. 6, no. 2, pp.
608–617, Sep. 2015.
[13] C. Wu, H. Mohsenian-Rad, and J. Huang, “Vehicle-to-aggregator
inter- action game,” IEEE Trans. Smart Grid, vol. 3, no. 1, pp.
434–442, Mar. 2012.
[14] W. Tushar, W. Saad, H. V. Poor, and D. B. Smith, “Economics of
electric vehicle charging: a game theoretic approach,” IEEE Trans.
Smart Grid, vol. 3, no. 4, pp. 1767–1778, Dec. 2012.
[15] S. Bahrami, and V. W.S. Wong, “A Potential Game Framework for
Charging PHEVs in Smart Grid,” in 2015 IEEE Pacific Rim Conference
on Communications, Computers and Signal Processing Conf. (PacRim),
pp. 28-33.
[16] C. Zhang, X. Chen, and A. Sumalee, “Robust Wardrop’s user
equilibrium assignment under stochastic demand and supply,”
Transport Res. B Methodol., vol. 45, no. 3, pp. 534–552, Mar.
2011.
[17] Z. Zhou, S. J. Moura, H. Zhang, X. Zhang, Q. Guo, and H. Sun,
“Power- traffic network equilibrium incorporating behavioral
theory: A potential game perspective,” Appl. Energy, vol. 289, no.
1, 116703, May 2021.
[18] D. Bertsimas, E. Litvinov, X. A. Sun, J. Zhao, and T. Zheng,
“Adap- tive robust optimization for the security constrained unit
commitment problem,” IEEE Trans. on Power Syst., vol. 28, no. 1,
pp. 52-63, Feb. 2013.
[19] W. Wei, S. Mei, L. Wu, M. Shahidehpour, and Y. Fang, “Optimal
traffic- power flow in urban electrified transportation networks,”
IEEE Trans. Smart Grid, vol. 8, no. 1, pp. 84-95, Jan. 2017.
[20] S.D. Manshadi, M.E. Khodayar, K. Abdelghany, and H. Uster,
“Wireless charging of electric vehicles in electricity and
transportation networks,” IEEE Trans. Smart Grid, vol. 9, no. 5,
pp. 4503-4512, Sep. 2018.
[21] W. Wei, S. Mei, L. Wu, J. Wang, and Y. Fang, “Robust operation
of distribution networks coupled with urban transportation
infrastructures,” IEEE Trans. Power Syst., vol. 32, no. 3, pp.
2118-2130, May 2017.
[22] C. He, L. Wu, T. Liu, W. Wei, and C. Wang, “Co-optimization
schedul- ing of interdependent power and gas systems with
electricity and gas uncertainties,” Energy, vol. 159, pp.
1003-1015, Sep. 2018.
[23] S. Dehghan, and N. Amjady, “Robust transmission and energy
storage expansion planning in wind farm-integrated power systems
considering transmission switching,” IEEE Trans. Sustain. Energy,
vol. 7, no. 2, pp. 765–774, May 2016.
[24] R. Zhou, X. Min, X. Tong, R. Chen, X. Li, and Z. Liu,
“Distributional robust optimization under moment uncertainty of
environment and economic dispatch for power system,” Proceedings of
the CSEE, vol. 35, no. 13, pp. 3248-3256, Aug. 2015.
[25] A. Zare, C. Chung, J. Zhan, and S. O. Faried, “A
distributionally ro- bust chance-constrained MILP model for
multistage distribution system planning with uncertain renewables
and loads,” IEEE Trans. Power Syst., vol. 33, no. 5, pp. 5248–5262,
Sep. 2018.
[26] Y. Cao, D. Li, Y. Zhang, Q. Tang, A. Khodaei, H. Zhang, and Z.
Han, “Optimal energy management for multi-microgrid under a
transac-
JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XX XXXX 12
tive energy framework with distributionally robust optimization,”
IEEE Trans. Smart Grid, early access, Sep. 2021.
[27] D. Pozo, A. Street, and V. Alexandre, “An ambiguity-averse
model for planning the transmission grid under uncertainty on
renewable dis- tributed generation,” in 2018 Proc. Power Syst.
Comput. Conf. (PSCC), Dublin, Ireland, pp. 1–7.
[28] A. Velloso, D. Pozo, and A. Street, “Distributionally robust
transmission expansion planning: A multi-scale uncertainty
approach,” IEEE Trans. Power Syst., vol. 35, no. 5, pp. 3353-3365,
Sep. 2020.
[29] C. Zhao, and Y. Guan, “Data-driven stochastic unit commitment
for integrating wind generation,” IEEE Trans. Power Syst., vol. 31,
no. 4, pp. 2587-2596, Jul. 2015.
[30] W. Wei, L. Wu, J. Wang, and S. Mei, “Expansion planning of
urban electrified transportation networks: A mixed-integer convex
program- ming approach,” IEEE Trans. Transport. Electrific., vol.
3, no. 1, pp. 210–224, Mar. 2017.
[31] “Bureau of Public Roads,” Traffic Assignment Manual, U.S.
Dept. Commerce, Washington, DC, USA, 1964.
[32] M. J. Beckmann, C. B. McGuire, and C. B. Winsten, “Studies in
the Economics of Transportation,” in New Haven, CT, USA: Yale Univ.
Press, Dec. 1956.
[33] Y. Sheffi, “Urban transportation networks: Equilibrium
Analysis with Mathematical Programming Methods,” Englewood Cliffs,
NJ: Prentice- Hall, pp. 339, 1986.
[34] W. Wei, J. Wang, and L. Wu, “Quantifying the impact of road
capacity loss on urban electrified transportation networks: an
optimization based approach,” Int. J. Transp. Sci. Technol., vol.
5, no. 4, pp. 268–288, Dec. 2016.
[35] J. Fortuny-Amat, and B. Mccarl, “A representation and economic
interpretation of a two-level programming problem,” J. Oper. Res.
Soc., vol. 32, no. 9, pp. 783-792, Sep. 1981.
[36] H. Xin, X. Jiang, Y. Jian, Y. Li, et al, “Distributed robust
energy management of a multimicrogrid system in the real-time
energy market,” IEEE Trans. Sustain. Energy, vol. 10, no. 1, pp.
396–406, 2019.
[37] D. Monderer, and L. Shapley, “Potential games,” Games Econ
Behav., vol. 14, no. 1, pp. 124–143, May 1996.
[38] Q. La, Y. Chew, and B. Soong, “Potential game theory
applications in radio resource allocation,” Springer, 2016.
[39] P. M. Esfahani and D. Kuhn, “Data-driven distributionally
robust op- timization using the wasserstein metric: Performance
guarantees and tractable reformulations,” Math. Program., vol. 171,
pp. 115–166, 2018.
[40] Y. Zhang, W. Liu, Z. Huang, F. Zheng, J. Le, and S. Zhu,
“Distri- butionally robust coordination optimization scheduling for
electricity- gas-transportation coupled system considering multiple
uncertainties,” Renewable Energy, vol. 163, pp. 2037-2052, Jan.
2021.
[41] W. Wiesemann, D. Kuhn, and M. Sim, “Distributionally Robust
Convex Optimization,” Oper. Res. the J. of the Oper. Res.,
2014.
[42] M. Zugno, and A. J. Conejo, “A Robust Optimization Approach to
Energy and Reserve Dispatch in Electricity Markets,” Euro. J. of
Oper. Res., vol. 247, no. 2, pp. 659–671, Dec. 2015.
[43] R. Li, M. Wang, M. Yang, X. Han, Q. Wu and W. Wang, “A
distribu- tionally robust model for reserve optimization
considering contingency probability uncertainty,” Int. J. Electr.
Power Energy Syst., vol. 134, 107174, Jan. 2022.
[44] A.M.Geoffrion, “Generalized Benders Decomposition,” J. of
Opti. The- ory and Appl., vol. 10, no. 4, Oct.1972.
[45] Y. Liu, Y. Wang, Y. Li, H. Gooi, and H. Xin, “Multi-Agent
Based Optimal Scheduling and Trading for Multi-Microgrids
Integrated with Urban Transportation Networks,” IEEE Trans. on
Power Syst., vol. 36, no. 3, pp. 2197 – 2210, Dec. 2020.
[46] B. Zeng, and L. Zhao, “Solving two-stage robust optimization
problems using a column-and-constraint generation method,” Oper.
Res. Lett., vol. 41, no. 5, pp. 457-461, Sep.2013.
I Introduction
II-A Transportation System Modeling
II-B Power System Modeling
IV-A Uncertainty Handling
IV-C Reformulation of the DRO Problem
V Solution Methodology
VI Case Study