journal of la distributionally robust day-ahead scheduling

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 traffic 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 noncooperative 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 constraints 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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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 constraint 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 upper/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 problem.
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 probability 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 distribution 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 distributionally robust day-ahead scheduling model of the power- transportation coupled network considering source-load uncertainties 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 represent the dynamic characteristics of active network management. 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
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.
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 demand, and the traffic flow distribution can be analyzed by the
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 different 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 maximize 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 network 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 problem 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 changing 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),
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 conservativeness 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 uncertainty 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 ambiguity 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 considering 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
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 equilibrium 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 scenario 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.
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)
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,
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 solutions 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
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 systems. 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 transportation 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.
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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

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