Distributionally Robust Day-ahead Scheduling for Power-traffic Network under a Potential Game Framework

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 $\omega$. According to Ref. [2], it is widely accepted that the value of $\omega$ is constant. 3) The heterogeneous 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. Every EV has to pass through and charge at one FCS before reaching the destination.

A TN can be represented by $\mathcal{G}_{TN}=(\mathcal{N},\mathcal{L})$, where $\mathcal{N}$ and $\mathcal{L}\subseteq\mathcal{N}\times\mathcal{N}$ denote the set of nodes and links, respectively. A pair $(a,b)\in\mathcal{L}$ 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. The set of O-D pairs is denoted by $\mathcal{R}$, specifically, $(o,d)\in\mathcal{R}$. The traffic demand (also called trip rate) of O-D pairs in time slot t can be described by $\boldsymbol{{q}}_{t}^{od}$ (1), which defines the number of vehicles in each time slot intending to travel from o to d.

2) O-D flow

Each O-D pair is connected by several paths, which consists of certain links [31]. The set of feasible routes p is denoted by $p\in\mathcal{P}_{od},\mathcal{P}_{od}=\{1,\cdots,N_{od}\}$, where $N_{od}$ denotes the number of the routes for each O-D pair. Let $f_{p,t}^{od}$ be the traffic flow of path p, and the traffic demand balance equation is described by (2). Eq. (3) denotes the non-negativity of path flows.

3) Link flow

In the transportation system, the set of all links with FCS is represented by $\mathcal{L}_{C}\subseteq\mathcal{L}$. The traffic link flow is equal to the total number of vehicles of the paths through it, which is described by (4).

In this paper, we assume that FCSs are located on some links. EVs can choose to refill in FCSs or pass by the bypass link. The bus index of distribution systems is marked as $j$, and we define the following constraints to ensure that all EVs are charged through the TN.

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 $t_{l,t}(x_{l,t})$ (8), which is also called the Bureau of Public Roads (BPR) function [32] and can reflect the delayed travel time in links accurately,

Furthermore, we assume that the charging demand of each EV in time slot $t$ is uniform, in other words, the power demand of EVs for each bus is proportional to the link flow. $\lambda_{j,t}$ is defined as the electricity price of FCS on bus $j$, and then the charging energy cost of a single vehicle is $\lambda_{j,t}e_{t}$, where $e_{t}$ is the charging demand of one vehicle. It is noted that the charging time of EV is neglected in travel cost due to the same charging efficiency. The total expense by a single EV on path p from o to d in time slot t can be calculated as

5) Wardropian traffic assignment

The balanced distribution of traffic flow is analyzed by the Wardrop user equilibrium (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 [33].

Then a TAP considering the charging cost is described as follows, in which each traveler wishes to minimize his travel expense.

According to classic transportation theory, the UE traffic state can be characterized as (13).

Proposition 1: The solution to problem (12) is an UE flow pattern satisfying (13).

Proof: The proof of this equivalency can be acquired by generalizing the method in [34]. In detail, the Lagrangian function of TAP for the equality constraint (2) and inequality constraint (3) can be formulated as

where $u_{t}^{od}$ and $v_{t,p}^{od}$ are the vectors of Lagrangian multipliers. The optimal solution of (12) satisfies the Karush-Kuhn-Tucher conditions as follows:

Stationarity: $\frac{\partial L_{\mathrm{TN}}}{\partial f_{p,t}^{od}}=0$.

Complementary slackness: $v_{t,p}^{od}f_{p,t}^{od}=0$.

Further considering that

Then $v_{t,p}^{od}=cost_{t,p}^{od}-u_{t}^{od}\geq 0$ and $v_{t,p}^{od}f_{p,t}^{od}=0$ can be obtained, which is equivalent to (13).∎

It is observed that satisfying condition (13) is equivalent to reaching the traffic UE state, and thus it can be considered as a complementary constraint for the optimization problem (12), whose UE state of the optimal solution can be ensured [35]. Meanwhile, the constraint (13) can be linearized as follows by the big-M method [36].

Furthermore, since the bivariate terms exist in Eq. (10), a piecewise linearization method [2] is adopted to transform it as (15).

Consequently, the linear optimization problem for TN can be constructed as follows:

The objective of the traffic network optimization problem (16) is to find the optimal traffic flow distribution at a certain electricity price.

In this study, there are a dispatchable generator (DG), energy storage (ES) unit and photovoltaic (PV) generation unit locating inside each FCS, which can be regarded as a MG [37]. Without loss of generality, we assume that each bus of power distribution network is connected to a MG. Demands of each MG include flexible demand response load and the charging load of EV users, which is depicted in Fig. 1. The application of ES can improve the flexibility of system power dispatch. The operating costs of MGs can be reduced by adjusting the electricity consumption schedule of the flexible demand response loads. Meanwhile, the demand response load can also obtain certain benefits by providing this service. Note that we assume MGs can sell their redundant energy to the main grid.

In the electrical power network, the index of MGs coupled with transportation network is denoted by $j$, which is also connected on the bus $j\in\mathcal{B}=\{1,2,\cdots,N_{b}\}$, where $N_{b}$ is the number of buses or MGs.

It should be noted that the adjacent node of bus $j$ is denoted by $i\in\mathcal{N}$, and subscript $ji$ indicates the transmission line from $j$ to $i$. The objective is to maximize the operation revenue of MG $j$. The constraints consist of the following five parts.

1) Transaction with the main grid

2) The generation constraint of DG

3) The ES

The Eq. (23) represents the remaining capacity of the ES in each period $t$. $\Delta t$ denotes the time interval, whose value is $1h$. Constraints (24) and (25) are the charge and discharge power limits of the ES, respectively. $P_{j,\max}^{\rm ES}$ is the maximum charge and discharge power, which is mainly affected by the capacity limitation of inverters. $v_{j,t}$ represents the charging and discharging state of the ES, The value of $v_{j,t}$ is $1$ when the state of the ES is charging, otherwise, it is $0$. Constraints (26) and (27) describe the $SOC$ limits, where $E_{\rm L}$ is the rated capacity of the ES device. $T$ is the scheduling period, and the value is $24h$. It is noted that the cost of ES equipment life loss is ignored in day-ahead power dispatch.

4) The flexible demand response load

5) The power flow constraint

For the power flow constraint of this network, the DC approximation of AC optimal power flow is applied to simplify this model. The fixed voltage magnitudes and small phase angle differences are assumed in the network. The reactive part of electrical power is neglected.

Eq. (34) models the active power flow, which is determined by Kirchhoff’s first law and the power flow equations and $p_{j,t}^{\rm PV}$ is the PV power output. The power on each node has to be balanced. In addition, the electricity demands of MG $j$ include the flexible demand response load and the charging demand, which depend on the traffic flow of the pertaining link. The active power flow on transmission line $ji$ is the multiply by voltage angle (in radians) differences $\theta_{j}-\theta_{i}$ and susceptance $b_{ji}$ as Eq. (36), where $b_{ji}$ is the power distribution factor for line $ji$ and $b_{ji}>0$. For the power flow limits (37), $F_{ji}^{\max}$ is the transmission capacity for line $ji$.

Based on the above five categories constraint, let vector $\boldsymbol{p}_{j,t}\triangleq\left\{p_{j,t}^{\rm buy},p_{j,t}^{\text{\rm sell}},p_{j,t}^{\rm DG},p_{j,t}^{\rm ES,D},p_{j,t}^{\rm ES,C},p_{j,t}^{d}\right\}$ be the decision vector of MG $j$ in time slot $t$. The objective of MG $j$ is to maximize the operation revenue, and the optimization problem is shown as follows. Note that the cost of PV generation is ignored.

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 (16). 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.

In the power network, MGs determine the power schedule and maximize their operation revenues by providing energy at the electricity prices. 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 ${p}_{j,t}^{\rm PV}$ and ${q}_{t}^{od}$, 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 $\Xi=\Big{\{}\{\mathcal{Q}\cup\mathcal{B}\},\{\{\mathcal{F}_{v}^{od}\}_{od\in\mathcal{R}},\{\mathcal{P}_{j}\}_{j\in\mathcal{B}}\},\{(-F_{{\rm TN}v})_{v\in\mathcal{Q}},(R_{j})_{j\in\mathcal{B}}\}\\ \Big{\}}$, and uncertain variables of the coupled network exist in the strategy sets. The components of game $\Xi$ are described as follows:

1) Customer Side (EVs)

The players noted as $\mathcal{Q}$ in the traffic network are the EV travelers corresponding to the travel demand in different O-D pairs. Player v belongs to $\mathcal{Q}\triangleq[0,{q}_{t}^{od}]$, which is an interval for the number of players. The strategy set consists of all routes connecting the O-D pair, in other words, the traffic flow on each path can be regarded as the strategy of traffic side, i.e., $\mathcal{F}_{v}^{od}=\left\{\boldsymbol{f}_{v,1}^{od},\boldsymbol{f}_{v,2}^{od},\cdots,\boldsymbol{f}_{v,N_{od}}^{od}\right\}$ , where $\boldsymbol{f}_{v}^{od}$ is the vector of the traffic flow on the O-D pair in a scheduling cycle. Each EV traveler aims to selfishly choose its route and FCS to maximize its traveling utility with uncertain travel demand, i.e., $\max\limits_{\boldsymbol{f}_{v}^{od}\in\mathcal{F}_{v}^{od}}\left(-\sum\limits_{t}\operatorname{cost}_{t,p}^{od}\right)$.

2) Supplier Side (MGs)

The players noted as $\mathcal{B}$ in the power network are MG operators, player j belongs to $\mathcal{B}\triangleq\{1,2,\cdots,N_{b}\}$, where $N_{b}$ is the number of MGs. The power decision vector for MG $j$ is $\boldsymbol{p}_{j}$ in one scheduling cycle. $\mathcal{P}_{j}=\{\boldsymbol{p}_{1},\boldsymbol{p}_{2},\cdots,\boldsymbol{p}_{N_{j}}\}$ is the strategy set, where $N_{j}$ is the number of MG $j$’s power generation strategies. Then the revenue maximization problem for MG $j$ with uncertain PV generation can be described by $\max\limits_{\boldsymbol{p}_{j}\in\mathcal{P}_{j}}R_{j}$.

The above game-theoretic perspective of this coupled network is shown in Fig. 2, where $\tilde{q}_{t}^{od}$ and $\tilde{p}_{j,t}^{\rm PV}$ 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 (16) and (38), 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 $q_{t}^{od}$ is determined, $f_{p,t}^{od}$ 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 $p_{j}^{\rm PV}$, which is only reflected in constraints.

Due to the autonomous decision-making of the vehicle, a potential game is formulated. 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:

Proof:

1) Variable Declaration

There are two kinds of decision vectors as follows. In transportation network, $\boldsymbol{f}_{v}^{od}=(f_{v,1}^{od},f_{v,2}^{od},\cdots,f_{v,24}^{od})$ is the traffic flow in the route selected by EV $v$. $\boldsymbol{f}_{-v}^{od}$ is the vector of the traffic flow in the route selected by all EVs except EV $v$, defined as $\boldsymbol{f}_{-v}^{od}=(\boldsymbol{f}_{v}^{od},\cdots,\boldsymbol{f}_{v-1}^{od},\boldsymbol{f}_{v+1}^{od},\cdots,\boldsymbol{f}_{q_{s}^{od}}^{od})$. $(\boldsymbol{f}_{-v},\boldsymbol{f}_{v}^{od})$ and $(\boldsymbol{f}_{-v}^{od},\hat{\boldsymbol{f}}_{v}^{od})$ are the arbitrary two strategies of EV $v$. In the same way, in the power network, $\boldsymbol{p}_{i}$ is the power profile of MG $i$. $\boldsymbol{p}_{-i}$ is the power profiles of all MGs except MG $i$ defined as $\boldsymbol{p}_{-i}=(\boldsymbol{p}_{1},\cdots,\boldsymbol{p}_{i-1},\boldsymbol{p}_{i+1},\cdots,\boldsymbol{p}_{N_{b}})$. $(\boldsymbol{p}_{-i},\boldsymbol{p}_{i})$ and $(\boldsymbol{p}_{-i},\hat{\boldsymbol{p}}_{i})$ are the arbitrary two strategies of MG $i$.

2) Transportation Network Side

We first let $L_{\mathrm{T}-\mathrm{P}}$ be the partial Lagrangian of $-\Phi$ associated with constraints (2) and (34),which are shown in Eq. (40). It should be noted that the charging price is determined by the marginal cost of power production of FCSs, viz. the dual variables of power balance constraints (34).

The Lagrangian for traffic side can be expressed as

By calculating the Lagrangian partial derivative, (40) and (41) yields

In addition, let $\boldsymbol{f}^{od}=(f_{1,1}^{od},\cdots,f_{i,24}^{od},\cdots,f_{q_{t}^{od},1}^{od},\\ \cdots,f_{q_{t}^{od},24}^{od})$ denote the travel flow profile vector of all EVs in the O-D pair. Considering the definition of ordinal potential game [39], for two strategy profiles $\boldsymbol{f}^{od}=(\boldsymbol{f}_{-v}^{od},\boldsymbol{f}_{v}^{od})$ and $\hat{\boldsymbol{f}^{od}}=(\boldsymbol{f}_{-v}^{od},\hat{\boldsymbol{f}}_{v}^{od})$, if $[-F_{{\rm TN}_{v}}(\boldsymbol{f}_{-v}^{od},\boldsymbol{f}_{v}^{od})]-[-F_{{\rm TN}_{v}}(\boldsymbol{f}_{-v}^{od},\hat{\boldsymbol{f}_{v}^{od}})]\geq 0$, then $\Phi(\boldsymbol{f}_{-v}^{od},\boldsymbol{f}_{v}^{od},\boldsymbol{p}_{i})-\Phi(\boldsymbol{f}_{-v}^{od},\hat{\boldsymbol{f}_{v}^{od}},\boldsymbol{p}_{i})\geq$$0$ can be derived due to the monotonicity and non-negativity of $F_{\rm TN}$. Thus $\Phi$ is a potential function for EV travelers.

3) Power network Side

In the power network side, the Lagrangian can be expressed as

The necessary and sufficient condition for potential function being obtained and the objective function of power network getting the minimum value simultaneously can be expressed as (44), which can be satisfied obviously.

In addition, for two strategy profiles $\boldsymbol{p}=(\boldsymbol{p}_{-i},\boldsymbol{p}_{i})$ and $\hat{\boldsymbol{p}}=(\boldsymbol{p}_{-i},\hat{\boldsymbol{p}}_{i})$, if $R_{i}(\boldsymbol{p}_{-i},\boldsymbol{p}_{i})-R_{i}(\boldsymbol{p}_{-i},\hat{\boldsymbol{p}_{i}})\geq$$0$, it can be proofed readily that ${\Phi}(\boldsymbol{p}_{-i},\boldsymbol{p}_{i},\boldsymbol{f}_{v}^{od})-{\Phi}(\boldsymbol{p}_{-i},\hat{\boldsymbol{p}_{i}},\boldsymbol{f}_{v}^{od})\geq$$0$. Thus $\Phi$ is a potential function for MGs. ∎

In the potential game $\Xi$, the utility and revenue function for each player can be mapped to the potential function ${\Phi}(f_{p,t}^{od},\boldsymbol{p}_{j,t})$. The Nash equilibrium of game $\Xi$ is equivalent to the set of optimal solutions of the potential function (39), where the feasible set of decision variables can be defined by constraints of the optimization problems (16) and (38). 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 [38].

The uncertainty of traffic demand for each O-D pair in time slot $t$ can be characterized as a box uncertainty set [20] in (45).

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.

According to the Wardrop UE Principle, there is a unique solution for the convex optimization problem (12), and the box uncertainty set $\tilde{f}_{p,t}^{od}\in{\rm BOX}(\underline{f}_{p,t}^{od},\overline{f}_{p,t}^{od})$ can be obtained by [20]

The link flow is monotonically increasing for the path flow according to Eq. (5). Then the box set of the link flow can be derived as

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 $j$ can be represented as

In addition, it is assumed that the available energy of PV generation fluctuates with the interval between $\underline{p}_{j,t}^{\rm PV}$ and $\overline{p}_{j,t}^{\rm PV}$ , and then the uncertainty set of PV power output can also be characterized by a box set as follows:

There are two common methods to establish an ambiguity set generally, including moment-based [25] and distance metric approach [40]. In this study, the moment-based method is adopted to construct a ambiguity set considering the mean and support information.

According to Ref. [26], the uncertain variable $\tilde{e}_{j,t}$ and $\tilde{p}_{j,t}^{\rm PV}$ can be written as

In this way, the ambiguity set $\mathscr{P}_{t}$ is defined as [41]