Table 1 Notations summary
From: Optimal joint operation of coupled transportation and power distribution urban networks
General parameters and variables | |||
|---|---|---|---|
\(C_{TN}\) | Objective function (travel time cost and associated cost of energy during a charging event - Cost of TN) | w | Parameter—Associated cost of travel time estimated in Euros |
\(\tau _{e}(t)\) | Latency function to calculate the travel time on link e during time period t | \(X^{cg}_{e,t}\) | Traffic congestion cost on link e during time period t |
\(\vartheta _{e,t}\) | Total aggregate of traffic flow on link e during time slot t | \(G_{ch}^{\{ev\}}\) | Aggregated charging costs of all EVs |
T | The predefined time period for optimization | \(\tau _{e}^{0}\) | Free flow traveling time on link e |
\(\alpha\) | Co-efficient for real traffic observation data | \(cap_e\) | Capacity of traffic flow on road link e |
\(C^{ch}_{j}\) | Associated cost (Euros) of electricity at the charging station connected to bus node j of PDN | \(\pi _{j}(t)\) | Locational Marginal Price (LMP) of bus j in PDN during time period t |
\(P_{ch}^{ev}\) | Average EV charging power (kW) for unit traffic flow | \(C^{sd}_{p,t}\) | Cost of travel on path p in the OD pair (s, d) during time period t |
\(\psi _{p}\) | Decision parameter to state if path p is chosen or not | \(T^{sd}_{p,t}\) | Time to travel on path p in the OD pair (s, d) during time spot t |
\(\psi _{e,p}\) | Decision variable which states if link e belongs to path p or not | \(f^{sd}_{p,t}\) | Traffic flow on path p in the OD pair (s, d) during time period t |
\(\eta _{sd}(t)\) | The traffic demand on OD pair (s,d) during time period t | \(x^{sd}_t\) | Binary variable if vehicle makes a trip from source s to destination d during time period t |
\(P_{max}^{ev}\) | Maximum charging power rate of electric vehicle ev | \(SoC_{min}Q\) | Minimum state of charge of the battery capacity Q |
\(SoC_{max}Q\) | Maximum state of charge of battery capacity Q | \(Cap_{ch}\) | Number of charging stations |
\(Cap_k^{ch}\) | Number of charging piles at charging station ch | \(P^{ev}_{{ch},k}\) | Charging Power (kW) of the electric vehicle ev in pile k of charging station ch |
\(l_{ch}\) | The load that the charging station can handle | \(N_{ch}\) | Power efficiency factor of charging station ch |
\(\pi _{ch}\) | Charging price (Euros) at charging station ch | \(\vartheta _{e,ch,k}\) | Traffic flow captured by the charging station ch in pile k through link e |
\(A_{ch}^{\{ev\}}\) | Amount of Energy (kWh) to be charged by all EVs at charging station ch | \(C_{PDN}\) | Objective function (energy cost of PDN for charging all EVs) |
\(\Phi _j\) | The cost (Euros) of generating power from bus node j | \(p^{gen}_x(t)\) | Average generated power (MW) by bus node x during time t |
\(\pi _{cs}(t)\) | Contract energy cost charged per kWh during time period t | \(\Gamma _{c}(t)\) | The purchase energy cost from main power grid excluding contract price, during time slot t |
\(\lambda _j\) | Set of all bus nodes which are at the end of the distribution line | \(P_{ij}(t)\) | Active power flow from bus node i to j during time period t |
\(r_{ij}\) | Reactance of active power line connecting bus node i to j | \(I_{ij}(t)\) | Current from bus node i to bus node j during time period t |
\(P_{jk}(t)\) | Average active Power flow (MW) from bus node j to pile k during time period t | \(p^{dem}_{x}(t)\) | Total active power demand (MW) at bus node x during time period t |
\(Q_{ij}(t)\) | Average reactive power flow (MVar) from bus node i to j during time period t | \(q^{gen}_{x}(t)\) | Average reactive power (MVar) generated at bus x during time slot t |
\(x_{ij}\) | Reactance of reactive power line connecting bus node i to j | \(Q_{jk}(t)\) | Average reactive power flow from bus node j to pile k during time period t |
\(q^{dem}_{j}(t)\) | Average reactive power (MVar) demand at bus node j during time period t | \(U_{x}(t)\) | Average voltage drop at bus node x during time period t |
\(U^f_{i}(t)\) | Lower bound of square voltage magnitude at bus node i during time slot t | \(U^r_{i}(t)\) | Upper bound of square voltage magnitude at bus node i during time slot t |
\(p^f_i\) | Lower limit of active power generation at bus node i | \(p^r_i\) | Upper limit of active power generation at bus node i |
\(q_{i}^{f}\) | Lower limit of reactive power generation bus node i | \(q_{i}^{r}\) | Upper limit of reactive power generation bus node i |
\(p^r_{i}(t)\) | Regular power demand at bus node i during time period t | \(\Psi\) | Unit traffic flow charging demand rate |
\(N^{ch}_b\) | The binary variable is set to 1 if charger b is connected to charging station ch | \(P_{sd}\) | Paths from origin s to destination d |
Indices | |||
t | Index for time period | e | Index for link |
p | Index for path | s, d | Index for OD pair |
i | Index for source bus node | j | Index for demand bus node |
ch | Index for charging station | ev | Index for electric vehicle |
k | Index for pile | Â | Â |