This research addresses the critical challenge of distribution network congestion exacerbated by the large-scale integration of electric vehicles (EVs) and renewable energy sources. Unlike conventional loads, electric vehicle loads exhibit unique spatiotemporal flexibility, enabling strategic charging/discharging scheduling to alleviate grid stress. However, uncertainties in distributed generation (e.g., wind/PV) and load patterns complicate congestion management. Here, I present a robust optimization framework leveraging dynamic tariffs (DTs) to coordinate electric vehicle behavior across coupled transportation-distribution networks, ensuring reliable grid operation under worst-case uncertainty scenarios.
1. Integrated Modeling of Transportation and Distribution Networks
1.1 Linear Dynamic Traffic Assignment for EV Flows
The transportation network is modeled as a graph G={A,N}G={A,N}, where AA denotes road segments (arcs) and NN represents nodes (junctions/FCSs). For EV travel demand between origin-destination (O-D) pair ii:fm,tod=∑d∈Diuad,j,tin,∀t,∀mfm,tod=d∈Di∑uad,j,tin,∀t,∀m∑tuad,j,tin=∑tvad,j,tout,∀j∈Pit∑uad,j,tin=t∑vad,j,tout,∀j∈Pi
where fm,todfm,tod is the total EV volume for O-D pair ii managed by aggregator mm at time tt, uad,j,tinuad,j,tin and vad,j,toutvad,j,tout denote EVs entering/exiting path jj, and DiDi is the set of paths for O-D pair ii.
Table 1: Key Parameters for Transportation Network Road Segments
| Road Segment | EV Capacity (NaNa) | Travel Time (τaτa, h) |
|---|---|---|
| T1–T2 | 200 | 5 |
| T2–T5 | 79 | 2 |
| T5–T6 | 82 | 2 |
| … | … | … |
Node-Specific Dynamics:
- Ordinary Nodes:vj,taprev=uj,tanext,∀n∈Nordvj,taprev=uj,tanext,∀n∈Nord
- FCS Nodes: EVs choose charging (uj,tcduj,tcd), discharging (uj,tdcuj,tdc), or bypassing (uj,tfreeuj,tfree):vj,ta+=uj,tcd+uj,tfreevj,ta+=uj,tcd+uj,tfreeuj,ta−=vj,tcd+uj,tfreeuj,ta−=vj,tcd+uj,tfreeQueue dynamics at FCS yy:xy,j,tqueue−xy,j,t−1queue=uj,tcd−vj,tcd,∀j∈Pyxy,j,tqueue−xy,j,t−1queue=uj,tcd−vj,tcd,∀j∈Py
1.2 State-of-Charge (SOC) Dynamics
EV SOC evolves based on travel and charging activities:
- On Roads:ea,j,tsoc−ea,j,t−1soc=eu,j,t−τain−ev,j,tout−catr⋅ua,j,t−τa,∀a,∀jea,j,tsoc−ea,j,t−1soc=eu,j,t−τain−ev,j,tout−catr⋅ua,j,t−τa,∀a,∀jwhere catrcatr is energy consumption per km, and τaτa is travel delay.
- At FCSs:ey,j,tsoc,cd−ey,j,t−1soc,cd=eu,j,tin,cd−ev,j,tout,cd+∑l∈LyplηΔtxl,j,tcdey,j,tsoc,cd−ey,j,t−1soc,cd=eu,j,tin,cd−ev,j,tout,cd+l∈Ly∑plηΔtxl,j,tcdwith plpl = charging/discharging power, ηη = efficiency.
Table 2: EV Travel Demand Parameters
| Origin | Destination | EV Volume (fi,todfi,tod) | Total Initial SOC (ei,tsocei,tsoc) |
|---|---|---|---|
| T9 | T1 | 5,000 | 2,000 |
| T4 | T10 | 5,000 | 2,000 |
2. Robust Congestion Management Model
2.1 Uncertainty Modeling
Wind/PV generation (Pr,tWT,Pr,tPVPr,tWT,Pr,tPV) and load (Pr,tConPr,tCon) uncertainties are bounded within intervals:ΔPr,tWT∈[−ξrWT,ξrWT],ΔPr,tPV∈[−ξrPV,ξrPV],ΔPr,tCon∈[−ξrCon,ξrCon]ΔPr,tWT∈[−ξrWT,ξrWT],ΔPr,tPV∈[−ξrPV,ξrPV],ΔPr,tCon∈[−ξrCon,ξrCon]
A robust control parameter Γr∈[0,1]Γr∈[0,1] adjusts conservatism:∣Pr,t∗−P‾r,t∗ξr∗∣≤Γr∗ξr∗Pr,t∗−Pr,t∗≤Γr∗
2.2 Distribution Network Constraints
Linearized AC power flow equations govern the distribution grid:∑s∈c(r)Ps,tl+Pr,tInj−∑k∈c(r)Pk,tl=Pr,tLoads∈c(r)∑Ps,tl+Pr,tInj−k∈c(r)∑Pk,tl=Pr,tLoadVr=Vs−Ps,tlRl+Qs,tlXlV0,∀lVr=Vs−V0Ps,tlRl+Qs,tlXl,∀l
Robust reformulation for worst-case uncertainty (min wind/PV, max load):∑s∈c(r)Ps,tl+(P‾r,tWT−ΓrWTξrWT)+(P‾r,tPV−ΓrPVξrPV)−∑k∈c(r)Pk,tl=(P‾r,tCon+ΓrConξrCon)+Pr,tEVs∈c(r)∑Ps,tl+(Pr,tWT−ΓrWTξrWT)+(Pr,tPV−ΓrPVξrPV)−k∈c(r)∑Pk,tl=(Pr,tCon+ΓrConξrCon)+Pr,tEV
Constraints include line flow and voltage limits:−flmax≤Plt≤flmax,Vmin≤Vr≤Vmax−flmax≤Plt≤flmax,Vmin≤Vr≤Vmax
Table 3: Distribution Line Flow Limits
| Line | Flow Limit (kW) |
|---|---|
| E5–F1 (L1) | 800 |
| E5–F2 (L2) | 300 |
| E5–F3 (L3) | 750 |
| E5–F4 (L4) | 350 |
2.3 Dynamic Tariff (DT) Formulation
The dynamic tariff RtRt reflects congestion severity, derived from dual variables (λt+,λt−λt+,λt−, γt+,γt−γt+,γt−) of line flow/voltage constraints:Rt=(λt+−λt−)+(γt+−γt−)Rt=(λt+−λt−)+(γt+−γt−)
DSO’s Objective: Minimize total EV travel time + charging cost:min∑m(Cmtime+Cmenergy)minm∑(Cmtime+Cmenergy)Cmenergy=(ct+β∑mpm,t)∑mpm,tCmenergy=(ct+βm∑pm,t)m∑pm,t
EVA’s Response (after receiving RtRt):min(Cmtime+(ct+β⋅pm,t+Rt)⋅pm,t)min(Cmtime+(ct+β⋅pm,t+Rt)⋅pm,t)
3. Simulation Analysis
3.1 Deterministic vs. Robust Management
- Deterministic Case: Without uncertainty, DTs manage congestion (Fig. 5).
*Table 4: DT Values Under Deterministic Strategy ($/kWh)*TimeF2F310:000.39730.340712:000.37100.323218:000.23780.2024 - With 10% Uncertainty: Deterministic strategy fails (Fig. 9), lines exceed limits.
3.2 Robust Optimization Results
Under worst-case uncertainty (10% fluctuation):
- Higher DTs (e.g., 10:00 at F2: $0.7738/kWh vs. $0.3973 deterministically).
- EVs shift paths/charging: 571.7 EVs from Path 3 to Paths 1–2 for O-D1; 396.5 EVs from Path 1 to Paths 2–3 for O-D2.
- Congestion avoided despite extreme fluctuations (Fig. 10).
*Table 5: Robust DT Values Under 10% Uncertainty ($/kWh)*
| Time | F2 | F3 |
|---|---|---|
| 10:00 | 0.7738 | 0.6521 |
| 12:00 | 0.7475 | 0.6346 |
| 18:00 | 0.6143 | 0.5137 |
3.3 Impact of Robustness Parameter ΓΓ
Increasing ΓΓ enhances robustness at higher costs:
| Uncertainty | Γ=0Γ=0 | Γ=0.5Γ=0.5 | Γ=1.0Γ=1.0 |
|---|---|---|---|
| 5% fluctuation | $4,537.4 | $5,102.1 | $5,692.8 |
| 10% fluctuation | $4,537.4 | $5,421.3 | $5,846.2 |
4. Conclusion
This work establishes a robust optimization-based congestion management framework for distribution networks with high electric vehicle penetration. Key contributions include:
- Linearized EV Flow and SOC Models: Captures spatiotemporal flexibility of electric vehicle mobility and charging.
- Dynamic Tariff Mechanism: DTs derived from dual variables provide real-time congestion signals, incentivizing EVs to shift loads.
- Uncertainty Handling: Box uncertainty sets and adjustable ΓΓ balance robustness and optimality.
- Decentralized Coordination: DSO computes DTs; EVAs optimize locally, ensuring scalability.
Simulations confirm that dynamic tariffs effectively mitigate congestion under 10% renewable/load fluctuations by leveraging electric vehicle flexibility. Higher ΓΓ values increase system robustness at the expense of higher EV operating costs, providing a tunable trade-off for grid operators. Future work will integrate stochastic optimization to reduce conservatism.