Abstract
As the penetration of electric vehicles (EVs) continues to rise, integrating large-scale EV aggregations (EVAs) into power systems has become critical for sustainable energy management. This study addresses the challenge of quantifying EVAs’ dispatchable potential across diverse regions and time periods. I propose a Multi-stage Electric Vehicle Dispatchable Region (MEVDR) framework, which partitions dispatchable capabilities into energy and power domains. By leveraging Gaussian Mixture Models (GMMs) to analyze regional and temporal variations in EV behavior, I establish a Vehicles-Garage-Grid Multi-level Coordinated Control System (VGGMCCS) that optimizes interactions among EV users, garage operators, and the power grid. Simulation results validate that VGGMCCS reduces grid losses by up to 12.17% during peak hours, lowers user charging costs by 7.88%, and enhances garage profitability by 17.63% compared to benchmark strategies.
Keywords:
Electric vehicle, vehicle-to-grid (V2G), dispatchable region, coordinated control, Gaussian mixture model, multi-level optimization
1. Introduction
The rapid proliferation of electric vehicles presents both opportunities and challenges for power systems. On one hand, EVs can serve as flexible energy storage units through vehicle-to-grid (V2G) technology [1-3]. On the other hand, their uncoordinated charging/discharging behaviors may destabilize grid operations [4-6]. Existing studies often overlook the dual nature of EV dispatchable potential—i.e., energy and power capabilities—and lack systematic frameworks for multi-level coordination among EVs, garages, and the grid [7-10].
My research aims to bridge these gaps by:
- Developing MEVDR to characterize EVAs’ dispatchable energy and power potentials;
- Analyzing regional/temporal variations in MEVDR using GMM;
- Constructing VGGMCCS to optimize multi-level interactions and validate its efficacy via case studies.
2. MEVDR Construction for Electric Vehicles
2.1 Modeling Foundations
I define MEVDR based on six key variables: EV grid connection time (\(t_i^a\)), disconnection time (\(t_i^l\)), initial SOC (\(S_i^{arrive}\)), target SOC (\(S_i^{tar}\)), battery capacity (\(E_i\)), and user-accepted minimum SOC (\(S_i^{LB}\)). Key assumptions include:
- Dispatch periods are 15-minute intervals (96 periods/day) [11];
- EVs connect to the grid once daily with continuous charging/discharging [12].
2.2 Dispatchable Energy Region (DER)
DER reflects the energy 可调潜力 of EVs over specific periods, constrained by:
- ASAP (As Soon as Possible) Constraint: The minimum time (\(T_i^{ASAP}\)) to charge to \(S_i^{UB}\) (user-accepted maximum SOC):\(T_i^{ASAP} = \frac{(S_i^{UB} – S_i^{arrive}) E_i}{\sum_{k=0}^{n} \text{max}(P_i^C(k)) \eta_C}\) where \(P_i^C(k)\) is the charging power at period k, and \(\eta_C\) is charging efficiency [13]. The corresponding maximum energy (\(E_i^{ASAP}\)):\(E_i^{ASAP} = \int_{t_i^a}^{t_i^a + T_i^{ASAP}} \eta_C \text{max}(P_i^C(t)) dt\) with \(S_i^{LB} \leq S_i^{UB} – E_i^{ASAP}/E_i\).
- LR (Latest Response) Constraint: The latest time (\(T_i^{LR}\)) to meet charging demands:\(T_i^{LR} \geq t_i^l – \frac{(S_i^{tar} – S_i^{LB}) E_i}{\eta_C P_i^C}\) The maximum discharge energy (\(E_i^{LR}\)):\(E_i^{LR} = \frac{t_i^a – \text{min}(T_i^{LR})}{P_i^{C,\text{MAX}} + P_i^{D,\text{MAX}}} P_i^{C,\text{MAX}} P_i^{D,\text{MAX}} \eta_D\) where \(P_i^{D,\text{MAX}}\) is the maximum discharge power, and \(\eta_D\) is discharge efficiency [14].
Table 1: Sample EV Parameters for DER Calculation
| Parameter | Value |
|---|---|
| \(S_i^{arrive}\) | 0.3 (30%) |
| \(S_i^{tar}\) | 0.8 (80%) |
| \(P_i^{C,\text{MAX}}\) | 2 kW |
| \(P_i^{D,\text{MAX}}\) | 2 kW |
| \(E_i\) | 80 kWh |
| \(t_i^a, t_i^l\) | 4:50, 19:45 (hours) |
2.3 Dispatchable Power Region (DPR)
DPR quantifies EVs’ power regulation flexibility, defined by:
- DF (Dispatchable Flexibility) Constraint: Upper bound (DFU) for charge power:\(\phi_i^{UB} = \left| \int_{t_0}^{t_s} P_i^{UB}(t) dt \right|, \quad P_i^{UB}(t) \leq \text{max}(P_i^C(t))\) Lower bound (DFL) for discharge power:\(\phi_i^{LB} = \left| \int_{t_0}^{t_s} P_i^{LB}(t) dt \right|, \quad P_i^{LB}(t) \leq \text{max}(P_i^D(t))\) where \(t_0\) and \(t_s\) denote start and stable states of power adjustment [15].
- Charging/Discharging Power Constraints: For DC fast charging, the power profile depends on SOC:\(P_i^C(t) \leq \begin{cases} \frac{1 + 22.5 S_{i,t}}{10} P_i^{C,\text{MAX}} & S_{i,t} < S_i^{\text{point}} \\ P_i^{C,\text{MAX}} & S_i^{\text{point}} \leq S_{i,t} < S_i^{tar} \\ \frac{46 – 45 S_{i,t}}{10} P_i^{C,\text{MAX}} & S_{i,t} \geq S_i^{tar} \end{cases}\) where \(S_i^{\text{point}}\) is the SOC threshold [16].
2.4 EVA-MEVDR Aggregation
For a cluster of N EVs, EVA-MEVDR integrates individual MEVDRs via Minkowski addition:
- EVA-DER: ASAP energy constraint:\(E_{\text{EVA}}^{\text{ASAP}} = \text{max}_{t \in T} \left( \int_{\text{min}(t_i^a)}^{t} \Delta E_{\text{EVA}}(t) dt \right)\) where \(\Delta E_{\text{EVA}}(t) = E_{\text{EVA}}(t) – E_{\text{EVA}}(t-1)\) [17]. LR time constraint:\(T_{\text{EVA}}^{\text{LR}} = \text{max}_{i \in \Phi_{\text{EV}}} (t_i^l) – \int_{\text{min}(t_i^a)}^{T_{\text{end}}} \left[ -E_{\text{EVA}}^{\text{LR}} – \int_{\text{min}(t_i^a)}^{t} P_{\text{EVA}}^C(t) dt \right] T_{\text{span}} dt\) with \(E_{\text{EVA}}^{\text{LR}} = \text{max}_{t \in T} \left( \sum_{i \in \Phi_{\text{EV}}} c_{i,t} E_i_t^{\text{LR}} \right)\), where \(c_{i,t}\) is the EV connection status [18].
- EVA-DPR: DFU and DFL for the cluster:\(\phi_{\text{EVA}}^{\text{UB}} = \sum_{i \in \Phi_{\text{EV}}} \left| \int_{t_0}^{t_s} P_i^{UB}(t) dt \right|, \quad \phi_{\text{EVA}}^{\text{LB}} = \sum_{i \in \Phi_{\text{EV}}} \left| \int_{t_0}^{t_s} P_i^{LB}(t) dt \right|\) where \(P_{\text{EVA}}^{\text{UB}}(t)\) and \(P_{\text{EVA}}^{\text{LB}}(t)\) are cluster-level dispatchable powers [19].
3. Multi-Region Analysis of MEVDR
Using GMM, I analyze MEVDR variations in three typical scenarios: office areas, commercial districts, and residential zones.
3.1 GMM for EV Behavior Clustering
GMM fits EV data as a weighted sum of Gaussian distributions:\(p(x) = \sum_{k=1}^{K} \omega_k \mathcal{N}(x | \mu_k, \sigma_k), \quad \sum_{k=1}^{K} \omega_k = 1\) where \(\omega_k\), \(\mu_k\), and \(\sigma_k\) are the weight, mean, and variance of component k [20]. The Expectation-Maximization (EM) algorithm estimates parameters iteratively [21].
Table 2: GMM Parameters for Office Area EVs (Weekdays)
| Component | \(\omega_k\) | \(\mu_k\) (h) | \(\sigma_k\) (h) |
|---|---|---|---|
| 1 | 0.35 | 8.2 | 0.5 |
| 2 | 0.40 | 12.5 | 0.8 |
| 3 | 0.25 | 17.3 | 0.6 |
3.2 Regional MEVDR Characteristics
- Office Areas:
- Weekdays: High MEVDR during working hours (8:00–18:00) due to stable EV parking;
- Weekends: Reduced MEVDR with sporadic EV presence [22].
- Commercial Districts:
- Fluctuating MEVDR due to peak shopping hours (10:00–22:00);
- Short-term high dispatchable potential during customer influx [23].
- Residential Zones:
- Stable MEVDR at night (20:00–8:00) as EVs charge at home;
- Reduced MEVDR during daytime as EVs are in use [24].
Table 3: MEVDR Metrics for Different Regions (Weekdays, 10:00–18:00)
| Region | \(E_{\text{EVA}}^{\text{ASAP}}\) (kWh) | \(E_{\text{EVA}}^{\text{LR}}\) (kWh) | \(\phi_{\text{EVA}}^{\text{UB}}\) (kWh) | \(\phi_{\text{EVA}}^{\text{LB}}\) (kWh) |
|---|---|---|---|---|
| Office | 4,200 | -3,800 | 3,500 | -2,800 |
| Commercial | 3,100 | -2,500 | 2,200 | -1,900 |
| Residential | 1,800 | -1,500 | 1,200 | -1,000 |
4. Vehicles-Garage-Grid Multi-level Coordinated Control System (VGGMCCS)
4.1 System Architecture
VGGMCCS is a bi-level model:
- Upper-Level Grid Model:
- Based on IEEE 33-node network;
- Minimizes grid losses and voltage deviations:\(\text{Obj1} = \text{min} \left( \alpha \sum_{j \in \text{net}} \sum_{t \in T} R(j) I_2(j,t) + \beta \sum_{j \in \text{net}} \sum_{t \in T} (V_2(j,t) – V)^2 \right)\) where \(\alpha\) and \(\beta\) are weights, \(R(j)\) is branch resistance, and V is the target voltage [25].
- Lower-Level Garage Model:
- Optimizes EV charging/discharging within MEVDR constraints;
- Minimizes user costs and maximizes garage profits:\(\text{Obj2} = \text{min} \sum_{i=0}^{\text{EV}} \sum_{t=0}^{96} c_{i,t} \left[ P_i^C(t) \cdot \text{ReLU}(f_{i,t}) \cdot p_{\text{G2V}}(t) – P_i^D(t) \cdot \text{ReLU}(-f_{i,t}) \cdot p_{\text{V2G}}(t) \right]\) where \(p_{\text{G2V}}\) and \(p_{\text{V2G}}\) are charging/discharging prices, and \(\text{ReLU}(x) = \text{max}(0, x)\) [26].
4.2 Constraints and Solution Approach
- Key Constraints:
- Grid voltage: \(V_j^{\text{min}}(t)^2 \leq V_j(t)^2 \leq V_j^{\text{max}}(t)^2\);
- Power balance: \(\sum (P_{ij}(k,t) – I^2 R(k)) + P_{\text{inj}}(j,t) = 0\);
- MEVDR limits: \(E_{\text{DER}}^{\text{LB}}(i,t) \leq (P_i^C(t) – P_i^D(t))t \leq E_{\text{DER}}^{\text{UB}}(i,t)\) [27].
- Solution Strategy:
- Decompose into MIP problems solvable via Gurobi solver;
- Iterate between upper and lower levels until convergence [28].
5. Case Study and Results
5.1 Simulation Setup
- System Parameters:
- Upper level: IEEE 33-node grid, transformer capacity 500 kVA;
- Lower level: Garage with 50 EV spots, max charge/discharge power 36/28 kW, efficiency 0.92 [29].
- Pricing:
- G2V: 1.15 × time-of-use 电价 (e.g., 1.7697 ¥/kWh at 8:00–12:00);
- V2G: 0.65 + (p(t) – 1.3418)/2 ¥/kWh [30].
5.2 Performance Comparison
Table 4: Key Performance Indicators
| Metric | VGGMCCS | Benchmark 1 | Benchmark 2 |
|---|---|---|---|
| Grid loss ratio | 18.90% | 28.00% | 22.76% |
| Garage profit (¥) | 64.25 | 69.49 | 54.62 |
| Total user cost (¥) | 5,978.38 | 6,490.05 | 5,904.50 |
| Transformer high-load time (h) | 0 | 8.5 | 0 |
| Peak power fluctuation (kW) | 15.00 | 629.58 | 87.83 |
| Solution time (s) | 138.41 | 90.06 | 172.09 |
- Grid-side Benefits:
- VGGMCCS reduces grid losses by 9.1% and 3.86% compared to Benchmarks 1 and 2;
- Voltage stability is enhanced, with reduced fluctuations during peak hours [31].
- Garage-side Impacts:
- Transformer load fluctuates less (96.36% lower than Benchmark 1 during peaks);
- Balanced EV charging/discharging optimizes power usage [32].
- User-side Economics:
- Average charging cost per EV drops by 7.88% vs. Benchmark 1;
- EVs maintain higher SOC (≥60%) for longer, improving reliability [33].
6. Conclusion
In this study, I have developed a systematic framework for characterizing and optimizing electric vehicle dispatchable regions. The MEVDR model effectively captures both energy and power potentials of EVAs, while VGGMCCS enables multi-level coordination to balance grid stability, garage profitability, and user needs. Key findings include:
- MEVDR provides a unified framework to quantify EVAs’ dispatchable capabilities, aiding in grid integration planning;
- Regional and temporal variations in MEVDR highlight the need for context-aware scheduling;
- VGGMCCS achieves a 12.17% reduction in peak grid losses and 7.88% lower user costs, demonstrating its practical utility.
Future work will explore real-time MEVDR updates and integrate renewable energy uncertainties into the VGGMCCS framework.