Abstract The distributed economic model predictive control (DEMPC) method for electric vehicle (EV) charging stations to participate in grid auxiliary frequency regulation. The goal is to address the challenge of ensuring economic feasibility while maintaining system performance in large-scale EV-assisted frequency regulation. By integrating EVs as flexible loads, the proposed DEMPC achieves distributed collaborative control across multi-area power grids through a single-layer structure managing a two-tiered hierarchy. Convex relaxation of economic cost functions and appropriate terminal cost functions are used to optimize controllers and guarantee asymptotic stability. Simulation results validate the effectiveness and superiority of the method compared to centralized and decentralized approaches.
1. Introduction With the depletion of fossil fuels and growing environmental concerns, the integration of renewable energy and electric vehicles (EVs) has become crucial for sustainable power systems. EVs, as flexible loads, can bidirectionally exchange power with the grid, acting as both consumers (during charging) and generators (during discharging) . This “source-load” duality makes them valuable for load frequency control (LFC), where they can quickly mitigate frequency fluctuations caused by the intermittency of renewable energy and imbalances between generation and demand .
Traditional LFC in multi-area power systems relies on a two-layer hierarchical control structure, which often neglects instantaneous economic optimization . Economic model predictive control (EMPC) offers a single-layer approach to combine economic load scheduling and LFC, enabling transient optimization . However, EMPC faces challenges in complex, large-scale grids due to high computational burdens and poor scalability when applied centrally . Distributed control strategies are thus necessary to handle the decentralized nature of multi-area grids while ensuring collaborative optimization and stability.
This study proposes a DEMPC method for EV aggregator stations to assist in LFC. The key objectives are to: 1) develop a multi-area grid model incorporating EVs; 2) design a distributed control framework for economic optimization and frequency regulation; 3) prove the asymptotic stability of the algorithm; and 4) validate performance through simulations.
2. Multi-Area Grid and EV Aggregator Models
2.1 Single-Area Equivalent Power Model
A four-area interconnected grid is considered, where each area includes a thermal power unit with a governor, turbine, and generator. The structure is shown in Figure 1 (conceptual), with power exchanges between areas denoted as \(P_{\text{tie,ij}}\) .
Governer Dynamics
The governor adjusts output in response to frequency 偏差 (\(\Delta f_i\)):\(\Delta \dot{X}_{gi} = -\frac{1}{T_{gi} R_{i}} \Delta f_{i} – \frac{1}{T_{gi}} \Delta X_{gi} + \frac{1}{T_{gi}} \Delta u_{i} \quad (1)\) where \(u_i\) is the control input from the EMPC controller, \(T_{BU}\) is the governor time constant, and \(R_i\) is the droop coefficient .
Turbine Dynamics
Turbine power (\(\Delta P_{gi}\)) follows a first-order lag:\(\Delta \dot{P}_{gi} = -\frac{1}{T_{ti}} \Delta P_{gi} + \frac{1}{T_{bi}} \Delta X_{gi} \quad (2)\) with \(T_{ti}\) as the turbine time constant .
Generator and Tie-Line Power
The generator’s frequency deviation and tie-line power are modeled as:\(\Delta P_{gi} – \Delta P_{di} – \Delta P_{\text{tie,i}} = (M_i^a s + D_i) \Delta f_i \quad (3)\)\(\Delta P_{\text{tie,ij}} = \frac{2\pi}{s} \sum_{\substack{j=1 \\ j \neq i}}^{M} T_{ij} (\Delta f_i – \Delta f_j) \quad (4)\) where \(M_i^a\) and \(D_i\) are inertia and damping coefficients, and \(T_{ij}\) is the synchronizing coefficient between areas .
2.2 EV Aggregator Model
An equivalent EV model for LFC is developed, considering charging/discharging power (\(\Delta P_e\)) and state of charge (SOC). The dynamic equation for EV power is:\(\Delta \dot{P}_{Ek} = \frac{1}{T_e} \Delta u_{Ek} – \frac{1}{T_e} \Delta P_{Ek} \quad (5)\) with \(T_e\) as the EV time constant and \(\Delta u_{Ek}\) as the control signal .
Constraints
- Power bounds: \(-\mu_{ek} \leq \Delta P_{Ek}(t) \leq \mu_{ek}\)
- Rate limits: \(-\delta_{ek} \leq \Delta \dot{P}_{Ek}(t) \leq \delta_{ek}\)
- SOC limits: \(S_{\text{OC,min}} \leq S_{\text{OC,k}} \leq S_{\text{OC,max}}\) .
Example Calculation: For a 100 MW grid requiring 5% 调频容量 (5 MW), assuming 50 kW EVs with 20% adjustable capacity (70–90% SOC), the number of EVs needed is:\(\text{Number of EVs} = \frac{5 \times 10^6 \, \text{W}}{50 \times 10^3 \, \text{W} \times 0.2} = 500 \, \text{units} \quad (6)\)
2.3 Combined Grid Model with LFC Controller
The overall LFC framework includes thermal units, EV aggregators, and load disturbances. The state space model combines equations (1)–(5) and tie-line dynamics, discretized as:\(x(k+1) = Ax(k) + Bu(k) + w(k) \quad (7)\) where x includes states like \(\Delta f_i\), \(\Delta P_{gi}\), and \(\Delta P_{Ek}\); u is the control input vector; and w represents disturbances .
3. Distributed Economic Model Predictive Control (DEMPC)
3.1 Economic Model Predictive Control (EMPC)
EMPC optimizes an economic cost function directly, unlike standard MPC, which focuses on tracking setpoints. The stage cost function for EMPC is:\(l(x, u) = \sum_{i=1}^{M} \left( F_{pi} + F_{ci} + F_{di} + F_{ei} \right) \quad (8)\) where:
- 发电成本 (Generation Cost):\(F_{pi}(k) = \frac{1}{2} a_i \Delta P_{gi}^2(k) + b_i \Delta P_{gi}(k) + c_i \quad (9)\)\(a_i, b_i, c_i\) are cost coefficients .
- 频率控制成本 (Frequency Control Cost):\(F_{ci}(k) = \Delta f_i(k) Q_{ci} \Delta f_i(k) \quad (10)\)\(Q_{ci}\) is a weighting coefficient to penalize frequency deviations .
- 联络线功率成本 (Tie-Line Power Cost):\(F_{di}(k) = \Delta P_{\text{tie,i}}(k) Q_{di} \Delta P_{\text{tie,i}}(k) \quad (11)\)\(Q_{di}\) weights tie-line power deviations .
- EV 调节成本 (EV Regulation Cost):\(F_{ei}(k) = a_{ei} \Delta P_{EV,i}^2(k) + b_{ei} \Delta P_{EV,i}(k) \quad (12)\)\(a_{ei}, b_{ei}\) represent battery degradation and electricity price costs .
3.2 Distributed Control Framework
For multi-area grids, DEMPC coordinates subarea controllers to minimize a collaborative cost function. Each subarea’s local cost function is:\(\phi_i(x, u_i) = \sum_{t=0}^{N-1} l_i(x_i(t|k), u_i(t|k)) + V_{fi}(x_i(N|k)) \quad (13)\) where \(V_{fi}\) is the terminal cost function. The collaborative cost function integrates local costs with neighboring subareas:\(V(x, u_i^{\alpha\beta}) = \min_{u_i} \left[ \alpha_i \phi_i(x, u_i) + \sum_{j \neq i} \alpha_j \phi_j(x, u_j^{p-1}) \right] \quad (14)\) with \(\alpha_i > 0\) and \(\sum \alpha_i = 1\), ensuring distributed optimization through iterative collaboration .
3.3 Optimization Problem
The DEMPC optimization problem for subarea i is:\(\begin{aligned} \min_{u_i} & \sum_{t=0}^{N-1} l_i(x_i(t|k), u_i(t|k)) + V_{fi}(x_i(N|k)) \\ \text{s.t.} & \quad x_i(k+1) = \sum_{j \in \mathcal{N}_i} A_{ij} x_j(k) + B_{ii} u_i(k) \\ & \quad x(0|k) = x_0, \quad x(N|k) \in X_f \quad (15) \end{aligned}\) where \(X_f\) is the terminal set ensuring power balance at steady state:\(\| P_{gi}(N|k) – P_{di}(N|k) – P_{\text{tie,i}}(N|k) + P_{\text{EV,i}}(N|k) \| \leq \epsilon \quad (16)\)
4. Stability Analysis
4.1 Assumptions and Lyapunov Theory
Assumption 1: The system is strictly dissipative with a supply rate \(s_i(x_i, u_i) = l_i(x_i, u_i) – l_i(x_{si}, u_{si})\), and there exists a Lyapunov function \(\lambda_i\) such that:\(\lambda_i(x_i(t+1|k)) – \lambda_i(x_i(t|k)) \leq -\rho_i(x_i(t|k)) + s_i(x_i(t|k), u_i(t|k)) \quad (17)\) where \(\rho_i(\cdot) > 0\) .
Assumption 2: A terminal controller \(K_N\) exists such that:\(V_{fi}(x_i(t+1|k)) – V_{fi}(x_i(t|k)) \leq -l_i(x_i(t|k), u_i(t|k)) + l_i(x_{si}, u_{si}) \quad (18)\)
4.2 Terminal Cost and Set Calculation
The terminal cost \(V_{fi}(x) = x^T P_f x + x^T p_f\) is designed using discrete linear quadratic regulator (DLQR) or linear matrix inequalities (LMI) to ensure stability. The terminal set \(X_f = \{ x | x^T P_f x \leq \alpha \}\) is computed to satisfy input and state constraints .
Theorem 1: Under Assumptions 1–2, the DEMPC system is asymptotically stable and converges to the steady state \(x_s\) .
5. Simulation Results
5.1 Setup
A four-area grid with 100 MW capacity per area is simulated. Key parameters are listed in Tables 1 and 2.
Table 1: Thermal Power Plant Parameters
| Area | \(T_{gi}\) (s) | \(T_{ti}\) (s) | \(M_i^a\) (s) | \(D_i\) (p.u./Hz) | \(R_i\) (Hz/p.u.) |
|---|---|---|---|---|---|
| 1 | 0.081 | 0.28 | 3.50 | 2.75 | 2.6 |
| 2 | 0.072 | 0.30 | 3.70 | 3.20 | 2.8 |
| 3 | 0.083 | 0.32 | 4.00 | 2.80 | 2.7 |
| 4 | 0.075 | 0.35 | 3.75 | 2.50 | 2.4 |
Table 2: EV Aggregator Parameters
| EV | \(T_e\) (s) | \(\delta_e\) (p.u./s) | \(\mu_e\) (p.u.) | \(E_{\text{max}}\) (kWh) | \(E_{\text{min}}\) (kWh) |
|---|---|---|---|---|---|
| 1 | 0.02 | 0.01 | 0.95 | 50 | 40 |
| 2 | 0.02 | 0.015 | 0.90 | 50 | 37.5 |
5.2 Results
Frequency Deviation
At \(t = 3 \, \text{s}\), a load disturbance of 0.005 p.u. is applied. Figure 2 (conceptual) shows frequency deviations converge to zero within 4 seconds under DEMPC, faster than decentralized EMPC and centralized EMPC (which has higher computational delay) .
Output Power and Tie-Line Power
Thermal unit output powers (Figure 3) adjust rapidly, with EVs providing auxiliary power (Figure 4). Tie-line powers (Figure 5) stabilize within 5 seconds, demonstrating effective inter-area coordination .
Computational Efficiency
DEMPC achieves a 30% reduction in computation time compared to centralized EMPC, making it suitable for large-scale grids .
6. Conclusion This study presents a DEMPC method for EV charging stations to participate in grid frequency regulation, addressing both economic optimization and system stability. By integrating EVs as flexible loads and using distributed control, the method efficiently coordinates multi-area grids, outperforming centralized and decentralized approaches in speed and computational efficiency. Lyapunov-based stability analysis and simulations validate the effectiveness of the proposed framework, highlighting its potential for sustainable power systems with high renewable energy penetration.