With the rapid development of electric vehicles, particularly in regions like China EV markets, DC microgrids are increasingly integrating large-scale electric cars. These vehicles can function as energy storage units by discharging power or as loads when charging from the grid. However, when numerous electric cars are connected to a DC microgrid, their collective charging and discharging behaviors significantly influence the system’s stability. This article analyzes these effects by developing comprehensive models of DC microgrid components, including photovoltaic units, DC bus voltage control units, and electric car charging/discharging units. Using impedance-based stability analysis, we evaluate the system’s behavior through Bode and Nyquist plots, providing insights into how electric car integration affects microgrid dynamics.
The growing adoption of electric cars worldwide, especially in China EV initiatives, underscores the importance of understanding their impact on power systems. Electric cars can provide grid services through vehicle-to-grid (V2G) technology, but their unpredictable charging patterns may lead to instability in DC microgrids. This study focuses on modeling these interactions and proposes methods to enhance stability through controlled charging and discharging strategies.
System Modeling of DC Microgrid with Electric Car Integration
To analyze the stability of a DC microgrid with electric car integration, we first develop models for key components: the photovoltaic unit, DC bus voltage control unit, electric car charging unit (modeled as a constant power load), and electric car discharging unit. Each component’s dynamics are described using state-space equations and transfer functions.
Photovoltaic Unit Model
The photovoltaic unit consists of a PV array connected to a Boost converter, which steps up the voltage to the DC bus. The state-space equations for the PV unit are derived as follows:
$$ C_{PV} \frac{dU_{PV}}{dt} = I_{PV} – I_{L_{PV}} $$
$$ L_{PV} \frac{dI_{L_{PV}}}{dt} = U_{PV} – R_{PV} I_{L_{PV}} – (1 – d_1) U_{bus} $$
$$ C_{s1} \frac{dU_{bus}}{dt} = (1 – d_1) I_{L_{PV}} – I_{PV_{DC}} $$
where \( U_{PV} \) and \( I_{PV} \) are the output voltage and current of the PV array, \( L_{PV} \) is the inductor, \( R_{PV} \) is the equivalent resistance, \( C_{PV} \) and \( C_{s1} \) are capacitances, \( U_{bus} \) is the bus voltage, \( I_{PV_{DC}} \) is the output current, and \( d_1 \) is the duty cycle of the Boost converter. Linearizing around a steady-state operating point and applying Laplace transforms yields small-signal models. The output impedance of the PV unit, \( Z_{PV} \), is given by:
$$ Z_{PV} = Z_2 – \frac{G_1 G_{PV} G_3}{1 + G_{PV} H_1} $$
where \( G_1 \), \( G_3 \), and \( H_1 \) are transfer functions related to voltage and current dynamics, and \( G_{PV} = K_{PV} + \frac{T_{PV}}{s} \) is the PI controller for maximum power point tracking (MPPT).
DC Bus Voltage Control Unit (Energy Storage Unit)
This unit uses a bidirectional DC-DC converter with a battery model, typically a first-order Thevenin equivalent for lithium-ion batteries common in electric cars. The state-space equations are:
$$ L_{BAT} \frac{dI_{L_{BAT}}}{dt} = U_{BAT} – R_{BAT} I_{L_{BAT}} – d_2 U_{bus} $$
$$ C_{s2} \frac{dU_{bus}}{dt} = d_2 I_{L_{BAT}} – I_{BAT_{DC}} $$
where \( U_{BAT} \) and \( I_{L_{BAT}} \) are the battery voltage and inductor current, \( R_{BAT} \) and \( L_{BAT} \) are filtering elements, \( C_{s2} \) is the output capacitance, \( d_2 \) is the duty cycle, and \( I_{BAT_{DC}} \) is the output current. The battery model includes an internal resistance \( r \), polarization resistance \( R_o \), and capacitance \( C_o \). The output impedance \( Z_{BAT} \) is derived as:
$$ Z_{BAT} = \frac{Z_3 – R_d P_1 G_9 G_{10} – G_6 G_9 G_{10}}{1 + P_1 G_9 G_{10}} $$
Here, \( G_6 \), \( G_9 \), and \( G_{10} \) are transfer functions for current and voltage control loops, and \( P_1 \) and \( R_d \) are controller parameters.
Electric Car Charging Unit (Constant Power Load Model)
When electric cars charge, they are modeled as constant power loads (CPLs) connected in parallel to the DC microgrid. For multiple electric cars, the equivalent power demand is summed. The model includes an input capacitor \( C_{CPL_i} \) for each load, with power \( P_{0_i} \) and voltage \( U_{CPL_i} \). The controlled current source is:
$$ I_{CPL_i} = \frac{P_{0_i}}{U_{CPL_i}} $$
The equivalent impedance for a CPL is negative, which can destabilize the system. The aggregate impedance for \( n \) charging electric cars is:
$$ Z_{CPL} = \frac{1}{\sum_{i=1}^n \left( \frac{1}{R_{CPL_i} + s L_{CPL_i} + \frac{1}{s C_{CPL_i} – \frac{P_{0_i}}{U_{CPL_i}^2}} \right)} $$
This model highlights how increasing the number of charging electric cars in China EV fleets can lead to instability due to the negative impedance characteristic.
Electric Car Discharging Unit
During discharging, electric cars act as distributed energy resources, providing power to the grid. The discharging unit uses a bidirectional converter with voltage and current control. The state-space equations are:
$$ L_F \frac{dI_{L_F}}{dt} = U_F – R_F I_{L_F} – (1 – d_3) U_{bus} $$
$$ C_{s3} \frac{dU_{bus}}{dt} = (1 – d_3) I_{L_F} – I_{F_{DC}} $$
where \( U_F \) and \( I_{L_F} \) are the input voltage and inductor current, \( R_F \) and \( L_F \) are filtering components, \( C_{s3} \) is the output capacitance, \( d_3 \) is the duty cycle, and \( I_{F_{DC}} \) is the output current. The output impedance \( Z_F \) is:
$$ Z_F = \frac{Z_4 – G_{12} G_{15} G_{16}}{1 + P_3 G_{15} G_{16}} $$
Here, \( G_{12} \), \( G_{15} \), and \( G_{16} \) are transfer functions, and \( P_3 \) is the voltage controller. Discharging electric cars introduce positive impedance, enhancing stability.
Stability Analysis Using Impedance Methods
To assess DC microgrid stability with electric car integration, we use an impedance-based approach. The overall system impedance \( Z(s) \) is the parallel combination of all unit impedances:
$$ Z = Z_{PV} \parallel Z_{BAT} \parallel Z_F \parallel Z_{bus} \parallel Z_{CPL} $$
where \( Z_{bus} \) represents the line impedance. Stability is determined by three criteria based on \( Z(s) \):
- The real part of \( Z(s) \) should be positive: \( \text{Re}[Z(s)] > 0 \).
- The phase angle in the Bode plot should lie within \( \pm 90^\circ \).
- The Nyquist plot of \( Z(s) \) should not enter the left-half plane.
This method accounts for bidirectional power flow, crucial for electric car charging and discharging scenarios. By analyzing Bode and Nyquist plots, we can predict instability trends as the number of electric cars changes.
Case Studies and Results
We conduct multiple case studies to evaluate the impact of electric car charging and discharging on DC microgrid stability. System parameters are based on typical values: PV generation capacity of 500 kW to 1000 kW, individual electric car charging power of 5 kW, and discharging power of 3.5 kW to 16 kW. The table below summarizes key parameters used in the analysis.
| Parameter | Value | Description |
|---|---|---|
| \( R_{PV} \) | 0.05 Ω | PV equivalent resistance |
| \( L_{PV} \) | 1.45 mH | PV inductor |
| \( C_{PV} \) | 1 mF | PV capacitance |
| \( C_{s1} \) | 3227 μF | Boost output capacitance |
| \( K_{PV}/T_{PV} \) | 0.005/0.001 | PV controller gains |
| \( R_{BAT} \) | 0.05 Ω | Battery resistance |
| \( L_{BAT} \) | 0.6 mH | Battery inductor |
| \( C_{s2} \) | 0.1 F | Battery output capacitance |
| \( U_{bus} \) | 400 V | DC bus voltage |
| Charging power per electric car | 5 kW | Typical for China EV models |
| Discharging power per electric car | 3.5–16 kW | Varies based on battery state |
Case Study 1: Impact of Increasing Charging Electric Cars
In this case, we analyze the system with PV generation fixed at 500 kW and increasing numbers of charging electric cars. The equivalent impedance \( Z(s) \) is computed for 10, 20, and 30 electric cars. The Nyquist plots show that as the number of charging electric cars rises, the curves shift toward the left-half plane, indicating reduced stability. Bode plots reveal phase angles exceeding \( \pm 90^\circ \) for higher counts, confirming instability. This demonstrates that large-scale charging of electric cars in DC microgrids can trigger instability, especially in dense China EV deployments.
Case Study 2: Stabilizing Effect of Discharging Electric Cars
Here, the DC microgrid is initially unstable with 100 charging electric cars (total load 500 kW). We then integrate discharging electric cars with a voltage of 200 V and output power of 16 kW per vehicle. The Nyquist plots move to the right-half plane after integration, and Bode phase angles fall within \( \pm 90^\circ \), indicating restored stability. This highlights how electric car discharging can mitigate instability, offering a viable strategy for grid support.
Case Study 3: Transition Between Charging and Discharging
With a total of 100 electric cars, we evaluate scenarios where cars switch from charging to discharging: 100 charging vs. 0 discharging, 75 charging vs. 25 discharging, and 50 charging vs. 50 discharging. As more electric cars discharge, the system impedance shifts positively, improving stability. The table below quantifies this effect through phase margin and stability status.
| Scenario | Charging Cars | Discharging Cars | Phase Margin | Stability |
|---|---|---|---|---|
| 1 | 100 | 0 | < 90° | Unstable |
| 2 | 75 | 25 | ≈ 90° | Marginal |
| 3 | 50 | 50 | > 90° | Stable |
Case Study 4: Effect of Increased PV Capacity
With PV capacity raised to 1000 kW, we test the system with 140, 150, and 160 charging electric cars (charging power up to 800 kW). Instability occurs when charging power reaches approximately 80% of PV generation (e.g., 160 cars). This threshold is critical for planning electric car integration in China EV projects, as exceeding it may require additional stabilization measures.
Mathematical Formulation of Stability Criteria
The impedance-based stability analysis relies on the Nyquist criterion applied to the system impedance \( Z(s) \). For a stable system, the number of encirclements of the critical point (-1, 0) in the Nyquist plot of \( Z(s) \) must be zero. The phase margin \( \phi_m \) is derived from the Bode plot as:
$$ \phi_m = 180^\circ + \angle Z(j\omega_c) $$
where \( \omega_c \) is the crossover frequency. A positive phase margin within \( \pm 90^\circ \) ensures stability. For electric car charging, the negative impedance of CPLs reduces the phase margin, while discharging adds positive resistance.
The equivalent impedance for multiple electric cars can be generalized. For \( n \) charging cars, the total CPL impedance is:
$$ Z_{CPL_{total}} = \frac{1}{\sum_{i=1}^n \frac{1}{Z_{CPL_i}}} $$
where \( Z_{CPL_i} = \frac{U_{CPL_i}^2}{P_{0_i}} \) in the small-signal sense. For discharging cars, the impedance is inductive-resistive, contributing positively to \( Z(s) \).
Conclusions and Implications for China EV Deployment
This analysis demonstrates that the charging and discharging behaviors of electric cars have profound effects on DC microgrid stability. Increasing the number of charging electric cars leads to instability due to negative impedance characteristics, while discharging electric cars enhance stability by providing positive impedance. The transition between charging and discharging modes can be leveraged as an effective strategy to maintain grid stability, particularly in regions with high China EV adoption.
Key findings include:
– Electric car charging should be managed to avoid exceeding 80% of the PV generation capacity to prevent instability.
– Discharging electric cars can stabilize microgrids, supporting the use of V2G technology.
– Impedance-based analysis provides a robust framework for evaluating these effects.
Future work should focus on probabilistic modeling of electric car behaviors and developing real-time control strategies for large-scale China EV integration. By optimizing charging and discharging schedules, DC microgrids can achieve greater resilience and support the growing fleet of electric cars.