In modern power systems, the increasing integration of power electronic devices has led to significant challenges in stability, particularly sub/super-synchronous oscillations. As an essential component of highly power-electronized AC distribution and consumption systems, electric drive systems, which consist of numerous asynchronous motor-driven devices, exhibit severe oscillation issues. These electric drive systems operate in two distinct modes: power consumption, where the system draws power from the grid, and power feed, where it injects power back. Field observations indicate that these modes differentially impact system stability, with oscillations often occurring in the 10–20 Hz range, potentially triggering protective shutdowns. This article, from my perspective as a researcher, delves into the detailed impedance modeling of electric drive systems, analyzes the mechanisms behind sub/super-synchronous oscillations, and examines key influencing factors such as operating mode, grid strength, control parameters, and AC voltage amplitude. Using phase margin sensitivity, we identify dominant factors and validate findings through time-domain simulations based on actual system data.
The electric drive system in focus is based on a three-phase AC-DC-AC converter driving a squirrel-cage induction motor, capable of four-quadrant operation. The core structure includes a machine-side converter system and a grid-side converter system. The machine-side system comprises the load, induction motor, and machine-side converter, employing indirect flux-oriented control with dual loops for speed and flux. The grid-side system includes the converter, filter, and transformer, also using dual-loop control for DC-link voltage and current regulation. During operation, the electric drive system switches between power consumption (e.g., during lifting) and power feed (e.g., during lowering), making its dynamics complex and mode-dependent. To analyze stability, we adopt an impedance-based approach, which simplifies the system into interconnected subsystems represented by frequency-domain port impedances. This method offers clear physical insights and facilitates stability assessment using criteria like the Generalized Nyquist Criterion (GNC).
To model the electric drive system accurately, we decompose it into three submodules for modular multiport modeling. Submodule 1 represents the induction motor, whose equivalent circuit in the dq-frame yields a small-signal impedance equation. The voltage and flux equations are given by:
$$ \begin{align*}
v_{sd} &= R_s i_{sd} + \frac{d\psi_{sd}}{dt} – \omega_s \psi_{sq}, \\
v_{sq} &= R_s i_{sq} + \frac{d\psi_{sq}}{dt} + \omega_s \psi_{sd}, \\
\psi_{sd} &= L_s i_{sd} + L_m i_{rd}, \\
\psi_{sq} &= L_s i_{sq} + L_m i_{rq},
\end{align*} $$
where \( v_{sd}, v_{sq} \) and \( i_{sd}, i_{sq} \) are stator voltage and current components, \( R_s \) and \( L_s \) are stator resistance and inductance, \( L_m \) is mutual inductance, \( \omega_s \) is synchronous speed, and \( \psi \) denotes flux. The rotor dynamics are incorporated via swing equations. The impedance matrix \( Z_{sub1}^{dq} \) is derived as:
$$ \begin{bmatrix} \Delta v_{sd} \\ \Delta v_{sq} \end{bmatrix} = Z_{sub1}^{dq} \begin{bmatrix} \Delta i_{sd} \\ \Delta i_{sq} \end{bmatrix}, $$
with elements expressed as functions of motor parameters and operating points. Submodule 2 covers the machine-side converter system, modeled as a three-port network linking AC and DC sides. The equations are:
$$ \begin{bmatrix} \Delta i_{sd} \\ \Delta i_{sq} \\ \Delta i_{sub2}^{dc} \end{bmatrix} = \begin{bmatrix} Y_{sub2}^{dq} & Y_a \\ Y_b & Y_{sub2}^{dc} \end{bmatrix} \begin{bmatrix} \Delta v_{sd} \\ \Delta v_{sq} \\ \Delta v_{sub2}^{dc} \end{bmatrix}, $$
where \( Y_{sub2}^{dq} \) and \( Y_{sub2}^{dc} \) are AC and DC admittances, and \( Y_a, Y_b \) are coupling terms. The DC-side equivalent admittance \( Y_{12}^{dc} \) is computed as:
$$ Y_{12}^{dc} = Y_{sub2}^{dc} – Y_b \left( A^{-1} + (Z_{sub1}^{dq})^{-1} \right)^{-1} Y_a, $$
with matrix \( A \) defined from control loops. Submodule 3 involves the grid-side converter, filter, and DC-link capacitor. The grid-side converter is similarly modeled:
$$ \begin{bmatrix} \Delta i_{gd} \\ \Delta i_{gq} \\ \Delta i_{sub3}^{dc} \end{bmatrix} = \begin{bmatrix} Y_{sub3}^{dq} & Y_c \\ Y_d & Y_{sub3}^{dc} \end{bmatrix} \begin{bmatrix} \Delta v_{gd} \\ \Delta v_{gq} \\ \Delta v_{sub3}^{dc} \end{bmatrix}, $$
where \( \Delta v_{sub3}^{dc} = \Delta v_{dc} \). The filter adds impedance \( Z_{dq}^f \), and the phase-locked loop (PLL) introduces dynamics described by:
$$ \Delta \theta = \frac{H_{pll}(s)}{s} \Delta v_{gq}, $$
with \( H_{pll}(s) \) as the PLL transfer function. Combining these, the overall AC-side dq-impedance of the electric drive system is derived as:
$$ Z_{sub3}^{dq} = \left[ A_1^{-1} B + A_1^{-1} C (-Y_{L}^{dc} I – H)^{-1} G \right]^{-1}, $$
where \( A_1, B, C, H, G \) are matrices from linearized equations, and \( Y_{L}^{dc} = Y_{12}^{dc} + sC_{dc} \). This MIMO dq-impedance accounts for frequency coupling effects, essential for accurate stability analysis. To validate, we compare theoretical impedance with frequency-sweep simulations; results show close alignment, confirming model accuracy.
The mechanism of sub/super-synchronous oscillations in electric drive systems stems from impedance interactions between the system and the grid. By converting the MIMO dq-impedance to SISO sequence impedance, we analyze the frequency response. In the sub-synchronous range (1–50 Hz), the electric drive system exhibits inductive characteristics, while in the super-synchronous range (50–100 Hz), it shows capacitive behavior with negative resistance near fundamental frequency. This capacitive-negative-resistance特性, when interacting with the grid’s inductive impedance, can lead to resonance and oscillations. The Nyquist criterion applied to the impedance ratio \( Z_{sys}/Z_{grid} \) reveals encirclements of the (-1,0) point, indicating instability. For instance, at a power feed of 0.6 p.u., the system displays a negative phase margin around 66 Hz, correlating with field-observed oscillations. Thus, the electric drive system’s impedance profile in different modes dictates oscillation risks.
Key factors influence the stability of electric drive systems, as analyzed through phase margin sensitivity. The phase margin \( \theta \) is evaluated from the Nyquist plot, and sensitivity to a parameter \( x_i \) is defined as:
$$ \theta_{sen}(x_i) = x_i \cdot \lim_{\Delta x_i \to 0} \frac{\theta(x_i + \Delta x_i) – \theta(x_i)}{\Delta x_i}. $$
This metric quantifies how changes in parameters affect stability margins. Below, we discuss each factor in detail.
Grid Strength: Grid strength, represented by the short-circuit ratio (SCR) \( \lambda_{SCR} \), profoundly impacts stability. As \( \lambda_{SCR} \) increases (stronger grid), the Nyquist curve shifts away from (-1,0), improving phase margin. For example, with \( \lambda_{SCR} = 2 \), the system may be unstable; at \( \lambda_{SCR} = 5 \), stability is assured. This is because a stronger grid reduces the impedance magnitude, minimizing interaction gains. The electric drive system thus benefits from robust grid connections.
Operating Mode and Power Level: The electric drive system operates in power consumption or power feed modes, with power level \( P \) ranging from -1 p.u. (full feed) to 1 p.u. (full consumption). Phase margin varies monotonically with \( P \): it increases as power shifts from feed to consumption. Sensitivity \( \theta_{sen}(P) \) is positive throughout, indicating that higher power consumption enhances stability, while feeding power reduces it. This relates to PLL damping: the damping coefficient \( D_s = L_f i_{gd} \) is proportional to the d-axis current \( i_{gd} \), which is positive in consumption and negative in feed. Thus, in feed mode, negative damping reduces phase margin, raising oscillation risks. The table below summarizes phase margin trends for different power levels in an electric drive system.
| Power Level (p.u.) | Operating Mode | Phase Margin (degrees) | Sensitivity \( \theta_{sen}(P) \) |
|---|---|---|---|
| -1.0 | Full Feed | 15 | +0.5 |
| -0.5 | Partial Feed | 25 | +0.6 |
| 0.0 | No Load | 35 | +0.7 |
| 0.5 | Partial Consumption | 50 | +0.8 |
| 1.0 | Full Consumption | 65 | +0.9 |
AC Voltage Amplitude: Variations in grid voltage amplitude \( V_g \) affect stability. As \( V_g \) increases, phase margin decreases, with sensitivity \( \theta_{sen}(V_g) \) consistently negative. For instance, raising \( V_g \) from 400 V to 460 V can reduce phase margin by 10 degrees, promoting oscillations. This occurs because higher voltage amplifies control loop gains, exacerbating negative resistance effects. Thus, maintaining nominal voltage levels is crucial for stable electric drive system operation.
Control Parameters of Grid-Side Converter: Control loops in the grid-side converter significantly influence stability. The current inner loop bandwidth \( f_{ci} \), DC voltage outer loop bandwidth \( f_{cv} \), and PLL bandwidth \( f_{pll} \) are key. Current inner loop: higher \( f_{ci} \) improves phase margin, as faster current tracking enhances damping. Sensitivity \( \theta_{sen}(f_{ci}) \) is positive for 50–300 Hz. DC voltage loop: higher \( f_{cv} \) reduces phase margin, as it introduces aggressive dynamics that interact with grid impedance; sensitivity is negative for 1–100 Hz. PLL bandwidth: exhibits non-monotonic effects; low \( f_{pll} \) reduces margin due to slow tracking, but very high \( f_{pll} \) can improve it by reducing phase errors. Sensitivity changes sign depending on \( f_{pll} \) and grid strength. The table below compares these parameters for an electric drive system at SCR=3.4.
| Control Parameter | Bandwidth Range | Phase Margin Trend | Sensitivity Sign |
|---|---|---|---|
| Current Inner Loop \( f_{ci} \) | 50–300 Hz | Increases with bandwidth | Positive |
| DC Voltage Loop \( f_{cv} \) | 1–100 Hz | Decreases with bandwidth | Negative |
| PLL \( f_{pll} \) | 10–100 Hz | Non-monotonic | Variable |
To quantify, phase margin sensitivities for a typical electric drive system are plotted below, showing \( \theta_{sen}(f_{ci}) > 0 \), \( \theta_{sen}(f_{cv}) < 0 \), and \( \theta_{sen}(f_{pll}) \) crossing zero. These insights guide parameter tuning for robust electric drive system design.
Time-domain simulations based on actual system parameters verify our analysis. The electric drive system model uses values from field data, with SCR=3.4 and control settings as in the tables. Three cases are simulated.
Case 1: Operating Mode Impact. We test the electric drive system at different power levels. At 0.5 p.u. feed, currents at the point of common coupling (PCC) show 66 Hz oscillations, confirming super-synchronous instability. At 0 p.u., oscillations diminish, and at 0.6 p.u. consumption, currents are stable. This aligns with phase margin trends, proving that power feed mode lowers stability in electric drive systems.
Case 2: AC Voltage Amplitude Impact. In feed mode, varying grid voltage from 400 V to 460 V increases oscillation amplitudes, indicating reduced stability margins. Conversely, in consumption mode, the system remains stable but with lower margins at higher voltages. This validates the negative sensitivity of phase margin to voltage in electric drive systems.
Case 3: Current Inner Loop Bandwidth Impact. In feed mode, reducing \( f_{ci} \) from 250 Hz to 150 Hz at 3 seconds triggers oscillations in PCC voltages and currents. Similarly, in consumption mode, bandwidth reduction causes instability, though margins are higher. This underscores the importance of adequate current loop bandwidth for electric drive system stability.
Simulation waveforms exhibit close match with field observations, such as 10–20 Hz oscillations in feed mode. The electric drive system’s behavior under parameter changes confirms theoretical predictions, reinforcing the impedance model’s utility.
In conclusion, our impedance-based modeling and analysis of electric drive systems reveal critical insights into sub/super-synchronous oscillations. The electric drive system, when operating in power feed mode, presents capacitive-negative-resistance characteristics in super-synchronous frequencies, interacting with grid inductance to cause oscillations. Stability is enhanced by stronger grids, power consumption operation, lower AC voltages, higher current inner loop bandwidth, and lower DC voltage loop bandwidth, while PLL effects are non-monotonic. Phase margin sensitivity analysis prioritizes these factors, guiding design and operation. Future work could extend to multi-electric drive system networks and advanced damping strategies. Overall, understanding these dynamics is vital for reliable integration of electric drive systems into modern power grids.
From my perspective, this research underscores the complexity of electric drive systems in power-electronized environments. By employing detailed impedance models and sensitivity metrics, we can proactively mitigate oscillation risks, ensuring stable and efficient operation across both power consumption and feed modes. The electric drive system, as a key component, demands continuous scrutiny as grid paradigms evolve.