The seamless integration of distributed generation (DG) resources, such as photovoltaic arrays and wind turbines, into the main utility grid is paramount for modern power systems. The critical interface enabling this bidirectional energy flow is the on grid inverter. Among various filter topologies used to mitigate the high-frequency switching harmonics produced by the inverter, the LCL filter has become the predominant choice due to its superior harmonic attenuation capability with smaller component sizes compared to a simple L filter. However, the inherent third-order nature of the LCL filter introduces a resonant peak that can destabilize the current control loop if not properly managed. This paper delves into advanced control strategies for LCL-type on grid inverters, focusing on sensor reduction, resonant damping, and high-performance current tracking, culminating in the presentation of a novel observer-based control scheme.
Introduction to Grid-Connected Inverter Systems
The primary function of an on grid inverter is to convert the DC power from a DG source into high-quality AC power that is synchronized with the grid’s voltage and frequency. The performance of the on grid inverter is evaluated based on several key metrics: low total harmonic distortion (THD) of the output current, high power factor, fast dynamic response to reference changes, and robust stability against grid disturbances and parameter variations. The control system of an on grid inverter is typically multi-looped, involving an outer power or DC-link voltage control loop and an inner current control loop. The inner current loop’s performance is most critical, as it directly dictates the quality of the injected power and the system’s stability.
LCL Filter Modeling and Resonant Challenge
The LCL filter, composed of an inverter-side inductor \(L_i\), a filter capacitor \(C\), and a grid-side inductor \(L_g\), provides a -60 dB/decade attenuation roll-off after its resonant frequency. The transfer function from the inverter output voltage \(u_v\) to the grid-side current \(i_g\) in the Laplace domain is given by:
$$ G_p(s) = \frac{i_g(s)}{u_v(s)} = \frac{1}{L_i L_g C s^3 + (L_i + L_g)s} $$
This transfer function reveals the fundamental challenge: the denominator lacks an \(s^2\) term, indicating a pair of purely imaginary poles at the resonant frequency \(\omega_{res} = \sqrt{(L_i + L_g)/(L_i L_g C)}\). This results in an infinite gain at \(\omega_{res}\), causing instability in a closed-loop system. The comparison of filter characteristics is summarized below.
| Filter Type | Order | Attenuation | Cost/Size | Stability |
|---|---|---|---|---|
| L Filter | 1st | -20 dB/dec | Large inductor | Inherently stable |
| LC Filter | 2nd | -40 dB/dec | Moderate | Risk of resonance |
| LCL Filter | 3rd | -60 dB/dec | Smaller components | Requires damping |
Damping Techniques for LCL Resonance
To stabilize the LCL-type on grid inverter, damping must be introduced. Methods are broadly classified into passive and active damping.
1. Passive Damping (PD): This involves adding physical resistors in series or parallel with the LCL filter components, typically with the capacitor. While simple and robust, PD leads to significant power losses, reduced efficiency, and potential thermal management issues, contradicting the high-efficiency goals of modern on grid inverters.
2. Active Damping (AD): AD techniques mimic the effect of a physical resistor using control algorithms, eliminating power loss. The most common approach is state feedback. By feeding back a filter state variable (or a combination) through a proper gain, the closed-loop poles can be moved away from the imaginary axis. Common feedback variables include capacitor current, capacitor voltage, or grid-side current. The required sensors for various AD schemes are a key cost and reliability factor.
| Feedback Variable | Sensor Required | Advantages | Disadvantages |
|---|---|---|---|
| Capacitor Current (\(i_c\)) | Current Sensor | Simple constant gain, effective | Needs accurate current sensor |
| Capacitor Voltage (\(u_c\)) | Voltage Sensor | No extra current sensor | Feedback gain may be frequency-dependent |
| Grid-side Current (\(i_g\)) | Current Sensor | Part of control loop | Complex digital implementation |
| Multi-state Feedback | Multiple Sensors | Optimal pole placement | High cost, complex tuning |
A conventional control scheme for an LCL-type on grid inverter often employs a dual-loop strategy: an inner active damping loop (e.g., capacitor current feedback) and an outer grid current control loop. This typically necessitates at least three sensors: grid voltage \(u_g\) for synchronization, grid current \(i_g\) for control, and capacitor current \(i_c\) for damping. Reducing this sensor count without compromising performance is a significant research objective.
State Reconfiguration and Full-Order Observer Design
To minimize sensors, advanced observer-based control strategies are employed. The goal is to estimate all necessary state variables using only a minimal set of measurements. A promising approach is based on state reconfiguration and an Extended State Observer (ESO).
Consider the state-space model of the LCL filter in the stationary abc-frame:
$$
\begin{aligned}
\dot{i}_v &= \frac{1}{L_i}(u_v – u_c) \\
\dot{u}_c &= \frac{1}{C}(i_v – i_g) \\
\dot{i}_g &= \frac{1}{L_g}(u_c – u_g)
\end{aligned}
$$
where \(i_v\) is the inverter-side current, \(u_c\) is the capacitor voltage, \(i_g\) is the grid-side current, and \(u_g\) is the grid voltage. The standard ESO is designed for integral-chain systems. The states in (3) are coupled. The core idea of state reconfiguration is to derive a new set of states from the measured variable (e.g., \(i_v\)) that naturally forms an integral chain. By repeatedly differentiating \(i_v\), we obtain:
$$
\begin{aligned}
\bar{x}_1 &= i_v \\
\bar{x}_2 &= \dot{\bar{x}}_1 = \frac{1}{L_i}(u_v – u_c) \\
\bar{x}_3 &= \dot{\bar{x}}_2 = \frac{1}{L_i}\dot{u}_v – \frac{1}{L_i C}(i_v – i_g) \\
\dot{\bar{x}}_3 &= \frac{1}{L_i}\ddot{u}_v – \frac{1}{L_i C L_g}u_g – \frac{L_i+L_g}{L_i L_g C} \bar{x}_2 + \frac{1}{L_i L_g C} u_v
\end{aligned}
$$
We can define the total disturbance \(f = \dot{\bar{x}}_3 – b_0 u_v\), where \(b_0 = 1/(L_i L_g C)\) is a known constant. Let \(\bar{x}_4 = f\). The reconstituted system is now:
$$
\begin{cases}
\dot{\bar{x}}_1 = \bar{x}_2 \\
\dot{\bar{x}}_2 = \bar{x}_3 \\
\dot{\bar{x}}_3 = \bar{x}_4 + b_0 u_v \\
\dot{\bar{x}}_4 = h \quad (\text{assume } h = \dot{f} \text{ is bounded})
\end{cases}
$$
This is a canonical form for a fourth-order ESO. A linear ESO can be constructed as:
$$
\begin{aligned}
\dot{z}_1 &= z_2 + \beta_1 (\bar{x}_1 – z_1) \\
\dot{z}_2 &= z_3 + \beta_2 (\bar{x}_1 – z_1) \\
\dot{z}_3 &= z_4 + \beta_3 (\bar{x}_1 – z_1) + b_0 u_v \\
\dot{z}_4 &= \beta_4 (\bar{x}_1 – z_1)
\end{aligned}
$$
where \(z_1, z_2, z_3, z_4\) are the estimates of \(\bar{x}_1, \bar{x}_2, \bar{x}_3, \bar{x}_4\), and \(\beta_1\) to \(\beta_4\) are observer gains. By placing all observer poles at \(-\omega_o\) (observer bandwidth), the gains are: \(\beta_1 = 4\omega_o, \beta_2 = 6\omega_o^2, \beta_3 = 4\omega_o^3, \beta_4 = \omega_o^4\). Once these estimates are obtained, the original physical states and the grid voltage can be recovered through algebraic transformations derived from the reconfiguration process:
$$
\begin{aligned}
\hat{i}_v &= z_1 \\
\hat{u}_c &= u_v – L_i z_2 \\
\hat{i}_g &= i_v – C L_i z_3 + C \dot{u}_v \\
\hat{u}_g &= u_v – L_i z_2 – L_g (i_v – C L_i z_3 + C \dot{u}_v) – L_i L_g C z_4
\end{aligned}
$$
Thus, using only the measurement of \(i_v\) and the known inverter output \(u_v\), the ESO can estimate the capacitor voltage, grid current, and crucially, the grid voltage, potentially eliminating the need for corresponding sensors.
Enhancement with Quasi-Resonant Extended State Observer (QR-ESO)
A standard linear ESO, designed with poles on the negative real axis, is excellent for estimating DC or slowly varying disturbances. However, the states of an on grid inverter system are sinusoidal signals at the fundamental grid frequency (50/60 Hz). Tracking a sinusoid with a standard ESO requires a very high bandwidth (\(\omega_o\)), which makes the observer sensitive to high-frequency measurement noise and can degrade system robustness.
The Quasi-Resonant Extended State Observer (QR-ESO) solves this by incorporating a resonant term at the fundamental frequency into the observer correction loop. This is achieved by adding a quasi-resonant (QR) filter in parallel with the proportional correction path for each estimated state. The transfer function of a QR filter is:
$$ G_{QR}(s) = \frac{K_r \omega_c s}{s^2 + \omega_c s + \omega_g^2} $$
where \(\omega_g\) is the grid frequency and \(\omega_c\) is the cutoff bandwidth of the resonant peak. The modified observer correction for the first state, for example, becomes \( l_1(\bar{x}_1 – z_1) + G_{QR}(s)(\bar{x}_1 – z_1) \).
This structure provides near-infinite gain at \(\omega_g\), enabling perfect tracking of the sinusoidal component without requiring excessively high observer gains across all frequencies. The QR-ESO significantly improves the estimation accuracy of grid voltage and currents for the LCL-type on grid inverter, leading to better synchronization and control performance.
| Observer Type | Tracking DC/Low-Freq | Tracking Sinusoid (ω_g) | Noise Immunity | Design Complexity |
|---|---|---|---|---|
| Standard ESO | Excellent | Requires high ω_o | Poor with high ω_o | Low |
| QR-ESO | Good | Excellent (infinite gain at ω_g) | Good (selective high gain) | Moderate |
Current Control in the Synchronous Reference Frame
For high-performance current control, the system is typically transformed into the synchronous rotating (dq) frame. This transformation converts AC sinusoidal references into DC quantities, allowing for zero steady-state error with PI controllers. However, the dynamics of the LCL filter in the dq-frame are coupled and complex. The simplified relationship, considering the estimated grid voltage feedforward, is often used for controller design:
$$ L_g \frac{d i_{g,dq}}{dt} = u_{c,dq} – u_{g,dq} – j\omega L_g i_{g,dq} $$
Even this simplified model shows cross-coupling (\(j\omega L_g i_{g,dq}\) term) between the d and q axes. Traditional PI control requires explicit decoupling networks to achieve independent control. Furthermore, the actual plant is the third-order LCL filter, and its internal dynamics, along with parameter uncertainties, act as additional disturbances.
Active Disturbance Rejection Control (ADRC) for Current Loop
Active Disturbance Rejection Control (ADRC) presents a powerful alternative. Its philosophy is to treat all unknown dynamics, cross-couplings, and external disturbances as a “total disturbance,” estimate it in real-time via an ESO, and cancel it out in the control law. This provides inherent decoupling and robust performance.
For the current loop of the on grid inverter, we can define a first-order system for each axis (d and q):
$$ \dot{i}_{g,d} = f_d + b u_{v,d} $$
$$ \dot{i}_{g,q} = f_q + b u_{v,q} $$
Here, \(b\) is a rough estimate of the system gain (e.g., \(1/L_i\)), and \(f_d\), \(f_q\) represent the total disturbance for each axis, which includes:
- The complex internal LCL dynamics beyond the simple integrator model.
- The cross-coupling term from the other axis (\(-\omega L_g i_{g,q}\) for d-axis, \(\omega L_g i_{g,d}\) for q-axis).
- Grid voltage harmonics and variations.
- Parameter inaccuracies (e.g., in \(L_g, L_i, C\)).
A linear ESO (often a second-order one for this first-order plant) is designed to estimate \(i_g\) and \(f\) simultaneously:
$$
\begin{aligned}
\dot{z}_{1,d} &= z_{2,d} + \lambda_1 (i_{g,d} – z_{1,d}) + b_0 u_{v,d} \\
\dot{z}_{2,d} &= \lambda_2 (i_{g,d} – z_{1,d})
\end{aligned}
$$
where \(z_{1,d} \approx i_{g,d}\) and \(z_{2,d} \approx f_d\). The control law is then:
$$ u_{v,d} = \frac{u_{0,d} – z_{2,d}}{b_0}, \quad \text{where } u_{0,d} = k_p (i_{g,d}^* – z_{1,d}) $$
Substituting this control law into the plant equation yields \(\dot{i}_{g,d} \approx k_p (i_{g,d}^* – i_{g,d})\), which is a simple, decoupled first-order system. The same structure is applied to the q-axis. The ADRC for the LCL-type on grid inverter elegantly handles decoupling and disturbance rejection in a unified framework, simplifying the controller design compared to the PI+decoupling approach.
| Control Method | Decoupling Mechanism | Disturbance Rejection | Parameter Sensitivity | Tuning Complexity |
|---|---|---|---|---|
| PI with Decoupling | Explicit feedforward of coupling terms | Relies on integral action, slow | Moderate to High | High (multiple gains) |
| ADRC | Implicit, via total disturbance estimation | Active, real-time estimation & cancellation | Low | Moderate (ω_c, ω_o, b0) |
Integrated Control Strategy: A Case Study
Combining the aforementioned concepts leads to a sophisticated, sensor-minimized control strategy for an LCL-type on grid inverter. The proposed integrated system operates as follows:
- Measurement: Only the inverter-side current \(i_v\) is measured.
- State Reconfiguration & QR-ESO: The measured \(i_v\) is used to drive the QR-ESO based on the reconstituted integral-chain model. The QR-ESO provides accurate estimates \(\hat{i}_g\), \(\hat{u}_c\), and \(\hat{u}_g\).
- Grid Synchronization: The estimated grid voltage \(\hat{u}_g\) is fed into a Phase-Locked Loop (PLL) to obtain the grid angle \(\theta\) for coordinate transformations.
- Active Damping: An active damping signal is generated by feeding back the estimated capacitor current (\(\hat{i}_c = \hat{i}_v – \hat{i}_g\)) through a proportional gain \(H\). This signal is added to the current controller output to damp the LCL resonance.
- Current Control: The estimated grid current \(\hat{i}_g\) is transformed to the dq-frame. The ADRC current controllers process the error between the reference \(i_{g,dq}^*\) and \(\hat{i}_{g,dq}\) to generate the voltage reference \(u_{v,dq}^*\).
- Modulation: The final voltage reference, sum of the ADRC output and the damping signal, is transformed back to the abc-frame and used by the SVPWM module to generate gate signals for the inverter.
This strategy effectively reduces the sensor count to one current sensor while maintaining robust performance. The QR-ESO ensures accurate estimation even under grid distortions, and the ADRC handles coupling and model uncertainties effectively. The parameters for such a system require careful tuning, often guided by frequency-domain analysis and simulation.
| Component | Parameter | Symbol | Typical Value/Rule |
|---|---|---|---|
| LCL Filter | Inverter-side Inductor | \(L_i\) | 2-5 mH |
| Grid-side Inductor | \(L_g\) | 1-3 mH | |
| Filter Capacitor | \(C\) | 5-30 μF | |
| QR-ESO | Observer Bandwidth | \(\omega_o\) | \(2\pi \cdot 200\) to \(2\pi \cdot 500\) rad/s |
| Control Gain | \(b_0\) | \(1/(L_i L_g C)\) | |
| Resonant Bandwidth | \(\omega_c\) | \(2\pi \cdot 5\) to \(2\pi \cdot 15\) rad/s | |
| ADRC | Controller Bandwidth | \(\omega_c\) | \(2\pi \cdot 50\) to \(2\pi \cdot 150\) rad/s |
| Observer Bandwidth | \(\omega_o\) | \(3 \sim 10 \times \omega_c\) | |
| Plant Gain Estimate | \(b\) | \(1/L_i\) | |
| Active Damping | Feedback Gain | \(H\) | 0.5 to 5 |
Performance Analysis and Future Trends
The integrated QR-ESO and ADRC strategy for LCL-type on grid inverters offers compelling advantages. The significant reduction in sensors lowers system cost and improves reliability by reducing potential points of failure. The quasi-resonant observer provides superior estimation fidelity for grid synchronization and state feedback under distorted grid conditions compared to standard observers. Furthermore, the ADRC’s inherent robustness simplifies the control design process, as it does not require a precise model and automatically manages cross-coupling.
Simulation and experimental studies typically demonstrate that such a system can maintain low THD (e.g., below 3%) in the grid current, exhibit fast dynamic response to step changes in current reference or grid voltage, and remain stable under wide variations of grid impedance and filter parameters. The stability margins, while slightly reduced due to the phase lag introduced by the observer, remain satisfactory with proper tuning.
Future research directions for advanced control of on grid inverters include:
- Wider Application of Advanced Observers: Further development of adaptive observers or sliding mode observers to handle severe and non-stationary grid faults.
- Sensorless Topologies: Pushing the boundary to fully sensorless control, possibly by estimating the DC-link current or using very-low-cost voltage sensors only.
- Integration with Grid-Support Functions: Seamlessly incorporating fault ride-through, reactive power support, and harmonic compensation capabilities into the ADRC/observer framework.
- AI-Enhanced Tuning: Utilizing machine learning algorithms to auto-tune the multiple parameters (bandwidths, gains) of the coupled QR-ESO and ADRC systems for optimal performance across different operating conditions.
In conclusion, the evolution of control strategies for LCL-type on grid inverters is moving towards highly integrated, intelligent, and robust solutions that maximize performance while minimizing hardware complexity and cost. The combination of state reconstruction, quasi-resonant extended state observation, and active disturbance rejection control represents a significant step forward in this journey, enabling more reliable and efficient integration of renewable energy into the power grid.