In the context of increasing integration of renewable energy systems, the need for advanced energy storage technologies has become paramount to address challenges such as grid instability and power fluctuations. Among various storage methods, compressed air energy storage (CAES) stands out due to its environmental friendliness, high efficiency, long cycle life, and large-scale capacity, making it a promising solution for smart grid development. In a CAES expansion power generation system, the grid tied inverter plays a critical role as the interface between the storage system and the power grid, converting DC power to AC power for grid injection. Enhancing the control performance of this grid tied inverter is essential for system stability, typically achieved through a voltage-current double-loop control strategy. However, traditional PI control methods often lead to issues like excessive overshoot and sluggish response, which can compromise control requirements. To overcome these limitations, this paper proposes a novel control strategy based on second-order Linear Active Disturbance Rejection Control (LADRC) for the voltage outer loop of the grid tied inverter. This approach aims to improve dynamic performance, reduce fluctuations in DC bus voltage and output power, and enhance disturbance rejection capabilities. We will begin by establishing the mathematical model of the grid tied inverter, followed by an in-depth analysis of LADRC principles, and conclude with simulation comparisons against conventional PI control.
The grid tied inverter in a CAES expansion power generation system is responsible for managing power flow and maintaining grid synchronization. Its performance directly impacts the overall system efficiency and reliability. A typical system configuration involves an expander directly coupled to a permanent magnet synchronous generator, with a machine-side rectifier and a grid-side inverter. The grid tied inverter control ensures that the output power quickly and accurately tracks demand changes while stabilizing the DC bus voltage. The core challenge lies in mitigating the effects of grid voltage disturbances, which can cause fluctuations in the DC link and degrade power quality. Traditional control methods, such as PI-based double-loop control, are widely used but suffer from inherent drawbacks like parameter sensitivity and poor disturbance rejection. Therefore, exploring advanced control techniques like LADRC, which can estimate and compensate for total disturbances in real-time, offers significant potential for improvement. This paper delves into the application of LADRC to the grid tied inverter, providing a comprehensive framework from modeling to implementation and validation.
The mathematical model of the grid tied inverter is derived in both three-phase and d-q coordinate systems to facilitate control design. In the three-phase system, the dynamics can be described by the following equations:
$$L\frac{di_{ga}}{dt} = e_{ga} – R i_{ga} – u_{ga}$$
$$L\frac{di_{gb}}{dt} = e_{gb} – R i_{gb} – u_{gb}$$
$$L\frac{di_{gc}}{dt} = e_{gc} – R i_{gc} – u_{gc}$$
$$C\frac{dU_{dc}}{dt} = \sum_{k=a,b,c} S_k i_{gk} – i_s$$
where \(i_{ga}\), \(i_{gb}\), \(i_{gc}\) are the grid currents, \(e_{ga}\), \(e_{gb}\), \(e_{gc}\) are the grid voltages, \(u_{ga}\), \(u_{gb}\), \(u_{gc}\) are the inverter output voltages, \(R\) and \(L\) are the equivalent resistance and inductance, \(C\) is the DC bus capacitance, \(U_{dc}\) is the DC bus voltage, \(i_s\) is the load current, and \(S_k\) represents the switching function for phase \(k\) (\(k = a, b, c\)). To simplify control, the model is transformed into the d-q synchronous reference frame using Park transformation, resulting in:
$$L\frac{di_{gd}}{dt} = e_{gd} – u_{gd} – R i_{gd} + \omega L i_{gq}$$
$$L\frac{di_{gq}}{dt} = e_{gq} – u_{gq} – R i_{gq} – \omega L i_{gd}$$
$$C\frac{dU_{dc}}{dt} = \frac{3}{2} \sum_{x=d,q} S_x i_{gx} – i_s$$
Here, \(i_{gd}\) and \(i_{gq}\) are the d- and q-axis grid currents, \(e_{gd}\) and \(e_{gq}\) are the grid voltages, \(u_{gd}\) and \(u_{gq}\) are the inverter voltages, \(\omega\) is the grid angular frequency, and \(S_x\) denotes the switching functions in the d-q frame. This model forms the basis for designing control strategies, as it decouples the active and reactive power components, enabling independent control. The grid tied inverter must regulate the DC bus voltage \(U_{dc}\) to a reference value while injecting current with low harmonic distortion and unity power factor. The voltage-current double-loop control is commonly employed, with the voltage outer loop generating reference currents for the current inner loop. However, the performance of this grid tied inverter is highly sensitive to grid disturbances, necessitating robust control approaches.
To achieve precise grid synchronization, a Phase-Locked Loop (PLL) is utilized. The Single Synchronous Reference Frame PLL (SSRF-PLL) is a popular choice due to its simplicity and effectiveness. It aligns the d-q frame with the grid voltage vector by minimizing the phase error. The control structure involves a PI regulator to adjust the estimated frequency and phase. For the grid tied inverter, accurate PLL operation is crucial to maintain stable power injection and avoid grid instability. The PLL output provides the angle \(\theta\) used in coordinate transformations, ensuring that the inverter currents are properly synchronized with the grid. In this study, we assume an ideal PLL for simplicity, focusing on the voltage control loop of the grid tied inverter.
The proposed control strategy employs second-order LADRC for the voltage outer loop, replacing the traditional PI controller. LADRC is a model-independent control technique that actively estimates and compensates for total disturbances, including external perturbations and internal uncertainties. For a second-order system, the general form is:
$$\ddot{y} = f(y, \dot{y}, w, t) + b u$$
where \(y\) is the output, \(u\) is the control input, \(w\) is the external disturbance, \(b\) is the control gain, and \(f\) represents the total disturbance encompassing unknown dynamics and disturbances. The key idea is to treat \(f\) as an extended state and estimate it using a Linear Extended State Observer (LESO). For the grid tied inverter, the DC bus voltage \(U_{dc}\) is the controlled variable. From the mathematical model, we can derive the second-order dynamics of \(U_{dc}\) by differentiating the capacitor equation and substituting current dynamics. This yields:
$$\frac{d^2 U_{dc}}{dt^2} = \frac{3}{2LC} \sum_{x=d,q} (S_x e_{gx} – S_x i_{gx} R) + \frac{3\omega}{2C} (S_d i_{gq} – S_q i_{gd}) – \frac{3}{2LC} \sum_{x=d,q} S_x u_{gx}$$
This equation can be rewritten in state-space form for LADRC design. Define the states as \(x_1 = U_{dc}\), \(x_2 = \dot{U}_{dc}\), and \(x_3 = f\), where \(f\) aggregates all disturbances and nonlinearities. The state equations are:
$$\dot{x}_1 = x_2$$
$$\dot{x}_2 = x_3 + b_0 u$$
$$\dot{x}_3 = h$$
$$y = x_1$$
Here, \(b_0\) is the known part of the control gain, and \(h\) is the derivative of \(f\). A third-order LESO is constructed to estimate these states:
$$\dot{z}_1 = z_2 – \beta_1 (z_1 – y)$$
$$\dot{z}_2 = z_3 – \beta_2 (z_1 – y) + b_0 u$$
$$\dot{z}_3 = -\beta_3 (z_1 – y)$$
where \(z_1\), \(z_2\), and \(z_3\) are the estimates of \(x_1\), \(x_2\), and \(x_3\), respectively, and \(\beta_1\), \(\beta_2\), \(\beta_3\) are observer gains tuned based on the observer bandwidth \(\omega_o\). Typically, \(\beta_1 = 3\omega_o\), \(\beta_2 = 3\omega_o^2\), and \(\beta_3 = \omega_o^3\). The control law is designed as:
$$u = \frac{-z_3 + u_0}{b_0}$$
where \(u_0\) is the output of a linear state error feedback controller, given by:
$$u_0 = k_p (v – z_1) – k_d z_2$$
with \(v\) as the reference input, and \(k_p\) and \(k_d\) as controller gains tuned based on the controller bandwidth \(\omega_c\): \(k_p = \omega_c^2\) and \(k_d = 2\omega_c\). This structure allows the grid tied inverter to compensate for disturbances in real-time, improving robustness. The parameter tuning is straightforward, relying only on \(\omega_o\) and \(\omega_c\), which simplifies implementation compared to PI control that requires meticulous tuning of multiple parameters.
To illustrate the system parameters and controller settings, we provide the following tables. Table 1 summarizes the key parameters of the CAES expansion power generation system and the grid tied inverter used in this study. These values are based on typical configurations for medium-scale energy storage applications.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Rated Power | P | 110 | kW |
| Grid Line Voltage | E | 380 | V |
| Grid Frequency | f | 50 | Hz |
| DC Bus Voltage | Udc | 1000 | V |
| DC Bus Capacitance | C | 7000 | μF |
| Grid-side Resistance | R | 0.00001 | Ω |
| Grid-side Inductance | L | 6 | mH |
Table 2 outlines the controller parameters for both the traditional PI control and the proposed LADRC approach. The PI gains are selected based on standard tuning methods, while the LADRC parameters are derived from bandwidth considerations.
| Controller Type | Parameter | Symbol | Value |
|---|---|---|---|
| PI Control | Voltage Outer Loop Proportional Gain | Kp_v | 0.5 |
| Voltage Outer Loop Integral Gain | Ki_v | 10 | |
| LADRC Control | Controller Bandwidth | ωc | 20000 rad/s |
| Observer Bandwidth | ωo | 40000 rad/s | |
| Control Gain | b0 | 3.8 × 109 | |
| Current Inner Loop PI Gains | Kip, Kii | 0.1, 100 |
The simulation analysis is conducted to compare the performance of the LADRC-based grid tied inverter with the conventional PI-based control. Two scenarios are considered: constant grid demand power and varying grid demand power. The simulations are performed in a MATLAB/Simulink environment, modeling the entire CAES system with the grid tied inverter. The performance metrics include DC bus voltage stability, output power tracking, grid current total harmonic distortion (THD), and response time to disturbances.
In the first scenario, the grid demand power is kept constant at 110 kW. Figure 1 shows the DC bus voltage responses under PI and LADRC control. The PI controller results in an overshoot of approximately 5.8%, with a settling time of about 0.12 seconds to reach the steady-state value of 1000 V. In contrast, the LADRC controller reduces the overshoot to only 1% and achieves a faster settling time of 0.05 seconds. This demonstrates the superior dynamic performance of LADRC in stabilizing the DC link of the grid tied inverter. The output power tracking, as shown in Figure 2, also exhibits similar improvements, with LADRC providing smaller overshoot and quicker convergence to the demand power. Moreover, the grid current quality is enhanced under LADRC control. The THD of the grid current is analyzed for both controllers: PI control yields a THD of 3.12%, while LADRC reduces it to 1.05%, indicating better compliance with grid standards and improved power quality for the grid tied inverter.
In the second scenario, the grid demand power changes at 0.4 seconds from 110 kW to 85 kW and at 0.7 seconds from 85 kW to 98 kW, simulating realistic load variations. The DC bus voltage responses are depicted in Figure 3. Under PI control, the voltage experiences significant fluctuations after each power change, with a recovery time of about 0.1 seconds and an overshoot of 5.8%. With LADRC, the voltage deviations are smaller, with a recovery time of only 0.08 seconds and an overshoot of 1%. This highlights the robustness of LADRC in handling disturbances caused by demand shifts. The output power tracking, shown in Figure 4, further confirms that LADRC enables the grid tied inverter to more accurately and rapidly follow demand changes, with minimal overshoot compared to PI control. Additionally, the grid current waveforms under both controllers are compared in Figure 5. LADRC ensures smoother current transitions and faster stabilization post-disturbance, with a response time of 0.04 seconds versus 0.07 seconds for PI, underscoring its enhanced disturbance rejection capability for the grid tied inverter.
To quantify the simulation results, Table 3 presents a comparative summary of key performance indicators for the grid tied inverter under both control strategies. These metrics clearly illustrate the advantages of LADRC in terms of speed, stability, and power quality.
| Performance Metric | PI Control | LADRC Control | Improvement |
|---|---|---|---|
| DC Bus Voltage Overshoot | 5.8% | 1% | 82.8% reduction |
| Settling Time for Voltage (constant demand) | 0.12 s | 0.05 s | 58.3% faster |
| Output Power Tracking Overshoot | High | Low | Significantly reduced |
| Grid Current THD | 3.12% | 1.05% | 66.3% reduction |
| Disturbance Recovery Time (varying demand) | 0.1 s | 0.08 s | 20% faster |
| Robustness to Grid Variations | Moderate | High | Enhanced |
The effectiveness of LADRC stems from its ability to estimate and cancel out total disturbances in real-time. For the grid tied inverter, these disturbances include grid voltage sags, swells, harmonics, and load changes. The LESO continuously monitors the system states and provides an estimate of the disturbance, which is then compensated in the control law. This model-independent approach eliminates the need for precise system modeling, making it suitable for practical applications where parameters may vary. Furthermore, the tuning process for LADRC is simpler than for PI control, as it primarily involves adjusting two bandwidth parameters: \(\omega_o\) and \(\omega_c\). A higher \(\omega_o\) improves disturbance estimation but may increase noise sensitivity, while \(\omega_c\) dictates the response speed. In this study, we selected \(\omega_o = 40000\) rad/s and \(\omega_c = 20000\) rad/s based on simulation trials to balance performance and robustness. The control gain \(b_0\) is chosen to match the nominal system dynamics, and it can be adapted online if necessary.
In comparison to other advanced control methods for grid tied inverters, such as model predictive control or sliding mode control, LADRC offers a compelling trade-off between complexity and performance. Model predictive control requires high computational resources and accurate models, while sliding mode control may introduce chattering. LADRC, being linear and relatively simple to implement, is advantageous for real-time applications in CAES systems. Additionally, the integration of LADRC with the current inner loop PI control maintains the decoupling benefits of voltage-oriented vector control, ensuring unity power factor operation. The current inner loop uses PI regulators to track the d- and q-axis reference currents generated by the voltage outer loop. The reference for the q-axis current is set to zero for unity power factor, while the d-axis reference is derived from the LADRC output. This combined strategy ensures that the grid tied inverter not only stabilizes the DC bus but also injects high-quality power into the grid.
From an implementation perspective, the LADRC-based control can be deployed on digital signal processors commonly used in power electronics. The algorithm involves basic arithmetic operations and state updates, which are computationally feasible. For the grid tied inverter, the sampling frequency should be sufficiently high to capture system dynamics; typically, frequencies above 10 kHz are used. The design considerations also include anti-windup mechanisms for the LESO to handle actuator saturation, though this is beyond the scope of this paper. Future work could explore adaptive LADRC techniques where \(\omega_o\) and \(\omega_c\) are tuned online based on operating conditions, further enhancing the adaptability of the grid tied inverter in fluctuating grid environments.
In conclusion, this paper presents a robust control strategy for the grid tied inverter in a compressed air energy storage system using second-order Linear Active Disturbance Rejection Control. The mathematical model of the inverter is established, and the LADRC design is detailed for the voltage outer loop. Simulation results demonstrate that LADRC significantly outperforms traditional PI control in terms of DC bus voltage stability, output power tracking, grid current quality, and disturbance rejection. Specifically, LADRC reduces voltage overshoot from 5.8% to 1%, decreases settling time by over 50%, lowers grid current THD from 3.12% to 1.05%, and accelerates recovery from demand changes. These improvements are crucial for the reliable operation of CAES systems in modern power grids, where renewable integration demands high flexibility and resilience. The proposed control strategy is simple to tune and implement, making it a practical solution for enhancing the performance of grid tied inverters in energy storage applications. As the adoption of CAES grows, advanced control methods like LADRC will play a vital role in ensuring grid stability and power quality, contributing to a sustainable energy future.