The global energy landscape is undergoing a significant transformation, driven by the urgent need to address environmental concerns and ensure sustainable energy supply. In this context, distributed generation systems, particularly microgrids integrating renewable sources like photovoltaic (PV) arrays, have emerged as a pivotal solution. The seamless and high-quality integration of these sources into the utility grid or their stable operation in islanded mode heavily relies on the performance of the power electronic interface—the solar inverter. The core challenge lies in designing robust control strategies that enable these solar inverters to precisely regulate active and reactive power flow, maintain voltage stability, and ensure resilience during grid disturbances, thereby safeguarding and enhancing overall microgrid power quality. This article delves into an advanced cascade control strategy centered on Internal Model Control (IMC) principles, designed to empower photovoltaic inverters with superior dynamic performance and autonomous regulation capabilities.
Conventional control techniques for solar inverters, such as PI-based vector control, often face limitations in dealing with system parameter variations, inherent coupling between control axes, and the need for a balance between fast response and robustness. Internal Model Control offers a structured and transparent framework for controller design, particularly effective for processes with well-defined dynamics. The proposed methodology applies this framework to both the inner current loop and the outer voltage loop of a two-level, three-phase, three-wire voltage source inverter (VSI)—the standard workhorse for PV system interconnection. We will construct a comprehensive mathematical model of the inverter-system interface, derive reference currents using instantaneous power theory, and detail the design of IMC-based controllers. The ultimate goal is to demonstrate a control strategy that not only achieves decoupled active and reactive power control but also enables the photovoltaic inverter to autonomously adjust its operation during grid voltage sags, contributing significantly to microgrid stability and power quality management.
The foundation of any high-performance control system is an accurate plant model. We consider a grid-connected configuration where the solar inverter is interfaced with the AC grid through an L-type filter. The DC side of the inverter is modeled as a current source (representing the PV array output) in parallel with a DC-link capacitor. In the three-phase stationary (abc) frame, the voltage equation across the filter is given by:
$$
\begin{bmatrix} v_{za} \\ v_{zb} \\ v_{zc} \end{bmatrix} – \begin{bmatrix} v_{la} \\ v_{lb} \\ v_{lc} \end{bmatrix} = R_f \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} + L_f \frac{d}{dt} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix}
$$
where \( v_{zabc} \) are the grid voltages, \( v_{labc} \) are the inverter output voltages, \( i_{abc} \) are the line currents, and \( R_f \) and \( L_f \) are the filter resistance and inductance, respectively. For control design, this model is transformed into the synchronous rotating (dq) frame using the Park transformation, aligning the d-axis with the grid voltage vector position obtained from a Phase-Locked Loop (PLL). Assuming a balanced three-wire system and a properly tuned PLL (\(v_{zd} = 0\)), the model simplifies to the decoupled form often used for controller design:
$$
\begin{bmatrix} v_{zq} \\ 0 \end{bmatrix} – \begin{bmatrix} v_{lq} \\ v_{ld} \end{bmatrix} = \begin{bmatrix} R_f & -\omega_e L_f \\ \omega_e L_f & R_f \end{bmatrix} \begin{bmatrix} i_q \\ i_d \end{bmatrix} + \begin{bmatrix} L_f & 0 \\ 0 & L_f \end{bmatrix} \frac{d}{dt} \begin{bmatrix} i_q \\ i_d \end{bmatrix}
$$
Here, \( \omega_e \) is the grid electrical angular frequency. The instantaneous active power \(P\) and reactive power \(Q\) injected into the grid are given by:
$$
P = \frac{3}{2} (v_{zq} i_q + v_{zd} i_d) = \frac{3}{2} v_{zq} i_q, \quad Q = \frac{3}{2} (v_{zq} i_d – v_{zd} i_q) = \frac{3}{2} v_{zq} i_d
$$
This leads to the fundamental reference current calculations for the controllers of the photovoltaic inverter:
$$
i_q^* = \frac{2}{3} \frac{P^*}{v_{zq}}, \quad i_d^* = \frac{2}{3} \frac{Q^*}{v_{zq}}
$$
These references, \(i_q^*\) and \(i_d^*\), serve as the primary command signals for the inner current control loops, tasked with governing the active and reactive power output of the solar inverter, respectively.
| Symbol | Description | Typical Value/Unit |
|---|---|---|
| \(v_{zq}, v_{zd}\) | Grid voltage in dq-frame | V |
| \(i_q, i_d\) | Grid current in dq-frame | A |
| \(i_q^*, i_d^*\) | Current references | A |
| \(P^*, Q^*\) | Active/Reactive power references | W, Var |
| \(R_f, L_f\) | Filter resistance and inductance | Ω, H |
| \(\omega_e\) | Grid electrical frequency | rad/s |
| \(V_{dc}\) | DC-link voltage | V |
| \(C_{dc}\) | DC-link capacitance | F |
The proposed cascade control structure features an outer voltage loop and an inner current loop. The inner loop’s objective is to force the actual currents \(i_q\) and \(i_d\) to track their references \(i_q^*\) and \(i_d^*\) with high bandwidth and zero steady-state error. From the voltage equation, we can derive the transfer function from the inverter voltage command (after decoupling) to the output current. The decoupling terms are introduced as:
$$
\begin{bmatrix} v_{lq} \\ v_{ld} \end{bmatrix} = \begin{bmatrix} -\hat{v}_{lq} + v_{zq} – \omega_e L_f i_d \\ -\hat{v}_{ld} + \omega_e L_f i_q \end{bmatrix}
$$
where \(\hat{v}_{lq}\) and \(\hat{v}_{ld}\) are the outputs of the current controllers. Substituting this back effectively cancels the cross-coupling terms, yielding two independent, first-order systems:
$$
\frac{I_q(s)}{\hat{V}_{lq}(s)} = \frac{I_d(s)}{\hat{V}_{ld}(s)} = G_p(s) = \frac{1}{L_f s + R_f}
$$
This plant model \(G_p(s)\) is a first-order lag system, ideal for IMC design. The standard IMC structure involves an internal model \(M(s)\) of the plant and a controller \(Q(s)\). For a first-order plant \(M(s) = K / (\tau s + 1)\), where \(K = 1/R_f\) and \(\tau = L_f/R_f\), and including a first-order filter \(F(s) = 1/(\lambda s + 1)\) for robustness, the IMC controller \(Q(s)\) is designed as the inverse of the minimum-phase part: \(Q(s) = M^{-1}(s)F(s) = (\tau s + 1) / [K(\lambda s + 1)]\). This IMC scheme is equivalent to a feedback controller \(C(s)\) given by:
$$
C(s) = \frac{Q(s)}{1 – M(s)Q(s)} = \frac{\tau s + 1}{K \lambda s}
$$
This is precisely the form of a PI controller: \(C(s) = K_{pc} + K_{ic}/s\), where the proportional and integral gains for the current controller of the photovoltaic inverter are:
$$
K_{pc} = \frac{L_f}{\lambda R_f}, \quad K_{ic} = \frac{R_f}{\lambda} = \frac{1}{\lambda}
$$
The tuning parameter \(\lambda\) directly dictates the closed-loop speed of response. A smaller \(\lambda\) yields a faster response but may increase sensitivity to noise and model mismatch. This systematic design ensures optimal tracking performance for the current loops of the solar inverter.
The outer loop’s primary objective is to regulate the DC-link voltage \(V_{dc}\). This is crucial because the DC-link capacitor acts as an energy buffer between the variable power from the PV source and the AC grid. The control variable is the active power reference \(P^*\) fed to the inner loop. The plant model for this loop relates the power imbalance at the DC-link to the voltage. The dynamics can be derived from the power balance equation: \(P_{pv} – P_{ac} = d(\frac{1}{2}C_{dc}V_{dc}^2)/dt\), where \(P_{pv}\) is the PV power and \(P_{ac}\) is the power injected into the grid. Linearizing around an operating point \(V_{dc0}\) and considering \(P_{ac}\) as the control input, we obtain a first-order model. However, considering the inner closed-loop current dynamics as part of the voltage plant often results in an effective second-order model. We model this outer plant as a second-order system with apparent time delay \(L\) representing computational and PWM delays:
$$
M_v(s) = \frac{K_v e^{-L s}}{\tau_v^2 s^2 + 2\xi_v \tau_v s + 1}
$$
Applying the IMC design procedure for such a model and using a first-order filter \(F_v(s)=1/(\lambda_v s + 1)\) leads to a feedback controller \(C_v(s)\) with a PID structure. For implementation simplicity and to enhance disturbance rejection, a feedforward path from the PV power measurement (\(P_{DC}\)) is added. The final control law for the active power reference becomes \(P^* = P_{DC} + P_c^*\), where \(P_c^*\) is the output of the voltage PI controller. The PI gains derived from the IMC design for the voltage controller in the solar inverter are:
$$
K_{pv} = 2\xi_v \tau_v \omega_{nv} C_{dc} V_{dc0}, \quad K_{iv} = \tau_v^2 \omega_{nv}^2 C_{dc} V_{dc0}
$$
where \(\omega_{nv}\) is the desired natural frequency of the closed-loop voltage control system. This design ensures stable DC-link voltage regulation with specified dynamic characteristics.
| Controller | Plant Model \(M(s)\) | IMC Filter \(F(s)\) | Equivalent Feedback Controller \(C(s)\) | Tuning Parameters |
|---|---|---|---|---|
| Current (q/d-axis) | \(\frac{1}{L_f s + R_f}\) | \(\frac{1}{\lambda_c s + 1}\) | \(\frac{L_f s + R_f}{\lambda_c R_f s}\) | \(\lambda_c\): Current loop bandwidth |
| DC Voltage | \(\frac{K_v e^{-L s}}{\tau_v^2 s^2 + 2\xi_v \tau_v s + 1}\) | \(\frac{1}{\lambda_v s + 1}\) | \(\frac{\tau_v^2 s^2 + 2\xi_v \tau_v s + 1}{K_v \lambda_v s}\) | \(\lambda_v, \omega_{nv}, \xi_v\): Voltage loop dynamics |
The efficacy of the proposed IMC-based control strategy for photovoltaic inverters is validated through detailed simulation studies. Two critical operational scenarios are examined: dynamic power tracking and autonomous response to grid voltage sags.
In the first scenario, the PV generation (simulated as a DC current source) is stepped from a low level (3A) to a high level (100A) and back. The results demonstrate that the solar inverter control system rapidly adjusts the AC-side currents (\(i_q\)) to deliver the corresponding active power to the grid, while the reactive power (\(Q\)) remains at its commanded setpoint (e.g., zero for unity power factor). Throughout this transient, the grid voltage remains undisturbed, and the DC-link voltage is maintained tightly at its reference value (e.g., 800V), showcasing the decoupled power control and excellent voltage regulation capabilities of the designed controllers. The transition is smooth with minimal overshoot, attributable to the optimized IMC tuning.
The second, more demanding, scenario tests the microgrid’s autonomous functionality. A 50% grid voltage sag is simulated for a duration of 200ms. The proposed control strategy enables the photovoltaic inverter to exhibit an intelligent response. Unlike rigid control schemes, the outer voltage controller, in conjunction with the current references calculation, allows the DC-link voltage to temporarily sag in a controlled manner, reducing the stress on the system. Consequently, the active power injection is automatically curtailed during the fault, aligning with the reduced power transfer capacity of the depressed grid voltage. Simultaneously, the controller can be configured to inject or absorb reactive current (\(i_d^*\)) to support grid voltage recovery, if required by grid codes. Upon voltage recovery, the system seamlessly ramps the power back to its pre-fault level and restores the DC-link voltage to its nominal value. This autonomous adjust-and-return behavior is crucial for microgrid resilience and is a direct result of the robust cascade control structure employing IMC principles.
| Test Scenario | Key Observation | Controller Action | Implication for Power Quality |
|---|---|---|---|
| Step Change in PV Power | Fast, well-damped active power (\(P\)) tracking. Stable \(V_{dc}\). Unaffected grid voltage. | Inner IMC current loop provides fast torque response. Outer IMC voltage loop manages energy balance. | Prevents voltage fluctuations during renewable generation swings. Enables precise power dispatch. |
| Grid Voltage Sag (50%) | Controlled reduction in \(P\) and \(V_{dc}\). Possible reactive current (\(Q\)) injection. Smooth recovery post-fault. | Voltage controller adjusts \(P^*\) based on sag. Current loop executes new \(i_q^*, i_d^*\) references instantly. | Provides Low Voltage Ride-Through (LVRT) capability. Supports grid voltage. Enhances microgrid stability and autonomy. |
In conclusion, this article has presented a comprehensive and systematic approach to power control for grid-connected photovoltaic inverters based on Internal Model Control theory. The cascade control structure, featuring an IMC-designed inner current loop and an IMC-designed outer voltage loop, provides a powerful framework for managing solar inverters. The strategy delivers precise, decoupled control of active and reactive power, essential for modern grid support functions. More importantly, it embeds an inherent intelligence that allows the inverter to autonomously adapt its operating point during grid disturbances like voltage sags, a critical feature for enhancing microgrid reliability and power quality. The mathematical derivation, controller tuning formulas, and simulation validation collectively demonstrate that IMC offers a transparent and effective methodology for designing high-performance controllers for photovoltaic inverters, contributing significantly to the development of smarter and more resilient renewable energy integration systems. The principles outlined can be extended to other distributed energy resource interfaces, solidifying IMC’s role as a valuable tool in the future energy grid.