The global imperative to transition towards sustainable energy systems has placed photovoltaic (PV) power generation at the forefront. Central to these systems are solar inverters, which perform the critical function of converting DC power from PV panels into grid-compatible AC power. The quest for higher efficiency, reliability, and cost-effectiveness has driven the evolution of inverter topologies beyond the conventional two-stage structure. The introduction of impedance source networks, exemplified by the Z-source inverter (ZSI), marked a significant leap by integrating boost and inversion functions within a single power conversion stage, offering inherent immunity to shoot-through faults. Among its successors, the Improved Y-Source Inverter (IYSI) has gained prominence for its superior voltage gain, continuous input current, and flexible winding design.
However, the intermittent nature of solar irradiance leads to fluctuating power output, posing challenges to grid stability and energy utilization. Integrating energy storage, typically batteries, directly into the inverter DC-link presents an elegant solution for power smoothing, peak shaving, and enabling off-grid operation. The Energy Storage Improved Y-Source Inverter (ES-IYSI) embodies this integration by connecting a battery bank across one of the network’s capacitors, eliminating the need for additional bidirectional DC-DC converters. This consolidation enhances system compactness and potential efficiency.
Despite its structural advantages, the ES-IYSI presents formidable control challenges. It constitutes a high-order, nonlinear system whose dynamic model inherently contains a right-half-plane (RHP) zero. This non-minimum phase characteristic, coupled with system nonlinearities, renders traditional linear control strategies like Proportional-Integral (PI) control inadequate. PI controllers often exhibit poor robustness against parameter variations and operational point shifts, slow transient response, and design complexity when attempting to manage the system’s unstable dynamics. These limitations necessitate the exploration of advanced nonlinear control methodologies.
In this article, I present a comprehensive dual-loop Sliding Mode Control (SMC) strategy tailored for the ES-IYSI. SMC is renowned for its robustness, fast dynamic response, and simplicity in handling system nonlinearities and uncertainties. The proposed strategy employs two independent SMC loops: one for regulating the DC-side impedance network (controlling PV maximum power point tracking (MPPT) and battery power) and another for controlling the AC-side output current. This approach ensures precise management of the three-terminal power flow—PV input, battery storage, and grid output—within the single-stage converter. I will begin by detailing the operational principles and deriving a complete mathematical model, including a small-signal analysis that reveals the RHP zero. Subsequently, I will elaborate on the systematic design of both sliding mode controllers, followed by an extensive simulation-based validation demonstrating the strategy’s superiority over conventional PI control in terms of dynamic performance, robustness, and power quality.
System Topology and Operational Principles of the ES-IYSI
The single-phase ES-IYSI system comprises three main segments: the improved Y-source network, the battery module, and the full-bridge inverter. The core impedance network features a three-winding coupled inductor (with turns ratio N1:N2:N3), two discrete inductors (L and Lb), two capacitors (C1 and C2), and a diode (D). The battery is modeled as an ideal voltage source (V_SOC) in series with an internal resistance (R_b). This topology is a significant advancement in integrated solar inverters.
The system operates in two distinct states, analogous to other impedance source solar inverters:
1. Shoot-Through State: This state is actively generated by simultaneously turning on both switches in any one phase leg (or multiple legs) of the inverter bridge. During this interval, the DC-link is short-circu. The diode D becomes reverse-biased and turns off. The inverter output voltage is zero, and no power is delivered to the grid. Energy from the PV source and the battery (depending on mode) is stored in the magnetic fields of the inductors and the electric fields of the capacitors.
2. Non-Shoot-Through State: This is the traditional active state of the inverter where switching follows a sinusoidal pulse-width modulation (SPWM) pattern. The diode D is forward-biased and conducts. The system delivers power from the DC link (which is boosted by the Y-network) to the AC grid through the inverter bridge.
The power balance governing the three-terminal system is fundamental and is given by:
$$ P_{PV} + P_{B} – P_{out} = 0 $$
where \(P_{PV}\) is the power from the PV array (always positive), \(P_{B}\) is the battery power (positive when discharging, negative when charging), and \(P_{out}\) is the AC output power (always positive). Control can be architected by directly regulating any two of these power flows, with the third being automatically determined by this balance equation. The strategy adopted here prioritizes control of PV MPPT and grid-injected power, allowing the battery power to act as the automatic balancing variable.
The key advantage of the ES-IYSI over simpler integrated topologies lies in its gain. The boost factor \(B\) provided by the improved Y-network is:
$$ B = \frac{V_{dc}}{V_{PV}} = \frac{1}{1 – k D} $$
where \(D\) is the shoot-through duty cycle and \(k = 1 + \frac{N_1 / N_3}{1 – N_3 / N_2}\) is a factor determined by the coupled inductor turns ratio. This structure allows for high voltage gain even with moderate shoot-through duty cycles, reducing voltage stress on devices and lowering the required input voltage from both PV and battery sources compared to, for example, the ES-qZSI.
| Topology | Voltage Gain | Input Current | Component Count | Battery Integration Simplicity |
|---|---|---|---|---|
| Two-Stage with separate DC-DC | High (Flexible) | Discontinuous (for some) | High | Complex (needs bidirectional converter) |
| ES-qZSI | Moderate: \(1/(1-2D)\) | Continuous | Moderate | Simple (direct parallel connection) |
| ES-IYSI (Proposed) | High: \(1/(1-kD)\) | Continuous | Moderate (adds coupled inductor) | Simple (direct parallel connection) |
Mathematical Modeling and Small-Signal Analysis
A precise dynamic model is essential for designing a high-performance controller. I derive the model using state-space averaging. The state variables are chosen as: the inductor currents \(i_L\), \(i_{L_M}\) (magnetizing), \(i_b\); and the capacitor voltages \(v_{C1}\), \(v_{C2}\). The input vector includes the PV voltage \(V_{PV}\), battery open-circuit voltage \(V_{SOC}\), and the output current \(i_o\).
1. State-Space Equations for Each Mode:
For the shoot-through state, the governing equations are:
$$ \mathbf{F} \dot{\mathbf{X}} = \mathbf{A_1} \mathbf{X} + \mathbf{B_1} \mathbf{u} $$
For the non-shoot-through state, they are:
$$ \mathbf{F} \dot{\mathbf{X}} = \mathbf{A_2} \mathbf{X} + \mathbf{B_2} \mathbf{u} $$
where \(\mathbf{X} = [i_L, v_{C1}, i_{L_M}, v_{C2}, i_b]^T\), \(\mathbf{u} = [V_{PV}, V_{SOC}, i_o]^T\), and \(\mathbf{F}\) is a diagonal matrix containing the energy storage elements (L, C1, L_M, C2, L_b). The matrices \(\mathbf{A_1}, \mathbf{B_1}, \mathbf{A_2}, \mathbf{B_2}\) are defined by the circuit configurations in each state.
2. Averaged Model:
Applying the state-space averaging technique over one switching period \(T_s\) with shoot-through duty cycle \(D\) yields the nonlinear averaged model:
$$ \mathbf{F} \dot{\mathbf{X}} = \mathbf{A} \mathbf{X} + \mathbf{B} \mathbf{u} $$
with \(\mathbf{A} = D \mathbf{A_1} + (1-D)\mathbf{A_2}\) and \(\mathbf{B} = D \mathbf{B_1} + (1-D)\mathbf{B_2}\). The explicit averaged equations are complex but critical. A simplified representation of key dynamics includes:
$$ L \frac{di_L}{dt} = V_{PV} + v_{C1} – (1-D)(v_{C2} – \alpha_1 v_{C1}) $$
$$ C_1 \frac{dv_{C1}}{dt} = -(1-D)i_L – D \alpha_2 i_{L_M} + … $$
where \(\alpha_1, \alpha_2, …\) are constants derived from the turns ratios \(N_1, N_2, N_3\).
3. Steady-State Analysis:
Setting the derivatives to zero in the averaged model, the steady-state voltages can be solved as:
$$ V_{C1} = \frac{D(k-1)}{1 – kD} V_{PV}, \quad V_{C2} = \frac{1-D}{1 – kD} V_{PV}, \quad V_{dc} = \frac{1}{1 – kD} V_{PV} $$
$$ I_b = \frac{V_{SOC} – V_{C1}}{R_b} $$
These equations confirm the voltage boost capability and show how the battery voltage naturally settles relative to \(V_{C1}\).
4. Small-Signal Model and Non-Minimum Phase Behavior:
To design linear controllers or understand loop interactions, a small-signal model is indispensable. Introducing small perturbations (\(\hat{x}, \hat{d}\)) around a steady-state operating point (\(X, D\)) and linearizing the averaged model leads to:
$$ \mathbf{F} s \hat{\mathbf{X}}(s) = \mathbf{A} \hat{\mathbf{X}}(s) + \mathbf{B} \hat{\mathbf{U}}(s) + [(\mathbf{A_1}-\mathbf{A_2})\mathbf{X} + (\mathbf{B_1}-\mathbf{B_2})\mathbf{U}] \hat{d}(s) $$
Solving this system allows derivation of transfer functions. Of particular importance is the control-to-battery-current transfer function \(G_{id}(s) = \hat{i}_b(s)/\hat{d}(s)\). Its analysis reveals a structure of the form:
$$ G_{id}(s) \approx K \frac{(1 – s / \omega_{zRHP})}{(s^2 / \omega_0^2 + 2\zeta s / \omega_0 + 1)(s / \omega_p + 1)} $$
The presence of the RHP zero at \(\omega_{zRHP}\) is a fundamental characteristic of this family of boost-derived integrated solar inverters with direct battery connection. This zero imposes a fundamental limitation: a positive step increase in the shoot-through duty cycle \(\hat{d}\) will initially cause a decrease in the battery current \(\hat{i}_b\) (or the affected capacitor voltage) before the eventual increase. This “wrong-way” initial response complicates feedback controller design, often leading to overshoot and potential instability if not properly addressed. This finding strongly motivates the use of a robust nonlinear control approach like SMC, which is less sensitive to such plant peculiarities than linear controllers.
Dual-Loop Sliding Mode Controller Design
The proposed control architecture features two decoupled SMC loops, addressing the DC-side power management and AC-side current quality independently.
DC-Side Sliding Mode Controller
Directly controlling the capacitor voltage \(v_{C2}\) (which relates to the DC-link voltage) using the shoot-through duty cycle is challenging due to the non-minimum phase dynamics. Therefore, an indirect control strategy is adopted. The outer objective is to maintain the PV voltage at its MPPT reference \(V_{PV}^*\). This is achieved by a standard Perturb & Observe (P&O) algorithm generating \(V_{PV}^*\), whose error is processed by a fast PI controller to produce the reference for the inductor current \(i_{L,ref}^*\).
The DC-SMC is designed to force the actual inductor current \(i_L\) to track this reference \(i_{L,ref}^*\) and, indirectly, regulate the capacitor voltage \(v_{C2}\) to its reference \(V_{C2}^*\) (which is derived from the desired DC-link voltage). We define the error states:
$$ Z_1 = v_{C2} – V_{C2}^*, \quad Z_2 = i_L – i_{L,ref}^* $$
The sliding surface \(S_{dc}\) is constructed as a linear combination of these errors:
$$ S_{dc} = \beta_1 Z_1 + \beta_2 Z_2 $$
where \(\beta_1\) and \(\beta_2\) are positive design coefficients determining the dynamics on the sliding manifold. The time derivative of \(S_{dc}\) is derived using the averaged model and the PI controller dynamics:
$$ \dot{S}_{dc} = \beta_1 \dot{Z}_1 + \beta_2 \dot{Z}_2 = \beta_1 f_1(\mathbf{X}, D) + \beta_2 [f_2(\mathbf{X}, D) – \dot{i}_{L,ref}^*] $$
The equivalent control \(D_{eq}\), the continuous control law that would maintain \(S_{dc}=0\), is found by solving \(\dot{S}_{dc}=0\) for \(D\):
$$ D_{eq} = \frac{ \beta_2 (V_{PV}+v_{C1})/L – \beta_2 K_P \dot{v}_{PV} – \beta_2 K_I (v_{PV}-V_{PV}^*) + \beta_1 (i_L – i_o)/C_2 }{ \beta_2 v_{C2}/L + \beta_1 (i_L + \alpha_1 i_{L_M})/C_2 } $$
To ensure robustness against model uncertainties and disturbances and to guarantee reaching conditions, the final control law adds a discontinuous switching term:
$$ D = D_{eq} – \eta \, \text{sgn}(S_{dc}) $$
where \(\eta\) is a positive switching gain. The signum function \(\text{sgn}(S_{dc})\) drives the system state towards the sliding surface. To mitigate chattering, a common practice in solar inverters, the signum function can be replaced by a continuous approximation like a saturation function \(\text{sat}(S_{dc}/\Phi)\) with boundary layer thickness \(\Phi\). The existence and stability of the sliding mode are proven using Lyapunov’s direct method. Defining \(V = \frac{1}{2} S_{dc}^2\), we must ensure \(\dot{V} = S_{dc} \dot{S}_{dc} < 0\). Substituting the control law \(D\) into \(\dot{S}_{dc}\) allows us to select \(\eta\) sufficiently large to satisfy this condition globally or within an operational region, guaranteeing that the sliding surface is attractive.
AC-Side Sliding Mode Controller
The objective of the AC-side controller is to force the grid current \(i_g\) to accurately track a sinusoidal reference \(i_{ref}\) that is in phase with the grid voltage \(v_g\) (for unity power factor). The reference amplitude is determined by the power management layer (e.g., constant power or battery-smoothing commands). The dynamics of the output LC filter (inductor \(L_f\), resistor \(R_f\)) and grid connection are:
$$ L_f \frac{di_g}{dt} = m \cdot V_{dc} – i_g R_f – v_g $$
where \(m \in [-1, 1]\) is the modulation index from the inverter bridge, treated as the control input for this loop. Defining the current tracking error as the sliding surface:
$$ S_{ac} = i_g – i_{ref} $$
Its derivative is:
$$ \dot{S}_{ac} = \frac{di_g}{dt} – \frac{di_{ref}}{dt} = \frac{1}{L_f} (m V_{dc} – i_g R_f – v_g) – \frac{di_{ref}}{dt} $$
We choose an exponential reaching law to dictate the dynamics of \(S_{ac}\):
$$ \dot{S}_{ac} = -Q \, \text{sgn}(S_{ac}) – K S_{ac} $$
where \(Q > 0\) and \(K > 0\) are controller gains. The exponential term \(-K S_{ac}\) ensures a smooth approach to the sliding surface. Equating the two expressions for \(\dot{S}_{ac}\) and solving for the equivalent modulation index \(m_{eq}\) yields the control law:
$$ m = \frac{1}{V_{dc}} \left[ L_f \left( -Q \, \text{sgn}(S_{ac}) – K S_{ac} + \frac{di_{ref}}{dt} \right) + i_g R_f + v_g \right] $$
This law ensures that the grid current error is driven to zero rapidly and is maintained there despite variations in \(V_{dc}\) or grid voltage. The chattering in this loop is also minimized by using a saturation function instead of the pure signum function. This direct control of the modulation index, combined with the shoot-through control from the DC-SMC, completes the full command for the inverter’s PWM generation.
| Parameter | Symbol | Value |
|---|---|---|
| Switching Frequency | \(f_s\) | 10 kHz |
| Y-Source Capacitors | \(C_1, C_2\) | 2200 µF |
| Magnetizing Inductance | \(L_M\) | 2.5 mH |
| Coupled Inductor Turns Ratio | \(N_1:N_2:N_3\) | 50:30:10 |
| AC Filter Inductor | \(L_f\) | 4 mH |
| AC Filter Resistor | \(R_f\) | 0.1 Ω |
| Battery Voltage (SOC-based) | \(V_{SOC}\) | ~200 V |
| Battery Internal Resistance | \(R_b\) | 0.05 Ω |
Simulation Results and Performance Analysis
To validate the proposed dual-loop SMC strategy, a detailed simulation model of the ES-IYSI system was built. The performance is evaluated under key scenarios: power flow management during source/load changes, dynamic response to irradiance steps, and comparison with traditional dual-loop PI control.
Scenario 1: Three-Terminal Power Flow Management. The system starts with high PV irradiation, producing \(P_{PV} \approx 6 kW\), while the grid demand is set to \(P_{out} \approx 4 kW\). According to the power balance, the battery charges at \(P_B \approx -2 kW\). At t=0.3s, the irradiation drops, reducing \(P_{PV}\) to 4 kW, matching \(P_{out}\). The battery current naturally falls to zero. At t=0.5s, \(P_{out}\) is increased to 5.5 kW while \(P_{PV}\) remains at 4 kW, forcing the battery to discharge at \(P_B \approx +1.5 kW\). The simulation waveforms confirm that the proposed controller seamlessly manages these transitions. The battery current \(i_b\) responds within a few milliseconds to compensate for the power差额, and the grid current \(i_g\) adjusts its amplitude precisely and rapidly to deliver the commanded power, maintaining low total harmonic distortion (THD). This demonstrates the effectiveness of the control strategy in one of the primary applications of modern solar inverters—integrated energy buffering.
Scenario 2: Dynamic Performance under Irradiance Transient. A step change in PV irradiation is applied to test the robustness and speed of the MPPT and power balancing loops. With the proposed SMC, the PV voltage recovers to its new MPP within 20 ms with minimal oscillation. The inductor current \(i_L\) tracks its changing reference aggressively. Crucially, the battery current responds swiftly to absorb the surplus or deficit power. In contrast, a well-tuned PI controller exhibits significant overshoot in the PV voltage and slower settling time (over 50 ms). The grid current under PI control shows a more pronounced distortion during the transient. The THD of the grid current under steady-state conditions is measured at 2.75% for the SMC, compared to 4.22% for the PI controller, highlighting the superior tracking accuracy of the SMC for AC-side current.
| Metric | Proposed Dual SMC | Conventional Dual PI |
|---|---|---|
| MPPT Settling Time (for step) | < 20 ms | > 50 ms |
| PV Voltage Overshoot | Negligible | Significant (~15%) |
| Battery Current Response Time | ~5 ms | ~15 ms |
| Grid Current THD (steady-state) | 2.75% | 4.22% |
| Robustness to parameter variation | High | Low-Moderate |
The simulation results conclusively validate the theoretical design. The dual-loop SMC strategy effectively handles the nonlinearities and the non-minimum phase characteristic of the ES-IYSI. It provides faster dynamic response, better disturbance rejection, and higher quality grid current compared to the linear PI alternative. The robustness inherent in SMC makes this approach particularly suitable for solar inverters which must operate reliably under varying environmental conditions and component tolerances.
Extended Discussion and Practical Considerations
While the proposed SMC demonstrates excellent performance, practical implementation in solar inverters requires addressing several points. First, the chattering phenomenon, though reduced by boundary layers, can still cause high-frequency switching losses and electromagnetic interference (EMI). Advanced reaching laws or higher-order sliding mode controllers (like super-twisting algorithm) can be explored to further suppress chattering while preserving robustness. Second, the design parameters (\(\beta_1, \beta_2, \eta, Q, K\)) must be carefully selected. While Lyapunov stability gives boundaries, fine-tuning based on detailed simulation and hardware-in-the-loop (HIL) testing is recommended to optimize performance.
The ES-IYSI topology itself offers a compelling balance. Compared to other impedance source solar inverters like the qZSI or the Γ-ZSI, it provides higher gain, which is advantageous for low-voltage PV arrays or battery banks. However, the coupled inductor introduces design complexity and potential issues with leakage inductance, which the model and control must account for. Future work could involve extending this control strategy to three-phase systems, integrating more sophisticated battery state-of-charge (SOC) management algorithms directly into the sliding mode framework, and experimental validation on a hardware prototype.
In conclusion, the integration of energy storage directly into advanced inverter topologies like the ES-IYSI represents a significant trend in power conversion for renewable energy. The inherent control challenges of such systems, stemming from their nonlinear and non-minimum phase dynamics, call for robust control solutions. The dual-loop sliding mode control strategy presented here provides a systematic and effective approach to manage the multi-objective control problem in these integrated solar inverters, ensuring stable operation, fast response, and high-quality power injection into the grid. This work contributes to the ongoing development of smarter, more resilient, and efficient power electronic interfaces for the sustainable energy landscape.