In modern power systems, the proliferation of distributed generation has elevated the importance of power electronic interfaces. Among these, the three-level Neutral-Point-Clamped (NPC) inverter stands out as a critical link connecting renewable energy sources to the grid, prized for its high efficiency and reduced harmonic distortion. The performance and stability of the entire power system are directly influenced by the control quality of this grid-connected inverter. While traditional control methods like Proportional-Integral (PI) control are prevalent, Deadbeat Predictive Current Control (DBPCC) has gained significant attention for its superior dynamic response and inherent simplicity of implementation. However, a fundamental challenge persists: the performance of model-based predictive controllers, including DBPCC, is highly sensitive to parameter mismatches between the controller’s internal model and the actual physical system. Variations in filter inductance, often caused by temperature drift, aging, or magnetic saturation, can severely degrade current tracking accuracy, leading to increased harmonic distortion and potential instability.
This work addresses this critical robustness issue by proposing a novel Model-Reference Adaptive Deadbeat Predictive Current Control (MRA-DBPCC) strategy for NPC-type grid-connected inverters. The core innovation lies in integrating the Popov hyperstability theory into the predictive control framework to achieve online parameter identification and compensation. Furthermore, to ensure the practical viability of the approach, the control scheme incorporates a Space Vector Pulse Width Modulation (SVPWM) strategy with a dynamic redistribution factor to actively suppress the inherent neutral-point potential imbalance in NPC inverters. Comprehensive simulation and experimental results validate that the proposed strategy significantly enhances system robustness against parameter variations, improves output current quality, and maintains excellent dynamic performance.
1. Mathematical Model and Conventional DBPCC
The topology of a three-phase, three-level NPC grid-connected inverter is characterized by a split DC-link with capacitors \(C_1\) and \(C_2\), and four switching devices per phase leg. The output is connected to the grid through an L-type filter (inductance \(L\) and equivalent resistance \(R\)). In the synchronous rotating \(dq\)-reference frame, the continuous-time state-space model of the system is given by:
$$ L\frac{di_d}{dt} = u_d – Ri_d – e_d + \omega L i_q $$
$$ L\frac{di_q}{dt} = u_q – Ri_q – e_q – \omega L i_d $$
where \(i_d, i_q\) are the grid current components, \(u_d, u_q\) are the inverter output voltage components, \(e_d, e_q\) are the grid voltage components, and \(\omega\) is the grid angular frequency. For digital implementation, this model must be discretized. Using the forward Euler method with a sufficiently small sampling period \(T_s\), the discrete-time model is obtained:
$$
\begin{bmatrix} i_d(k+1) \\ i_q(k+1) \end{bmatrix} =
\begin{bmatrix}
1 – \frac{R T_s}{L} & \omega T_s \\
-\omega T_s & 1 – \frac{R T_s}{L}
\end{bmatrix}
\begin{bmatrix} i_d(k) \\ i_q(k) \end{bmatrix}
+
\frac{T_s}{L}
\begin{bmatrix}
u_d(k) – e_d(k) \\
u_q(k) – e_q(k)
\end{bmatrix}
$$
The principle of conventional DBPCC is to calculate the voltage command at time \(k\) such that the predicted current at time \(k+1\) equals its reference value \(i_d^*(k), i_q^*(k)\). By setting \(i_d(k+1)=i_d^*(k)\) and \(i_q(k+1)=i_q^*(k)\) in the equation above, the ideal control law is derived:
$$ u_d(k) = \frac{L}{T_s} \left[ i_d^*(k) – \left(1 – \frac{R T_s}{L}\right)i_d(k) – \omega T_s i_q(k) \right] + e_d(k) $$
$$ u_q(k) = \frac{L}{T_s} \left[ i_q^*(k) – \left(1 – \frac{R T_s}{L}\right)i_q(k) + \omega T_s i_d(k) \right] + e_q(k) $$
However, digital control systems introduce an inherent computation and modulation delay of one sampling period. To compensate for this, a two-step prediction is commonly employed. The control objective becomes forcing the current at time \(k+2\) to track the reference at time \(k\). The modified predictive model and the final voltage command equations, after compensating for this delay, are crucial for achieving good performance in a practical grid-connected inverter.
2. Robustness Analysis and the Need for Adaptation
The control law in Eq. (3) is explicitly dependent on the nominal values of the filter parameters \(L\) and \(R\). Let \(L_n\) and \(R_n\) denote the nominal values used in the controller, and \(L_a\) and \(R_a\) denote the actual physical values. A parameter mismatch occurs when \(L_n \neq L_a\) or \(R_n \neq R_a\). Substituting the actual plant dynamics into the control law designed with nominal parameters reveals an equivalent voltage disturbance \(v_d, v_q\) injected into the system:
$$
\begin{bmatrix} i_d(k+1) \\ i_q(k+1) \end{bmatrix} = \mathbf{F}_n \begin{bmatrix} i_d(k) \\ i_q(k) \end{bmatrix} + \mathbf{G}_n \begin{bmatrix} u_d(k)-e_d(k) \\ u_q(k)-e_q(k) \end{bmatrix} – \begin{bmatrix} v_d \\ v_q \end{bmatrix}
$$
where the disturbance is a function of the parameter errors \(\Delta L = L_n – L_a\) and \(\Delta R = R_n – R_a\), as well as the states and inputs. This disturbance directly leads to a steady-state current tracking error and degraded transient response. A simplified stability analysis, ignoring the coupling term and the relatively small resistance effect (\(RT_s/L << 1\)), provides clear insight. The closed-loop transfer function from reference to actual current in the q-axis becomes:
$$ \frac{i_q(z)}{i_q^*(z)} = \frac{L_n / L_a}{z + (L_n / L_a) – 1} $$
The pole of this system is \(z = 1 – L_n/L_a\). For stability, the pole must lie inside the unit circle (\(|z| < 1\)). This imposes the condition \(0 < L_n / L_a < 2\). If the controller overestimates the inductance by more than a factor of two (\(L_n > 2L_a\)), the system becomes unstable. This highlights the critical vulnerability of standard DBPCC to parameter inaccuracy in a grid-connected inverter. The following table summarizes the impact of typical parameter variations on system performance.
| Parameter Variation | Primary Impact on DBPCC | Consequence for Grid Current |
|---|---|---|
| \(L_n > L_a\) (Overestimation) | Reduced control gain, sluggish response. | Increased settling time, possible overshoot. |
| \(L_n < L_a\) (Underestimation) | Increased control gain, aggressive response. | Overshoot, oscillation, potential instability if severe. |
| \(L_n > 2L_a\) (Severe Overestimation) | Pole moves outside unit circle. | System instability. |
| \(R_n \neq R_a\) | Steady-state offset in predicted voltage. | Steady-state current tracking error. |
3. Proposed Model-Reference Adaptive DBPCC (MRA-DBPCC)
3.1 System Structure and Reference Model
To mitigate the sensitivity to parameter mismatch, a Model-Reference Adaptive Control (MRAC) structure is integrated with the DBPCC strategy. The core idea is to define a reference model that exhibits the desired dynamic response and then design an adaptive mechanism that adjusts the controller’s internal parameters so that the actual plant output tracks the reference model output. This approach directly minimizes the tracking error caused by model inaccuracies, rather than relying on explicit but often noisy parameter identification.
For the NPC grid-connected inverter, the continuous-time reference model in the \(dq\)-frame is defined using the *desired* or nominal parameters \(L\) and \(R\):
Reference Model:
$$ p \begin{bmatrix} i_d \\ i_q \end{bmatrix} =
\begin{bmatrix}
-R/L & \omega \\
-\omega & -R/L
\end{bmatrix}
\begin{bmatrix} i_d \\ i_q \end{bmatrix} +
\begin{bmatrix}
1/L & 0 \\
0 & 1/L
\end{bmatrix}
\begin{bmatrix} u_d – e_d \\ u_q – e_q \end{bmatrix} $$
where \(p\) is the differential operator. Let \(a = R/L\) and \(b = 1/L\). The reference model can be written compactly as \(p\mathbf{I} = \mathbf{A}\mathbf{I} + \mathbf{B}\mathbf{U}\), where \(\mathbf{I}=[i_d, i_q]^T\) and \(\mathbf{U}=[u_d-e_d, u_q-e_q]^T\).
3.2 Adjustable Model and Error Dynamics
In parallel, an adjustable model runs within the adaptive controller. This model has the same structure as the reference model but uses adjustable parameters \(\hat{a}(t)\) and \(\hat{b}(t)\), which correspond to estimates of \(a\) and \(b\).
Adjustable Model:
$$ p \begin{bmatrix} \hat{i}_d \\ \hat{i}_q \end{bmatrix} =
\begin{bmatrix}
-\hat{a} & \omega \\
-\omega & -\hat{a}
\end{bmatrix}
\begin{bmatrix} \hat{i}_d \\ \hat{i}_q \end{bmatrix} +
\begin{bmatrix}
\hat{b} & 0 \\
0 & \hat{b}
\end{bmatrix}
\begin{bmatrix} u_d – e_d \\ u_q – e_q \end{bmatrix} $$
or \(p\mathbf{\hat{I}} = \mathbf{\hat{A}}\mathbf{\hat{I}} + \mathbf{\hat{B}}\mathbf{U}\).
The goal of adaptation is to drive the output error \(\mathbf{e} = \mathbf{I} – \mathbf{\hat{I}} = [e_d^{err}, e_q^{err}]^T\) to zero. Subtracting the adjustable model equation from the reference model equation yields the error dynamics:
$$ p\mathbf{e} = \mathbf{A}\mathbf{e} + (\mathbf{A} – \mathbf{\hat{A}})\mathbf{\hat{I}} + (\mathbf{B} – \mathbf{\hat{B}})\mathbf{U} $$
$$ p\mathbf{e} = \mathbf{A}\mathbf{e} – \mathbf{m} $$
where \(\mathbf{m} = -(\Delta\mathbf{A}\mathbf{\hat{I}} + \Delta\mathbf{B}\mathbf{U})\) is a nonlinear function of the parameter errors \(\Delta a = \hat{a}-a\) and \(\Delta b = \hat{b}-b\), the adjustable model state \(\mathbf{\hat{I}}\), and the input \(\mathbf{U}\).
3.3 Adaptive Law Design via Popov Hyperstability Theory
The Popov hyperstability theory provides a powerful framework for designing stable adaptation laws for nonlinear systems. The overall system is viewed as a feedback interconnection of a linear time-invariant (LTI) forward block and a nonlinear time-varying feedback block containing the parameter adaptation mechanism. For stability, the forward path transfer matrix must be strictly positive real (SPR), and the feedback path must satisfy the Popov integral inequality.
The forward path in our case is described by \(p\mathbf{e} = \mathbf{A}\mathbf{e} – \mathbf{m}\), with output \(\mathbf{e}\). To ensure the SPR condition is met, a compensation gain matrix \(\mathbf{D}\) (chosen as the identity matrix \(\mathbf{I}\)) is added. The feedback path is defined by \(\mathbf{m} = \phi(\mathbf{\hat{I}}, \mathbf{U}, \Delta a, \Delta b)\).
Applying the Popov hyperstability criterion, we derive adaptation laws with a proportional-integral (PI) structure to ensure convergence and zero steady-state error. The adaptation laws for the estimated parameters are:
$$ \hat{a}(t) = -\left( k_{p1} + \frac{k_{i1}}{s} \right) \left( \hat{i}_d e_d^{err} + \hat{i}_q e_q^{err} \right) + \hat{a}(0) $$
$$ \hat{b}(t) = -\left( k_{p2} + \frac{k_{i2}}{s} \right) \left( (u_d-e_d) e_d^{err} + (u_q-e_q) e_q^{err} \right) + \hat{b}(0) $$
where \(k_{p1}, k_{i1}, k_{p2}, k_{i2} > 0\) are adaptive gains, and \(\hat{a}(0), \hat{b}(0)\) are initial estimates. From these, the real-time estimates of the physical filter parameters for the grid-connected inverter are recovered:
$$ \hat{L}(t) = \frac{1}{\hat{b}(t)}, \quad \hat{R}(t) = \frac{\hat{a}(t)}{\hat{b}(t)} $$
These continuously updated parameters \(\hat{L}(t)\) and \(\hat{R}(t)\) are then fed directly into the DBPCC voltage calculation equations (e.g., Eq. (3) or its delayed-compensation version), replacing the fixed nominal values \(L_n\) and \(R_n\). This closes the adaptive loop, allowing the predictive controller to self-correct in the face of parameter variations.
4. Neutral-Point Potential Balancing with Dynamic SVPWM
A key practical challenge in NPC inverter operation is the balancing of the DC-link capacitor voltages \(V_{C1}\) and \(V_{C2}\). An imbalance leads to increased voltage stress on devices and higher output voltage distortion. The imbalance is caused by the neutral-point current \(i_{np}\), which flows into or out of the midpoint \(O\) depending on the switching states. Among the 27 switching states of a three-level inverter, only the medium vectors and the so-called “small” vectors affect \(i_{np}\). Crucially, each small vector has a redundant switching state that produces an opposite \(i_{np}\).
Standard SVPWM can be modified to exploit this redundancy. In the proposed strategy, the dwell times for the two redundant small vectors used in a sampling period are dynamically redistributed using a factor \(m\) (\(-1 \le m \le 1\)). The goal is to inject a compensating charge \(\Delta Q_{comp}\) in the current cycle to cancel the charge imbalance \(\Delta Q = C \cdot \Delta V_{dc}\) from the previous cycle, where \(\Delta V_{dc} = V_{C1} – V_{C2}\). Based on the principle of charge conservation, the dynamic factor \(m\) is calculated as:
$$ m = -\frac{C \Delta V_{dc} + T_c i_b – T_b i_c}{T_a i_a} $$
where \(T_a, T_b, T_c\) are the calculated dwell times for the active vectors in a given sector, and \(i_a, i_b, i_c\) are the measured phase currents. This value of \(m\) is then used to redistribute the dwell times between the redundant small vector pairs within the SVPWM algorithm. This integrated approach ensures that the modulation stage not only synthesizes the desired voltage vector from the MRA-DBPCC controller but also actively stabilizes the DC-link midpoint potential of the grid-connected inverter.
5. Simulation and Experimental Validation
5.1 Simulation Setup and Parameters
The proposed MRA-DBPCC strategy with dynamic SVPWM was validated through simulation in MATLAB/Simulink and experimentally on a 5 kW NPC inverter prototype. The system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link Voltage | \(U_{dc}\) | 550 V |
| Grid Phase Voltage (RMS) | \(e\) | 220 V |
| Grid Frequency | \(f\) | 50 Hz |
| Nominal Filter Inductance | \(L_n\) | 10 mH |
| Nominal Filter Resistance | \(R_n\) | 0.5 Ω |
| DC-link Capacitance | \(C_1, C_2\) | 450 μF |
| Switching/Sampling Frequency | \(f_s = 1/T_s\) | 20 kHz |
5.2 Performance Under Parameter Mismatch
First, the performance was evaluated under a severe inductance mismatch. With the nominal controller set to \(L_n = 10\) mH, the actual inductance was stepped from 10 mH to 7 mH (-30%) and then to 13 mH (+30%). The results starkly contrast the conventional DBPCC and the proposed MRA-DBPCC.
Conventional DBPCC: The current waveform showed significant distortion and tracking error immediately after the parameter change. The Total Harmonic Distortion (THD) increased from 1.75% (matched) to 3.86% under mismatch.
Proposed MRA-DBPCC: The adaptive mechanism quickly identified the change and adjusted the model parameters. The output current maintained high quality with minimal transient disturbance. The THD only increased slightly to 2.57% under the same mismatch conditions, demonstrating superior robustness. The parameter estimator accurately tracked the actual inductance value with an error of less than 1.5%.
5.3 Dynamic Response Comparison
The dynamic performance was tested by applying a step change in the current reference from 20 A to 10 A. The dynamic response time is defined as the time for the current to settle within ±2% of the new reference.
Conventional DBPCC: Exhibited a response time of 328 μs. With parameter mismatch, this response degraded further, showing overshoot or sluggishness.
Proposed MRA-DBPCC: Achieved a faster and cleaner response with a settling time of 192 μs, which is approximately 34% faster than the conventional method. This improvement stems from the accurate real-time model provided by the adaptive scheme, allowing the predictive controller to calculate the precise voltage needed without error.
5.4 Neutral-Point Balancing Performance
The effectiveness of the dynamic SVPWM strategy was evaluated by observing the capacitor voltages \(V_{C1}\) and \(V_{C2}\). Without the balancing algorithm, the neutral-point voltage exhibited significant drift and low-frequency oscillation, with an imbalance (\(\Delta V_{dc}\)) of several volts. After enabling the proposed balancing strategy, the imbalance was actively controlled and confined to a band within ±0.6 V. This effective balancing contributes to the overall stability and output waveform quality of the grid-connected inverter.
5.5 Experimental Verification
Experimental results on the 5 kW prototype confirmed the simulation findings. Under matched parameters, both methods produced good current waveforms, with MRA-DBPCC yielding a slightly lower THD (2.96% vs. 3.27%). When an intentional inductance mismatch was introduced (changing from 10 mH to 7 mH via a relay-switched external inductor), the conventional DBPCC current became noticeably distorted (THD 4.36%), while the MRA-DBPCC current remained sinusoidal (THD 3.15%). The phase alignment between grid voltage and current was also better maintained with the adaptive strategy, ensuring a high power factor. The experimental dynamic response results further solidified the advantage of the proposed method.
6. Conclusion
This work has presented a comprehensive adaptive control solution to address the critical robustness issue in deadbeat predictive current control for NPC grid-connected inverters. The proposed MRA-DBPCC strategy, founded on the Popov hyperstability theory, successfully integrates online parameter adaptation with a delay-compensated predictive controller. By continuously identifying the actual filter inductance and resistance, the controller compensates for model mismatches in real-time. The complementary dynamic SVPWM strategy ensures stable operation by actively suppressing the inherent neutral-point potential imbalance.
The key outcomes are: 1) The system maintains high-performance current tracking with low THD even under substantial (±30%) parameter variations, whereas conventional DBPCC degrades significantly. 2) The dynamic response is improved by approximately 34% due to the accurate system model used for prediction. 3) The integrated approach provides a practical and robust control framework for high-power quality grid integration. Future work will focus on extending the adaptation mechanism to handle simultaneous multiple parameter perturbations and integrating robustness against grid impedance variations, further enhancing the applicability of adaptive predictive control in complex and variable grid environments for modern grid-connected inverters.