In electrical power systems, the grid frequency typically remains relatively stable, but in systems like microgrids, isolated grids, or grids with high penetration of renewable energy sources, the frequency might deviate from the standard value. This can result from variations in generation, load, or the presence of intermittent renewable sources such as solar and wind. The concept of digital resonant control involves designing control algorithms that can detect and mitigate resonant conditions within the system. Resonance occurs when the natural frequency of the power converter circuit matches the frequency of the grid fluctuations. This can lead to amplified voltage or current oscillations, potentially causing instability, voltage distortion, and even damage to the converter and connected components. The control system continuously monitors the grid frequency, which might be non-constant due to variable generation and load conditions.

The control algorithm analyzes the system’s response to the varying grid frequency to detect any signs of resonance. This involves monitoring the voltage and current waveforms and identifying harmonic components that might indicate resonance. Once resonance is detected, the control algorithm adjusts the converter’s control parameters in real-time to counteract the effects of resonance. This might involve tuning the converter’s output impedance or adjusting the control loop gains. The control strategy can incorporate damping techniques and filtering to reduce the impact of resonance on the system. This could include adding virtual resistors or filters to dampen oscillations and prevent them from increasing.

In addition to resonance control, the algorithm might include synchronization techniques to ensure that the converter output is synchronized with the variable grid frequency, maintaining stable operation and minimizing voltage and current distortions. Digital resonant control is particularly valuable in scenarios where traditional control methods might struggle to maintain stability under varying grid frequency conditions. Its adaptive nature allows power converters to respond in real-time to changes in the system dynamics, thereby improving overall system performance, efficiency, and reliability. This control strategy is crucial for the successful integration of renewable energy sources into the grid, as these sources often introduce variability in the grid frequency. Digital resonant control enhances the stability and robustness of power electronic systems by effectively managing resonant conditions, ensuring smooth operation even in challenging grid conditions.

Renewable energy has the ability to cope with ever-increasing energy needs. It supplies electric power directly to the consumer or connects to the main grid. Solar photovoltaic (PV) and wind energy-based power systems Dong et al. (2023); Desalegn et al. (2023); Khan et al. (2022) are inherently stochastic in nature due to changing weather conditions such as irradiance and temperature variations in the case of the solar PV system and wind speed variations in the case of the wind energy conversion system. PV systems and wind integrated systems directly supply energy to the consumers such as residential, commercial, or industrial loads or connect to the main grid using DC–DC or DC–AC converters. There are many associated challenges such as grid stability and bi-directional power flow control in hybrid energy systems (HESs) in electric vehicles that use multi-port bi-directional converters (Malik et al., 2023a; Malik et al., 2023b). With variable weather conditions, the PV system suffers low voltage at the output; in this scenario, the DC–DC boost converter becomes a very necessary part to get high voltage gain Malik et al. (2017). The frequency of the output voltages must also be tightly regulated before injecting the output power into the grid Yasmeena and Das (2015)-Chen et al. (2021) from these renewable systems.

Recently, concentrated photovoltaic (CPV) has been given preference over conventional photovoltaic due to its high efficiency. The solar cell area can be augmented with optical concentrators Iqbal et al. (2023), thereby improving efficiency of renewable solar-powered systems. Short-term load forecasting technologies can also be integrated with modern photovoltaic-based energy conversion system to further improve in the planning of energy demand, smart and efficient economic dispatch, and energy management of electrical power Javed et al. (2021)-Wei et al. (2023). However, without efficient grid synchronization Lin et al. (2023, 2022) and load frequency control (LFC), it degrades power quality and increases total harmonic distortion (THD) in the power network Ahsan et al. (2021); Liu et al. (2023) or drive systems coupled with it Liu and Liu (2021). To solve this problem, various classical control system approaches, such as proportional integral (PI) controller, deadbeat controllers, hysteresis controllers, repetitive controllers, and current controllers based on resonant filters, are employed with certain limitations such as hardware complexity and non-zero steady-state error performance Bourguiba et al. (2016) - Nazir et al. (2020).

A proportional resonant controller (PRC) is a control technique that combines the principles of proportional control with the ability to address resonant frequencies in a system. PRCs have garnered significant attention due to their effectiveness in dealing with resonance-related issues in various applications. This literature review provides an overview of key research findings, advancements, and applications related to PRCs. PRCs combine the features of proportional control with the capability to address resonant frequencies, making them well-suited for applications requiring precise control in the presence of resonant phenomena. Resonant controllers can be used to manage harmonic distortion in power electronics systems, demonstrating their ability to selectively attenuate specific frequency components in control signals while maintaining proportional control action. PRCs can be applied to grid-connected converters and voltage source inverters, aiming to enhance the power quality by suppressing current harmonics and damping resonances, thereby improving the dynamic response and stability of renewable energy systems.

In the realm of motor drives and motion control, PRCs have demonstrated their potential to mitigate mechanical resonances. Advancements in PRCs have focused on refining control strategies and tuning methodologies. Adaptive tuning techniques have been explored to optimize PRC parameters in real-time, adapting to changes in system dynamics and load conditions. Moreover, the integration of PRCs with advanced control algorithms, such as predictive control and sliding mode control, has been investigated to achieve higher accuracy and robustness.

The prevalence of digital control systems has led to extensive research on the digital implementation of PRCs. Zero-order hold and bilinear transformation techniques have been employed to adapt continuous-time PRC designs for digital platforms. Recent research has expanded the concept of PRCs to hybrid and multi-resonant control strategies. Researchers have integrated PRCs with other control techniques, such as repetitive control, to address multiple resonant frequencies in complex systems. These hybrid approaches aim to enhance control performance by effectively managing multiple resonance-related challenges.

The PRC is one of the most promising approaches that has recently gained considerable attention in regulating sinusoidal current and voltage signals in the case of grid-tied inverters Kuperman (2015). The PR controller is based on the internal model principle (IMP), which has the capability of providing a high-quality sinusoidal signal in the case of grid-connected inverters Quan and Cai (2010). In fact, the PR controller is the real alternative to conventional PI controllers that operate in positive and negative synchronous frame sequences Wang et al. (2018); Tarraso et al. (2017). Only the integration of the PR controller and PI controllers is different. The PR controller integrates at resonant frequency without producing phase shift and steady-state error.

In the synchronous frame of reference, PR controller performance and transient response are almost similar to those of traditional PI controller D. Zammit and Apap (2014). However, the performance of the PR controller is much better when controlling sinusoidal signals. The PR controller overcomes the shortcomings of the PI controller, such as the poor and sluggish tracking of sinusoidal reference signals and disturbance rejection Lakshmi and Naik (2017). Other advantages of the PR controller over the conventional PI controller are lower complexity and computational burden, direct connection to single-phase PWM inverters, and lower sensitivity Essaghir et al. (2018).

PR controller applications are widely applied to photovoltaic inverters, wind turbines, power converters, and uninterrupted power supplies Buso and Mattavelli (2015)- Husev et al. (2020). The PR controller is tuned to the fundamental frequency of the system to ideally provide infinite gain at the resonant frequency and almost no attenuation outside this frequency. Therefore, the PR controller can also be combined with a notch filter to compensate harmonics and to achieve good transient and steady-state objectives Rufa’I et al. (2019). Parvez et al. (2020) used the PR controller in the main control loop of an uninterrupted power supply inverter for low steady-state error, improved dynamic response, better output regulation, and lower THD. The dual integration method is commonly used to implement the PR controller due to the ease of frequency adaptation Das and Barai (2021).

In Mohammed et al. (2017), various control strategies are explored for integrating and managing renewable power injection into the grid. These strategies significantly contribute to enhancing power system stability, improving power quality, and overall system performance. Achievements include maintaining low total harmonic distortion (THD), achieving high-power factor, and demonstrating rapid dynamic responses. Additionally, the utilization of an algorithm for tracking the maximum power point (MPPT) in grid-connected photovoltaic inverters, along with the creation of a precise phase-locked loop (PLL) for phase angle detection, amplitude regulation, and voltage frequency control, plays a crucial role in ensuring accurate generation of reference signals.

The work presented by Li et al. (2021) introduces a sophisticated quasi-proportional integral resonant (QPIR) controller tailored for a three-phase LCL grid-connected inverter with integrated active damping. In contrast to conventional QPR controllers, this enhanced QPIR controller exhibits remarkable precision in its regulation of grid-connected voltages and currents. The innovative QPIR controller effectively attains autonomous control over both active and reactive power in the context of a three-phase grid-connected inverter. Notably, this controller excels in ensuring the simultaneous accuracy and swiftness of control over grid-connected currents, rendering it particularly pertinent for applications emphasizing high-power factors and aiming to optimize DC-side utilization efficiency. It is important to note, however, that while this controller demonstrates considerable promise, there are limitations preventing its direct applicability to practical systems.

In Alathamneh et al. (2022), the focus is on enhancing the stability of a grid-connected inverter operating under weak grid conditions. The study reveals that problematic negative resistance areas arise due to interactions between the phase-locked loop (PLL) and the grid voltage feedforward (GVF) elements. To counter this instability originating from the PLL and GVF, the research introduces a customized stability control method for grid-connected inverters. This approach revolves around modifying the impedance in the q-axis. Specifically, the proposed method integrates q-axis voltage directly into the q-axis current regulator’s output through a transfer link. This deliberate step effectively counteracts the adverse effects of both the PLL and GVF on the q-axis output impedance of the grid-connected inverter. However, it is important to note that this analysis is confined to a single inverter system.

In Gopinath and Vijayakumar (2022), a multi-level inverter designed for grid-connected converters is presented. This inverter tackles issues such as leakage current reduction, voltage enhancement, self-voltage balancing, and minimizing capacitor voltage stress. The suggested design proves to be a strong candidate for applications involving grid-connected photovoltaic systems.

Most of the prior research has focused on continuous-time design of the PR controller to achieve the desired performance, but these controllers cannot be applied directly to digital systems. Therefore, for the digital system, it is necessary to implement the PR controller digitally. Different discretization methods such as zero-order hold (ZOH), first-order hold (FOH), impulse invariant (II), and the zero pole matching (ZPM) method are used for digital implementation of the PR controller Yepes (2011). However, discretization of PR is not an easy task, as it causes displacement of poles and zeros of the controller. The displacement of zeros poses problems of stability, whereas the displacement of poles causes a resonant frequency shift, leading to steady-state error. Since sinusoidal voltage is the basic electrical variable of the power system, the main goal of this work is to propose an efficient discretization method used for digital implementation of the PR controller that can effectively track the frequency of the desired signal with zero steady-state error.

The remainder of the paper is organized as follows. Section 4 presents modeling of the single-phase PWM inverter. Section 5 presents the digital control techniques for the single-phase PWM inverter. Section 6 discusses in detail the discretization methods used for digital implementation of the PR controller and its effects on poles and zeros of the controller. It also provides three efficient discretization methods that can be used for PR controllers. Simulation results and discussion are given in Section 7. Section 8 concludes the paper.

A single-phase PWM inverter with the control loop is shown in Figure 1. It consists of DC input source, inverter bridge, low-pass LC filter, non-linear rectifier load, and PWM pulse generator. E _n represents the nominal value of the DC input source. The inverter bridge consists of four insulated gate bipolar transistor (IGBT) switches from S ₁ to S ₄ with protection diodes. The switches generate an alternating signal at the output, while the purpose of protection diodes is to provide a reverse path for current flow. The low-pass filter consists of an inductor L _n and a capacitor C _n , and the values of L _n and C _n are set to achieve the desired frequency response. The inverter may be standalone or grid-connected. In the standalone case, the generated AC power is supplied directly to consumers, while in the case of the grid-connected inverter, the AC output is delivered to the grid. The switching pattern of the inverter is decided by the PWM pulse generator. The PWM pulse generator is controlled by the controller circuit of the inverter. The desired voltage with nominal frequency is generated by efficiently controlling the PWM pulses to the input of the inverter Nazir et al. (2019).State space equations of the PWM inverter are given by Cha et al. (2015). v ̇ c v ̈ c = 0 1 − 1 L n C n − 1 R n C n v c v ̇ c + 0 1 L n C n v i n , (1) where E _n , v _c , C _n , L _n , and R _n represent the inverter input voltage, pulse width modulated output voltage, nominal values of inductor, capacitor, and load resistor, respectively. The state space equation x ̇ = A x + B u can be written in the sampled form as x k + 1 = e A τ x k + ∫ 0 τ e A T − τ B u τ d τ . (2) Therefore, the discrete time version of Eq. 1 is given by Phillips et al. (2015). v c k + 1 v ̇ c k + 1 = Φ 11 Φ 12 Φ 21 Φ 22 v c k v ̇ c k + g 1 g 2 Δ T , (3) where g ₁ = E _n T/2L _n C _n , g 2 = E n / L n C n − E n / 2 L n C n 2 R n , Φ₁₁ = 1 − T ²/2L _n C _n , Φ₁₂ = −T/L _n C _n + T ²/2L _n C _n R _n , Φ₂₁ = −T/L _n C _n + T ²/2L _n C _n R _n , and Φ 22 = 1 − T / R n C n − T 2 / 2 L C + T 2 / 2 R n 2 C n 2

The PWM generator block, which is the built-in block of Simulink/MATLAB, is used to generate PWM pulses. The triangular carrier signal of the specified carrier frequency is compared with the sinusoidal modulating signal. The modulating signal is shown in Figure 2A. When the modulating signal is greater than the carrier pulse, 1 is high and 0 is low, otherwise. The PWM pulse for the other switch in the same leg is exactly the inverse of pulse 1, as can be seen in the first and second waveform of Figure 2B. For the switches of the second leg of the inverter, the PWM signals are generated by the comparison of the same triangular waveform with the negative of the modulating signal used for arm 1 (180 degrees phase shift).

The transfer function of the continuous time PR controller is given by G s = K p + K i s s 2 + h 2 w 1 2 , (8) where K _p , K _i , and h represent proportional gain, integral gain, and harmonics order, respectively. The double integration-based controller, due to the ease of frequency adaptation, is used for implementation of the PR controller Das and Barai (2021). Figure 3 presents the dual integration-based implementation of the continuous time resonant controller, where the input hw ₁ represents the desired resonant frequency to be tracked Parvez et al. (2016). However, if various frequencies need to be tracked simultaneously, several resonant controllers can be implemented in parallel, and the transfer function of multiple resonant controllers is G c s = K p + ∑ h n h K h s s 2 + h 2 w 1 2 , (9) where n _h is the highest harmonic order and K _h is the integral gain at h ^th harmonic.

The phase and magnitude response of a PR controller (314.16 rad/sec= 50 Hz) is presented in Figure 4. It can be observed that the PR controller ideally provides very high gain at the resonant frequency, i.e., 50 Hz only.

The transfer function of the PR controller, given by (9), can be discretized using various methods, such as ZOH, FOH, II, ZPM, forward Euler (FE), backward Euler (BE), Tustin (T), and Tustin with pre-warping (TP) Yepes et al. (2009). The PR controller is very sensitive to the discretization process due to its narrow band at resonance frequency Yepes et al. (2010). The two integrators in Figure 3, the forward and backward, are commonly discretized by using the Euler forward and backward discretization method Ghoshal and John (2017). Djordje in Stojic (2016) also suggested using modified Tustin or backward Euler simultaneously for discretization of both integrators. However, ZOH, FOH, II, and ZPM methods are simultaneously applied to both integrators. Table 1 shows different discretization methods and their corresponding transfer function used for digital implementation of the forward integrator. The integral part of the PR controller can be designated by R _h (s).

As some of the discretization methods given in Table 1 have same denominators and only the numerators are different, therefore, based on the denominator, the transfer functions used for digital implementation of PR controllers can be classified into five groups given in Table 2 Yepes et al. (2011).

The system described by Figure 1 has been simulated using MATLAB Simulink ^TM . The simulation parameters are given in Table 3. The variable frequency reference signal needs to be measured and provided to the controller in practical implementations. However, in this work, the variable frequency reference signal is directly generated and provided to the controller. Steady-state error and THD of output voltage for non-linear rectifier load are the parameters to be observed. The tracking capability of the conventional deadbeat controller alone and in combination with three types of digital PR controllers is examined in the presence of the variable frequency reference signal.

First of all, steady-state error and THD of the deadbeat controlled single-phase inverter are examined by continuously updating the delay element to the nearest integer value, and then these parameters are investigated for three types of digital PR controllers plugged into the deadbeat controlled inverter. Controllers are initially tuned at 50 Hz nominal frequency, while the frequency of the reference signal is varying between 49 Hz and 51 Hz with a step size of 0.1 Hz. It is observed from Figure 12 and Figure 13 that the deadbeat controller is unable to track the reference signal efficiently because the corresponding error and THD of output voltage are approximately 5 V and 5%, respectively. This is not acceptable for hardware implementation as practical implementation incurs much higher error than simulated work. Other three types of digital PR controllers track the reference signal with lesser THD and relatively small steady-state error. However, the performance of the ZPM-based digital PR controller is more satisfactory (nearly constant) among the three types of digital PR controllers.

The tracking efficiency of the controllers is checked for three selected frequencies of the reference signal.

The PR controller offers the advantage of effective regulation of sinusoidal current and voltage, achieving minimal steady-state errors. However, its direct application to digital systems is not straightforward due to power systems primarily functioning in discrete time. Conventional PR controllers cannot be directly transposed to digital setups. Various discretization methods are available for the digital implementation of PR controllers. Nevertheless, the impact of these methods on the desired controller gain when dealing with varying reference frequencies is not extensively researched. Considering factors such as renewable energy intermittency, distributed generation, and source–load imbalances, the grid frequency slightly deviates around its nominal value. Evaluating how these frequency variations influence the performance of resonant-controlled power converters is essential. This study involves the digital discretization of PR controllers using techniques such as ZOH, II method, ZPM, and deadbeat control. The investigation assesses the performance of these PR controllers under the influence of a variable frequency reference signal, simulating real-world grid conditions. The study focuses on a single-phase PWM inverter, a crucial component of modern power systems.

The primary goal in this research was to identify an efficient digital PR controller capable of accurately tracking both constant and variable reference frequency signals while minimizing THD and achieving near-zero steady-state errors. Simulation results demonstrate that employing only the deadbeat controller yields a steady-state error of 4.9 V and a THD of 4.82%. However, the proposed ZPM-based digital PR controller significantly improves these metrics, reducing the steady-state error to 0.12 V and THD to 0.45%. This outcome highlights the superiority of the proposed method.

The challenge of regulating current and voltage in a variable-frequency inverter is effectively addressed using the PR controller. Notably, the discretization process directly impacts the controller’s poles and zeros. Pole displacement alters the resonant frequency position where extremely high gain is required, while controller zeros impact the stability. Based on the denominator of the controller’s transfer function, discretization methods for digital PR implementation fall into five groups: A, B, C, D, and E. Group E, encompassing ZOH, II, and ZPM techniques, demonstrates satisfactory pole and zero mapping, even with non-integer reference frequencies. Simulation outcomes highlight that the three digital PR controllers from group E perform well, maintaining THD and error below 5%. Among these, the ZPM-based digital PR controller exhibits superior performance. Consequently, the ZPM-based digital PR controller is recommended for practical applications under variable frequency conditions.

The literature shows that the PR controller is capable of tracking sinusoidal signals of known frequency and rejecting disturbances with low steady-state error. Variable frequency reference signals are considered for this research work, and results have shown that the digital PR controllers of group E accurately track the variable frequency reference signal. Rejection of disturbance using PR shows good results as well. However, the disturbance rejection performance is same in the fixed frequency and variable frequency environment. Simulations were carried out on a stand-alone converter with non-linear loads instead of grid-connected converters for safety and other reasons. Due to limited funds and available resources, practical implementation was not possible, and we prepared this manuscript based on stand-alone converter’s simulations. Moreover, the reference frequency variation and range (49 Hz–51 Hz) was kept the same as for the grid-tied converter case to closely match the two systems. Results have not been included in this manuscript to avoid duplication. For implementing a prototype in future work, we need a specialized laboratory setup that uses a controller kit from dSpace for real-time implementation, which is not available due to limited funds. However, we plan to implement it in our future research work by looking at its alternatives.