A frequency-accurate discrete proportional–resonant control strategy for grid simulators

1 Introduction

With the increasing penetration of renewable-energy generation and distributed energy-storage systems, power electronic converters have become key interfaces between distributed energy resources and utility grids. Converter-interfaced equipment must operate reliably under abnormal grid conditions, including voltage sags, frequency deviations, harmonic distortion, phase jumps, and three-phase unbalance. Before such equipment is connected to practical power systems, its grid adaptability and dynamic performance should therefore be evaluated under controllable and repeatable operating conditions. Grid simulators, also referred to as grid emulators, can reproduce prescribed normal and abnormal grid-voltage waveforms and have consequently become important test platforms for renewable-energy and energy-storage converters (Ma et al., 2021; Li et al., 2024). In the present study, energy-storage converters are considered only as representative devices under test; no specific battery, supercapacitor, or other energy-storage technology is modeled. The present validation focuses on the voltage-waveform-generation capability of the grid simulator rather than on the internal dynamics of a particular energy-storage system.

Converter-based grid simulators provide flexible voltage-waveform generation, bidirectional power-flow capability, and high controllability. Representative implementations include back-to-back voltage-source converters, cascaded H-bridge converters, and modular multilevel converters (Eloy-García et al., 2013; Li et al., 2016; Zhang et al., 2024). Reconfigurable power-electronic platforms can also emulate dynamic source and load characteristics for system-level testing (Wang et al., 2016). In an AC–DC–AC grid simulator, the front-end PWM rectifier regulates the common DC-link voltage and enables bidirectional power exchange, whereas the inverter stage synthesizes the prescribed grid-voltage waveform. Three independently controlled single-phase inverter units can further reproduce balanced and unbalanced phase-voltage conditions.

The voltage-control performance of the inverter stage directly affects the waveform-reproduction accuracy of the grid simulator. Proportional–integral control is suitable for regulating DC quantities or variables transformed into a synchronous reference frame, but it does not inherently eliminate the steady-state error when sinusoidal references are tracked directly in the stationary frame. A proportional–resonant controller provides high gain at a selected resonant frequency according to the internal-model principle and is therefore well suited to stationary-frame AC-voltage and current regulation (Zmood and Holmes, 2003; Teodorescu et al., 2006; Ciobotaru et al., 2006).

Although PR controllers are generally designed in the continuous-time domain, practical implementation requires digital discretization. The selected discretization method determines the pole–zero mapping of the resonant term in the z-domain. A deviation in the resonant-pole angle shifts the mapped discrete resonant frequency away from its specified continuous-domain value, while differences in the numerator polynomial alter the zero locations and off-resonant phase characteristics. Digital PR implementations should therefore be evaluated in terms of pole location, resonant-frequency preservation, phase response, and closed-loop performance (Yepes et al., 2010; Lin et al., 2022; Ijaz et al., 2023).

Resonant controllers have been applied to grid simulators for reproducing voltage sags, unbalanced voltages, harmonics, flicker, and frequency variations (Eloy-García et al., 2013). Modified-Tustin and frequency-prewarped implementations have also been investigated to reduce digital resonant-frequency errors (Stojic, 2016), and recent studies have extended discretization comparisons to PR-controlled three-phase converters (Nguyen, 2025). However, the relationship between resonant-frequency mapping accuracy and the actual closed-loop waveform-reproduction performance of a grid simulator has received limited quantitative attention under identical controller gains, power-stage parameters, and sampling conditions.

Standard Tustin discretization is widely used because of its bilinear mapping and straightforward implementation. Its nonlinear frequency mapping, however, introduces a deviation between the specified continuous resonant frequency and the mapped discrete resonant frequency. Frequency pre-warping eliminates this deviation at the selected design frequency. When the resonant frequency is much lower than the controller sampling frequency, the standard-Tustin mapping error may already be very small, and exact frequency preservation may therefore have only a limited influence on the practical closed-loop response. A controlled comparison under identical operating conditions is required to quantify this influence.

This study presents an application-oriented evaluation and implementation procedure for discrete PR voltage control in the inverter stage of an AC–DC–AC grid simulator. The main contributions are as follows:

• Several commonly used discretization methods are compared in terms of resonant-pole mapping, resonant-frequency preservation, and off-resonant phase characteristics. The effects of denominator-based pole mapping and numerator-dependent zero locations are distinguished.
• Tustin discretization with frequency pre-warping is implemented in a dual-loop voltage-control structure and compared with standard Tustin discretization. The corresponding resonant terms, R tp z and R t z, and their numerical coefficients are provided. Both controllers use identical proportional and resonant gains, so the comparison isolates the influence of frequency pre-warping.
• The two implementations are evaluated under identical power-stage and control conditions. The assessment includes nominal closed-loop stability, limited parameter sensitivity, rated steady-state tracking, low-order and wideband voltage THD, sinusoidal tracking from 45 to 55 Hz, and voltage-sag response. Three-phase unbalanced-voltage reproduction is additionally demonstrated as a functional capability of the independently controlled inverter units.

2 Mathematical model of the grid simulator

2.1 AC–DC–AC grid-simulator topology

The AC–DC–AC grid-simulator topology considered in this study is shown in Figure 1. The utility grid is connected to a three-phase PWM rectifier through a grid-side filter, where u g and i g denote the three-phase grid voltage and grid current, respectively. The PWM rectifier regulates the common DC-link voltage U d c and enables bidirectional active-power exchange between the utility grid and the DC link. Because the present study focuses on inverter-side voltage control, the regulated DC-link voltage is treated as the DC input of the inverter stage in the subsequent modeling and simulation analysis.

The inverter stage consists of three independently controlled single-phase H-bridge inverter units corresponding to phases A, B, and C. Each inverter unit generates its prescribed phase voltage through an LC output filter. Under balanced operation, the three phase-voltage references have equal amplitudes and a common frequency, with a phase displacement of 120 °. Owing to the independent control of the three inverter units, the amplitude and phase of each phase voltage can also be adjusted separately, allowing balanced and unbalanced grid-voltage conditions to be reproduced. The following analysis therefore focuses on the mathematical model and voltage-control structure of one representative single-phase inverter unit.

2.2 Single-phase inverter model

The single-phase full-bridge inverter and its LC output filter are shown in Figure 2. The common DC-link voltage is denoted by U d c, and C d is the DC-link capacitor. The inductance L and capacitance C form the output low-pass filter, while r represents the equivalent series resistance of the inductor branch. The parameter r is retained in the generalized averaged equations to represent a possible equivalent series resistance of the output-filter inductor. In the implemented switching-level model, however, the 46 μ H output-filter inductor is represented by a pure L branch without a series-resistance parameter. The averaged bridge-output voltage and the filtered output voltage are denoted by u i and u o, respectively. The variables i L, i C, and i o denote the filter-inductor current, filter-capacitor current, and output current.

Applying Kirchhoff’s voltage and current laws to the averaged inverter model gives Equation 1.

L d i L t d t = u i t − u o t − r i L t C d u o t d t = i L t − i o t (1)

After applying the Laplace transform, the corresponding relationships are given in Equation 2.

U i s − U o s = L s + r I L s I L s − I o s = C s U o s (2)

Eliminating I L s from Equation 2 yields Equation 3

L C s 2 + r C s + 1 U o s = U i s − L s + r I o s (3)

Therefore, the output voltage can be expressed by Equation 4, where G_p(s) and Z_o(s) are defined in Equation 5.where

U o s = G p s U i s − Z o s I o s (4)

G p s = 1 L C s 2 + r C s + 1 Z o s = L s + r L C s 2 + r C s + 1 (5)

Here, G p s represents the control-to-output dynamics of the LC-filtered inverter, whereas Z o s describes the influence of the output-current disturbance on the output voltage. Because i o depends on the connected load or device under test, it is treated as an external disturbance in the voltage-control design. Within the linear modulation range, the switching bridge is represented by the equivalent PWM gain K PWM, which is included in the voltage-control structure introduced in the following section.

3 PR voltage control and digital implementation

3.1 Dual-loop voltage-control structure

The dual-loop voltage-control structure adopted for each single-phase inverter unit is shown in Figure 3. The outer voltage loop regulates the output voltage, while the inner loop uses the filter-inductor current to shape the current dynamics. Load-current feedforward is included in the current-reference path so that changes in the output current can be compensated without relying solely on the voltage-feedback loop.

Inductor-current feedback is adopted to provide direct control of the filter-inductor current, while measured load-current feedforward is introduced to reduce the equivalent output impedance and improve load-disturbance rejection (Deng et al., 2008; Lin et al., 2021). The outer voltage loop is implemented using a PR controller to achieve accurate sinusoidal-voltage tracking at the designed fundamental frequency (Zmood and Holmes, 2003; Teodorescu et al., 2006).

The output-voltage error is defined by Equation 6.

E v s = U ref s − U o s (6)

The outer voltage controller G v s generates the current reference according to Equation 7.

I ref s = G v s E v s (7)

As illustrated in Figure 3, the measured load current is added to the current-reference path through the feedforward coefficient K ff, while the measured inductor current is subtracted through the feedback coefficient K if. The input of the inner current-control element is therefore E i s = I ref s + K ff I o s − K if I L s.

The corresponding control signal is given by Equation 8.where G i s denotes the inner current-control element. In the implemented model, no additional dynamic inner current controller is used; equivalently, G i s = 1. The output of the outer PR controller is combined algebraically with load-current feedforward and inductor-current feedback. The load-current feedforward and inductor-current feedback coefficients are K ff = K if = 1 / 200 = 0.005. The resulting modulation command is limited to [− 1.2 , 1.2 and compared with a unit-peak bipolar triangular carrier at 5 kHz. Within the linear modulation range, the equivalent averaged full-bridge gain is K PWM = U d c. No separate 1 / U d c normalization block is implemented in the switching-level model.

U c s = G i s E i s , G i s = 1 (8)

To achieve zero steady-state error for a sinusoidal reference at the selected fundamental frequency, the outer-loop controller is implemented as an ideal proportional–resonant controller, as shown in Equation 9.The resonant term R(s) is defined in Equation 10.

G v s = G PR s = K p + K r R s (9)

R s = s s 2 + ω 0 2 (10)

Here, K p is the proportional gain, K r is the resonant gain, and ω 0 is the designed resonant angular frequency. Under the ideal linear model, the resonant term provides theoretically infinite gain at ω = ω 0 and eliminates the fundamental-frequency steady-state tracking error. In the switching-level system, the remaining instantaneous waveform error also contains switching ripple and harmonic components.

Because the controller is executed digitally, the continuous resonant term R s must be converted into a discrete form. Different discretization methods produce different pole locations, frequency mappings, and phase characteristics. These differences are examined in the following section before the final digital implementation is selected.

3.2 Comparison of PR discretization methods

The continuous resonant term must be discretized before digital implementation. Different s-to-z mappings produce different resonant-pole locations, mapped resonant frequencies, and zero-dependent phase characteristics. Table 1 summarizes the discrete resonant terms considered in this study based on established resonant-controller discretization analyses (Yepes et al., 2010; Ijaz et al., 2023). Previous studies have shown that inaccurate pole mapping can shift the resonant peak, whereas differences in the numerator and zero locations can modify off-resonant behavior and closed-loop design (Yepes et al., 2010; Lin et al., 2022).

TABLE 1. Discrete resonant-term expressions obtained using different discretization methods.

For convenience, the normalized resonant angle is defined by Equation 11.where T s is the controller sampling period. A general second-order discrete resonant term can be expressed by Equation 12.where the subscript X denotes the discretization method. The poles of R X z are determined by Equation 13.

θ 0 = ω 0 T s (11)

R X z = b 0 , X + b 1 , X z − 1 + b 2 , X z − 2 1 + a 1 , X z − 1 + a 2 , X z − 2 (12)

1 + a 1 , X z − 1 + a 2 , X z − 2 = 0 (13)

Figure 4 compares the resonant-pole loci of the five discretization groups at selected resonant frequencies of 50, 150, 250, 350, 450, 550, 650, 750, and 850Hz, with the controller sampling frequency fixed at 10 kHz. These frequencies are used only as representative evaluation points for illustrating the frequency-dependent discretization characteristics. Group A represents forward Euler discretization, Group B represents backward Euler discretization, Group C represents standard Tustin discretization, Group D represents forward–backward Euler discretization, and Group E comprises zero-order hold, Tustin discretization with frequency pre-warping, and impulse-invariant discretization.

The forward-Euler poles are located outside the unit circle over the investigated range, whereas the backward-Euler poles are mapped inside the unit circle and therefore introduce numerical damping. The poles obtained using standard Tustin and forward–backward Euler remain on the unit circle, but their angular positions do not generally coincide with the specified resonant frequencies. The Group-E methods produce the pole pair given in Equation 14.and therefore preserve the specified resonant pole angle exactly.

z p , 1 , 2 = e ± j θ 0 (14)

The resonant frequency represented by a discrete pole z p can be evaluated using Equation 15.

ω ^ r = ∠ z p T s (15)

For the standard Tustin transformation, the mapped resonant frequency is given by Equation 16.which differs slightly from ω 0 because of the nonlinear frequency mapping of the bilinear transformation. By contrast, frequency pre-warping gives the exact mapping shown in Equation 17.

ω ^ r , t = 2 T s tan − 1 ω 0 T s 2 (16)

ω ^ r , tp = ω 0 (17)

In Figure 5, f set denotes the specified resonant frequency, f ^ r denotes the corresponding mapped discrete resonant frequency, and Δ f = f ^ r − f set denotes the frequency-mapping error. Figure 5 compares the specified and mapped resonant frequencies of the representative discretization groups at a controller sampling frequency of 10 kHz. Standard Tustin discretization maps the specified resonant frequency to a slightly lower value, whereas forward–backward Euler discretization produces a slightly higher mapped frequency. In contrast, the Group E methods preserve the specified resonant frequency exactly. The frequency-mapping deviations of Groups C and D increase with the specified resonant frequency. At the designed resonant frequency of 50 Hz, the corresponding mapping errors are − 0.004112 Hz for standard Tustin discretization, + 0.002056 Hz for forward–backward Euler discretization, and 0 Hz for the Group E methods. These errors remain very small at the fundamental operating frequency because the resonant frequency is substantially lower than the controller sampling frequency.

Although the Group-E methods preserve the same resonant-pole angle, their numerator polynomials and zero locations are different. Consequently, identical pole mappings do not necessarily produce identical off-resonant magnitude and phase characteristics. Figure 6 compares the frequency responses of the continuous resonant term and the Group-E discrete implementations at a representative resonant frequency of 350 Hz and a controller sampling frequency of 10 kHz. This representative frequency is selected only to make the differences more visible and does not correspond to the fundamental operating frequency of the grid simulator.

Standard Tustin discretization is attractive because of its bilinear mapping and straightforward implementation, but its nonlinear frequency mapping can shift the discrete resonant frequency. Frequency pre-warping compensates for this mapping error at the selected design frequency and is therefore appropriate when exact resonant-frequency preservation is required (Yepes et al., 2010; Stojic, 2016). Based on the pole-mapping, frequency-preservation, and phase-response comparisons presented in Figures 4–6, Tustin discretization with frequency pre-warping is selected for the implemented controller, while standard Tustin discretization is retained as the comparison baseline. The corresponding discrete resonant terms, R tp z and R t z are specified in Section 3.3.

3.3 Implemented discrete PR controllers

The two controllers used in the subsequent comparison have the same proportional and resonant gains and differ only in the discretization of the resonant term. Their common implementation is given by Equation 18.where R tp z denotes the resonant term obtained using Tustin with frequency pre-warping, and R t z denotes the resonant term obtained using standard Tustin discretization. The proportional and resonant branches are implemented separately, so the coefficients reported in Table 2 correspond only to R X z.

G PR , X z = K p + K r R X z , X ∈ tp , t (18)

TABLE 2. Numerical coefficients of the implemented discrete resonant terms.

The coefficients correspond to the resonant term R X z at f 0 = 50 Hz and T s = 100 μ s. Both controllers use K p = 0.07 and K r = 50, implemented in separate proportional and resonant branches.

For the pre-warped Tustin implementation, the resonant term is given by Equation 19.

R tp z = sin ω 0 T s 2 ω 0 1 − z − 2 1 − 2 ⁡ cos ω 0 T s z − 1 + z − 2 (19)

For the standard Tustin implementation, substituting s = 2 T s 1 − z − 1 1 + z − 1 into the continuous resonant term gives Equation 20.

R t z = 2 T s 1 − z − 2 4 + ω 0 2 T s 2 + 2 ω 0 2 T s 2 − 8 z − 1 + 4 + ω 0 2 T s 2 z − 2 (20)

Both controllers use K p = 0.07 , K r = 50 with the designed resonant frequency and sampling conditions f 0 = 50 Hz , ω 0 = 2 π f 0 , f s = 10 kHz , T s = 100 μ s. The common controller gains were selected under the nominal switching-level operating condition to obtain stable 50 Hz sinusoidal tracking with acceptable waveform quality. The gains were not optimized separately for the two discretization methods and were kept fixed in all subsequent controller comparisons and parameter-variation tests.

The implemented discrete resonant terms are expressed in the common form given by Equation 21.and their numerical coefficients are listed in Table 2. The coefficient differences between the two implementations are very small because the designed resonant frequency is much lower than the controller sampling frequency.

R X z = b 0 , X + b 1 , X z − 1 + b 2 , X z − 2 1 + a 1 , X z − 1 + a 2 , X z − 2 (21)

For the pre-warped implementation, the resonant poles are located at z p , 1 , 2 = e ± j ω 0 T s and the mapped resonant frequency is therefore exactly f ^ r , tp = 50.0000000 Hz.

For the standard Tustin implementation, the mapped resonant frequency is given by Equation 22.The corresponding frequency-mapping error is given by Equation 23.

f ^ r , t = 1 π T s tan − 1 ω 0 T s 2 = 49.9958888 Hz (22)

Δ f t = f ^ r , t − f 0 = − 0.004112 Hz (23)

This small deviation indicates that the two controllers are expected to exhibit closely matched behavior around the rated 50 Hz operating point. Their comparison is therefore used to quantify the practical influence of exact resonant-frequency mapping rather than to assume a substantial performance difference in advance.

4 Simulation results and discussion

4.1 Simulation setup and stability assessment

A switching-level simulation model of the AC–DC–AC grid simulator was established in MATLAB/Simulink. The two investigated controllers use the same power circuit, dual-loop control structure, controller gains, sampling frequency, and PWM switching frequency. They differ only in the discrete implementation of the resonant term. For conciseness, the Tustin PR controller with frequency pre-warping is denoted as TP-PR, whereas the standard Tustin PR controller is denoted as T-PR. The main electrical and control parameters are listed in Table 3. All switching-level results presented in Figures 7–10 and Table 4 were generated using the final parameter values reported in Table 3. In both the TP-PR and T-PR models, the output-filter inductance was implemented as a pure 46 μ H L branch without a series-resistance parameter. The nominal single-phase load power is P 0 = U o , rms 2 R 0 ≈ 166.9 kW. In the load-level sensitivity analysis, the load resistance was varied according to R load = R 0 / P out / P 0. The controller sampling frequency was set to 10 kHz, which is 200 times the 50 Hz fundamental frequency and twice the 5 kHz PWM switching frequency. The same sampling and switching frequencies were used for both controllers so that their comparison isolates the effect of resonant-term discretization. The switching bridge was represented by ideal controllable semiconductor switches.

TABLE 3. Main electrical and control parameters of the simulated grid simulator.

TABLE 4. Quantitative performance comparison of the two discrete PR controllers.

THD 2 − 20 and THD 2 − 200 are calculated using the 2nd–20th and 2nd–200th harmonics, respectively. max ∣ φ ∣ is evaluated at the tested frequencies from 45 to 55 Hz. E 1 cyc denotes the relative deviation of the forward one-cycle output-voltage RMS over the first complete post-transition cycle from the final target RMS voltage. It is an offline cycle-level indicator rather than a causal settling-time measure.

The closed-loop stability assessment was performed using a reduced averaged model corresponding to the implemented current-feedback, load-current-feedforward, and PWM structure. This model is used to evaluate the influence of the discrete PR implementation and the selected parameter variations; it is not intended to reproduce the switching and saturation nonlinearities of the detailed time-domain model.

For a resistive load R load, the equivalent continuous-time plant from the PR-controller output to the output voltage can be written as Equation 24.

P ad s = K PWM L C s 2 + L R load + r C + K PWM K if C s + 1 + r R load + K PWM K if − K ff R load (24)

The expression retains r only to preserve the general form of the averaged plant. The audited TP-PR and T-PR switching-level models both use a pure 46 μ H output-filter L branch without a series-resistance parameter. Accordingly, r = 0 was used consistently in both the switching-level simulations and the reduced averaged pole assessment. All pole-magnitude results reported in Table 5 correspond to this stated reduced-model assumption.

TABLE 5. Closed-loop pole magnitudes under baseline and parameter-varied conditions.

Each parameter was varied independently while all other parameters and controller coefficients were held at their baseline values in the reduced averaged model. The output-filter inductance was represented by the same pure L branch, corresponding to r = 0, in both the switching-level simulations and the reduced averaged pole assessment. All 40 evaluated closed-loop cases satisfied ρ max < 1.

The reduced averaged plant was discretized by the zero-order-hold method at T s = 100 μ s: P ad z = N P z D P z.

The discrete PR controller is written as shown in Equation 25.the closed-loop characteristic polynomial is obtained as D cl , X z = D G , X z D P z + N G , X z N P z.

G PR , X Z = N G , X Z D G , X Z , X ∈ tp , t (25)

The closed-loop poles are the roots of D cl , X z = 0, and stability is evaluated using their maximum magnitude.

The maximum closed-loop pole magnitude is defined by Equation 26.where z i denotes the ith closed-loop pole. The discrete closed-loop system is asymptotically stable when ρ max ⁡ , X < 1.

ρ max ⁡ , X = max i z i , X (26)

Under the baseline condition of the reduced averaged model, the maximum closed-loop pole magnitudes of the TP-PR and T-PR systems are 0.980349588 and 0.980351156, respectively. Both values are below unity, confirming baseline stability of the reduced averaged models. The difference between the two values is small, which is consistent with the small difference between their discrete resonant coefficients at the rated 50 Hz operating point.

A limited parameter-sensitivity analysis was then conducted using the single-parameter variation method. The filter inductance, filter capacitance, DC-link voltage, and output load level were independently varied from 0.8 to 1.2 p.u. At each operating point, all remaining circuit parameters and controller coefficients were maintained at their nominal values. The evaluated values were specified according to Equation 27.where x denotes the parameter being varied and x 0 denotes its nominal value. The resulting maximum pole-magnitude ranges are summarized in Table 5.

x x 0 = 0.8 , 0.9 , 1.0 , 1.1 , 1.2 (27)

The largest pole magnitude observed over all evaluated cases was 0.980594615, obtained for the T-PR system during the DC-link-voltage scan. Therefore, all evaluated closed-loop poles remained inside the unit circle throughout the investigated parameter ranges. The TP-PR and T-PR systems also exhibited closely matched pole-magnitude ranges, indicating that frequency pre-warping does not produce a meaningful change in the assessed closed-loop stability characteristics under the considered operating conditions.

4.2 Rated steady-state and frequency-range tracking performance

The steady-state interval from 0.06 to 0.14 s was selected for analysis. This interval contains four complete 50 Hz cycles. The TP-PR and T-PR waveforms were interpolated onto a common grid containing 80,000 uniformly spaced samples, corresponding to an evaluation sampling frequency of 1 MHz. Because the selected interval contains an integer number of fundamental cycles, a rectangular window was used without spectral leakage at the harmonic frequencies. The resulting frequency resolution was 12.5 Hz.

The normalized RMS tracking error is defined by Equation 28.

E RMS = 1 N ∑ k = 1 N u ref k − u o k 2 U ref , rms × 100 % (28)

Here, E RMS is the RMS value of the instantaneous waveform-tracking error normalized by the reference RMS voltage. It is not the relative difference between the RMS magnitudes of the output and reference voltages. N is the number of samples and U ref , rms is the RMS value of the reference voltage.

The voltage THD calculated from the second to the H th harmonic is defined by Equation 29.where U 1 is the RMS value of the fundamental component and U h is the RMS value of the hth harmonic. Two frequency ranges were evaluated. The low-order index THD 2 − 20 includes the 2nd–20th harmonics, whereas the wideband index THD 2 − 200 extends the analysis to 10 kHz and therefore covers the vicinity of the 5 kHz switching frequency.

THD 2 − H = ∑ h = 2 H U h 2 U 1 × 100 % (29)

Figure 7 shows that the TP-PR and T-PR output waveforms and spectra nearly overlap at the rated 50 Hz operating point. The quantitative results in Table 4 confirm that the two implementations exhibit nearly identical normalized RMS tracking errors and voltage THD values.

To evaluate the closed-loop tracking characteristics over a wider frequency range, the reference frequency was successively set to 45, 47.5, 50, 52.5, 55 Hz.

The designed resonant frequency of both controllers remained fixed at 50 Hz throughout the test, and all other circuit and controller parameters were maintained at their nominal values. At each reference frequency, the steady-state reference and output voltages were fitted to the sinusoidal function given in Equation 30.

u t = A ⁡ sin 2 π f t + ϕ (30)

The closed-loop magnitude and phase responses were then calculated as M f = 20 ⁡ log 10 A o f A ref f and φ f ≡ ϕ o f − ϕ ref f.

As shown in Figure 8, the magnitude deviations of both controllers remain below 0.016 dB, and their absolute phase deviations remain below 0.073 ∘ over the tested 45–55 Hz range. The two response curves are nearly indistinguishable.

The maximum magnitude and phase differences between the two controllers over the tested frequencies are only 0.00017 dB and 0.00090 °, respectively. These results demonstrate that both implementations provide closely matched closed-loop tracking performance from 45 to 55 Hz.

Frequency pre-warping ensures that the resonant pole angle corresponds exactly to the specified 50 Hz resonant frequency. However, because the standard-Tustin frequency-mapping error is only −0.004112 Hz under the adopted sampling conditions, this exact mapping does not result in a substantial improvement in the observed closed-loop tracking performance. The practical contribution of pre-warping in the present operating range is therefore frequency-mapping accuracy rather than a pronounced enhancement in steady-state voltage quality or closed-loop dynamic response.

4.3 Voltage-sag response and unbalanced-voltage reproduction

The transient tracking capability of the two discrete PR controllers was evaluated using a voltage-sag test. The phase-voltage reference was reduced from 100% to 40% of the rated value, corresponding approximately to a change from 220 to 88 V RMS. The implemented sinusoidal reference used a peak value of 311 V. To avoid introducing an ideal discontinuity, the reference amplitude was changed linearly over 1 ms, from t = 0.150 s to t = 0.151 s. All circuit parameters and controller coefficients were maintained at their nominal values.

The instantaneous voltage-tracking error is defined as e u t = u ref t − u o t .

The maximum normalized tracking error is defined by Equation 31.where U rated , pk = 2 U rated , rms .

E max ⁡ , sag = max 0.150 ≤ t ≤ 0.171 e u t U rated , pk × 100 % (31)

To characterize the RMS-voltage deviation of the first complete cycle following the commanded amplitude transition, a forward one-cycle RMS value of the output voltage is defined by Equation 32.

U o , rms fw t = 1 T 0 ∫ t t + T 0 u o 2 τ d τ , T 0 = 1 f 0 (32)

The RMS-voltage deviation over the first complete fundamental cycle following the commanded amplitude transition is defined by Equation 33.where t tr = 0.151 s and U final , rms = 0.4 × 311 2 = 87.9641 V .

E 1 cyc = U o , rms fw t tr − U final , rms U final , rms × 100 % (33)

Because U o , rms f w t tr is calculated using the complete waveform over the interval from t tr to t tr + T 0, E 1 cyc is an offline cycle-level voltage-deviation indicator and is not interpreted as a causal settling or physical recovery time.

Figure 9 shows the voltage-sag responses of the two controllers. Their reference signals are identical, and the TP-PR and T-PR output voltages nearly overlap throughout the transition. The maximum normalized tracking errors are 5.5929% for TP-PR and 5.5927% for T-PR.

The forward one-cycle RMS output voltages evaluated over the first complete post-transition cycle are 88.0453 V for TP-PR and 88.0402 V for T-PR. Relative to the final target RMS voltage of 87.9641 V, the corresponding first-cycle RMS deviations are 0.0924% and 0.0866%, respectively. Both values are below the adopted 2% voltage-tolerance level. This cycle-level offline indicator confirms that the first complete post-transition voltage cycle satisfies the prescribed RMS-voltage tolerance, but it is not interpreted as a sub-millisecond physical settling time.

The capability of the grid simulator to reproduce an unbalanced three-phase voltage condition was then examined using the TP-PR controller. Because the three single-phase inverter units are independently controlled, different phase-voltage references can be assigned without modifying the individual voltage-control structures. In this test, the three-phase reference contained a 10% negative-sequence fundamental component and an 8% zero-sequence fundamental component in addition to the positive-sequence component.

The A-phase initial angles of the positive-, negative-, and zero-sequence components were all set to 0 ∘. The negative-sequence component used the reverse phase order, whereas the zero-sequence components of the three phases were in phase. With a positive-sequence phase-voltage peak of 311 V, the negative- and zero-sequence peaks were 31.10 and 24.88 V, respectively. The resulting phase-reference peaks were approximately 366.98, 283.06, and 283.06 V for phases A, B, and C, respectively, with corresponding fundamental phase angles of approximately 0 ∘, − 121.09 ∘, and 121.09 ∘.

As shown in Figure 10, the three output phase voltages follow the prescribed unbalanced references, including the unequal amplitudes and modified phase relationships produced by the negative- and zero-sequence components. This test is included as a functional demonstration of sequence-component reproduction by the independently controlled inverter units rather than as a comparison between the TP-PR and T-PR implementations.

5 Conclusion

This study investigated the digital implementation of proportional–resonant voltage control for the inverter stage of an AC–DC–AC grid simulator. Several established discretization methods were compared in terms of resonant-pole location, resonant-frequency mapping, and off-resonant phase characteristics. Zero-order hold, impulse-invariant discretization, and Tustin discretization with frequency pre-warping preserve the specified resonant pole angle, whereas standard Tustin discretization introduces nonlinear frequency mapping.

Tustin discretization with frequency pre-warping was selected for the frequency-accurate implementation and was compared with standard Tustin discretization using identical proportional and resonant gains. At a designed resonant frequency of 50 Hz and a controller sampling frequency of 10 kHz, the TP-PR implementation preserves the specified resonant frequency, whereas the T-PR implementation produces a mapped resonant frequency of 49.995888 Hz, corresponding to a deviation of − 0.004112 Hz.

The corresponding reduced averaged closed-loop models remained stable under the baseline condition and under independent variations of the filter inductance, filter capacitance, DC-link voltage, and load level from 0.8 to 1.2 p.u. All 40 evaluated cases satisfied ρ max < 1. Switching-level simulations further showed that the TP-PR and T-PR implementations provide nearly identical rated steady-state tracking, low-order and wideband voltage THD, sinusoidal tracking over the investigated 45–55 Hz range, and voltage-sag response. The independently controlled inverter units also reproduced the prescribed three-phase unbalanced-voltage condition.

The present validation is limited to a detailed MATLAB/Simulink switching-level model. Experimental and hardware-in-the-loop verification has not yet been performed. Practical implementation effects, including controller-coefficient quantization, computation and PWM-update timing, dead time, measurement noise, DC-link-voltage fluctuation, and switching-device nonlinearities, are not represented in the present results. Future work will evaluate these effects through real-time hardware-in-the-loop and prototype experiments.

The results indicate that frequency pre-warping provides exact and predictable resonant-frequency mapping. Under the investigated 50 Hz operating condition, however, the standard-Tustin mapping error is already very small because the resonant frequency is much lower than the controller sampling frequency. Consequently, the exact mapping of the TP-PR implementation does not produce a substantial change in the evaluated closed-loop waveform quality or transient response. Its practical significance in the present grid-simulator application is therefore the elimination of the discretization-induced resonant-frequency deviation and the resulting consistency between the specified and implemented resonant frequencies. These conclusions are limited to the adopted controller gains, sampling conditions, power-stage parameters, and investigated operating ranges.

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Footnotes

Edited by: Aleksandra Rybak, Silesian University of Technology, Poland

Reviewed by: Fateh Krim, University Ferhat Abbas of Setif, Algeria

Anandita Chowdhury, Sardar Vallabhbhai National Institute of Technology Surat, India