Non-singular Fast Terminal Sliding Mode Control Integrated with Proportional Multi-Resonant-Based Controller for Multifunctional Grid-Tied LCL-Filtered Inverter

The utilisation of voltage source inverters (VSIs) in grid-tied inverter (GTI) applications has become prevalent, driven by the increasing demand for sustainable energy sources. However, the presence of harmonic currents poses a threat to the efficacy of energy generation, transmission and utilisation. To address these issues, an advanced power electronic device known as an active power filter (APF) has been developed to generate compensating currents to mitigate the harmful effects of harmonic currents on the electric power system (Dehghani et al., 2020; Trinh and Lee, 2012).

Regarding APF, the primary objective of the control algorithm is to determine the reference compensation currents that are subsequently introduced into the grid at the point of APF connection (Maciążek, 2022). In an optimal scenario, the source current transforms into a sinusoidal active current while all adverse components are confined within the APF-load circuit. The high level of dynamics necessitated by this process typically entails time-domain simulation (Bosch et al., 2017; Sozanski and Szczesniak, 2023), although frequency-domain control algorithms can also be employed (Alali et al., 2019; Lou et al., 2022).

To meet required standards and establish sinusoidal currents between the inverter and the grid, implementing a filter is essential to mitigate the effects of PWM (Pulse Width Modulation) switching harmonics (Lou et al., 2022, IEEE Std 519-1992). Compared to L filter, LCL filter offers numerous advantages, including superior attenuation capabilities at lower switching frequencies due to their smaller inductance size (Mondal and Karuppaswamy, 2024; Satpathy and De, 2024; Zheng et al., 2018). However, LCL filters present inherent complexities from a control perspective. Resonance issues between filter components can introduce complex conjugate poles, potentially compromising system stability. Therefore, identifying an appropriate control strategy to ensure desirable system dynamics is critical.

To address resonance issues in the LCL filter, two general approaches have been employed: passive damping and active damping. Passive damping places low-value resistors within the LCL filter, a straightforward but less efficient solution (Liu et al., 2019). In contrast, active damping utilises virtual resistors and requires sensors for voltage or current measurement. Although active damping is more reliable than passive damping, it becomes unstable due to grid parameters variations and fluctuating resonance frequencies (Huang et al., 2023; Kołek and Firlit, 2021; Lorzadeh et al., 2016).

Different current control techniques utilising harmonic extraction as input demonstrate distinct approaches, each with its own advantages and limitations. The selection of an appropriate technique depends on factors such as the load’s characteristics, robustness and ease of implementation.

Hou et al. (2024) described a methodology for identifying the most efficient control mechanism for a new controller. Their study provides a clear resolution to the Karush–Kuhn–Tucker problem and simplifies computations to reduce complexity. However, it still requires numerous sensors and involves certain intricacies.

Sozanski and Szczesniak (2023) presented a novel approach for controlling a shunt APF using the sliding discrete Fourier transform (SDFT) technique. This method leverages the conventional SDFT algorithm while innovatively switching between two parallel SDFT algorithms. Although this technique requires a substantial number of mathematical operations, it can still be implemented using readily available microcontrollers.

Intelligent current techniques (ICT) rely on expert knowledge to extract information from signal processing (Boopathi and Indragandhi, 2023; Chen et al., 2022). To enhance the performance of APF for harmonic compensation, Hou et al. (2024) proposed a control strategy for APF reference compensation current based on a Recurrent Wavelet Fuzzy Neural Network (RWFNN). Implementing ICT requires a deep understanding of the functional and behavioural aspects of APF systems.

Linear controllers, such as proportional-integral (PI) controllers, are frequently used to regulate the current loop of an LCL GTI (Cha et al., 2009; Kołek and Firlit, 2021). These controllers are widely adopted due to their simplicity and effectiveness. However, they come with limitations, notably instability issues and challenges in reference tracking (Trinh and Lee, 2012). Proportional-resonant (PR) controllers offer significant improvements in reference tracking and can address some shortcomings of conventional PI controllers (Alathamneh et al., 2022; Kołek and Firlit, 2021). However, PR controllers only compensate for a single frequency (Avci and Ucar, 2020; Trinh and Lee, 2012). To overcome these limitations, proportional multi-resonant (PMR) controllers are employed. A PMR controller combines a proportional controller, a resonant controller tuned to the fundamental frequency and additional resonant controller to mitigate all harmonics, including those caused by non-linear loads (Hao et al., 2012; Utkin, 1978). Non-linear controllers, particularly sliding mode controllers (SMCs), exhibit desirable characteristics such as fast dynamic response and robustness to parameter variations when compared to conventional controllers (Guzman et al., 2016; Huang et al., 2019; Rabbani et al., 2023). The sliding surface of the LCL GTI typically involves a linear combination of variables encompassing the inverter current error, capacitor voltage error, grid current error and their respective integrals or derivatives (Altin et al., 2018; Ammar et al., 2024; Boukattaya et al., 2018).

In this study, an SMC strategy in the natural frame is introduced for a three-phase grid-tied system with LCL filters. The derivation of the sliding surface function is based on the error of the capacitor voltage and its derivative. Due to the inherent active damping effect of capacitor voltage measurement on the system, resonance damping occurs automatically, eliminating the need for a virtual resistor in the closed-loop system. The generation of the capacitor voltage reference is facilitated by a PMR controller, ensuring zero steady-state error in both the fundamental and harmonic components of the grid current. Furthermore, an interconnected non-linear load is linked to the grid, enabling the inverter to function as an APF. The primary advantage of this methodology is its intrinsic ability to achieve sinusoidal grid currents with low total harmonic distortion (THD) without the necessity of compensating for grid harmonics to obtain a sine wave output with minimal harmonic distortion (Hao et al., 2012). The novelties of the proposed method are as follows: -

Developing NFTSMC for controlling a three-phase GTI with an LCL filter, demonstrating the suitability of NFTSMC for systems with inherent uncertainties and external disturbances, and ensuring robust and precise control.

The rest of the paper is organised as follows. Section 2 presents a brief modelling of the GTI with LCL filter. The NTFSMC and PMR controllers are introduced and designed in Sections 3 and 4. Simulation results are discussed in Section 5, and the paper is concluded in Section 6.

Figure 1 illustrates the configuration of a three-phase VSI equipped with an LCL filter. The LCL filter consists of the inverter-side inductor L₁, the filter capacitor C and the grid-side inductor L₂. The resistances r₁ and r₂ represent the parasitic series resistances of the inductors L₁ and L₂, respectively. The VSI comprises six switching devices, which are governed by a controller-generated signal. The state-space equations for the VSI can be derived from the circuit as shown below: (1) L1di1dt+r1i1=ΓukVdc−vc, {L_1}{{d{i_1}} \over {dt}} + {r_1}{i_1} = \Gamma {u_k}{V_{dc}} - {v_c}, (2) L2di2dt+r2i2=vc−vgk, {L_2}{{d{i_2}} \over {dt}} + {r_2}{i_2} = {v_c} - {v_{gk}}, (3) Cdvcdt=i1−i2. C{{d{v_c}} \over {dt}} = {i_1} - {i_2}. where Γ=2−1−1−12−1−1−12, i1=i1ai1bi1cT, i2=i2ai2bi2cT, vc=vcavcbvccT \Gamma = \left[ {\matrix{ 2 & { - 1} & { - 1} \cr { - 1} & 2 & { - 1} \cr { - 1} & { - 1} & 2 \cr } } \right],\;{i_1} = {\left[ {\matrix{ {{i_{1a}}} & {{i_{1b}}} & {{i_{1c}}} \cr } } \right]^T},\;{i_2} = {\left[ {\matrix{ {{i_{2a}}} & {{i_{2b}}} & {{i_{2c}}} \cr } } \right]^T},\;{v_c} = {\left[ {\matrix{ {{v_{ca}}} & {{v_{cb}}} & {{v_{cc}}} \cr } } \right]^T} and v_gk = V_g cos (ωt + φ_k) with φ_k takes 0 ‘− 2π/3 and −4π/3 for phases a, b and c, respectively. u_k is the bipolar switching function defined as (4) uk=1, upper q closed−1, lower q closed and k=a,b,c {u_k} = \left\{ {\matrix{ {1,\;{\rm{upper\,q\,closed}}} \cr { - 1,\;{\rm{lower\,q\,closed}}} \cr } } \right.\;\;\;{\rm{and}}\;k = a,b,c and the reference current i2,k* i_{2,k}^* , which consist of refrence current i2s,k* \left( {i_{2s,k}^*} \right) and harmonic detected current of non-linear load (i_lh,k), is defined as (5) i2,k*=i2s,k*+ilh,k=I2*cosωt+φk+θ+ilh,k i_{2,k}^* = i_{2s,k}^* + {i_{lh,k}} = I_2^*{\rm{\;cos\;}}\left( {\omega t + {\varphi _k} + \theta } \right) + {i_{lh,k}}

Figure 1.

Three-phase LCL-filtered GTI with the proposed control scheme. GTI, grid-tied inverter; HCC, hysteresis current controller.

It should be noted that the generation of vc* v_c^* is performed by PMR and will be discussed in Section 4.

The sliding surface function vector can be defined as (Zhang and Fei, 2023) (6) S=x1+k1x1αsignx1+k2x2βsignx2 S = {x_1} + {k_1}{\left| {{x_1}} \right|^\alpha }{\rm{sign}}\left( {{x_1}} \right) + {k_2}{\left| {{x_2}} \right|^\beta }{\rm{sign}}\left( {{x_2}} \right) where k₁ and k₂ are positive constants 1 < β < 2 and β < α. x₁ and x₂ are calculated by Eqs (7) and (8) (7) x1=vc*−vc {x_1} = v_c^* - {v_c} (8) x2=v˙c*−v˙c {x_2} = \dot v_c^* - {\dot v_c}

During the sliding regime, the dynamics of the converter are forced to slide along the sliding surface. To design an effective SMC system, it is necessary to satisfy the following condition (Bacha et al., 2014). (9) SS˙<0 S\dot S < 0 where S˙ \dot S is the time derivative of S and it is presented in Eq. (10). (10) S˙=x2+αk1x1α−1x2+βk2x2β−1x˙2 \dot S = {x_2} + \alpha {k_1}{\left| {{x_1}} \right|^{\alpha - 1}}{x_2} + \beta {k_2}{\left| {{x_2}} \right|^{\beta - 1}}{\dot x_2}

According to Eqs (7) and (8), the derivative of x₁ can be obtained (11) x˙1=x2 {\dot x_1} = {x_2}

By substitution Eq. (3) into Eq. (8) gives us (12) x2=dvc*dt−dvcdt=dvc*dt−1Ci1−i2 {x_2} = {{dv_c^*} \over {dt}} - {{d{v_c}} \over {dt}} = {{dv_c^*} \over {dt}} - {1 \over C}\left[ {{i_1} - {i_2}} \right]

Taking a derivative of Eq. (12) results in (13) x˙2=d2vc*dt2−1Cdi1dt−di2dt {\dot x_2} = {{{d^2}v_c^*} \over {d{t^2}}} - {1 \over C}\left( {{{d{i_1}} \over {dt}} - {{d{i_2}} \over {dt}}} \right)

Substituting di1dt {{d{i_1}} \over {dt}} and di2dt {{d{i_2}} \over {dt}} from Eqs (1) and (2) in Eq. (13) yields (14) x˙2=d2vc*dt2−1L1CuVdc+1L1Cvc+r1L1Ci1+1L2Cvc−1L2Cvg−r2L2Ci2 {\dot x_2} = {{{d^2}v_c^*} \over {d{t^2}}} - {1 \over {{L_1}C}}u{V_{dc}} + {1 \over {{L_1}C}}{v_c} + {{{r_1}} \over {{L_1}C}}{i_1} + {1 \over {{L_2}C}}{v_c} - {1 \over {{L_2}C}}{v_g} - {{{r_2}} \over {{L_2}C}}{i_2}

Now, substituting i₁ instead of i₂ + C dv_c/dt yields (15) x˙2=d2vc*dt2−1L1CuVdc+1L1Cvc+r1L1Ci2+r1L1dvcdt+1L2Cvc−1L2Cvg−r2L2Ci2 {\dot x_2} = {{{d^2}v_c^*} \over {d{t^2}}} - {1 \over {{L_1}C}}u{V_{dc}} + {1 \over {{L_1}C}}{v_c} + {{{r_1}} \over {{L_1}C}}{i_2} + {{{r_1}} \over {{L_1}}}{{d{v_c}} \over {dt}} + {1 \over {{L_2}C}}{v_c} - {1 \over {{L_2}C}}{v_g} - {{{r_2}} \over {{L_2}C}}{i_2}

By replacing dv_c/dt with dvc*/dt−x2 dv_c^*/dt - {x_2} and adding the terms 1L1Cvc*−1L1Cvc* {1 \over {{L_1}C}}v_c^* - {1 \over {{L_1}C}}v_c^* and 1L1Cvc*−1L1Cvc* {1 \over {{L_1}C}}v_c^* - {1 \over {{L_1}C}}v_c^* into Eq. (15), we finally obtain (16) x˙2=ω12uVdc−ωr2x1−r1L1x2+D {\dot x_2} = \omega _1^2u{V_{dc}} - \omega _r^2{x_1} - {{{r_1}} \over {{L_1}}}{x_2} + D where ω1=1/L1C, ω2=1/L2C, ωr=L1+L2/L1L2C {\omega _1} = 1/\sqrt {{L_1}C} ,\;{\omega _2} = 1/\sqrt {{L_2}C} ,\;{\omega _r} = \sqrt {\left( {{L_1} + {L_2}} \right)/{L_1}{L_2}C} and D is disturbance term which can be obtained from (17) D=r1ω12i2−r2ω22i2+d2vc*dt2+r1L1dvc*dt+ωr2vc*−ω22vg D = {r_1}\omega _1^2{i_2} - {r_2}\omega _2^2{i_2} + {{{d^2}v_c^*} \over {d{t^2}}} + {{{r_1}} \over {{L_1}}}{{dv_c^*} \over {dt}} + \omega _r^2v_c^* - \omega _2^2{v_g}

Substitution of Eqs (12) and (17) in Eq. (10) results in (18) S˙=1+αk1x1α−1−r1L1βk2x2β−1x2+βk2x2β−1ω12uVdc−ωr2x1+D \dot S = \left( {1 + \alpha {k_1}{{\left| {{x_1}} \right|}^{\alpha - 1}} - {{{r_1}} \over {{L_1}}}\beta {k_2}{{\left| {{x_2}} \right|}^{\beta - 1}}} \right){x_2} + \beta {k_2}{\left| {{x_2}} \right|^{\beta - 1}}\left( {\omega _1^2u{V_{dc}} - \omega _r^2{x_1} + D} \right)

Now, defining the control variable (19) u=−signS u = - sign\left( S \right)

By Eqs (19) and (18) existing condition obtained (20) S˙=1+αk1x1α−1−r1L1βk2x2β−1x2+βk2x2β−1ω12uVdc−ωr2x1+D \dot S = \left( {1 + \alpha {k_1}{{\left| {{x_1}} \right|}^{\alpha - 1}} - {{{r_1}} \over {{L_1}}}\beta {k_2}{{\left| {{x_2}} \right|}^{\beta - 1}}} \right){x_2} + \beta {k_2}{\left| {{x_2}} \right|^{\beta - 1}}\left( {\omega _1^2u{V_{dc}} - \omega _r^2{x_1} + D} \right) (1)

When S < 0 → u = 1; (21) 1+αk1x1α−1−r1L1βk2x2β−1x2+βk2x2β−1ω12Vdc−ωr2x1+D>0 \left( {1 + \alpha {k_1}{{\left| {{x_1}} \right|}^{\alpha - 1}} - {{{r_1}} \over {{L_1}}}\beta {k_2}{{\left| {{x_2}} \right|}^{\beta - 1}}} \right){x_2} + \beta {k_2}{\left| {{x_2}} \right|^{\beta - 1}}\left( {\omega _1^2{V_{dc}} - \omega _r^2{x_1} + D} \right) > 0

The above inequality assures that the state variables remain on the sliding surface while converging towards the origin. During the sliding motion (S = 0 and S˙=0 \dot S = 0 ), Eqs (21) and (22) define the operational boundaries of the SMC system.

To mitigate the adverse effects of high-frequency switching and enhance the efficacy of the SMC, a sign function was employed as a replacement for the hysteresis function. (23) u=1, S<+h−1,S>h u = \left\{ {\matrix{ {1,\;S < + h} \hfill \cr { - 1,S > h} \hfill \cr } } \right. where +h and −h are the higher and lower bands of the hysteresis function.

The stability of the proposed reaching law can be evaluated using Lyapunov’s stability criterion. For this purpose, the following function V is defined as the Lyapunov function: (24) V=0.5 S2<0 V = 0.5\;{S^2} < 0

By differentiating V with respect to time and applying Eq. (10), the following results are obtained. (25) V˙=SS˙=Sx2+αk1x1α−1x2+βk2x2β−1x˙2 \dot V = S\dot S = S\left( {{x_2} + \alpha {k_1}{{\left| {{x_1}} \right|}^{\alpha - 1}}{x_2} + \beta {k_2}{{\left| {{x_2}} \right|}^{\beta - 1}}{{\dot x}_2}} \right)

Assuming that the state variables approach zero (x_1a ≅ x_2a ≅ 0) during the sliding mode operation, the derivative of the sliding surface function for a phase can be expressed as follows: (26) S˙a≅ω12Vs62ua−ub−uc+Da {\dot S_a} \cong {{\omega _1^2{V_s}} \over 6}\left( {2{u_a} - {u_b} - {u_c}} \right) + {D_a} with (27) Da=d2vca*dt2+r1L1dvca*dt+ωr2vca*+r1ω12−r2ω22i2sa*−ω22vga {D_a} = {{{d^2}v_{ca}^*} \over {d{t^2}}} + {{{r_1}} \over {{L_1}}}{{dv_{ca}^*} \over {dt}} + \omega _r^2v_{ca}^* + \left( {{r_1}\omega _1^2 - {r_2}\omega _2^2} \right)i_{2sa}^* - \omega _2^2{v_{ga}}

The voltage of the reference capacitor can be determined using the equations v_ga = V_g cos ωt, as indicated in Eqs (5) and (2), as follows. (28) vca*=−ωL2I2*sinωt+r2I2*+Vgcosωt v_{ca}^* = - \omega {L_2}I_2^*{\rm{\;sin\;}}\omega t + \left( {{r_2}I_2^* + {V_g}{\rm{\;cos\;}}\omega t} \right)

By substituting the undistorted grid voltage equation v_ga =V_g cos ωt and Eqs (5) and (27) into Eq. (26), the following outcome is obtained. (29) Da=M1asinωt+M2acosωt=Mdasinωt+γ {D_a} = {M_{1a}}{\rm{\;sin\;}}\omega t + {M_{2a}}{\rm{\;cos\;}}\omega t = {M_{da}}{\rm{\;sin\;}}\left( {\omega t + \gamma } \right) where (30) M1a=ωI2*ω2L2−ωr2L2−r1r2L1−ωr1L1Vg {M_{1a}} = \omega I_2^*\left( {{\omega ^2}{L_2} - \omega _r^2{L_2} - {{{r_1}{r_2}} \over {{L_1}}}} \right) - {{\omega {r_1}} \over {{L_1}}}{V_g} (31) M2a=I2*ω12r1+ω12r2−ω2r2−ω2r1L2L1−Vgω2−ω12 {M_{2a}} = I_2^*\left( {\omega _1^2{r_1} + \omega _1^2{r_2} - {\omega ^2}{r_2} - {{{\omega ^2}{r_1}{L_2}} \over {{L_1}}}} \right) - {V_g}\left( {{\omega ^2} - \omega _1^2} \right) (32) Mda=M1a2+M2a2, γ=tan−1M2aM1a {M_{da}} = \sqrt {M_{1a}^2 + M_{2a}^2} ,\;\gamma = {\rm{ta}}{{\rm{n}}^{ - 1}}\left( {{{{M_{2a}}} \over {{M_{1a}}}}} \right)

Substituting Eqs (21) and (26) into Eq. (24) result in (33) Saω12Vs3signSa−ω12Vs6signSb−ω12Vs6signSc−Da<0 {S_a}\left[ {{{\omega _1^2{V_s}} \over 3}{\rm{sign}}\left( {{S_a}} \right) - {{\omega _1^2{V_s}} \over 6}{\rm{sign}}\left( {{S_b}} \right) - {{\omega _1^2{V_s}} \over 6}{\rm{sign}}\left( {{S_c}} \right) - {D_a}} \right] < 0

Assuming S_b > 0 and S_c < 0 and utilising the equations |S_a| = S_asign(S_a) and S_a = |S_a|sign(S_a), we derive the following results (34) 13Saω12Vs−sign3D<0 {1 \over 3}\left| {{S_a}} \right|\left[ {\omega _1^2{V_s} - {\rm{sign}}\left( {3D} \right)} \right] < 0

The inequality in Eq. (34) is valid provided that the subsequent condition is met (35) 3Da>ω12Vs \left| {3{D_a}} \right| > \omega _1^2{V_s}

Eq. (35) can be stated as (36) −ω12Vs<3Da<ω12Vs - \omega _1^2{V_s} < 3{D_a} < \omega _1^2{V_s}

Now, dividing both sides by ω12Vs \omega _1^2{V_s} gives (37) −1<3Daω12Vs<1 - 1 < {{3{D_a}} \over {\omega _1^2{V_s}}} < 1

Substituting Eq. (29) into Eq. (37) and using γ ≅ −90° gives (38) −1<3Mdaω12Vscosωt<1 - 1 < {{3{M_{da}}} \over {\omega _1^2{V_s}}}{\rm{\;cos\;}}\omega t < 1

By substituting the parameters utilised in the simulation study, it is observed that the condition expressed in Eq. (38) is consistently satisfied within the interval where S_b > 0 and S_c < 0. Furthermore, the analogous control for the remaining legs can be derived through a similar approach.

Following the development of the inner loop controller, the PMR-based outer current loop is evaluated for its capability to compensate for low-order harmonics. This approach simplifies the controller design and reduces the computational burden within the inverter system, as it avoids the need for axis transformations or non-linear control strategies. The PMR controller generates vc* v_c^* , ensuring zero steady-state error in the grid current. When the network current error i2*−i2 \left( {i_2^* - {i_2}} \right) is applied to the input of the PMR controller, the output vc* v_c^* is obtained, ensuring that i₂ follows i2s,k*+ilh,k i_{2s,k}^* + {i_{lh,k}} .

The transfer function of the PR controller (G_PR(s)) is defined in Eq. (39), where K_p, K_i, w₀ and w_c denote the proportional part, resonant gain, fundamental frequency and cut-off frequency of the PR controller, respectively. Although G_PR(s) exhibits high gain exclusively at the fundamental frequency, it can be expanded to address multiple low-order harmonics (h) as shown in Eq. (40). Here, K_p represents the proportional part of the PMR controller, and h signifies the harmonic order where the frequency determines the PMR controller’s resonant frequency (w_h). By employing the PMR controller, each harmonic component in the output current can be effectively and individually suppressed, achieving zero steady-state error. (39) GPRs=Kp+2KiωcSS2+2ωcS+ω02 {G_{{\rm{PR}}}}\left( s \right) = {K_p} + {{2{K_i}{\omega _c}S} \over {{S^2} + 2{\omega _c}S + \omega _0^2}} (40) GPMRs=Kp+∑h=1,5,7,92KiωchSS2+2ωchS+ωh2 {G_{{\rm{PMR}}}}\left( s \right) = {K_p} + \sum\limits_{h = 1,5,7,9} {{{2{K_i}{\omega _{ch}}S} \over {{S^2} + 2{\omega _{ch}}S + \omega _h^2}}}

The validation of the system, as illustrated in Figure 1, was performed through simulation using the parameters specified in Table 1. A Hysteresis current controller (HCC) was employed for this purpose, with the simulation carried out in Matlab/Simulink (The MathWorks, Inc., Natick, MA, USA). The objective of this study is to evaluate the performance of a control structure in which a dependent current source is utilised as a non-linear load. The G_PMR block, which represents the PMR controller, is used to generate the reference for the capacitor voltage.

Figure 2 illustrates the steady-state performance of the proposed system under a highly non-linear load. The grid current and voltage maintain consistent phase alignment, demonstrating the system’s stability. Additionally, the LCL filter effectively suppresses switching harmonics, with no resonance observed in the grid current.

Figure 2.

Steady-state waveforms of GTI (a) grid current (i_2a, i_2b, i_2c) (b) load current (i_la, i_lb, i_lc) (c) grid voltage (v_ga, v_gb, v_gc). GTI, grid-tied inverter.

To evaluate the tracking capability of the controller, a step change was applied to the current reference value, and the results are shown in Figure 3. Specifically, at t = 0.1 s, the current set point was increased by 100% from its initial peak value of 20 A to 40 A. The results depicted in Figure 3 highlight the controller’s rapid response and its ability to track the reference value accurately with short settling time.

Figure 3.

Grid current (i_2a, i_2b, i_2c) responses to the variation of reference grid current.

The resilience of the proposed model against parameter variations was also assessed through simulations with altered grid-side inductance values: a 20% decrease (0.549 mH) and a 500% increase (3.433 mH). The outcomes presented in Figure 4 confirm the robustness of the model under both increased and decreased inductance conditions. Notably, THD of the grid current was measured at 0.33% and 0.53% for the cases shown in Figures 4(a) and 4(b), respectively. These values remain well within the IEEE standard limits (<5%), demonstrating the system’s robustness and ability to maintain performance despite significant deviations in inductance.

Figure 4.

Fast Fourier transform of Grid-side current (i_2a) for two different values of L2: (a) 20% decrease and (b) 500% increase. THD, total harmonic distortion.

The proposed controller is capable of functioning reliably even when the grid voltage is distorted. The performance of the VSI under such conditions is illustrated in Figure 5. It is noteworthy that a dedicated PLL (phase-locked loop) algorithm for extracting positive and negative sequence grid voltage components is not a prerequisite in this scenario. The THD recorded was 0.86%, which is still compliant with the criteria outlined in IEEE.Std519-2014. These results demonstrate the resilience and dependability of the proposed controller, rendering it suitable for deployment in diverse applications where grid voltage balance cannot be guaranteed.

Figure 5.

Simulation results under distorted grid voltage conditions: (a) grid voltage (b) grid current.

The three-phase grid currents (i_2a, i_2a, i_2c) and the load current variation following compensation are depicted in Figure 6. The results demonstrate the effectiveness of the controller, as evidenced by the grid currents maintaining a sinusoidal waveform even under significant changes in load conditions. Notably, at t = 0.1 s, the current of the non-linear load experiences a two-fold increase. Despite this abrupt and substantial rise in load current, the grid currents remain unaffected in terms of waveform distortion.

Figure 6.

Load step change at t = 0.1 s. (a) load current (i_la, i_lb, i_lc) and (b) grid current (i_2a, i_2b, i_2c).

Figure 7 compares the dynamic performance of the NFTSMC with that of conventional SMC. The results reveal that NFTSMC exhibits a faster dynamic response, enabling the system to reach the desired values more quickly, under sudden load changes or step inputs. Additionally, NFTSMC achieves lower steady-state error due to its design, which eliminates chattering, a common issue in traditional SMC, and ensures smoother control actions. Furthermore, the results indicate enhanced precision in current tracking within the dq frame.

Figure 7.

Comparison of the proposed NTFSMC with ordinary SMC under step change. SMC, sliding mode controller.

Figure 8 illustrates the fast Fourier transform of the grid-side current (i₂) when DC voltage is series with 60-Ω resistor. This demonstrates that the system operates effectively, even in the presence of non-ideal DC voltage conditions.

Figure 8.

Fast Fourier transform of grid-side current (i_2a) for Non-ideal DC voltage. THD, total harmonic distortion.

Figure 9 illustrates that an increase in the hysteresis band results in a corresponding decrease in the switching frequency; however, the dynamic response remains unchanged within this domain. It is important to note that a reduction in frequency adversely affects the THD of the grid current (i₂), increasing it from 0.66 to 2.49.

Figure 9.

Fast Fourier transform switching pulses (a) h = 2,000 (b) h = 1,000,000. THD, total harmonic distortion.

Moreover, the system exhibits robust performance when exposed to non-linear and non-symmetrical loads. The empirical evidence presented in Figure 10 confirms that the system functions adequately under these adverse conditions, achieving a THD of 1.14 in the grid current, which is indicative of a highly favourable outcome.

Figure 10.

Non-symmetrical distortions in C phase. (a) non-symmetrical load and (b) sinusoidal grid current.

This study also includes a comparative analysis of the NFTSMC and PMR controlled system with other related methods, as summarised Table 2. While the modulation techniques described in Sozanski and Szczesniak (2023), Huang et al. (2023) and Zhang and Fei (2023) exhibit superior performance in certain aspects, the proposed method offers distinct advantages, such as reduced complexity and computational burden, alongside promising dynamic behaviour and robustness. Unlike the methods in Huang et al. (2023) and Zhang and Fei (2023), which requires an explicit model of the system for implementation, the proposed method operates without such a model. This characteristic enhances its resilience to disturbances, further solidifying its suitability for real-world applications.

This study introduces a novel control strategy for a three-phase grid-connected VSI with an LCL filter, leveraging an NFTSMC and a PMR approach. A key feature of the proposed control strategy is the generation of three-phase reference capacitor voltages as a critical component. To eliminate the need for derivative operations through the use of the PMR controller. This design ensures that the system operates independently of grid inductance, thereby enhancing robustness and reliability. The proposed method effectively addresses the challenges posed by non-linear loads, significantly enhancing power quality by minimising THD and achieving rapid tracking of reference currents. Moreover, the absence of system modelling requirements simplifies implementation while ensuring dynamic performance and robustness. Simulation results validate the controller’s ability to maintain sinusoidal grid currents and compensate for harmonic distortions under various operating conditions. The PMR controller’s intrinsic active damping capability further enhances system stability, removing the need for additional passive components. This work resolves several challenges in APF design and lays a strong foundation for future advancements. Potential avenues for future research include: -

Integration of intelligent control methods: Explore the use of machine learning algorithms to optimise real-time performance and adaptability of APF in dynamic environments.