Inverter harmonic perturbations rejection in renewable energy conversion systems applying a super-twisting algorithm

Grid background and dead-time harmonic perturbations pose challenges in maintaining power quality and stability in the grid integration of renewable energy conversion systems. Therefore, a super-twisting algorithm with the ability to reject harmonic perturbations on three-phase grid-tied inverters is proposed. The algorithm design is related to dead-time and grid background harmonics so that robust performance is achieved under these perturbations. To highlight the advantages of the proposed algorithm, it is compared against proportional-integral (PI) control in several dead-time and grid background harmonic sweeps. The maximum inverter current distortion obtained with PI control in simulation is 15.53%. However, by using the proposed algorithm instead, the maximum current distortion is reduced to 1.8%. Dead-time is experimentally swept by using a permanent magnet motor as a harmonic-free grid. Experimental results of grid background harmonics are obtained by injecting reactive energy into a grid with 2.19% voltage distortion. When PI control is used in the experimental setup, the maximum inverter current distortion obtained is 11.4%. However, with the proposed algorithm, the maximum current distortion is reduced to 1.5%, which complies with the standard IEEE 1547-2018.

Grid background harmonics are present in all electric power systems due to the non-linear loads demanding power from the grid [4][5][6][7]. The standard IEEE 519 [16] stipulates limits for grid background harmonics at the point of common coupling. High levels of these harmonics have severe consequences on RECS, such as distortion of the inverter currents, equipment overheating, and failed operations of electronic equipment [1][2][3][4][5][6][7][8][9]. In addition, dead time, which prevents shoot-through in RECS inverters, also distorts the inverter currents [10][11][12][13]. Even with the availability of high-speed switching devices, dead-time effects cannot be ignored [10]. Consequently, dead-time and grid background harmonics can cause serious power quality problems in RECS [1,8,11,12]. For these reasons, harmonic perturbations affecting the inverter in RECSs need careful consideration.
The total rated-current distortion (TRD), defined by Equation (2) in the standard IEEE 1547 [17], permits to measure power quality in RECSs. In this standard, the maximum permissible TRD is 5%. Thus, RECSs should deliver high-quality power despite harmonic perturbations affecting the inverter.
Decreasing TRD is achieved by increasing the harmonic rejection properties of the power filter [5,18] or by increasing the bandwidth of the control system [2,3,5]. By using more efficient transistors, it is possible to increase the inverter switching frequency [19][20][21]. The switching frequency increase allows the use of power filters with smaller impedance and size, which is a trend at present with the introduction of high power density inverters [22]. However, the reduction of the filter impedance Grid-tied inverters are exposed to harmonic perturbations makes the inverter more susceptible to harmonic perturbations (this statement is explained in Section 2). Due to its potential to increase energy quality, the search for control algorithms that can reject harmonic perturbations is a topic of interest [2][3][4][5][6][7][8][10][11][12].
Conventionally, proportional-integral (PI) regulators are used to control the RECS internal states. However, PI control has low harmonic perturbations rejection capacity. Consequently, PI control cannot reject completely the effects that dead-time and grid background harmonics produced on the inverter currents [4,8,9]. For instance, the study of a commercial solar inverter performed in [1] shows how a 3% total harmonic distortion (THD) of grid background voltage can induce up to 20% TRD in the inverter output currents.
There are several harmonic rejection algorithms for gridtied inverters, which fall into two main categories: Selective and non-selective algorithms [9]. PI control with resonant action (PI+R) is the most popular selective algorithm [7,8]. However, PI+R increases the computational burden because one resonant action is needed to reject each one of the unwanted harmonic components. PI control with repetitive action (PI+RC) is a nonselective algorithm that can reject all harmonic perturbation components with one controller [4,23]. Nevertheless, PI+RC performance is degraded with grid frequency variations. Thus, PI+RC with frequency adapting capabilities is preferred, even though such enhancements increase the computational burden too [24][25][26].
Due to its fast dynamics and its robustness, an alternative to reject harmonic perturbations in three-phase grid-tied inverters is the sliding mode control (SMC). In [27], a grid-tied inverter working at unity power factor using SMC is presented. An adaptive integral SMC for a three-phase grid-tied inverter is proposed in [28,29]. An adaptive fuzzy SMC for a threephase active power filter is presented in [30], which permits a fast response and chattering reduction. A neutral-point-clamped inverter using SMC is presented in [31], where the power quality improvement due to SMC is reported. Space vector modulation based on SMC for a grid-tied inverter is designed in [32]. Although effective, all the aforementioned applications of SMC are based on first-order sliding mode theory, in which chattering is a severe complication. Commonly, chattering is attenuated by approximating the sign function at the expense of control robustness decrease [33]. Second-order SMC algorithms reduce the undesirable chattering effects, and they preserve control robustness [33]. The super-twisting SMC (ST-SMC) is one of the most popular second-order algorithms, as it does not rely on any time derivative. An ST-SMC is designed in [34,35], where it is used in a single-phase inverter to deliver high-quality power into the grid. An application of an ST-SMC algorithm for a three-phase inverter is designed in [36], where the controller gains are adaptively computed in real time.
The main contribution in this study is the proposition of an ST-SMC as a non-selective harmonic perturbation rejection algorithm in three-phase grid-tied inverters. The ST-SMC is robust against grid frequency variations (an existing problem in repetitive control). Moreover, in contrast to resonant control, all harmonic perturbations are rejected by one ST-SMC. The ST-SMC is compared against PI control, in all experimental and simulated situation proposed. In simulation, the dead time is swept from 0 to 2 µs, and the grid background harmonic component is swept from the second to the 25 th . In the experimental setup, the dead time is swept from 0.4 to 2 µs by using a permanent magnet synchronous motor (PMSM) as the experimental harmonic-free grid. Finally, a grid polluted with 2.19% THD is used experimentally. The results show how by using PI control under harmonic perturbations, the inverter TRD increases up to 15.58%, whereas the ST-SMC keeps the inverter TRD below 1.8% so that RECSs meet the standard IEEE 1547.

PROBLEMS STATEMENT
Figure 2(a) shows the vector control diagram of a grid-tied inverter with an L-filter, where PI regulators are used for the synchronous current control. Figure 2(b) shows the d-axis PI control system, where the grid impedance is neglected, e d stands for harmonics perturbations, and i d is the d-axis inverter current. Figure 2(b) is rearranged into Figure 2( FIGURE 3 Bode plot of the proportional-integral (PI) control system admittance reference is set to zero (i.e. i * d = 0) to illustrate the effect that e d has on i d . Bode plots of the transfer function are shown in Figure 3(c), using the parameters listed in Appendix A, and the L-filter admittance for different inductance values are shown in Figure 3. Observe in Figure 3 how the perturbations can be attenuated by increasing any of the following: The L-filter inductance or the control system bandwidth. Although increasing the inductance will increase the inverter cost and weight [5,18], the easiest solution is to increase the controller bandwidth, where increasing the bandwidth of a PI controller will decrease the system stability [8,37]. This bandwidth limitation of PI controllers makes evident the need for control alternatives, other than just PI control, which is able to reject harmonic perturbations in grid-tied inverters [8].

Three-phase inverter general mathematical model
The inverter general mathematical model in the synchronous reference frame (dq frame) is given as follows: where A ∈ R 2n×2n is the state matrix, B ∈ R 2n×2 is the input matrix, i ∈ R 2n is the state vector, v ∈ R 2 is the inverter output voltage vector, e ∈ R 2n represents external perturbations, and n stands for the inverter filter order (e.g. for an L-filter: n = 1, for an LCL-filter: n = 3). Uncertainty is included by assuming that A, and B are comprised by A =Ā + ΔA; and B =B + ΔB Also, vector uncertainty of i, v, and e is accounted for in the following expressions: i =ī + Δi; v =v + Δv; and e =ē + Δe whereĀ,B,ī,v andē are ideal matrices and vectors, which are all known by the control system. ΔA, ΔB, Δi, Δv and Δe represent the uncertainty and perturbations affecting the inverter, which are unknown by the control system. The description of Equations (2) and (3) is summarised in Table 1. By substituting Equations (2) and (3) in (1), the inverter model is transformed as follows: where ∈ R 2n includes all the uncertainty and perturbations affecting the inverter. A common practice in control system design is to assume that is equal to zero. However, this assumption can be dangerous, as is never equal to zero. Vector is decomposed into the next two vectors: where ∈ R 2n is the projection of all inverter perturbations and uncertainties onto the input matrix B, and ∈ R 2n is orthogonal to B. Vector ı includes all perturbations that can be rejected by SMC; hence, it is called: Matched perturbation vector. On the contrary, represent all perturbations that cannot be rejected completely by SMC. For this reason, is called an unmatched perturbation vector. SMC algorithms are robust only if the perturbations are in the form of ı (see [33]).

Three-phase inverter with L-filter
The most interesting case for Equation (1) is when n = 1. This case describes an inverter with L-filter (see Figure 2). The matrices are (see [38]): where is the grid rated frequency, R and L are the rated resistance and inductance of the L-filter. This inverter is interesting, as the matrix B is full rank. Thus, all perturbations affecting this inverter can be rejected, that is, By taking advantage of this property, the present study proposes an SMC algorithm for the inverter with an L-filter that presents robustness to all perturbation, such as parameter variation, background harmonics, dead time, and so forth.

3.1.2
Three-phase inverter with LCL-filter When n = 3, Equation (1) describes the inverter with LCL-filter as shown in Figure 4. The equations describing this inverter are presented in [38]. Due to its higher dynamics, the input matrix Grid-tied inverter with LCL filter B in this inverter is not a square matrix. Therefore, some perturbations will be matched perturbations, while others will be unmatched. Take for instance the most common perturbations affecting any grid-tied inverter, that is, grid background harmonics denoted as e, and dead-time harmonics denoted by Δv. Base on the LCL inverter model, see [38], the perturbations are expressed as In this case, dead-time effects can be rejected completely by SMC, but a complete rejection of grid background harmonics is impossible, and only attenuation is achievable [33]. More complex alternatives, such as SMC enhanced with resonant action can improve the harmonic rejection properties of the LCL-filter inverter [39]. However, a study of such alternatives is out of the scope of this study.

Vector sliding surface
Vector sliding surface is defined in vector form as whereī is a current measurement, i * is the control reference.

Vector sign function
Vector sign function is defined from Equation (9) as

ST-SMC algorithmBased on Equations
where A, B are given by Equations (2) and (6), i is the inverter current, e is the grid voltage,v is the ideal inverter voltage, Δv is the dead-time distortion, k P , k i , k 1 , k 2 are controller gains, u is the control system integral action, andē is the measurement of grid voltage. Here,ē increases the ST-SMC performance. A PI control with the feedforward ofē is a common practice [37], but this alternative is not enough to reject grid harmonics because it does not consider dead-time distortion and the analogue-to-digital conversion (ADC) process errors [40,41].
Dc voltage and inverter currents are also processed by an ADC (see Figure 5). The phase lock loop shown in Figure 5 is the same as in [42]. A PI controller regulates the dc voltage (designed also in [42]), in which a low-pass filter (LPF) rejects the noise introduced by the ADC. For stability reason, LPFs are not used for the currents; instead, the sampling frequency is twice the switching frequency to avoid aliasing [41].

Controller stability demonstration
The stability demonstration of Equation (12) is carried out by defining the potential and kinetic energy of Equation (9). The objective of this demonstration is to determine the controller gains that guarantee the convergence to zero of Equation (9) in the presence of perturbations and uncertainties so that robust control is achieved. The definition of the potential energy will be obtained from the second time derivative of Equation (9). The first time derivative of Equation (9), substituting Equation (12(a)), is expressed as follows: By substitutingv in Equation (13) for Equation (12(b)), it gives as a result: where ı is the matched perturbations vector, stands for the unknown uncertainties, and f (x) = √ |x| sign(x). Observe how ı contains all the perturbations that affect the inverter.
By taking the time derivative of Equation (14(a)), substituting Equation (12(a)), a second-order matrix differential equation is obtained: is the Jacobian matrix of f (x), given by the following positive definite matrix:

Kinetic energy
Kinetic energy is defined as the following quadratic form:

Potential energy
Potential energy is defined as the line integral of Equation (15(b)), that is, where l is a dummy variable used for integration purposes. By assuming that depend only on time, it can come out of the integral. Thus, the evaluation of Equation (18) yields: where ‖x‖ 1 is the 1-norm [42].

Lyapunov function
The Lyapunov function is defined as the addition of Equations (17) and (19), which gives as a To be a valid Lyapunov function, Equation (20) must fulfil two conditions: It must be positive definite and its time derivative must be negative definite [33]. Expression (21) is obtained by replacing Equation (15) in the time derivate of Equation (20). Since k p , k i , k 1 , k 2 multiply positive values, it is sufficient that k p , k i , k 1 , k 2 > 0 so that the system is stable. But the ST-SMC must perform robustly in the presence of harmonic perturbations; thus, the gains k p , k i , k 1 , k 2 will be designed carefully next.

Linear gains k p and k i
The linear controller gives stability to the system when the error is big [36]. In order to design it, the next two expressions, taken from [42], are used: where c is the crossover frequency that sets the controller speed of response, given in radians per second, and PM is the phase margin that sets the controller stability, given in radians. k p , k i are designed by assuming that c = 1000 [rad∕s], PM = ∕3 [rad], and using the L-filter parameters listed in Appendix A.

Gain k 1
There is not yet a direct relationship between the gain k 1 and the inverter current distortion. However, it has been observed that increasing the value of k 1 increases the chattering frequency to higher values, and thus improving the algorithm performance. However, selecting a high value for k 1 will increase chattering amplitude, so a compromise between these two phenomena has to be done. The proposed gain k 1 that gives good results is listed in Appendix A.

4.2.6
Gain k 2 After x reaches zero, it is necessary to maintain it at zero all the time so that robustness is assured. Since all values in Equation (21) are positive except the time derivative of , the gain k 2 is designed to overcome this time derivative. This is achieved by taking the worst-case scenario, that is, when the time derivative of is at its maximum negative value, which is expressed mathematically by the following inequality: where Z n is the n th harmonic component of , is the grid angular frequency, and ‖ ⋅ ‖ 2 is the 2-norm [43]. The vector Z n is comprised of dead-time harmonic components (D n ), the grid background harmonics components (G n ), and the unknown perturbation harmonic components (K n ). As a result, the gain k 2 is computed as The term ‖K n ‖ 2 is the unknown perturbations harmonic components 2-norm, given by Equation (14(c)). Even though it is unknown, this perturbation is assumed upper bounded. Therefore, an upper bound value is proposed as follows: Finally, by adding Equations (26) to (28), the gain k 2 is obtained. The value of k 2 is given in Appendix A.

SIMULATION RESULTS
Simulations are carried out in MATLAB, where the simulated inverter is comprised of 6 Insulated Gate Bipolar Transistor (IGBT), and it is switched at 40 kHz using asymmetrical regular sample of Pulse Width Modulation (PWM) [44]. The control system is digitally implemented using C code, sampled at 80 kHz. In order  to observe the complete effect of grid background harmonics on the inverter currents, the feedforward correction termē, see Figure 5, is assumed to be equal to zero for all the simulations. In all the following simulations the inverter injects 15 amperes of q-axis current (reactive power) into the grid because the d-axis current regulates the dc-link voltage (see Figure 5). The system parameters and control gains are listed in Appendix A.

5.1
Grid polluted with 5th harmonic component

PI control results
PI control is simulated using the vector control diagram shown in Figure 2(a) and the grid voltages shown in Figure 6(a). Observe in Figure 6(a) that after the 0.7 s, the grid voltages are polluted with 5% THD of negative sequence 5 th harmonic component. The inverter currents obtained are shown in Figure 6(b), which shows two cases: When energy is injected into a harmonics-free grid and when energy is injected into a grid polluted with harmonics. Figure 6(c) shows the harmonic spectrum of the currents injected into a harmonic-free grid, in which the TRD is only 1.6%. Figure 6(d) shows the harmonic spectrum of the currents when the grid is polluted with 5% THD. Compare Figures 6(c) with (d) and observe how the TRD increases from 1.6% to 15.53 % because of the grid background harmonics.

ST-SMC algorithm results
The proposed ST-SMC algorithm is simulated using the vector control diagram shown in Figure 5. The grid voltages used are shown in Figure 7(a), which are polluted with 5% THD of negative sequence 5 th harmonic component. The inverter currents obtained are shown in Figure 7(b), which shows two cases: When PI control is used and when the ST-SMC algorithm is used. Observe how, as opposed to PI control, the ST-SMC rejects completely the effects of grid background harmonics on the currents. Thus, the inverter TRD is reduced from 15.53% to 1.90%. Also, observe in Figure 7

Grid harmonic sweep
A harmonic sweep is carried out by polluting the grid voltages with 5% THD of harmonics in the order between the 2nd and the 25th. Positive and negative sequence components are considered because they are present in the system in the case of the appearance of unbalanced harmonics. The results of the harmonic sweep with PI control are presented in Figure 8(a), where the inverter output current TRD is different for different grid harmonic components. A maximum 15.53% and a minimum 5% TRD are observed. Figure 8(b) shows the inverter output current TRD when ST-SMC is used. The proposed algorithm is able to suppress any grid harmonic component, and it keeps the TRD below 1.8 %. However, the negative sequence 2nd harmonic component surprisingly produces the highest TRD. Therefore, the 2nd harmonic component may be more difficult to reject than any other harmonic components of the grid voltages.

Dead-time harmonic rejection
The PI controller is compared against the ST-SMC algorithm under 1 µs of dead time. In this simulation, the inverter is switched at 40 kHz, and it is tied to a harmonic-free grid with voltages as shown in Figure 9(a). The inverter output currents are shown in Figure 9(b) at the instant when the control structure is changed from PI control to the proposed ST-SMC. Observe in Figure 9(b) how the ST-SMC algorithm rejects completely the effects of dead-time harmonics on the inverter currents. Thus, the inverter TRD is reduced from 7.61% to 1.44%; also, compare Figures 9(c) with (d). Figure 10 shows a dead-time sweep from 0 to 2 µs, where the ST-SMC algorithm is compared against PI control. Observe how in both situations the inverter TRD increases as the deadtime period increases. Note how with PI control, the inverter TRD increases to such a degree that 12.79% is reached when deadtime is equal to 2 µs. On the contrary, observe how the ST-SMC keeps the TRD lower than 1.56% for all dead-time periods.

6
EXPERIMENTAL RESULTS

Experimental setup
The laboratory equipment used to obtain experimental results is shown in Figure 11. A connection diagram is shown in   Figure 13, where test 1 consist of a dead-time sweep, and test 2 is the reactive power injection into the grid. The control system is digitally implemented using C code sampled at 80 kHz. All the integrators and filters are digitally implemented using the Tustin approximation [41]. The laboratory grid voltage is used as a three-phase source. The PMSM 1 is used as a harmonic-free grid, and the PMSM 2 is controlled by the BTBC 2 to rotate at 900 rpms so as to obtain 60 Hz three-phase voltages at the PMSM 1 terminals.6.2 Experimental rejection of dead-time harmonics In this experiment, the PMSM 2 is set to rotate at 900 rpm so that a harmonic-free back-electromotive force (BEMF) is induced in PMSM 1 terminals (see Figure 14(a)). This BEMF is used as a harmonic-free three-phase grid. The line-to-line rms voltage of this BEMF is 77 V, and its THD is less than 1% (see Figure 14(b)).
The BEMF magnitude of PMSM 1 is different from the grid voltage used in simulations. However, according to Equation (B2) in Appendix B, dead-time distortion only depends on dc voltage, switching frequency and dead-time period. Therefore, the experiment and simulation results will be approximately the same due to dc voltage, switching frequency and dead-time  Active power equal to 1.15 kW is injected into PMSM 1 so that the experimental currents are equal to the simulation currents. The dead-time period is equal to 1 µs. The system parameter used is listed in Appendix A. The currents resulting from this experiment are shown in Figure 15(a), which shows two cases: Active power injection with PI control, and active power injection with the proposed ST-SMC. The harmonic spectrum of the currents that result from the use of PI control is shown in Figure 15(b), where the TRD obtained is 6.73%. Observe in Figure 15(b) how there are strong 5 th , 7 th , 11 th , and 13 th harmonics components present, in addition to some 2 nd , 4 th , 6 th , and 8 th harmonic components. The appearance of even harmonic components in the inverter currents is mainly caused by asymmetries in the system, such as impedance unbalance, voltage unbalance, and so forth [14]. Although similar, this result contradicts what simulations predicted because only odd harmonics appear in the simulation results shown in Figure 9(c). When the proposed ST-SMC algorithm is used instead of PI control, the inverter TRD is reduced from 6.73% to 1.04% (see Figure 15(c)). Observe also in Figure 15(c) how the ST-SMC algorithm rejects odd and even harmonic components, which is one benefit of the ST-SMC algorithm. Now, compare Figures 15(c) with 9(d) and observe the similarity between experimental and simulation results.  Figure 17 presents the inverter synchronous currents responding to a step control reference. The results obtained using the ST-SMC algorithm are shown in Figure 17(a), where the inverter currents respond in 1 ms. The results obtained using PI control are shown in Figure 17(b), where the inverter currents respond in 3 ms. Oscillations, as a consequence of using PI control under dead-time effects, are present in the inverter currents shown in Figure 17(b). The proposed ST-SMC algorithm is only three times faster than the PI controller. However, dead-time harmonics are rejected because of their robustness property. Therefore, in Figure 17(a), only dc current is obtained. The inverter currents in abc reference frame using the ST-SMC algorithm and PI control are shown in Figures 18(a) and (b), respectively. Note how the currents in Figure 18(b) are distorted, while the currents shown in Figure 18(a) are sinusoidal. Low current distortion is the main benefit of the ST-SMC algorithm, although some high-frequency chattering is present in the currents shown in Figures 17(a) Figure 19(a) shows these grid voltages, where some distortion is visible. Figure 19(b) shows the harmonic spectrum of the grid voltages; note how the 3 rd , 5 th , and 7 th unbalanced harmonic components are present. In this experiment, the dead time is reduced to 0.4 µs (minimum dead time safe for the inverter used), and the switching frequency is reduced to 20 kHz. This is done to reduce the dead time as much as possible so that dead-time effects are negligible in the inverter currents. Reactive power equal to 2 kVAR is injected into the grid. Figure 20(a) shows the inverter currents in which two cases are shown: When PI control is used and when ST-SMC algorithm is used. When PI control is used, the inverter currents Observe how in contrast to PI control, the ST-SMC rejects the grid background harmonics efficiently, and thus the inverter TRD is reduced from 8.28% to 0.88%. The harmonic spectrum of the inverter currents, obtained using PI control, is shown in Figure 20(b). The harmonic spectrum of the inverter currents obtained using PI control is shown in Figure 20

CONCLUSION
The proposed ST-SMC presented robust performance under the appearance of background harmonics, grid frequency variations, dead-time harmonics, and system parameter variations. By using this algorithm as the inverter current control loop, in simulations and experiments, the power quality of a RECS was increased in comparison to PI control. With PI control, the inverter output current TRD increased up to 15.5%. In contrast, the proposed ST-SMC algorithm increased the energy quality by keeping the TRD of the inverter output current lower than 1.8% in the presence of 5% THD of individual grid harmonics as dictated by the standard IEEE 519. Also, excellent experimental performance was achieved in the presence of inverter switching dead-time harmonics, meeting the standard IEEE 1547. Based on the experimental results obtained in this study, the energy delivered by RECSs three-phase inverters, equipped with an Lfilter and the proposed ST-SMC algorithm, can be increased despite the presence of dead time, grid background voltage, and unknown uncertainties. The results obtained were independent of the L-filter inductance used, so an L-filter with low inductance can be used in high power density inverters. Additionally, since the ST-SMC is independent of the system frequency, this algorithm can be used to control the torque and flux in PMSM drives, where the harmonic perturbations are of variable frequency. Finally, the study of harmonic perturbations rejection capability of the ST-SMC in LCL grid-tied inverter is left as a future work.  where v abc is the three-phase vector of the inverter voltages, which is measured from the phases to the half dc-bus voltage (phase voltage),v abc is the ideal PWM phase voltage, and Δv abc is the dead-time phase voltage. Fourier analysis of Equation (B1) is complex [13]; thus, an approximate Fourier analysis, similar to [14], is preferred. Therefore, the dead time is described by where T d is the dead-time period, T s is the switching period, v dc is the dc-bus voltage, sign(i x ) is equal to 1 if i x > 0 and −1 otherwise (where x = a, b, c). The complex Fourier series coefficients of phase a in Equation (B2) are given by where ∅ is the phase delay of the inverter currents. Fourier coefficients of phase b and c are given by Equation (B3) displaced by a factor e ± j 2 n∕3 . Based on this assumption and using Equation (B3), Expression (B2) has the following equivalent form: The Clark transform used is given by where Δv is the dead-time distortion in reference frame, which is given by the following expression: In order to take Equation (B6) into the dq frame, the Park transform is given in the complex form as the following matrix: By multiplying Equations (B6) and (B7), then dead-time voltage in dq frame is obtained as follows: where Δv is the dq dead time, and Λ n is the following vector: [ a n e j 0 (n+1)t + b n e − j 0 (n−1)t ja n e j 0 (n+1)t − jb n e − j 0 (n−1)t ] with the parameters a n , and b n expressed as follows: Giving as a result: ] a n e j 0 (n+1)t + b n e j 0 (n−1)t ) (B11) Expression (B11) is divided into positive (Δv + ) sequence and negative (Δv − ) sequence dead-time voltages, that is, n=−∞ C n−1 a n−1 e j 0 nt (B12a) where the sub-indices in Equations (B12(a)) and (B12(a)) have been shifted by −1 and by +1, respectively, without affecting the final result. By adding Equation (B12(a)) and (B12(b)), the final expression for the dead-time harmonics in dq frame is obtained as ] C n−1 a n−1 + 1 2 which has a 2-norm equal to

Grid background harmonics
Limits for grid harmonic components are listed in the standard IEEE 519 [16], This standard states that for systems operating below 1 kV, the maximum THD expected is 8% with a maximum 5% individual THD for each component. In this study, a 10% THD with 5% of the 5 th , 7 th , 11 th and 13 th harmonic components are considered. This is expressed as { 0.05e j n |n| = 5, 7, 11, and 13 0 otherwise (B16) where G n represent the grid background harmonics components, V LL is the grid rated line-to-line rms voltage, and n is the n th harmonic phase angle. The dq grid voltage decomposition is obtained by substituting C n in Equation (B4) for G n in Equation (B16) and exactly following the same procedure described from Equations (B4) to (B15), which yields the 2-norm of the grid background harmonics, which is expressed by where G n is the vector that contains the grid background harmonics Fourier coefficients in dq reference frame.