Spectrum Estimation of Input Current Ripple on a Wide Class of Multilevel Grid-Tied Converters

Multilevel (ML) converters are frequently used to implement grid-tied ac–dc conversion systems. Their design may benefit from multiobjective optimization techniques, which typically involves time-consuming circuit simulations in order to obtain input current estimations suitable for input inductor and electromagnetic interference filter design. Herein, a closed-form expression of the input current ripple is derived to ease harmonic content estimations. The proposed approach separates the fundamental grid-current component from its ripple and models the latter like an amplitude modulation, where the modulating signal is its envelope and the carrier is the triangular current waveform. First, a general waveform analysis of ML converters is performed to derive the voltage across the grid-side inductor, then the associated current ripple is modeled. Experimental results on an ML converter prototype are reported to validate the analytical results.


I. INTRODUCTION
I N INDUSTRIAL applications, the energy supply chain may involve high-power ac-dc converter systems interlinking medium/high-voltage (MV/HV) ac transmission lines and inhouse ac or dc loads connected by means of a second conversion stage. Multilevel (ML) converter topologies are widely used at the ac-grid side by virtue of their features: First, they are often implemented with input-series connected modules sharing the MV/HV voltage and, hence, the input power, allowing a significant increase in efficiency respect other solutions (e.g., two-level inverter); second, they provide high performance in terms of harmonic distortion and EMI compliance mostly related to conducted emissions [1], [2].
A typical application is on power factor correction (PFC) systems, where the aim is to increase the power factor by exchanging sinusoidal currents that are in-phase with the grid voltage in order to minimize the reactive power circulating through the distribution grid. As shown in Fig. 1, the converter control shapes the input voltage v m (t) through the filter inductance L g to absorb the desired input currentī g in-phase with the input grid voltagē v gv wherev g andv m are the phasor representations of the grid voltage v g (t) and the fundamental component of the ML-converter voltage v m (t), respectively, defined as v x (t) = {v x e j2πf g t } with f g the grid frequency. To generate (1), ML converters exploit an N L -level voltage waveform. In the following analysis, a particular class of ML converters is considered where the voltage levels are equally spaced by a quantity V N , and the generated voltage can switch only between two consecutive levels in a switching period T m of the converter. An example of the generated voltage by an 11-level converter and the absorbed current from the grid in a positive half-wave are displayed in Fig. 2(a) and (b). The average values · T m computed within a switching period T m are also indicated, where · T denotes the moving average operator, that is, Fig. 2(c) shows the generic waveforms v m and i g within of two consecutive switching periods T m .
In the design process, especially when multiobjective optimization approaches are used, the knowledge of the harmonic content of grid current i g is crucial for the design of filter inductor L g and input EMI filter. The power losses of the filter components can be easily calculated by the superposition of losses related to each frequency of the current spectrum (e.g., winding and core losses in the inductors).
In the literature, the grid current spectrum is often obtained by computer simulation, which is typically very time-consuming. In fact, the minimum spectrum resolution should equal f g in order to correctly appreciate the sideband harmonics; then, the simulation time should cover at least 20 ms of steady-state operation while to reach the frequency range of conducted emissions (i.e.,  should be lower than 16.7 ns (i.e., 1/60 MSps) [3], [4], [5]. On this basis, each design of a multiobjective optimization would collect ࣙ 1.2 G samples of the input current to ensure compliance with conducted emissions standards. Alternatively, double Fourier integral analysis [6] on the grid current i g is possible too, but this approach is always solved numerically and requires a high computation cost when the number of voltage levels increases [7], [8].
In the following, a spectrum model to estimate the input current ripple on the considered class of ML converters is proposed. The rest of this letter is organized as follows. Specifically, Section II provides the theoretical basis of the presented model. Section III demonstrates the model considering experimental results from an ML-converter prototype. Finally, Section IV concludes this letter.

II. SPECTRUM MODEL OF ML CURRENT RIPPLE
Two main assumptions have been done in the considered class of ML converters: 1) The voltage waveform v m produced by the ML converter can be associated with a staircase waveform whose step amplitude is constant and constrained to switch between two consecutive voltage levels of the allowed N L for each switching period T m (e.g., V N and 2V N , 2V N and 3V N , etc.); 2) the voltage pulse changes position when ML-converter average voltage moves through next couple of consecutive voltage levels such that the current ripple waveform is mirrored around its average value i g T m , as shown in Fig. 2(c). These assumptions are valid for example in interleaved-PWM ML converters [9]. This approach is independent of the selected modulation scheme and topology of the ML converter: The number of modules can determine both the possible voltage levels N L , and the switching period T m using interleaving PWM strategies to multiply the switching frequency of the single module.
Considering the general voltage waveform depicted in Fig. 2(c) in the switching period T m , it is characterized by a voltage pulse of amplitude V N and width dT m that leans on an offset voltage k s V N . Since the voltage levels must be bipolar, k s can assume different values depending on the total number of levels N L allowed by the ML converter. For the converter class discussed herein, k s is limited in the following set K, depending on the parity of N L as To establish the modulation parameters of the voltage waveform (i.e., the duty cycle d and the base level voltage k s V N ), the functions (·) S and (·) f are now defined as where the operator (x) S maps the argument into the closest s value in the set S, which is lower in a module to x. For the sake of clarity, Fig. 3 shows how the two functions (·) K and (·) f map a real number and which are their domain and codomain depending on the number of levels N L . From Fig. 2(c), the average voltage in the switching period T m can be expressed as where the sign is concordant with the v m one. Using (3) in (4), the time trend of k and d values during a grid voltage period can be calculated as permitting to solve the input current spectrum and a closed form for its estimation. The absolute notation on d(t) function allows us to relate its value to the meaning of duty-cycle of the voltage pulse depicted in Fig. 2(c). Defining the high-frequency (HF) current component i g,HF as the difference between the grid current i g and the fundamental harmonic component, namely, i g,HF = i g − i g T m , the current ripple can be calculated from the voltage across the grid-tied filter inductor v L . Assuming a quasi-stationary condition, that is, the converter dynamics are neglected, being the operating point changing very slowly with respect to the switching period, the inductor voltage is where Remarkably, the average voltage of ML converter is related to the grid current i g by (1) allowing the further step on Assuming negligible the sinusoidal term, namely ω g L g I g V N , the voltage across the filter inductor can be just approximated to HF switching component of This assumption is satisfied commonly in industrial applications where used values of the filter inductor bring ω g L g I g /V g < 0.05 . . . 0.15 [1]. As an initial consideration, this relationship defines a maximum value of N L for which the spectrum approximation can be adopted. Exploiting the volt-second balance on the inductor L g under the hypothesis above, the current ripple amplitude ΔI g,HF is as displayed in Fig. 2(c). The ML converter is modulated such that v m (t) T m ≈ v g (t) due to the negligible voltage drop on the filter inductor. Then, the effective duty cycle d can be derived from the grid voltage as follows: where U g = V g /V N , which is the grid voltage amplitude V g normalized with respect to the step voltage V N . Substituting (9) into (8), the spectrum of the ripple current ΔI g,HF is given by ΔI g,HF = F{ΔI g,HF }, where F(·) indicates the Fourier transform operator. The presence of (·) f in (9) impedes a straightforward estimation of the inductor current ripple because it is a nonlinear function and cannot be computed in a closed form. At this point, double Fourier integral analysis is the only tool to cope with the spectrum of (8) requiring a significant computation effort. However, the term d(1 − d) in (8) is a parabolic function that can be well approximated by a sinusoidal term The two functions reported in (10) are plotted in Fig. 4 depending on d value together with their difference Δf . The graph shows up the curves are close and the mismatch can be considered negligible in the estimation of the current ripple envelope. The same similarity is also reflected on the Fourier series expansions  of (10) In fact, looking at (11), the components of a parabolic term differ in module with respect to the sinusoidal one by 5% for the dc and the first-order harmonic components while by 15% to 20% for other components. This approximation leads the following estimate ΔI g,HF of the amplitude of the current ripple (8): The next step should be the substitution of (9) into (12) but this brings a nonlinearity due to the function (·) f again. However, exploiting the periodicity of sine in (10), the (·) f operator can be removed because the sine function already maps periodically its argument to the interval [−1, 1] taking into account also the phase change of the current ripple waveform between two consecutive levels. Moving from these premises, (12) becomes where the nonlinear function (·) f is substituted by the function Γ, which distinguishes the two cases related to the parity of N L , since (2) implies a 1/2 offset on d argument if N L is even, as shown in Fig. 3(b). Fig. 5 reports both the HF current component i g,HF (t) and its estimated envelope ΔI g,HF (t) for the 11-level converter in the positive grid half-wave of Fig. 2(a). Fig. 5 shows a close matching between (13) and the envelope of the HF current ripple of i g . Remarkably, (13) describes a phase-modulated (PM) signal, then, by the relevant modulation theory, the spectrum in terms of the Fourier series of the given signal can be found. The sine and cosine terms of (13) can be expanded using the Jacobi-Anger expansion [10], that is where J ν (z) is a first-kind Bessel function. Using (14) in (13), the resulting PM signal is composed of only the odd or even components around the fundamental harmonic (i.e., the grid component ω g = 2πf g ) accordingly to the N L parity. The magnitude of the nth spectral component of the estimated current ripple amplitude ΔI g,HF can be deduced by (13) as where n is odd or even based on the parity of N L . Equation (13) describes the envelope of the HF current ripple as the modulating signal in an amplitude modulation where the carrier is the triangular wave of the inductor current. Since i g,HF is a function of the duty-cycle d, also the harmonic components of its spectrum are affected by this dependence. Starting from Fig. 2(c), the HF current i g,HF in a switching period T m can be approximated to a scaled triangular wave as where τ = t/T m and d is specified by (9) at the considered instant. It is clear that (16) describes the current waveform in only a time frame of the grid period. Useful simplification can be also done to calculate the i g,HF carrier spectrum looking to the Fourier series coefficients of (16) reported herein, rather than substitute d with (9) merely It is worth noting that the module of the first harmonic (i.e., p = 1) is more than double other high-order harmonics and its amplitude is quite flat as d varies. This permits us to consider that the carrier energy is almost entirely concentrated in this first harmonic, allowing us to approximate the HF current spectrum as (15) shifted around the carrier frequency f m and scaled by (17) for p = 1. Since the codomain of the d-dependent term in (17) is [π, 4], the worst case is considered (i.e., d = 0.5) to determine the estimated HF current ripple spectrum i g,HF . Then, the magnitude of nth spectral component of i g,HF is which is the aimed at approximating the single-sideband harmonics around f m in a closed form. The spectrum description provided by (18) allows different simplified approaches in the design of ML converters as follows: 1) calculating the total rms current associated with the harmonics around the switching frequency f m for the bandwidth required by EMI standards, Raggl et al. [11] and Liserre et al. [12] define a design procedure for the differential mode filter; 2) starting from (18) is possible to estimate the current spectrum through the common EMI receivers (e.g., peak, quasi-peak, or average) to evaluate the compliance with EMI standards [13]; 3) the knowledge of the rms current for each HF harmonics allows the estimation of conduction losses in the grid-tied inductor calculating the ac resistance at each frequency. The same approach can be used in the calculation of the magnetic core losses by applying Steinmetz's equation at each current harmonics for the major and minor B-H loop [14], [15].

III. EXPERIMENTAL VALIDATION
A single-phase ML converter is considered. The prototype is based on an Imperix B-Board PRO controller and several half-bridge modules. The main core of the converter is implemented by four series-connected full-bridges controlled with classical interleaved PWM modulation. Each module is supplied at V N = 25 V and switches at 10 kHz. Different configurations of the ML converter have been considered, in terms of a number of voltage levels N L and modulation index U g . A subset of experimental configurations is reported in Table I, where the four highest harmonics of i g around the effective switching frequency f m are shown in descending order.The ith peak is located at f = f m ± n i f g and is identified by the measured value I ∧ i and the estimated one I ∧ i = I n i g,HF calculated from (18). To measure estimation effectiveness, the percentage of relative error Δ % i is referred to herein where the difference between the magnitudes of estimated and the measured harmonics is normalized with the dominant component I ∧ 1 . This allows us to weight the estimation error through the side-band harmonics, keeping the focus both on the shape of the estimated spectrum and on the maximum peak for EMI compliance. From the experimental measurements, it is notable that the maximum error for the highest peak I ∧ 1 is always less than 3% while for the other harmonics is limited to 10%. Fig. 6(a) and (c) displays the grid current spectrum around f m of the configurations highlighted in Table I and shows that the peaks given by (18) well approximate the measured ones. Fig. 6(d) and (f) shows that the frequency dependence of the magnitude expressed in (17) by the term p = f/f m is well modeled and   Table I. I ∧ 1 f 2 m /f 2 can be considered as an upper limit for f < 20f m . A further outcome of this approach is a rough estimation of HF rms current going through Parseval's theorem, which is useful for the evaluation of the winding and core losses in the inductor.

IV. CONCLUSION
This letter proposes a simple yet accurate method to estimate the input current spectrum of the ML converter. The model is validated by means of simulations and experimental measurements on an ML prototype implemented. Differently from the literature where the simulation approach is widely used, the model discussed herein allows us to a straightforward closed-form expression useful to perform analyses relevant to, for example, the IEEE 519-2014 standard compliance and the HF rms input current that impact the design of the EMI and input inductor filters.

ACKNOWLEDGMENT
The author would like to thank Ph.D. students Hossein Abedini and Andrea Lauri for their valuable contribution concerning the set-up of the ML prototype.