Multiband SHEPWM Control Technology Based on Walsh Functions

Abstract

The drawback of modulation ratio limitation in selective harmonic elimination pulse width modulation (SHEPWM) technology based on Walsh function transformation. In order to solve this problem, a multiband SHEPWM control technology method for a multilevel inverter based on Walsh functions is proposed. By analyzing multiband SHEPWM for multilevel inverter voltage waveforms under a Fourier series transform, the unified nonlinear multiband SHEPWM equations for a multilevel inverter with a modulation index varying from 0 to 1 can be generalized. The equations can be solved by Walsh function transform. In this way, the difficulties faced in expanding the modulation index, online calculation, and real-time control are resolved simultaneously. A seven-level inverter is taken as the example, for which the piecewise linear equations of Walsh functions under different bands and the trajectories of switching angles are given. Meanwhile, the conditions for multiple sets of solutions at such a point where the modulation index is switched over are also taken into account. The feasibility of the proposed method is verified by simulation based on MATLAB and SIMULINK. Finally, the feasibility of the practical application is proved by the experiment based on Digital Signal Processor (DSP).

1. Introduction

The cascaded H-bridge (CHB) multilevel inverter is one of the popular converter topologies used in high-power medium-voltage (MV) drives [1]. It has many advantages, such as its high voltage level, low voltage change rate, and its easy modularization [2]. The pulse width modulation schemes for multilevel inverters can be generally classified into carrier-modulated sine pulse width modulation (SPWM) [3], voltage space vector pulse width modulation (SVPWM) [4], and selected harmonics elimination pulse width modulation (SHEPWM) [5,6,7]. The SHEPWM scheme has several distinct advantages, such as having a low switching frequency, high output quality, low switching losses, and high inverter efficiency. The core issue of the SHEPWM control is the solution of nonlinear equations of the multilevel inverter, including the selection of initial values and the selection of iterative algorithms. In addition, the traditional trapezoid wave modulation SHEPWM for multilevel inverter control technology is limited by the modulation index, which generally cannot randomly change from 0 to 1. When the modulation index is relatively low, the SHEPWM nonlinear equations may have no solutions.

At present, most papers use the Newtonian iterative method to solve nonlinear equations, but this needs to find the initial value, which is relatively close to the accurate value, in order to make the algorithm converge, otherwise the iterative algorithm is be convergent. The polynomial synthesis method and the power method are proposed to solve nonlinear equations in [4,5]. However. these methods are too complicated and need to be derived for different levels, which makes it hard for them to be widely used. The homotopic algorithm is applied to solve the SHEPWM nonlinear equations in [7], which raises the rate of convergence of the iterative process to a certain extent. However, the dependence on the initial value of the iterative process is still strong and there is no guarantee that the iteration can be convergent. A SHEPWM function is proposed based on the Walsh functional transform in [8] by analyzing the relationship between the Walsh domain and the Fourier domain, and by transforming the nonlinear transcendental equations of the Fourier domain into the Walsh function under the domain of the piecewise linear system of equations. The calculation steps of the conversion process are improved and the recursive methods used between the harmonic elimination angles and the fundamental wave voltage section amplitude are proposed in [9,10,11,12].

To obtain the initial value, the trial and error method is used in [8]. If there is a lack of evidence when selecting the initial value, the efficiency of the iteration is low. Sometimes equations may also be divergent. The triangular carrier SPWM method is used to get the initial solution in [9]. In fact, this is the method used to solve the reference sine function and a series of piecewise linear functions. However, the calculation process is very troublesome. The approach of Wu is introduced to find the solutions for the initial values of the SHEPWM nonlinear equations in [10], which is complex and hard to achieve.

To expand the modulation index of SHEPWM for a multilevel inverter, the method of reducing a pulse to extend the modulation index to 0.2 is used in [11]. However, the freedom degrees and the number of harmonics eliminated will be reduced at the same time. The polarity reverse method is proposed to expand the modulation index in [12], however the energy consumption control of the DC link needs to be increased to solve the problem of the energy feedback caused by its structure. The six sections of the ladder-type wave switching model are proposed in [13], but the problem of the energy feedback also exists and the working frequency of the switch cannot stay the same.

As it is easy to solve nonlinear equations by using Walsh functions and they have a low dependence on the initial solution, a kind of transformation method is proposed, for which the multilevel inverter modulation index of the multiband SHEPWM modulation scheme [14] can be broadened and combined with Walsh functions. The SHEPWM voltage waveforms of the multilevel inverter in different waveform band modes are analyzed and the multiband SHEPWM solution model with modulation indexes ranging from high to low is established. Finally, the unity SHEPWM equations of the piecewise linear system of equations can be solved using the Walsh function transformation.

This paper is organized as follows: The harmonic elimination model of the multiband SHEPWM is presented in Section 2. Section 3 introduces the Walsh function. The Walsh function transform is proposed to solve the angle of the multiband equation in Section 4. Simulations and experimental results are presented in Section 5 and Section 6, respectively, to validate the performance of the proposed method. Finally, the conclusions are summarized in Section 7.

2. Harmonic Elimination Model for Multiband SHEPWM

The topology structure of the H-bridge cascaded seven-level inverters is shown in Figure 1. It is composed of a series connection of three H-bridge cells. Each H-bridge unit is powered by a separate DC power supply. Assuming that the input voltages of the three H-bridge cells are V_dc1 = V_dc2 = V_dc3 = E, the corresponding output voltages of each cell are V_H1, V_H2, V_H3. The generated output voltage of each cell is composed of the three voltage levels +E, −E, and 0, using a different combination of the four switches.

Table 1. The outputs corresponding to different output state functions.

S1S2S3	VH1	VH2	VH3	VAN
111	E	E	E	3E
110	E	E	0	2E
101	E	0	E
011	0	E	E
100	E	0	0	E
010	0	E	0
001	0	0	E
000	0	0	0	0
−100	−E	0	0	−E
0−10	0	−E	0
00−1	0	0	−E
−1−10	−E	−E	0	−2E
−10−1	−E	0	−E
0−1−1	0	−E	−E
−1−1−1	−E	−E	−E	−3E

shows the output voltage of the inverter under different output functions. Where S₁, S₂ and S₃ are the output level function of H1, H2 and H2 respectively. It can be seen from the table that the phase voltage V_AN of the three-element CHB inverter canh1 have a total of seven levels of ±3E, ±2E, ±E, and 0. In order to avoid energy circulation, the output states of the opposite polarity of the output voltage of each unit are removed in Table 1. It can be seen from the table that when the output levels of the phase voltage V_AN are ±2E and ±E, each unit has a redundant output state function.

The multiband SHEPWM multilevel inverter waveform is shown in Figure 2.

The graph of the seven−level phase voltage is shown in Figure 2a, which is transformed by a Fourier transform as follows:

$$u(t)=c_0+{\displaystyle \sum _{n=1}^{\infty }(c_n\cos (n\omega t)+b_n\sin (n\omega t))}$$

(1)

where

$$\{\begin{array}{l}{c}_0=\frac{1}{T}{\displaystyle {\int }_0^{T}u(t)}dt\\ c_n=\frac{2}{T}{\displaystyle {\int }_0^{T}u(t)\cos (n\omega t)}dt,n=1,2,3\cdots \end{array}$$

(2)

Expanding and analyzing bn in Equation (1) we get:

$$u(\omega t)=\{\begin{array}{c}{b}_n=0,\begin{array}{ccc}n& \mathrm{is}& \mathrm{even}\end{array}\\ b_n=\frac{4E}{n\pi }{\displaystyle \sum _{k=1}^N{p}_k\cos (n{\theta }_k),\begin{array}{ccc}n& is& odd\end{array}}\end{array}$$

(3)

The switching angles denoted by ${\theta }_k$ (in radians) are $0^0\le {\theta }_1<{\theta }_2<\cdots <{\theta }_N\le {90}^0$. Here, E is the voltage value for each unit of dc; b_n is the harmonic coefficient; p_k is marked + 1 when it rises at the point of ${\theta }_k$ and −1 when it decline. The graph of the half-wave symmetry waveform of the seven-level ladder-type modulation is shown in Figure 2b, in which the output voltages of each unit H-bridge are V_H1, V_H2, and V_H3. According to Equation (3), a harmonic elimination of nonlinear equations can be established. There are no solutions when the modulation index is lower than 0.4, as the five- and seven-fold harmonics are eliminated by the numerical algorithm. According to the above situation, a multiband modulation scheme is proposed in this paper by broadening the index of modulation. The modulation index can be broadened as shown in Figure 2c,d.

By observing the staircase waveform, it is easy to find that the waveform transformation rules between different wavebands are used to fold down the waveform with the dotted line at the top. The applicable modulation indexes shown in Figure 2b to Figure 2d for modulation modes A, B, and C are [0.4, 1], [0.3, 0.3], and [0, 0.3], respectively. The modulation index can be broadened to 0–1 according to the above multiband modulation method. The correct switch angles under different modulation indexes can be obtained by establishing SHEPWM nonlinear equations in accordance with the waveforms of different wavebands, for which the modulation index has different ranges.

3. Walsh Functions

The Walsh function is a standard orthogonal function system. It has only two values, 0 and 1, but the value of the function changes between 0 and 1, so the theoretical analysis is more complex than the trigonometric function. In recent years, due to the rapid development of new disciplines and technologies, Walsh function theory has been widely used in radar, communication, medicine, imaging, and other fields. The introduction of the Walsh function into the electrical field is of great significance.

The Walsh functions are a set of complete orthogonal function systems that can be expressed as Wal(n,t), where n is usually the zero crossing point (ZCP) number under time T of a certain order for the definition of Walsh functions, where t is the normalized time spacing of T.

The quarter-cycle waveform normalized by Figure 2a is shown in Figure 3, where T (0–1) is divided into 2^m uniform areas, c is the normalized value of θ₁, and the interval number of the rising edge is marked with N. Similar to Fourier expressions, the Walsh functions are obtained using the waveform transformed by the Walsh function, which are shown below:

$$f(t)=a_0+W_{\mathrm{a}1}(0,t)+{\displaystyle \sum _{i=1}^{N-1}{a}_i{W}_{\mathrm{a}1}(i,t)}$$

(4)

$$\{\begin{array}{l}{a}_0=\frac{2}{T}{\displaystyle {\int }_0^{T}f(t)W_{\mathrm{a}1}(0,t)dt,i=1,2,3,\cdots }\\ a_1=\frac{1}{T}{\displaystyle {\int }_0^{T}f(t)W_{\mathrm{a}1}(i,t)dt,i=1,2,3,\cdots }\end{array}$$

(5)

Equation (4) is simplified according to the symmetry as follows:

$$f(t)={\displaystyle \sum _{i=1}^{N/2}{W}_{4i-3}{W}_{\mathrm{a}1}(4i-3,t)}$$

(6)

In Equation (6), $W_{4i-3}=F^w(4i-3)$ is the coefficient of $W_{\mathrm{a}1}(4i-3,t),$ meaning that the corresponding waveform of the Walsh functions is f(t). A quarter cycle of the normalized three-level PWM waveform is shown in Figure 2. According to Figure 2, Equation (7) can be deduced as follows:

$$\begin{array}{ll}{F}^w(4i-3)& ={\displaystyle {\int }_0^{1}f(t)W_{\mathrm{a}1}(4i-3,t)dt}\\ & =\frac{1}{2^{m-2}}{\displaystyle \sum _{n=N+1}^{2^m/4}{W}_{\mathrm{a}1}(4i-3,n)}+(\frac{N}{2^{m-2}}-4c)W_{\mathrm{a}1}(4i-3,N),i=1,2,3,\cdots \end{array}$$

(7)

As seen in Equations (1) and (4), some transformations can be taken to convert the transcendental equations under the domain of the Fourier transform into piecewise linear equations under the domain of the Walsh function. According to the Equation (3), we can get:

$$u(wt)=f(wt)={\displaystyle \sum _{n=1}^{\infty }{b}_{2n-1}\sin [(2n-1)wt]}$$

(8)

$$b_{2n-1}=\frac{8}{\pi }{\displaystyle {\int }_0^{\pi /4}f(wt)\sin [(2n-1)wt]d(wt)},n=1,2,3,\cdots$$

(9)

Under the same angular frequency, Equation (6) can be developed under the domain of the Walsh function to infinity, Equation (8) can be made equivalent with it under the Fourier transform, then Equation (6) can be brought into Equation (9), and we can get:

$$b_{2n-1}=\frac{8}{\pi }{\displaystyle {\int }_0^{\pi /4}{\displaystyle \sum _{i=1}^{\infty }{W}_{4i-3}{W}_{\mathrm{a}1}(4i-3,wt)}\sin [(2n-1)wt]d(wt)},n=0,1,2,3,\cdots$$

(10)

Assuming that:

$$B_{2n-1}(W_{\mathrm{a}1}(4i-3,wt))=\frac{8}{\pi }{\displaystyle {\int }_0^{\pi /4}{W}_{\mathrm{a}1}(4i-3,wt)\sin [(2n-1)wt]d(wt)}$$

(11)

Equation (11) is the Fourier coefficient of the (2n−1) harmonic for the Walsh function $W_{\mathrm{a}1}(4i-3,wt),$ then we plug in Equation (10) into the deformation:

$$b_{2n-1}={\displaystyle \sum _{i=1}^{\infty }{W}_{4i-3}{B}_{2n-1}[W_{\mathrm{a}1}(4i-3,wt)]}$$

(12)

Equation (11) is complex to solve, while a nonrecursive formula of the Walsh function Fourier transformation is given in [15,16]. For k = 2i−1, i = 1,2,3,…, m = 4n−3, n = 1,2,3,…:

$$B_k(W_{\mathrm{a}1}(m,wt))=2{(-1)}^{g_0}{(-j)}^d\times [{\displaystyle \prod _{x=0}^{M-1}\cos (\frac{\pi k}{2^{x+1}}-\frac{\pi g_x}{2})]}\frac{\sin (\pi k/2^M)}{\pi k/2^M}$$

(13)

In which j = (−1)^1/2 records only plus or minus of (−j)^d when calculating, while the variable instructions of Equation (13) are as follows:

• Record the order of the Walsh function as m, making its expressions for e binary;
• Convert e as gray code g, the record the x bite of the gray code for g_x;
• M is the total number of g, while d is the total number of 1 contained in g.

4. Solving Equations

Summarizing Equation (11):

[b] = [U] = [B][F]

(14)

In this process, [b] is the harmonic coefficient matrix, which is replaced by [U] for simplification. The matrixes [B] and [F] are the Walsh transform matrix and Walsh coefficient matrix.

We assume that the harmonic coefficient U_k of the k time eliminated harmonic component is zero, which satisfies the fundamental current amplitude at the same time. We then establish a harmonic elimination system for linear equations and solve the switch angle, the calculation steps for which are shown as follows:

• Determine the total interval number N according to the precision of the harmonic elimination, give a set of relatively accurate initial values, determine the interval after normalization, and serve the initial conditions;
• According to interval number N and the harmonic number k, calculate the Walsh transform matrix [B];
• According to interval number N and the determined initial conditions, calculate the Walsh coefficient matrix [F];
• According to Equation (14), establish the piecewise linear system of equations;
• Solve the piecewise linear system of equations and get the corresponding number of switch angles.

Taking the seven-level waveband A as an example, the accuracy of the interval number N is shown as 64 [16], the initial switch angles for a₁, a₂, and a₃ are 11.52°, 28.728°, and 57.096°. The interval numbers of the switch angles are 3, 6, and 11, respectively. The scope of the initial angles is shown as:

2 × 360/64 ≤ a₁ ≤ 3 × 360/64, 5 × 360/64 ≤ a₂ ≤ 6 × 360/64, 10 × 360/64 ≤ a₃ ≤ 11 × 360/64

(15)

For the seven-level inverter’s three-phase system, five and seven-fold harmonics are generally eliminated. The Walsh coefficient matrix [F] can be obtained by the selection of the initial conditions and by using Equation (7), as shown in

Table 2. Walsh coefficient matrix [F].

Fw(1)	−4a1 − 4a2 − 4a3 + 3	Fw(5)	−4a1 − 4a2 + 4a3 − 1
Fw(9)	−4a1 + 4a2 + 4a3 − 1	Fw(13)	−4a1 + 4a2 − 4a3
Fw(17)	4a1 + 4a2 + 4a3 − 5/4	Fw(21)	4a1 + 4a2 − 4a3 + 1/4
Fw(25)	4a1 − 4a2 − 4a3 + 3/4	Fw(29)	4a1 − 4a2 + 4a3 − 3/4
Fw(33)	4a1 + 4a2 + 4a2 − 9/8	Fw(37)	4a1 + 4a2 − 4a3 + 1/8
Fw(41)	4a1 − 4a2 − 4a3 + 7/8	Fw(45)	4a1 − 4a2 + 4a3 − 3/8
Fw(49)	−4a1 − 4a2 − 4a3 + 9/8	Fw(53)	−4a1 − 4a2 + 4a3 − 1/8
Fw(57)	−4a1 + 4a2 + 4a3 − 7/8	Fw(61)	−4a1 + 4a2 − 4a3 + 3/8

. The corresponding coefficient of the Walsh function is F^w(4i−3), where i = 1,2,3…, and using the interval number N and the eliminated harmonic number k, the Walsh transform matrix [B] can be obtained, as shown in

Table 3. Walsh coefficient matrix [B].

k	Wal(1,t)	Wal(5,t)	Wal(9,t)	Wal(13,t)	Wal(17,t)	Wal(21,t)	Wal(25,t)	Wal(29,t)	Wal(33,t)	Wal(37,t)	Wal(41,t)	Wal(45,t)	Wal(49,t)	Wal(53,t)	Wal(57,t)	Wal(61,t)
1	1.2732	−0.5274	−0.1049	−0.2533	−0.0249	0.0103	−0.0519	−0.1254	−0.0062	0.0026	0.0005	0.0012	−0.0124	0.0052	−0.0259	−0.0626
5	0.2546	0.6148	0.9201	−0.3811	−0.2037	−0.4918	0.3286	−0.1361	−0.0341	−0.0823	−0.1232	0.0510	−0.0955	−0.2305	0.1540	−0.0638
7	0.1819	−0.0753	0.3788	0.9144	−0.7505	0.3108	0.0618	0.1493	−0.0534	0.0221	−0.1112	−0.2685	−0.3272	0.1355	0.0270	0.0651
11	0.1157	0.2794	0.4182	−0.1732	0.3241	0.7824	−0.5228	0.2166	−0.1298	−0.3134	−0.4690	0.1943	0.1038	0.2507	−0.1675	0.0694
13	0.0979	0.2365	−0.1580	0.0654	0.2157	0.5208	0.7795	−0.3229	−0.2395	−0.5781	0.3863	−0.1600	0.0485	0.1172	0.1754	−0.0726

. As space is limited, only some of the data are shown.

Combined with Equation (14):

U₁ = 4.5327 − 1.9431a₁ − 4.1112a₂ − 6.8591a₃, U₅ = − 0.6268 − 7.4570a₁ − 3.3862a₂ + 7.1595a₃,
U₇ = 0.8745 − 7.7586a₁ + 4.6724a₂ − 6.3000a₃

(16)

Assuming that the harmonic coefficients U₅ and U₇ are zero, according to Equation (16):

a₁ = −0.0140U₁ + 0.0748, a₂ = −0.1186U₁ + 0.4422, a₃ = −0.0707U₁ + 0.3746

(17)

According to Equations (15) and (17), we get 1.4172 ≤ U₁ ≤ 1.4523, which meets the fundamental current amplitude conditions. Thus, the first set of solutions can be obtained. Then, the limit value, which met Equation (17), should be substituted into Equation (16), and the other set of switch angles can be obtained. After determining the range, the process above can be repeated and another set of solutions can be obtained. All switch angles can also be obtained.

Taking the seven-level inverter multiband ladder-type modulation as an example, the switching angles for SHEPWM can be solved using Walsh transforms. The selected interval number N is 64 and the eliminated 5th and 7th harmonic are calculated in accordance with the above example. Along with the multiband modulation mode, as shown in Figure 2, the harmonic elimination switch angles of A, B, and C mode can be obtained, as shown in

Table 4. Results of A waveband switching angles.

Range of Fundamental Amplitude(U1)	Switch Angels (a1,a2,a3) after Normalization	Range of Fundamental Amplitude (U1)	Switch Angels (a1,a2,a3) after Normalization
3.2202–3.1742	a1 = 0.2442U1 − 0.7395	2.4257–2.2879	a1 = −0.1134U1 + 0.3532
	a2 = −0.3261U1 + 1.0976		a2 = −0.0096U1 + 0.1747
	a3 = −0.0928U1 + 0.4430		a3 = −0.0653U1 + 0.3352
3.1738–3.0868	a1 = 0.0288U1 − 0.0557	2.2870–2.2762	a1 = −0.1197U1 + 0.3675
	a2 = −0.1433U1 + 0.5173		a2 = 0.0880U1 − 0.0484
	a3 = −0.0881U1 + 0.4282		a3 = −0.1376U1 + 0.5007
3.0839–3.0736	a1 = 0.0731U1 − 0.1933	2.2769–2.1240	a1 = −0.0734U1 + 0.2623
	a2 = −0.1749U1 + 0.6157		a2 = 0.0186U1 + 0.1093
	a3 = −0.0793U1 + 0.4008		a3 = −0.1022U1 + 0.4202
3.0698–2.9380	a1 = −0.0140U1 + 0.0748	2.1236–1.9870	a1 = −0.0212U1 + 0.1515
	a2 = −0.1186U1 + 0.4422		a2 = −0.0297U1 + 0.2120
	a3 = −0.0707U1 + 0.3746		a3 = −0.0912U1 + 0.3968
2.9363–2.8491	a1 = −0.0588U1 + 0.2064	1.9825–1.9538	a1 = −0.0214U1 + 0.1518
	a2 = −0.1060U1 + 0.4050		a2 = −0.0215U1 + 0.1956
	a3 = −0.0555U1 + 0.3300		a3 = −0.0963U1 + 0.4069
2.8485–2.7871	a1 = −0.0425U1 + 0.1599	1.9526–1.9324	a1 = 0.0270U1 + 0.0572
	a2 = −0.1037U1 + 0.3984		a2 = −0.0757U1 + 0.3015
	a3 = −0.0586U1 + 0.3388		a3 = −0.0833U1 + 0.3814
2.7887–2.7219	a1 = −0.0826U1 + 0.2717	1.9270–1.9169	a1 = 0.0305U1 + 0.0506
	a2 = −0.1201U1 + 0.4443		a2 = −0.0680U1 + 0.2866
	a3 = −0.0269U1 + 0.2504		a3 = −0.0898U1 + 0.3936
2.7212–2.6417	a1 = −0.0738U1 + 0.2477	1.9172–1.8738	a1 = −0.0101U1 + 0.1283
	a2 = −0.0974U1 + 0.3823		a2 = −0.0574U1 + 0.2663
	a3 = −0.0385U1 + 0.2820		a3 = −0.0706U1 + 0.3572
2.6424–2.5541	a1 = −0.1099U1 + 0.3432	1.8994–1.7432	a1 = −0.0077U1 + 0.1240
	a2 = −0.1373U1 + 0.4878		a2 = −0.0555U1 + 0.2627
	a3 = 0.0151U1 + 0.1402		a3 = −0.0731U1 + 0.3618
2.5545–2.4011	a1 = −0.1019U1 + 0.3228	1.7439–1.5053	a1 = 0.0038U1 + 0.1038
	a2 = −0.0762U1 + 0.3320		a2 = −0.0725U1 + 0.2924
	a3 = −0.0276U1 + 0.2494		a3 = −0.0655U1 + 0.3486
2.5096–2.4252	a1 = −0.1314U1 + 0.3968	-	-
	a2 = −0.1281U1 + 0.4621		-
	a3 = 0.0376U1 + 0.0856		-

,

Table 5. Results of B waveband switching angles.

Range of Fundamental Amplitude (U1)	Switch Angels (a1,a2,a3) after Normalization
1.4410–1.4066	a1 = −0.0201U1 + 0.1430, a2 = −0.0669U1 + 0.2816, a3 = 0.0517U1 + 0.1755
1.4288–1.2282	a1 = −0.0332U1 + 0.1623, a2 = −0.0779U1 + 0.2988, a3 = 0.0294U1 + 0.2062
1.2268–1.1439	a1 = −0.0410U1 + 0.1719, a2 = −0.1216U1 + 0.3523, a3 = −0.0205U1 + 0.2674
1.1417–1.0902	a1 = −0.0381U1 + 0.1685, a2 = −0.1014U1 + 0.3293, a3 = −0.0015U1 + 0.2458
1.0904–1.0538	a1 = −0.0415U1 + 0.1722, a2 = −0.2549U1 + 0.4967, a3 = −0.1580U1 + 0.4165

and

Table 6. Results of C waveband switching angles when N = 64.

Range of Fundamental Amplitude (U1)	Switch Angels (a1,a2,a3) after Normalization
1.1416–1.0473	a1 = 0.0384U1 − 0.0022, a2 = 0.1656U1 + 0.0297, a3 = 0.0265U1 + 0.1995
1.0481–0.9226	a1 = 0.0318U1 + 0.0047, a2 = 0.0816U1 + 0.1176, a3 = −0.0565U1 + 0.2865
0.9219–0.8648	a1 = 0.0298U1 + 0.0065, a2 = 0.0947U1 + 0.1056, a3 = −0.0432U1 + 0.2742
0.8658–0.8030	a1 = 0.0165U1 + 0.0180, a2 = 0.0745U1 + 0.1230, a3 = −0.0618U1 + 0.2902
0.8062–0.6579	a1 = 0.0209U1 + 0.0144, a2 = 0.0749U1 + 0.1226, a3 = −0.060 5U1 + 0.2893
0.6560–0.6484	a1 = −0.0035U1 + 0.0304, a2 = 0.0710U1 + 0.1253, a3 = −0.0637U1 + 0.2913

.

The linear equations transformed by Walsh functions cannot be solved when the seven-level inverter modulation index of 64 under the C band mode is below 0.170, as shown in Table 5. To solve this problem, the solution involving expanding the interval number is adopted, so that the interval number N is taken as 128 and the C band mode switch angles can be evaluated, as shown in

Table 7. Results of C waveband switching angles when N = 128.

Range of Fundamental Amplitude(U1)	Switch Angels (a1,a2,a3) after Normalization	Range of Fundamental Amplitude (U1)	Switch Angels (a1,a2,a3) after Normalization
1.1869–1.1867	a1 = −41.3455U1 + 49.0735	0.8708–0.8319	a1 = −0.2008U1 + 0.2608
	a2 = −7.4786U1 + 8.9118		a2 = −0.1062U1 + 0.2033
	a3 = −1.8609U1 + 2.2777		a3 = −0.1063U1 + 0.2415
1.1867–1.1865	a1 = −15.2520U1 + 18.1081	0.8319–0.8234	a1 = −0.3251U1 + 0.3642
	a2 = −7.9191U1 + 9.4347		a2 = −0.2547U1 + 0.3269
	a3 = −1.8210U1 + 2.2305		a3 = −0.1275U1 + 0.2591
1.1864–1.1862	a1 = −9.4425U1 + 11.2151	0.8234–0.8094	a1 = −0.2590U1 + 0.3098
	a2 = −5.3968U1 + 6.4420		a2 = −0.2145U1 + 0.2938
	a3 = −2.1462U1 + 2.6162		a3 = −0.1514U1 + 0.2788
1.1862–1.1860	a1 = −6.7262U1 + 7.9930	0.8090–0.8033	a1 = −0.2296U1 + 0.2860
	a2 = −3.3420U1 + 4.0046		a2 = −0.1563U1 + 0.2467
	a3 = −1.1600U1 + 1.4463		a3 = −0.1191U1 + 0.2526
1.1860–1.1845	a1 = −5.0229U1 + 5.9730	0.8033–0.7915	a1 = −0.3709U1 + 0.3995
	a2 = −3.8515U1 + 4.6089		a2 = −0.3290U1 + 0.3854
	a3 = −1.1952U1 + 1.4882		a3 = −0.1464U1 + 0.2745
1.1844–1.1822	a1 = −3.0451U1 + 3.6302	0.7794–0.7616	a1 = −0.2836U1 + 0.3304
	a2 = −3.3748U1 + 4.0443		a2 = −0.2769U1 + 0.3442
	a3 = −1.4747U1 + 1.8192		a3 = −0.1789U1 + 0.3003
1.1821–1.1818	a1 = −1.9737U1 + 2.3637	0.7795–0.7706	a1 = −0.4671U1 + 0.4735
	a2 = −3.0006U1 + 3.6017		a2 = −0.5118U1 + 0.5272
	a3 = −1.9294U1 + 2.3566		a3 = −0.2255U1 + 0.3366
1.1818–1.1811	a1 = −2.1677U1 + 2.5930	0.7707–0.7666	a1 = −0.3392U1 + 0.3749
	a2 = −3.5741U1 + 4.2796		a2 = −0.4585U1 + 0.4862
	a3 = −2.0826U1 + 2.5378		a3 = −0.2967U1 + 0.3915
1.1810–1.1789	a1 = −1.4996U1 + 1.8038	0.7666–0.7582	a1 = −0.2816U1 + 0.3307
	a2 = −1.9516U1 + 2.3632		a2 = −0.3179U1 + 0.3784
	a3 = −1.0120U1 + 1.2733		a3 = −0.2089U1 + 0.3242
1.1790–1.1757	a1 = −0.9903U1 + 1.2034	0.7582–0.7522	a1 = −0.4541U1 + 0.4615
	a2 = −1.8240U1 + 2.2130		a2 = −0.5405U1 + 0.5472
	a3 = −1.3131U1 + 1.6283		a3 = −0.2542U1 + 0.3586
1.1756–1.1748	a1 = −1.1810U1 + 1.4275	0.7521–0.7385	a1 = −0.3132U1 + 0.3555
	a2 = −2.1990U1 + 2.6537		a2 = −0.4749U1 + 0.4978
	a3 = −1.4225U1 + 1.7569		a3 = −0.3257U1 + 0.4124
1.1749–1.1748	a1 = −0.7821U1 + 0.9590	0.7383–0.7357	a1 = −0.2448U1 + 0.3051
	a2 = −2.8460U1 + 3.4141		a2 = −0.3101U1 + 0.3761
	a3 = −2.4265U1 + 2.9366		a3 = −0.2246U1 + 0.3377
1.1747–1.1681	a1 = −0.5533U1 + 0.6901	0.7356–0.7350	a1 = −0.3861U1 + 0.4090
	a2 = −1.1432U1 + 1.4135		a2 = −0.4911U1 + 0.5094
	a3 = −0.9186U1 + 1.1650		a3 = −0.2604U1 + 0.3640
1.1682–1.1670	a1 = −0.3997U1 + 0.5107	0.7350–0.7165	a1 = −0.2548U1 + 0.3126
	a2 = −1.6291U1 + 1.9812		a2 = −0.4231U1 + 0.4594
	a3 = −1.5353U1 + 1.8855		a3 = −0.3206U1 + 0.4083
1.1671–1.1586	a1 = −0.2959U1 + 0.3897	0.7163–0.7136	a1 = −0.1849U1 + 0.2625
	a2 = −0.6242U1 + 0.8085		a2 = −0.4672U1 + 0.4909
	a3 = −0.6143U1 + 0.8107		a3 = −0.4319U1 + 0.4879
1.1581–1.1577	a1 = −0.3442U1 + 0.4455	0.7139–0.6966	a1 = −0.1378U1 + 0.2288
	a2 = −0.7335U1 + 0.9351		a2 = −0.2474U1 + 0.3341
	a3 = −0.6575U1 + 0.8607		a3 = −0.2562U1 + 0.3626
1.1575–1.1558	a1 = −0.2594U1 + 0.3473	0.6967–0.6895	a1 = −0.1893U1 + 0.2647
	a2 = −1.0795U1 + 1.3355		a2 = −0.3303U1 + 0.3918
	a3 = −1.0788U1 + 1.3484		a3 = −0.2852U1 + 0.3828
1.1558–1.1411	a1 = −0.1997U1 + 0.2785	0.6896–0.6860	a1 = −0.1361U1 + 0.2280
	a2 = −0.3944U1 + 0.5438		a2 = −0.3626U1 + 0.4141
	a3 = −0.4443U1 + 0.6151		a3 = −0.3691U1 + 0.4407
1.1411–1. 1393	a1 = −0.1654U1 + 0.2393	0.6858–0.6499	a1 = −0.0994U1 + 0.2028
	a2 = −0.6211U1 + 0.8025		a2 = −0.1763U1 + 0.2863
	a3 = −0.7001U1 + 0.9070		a3 = −0.2177U1 + 0.3368
1.1392–1.1133	a1 = −0.1371U1 + 0.2070	0.6501–0.6486	a1 = −0.0714U1 + 0.1847
	a2 = −0.2000U1 + 0.3228		a2 = −0.0816U1 + 0.2248
	a3 = −0.3019U1 + 0.4533		a3 = −0.1495U1 + 0.2925
1.1133–1.1101	a1 = −0.1155U1 + 0.1829	0.6487–0.6109	a1 = −0.0602U1 + 0.1774
	a2 = −0.0616U1 + 0.1687		a2 = −0.0856U1 + 0.2274
	a3 = −0.1802U1 + 0.3178		a3 = −0.1642U1 + 0.3020
1.1109–1.0968	a1 = −0.1309U1 + 0.2001	0.6112–0.6028	a1 = −0.0762U1 + 0.1872
	a2 = −0.0965U1 + 0.2074		a2 = −0.1152U1 + 0.2455
	a3 = −0.1935U1 + 0.3326		a3 = −0.1768U1 + 0.3097
1.0964–1.0759	a1 = −0.1214U1 + 0.1897	0.5390–0.4566	a1 = −0.0435U1 + 0.1683
	a2 = −0.1176U1 + 0.2305		a2 = −8.9980e−4U1 + 0.1792
	a3 = −0.2228U1 + 0.3647		a3 = −0.0938U1 + 0.2615
1.0761–1.0435	a1 = −0.1081U1 + 0.1753	0.4573–0.4563	a1 = −0.0594U1 + 0.1756
	a2 = −0.0209U1 + 0.1265		a2 = −0.0218U1 + 0.1888
	a3 = −0.1367U1 + 0.2721		a3 = −0.0983U1 + 0.2636
1.0442–1.0215	a1 = −0.1268U1 + 0.1949	0.4562–0.3584	a1 = −0.0450U1 + 0.1690
	a2 = −0.0605U1 + 0.1679		a2 = 0.0106U1 + 0.1740
	a3 = −0.1531U1 + 0.2892		a3 = −0.0799U1 + 0.2552
1.0212–0.9777	a1 = −0.1128U1 + 0.1806	0.3585–0.2635	a1 = −0.0351U1 + 0.1655
	a2 = −0.0041U1 + 0.1103		a2 = 0.0311U1 + 0.1667
	a3 = −0.1081U1 + 0.2432		a3 = −0.0690U1 + 0.2513
0.9780–0.9529	a1 = −0.1416U1 + 0.2088	0.2641–0.2458	a1 = −0.0551U1 + 0.1708
	a2 = −0.0499U1 + 0.1550		a2 = 0.0086U1 + 0.1726
	a3 = −0.1215U1 + 0.2564		a3 = −0.0713U1 + 0.2519
0.9517–0.9185	a1 = −0.1254U1 + 0.1933	0.2453–0.1334	a1 = −0.0451U1 + 0.1683
	a2 = −0.0085U1 + 0.1156		a2 = 0.0253U1 + 0.1685
	a3 = −0.0937U1 + 0.2298		a3 = −0.0645U1 + 0.2502
0.9185–0.8943	a1 = −0.1720U1 + 0.2361	0.1347–0.1254	a1 = −0.0242U1 + 0.1655
	a2 = −0.0698U1 + 0.1718		a2 = 0.0451U1 + 0.1658
	a3 = −0.1055U1 + 0.2407		a3 = −0.0655U1 + 0.2504
0.8952–0.8762	a1 = −0.1592U1 + 0.2246	0.1246–0.0029	a1 = −0.0128U1 + 0.1641
	a2 = −0.0656U1 + 0.1681		a2 = 0.0593U1 + 0.1641
	a3 = −0.1134U1 + 0.2478		a3 = −0.0627U1 + 0.2500
0.8757–0.8709	a1 = −0.1444U1 + 0.2117	-	-
	a2 = −0.0297U1 + 0.1367		-
	a3 = −0.0896U1 + 0.2269		-

.

As is known, after the interval number N changes from 64 to 128, the number of modulation segments increases dramatically. To get a lower modulation index for the switch angles, the precision of the switch angles is improved as well, but the calculation is increased with the corresponding computation.

5. Simulation Result

Based on the data in the above tables, this paper draws the change track of the angle of the harmonic elimination switch with the amplitude of the fundamental voltage, as shown in Figure 4.

It can be seen from Figure 4 that the correct switching angle can be obtained for each band of the modulation index in the range of 0–1. Moreover, it can be found that there are multiple sets of switch angle trajectories in each cell near each switch band. These redundant switch angles can increase the flexibility of the switch angle selection and improve the control performance in the actual control.

To test and verify the correctness of the above theoretical analysis and the solving solution, MATLAB and SIMULINK software were used for the seven-level inverter of the multiband SHEPWM based on the Walsh function transform to build the model and do the experiment. The simulation parameters were as follows: the independent dc voltage source was 50; A, B, and C bands took fundamental voltage amplitude U₁ as 3.15, 1.32, and 1.0, respectively, when choosing the switch angles. The output voltage waveforms and spectrum distribution are shown in Figure 5, Figure 6 and Figure 7.

Figure 5 shows the waveform of the output phase voltage V_AN and line voltage V_AB in the A band when the fundamental voltage amplitude is 3.15, along with the Fast Fourier Transform (FFT) analysis.

Here, V_AN is the A phase voltage, V_AB is the line voltage, V_ANi is the A phase voltage harmonic, V_ABi is the A phase line voltage harmonic, V_AN1 is the fundamental wave of the A phase, and V_ABi is the fundamental wave of the line voltage between phase A and phase B.

According to Figure 5a, it can be seen that the phase voltage V_AN is 7 levels and the line voltage V_AB is 13 levels, which conforms to the design of the A band. From Figure 5b, it can be seen that the Total Harmonic Distortion (THD) of the phase voltage V_AN is 13.32%, in which the 5th and 7th harmonics are eliminated. The THD of the line voltage V_AB is 8.84%, in which the 3rd, 5th, and 7th harmonics are eliminated, which proves the correctness of the switch angle calculation. At the same time, a high-quality voltage waveform can be obtained under the SHEPWM modulation.

Figure 6 shows the waveform of the output phase voltage V_AN and the line voltage V_AB in the B band when the fundamental voltage amplitude is 1.32, along with the FFT analysis.

According to Figure 6a, it can be seen that the phase voltage V_AN is 5 levels, while the line voltage V_AB is 7 levels, which conforms to the design of the B band. From Figure 6b, it can be seen that the THD of the phase voltage V_AN is 52.57%, in which the 5th and 7th harmonics are eliminated. The THD of line voltage V_AB is 23.87%, in which the 3rd, 5th, and 7th harmonics are eliminated, which proves the correctness of the switch angle calculation.

Figure 7 shows the waveform of the output phase voltage V_AN and the line voltage V_AB in the C band when the fundamental voltage amplitude is 1.0, along with the FFT analysis.

As shown in the Figure 7a, the phase voltage V_AN is 3 levels, while the line voltage V_AB is 5 levels, which conforms to the design of the C band. From Figure 7b, it can be seen that the THD of phase voltage V_AN is 47.48%, in which the 5th and 7thharmonics are eliminated. The THD of line voltage V_AB is 32.66%, in which the 3rd, 5th, and 7th times harmonics are eliminated, which proves the correctness of the switch angle calculation.

As shown in Figure 5, Figure 6 and Figure 7, the 5th and 7th harmonics have basically been eliminated, which can be proved using the control technology of the SHEPWM multilevel inverter based on the Walsh transform, combined with multiband modulation, which is feasible in this paper. Because of the limitation of the calculation accuracy and the precision of the numerical algorithm needed to solve the switch angles, the harmonic cannot be completely eliminated. Usually it will have a small amount of deviation, and the calculation of the accuracy of the switch angles under different modulation indexes is not the same.

In this paper, the effects on harmonic elimination under various modulations based on multilevel inverters have been studied, and 5th and 7th harmonic indexes have been listed. The amplitudes of 5th and 7th harmonics, which have been completely eliminated, theoretically are actually not zero, as shown in Figure 8a. The 5th harmonic indexes based on 64 and 128 phase voltage interval schemes under the C band model are given in Figure 8b.

As shown in the Figure 8a, although the 5th and 7th harmonics have not been completely eliminated, there is a very small amount of error. The values are different under various modulation indexes. The errors indicate that the switch angles of the SHEPWM are not accurate, but they can be completely ignored in actual engineering control. From Figure 8b, the harmonic indexes in the 128-interval scheme are lower than the 64-interval scheme overall. This proves that the larger the interval scheme that is selected, the more precise the switch angels are and the better harmonic elimination effect it has.

6. Experiment Verification

In order to prove the feasibility of the algorithm proposed above and the harmonic elimination effect in the practical engineering, an inverter experiment platform is set up, whose parameters are reported in

Table 8. Key parameters of the experiment.

Paremeter	Value
Number of H-bridges	3
DC bus voltage	22 V
Load	40 Ω
Output voltage frequency	50 Hz

. An interface board is designed to do the isolated signal conditioning required for Analog to Digital Converter (ADC) module of a Digital Signal Processor(DSP)controller. The utilized processor is a TMS320F2812(Texas Instruments, Texas, USA) controller used to perform all necessary functions in the discrete time domain.

The output phase voltage V_AN of the inverter under the SHEPWM modulation when the modulation index is 0.75 and its spectrum are shown in Figure 9a,b, respectively. Figure 9c,d show the output phase voltage V_AN of the inverter when the modulation is 0.46 and its spectrum, respectively. Figure 9e,f show the output phase voltage V_AN of the inverter when the modulation is 0.27 and its spectrum, respectively.

When the modulation index is 0.75, it corresponds to the SHEPWM modulation in the A band. Figure 9a shows that the output phase voltage is 7 levels. In addition, Figure 9b shows that the 5th and 7th harmonics are basically eliminated and the THD of the phase voltage V_AN is 34.9%.

When the modulation index is 0.46, it corresponds to the SHEPWM modulation in the B band. Figure 9b shows that the output phase voltage is 5 levels. In addition, Figure 9e shows that the 5thand 7th harmonics are basically eliminated and the THD of the phase voltage V_AN is 43.1%.

When the modulation index is 0.27, it corresponds to the SHEPWM modulation in the C band. Figure 9c shows the output phase voltage is 3 levels. In addition, Figure 9f shows that the 5th and 7th harmonics are basically eliminated and the THD of the phase voltage V_AN is 63.2%.

Overall, the correctness of the switch angle obtained by the multilevel inverter for the multiband SHEPWM control technology based on the Walsh function is proven. In addition, the multiband SHEPWM control technology of the multilevel inverter based on the Walsh function proposed in this paper can solve the problem of the limited modulation index in the specific harmonic elimination PWM technology effectively.

7. Conclusions

The following conclusions can be summarized from the analysis and research for the multiband SHEPWM control method based on the Walsh function transform for the multilevel inverter.

• By combining the Walsh function transformation with the multiband SHEPWM control method for the multilevel inverter, the regulation operation range of the inverter can be expanded to 0–1. At the same time, the traditional SHEPWM nonlinear equations are transformed into piecewise linear equations, and the online calculation of the switching angle of the SHEPWM multilevel inverter is obtained, which makes real-time harmonic elimination control possible.
• The switch angles cannot be solved with the fundamental current amplitude, which is below 0.656, by using 64-interval accuracy for the Walsh function transformation, but the switch angles under the lower precision modulation index can be solved by using the 128-interval Walsh function transformation.
• The simulation results show that the 5th and 7th harmonics of each band modulation mode are nearly eliminated, which proves that the method based on the Walsh function is correct and feasible.