Optimized Dual Randomized PWM Technique for ...

Optimized Dual Randomized PWM Technique for Reducing Conducted EMI in DC-DC Converters A. Boudouda, D. Sekki, N. Boudjerda, B. Nekhoul, K. El Khamlichi Drissi and K. Kerroum

 Abstract--This paper deals with the control of the full bridge dcdc voltage converter by the Random Pulse Width Modulation (RPWM) technique. We propose a RPWM scheme based on a triangular carrier having two randomized parameters: the period T and the peak position  (half period)  First, we propose the modulating principle, then the voltage analysis based on Power Spectral Density (PSD) shows the advantage from the point of view of Electro-Magnetic Compatibility (EMC) of this scheme compared to conventional RPWM schemes having only one randomized parameter. Moreover, this analysis reveals the existence of an optimal value of the variation interval of parameter  for a maximum spread of the PSD. Once the optimization problem modeled, the resolution is performed using two powerful nonlinear methods: the trust region method and the simplex algorithm. Index Terms--EMC in power electronics; spread spectrum; random PWM; optimized RPWM scheme; Simplex; Trust region.

I. INTRODUCTION

N

owadays, much of the electrical energy is consumed through static converters, which allow adjusting this energy to the desired form. The use of PWM technique for the control of power converters allows adjusting the useful component of the voltage and eliminating some unwanted harmonics [1]. Thus, it is required for power converters to provide the desired electrical functionality and to meet international standards of Electro-Magnetic Compatibility (EMC) by reducing conducted and radiated emissions [2]. In order to better meet the EMC standards for conducted disturbances, we can use the filtering technique (passive and/or active power filters). Furthermore, RPWM technique is one of the most effective and least-cost solutions: it allows spreading the power spectrum over a wide frequency range while significantly reducing its amplitude and this constitutes a significant EMC advantage without any additional hardware. Several works regarding this new control technique have been published recently, two conventional RPWM schemes are proposed: the scheme in which the switching period is

randomized (Randomized Carrier Frequency Modulation: RCFM) and the scheme in which the period is kept constant and the pulse position is randomized (Randomized Pulse Position Modulation: RPPM) in both DC-DC and DC-AC [3] [5]. It has been shown that RCFM scheme allows a better spreading of the spectrum then RPPM scheme [6]. However, to obtain a maximum spread, the combination of these two schemes has been proposed (RCFM-RPPM) that we call dual RPWM scheme [5]-[6]. We propose in this paper an optimized scheme with two randomized parameters “optimized RCFMRPPM”. The switching functions are generated by comparing a carrier having two random parameters with two deterministic reference signals: one switching function for each arm of the converter. First, we propose the modulating principle of this technique. Then an analytical model of the Power Spectral Density (PSD) of the output voltage is developed and validated. This model is expressed directly in terms of the two random parameters of the carrier. Note that the simple schemes (RCFM and RPPM) are deduced directly from the proposed general model. The PSD analysis shows that the proposed dual RPWM scheme allows a better spread shape of PSD compared to the simple randomization schemes that is the desired EMC advantage. In addition, a parametric study shows that there is an optimum shape of the PSD based on statistical parameters, The treatment of this problem with two powerful nonlinear methods (Trust Region method and Simplex algorithm), gives the same results. II. MODULATING PRINCIPLE The structure of the converter is given in Fig. 1; it requires two switching functions qa et qb.

Load

E u qa

A. Boudouda, D. Sekki, N. Boudjerda and B. Nekhoul are with Departement of Electrical Engineering, Jijel University, LAMEL Laboratory Jijel, Algeria (e-mail: [email protected]). K. El Khamlichi Drissi and K. Kerroum are with the Department of Electrical Engineering, Blaise Pascal University, LASMEA Laboratory Clermont Ferrand, France (e-mail: [email protected]).

qb

qa

qb

Fig.1. Full Bridge DC-DC Voltage Converter

The modulating principle is illustrated in Fig.2. The two switching functions qa and qb are obtained by comparing two deterministic reference signals ra and rb of magnitudes da and db respectively, to a single randomized carrier c (Fig.2.). Generally, the magnitudes da and db are taken as follows: 0  da  1 0  db  1

(1)

d a  db  1

The output voltage u can be expressed using the switching functions qa and qb as follows:

u  (qa  qb ) E

From Fig.2, a random variation of parameter a between 0 and (1 - da), “respectively b between 0 and (1 - db)” is obtained by using a carrier with random parameter  between 0 and 1, which gives:

 a   1  d a    b   1  d b 

Thus, the average value U0 of the output voltage u is:

U 0  (2d a  1) E

(2) Note that a variation of da between 0 and 1 allows adjustment of the voltage U0 between –E and +E. mTm

1

c

ra

da rb db

qa

tm amTm

Tm daTm

1

qa

The use of parameter  instead of parameters a and b allows defining the switching functions qa and qb and the output voltage u directly in term of the random parameters T and  of the carrier, therefore the voltage analysis can be performed directly from these two parameters. In addition, it provides only a single random variable “” instead of the two parameters “a” and “b”.

t qb

bmTm



W  f    R e  j2 f d

dbTm

(5)



1

qb u

(4)

III. MATHEMATICAL MODEL OF PSD A rigorous study of the spectral content of the output voltage is based on the Power Spectral Density (PSD) of random signals, defined as the Fourier transform of the autocorrelation of the considered signal:

t

tm+1

(3)

And:

u=E(qa - qb)

t

R   lim

1 T0  2T0

E t Fig.2. Modulating principle

Each switching function is completely characterized by: The switching period T.  The duty cycle d.  The delay report .  Theoretically, the three parameters (T,  and d) should be randomized in a separated or a combined way. However, in industrial applications, the duty cycle d is generally deduced from a reference signal and allows the control of output voltage. Therefore, only T and are really randomized, resulting in the four (4) following configurations (Tab. I): Table I. RPWM schemes

T0

 ut ut   dt

(6)

T0

R  and W  f  are respectively the auto-correlation function and the power spectral density of the signal u t  .

For a Wide Sense Stationary (WSS) pulse signal u t  , the

PSD can be expressed as follows [4]-[6]: W  f   lim

N 

 1  N E  U m  f U m* k  f  T  k  N 

(7)

U m  f  and U m*  k  f  are respectively the Fourier transform

of the signal u t  during the switching period Tm and its conjugate during the switching period Tm+k. In relative magnitudes (per unit system), the expression (3) of the output voltage becomes: (8) u  qa  qb

PWM scheme



T

DPWM

 = 0.5 Fixed :

fixed

RPPM

randomized

fixed

RCFM

Fixed : = 0.5

randomized

qam  f  and qbm  f  are respectively the Fourier transforms of

RCFM-RPPM

randomized

randomized

qa and qb during the switching period Tm .

And its Fourier transform over a switching period Tm is then:

U m  f   qam  f   qbm  f 

(9)

After some mathematical transformations, the PSD of the output voltage for the scheme RCFM-RPPM can be written as follows [5]: E U  f  2   T  1   j 2fT * (10)     E U f e E U f Wf    T ,  T ,  2 T f   2Real    1  ET e j 2fT







   



Where:





  j 2f 1daT  jfd aT e sin fd a T e Uf     e  j 2f 1db T e  jfdbT sin fd T b 



    



a. Computed PSD

(11) 0

ET   : Expected value related to the random variable T . ET ,    : Expected value related to the random variables

IV. VALIDATION OF THE MATHEMATICAL MODEL OF PSD The validation of the mathematical model of the PSD for the three RPWM schemes is performed by comparing the PSD calculated analytically with that estimated by the Welch method which can be obtained after simulation of the converter [7]. Moreover, for simple schemes (RCFM and RPPM), measurement results published in the literature are also used [1] for the comparison which is achieved with the following conditions:

PSD [dB/Hz]

T and  . From expressions (10) and (11), the simple schemes RCFM and RPPM can be obtained as particular cases: for RCFM scheme, the parameter  is constant ( = 0.5) and for RPPM, the period T is constant.

-20 -40 -60 -80 0

10

20 Frequency [kHz]

30

b. Estimated PSD (Welch method)

 Input voltage: E  200 volts .  Switching period T: randomized according to the uniform probability law between two values: Tmin = 167 s and Tmax = 250 s.  is randomized according to the uniform distribution in the interval:





    1  R 2,  1  R 2

c. Measured PSD [1] Fig. 3. Validation for RPPM scheme

  0.5 : Statistical average R : Randomness level, it allows defining the variation interval of parameter , theoretically the maximum value is R max  2 .

 

Fig.3 and Fig.4 show very good agreements between the estimate, the analytical calculation and the measurement for RPPM and RCFM schemes respectively while Fig.5 shows a very good agreement between the estimation and the calculation for RCFM-RPPM scheme thereby validating our proposed model.

a. Computed PSD

40

b. Estimated PSD (Welch method)

V. EMC ADVANTAGE OF THE RCFM-RPPM SCHEME Fig. 6 shows the PSDs of the output voltage obtained while varying R for three values of RT (RT = 0.2, RT = 0.3 and RT = 0.4) at one functioning point corresponding to the following duty ratio d = da – db = 0.75 – 0.25 = 0.5. The effect of RT appears clearly: important values of RT give better spread PSD. In addition, the dual randomized scheme (RCFM-RPPM) adds a significant spread to the PSD compared to the simple scheme (RCFM). This spreading is also accompanied by a decrease in peaks. Nevertheless, a significant increase in R (Fig. 6) causes an increase in the peak of the PSD around Fs (Fs: Average frequency modulation) and a decrease in the peak of the PSD around 2Fs. Knowing that our purpose is to spread best the PSD and to reduce its peaks (in order to meet EMC standards), a compromise between the two peaks of the PSD (at Fs and 2Fs) can be achieved with an optimal value of R. The optimization problem is posed and solved in the following paragraph.

c. Measured PSD [1] Fig. 4. Validation for RCFM scheme

a. PSD for RT = 0.2 and (R = 0, 1 et 2)

a. Computed PSD

b. PSD for RT = 0.3 and (R = 0, 1 et 2)

b. Estimated PSD (Welch method) Fig. 5. Validation for RCFM-RPPM scheme

c. PSD for RT = 0.4 and (R = 0, 1 et 2) Fig. 6. Effects of RT and R on PSD shape

VI. OPTIMISATION OF PARAMETER R Our purpose is to find an optimal value of the randomness level R (optimization parameter) in order to achieve the best spread of the PSD. R is subject to the following constraints of



domain boundaries: 0  R  2



Generally, an optimization problem subject to the constraints of domain boundaries can be written as follows: n  Min F x      l  xk  u k  1.......n

(12)

Where:  x : n-dimensional vector (xk, k = 1,…,n) representing the parameters of the problem to optimize.  l and u : Lower and upper limits of the search area (domain boundaries).   n : Search domain bounded by the limits l and u.  F x  : Objective function or optimization criterion.

coefficients in the objective function so as to give more important to the frequency Fs compared to 2Fs as follows: Min F x   a *W R , FS 2  b *W R ,2 FS 2  (15)   0  R  2



m

2

x



Where:

f ( x)  [ f1 x  f 2 x ...............f m x ]T : Vector containing m non-linear functions. f x  2 : Second order norm of vector f (x) . x   n : Vector of optimization parameters. A. Mathematical Formulation The parameter 

min ,

varies randomly in the interval

max  , with βmin  0 and βmax  1 . By introducing the

statistical mean  and the randomness level R, we obtain:



R



2

   1 

,

1

R   2 

Where:   0.5 et 0  R  2

In expression 13, f x  is a vector of dimension m = 2, it represents the PSD at frequencies Fs and 2Fs and x represents the searched value of R. We can then consider the problem as an optimization problem in the sense of nonlinear least squares with constraints of the search area limits as follows:  Unweighted problem :







2 2  Min F x   W R  , FS  W R  ,2 FS   0  R   2



(14)

 Weighted problem : Usually the low frequencies are more harmful for inductive loads, then we can introduce weighting

1

T f 2



 

     

       

(16)

Note: In the expression (16) of PSD, the frequency f takes only two values (Fs and 2Fs) and the variable Ris introduced while replacing the expectations related to the random variable , ET , .. by their expressions.



(13)

i 1



  2  ET  U R  , f      E U R , f e j 2fT E U * R , f   T ,   2Real  T ,    1  ET e j 2fT  



In our case, the criterion F is chosen with the least square (quadratic). It is defined by the sum of m non-linear functions squares. Thus, the resolution of a least square optimization

minn F x , F x   f x  2   f i 2 x  m  n



 a and b: weighting coefficients, in practice the impedance of an inductive load is proportional to the frequency, thus its judicious to choose the ratio a/b inverse of the ratio between the frequencies Fs and 2Fs, which gives a = 2 and b = 1.  W : PSD of the voltage evaluated at frequencies Fs and 2Fs using the general expression:



minimizes the objective function as follows, [8]:

 

Where:  R  : Optimization parameter of the objective function.

W R , f 

problem consists of searching the vector x*   n which





B. Optimization methods To solve this optimization problem, we propose two algorithms: the trust region method, available in MATLAB as the utility named lsqnonlin and to consolidate the results, we use the simplex algorithm (Nelder and Mead) available in MATLAB as the utility named fminsearch. These two algorithms are well suited to our problem, which is nonlinear with the constraints of the domain limits. B.1. Trust region method This method involves replacing the resolution of the problem posed by equation (12) by solving a succession of simpler sub-problems where the optimization criterion F is replaced by a simpler function m called model function, which reflects the behavior of F in a neighborhood N around the point x, this neighborhood is called trust region and is generally spherical or ellipsoidal [9]. Originally, this algorithm is used in nonlinear optimization problems for both cases: without constraints and with constraints of the domain limits [10]. B.2. Simplex Algorithm This algorithm is based on the concept of direct search: it attempts to solve the problem by using directly the objective function value without using its derivatives. It is suitable for strongly nonlinear optimization problems without constraints [11]. However, it may take into account the domain limits. A simplex is a geometric figure of n-dimension, created from (n + 1) points, each dimension corresponds to a parameter of the

optimization problem. The minimum is sought by changing the simplex through standard operations: reflection, expansion and contraction, [12]. C. Numerical results of optimization For fixed RT , we search the optimal value of R  in the





interval 0, 2 with an arbitrary initial value R0  0, 2 . The obtained results are given in tables II, III, IV and V. Note that these results remain unchanged for any value of R  0 , demonstrating the robustness of these algorithms. The two algorithms give almost the same results (difference less than 0.1%), which reinforce the obtained results.  Without frequency weighting (a = b = 1) TABLE II. OBTAINED RESULTS (TRUST REGION)

RT 0.2 0.3

Optimal R 1.1755 1.1791



W R , Fs



0.1649 0.0815



W R ,2Fs

0.2205 0.1150

TABLE III. OBTAINED RESULTS (SIMPLEX)

RT 0.2 0.3



Optimal R 1.1748 1.1782

W R , Fs





0.1645 0.0813



W R ,2Fs



[1]

 With frequency weighting (a = 2b = 2)

RT 0.2 0.3

Optimal R 1.0238 1.0167



W R , Fs



0.1036 0.0509



W R ,2Fs

RT 0.2 0.3

Optimal R 1.0241 1.0177



W R , Fs

0.1037 0.0511



0.2847 0.1457

TABLE V. OBTAINED RESULTS (SIMPLEX)





W R ,2Fs



0.2845 0.1455

To further support our results, we compare in Fig.7, the variation of the PSD around the optimum value of R for RT fixed. We see clearly (Fig.7) the compromise reached between the two peaks at Fs and at 2 Fs .

a. PSD in volt²/Hz

VII. CONCLUSION The purpose of this paper is the reduction of conducted EMI in DC-DC converters by an optimized dual RPWM technique. A modulating principle based on a carrier with two randomized parameters is proposed for both simple and dual RPWM techniques. To make a rigorous analysis of the output tension, we have developed and validated a mathematical model of the power spectral density of this voltage. A parametric study reveals the existence of optimal statistical parameters. This problem has been treated with two powerful nonlinear methods. VIII. REFERENCES

0.2207 0.1151

TABLE IV. OBTAINED RESULTS (TRUST REGION)

b. PSD in dB/Hz Fig.7. Comparison of PSDs

M. M. Bech, "Random Pulse-Width Modulation Techniques for Power Electronic Converters", Ph.D. Thesis, Aalborg University, Denmark, 2000. [2] R. Redl, "Power Electronics and Electromagnetic Compatibility", IEEE proceedings of PESC’96, Record, 27th Annual IEEE Conf, Vol. 1, June, 1996, pp. 15-21. [3] K. K. Tse, Henry Shu-hung Chung, S. Y. Hui and H. C. So, "A Comparative Investigation on the Use of Random Modulation Schemes for DC/DC Converters", IEEE Trans. On Industrial Electronics, Vol. 47, N0. 2, April 2000, pp. 253-263. [4] R. L. Kirlin, M. M. Bech and A. M. Trzynadlowski, "Analysis of Power and Power Spectral Density in PWM Inverters with Randomized Switching Frequency", IEEE, Trans. On Industrial Electron. Vol. 49, N0. 2, April 2002, pp. 486-499. [5] N. Boudjerda, M. Melit, B. Nekhoul, K. El Khamlichi Drissi and K. Kerroum, "Reduction of Conducted Perturbations in DC-DC Voltage Converters by a Dual Randomized PWM Scheme", Journal of Communications Software and Systems, Vol. 5, No.1, March, 2009. [6] N. Boudjerda, "Réduction des Perturbations Conduites dans les Convertisseurs de l’Electronique de Puissance par une Commande en MLI Aléatoire", Thèse de doctorat en sciences, Université Ferhat Abbas-Setif, Juillet 2007. [7] P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging over Short, Modified Periodograms", IEEE Trans, On Audio and Electroacoustic, Vol.Au-15, June 1967, pp.70-73. [8] Dennis, J.E., Jr., "Nonlinear Least-Squares", State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, 1977, pp. 269–312. [9] J. F. Rodriguez, J. E. Renaud, B. A. Wujek and R. V. Tappeta, "trust region model management in multidisciplinary design optimization", Journal of Computational and Applied Mathematics, Vol.124, Issues 12, December 2000, pp.139-154. [10] T. F. Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds", SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445. [11] R. M. Lewis, V. Torczon and M. W. Trosset, "Direct search methods: then and now", Journal of Computational and Applied Mathematics, Vol 124, Issues 1-2, December 2000, pp.191-207. [12] J.A. Nelder, R. Mead, "A simplex method for function minimization", Computer Journal, vol.7, 1965, pp.308-312.