A ripple current minimisation based single phase PWM inverter

Transcription

1 Sohag University, Faculty of Engineering From the SelectedWorks of Khairy F. A. Sayed Dr 014 A ripple current minimisation based single phase PWM inverter Khairy F. A. Sayed, Dr, Sohag University, Faculty of Engineering Available at:

2 Int. J. Power Electronics, Vol. 6, No. 3, A ripple current minimisation based single phase PWM inverter Khairy Sayed* Electrical Engineering Department, Sohag University, Sohag, Egypt khairy_fathy@yahoo.ca *Corresponding author Mazen Abdel-Salam and Adel Ahmed Electrical Engineering Department, Assiut University, Assiut, Egypt mazen000as@yahoo.com a_ahmed65@yahoo.de Mahmoud Ahmed Mechanical Engineering Department, Assiut University, Assiut, Egypt aminism@aun.edu.eg Abstract: This paper is aimed at improving the output voltage waveform of a single phase PWM inverter. Two approaches is proposed, the first approach is based on selected harmonic elimination (SHE) of order up to 7th harmonic, for minimising harmonic distortion and modulating amplitude of the fundamental component of the output voltage waveform. For the first time, the Levenberg Marquardt algorithm (LMA) is used for determining the switching angles of the inverter switches. The second approach is based on ripple current minimisation using LMA. A simulation model is developed using PSIM for the inverter to verify the proposed approaches. An experimental system was implemented to demonstrate the effectiveness of the proposed approaches by using PIC16F877 microcontroller. Analysis of the voltage THD as influenced by the amplitude modulation index is made using MATLAB based on the computed switching angles. Keywords: single phase PWM inverter; LMA; Levenberg Marquardt algorithm; switching angles; ripple current minimisation; selected harmonic elimination. Reference to this paper should be made as follows: Sayed, K., Abdel-Salam, M., Ahmed, A. and Ahmed, M. (014) A ripple current minimisation based single phase PWM inverter, Int. J. Power Electronics, Vol. 6, No. 3, pp Biographical notes: Khairy Sayed received his BS degree in Electrical Power and Machines in 1997 from Assiut University, Assiut, Egypt. He got Masters degree at the Electrical Energy Saving Research Center, Graduate School of Electrical Engineering, Kyungnam University, Masan, Korea, 007. He got PhD degree from Assiut University, 013. He is working as a Lecturer in the Copyright 014 Inderscience Enterprises Ltd.

3 0 K. Sayed et al. Department of Electrical Engineering, Sohag University, Egypt. His research interests include soft switching DC-DC power converter topologies, high frequency inverter applications, renewable energy related power conditioners and control schemes. Mazen Abdel-Salam is working as a Professor, Department of Electrical Engineering, Faculty of Electrical Engineering at Assiut University. He was an Alexander-Von-Humboldt Fellow in the Electrical Engineering Department, Technical University of Munich, Germany during In September 1979, he worked as a Researcher with General Electric Company, Pittsfield, MA, USA. In January 198, He has been elected Fellow of IEEE, Institute of Electrical and Electronics Engineers, in 199, New York, USA, Fellow of IEE, Institution of Electrical Engineers, in 199, London, England, UK and Alexander-von-Humboldt Fellow in 1977, Bonn, West, Germany, Fellow of IOP, Institute of Physics in 00, Bristol, UK, Fellow of JSPS, Japanese Society for Promotion of Science, Tokyo, Japan in Adel Ahmed received the BSc and MSc degrees in Electrical Engineering from Assiut University, Egypt, in 1987 and 1993, respectively. He received the Dr.- Ing. degree from Technical University of Hamburg-Harburg (TUHH), Hamburg, Germany, in 001. He is currently a Professor of Electrical Engineering at Assiut University. In 006, he was a researcher at TUHH, Germany. His research activities include digital calculation of electric fields, corona studies, investigations of high-voltage phenomena, low-voltage distribution networks, computer applications in large power systems and applications of renewable energy systems. Mahmoud Ahmed received his BS and MS from Assiut University, Egypt in 1983 and 1987 respectively, and his PhD from Mississippi State University, in He is currently a Professor of Mechanical Engineering at Assiut University. His research interests are in renewable energy systems, numerical simulation of thermo fluid systems, and micro-nanoflows. 1 Introduction Some non-linear electric loads connected to power distribution lines through PWM inverters inject signal distortion to these lines. Such distortion may cause problems with the other loads. Harmonics can shorten the life of the appliances by voltage stress and increased heating of electrical insulation (Bose, 007; Rashid, 001). The effect of harmonics in the supply lines increases distribution losses and generally degrades the quality of the supply. To maintain the supply quality at an acceptable level, national standards such as IEC 555- and IEEE 519 recommend limits for harmonic levels at various points within the system. A frequently quoted limit is 5% for the total harmonic voltage distortion Several methods are applied to reduce voltage harmonics at the output of PWM inverters. One method is by using passive circuits and the other is by active shaping of the input line current. The traditional method is by using shunt passive LC filters which have been widely used with particular values of L and C for each harmonic order. However, LC passive filters for low frequency harmonics have significant disadvantages,

4 A ripple current minimisation based single phase PWM inverter 03 such as increase of inverter physical size with a subsequent increase of the inverter cost, more losses and significant dependence of the output voltage on the load power. Therefore, attention has been turned to active filters (AF) (Mohan et al., 1995) which are electronic circuits used to shape the input current to follow a sinusoidal waveform at near unity power factor. The active filter eliminates the harmonics by injecting its compensating current directly into the ac power lines. The disadvantages of AF are difficulty of use for high capacity applications, complicated control circuit, reduced efficiency due to additional losses and more generation of high order harmonic currents which can distort telecommunication systems (Patel and Hoft, 1973). PWM techniques are extensively used for eliminating harmful low-order current and voltage harmonics in input and output of static power inverters. PWM-based inverters generally employ the selected harmonic elimination (SHE) technique (Bose, 007; Rashid, 001; Mohan et al., 1995; Patel and Hoft, 1973). The main advantage of using selected harmonic elimination is a reduction of the total harmonic distortion (THD). However, the dc input current in the PWM inverters contains the second harmonic, which in turn contributes to the generation of a second voltage harmonic in the dc bus (Rashid, 001). A method to reduce the THD of the output voltage of the PWM inverter was proposed (Choe and Park, 1988) by computing the switching angles of inverter switches using a simple modified reference approach (MRA) (Ray et al., 008). Figure 1 Single phase PWM inverter Q 1 Q DC Soure DC link Capacitor L f C f Load Q 4 Q 3 Gate Driver Voltage Sensing Resistor Controller A combination of selective harmonic elimination with pulse-width modulation (SHE-PWM) technique for a high-power inverter used in constant-frequency utility applications was reported (Dahidah and Agelidis, 008). The objective is to find the switching angles that produce the fundamental component as well as harmonics different from the selected low-order ones; therefore, the output of the inverter is decomposed into

5 04 K. Sayed et al. its Fourier transform equivalent, and the coefficients of the selected harmonics are equal to zero. This ends up with formation of a set of transcendental equations whose solution determines the switching angles. The solution of the transcendental equations was done using Newton Raphson algorithm (NRA) (Chiasson et al., 004; Czarkowski et al., 00). However, NRA s iteration is very sensitive to initial values. The initial solution not only affects convergence of the iterative technique but also the quality of the solution in terms of reduced THD (Jabr, 006). A method for voltage harmonic elimination in single phase PWM inverter using genetic algorithms (GAs) was developed (Schutten and Torrey, 1995; Rahman, 001; Al-Othman et al., 007). The solution method of the nonlinear equations was applied to calculate the appropriate switching angles, for offline elimination of the third, fifth and seventh harmonics. The GA method requires a large number of iterations in comparison with the Newton Raphson method (Butun et al., 006). Particle swarm optimisation (PSO) was presented as a non-traditional solution to the problem of selective harmonic elimination (SHE) in induction motor drives fed by three-phase voltage-source inverters (VSI) (Azab and Awadallah, 013; Azab, 010). A PWM inverter using the Walsh function harmonic elimination method was presented (Liang et al., 1997) where the switching angles are optimised by linearising the describing non-linear transcendental equations. This paper is aimed at improving the output voltage waveform of a single phase PWM inverter. Two approaches are proposed: the first approach is based on selected harmonic elimination (SHE) of order up to 7th harmonic, for minimising harmonic distortion and modulating amplitude of the fundamental component of the output voltage waveform. For the first time the Levenberg Marquardt algorithm LMA (Levenberg, 1944; Marquardt, 1963) is used for determining the SHE switching angles of the inverter switches. The LMA is more robust than the GNA and NRA, where the rate of convergence to the final solution is faster. The second approach is based on ripple current minimisation using LMA. Even the two approaches have the same goal in minimising the harmonic distortion but the methodology is different. The high-frequency harmonic components are easily removed using additional passive filter circuits. The two approaches are checked experimentally by implementing the proposed inverter with a PIC microcontroller using a look-up table of switching angles at different values of PWM amplitude modulation index. Proposed method.1 Proposed approach based on the SHE The proposed method is based on the use of LM algorithm for determining the switching angles α iteratively. The SHE pulse angles have been determined using LMA for different values of PWM amplitude modulation index m a. The modulation of the amplitude of the output voltage is accomplished by modifying the switching angles. If m a < 1, the amplitude of the fundamental component of the output V 1 is linearly proportional to m a and can be expressed as: m V 1 a (1) V dc

6 A ripple current minimisation based single phase PWM inverter 05 Figure shows the output voltage waveform of the single phase PWM inverter. The undesirable lower order harmonics of this waveform can be eliminated and the fundamental voltage can be controlled using the selected harmonic elimination SHE (Bose, 007). The voltage pulses are created using predetermined switching angles (α 1, α, α 3, α k ). Figure Voltage wave for SHE PWM The general Fourier series of the voltage wave can be given as: k k () vt () a cosk t b sink t k1 Due to the symmetry of the waveform, only odd harmonics with sine component are present, therefore, a 0 k k k1 vt () b sinkt, where / / 4 4 bk v()sin t ktdt Vdcsinktdt (3) 0 0 and using the general relation 1 sin ktdt cos k1cos k (4) k 1 4 4V dc Then bk sin ktdt sin ktdt 1 3 One attains b k equation as follows: 4V dc bk k N K1 ( 1) cosk K (5) K1

7 06 K. Sayed et al. The ac output voltage should feature N pulses per half-cycle in order to adjust the fundamental component and eliminate N 1 harmonics. For instance, N equal 4 in order to eliminate the third, fifth and seventh harmonics and to perform fundamental magnitude control. 4V dc b cos(1 ) cos(1 ) cos(1 ) cos(1 ) (6) 4V dc b cos(3 ) cos(3 ) cos(3 ) cos(3 ) (7) 3 4V dc b cos(5 ) cos(5 ) cos(5 ) cos(5 ) (8) 5 4V dc b cos(7 ) cos(7 ) cos(7 ) cos(7 ) (9) 7 where b 1 =V o1 and to eliminate the 3rd, 5th and 7th harmonics b 3, b 5, b 7 are equal zero, the equations to be solved are: Vo 1 cos( 1) cos( ) cos( 3) cos( 4) 4 V cos(3 ) cos(3 ) cos(3 ) cos(3 ) cos(5 ) cos(5 ) cos(5 ) cos(5 ) dc cos(7 ) cos(7 ) cos(7 ) cos(7 ) 0 (10) Equations (6) through (9) represent a system of four non-linear equations, f1( 1,, 3, 4) cos( 1) cos( ) cos( 3) cos( 4) Vo 1/(4 Vdc) f (,,, ) cos(3 ) cos(3 ) cos(3 ) cos(3 ) f (,,, ) cos(5 ) cos(5 ) cos(5 ) cos(5 ) f (,,, ) cos(7 ) cos(7 ) cos(7 ) cos(7 ) (11) An iteration method is used for solving the above system of equations as described in Appendix A. A set of initial values of the switching angles is proposed and /. Proposed approach based on ripple current minimisation One disadvantage of SHE is that the elimination of lower order harmonics boosts the next higher order harmonics (Bose, 007). Since the harmonic distortion is dictated by the rms ripple current I ripple which should be minimised instead of emphasising on the individual harmonics; I I I I I (1) ripple

8 A ripple current minimisation based single phase PWM inverter 07 1 V k Iripple k3,5,7,.. Z (13) k where I 5, I 7, = rms harmonic currents k = order of harmonic V k = peak value of kth-order harmonic Z k = the load impedance at kth harmonic Z R ( kl) k L = effective load inductance ω = fundamental frequency where R=effective load (line resistance) resistance For a given number of pulses, I ripple can be expressed as a function of angles α by substituting V k in terms of the input V dc. The AC output current should feature N pulses per half-cycle in order to adjust the fundamental component and eliminate N 1 harmonics. Therefore, N is equal 4 in order to eliminate the third, fifth and seventh harmonics and to perform fundamental magnitude control. 4V dc Z1 b cos( ) cos( ) cos( ) cos( ) (14) 4V dc Z3 b cos(3 ) cos(3 ) cos(3 ) cos(3 ) (15) 4V dc Z5 b cos(5 ) cos(5 ) cos(5 ) cos(5 ) (16) 4V dc Z7 b cos(7 ) cos(7 ) cos(7 ) cos(7 ) (17) where b 1 =V o1. In order to eliminate the 3rd, 5th and 7th harmonics, the coefficients b 3, b 5 and b 7 are equated to zero and the resulting four non-linear equations to be solved. The general Fourier series coefficients of the voltage wave can be given as, N 4V dc K1 bk ( 1) cosk K k (18) K1 The ripple current is expressed as: I ripple N 1 4V dc cos n k k5,7,11,... K1 k Zk (19) I ripple N 4V 1 1 dc cos n k k5,7,11,... K1 kzk (0)

9 08 K. Sayed et al. For ripple current minimisation (RCM), the four non-linear equations based on equation (5) through (8) are formulated as: 1 1 I (,,, ) cos( ) cos( ) cos( ) cos( ) * V /(4* V ) o1 dc Z 1 Z 1 I I I 1 (,,, ) cos(3 ) cos(3 ) cos(3 ) cos(3 ) Z3 1 (,,, ) cos(5 ) cos(5 ) cos(5 ) cos(5 ) Z5 1 (,,, ) cos(7 ) cos(7 ) cos(7 ) cos(7 ) Z7 These equations are simplified as: V V o1 dc cos( ) cos( ) cos( ) cos( ) * /(4* ) 0 cos(3 ) cos(3 ) cos(3 ) cos(3 ) cos(5 ) cos(5 ) cos(5 ) cos(5 ) cos(7 ) cos(7 ) cos(7 ) cos(7 ) 0 (1) Solution of the set of non-linear equations using LM algorithm determines the values of switching angles. This set of equations is different from the set (10) (11) which was solved in the first approach. For a given number of notch angles, I ripple is found as a function of α angles. The switching angles α can be then iterated in a computer program so as to minimise I ripple for a certain desired output voltage..3 Ripple current minimisation in three-phase PWM inverters In three-phase systems the line output voltages are balanced and 10 out of phase, the third harmonic and its multiples (9,15, ), which could be present in the phase voltages (V an, V bn, and V cn ), will not be present in the load voltages (V ab, V bc, and V ca ). Therefore, these harmonics are not required to be eliminated, thus the chopping angles are used to eliminate only the harmonics at frequencies 5; 7; 11; 13; as required. The expressions to eliminate a given number of harmonics are the same as those used in single-phase inverters. For instance, to eliminate the fifth and seventh harmonics and perform fundamental magnitude control (N =3), the equations to be solved are: V an1 cos( 1) cos( ) cos( 3) cos( 4) /4 Vdc cos(5 ) cos(5 ) cos(5 ) (1/) 1 3 cos(7 ) cos(7 ) cos(7 ) 1/ 1 3 () This method is applicable for three-phase and multi-level PWM inverters.

10 A ripple current minimisation based single phase PWM inverter 09 3 Total harmonic distortion (THD) The THD as a performance index can be calculated using: THD k3 H H 1 k (3) where H k is the amplitude of the kth harmonic 4V dc As vt () Vk sinkot and Vk cos kk, equation (3) can be written as: k k, odd THD k3 1 (cos k1 cos k cos k3 cos k4 ) k cos cos cos cos (4) 4 Output filter for high order harmonics An overview of the design of a voltage source inverter including an LC output filter is presented. The parameters of the filter are chosen to minimise output voltage harmonics. The resonant (or cut-off) frequency is expressed by: 1 r, LC f f 1 fr (5) LC f f When sizing the values of L and C, use bigger C and smaller L, or the other way round. This depends on the type of load, whether inductive or purely resistive. The choice of the individual values of L f and C f is a remaining degree of freedom. The filter inductor is selected as 0.9 mh. By substituting the inductance value into (5) the capacitance C f is calculated as 110 μf at cut-off frequency of 506 Hz. 5 Results and discussion A MATLAB program is used to computes the switching angles for 0 VAC, 50 Hz Single phase PWM inverter feeding an inductive load (R = 10 Ω, L = 5 mh). In this program, the LMFSOLVE function (see exchange/16063) is used. Figure 3a shows how the computed switching angles change with the amplitude modulation index (range from 0. to 1.) with a trend that agrees with that reported before (Bose, 007), with a fewer number of iterations. In case of threephase inverters the switching angles are plotted in Figure 3b.

11 10 K. Sayed et al. Figure 3 (a) Switching angles for RCM of third, fifth and seventh harmonic, (b) switching angles for RCM of fifth and seventh harmonic in three-phase system, and (c) the calculated switching angles to remove up to 19th harmonics (see online version for colours) 90 Alpha [Degrees] Alpha 1 Alpha Alpha 3 Alpha Modulation index 90 (a) Alpha [Deg.] Alpha 1 Alpha Alpha Modulation index (b)

12 A ripple current minimisation based single phase PWM inverter 11 Figure 3 (a) Switching angles for RCM of third, fifth and seventh harmonic, (b) switching angles for RCM of fifth and seventh harmonic in three-phase system, and (c) the calculated switching angles to remove up to 19th harmonics (see online version for colours) (continued) (c) The capability and superiority of the LMA was confirmed by trying a large number of angles (10 angles) for eliminating 3rd, 5th, 7th, 9th, 11th, 13th, 15th, 17th, 19th harmonics. The calculated switching angles are shown in Figure 3c. A simulation model is developed using power electronics simulation program PSIM for single-phase voltage source inverter with PWM pattern control for elimination of 3rd, 5th, 7th harmonics by applying the computed switching angles in Table 1 to the MOSFET gates. The switching angles based on the SHE are slightly different from to those obtained using RCM ripple current minimisation, Table 1. The use of LM algorithm results in reduction in the current THD from.5133 to.3036 in case of SHE as shown in Table. In case of using ripple current minimisation the current THD reduces to.05%. (these calculation are done using 300 Vdc input voltage and modulation index=0.8). The output voltage and current waveforms are shown in Figure 4. The output voltage and current with resistive load and without filter are shown in Figure 4a. The voltage and current waveform with using LC filter and inductive load is almost sinusoidal because it is filtered by the LC filter to remove the high frequency voltage harmonics, Figure 4b. The Output voltage and current with C filter are shown in Figure 4c.

13 1 K. Sayed et al. Table 1 Switching angles during л period Switch Switching angles [degrees] based on SHE Q1, Q Q, Q No. of iterations = 16, time = sec Switch Switching angles [degrees] based on RCM Q1, Q Q, Q No. of iterations = 18, time = sec Table THD at different cases THD without SHE % THD without 3, 5, 7 harmonics.5133% THD without 3, 5, 7 harmonics but with SHE using LM algorithm.3036% THD with ripple current minimising using LM algorithm.0545% Figure 4 Output voltage [V] and current [A] with (a) resistive load, (b) LC filter, and (c) C filter (a)

14 A ripple current minimisation based single phase PWM inverter 13 Figure 4 Output voltage [V] and current [A] with (a) resistive load, (b) LC filter, and (c) C filter (continued) (b) (c)

15 14 K. Sayed et al. Table 3 shows the calculated switching angles based on the LMA and NRA. As shown in table, number of iterations based on the two algorithms is different with faster calculation speed based on LMA. Time of computing is determined based on.1 GHz processor. Table 3 Switching angles during quarter cycle period Switching angles in radians Initial values NRA LMA No. of iterations 0 6 Time of computing on.1 GHz processor [sec] The FFT analysis is done for the output voltage as shown in Figure 5. The low pass filter and the nature of the highly inductive load take care of the higher order of harmonics. Analysis of the relationship between the voltage THD and the amplitude modulation index is done by MATLAB using the calculated switching angles. The voltage THD increases dramatically with decreasing the modulation index, Figure 6, whereas, the corresponding increase of the voltage THD is smaller on using the SHE. The THD on using RCM assumes lower values when compared with the SHE approach. Figure 5 Output voltage FFT analysis (see online version for colours) SHE RCM SHE and LC filter

16 A ripple current minimisation based single phase PWM inverter 15 Figure 6 Total harmonic distortion of output voltage versus amplitude modulation index (see online version for colours) wihout SHE with SHE with RCM 50 THD [%] Modulation index 6 Experimental set-up and verification A PIC16F877 microcontroller has been employed to implement the control scheme for the investigated inverter. The output of inverter is given by: VoVd masint, where V d is DC link voltage, m a being amplitude modulation index. With open loop the ( ma sin t ) is stored in lookup table. In order to make the output AC voltage adjustable for regulation purpose, either change V d or m a. The PWM pattern is selected based on the amplitude modulation index. For sensing the instantaneous voltage of the AC waveform, the inverter output voltage is reduced to a reasonable value. The feedback signal after being converted to digital one is applied to the microcontroller. This digital value is compared with a reference value by the microcontroller, a particular PWM switching angles pattern is selected and switch gatings are updated. The implemented PWM generation algorithm is shown in Figure 7. The switching angles are implemented using a look-up table inside the PIC microcontroller. Figure 8 shows the experimental set-up for the proposed inverter. The circuit is composed of four MOSFET switches (IRFPC60, 16 A and 600 V). An optocoupler TLP 50 is used to drive switches. A bootstrap capacitor is connected from the supply positive DC rail (V+) to the output voltage. Due to the charge storage characteristics of a capacitor, the bootstrap voltage will rise above (V+) providing the needed gate drive voltage.

17 16 K. Sayed et al. Figure 7 The flow chart for programming the inverter-control algorithm Start Initialization of param eters R ead initial look -up T able M easure V out C om pare w ith reference value Yes Vout = Vref? No Yes Vout > Vref? No D eterm ine m a value D eterm ine m a value R ead look -up values O utput corresponding PW M control signals Figure 8 Experimental set-up for the proposed inverter (see online version for colours) TLP50 Bootstrap capacitor MOSFET Bootstrap diode

18 A ripple current minimisation based single phase PWM inverter 17 Figure 9 shows a block diagram of PWM inverter control system. Figure 10 represents the experimental results of switching signals using PIC 16F877 microcontroller. Gate drive signal is generated using PIC microcontroller, and then the signal is passed to TLP50 optocoupler to drive the MOSFET switch. Figure 11 shows the inverter output voltage for resistive load with and without filter. In Figure 11a, the low order harmonics are removed while in Figure 11b, the low and high order harmonics are removed due to LC filter. This is why the output voltage waveform is almost sinusoidal. Figure 1 shows the inverter output voltage with 0.86 power factor inductive load. The current signal leads the voltage because of using a capacitor filter of 10 uf. The current waveform is improved by using the C filter with little improvement in the voltage waveform, Figure 1. The experimental output waveform is similar to the waveform obtained by PSIM simulation, Figure 4. Figure 9 Control block diagram interlocks DC Voltage Controller driver Power Stage AC output Isolation V FDBK Voltage Sensor I FDBK Current Sensor Figure 10 SHE PWM control signal using PIC microcontroller (see online version for colours)

19 18 K. Sayed et al. Figure 11 Inverter output voltage with resistive load (see online version for colours) (a) Without LC filter 50 V/div 50 V/div (b) With LC filter Figure 1 Inverter output voltage and current with inductive load and C filter (see online version for colours)

20 A ripple current minimisation based single phase PWM inverter 19 7 Conclusion A new method based on LMA for calculation of switching angles for selective harmonic is proposed. The proposed method based on the LMA converges faster than the GNA and NRA. A single phase PWM inverter with PIC microcontroller was built on the bench and selected results were verified experimentally by applying the corresponding switching angles to the inverter circuit, through the control scheme. The relationship between the voltage THD and the amplitude modulation index is investigated using the calculated switching angles. The voltage THD increases dramatically with decreasing the modulation index, with high values on using the SHE in comparison with RCM. The switching angles results obtained using SHE or RCM approaches are verified by PSIM where the estimated switching angles are applied to the gating signals of a single phase inverter in the PSIM simulation file. References Al-Othman, A.K., Ahmed, N.A., Al-Kandari, A.M. and Ebraheem, H.K. (007) Selective harmonic elimination of PWM AC/AC voltage controller using hybrid RGAPS approach, World Academy of Science, Engineering and Technology, Vol. 9, pp Azab, M. (010) Particle swarm optimisation-based solutions for selective harmonic elimination in single-phase PWM inverters, International Journal of Power Electronics, Vol., No., pp Azab, M. and Awadallah, M.A. (013) Selective harmonic elimination in VSI fed induction motor drives using swarm and genetic optimisation, International Journal of Power Electronics, Vol. 5, No. 1. Bose, B.K. (007) Modern Power Electronics and AC Drive, Prentice-Hall of India. Butun, E., Erfidan, T. and Urgun, S. (006) Improved power factor in a low-cost PWM single phase inverter using genetic algorithms, Energy Conversion and Management, Vol. 47, pp Chiasson, J.N., Tolbert, L.M., McKenzie, K.J. and Zhong, D. (004) A complete solution to the harmonic elimination problem, IEEE Transactions on Power Electronics, Vol. 19, pp Choe, G-H. and Park, M-H. (1988) A new injection method for AC harmonic elimination by active power filter, IEEE Transactions on Industrial Electronics, Vol. 35, pp Czarkowski, D., Chudnovsky, D.V., Chudnovsky, G.V. and Selesnick, I.W. (00) Solving the optimal PWM problem for single-phase inverters, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 49, pp Dahidah, M.S.A. and Agelidis, V.G. (008) Single-carrier sinusoidal PWM-equivalent selective harmonic elimination for a five-level voltage source converter, Electric Power Systems Research, Vol. 78, pp Jabr, R.A. (006) Solution trajectories of the harmonic-elimination problem, IEE Proceedings on Electric Power Applications, Vol. 153, No. 1, pp Levenberg, K. (1944) A method for the solution of certain problems in least squares, Quarterly of Applied Mathematics, Vol., pp

21 0 K. Sayed et al. Liang, T-J., O Connell, R.M. and Hoft, R.G. (1997) Inverter harmonic reduction using Walsh function harmonic elimination method, IEEE Transactions on Power Electronics, Vol. 1, pp Marquardt, D. (1963) An algorithm for least-squares estimation of nonlinear parameters, SIAM Journal on Applied Mathematics, Vol. 11, pp Mohan, N., Undeland, T.M. and Robbins, W.P. (1995) Power Electronics: Converters, Applications and Design, nd ed., John Wiley and Sons, NY. Patel, H.S. and Hoft, R.G. (1973) Generalized techniques of harmonic elimination and voltage control in thyristor inverters: Part I Harmonic elimination, IEEE Transactions on Industry Applications, Vol. IA-9, No. 3, pp Rahman, A. (001) A flexible way to generate PWM-SHE switching patterns using genetic algorithms, Proceedings of IEEE Applied Power Electronics Conference and Exposition (APEC), pp Rashid, M.H. (001) Power Electronics Handbook, Academic Press, USA. Ray, R.N., Chatterjee, D. and Goswamie, S.K. (008) A modified reference approach for harmonic elimination in pulse-width modulation inverter suitable for distributed generations, Electric Power Components and Systems, Vol. 36, No. 8, pp Schutten, M.J. and Torrey, D.A. (1995) Genetic algorithms for control of power converters, Proceedings of IEEE Power Electronics Specialists Conference, pp

22 Appendix A A ripple current minimisation based single phase PWM inverter 1 A1 Solution based on Newton Raphson algorithm (NRA) The system arranged for the nth iteration is as follows: 1 n1 f1( 1 n, n, 3 n, 4n) 1 n 1 n1 ( J( 1 n, n, 3 n, 4n)) f( 1 n, n, 3 n, 4n) n 3n 1 f3( 1 n, n, 3 n, 4n) 3n f (,,, ) 4n1 4 1n n 3n 4n 4n The matrix J is the Jacobian matrix at the nth iteration as: sin( 1 n) sin( n) sin( 3n) sin( 4n) 3sin(3 1 n) 3sin(3 n) 3sin(3 3n) 3sin(3 4 ) n J( 1 n, n, 3n, n4) 5sin(5 1 n) 5sin(5 n) 5sin(5 3n) 5sin(5 4n) 7sin(7 1 n) 7sin(7 n) 7sin(7 3n) 7sin(7 4n) and (11) (1) (13) in 1 in % error i 100 in where i = 1,,3,4. The process is iterated until the percent error (%error) converges to a predetermined value by using the last values as new initial values of the switching angles. The convergence depends on how close the initial values of the first iteration to the final solution. A Solution based on Levenberg Marquadt algorithm (LMA) Levenberg Marquardt is a popular alternative to the Gauss Newton method of finding the minimum of a function F(x) that is a sum of squares of nonlinear functions. In the present work, the sum of squares of the ripple harmonic current components is considered a nonlinear function of the switching angles α to be minimised with respect to the parameter vector α. 1 F( x) [ ( )] m fi x or i1 I 1 (14) ripple ( ) Ii i3,5,7,... In the proposed system, a ripple current function I r (α) is to be minimised with respect to the parameter vector α. Then the Newton s method would be: 1 Iripple ( ) Ir ( ) (15)

23 K. Sayed et al. Assume that I r (α) is the sum of square of the harmonic components up to N, where N is the number of pulses per half cycle i.e., I N r( ) Ii ( ) i1 Let the Jacobian of I r (α) is denoted as J(α): I1( ) I1( ) I1( ) I1( ) I3( ) I3( ) I3( ) I3( ) J ( ) I5( ) I5( ) I5( ) I5( ) I7( ) I7( ) I7( ) I7( ) where, N = 4 and (16) N ( ) i( ) i( ) i1 S I I For the Gauss Newton method it is assumed that S(α) 0 and the ith update of equation (16) becomes: 1 T T J ( ) J( ) J ( ) I i ( ) (17) The LM modification to the Gauss-Newton method is: 1 T T J ( ) J( ) I J ( ) I i ( ) (18) The primary application of the LMA algorithm is to optimise the parameters vector α so that the sum of the squares of the deviations α becomes minimal. Like other numeric minimisation algorithms, the LMA algorithm is an iterative procedure. To start a minimisation, one has to provide an initial guess for the parameter vector, α. The LM algorithm converges more quickly if the initial guess is already somewhat close to the final solution. The choice of the initial condition of switching angles to eliminate particular harmonics is explained in Appendix B.

24 Appendix B A ripple current minimisation based single phase PWM inverter 3 Initial condition calculation The amplitude of the fundamental frequency for square wave output from full bridge inverter is determined by the DC input voltage. A controlled output can be produced by modifying the switching scheme. An output voltage has intervals corresponding to zero output as well as +Vdc and Vdc as shown in Figure B1. The output voltage can be controlled by adjusting interval α on both sides of the pulse where the output is zero. Figure B1 Controlled output voltage from full bridge inverter The rms value of the voltage waveform in Figure B1 is equal: Vrms Vdc 1 The Fourier series of the waveform is expressed as: vo() t Vnsin( not) n, odd Taking advantage of half-wave symmetry, the amplitudes are: 4V dc Vn Vdcsin( not) d( ot) cos( n) (5) n The amplitude of each harmonic of the output is a function of α. In particular, the amplitude of the fundamental is controlled by adjusting α. 4V V1 dc cos( ) Harmonic contents can be controlled by adjusting α. If α = 30, for example, V 3 = 0. This is a significant because the third harmonic can be eliminated from the output voltage and current. The nth harmonic can be eliminated by choosing a value of α which makes the cosine term in equation (5) equal to zero, i.e. α = 90/n. To eliminate the 3rd harmonic α is to be chosen equals to 30. To eliminate the 5th harmonic α is to be chosen equals to 18. To eliminate both the 3rd and 5th, α is to be chosen between 18 and 30. In the present case, α 1, α, α 3 and α 4 take the initial values such that 18< α 1 < α < α 3 <α 4 <90.