An Alternative Pulse-Width Modulation Scheme for Automotive Inverters | MTH 844, Study Guides, Projects, Research of Mathematics

Download An Alternative Pulse-Width Modulation Scheme for Automotive Inverters | MTH 844 and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! An Alternative Pulse-Width Modulation Scheme for Automotive Inverters Bulent Buyukbozkirli Hafid Chrifi-Alaoui May, 2001 Abstract: Future automotive 42-Volt DC electrical systems will use pulse-width modulated inverters to convert the bus voltage to a 3-phase AC voltage with desired amplitude and frequency. In this report we develop a method using Genetic Algorithms that finds switching algorithms that reduce the number of filter capacitors needed at the DC side of the inverter, thus decreasing the production cost of the inverters significantly while maintaining acceptable AC bus power quality. Work done for McCleer Power, Inc. under the direction of C. R. MacCluer and P. J. McCleer, in partial fulfillment of the requirements of Michigan State University MTH 844. Table of Contents Introduction .........................................................................................................1 A Simplified Example ……………………...............….......................................3 Genetic Algorithms ……………………………………………………………..4 Simulation Results …………………………………………….………………..7 Application of Genetic Algorithms ……………………………………………..11 Conclusion …..………………………………………...………………………..15 Acknowledgments …………………………………………..………………….16 References …..…………………………………………..……….…………..…17 Appendix A. …………………………...………………………….………….....18 of the other switch is delayed by a blanking time, which introduces low-order harmonics in the output. 2 A Simplified Example In this section we analyze a simplified half-wave version of Figure 2 as shown in Figure 3. Our goal is to illustrate with this simple case that it is possible to find, by means of Genetic Algorithms, a switching pattern that satisfies desired specifications, yet decreases capacitor currents. The basics of Genetic Algorithms are described in Section 3. In our model, Figure 3, a resistive load is used for simplicity. The resistance r represents the wiring losses. All the values of the circuit elements are normalized so that the battery voltage is 1 volt. The voltage Vc across the capacitor satisfies the differential equation rC V V RCrCdt dV DC c c        11 , (1) where R= when the switch is open. The solution to (1) is t RrC DCc AeVRr R tV           111 )( , (2) where the constant A is determined by the initial value of Vc. Setting RG /1 , rg /1 and finding the value of A in terms of the initial voltage Vc(0), (2) becomes 3 R=2      i r i C iR + V C - + - r=0.1 V DC = 1 volt V R + Figure 3 Current chopping by ideal switch t C Gg DCcDCc eVrG VV rG tV )( ) 1 1 )0(( 1 1 )(       . (3) Here we take the conductivity G=0.5 when the switch is ON, and G=0 when the switch is OFF. Using solution (3) for the voltage it is quite easy to find the currents through the resistance R and r, since we have RVi cR / when the switch is on (otherwise zero), and   rVVi cDCr / . Section 4 contains the results of the simulation of this circuit using MATLAB. 3 Genetic Algorithms Genetic Algorithms (GAs), developed by J. H. Holland in 1975, [2][3], are search algorithms based on the dynamics of natural selection and genetics. They represent the evolution of a given population in the parameter space towards an optimal configuration, the only one that is able to “survive” in relation with the given experimental data. More information and the mathematical foundations of GAs can be found in [4]. GAs have been applied in the control design of PWM digital controllers [5], but never to our knowledge to finding the optimal switching pattern of the power inverters as presented in this report. The wide interest in GAs is due to their computational simplicity and their capability of detecting global optima in the presence of many local minima. Genetic algorithms are superior to other optimization and search procedures in four ways:  Direct use of a coding: GAs require the natural parameter set of the optimization problem to be coded as a finite-length string over some finite alphabet. Thus, they work with a coding of the parameter set, not the parameters themselves.  Search from a population: In many optimization methods, we move from a single point in the decision space to the next using some transition rule. By contrast, GAs work from a rich database of points simultaneously (a population of strings), climbing many peaks in parallel; thus, the probability of finding a false peak is reduced over methods that go point-to-point.  Blindness to auxiliary information: To perform an effective search for better and better structures, GAs exploit all the information directly from the fitness 4 function (objective function) associated with individual strings and therefore do not require any auxiliary information or specific assumptions about the parameter space (such as continuity or differentiability).  Randomized operators: GAs use probabilistic transition rules to guide their search, yet they are not a simple random search. Random choice is used as a tool to guide a search toward regions of the search space with likely improvement. In order to illustrate how the genetic algorithm works, we consider the following simplified example: Suppose that the switch of our circuit turns on 5 times during each period of the output frequency. We want to determine the start time and duration of each turn-on period. So, we can take the time difference between the start times of each turn-on period as our parameter space and ask the genetic algorithm to find optimum turn-on time values for us. Thus, each member of our population is represented by a string (chromosome) of 5 integers, such as 3-6-2-7-4. Hence, “3” here means that the first turn-on will take place after 3 units from the start of the period, and “6” means that the second turn- on will be after 6 units from the end of the first turn-on period and so on. Each integer in this string, called the “field,” may take values in a specified discrete range (in this example, we use 3-bit values: 0,1,2,…,7). To find the actual time lengths, one needs to normalize the string 3-6-2-7-4 so that the addition of its entries equals to one period of the output signal. Genetic algorithm starts with an initial population containing many such strings. Successive generations are produced by means of three fundamental operators from the previous generation: reproduction, crossover, and mutation. To illustrate these operators, let us take an initial population of only 4 strings, for simplicity, 3-6-2-7-4 4-2-3-3-1 6-3-2-5-2 7-4-5-1-5 . A fitness value is assigned to each of these strings, which is a measure of how good a string is. In other words, it is the figure of merit that we want to maximize. Assume that we have the following fitness values for these members as shown in Table 1. 5 First we find a symmetric PWM pattern assuming that the current though R remains fixed at the value A=0.5 amperes during the ON-intervals. The duration of ON and OFF intervals are found as follows: The time interval from t=0 to t=0.005 is divided into 50 equal sections. Then, for each section, the duration of ON-interval is calculated which would supply the same amount of energy to the load as the ideal sinusoidal power P(t) would give during the given section. We will call this switching pattern as Symmetric Linear Partition pattern. Figure 5 illustrates this pattern on the graph of the corresponding power through R as a square-like graph. The sinusoidal graph in this figure is the graph of the desired ideal power. Note that, as the ideal power increases, the length of the ON- interval increases since one needs more integrated power to supply the energy that would be supplied in the ideal case. In Figure 6(a) we see the graphs of the capacitor voltage VC when the above ON- OFF pattern is applied. Here, we have a continuous graph that is either exponential growth or decay depending on the position of the switch (as expected). The corresponding battery current ir behaves similarly as seen in Figure 6(b). We observe similar ripples on ir. 8 0 1 2 3 4 5 x 10-3 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 watts seconds Figure 5 The synthesized and the ideal power through the resistance R.. Figure 7 contains the graphs of both the current estimated, (which was assumed to be constant at iR=0.5 ampere for ON-intervals), and the graph of the actual iR when the switching pattern obtained above is applied. We observe that the actual current fluctuates around 0.49 ampere with small exponential increase or decrease during ON-intervals. 9 0 1 2 3 4 5 x 10 -3 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 seconds volts 0 1 2 3 4 5 x 10 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 seconds amps Figure 6 (a) Voltage V C across the capacitor (b) Current i r through the battery The difference between the estimated and the actual load current causes a loss in the desired power quality. Because of this loss, the actual energy supplied to the load R is less then the desired energy. Using the formula dttRA dtiVdttRA lossenergypercentage Rc       22 22 sin sin 100 , where the integrals are taken over one period of tA sin , we have found this error to be 6.89%. 10 Figure 7 0 0.2 0.4 0.6 0.8 1 1.2 x 10 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 seconds amps seconds amps (a) Estimated load current i R 0 1 2 3 4 5 x 10 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 amps seconds (b) Actual load current i R . In this case, we calculate the required duration of on-intervals, by a MATLAB routine, so that the desired output power requirement (within 10% of the ideal power, which would be supplied if the output current were an ideal sinusoidal signal) is satisfied. Then, using this pattern of ON and OFF intervals, the routine assigns the fitness of the string as only Fit1 of the first case. After 20 generations, with 200 strings in each generation, we have obtained the results shown in Figure 11. In both cases, Figures 9 and 11, we observe that the maximum fitness for each generation is getting better as the new generations are produced. We also observe an overall increase in the minimum fitness of each population, which means that there is an 13 ... ONOFF OFF OFF OFF OFF ON ONOFF 1st field (gene) = 4 50th field (gene) = 3 String: (Chromosome) 2nd field (gene) = 14 Figure 10 Strings (chromosomes) of Case 2: Symmetric Nonlinear Partition 0 5 10 15 20 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 minimum fitness maximum fitness Fitness F= Fit 1 generation # Figure 11 GALOPP results for Case 2 improvement in the average member of each generation when the new generations are produced. In Case 1 (Figure 9) the maximum fitness value possibly did not converge to its best value since we see a regular increase in every 5 to 6 generations. Table 2 below compares the commonly used symmetric linear partition of section 4 with Case 1 and Case 2. The second column is the percentage energy loss which is caused by the difference between the estimated current through the load R and the actual current obtained by the given ON-OFF switching design scheme. The percentage energy error for each case is within acceptable limits (within 10%). The third column shows that the switching pattern of Case 2 is better than the other two, as far as the amount of capacitor current is concerned. Case 1 was a very hard situation for GALOPP to work on since a switching pattern which is “good” for power quality may not usually be a “good” one for the capacitor current, thus making it quite difficult to improve both Fit1 and Fit2 at the same time. In the last column, we have the values measuring how well the output power imitates tRA 22 sin . Here, we see that the switching pattern of Case 1 is not successful at all. On the other hand, when we compare Case 2 with linear partition case, we see that the improvement in the capacitor current is obtained at the expense of output power quality. However, Case 2 produces an output power that is quite close to the quality of the standard one. Although we have not looked at the Fourier analysis of the output power signal, we expect to see a higher harmonic distortion for Case 2 than the symmetric linear partition case. Table 2 Comparison of the three switching schemes % energy loss Fit1 (Capacitor current) Fit2 (Output power quality) Symmetric Linear Partition of section 4 6.8893 1.1920 26.5626 Case 1 Non-Symmetric Nonlinear Partition 6.5420 0.8549 0.8569 Case 2 Symmetric Nonlinear Partition 7.5295 1.3092 19.2824 14 In Figure 12, we see the comparison of all three cases over each generation for the capacitor current fitness Fit1 only. This graph clearly shows that Nonlinear Symmetric Partitions obtained from GA application are better than the other two cases, as far as the capacitor current is concerned. Even in the first population that was selected randomly, we have a member that is better than the standard Symmetric Linear Partition. The following generations of Case 1 produce much betters elements. 6 Conclusion and Future work In our simplified model, we achieved an improvement in the capacitor current at the expense of output power quality. The amount of this tradeoff can be controlled by the GA application. Even though we have started with a random selection of the initial population, 15 0 4 8 12 16 20 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 (Case 2) Symmetric Nonlinear Partition (Case 1) Non-symmetric Nonlinear Partition Symmetric Linear Partition (standard) Generation # Fi t 1 Figure 12 Comparison of GALOPP results of Case 1 and Case to with the standard pattern 8 References [1] N. Mohan, T. M. Undeland, W. P. Robbins, Power Electronics, second edition, New York: John Wiley & Sons. [2] J. H. Holland, Genetic algorithms and the optimal allocations of trials, SIAM Journal of Computing, 2 (2), 88-105, 1973. [3] J. H. Holland, Adaptation in natural and artificial systems, Ann Arbor: The University of Michigan Press, 1975. [4] David E. Goldberg, Genetic Algorithms, in Search, Optimization & Machine Learning, Addison-Wesley, 1989. [5] L.S. Shieh, W. Wang, J.S.H. Tsai, Optimal digital design of hybrid uncertain systems using genetic algorithms, IEE Proceedings online no. 19990082, 1999. [6] GALOPP , http://garage.cps.msu.edu/software/software-index.html 18 Appendix A The circuit used in the analysis part of the project, is in fact a simplified version of the actual circuit, a picture of more realistic version of the actual circuit is shonw in Fig A1: Figure A1 The induction motor equivalent circuit is introduced in the new version as a more realistic load. The two switches in the new version, allow the current to recirculate through diodes. When transistors are switched off, two cases are introduced: The first case is when switch 1 is ON and switch 2 is OFF. Fig A2 describes the equivalent circuit. Figure A2 In general U=Z*I (see Figure A3.) 19 Figure A3 Now, we rewrite the circuit of Figure A2 as shown below in Figure A4 Figure A4 Z=R1 + L1s+ sLsLR 233 11 1   and U = ( R1 + L1s + 2 3223 323 sLLsLR sLsLR   )*I, That is : ( R2L2s + L2L3s2 ) * U = ( R1 + L1s )*( R3L2s + L2L3s2 ) + R3 + L2s + L3s; And so has the time-domain model 20 The response equation for the first component is given in analysis part of the project. When analyzing the second component, we will concentrate on the current in the first resistor R1. The state equations are as follows: i = i2 + i3, (2.1) L1 dt di + R2 i = L2 dt di2 , (2.2) L3 dt di3 - R3i3 = L2 dt di2 . (2.3) Substitute ( 2.1) in (2.2) and (2.3) to get -L1 dt di + R2 i = L2 dt di - L2 dt di3 (2.4) L3 dt di3 - R3i3 = L2 dt di - L2 dt di3 (2.5) From (2) we can get dt di3 as a function of i L2 dt di3 = (L2 +L1) dt di + R2 i. (2.6) That is dt di3 = 2 12 L LL  dt di + R2 i. (2.7) On the other hand we can write (2.5) as (L3 + L2 ) dt di3 - R3i3 = - L2 dt di . (2.8) Substituting dt di3 from (2.7) into (2.8) yields to (L3 + L2 ) ( 2 12 L LL  dt di + R2 i ) - R3i3 = - L2 dt di . (2.9) Which yields to i3 = 3 1 R [ (L3 + L2 ) ( 2 12 L LL  dt di + R2 i ) + L2 dt diz ] = [ 23 1223 ))(( LR LLLL  + 3 2 R L ] dt di + 3 223 )( R RLL  i. (2.10) 23 At the end we can use equation (2) to write a state equation in the parameter of interest i3 ( 23 1223 ))(( LR LLLL  + 3 2 R L ) 2 2 dt id + 3 223 )( R RLL  dt di = 2 12 L LL  dt di + R2 i. (2.11) Giving the equation in standard form ( 23 1223 ))(( LR LLLL  + 3 2 R L ) 2 2 dt id +[ 3 223 )( R RLL  - 2 12 L LL  ] dt di - R2 i = 0. (2.12) The two cases studied in this section yield to ordinary differential equation with constant coefficients. A case easy to solve, still, in the first case the ODE is of order four, so we need four independent initial conditions to get an exact answer, that is we need to keep track of four independent variables from the previous cycle. And the same we need to keep track of three independent variables from the first case, to get initial conditions for the last two differential equations. 24