Filter specifications
1. Introduction
Continuous technological development facilitates the increase in the number of nonlinear loads that significantly affect the power quality in a power system and, consequently, the quality of the electric power delivered to other customers. DC and AC variable speed drives and arc furnaces are ranked among the most commonly used large power nonlinear loads.
DC drives can be a significant plant load in many industries. They are commonly used in the oil, chemical, metal and mining industries. These drives are still the most common large power type of motor speed control for applications requiring very fine control over wide speed ranges with high torques. Power factor correction is particularly important for this drives because of relatively poor power factor, especially when the motor is at reduced speeds. Additional transformer capacity is required to handle the poor power factor conditions and more utilities are charging a power factor penalty that can significantly impact the total bill for the facility. The DC drives also generate significant harmonic currents. The harmonics make power factor correction more complicated. Power factor correction capacitors can cause resonant conditions which magnify the harmonic currents and cause excessive distortion levels. For the same reasons arc furnaces are very difficult loads for a supplier and for the customer they are very difficult objects of reactive power compensation and harmonics filtering.
One of the most common methods to prevent adverse effects of nonlinear loads on the power network is the use of passive filters. However, different configurations should be considered before making the final design decision. Among the performance criteria are current and voltage ratings of the filter components, and the effect of filter and system contingency conditions. Before any filter scheme is specified, a power factor study should be done to determine if any reactive compensation requirements are needed. If power factor correction is not necessary, then a minimum power filter can be designed; one that can handle the fundamental and harmonic currents and voltages without consideration for reactive power output. Sometimes, more than one tuned filter is needed. The filter design practice requires that the capacitor and the reactor impedance be predetermined. For engineers not knowing the appropriate initial estimates, the process has to be repeated until all the proper values are found. This trialanderror approach can become complex as more filters are included in the systems.
While the effectiveness of a filter installation depends on the degree of harmonic suppression, it also involves consideration of alternate system configurations. As the supplying utility reconfigures its system, the impedance, looking back to the source from the plant’s standpoint, will change. Similar effects will be seen with the plant running under light versus heavy loading conditions, with splitbus operation, etc. Therefore, the filtering scheme must be tested under all reasonable operating configurations.
The general procedure in analyzing any harmonic problem is to identify the worst harmonic condition, design a suppression scheme and recheck for other conditions. Analysis of impedance vs frequency dependencies for all reasonable operating contingencies is commonly used practice. A frequency scan should be made at each problem node in the system, with harmonic injection at each point where harmonic sources exist. This allows easy evaluation of the effects of system changes on the effective tuning. Of particular importance is the variability of parallel resonance points with regard to changing system parameters. This problem is illustrated by the practical example.
In a most classic cases all filter considerations are carried out under the following simplifying assumptions: (i) the harmonic source is an ideal current source; (ii) the filter inductance
2. Singletuned single branch filter
Many passive
Where
The schematic diagram of an example group of filters in a large industrial installation and characteristics illustrating the line current variations and the 5th harmonic voltage variations in result of connecting ONLY the 5th harmonic filter are shown in Fig. 2. The figure also shows the 7th harmonic voltage variations prior to and after connecting the 5th harmonic filter. The 5th harmonic filter selectivity is evident ― its connection has practically no influence on the 7th harmonic value.
Relations (2) allow determining parameters of a group of singletuned filters taking into account their interaction, as well as choosing the frequencies for which the impedance frequency characteristic of the filter bank attains maxima, where (the filters' resistances R_{i} 0): C_{i}  the filters' capacitances;
2.1. Example 1
An example application of the method will be the design of singletuned filters (two singletuned filters) for DC motor (Fig. 3). The basis for design is modelling of the whole supplying system. The system may comprise nonlinear components and analysis of the filters can take into account their own resistance, which depends on the selected components values. Generally speaking, the model can be detailed without simplifications.
Parameters of the singletuned filters group were determined by means of the Genetic Algorithm minimising the voltage harmonic distortion factor with limitation of the phase shift angle between fundamental harmonics of the current and voltage
The genetic algorithm objective is to find the capacitance values of two singletuned filters tuned to harmonics
Basic characteristics of the power system, before and after connecting the filters, are tabulated in Fig. 4 (
In industry many of the supply systems consist of a combination of tuned filters and a capacitor bank. Depending on the system configuration the capacitor bank can lead to magnification or attenuation of the filters loading. Filter detuning significantly affects this phenomenon. Therefore, specifying harmonic filters requires considerable care under analysis of possible system configurations for avoidance of harmonic problems.
3. Parallel operation of filters
3.1. Example 2 – description of the system
Fig. 5 shows a oneline diagram of a mining power supply system which will be used to analyze operation characteristics of the single tuned harmonic filters in a power supply system including power factor correction capacitor banks. System contains two sets of powerful DC skip drives as harmonic loads connected to sections A and B. The drives are fed from sixpulse converters. As a result, there is significant harmonic current generation and the plant power factor without compensation is quite low.
Shunt capacitors 2×1.5 MVA connected to main sections 1 and 2 to partially correct the power factor but this can cause harmonic problems due to resonance conditions. The sections A and B can be supplying from the main section 1 or 2. Four singletuned filters (5^{th}, 7^{th}, 11^{th}, and 13^{th} harmonic order) have been added to the sections A and B to limit harmonic problems and improve reactive compensation. Specifications of the harmonic filters are shown in the Table 1.
Allowable current limit for filter capacitors is 130% of nominal RMS value and voltage limit 110%. The ironcore reactors take up less space comparatively to aircore reactor and make use of a threephase core. Reactors built on these cores weigh less, take up less space, have lower losses, and cost less than three singlephase reactors of equal capability. Reactors are manufactured with multigap cores of cold laminated steel to ensure low tuning tolerance. The primary draw back to ironcore reactors is that they saturate.




F5 
Bank rating  2×500  kvar  Nominal voltage  7.2  kV 
Nominal voltage  6.6  kV  Nominal current  120.0  A  
Nominal current  87.4  A  S.c. current  14.0  kA  
Capacitance  73.1  μF  Inductance  6.0  mH  
Cap. tolerance  5…+10  %  Inductance tolerance  ± 5  %  
F7 
Bank rating  2×400  kvar  Nominal voltage  7.2  kV 
Nominal voltage  6.6  kV  Nominal current  100.0  A  
Nominal current  70.0  A  S.c. current  14.0  kA  
Capacitance  58.4  μF  Inductance  3.54  mH  
Cap. tolerance  5…+10  %  Inductance tolerance  ± 5  %  
F11 
Bank rating  2×500  kvar  Nominal voltage  7.2  kV 
Nominal voltage  6.6  kV  Nominal current  130.0  A  
Nominal current  87.4  A  S.c. current  14.0  kA  
Capacitance  73.1  μF  Inductance  1.16  mH  
Cap. tolerance  5…+10  %  Inductance tolerance  ± 5  %  
F13 
Bank rating  2×500  kvar  Nominal voltage  7.2  kV 
Nominal voltage  6.6  kV  Nominal current  130.0  A  
Nominal current  87.4  A  S.c. current  14.0  kA  
Capacitance  73.1  μF  Inductance  0.82  mH  
Cap. tolerance  5…+10  %  Inductance tolerance  ± 5  % 
The saturation level is dependent upon the fundamental current and the harmonic currents that the reactor will carry. There is not standard for rating harmonic filter reactors and therefore, it is difficult to evaluate reactors from different manufacturers. For example, some reactor manufacturers base their core designs (cross sectional area of core) on RMS flux, while other will based it on peak flux (with the harmonic flux directly adding). There is a significant difference between these two design criteria. For evaluation purposes, reactor weight and temperature rise are a primary indication of the amount of iron that is used. The second feature of the reactors is considerable frequency dependency of eddy currents loss in the winding.
Equation (1) shows that the relative resonant frequency
where:
Assuming
This means that the analysed filter circuits have the following possible ranges of relative resonant frequency
It is obvious that the detuning of higher order filter is more sensitive for the same filter capacitance or inductance drift than detuning of lower order filter, as value of resonant frequency
3.2. Filter characteristics analysis
In order to demonstrate filter circuits behavior under all reasonable operating configurations and get numerical results for comparison purposes, computer simulations have been performed using frequency and time domain software.
Measurements performed at the facility were used to characterize the DC drive load and obtain true source data for computer analysis of the filter characteristics. For example, Fig. 6 shows the DC drive current and its harmonic spectrum in the supply system consisting of 5^{th} order filter under isolated operation of the section A.
Harmonic currents in the supply system components are listed in Table 2. There are obvious important findings from these measurements: 1) noncharacteristics current harmonics are present due to irregularities in the conduction of the converter devices, unbalanced phase voltages and other reasons; 2) there is resonance condition near 4^{th} harmonic in the system configuration with 5^{th} filter connected. Similar measurements also provided for the system with other filter sets.
Analysis of the system response is important because the system impedance vs frequency characteristics determine the voltage distortion that will result from the DC drive harmonic currents. For the purposes of harmonic analysis, the DC drive loads can be represented as sources of harmonic currents. The system looks stiff to these loads and the current waveform is relatively independent of the voltage distortion at the drive location. This assumption of a harmonic current source permits the system response characteristics to be evaluated separately from the DC drive characteristics.
In Fig. 7 are depicted the worst case of frequency scan for system impedance looking from the section A with several filters connected as concerns 5^{th} harmonic filter loading. These conditions occur with upper limit (see (3)) of filter reactor and capacitor rating variations. Proximity of the frequency response resonance peaks to 4^{th} and 5^{th} harmonics produces significant magnification the harmonic currents in the 5^{th} filter and feeder circuits.



1  241,49  100,0  309,68  100,0  157,87  100,0 
2  8,49  3,5  8,04  2,6  0,61  0,4 
3  9,75  4,0  8,03  2,6  1,77  1,1 
4  41,62  17,2  7,51  2,4  39  24,7 
5  27,02  11,2  68,09  22,0  42,54  26,9 
6  4,31  1,8  6,75  2,2  2,42  1,5 
7  25,43  10,5  30,21  9,8  5,28  3,3 
8  0,74  0,3  0,85  0,3  0,28  0,2 
9  2,81  1,2  3,36  1,1  0,57  0,4 
10  2,92  1,2  3,49  1,1  0,57  0,4 
11  22,09  9,1  26,36  8,5  4,33  2,7 
12  4,16  1,7  5,04  1,6  0,89  0,6 
13  12,7  5,3  15,43  5,0  2,77  1,8 
14  1,21  0,5  1,49  0,5  0,27  0,2 
15  2,79  1,2  3,37  1,1  0,59  0,4 
16  2,84  1,2  3,41  1,1  0,59  0,4 
17  12,14  5,0  14,57  4,7  2,48  1,6 
18  3,52  1,5  4,28  1,4  0,73  0,5 
19  7,35  3,0  8,74  2,8  1,38  0,9 
Harmonic current magnification in a filter circuit can be defined by the following factor:
and for the feeder circuit similarly:
where:
The harmonic magnification factor allows estimating harmonic current in a filter or feeder circuit for several system configurations relative to source harmonic current. A value less than 1.0 means that only a part of the source harmonic current flows in the circuit branch.
Calculated values of harmonic magnification factors for analyse 5th filter loading in the several system configurations are listed in Table 3. Column “Upper deviation limits” with 2×1.5 Mvar capacitors corresponds to the Fig. 7. The significant 4th and 5th harmonics magnification can be observed from the Table 3 in the 5th filter and feeder circuits in the case of 2×1.5 Mvar capacitors connected. It can cause the filter overload and allowable system voltage distortion exceeding. On the other hand when lower deviation of the filter parameters the magnification factors are considerably less. Switching off the 2×1.5 Mvar capacitors reduces 5th harmonic magnification in the circuits to acceptable levels, but 4th harmonic is magnificated considerably more due to close to resonant peak.









With cap. 2×1.5 Mvar  β_{F4}  β_{F5}  β_{S4}  β_{S5}  β_{F4}  β_{F5}  β_{S4}  β_{S5} 
F5  4.2  0.8  9.3  1.7  0.5  1.1  2.6  0.4 
F5+F7  14.4  1.4  32.2  2.8  0.6  1.0  3.3  0.4 
F5+F7+F11  4.8  2.8  10.2  5.5  0.8  0.9  4.5  0.4 
F5+F7+F11+F13  2.4  18.6  5.2  36.7  1.4  0.8  7.4  0.3 
Without cap. 2×1.5 Mvar  
F5  0.8  0.3  1.9  0.7  0.2  1.6  1.3  0.6 
F5+F7  1.1  0.4  2.4  0.8  0.3  1.5  1.4  0.6 
F5+F7+F11  1.5  0.5  3.2  1.0  0.3  1.4  1.6  0.5 
F5+F7+F11+F13  2.1  0.6  4.5  1.1  0.3  1.3  1.8  0.5 
The calculated harmonic current magnification factors in filter circuits in the possible filter configurations are depicted in the Table 4. It is here noted that harmonic loading of the filters in the system without 2×1.5 Mvar capacitors depends on the filter configuration and filter detuning. It is well known that the series LC circuit has the lowest impedance at its resonant frequency. Below the resonant frequency the circuit behaves as a capacitor and above the resonant frequency as a reactor. When a filter is slightly undertuned to desired harmonic frequency it has lower harmonic absorbing as a result of the harmonic current dividing between the filter and system inductances. If the filter is slightly overtuned than parallel resonant circuit created of the filter capacitance and system inductance will magnify the source harmonic current. Regularity of the phenomena for the analyzed system with multiply filter circuits one can see in the bottom part of the Table 4 for the system configuration without capacitors 2×1.5 Mvar.
Switching in capacitors 2×1.5 Mvar to the bus section changes the filters loading due to parallel resonant circuit created of the capacitors and system impedances. The resonant frequency of the system looking from the section A with several connected filters depends on the number of the filters and specifies the filter loading.








With cap. 2×1.5 Mvar  Up  Lo  Up  Lo  Up  Lo  Up  Lo 
F5  0.8  1.1             
F5+F7  1.4  1.0  0.1  0.1         
F5+F7+F11  2.8  0.9  0.1  0.1  0.6  1.9     
F5+F7+F11+F13  18.6  0.8  0.1  0.1  1.3  1.3  0.6  3.2 
Without cap. 2×1.5 Mvar  
F5  0.3  1.6             
F5+F7  0.4  1.5  0.5  3.0         
F5+F7+F11  0.5  1.4  0.7  1.9  0.7  1.3     
F5+F7+F11+F13  0.6  1.3  0.8  1.5  2.5  1.0  0.7  2.1 
Figure 8 shows current waveforms and its harmonic spectrums for parallel 11^{th} and 13^{th} harmonic filters in the analyzed system obtain from time domain computer simulation of the system. The first observation of these two cases is significant harmonic overloading of the filters. In the case in question of filter ironcore reactor the phenomenon can cause the reactor temperature rise and its failure.
The most representative cases of the parallel filter configurations (e.g. when feeding sections A and B from section 1) are depicted in the Table 5. Two parallel the same order filters have opposite resonance detuning with upper and lower parameter deviation limits. From analysis of the Table 5 it is seen that opposite resonance detuning of the same order filters can cause considerable filter overload. As it has been noted earlier the higher order harmonic filters are more sensitive to filter component parameter variations from the detuning point of view. Furthermore, resonance detuning of the same order filters in the some system configurations can cause parallel system resonance peaks close to characteristic harmonic.
It should be quite clear from the above presented example that specifying harmonic filters and power factor correction requires considerable care and attention to detail. Main results of the investigation are follows:
it is a bad practice to add filter circuits to existing power factor correction capacitors,
improper design of the filter resonant point considering capacitor and reactor manufacturing tolerance and operation conditions can cause significant harmonic overloading of the filter,
it is desirable to avoid the parallel operation of the same order filters in the system.








With cap. 2×1.5 Mvar  Up  Lo  Up  Lo  Up  Lo  Up  Lo 
2×(F5+F7)*  0.55  0.55  0.50  0.50         
2×(F5+F7)  0.15  0.91  0.10  0.21         
2×(F5+F7+F11+F13)  0.18  1.05  0.10  0.16  0.57  1.42  2.92  4.82 
(F5+F7)+2×(F11+F13)  18.34    0.11    0.55  1.44  3.02  5.01 
Without cap. 2×1.5 Mvar  
2×(F5+F7)*  0.41  0.41  0.50  0.50         
2×(F5+F7)  0.13  0.77  5.40  9.44  
2×(F5+F7+F11+F13)  0.15  0.86  1.21  2.11  0.50  1.23  3.12  5.13 
(F5+F7)+2×(F11+F13)  0.84    1.23    0.48  1.20  3.11  5.04 
4. Doubletuned filter
Doubletuned resonant filters are sometimes used for harmonic elimination of very high power converter systems (e.g. HVDC systems). Just like any other technical solution they also have their disadvantages (e.g. more difficult tuning process, higher sensitivity of frequency characteristic to changes in components values) and advantages (e.g. lower power losses at fundamental frequency, reduced number of reactors across which the line voltage is maintained, compact structure, single breaker) versus singletuned filters. Such filters prove economically feasible exclusively for very large power installations and therefore they are not commonly used for industrial applications. There are, however, rare cases in which the use of such filter is justified. The doubletuned filter structure and its frequency characteristics are shown in Fig. 9. There are also the relations used to determine its parameters.
4.1. Example 3
As an example let us design a doubletuned filter (consider alternative configurations presented in Fig. 10) with parameters:
Figure 11 shows graphic window of the programme developed by authors in the Matlab environment for optimisation of doubletuned filter. Ranges of filter parameters seeking are visible in the upper part of the widow, below the found characteristic is displayed, and basic parameters of the found solution are shown in the lowest part.
The range of variability of decision variables:
Table 6 provides results of a doubletuned filter (Fig. 9 and 10) optimisation. The solutions are similar to each other (in terms of their values). It is noticeable that genetic algorithm is aiming to minimize the influence of additional resistances, that is to make the filter structures similar to the basic structure from Fig. 9. It means that additional resistances worsen the quality of filtering. The obtained result ensues from the applied optimisation method, i.e. optimisation of the frequency characteristic shape.












85.52  85.52  85.52  85.53  85.53  85.53  85.53 

732.21  732.73  732.72  732.71  732.71  732.72  731.90 

3.481  3.481  3.482  3.482  3.482  3.482  3.482 

0.384  0.384  0.384  0.384  0.384  0.384  0.385 

10.93  10.94  10.94  10.93  10.94  10.94  10.94 

1.207  1.207  1.207  1.207  1.207  1.207  1.208 

7.44  7.44  7.44  7.44  7.44  7.44  7.44 

0.869  0.868  0.868  0.868  0.868  0.868  0.870 

36  36  36  36  36  36  36 

35.82  35.8  35.83  35.85  35.79  35,85  35.86 

252.76  252.58  252.53  252.47  252.59  252.53  252.87 

40  40  40.04  40.01  40  40.01  40 

1  1  1  1  1  1  1 

546.09  546.06  546.10  546,11  546.07  546.11  546.11 

  1  1  1    1   

1    1  1  0     
5. Ctype filter
The principal disadvantage of the majority of filtercompensating device structures is the poor filtering of high frequencies. To eliminate this disadvantage are usually used broadband (damped) filters of the first, second or third order; the Ctype filter is included in the category of broadband filters [1, 2, 10]. Broadband filters have one more advantage, substantial for their cooperation with power electronic converters: they damp commutation notches more effectively than single branch filters  they have a much broader bandwidth. They also more effectively eliminate interharmonic components (in sidebands adjacent to characteristic harmonics) generated by static frequency converters. In the Ctype filter in which the
5.1. Example 4
In result of the arc furnace modernization (Fig. 13.) its power and consequently the level of loadgenerated harmonics have increased. It was, therefore, decided to expand the existing reactive power compensation and harmonic mitigation system. Prior to the modernization the system comprised two parallel, singletuned 3rd harmonic filters that were the cause of a slight increase in the voltage 2nd harmonic.
Considering the system expansion the designed Ctype filter should be tuned to the 2nd harmonic. Although currently the 2nd harmonic level in the existing system does not exceed the limit, connection of new loads may increase the 2nd harmonic to an unacceptable level.
5.1.1. Traditional approach
The filter impedance is given by (Fig. 12) [1]:
The
hence
The Ctype filter is tuned to the resonance angular frequency
hence
The filter reactive power (Q_{F}) for the fundamental harmonic is given by the relation:
that is:
Distribution of the loadgenerated harmonic current between the filter tuned to that harmonic and the system is:
Summarizing, the Ctype filter parameters can be determined from above formulas. For the arc furnace power supply system (Fig. 13) and the design requirements:
The Ctype filter parameters are:
Figure 14a shows frequencyimpedance characteristics of: the power network, the resultant impedance of two singletuned 3^{rd} harmonic filters, and the Ctype filter impedance. Fig. 14b shows frequencyimpedance characteristics of: the network, the resultant impedance of the network and two 3^{rd} harmonic filters, and the resultant impedance of the network, two 3^{rd} harmonic filters and the Ctype filter.
Data listed in Table 7 demonstrate that connecting the Ctype filter results in the expected reduction of the 2^{nd} voltage harmonic in the supply system, whereas other harmonics are reduced to a small extent. Further reduction of the second harmonic can be achieved by improving the Ctype filter quality factor










Busbars voltage harmonics without filters  1,76  3.01  1.66  2.88  1.12  1.75  1.00  1,12  5.87 
Busbars voltage harmonics with two 3rd harmonic filters  2.47  0,27  0.95  1.87  0.78  1.24  0.72  0.81  4.07 
Busbars voltage harmonics with two 3rd harmonic filters and the Ctype filter  1.32  0.27  0.91  1.78  0.74  1.18  0.69  0.78  3.37 
Figure 15a shows the Ctype filter frequency characteristics for different filter quality factors, figure 15b illustrates the relation between the resistance

38.5  44.4  50.0  51.9  57.1  66.6  75.0  83.3  91.0 

61.5  55.6  50.0  48.1  42.9  33.3  25.0  16.7  9.0 

1.60  1.25  1.00  0.93  0.75  0.50  0.33  0.25  0.10 

172  221  276.86  300  350  555  840  1111  2778 
Seemingly, the most advantageous solution is to increase the filter resistance
5.1.2. Genetic approach
The goal of genetic algorithm is to seek the Ctype filter capacitance (
The range of variability of decision variables:
According with the achieved results the total capacitance
Measurements in the power system, configured according to the above specification, were carried out in order to check the correctness of the system operation. The instruments locations were (fig. 13):
Figures 16 – 18 illustrate voltage and current waveforms recorded at the 110kV side, the arc furnace supply voltage the arc furnace and the Ctype filter currents and total harmonic voltage distortion factor THD_{U} at both: the 30kV and 110kV side. The measurements have demonstrated that the Ctype filter performance has met the requirements, i.e. it attains the expected reduction of reactive power, ensures the second harmonic reduction in the power system and harmonic distortion THD_{U} reduction by means of high harmonics mitigation. The measurements verified the proposed method and the Ctype filter designed using this method operates according to the requirements.
6. Conclusion
This chapter presents several selected cases of power electronic systems analysis with respect to high harmonics occurrence and reactive power compensation. For these cases are proposed classical solutions, i.e. power passive filters which still are a basic and the simplest method for high harmonics mitigation. Analytical formulas that enable to determine basic parameters of various filters' structures and a group of singletuned filters are provided.
Also a method for passive filters' design employing artificial intelligence, which incorporates genetic algorithms, is presented. It has been proved that this method can be the attractive tool to solve some kinds of power quality problems. The results obtained using GA are very close to those obtained with the analytical method. Hence the conclusion that genetic algorithms can be an efficient tool for passive filters design. The advantage of the method employing genetic algorithms is the possibility of multicriterial optimisation and taking into account at the design stage different (e.g. voltage or current) constraints. It also can be applied to filters of various structures and degrees of complexity and can account for filters' resistance that may influence the filter resonance frequency. In other words, genetic algorithm can be a useful design tool in cases where the system analysis is too complex or even not possible.





Furnace and filters in operation  off  on  on  on  on  on  off 
U_{RMS} [kV]  18.29  17.59  65.04  66.58  
I_{RMS} [A]    2383  391  398  385  602  86.5 
P [MW]    93.75  0.083  0.234  0.198  105  12.14 
Q [MVAr]    71.19  19.55  19.84  19.68  28.2  6.52 
S [MVA]    125.7  20.65  21.0  20.34  117.5  17.25 
PF    0.744  0.0043  0.011  0.001  0.89  0.57 
THD_{U} [%]  1.56  2.45  1.92  1.58  
THD_{I} [%]    6.44  8.04  12.04  10.62  4.64  6.61 
I_{(1)RMS} [A]    2357  387  376  373  594  85.46 
U_{(1)RMS} [kV]  18.29  17.57  65.0  66.57  
U_{(2)RMS} [%]  0.06  0.73  0.42  0.07  
U_{(3)RMS} [%]  0.57  0.61  0.43  0.43  
U_{(4)RMS} [%]  0.04  0.34  0.19  0.04  
U_{(5)RMS} [%]  1.15  1.51  1.22  1.25  
I_{(2)RMS} [A]    58.7  32  10.8  9.8  15  2.21 
I_{(3)RMS} [A]    97.3  3.5  44.3  38.8  9  4.8 
I_{(4)RMS} [A]    23.1  1.5  5.1  4.5  3.7  0.6 
I_{(5)RMS} [A]    71.2  3.7  11.9  11.2  12.5  3.5 
Pst [%]  1  16.66  9.17  1.02 
Appendix A  Genetic Algorithms
Genetic Algorithms (GA) are stochastic global search method, mimicking the natural biological evolution. It has been noted that natural evolution is done at the chromosome level, and not directly to individuals. In order to find the best individual, genetic operators apply to the population of potential solutions, the principle of survival of the fittest individual. In every generation, new solutions arise in the selection process in conjunction with the operators of crossover and mutation. This process leads to the evolution of individuals that are better suited to be the existing environment in which they live.
GA popularity is due to its features. They: (i) don’t process the parameters of the problem directly but they use their coded form; (ii) start searching not in a single point but in a group of points; (iii) they use only the goal function and not the derivatives or other auxiliary information; (iv) use probabilistic and not deterministic rules of choice. These features consists in effect on the usability of Genetic Algorithms and hence their advantages over other commonly used techniques for searching for the optimal solution. There is a high probability that the AG does not get bogged in a local optimum.
An important term in genetic algorithms is the objective function. It is on the basis of all the individuals in the population are evaluated and on the basis of a new generation of solutions is created. Each iteration of the genetic algorithm creates a new generation. Figure 20. shows the basic block diagram of a Genetic Algorithm.
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Dugan R., McGranaghan M., Electrical power systems quality, McGrawHill, 2002  2.
YaowMing Ch., Passive filter design using genetic algorithms, IEEE Transactions on Industrial Electronics, vol. 50, no. 1, February 2003  3.
Younes M., Benhamida, Genetic algorithmparticle swarm optimization (GAPSO) for economic load dispatch, Electrical Review 10/2011, 369372  4.
Zajczyk R., Nadarzyński M., Elimination of the higher current harmonics by means of transverse filters, Electrical Review 10/2004, 963966  5.
Hanzelka Z., Klempka R., Application of genetic algorithm in double tuned filters design, EPE01, Graz 2729 VIII 2001  6.
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