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TEMPUS ENERGY: POWER QUALITY, HARMONICS 1: The use of a frequency spectrum and the THD By using Fourier analysis, every periodic signal can be considered as the sum of a sinusoidal first order harmonic and sinusoidal harmonics of higher order. The first order harmonic has the same frequency as the original periodic signal. An n-th order harmonic is a sine whose frequency equals π times the frequency of the original signal. In case π(π‘) has a period π (with π = 2πβπ), π(π‘) can be expressed as β π(π‘) = πΌ0 + β πΌπ π ππ(πππ‘ + ππΌπ ) π=1 with appropriate values for πΌ0 , πΌπ and ππΌπ (for π = 1,2,3,4, β¦ ). When considering a single phase rectifier bridge having a sufficiently large inductor at DC side and assuming the rectifier diodes are ideal, the current extracted from the grid is a square wave having a frequency of 50 π»π§. Suppose the square wave current has an amplitude π΅. When applying the Fourier analysis to this square wave, πΌ0 = 0 (the current does not contain a DC component), πΌπ = 0 for even valued π = 2,4, 6, .. and for odd valued π = 1,3, 5, .. πΌπ = 1 4π΅ . π π The higher the order π of the harmonic, the smaller its amplitude (which is commonly the case). In theory, an infinite number of harmonics are needed to obtain the original signal π(π‘) but in reality by adding a finite number of harmonics a good approximation is obtained. The sum of these harmonics up to the seventh order approximates the square wave as visualized in Figure 1. By increasing the number of harmonics which are added, the better the approximation of the square wave. Figure 1: Harmonic analysis of a square wave Instead of visualizing the harmonic contents of the square wave current in the time domain as has been done in Figure 1, it is also possible to visualize this contents in the frequency domain by using a frequency spectrum. Figure 2 visualizes the frequency spectrum of the square wave current showing there is no DC component, there is a first order harmonic having a 50 π»π§ frequency, there is a third order harmonic having a 150 π»π§ frequency, there is a fifth order harmonic having a 250 π»π§ frequency, there is a seventh order harmonic having a 350 π»π§ frequencyβ¦ Figure 2: Frequency spectrum of a square wave The Total Harmonic Distortion (THD) indicates the distortion in comparison with the ideal sinusoidal shape. More precisely ππ»π·πΌ = 2 βββ π=2 πΌπ . πΌ1 The smaller this number, the better the sinusoidal shape is approximated. The grid can be modeled by its Thévenin equivalent circuit as visualized in Figure 3. Due to the current extracted from the grid, a voltage drop appears across this grid impedance implying the voltage π’ππππ (π‘) does not equal π’π1 (π‘). In case the current ππ1 (π‘) extracted from the grid is not a sine, the voltage drop across the grid impedance is also not a sine implying also π’ππππ (π‘) is not a sine anymore (although π’π1 (π‘) is assumed to be a sine). Figure 3: Thévenin equivalent circuit of the grid This means π’ππππ (π‘) can be expressed as β π’ππππ (π‘) = π0 + β ππ π ππ(πππ‘ + πππ ) π=1 with appropriate values for π0 , ππ and πππ (for π = 1,2,3,4, β¦ ). The Total Harmonic Distortion (THD) indicates the distortion in comparison with the ideal sinusoidal shape. More precisely 2 βββ π=2 ππ ππ»π·π = . π1 In general, ππ»π·π is much smaller than ππ»π·πΌ since the distortion of the grid voltage is much smaller than the distortion of the current. 2: Non linear loads: examples The number of non linear loads is countless, but a few examples will give an idea. 2.1: Transformer Due to the non linear behavior of the ferromagnetic material in a transformer, the transformer consumes a non sinusoidal magnetization current (also when the transformer is supplied with a sinusoidal voltage). This magnetization current is the current consumed by the primary winding of the transformer in case the secondary winding is open. The primary winding of the transformer behaves as a series RL circuit. Since this magnetization current is small, the voltage drop across the resistance of this series RL circuit can be neglected. This implies the applied grid voltage π’(π‘) (which is assumed to be a sine here) stands across this inductance of the series RL circuit. This causes the magnetization current implying there is a magnetic flux π(π‘). In case the primary winding has π1 turns, due to Faradayβs Law π’(π‘) = β π1 π π(π‘) . ππ‘ Since π’(π‘) is a sine, also the magnetic flux π(π‘) is a sine. This implies also the magnetic induction π΅(π‘) = π(π‘) π΄πΆ in the core material is a sine (π΄πΆ is the area of the magnetic core material). Although the magnetic induction is a sine, due to saturation effects, the magnetic field intensity π» and therefore also the magnetization current are not sinusoidal. This is visualized in Figure 4. As the amplitude of the supply voltage π’(π‘) increases, the ferromagnetic core material saturates and more distortion of the magnetization current is obtained. In the linear part of the magnetization curve (BH curve), the current remains (almost) sinusoidal. The magnetization current in Figure 4 contains a first order harmonic, a third order harmonic, a fifth order harmonic, β¦ Figure 4: Magnetization curve of a transformer When the hysteresis phenomenon is taken into account, the zero crossing of the magnetization current will be shifted in comparison with the zero crossing of the flux π(π‘) and the induction π΅(π‘). The positive and the negative alternation of the current have a different shape. Even inside the positive and the negative alternation of the current there is a non-symmetry. Due to this nonsymmetry, the spectrum does not only contain odd harmonics. The spectrum also contains even harmonics. Figure 5: Magnetization current of a transformer 2.2: Six pulse rectifier bridge As already mentioned, power electronic devices are an important source of harmonics in the current. This is not only the case when considering single phase rectifier bridges, also three phase rectifier bridges are important sources of harmonics. Figure 6 visualizes a three phase rectifier bridge where due to an inductor at DC side (smoothing reactor) the DC current is (almost) constant. The rectifier bridge of Figure 6 is a six pulse rectifier bridge. Each 20 ππ (period of the grid voltage), the output voltage contains six ripples. Figure 6: Three phase rectifier bridge with smoothing reactor In case the diodes or the thyristors of the rectifier bridge are idealized, the current consumed by the rectifier bridge is visualized in Figure 7. The corresponding spectrum is visualized in Figure 8. The current of such a six pulse rectifier bridge contains a first order harmonic of 50 π»π§ , a fifth order harmonic of 250 π»π§, a seventh order harmonic of 350 π»π§, an eleventh order harmonic of 550 π»π§β¦ Figure 6: Grid current of a three phase rectifier bridge Figure 7: Spectrum of the grid current of a three phase rectifier bridge In case of a six pulse rectifier bridge, the current contains harmonics of order 6π ±1 for all integer values π β₯ 0. Especially when considering lower powers, the smoothing reactor in Figure 6 is often omitted (reactors are heavy and they are rather expensive). In Figure 8, a capacitor is placed in parallel with the DC output of the rectifier to obtain a constant DC voltage. In such a situation, the DC current is not constant implying the shape of the grid current changes as visualized in Figure 9. Figure 10 visualizes the spectrum of the grid current visualized in Figure 9. Notice the THD of the grid current in Figure 9 and Figure 10 is higher than the THD of the grid current in Figure 6 and Figure 7. Figure 8: Three phase rectifier bridge with a capacitor at DC side Figure 9: Grid current of a three phase rectifier bridge with capacitor at DC side Figure 10: Spectrum of the grid current of a three phase rectifier bridge with capacitor at DC side 2.3: Twelve pulse rectifier bridge Figure 11 visualizes a twelve pulse rectifier bridge. A twelve pulse rectifier bridge contains twice a six pulse rectifier bridge. The three phase rectifiers are fed using a three phase transformer having one primary winding and two secondary windings. The first secondary winding has delta connected windings and the second secondary winding has wye connected windings. By combining these delta and wye connected windings, phase differences of 30° are obtained between the secondary voltages of these windings. Figure 11: Twelve pulse rectifier bridge By cascading the DC voltages of the six pulse rectifier bridges, the total DC voltage is the sum of the two six pulse rectified DC voltages. Since these DC voltages are shifted over an angle of 30°, the total voltage contains twelve ripples each 20 ππ . This implies indeed a twelve pulse rectifier is obtained. Both six pulse rectifiers require the same current as visualized in Figure 12 but these currents are shifted over an angle of 30° . Figure 12 visualizes both individual currents and the total current which flows at primary side of the transformer. Notice this total current is a much better approximation of a sine implying its THD is much lower. Figure 12: Current consumed by a twelve pulse three phase rectifier bridge Figure 13 visualizes the spectrum of the total current consumed by the twelve pulse rectifier bridge. This shows the current contains a first order harmonic, eleventh and thirteenth order harmonics, β¦ In comparison with Figure 7, the fifth and the seventh order harmonics have disappeared (the THD of the current in Figure 12 and Figure 13 is lower than the THD of the current in Figure 6 and Figure 7). In case of a twelve pulse rectifier bridge, the current contains harmonics of order 12 π ± 1 for all integer values π β₯ 0. Figure 13: Spectrum of the current consumed by a twelve pulse rectifier bridge 2.4: Electric lamps Also electric lamps are important sources of harmonic pollution in the electrical grid. Traditionally incandescent lamps are commonly used and they are linear loads i.e. when fed by a sinusoidal voltage also the current is sinusoidal. Figure 14 visualizes the current consumed by an incandescent lamp. As visualized in Figure 15, also a halogen lamp consumes a current which is almost a sine when fed by the public electricity grid. Figure 14: Current consumed by an incandescent lamp Figure 15: Current consumed by a halogen lamp Fluorescence lamps are frequently used since they produce approximately 60 lumens per watt which is approximately four times the efficiency of a typical incandescent lamp. Figure 16 visualizes the current consumed by a fluorescence lamp in case a classical inductive ballast has been used. Figure 17 visualizes the current consumed by a fluorescence lamp in case an electronic ballast has been used. Figure 16: Current consumed by a fluorescence lamp having an inductive ballast Figure 17: Current consumed by a fluorescence lamp having an electronic ballast A lot of countries promote the use of low energy light bulbs (compact fluorescent lamps) to replace the classical incandescent lamp. In a lot of countries, the use of an incandescent lamp is even banned in order to reduce electrical power consumption (which reduces the dependency on fossil fuels and CO2 exhaust). Figure 18 visualizes the current consumed by a low energy light bulb. It is clear this current has a non sinusoidal shape i.e. the current contains a lot of harmonics causing harmonic pollution in the grid. Figure 19 visualizes the current consumed by a LED light which is also non sinusoidal and causes harmonic pollution of the grid. Figure 18: Current consumed by a low energy light bulb Figure 19: Current consumed by a LED lamp 3: Consequences of harmonic pollution in an electrical grid When considering harmonics in an electrical grid, Figure 3 shows that harmonics in the current imply harmonics in the grid voltage. At the other hand, if the grid voltage contains harmonics this implies harmonics in the current even when the load is linear. This implies current harmonics and voltage harmonics are related with each other and for this reason the present section will sometimes focus on current harmonics and sometimes on voltage harmonics. 3.1: Skin and proximity effect An alternating current has a larger current density near the surface of the conductor than at its core. The electric current tends to flow at the βskinβ of the conductor which is called skin effect (see Figure 20). The skin effect causes the effective resistance of the conductor to increase (according to the law of Pouillet). Due to this increased resistance, there will be more heat dissipation in the conductor. When the frequency increases, the influence of the skin effect will be larger. Figure 20: Current density due to the skin effect The increase of the resistance depends on the radius π of the conductor, the specific resistance Ο of the conductor material, the permeability ΞΌ of the conductor material and the frequency π (with pulsation π = 2ππ). When π 0 is the DC resistance, than π is the resistance with pulsation π given by 2 4 6 π 1 πππ 2 1 πππ 2 11 πππ 2 =1+ ( ) β ( ) + ( ) β β¦ π 0 12 4π 180 4π 26880 4π in case πππ 2 < 2. 4π This expression allows to calculate π βπ 0 in case of a 50 π»π§ and a 250 π»π§ current as a function of the cross section of the conductor as visualized in Figure 21. The skin effect is more important when the cross section of the conductor increases and when the frequency increases. The higher order harmonics βseeβ a larger resistance which increases the heat dissipation in the conductors of the cable. With an increasing order of the harmonic, the more important the skin effect. Figure 21: Skin effect in a conductor The proximity effect describes that the current in a first conductor influences the current densities in a second conductor. Only a part of the cable cross section is used implying the resistance and the joule losses increase. As the frequency increases, the importance of the proximity effect increases too. With an increasing order of the harmonic, the more important the proximity effect. Figure 22 visualizes the proximity effect for two parallel conductors. On the left the situation where the two currents are flowing in the opposite direction and on the right the situation where the two currents are flowing in the same direction. Figure 22: Current density due to the proximity effect 3.2: The neutral conductor Consider a three phase grid containing phase conductors πΏ1 , πΏ2 , πΏ3 and a neutral conductor π. There are three identical single phase non linear loads π1 , π2 and π3 as visualized in Figure 23. Suppose these single phase non linear loads are diode rectifier bridges having a smoothing inductor at DC side implying they consume (in the idealized case) a square wave current (πΌπ1 (π‘), πΌπ2 (π‘), πΌπ3 (π‘)). As already studied, such a square wave current contains a first order harmonic, a third order harmonic, a fifth order harmonic, a seventh order harmonic, β¦ Figure 23: Three phase grid with neutral conductor Figure 24 visualizes these phase currents πΌπ1 (π‘), πΌπ2 (π‘) and πΌπ3 (π‘) in case only the first order harmonic, the third order harmonic, the fifth order harmonic and the seventh order harmonic are taken into consideration. Figure 24 also visualizes the current in the neutral conductor which equals πΌπ (π‘) = πΌπ1 (π‘) + πΌπ2 (π‘) + πΌπ3 (π‘). Notice the first order harmonic, the fifth order harmonic and the seventh order harmonic do not appear in the current of the neutral conductor due to the phase shifts between these harmonics in the phase currents. But the third order harmonics in the three phase conductors have the same phase i.e. they are homopolar. The sum of these third order harmonics in the neutral conductor is three times the third order harmonic in one single phase conductor. The same homopolar behavior applies for all harmonics having an order which is a multiple of three. Figure 24: Currents in a three phase grid with neutral conductor Due to this current in the neutral conductor, additional heat dissipation occurs in the grid cable. There can be a considerable heat dissipation in the neutral conductor since: - the cross-section of the neutral conductor is often smaller than the cross section of the phase conductors, the current in the neutral conductor can be quite large, the 150 π»π§ frequency (or higher) implies a noticeable skin effect. Traditionally, the cross section of the neutral conductor is smaller than the cross section of the phase conductors. Indeed, when considering a symmetric three phase grid with sinusoidal currents the current in the neutral conductor equals zero. In general, the non symmetry and the harmonic pollution in the grid is limited implying the current in the neutral conductor is sufficiently small to make the cross section of the neutral conductor smaller (often the cross section of the neutral conductor is half the cross section of a phase conductor). In case the loads consume a large third order harmonic (or another homopolar harmonic), a large current is obtained in the neutral conductor. To limit the heat dissipation in the neutral conductor, it can be necessary to give the neutral conductor the same cross section as the phase conductors. Sometimes, it can even be necessary to give the neutral conductor a larger cross section than the phase conductors. Of course, instead of realizing a neutral conductor with an extra large cross section the question arises whether it is possible to avoid the harmonic pollution of the grid i.e. to reduce the harmonic contents of the currents. 3.3: Transformers When feeding a transformer with a non sinusoidal voltage, the current harmonics in the primary and secondary windings will cause additional copper losses (skin effect and proximity effect has to be taken into account). Due to the harmonics, also additional iron losses will occur. A distinction can be made between hysteresis losses and eddy current losses. Suppose a harmonic current of the n-th order ππ (π‘) = β2 πΌπ π ππ(πππ‘) is flowing (π = 314 πππ/π ). In case π is increasing, and πΌπ does not depend on π, the hysteresis losses are proportional with π i.e. with the frequency of the harmonic. The eddy current losses will be proportional with π2 i.e. with the square of the frequency of the harmonic. When feeding the transformer with a non sinusoidal voltage π’(π‘) containing harmonics, not only current harmonics will flow. The frequencies of the harmonics will also be present in the magnetic flux of the core (and the magnetic induction). Indeed, due to Faradayβs Law π’(π‘) = β π1 π π(π‘) ππ‘ the flux π(π‘) contains the same frequencies as π’(π‘). A flux change with a frequency π causes magnetostriction of the core material causing noise having a frequency 2π. In case there is only a 50 π»π§ voltage only a 100 π»π§ hum will appear but a fifth order harmonic will cause a 500 π»π§ noise, a seventh order harmonic will cause a 700 π»π§ noiseβ¦ When considering a single phase transformer, all voltage harmonics will be transformed from the primary winding to the secondary winding (whether you like it or not). When considering a three phase transformer the non homopolar harmonics will be transformed from the primary winding to the secondary winding. When considering the homopolar harmonics, it depends on the winding configurations (delta, wye, with or without neutral conductor) whether these homopolar voltage harmonics will be transformed from the primary winding to the secondary winding. 3.4: Induction motors Suppose an induction motor is fed by a polluted grid i.e. by a grid voltage containing harmonics. Due to these harmonics, there are additional copper losses and additional iron losses similar with the additional losses of a transformer. The harmonics also cause additional noise due to the magnetostriction effect of the ferromagnetic material. Moreover, these harmonics have an influence on the motor torque. In order to prove the effect on the mechanical torque of the induction machine, it is important that the harmonics in the applied voltage cause harmonics in the currents. The first order harmonic generates a magnetic rotating field having a speed (expressed in rpm) 60πβπ where π is the grid frequency and π is the number of pole pairs. In general, an n-th order harmonic generates a magnetic rotating field having as speed 60 π π . π These rotating magnetic fields have a direction which depends on π. Suppose a three phase voltage contains a first order harmonic and a seventh order harmonic both generating a magnetic rotating field in the same direction. This implies there are two torque-speed characteristics. The first torque speed characteristic has a synchronous speed ππ = 60 πβπ and the second torque speed characteristic has a synchronous speed 7ππ as visualized in Figure 25. Figure 25: Torque speed characteristics due to first and seventh order harmonics As visualized in Figure 25, the rotor rotates with a speed π which is somewhat smaller than ππ . The total motor torque is the sum of the torques generated due to the first and the second torque speed characteristic. When comparing with ππ , the rotor has a slip π 1 = ππ β π ππ and when comparing with 7ππ , the rotor has a slip π 7 = 7ππ β π . 7 ππ There are two stator rotating magnetic fields, the first one has the synchronous speed ππ and the second one has the synchronous speed 7ππ . This first stator rotating magnetic field generates voltages and currents having a frequency π 1 50 π»π§. This generates a rotor rotating magnetic field having a speed π 1 ππ with respect to the rotor. Since the rotor has a speed (1 β π 1 ) ππ , the rotor rotating field has an absolute speed ππ . The second stator rotating magnetic field generates voltages and currents having a frequency π 7 350 π»π§. This generates a rotor rotating magnetic field having a speed π 7 7ππ with respect to the rotor. Since the rotor has a speed (1 β π 7 )(7ππ ) = (1 β π 1 )ππ , the rotor rotating field has an absolute speed 7ππ . This means there are four rotating magnetic fields: - Rotating field 1: stator field with speed ππ Rotating field 2: stator field with speed 7ππ Rotating field 3: rotor field with speed ππ Rotating field 4: rotor field with speed 7ππ . The interaction of the rotating fields 1 and 3 gives a constant torque. This is the desired useful torque (coming from the first torque speed characteristic). The interaction of the rotating fields 2 and 4 also gives a constant torque. This is the torque coming from the second torque speed characteristic. This additional torque is not a problem but it causes a lot of heat losses since the slip π 7 is close to unity. The interaction between the rotating fields 1 and 4 gives a pulsating torque (which is on average zero). The interaction between the rotating fields 2 and 3 gives a pulsating torque (which is on average zero). Due to these pulsating torques, the torque of the induction motor is no longer constant. 3.5: Miscellaneous Harmonics do not only affect the conductors, transformers and electrical machines. Actually the presence of harmonics has much more effects on all kind of grid components. In the present paragraph, just a few additional examples are given i.e. we consider capacitors, protective devices and measuring devices. Capacitors placed in parallel with the grid voltage in order to improve the power factor have an impedance which is decreasing as the order π increases. Even if the n-th order voltage harmonic is rather small, the n-th order harmonic current in the capacitor can be relatively large. The harmonics in the current increase the RMS-value of this current. Due to this increased RMSvalue, the joule losses in the fuse increase and the melting time decreases. When using an electronic protection device instead of a classical fuse, quite often the peak value πΌππππ of the current is detected using analog electronics. When the current is a sine, its RMS-value equals πΌπ ππ = πΌππππ β2 . Due to harmonics, πΌππππ can be large. This implies using πΌπ ππ = πΌππππ /β2, the estimate of πΌπ ππ can be too large. Using digital electronics, it is possible to sample the current signal and a true RMS measurement can be obtained. An analog multimeter (DβArsonval moving coil meter) physically measures the mean value of the current or voltage. The scale is calibrated for 1.11 π’ππππ or 1.11 πππππ which is the RMS-value when the voltage or current is a sine. In case the voltage/current is not a sine, one still measures 1.11 π’ππππ or 1.11 πππππ but this is not the RMS value. When using power meters to measure active power, reactive power and apparent power in the presence of harmonics, it is always important to pay attention. Different (digital) measurement devices can give different results. Especially when considering reactive and apparent power, different theoretical definitions are used which give different results. In case these measurements are used to bill the electrical power/energy consumption, discussions about the theoretical definitions and appropriate measurement techniques are relevant. 4: Reducing harmonic pollution in an electrical grid 4.1: Rectifiers and frequency converter By using power electronic devices which generate less current harmonics, the harmonic pollution in the grid can be reduced. It can be appropriate to use a twelve pulse rectifier bridge instead of a six pulse rectifier bridge. Notice however a twelve pulse rectifier bridge is much more expensive than a six pulse rectifier bridge implying it will only be used for large powers. For very large powers, it might even be a good practice to use a 24 pulse rectifier bridge as visualized in Figure 26 in order to obtain a further reduction of the THD of the consumed grid current (in Figure 26 there is also an inverter implying a frequency converter is obtained). Figure 26: 24 pulse rectifier bridge A 24 pulse rectifier bridge contains four times a six pulse rectifier bridge. Each six pulse rectifier bridge is fed by a three phase voltage and due to the transformers there are phase differences of 15° between these voltages. By cascading the DC voltages of the six pulse rectifier bridges, the total DC voltage is the sum of the four rectified DC voltages. Since these DC voltages are shifted over an angle of 15°, the total voltage contains 24 ripples each 20 ππ . This implies indeed a 24 pulse rectifier is obtained. Instead of using diode or thyristor rectifier bridges, it is also possible to use an Active Front End (AFE). Figure 27 visualizes a frequency converter containing an AFE, a capacitor C and an IGBT inverter feeding an induction motor. Such an AFE contains six IGBT switches (S1, S2, S3, S4, S5 and S6), freewheeling diodes (D1, D2, D3, D4, D5 and D6) and an inductor in each phase. These inductors πΏ1 , πΏ2 and πΏ3 are crucial in the operation of the AFE. Based on a switching frequency in the ππ»π§ range, the phase currents are compared with the desired sinusoidal current. When a current is too small, due to an appropriate switching of the IGBTs a positive voltage ππΏ appears over the inductor. Based on ππΏ (π‘) = πΏ π π(π‘) , ππ‘ the current increases. When a current is too large, due to an appropriate switching of the IGBTs a negative voltage ππΏ appears over the inductor implying the current decreases. By using a switching frequency in the ππ»π§ range, it is possible to obtain a good approximation of the desired 50 π»π§ current. Figure 27 visualizes the current consumed by an AFE. Notice the sine shape is well approximated but also notice the effect of the switching frequency in the ππ»π§ range. Figure 27: AFE current Figure 28 visualizes the spectrum of the AFE current of Figure 27. Notice the first order harmonic (50 π»π§) and small harmonics having frequencies below 1 ππ»π§. Notice the impact of the switching frequency in the spectrum around 3 ππ»π§ and especially 6 ππ»π§. Figure 28: Frequency spectrum of the AFE current Figure 29 shows a close-up of the frequency spectrum visualized in Figure 28. Notice the first order harmonic (50 π»π§) and very small harmonics giving a small THD when only harmonics up to the 40-th order are taken into consideration (IEC-61000-3-12). Actually only the seventh order harmonic has some importance. 2 ββ40 π=2 πΌπ ππ»π·πΌ = πΌ1 . Figure 29: Frequency spectrum of the AFE current 4.2: The use of passive filters