POWER QUALITY. 1. Introduction. ELEC9713 Industrial and Commercial Power Systems

Transcription

1 ELEC9713 Industrial and Commercial Power Systems POWER QUALITY 1. Introduction The quality of the electrical supply is an important consideration for both power utilities and consumers. Electrical engineers have always been concerned about power quality. They see it as anything that affects voltage, current, and frequency of power being supplied to end-users. A power quality (PQ) problem is defined as any problem that causes voltage, current, or frequency deviations in the supply and may result in failure or mal-operation of end-user equipment. It should be noted that in the majority of cases, power quality actually refers to the quality of the voltage. This is because the supply distribution system can only control the quality of the voltage but it has no control over the currents drawn by the loads. Therefore, PQ standards are mostly aimed at specifying the requirements on the supply voltage. Although such standards are often used as benchmarks, there is no agreed definition on how to accurately quantify power quality. The ultimate measure is determined by the performance and productivity of end-user equipment. ELEC9713: Power Quality page 1/41

2 Historically, PQ and reliability were synonymous. In early days, the main concern was about keeping the lights on. Various measures were applied to protect power system: use of surge arresters and circuit breakers, redundancy, computers checking power flow and transient stability, etc. Since the late 1980s, the emphasis has shifted from reliability concern at generation, transmission and distribution level to concern about PQ at the customer or end-user level. There are four major factors that cause an increased need to solve and prevent power quality problems: 1) Increased use of power quality-sensitive equipment such as computers, microprocessors, consumer electronic and telecommunication appliances, etc. Electronic devices do not require much energy or significant over-voltage to cause insulation breakdown. These values are decreasing with increasing reduction in the scale of micro-circuit elements. Studies were performed to determine effect of variations in voltage levels and durations. The Information Technology Industry Council (formerly Computer and Business Equipment Manufacturers Association) developed the ITIC (CBEMA) tolerance envelope. It describes an ac input voltage envelope which can be tolerated by most information technology (IT) equipment without loss of function. The curve is considered as a design objective for computer designers. The acceptable or function region is bracketed by an upper over-voltage ELEC9713: Power Quality page /41

3 curve (prohibited region) and a lower under-voltage curve (no damage region). It can be seen that there is a very strong dependence on the duration, i.e. more tolerant if duration is short. Note that the tolerance envelope shown only applies to IT equipment. Other equipment may and generally does have an entirely different sensitivity characteristic. ) Increased use of equipment that generates power quality problems: harmonic distortion is produced from nonlinear loads such as adjustable-speed drives, electronic ballasts for fluorescent lamps, arc welders. Adjustable- ELEC9713: Power Quality page 3/41

4 speed drives have become one of the most popular technologies for saving energy in industry. They use power electronic switching to control the motor speed to match the load requirement. 3) Increased inter-connectedness of power system: problem can propagate and is difficult to isolate. Harmonics and flicker are examples of power quality problems that can be transferred from a utility to another through interconnection. 4) Deregulation of the power industry: change from the monolithic structure of a single full-service, vertically integrated supply authority to competitive, decentralized supply industry. Complications arise because different companies will supply generation, transmission and distribution services. Deregulation has been in effect in many parts of the world including Australia and the USA. ELEC9713: Power Quality page 4/41

5 Summary of power quality problems [Ref: R.C. Dugan]. Reliability measures for distribution systems One reliability measure is the supply availability, defined as the time that supply is available to customers. Expressed as a percentage, with 100% indicating no supply interruption. ELEC9713: Power Quality page 5/41

6 In 000/001, average number of minutes EnergyAustralia's 1.4 million electricity customers were without supply was 101 minutes. This corresponds to a supply availability of 99.98%, i.e. supply availability is less than 4 nine s. Generally, power companies aim to deliver 4-nine or higher availability. Note that: 4 nine s total outage of 5 minutes in one year 6 nine s total outage of 31 seconds in one year 8 nine s total outage of less than 1 second in a year Another measurement parameter is the average duration (in minutes) lost per customer per year, generally referred to as the Reliability Index or SAIDI (System Average Interruption Duration Index). This is one of the most common methods of assessing and quantifying the reliability of supply, and is used as a means of benchmarking distribution companies. However, this is an average figure and it excludes interruptions of 1 minute or less, or interruption resulting from transmission grid failures or major storms! Other commonly accepted measures for network reliability are: SAIFI (System Average Interruption Frequency Index): average number of interruptions that a customer experiences each year. CAIDI (Customer Average Interruption Duration Index): average duration (in minutes) that a customer is without power when affected by an interruption to supply. EnergyAustralia s reliability objective is to limit the number of sustained normalised interruptions in any financial year to 9 interruptions for CBD/urban customers and 15 interruptions for rural customers. ELEC9713: Power Quality page 6/41

7 . General classes of power quality problems The Institute of Electrical and Electronics Engineers Standards Coordinating Committee (IEEE SCC) has led the main effort in USA to define power quality standards. In Europe, the work is done by the International Electrotechnical Commission (IEC) and the Congress Internationale des Grand Reseaux Electrique a Haut Tension (CIGRÉ). Principal phenomena causing electromagnetic disturbances as classified by the IEC ELEC9713: Power Quality page 7/41

8 Categories and characteristics of power system electromagnetic phenomena [Ref: R.C. Dugan]. ELEC9713: Power Quality page 8/41

9 EnergyAustralia adopts a somewhat simplified classification system for power quality problems. It is illustrated in the figure below:.1 Transients [Ref. Electricity Network Operation Standards] Denote events that are undesirable and momentary in nature. A transient, broadly defined, is that part of the change in a variable that disappears during transition from one steady state operating condition to another. Broadly speaking, transients can be classified into categories: impulsive and oscillatory. These terms reflect the waveshape of a current or voltage transient. (1) Impulsive transients An impulsive transient is a sudden, non-power frequency change in the steady-state condition of voltage, current, or ELEC9713: Power Quality page 9/41

10 both, that is unidirectional in polarity (either +ve or -ve). It is characterized by rise and decay times, which can also be revealed by spectral content. The most common cause is lightning. Standardised lightning waveforms are characterized as below: Voltage impulse wave of 1./50μs: crest is reached in 1.μs. The wave decays to half the crest in 50μs and completely dissipates in μs. Current impulse wave of 8/0μs: crest is reached in 8μs and wave decays to half the crest magnitude in 0μs. Due to high frequencies involved, the shape of impulsive transients can be changed quickly by circuit components and may have significantly different characteristics when viewed from different parts of the power system. They are generally not conducted far from the source where they enter the power system, although they may in some cases, be conducted for quite some distance along utility lines. () Oscillatory transients An oscillatory transient is a sudden, non-power frequency change in the steady-state condition of voltage, current, or both, that includes both positive and negative values. Based on the spectral content, there are 3 classes: High-frequency oscillatory transient: primary frequency component greater than 500kHz and a typical duration ELEC9713: Power Quality page 10/41

11 measured in micro-seconds. It is often the result of a local system response to an impulsive transient. Medium-frequency oscillatory transient: primary frequency component between 5kHz and 500kHz with duration measured in tens of micro-seconds. Examples include back-to-back capacitor energisation, cable switching. It can also be the result of system response to an impulsive transient. Low-frequency oscillatory transient: primary frequency component less than 5kHz, and duration measured from 0.3 to 50ms. This category is frequently encountered on utility sub-transmission and distribution systems and is caused by many types of events. The most frequent is capacitor bank energisation, the resulting transient has typical primary frequency between 300 and 900Hz with peak magnitude between 1.3 to 1.5 pu (can approach pu) and duration between 0.5 and 3 cycles. Oscillatory transients with principal frequencies less than 300Hz can also be found on distribution systems, generally associated with transformer energisation, ferro-resonance and series capacitors. Sometimes, transients (and other disturbances) are also categorized according to their mode: common mode or normal mode, depending on whether they appear between line or neutral and ground, or between line and neutral. ELEC9713: Power Quality page 11/41

12 . Long-duration voltage variations Long duration voltage variations comprise deviations for longer than one minute. There are three types: overvoltage, undervoltage and sustained interruption. (1) Overvoltage An increase in the rms ac voltage greater than 110% at power frequency for duration longer than 1 minute. Usually due to load switching (eg. switching off a large load, energizing a capacitor bank). Overvoltages result because system is either too weak or the desired voltage regulation or controls are inadequate. Incorrect tap settings on transformers can also result in system overvoltages. () Undervoltage A decrease in rms ac voltage to less than 90% at power frequency for duration longer than 1 minute. Too much load on system (during very hot or cold weather), for example, or loss of a major transmission line serving a region can cause undervoltages. End-users recognize when lights dim and motors slow down. Sometimes utilities deliberately cause undervoltages to reduce load during heavy load conditions. The term brownout is often used to describe sustained periods of ELEC9713: Power Quality page 1/41

13 undervoltage initiated as a specific utility dispatch strategy to reduce power demand. (3) Sustained interruptions When voltage is 0 for duration longer than 1 minute. Usually a permanent event and thus require human intervention to repair the system for restoration. No relation to usage of the term outage. Outage does not refer to a specific phenomenon, but rather to the state of a component in a system that has failed to function as expected..3 Short-duration voltage variations There are 3 types of short-duration voltage variations: interruption, sag and swell. Each type can be designated as instantaneous ( sec), momentary (0.5 3 sec) or temporary (3-60 sec). (1) Interruption Occur when voltage decreases to less than 0.1 pu for a period of time not exceeding 1 minute. Can be result of power system fault, equipment failure, control malfunction. Duration of interruption due to system fault is determined by operating time of protective devices. Instantaneous reclosing limits the interruption to < 0.5s ELEC9713: Power Quality page 13/41

14 whereas delayed reclosing may cause a momentary or temporary interruption. () Voltage sags Decrease to between 0.1 and 0.9 pu in rms voltage at the power frequency for durations from 0.5 cycles to 1 minute. Also called voltage dips. Generally, sags are less than 1s. Associated with faults which inevitably occur on the distribution/transmission network and high current faults flow for a fraction of a second until cleared by protective devices. Occasionally, faults are caused by failure of network equipment. More often, equipment is damaged by branches or trees falling on power lines, vehicles colliding with poles, bird and animal damage, or people digging or boring into underground cables. When such an incident occurs, most customers will only experience a dip in voltage for a fraction of a second until the fault is cleared by the protective devices. Mostly, this will not be noticed and should not affect equipment. However, customers connected directly to the affected line or circuit will experience an interruption. In some cases, supply to these customers may be restored by automatic switching or reclosing within a few seconds. Precise voltage levels that are reached during faults depend on many variables nature of fault, where it occurs, location of customer relative to the fault, etc. In most cases, duration of disturbance is less than 1 second. ELEC9713: Power Quality page 14/41

15 Once fault has been isolated, voltage on the rest of network will return to normal, but for customers connected to the faulted section, supply will be interrupted until the fault is corrected, or until alternative supply (where available) is switched in. Voltage sags can also be caused by energization of heavy loads or starting of large motors. An induction motor will draw 6 to 10 times its full-load current during starting. Voltage Dips Average EnergyAustralia s Objective Dips down to % of nominal Volts <30% 30-50% 50-70% 70-80% 80-90% Number of Dips (per year) The figure below shows the overall voltage sag performance of typical EnergyAustralia distribution networks. For example, voltage sag to 0% may persist between s for urban supply but is less ( s) for CBD supply. The ITIC dotted line is the voltage tolerance envelope by the Information Technology Industry Council of America. For example, equipment that met the ITIC requirement should be immune to voltage drop to 70% for at least 0.5s. ELEC9713: Power Quality page 15/41

16 (3) Voltage swells Increase to between 1.1 and 1.8 pu in rms voltage at the power frequency for duration from 0.5 cycles to 1 minute. Also called momentary overvoltages. As with sags, swells are usually associated with system fault conditions, but they occur less frequently. An example is the temporary voltage rise on unfaulted phases during a single-line to ground (SLG) fault such as a tree striking a live conductor. ELEC9713: Power Quality page 16/41

17 .4 Voltage imbalance (or voltage unbalance) Can result from either unbalanced network impedances or unequal distribution of single-phase loads. Balanced impedances under normal operating conditions are achievable by appropriate design and construction practices. Consequently, means of controlling unbalance is the balancing of three phase loads and the distribution of single phase loads. According to NEMA (National Electrical Manufacturers Association of USA) Standard, voltage unbalance is defined as: ΔVmax Percent unbalance = 100% V where: Δ V max = maximum voltage deviation from V ave V = average value of the three (phase) voltages ave Alternatively, voltage unbalance can also be defined using symmetrical components: where: ave V Percent unbalance = 100% V V = negative sequence voltage V 1 = positive sequence voltage V = zero sequence voltage 0 Note that the above definition is actually the negative sequence voltage unbalance factor. The zero sequence 1 ELEC9713: Power Quality page 17/41

18 voltage unbalance factor ( = V0 V 1 ) is not often used. This is because zero sequence currents cannot flow in a three-wire system (no neutral) as found in many three-phase induction motors. On the other hand, the negative sequence unbalance factor is important because it indicates the level of voltage that causes an induction motor to turn in a direction opposite to that produced by the positive sequence voltage. The negative sequence unbalance factor can also be computed more conveniently by using the three line-line voltage readings: Percent unbalance 1 3 6β = 100% β where: β = V + V + V ab bc ca ( Vab + Vbc + Vca ) This is known as the IEC definition. In both definitions (NEMA and IEC), line-neutral voltages are not used because the zero sequence components can give incorrect results. There is general acceptance internationally that the limiting criteria for unbalance is based on the amount that motors can tolerate. The acceptable level is %. On a 40/415 volt low-voltage distribution network, % unbalance would equate to the highest and lowest phase voltages differing by approximately 6.7%, e.g. 3, 40 and 48 volts. ELEC9713: Power Quality page 18/41

19 EnergyAustralia s objective is not to exceed: 6% difference between highest and lowest phase or line steady state voltage (5 minute average) for LV network. 3% difference between highest and lowest steady state line voltage (5 minute average) for HV network. Large unbalanced voltages can cause overheating and mal- of certain types of 3φ motors. Can also cause operation network problems such as mal-operation of protection relays and voltage regulation equipment, and the generation of noncharacteristic harmonics from power electronic loads..5 Waveform distortion Defined as a steady-state deviation from an ideal sine wave of power frequency. There are five primary types of waveform distortion: dc offset, harmonics, inter-harmonics, notching and noise. (1) DC offset Presence of a DC voltage or current in an AC system. Can occur as results of a geomagnetic disturbance or caused by equipment that has different operating characteristics in each half of the voltage cycle (e.g. halfwave rectification such as that found in incandescent light bulb extenders). ELEC9713: Power Quality page 19/41

20 Even a small amount of DC offset current is undesirable as it can result in corrosion of the network and customer s earthing system. Equipment which is known to cause DC current is required to use an isolating transformer. It is not possible to eliminate DC current from the distribution network as these will occur in small quantities from some types of lighting equipment, domestic appliances, including rectifiers with mismatched components. The DC current contribution from these sources is limited by AS3100 Approval and Test Specification Definitions and general requirements for electrical materials and equipment. According to NSW Electrolysis Committee, DC voltage component of the neutral conductor with respect to earth could reach ± 10 volts. () Harmonics Sinusoidal voltages at frequencies that are integral multiples of 50Hz (e.g. 150Hz is third harmonic). Distorted periodic waveforms can be decomposed into a sum of the fundamental frequency (50Hz) sine wave and high frequency harmonics. Harmonics are caused by non-linear devices and loads on the system. Limits need to be placed on the generation of harmonic distortion so that appliances and equipment of other customers, and network components themselves, are not excessively interfered with or damaged. The amount of ELEC9713: Power Quality page 0/41

21 distortion allowed is specified in AS/NZS (Electromagnetic compatibility - Limits - Assessment of emission limits for distorting loads in MV and HV power systems). In general, utilities would seek to restrict their contribution at the point of common coupling to 30-50% of these limits to allow for any existing background level and the further contribution of additional customers. (3) Inter-harmonics Caused by waveforms that have frequency components that are not integral multiples of the fundamental frequency (50Hz). They can appear as discrete frequencies or as a wideband spectrum. Main causes are static frequency converters, cycloconverters, induction motors, arcing devices. Can cause problems such as light flicker, audible noise in TV sets, radios and audio equipment, and vibration in rotating induction machines. The allowable limits are specified in IEC Std N ote: Another source of harmonics and inter-harmonics is the mains signaling voltages, injected onto the network by the utility, primarily for the purpose of switching customers time controlled tariff equipment (e.g. Off Peak tariffs). These are at nominal frequencies of 49, 750 and 1050Hz. In addition, Hz frequencies may possibly be used in the future. ELEC9713: Power Quality page 1/41

22 The optimum signaling voltage is a compromise to ensure effective operation of load control relays without causing significant interference. The signaling system is designed to provide signaling voltages less than 10 volts. (4) Notching It is a periodic voltage disturbance caused by the switching operation of power electronic devices when the current is commutated from one phase to another (two phases of supply are effectively short-circuited for a short time). The extent of distortion is determined by the depth and width of the notch which in turn are dependent on firing delay angle, commutation overlap period and impedances of the supply. Since the frequency components due to notching can be quite high, measuring instruments normally used for harmonic analysis may not be able to detect this effect. Present Australian standards limit the notch depth to 0% of the peak supply voltage at the point of common coupling. (5) Noise Defined as any unwanted signals that cannot be classified as harmonic distortion or transients and have broadband spectral content lower than 00kHz. ELEC9713: Power Quality page /41

23 Caused by power electronic devices, control circuits, arcing equipment, switching power supplies. Improper grounding can exacerbate the problem. In Australia, the acceptable limits are set by the Spectrum Management Agency. ( 6) Voltage/current differences between neutral and earth It is important to maintain the voltage difference between neutral and earth within specified limit (typically 10V) to minimise the voltage unbalance, and to prevent corrosion to the earthing system (DC current flow).. 6 Voltage fluctuation Rapid changes in voltage within the allowable limits of the nominal voltage, e.g. 0.9 to 1.1 pu for ANSI C For low-voltage networks, utilities such as EnergyAustralia aim to maintain the steady-state phase-neutral voltage (tenminute average) within ± 10% of the nominal voltage of 40V (i.e. between 64V and 16V), at the consumer s terminals [Electricity Network Operation Standards, July 004]. If possible, it will aim for the 6-53V range. Voltage drops occur also within customer installations, between customer s terminals and the customer s equipment. AS/NZS 3000:000 Electrical Installations (known as the Australian New Zealand Wiring Rules) sets a limit on the ELEC9713: Power Quality page 3/41

24 voltage drop within customer installation of 5%, which should be added to the voltage range of the network. AS specifies the new standard nominal voltage to be 30 volts (reduced from 40 volts). The tolerance range will be +10% to -6% which means that the actual supply voltage range will not necessarily be lowered. Devices like electric arc furnaces and welders that have continuous, rapid changes in load current cause voltage fluctuations. This can cause lamps to blink rapidly, often referred to as flicker and is visible to human eyes at flickering frequencies of 6-8Hz..7 Power frequency variations Defined as deviation of power system fundamental frequency from its nominal value (50Hz or 60Hz). Power system frequency is directly related to generator rotating speed. There are slight variations in frequency as dynamic balance between load and generation changes. The size of frequency shift and its duration depends on load characteristics and response of generation control system. Frequency variation that exceeds acceptable limits can be caused by faults on the bulk power transmission system, a large block of load being disconnected, or a large source of generation going off-line. ELEC9713: Power Quality page 4/41

25 In Australia, frequency standards are set by NEMMCO (National Electricity Market Management Co Pty Ltd). NSW Frequency Control Standard As frequency variation will affect accuracy of some clocks, there has been a policy in NSW of maintaining electrical time within 3 seconds of Australian Eastern Standard or Daylight Saving Time. To conform to that figure, discrepancies are periodically corrected by means of control equipment which offsets the frequency excursion within the normal band. ELEC9713: Power Quality page 5/41

26 Appendix A: Symmetrical components A.1 Symmetrical components sequences Positive sequence X, X, X ( 1a 1b 1c) a-b-c Negative sequence X, X, X ( a b c) a-c-b Zero sequence X, X, X ( 0a 0b 0c) X 1c X 1a X a X c X 0a X 0b X 0c X 1b X b X = X = X 1a 1b 1c 0 a θ b = 0 b θ c = θ θ X1a = X1a θ1a X1b = a X1a X = ax 1c 1a X = X = X a b c 0 a θ b = 0 b θ c = θ 10 θ 10 Xa = Xa θa X = ax X b a c = a Xa An unbalanced 3φ sequence ( X a, Xb, X c) X = X = X 0a 0b 0c 0 0a θ0b = 0 0 0b θ0c = 0 θ θ = θ X 0b = 1X 0a X = 1X X0a X0a 0a 0c 0a X c X b X a Xa = Xa θa Xb = Xb θb Xc = Xc θc X X X a b c 0 θ θ θ θ θ θ 10 a b b c c a ELEC9713: Power Quality page 6/41

27 A. Use of the a operator Define: Thus: π j 0 3 a = 1 10 = 1e = j a = 1 40 = 0.5 j a = = a = = 1 60 = 0.5 j a * = 1 10 = a a -a -a 3 1 a 3 a -a Also: 0 1+ a = 1 60 = a 0 1 a = 3 30 = 1.5 j a = 1 60 = a 0 1 a = 3 30 = j0.866 a+ a = 1 a a = j 3 1+ a+ a = 0 In a balanced three-phase supply system, all the three lineneutral voltages (and currents) are equal in magnitude and phase displaced by 10 degrees. For a balanced 3φ system with positive phase sequence a-b-c: ELEC9713: Power Quality page 7/41

28 V V bn cn Vab = ( 1 a ) = a Van bc = ( ) V a a V = av V = ( a 1) V an ca V an an an A.3 Analysis of unbalanced 3-phase sequence It can be proven that, given any unbalanced 3-φ sequence X, X, X, can find +ve, -ve and zero sequences ( a b c) ( X1a, X1b, X 1c), ( X a, Xb, X c), and ( X 0a, X0b, X 0c) which sum to it: i.e. that is X = X + X + X X = X0 + X1 + X X = X + X + X a 0a 1a a b b b b c 0c 1c c X = X + X + X Or in matrix form: a 0a 1a a b = 0a + 1a + a c = 0a + 1a + a X X a X ax X X ax a X X X a 0a Xb = 1 a a X1a X c X a 1 a a It then follows that the symmetrical components for X, X, X are given by: ( a b c) ELEC9713: Power Quality page 8/41

29 X X 0a a 1 X1a = 1 a a Xb 3 X a 1 X c a a X1 = X + ax + a X 3 1 X = X + a X + ax 3 i.e. X 0a = ( Xa + Xb + Xc) Example: ( ) ( ) a a b c a a b c X1c X 1a X a X c + + X 0a X 0c X 0b X 1b X b X 0a X a X c X a = X 1a X b ELEC9713: Power Quality page 9/41

30 Appendix B: Harmonic distortion In general, the sine wave voltage generated from power stations has very little distortion and remains so over the transmission networks. Waveform distortion increases as we move closer to the loads. Most distortion is periodic, i.e. the same in every 50Hz cycle. Th is results in harmonic distortion: presence of frequency components which are integral multiples of the fundamental (50Hz) frequency. Harmonic distortion is caused by the presence of non-linear devices. A non-linear device is one in which the current is not proportional to the applied voltage. In power system networks, the primary sources are the transformers due to their non-linear magnetizing impedance. The end-user sector such as industrial users with adjustable speed drives, arc furnaces, induction furnaces, etc. suffers more from harmonic problems than the utility sector. Fig.B1: Current distortion caused by non-linear resistance. ELEC9713: Power Quality page 30/41

31 Harmonic problems can be due to either voltage or current distortion or both. Nonlinear loads appear to be sources of harmonic current in shunt with and injecting harmonic currents into the power system. We can treat these harmonicproducing loads simply as current sources. Fig.B: Harmonic current flowing through system impedance results in harmonic voltage at the load. In Fig.B, voltage distortion is the result of distorted currents passing through the linear, series impedance of the power system. Although the source bus contains only 50Hz voltage, harmonic currents flowing in the system impedance cause a voltage drop for each harmonic. This results in voltage harmonics appearing at the load bus. While load current harmonics ultimately cause voltage distortion, the load has no control over voltage distortion. The same load in different locations on the system will result in different voltage distortion values. Thus, harmonic control (IEEE Standard ) can be divided into: Limiting the harmonic current injected into the system which takes place at end-use application. Limiting the voltage distortion by the utility which has control over the system impedance. ELEC9713: Power Quality page 31/41

32 By popular convention, the term harmonics refers to harmonic current when dealing with load apparatus and harmonic voltage when dealing with utility system. B.1 Fourier analysis Any periodic waveform can be expressed as a Fourier series. This is a sum of pure sine waves in which each sinusoid is a harmonic of the fundamental frequency of the distorted wave. The advantage of using Fourier series is that it is easier to find the system response to a sinusoidal input signal. Conventional steady-state analysis technique can be used and the system is analysed separately for each harmonic. The outputs at each frequency are then combined to form the total system response. Fig.B: Fourier series representation of a distorted periodic wave. ELEC9713: Power Quality page 3/41

33 Note that the distorted waveforms of most common harmonic-producing devices have identical shapes in the two positive and negative half-cycles. The Fourier series, in this case, contains only the odd harmonics. Also for power system analysis, effects of higher-order harmonics (usually above the 5 th ) are negligible and thus can be ignored. An arbitrary waveform with a period T can be expressed as: () = + ( cos ω + sin ω ) f t a a n t b n t 0 n n n= 1,, K where: a 0 1 T = T 0 T () f t dt an = f () t cos( nωt) dt T 0 T bn = f t n t T 0 () sin( ω ) dt Also, f(t) can be further expressed as: () = + cos( ω + θ ) f t a c n t 0 n= 1,, K n n where: n b cn = a + n b θ n = a tan n a n In general, the AC supply voltage and current have no DC components and thus are expressed as: ELEC9713: Power Quality page 33/41

34 ( ω θ ) ( ω θ ) ( ω β ) ( ω β ) v= V1cos t+ 1 + Vcos t+ + L i = I1cos t+ 1 + Icos t+ + L The phase angle between the n-th harmonic components of the voltage and current is: φn = θn βn In particular, the phase angle of the fundamental is called the displacement angle: φ = θ β The effective (rms) values of voltage and currents are: 1 T 0 1 T () 1 V = v t dt V V T = + + L 0 () 1 I = i t dt I I T = + + L B. Total harmonic distortion There are several measures commonly used for indicating the harmonic content of a waveform with a single parameter. The most common is the total harmonic distortion (THD). It is a measure of the effective value of harmonic components of a distorted waveform. For voltage waveforms, the THD is defined as: ELEC9713: Power Quality page 34/41

35 THD V Vn = n= V 1 100% Note that this is the definition adopted by the IEEE. For the IEC, THD is calculated as a percentage of the total rms (not the rms of just the fundamental). The THD I for current is expressed in a similar manner. THD is related to the rms value of the total waveform as follows: THDV = n = 1 1+ V V V n= THD is a useful quantity in some applications, e.g. it provides measure of how much extra heat is dissipated when distorted voltage is applied across a resistive load. However, it is not a good indicator of voltage stress on a capacitor because that is related to peak value of voltage waveform. Harmonic voltages are referenced to fundamental value of the waveform at the time of the sample. Because voltage varies only a few percent, THD is a meaningful number. V This is not the case for current. A small current may have a high THD I but not be a problem. However it can be misinterpreted as a serious power quality problem. This can be avoided by referring THD I to the fundamental of the peak demand current rather than fundamental of present sample. This is called total demand distortion (TDD). ELEC9713: Power Quality page 35/41

36 The crest factor is another quantity that is sometimes used to indicate the extent of harmonic distortion: CF peak of waveform its rms value For a pure sinusoidal voltage (or current): CF = = Harmonic distortion would cause CF to deviate from this value. For example, a square wave has a CF = 1. B.3 Effect of harmonics on power factor The formal definition of power factor is: where: pf P S 1 T Average power: P= v()() t i t dt T 0 = VI cosφ + VI cosφ +L Apparent power: S = VI If there are no harmonic distortions (only fundamental): = cos( ω + θ ) ; i = I cos( ωt+ β ) v V t so: P= V1I1cosφ1 ; S = V1I1 pf = cosφ1 Thus for single-frequency, difference between real power and apparent power is due to the displacement (angleφ 1 ) between the voltage and current waveforms. ELEC9713: Power Quality page 36/41

37 If there are harmonic distortions: P P P pf = = = S VI THD V THD I V1 1+ I P 1 = VI 1 1 THDV THDI The first component is called the displacement power factor: pf = disp P VI 1 1 The second component resulting from harmonic distortions is called the distortion power factor: 1 pfdist = THDV THDI The t otal power factor (also called true power factor) is: pftrue = pfdisp pfdist Observe that: pftrue pfdisp Power quality monitoring instruments commonly provide measurements of these different types of power factor. Devices such as switch-mode power supplies and pulse-width modulated (PWM) adjustable-speed drives have a near- ELEC9713: Power Quality page 37/41

38 unity pf disp, but pf true may be pf true gives an indication of how large the power system must be built to supply a given load. Relying on only pf would give a false disp sense of security that the supply is adequate. Note that traditionally, cosφ is often referred to as the power factor. This actually is pf disp using fundamental frequency components only, i.e. that φ refers to the angle between the fundamental voltage and fundamental current. Also note that in reality, the voltage distortion is normally much less than the current distortion. If v(t) is entirely fundamental frequency: P= VI 1 1cosφ1 which indicates the average power is a function only of the fundamental frequency quantities. Because voltage distortion is generally very low (< 5%), this is a good approximation regardless of how distorted the current is. B.4 Harmonic resonance LC circuits can develop resonance. A common example is the combination of power factor correction (PFC) capacitors with the inductance of the power system. If the resonant frequency occurs near a harmonic frequency, even a small harmonic current can result in very high voltage and current. Consider the circuit below. For simplicity, ignore all other customer loads at the point of common coupling (pcc). ELEC9713: Power Quality page 38/41

39 jx s pcc pfc capacitor Ideal sinusoidal voltage source -jx c non-linear load Equivalent circuits: jx s jnx s V 1 -jx c I 1 -jx c n I n fundamental n-th harmonic The impedance seen by the n-th harmonic current source is: jx c Zn = ( jnxs) n Thus: nx cx s Z n = nx X Parallel resonance develops when s c ( nxs Xc) = 0 which makes Z n become infinite. If L is the system inductance and C is the pfc capacitance then the resonant frequency is: 1 f r = π LC It can also easily be shown that: ELEC9713: Power Quality page 39/41

40 where: n X X c = = s FL Q c FL = fault level at pcc Q c = VAr rating of capacitor. Some important observations: At resonance, even a small harmonic current can result in very high voltage at the pcc. This high voltage means current in X s and X c can be much larger than source harmonic current I n. Resonant frequency is lower for larger capacitor bank. Note that in practice, there always exists some small resistance in the system and thus: jx c Zn = ( Rs + jnxs) n Hence: nx cx s jrsx c nx cx s Zn = nr + j n X X nr + j n X X ( ) ( ) s s c s s c X cx s So at resonance: Zn = Rs Thus, the harmonic voltage developed is still substantial but somewhat reduced. Parallel resonance can be avoided by detuning the capacitor bank with a series inductance (L d ). Furthermore, this combination also acts as a filter preventing those ELEC9713: Power Quality page 40/41

41 harmonic currents near its series resonance frequency from spreading throughout the system. jnx s pcc jnx d -jx c n I n Z n = jx c ( Rs + jnxs) jnxd + n jx c Rs + jnxs + jnxd + n ( ) This results in series resonance at harmonic order: X c ns = X d and parallel resonance at harmonic order: n p = X s X c + X d It can be seen that: np < ns. The technique involves selecting n p less than the lowest order harmonic present (to prevent parallel resonance). The capacitor bank acts as a lowimpedance shunt filter at series resonance, absorbing those harmonics close to order n s. ELEC9713: Power Quality page 41/41