Harmonics Analysis and Calculation in Power System Networks

- The harmonic distortion in the power system is increasing with wide use of nonlinear loads such as wave rectifiers, static VAR compensator, and solid-state controlled devices.Thus, it is important to analyze and evaluate the various harmonic problems in the power system prior to their occurrence. This paper presents a technique to analyze propagation of harmonic current and voltage in power system networks, and determine the location and magnitude of the maximum harmonic current and voltage in the network .Frequency dependent models of a power system elements and loads have been developed for the appropriate range of frequency. The harmonic source representation is described. The location of harmonic source, and the system loads are all found to be factors in determining the magnitude of harmonic currents and voltages flow in the system


Introduction
The problems of power system harmonics are not a new.It is as old as power network itself, at past time the major concerns were the effects of harmonics on a synchronous and induction motors, communication equipment interface and shunt capacitor failure .Number of precaution was taken by the manufactures such as the designing of the equipment more tolerable to the effect of the harmonic [1] .Utilities were able to eliminate or reduce the harmonics flowing in the power system by means of harmonic filters.Recently, the problem of harmonics is becoming again a current concern with the widespread use of solid state power electronics devices.The most common devices being introduced into industrial plants include variable speed ac motor drive, variable speed dc motor drive.There are a number of advantages when using these new devices.The advantages are normally a reduced cost of operation in the form of lower energy costs, better efficiency, less maintenance, and dependable operation .Each of these devices inject a nonsinusoidal current waveform into the plant distribution system and transmission network depending upon the size of the device and the system parameters.Other sources of harmonics include static var system (SVS) which have largely replaced conventional synchronous condenser [2].The linear nature of the loads has shown a trend to change to highly nonlinear characteristics with a significant increase in primarily current harmonics.It should be recommended practice to analyze an electrical system before addition of large power electronic devices to determine the effect on the other parts of the power system After the wave shape of both the current and voltage have been determined, the harmonic content can be easily analyzed.The analysis to determine the magnitude and the order of the harmonics included in the waveform is based on Fourier transform theory.

Concept of Distortion Waveforms
To understand the distortion phenomena, it is necessary to analyze the distorted waveform by a process called harmonic analysis .It allows us to express the distorted waveform as a sum of dc component, fundamental sine wave of the distorted waveform and series of pure sine waves .These sine waves have different magnitudes and their frequencies are integer multiple of the fundamental distorted waveform.This expression is called Fourier series representation.
A distorted waveform can be analyzed using Fourier series representation applies following equations [3] Where when the positive and negative half cycles of a waveform have identical shapes with respect to the vertical axis and the average value of its period equal to zero(dc component equal zero),the Fourier series contains only odd harmonic component .These waveforms are called symmetrical waveform.The harmonic component can be plotted using time domain.Figure (1) shows the summation of the fundamental (50Hz) and third harmonic (150Hz) [2,3].

Development of Frequency Dependent Model for Network Elements
To assemble the elements of a system into a bus impedance matrix at each harmonic, frequency dependent model for each element must be developed.

1-PassiveElements
All passive elements are considered to behave linearly with frequency, such as resistor, inductor and capacitor .The following characteristics are observed by these elements [2]: R=constant (3) Where X L and X C are the fundamental frequency inductive and capacitive reactances respectively.

2-Generators
For the purpose of determining the network harmonic admittance, the generators can be modeled as series combination of resistance and inductive reactance [3]: (5) Where R is derived from the machines power losses and X d '' is the generator ' s sub transient reactance.negative sequence fundamental frequency stator current produce flux rotate at twice synchronous speed .The resultant flux is forced into paths flow permeability which do not link any rotor circuit.These paths are characterized by the sub transient reactance [4].When harmonic currents flow from network into the stator windings of a generator, they create a flux rotating at a speed greater than the speed of the rotor .This is very similar to the action of negative sequence currents in a synchronous machines [5], therefore use sub transient reactance for generator.

3-Transformers
These are considered to be liner elements and modeled as series impedance of resistance and inductive reactance with resistance R adjusted for skin effects.The series harmonic impedance is [2, 3]:

+ =
Where R derived from the transformer power losses.The transformer's saturation effect could be represented by means of a harmonic current injection source.

4-Transmission Lines
A transmission line consist of distributed inductance and capacitance, which affect the magnetic and electrostatic condition of the line ,and resistance and conductance which affect the losses .

5-Capacitors Bank
These are considered to be passive elements [2,3], where

Where
The magnitude of V L obtained from fundamental frequency information of conventional power flow solution.

6-Linear load
There are no generally acceptable load equivalent for harmonic analysis .in each case the derivation of equivalent impedance harmonic bandwidths from P (active) and Q (reactive) power flows will need extra information on the actual composition of the load .The aggregate nature of the load makes it difficult to establish models based purely on theoretical analysis .Attempts to deduce models from measurements have been made.There are three type loads model as below [2, 3]:

Where
(20 Models (ii) and (iii) with a coefficients are derived from measurements on medium voltage.Where the magnitude of V L obtained from fundamental frequency information of conventional power flow solution.

7-Non-linear load
These are represented by either a harmonic current injection source or by harmonic voltage source .Harmonic current source are used to represent the harmonic contribution of static var compensator (SVC), rectifier and electronic appliance .A static var compensator is represented by harmonic current injection given by I h = (%h) I 1Ф , where (%h) is a percentage of fundamental SVC current, the SVC current at fundamental frequency given by [2, 8]: ( Where θ and V L obtained from information of conventional power flow solution , and π/2 is required phase shift since the current leads or lags the voltage by 90.Therefor the need to solve the conventional fundamental frequency power flow to use the information in determine the harmonic current injected by source of harmonic (I 1Ф ), and determine the linear load parameter (R, X L , and X S ) .Therefore the results of fundamental frequency power flow solution effect on the magnitude of harmonic source current.The harmonic voltage sources are arc furnaces, and pulse width modulation (PWM) converter [3].

Harmonic Voltage and Current Flow in Power System Networks
The magnitude of the currents generated and impedance matrices for various frequencies present are obtained from system and source of harmonics data which are normally known.For these harmonics, the reference node for the impedance matrix will be at ground potential, so the system is solved at each frequency of interest using [2, 5] If a source of harmonic were placed at node m of an n node system, the matrix for harmonic voltage would be [5] (21) Where I h =magnitude of harmonic current entering the system at node m Therefore the harmonic voltages are ( 27) Where i is an integer from one to n.Since the impedance matrix and I h change for each frequency, the harmonic voltages will also vary.Knowing the harmonic voltages at every node, it is possible to determine the harmonic current flow on any link ( 28) If there is more than one harmonic source in the system, the harmonic current in each element is calculated for each source separately .All the currents associated with particular elements are then summed vectorially [5,9].

Block Diagram of Harmonic Analysis
Figure (4) show the processes used to analyze and determine the harmonic voltages and currents at all nodes and transmission lines in the power system networks.The harmonic analysis approach proceeds along the following steps.
step1: enter the transmission lines parameters(R, X, and B), active and reactive power for each load bus, the voltage for generation bus, and the generators , impedance.
Step2: A conventional power flow solution is carried out for the system under study to drive fundamental frequency information for the voltage magnitudes and angles at all nodes of the network for use this information to calculate the parameters of linear loads impedance (R, X L , and X S ), and use this information for calculate the fundamental frequency current of harmonic source (TCR) from equation (20).
Step3: develop the harmonics models for each component of network (transmission lines, linear loads, generators, transformer, and harmonic source).
Step4: form the harmonic admittance matrices [Y h ] and harmonic impedance matrix [Z h ] for system under study.
Step5: determine the harmonic voltages for all buses and harmonics currents flowing in all transmission lines in network by using the equations from ( 24) to (28).
Step 6: plot the results

System under Study
Figure ( 5) shows a one line diagram of the system under study, buses one through five are 138 KV and the buses six through twelve are 69 KV .thetransmission line and branch data are given in table (1),all base on (100MVA).The harmonics source current was static var compensator simulated by thyristor controlled reactor (TCR) as shown in figure(6)[9,10] ,and the maximum amplitude of harmonic current at full conduction angle as a percentage of the fundamental frequency current for delta connection of TCR are listed in table (2) [8,10].The (TCR) is used in load compensation in electrical power system ,load compensation is the management of reactive power to improvement the power factor, and voltage regulation of power system, the compensating equipment usually being installed near to the load.It is also used in compensation of transmission system in order to increase its power transmission capacity [2,8].TCR can be represented as a constant reactive power load in conventional power flow solution and a source of harmonic in harmonic analysis [9].The loads (load3, load4, load5, load7, load8, load9, load10, load11, and load 12) in figure (5) are modeled by using the (26) linear load model type (i), (ii), and (iii).

Result
Case 1: In this case (TCR) located at bus3 and the system's loads model type (iii).It is observed from data listed in table (3), that the bus3 and buses connected directly and near bus 3 carry the largest harmonic voltages and currents.The harmonic bus voltage are equal to harmonic injected current product with impedance line between the bus and the bus that the harmonic source connect to it, the voltage harmonic at bus3 equal to product of harmonic injected current with driving point impedance of bus3,it is the largest impedance of other buses, therefore bus3 carry the largest harmonic voltages in network .The current injected to the system at fifth harmonic it is larger than other harmonic order, therefore the maximum harmonic voltage for all buses occur at the fifth harmonic order.The figures ( 7) to (10) show the spectrum of four largest harmonic voltages, and figures (15) to ( 18) show the distortion waveform in time domain for four largest harmonic voltages in the System at buses (3, 5, 2, and 4).It is observed from data listed in table (4) that the highest harmonic current flowing in the transmission line directly connected at the bus 3 and the lines near it because these lines carry largest harmonic voltage, figures from ( 11) to ( 14) show spectrum of four largest harmonic currents in system lines4-5, 2-3, 3-4, and 2-4, and the figures from ( 19) to ( 22) show the distorted waveforms of these currents in time domain.Since the current flowing on a line is a function of the impedance of that line and the voltage across that impedance, then it is important to observe the voltages across these lines and their series impedances change with harmonic order, therefore the peak of harmonics currents do not appear at the same harmonic order for all lines.Table (4) shows that the largest harmonic currents occur at fifth harmonic order, because the largest harmonic voltages occur at fifth harmonic.
Case2: In this case (TCR) located at bus12 and the system's loads model type (iii).It is observe from data in table (5) that the highest harmonic voltages occur at bus12 directly connected to the TCR and the buses directly connected to bus 12 at 7, and 11 as shown in figures (23), and ( 24), also it is shows that the maximum harmonic voltage occurs at fifth harmonic order for all buses.Table (6) shows that the largest harmonic currents flowing on the transmission lines direct connected to bus 12 were placed the TCR, figures (25), and ( 26) show the spectrum of maximum harmonic current flowing on lines12-7, and 12-11. Case3: In this case (TCR) located at bus3 and the system's loads model type (ii).It is observed from table (7), bus3 and buses directly connected and near it carry largest harmonics voltages, also it is observed that the maximum harmonic voltage for all buses occur at the fifth harmonic, the data listed in table (8) explain that the highest harmonic current flowing in line directly connected to bus3.

Case4:
In this case (TCR) located at bus12 and the system ' s loads model type (ii).It is observed from table (9)that the largest harmonic voltage occurs at the bus12 directly connected to the TCR ,and the buses directly connected to bus12 .Table (10 )shows the largest harmonic current flowing in line 12-7 and 12-11,lines connected to bus 12.
Case5: In this case (TCR) located at bus3 and the system ' s loads model type (i).It is observed from table ( 11) and ( 12) that the highest harmonic voltages occur at bus3 and the bus directly connected near to bus3, and the largest current flowing on lines directly connected and near to bus3 lines4-5, 2-3, 3-4, and line 2-4.

Case6:
In this case (TCR) located at bus12 and the system ' s loads model type (i).It is observed from table (13) that the largest harmonic voltage occurs at the bus12 directly connected to the TCR, and the buses directly connected and near of bus12.Table (14) shows the largest harmonic current flowing in lines connected to bus 12, lines 12-7 and 12-11.

Conclusion
From the results of the six cases studied, it appears that the harmonic current flows on system transmission lines can exceed the current injected by a harmonic source at any frequency, if this frequency causes large harmonic voltage magnitude.The maximum harmonic voltages and currents flowing in transmission lines occur at the bus directly connected to (TCR) and the lines and buses directly connected to this bus.The highest injected harmonic current at fifth harmonic order for all cases study, therefore the highest harmonic voltage and harmonic current occur at fifth harmonic.As the harmonics frequencies increase, the magnitude of the harmonic injected current decrease therefore, in order to obtain a large harmonic current flow at the higher orders, the amplification of injected for a line would have to be very large .It was found that this happens when a transmission line reaches a resonant frequency.The harmonic injected current in case of TCR connected to bus12 larger than TCR connected to bus 3, therefore the highest harmonic voltage and current for TCR located at bus12 larger than highest harmonic voltage and current in case TCR located at bus3 for all types of loads.Eng.&Technology, Vol., No., 2008 Harmonics at fundamental frequency (Ω) V j : Harmonic voltage at node j (KV) V k : Harmonic voltage at node k (KV) V L : RMS line voltage (KV) X C : Capacitive reactance at fundamental frequency (Ω) X d '' : Generator sub transient reactance (Ω) X L : Inductive reactance at fundamental frequency (Ω) Y ' : Shunt harmonic admittance of transmission line(S) Y 1 , Y 2 : Harmonic admittance of long transmission Line(S) Y h : Harmonic admittance(S) Z ' : Series harmonic impedance of transmission line(Ω) Z h : Harmonic impedance (Ω) Z im : Harmonic impedance between node i and node m (Ω) Z jk : Harmonic impedance of link between node j and k (Ω) Z o : Characteristics impedance of transmission line (Ω) Z t : Harmonic impedance of transformer (Ω) ω: Angular velocity (rad/sec) γ : Propagation constant : Length of transmission line (Km) l θ: Fundamental voltage angle (degree) modified series impedance and shunt capacitance by an (equivalent π model) derived from (nominal π model) as shown in figure (3) [6, 7].

Figure ( 10 Figure
Figure (10) harmonic voltage spectrum in per unit at bus 2

FigureFigureFigure
Figure (14) harmonic current spectrum in per unit flowing in line 2-4

Figure
Figure (23) harmonic voltage spectrum in per unit at bus 7

Table ( 4
) harmonic current in per unit flowing on transmission lines Analysis and Calculation in Power System Networks

Table ( 6
) harmonic current in per unit flowing on transmission lines

Table ( 10
) harmonic current in per unit flowing on transmission lines Table (14) harmonic current in per unit flowing on transmission lines