Harmonic Load Flow Analysis

Table of Content

Power Systems of today are very large, complex interconnected network of various components. Major components of a power system can be subdivided into: a. Generation b. Transmission and Sub transmission c. Distribution

Conventional ac electric power systems are designed to operate with sinusoidal voltages and currents. However, nonlinear and electronically switched loads distort steady state ac voltage and current waveforms. It is important to calculate these distortions for safe and reliable operation of the power system. Periodically distorted waveforms can be studied by examining the harmonic components of the waveforms. The text here presents method of 3 – phase harmonic analysis for industrial distribution systems with the aid of a personal computer using the harmonic analysis software.

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This software uses C++ for analysis and Visual C++ based GUI as front end. It performs following analyses. * Load Flow Analysis * 3 – Phase Harmonic Analysis The software allows the user to draw and model a distribution network and input the injected harmonic currents. It performs load flow analysis using backward- forward sweep method. Same method is used for 3- phase harmonic analysis. Finally harmonic voltages and total harmonic distortion (THD) at various buses is calculated. Objective of this project was to build software for designing a 3- phase distribution system and its harmonic load flow analysis.

Backward- Forward Sweep method was used due to its efficacy in radial distribution systems. This software works for a non branched network and can be easily extended into a complete radial network.


A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i. e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f . . . etc. The harmonics have the property that they are all periodic at the fundamental frequency; therefore the sum of harmonics is also periodic at that frequency.

In relation to power systems, harmonics are used to represent the multiple frequencies of voltage and current waveforms. The following sections discuss the concept of power system harmonics, their sources and effects.

Concept of Power System Harmonics

Nonlinear and switched loads and sources can cause distortion of the nominal sinusoidal current and voltage waveform in an ac power system. In this section, basic definitions and concepts associated with the analysis of periodic steady state waveform distortion are discussed.

Fourier Series and Harmonics

Under periodic steady state conditions, distorted voltage and current waveforms can be expressed in the form of a Fourier Series. The Fourier series for a periodic function f(t) with fundamental frequency ? can be presented as: ft= C0+ n=1? Cncos(n? t+? n) Equation 1 Fourier series for a periodic function The coefficients Cn and phase angles n for n-th harmonic are given by: Cn= An2+ Bn2 Equation 2 ?n= tan-1-BnAn Equation 3 where T=2 /and C0 is the dc component of the function. The rms value of ft is defined as: An= 2T0Tf(t)cos(n? t)dt Equation 4 Bn= 2T0Tf(t)sin(n t)dt Equation 5 C0= 1T0Tf(t)dt

Equation 6 RMS= C02+ n=1? Cn22 Equation 7 RMS value In general, one can think of devices that produce distortion as exhibiting a nonlinear relationship between voltage and current. Such nonlinear relationships can lead to several forms of distortion as summarized below: * A periodic steady state exists and the distorted waveform can be expressed as a Fourier series with a fundamental frequency equal to the power frequency. * A periodic steady state exists and the distorted waveform can be expressed as a Fourier series with fundamental frequency that is a sub multiple of power frequency.

The waveform is aperiodic but perhaps almost periodic. A trigonometric series expansion may still exist (as an exact representation or as an approximation). The first case is commonly encountered in harmonic studies. There are several advantages to decomposing waveforms in terms of harmonics. Harmonics have a physical interpretation and an intuitive appeal. As discussed later, the transmission network is usually modeled as a linear system. Thus the propagation of each harmonic can be studied independent of others in the frequency domain.

Generally, the number of harmonics to be considered is small which simplifies computation. Consequences such as losses can be related to harmonic components and measures of waveform quality can be developed in terms of harmonic amplitudes. Certain types of pulsed or modulated loads and integral cycle controllers can create waveforms corresponding to the second category. The Fourier representation, when applicable, can be advantageous for the reasons cited above and measures of waveform quality can be adapted to such systems, although standards do not yet exist.

Some practical situations correspond to the third case. For example, dc arc furnaces consist of a conventional rectifier input but the underlying process of melting is not a periodic process. When reference is made to harmonics in this instance it corresponds to the periodic waveform that would be obtained if furnace conditions were to be maintained constant over a period of time. While such modeling obviously does not predict the exact response, it can, to a certain extent, lend insight into some of the potential problems caused by the distortion producing devices.

Distortion Indices

The most commonly used measure of deviation of a periodic waveform from a sine wave is called total harmonic distortion (THD) or distortion factor. THD=1C1n=2 Cn2 Equation 8 Total Harmonic Distortion The term distortion factor is more appropriate when the summation in the equation above is taken over a selected number of harmonics. IEEE Std. 519 [l], specifies limits on voltage and current THD for ‘Low Voltage, General Distribution, General Sub transmission, and High Voltage systems and Dispersed Generation and Cogeneration’.

Other distortion factors such as Telephone Influence Factor (TIF), the C-message weighted indices are also used. Only THD is discussed in this report.

Characteristics of Harmonics in Power Systems

Most devices operate in an identical manner in the positive and negative half cycle, thus eliminating even order harmonics. In balanced three-phase systems, under balanced operating conditions, harmonics in each phase have specific phase relationships. For example, in the case of the third harmonic, phase b currents would lag those in phase a by 3×1200 or 3600, and those in phase c would lead by the same amount.

Thus, the third harmonics are in phase and appear as zero-sequence components. As such, in a grounded wye system these harmonics flow in the lines and neutral/ground circuits, while in delta or ungrounded systems they cannot exist in line current at all. Similar analysis shows that fifth harmonics appear to be of negative sequence, seventh are of positive sequence, etc. Therefore, the impedances and manner of connection of rotating machines, transmission lines, and transformers must be modeled carefully for each harmonic.

The harmonics produced by many devices, particularly solid-state power converters are well-defined ‘characteristic harmonics’. An ideal, p-pulse, line commutated, converter, for example, produces ac side harmonic currents of order np+1, n=1, 2, 3…. The interpretations discussed above do not apply to the unbalanced cases. When supplied with unbalanced voltage, most three-phase power electronic converters can generate non characteristic harmonics. In many cases, the three-phase harmonics do not follow the sequence order of the balanced cases.

Furthermore, the nature of some harmonic problems requires the assessment of unbalanced harmonics. For example, zero sequence harmonic currents generally cause much more interference with telephone circuits than positive or negative sequence harmonics. Systems with unbalanced loads and components need to be studied using a three-phase model with proper representation of neutral and ground circuits.


Prior to the appearance of power semiconductors, the main sources of waveform distortion were electric arc furnaces, accumulated effect of fluorescent lamps, and to lesser extent electrical machines and transformers.

The increasing use of power electronic devices for the control of power apparatus and systems has been the reason for greater concern about waveform distortion in recent times.


Each element of the power system must be examined for its sensitivity to harmonics as a basis for recommendations on the allowable levels. The main effects of voltage and current harmonics within the power system are:

  • The possibility of amplification of harmonic levels resulting from series and parallel resonances.
  • A reduction in the efficiency of the generation, transmission and utilization of electric energy.
  • Ageing of insulation of electric power plant components with consequent shortening of their usual life.
  • Malfunctioning of system or plant components.

Among the possible external effects of harmonics is degradation in communication systems performance, excessive audible noise and harmonic-induced voltage and currents.

Thermal losses in harmonic environment

Harmonics increase the equipment copper, iron and dielectric losses and cause thermal stress. The per unit increase in the copper loss is obtained from the total voltage distortion. ?PRpu= THD2v Iron losses consist of eddy-current and hysteresis loss and result in reducing the efficiency and increasing the core temperature thus limiting the output.

Dielectric loss in a capacitor or insulation loss in a cable is due to the fact that there is not practically any capacitor whose current leads the voltage by 900.

Effects on power system equipment

Harmonics result in increased losses and equipment loss-of-life. Triplen harmonics result in the neutral carrying a current which might equal or exceed the phase currents even if the loads are balanced. Moreover, harmonics-caused resonance might damage equipment and interfere with protective relays, metering devices, control and communication circuits, and customer electronic equipments.  Capacitor banks.  Capacitors are overloaded by harmonic currents, since their reactance decreases with frequency they act as sinks for harmonic currents. Also harmonic voltages produce large currents causing capacitor fuses to be blown. Harmonics tend to increase dielectric loss.  Capacitors combine with source inductance to form a parallel resonant circuit. In presence of resonance, harmonics are amplified. The resulting voltages highly exceed the voltage rating and result in blown fuses.


Transformers operating in harmonic environment suffer from: Increased load losses due to copper and stray losses.  Increased hysteresis and eddy-current loss.  The possibility of resonance between transformer inductance and power factor correction capacitors.  Increased insulation stress due t o increased peak voltage. Rotating machines. Copper and iron losses are increased resulting in heating,  Pulsating torques are produced due to the interaction of the harmonics-generated magnetic fields and the fundamental.


Harmonic load flow analysis of the 3- phase system fundamental load flow analysis and harmonic analysis. Fundamental load flow analysis is used to evaluate the fundamental voltages at various buses within a specified tolerance value. Harmonic analysis uses the injected harmonic currents to evaluate harmonic voltages and total harmonic distortion at various buses.


In power engineering, load-flow study is an important tool involving numerical analysis applied to a power system. Unlike traditional circuit analysis, a load flow study usually uses simplified notation such as a one-line diagram and per-unit system, and focuses on various forms of AC power (i. e. reactive, real, and apparent) rather than voltage and current. It analyses the power systems in normal steady-state operation. There exist a number of software implementations of power flow studies. The great importance of power flow or load-flow studies is in the planning the future expansion of power systems as well as in determining the best operation of existing systems. The principal information obtained from the power flow study is the magnitude and phase angle of the voltage at each bus and the real and reactive power flowing in each line.

Load flow problem formulation

The goal of a power flow study is to obtain complete voltage angle and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions. Once this information is known, real and reactive power flow on each branch as well as generator reactive power output can be analytically determined. Due to the nonlinear nature of this problem, numerical methods are employed to obtain a solution that is within an acceptable tolerance. The solution to the power flow problem begins with identifying the known and unknown variables in the system.

The known and unknown variables are dependent on the type of bus. A bus without any generators connected to it is called a Load Bus. With one exception, a bus with at least one generator connected to it is called a Generator Bus. The exception is one arbitrarily-selected bus that has a generator. This bus is referred to as the Slack Bus. In the power flow problem, it is assumed that the real power PD and reactive power QD at each Load Bus are known. For this reason, Load Buses are also known as PQ Buses. For Generator Buses, it is assumed that the real power generated PG and the voltage magnitude |V| is known.

For the Slack Bus, it is assumed that the voltage magnitude |V| and voltage phase ? are known. Therefore, for each Load Bus, the voltage magnitude and angle are unknown and must be solved for; for each Generator Bus, the voltage angle must be solved for; there are no variables that must be solved for the Slack Bus. In a system with N buses and R generators, there are then 2(N ? 1) ? (R ? 1) unknowns. In order to solve for the 2(N ? 1) ? (R ? 1) unknowns, there must be 2(N ? 1) ? (R ? 1) equations that do not introduce any new unknown variables.

The possible equations to use are power balance equations, which can be written for real and reactive power for each bus. The real power balance equation is: 0= -Pi+ k=1NViVK(Gikcos? ik+ Biksin? ik ) Equation 9 Real power balance where Pi is the net power injected at bus i, Gik is the real part of the element in the Ybus corresponding to the ith row and kth column, Bik is the imaginary part of the element in the Ybus corresponding to the ith row and kth column and ? ik is the difference in voltage angle between the ith and kth buses.

The reactive power balance equation is: 0= -Qi+ k=1NViVK(Giksin? k- Bikcos? ik ) Equation 10 Reactive power balance where Qi is the net reactive power injected at bus i. Equations included are the real and reactive power balance equations for each Load Bus and the real power balance equation for each Generator Bus. Only the real power balance equation is written for a Generator Bus because the net reactive power injected is not assumed to be known and therefore including the reactive power balance equation would result in an additional unknown variable. For similar reasons, there are no equations written for the Slack Bus.

Backward forward sweep method

Load flow problem formulated above can be solved using different methods such as Newton Raphson method, Gauss- Seidel method etc. In this report backward forward sweep method is used to solve the three phase fundamental load flow problem. This particular method is useful for radial distribution systems because it reaches the desired tolerance level in less number of iterations. Moreover, 3- phase impedance matrix is not to be built for the entire network, instead 3- phase impedance matrix is built for each feeder as shown below which is easier. Also, the impedance matrix need not be inverted saving considerable computational effort.

Since the report deals with a distribution system the buses in the network are PQ buses. There are no PV buses as it is assumed that no generators are present in the network. Feeder line modeling The following figure shows the three phase transmission series impedance. A 4 ? 4 matrix is formed using Carson’s formulation, which takes into account the self and mutual coupling terms. Ig Va In Vn Zag-n Zbg-n Zcg-n Ia Ib Ic Zaa-n Zab-n Zac-n Zba-n Zbb-n Zbc-n Zca-n Zcb-n Zcc-n ?Va ?Vb ?Vc ?Vg Zga-n Zgb-n Zgc-n = Zgg-n Ig Iabc ZA ZB ?Vg ?Vabc Ig ZC ZD =

Using Kron’s reduction the 4 4 matrix can be converted into a 3? 3 matrix as shown below. Zaa-n| Zab-n| Zac-n| Zba-n| Zbb-n| Zac-n| Zca-n| Zcb-n| Zcc-n| [Zabc] = [ZA]- [ZB][ZD]-1[ZC] = The 3? 3 impedance matrix of the feeder line is used for calculations. In the software a 3 phase balanced line is modeled. Hence the per unit length reactance and resistance values are taken as constant reference values. However the length of the different feeders in the network can be varied. Algorithm The root node is taken as the slack node with known magnitude and angle, and the initial voltage of all nodes is taken equal to the root node voltage.

The iterative algorithm for solving the radial system consists of three steps. At iteration k:

Nodal current calculation IiaIibIic(k)=SiaVia(k-1)*SibVib(k-1)*SicVic(k-1) Equation 11 Nodal current where Iia , Iib , Iic are current injections at node I corresponding to constant power load elements. Sia, Sib, Sic are scheduled (known) power injections at node i Via , Vib ,Vic are voltages at node i

Backward Sweep Backward sweep is used to sum up line section current; starting from the line section in the last layer and moving towards the root node, the current in section l is JlaJlbJlck= -IjaIjbIjck+m MJmaJmbJmck Equation 12 Backward Sweep where Jia ,Jib and Jic are the current flows in the line section l and M is the set of line sections connected to node j. The negative sign before injected currents is to keep consistent with nodal current calculation formula.

Forward Sweep Forward sweep is used to update nodal voltage; starting from the first layer and moving towards the last layer. At each node, the change in the nodal voltage with respect to previous iteration is calculated for each phase. If this deviation exceeds the provided tolerance value in any case the backward forward sweep is carried out again.

VjaVjbVjck= ViaVibVick- zaa,nzab,nzac,nzba,nzbb,nzbc,nzca,nzcb,nzcc,nJlaJlbJlck Equation 13 Forward Sweep The following flowchart depicts the steps to be carried out for fundamental load flow analysis.


The report presents a fast harmonic load flow method for industrial distribution systems using forward backward sweep technique.

The proposed method saves computational time and accomplishes real- time harmonic analysis. Based on the network structures of distribution systems, current- injection formulation and the Kirchoff’s law, the relationships between the bus voltages, branch currents and harmonic sources can be formulated and then the harmonic voltage for each bus can be calculated using simple forward backward sweep techniques. The total harmonic distortion can also be calculated easily. The presented method is very efficient since the computational time for the LU decomposition of the Jacobian matrix (or Admittance matrix) can be saved.

The conventional harmonic load flow methods use load flow program, employ the frequency- based component model, update the Jacobian matrix (or Admittance matrix), decompose the matrix and rerun the load flow program for each harmonic order. However the decomposition of the Jacobian matrix (or Admittance matrix) is a time consuming process making the conventional methods difficult for real time analysis. The fast harmonic load flow method saves computational time making it suitable for real time analysis.

Components model

The feeder is modeled as mentioned above in the load flow section.

The reactance matrix is however multiplied with the harmonic order for different harmonics. Harmonic sources are modeled as constant current injection sources. Harmonic data is obtained from the harmonic analyzers. In industrial distribution systems, the industrial loads are usually driven by six pulse converters. Since the six pulse converters produce harmonics of the order 6k 1, the dominant harmonic components in the system are of order 5, 7, 11, 13 etc. Hence in the software harmonics of order 5, 7, 11, 13, 17, 19, 23 and 25 are considered.

The ac source transformers in a plant are installed with wye- delta and delta-wye alternatively to eliminate the harmonic components of order 3n.


The algorithm consists of backward current sweep and forward voltage implementation. Backward sweep is used to sum up line section current; starting from the line section in the last layer and moving towards the root node, the current in section l is JlaJlbJlc(k)= -IjaIjbIjc(k)+m? MJmaJmbJmc(k) Equation 14 Backward Sweep where Jia ,Jib and Jic are the current flows in the line section l and M is the set of line sections connected to node j.

The negative sign before injected currents is to keep consistent with nodal current calculation formula. Forward Sweep Forward sweep is used to update nodal voltage; starting from the first layer and moving towards the last layer. At each node, the change in the nodal voltage with respect to previous iteration is calculated for each phase. If this deviation exceeds the provided tolerance value in any case the backward forward sweep is carried out again. VjaVjbVjck= ViaVibVick- zaa,nzab,nzac,nzba,nzbb,nzbc,nzca,nzcb,nzcc,nJlaJlbJlck Equation 15 Forward Sweep Calculation of THD

Total harmonic distortion of voltages at each bus is calculated using the formula: THD=1V1n=2? Vh2 Equation 16 Voltage THD The following flow chart represents the steps carried for harmonic load flow analysis. Start Read 3 phase harmonic currents for h=5,7,11,13,17,19,23,25 Calculate branch currents using backward sweep Calculate bus voltages using forward sweep Calculate THD for each bus h=25


Theory of various analyses and their computer implementation algorithms have been discussed in details in previous section.

The software implementation of these algorithms has been done using C++ language and the GUI has been developed in Visual C++ 6. 0. The complete software has two main components. They are – 1. Display and control using Visual C++6. 0 This is mainly a Graphical user Interface (GUI) which is developed using Visual C++ 6. 0 and is used for display purposes as well as for control I. e. simulate the distribution system model and further to carry out the Harmonic Load Flow analysis on the simulated system. This is, thus, the front-end of our application. 2. Running Analysis using C++ language

C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. As one of the most popular programming languages ever created, C++ is widely used in the software industry. Several groups provide both free and proprietary C++ compiler software, including the GNU Project, Microsoft, Intel and Borland making our software an open source.


An interactive GUI and a reliable programming language led to the choice of Visual C++ and C++ language. 1. Visual C++ 6. 0

When talking about developing a GUI the names that instantly come to the mind are JAVA and Visual C++. VC++ 6. 0 is quite popular and often used to maintain legacy projects. Further it is easier to develop a GUI in it and was easier to learn. Since the background analysis program was already developed in C++ language so the choice was obvious as it supports the developed C++ code and smoother inter process communication as both uses the same compiler. It has tools for developing and debugging C++ code, especially code written for the Microsoft Windows API, the DirectX API, and the Microsoft .

NET Framework. So, we decided to use Microsoft Visual 6. 0 Enterprise edition and used help at MSDN, a great site for developers. 2. C++ language It is a favored language by all programmers for development of any open source software application. Prior knowledge of the language and the courses taken on C++ language before led to an easy decision of developing the software in this language. The Integrated Development environment used for the development was NetBeans IDE 6. 8, the latest release of NetBeans.

It lets C/C++ developers use their specified set of compilers and tools in conjunction with NetBeans IDE to build native applications for Windows, Mac OS X, Linux, and Solaris. It is easy to use and has features of auto-completion, listing the classes defined along with their variables and methods thus giving a clear insight of current development status of the software. The software models the distribution system as non branched network i. e. a radial network without any branches, though any branched network can be derived from the same network and the same can be extended for implementing a normal radial network. A flow diagram is shown to understand the control flow in the software.

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