Arnold, C.P., Watson, N.R. Power System Analysis Software The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton CRC Press llc. 2000
Arnold, C.P., Watson, N.R. “Power System Analysis Software” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
68 Power system analysis Software 68.1 Introduction .2 Early Analysis Programs Load Flow(Power Flow).Fault Analysis. Transient Stability Fast Transients. Reliability. Economic Dispatch and Unit Commitment 8.3 The Second Generation of programs Graphics. Protection. Other Uses for Load Flow Analysis Extensions to Transient Stability Analysis. Voltage Collapse SCADA.Power Quality. Finite Element Analysis C P Arnold and Grounding. Other Programs N.R. Watson 68.4 Further Development of Programs University of canterbury. New zealand 68.5 Conclusions 68.1 Introduction Power system software can be grouped in many different ways, e.g., functionality, computer platform, etc. but here it is grouped by end user. There are four major groups of end users for the software major utilities of ele consultants Large comprehensive program packages are required by utilities. They are complex, with many different anctions and must have very easy input/output(IO). They serve the needs of a single electrical system and may be tailor-made for the customer. They can be integrated with the electrical system using SCADA (Super visory Control And Data Acquisition). It is not within the scope of this chapter to discuss the merits of these programs. Suffice to say that the component programs used in these packages usually have the same generic/development roots as the programs used by the other three end user groups The programs used by the other three groups have usually been initially created in the universities. They start life as research programs and later are used for teaching and/or consultancy programs Where the consultant is also an academic, then the programs may well retain their crude research style IO. However, if they are to be used by others who are not so familiar with the algorithms, then usually they are modified to make them more user friendly. Once this is achieved, the programs become commercial and are used by consultants and utilities. These are the types of programs that are now so commonly seen in the engineering journals quite often bundled together in a generic package c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 68 Power System Analysis Software 68.1 Introduction 68.2 Early Analysis Programs Load Flow (Power Flow) • Fault Analysis • Transient Stability • Fast Transients • Reliability • Economic Dispatch and Unit Commitment 68.3 The Second Generation of Programs Graphics • Protection • Other Uses for Load Flow Analysis • Extensions to Transient Stability Analysis • Voltage Collapse • SCADA • Power Quality • Finite Element Analysis • Grounding • Other Programs 68.4 Further Development of Programs Program Suites 68.5 Conclusions 68.1 Introduction Power system software can be grouped in many different ways, e.g., functionality, computer platform, etc. but here it is grouped by end user. There are four major groups of end users for the software: • major utilities • small utilities, and industry consumers of electricity • consultants • universities Large comprehensive program packages are required by utilities. They are complex, with many different functions and must have very easy input/output (IO). They serve the needs of a single electrical system and may be tailor-made for the customer. They can be integrated with the electrical system using SCADA (Supervisory Control And Data Acquisition). It is not within the scope of this chapter to discuss the merits of these programs. Suffice to say that the component programs used in these packages usually have the same generic/development roots as the programs used by the other three end user groups. The programs used by the other three groups have usually been initially created in the universities. They start life as research programs and later are used for teaching and/or consultancy programs. Where the consultant is also an academic, then the programs may well retain their crude research style IO. However, if they are to be used by others who are not so familiar with the algorithms, then usually they are modified to make them more user friendly. Once this is achieved, the programs become commercial and are used by consultants, industry, and utilities. These are the types of programs that are now so commonly seen in the engineering journals quite often bundled together in a generic package. C.P. Arnold and N.R. Watson University of Canterbury, New Zealand
68.2 Early Analysis Programs Two of the earliest programs to be developed for power system analysis were the fault and load flow(power flow) programs. Both were originally produced in the late 1950s. Many programs in use today are either based on these two types of program or have one or the other embedded in them Load Flow(Power Flow) The need to know the flow patterns and voltage profiles in a network was the driving force behind the development of load flow programs Although the network is linear, load flow analysis is iterative because of nodal(busbar)constraints. At most busbars the active and reactive powers being delivered to customers are known but the voltage level is not far as the load flow analysis is concerned, these busbars are referred to as PQ buses. The generators are scheduled to deliver a specific active power to the system and usually the voltage magnitude of the generator terminals is fixed by automatic voltage regulation. These busbars are known as PV buses. losses in the system cannot be determined before the load flow solution, one generator busbar only has its voltage magnitude specified. In order to give the required two specifications per node, this bus also has its voltage angle defined to some arbitrary value, usually zero. This busbar is known as the slack bus. The slack bus is a mathematical requirement for the program and has no exact equivalent in reality. However, in operating practice, the total load plus the losses are not known. When a system is not in power balance, i. e, when the input power does not equal the load power plus losses, the imbalance modifies the rotational energy stored in the system. The system frequency thus rises if the input power is too large and falls if the input power is too little. Usually a generating station and probably one machine is given the task of keeping the frequency constant by varying the input power. This control of the power entering a node can be seen to be similar to the slack bus. The algorithms first adopted had the advantages of simple programming and minimum storage bu t were slow to converge requiring many iterations. The introduction of ordered elimination, which gives implicit inversion of the network matrix, and sparsity programming techniques, which reduces storage requirement allowed much better algorithms to be used. The Newton-Raphson method gave convergence to the solution in only a few iterations Using Newtonian methods of specifying the problem, a Jacobian matrix containing the partial derivatives of the system at each node can be constructed. The solution by this method has quadratic onvergence. This method was followed quite quickly by the Fast Decoupled Newton-Raphson method. This exploited the fact that under normal operating conditions, and providing that the network is predominately reactive, the voltage angles are not affected by reactive power flow and voltage magnitudes are not effected by real power flow. The Fast Decoupled method requires more iterations to converge but each iteration uses less omputational effort than the Newton Raphson method. A further advantage of this method is the robustness of the algorithm Further refinements can be added to a load flow program to make it give more realistic results. Transformer n-load tap changers, voltage limits, active and reactive power limits, plus control of the voltage magnitudes at buses other than the local bus help to bring the results close to reality. Application of these limits can slow The problem of obtaining an accurate, load flow solution, with a guaranteed and fast convergence has resulted in more technical papers than any other analysis topic. This is understandable when it is realized that the load flow solution is required during the running of many other types of power system analyses. While improvements have been made, there has been no major breakthrough in performance. It is doubtful if such an achievement overall time of the anal quired to prepare the data and process the results represents a significant part of the is possible as the time re Fault Analysis A fault analysis program derives from the need to adequately rate switchgear and other busbar equipment for the maximum possible fault current that could flow through them c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 68.2 Early Analysis Programs Two of the earliest programs to be developed for power system analysis were the fault and load flow (power flow) programs. Both were originally produced in the late 1950s. Many programs in use today are either based on these two types of program or have one or the other embedded in them. Load Flow (Power Flow) The need to know the flow patterns and voltage profiles in a network was the driving force behind the development of load flow programs. Although the network is linear, load flow analysis is iterative because of nodal (busbar) constraints. At most busbars the active and reactive powers being delivered to customers are known but the voltage level is not. As far as the load flow analysis is concerned, these busbars are referred to as PQ buses. The generators are scheduled to deliver a specific active power to the system and usually the voltage magnitude of the generator terminals is fixed by automatic voltage regulation. These busbars are known as PV buses. As losses in the system cannot be determined before the load flow solution, one generator busbar only has its voltage magnitude specified. In order to give the required two specifications per node, this bus also has its voltage angle defined to some arbitrary value, usually zero. This busbar is known as the slack bus. The slack bus is a mathematical requirement for the program and has no exact equivalent in reality. However, in operating practice, the total load plus the losses are not known. When a system is not in power balance, i.e., when the input power does not equal the load power plus losses, the imbalance modifies the rotational energy stored in the system. The system frequency thus rises if the input power is too large and falls if the input power is too little. Usually a generating station and probably one machine is given the task of keeping the frequency constant by varying the input power. This control of the power entering a node can be seen to be similar to the slack bus. The algorithms first adopted had the advantages of simple programming and minimum storage but were slow to converge requiring many iterations. The introduction of ordered elimination, which gives implicit inversion of the network matrix, and sparsity programming techniques, which reduces storage requirements, allowed much better algorithms to be used. The Newton-Raphson method gave convergence to the solution in only a few iterations. Using Newtonian methods of specifying the problem, a Jacobian matrix containing the partial derivatives of the system at each node can be constructed. The solution by this method has quadratic convergence. This method was followed quite quickly by the Fast Decoupled Newton-Raphson method. This exploited the fact that under normal operating conditions, and providing that the network is predominately reactive, the voltage angles are not affected by reactive power flow and voltage magnitudes are not effected by real power flow. The Fast Decoupled method requires more iterations to converge but each iteration uses less computational effort than the Newton Raphson method. A further advantage of this method is the robustness of the algorithm. Further refinements can be added to a load flow program to make it give more realistic results. Transformer on-load tap changers, voltage limits, active and reactive power limits, plus control of the voltage magnitudes at buses other than the local bus help to bring the results close to reality. Application of these limits can slow down convergence. The problem of obtaining an accurate, load flow solution, with a guaranteed and fast convergence has resulted in more technical papers than any other analysis topic. This is understandable when it is realized that the load flow solution is required during the running of many other types of power system analyses. While improvements have been made, there has been no major breakthrough in performance. It is doubtful if such an achievement is possible as the time required to prepare the data and process the results represents a significant part of the overall time of the analysis. Fault Analysis A fault analysis program derives from the need to adequately rate switchgear and other busbar equipment for the maximum possible fault current that could flow through them
Initially only three-phase faults were considered and it was assumed that all busbars were operating at unity per unit voltage prior to the fault occurring. The load current flowing prior to the fault was also neglected By using the results of a load flow prior to performing the fault analysis, the load currents can be added to the fault currents allowing a more accurate determination of the total currents flowing in the system. Unbalanced faults can be included by using symmetrical components. The negative sequence network is similar to the positive sequence network but the zero sequence network can be quite different primarily because of ground impedance and transformer winding configurations Transient Stability After a disturbance, due usually to a network fault, the synchronous machine's electrical loading changes and the machines speed up(under very light loading conditions they can slow down). Each machine will react differently depending on its proximity to the fault, its initial loading and its time constants. This means that the angular positions of the rotors relative to each other change. If any angle exceeds a certain threshold(usually between 140@and 160%)the machine will no longer be able to maintain synchronism. This almost always results in its removal from service Early work on transient stability had concentrated on the reaction of one synchronous machine coupled to a very large system through a transmission line. The large system can be assumed to be infinite with respect to the single machine and hence can be modeled as a pure voltage source. The synchronous machine is modeled by the three phase windings of the stator plus windings on the rotor representing the field winding and the eddy current paths. These are resolved into two axes, one in line with the direct axis of the rotor and the other line with the quadrature axis situated 90%(electrical) from the direct axis. The field winding is on the direct axis Equations can be developed which determine the voltage in any winding depending on the current flows in all the other windings. A full set of differential equations can be produced which allows the response of the machine to various electrical disturbances to be found. The variables must include rotor angle and rotor speed which can be evaluated from a knowledge of the power from the turbine into, and power to the system out of the machine. The great disadvantage with this type of analysis is that the rotor position is constantly changing as it rotates. As most of the equations involve trigonometrical functions relating to stator and rotor windings, the matrices must be constantly reevaluated. In the most severe cases of network faults the results, once the dc transients decay, are balanced. Further, on removal of the fault the network is considered to be balanced.There is thus much computational effort involved in obtaining detailed information for each of the three phases which is of little value to the power system engineer. By contrast, this type of analysis is very important to machine designers. However, programs have been written for multi-machine systems using this method. Several power system catastrophes in the U.S. and Europe in the 1960s gave a major boost to dev transient stability programs. What was required was a simpler and more efficient method of representing the machines in large power systems Initially, transient stability programs all ran in the time domain. A set of differential equations is developed describe the dynamic behavior of the synchronous machines. These are linked together by algebraic equations for the network and any other part of the system that has a very fast response, i.e, an insignificant time constant, relative to the synchronous machines. All the machine equations are written in the direct and quadrature axes of the rotor so that they are constant regardless of the rotor position. The network is written in the real and imaginary axes similar to that used by the load flow and faults programs. The transposition between these axes only requires knowledge of the rotor angle relative to the synchronously rotating frame of reference of the Later work involved looking at the response of the system, not to major disturbances but to the build-up of oscillations due to small disturbances and poorly set control systems. As the time involved for these disturbances to occur can be large, time domain solutions are not suitable and frequency domain models of the system were produced. Lyapunov functions have also been used, but good models have been difficult to produce. However, they are now of sufficiently good quality to compete with time domain models where quick estimates of stability are needed such as in the day to day operation of a system. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Initially only three-phase faults were considered and it was assumed that all busbars were operating at unity per unit voltage prior to the fault occurring. The load current flowing prior to the fault was also neglected. By using the results of a load flow prior to performing the fault analysis, the load currents can be added to the fault currents allowing a more accurate determination of the total currents flowing in the system. Unbalanced faults can be included by using symmetrical components. The negative sequence network is similar to the positive sequence network but the zero sequence network can be quite different primarily because of ground impedance and transformer winding configurations. Transient Stability After a disturbance, due usually to a network fault, the synchronous machine’s electrical loading changes and the machines speed up (under very light loading conditions they can slow down). Each machine will react differently depending on its proximity to the fault, its initial loading and its time constants. This means that the angular positions of the rotors relative to each other change. If any angle exceeds a certain threshold (usually between 140° and 160°) the machine will no longer be able to maintain synchronism. This almost always results in its removal from service. Early work on transient stability had concentrated on the reaction of one synchronous machine coupled to a very large system through a transmission line. The large system can be assumed to be infinite with respect to the single machine and hence can be modeled as a pure voltage source. The synchronous machine is modeled by the three phase windings of the stator plus windings on the rotor representing the field winding and the eddy current paths. These are resolved into two axes, one in line with the direct axis of the rotor and the other in line with the quadrature axis situated 90° (electrical) from the direct axis. The field winding is on the direct axis. Equations can be developed which determine the voltage in any winding depending on the current flows in all the other windings. A full set of differential equations can be produced which allows the response of the machine to various electrical disturbances to be found. The variables must include rotor angle and rotor speed which can be evaluated from a knowledge of the power from the turbine into, and power to the system out of the machine. The great disadvantage with this type of analysis is that the rotor position is constantly changing as it rotates. As most of the equations involve trigonometrical functions relating to stator and rotor windings, the matrices must be constantly reevaluated. In the most severe cases of network faults the results, once the dc transients decay, are balanced. Further, on removal of the fault the network is considered to be balanced. There is thus much computational effort involved in obtaining detailed information for each of the three phases which is of little value to the power system engineer. By contrast, this type of analysis is very important to machine designers. However, programs have been written for multi-machine systems using this method. Several power system catastrophes in the U.S. and Europe in the 1960s gave a major boost to developing transient stability programs. What was required was a simpler and more efficient method of representing the machines in large power systems. Initially, transient stability programs all ran in the time domain. A set of differential equations is developed to describe the dynamic behavior of the synchronous machines. These are linked together by algebraic equations for the network and any other part of the system that has a very fast response, i.e., an insignificant time constant, relative to the synchronous machines. All the machine equations are written in the direct and quadrature axes of the rotor so that they are constant regardless of the rotor position. The network is written in the real and imaginary axes similar to that used by the load flow and faults programs. The transposition between these axes only requires knowledge of the rotor angle relative to the synchronously rotating frame of reference of the network. Later work involved looking at the response of the system, not to major disturbances but to the build-up of oscillations due to small disturbances and poorly set control systems. As the time involved for these disturbances to occur can be large, time domain solutions are not suitable and frequency domain models of the system were produced. Lyapunov functions have also been used, but good models have been difficult to produce. However, they are now of sufficiently good quality to compete with time domain models where quick estimates of stability are needed such as in the day to day operation of a system
CHARLES PROTEUS STEINMETZ (1865-1923) C harles Steinmetz(1865-1923)came to the United States in 1889 from breslau Germany, where he was a student at the University of Breslau. He joined the inventor Rudolf Eickemeyer in building electric appara tus at Yonkers, New York, and at age 27 he for- ulated the law of hysteresis, which made it possible to reduce the loss of efficiency in elec trical apparatus. When Eickemeyer's firm was bought by General Electric, Steinmetz joined the new company, beginning a 31-year relationship that ended only with his death. His improvements in methods of making cal- culations of current in alternating current cir- cuits revolutionized power engineering, and his theory of electrical transients stood as another important contribution. In the midst of his GE career, Steinmetz was also a professor at Union College and a vocal champion of civic and polit- Charles Proteus Steinmetz(1865-1923) ical causes. Courtesy of the IEEE Center for the History of Electrical Engineering. Fast Transients While the transient stability program assumed a fast transient response was equivalent to an instantaneous response and only concentrated on the slower response of the synchronous machines, the requirement to model the fast transient response of traveling waves on transmission lines brought about the development of programs that treated variables with large time constants as if they were constants and modeled the variables with very small time constants by differential equations. The program is based on the equations governing voltage and current wave propagation along a lossless line Attenuation is then included using suitable lumped resistances. A major feature of the method is that inductance and capacitance can both be represented by resistance in parallel with a current source. This allows a purely resistive network to be formed Whereas, with the most other programs, source code was treated as intellectual property, the development of the fast transient program was done by many different researchers who pooled their ideas and programs An electromagnetic transient program developed quickly and it probably became the first power systems analysis tool to be used for many different purposes throughout the world. From this base, numerous commercial In parallel with the development of electromagnetic transient programs, several state variable programs were produced to examine the fast transient behavior of parts of the electrical system, such as ac transmission lines and HVdc transmission systems. As these programs were specifically designed for the purpose they were intended, it gave them certain advantages over the general purpose electromagnetic transient program c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Fast Transients While the transient stability program assumed a fast transient response was equivalent to an instantaneous response and only concentrated on the slower response of the synchronous machines, the requirement to model the fast transient response of traveling waves on transmission lines brought about the development of programs that treated variables with large time constants as if they were constants and modeled the variables with very small time constants by differential equations. The program is based on the equations governing voltage and current wave propagation along a lossless line. Attenuation is then included using suitable lumped resistances. A major feature of the method is that inductance and capacitance can both be represented by resistance in parallel with a current source. This allows a purely resistive network to be formed. Whereas, with the most other programs, source code was treated as intellectual property, the development of the fast transient program was done by many different researchers who pooled their ideas and programs. An electromagnetic transient program developed quickly and it probably became the first power systems analysis tool to be used for many different purposes throughout the world. From this base, numerous commercial packages have been developed. In parallel with the development of electromagnetic transient programs, several state variable programs were produced to examine the fast transient behavior of parts of the electrical system, such as ac transmission lines and HVdc transmission systems. As these programs were specifically designed for the purpose they were intended, it gave them certain advantages over the general purpose electromagnetic transient program. CHARLES PROTEUS STEINMETZ (1865–1923) harles Steinmetz (1865–1923) came to the United States in 1889 from Breslau, Germany, where he was a student at the University of Breslau. He joined the inventor Rudolf Eickemeyer in building electric apparatus at Yonkers, New York, and at age 27 he formulated the law of hysteresis, which made it possible to reduce the loss of efficiency in electrical apparatus. When Eickemeyer’s firm was bought by General Electric, Steinmetz joined the new company, beginning a 31-year relationship that ended only with his death. His improvements in methods of making calculations of current in alternating current circuits revolutionized power engineering, and his theory of electrical transients stood as another important contribution. In the midst of his GE career, Steinmetz was also a professor at Union College and a vocal champion of civic and political causes. (Courtesy of the IEEE Center for the History of Electrical Engineering.) C
eliability f constant concern to the operators of power systems is the reliability of equipment. This has become more aportant as systems are run harder In the past, reliability was ensured by building in reserve equipment which was either connected in parallel with other similar devices or could be easily connected in the event of a failure Not only that, knowledge of the capabilities of materials has increased so that equipment can be built with a more certain level of reliability. However, reliability of a system is governed by the reliability of all the parts and their configuration. Much work has been done on the determination of the reliability of power systems but work is still being done to fully model power system components and integrate them into system reliability models The information that is obtained from reliability analysis is very much governed by the nature of the sys tem The accepted breakdown of a power system containing generation, transmission, and distribution is into three hierarchical levels. The first level is for the generation facilities alone, the second level contains generation and transmission,while the third level contains generation, transmission, and distribution facilities. Much of the early work was focused on the generation facilities. The reasons for this was that, first, more information was available about the generation; second, the size of the problem was smaller; and, third, the emphasis of power systems was placed in generation. With the onset of deregulation, distribution and customer requirements are ed paramount. At the generation and transmission levels, the loss of load expectation and frequency and duration evaluation re prime reliability indicators. a power system component may well have several derated states along with the fully operational and non-operational states. Recursive techniques are available to construct the system models and they can include multi-state components The usual method for evaluating reliability indices at the distribution level, such duration per customer per year, is an analytical approach based on a failure modes assessment and the use of equations for series and parallel networks. Economic Dispatch and Unit Commitment Many programs are devoted to power system operational problems and the minimization of the cost of production and delivery of energy is of great importance. Two types of program which deal with this problem are economic dispatch and unit commitment. Economic dispatch uses optimization techniques to determine the level of power each generator(unit)must supply to the system in order to meet the demand. Each unit must have its generating costs, which will be nonlinear functions of energy, defined along with the units operational maximum and minimum power limits The transmission losses of the system must also be taken into account to ensure an overall minimum cost Unit commitment calculates the necessary generating units that should be connected (committed)at any time in order to supply the demand and losses plus allow sufficient reserve increase or accidental loss of a generating unit. Several operating restrictions must be taken into account when determining which machines to commit or decommit. These include maximum and minimum running times for a unit and the time needed to commit a unit. Fuel availability constraints must also be considered. For ample, there may be limited fuel reserves such as coal stocks or water in the dam. Other fuel constraints may be minimum water flows below the dam or agreements to purchase minimum amounts of fuel. Determining Init commitment for a specific time cannot be evaluated without consideration of the past operational con- figuration or the future operating demands. 68.3 The Second Generation of programs It is not the intention to suggest that only the above programs were being produced initially. However, most of the other programs remained as either research tools or one-off analysis programs. The advent of the PC gave a universal platform on which most users and programs could come together. This process was further assisted when windowing reduced the need for such a high level of computer literacy on the part of users. For c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Reliability Of constant concern to the operators of power systems is the reliability of equipment. This has become more important as systems are run harder. In the past, reliability was ensured by building in reserve equipment which was either connected in parallel with other similar devices or could be easily connected in the event of a failure. Not only that, knowledge of the capabilities of materials has increased so that equipment can be built with a more certain level of reliability. However, reliability of a system is governed by the reliability of all the parts and their configuration. Much work has been done on the determination of the reliability of power systems but work is still being done to fully model power system components and integrate them into system reliability models. The information that is obtained from reliability analysis is very much governed by the nature of the system. The accepted breakdown of a power system containing generation, transmission, and distribution is into three hierarchical levels. The first level is for the generation facilities alone, the second level contains generation and transmission, while the third level contains generation, transmission, and distribution facilities. Much of the early work was focused on the generation facilities. The reasons for this was that, first, more information was available about the generation; second, the size of the problem was smaller; and, third, the emphasis of power systems was placed in generation. With the onset of deregulation, distribution and customer requirements are now considered paramount. At the generation and transmission levels, the loss of load expectation and frequency and duration evaluation are prime reliability indicators. A power system component may well have several derated states along with the fully operational and non-operational states. Recursive techniques are available to construct the system models and they can include multi-state components. The usual method for evaluating reliability indices at the distribution level, such as the average interruption duration per customer per year, is an analytical approach based on a failure modes assessment and the use of equations for series and parallel networks. Economic Dispatch and Unit Commitment Many programs are devoted to power system operational problems and the minimization of the cost of production and delivery of energy is of great importance. Two types of program which deal with this problem are economic dispatch and unit commitment. Economic dispatch uses optimization techniques to determine the level of power each generator (unit) must supply to the system in order to meet the demand. Each unit must have its generating costs, which will be nonlinear functions of energy, defined along with the units operational maximum and minimum power limits. The transmission losses of the system must also be taken into account to ensure an overall minimum cost. Unit commitment calculates the necessary generating units that should be connected (committed) at any time in order to supply the demand and losses plus allow sufficient reserve capability to withstand a load increase or accidental loss of a generating unit. Several operating restrictions must be taken into account when determining which machines to commit or decommit. These include maximum and minimum running times for a unit and the time needed to commit a unit. Fuel availability constraints must also be considered. For example, there may be limited fuel reserves such as coal stocks or water in the dam. Other fuel constraints may be minimum water flows below the dam or agreements to purchase minimum amounts of fuel. Determining unit commitment for a specific time cannot be evaluated without consideration of the past operational con- figuration or the future operating demands. 68.3 The Second Generation of Programs It is not the intention to suggest that only the above programs were being produced initially. However, most of the other programs remained as either research tools or one-off analysis programs. The advent of the PC gave a universal platform on which most users and programs could come together. This process was further assisted when windowing reduced the need for such a high level of computer literacy on the part of users. For
example, electromagnetic transient programs generality, which made it so successful, is also a handicap and it quires good programming skill to utilize it fully. This has lead to several commercial programs that are loosely based on the methods of analysis first used in by the electromagnetic transient program. They have the advantage of a much improved user interface. Not all software is run on PCs. Apart from the Macintosh, which has a similar capability to a PC but which is less popular with engineers, more powerful workstations are available usually based on the Unix operating system. Mini computers and mainframe computers are also still in general use in universities and industry even Hardware and software for power system operation and control required at utility control centers is usually sold as a total package. These systems, although excellent, can only be alluded to here as the information is proprietary. The justification for a particular configuration requires input from many diverse groups within the utility. graphics Two areas of improvement that stand out in this second wave of generally available programs are both asso with the graphical capabilities of computers. A good diagram can be more easily understood than many of text or tables The ability to produce graphical output of the results of an analysis has made the use of computers in all engineering fields, not just power system analysis, much easier. Tabulated results are never easy to interpret. They are also often given to a greater degree of accuracy than the input data warrants a graph of the results, where appropriate, can make the results very easy to interpret and if there is also an ability to graph any variable with any other, or two if three dimensions can be utilized, then new and possibly significant information can g New packages became available for business and engineering which were based on either the spreadsheet or atabase principle. These also had the ability to produce graphical output. It was no longer essential to know programming language to do even quite complex engineering analysis. The programming was usually inef- ficient and obtaining results was more laborious, e.g., each iteration had to be started by hand. But, as engineers had to use these packages for other work, they became very convenient tools A word of caution here--be careful that the results are graphed in an appropriate manner. Most spreadsheet packages have very limited x-axis(horizontal)manipulation. Provided the x-axis data comes in regular steps, the results are acceptable. However, we have seen instances where very distorted graphs have been presented Apart from the graphical interpretation of results, there are now several good packages that allow the analy to enter the data graphically. It is a great advantage to be able to develop a one-line, or three-phase, diagram of a network directly with the computer. All the relevant system components can be included. Parameter data ill require entry in a more orthodox manner but by merely clicking on a component, a data form for that component can be made available. The chances of omitting a component are greatly reduced with this type of data entry. Further, the same system diagram can be used to show the results of some analyses An extension of the network diagram input is to make the diagram relate to the actual topography. In these cases the actual routes of transmission lines are shown and can be geographical maps. The lines in these cases have their lengths automatically established and, if the line char- acteristics are known, the line parameters can be calculated These topographical diagrams are an invaluable aid for power reticulation problems, for example, the minimum route length of reticulation given all the points of supply and the route constraints. Other optim zation algorithms include determination of line sizes and switching operations The analysis techniques can be either linear or nonlinear. If successful, the nonlinear algorithm is more accura ate but these techniques suffer from larger data storage requirements, greater computational time, and possible divergence. There are various possible optimization techniques that can and have been applied to this problem. There is no definitive answer and each type of problem may require a different choice The capability chart represents a method of graphically displaying power system performance. These charts are drawn on the complex power plane and define the real and reactive power that may be supplied from a c 2000 by CRC Press LLC
© 2000 by CRC Press LLC example, electromagnetic transient program's generality, which made it so successful, is also a handicap and it requires good programming skill to utilize it fully. This has lead to several commercial programs that are loosely based on the methods of analysis first used in by the electromagnetic transient program. They have the advantage of a much improved user interface. Not all software is run on PCs. Apart from the Macintosh, which has a similar capability to a PC but which is less popular with engineers, more powerful workstations are available usually based on the Unix operating system. Mini computers and mainframe computers are also still in general use in universities and industry even though it had been thought that they would be totally superseded. Hardware and software for power system operation and control required at utility control centers is usually sold as a total package. These systems, although excellent, can only be alluded to here as the information is proprietary. The justification for a particular configuration requires input from many diverse groups within the utility. Graphics Two areas of improvement that stand out in this second wave of generally available programs are both associated with the graphical capabilities of computers. A good diagram can be more easily understood than many pages of text or tables. The ability to produce graphical output of the results of an analysis has made the use of computers in all engineering fields, not just power system analysis, much easier. Tabulated results are never easy to interpret. They are also often given to a greater degree of accuracy than the input data warrants. A graph of the results, where appropriate, can make the results very easy to interpret and if there is also an ability to graph any variable with any other, or two if three dimensions can be utilized, then new and possibly significant information can be quickly assimilated. New packages became available for business and engineering which were based on either the spreadsheet or database principle. These also had the ability to produce graphical output. It was no longer essential to know a programming language to do even quite complex engineering analysis. The programming was usually inef- ficient and obtaining results was more laborious, e.g., each iteration had to be started by hand. But, as engineers had to use these packages for other work, they became very convenient tools. A word of caution here—be careful that the results are graphed in an appropriate manner. Most spreadsheet packages have very limited x-axis (horizontal) manipulation. Provided the x-axis data comes in regular steps, the results are acceptable. However, we have seen instances where very distorted graphs have been presented because of this problem. Apart from the graphical interpretation of results, there are now several good packages that allow the analyst to enter the data graphically. It is a great advantage to be able to develop a one-line, or three-phase, diagram of a network directly with the computer. All the relevant system components can be included. Parameter data still require entry in a more orthodox manner but by merely clicking on a component, a data form for that component can be made available. The chances of omitting a component are greatly reduced with this type of data entry. Further, the same system diagram can be used to show the results of some analyses. An extension of the network diagram input is to make the diagram relate to the actual topography. In these cases, the actual routes of transmission lines are shown and can be superimposed on computer generated geographical maps. The lines in these cases have their lengths automatically established and, if the line characteristics are known, the line parameters can be calculated. These topographical diagrams are an invaluable aid for power reticulation problems, for example, the minimum route length of reticulation given all the points of supply and the route constraints. Other optimization algorithms include determination of line sizes and switching operations. The analysis techniques can be either linear or nonlinear. If successful, the nonlinear algorithm is more accurate but these techniques suffer from larger data storage requirements, greater computational time, and possible divergence. There are various possible optimization techniques that can and have been applied to this problem. There is no definitive answer and each type of problem may require a different choice. The capability chart represents a method of graphically displaying power system performance. These charts are drawn on the complex power plane and define the real and reactive power that may be supplied from a
point in the system during steady state operation. The power available is depict Ion o n the plane and the boundaries of the region represent the critical operating limits of the system. The best known example of a capability chart is the operating chart of a synchronous machine. The power available from the generator is restricted by limiting values of the rotor current, stator current, turbine power(if a generator), and synchronous stability limits. Capability charts have been produced for transmission lines and Hvdc converters Where the capability chart is extended to cover more than one power system component, the two-dimensional capability chart associated with a single busbar can be regarded as being a single slice of an overall 2n dimensional capability chart for the n busbars that make up a general power system. If the system is small, a contour plotting approach can be used to gradually trace out the locus on the complex power plane. a load flow algorithm used to iteratively solve the operating equations at each point on the contour, without having to resort to an explicit closed form solution The good contour behavior near the operating region has allowed a faster method to be adopted. A seed load flow solution, corresponding to the nominal system state, is obtained to begin drawing the chart. A region rowing process is then used to locate the region in which all constrained variables are less than 10% beyond their limits. This process is similar to a technique used in computer vision systems to recognize shapes of objects. The region grows by investigating the six nearest lattice vertices to any unconstrained vertex. Linear interpolation along the edges between vertices is then used to estimate the points of intersection between the contour and the lattice. This method has a second advantage in that it can detect holes and islands in the chart. However, it should be noted that these regions are purely speculative and have not been found in practice. Protection The need to analyze protection schemes has resulted in the development of protection coordination programs. Protection schemes can be divided into two major groupings: unit and non-unit schemes The first group contains schemes that protect a specific area of the system, i.e., a transformer, transmission line, generator, or busbar. The most obvious example of unit protection schemes is based on Kirchhoffs current law-the sum of the currents entering an area of the system must be zero. Any deviation from this must indicate an abnormal current path. In these schemes, the effects of any disturbance or operating condition outside the rea of interest are totally ignored and the protection must be designed to be stable above the maximum possible fault current that could flow through the protected area. Schemes can be made to extend across all sides of a transformer to account for the different currents at different voltage levels. Any analysis of these schemes ar thus of more concern to the protection equipment manufacturers. The non-unit schemes, while also intended to protect specific areas, have no fixed boundaries. As well as protecting their own designated areas, the protective zones can overlap into other areas. While this can be very beneficial for backup purposes, there can be a tendency for too great an area to be isolated if a fault is detected by different non-unit schemes. The most simple of these schemes measures current and incorporates an inverse time characteristic into the protection operation to allow protection nearer to the fault to operate first. while this is relatively straightforward for radial schemes, in networks, where the current paths can be quite different lepending on operating and maintenance strategies, protection can be difficult to set and optimum settings are probably impossible to achieve. It is in these areas where protection software has become useful to manu- facturers consultants, and utilities The very nature of protection schemes has changed from electromechanical devices, through electronic equivalents of the old devices, to highly sophisticated system analyzers. They are computers in their own right and thus can be developed almost entirely by computer analysis techniques. Other Uses for Load Flow analysis It has already been demonstrated that load flow analysis is necessary in determining the economic operation of the power system and it can also be used in the production of capability charts. Many other types of analyses require load flow to be embedded in the program. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC point in the system during steady state operation. The power available is depicted as a region on the plane and the boundaries of the region represent the critical operating limits of the system. The best known example of a capability chart is the operating chart of a synchronous machine. The power available from the generator is restricted by limiting values of the rotor current, stator current, turbine power (if a generator), and synchronous stability limits. Capability charts have been produced for transmission lines and HVdc converters. Where the capability chart is extended to cover more than one power system component, the two-dimensional capability chart associated with a single busbar can be regarded as being a single slice of an overall 2n dimensional capability chart for the n busbars that make up a general power system. If the system is small, a contour plotting approach can be used to gradually trace out the locus on the complex power plane. A load flow algorithm is used to iteratively solve the operating equations at each point on the contour, without having to resort to an explicit closed form solution. The good contour behavior near the operating region has allowed a faster method to be adopted. A seed load flow solution, corresponding to the nominal system state, is obtained to begin drawing the chart. A region growing process is then used to locate the region in which all constrained variables are less than 10% beyond their limits. This process is similar to a technique used in computer vision systems to recognize shapes of objects. The region grows by investigating the six nearest lattice vertices to any unconstrained vertex. Linear interpolation along the edges between vertices is then used to estimate the points of intersection between the contour and the lattice. This method has a second advantage in that it can detect holes and islands in the chart. However, it should be noted that these regions are purely speculative and have not been found in practice. Protection The need to analyze protection schemes has resulted in the development of protection coordination programs. Protection schemes can be divided into two major groupings: unit and non-unit schemes. The first group contains schemes that protect a specific area of the system, i.e., a transformer, transmission line, generator, or busbar. The most obvious example of unit protection schemes is based on Kirchhoff 's current law—the sum of the currents entering an area of the system must be zero. Any deviation from this must indicate an abnormal current path. In these schemes, the effects of any disturbance or operating condition outside the area of interest are totally ignored and the protection must be designed to be stable above the maximum possible fault current that could flow through the protected area. Schemes can be made to extend across all sides of a transformer to account for the different currents at different voltage levels. Any analysis of these schemes are thus of more concern to the protection equipment manufacturers. The non-unit schemes, while also intended to protect specific areas, have no fixed boundaries. As well as protecting their own designated areas, the protective zones can overlap into other areas. While this can be very beneficial for backup purposes, there can be a tendency for too great an area to be isolated if a fault is detected by different non-unit schemes. The most simple of these schemes measures current and incorporates an inverse time characteristic into the protection operation to allow protection nearer to the fault to operate first. While this is relatively straightforward for radial schemes, in networks, where the current paths can be quite different depending on operating and maintenance strategies, protection can be difficult to set and optimum settings are probably impossible to achieve. It is in these areas where protection software has become useful to manufacturers, consultants, and utilities. The very nature of protection schemes has changed from electromechanical devices, through electronic equivalents of the old devices, to highly sophisticated system analyzers. They are computers in their own right and thus can be developed almost entirely by computer analysis techniques. Other Uses for Load Flow Analysis It has already been demonstrated that load flow analysis is necessary in determining the economic operation of the power system and it can also be used in the production of capability charts. Many other types of analyses require load flow to be embedded in the program
As a follow on from the basic load flow analysis, where significant unbalanced load or unbalanced transmis sion causes problems, a three-phase load flow may be required to study their effects. These programs require each phase to be represented separately and mutual coupling between phases to be taken into account. Trans former winding connections must be correctly represented and the mutual coupling between transmission lines on the same tower or on the same right-of-way must also be included Motor starting can be evaluated using a transient stability program but in many cases this level of analysis unnecessary. The voltage dip associated with motor start up can be determined very precisely by a conventional load flow program with a motor starting module. Optimal power system operation requires the best use of resources subject to a number of constraints over any specified time period. The problem consists of minimizing a scalar objective function (normally a cost criterion) through the optimal control of a vector of control parameters. This is subject to the equality nstraints of the load flow equations, inequality constraints on the control parameters, and inequality con straints of dependent variables and dependent functions. The programs to do this analysis are usually referred to as optimal power flow(OPF)programs Often optimal operation conflicts with the security requirements of the system. Load flow studies are used assess security(security assessment). This can be viewed as two separate functions. First, there is a need to detect any operating limit violations through continuous monitoring of the branch flows and nodal voltages Second, there is a need to determine the effects of branch outages(contingency analysis). To reduce this to a manageable level, the list of contingencies is reduced by judicial elimination of most of the cases that are not expected to cause violations. From this the possible overloading of equipment can be forecast. The program should be designed to accommodate the condition where generation cannot meet the load because of network islanding. The conflicting requirements of system optimization and security require that they be considered together The more recent versions of OPF interface with contingency analysis and the computational requirement enormous Extensions to Transient Stability analysis Transient stability programs have been extended to include many other system components, including FACTS (flexible ac transmission systems)and dc converters. FACTS may be either shunt or branch devices. Shunt devices usually attempt to control busbar voltage by varying their shunt susceptance. The device is therefore relatively simple to implement in a time domain program Series devices may be associated with transformers. Stability improvement is achieved by injecting a quadrature component of voltage derived from the other two phases rather than by a tap changer which injects a direct component of voltage. Fast acting power electronics can inject either or a combination of both direct and quadrature voltage to help maintain voltage levels and improve stability margins Dc converters for HVdc links and rectifier loads have received much attention. The converter controls are very fast acting and therefore a quasi steady state(QSS)model can be considered accurate. That is, the model of the converter terminals contains no dynamic equations and in effect the link behaves as if it was in ste state for every time solution of the ac system. While this may be so some time after a fault has been remot during and just after a fault the converters may well suffer from commutation failure or fire through. These events cannot be predicted or modeled with a QSS model. In this case, an appropriate method of analysis to combine a state variable model of the converter, which can model the firing of the individual valves, with a conventional multi-machine transient stability program containing a QSS model. During the period of maxi- mum disturbance, the two models can operate together. Information about the overall system response is passed to the state variable model at regular intervals. Similarly the results from the detailed converter model are passed to the multi machine model overriding its own QSS model. As the disturbance reduces, the results from the two different converter models converge and it is then only necessary to run the computationally inexpensive QSS model within the multi machine transient stability program c 2000 by CRC Press LLC
© 2000 by CRC Press LLC As a follow on from the basic load flow analysis, where significant unbalanced load or unbalanced transmission causes problems, a three-phase load flow may be required to study their effects. These programs require each phase to be represented separately and mutual coupling between phases to be taken into account. Transformer winding connections must be correctly represented and the mutual coupling between transmission lines on the same tower or on the same right-of-way must also be included. Motor starting can be evaluated using a transient stability program but in many cases this level of analysis is unnecessary. The voltage dip associated with motor start up can be determined very precisely by a conventional load flow program with a motor starting module. Optimal power system operation requires the best use of resources subject to a number of constraints over any specified time period. The problem consists of minimizing a scalar objective function (normally a cost criterion) through the optimal control of a vector of control parameters. This is subject to the equality constraints of the load flow equations, inequality constraints on the control parameters, and inequality constraints of dependent variables and dependent functions. The programs to do this analysis are usually referred to as optimal power flow (OPF) programs. Often optimal operation conflicts with the security requirements of the system. Load flow studies are used to assess security (security assessment). This can be viewed as two separate functions. First, there is a need to detect any operating limit violations through continuous monitoring of the branch flows and nodal voltages. Second, there is a need to determine the effects of branch outages (contingency analysis). To reduce this to a manageable level, the list of contingencies is reduced by judicial elimination of most of the cases that are not expected to cause violations. From this the possible overloading of equipment can be forecast. The program should be designed to accommodate the condition where generation cannot meet the load because of network islanding. The conflicting requirements of system optimization and security require that they be considered together. The more recent versions of OPF interface with contingency analysis and the computational requirements are enormous. Extensions to Transient Stability Analysis Transient stability programs have been extended to include many other system components, including FACTS (flexible ac transmission systems) and dc converters. FACTS may be either shunt or branch devices. Shunt devices usually attempt to control busbar voltage by varying their shunt susceptance. The device is therefore relatively simple to implement in a time domain program. Series devices may be associated with transformers. Stability improvement is achieved by injecting a quadrature component of voltage derived from the other two phases rather than by a tap changer which injects a direct component of voltage. Fast acting power electronics can inject either or a combination of both direct and quadrature voltage to help maintain voltage levels and improve stability margins. Dc converters for HVdc links and rectifier loads have received much attention. The converter controls are very fast acting and therefore a quasi steady state (QSS) model can be considered accurate. That is, the model of the converter terminals contains no dynamic equations and in effect the link behaves as if it was in steady state for every time solution of the ac system. While this may be so some time after a fault has been removed, during and just after a fault the converters may well suffer from commutation failure or fire through. These events cannot be predicted or modeled with a QSS model. In this case, an appropriate method of analysis is to combine a state variable model of the converter, which can model the firing of the individual valves, with a conventional multi-machine transient stability program containing a QSS model. During the period of maximum disturbance, the two models can operate together. Information about the overall system response is passed to the state variable model at regular intervals. Similarly the results from the detailed converter model are passed to the multi machine model overriding its own QSS model. As the disturbance reduces, the results from the two different converter models converge and it is then only necessary to run the computationally inexpensive QSS model within the multi machine transient stability program
Voltage Collapse Steady state analysis of the problem of voltage instability and voltage collapse are often based on load flow analysis programs. However, time solutions can provide further insight into the problem A transient stability program can be extended to include induction machines which are associated with many of the voltage collapse problems. In these studies, it is the stability of the motors that are examined rather than the stability of the synchronous machines. The asynchronous nature of the induction machine means that rotor angle is not a concern, but instead the capability of the machines to recover after a fault has depressed the voltage and allowed the machines to slow down. The re-accelerating machines draw more reactive current which can hold the terminal voltage down below that necessary to allow recovery. Similarly starting a machine will depress the voltage which affects other induction machines which further lowers the voltage. However, voltage collapse can also be due to longer term problems. Transient stability programs then need to take into account controls that are usually ignored. These include automatic transformer tap adjustment and generator excitation limiters which control the long-term reactive power output to keep the field currents nin their rated values The equipment that can contribute to voltage collapse must also be more carefully modeled. Simple imped ance models for loads(P=PoVQ=Q V2)are no longer adequate. An improvement can be obtained by replacing the (mathematical) power 2 in the equations by more suitable values. Along with the induction machine models, the load characteristics can be further refined by including saturation effects. SCADA SCADA(Supervisory Control And Data Acquisition)has been an integral part of system control for many years. A control center now has much real time information available so that human and computer decisions about system operation can be made with a high degree of confidence. In order to achieve high quality input data, algorithms have been developed to estimate the state of a system based on the available on-line data ( state estimation). These methods are based on weighted least squares techniques to find the best state vector to fit the scatter of data. This becomes a major problem when conflicting information is received. However, as more data becomes available, the reliability of the estimate can be improved. ower Quality One form of poor power quality which has received a large amount of attention is the high level of harmonics that can exist and there are numerous harmonic analysis programs now available Recently, the harmonic levels of both currents and voltages have increased considerably due to the increasing use of non-linear loads such as arc furnaces, HVdc converters, FACTS equipment, dc motor drives, and ac motor speed control. Moreover, commercial sector loads now contain often unacceptable levels of harmon due to widespread use of equipment with rectifier-fed power supplies with capacitor output smoothing(e.g computer power supplies and fluorescent lighting). The need to conserve energy has resulted in energy efficient designs that exacerbate the generation of harmonics. Although each source only contributes a very small level of harmonics, due to their small power ratings, widespread use of small non-linear devices may create harmonic problems which are more difficult to remedy than one large harmonic source Harmonic analysis algorithms vary greatly in their algorithms and features; however, almost all use the frequency domain. The most common technique is the direct method (also known as current injection method Spectral analysis of the current waveform of the non-linear components is performed and entered into the program. The network data is used to assemble a system admittance matrix for each frequency of interest. Thi set of linear equations is solved for each frequency to determine the node voltages and, hence, current flow hroughout the system. This method assumes the non-linear component is an ideal harmonic current source. The next more advanced technique is to model the relationship between the harmonic currents injected by a component to its terminal voltage waveform. This then requires an iterative algorithm, which does require c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Voltage Collapse Steady state analysis of the problem of voltage instability and voltage collapse are often based on load flow analysis programs. However, time solutions can provide further insight into the problem. A transient stability program can be extended to include induction machines which are associated with many of the voltage collapse problems. In these studies, it is the stability of the motors that are examined rather than the stability of the synchronous machines. The asynchronous nature of the induction machine means that rotor angle is not a concern, but instead the capability of the machines to recover after a fault has depressed the voltage and allowed the machines to slow down. The re-accelerating machines draw more reactive current which can hold the terminal voltage down below that necessary to allow recovery. Similarly starting a machine will depress the voltage which affects other induction machines which further lowers the voltage. However, voltage collapse can also be due to longer term problems. Transient stability programs then need to take into account controls that are usually ignored. These include automatic transformer tap adjustment and generator excitation limiters which control the long-term reactive power output to keep the field currents within their rated values. The equipment that can contribute to voltage collapse must also be more carefully modeled. Simple impedance models for loads (P = PoV2 ; Q = QoV2) are no longer adequate. An improvement can be obtained by replacing the (mathematical) power 2 in the equations by more suitable values. Along with the induction machine models, the load characteristics can be further refined by including saturation effects. SCADA SCADA (Supervisory Control And Data Acquisition) has been an integral part of system control for many years. A control center now has much real time information available so that human and computer decisions about system operation can be made with a high degree of confidence. In order to achieve high quality input data, algorithms have been developed to estimate the state of a system based on the available on-line data (state estimation). These methods are based on weighted least squares techniques to find the best state vector to fit the scatter of data. This becomes a major problem when conflicting information is received. However, as more data becomes available, the reliability of the estimate can be improved. Power Quality One form of poor power quality which has received a large amount of attention is the high level of harmonics that can exist and there are numerous harmonic analysis programs now available. Recently, the harmonic levels of both currents and voltages have increased considerably due to the increasing use of non-linear loads such as arc furnaces, HVdc converters, FACTS equipment, dc motor drives, and ac motor speed control. Moreover, commercial sector loads now contain often unacceptable levels of harmonics due to widespread use of equipment with rectifier-fed power supplies with capacitor output smoothing (e.g., computer power supplies and fluorescent lighting). The need to conserve energy has resulted in energy efficient designs that exacerbate the generation of harmonics. Although each source only contributes a very small level of harmonics, due to their small power ratings, widespread use of small non-linear devices may create harmonic problems which are more difficult to remedy than one large harmonic source. Harmonic analysis algorithms vary greatly in their algorithms and features; however, almost all use the frequency domain. The most common technique is the direct method (also known as current injection method). Spectral analysis of the current waveform of the non-linear components is performed and entered into the program. The network data is used to assemble a system admittance matrix for each frequency of interest. This set of linear equations is solved for each frequency to determine the node voltages and, hence, current flow throughout the system. This method assumes the non-linear component is an ideal harmonic current source. The next more advanced technique is to model the relationship between the harmonic currents injected by a component to its terminal voltage waveform. This then requires an iterative algorithm, which does require
