UNIVERSITY OF NAIROBI DE-COUPLED LOAD FLOW STUDY METHOD

Transcription

1 UNIVERSITY OF NAIROBI SCHOOL OF ENGINEERING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING DE-COUPLED LOAD FLOW STUDY METHOD PROJECT INDEX: PRJ (71) BY KETER SAMSON KIPKIRUI F17/30052/2009 SUPERVISOR: DR.N.O. ABUNGU EXAMINER: Prof. MBUTHIA This Project report submitted in partial fulfillment of the Requirement for the award of the degree Bachelor of Science in Electrical and Electronic Engineering of the University Of Nairobi. Of SUBMITTED ON: 28 TH APRIL

2 DECLARATION OF ORIGINALITY NAME OF STUDENT: KETER SAMSON KIPKIRUI REGISTRATION NUMBER: F17/30052/2009 COLLEGE: Architecture and Engineering FACULTY/SCHOOL/INSTITUTE: Engineering DEPARTMENT: Electrical and Information Engineering COURSE NAME: Bachelor of Science in Electrical and Electronic Engineering TITLE OF WORK: DE-COUPLED LOAD FLOW STUDY METHOD 1) I understand what plagiarism is and I am aware of the university policy in this regard. 2) I declare that this final year project report is my original work and has not been submitted elsewhere for examination, award of a degree or publication. Where other people s work or my own work has been used, this has properly been acknowledged and referenced in accordance with the University of Nairobi s requirements. 3) I have not sought or used the services of any professional agencies to produce this work. 4) I have not allowed, and shall not allow anyone to copy my work with the intention of passing it off as his/her own work. 5) I understand that any false claim in respect of this work shall result in disciplinary action, in accordance with University anti-plagiarism policy. Signature: Date: 2

3 DEDICATION To my family for the endless support and bringing the best of me at early age i

4 ACKNOWLEDGEMENTS I would like to express my deep gratitude to Dr. N. O. Abungu my project supervisor, for his patient guidance, enthusiastic encouragement and useful critiques of this research work. I would like to express my great appreciation to Mr. Peter Musau for his valuable and constructive suggestion during planning and development of this project work. His willingness to give his time so generously and keeping my progress on schedule has been very much appreciated. Special thanks to the Dean-Faculty of Engineering; Chairman-Department of Electrical and Information Engineering and all my lecturers at the University of Nairobi for their support which contributed greatly to the provision of knowledge as well as the completion of this project Sincere thanks should be given to all my friends and especially Mark Musembi and Erick Mbugua for the special insights and valuable ideas that help me in understanding load flow problem and MATLAB programming, May God s blessing always be with them. I thank God for His guidance, faithfulness throughout my life as a student and for giving me peace throughout my final academic year. I also extend my appreciation to my parents for their continued support and encouragement throughout my studies. ii

5 DECLARATION AND CERTIFICATION This BSc. work is my original work and has not been presented for a degree award in this or any other university... KETER SAMSON KIPKIRUI F17/30052/2009 This report has been submitted to the Department of Electrical and Information Eng., University of Nairobi with my approval as supervisor: Dr. Nicodemus Abungu Odero Date: iii

6 LIST OF ABBREVIATIONS DLF FDLF GS IEEE MATLAB MVA MVAR MW: NR P.U PSAT P-V P-Q Decoupled Load Flow Fast Decoupled Load Flow Gauss-Siedel Load Flow Method Institute of Electrical and Electronics Engineering Matrix Laboratory Mega Voltage Ampere Reactive Power in Mega watts Real power in Mega Watts Newton Raphson Method Per Unit Power System Analysis Toolbox Voltage Controlled Bus Load Bus iv

7 TABLE OF CONTENT Contents DECLARATION OF ORIGINALITY... 2 DEDICATION... i ACKNOWLEDGEMENTS... ii DECLARATION AND CERTIFICATION... iii LIST OF ABBREVIATIONS... iv TABLE OF CONTENT... v LIST OF FIGURES... vii LIST OF TABLES... viii ABSTRACT... ix CHAPTER INTRODUCTION Load flow studies Constraints on load flow solution Solution to Load flow survey of earlier work Problem statement Objectives Organization of the Report CHAPTER LITERATURE REVIEW Load flow study Importance of load flow studies Load flow Analysis Methods of load flow analysis Load Flow Methods Convergence procedure Acceleration of convergence Algorithm modification when PV Buses are also present Comparison of Load Flow Methods CHAPTER METHODOLOGY Computational procedure for decoupled load flow method [1] Design Flow Chart IEEE 14 Bus Test Network Load Flow Data Assembling load flow MATLAB data CHAPTER RESULTS, ANALYSIS AND DISCUSSION Results Analysis, Discussion and Validation Performance Analysis Comparative Results Charts and Graphs CHAPTER CONCLUSION AND RECOMMENDATION Conclusion Recommendations for Future Work v

8 REFERENCES APPENDIX PROGRAM LISTING vi

9 LIST OF FIGURES FIGURE 2.1: Π LINE FLOW REPRESENTATION FIGURE 3.1: DECOUPLED LOAD FLOW CHART-[1] FIGURE 3.2: IEEE 14 BUS SYSTEM [7] FIGURE 3.3: DIAGRAM OF A TWO-WINDING TRANSFORMER CIRCUIT [16] FIGURE 3.4:ONE LINE DIAGRAM FOR 14 BUS TEST SYSTEM FIGURE 4.1: NEWTON RAPHSON VOLTAGE PROFILE FIGURE 4.2: DECOUPLED LOAD FLOW VOLTAGE PROFILE FIGURE 4.3: ANGLE PROFILE FOR DLF AND NR FIGURE 4.4: DLF REAL AND REACTIVE POWER FLOW FIGURE 4.5: DLF AND NR POWER FLOW FIGURE 4.6: DLF LINE LOSSES FIGURE 4.5:S LINE LOSSES vii

10 LIST OF TABLES TABLE 2.1: SUMMARY OF BUS VARIABLES TABLE 3.1: BUS DATA TABLE 3.2: LINE DATA TABLE 4.1: BUS VOLTAGES, POWER GENERATED AND LOAD AFTER CONVERGENCE OF DECOUPLED LOAD FLOW TABLE 4.2: REAL AND REACTIVE POWER FLOW OVER DIFFERENT LINES AND LOSSES TABLE 4.3: VOLTAGE, ANGLE, GENERATION AND LOAD POWER COMPARISON BETWEEN DLF AND NR TABLE 4.4: REAL AND COMPLEX BUS POWER COMPARISON FOR THE DLF AND NR METHOD.. 54 TABLE4.5: PSAT SIMULATED RESULTS TABLE 4.6: DATA USED TO SHOW RELATIVE ACCURACY OF THE RESULTS OF EACH METHOD.. 58 viii

11 ABSTRACT Load flow study is the analysis of a network under steady state operation subjected to inequality constraints in which the system operates. Load flow analysis is the backbone of power system analysis and design. They are necessary for planning, operation, economic scheduling and exchange of power between utilities. The principal information of power flow analysis is to find the magnitude and phase angle of voltage at each bus and the real and reactive power flows in each transmission lines. Therefore, load flow analysis is an importance tool involving numerical analysis applied to a power system. In this analysis, iterative techniques are used because there are no known analytical method to solve the load flow problem. This iterative techniques includes; Gauss Siedel, Newton Raphson, Decoupled method and Fast Decoupled method. Load flow analysis is difficult and time consuming to perform by hand. The Decoupled load flow method in detail; Formulation of static load flow equations and computational algorithm is clearly discussed. The objective of this project is to develop a load flow program based on Decoupled method that will ease the analysis of load flow problem. MATLAB software was used as a programming platform. The program was run on an IEEE 14-bus system test network and the results compared with those from other methods, i.e. Newton Raphson method and finally, validated by simulated results from Power System Analysis (PSAT) simulation software. The load flow results obtained were analyzed and discussed. Both the decoupled load flow and Newton-Raphson methods gave similar results. However, the decoupled method converged faster than the Newton-Raphson method. The bus voltage magnitudes, angles of each bus along with power generated and consumed at each bus has been tabulated in Table 4.1 and 4.2. It is seen from this tables that the total power generated is MW whereas the total power consumed is 259 MW. This indicates that there is a line loss of about MW for all the lines put together. For Newton Raphson method, the total power generated were MW whereas the power demand were 259 MW thus a loss of MW. The power loss as obtained from PSAT was MW. The results indeed, compares very well. Therefore the decoupled load flow method was verified to be effective and reliable method of obtaining optimum solution for a load flow problem. ix

12 CHAPTER 1 INTRODUCTION 1.1 Load flow studies Load flow solution is a solution of the network under steady state operation subjected to certain inequality constraints under which the system operates. Load flow studies are important in planning and designing future expansion of power systems. The study gives steady state solutions of the voltages at all the buses, for a particular load condition. Different steady state solutions can be obtained, for different operating conditions, to help in planning, design and operation of the power system [1],[9]. Generally, load flow studies are limited to the transmission system, which involves bulk power transmission. The load at the buses is assumed to be known. Load flow studies throw light on some of the important aspects of the system operation, such as: violation of voltage magnitudes at the buses, overloading of lines, overloading of generators, stability margin reduction, indicated by power angle differences between buses linked by a line, effect of contingencies like line voltages, emergency shutdown of generators, etc. Load flow studies are required for deciding the economic operation of the power system. They are also required in transient stability studies. Hence, load flow studies play a vital role in power system studies. Thus the load flow problem consists of finding the power flows (real and reactive) and voltages of a network for given bus conditions. At each bus, there are four quantities of interest to be known for further analysis: the real and reactive power, the voltage magnitude and its phase angle. Because of the nonlinearity of the algebraic equations, describing the given power system, their solutions are obviously, based on the iterative methods only. 1.2 Constraints on load flow solution The constraints placed on the load flow solutions could be: The Kirchhoff s relations holding well, Capability limits of reactive power sources, Tap-setting range of tap-changing transformers, Specified power interchange between interconnected systems, Selection of initial values, acceleration factor and convergence limit. In load flow analysis, an electrical power system network consists of hundreds of buses and branches with impedances specified in per unit on a common MVA base. Performance of power system network both in normal operating conditions and under fault should be continuously analyzed [3]. For optimal operation of an electrical power system requires that; 10

13 Generation must supply the load plus losses, The bus voltage magnitudes must remain close to rated values, generators must operate within specified real and reactive power limits and that transmission lines and transformers should not be overloaded for long periods. [2]; Load flow study covers a wide range of time constants which include steady state and transient conditions. The symmetrical steady state operation of an electrical power system is the most important mode of operation since it ensures supply of real and reactive power demanded by various loads, the frequency and bus voltages being maintained within specified tolerances and with optimum economy [4]. Load flow deals with the flow of electrical power from one or more sources to loads consuming energy through available paths as commonly shown in a one line diagram [3]. Electric energy flow in a network divides among branches according to their respective impedances until a voltage balance is reached in accordance to Kirchhoff s Laws [5]. The flow will shift anytime the circuit configuration is changed or modified, generation is shifted or load requirements changes. 1.3 Solution to Load flow Load flow study is the determination of steady-state conditions of a power system for a specified power generation and load demand. The load flow problem is the computation of voltage magnitude and phase angle at each bus and also active and reactive flows in a power system. Load flow analysis is performed extensively both for system planning purposes, to analyze alternative plans of future systems operation and to evaluate different operating conditions of existing systems. In static contingency analysis, load flow study is used to assess the effect of branch or generator outages. In transfer capability analysis, repetitive power flow analysis is performed to calculate the power transfer limits. In load flow analysis, it is normal to assume that the system is balanced and that the network is composed of constant, linear, lumped-parameter branches. In the most basic form of the power flow, transformer taps are assumed to be fixed [1]. This assumption is relaxed in commercial load flow. Therefore, nodal analysis is generally used to describe the network. However, because the injection and demand at bus bars is generally specified in terms of real and reactive power, the overall problem is nonlinear. Accordingly, the load flow problem is a set of simultaneous nonlinear algebraic equations. Numerical techniques are required to solve this set of equations [2]. 11

14 Traditional solutions of the load-flow problems follow an iterative process by assigning estimated values to the unknown bus voltages and angles and calculating a new value for each bus voltage and angle from the estimated values at the other buses, the real power specified, and the specified reactive power or voltage magnitude given in some buses. A new set of values for voltage and angle are thus obtained for each bus and still used to calculate another set of bus voltages and angles in a sequential algorithm. The iterative process is repeated until the changes at each bus are less than the specified tolerance value, ( <ε<0.0001). Load flow analysis has become in recent years one of the major areas of research in electrical engineering. However load flow study is a difficult task. First, the load distribution network is a complex system and exhibits lots computational procedure hence time consuming. Secondly, there are losses in electrical network distribution hence quantification and minimization of losses is important because it will determine the economic operation of the power system. 1.4 Survey of Earlier Work Over the years, the direction of research has shifted, replacing old approaches with newer and more efficient ones. Apparently due to their limited success, a number of old approaches seem to no longer in use. These include such methods as Runge-Kutta, Iwamoto, and Ward and Hale methods load-flow study methods. There is also considerably less emphasis on methods such as AC and DC Decoupled methods, Gauss-Seidel load-flow study. The rapidly increasing power of the personal computer is making it possible to apply more complicated solution techniques methods based on few and faster iterations technique such as Newton-Raphson (NR), Decoupled load flow and Fast Decoupled Load Flow methods [6, 10]. For large scale power transmission system, decoupled load flow has been found to be an alternative strategy for improving the computational efficiency and reducing computer storage requirements. This method uses an approximate version of NR procedure. The DLF requires more iterations than NR method, but, requires considerably less time per iterations and thus power flow solution is obtain rapidly. This technique is very useful tool in contingency analysis where numerous outages are to be simulated or when a power flow solution is required for line control. Fast Decoupled load flow method is a variation on Newton-Raphson that exploits the approximate decoupling of active and reactive flows in well-behaved power networks, and additionally fixes the value of the Jacobean during the iteration in order to avoid costly matrix decompositions. It is achieved by only inverting the Jacobean matrix once within its algorithm. It has 3 assumptions. First, the conductance between the buses is zero. Second, the magnitude 12

15 of the bus voltage is one per unit. Thirdly, the sine of phases between buses is zero, whereas the cosine of phases is 1. In reality the decoupled load flow can return the answer within seconds, whereas a Newton Raphson method takes much longer to return an answer. The decoupled load flow is a computer-driven method, in the sense that it is not necessary for the researcher to calculate manually each and every computational procedures in order to arrive at the final solution of the load flow problem instead a computer based algorithm can be used to solve large and complex load flow problems with an ease. 1.5 Problem Statement The purpose of this work is to understand the theory of load flow analysis and to develop a reliable and effective program based on Decoupled Load Flow study method. MATLAB 7.6(r2013b) software was used as a programming platform and PSAT (Power System Analysis Toolbox) as a validating tool. The proposed Decoupled load flow method should accurately calculate and analyze a well-conditioned load flow study with minimal losses on the buses, branches and the minimal number of iteration required for convergence. The effectiveness of the decoupled program is tested on IEEE 14 bus test network to give reliable results. To achieve this relevant input variables are to be identified, formulated and gathered from the load flow data. 1.4 Objectives The objective of the research can be stated as follows: To understand Decoupled load flow study method and use it to find the optimal solution for load flow of a 14 bus test network. To develop a decoupled load flow program using MATLAB as programming platform 13

16 1.6 Organization of the Report This project report has been arranged into five chapters. In chapter 1, general introduction to load flow is made, it also addresses the load flow constraints, statement of problem and objectives. Finally, organization of the report is also presented in this chapter In Chapter 2, a literature review of electrical power flow study has been conducted followed by the load flow study methods which are the Gauss Siedel method, Newton-Raphson method, Decoupled load flow and finally, fast decoupled load flow. Each subject has been independently broken down and addressed separately in detail. Further, the Decoupled load flow was expounded on how it is used to solve load flow problem. In Chapter 3, the algorithm and the flow chart of decoupled load flow was discussed. The IEEE test network and data of the 14 bus network was featured in this chapter. The Validating tool, Power Analysis Tool Box (PSAT) is also featured. Data from the field as well as other sources are introduced, analyzed, interpreted and validated. The data has been plotted using MATLAB for easier analysis and validated by (PSAT. The selection of the data was as a result of research n determining the most suitable input variables for the decoupled load flow method. The step by step process of calculation and simulation of decoupled load flow study. In Chapter 4, the results are discussed and analyzed giving brief explanations of what can be drawn from the output of the Decoupled load flow method which includes number of iteration required for solution to converge and its level of accuracy in making a load flow analysis. In Chapter 5, conclusion and recommendation of the report. Recommendation of the report by giving a review of the study in the preceding chapters and identifies some problems for future work in this area. 14

17 CHAPTER 2 LITERATURE REVIEW 2.1 Load flow study In power engineering, the power flow study, also known as load-flow study, is an analysis of the voltages, currents and power flows in a power system under the steady conditions. Load flow is an important tool involving numerical analysis applied to a power system [9]. It usually uses simplified notation such as a one-line diagram and per-unit system, and focuses on various components of Alternating Current AC power i.e.: voltages, voltage angles, real power and reactive power. The study is based on normal operation of a power system and operating under balanced conditions [11]. Conducting a load flow study helps ensure that the power system is adequately designed to satisfy the required performance criteria. A properly designed system helps contain initial capital investment and future operating costs. It also helps develop equipment specification guidelines, optimize circuit usage, minimize KW and KVAR losses and identify transformer tap settings.. The principal information obtain from a power flow study is the magnitude and the phase angle of the voltage at each bus and the real and reactive power flowing in each line and line loses [9] This information obtained is important for the continuous monitoring of the current state of the system and for analyzing the effectiveness of alternative plans for future system expansion to meet increased load demand [1], [6]. When the current state of the system is monitored and found to be unsatisfactory e.g. if voltage at bus is too low, then a control action is taken to correct the voltage e.g. put HVDC or use compensation. The load continues to increase and hence the system needs to be expanded regularly. For the continuously increasing load demand plans has to be made to match with generation facility. 2.2 Importance of load flow studies Load flow studies are performed in major areas of power system development and operation because of the following rationale; 1. Planning: Necessary for planning, economic scheduling, and control of an existing system as well as planning its future expansion. This is the future development of a system in which load flows are used to study the effects and feasibility of changes in network configuration such as the removal or addition of lines, new generation units, or increased loads due to a growing consumer demand. Load flow is central to the 15

18 stability analysis performed on the proposed system. System security is also determined and multiple load flows are performed -to evaluate contingencies. 2. Operation and Control, the configuration of the network changes due to loss of generation units or transmission circuits, or the change in demand of consumer load. Load flow studies are used to evaluate these changes and compensate for high or low bus voltages by the addition or removal of static capacitors, the altering of ratios of transformers or by changing the reactive power of synchronous condensers or generator units. Stability analysis and system security Studies are also performed. 3. The Economic Operation: As loads change throughout the day there is a need to determine the best generating pattern to minimize costs of operation and provide the best voltage regulation. Load flow is used to obtain the optimum settings of transformer taps, shunt capacitance and unit generation; subject to the operational constraints of equipment in the system 4. Load-flow studies are performed to determine the steady-state operation of an electric power system. It calculates the voltage drop on each feeder, the voltage at each bus, and the power flow in all branch and feeder circuits. 5. Determine if system voltages remain within specified limits under various contingency conditions, and whether equipment such as transformers and conductors are overloaded. 6. Load-flow studies are often used to identify the need for additional generation, capacitive, or inductive VAR support, or the placement of capacitors and/or reactors to maintain system voltages within specified limits. 7. Losses in each branch and total system power losses are also calculated. 2.3 Load flow Analysis The goal of load flow analysis 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. Load flow uses a mathematical algorithm of successive approximation by iteration, or the repeated application of calculation steps on the non-linear load flow equations [4]. These steps represent a process of trial and error that starts with assuming one array of numbers for the entire system, comparing the relationships among the numbers to the 16

19 laws of power flow equations, and then repeatedly adjusting the numbers until the entire array is consistent with both physical law and the conditions stipulated by the user. In practice, this is a computer program to which the operator gives certain input information about the power system, and which then provides output that completes the picture of what is happening in the system. There are variations on what types of information are chosen as input and output. Typically the input data is divided into: Line data, Bus data, Generator data, Transformer data and Load data. This data is included with every load flow output file in order to document the system, load configuration that the solution applies for. The load flow study have a predefined set of criteria that the system evaluated must meet. These criteria are not exception of: Voltage criteria in which bus voltages must be within their limits. Power flows on cables and transformers must be within equipment ratings. Generator reactive outputs must be within the limits defined by the generator capability curve Types of Variables Basically load-flow analysis deals with known real and reactive power flows at each bus, and those voltage magnitudes that are explicitly known, and from this information calculating the remaining voltage magnitudes and all the voltage angles is made possible [3], [4], and [9]. Owing to the nonlinear nature of the load-flow problem, it may be impossible to find one unique solution because more than one answer is mathematically consistent with the given configuration. However, it is usually straightforward in such cases to identify the true solution among the mathematical possibilities based on physical plausibility and common sense. Conversely, there may be no solution at all because the given information was hypothetical and does not correspond to any situation that is physically possible. Still, it is true in principle and most important for a general conceptual understanding that two variables per node are needed to determine everything that is happening in the system. In practice, current is not known at all; the currents through the various circuit branches turn out to be the last thing that are calculated once the load-flow analysis has been completed. Voltage is known explicitly for some buses but not for others. More typically, what is known is the amount of power going into or out of a bus. Load-flow analysis consists of taking all the known real and reactive power flows at each bus, and those voltage magnitudes that are explicitly known, and from this information calculating the remaining voltage magnitudes and all the voltage angles, this is the hard part. The easy part 17

20 is to calculate the current magnitudes and angles from the voltages, knowing how to calculate real and reactive power from voltage and current, power is basically the product of voltage and current, and the relative phase angle between voltage and current determines the respective contributions of real and reactive power. Conversely, one can deduce voltage or current magnitude and angle if real and reactive power is given, but it is far more difficult to work out mathematically in this direction. This is because each value of real and reactive power would be consistent with many different possible combinations of voltages and currents. In order to choose the correct ones, one has to check each node in relation to its neighboring nodes in the circuit and find a set of voltages and currents that are consistent all the way around the system Load flow problem formulation The complex power injected by the source into the h bus of a power system is [1, 9]. = + = = 1, 2,. (2.1) Where the voltage at the h bus and with respect to ground and is the source current injected into the bus. The load flow problem is handled more conveniently by use of rather than. Therefore, taking the complex conjugate of Eq. (2.1), hence = ; = 1, 2, (2.2b) Equating real and imaginary parts ( ) = (2.3 ) ( ) = (2.3 ) In polar form Real and reactive powers can be expressed as = = ( ) = cos( + ) ; = 1, 2, (2.4) ( ) = sin( + ) ; = 1, 2, (2.5) 18

21 Equations (2.4) and (2.5) represent 2n power flow equations at n buses of a power system (n real power flow equations and n reactive power flow equations). Each bus is characterized by four variables:,, and. Resulting in a total of 4n variables. Equations (2.4) and (2.5) can be solved for 2n variables if are specified. Practical considerations allow a power system analyst to fix a priori two variables at each bus. The solution for the remaining 2n bus variables is rendered difficult by the fact that Equations. (2.4) and (2.5) are non-linear algebraic equations (buses voltages are involved in product form and sine and cosine terms are present) and therefore, explicit solution is not possible. Solution can only be obtained by iterative numerical techniques. Depending upon which two variables are specified a priori, the buses are classified into three categories Types of Buses Load flow analysis buses are represented as nodes, but there are many types of buses (typically 3) which should be known for better understanding. [1].The three main types of buses are [9, 5]: 1. Load buses. This is a bus without any generators connected to it, both Power generated and reactive power generated are zero and the real power ;and reactive power drawn from the system by the load (negative inputs into the system) are known from historical record, load forecast, or measurement. Quite often in practice only the real power is known and the reactive power is based on an assumed power factor such as 0.85 or higher. 2. Voltage-controlled buses (P-V). Any bus of the system at which the voltage magnitude is kept constant is said to be voltage controlled. At each bus to which there is a generator connected, the megawatt generation can be controlled by adjusting the prime mover, and the voltage magnitude can be controlled by adjusting the generator excitation. Therefore, at each generator bus the power generated and voltage magnitude are specified. With Power demand of the bus also known, we can define mismatch.generator reactive power required to support the scheduled voltage magnitude cannot be known in advance, and so reactive power mismatch is not defined. Therefore, at a generator bus voltage angle is the unknown quantity to be determined [9]. 3. Slack bus. The bus is also known as swing or reference bus. The known qualities are the voltage magnitude at the bus and the voltage angle. The voltage of the slack bus serves as reference for the angles of all the other bus voltages where the usual practice is set. The unknown quantities are the active and reactive power and at this bus and mismatches are 19

22 therefore not defined for the slack bus [9], [1].The slack bus is usually designated as bus 1 and there is only one type of this bus in a power system Need for a slack bus Unlike the other two buses which represent physical systems conditions, this bus is more a mathematical requirement. It is needed to provide a reference angle to which all the other angles are referred [2]. Also in a load flow study active and reactive power cannot be fixed a priori at all the buses as the net complex power flow into the network is not known in advance. This is because the system power loses are unknown till the load flow solution is completed [9]. In order that the variations in real and reactive power at the slack bus during the interactive process are a small percentage of the generating capacity, the slack bus is normally selected as the bus connected with the largest generating station [1].. Realpower = Total Total generation generation load = I R (2.6 ).. = ( 2.6 ) Real power losses are loses in the transmission lines and transformers of the network. Individual currents in various transmission lines of the network cannot be calculated until after the voltage magnitude and angle are known at all the buses of the system and hence is initially unknown. Accounts for the combined MVARS associated with line charging, shunt capacitors and reactors at buses and the losses in the series reactance of the transmission lines. It is given by the difference between the total MVARS supplied by the generator at the buses and the MVARS received by the loads [9]. After load flow problem has been solved the difference (slack) between the total specified power going into the system at lay the other buses and the total output power plus losses are assigned to the slack bus [9]. 20

23 Table 2.1: Summary of bus variables Bus type Specified variables Unknown variables Slack or reference bus,, Generator or PV bus,, Load or PQ bus,, Variable types and Limits Variable types Control variables (excepting slack bus), or Non-control variables and State variables and _ Variable limits (i) Voltage magnitude must satisfy the inequality (2.7) The power system equipment is designed to operate at fixed voltages with allowable variations of ± (5 10)% of the rated values [7]. (ii) Certain of the (state variables) must satisfy the inequality constraint of Power angle. (2.8) This constraint limits the maximum permissible power angle of transmission line connecting buses and and is imposed by considerations of system stability [1]. (iii)owing to physical limitations of P and Q generation sources, and are Constrained as follows Power limits:,, (2.9),, (2.10) Power Balance Equations Power balance equations it is, of course obvious that the total generation of real and reactive power must equal the total load demand plus losses, i.e. [4, 10]. = + (2.11) = + (2.12) 21

24 Where and are system real and reactive power loss, respectively. This leads to optimal sharing of active and reactive power generation between sources. Once and V are known, the voltage angle and magnitude at every bus, it can be easy to find the current through every transmission link; it becomes a simple matter of applying law to each individual link. (In fact, these currents have to be found simultaneously in order to compute the line losses, so that by the time the program announces s and V s, all the hard work is done.) Depending on how the output of a load-flow program is formatted, it may state only the basic output variables, as in it may explicitly state the currents for all transmission links in amperes, or it may express the flow on each transmission link in terms of an amount of real and reactive power owing, in megawatts (MW) and (MVAR) Static load flow solution The following assumptions and approximations are made in the load flow Esq. (2.4) and (2.5). i. Line resistances being small are neglected (shunt conductance of overhead lines is always negligible), i.e. PL, the active power loss of the system is zero. Thus in Esq. (2.4) and (2.5) 90 and 90. ii. ( ) Is small (< /6) so that sin( ) ( ). This is justified from considerations of stability. iii. All buses other than the slack bus (numbered as bus 1) are PV buses, i.e. voltage magnitudes at all the buses including the slack bus are specified. Equations (2.4) and (2.5) then reduce to = ( ) ; = 1, 2, (2.13) = cos( ) + ; = 1, 2, (2.14) Since s are specified, Eq. (2.13) represents a set of linear algebraic equations in s which are (n-1) in number as is specified at the slack bus ( = 0). Nth equation corresponding to slack bus (n=1) is redundant as the real power injected at this bus is now fully specified as Equations (2.13) can be solved explicitly (non-iteratively) for,,,, which when substituted in Eq. (2.14), yields s, the reactive power bus injecting. It may be noted that the assumptions have decoupled Esq. (2.13) and (2.14) so that these need not be solved simultaneously but can be solved sequentially [solution of Eq. (2.14) follows immediately upon 22

25 simultaneous solution of Eq. (2.14)]. Since the solution is non-iterative and the dimension is reduced to (n-1) from 2n, it is computational highly economical General building rules of YBUS 1 Self-admittance of node, equals the algebraic sum of all the admittances connected to node. 2 Mutual admittance between nodes and,, equals the negative sum of all admittances connecting nodes and k. 3 = Characteristics of YBUS 1. It is symmetric 2. It is very sparse (>90% for more than 100 buses) 2.4 Methods of load flow analysis The numerical analysis involving the solution of algebraic simultaneous equations forms the basis for solution of the performance equations in computer aided electrical power system analyses, such as during linear graph analysis, load flow analysis (nonlinear equations), transient stability studies (differential equations), etc. Hence, it is necessary to review the general forms of the various solution methods with respect to all forms of equations as under. There are various methods in which the load flows can be done. Some of them include Gauss- Seidel, Newton Raphson, Decoupled load flow Fast decoupled load flow and various other novel methods are being proposed. In this project we made use of the decoupled load flow method which is one of the most basic methods of load flow analysis introduced in power system analysis. The more successful methods of load flow solution are based on the admittance matrix [y] representation of a system. The advantages gained are ease of problem and data preparation and changes made to the system do not involve the recalculation of all network elements. The admittance matrix is sparse for a practical power system, i.e. it has only a few non-zero elements for large systems. By contrast the impedance matrix [Z] of a system (which is the inverse of the admittance matrix) is full, and changes in system configuration affect the whole of the matrix. The first practical digital solution methods for load flow were the Y matrix--iterative methods, these were suitable because of the low storage requirements, but had the disadvantage of converging slimly or not at all. Z matrix methods were developed which overcame the reliability problem but a sacrifice was made of storage and speed with large systems. The Newton-Raphson method was developed this time and was found to have very strong 23

26 convergence. The current problems faced in the development of load flow are: an ever increasing size of systems to be solved, on-line applications for automatic control, and system optimization. Newer and modified methods of load flow have been developed to overcome these problems Properties of load flow solution method. High computational speed. This is especially important when dealing with large systems, real time applications (on-line), multiple case load flows such as in system security assessment, and also in interactive applications. Low computer storage. This is important for large systems and in the use of computers with small core storage availability, e.g. mini-computers for on-line application. Reliability of solution. It is necessary that a solution be obtained for ill-conditioned problems, in outage studies and for real time applications. Versatility. An ability on the part of load flow to handle conventional and special features (e.g. the adjustment of tap ratios on transformers; different representations of power system apparatus), and its suitability for incorporation into more complicated processes. Simplicity. The ease of coding a computer program of the load flow algorithm The type of solution required from a load flow also determines the method used: accurate or approximate unadjusted or adjusted offline or on line single case or multiple cases 2.5 Load Flow Methods Gauss-Seidel Method The Gauss-Siedel (GS) method is an iterative algorithm for solving a set of non-linear algebraic equations [6, 1]. To start with, a solution vector is assumed, based on guidance from practical experience in a physical situation. One of the equations is the used to obtain the revised value of a particular variable by substituting in it the present values of the remaining values. The solution vector is immediately updated in respect of these variables. The process is then repeated for all the variables thereby completing one iteration. The iterative process is repeated till the solution vector converges within prescribed accuracy. The convergence is quite sensitive to the starting values assumed. Fortunately, in load flow study a starting vector close to the final solution can be easily identified with previous experience 24

27 To explain how the GS method is applied to obtain the load flow solution, let it be assumed that all the buses other than the slack bus are PQ buses. We shall see later that the method can be easily adopted to include PV buses as well. The slack bus voltage being specified, there are (n-1) bus voltage starting values of whose magnitudes and angles are assumed. These values are then updated through an iterative process. During the course of iteration, the revised voltage at the h bus is obtained as follows: From equation (2.27) = ( ) (2.15) = 1 Substituting for from equation (2.38) into (2.39) (2.16) = 1 ; = 2, 3,, (2.17) The voltages substituted in the right hand side of Eq. (2.17) are the most recently calculated (updated) values for the corresponding buses. During each iteration voltages at buses =1, 2, 3 n are sequentially updated through use of Eq. (2.17). V1, the slack bus voltage being fixed is not required to be updated. Iterations are repeated till no bus voltage magnitude changes by more than a prescribed value during iteration. The computation process is then said to converge to a solution. If instead of updating voltages at every step of iteration updating is carried out at the end of a complete iteration, the process is known as the Gauss iterative method. It is much slower to converge and may sometimes fail to do so Algorithm for load flow solution Presently we shall continue to consider the case where all the buses other than the slack are PQ buses. The steps of a computational algorithm are given below: 1. With the load profile known at each bus i.e.p, Q are known, allocate e P and Q to all generating stations. While active and reactive generations are allocated to the slack bus, these are permitted to vary during iterative computation. This is necessary as voltage magnitude and angle are specified at this bus (only two variables can be specified at any bus) With this step, bus injections (P + jq) are known at all buses other than the slack bus. 25

28 2. Assembly of bus admittance matrix YBUS: with the line and shunt admittance data stored in the computer, YBUS is assembled by using the rule for self and mutual admittances. Alternatively YBUS is assembled using Eq. (2.4), where input data is the form of primitive matrix Y and singular connection matrix A. 3. Iterative computation of bus voltages (V ;i= 2, 3., n): to start the iterations a set of initial voltage values is assumed. Since, in a power system the voltage is not too wide, it is normal practices to use a flat voltage start, i.e., initially all voltages are set to (1 + j0) except the voltage of the slack bus which is fixed. It should be noted that (n 1) equation (2.40) complex numbers are to be solved iteratively for finding(n 1) complex voltagesv, V,., V. If complex number operation are not available in computer, Equation (2.40) can be converted into 2(n 1) equations in real unknowns (e, f or V, δ ) by writing V = e + jf = V e (2.18) A significant reduction in the computer time can be achieved by performing in advance all the arithmetic operations that do not change with iterations. Define = = 2, 3,.., (2.19) = = 2, 3,.., ; (2.20) Now for the ( = 2, 3,.., ; + 1) h iteration, the voltage Eq. (2.17) becomes ( ) = ( ( ) ) ( ) ( ) = 2, 3,., (2.21) The iterative process is continued till the change in magnitude of bus voltage, ( ) = ( ) ( ) < ; = 2, 3,., (2.22) 4 Computation of slack bus power: substitution of all bus voltages computed in step 3 along with V yieldss = P jq. 5 Computation of line flows and line losses: this is the last step in the load flow analysis wherein the power flows on the various lines of the network are computed [1, 4]. Consider the lines connecting buses and k. The line and the transformers at each end can be represented by a circuit with series admittance y and two shunt admittances y and y as shown in Fig (2) 26

29 Bus Bus k Figure 2.1: π line flow representation The current field fed by bus into the line can be expressed as = + (2.23) = ( ) (2.24) = (2.25) From Eqns. (2.23), (2.24) and (2.25), we get, = ( ) + (2.26) The power fed into the line from bus is: = + (2.27) + = Using Eqns. (2.26) and (2.28), we get + = [( ) + ( )] + = ( ) + = ( ) + ( ) (2.28) = + ( 2.29) Similarly, power fed into the line from bus k is = + (2.30) Now = = (2.32) From Eqns. (2.30) and (2.32), we get = + + (2.33) = <, = <, = < = = [ cos + cos( + ) 27

30 sin sin( + ) (2.34) = cos + cos( + ) (2.35) = sin sin( + ) (2.36) Similarly power flows from k to can be written as: = cos + cos( + ) (2.37) = sin sin( + ) (2.38) Now real power loss in the line ( ) is the sum of the real power flows determined from Eqn. (2.35) and (2.37) = + = cos + cos( + ) cos + cos( + ) = ( + ) cos + [cos{ ( )} + cos{ + ( )}] = ( + ) cos + 2 cos cos( ) = [2 cos( ) ] cos (2.39) Let = + = cos = sin = [ 2 cos( ) ] (2.40) Reactive power loss in the line ( ) is the sum of the reactive power flows determined from Equations (2.36) and (2.38), i.e. = + = sin sin( + ) + sin sin( + ) = ( + ) [sin( + ) + sin( + )] + = ( + ) 2 cos( ) + = [ + 2 cos( )] + (2.41) The power loss in the ( ) h line is the sum of the power flows determined from equation (2.40) and (2.41). Total transmission loss can be computed by summing all the line flows ( + ) for all 28

31 ,. it may be noted that the slack bus power can also be found by summing the flows on the lines terminating at the slack bus Newton-Raphson Method Newton-Raphson is an iterative method which approximates the set of non-linear simultaneous equations to set of linear equations using Taylor s series expansion and the terms are restricted to first order approximation [6, 1]. Given a set of nonlinear equations = (,, ) = (,, ) (2.42) = (,, ) And the initial estimate for the solution vector ( ), ( ),, ( ) Assuming,,.., are the corrections required for ( ), ( ),. ( ) respectively, so that the equation (2.16) are solved i.e. = ( ( ) +, ( ) +,, ( ) + ) = ( ( ) +, ( ) +,, ( ) + ) (2.43) = ( ( ) +, ( ) +,, ( ) + ) Each equation of set can be expanded by Taylor s series for a function of two or more variables. For example, the following is obtained for the first equation. = ( ( ) +, ( ) +,, ( ) + ) = ( ), ( ),., ( ) Where a function of is higher powers of,,, and second, third, derivatives of the function. Neglecting, the linear set of equations resulting is as follows: 29

32 = ( ), ( ),., ( ) = ( ), ( ),., ( ) (2.44) = ( ), ( ),., ( ) Or ( ) ( ) ( ),,., ( ) ( ) ( ),,., ( ) ( ) ( ) =,,., D=JR Where J is the Jacobean for the functions and R is the change vector.eqn (2.45) May be written in iterative form i.e. (2.45) ( ) = ( ) ( ) (2.46) ( ) = ( ) ( ) ( ) (2.47) The new values for ( ) = ( ) + ( ) s are calculated from The process is repeated until two successive values for each (2.48) differ only by a specified tolerance. In this process J can be evaluated in each iteration may be evaluated only once provided are changing slowly. Because of quadratic convergence, Newton s method is mathematically superior to Gauss-Seidel method and is less prone to divergence with illconditioned problems. Newton-Raphson method is more efficient and practical for large power systems. Main advantage of this method is the number of iterations required to obtain a solution is independent of the size of the problem and computationally it is very fast [5]. Here load flow problem is formulated in polar form. Rewriting equations (2.4) and (2.5) = cos( + ) (2.49) = sin( + ) (2.50) 30

33 Equations (2.49) and (2.50) constitute a set of nonlinear algebraic equations in terms of the independent variables, voltage magnitude in per unit and phase angles in radians; it can be easily observed that the two equations for each load bus given by equation (2.49) and (2.50) and one equation for each voltage controlled bus, given by equation. (2.49). Expanding equation (2.49) and (2.50) in Taylor-series and neglecting higher-order terms. We obtain, ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2.51) In the above equation, bus-1 is assumed to be the slack bus. Eqn. (2.51) can be written in short form i.e. = (2.52) Decoupled load flow solution An important characteristic of any practical electric power transmissions system operating in steady state is the strong interdependence between real powers and bus voltages angles and between reactive powers and voltage magnitudes.this interesting property of weak coupling between P - and Q-V variables gave the necessary motivation in developing the decoupled load flow (DLF) method, in which P and Q-V problem are solved separately.in any conventional Newton method, half of the elements of the Jacobean matrix represent the weak coupling referred to above, and therefore may be ignored' Any such approximation reduces the true quadratic convergence to geometric one, but-there are compensating computational benefits large number of decoupled algorithms have been developed in the literature[1]. Transmission lines of power systems have a very low R/X ratio [3, 10]. For such system, real power mismatch are less sensitive to changes in the voltage magnitude and very sensitive to changes in phase angle. Similarly, reactive power mismatch is less sensitive to changes in angle and very much sensitive on changes in voltage magnitude. Therefore, it is reasonable to set elements and of the Jacobian matrix to zero. Therefore, eqn (2.52) reduces to 31

34 = 0 0 (2.53) Or = (2.54) = (2.55) For voltage controlled buses, the voltage magnitudes are known. Therefore, if m buses of the system are voltage controlled, is of the order (n-1) (n-1) and is of the order (n-1- m) (n-1-m). Now the diagonal elements of are = sin( + ) (2.56) Off-diagonal elements of are = sin( + ) (2.57) The diagonal elements of are = 2 sin( + ) (2.58) = sin( + ) (2.59) The terms ( ) and ( ) are the difference between the scheduled and calculated values at bus known as power residuals, given by ( ) ( ) = ( ) ( ) ( ) = ( ) (2.60) (2.61) The new estimates for bus voltage magnitudes and angles are, ( ) = ( ) + ( ) (2.62) ( ) = ( ) + ( ) (2.63) The main advantage of the Decoupled Load Flow (DLF) as compared to the NR method is its reduced memory requirement in storing the Jacobean. There is not much of an advantage from the point of view of speed since the time per iteration of the DLF is almost the same as that of NR method and it always takes more number of iterations to converge because of the approximation. 32

35 2.5.5 Fast decoupled load flow solution Further physically justifiable simplifications may be carried out to achieve some speed advantage without much loss in accuracy of solution using (DLF) model [1]. The result is a simple, faster and more reliable than the (NR) method called the fast decoupled load flow (FDLF) method [1].Sub-matrices can be further simplified, using the guidelines given below to eliminate the need for re-computing of the sub-matrices during each iteration [6]; i. Some terms in each element are relatively small and can be eliminated. ii. iii. iv. The remaining equations consist of constant terms and one variable term. The one variable term can be moved and coupled with the change in power variable. The resultant is a Jacobean with constant term elements. The equation for the diagonal elements of H as given by equation 2.58 can be written as [11]; = ( + ) (2.64 ) Using the above equation 2.64, and since from the (SLFEs) equation 2.14; We can write equation (2.64a) as; = (2.64 ) But = (2.65) Where is the imaginary part of the diagonal elements of the bus admittance matrix ( ) Also in a practical power system, may be neglected in the equation because [12]; (2.68) Further simplification is obtained by assuming, = (2.69) With these assumptions, the equation 2.41(b) reduces to, P δ = V B (2.64c) The off-diagonal elements of H described below (as given earlier by equation (2.57) 33

36 = ( + ) (2.57) The following assumptions are made to simplify it [3, 6]; Under normal operating conditions of a power system, ( ) is quite small ( 0) and hence; (θ + δ δ ) θ = 1.0 The equation can therefore be written as; P δ = V B (2.65) The diagonal elements of as was given earlier by equation 2.58; = 2 sin ( + ) (2.58) Can be written as (for k=i) = sin ( + ) (2.66 ) But from (SLFEs) equation 2.14; = ( + ) sin = And; Q = V B V (2.66b) The equation for the off-diagonal elements of L below (as given earlier by the equation) But; = sin( + ) (2.59) 34

37 Under normal operating conditions of a power system, ( ) is quite small ( 0) and hence; (θ = + δ δ ) θ Q = V B V (2.67) From the analysis done it can be observed that the equation for The equation (2.64c) for the diagonal elements of H is equal to the equation (2.66b) for the diagonal elements ofl. The equation 2.65 off-diagonal elements H are equal to the equation 2.68 for the off-diagonal elements of L[1]. With all the simplifications made, the resultant FDLF equations in matrix form become ΔP V = B [Δδ] (2.68) ΔQ V = B [Δ V ] (2.69) Where and are the imaginary part of the bus admittance matrix, such that contains all buses admittances except those related to the slack bus, and is deprived from all voltage-controlled buses related admittances. They are real, sparse and have the features of H L respectively. Since they contain only admittances they are constant which need to be inverted only once at the beginning of the study [12, 1]. We can then write; Where; And; [ΔP] = [H][Δδ] (2.70) [ΔQ] = [L][Δ V ] (2.71) H = V B (2.72) H = V B (2.73) L = V B (2.74) 35

38 = (2.75) To obtain the corrections to the initial estimates the equations below are used; Δδ = B ΔP V (2.76) Δ V = b B ΔQ V (2.77) The simplified (FDLF) equations are solved alternatively always employing the most recent voltage values. One iteration is called (1 ) (1 ) iteration [1]. This implies One solution for [ ] to update [ ] One solution for [ ] to update [ ] Separate convergence tests are applied for the real and reactive power mismatches as follows; Max [ ] where are tolerance Max [ ] 2.6 Convergence procedure The updated voltages immediately replace the previous values in the solution of the subsequent equations. This process is continued until changes of bus voltages between successive iterations are within a specified accuracy, define [1]. = ( ) ( ), = 1, 2,., If, then the solution has converged. Is pre-specified. Usually = may be considered. Another convergence criterion is the maximum difference of mismatch of real and reactive power between successive iterations. Define = = ( ) ( ) ( ) ( ) If and, the solution has converged. In this case may be taken as or Acceleration of convergence Convergence in the GS method can be sometimes be speeded up by the use of the use of acceleration factor, since the method is slow and it requires a large number of iterations before a solution is obtained [3, 10]. The process of convergence can be speeded up if the voltage 36

39 correction during iterative process is modified. For the h bus, the accelerated value of voltage at the ( + 1) h iteration is given by ( ) ( ) = ( ) + ( ( ) ( ) ) Where a real number is is called the acceleration factor. A suitable value of for any system can be obtained by trial load flow studies. A generally recommended value is Wrong choice of might indeed slow down convergence or even cause the method to divergence Algorithm modification when PV Buses are also present At the PV buses and are specified and and are the unknowns to be determined. Therefore, the values of and are to be updated in every Gauss Siedel iteration through appropriate bus equations. This is accomplished in the following steps for the h PV buses. From Equation. = (2.78) The revised value of is obtained from the above equation by substituting most updated values of voltages on the right hand side. In fact, for the ( + 1) h iteration one can write from the above equation = ( ( ) ) ( ) + ( ( ) ) ( ) (2.79) 2. The revised value of is obatained from Eq. (2.72) immediately following step 1. Thus ( ) =< ( ) = Where ( ( +1) 1 ( +1) ( ) ) =1 = +1 (2.80) ) ( ) = ( ) (2.81) As explained already, physical limitations of Q generation require that Q demand at any bus must be in the range. If at any stage during the computation, Q at any bus goes outside these limits, it is fixed at or as the case may be, and the bus voltage specification is dropped, i.e. the bus is now treated like a PQ bus. Thus step 1 above branches out to step 3 below. 37

40 3. If ( ) <, set ( ) =, and treat bus as a PQ bus. Compute ( ) and ( ) from Eqs (2.68) and (2.46) respectively. If ( ) >,, set ( ) =, and treat bus I as PQ bus. Compute ( ) and ( ) from Eqs (2.68) and (2.46), respectively. It is assumed that out of buses, the first is slack as usual, and then 2, 3,., are PV buses and the remaining + 1,.., are PQ buses. 2.9 Comparison of Methods Load Flow In this part comparison is made on GS and NR methods when both use YBUS as the network model [1, 2]. It is experienced that the GS method works well when programmed using rectangular coordinates, whereas NR requires more memory when rectangular coordinates are used. Hence, polar coordinates are preferred for the NR method. The GS method requires the fewest number of arithmetic operations to complete iteration. This is due to the sparsity of the network matrix and the simplicity of the solution techniques. Consequently, this method requires less time per iteration. With NR method, the elements of the Jacobeans are to be computed in each iteration, so time is considerably longer. For typical larger systems, the time per iteration in both these methods increases almost directly as the number of buses of the network. The rate of convergence of GS method is slow i.e. linear convergence, requiring considerably greater number of iterations than the NR method which has a quadratic convergence characteristics to obtain a solution and hence NR is the best. In addition, the number of iterations for the GS method increases directly as the number of buses of the network, whereas the numbers of buses for the NR method remain practically constant, independent of the system size. NR methods need 3 to 5 to reach an acceptable solution for a large system. In GS method and other methods, convergence is affected by the choice of the slack bus and the presence of series capacitor, but the sensitivity of the NR method is minimal to these factors which cause poor convergence. Therefore for the large systems the NR method is faster, more accurate and more reliable than the GS method or any other known method [2]. In facts it works for any size and kind of problem and is able to solve a wider variety of ill-conditioned problems. Its programming logic is considerable more complex and it has the disadvantage of requiring a large computer memory even when a compact storage scheme is used for the Jacobean and admittance matrices. In fact, it can be made faster by adopting the scheme of optimally renumbered buses. 38

41 The method is probably best suited for optimal load flow studies because of its high accuracy which is restricted only by round-off errors. The chief advantage of the GS method is the ease of programming and most efficient utilization of core memory. It is, however, restricted in use in small system because its convergence is never guaranteed and longer time needed for solution of large power networks. Thus the NR method is therefore more suitable than the GS method for all but very small system. The main computational [5] effort of the decoupled method a part from initially factorizing and matrices is the calculation at each iteration of the mismatch vectors [ ] and[ ]. This is much less computation than is required by the NR method where the full Jacobean J is built and factorized each iteration. Typically NR iteration takes around five times as long as a fast decoupled iteration. However the decoupled method requires more iterations than the NR method, taking in the order of two times as many iterations for normal power systems with normal loading conditions. Consequently the decoupled method is much faster for normal systems and for moderate accuracy. Under these circumstances it is also very reliable. However, if the system is stressed i.e. is operating close to its limits, or if it contains a significant proportion of lines with high ratios, then convergence of the fast decoupled method can become slow and unreliable. This occurs because the assumptions upon which the fast decoupled method is based are no longer valid. Because the NR technique does not rely on any assumptions, it is more robust and will often converge reliably in situations where the fast decoupled method would not converge. For high accuracy NR method is more suitable than the fast decoupled method. The NR method recalculates the elements of Jacobean J at each iterations, so near the solution point, and updates always drive the process closer and closer toward that solution point. Fast decoupled method use an approximate relationship between, and,. It therefore cannot be guaranteed that at each iteration the updates, will drive the process closer to the solution point. The values of, obtained at each iteration may bounce around the actual solution point. Convergence is slowed, and in fact not occurs at all. The decoupled load flow method has an advantage over the NR method if storage requirements are critical. Because the fast decoupled method does not store the J and N sub-matrices, its storage requirements are typically only 60% of those of the NR method. 39

42 CHAPTER 3 METHODOLOGY 3.1Computational procedure for decoupled load flow method [1]. The algorithm written according to the equations derived in the previous section is as follows: Step 1: Creation of the bus admittance according to the lines data given by the IEEE standard bus test systems. Step 2: Detection of all kinds and numbers of buses according to the bus data given by the IEEE standard bus test systems, setting all bus voltages to an initial value of 1.0 P.U, all voltage angles to 0, and the iteration counter to 0. Step 3: Creation of the and according to equations (2.71) and (2.72). Step 4: If max (, ) accuracy Then go to step 6 Else 1. Calculation of the H and L elements. 2. Calculation of the real and reactive power at each bus, and checking if MVAR of generator buses are within the limits, otherwise update the voltage magnitude at these buses by±2%. 3. Calculation of the power residuals, and. 4. Calculation of the bus voltage and voltage angle updates and according to equations (2.72) and (2.73). 5. Update of the voltage magnitude V and the voltage angle at each bus. 6. Increment of the iteration counter = + 1 Step 5: If maximum number of iteration Then go to step 4 Else print out solution did not converge and go to step 6 Step 6: Print out of the power flow solution, computation and display of the line flow and losses. The update of this algorithm was based on the weak coupling between and, and between and, explained in the previous section. 40

43 3.2 Design Flow Chart Start Increment iteration p=p+1 Read data and build Determine Set =1< for all buses Set iteration count p=0 ( ) = + Set bus count =1 Compute compute and Is =1 or slack bus? Assemble jacobianj Are all max Δ Determine NO Yes NO Within tolerance? Yes And print line flows, Calc. bus current Det. maximum Stop Is bus Increment bus count, Yes P-V P-V or P-Q P-Q Is =n Determine Determine NO Last node? and and Is Yes Is Yes Determine =, Compute NO Compute NO =, =, Figure 3.1: Decoupled load flow chart-[1] 41

44 3.3 IEEE 14 Bus Test Network Test network system is widely used in power system research and education. It is imperative to understand the importance of using the standard test network. This is very vital because; Practical power systems data are partially confidential, also the dynamic and static data of the system are not well documented, more so, Calculations of numerous scenarios are difficult due to large set of data and the lack of software capabilities for handling large set of data less generic results from practical power system The 14 bus system consists of five synchronous machines with IEEE type; 1 exciter, four of which are synchronous compensators used only for reactive power support. There are nine load buses in the system totaling to 259MW and 81.3 MVAR. The dynamic and static data of the system can be found. The system is widely used for voltage stability as well as low frequency oscillatory stability analysis. The 14 bus test case does not have line limits compared to other systems. It has also a low base voltage and an overabundance of voltage control capability. 42

45 Figure 3.2: IEEE 14 bus system [7]. 43

46 3.4 Load Flow Data Bus data The bus data provided for the IEEE-14bus system is given in the table 3.1 below. Table 3.1: Bus Data Bus Bus Volt. Angl Load Generator Inj. No. code Mag. e MW MVA MW MVA Q Q MVAR Deg. R Limits of the MVAR demand must be specified. The 14 bus test system being used has four generator buses 2, 3, 6 and 8. Apart from bus number 8, the rest of the generator buses have loads tapped from them. To identify the P-V buses from the rest of the bus types in the system given, they are coded 2. PQ this type means to be used for load buses. The loads are entered positive in inputting megawatts and MVAR; negative in outputting megawatts and MVAR by the power system. For this bus, initial voltage estimations must be specified. This is usually 1 and 0 for voltage magnitude and phase angle, respectively. The system has nine P-Q buses 4, 5, 7, They are coded 0. The bus data table 3-1 provides information on; The value of the loads that are tapped from the system and to which buses they are connected. The capacity of the generators that supply the system and to which buses they are connected. The voltage magnitude and phase angles at the buses. 44

47 The maximum and minimum reactive power limits for the generators. Amount of injected MVAR at the buses Line data The line data table 3.2 below provides the values for the resistance, reactance and half Susceptance in Per Unit., of the transmission lines connecting the buses in the system. This information is necessary for building the Y matrix. The other information provided by the line data table is the tap settings of the transformers connected between the lines. Table 3.2: Line data Sending end bus Receiving end bus Resistance (r) per unit Reactance (x) P.U Half Susceptance (B/2) P.U Transformer tap (a) The network of the power system network has its transmission lines modeled in standard π (Pi) model. The impedance of a line is represented as a series impedance Z. the line charging effects are divided between the two shunt arms each with an admittance of Y 2 [9]. The admittance is made up of a resistance R and a reactance X. That is; Z = R + JX (3.1) 45

48 3.4.3 Transformer Data Two-winding transformer or three-winding transformer data is included in last column of line data structure. At each line, 1 must be entered in this column due to no transformers on this transmission line. The lines may be entered in any sequence or order with the only restriction Being that if the entry is a transformer, the left bus number is defined as the tap side of the transformer. For a two-winding transformer, which is the also basic component of three-winding transformer, represented by the equivalent PI circuit as shown in Figure 3. The transformer tap ratio is setting as 1:k. The branch admittance elements can be calculated from its PI equivalent circuit. Figure 3.3: Diagram of a two-winding transformer circuit [16]. The branch self-admittance of bus is obtained by the following equation. The branch self-admittance of bus j is obtained by the following equation. There are several ways or steps of doing decoupled load flow analysis, the most important is outlined in four steps as below; 1) Assembling of load flow MATLAB data. (IEEE Data was used) 2) Running the MATLAB assembled code. 3) Creating a Power System Analysis Tool Simulink diagram. 46

49 4) Simulating the one line diagram for results validation 3.5 Assembling load flow MATLAB data. The bus data and the line data input were assembled on a MATLAB.. A matrix composed of 14 rows and 11 columns was used to input bus data and a matrix composed of 20 rows and 6 columns was used to input line data with the input vectors oriented column wise. To introduce it to MATLAB workspace the following command were used to call the functions: = ( ); = ( ); This two command functions will input the data that will be analyzed by the written MATLAB code Running the MATLAB code. After all the. containing MATLAB data are in the current path of workspace directory, the run button on the toolbar menu was clicked to simulate the code. The output results obtained from the workspace were tabulated on the Tables Creating PSAT one line diagram graphical user interface is as shown in the figure below All components that constitute the one line diagram were assembled to form the load flow system using PSAT simulator. Some of these components are; generators, loads, buses, transmission lines and transformers. The result diagram is referred as the diagram, which represents a simple model of a real system to be studied. This helps in simplifying simulation of the entire power flow system. Once a one line diagram has been drawn, extra data entry can be done to the one line diagram for the desired objective to be obtained. PSAT Simulating PSAT one line diagram Once the Simulink single line diagram was fed with all the data required, it was loaded to PSAT software through load file menu. Once the running the load flow command on the Graphical user interface. The file is loaded, it was triggered to run by single line diagram was simulated of which the various results were tabulated in Tables below. The results were compared with the MATLAB results, finally the comparison in Tables (that follow.) were used to analyze decoupled load flow method as a tool of evaluating load flow study. 47

50 Figure 4.5: diagram for 14 bus test system [22]. 48