Automatic Control & Systems Engineering.

Transcription

1 Automatic Control & Systems Engineering. RECEDING-HORIZON CONTROL & OPTIMISATION OF POWER NETWORKS Rinu Ravikumar Vasudeva Panicker August 2013 Supervisor: Dr Paul Trodden A dissertation submitted in partial fulfilment of the requirements for the degree of Degree of Master of Control Systems Engineering

2 EXECUTIVE SUMMARY INTRODUCTION/BACKGROUND As power systems are undergoing major changes in methods of generation as well as increased complexity due to deregulation, new control strategies are to be implemented in order to optimise the generation as well as to maintain the power system stability. The introduction of Flexible AC Transmission Systems (FACTS) devices has improved the overall system performance as well as stability. An effective control strategy would make use of integrated power system to improve generation & system stability. AIMS AND OBJECTIVES The aim of this project is to develop control algorithm for optimal control of power generated as well as use of SVCs to provide voltage regulation in the transmission network. The following are the objectives of this project: 1. Study of power transmission networks. 2. Study of SVCs characteristics and voltage regulation based on SVC. 3. Examine test network & available data. 4. Development of General AC OPF to minimize the cost of power generation and validate the model developed. 5. Modify AC OPF developed in objective 4 to include SVCs and accomplish voltage control in the test network. 6. Study the load variations over period of time in test networks and formulate a receding horizon problem of AC OPF with SVCs for predictive control. ACHIEVEMENTS Optimal power flow equations for AC system was developed and this AC Optimal Power Flow problem has been solved & validated. SVC was included in the system so as to improve voltage regulation in transmission lines by efficient reactive power I

3 compensation. Based on the AC OPF with SVC formulated, a receding horizon optimisation problem was formulated and solved in order to produce optimal power & improve voltage regulation for system with load variations. CONCLUSIONS / RECOMMENDATIONS Formation of receding-horizon AC OPF with SVC has produced optimal power generation as well as improved the voltage regulation in the system. The optimisation problem are made use in planning & operation of power systems. Receding-horizon provides an opportunity to combine higher decision levels (OPF) & lower decision levels (voltage & frequency control). Once the future variations in load as well as other variations in generation( for example wind power, where the exact amount of power generated cannot be predicted) when solving OPF, we could possibly find solutions which will require less corrective control measures when later. The optimisation problem formulated can be used on a larger network with different generating units (run from renewable & non-renewable sources). Additionally considerations can be given to power trade between different operators. Other security constraints such as frequency & rotor angle stability could also be considered for varying loads in the system. Various FACTS devices to improve system performance can be modelled & used in the optimisation problem. Moreover an optimal reactive power flow (ORPF) equation can also be created & could be solved simultaneously with OPF. A multi objective optimisation problem can be formed by combining ideas of cost effective power generation and system security. II

4 ABSTRACT This dissertation is devoted to the development of techniques on AC power transmission network which ensures economic & optimal generation and transmission of power in a growing complex power systems network. The dissertation begins with introduction to power generation & transmission system and further to specific study of test network. The system is analysed and a general optimisation problem is formulated for AC transmission network which focuses on optimal power generation minimizing the cost of power generation within generating units. The second part will be dealing with security constraint voltage control via use of Static VAR compensators (SVC) so as to maintain the system voltage at each bus in the system within the limits under various operating conditions. The final part of the dissertation will be about formulation of receding horizon optimal power flow equation which would enable us to predict necessary control actions to be produced for different demands in the network over a period of time by using a trend in load variation. III

5 ACKNOWLEDGEMENTS I would like to take this opportunity to thank Dr Paul Trodden for his continuous support & constructive criticism throughout the project. I also express my gratefulness to all staff members of The University of Sheffield for their guidance & support throughout the course. I am also thankful to all my classmates & friends for always being there to support me. I would like to thank my parents who gave me this opportunity. IV

6 TABLE OF CONTENTS Chapter 1 - Introduction Background and Motivation Problem Definition Aims and Objectives Project Management... 4 Chapter 2 - Literature Review Operation & Planning of Power Systems Electrical Power Transmission Systems Power Flow Analysis Voltage Instability Voltage Control in Power Transmission Systems & FACTS Devices Optimisation Problem AMPL A Modelling Language for Mathematical Programming MATPOWER Test Network Chapter 3 - AC Optimal Power Flow Formulation of AC OPF Objective Function Constraints Application of KCL to Each Bus in the System Variables Program for AC OPF Using Matlab & AMPL Optimisation Results & Comparison with Matpower Chapter 4 - AC Optimisation with Voltage Control via SVC Analysis of Bus Voltage Fluctuations Voltage Regulation SVC Static VAR Compensator V

7 4.4. AC OPF with SVC Objective Function Constraints Variables Program for AC OPF with SVC Using Matlab & AMPL Results of AC OPF with SVC Stressed Condition & Varying Load Chapter 5 - Receding Horizon/Multi Step Optimisation Problem Load Variations in Power System Optimisation Problem with Varying Load Multi Step Optimisation Problem Model Testing & Validation Power Generation & AGC Ramp Rate Multi Step Optimisation Problem with Constraint Objective function Constraints Program for Receding Horizon AC OPF with SVC Results & Analysis of Multi-Step AC OPF with SVC Chapter 6 - Conclusion & Future Work REFERENCES APPENDIX VI

8 LIST OF FIGURES Figure 1-1: Fundamental Structure Of Power Systems Network... 1 Figure 2-1: Schematic Of Power System... 7 Figure 2-2: Example Of A Simple Transmission System Figure 2-3: FACTS Devices & Conventional Methods Figure 2-4: IEEE 9-Bus System Figure 3-1: AC OPF Block Diagram Figure 3-2: Cost Vs Real Power Generated For Thermal Plant Figure 3-3: Constraints For AC OPF Figure 3-4: Basic Idea On KCL Constraint Figure 3-5: General System & KCL Figure 3-6: Nodal Analysis Of KCL Figure 3-7: Π - Model Of Transmission Line Figure 3-8: Π - Model Of Transmission Line With Transformer Figure 3-9: P-Q Curve & Transmission Line Constraint Figure 3-10: OPF Flowchart For Programming Figure 3-11: Build Problem For AC OPF Figure 3-12: AMPL Model For AC OPF Figure 3-13: AMPL Instructions To Run & Print Figure 3-14: Matlab File Managing & Optimisation Using AMPL Figure 3-15: AC OPF Results With AMPL Figure 3-16: AC OPF Results With Matpower Figure 4-1: Bus Voltage Across System Bus Figure 4-2: Sending & Recieving End Voltage Levels Figure 4-3: Plot Of Voltage Regulation Figure 4-4: SVC General Arrangement Figure 4-5: Scheme Of SVC In Transmission Line Figure 4-6: Model Of SVC Figure 4-7: System With SVC Figure 4-8: Constraints For AC OPF With SVC Figure 4-9: System With Svc & KCL Figure 4-10: Π Model Of Transmission Line With SVC Figure 4-11: Π Model Of Transmission Line With Transformer & SVC VII

9 Figure 4-12: Build Problem For AC OPF With SVC Figure 4-13: AMPL Model For AC OPF With SVC Figure 4-14: AMPL Instructions To Run & Print Figure 4-15: AC OPF With SVC Optimisation Results Figure 4-16: Bus Voltage With SVC Figure 4-17: Sending & Receiving End Voltages With SVC Figure 4-18: Voltage Regulation With SVC Figure 4-19: Comparison Of VR In Systems With & Without SVC Figure 4-20: Generation Cost With Varying Load & Generators Stressed Figure 5-1: Load Variations With Time Figure 5-2: Build Problem For AC OPF With SVC & Varying Load Figure 5-3: AMPL Model For AC OPF With SVC & Varying Load Figure 5-4: AMPL Instructions To Run & Print AC OPF With SVC & Varying Load Figure 5-5: Matlab File For Managing & Optimisation Of AC OPF With SVC & Varying Load Figure 5-6: Results Of Multi - Step Optimisation Figure 5-7: Result Of AC OPF With SVC Optimisation With 30% Pd Figure 5-8: Result Of AC OPF With SVC Optimisation With 100% Pd Figure 5-9: Result Of AC OPF With SVC Optimisation With 80% Pd Figure 5-10: Receding-Horizon AC OPF With SVC Constraint Block Diagram Figure 5-11: Basic Idea On AGC Ramp Rate Figure 5-12: Build Problem For Multi-Step AC OPF With SVC Figure 5-13: AMPL Model For Multi-Step AC OPF With SVC Figure 5-14: Hourly Generated Power & Cost With AGC Case Figure 5-15: Voltage Regulation For AGC Case Figure 5-16 Hourly Generated Power & Cost With AGC Case Figure 5-17: Voltage Regulation For AGC Case Figure 5-18: Hourly Generated Power & Cost With AGC Case VIII

10 Notations The notation used throughout this dissertation is stated below for quick reference. Sets B L G D Gbs Dbs T Set of buses in the test network. Set of lines in the test network. Set of generators in the test network. Set of loads/demands in the test network. Set of generators at each bus in the test network. Set of demands/loads at each bus in the test network. Set of times at which system load changes Parameters PD QD Real power demand at each bus in the test network (p.u) Reactive power demand at each bus in the test network (p.u) PG max Maximum real power generation capacity of generator (p.u) PG min Minimum real power generation capacity of generator (p.u) QG max Maximum reactive power generation capacity of generator (p.u) QG min Minimum reactive power generation capacity of generator (p.u) SL max Maximum apparent power rating of transmission lines (p.u) V max Maximum voltage limit of bus (p.u) IX

11 V min Minimum voltage limit of bus (p.u) R L Resistance of transmission line (p.u) X L Reactance of transmission line (p.u) Z L Impedance of transmission line (p.u) = R L + jx L G L Conductance of transmission line (p.u) B L Susceptance of transmission line (p.u) Y L Admittance of transmission line (p.u) = G L + jb L = 1/Z L BC Total line charging/shunt capacitance of transmission line (p.u) b SVC range Susceptance range of Static VAR compensator (p.u) Tap Deltashift Transformer off nominal turns ratio Transformer phase shift angle (radians) c 0 Input-output characteristics of power generation units, equivalent to the fuel consumption rate of generation unit without producing power. c 1, c 2 basemva A Input-output characteristics of power generation units. Base unit of apparent power for p.u calculations. Line bus matrix (specifies line, from bus & to bus) Variables pg Real power output generated from the generator. X

12 plkm plmk qg qlkm qlmk delta v Real power injected into line from bus k to bus m. Real power injected into line from bus m to bus k. Reactive power output generated from the generator. Reactive power injected into line from bus k to bus m. Reactive power injected into line from bus m to bus k. Voltage phase at bus end. Voltage at bus. b SVC Susceptance of static VAR compensator connected to line. Q SVC Reactive power injected/absorbed by the static VAR compensator. GS, BS Shunt conductance & Susceptance of line Functions Cost Cost function for power generation units Indices l b g d t Index of lines. Index of buses. Index of generators. Index of demands. Index of Time. XI

13 Abbreviations AC AGC AMPL AVR CPLEX DC FACTS FC IEEE IPOPT KCL kv KVL LTC Max Min MVA MVAR Alternating Current Automatic Generation Controls A Modelling Language for Mathematical Programming Automatic Voltage Regulator Optimisation tool Direct Current Flexible AC transmission systems Fixed Capacitor Institute of Electrical and Electronics Engineers Interior Point OPTimizer Kirchhoff s Current Law Kilo Volt Kirchhoff s Voltage Law On Load Tap changing transformer (In phase transformer) Maximum Minimum Mega Volt-Ampere Mega Volt-Ampere Reactance XII

14 MW OPF ORPF P.F P.U PST RTS SVC TCR TSC TSR VR Mega Watts Optimal Power Flow Optimal Reactive Power Flow Power factor Per Unit Phase Shifting Transformer Reliability Test System Static VAR Compensator Thyristor Controlled Reactor Thyristor Switched Capacitor Thyristor Switched Reactor Voltage regulation XIII

15 Chapter 1 - Introduction 1.1. Background and Motivation The electric power industry is undergoing the most profound technical, economic and organisational changes since its inception some one hundred years ago. In the past years the electric power industry was characterized by a vertically integrated structure, consisting of power generation, transmission/distribution and trading (Xiao-Ping Zhang, Flexible AC Transmission Systems: Modelling & Control, 2006). The restructuring, privatization & deregulation process has resulted in the unbundling of this organizational structure. In a global level it is estimated that in the next 20 years, the power demand would increase to 250% and in industrialised countries demand would increase by 37% (Global Insight 2008, Siemens E ST MOP 10/2008). The fundamental structure of electric power systems is as shown below FIGURE 1-1: FUNDAMENTAL STRUCTURE OF POWER SYSTEMS NETWORK Electrical power generation currently uses a combination of renewable (e.g. solar, wind power) & non-renewable sources (e.g. thermal, nuclear plants) of energy. Beyond integration of different power generation units there arises a question of optimal control considering minimization of resources or the cost. This is one of the areas which are of high interest today. The electric network (transmission & distribution systems) which carries generated power from the generating stations to the load end plays a vital role in electric energy business. The transmission network has a limit on the amount that can be carried, generally in association with heating. Due to the deregulation in the power system industry, the systems are being stressed & pushed to their upper operating limits. As the system becomes complex, the cost involved in the operation also increase. 1

16 The operation & planning approaches in the power system sector is undergoing reengineering. The idea of power flowing purely from generating plants to the customer is no longer valid. These factors makes the system more unpredictable system & therefore requires new equipment to handle these complex situation. Additionally the distribution network must take new functioning of balancing power flows in the network. It is also to be noted that the capacity of elements in the system may not be sufficient to handle these situations. In order to handle these situation the growing power electronics industry has provided us with family of devices commonly described by the term 'Flexible AC Transmission Systems' or 'FACTS devices'. FACTS-devices can be used to boost the capacity of transmission network, improve the stability and dynamic behaviour or ensure better power quality in modern power systems. Main capabilities of FACTS devices are reactive power compensation, voltage control and power flow control. Due to their controllable power electronics, FACTS-devices always provide fast control actions in comparison to conventional devices. This dissertation offers a basic understanding of control approaches to power generation & distribution. A mathematical model of the AC power network is created from the data provided in the test network which intends to optimise the power generated. The next stage is to control SVC so as to maintain the system voltage within limits under different conditions of load. The final stage would be to formulate a receding-horizon optimisation problem which focus on to predict control actions necessary based on load variations throughout a specific period of system operation Problem Definition Present day power systems are stressed to close their stability limits. It is required that the system should be stable & secure in the event of any credible contingency. Optimal power flow can be obtained when the generated power supplies loads in the systems at lower generation cost. 2

17 Voltage instability can be a major factor causing power system instability and blackouts in the worst scenario. Voltage instability is a case when the receiving end voltage increases or decreases beyond the limits and cannot be restored even by use of compensators or AVRs. Voltage collapse is the process by which the voltage falls to a low, unacceptable value as a result of an avalanche of events accompanying voltage instability (Taylor, 1994). Basic form of voltage control is to maintain the bus voltage within specific limits so as to facilitate the smooth and safe operation of the system. This could be done using reactive power compensation & could be controlled to a certain extent by timely and optimal control actions. As in many physical systems, time-delay exists in electrical systems too. The effect of time-delays causes considerable problems in effective control. A possible solution to this would be predict the systems performance beyond the time delay. Such a mechanism could be used in the power transmission systems where we could use the general trend in load variations over some specific period of time and based on these, calculations and necessary control actions could be obtained for that instant. Such a design leads to the formulation of receding-horizon problem where we tend to optimize the cost of power generation as well as maintain the system security factor bus voltages within the specific limits Aims and Objectives The aim of this project is to develop control algorithm for optimal control of power generated as well as use of SVCs to provide voltage regulation in the transmission network. An AC OPF problem is developed for optimal power generation & voltage control. Furthermore a receding horizon problem is formulated for model based predictive control. The following are the objectives of this project: 1. Study of power transmission networks. 2. Study of SVCs characteristics and voltage regulation based on SVC. 3. Examine test network & available data. 3

18 4. Development of General AC OPF to minimize the cost of power generation and validate the model developed. 5. Modify AC OPF developed in objective 4 to include SVCs and accomplish voltage control in the test network. 6. Study the load variations over period of time in test networks and formulate a receding horizon problem of AC OPF with SVCs for predictive control Project Management Major steps involved & the time taken to accomplish each task is shown in the Gantt chart shown below Major steps involved in the project were: Formation of AC OPF: Analysing the test network, running AMPL & Matpower on Matlab, Study of OPF, Formulation of OPF & validating with Matpower results. Formation of AC OPF with SVC: Study of characteristics & working of SVC, Modelling SVC, Analysing the optimised test network for voltage regulation in line, Sizing & placing of SVC, Formulation of AC OPF with SVC & testing. Formulation of receding-horizon AC OPF with SVC: Study of load changes in power systems, Formulation of multi-step OPF in AMPL & testing, Study on generator ramp rates, Formulation of multi-step AC OPF with SVC & testing. 4

19 Chapter 2 - Literature Review The following topics are discussed Operation & planning of power systems, power transmission systems, power flow/load flow analysis, Analysis of bus parameters, causes of voltage fluctuations in transmission networks, conventional methods of voltage control & FACTS devices & basic principle of operation, general structure of optimisation problems, optimisation via software AMPL, general structure of optimisation problem in AMPL and solvers for AMPL, MATPOWER, Test network used and analysis Operation & Planning of Power Systems Operation of power systems involves basically with optimal power generation & minimising power outages. The load on the system can be classified into three types as base load which are always present on the system, intermediate loads which occur at regular intervals of time & peak loads which operate for a short time. Based on these we have generating units designed for taking up various load types. Based on load studies, a pattern or trend in the load variations over time can be obtained which can be used to predict the amount of power to be generated. Associated with power system are stability issues which can be broadly classified into three types voltage instability, frequency instability & rotor angle instability (Montefiore Institute, EEE). Rotor angle stability concerns with the ability of generator to maintain synchronism. Frequency stability relates to the ability of system to maintain frequency within limits. Voltage instability relates to the ability of system to maintain voltages within specified limits. Whenever there is a disturbance acting on the power system, it must be able to retain frequency, rotor angle & voltage stability. Unit Commitment: With the trend in load variations, future system loads are anticipated. These loads may be for a week or could be on a daily basis. Based on these assumed loads, a schedule or sequence for generating units are made. Various factors such as generation cost of units, maintenance factors, run hours, start-up & shut down times are considered when deciding the units to be switched on. As the 5

20 generation units are selected for a particular time frame, the next process would be to determine the amount of power to be generated from each unit (Zhu). Economic Dispatch (ED): Deals with the problem of generation with lowest cost. When there are number of generating units with different cost characteristics, we need to pick out the units & determine the power to be produced from each unit with lower cost of generation so as to reduce the overall cost of power production. The economic dispatch problem may not consider the transmission constraints or power flow constraints during transmission to load centres. The produced results of economic dispatch may be lower than the actual generation cost which would also supply for losses also in the system (Zhu). Optimal Power Flow (OPF): Optimal power flow problem would focus on optimal power generation at a minimised cost considering into account the power flow constraints or transmission loss in the system unlike the economic dispatch problem. As the unit commitment is decided for a day or a week, OPF can be calculated on an hourly basis to determine the amount of power to be generated from each unit. Apart from providing for real power, reactive power can be considered to be optimise. The optimal reactive power flow (ORPF) can be used to determine the reactive power to be injected or removed from the system based on the voltage fluctuations in the system. Automatic Generation Control (AGC): Once the OPF has been solved, the AGC can be set to control the frequency fluctuations in the system. The system frequency varies with the load. As load increase, the system frequency drops and as load decreases, frequency increases. The generator frequency depends on the generator speed which in turn depends on the speed of prime mover. AGC sets control signals to governors of prime-movers so as to control the generator frequency and hence the system frequency. AGC controls the system frequency on a time frame of minutes. 6

21 2.2. Electrical Power Transmission Systems Power transmission systems forms are systems responsible for delivering power from generating stations to load/customer end. Power transmission systems may be HVAC or HVDC. For AC systems power is transmitted as 3 phase power at high voltage supply so as to reduce the losses involved in transmission and also to reduce the size of conductor used in transmission. Inter connected power transmission lines form a power transmission network or grid. A general schematic diagram of power system is shown in figure 2-1: FIGURE 2-1: SCHEMATIC OF POWER SYSTEM The schematic diagram of power transmission network is shown in figure 2-2 with major components Power Flow Analysis Power flow or load flow analysis can be defined as the set of solutions that gives the values of currents, voltages, real & reactive power flows at each bus in the network (Zhu). The relation between real power & reactive power consumption at bus or the generated power & voltage magnitude at the transmission bus.are non-linear. Power flow analysis provides solutions to problems in transmission systems for a particular set of loads or generator outputs. The general components/parameters involved in power flow analysis are: 1. Real power (P) Includes the real power generated at the transmitting end of bus, power lost in transmission line & the power consumed at the load end. 7

22 2. Reactive power (Q) Includes the real power generated at the generator side, power absorbed/injected from compensation devices in transmission lines, power lost in transmission & reactive power consumed by the load. 3. Bus voltage (V) Voltage magnitude at each bus in the system. It is required that the voltage at each bus in the system must be maintained within specified limits at all time of operation. Voltage stability is one of the main causes of power system failures. 4. Angle (θ) Defines the angle of voltage at each bus in comparison to a reference bus called slack bus. The above four forms variables in a power flow problem. It is required for power flow analysis at each bus in the system, it is required that at least two of the above mentioned variables are to be known. Depending on available data of each bus, buses in the system can be classified as 1. PQ bus Magnitudes of real and reactive power are known. Voltage magnitude and voltage angle are unknown. 2. PV bus Magnitudes of real power & voltages are known. Basically a bus connected to a generator is a PV bus. Reactive power and voltage angle are unknown. 3. Slack bus It is a reference bus which forms a basis of comparison for other buses in the system. Also known as swing bus. Generally slack bus is a generator bus where magnitude & angle of voltage are unknown. Normally there is only one slack bus in the power flow analysis however when dealing with complex systems we can use distributed slack buses. In power flow analysis we use per unit system (refer Appendix-A, section A1) for calculations. Power flow analysis helps in study of short circuit fault analysis, unit commitment, economic load dispatch, & optimal power flow by which provides the most economical power generation solution. 8

23 Some methods to perform load flow analysis are listed below: 1. Newton Raphson 2. Gauss Seidel method 3. DC power flow method 4. AC power flow method 5. Decoupling methods 6. Interior point method 2.4. Voltage Instability Voltage stability is one of the prime factors concerning the effective as well as stable operation of power system. From the past events around the globe, it is seen that voltage stability & subsequent voltage collapse have led to major problems in the power industry including blackouts (e.g. US-Canada blackout, Danish/Swedish blackouts) (Bialek). Voltage instability stems from the attempt of load dynamics to restore power consumption beyond the capability of the combined transmission and generation system (Glavic). Voltage collapse is a process where voltage instability and other events leads to unacceptable low voltage levels in the whole system or part of the system. Voltage stability and angle stability are interlaced & determines system stability. Formation of one may lead to another and as a consequence will affect the overall power system stability. Voltage instability may arise due to one or more of the following reasons: 1. Line or generator outage 2. Due to heavily loaded line 3. Operation of system closer to limits 4. Large distance between generation and load units 5. Poor co-ordination between different protective & control elements of the power system 6. Over usage of shunt capacitor compensation. 9

24 Under voltage at the load end may cause degradation in the equipment performance. Similarly over voltage may lead to magnetic saturation, harmonics generation and insulation breakdown in electrical equipment. Therefore it is necessary to maintain voltage at the consumer end within specified limits at all times. In general the tolerance for voltage variation at the receiving end of the transmission line may be ±5% of sending end voltage (National Electric Code, 2011). Power transmission & distribution system always have a reactive part associated with it because of the inductance and capacitance associated with the line. This reactive part may lead voltage instability in a transmission system which can be defined as a state in which the system voltage/bus voltage may exceed certain specified limits. One of the main reasons behind voltage instability can be a sudden change in load on the system. In case the load on the system increase suddenly the voltage at the load end may fall, in the other case when the load is switched off from the system suddenly it may lead to development of excess voltage. Both cases can cause voltage instability. This may be explained as stated below: Consider a transmission line as below supplying a single load from a single generator, FIGURE 2-2: EXAMPLE OF A SIMPLE TRANSMISSION SYSTEM The components of the transmission system can be generalized to compose of resistance R L & reactance X L. In reality the resistance of line is very small in comparison to the reactance of the line. Therefore we can approximate the impedance of line to be equal to the reactance of line. From the above diagram we can relate sending end voltage (V S ) and receiving end voltage (V R ) as 10

25 V S = V R + ji L X L EQN 2.4/A Depending on power factor (p.f) of the load we will have 3 conditions load with lagging p.f, load with leading p.f and load with unity p.f For lagging load(inductive), the reactance of the load can considered positive & total impedance X = X L + reactance of load, therefore as the load increase, receiving end voltage can be written as, V R = V S - ji L X EQN 2.4/B From the above EQN 2.4/B, it can be seen that the receiving end voltage decreases for lagging load. For leading load (capacitive), the reactance of the load can considered negative & total impedance X = X L reactance of load, therefore as the load increase, receiving end voltage can be written as, V R = V S + ji L X EQN 2.4/C From the above EQN 2.4/C, it can be seen that the receiving end voltage increases for leading load. For load with unity p.f(resistive), the reactance of the load can be considered zero, therefore as the load increase, receiving end voltage can be written as, V R = V S - ji L X L EQN 2.4/D From the above EQN 2.4/D, it can be seen that the receiving end voltage decreases for resistive load. As with the above three conditions we can see that voltage fluctuates at the receiving end of the transmission line. 11

26 2.5. Voltage Control in Power Transmission Systems & FACTS Devices Transmission lines are associated with resistance & reactance. The loads connected at the receiving end of the line are varying. Depending on these load conditions the amount of current flowing through the transmission lines and associated loss may vary. Since reactance is larger than resistance, there is considerable amount of reactive power loss which may lead to voltage fluctuations at the receiving end of the line. Conventional methods of voltage control in power transmission lines are use of tap changing transformers, phase shifting transformers, transposition of conductors to modify the reactance, use of capacitor & inductor banks etc. However these device consist of mechanical components and speed of response is low. Introduction of high power electronic devices have led to a new field - Flexible AC Transmission Systems (FACTS). It can be said as the name given to a family of devices with a view on improving the power quality in transmission systems. General concepts of FACTS devices are to modify the reactance of the line in series compensated FACTS devices and to modify the reactive power in the transmission line for shunt compensated FACTS devices. A comparison of voltage regulation techniques using conventional techniques and FACTS devices are shown below FIGURE 2-3: FACTS DEVICES & CONVENTIONAL METHODS 12

27 2.6. Optimisation Problem An optimisation problem may consist of three components objective function, constraints & decision variables. Generally the aim would be to minimize or maximize the objective function (e.g. minimize loss or maximize profit). Maximising or minimising the objective function must obey some rules or equations which are formed in the constraint set. The constraints can be equality or inequality constraint. Decision variables are varied according to our objective of minimising or maximising the objective function all within the constraints AMPL A Modelling Language for Mathematical Programming AMPL provides a simple language for mathematical programming in order to model various systems. There are various solvers available to solve with models in AMPL based on the type of mathematical model considered. The same model can be used on a wide range of solvers without altering the structure of the mathematical model. Some examples include CPLEX, IPOPT for linear & nonlinear equations. Moreover AMPL can be easily interfaced with Matlab and the results from AMPL can be used with Matlab to perform other computations. In order to use AMPL to solve optimisation problem we build the model in model file (.mod). Along with the model, parameters and sets of data are specified in a data file (.dat). This along with the run file containing list of commands to solve the optimisation problem is fed into AMPL. The solver used to solve the optimisation problem is IPOPT Interior Point optimizer because of the non-linear of the optimisation problem MATPOWER MATPOWER provides a set of matlab files dedicated to solve various power flow & optimisation problems in power network. MATPOWER contains inbuilt AC OPF for wide range of test networks included within the program. MATPOWER uses MATLAB Interior Point Solver MIPS for the AC OPF problem. Matpower version 4.1 was used to optimise the IEEE 9-bus RTS for AC OPF. The results of simulation was used to compare and verify results from the AMPL optimisation model created in 13

28 Chapter 3. For further details & coding to use on MATPOWER for AC OPF, refer appendix-a, section A Test Network The test network used in this project is a standard 9-bus IEEE Reliability Test System (RTS). The test file is provided in PTI format in an M-file. Additional data on the format type is shown in appendix B4. The system consists of nine buses connecting nine lines. The total number of generators in the system is three supplying power for three loads. For further details on network parameters refer appendix-a, section A3. A schematic diagram of the test network is shown below: 14

29 FIGURE 2-4: IEEE 9-BUS SYSTEM 15

30 Chapter 3 - AC Optimal Power Flow The following topics are discussed - formation of optimisation problem for AC transmission network, explanation of objective functions & different constraints involved, basic structure of AC OPF with AMPL and working, discussion on AC OPF results and validation with MATPOWER Formulation of AC OPF The AC Optimal Power Flow equation is formulated with a view to produce a model of the AC power generation & transmission system. By optimal we mean to reduce the cost which may include cost of fuel, maintenance, operation etc. of the power system as well as maintaining factors such as power quality (which may include voltage regulation, power factor etc.), emissions factors (as we have different sources to produce electric power broadly classified into renewable and nonrenewable sources). Power flow solutions to a network of electric power systems may produce a set of measurements of voltages (V), currents (A), real power (W) and reactive power (VAR). Generally in AC power flow equations the relationship between voltage and current is non linear same is the case with real power and reactive power, therefore the AC OPF is a non-linear optimisation problem. FIGURE 3-1: AC OPF BLOCK DIAGRAM 16

31 The general structure of the AC OPF problem consist of an objective function - a cost function of generated power, constraints which may include equality as well as inequality constrains and variables. A general block diagram of the AC OPF is shown below in figure 3-1. The AC OPF problem starts with the producing the cost function which is to be minimised by varying the variables depending on the different constraints formulated from the transmission & generation limits Objective Function Objective function in case of AC OPF is taken to be cost function. The cost function relates the actual power output from each generators and the cost involved. Cost may not just include the prices of fuel consumed but maintenance cost, operational charges, labour charges etc. Cost itself can be subdivided into two types which may be fixed cost and varying cost term varying depends on the amount if power generated. Fixed cost in general may include the operational, maintenance & labour charges whereas the variable cost consist of fuel prices. These characteristics are different for different generators with different power outputs and fuel/input intake characteristics. Renewable and non-renewable resources may be involved in power production. Within renewable and non-renewable resources we have more classifications. All these contribute to different characteristics of power generation. These factors are also to be considered. Based on these objective function for AC OPF can be formulated as: Minimise cost = c 2 * (pg) 2 + c 1 * (pg) + c 0 EQN 3.2/A Where c 0, c 1 & c 2 : pg: constants for each generator depending on real power output. real power (kw) generated by each generator in the power system network As from the above EQN 3.2/A (Zhu) it can be seen that the cost function is non-linear. The parameters c 2 and c 1 are linked directly to the real power output of the 17

32 system where as parameter c 0 is independent. Parameter c 0 is provided by the cost of running the generator without providing power to any load. These values are obtained by studying the performance, operating the generators under different conditions of load & and from design data of generators. Based on these data EQN-3.2/A can be formulated. Cost vs real power output of a thermal plant is shown in figure 3-2 below for reference. The parameter c is marked for reference. The complete set of values for parameters c 0, c 1 & c 2 for all generators in the 9 bus test network is provided in refer appendix-a, section A6. FIGURE 3-2: COST VS REAL POWER GENERATED FOR THERMAL PLANT 3.3. Constraints Constraints for AC OPF under consideration can be broadly classified as equality & inequality constraints produced by limitations of generators, lines, analysing parameters at each bus in the network. It is to be noted that all parameters specified for the AC OPF has to be considered in per unit values using the base MVA specified. Below given is a block diagram consisting of all constraints involved in AC OPF formation is shown in figure

33 FIGURE 3-3: CONSTRAINTS FOR AC OPF Equality Constraints Equality constraints are considered by applying Kirchhoff s Voltage Law & Current Law to each line & bus respectively in the system. The same can be applied for real and reactive power flows in the system Application of KCL to Each Bus in the System As per KCL the total current at a node is zero, which implies current entering the node is equal to current leaving the node. The same can be applied in case of power. We will be using power flow analysis at each node/bus because the value of current at each bus is unknown and every bus in the system is specified with at least two parameters out of four P, Q, V & θ. Each bus in the system can be treated as a node. The basic idea behind introducing KCL in the problem is to generate minimum power to compensate for losses in transmission and to supply demands. The figure given below depicts the same. 19

34 FIGURE 3-4: BASIC IDEA ON KCL CONSTRAINT From the above figure, we can write, Power Generated = Power lost in transmission + Power demand EQN /A The above equation applies to real power & reactive power. If we could reduce the power lost in transmission, we could supply for demand with smaller power generation and hence reduce the cost explained in section 3.2. Consider the figure given below, FIGURE 3-5: GENERAL SYSTEM & KCL From the above figure we see that plmk & qlmk represents the power injected into line l from bus-m. But we could consider these power to be absorbed instead of injected there by representing power at bus m for the load to be plmk & -qlmk. We can say that power injected from generator bus k supplies to load at load bus m. Using this we can formulate the real & reactive power losses in line as, plkm = - plmk + Real power lost in transmission EQN /B 20

35 qlkm = - qlmk + Reactive power lost in transmission EQN /C Nodal analysis of the same is shown in figure below, FIGURE 3-6: NODAL ANALYSIS OF KCL From the above two equations /B & C, we can find the amount of losses in the system and can be substituted in EQN /A to form the two equations given below which forms the two KCL constraints for the AC OPF. pg = PD + (plkm + plmk) qg = QD + (qlkm + qlmk) EQN /D EQN /E The above two equations, EQN /D & E forms KCL constraints on each bus in the system Application of KVL to Each Line in the System As per KVL the total emf or voltage in a mesh equals zero which allows us to calculate voltage drops in the circuit. The same principle can be applied to calculate power drop. Consider a π model of transmission line l connecting buses k and m with a total impedance of Z L and total shunt Susceptance of BC divided into two and shown at two ends of the transmission line as shown below in figure 3-7, The general equation for real & reactive power flow in a transmission line can be represented as (Jody Verboomen) with EQN /A ( ) EQN /B 21

36 FIGURE 3-7: Π - MODEL OF TRANSMISSION LINE The real & reactive power injected from bus - k to bus m can be written as, ( ) ( ) EQN /C ( ) ( ) ( ) EQN /D The transmission line may contain an on load (in phase) or phase shifting transformer at the sending end of line so as to control the power flow as per the above equations /A & B. Considering possibility of transformer in the line and remodelling the system as show in figure 3-8 below, FIGURE 3-8: Π - MODEL OF TRANSMISSION LINE WITH TRANSFORMER 22

37 Considering a fictional node n at the secondary of transformer in the above figure, we have voltage magnitudes V k & V n as well as voltage angles &. Considering the complex voltage at bus k and node - n, we have, EQN /E For on load tap changing transformer or in phase transformer (LTC), Therefore the above EQN /E becomes, EQN /F For phase shifting (PST) in line, Therefore the above EQN /E becomes, ( ) ( ) EQN /G The off nominal tap ratio, Tap & the transformer phase shift angle, Deltashift (specified in degrees converted to radians for use in power calculations) is available within the test network data. We can use EQN /G to specify fiction node voltage for LTC also by taking Deltashift = 0. Writing real and reactive power 23

38 injected from bus k to bus m by using EQN /C and figure 3-8. (Note: In figure the power transmission or injection is considered after the transformer, taking the transformer to be lossless and therefore from end voltage will be taken at node n. Since this voltage is unknown we can use EQN /G to write node n voltage in terms of bus k voltage of the test network) ( ) ( ) ( ) ( ) ( ) EQN /H ( ) ( ) ( ) ( ) ( ) ( ) EQN /I Similarly power injected in from bus - m to bus k through line l can be written as ( ) ( ) ( ) ( ) ( ) EQN /J ( ) ( ) ( ) ( ) ( ) EQN /K The above given EQN /H to K represents the equality constraints imposed by KVL on the transmission lines in the test network Inequality Constraints Inequality constraints considered are generator power limits real & reactive power limits, line power limits apparent power limits, bus voltage magnitude limits. 24

39 Generator Power Limits The test network consists of three generators providing power to the loads and losses in the system. Each generator is of different characteristics. Clearly every generator has its limits of producing power. This can be seen from the capacity curve provided by the manufacturer apart from the standard MVA rating, voltage rating and p.f (which provides the limit of power capacity without overheating). Based on these values of real & reactive power, we create the per unit values and the generated power has to be within these constraints. The real power upper limit of generator is brought out by the prime mover limitation (depending on different sources of energy and different prime movers). The reactive power upper limit is brought out by three factors armature current limit, field current limit and the heating limits. Real Power Generated Min real power limit of generator Real Power Generated Max real power limit of generator EQN /A EQN /B Reactive Power Generated Min reactive power limit of generator EQN /C Reactive Power Generated Max reactive power limit of generator EQN /D Sample capability curve of a generators is shown in Appendix-A, section A Line Flow Limits The power transmission lines are designed to transmit specific amount of power. It is to be made sure that the injected power from a bus into the line from transmitting point of view or power injected from a line into a bus at the receiving point of view are always maintained within limits of the line. The transmission system will be designed to transmit specific amount of power along with other constraints such as voltage drop in line, length of line size of constraints etc. As specified in section the real and reactive power of the generators vary within their upper and lower limits so as to minimize the cost and so is the load as well as the power 25

40 injected or removed from the line. These variables (power injected or absorbed from the line) should obey the constraint set by the transmission line given below: Apparent Power Carrying Capacity of Line Real Power injected + j Reactive Power injected EQN /A The above equation is valid for power injected from bus k as well as power injected from bus m. FIGURE 3-9: P-Q CURVE & TRANSMISSION LINE CONSTRAINT Voltage Magnitude Limits The bus voltage are to be maintained at certain limits. The generator terminals a constant voltage during power generation (theoretically). In practise we may use various control techniques & equipment (e.g. AVR) to ensure this during different operating conditions. The power injected at the sending end bus and the power absorbed at the receiving bus should maintain the same voltage (tolerance of ±5% is allowed). The IEEE 9 bus test networks provides data on the minimum and maximum limit of voltage variation allowed for each bus in the system. Overvoltage & under voltage may cause malfunction and overheating of machines. Apart from these large 26

41 deviation in voltage may result in voltage instability. The voltage magnitude constraint is shown below: Bus Voltage Min Bus Voltage Limit Bus Voltage Max Bus Voltage Limit EQN /A EQN /B 3.5. Variables The decision variables involved in AC OPF problem are as listed below, 1. Real power generated from the generating units (pg) 2. Reactive power generated from the generating units(qg) 3. Real power injection between buses, plkm & plmk resulting in real power loss in transmission, plkm + plmk 4. Reactive power injection between buses, qlkm & qlmk resulting in reactive power loss in transmission, qlkm + qlmk. 5. Bus voltage magnitudes, V controlled within specified upper & lower limits of the test network 6. Bus voltage angle, Delta 3.6. Program for AC OPF Using Matlab & AMPL The AC OPF is programmed using Matlab & AMPL. AMPL provides simple language to structure the optimisation problem which could be used with different solvers depending on the type of optimisation problem. We will use IPOPT solver to solve this non-linear problem. The flowchart given below shows how matlab is used to solve the optimisation problem shown in figure 3-10 below, 27

42 FIGURE 3-10: OPF FLOWCHART FOR PROGRAMMING The optimisation begins by running the file run_ac_opf.m which loads test network 9 bus test network data into Matlab. Data for AMPL is built from this case file. FIGURE 3-11: BUILD PROBLEM FOR AC OPF 28

43 The mathematical model of the AC OPF is written into model file AC_OPF.mod and used as AMPL language to convey optimisation problem to the solver. Refer appendix B, sectionb1 for model file. This is shown in figure 3-12 below, FIGURE 3-12: AMPL MODEL FOR AC OPF The run file AC_OPF.run includes all commands to select the solver for AMPL and to run the optimisation. The print file AC_OPF.print specifies how the optimised results/variables values are written into data file shown in figure The interface file to switch data between Matlab and AMPL is provided in AC_OPF.m which writes all model file, data file, run & print files into AMPL using auxiliary program amplwrite.m and brings back the results from AMPL into Matlab for further computation (shown in figure 3-14). 29

44 FIGURE 3-13: AMPL INSTRUCTIONS TO RUN & PRINT FIGURE 3-14: MATLAB FILE MANAGING & OPTIMISATION USING AMPL 30

45 3.7. Optimisation Results & Comparison with Matpower The optimisation problem formulated in sections 3.2 to 3.5 is run and the following results are obtained FIGURE 3-15: AC OPF RESULTS WITH AMPL The optimisation problem for AC OPF for the 9 bus IEEE RTS is run in Matlab via Matpower s inbuilt AC OPF solver. The following results were obtained. FIGURE 3-16: AC OPF RESULTS WITH MATPOWER By comparing the results of AC OPF from the model built in AMPL (figure 3-16) as well as the inbuilt optimisation tool in Matpower, we can confirm the model of AC OPF problem formulated in AMPL is valid. The results above shows the total power generated from all the generators in the nine bus IEEE RTS and the cost after the optimisation process. Each of the generators in the system has different power outputs & cost characteristics. Detailed codes for running the optimisation process in Matpower as well as the complete results for AC OPF is listed in refer appendix-a, section A2. 31

46 Chapter 4 - AC Optimisation with Voltage Control via SVC The following topics are discussed - analysis of bus voltage fluctuations, voltage regulation & selection of bus for VR improvement, SVC working and mathematical modelling techniques, Addition of SVC to AC OPF, explanation of additional constraints introduced in AC OPF for voltage control using SVC, comparison of optimisation results with and without SVC, optimisation with variable load and stressed condition Analysis of Bus Voltage Fluctuations In section 3.7 we have analysed the cost and power generation after the optimisation process. However apart from producing and transmitting power optimally, there are other factors such as power quality terminal voltage to be maintained at specified limits. Voltage at the sending bus may not match with the receiving bus. However a tolerance factor of ±5% is allowed in real world system. Figure 4-1 shows bus voltage after the optimisation process discussed in chapter 3. FIGURE 4-1: BUS VOLTAGE ACROSS SYSTEM BUS Depending on how the lines are linked with different buses in the system, we can produce a graph comparing the voltage variation between the sending and receiving 32

47 end of the buses. Figure 4-2 shows a comparison of voltages at the sending and receiving end of the bus for all lines in the system. FIGURE 4-2: SENDING & RECIEVING END VOLTAGE LEVELS 4.2. Voltage Regulation Voltage regulation can be defined as difference in voltage levels of sending and receiving end expressed in percentage of sending end voltage. A general expression of voltage regulation can be written as, VR = ((Vs-Vr)/Vs) *100 %...EQN 4.2/A Voltage regulation can be used to study the deviation in voltage levels in power transmission lines. It essentially provides the ability of the transmission system to provide a nearly constant value for voltage across the transmission network over a wide range of conditions. The below given table provides data on values of sending end voltage, receiving end voltage & voltage regulation for the optimised system in AC OPF discussed in Chapter 3. 33

48 TABLE - 4A: LIST OF LINES, BUSES, VOLTAGES & VOLTAGE REGULATION LINE REFERENCE FROM BUS REFERENCE TO BUS REFERENCE FROM BUS VOLTAGE TO BUS VOLTAGE VOLTAGE REGULATION The plot of voltage regulation over each line in the power system is shown in figure 4-3 given below. FIGURE 4-3: PLOT OF VOLTAGE REGULATION Different voltage regulation devices are used depending on situation and control strategy needed some examples have been listed in chapter-2. Another important factor is deciding the size and location of compensation devices SVC Static VAR Compensator SVCs are shunt reactive power compensation FACTS devices. SVC uses power electronics to control flow of power and improve the stability of power systems. In 34

49 electrical systems, reactive power is generated and consumed at various points. As load varies with time, the reactive power changes in time and this causes fluctuations in voltage. SVC can continuously provide & remove reactive power from the system to so as to keep the voltage within limits. It is also possible for SVC devices to provide damp real power oscillations by varying the voltage profile (Xiao-Ping Zhang, 2006). This form of reactive power compensation finds extensive applications today where power production is distributed. The main aspects for using SVCs in power transmission systems are the following, 1. Absorb & inject reactive power into & from the system to damp fluctuations of reactive power in the system there by preventing voltage fluctuations or collapse. 2. When installed at the consumed end, it improves p.f and hence power quality. 3. Improves static or transient stability Building Blocks of SVC & Characteristics Main components of SVC are capacitor banks switched by thyristors and reactor banks which are either switched or controlled continuously by thyristors. The general arrangements are shown below in figure 4-4: FIGURE 4-4: SVC GENERAL ARRANGEMENT 35

50 For the capacitors banks in SVCs may be thyristor switched or fixed capacitor meaning the capacitive Susceptance to vary in steps. The inductor or reactor may be thyristor switched making inductive Susceptance to vary in steps or could be thyristor controlled making inductive Susceptance to vary continuously over the range of inductances. Generally when the line voltage are above 69kV, a step down transformer is used to connect the SVCs to the line so as to reduce cost involved in SVC manufacture. With the development of Power Electronics, in future we could possibly avoid the use of this transformer and making direct connection to the HT line. As from the above configurations, we can see that the SVC can vary its performance form completely capacitive to completely inductive. Considering min maximum Susceptance when SVC is capacitive to be b SVC & maximum Susceptance when SVC is inductive to be b max SVC. Therefore the total Susceptance bsvc introduced by SVC can be varying as shown below, b SVC min b SVC b SVC max EQN 4.4/A The total power injected/removed by SVC can be expressed in terms of the voltage as, Q SVC = b SVC * V 2 EQN 4.4/B When SVC is capacitive the reactive power is generated in the SVC and injected into the network. When SVC is inductive, the reactive power is removed from the network and absorbed in the SVC. Refer Appendix A, section A5 for V- I characteristics of SVC, when operated in voltage regulation mode Modelling of SVC Using the logic behind SVC operation discussed in 4.1 to 4.4 we will be introducing SVC into the transmission line as shown in below, 36

51 FIGURE 4-5: SCHEME OF SVC IN TRANSMISSION LINE The step down transformer connecting the HT line and the SVC substation has been ignored in this case for simplicity. The SVC can be thought of as a parallel connection of two variable inductor and capacitor. The total Susceptance of the SVC can be assumed to be sum of Susceptance of inductor and Susceptance of capacitor (taken negative). Similarly the total reactive power generated or absorbed by the SVC is the sum of inductive and capacitive reactive power. The model can be further simplified by considering the maximum inductance and maximum capacitance to be the same specified by b range SVC. The simplified model of the SVC as a variable Susceptance shunt is shown in figure 4-6 given below, FIGURE 4-6: MODEL OF SVC 37

52 4.4. AC OPF with SVC The AC OPF formulated in chapter 3 section 3.1 will be used as a foundation to introduce the SVC into the power network. There are two main factors regarding use of SVC in the transmission line within the 9-bus test network - placement of SVC & sizing of SVC device. The above stated two are optimisation problems within itself. However as we are focussed on to providing voltage regulation to AC OPF problem, we will simple technique using voltage regulation for placement of SVC in the transmission line. We will assume SVC to be variable from 100MVAr capacitive to 100MVAr inductive there by solving the sizing problem. FIGURE 4-7: SYSTEM WITH SVC Considering voltage regulation discussed in section 4.2, table - 4A and figure 4.3 we can see that the maximum voltage fluctuation or voltage regulation is high for 38

53 line-8 in our 9 bus test network. From the test network data, it can be seen presence of load at bus 9 and therefore we place the SVC in line 8 towards the receiving end of the line. Analysing line 8 using node-admittance matrix as shown in figure Objective Function As discussed in section 3.2 of chapter 3 the objective function for the AC OPF problem with SVC will be to minimize the cost function & power generated. The cost function is given in equation EQN - 3.2/A as Minimise cost = c 2 * (pg) 2 + c 1 * (pg) + c Constraints Constraints for AC OPF with SVC can be broadly classified into equality and inequality constraints. The block diagram shown in figure 4-8 below gives a general idea on how constraints are considered here. FIGURE 4-8: CONSTRAINTS FOR AC OPF WITH SVC 39

54 Equality Constraints The equality constraints for AC OPF with SVC are similar to those of AC OPF discussed in section using KVL & KCL Application of KCL on Each Bus in the System KCL can be applied to the AC system with SVC in terms of the real and reactive power injected into the line. Consider the figure 4-9 shown below. As from the figure it can be seen that reactive power injected from bus k is combined with SVC reactive power (could be absorbing reactive power or injecting reactive power depending on the Susceptance b SVC ) FIGURE 4-9: SYSTEM WITH SVC & KCL From the above diagram and applying KCL, we can formulate the following as in section , pg = PD + (plkm + plmk) qg = QD + (qlkm + qlmk) EQN /A EQN /B The above EQN /A & B forms the KCL equality constraint for AC OPF with SVC. 40

55 Application of KVL on Each Line in the System KVL rule can be applied to AC transmission line with SVC so as to calculate the real & reactive power injected in from the buses. Consider the π model of transmission line as shown in figure 4-10 below, FIGURE 4-10: Π MODEL OF TRANSMISSION LINE WITH SVC As discussed in section, we have placed the SVC close to the receiving end of the transmission line for voltage control. The below given figure 4-11 shows the complete system by considering presence of an LTC or PST at the sending end of the transmission line. FIGURE 4-11: Π MODEL OF TRANSMISSION LINE WITH TRANSFORMER & SVC 41

56 From the above diagram and section , we can formulate the following, ( ) ( ) ( ) ( ) ( ) EQN /A ( ) ( ) ( ) ( ) ( ) ( ) EQN /B Similarly power injected in from bus - m to bus k through line l can be written as ( ) ( ) ( ) ( ) ( ) EQN /C ( ) ( ) ( ) ( ) ( ) EQN /D The above given EQN /A to D represents the equality constraints imposed by KVL on the transmission lines with SVC in the test network SVC Reactive power The Susceptance of SVC can be varied from fully capacitive to fully inductive. When SVC is capacitive it injects reactive power into the line. When SVC is inductive, it absorbs the reactive power from the line. The total reactive power absorbed/injected by the SVC can be formulated as, ( ) EQN /A Where V specifies the bus voltage to which the SVC is connected. 42

57 The above EQN /A specifies the equality constraint for SVC reactive power injection/absorption Inequality Constraints Inequality constraints considered are similar to those discussed in section Generator Constraints Refer section , EQN /A, /B, /C & /D Line Flow Limits Refer section , EQN /A Voltage Magnitude Limits Refer section , EQN /A & /B SVC Susceptance Limits As discussed in section 4.3.1, & 4.4 regarding the SVC design we have considered SVC to vary its performance from fully capacitive to fully inductive with an upper & lower reactance limits fo100mvar (inductive) & -100MVAr (capacitive). This range of Susceptance is specified as b range SVC in per unit values. The constraints can be formulated as, b SVC - 1 * b SVC range (Max capacitive Susceptance) b SVC b SVC range (Max inductive Susceptance) EQN /A EQN /B 4.7. Variables The decision variables involved in AC OPF problem are as listed below, 1. Real power generated from the generating units, pg 2. Reactive power generated from the generating units, qg 3. Real power injection between buses, plkm & plmk resulting in real power loss in transmission, plkm + plmk 43

58 4. Reactive power injection between buses, qlkm & qlmk resulting in reactive power loss in transmission, qlkm + qlmk. 5. Bus voltage magnitudes, V controlled within specified upper & lower limits of the test network 6. Bus voltage angle, Delta 7. SVC Susceptance, b SVC 8. SVC Reactive power injected/absorbed, Q SVC Program for AC OPF with SVC Using Matlab & AMPL The program for AC OPF with SVC is written using Matlab & AMPL as explained in section 3.5 with the addition of components to include SVC. The flowcharts shown below highlights the additional components. FIGURE 4-12: BUILD PROBLEM FOR AC OPF WITH SVC 44

59 FIGURE 4-13: AMPL MODEL FOR AC OPF WITH SVC Refer appendix B, section B2 for model file. Additional print commands for b SVC and Q SVC and is shown in the flowchart shown in figure 4-14 below FIGURE 4-14: AMPL INSTRUCTIONS TO RUN & PRINT 45

60 4.9. Results of AC OPF with SVC The AC OPF with SVC discussed in sections 4.5 to 4.8 is run in Matlab and the following results are obtained. FIGURE 4-15: AC OPF WITH SVC OPTIMISATION RESULTS Figure 4-15 shows the optimal power produced as well as the cost involved. Comparing this with the AC OPF for the same load (discussed in chapter 3, section 3.7 we can see that the results are valid. FIGURE 4-16: BUS VOLTAGE WITH SVC Apart from providing optimal power, we have added SVC to line - 8 in order to improve the voltage regulation or to make the receiving end voltage close to the sending end voltage. The bus voltages & a comparison of sending & receiving end voltages of each line are shown in figures 4-16 &

61 FIGURE 4-17: SENDING & RECEIVING END VOLTAGES WITH SVC Voltage regulation of all transmission lines in the system are shown in figure 4-18 shown below FIGURE 4-18: VOLTAGE REGULATION WITH SVC 47

62 Table - 4B shows the actual values of voltages at the sending & receiving end of each lines along with the voltage regulation. TABLE 4B: LIST OF LINES, BUSES, VOLTAGES & VOLTAGE REGULATION (WITH SVC) LINE REFERENCE FROM BUS REFERENCE TO BUS REFERENCE FROM BUS VOLTAGE TO BUS VOLTAGE VOLTAGE REGULATION In order to study the effects of SVC placed in line 8, a comparison can be made between voltage regulation of AC OPF systems designed in chapter 3 & AC OPF systems designed with SVC. The figure 4-19 shows a comparison between voltage regulations in the two systems. It can be seen that with the same amount of optimal power & cost, there has been considerable improvement in the voltage regulation. The table 4C highlights the effect of SVC on line 8 and is shown below, TABLE 4C: LIST OF LINES, VOLTAGE REGULATION WIHOUT & WITH SVC LINE REFERENCE VOLTAGE REGULATION WITHOUT SVC WITH SVC

63 FIGURE 4-19: COMPARISON OF VR IN SYSTEMS WITH & WITHOUT SVC NOTE: SVC is only included on line-8 as discussed in section Stressed Condition & Varying Load The above designed AC OPF (chapter 3 & chapter -4) was considered for full load (100% PD). In electrical power system the load varies continuously with time. We will discuss such an instance when the load varies from 10% to 100% PD in 10 steps. The cost of generation is calculated for each system (with & without SVC) by varying the load in the build problem and running the code for each case. In order to consider the effect of SVC in the system & power generated, we would consider the case where the generators are stressed. The stressed condition is considered as the reduction in the ability of all generators to absorb reactive power. As the load varies the real power generated as well as the reactive power (generated or absorbed) will vary. When load is removed from the system, the voltage at the bus terminals increase. To restrict the voltage at the bus terminals, the reactive power has to be removed. This reactive power is to be absorbed by the generator. For the generator (considering synchronous generator) to 49

64 absorb the reactive power, it has to be operated in under excited mode. The P G -Q G curve (or capability curve shown in Appendix A, section A6) determines the variation of real power generated & reactive power (consumed or produced) by the generator. Based on the capability curve, the generator may tend to increase its real power generation so as to cope up with the increased absorption of reactive power so as to control the bus voltages. This will lead to imperfect power generation and cause voltage collapse. The optimisation problem becomes infeasible. Infeasibility occurs for all loads up to 50 % of system load, PD. Considering the stressed condition on the system with SVC installed, the optimisation problem becomes feasible and the power is generated at a lower rate close to the normal condition of the generators. SVC absorbs the reactive power once the generators has reached its reactive power saturation. This helps in not only optimal power production but also maintains the bus voltage within the limits. The figure 4-20 given below shows a comparison of generation cost for varying load when the system is stressed. FIGURE 4-20: GENERATION COST WITH VARYING LOAD & GENERATORS STRESSED 50

65 The table given below shows the values (p.u) of real & reactive power, bus voltages & SVC absorbed power for systems with & without SVC when working in normal & stressed condition for a 40% of system load, PD. PARAMETER Pg (listed in order of generator references) Qg (listed in order of generator references) V_bus (listed in order of bus reference 1 to 9) TABLE 4D: COMPARISON OF SYSTEMS WITH & WITHOUT SVC WIHTOUT SVC WITH SVC 40% LOAD (NORMAL) 40% LOAD (STRESSED) 40% LOAD (STRESSED) QSVC NO SVC

66 Chapter 5 - Receding Horizon/Multi Step Optimisation Problem The following topics are discussed - study of general trend in load variations, optimisation problem with varying load, formulation of multi-step AC OPF with SVC on varying load, programming multi step optimisation problem in AMPL, testing & validation of multi-step AC OPF with SVC on varying load, power generation & AGC ramp rate, introduction of AGC ramp rate to multi step optimisation, programming with AMPL & analysis of test results Load Variations in Power System In power systems, the load on the system keeps changing with time. Generally pattern or trend can be made from continuously observing load changes. Study of such trends in load changes have been used to decide the load variations for the receding horizon optimisation problem.we will be considering load changes over three instances as shown in the figure 5-1 below FIGURE 5-1: LOAD VARIATIONS WITH TIME From the above figure we can see that the load, PD specified in 9 bus IEEE RTS system is varied during instants t = 1, 2 & 3 by 30, 100 & 80% respectively. This variation of load will be used in the receding horizon optimisation problem discussed in the next section of this chapter. 52

67 5.2. Optimisation Problem with Varying Load We will be considering load variation with time as discussed in section 5.1. As a result of these load variation we need to vary the power generated at different time intervals. The objective here will be compute the optimal power generation at different time intervals as well as to improve voltage regulation using SVC at these time intervals. In this optimisation problem the power generated from each generator and hence the overall power generated & generation cost during each time instant will be minimized depending on the constraints. Computation at different time interval will be similar to optimisation problem discussed in chapter 4, except for various values of loads during each time instant Objective Function, Constraints & Variables The objective function, constraints & variables for this optimisation problem are same as that discussed in Chapter 3. The variation is that we consider multi step optimisation, i.e. the objective function is optimised for different time instants Objective function The objective function here is to minimise the real power generated & cost based on constraints as well as improve voltage regulation using SVC. The objective function is given below: Minimise cost = c 2 * (pg) 2 + c 1 * (pg) + c 0 This objective function will be optimised for different time instants and according to the load at these instants Constraints The constraints considered here are same as listed in section 4.6 of chapter 4. These are briefly discussed below, Equality Constraints are listed below: 53

68 Application of KCL on Each Bus in the System pg = PD + (plkm + plmk) qg = QD + (qlkm + qlmk) EQN /A EQN /B Application of KVL on Each Line in the System ( ) ( ) ( ) ( ) ( ) EQN /C ( ) ( ) ( ) ( ) ( ) ( ) EQN /D ( ) ( ) ( ) ( ) ( ) EQN /E ( ) ( ) ( ) ( ) ( ) EQN /F SVC Reactive power ( ) EQN /G Inequality Constraints are listed below: Generator Constraints Real Power Generated Min real power limit of generator EQN /H 54

69 Real Power Generated Max real power limit of generator EQN /I Reactive Power Generated Min reactive power limit of generator EQN /J Reactive Power Generated Max reactive power limit of generator EQN /K Line Flow Limits Apparent Power Carrying Capacity of Line Real Power injected + j Reactive Power injected EQN /L Voltage Magnitude Limits Bus Voltage Min Bus Voltage Limit Bus Voltage Max Bus Voltage Limit EQN /M EQN /N SVC Susceptance Limits b SVC - 1 * b SVC range (Max capacitive Susceptance) b SVC b SVC range (Max inductive Susceptance) EQN /O EQN /P Variables The decision variables involved in AC OPF problem are as listed below, 1. Real power generated from the generating units, pg 2. Reactive power generated from the generating units, qg 3. Real power injection between buses, plkm & plmk resulting in real power loss in transmission, plkm + plmk 4. Reactive power injection between buses, qlkm & qlmk resulting in reactive power loss in transmission, qlkm + qlmk. 5. Bus voltage magnitudes, V controlled within specified upper & lower limits of the test network 6. Bus voltage angle, Delta 55

70 7. SVC Susceptance, b SVC 8. SVC Reactive power injected/absorbed, Q SVC Program for AC OPF with SVC & Varying Load The AC OPF is programmed using Matlab & AMPL similar to the program discussed in section 4.8. But here the program is structured by consider the additional time factor for certain parameters. The basic flow chart on how Matlab is used to run the optimisation problem is as shown in figure In order to build data which includes time varying loads real (PD) & reactive (QD) demands we will specify a new set of data time (T). This is shown in the below in figure 5-2, FIGURE 5-2: BUILD PROBLEM FOR AC OPF WITH SVC & VARYING LOAD The set T contains data on different time instant considered. Here we consider three time instants. Based on the different time instants (T = [1 2 3]) we loads varying for each of these time instants specified in PD & QD. This is highlighted in the above flowchart. (Note: The student version of AMPL has a limitation on the number of variables and constraints calculated to 300 variables and 300 constraints. As we consider time 56

71 instants beyond three, we exceed this limitation & hence cannot proceed with computation) FIGURE 5-3: AMPL MODEL FOR AC OPF WITH SVC & VARYING LOAD The figure 5-3 above shows the AMPL model of AC OPF with SVC & varying loads. The load PD & QD are specified in terms of time instants as shown in previous flowchart. Based on the time set T, we define all variables. The optimisation problem will be solved for each time instant separately and the values of all variables are obtained as per the time instant. The below given flowchart (figure 5-4) shows run & print file, 57

72 FIGURE 5-4: AMPL INSTRUCTIONS TO RUN & PRINT AC OPF WITH SVC & VARYING LOAD The interface file is modified to segregate the data from AMPL based on the instant. Once segregated, the values of variables are used to compute the power generated & cost of generation for each instant. This is shown in the flowchart (figure 5-5) below, FIGURE 5-5: MATLAB FILE FOR MANAGING & OPTIMISATION OF AC OPF WITH SVC & VARYING LOAD 58

73 5.3. Multi Step Optimisation Problem Model Testing & Validation As we have formulated the multi step optimisation problem, we could now check and validate the system. The system for multi step optimisation with SVC & load variations as discussed in section 5.1 is run and the results obtained is shown below in figure 5-6, FIGURE 5-6: RESULTS OF MULTI - STEP OPTIMISATION The system described in chapter 4, section 4.4 will be used to test the multi step optimisation problem. The system described in chapter 4 was run using 100% load, PD. This load is varied to 30%, 100% and 80% & the results of optimisation produced are compared against figure 5-6. The results of AC OPF with SVC under various load conditions are shown below, AC OPF with SVC result for 30% load (for t =1), FIGURE 5-7: RESULT OF AC OPF WITH SVC OPTIMISATION WITH 30% PD AC OPF with SVC result for 100% load (for t =2), FIGURE 5-8: RESULT OF AC OPF WITH SVC OPTIMISATION WITH 100% PD 59

74 AC OPF with SVC result for 80% load (for t =3), FIGURE 5-9: RESULT OF AC OPF WITH SVC OPTIMISATION WITH 80% PD Upon comparing results of multi step optimisation (Figure 5-6) and AC OPF with SVC (figures 5-7, 5-8 & 5-9) we can see that the multi step optimisation problem model is valid for varying loads. The next step will be to add in constraint where the power variation during each time instants is limited by the generator constraint Power Generation & AGC Ramp Rate Automatic Generation Control (AGC) are used in power system to control the output of various generating units. As we have discussed in the previous chapters our objective is to minimize the cost of generation as well as keeping the system stable (voltage stability). The AGC are responsible to keep a close balance between the load on the system (including losses in transmission and the actual load) and the power being generated. In electrical power system the load on the system varies at different instants of time. As load on the system increase and power generated is less than required (or vice versa) the system will be driven to instability. Based on these changes in load, it is required to vary the amount of power generated (in an optimal way to keep the generation cost minimum). The AGC generally concerns with the variation of power with variations in load at different instants of time. We will be dealing with one of the factors in AGC which is in direct relation to how power generated is varied, so that we may generate optimal power even the when the load is varying. As we have designed the optimisation problem in section 5.2 & 5.3 to calculate power at different instants of time based on load changes, we also need to consider the fact how generators could cope up with these variations. One of the factors that directly involves with the ability of generators to produce power 60

75 according to the trend in load is the Ramp rate. Ramp rate can be defined as the rate of change in power produced (generally specified in MW) per minute MW/minute. Ramp rate details (specified as RAMP_AGC in column 17 of generator data) for real power is available for 24 bus IEEE RTS. Data from this system will be used to produce ramp rate of generators in our 9 bus test network. Sl No: GENERATOR REFERENCE TABLE 5.4A: IEEE 24 BUS RTS GENERATOR DATA PMAX (MW) PMIN (MW) QMAX (MVAr) QMIN (MVAr) RAMP RATE AGC (MW/MIN) Time taken to reach Pmax (min) 1 1,2,5, , , ,10, ,17,18,19, ,22,31, , ,26,27,28, 29, The complete set of generator data for IEEE 24 bus & 9 bus RTS systems are given in appendix A, section A4. Upon analysis of 24 bus RTS, we find generator data from which we can compute the time taken for each generator to reach maximum power PGmax in minutes. We would be using this data to compute approximate values of ramp rate for the generators in 9 bus IEEE RTS. This observation & calculation is depicted in table 5.4A shown above. The below given table 5.4B shows results of analysis performed on 9 bus IEEE RTS. 61

76 TABLE 5.4B: IEEE 9 BUS RTS GENERATOR DATA Sl No: GENERATOR REFERENCE PMAX (MW) PMIN (MW) QMAX (MVAr) QMIN (MVAr) RAMP RATE AGC (MW/MIN) Unavailable A comparative study has been performed to produce an approximate value for ramp rate for the 9 bus IEEE RTS. We have compared the capacity of generators (Pmax) as well as the ramp rate specified for 24 bus IEEE RTS in table 5.4A against the capacity (Pmax) of generators for 9 bus IEEE RTS to produce an approximate value of ramp rate for generators in the 9 bus IEEE RTS. Additionally, a stressed condition for generators are also considered by reducing the ramp rate. The approximate value as well the stressed condition values for ramp rate are as shown in table 5.4C given below Sl No: GENERATOR REFERENCE TABLE 5.4C: IEEE 9 BUS RTS GENERATOR RAMP RATE DATA PMAX (MW) TIME TAKEN TO REACH PMAX (MIN) RAMP RATE AGC (MW/MIN) CASE-1* CASE-2 CASE - 1* CASE * APPROXIMATED Ramp rate from comparitive study [Note: Case 1 represents the ramp rate considered from comparative study whereas case 2 represents the stressed condition by assuming each generator takes double the time to reach it maximum generation capacity, P max ] The ramp rate is specified in MW/min & is to be converted into the per unit value of before it can be used for AC OPF. This is done by dividing the obtained ramp rate in MW/min by the base MVA 62

77 5.5. Multi Step Optimisation Problem with Constraint We have formulated the AC OPF with SVC for varying load in section 5.2. Load variations are considered for every hour specified in section 5.1. In actual power systems, AGC imposes ramp rate constraint on the power generated by the generating units. We would use this as a constraint to form the receding horizon or multi step optimisation problem. The optimisation problem would be feasible only if the all the generators in the system can supply power (in one hour) by staying within ramp rate limits for an hour. The following sections of this chapter describes the objective functions, constraints & variables involved in the Multi step AC OPF with SVC voltage control Objective function The objective function here is to minimise the real power generated & cost based on constraints as well as improve voltage regulation using SVC. The objective function is given below: Minimise cost = c 2 * (pg) 2 + c 1 * (pg) + c 0 This objective function will be optimised for different time instants and according to the load at these instants Constraints The constraints considered here are same as listed in section 4.6 of chapter 4 with addition of AGC ramp rate constraint. Generally constraints can be split into equality & inequality constraints. The figure 5-10 given below provides an overview of all constraints involved. 63

78 FIGURE 5-10: RECEDING-HORIZON AC OPF WITH SVC CONSTRAINT BLOCK DIAGRAM Equality Constraints Application of KCL on Each Bus in the System Refer EQN /A & /B Application of KVL on Each Line in the System Refer EQN /C, /D, /E & /F SVC Reactive power Refer EQN /G Inequality Constraints Generator Constraints Refer EQN /H, /I, /J & /K 64

79 Line Flow Limits Refer EQN /L Voltage Magnitude Limits Refer EQN /M & /N SVC Susceptance Limits Refer EQN /OM & /P AGC Ramp Rate Constraint In order to explain the AGC ramp rate constraint we will consider an example situation. Consider a system consisting of a load and single generator. The load variation as well as the power output from the generator is shown in the graph below Power Generation vs. Time. Let the load variation occur every hour and this time change is denoted by t, t +1, t +2 etc & accordingly let the power output from the generator be P t, P t+1, P t+2 etc, Consider the instance from t to t +1. The load variation during this time frame can be written as P t+1 - P t The rate at which the load has changed over an hour can be written as Now considering the ramp rate of the generator with reference to the graph shown below. Ramp rate can be defined as the increase in real power output of the generator over a minute. The ramp rate can be represented as, 65

80 FIGURE 5-11: BASIC IDEA ON AGC RAMP RATE For the generator to maintain power supply to the load, it must vary its power generated at a rate faster than the rate at which load varies. In other words the ramp rate of generator over an hour must be greater than or equal to the rate of load variation over an hour. This can be expressed in general form, Change of generated power output over an hour Ramp rate (MW/min) * 60 EQN /A Every generator in the system differ in their characteristics and ramp rates. Based on this the AGC constraints are to be applied for each generator in the system. The above EQN /A forms the AGC ramp rate constraint for the optimisation problem Variables Refer section

81 5.8. Program for Receding Horizon AC OPF with SVC Program for receding horizon AC OPF with SVC will be built from the program discussed in section for the AC OPF with SVC & varying load.. The entire optimisation program is run with basic structure shown in figure 3-11 in chapter 3. Additional constraint considered is the AGC ramp rate for each generator discussed in section 5.5. This additional parameter is included in the build problem section and is shown in figure 5-12 given below. FIGURE 5-12: BUILD PROBLEM FOR MULTI-STEP AC OPF WITH SVC Additional constraint for the multi-step optimisation is derived from the AGC ramp rate and is included in the program. This basically compares the variation of power at different time instants specified in the time set T. This constraint is highlighted in the flowchart given below which forms the model of the OPF problem in AMPL. Refer appendix B, section B3 for model file. 67

82 FIGURE 5-13: AMPL MODEL FOR MULTI-STEP AC OPF WITH SVC 5.9. Results & Analysis of Multi-Step AC OPF with SVC The multi-step/receding horizon optimisation problem formulated in sections 5.6 to 5.8 is run using Matab. We have considered two cases with different AGC ramp rate as shown in table 5.4C. The results are as shown below, For case -1, with generator ramp rates of 4.17, 10 & 4.5, we have power produced & cost of generation as shown in figure 5-14 given below, 68

83 FIGURE 5-14: HOURLY GENERATED POWER & COST WITH AGC CASE-1 The voltage regulation for every bus in the system through instants 1, 2 & 3 for case 1 are shown in figure 5-15 given below FIGURE 5-15: VOLTAGE REGULATION FOR AGC CASE 1 For case -2, with generator ramp rates of 2.08, 5 & 2.25, we have power produced & cost of generation as shown in figure 5-16 below, FIGURE 5-16 HOURLY GENERATED POWER & COST WITH AGC CASE-2 69

84 FIGURE 5-17: VOLTAGE REGULATION FOR AGC CASE 2 The voltage regulation for every bus in the system through instants 1, 2 & 3 for case 2 are shown in figure 5-17 given above. A table comparing voltage regulations of systems with & without SVC for AGC case 1 & AGC case 2 is shown in Appendix A, section A7. As from the above two cases of AGC ramp rates we can see that all the generators in the system when working together can take up the load and produce enough power to supply changing load. In order to study the variation of power consider another case with generator 2 & 3 with lower ramp rates. Hence the new ramp rates are [2.08,1,1] and considering it as case 3. The following results were obtained with this conditions. FIGURE 5-18: HOURLY GENERATED POWER & COST WITH AGC CASE-3 70

85 A comparison study between AGC case 2 & case -3along with variations in generating cost is shown in table 5.4D given below PARAMETERS TABLE 5.4D: COMPARISON ON COST & POWER GENERATED AGC CASE-2 & LOAD VARIATIONS AGC CASE-3 & LOAD VARIATIONS t=1 t=2 t=3 t=1 t=2 t=3 Gen. 1 pg Gen.2 pg Gen.3 pg Total Generation Cost From the above table it can be seen that price variation occurs for loads at t-1 & t=3. The reason for additional cost is because generator 3 power production is varied (lowered ramp rates) in case 3, therefore other generators 1 & 2 will have to produce the additional power. The cost characteristics of all generators are differ & are specified by the parameters c 0, c 1 & c 2. These results in the variation of cost even though total power produced is the same. 71

86 Chapter 6 - Conclusion & Future Work This chapter summarises all the work performed in the dissertation. Furthermore development of the system are also discussed in future work section. Conclusion Formulation of AC optimal power flow has been performed & results produced shows how optimal power can be produced by controlling various parameters in the power system. The results were verified to be true by comparing with Matpower test results on the same system. Apart from producing power at an optimal rate, system security or power stability was considered. We have considered the problem of voltage instability in the system & shunt reactive power compensation was made use by Static VAR compensators. The placing & sizing of SVC are optimisation problem on its own. A simple strategy using voltage regulation has been used to determine the placing of SVC, The effect of SVC in the system has been analysed by comparing the voltage regulation of each line in the system. It is shown that by using SVC in line 8 in the 9 bus IEEE RTS the voltage regulation on line 8 is improved. Furthermore the effect of SVC in case of load variations along with stressed conditions of generators has been considered. A comparative analysis of systems with & without SVC shows that the system is stabilised for lower loads and generation cost is reduced when the system is loaded to 50% of full load. As with power systems, the load on the system varies with time. This is taken into consideration & multi-step optimisation or a receding-horizon problem is created for the system. The basic requirement here was to produce optimal power during the load changes and also to keep the power generated within the ramp rates of AGC for the generating units. As the ramp rates of generators in AGC was not available for the 9 bus system, approximate ramp rates were considered by a comparative study with ramp rates of generators in 24 bus IEEE RTS. We have formulated different cases of ramp rates and investigated the performance of the receding horizon problem. A stressed case when the ramp rates of generators were low are also considered and the 72

87 results were analysed for the variations in generation cost even though load variations were same as in the previous case. With the increasing demand in power across systems in the world as well as integration of different types of generating units, we require an optimal system for power generation and at the same time ensuring reduced power system failures. The receding horizon AC OPF with SVC designed in this dissertation forms a basic logic to generate power at a lower generation cost as well as ensuring voltage stability. The optimisation problem are made use in planning & operation of power systems. Receding-horizon provides an opportunity to combine higher decision levels (OPF) & lower decision levels (voltage & frequency control). Once the future variations in load as well as other variations in generation( for example wind power, where the exact amount of power generated cannot be predicted) when solving OPF, we could possibly find solutions which will require less corrective control measures when later. Future Work The optimisation problems listed in this dissertation forms a foundation to optimal power flow in AC power systems. We have considered the 9 bus IEEE RTS for building an optimisation problem so as to minimise the cost of generation as well as to regulate the bus voltages using SVCs. A larger network with different types of generating units such as nuclear, thermal, hydroelectric, renewable resources like solar, wind energy etc. can be modelled as generating units. Each unit has different characteristics in terms of cost factors, power output, reactive power absorption/injection, start-up times, maintenance schedule etc. All these factors can be taken into consideration. Also considerations can be given to power trade between different operators. The modelled systems are to be tested with different operating conditions matching the real world system. Apart from optimal power generation other factors such as power quality & system security has to be taken into account. In this dissertation we have considered 73

88 SVC to control system voltage. Similarly we could consider frequency instability & rotor angle instability, analyse the system performance with forced outage or instability. The use of different FACTS devices as well as auxiliary devices in the system are to be taken into consideration. Moreover an optimal reactive power flow (ORPF) equation can also be created & could be solved simultaneously with OPF. A multi objective optimisation problem can be formed by combining ideas of cost effective power generation and system security. 74

89 REFERENCES Andersson, G. (2008). Modelling and Analysis of Electric Power Systems. Zurich. Bialek, J. W. (n.d.). Recent blackouts in US, UK and Europe: Is it contagious?. Edinburgh: University of Edinburgh. Dom ınguez-garc ıa, A. D. (n.d.). Models for Impact Assessment of Wind-Based Power Generation on Frequency Control. Ghosh, P. A. (n.d.). Power System Analysis. NPTEL. Glavic, D. M. (n.d.). POWER SYSTEM VOLTAGE STABILITY. University of Liège. Global Insight 2008, Siemens E ST MOP 10/2008. (n.d.). Global Insight 2008, Siemens E ST MOP 10/2008. Jody Verboomen, D. V. (n.d.). Phase Shifting Transformers: Principles and Applications. Kundur, P. (n.d.). Power System Dynamics & Stability. Matpower: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education," Power Systems, IEEE Transactions on, vol. 26, no. 1, pp. 12. (2011). McCalley, J. D. (n.d.). Introduction to System Operation, Optimization, and Control. Iowa State University. Montefiore Institute, EEE. (n.d.). Power System Angle Stability. National Electric Code. (2011). Paul Trodden, W. A. (n.d.). Local Solutions of Optimal Power Flow. Taylor, C. W. (1994). Power System Voltage Stability. McGraw-Hill. 75

90 Xiao-Ping Zhang, C. R. (2006). Flexible AC Transmission Systems: Modelling & Control. Xiao-Ping Zhang, C. R. (n.d.). Flexible AC Transmission Systems: Modelling & Control. Zhu, J. (n.d.). OPTIMIZATION OF POWER SYSTEM OPERATION. IEEE Press. 76

91 APPENDIX A1. THE PER UNIT SYSTEM NOTE: Base power used is apparent power. A2. MATPOWER CODE Please follow the below given link to download the latest version of Matpower. Instructions to use Matpower: 1. Upon download change Matlab directory to the folder containing Matpower. 2. Test run Matpower by using command - test_matpower 3. Once test is complete load the AC optimisation problem for 9 bus IEEE RTS using command - runopf('case9') 4. The following results are displayed on screen. 77

92 A3. TEST NETWORK DETAILS The test network used for the dissertation is 9 bus IEEE RTS. Details on generator, bus, line (branch) & generator costs are as given below. Generator Details Gen_status: 1 represents generator in service 78

93 Bus Details Bus Type: 1 for PQ, 2 for PV, 3 for slack bus, 4 for isolated bus. Branch Details Generator Cost Data 79

94 A4. 9 BUS IEEE RTS WITH SVC 80

95 A5. SVC V-I CHARACTERISTIC CURVE A6. GENERATOR CAPABILITY CURVE (PG-QG CURVE) 81