Improving Low Voltage Ride- Through Requirements (LVRT) Based on Hybrid PMU, Conventional Measurements in Wind Power Systems

Transcription

1 Improving Low Voltage Ride- Through Requirements (LVRT) Based on Hybrid PMU, Conventional Measurements in Wind Power Systems Förbättra Lågspänning Rider Genom Krav (LVDT) Baserat på Hybrid PMU, Konventionella Mätningar i Vindkraftsystemet Chinedum John C. Ekechukwu Faculty of Health, Science and Technology Master s program in Electrical Engineering Degree Project of 30 credit points Internal Supervisor: Arild Moldsvor, Karlstad University, Sweden Examiner: Jorge Solis, Karlstad University, Sweden Date: Serial Number:

2 Abstract Previously, conventional state estimation techniques have been used for state estimation in power systems. These conventional methods are based on steady state models. As a result of this, power system dynamics during disturbances or transient conditions are not adequately captured. This makes it challenging for operators in control centers to perform visual tracking of the system, proper fault diagnosis and even take adequate preemptive control measures to ensure system stability during voltage dips. Another challenge is that power systems are nonlinear in nature. There are multiple power components in operation at any given time making the system highly dynamic in nature. Consequently, the need to study and implement better dynamic estimation tools that capture system dynamics during disturbances and transient conditions is necessary. For this thesis work, we present the Unscented Kalman Filter (UKF) algorithm which integrates Unscented Transformation (UT) to Kalman filtering. Our algorithm takes as input the output of a synchronous machine modeled in MATLAB/Simulink as well as data from a PMU device assumed to be installed at the terminal bus of the synchronous machine, and estimate the dynamic states of the system using a Kalman Filter. We have presented a detailed and analytical study of our proposed algorithm in estimating two dynamic states of the synchronous machine, rotor angle and rotor speed. Our study and results shows that our proposed methodology has better efficiency when compared to the results of the Extended Kalman Filter (EKF) algorithm in estimating dynamic states of a power system. Our results are presented and analyzed on the basis of how accurately the algorithm estimates the system states following various simulated transient and small-signal disturbances. Keywords: State estimation, Power systems, Unscented Kalman filters, Phasor Measurement units (PMUs) i

3 Acknowledgments This thesis has been made possible because of the invaluable support, guidance and encouragement that I have received from a number of individuals. Without listing such persons in any particular order, I thank you my supervisor, Professor Arild Moldsvor who has been supportive through the period during which I have been writing this thesis. I also want to thank my examiner, Professor Jorge Solis, for his contributions toward successful completion of this work. Thanks to all the professors, staff and students of Karlstad University who have influenced me in one way or another while I have been studying at Karlstad University. Special thanks and profound gratitude to the following people: my parents, Engr. and Mrs. Lucius Ekechukwu who have given me immeasurable support and encouragement beyond words; my siblings, my adorable Loretta and my brother in-law; Engr. Chibike Nnorom. You all have encouraged and supported me in every way possible. Your various roles in my life, before and during my studies as well as through the development of my career thus far, continue to humble me. Words fail me in expressing my gratitude. I would also love to thank my friends, both in Karlstad and around the world. For those in Karlstad, thank you all for making my transition a smooth one when I arrived to Sweden initially. You all remained very supportive. Above all, I will forever bless God and continue to thank Him for my life. ii

4 Contents Abstract... i Acknowledgments...ii List of Figures... v List of Abbreviations... vi 1 Introduction Project Motivation Problem definition Kalman Filtering Main Contribution and objective Thesis outline Wind Power Systems Background Wind power systems requirements and limitations Low Voltage Ride through (LVRT) Power System Stability Modeling the wind power system Measurement acquisition methods in power systems Conventional/SCADA measurements Measurement model and assumptions Synchronized Phasor Measurements Units (PMUs) Measurement model and assumptions Mixed (PMU and Conventional/SCADA) measurements State Estimation Static State Estimation (SSE) Forecasting Aided State Estimation (FASE) Incorporating measurements to FASE Proposed methodology Unscented Transformation (UT) The Unscented Kalman Filter (UKF) UKF based state estimation in power systems (UKF/SE) Simulations and Results Implementation of our proposed methodology Results iii

5 6.3 Discussion Conclusions and future work Conclusions Future work References Appendix A: Holt s two initialization Appendix B: Gradient calculation for EKF algorithm method Appendix C: Machine parameters used and states to be estimated Appendix D: UKF algorithm iv

6 List of Figures Fig. 2.1: Required ride through capability of wind farms for Super grid Voltage dips of duration greater than 140ms... 6 Fig. 2.2: Single machine infinite bus system Fig. 2.3: Schematic description of the powers and torque in synchronous machines Fig. 3.1: Elements of the EMS/SCADA system Fig. 3.2: Two-port π-model of transmission line Fig. 3.3: Phasor illustration Fig. 3.4: Hierarchical placement of phasor measurement units Fig. 3.5: Two-port π network Fig. 5.1: Unscented Transformation, showing sigma points being transformed Fig. 5.2: Unscented Transform (UT) block diagram Fig. 6.1: Proposed block diagram of UKF estimator Fig. 6.2: Top level of the Implmentation block in Simulink Fig. 6.3: Layout of the synchronous machine connected to the transmission line Fig. 6.4: Mechanical part Sub-model of the synchronous machine Fig. 6.5: Electrical part Sub-model of the synchronous machine Fig. 6.6: Powergui machine initialization tool Fig. 6.7: Output power Pt measured at the terminal bus Fig. 6.8: Generator rotor angle Fig. 6.9: Generator rotor speed Fig. 6.10(a): Real rotor angle Vs estimated angle using the UKF algorithm Fig. 6.10(b): Real rotor angle Vs estimated angle using the EKF algorithm Fig. 6.11(a): Real rotor speed Vs estimated speed using the UKF algorithm Fig. 6.11(b): Real rotor speed Vs estimated speed using the EKF algorithm Fig. 6.12(a): Rotor angle response with simulated fault cleared in 0.07s for the UKF algorithm Fig. 6.12(b): Rotor angle response with simulated fault cleared in 0.07s for the EKF algorithm Fig. 6.13: E fd Step input v

7 List of Abbreviations AMI Advanced Metering Infrastructure CPU Central Processing Unit DGs Distributed Generators DS Dynamic State DSE Dynamic State Estimators DR Distributed Resources DFIG Doubly Fed Induction Generator EMS Energy Management Systems EMS Energy Management Systems EKF Extended Kalman Filter FASE Forecasting-Aided State Estimation GPS Global Positioning System GLFS Generation and Load Forecast System HVDC High Voltage Direct Current LVRT Low Voltage Ride through MASE Multi Area State Estimation OPF Optimal Power-Flow ODEs Ordinary Differential Equations PMUs Phasor Measuring Units PDF Probability Density Function RTUs Remote Terminal Units SMIB Single Machine Infinite Bus SV State Variable SCADA Supervisory Control and Data Acquisition SEA State Estimation Algorithms SSE Static State Estimation SR-UKF Square-Root Unscented Kalman Filter UKF Unscented Kalman Filter UT Unscented Transformation WLS Weighted Least Squared vi

8 1 Introduction With the emergence of distributed generation and the need to tie distributed generators (DGs) to the smart grid while maintaining power systems stability, reliability and security continues to be of great concern in today s energy development across the world. An important angle to this requires adequate and continuous knowledge of the power systems state. Energy management systems (EMS) play significant roles in obtaining system state through monitoring and state estimation. State estimations in the past have been performed by static approach which is based on the weighted least squared (WLS) method. Although this approach is simple to implement and has fast convergence since it uses a single set of measurement, its ability to predict future operating points of the system remains limited. These have led to several cases of severe system instability and in some cases complete system collapse or blackout, as operators could not foresee impending system contingencies and risks to the system. Secondly, such static estimators must be reinitialized for every new measurement without using predictions from previous state estimators [1]. In addition, the quasi steady-state nature of the system network due to slow dynamic load changes makes state estimation quite a challenging task. For this reason, estimators which take into account the changing dynamics of the system are required and have been developed for state estimation in recent times. Such estimators are called Dynamic State Estimators (DSE). DSEs have the ability to predict the systems state vector one step ahead, providing a foresight to potential contingencies and security risks. They also take multiple samples of measurements which provide fast and accurate estimation for each time sample [1]. In addition, they provide state estimates at a particular time instant based on past and present measurements. 1.1 Project Motivation The motivation for this thesis is to investigate how these DSE are used for power system estimation on a broad view and more closely for the estimation of the synchronous generator states, which is presented in this thesis as a part of the power system. State estimation requires adequate and accurate measurements of system parameters and or states that can be measured so as to facilitate estimation of states which cannot be directly measured. Measuring devices such as Phasor Measuring Units (PMUs), which were developed in the early 80s to supplement measurements provided by conventional means, have greatly improved state estimation over time considering they offer near real-time monitoring of the system. Their measurements are synchronized through the GPS space-based satellite navigation system. They have added a new dimension to state estimation. 1

9 We are also motivated to investigate and present how hybrid PMUs, Conventional measurements can be integrated to our proposed forecasted estimation scheme, the Unscented Kalman Filter (UKF), for state estimation. 1.2 Problem definition The main task to be accomplished is to estimate the rotor angle and rotor speed of the synchronous generator modeled in MATLAB/Simulink with a single machine infinite bus system (SMIB). Considering that all states cannot be directly measured, we are saddled with the responsibility of developing a state estimation algorithm as well as filtering out the synchronous machine s rotor angle and rotor speed in the presence of simulated small disturbances and transient faults. Thus, given a number of control inputs and measurable observations, this thesis work focuses on estimating the dynamic states of the synchronous machine during simulated system disturbances. 1.3 Kalman Filtering In this section, we present an overview of the Kalman filter so as to lay a foundation for better understanding of the proposed filtering method and to show the motivation for the choice of the proposed methodology. Kalman filters have remained one of the best filtering methods for estimating dynamic states in a dynamic system using noisy measurements. They are considered to be recursive filters. They are known as linear quadratic estimators and when mixed with linear quadratic regulator, they solve linear quadratic Gaussian control problems. The Kalman filter is named after Rudolph E. Kalman, who published his famous paper describing a recursive solution to the discrete-data linear filtering problem in Kalman filtering was then used in the 70 s when the term dynamic state estimation was first introduced [2], [3] in combination with a steady state power system, to track the static states, bus voltages and phase angle. Kalman filters were used to improve the computational performance of the traditional steady state estimation process used for power system applications. Kalman filters are a set of mathematical equations used to implement a predictor-corrector type estimator which is optimal considering that it minimizes the estimated error covariance when some assumed conditions are met [4]. Kalman filtering is generally divided into two steps: Predict and Update steps. The predict step, also known as time update, uses previous state estimate to predict present estimates of the state variable. It is responsible for forward projection in time of the state estimate so as to obtain a priori estimate for the next time step. It is considered the predictor equation in the predictor-corrector relationship. The update step which is also called the measurement update, utilizes measurements obtained at the present time step to correct the 2

10 estimated state variables. They are responsible for the feedback, i.e., incorporating the new measurements into the a priori estimate to obtain an improved a posteriori estimate. It is considered the corrector equation. Unlike the Weiner filter which operates on all data directly for each estimate, the Kalman filter repeats the process after each time and measurement update, using the previous a posteriori estimates used to predict the new a priori estimate. This recursive nature of the Kalman filter makes it a unique filtering method. Considering that most dynamic processes in power systems are non- linear in nature, there is the need for a filter which estimates these non-linear processes. The Extended Kaman filter (EKF) can be used in place of the discrete Kalman filter considering its ability to linearize about the current mean. Assuming now that the process is no longer governed by linear equations but by non-linear stochastic differential equations [4], the nonlinear functions thus relates the states at time instant say 1 to states at the present time. The nonlinear measurement functions relate the states at time instant to measurements at time instant. In real time estimation or practice, the value of the noise is of course not known. As such, approximate values of the state and measurement vector can be computed. However, the Jacobian matrices of EKF filters are different for each time step and as such should be recomputed at each time step. Of particular interest is the fact that the EKF losses the distributions of its random variables after transformation due to linearization. Although the EKF is still a useful tool for filtering purposes, the novel Unscented Kalman filter (UKF) which is a more iteratively accurate tool for filtering has been proposed to be used for this thesis work. It is known for the fact that it preserves the distributions throughout the nonlinear transformation. The UKF is an efficient discretetime recursive filter. It is based on unscented transformation (UT). Unlike the EKF, the UKF does not linearize the non-linear equations giving it a major advantage over the EKF. 1.4 Main Contribution and objective We have modeled the power system in MATLAB/Simulink. As we have just previously established, the proposed UKF algorithm is built upon the Kalman filtering process. We have made modifications to the standard UKF algorithm to perform dynamic state estimation in power systems as is the focus in this thesis work. Our algorithm tracks the changes in the state vector of an SMIB system while we simulate disturbances in the system. In contrast to a similar prior publication in [6], we have provided a second order model of the synchronous machine using MATLAB/Simulink. We have also presented a detailed description of the implementation of the UKF algorithm using MATLAB s embedded function block. A rigorous description of the algorithm s codes is presented and two dynamic states are estimated based on the second order model. 3

11 Incidentally, many papers have treated power systems state estimation but not much emphasis have been placed on the dynamic states. We thus seek to apply the proposed algorithm to state estimation in dynamical systems and compare its efficiency to the EKF filtering algorithm [7]. Our results will show that the proposed algorithm is very relevant for stability analysis in wind energy systems and today s smart grid developments. 1.5 Thesis outline The remaining chapters of the thesis are as follows: In chapter 2, we discuss wind power systems with emphasis on conditions and requirements necessary for as well as limitations in integrating wind driven systems to the smart grid. We also define and discuss stability conditions and finally model the wind power system. In chapter 3, we will discuss some measuring techniques and devices. In chapter 4, we will present state estimation, giving background knowledge on the evolution of state estimation. We will also introduce the FASE system. In chapter 5, we will discuss our proposed methodology. In Chapter 6 we will present our simulations and results as well as discussions. We conclude in chapter 7 with future works and conclusions. 4

12 2 Wind Power Systems 2.1 Background The need to meet the challenging demands for sustainable and reliable energy across the globe has kept engineers and scientist on continuous research for new and improved energy resources. This constant search led to the discovery of Distributed Resources (DR) such as wind, solar photo-voltaic, solar thermal, small hydro, micro-turbines etc and energy storage systems. With the emergence of such improved energy generation technologies came the challenge of improving control measures to maintain power system stability and security during interconnections. This necessitated the need for stringent installation and connection requirements and conditions for the wind farms. 2.2 Wind power systems requirements and limitations Wind energy installation requirements are summarized in [8], where the impact of the wind installations on the power grid is discussed. Recommendations for power quality limits that need not be exceeded during the operation of wind installations are also discussed in that literature. However, we will dive a little into such requirements and demands for the purpose of the background knowledge needed for our work. Another important aspect of wind power system involves proper synchronization of these wind installations to the grid without compromising system stability and security. In [9], a proposed definition for power system stability states Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact. We will give further discussions on stability in a later paragraph Low Voltage Ride through (LVRT) Recent grid codes require that wind farms remain connected to the grid during severe grid disturbances, ensuring fast restoration of active power to the pre-fault level as soon as the fault is cleared and in certain cases produce reactive current in order to support grid voltage during disturbances [10]. During such disturbances or faults on the grid, there are usually voltage dips, depending on the severity of such faults. These codes require wind farms, especially those connected to HV grids, to be able to withstand voltage dips to a certain percentage of the nominal voltage for a specified duration. Such fault ride through requirements is known as Low Voltage Ride through (LVRT) requirements. 5

13 Voltage dips resulting from transient faults inevitably cause torque and power transient stability problems in the wind turbine, for instance that of the DFIG, due to the asynchronous nature of the turbines. These problems do not depend on the rotor angle but on voltage issues and fault ride through capability of the generators. During these faults, the short circuit current contributions of the wind generators are low. In the past, it was acceptable to have wind turbines disconnect during faults in the network. However, the impact of such disconnections on system stability has remained adverse. The required fault ride through behavior of a wind farm can be summarized into four requirements [10]: 1) For system faults that last up to 140 ms the wind farm has to remain connected to the network. For super grid voltage dips of duration greater than 140 ms the wind farm has to remain connected to the system for any dip-duration on or above the heavy black line shown in Figure ) During system faults and voltage sags, a wind farm has to supply maximum reactive current to the Grid System without exceeding the transient rating of the plant. 3) For system faults that last up to 140ms, upon the restoration of voltage to 90% of nominal, a wind farm has to supply active power to at least 90% of its pre-fault value within 0.5 sec. For voltage dips of duration greater than 140 ms, a wind farm has to supply active power to at least 90% of its pre-fault value within 1 sec of restoration of voltage to 90% of nominal. 4) During voltage dips lasting more than 140ms the active power output of a wind farm has to be retained at least in proportion to the retained balanced super grid voltage. Fig. 2.1: Required ride through capability of wind farms for Super grid Voltage dips of duration greater than 140ms 6

14 However, for scenarios where less than 5% of the turbines are running or under very high wind speed conditions where more than 50% of the turbines have been shut down, a wind farm is permitted to trip or go off the grid Power System Stability The major analytical tools for supervisory, control and planning of the electric power grid are the power-flow analysis and stability programs. Power-flow programs include conventional power-flow analysis and optimal power-flow (OPF) analysis, to mention a few. However, in this sub-chapter we will concentrate on the stability programs. The need to maintain power system stability, during or off fault conditions, remains an integral part of the power systems generation, transmission and distribution schemes. The behaviour of the system following a disturbance is of interest in stability analysis programs. Studies of this behaviour are called transient stability analysis. To achieve a better overview and structure of stability analysis of power systems, it is important to categorise stability into two main categories [11]: small signal (or small-disturbance) stability and transient stability. Small signal stability is the ability of the system to remain in synchronism under small disturbances. These disturbances are as a result of small variations in the loads and generations. Instability that may result in this case is: 1) steady increase in the rotor angle as a result of insufficient synchronizing torque, or 2) increased amplitude of rotor oscillations as a result of insufficient damping torque. Transient stability is the ability of the system to remain in synchronism under severe transient disturbances. Instability that may occur involves large excursions of the generator rotor angles resulting from the nonlinear power-angle relationship of the rotors of the synchronous machine. Thus, we can define stability in this sense as the ability of the system machines to recover from disturbances and still maintain synchronism. In analysing disturbances, the contingencies used are the various types of short circuits: phase-toground, phase-to-phase-to-ground, or three-phase. These contingencies will be used to analyse the estimation method proposed to be used for this in a later chapter. Categorising stability into the two mentioned categories makes analysis easier as well as understanding the nature of stability challenges or problems. As earlier mentioned, the dynamic condition of the power system can be initiated by disturbances in the system. Considering the nature of the parameters, such disturbances could be as a result of changes in the line impedance during short circuit on a transmission line, switching on of a large block of loads or opening of a line. However, the dynamic behavior of the system after disturbance depends on how large such disturbances are, considering that the power system is nonlinear. The 7

15 following comments from [10] give further insights to some crucial aspects of the definition of stability earlier given: 1) It is not necessary that the system re-gains the same steady state operating equilibrium as prior to the disturbance. This would be the case when for instance; the disturbance has caused any power system component (line, generator, etc.) to trip. Voltages and power flows will not remain the same after the disturbance due to changes in the system topology or structure. 2) It is important that the final steady state operating equilibrium after the fault is acceptable. Else, protection or control actions could introduce new disturbances that might further influence the stability of the system. Acceptable operating conditions must be clearly defined for the power system under study. Power system stability can thus be discussed under three main classifications: Rotor Angular or Synchronous Stability, Frequency Stability and Voltage Stability Rotor angular or synchronous stability The sum of the total power output fed into the power system (or infinite bus system as is used to model the power system in this work) is equal to the sum of the active power consumed by the loads, including losses in the system. However, the power fed into the generators by their prime movers, such as the wind, steam, gas or hydro turbines are not always equal to the power consumed by the loads. If such an imbalance occurs, perhaps due to a three phase fault, the rotating parts of the generator act as energy buffers and thus increase or decrease their kinetic energy. This increase or decrease could cause the generators to fall out of synchrony since they have been accelerated during the fault, depending on how close the fault is to the generator or how severe the disturbance or fault is. Rotor angle stability is thus the ability for the synchronous machines of a power system to maintain synchrony after a disturbance to the system Frequency stability In comparison to the previously described scenario for rotor angular instability, if the power fed from the prime movers is less that that consumed by the loads, including losses, such imbalance results in frequency instability in the system. The kinetic energy stored in the rotating parts of the synchronous machine and other rotating electrical machines will compensate for the frequency imbalance by regulating the active power input from the prime movers for cases where the imbalance is not too large. For large deviations in the frequency imbalance, the generators in most cases are cut off because such imbalances cause detrimental oscillations in the turbines. This is why in most control rooms in power generating stations, the system frequency readings are usually displayed boldly and 8

16 are closely monitored. This form of instability is termed frequency instability and it plays an important role in power systems generation and transmission Voltage stability We have just stated in the previous subchapter that the generators are very important for the analysis of angular instability. It is often times said that these are the driving force in instability. However, more detailed studies and analysis show that the loads on the system are more often the driving force in terms of voltage instability. 2.3 Modeling the wind power system In modeling the wind power systems, it is almost impossible to develop models that perfectly describe all dynamics of a power system and be sure that they can be used practically. This is true for two reasons: first, such models will require so much uniquely defined parameter data for the whole system. Hence, only models that capture adequately the specific system dynamics and interaction to be analyzed or investigated are used. Secondly, analysis of such a system would result in great or huge amount of computational work and may be almost impossible. The ability to make informed analysis, review and take necessary actions is an integral part of power systems engineering. It is important to have as precise as possible, input information for power system analysis softwares so as to ensure little or no errors in system modeling and design. When modeling the power system or any other system, care should be taken such that the models chosen are not too simple, as such, misrepresenting the interactions and processes within the system. It is also very detrimental if wrong parameter data are used in modeling the system. In principle, these design parameter data describing the system should be available for soft ware developers. But in reality, there are parameters that may be genuinely unknown, for instance, the ground resistivity under a power line. The importance of having adequate and sufficient parameter data for engineering systems modeling and design cannot be overemphasized and as such it could be very important, though costly in practice, to keep and update data bases for these parameters. Today, such practice is called data engineering. The working principle of the wind farm requires that the wind is harnessed through the generator s rotor blades, which are connected to the wind turbine through a shaft. This mechanical energy is subsequently converted to electrical energy using electrical generators. Energy is transferred through the grid transmission lines and subsequently distributed for consumption. For the sake of simplification, we will assume a variable speed wind turbine generator model based on the DFIG technology and assume its state estimates to be a part of those of a larger power system. The order of the synchronous generator model used determines the number of states to be estimated and of 9

17 course the level of complexity of the estimation process. Several orders of a generator model have been proposed and discussed by various authors. In [12], a third order SMIB model is used. In [13], a sixth-order power system assuming a third order model for the synchronous machine was used. In [14], a fourth-order nonlinear model of the synchronous machine was used. In [6], a fourth-order nonlinear model for the synchronous generator was presented. Our contribution to this thesis is observed in our choice of a second-order classical model for the synchronous generator, in contrast to the previously proposed model orders. Compared to the model order presented in [6] and other higher order models presented in other publications, the effect of damper windings and stator dynamics are neglected in our work considering that we are not interested in a very fast dynamic (sub-transient) system. The choice of a second-order model and hence the number of states to be estimated is motivated by the fact that an increase in the order of the overall system model indicates that the size of the system that can be simulated is limited. In other words, in increasing the order of the overall system model, the size of the system that can be simulated is limited [11]. So with a minimal order, we do not place a limit on the size of the overall system that can be simulated. Power systems analysis in recent times has been based on two kinds of machine bus systems. They include the Single Machine Infinite Bus (SMIB) system and the Multi machine infinite Bus (MMIB) system. The SMIB is subdivided into classical and detailed models. For the purpose of simplification, we will use the classical model to represent the generator connected to an infinite bus in an SMIB power system. The concepts developed for the SMIB system can be easily applied to a two machines or Multi machine power system as well when they are reduced to a SMIB system. Considering the SMIB system described with figure 2.2, we will assume that we are not interested in very fast dynamic (sub-transient) conditions and as such, neglect the damper winding and stator dynamics of the synchronous generator. But the effect of the damper winding is considered in the rotor-damping factor [11]. Hence for this study, modeling of the generator will be done considering transient conditions. Fig. 2.2: Single machine infinite bus system 10

18 Power system transient stability assessment involves evaluating the system s ability to remain synchronized after undergoing disturbances as well as proposing adequate remedial actions when there is the need for such [15]. In transient stability assessment, state variables (SV) associated with transient stability can be divided into Machine state variables, which has a minimum of two mechanical SVs, machine rotor angle and machine rotor speed, and at least six electrical SVs, among other variables; Load state variables, which include static loads and dynamic loads SVs and; Special devices such as FACTS,HVDC links,svcs, SVs As such, for real-time transient stability monitoring and control, the rotor angle and speed SVs play very important roles. If they can be considerably and accurately estimated, they can be exploited to monitor real time loss of synchrony and devise automatic closed loop stabilization schemes [16]. We will begin modeling the synchronous machine by first defining a few important parameters of the machine model. The total moment of inertia of the synchronous machine,, is the sum of all moments of inertia of all rotating parts of a synchronous machine, i.e., moments of inertia of the rotor, turbines, shafts etc. The inertia constant, of a synchronous machine [17] is thus described as (2.1) = 0.5 (2.1) Where the numerator is the total kinetic energy stored in the synchronous machine at steady state and is the MVA rating of the machine. The inertia constant is measured in seconds. It gives us the time taken to bring the machine from its synchronous speed to a standstill if the rated power is extracted from it with no mechanical power fed in. Its value varies within a much smaller range than the value of for different machines. As we mentioned earlier, in steady state the synchronous machines rotate with similar electrical angular velocities but during disturbances, the generators kinetic energy tend to increase or decrease and as such they lose synchrony. Figure 2.3 [17] illustrates the electromechanical description of the synchronous machines: Fig. 2.3: Schematic description of the powers and torque in synchronous machines 11

19 The rotor dynamics for the synchronous machine can be described with [17] the second order ordinary differential equation (ODE) presented in (2.2) = (2.2) where = = is the mechanical power acting on the rotor (W) = is the electrical power acting on the rotor (W) = = Inertia constant. Both equations of (2.2) are alternative forms of each other where the second equation of (2.2), the power form, is obtained by multiplying both sides of the first part of (2.2) by. Higher order ODEs can be written as first order ODEs and are used for describing multi machine systems. Using the classical model for the synchronous machine (generator), which ignores the saliency of the round rotor, only the quadrature transient reactance is considered, assuming that the direct and quadrature components are equal. We will also assume the transmission line resistance, is zero (0). Hence the total active power from the generator is delivered to the infinite bus, i.e. =. The generator s voltage is represented by while the infinite bus voltage is represented with. is the rotor angle which represents the angle by which generator voltage leads the infinite bus voltage. It can also be described as the angle by which the -axis component of the internal generator voltage leads the terminal bus voltage of the machine ( ). For the purpose of clarity, we will use from here on. = (2.3) Whenever the system experiences disturbance, the magnitude of the generator voltage remains constant at its pre-disturbance value while the rotor angle changes as the generator rotor speed deviates from synchronous speed. Thus we can introduce the relation =. If a detailed model is to be considered, the field coil on the direct axis (d-axis) and damper coil on the quadrature axis (q-axis) are used. is the d-axis transient reactance. Taking to be the reference phasor, a third order nonlinear differential equation [18] can be used to describe the synchronous machine connected to an infinite bus through two parallel transmission lines, each with impedance as described in figure

20 However, since we are considering the classical model, we can thus introduce the relation = to (2.2) and re-write it as (2.4): = = 1 2 ( ) (2.4) where is the damping factor. = shows the relationship between the relative angular velocity and the synchronous rotating system. This relative angular velocity is of interest when rotor oscillations are being studied while the absolute angular velocity is of interest when studying frequency stability. We can rewrite (2.4) in terms of system states and inputs as (2.5a) and (2.5b): = [ ] = [ ] (2.5a) as such, = [ ] = [ ] where = = ( ) is the generator rotor angle (first state), is the machine nominal or base synchronous speed is the generator rotor speed in pu (second state), is inertia constant is the electrical output torque in pu, is the mechanical input torque from the turbine in pu, (2.5b) is the control voltage or exciter output voltage as seen from the armature. It is in pu. The electrical power output will be equal to the terminal electric power measured at the generator bus if we assume the stator resistance to be zero ( = 0) and = 1pu. Thus, = + (2.6) = 0, = + where are the internal -and -axis voltages of the machine respectively. 13

21 The rotor angle and the rotor speed are available measurements. However the dynamics of the internal voltages cannot be measured because they are lumped variables that aggregates many voltage dynamics around the machine principal axis [19]. However, pseudo measurements technology can be used to acquire such variables. For instance the output voltage measured at the terminal bus and the injected current to the generator are used to solve some observation equations which are nonlinear functions of these variables. Thus using measurable rotor angle, output voltage can be obtained[6]: Solving further, we get = = = = +. (2.7a) (2.7b) Also, the -and -axis currents are given as: = cos = sin (2.8a) Where is assumed to be measureable. After further calculations, we get = +. (2.8b) Substituting the variables and in (2.7) with the states as described in (2.5a), we obtain = cos = sin Substituting and in (2.6) and solving, we obtain the electrical output power at the terminal bus as (2.10) (2.9) sin 2 (2.10) In terms of the states, is re-written as 14

22 sin 2 (2.11) Using (2.5a), (2.5b), (2.9) and (2.11), the vectors for the state and input parameters are described in equation (2.12a) as: = =, = = (2.12a) The second order dynamic model for the synchronous machine can is thus described with (2.12b) as = = 1 2 ( ( sin 2 ) ) = sin 2 (2.12) A compact representation for the dynamic model described above can be summarized as, = (, ) = (, ) = (,, ) = h(,, ) (2.13) = h (, ) The terminal bus signals, and, used for online state estimation can be measured directly from the PMUs placed at the generator bus terminal. Considering that the PMU measures the voltage magnitude and angle as well as the line injection and flow current magnitudes and angles, these quantities are thus used to calculate the terminal bus signals which in turn are used as inputs for the UKF algorithm proposed to be used for this thesis. As will be seen later, the output measurement is directly fed to our embedded MATLAB function block as an input for the UKF algorithm. We had earlier said that another synchronous machine model used for power network studies was the multi-machine classical model. In this model the generators are represented by constant voltage behind transient reactance, constant impedance loads etc. When uniform damping is considered for this model, the generators motion is described with the following set of equations [20]: = + =, = 1,2,, (2.14a) 15

23 where the acronym COI means center of inertia. When damping is not considered, the motion of the system is described as where = =, = 1,2,, (2.14b) where is the displacement angle and is the angular velocity described as = and = respectively, is angle of voltage behind transient reactance, indicative of generator rotor position, is rotor speed, is generator inertia constant, is damping coefficient. As mentioned earlier, for the purpose of this thesis, we will be using the classical model of the singlemachine infinite bus (SMIB) system to represent the power system. As such, detailed study of other models, which we have briefly described in this work, can be done with the given references. When we discuss dynamic state estimation in power system, the dynamic states of the system are considered rather than the static states such as the voltage magnitude and angle at the buses. There are of course several dynamic states in a power system. The choice of the states to be estimated also depends on the scope or extent of the period of study and analysis to be done. The states are represented by a set of non-linear differential equations and an increase in the number of equations increases the order of the overall system model. Recalling that an increase in the order of the overall system model limits the size of the system that can be simulated, we are hence motivated to limit our studies to two dynamic states; the generator rotor angle and generator speed. Our objective in dynamic state estimation is to estimate the system states using a second order, highly non-linear differential-algebraic equation representation of the system, which has been earlier presented in this chapter. In chapter four, we give an overview of state estimation and then in chapter five we present our proposed methodology. But before these, it is imperative that we discuss some measurement acquisition methods in power system. 16

24 3 Measurement acquisition methods in power systems Monitoring the power system is done by obtaining and processing measurements from the system. These measurements are used to obtain measurement models. Measurement models show the relationship between the measurements obtained from the system and the estimated or calculated variables. These variables are divided into state variables and dependent variables. State variables are a part of the set of problem variables which describe the network. These variables are connected by a set of network equations, called network model. The network model expresses the relationship between branch active power flows and bus voltage angle. They also show the active power balance at network busses. State variables describe the system completely, i.e., if the states are known, the remaining dependent variables can be calculated using the network model equations. These set of state variables is minimum. This implies that if they are removed, the other dependent variables cannot be calculated. In other words, when operators have correct network model gotten via measurements obtained, the system is monitored adequately. Voltage magnitudes and phase angles enable us effectively calculate all other quantities such as power flows, loads, generations etc. As such, voltage magnitudes and their phase angles are considered state variables. The measurements used to estimate these state variables could be corrupted as a result of faulty sensors or measuring devices. As such, there is the need to make the best guess for the state variables given the noisy measurements. Traditionally, for static systems, the Weighted Least square (WLS) technique is used for this task where the system is linear (noniterative). In order to run the state estimator, we must know how the transmission lines are connected to the load and generation buses. We call this information the network topology or configuration. Since the breakers and disconnect switches in any substation can cause the network topology to change, a program must be provided that reads the telemetric breaker/switch status indications and restructures the electrical model of the system. Proper knowledge of the systems network topology improves power system monitoring, as operating conditions of the system at any given time is known when the network model and complex phasor voltages of every system bus are known [21]. The set of complex voltages is called the static state of the system since it fully specifies the system. The state estimation function is thus responsible for monitoring the system state. It does this by processing redundant measurements to get best estimates of the systems current operating state. State estimators also serve the purpose of filtering incorrect or corrupted measurements, data etc., received by the Supervisory Control and Data Acquisition (SCADA). 17

25 3.1 Conventional/SCADA measurements Various tools/elements such as the Energy Management System (EMS) and Supervisory Control and Data Acquisition (SCADA) systems have been put in place for monitoring, supervisory control, data acquisition and delivery among other things. They also ensure that power network regulatory conditions are met. A few of such elements are listed below: Energy Management Systems (EMS) State Estimation Algorithms (SEA) Supervisory Control and Data Acquisition(SCADA) Remote Terminal Units (RTUs) Advanced Metering Infrastructure (AMI) Generation and Load Forecast System(GLFS) These tools, in addition to planning and analysis functions, form the EMS/SCADA system. A basic understanding of the relationship between the EMS/SCADA system and state estimation is illustrated with figure 3.1 [22]. The data acquisition function receives real-time measurement from remote terminal units (RTUs) and phasor data concentrator elements installed at various parts in the network. Fig. 3.1: Elements of the EMS/SCADA system 18

26 3.1.1 Measurement model and assumptions Most commonly used measurements for state estimation are line real and reactive power flows, real and reactive bus power injection, line current flow magnitudes and bus voltage magnitudes. These measurements can be expressed in terms of state variables given the state vector R i.e., for buses, there are voltage magnitudes and ( 1) phase angle. The state vector for a static system can thus be described as = [,,,,,, ] (3.1) where is voltage magnitude and is phase angle at bus. It should be noted that one of the buses id selected as a reference or slack bus resulting in a ( 1) phase angle. It is safe to assume that the complex phasor voltage measured, = cos +, has a magnitude 1 p.u. and voltage phase angle between the adjacent busses is arbitrarily small. At this juncture, it is imperative that we remind ourselves of a few fundamentals to enable us understand power expressions in a transmission line. One of such is Ohm s law which states that the current through a conductor between two points and are directly proportional to the potential difference across the two points. It is mathematically expressed as = = (3.2) where are complex voltage and current phasors. is complex impedance expressed as = +. R is resistance to flow of direct current (DC) and is reactance, i.e., resistance to the flow of alternating current. Re-writing equation (4.2) in, we have = ( ) (3.3) where admittance (inverse of impedance) is expressed as = +. is conductance and is susceptance. The flow of current within the network is described by Gustav Kirchhoff s law which states that the algebraic sum of currents in a network of conductors meeting at a point is zero. It is expressed mathematically as = 0 (3.4) If we represent the transmission line with the two-port -model described in figure 3.2 [21] for an N bus system, the set of nodal equations expressing 19

27 Fig. 3.2: Two-port -model of transmission line connected between bus and Kirchhoff s current law at each bus, is given as = = = (3.5) Where C is the vector of net current injections at each bus C is voltage phasor at each bus, with = Y is admittance matrix with each entry given as 0 = + = + = h (3.6) where is admittance of the line from node, is the set of all nodes connected to node, is the sum of all shunt admittances connected to node. The conventional measurements for static estimations are given by the real power and reactive power injection at bus, real power line flow and reactive power line flow between bus and. They are expressed as Power injection at bus : = ( cos + ) (3.7) 20

28 = ( cos ) (3.8) Power flow from bus : = + ) ( cos + (3.9) = + ) ( sin + (3.10) These conventional measurements are obtained from SCADA systems and are related to the state vector described in equation (3.1) by the over determined system of non linear equations given as (3.11) = h() + (3.11) with describing the mean Gaussian measurement noise vector with covariance matrix R. 3.2 Synchronized Phasor Measurements Units (PMUs) PMUs are measurement devices capable of providing near real-time measurements of positive sequence voltage and current phasors at monitored and adjacent buses, utilizing the satellite-based GPS system. The measured current and voltage signals are oversampled and conditioned using analogue and digital anti-aliasing filters. The sampling clock is phase-locked with the GPS system, providing synchronized measurement up to 0.2 micro seconds. The IEEE 1344 standard requires that the PMU s reliability exceeds percent [23]. The phasor representation is illustrated with figure 3.3 Fig. 3.3: Phasor illustration Measurement model and assumptions In wide area monitoring, measurement data is collected at various part of the power system and transmitted to a central location for analysis and control of the systems operating conditions. The 21

29 SCADA system transmits these measurement data provided by RTUs which are placed at various parts of the network. However, the SCADA system captures only steady state conditions of the power system and is limited by its time skew making it difficult to get real-time condition of the power network. To cushion these effects, PMUs are installed at strategic places in the network such as at substations and they provide time-stamped measurements. Figure 3.4 [22] illustrates the placement of PMUs within the system and how the measurements gotten from the PMUs are transmitted to the SCADA system. Fig 3.4: Hierarchical placement of phasor measurement units. Let us consider the two-port pie network figure 3.5 [22] of a transmission line. We would assume that we have placed a PMU at bus. The complex voltage phasors at buses is respectively. Let the state vector be denoted as x. We thus have for this system = R R I I (3.12) 22

30 Fig. 3.5: Two-port π network. The voltage measurement from the PMU at bus is given as = R I + = (3.13) = + For the current measurements at bus, using the PMU, we have = R I + (3.14) Where and are the vector of PMU voltage and current measurements respectively while and are their corresponding zero-mean Gaussian measurement noise. Applying Kirchhoff s law, we derive the relationship between the current and the state vector (3.15): = + = + I + ( + I ) (3.15) Simplifying and separating the real and imaginary parts, we obtain (3.16) and (3.17) R = + R R + I + I (3.16) I = + R R + + I I (3.17) Combining equations (3.16) and (3.17), we obtain (3.18) 23

31 R I = ( + ) ( + ) ( + ) ( + ) (3.18) Substituting (3.19) into (3.14), we obtain (3.20) Combining (4.13) and (4.20), we get (3.21) = (3.19) = + (3.20) = + (3.21) = + (3.22) Where is the measurement vector of PMU measurements with voltage measurements and current measurements, R () is matrix containing rows of vectors with zeros but having a one in the column where representing the bus where a PMU is installed. R () is the matrix of admittances while e is the zero-mean Gaussian measurement noise which occurs when taking measurements. 3.2 Mixed (PMU and Conventional/SCADA) measurements In networks where sufficient numbers of PMUs are placed, the system is usually completely visible and as such estimation can be done strictly with PMU measurements. They do not need iterative solutions as the PMU readings are direct readings of all system buses. Such systems are regarded as linear systems. However, this is a challenge today due to the cost implications of installing sufficient PMUs for this purpose. Today, state estimators are used which make use of SCADA and PMU measurements without necessarily changing the already existing SCADA structure in place in within the network. Various literatures have considered various methods of combining both measurements for state estimation. An optimal method/algorithm for PMU placement for hybrid SCADA/PMU based state estimation will be considered later on in this thesis work. There are two methods [24] of integrating PMU measurements to the state estimation process. One method is to use an estimator which uses a mixture of PMU and traditional power flow measurements. This method has an advantage of relatively better performance in terms of accuracy and redundancy. The other method uses a two-stage scheme. It is called hierarchical state estimator. State estimation is first done using the SCADA measurements. The estimates are then improved in the second stage by using another estimator that uses only PMU measurements. This method of estimation has the advantage of leaving the existing SCADA software/structure in place. In this method, the state estimate x (as given by equation 3.23) is converted to voltage phasors = () and subsequently added as an additional measurement to a linear measurement model [22] as described below 24

32 = + (3.23) Where sifts out the needed phasors and is the noise vector associated with the transformed conventional measurements. The weighted least square approach can be used to solve for the unknown phasor. 25

33 4 State Estimation A background understanding of the evolution of state estimation is important in understanding the goals of this thesis. This chapter starts by discussing briefly the evolution of some state estimation techniques, with an understanding that they are built basically on conventional power-flow, powerinjection and bus-bar voltage-magnitude measurements. Thereafter, we will go on to give a brief discus on maximum likelihood estimation before we proceed to our main focus of estimation in power systems in the next chapter. State estimation (SE) schemes can be classified into three distinct frameworks [22]. They include Static state estimation (SSE) Forecasting-aided state estimation (FASE) Multi area state estimation (MASE) Conventional measuring and monitoring technologies, such as implemented by SCADA systems, can only take non-synchronized measurements every two to four seconds. As such, state estimation in previous years could only be assumed to be static which is based on the WLS method. The state estimates of the SSE are updated once every few minutes so as to reduce computational complexity during SE. Consequently, the SSE process cannot be considered an optimal process for real time monitoring of the network. Also, the SSE depends on single set of measurements taken at one snap shot of the system. This means that the states of previous measurement instances are disregarded when estimating a new state. Hence, even when the system is fully observable and the state estimates are within appreciable limits, the SSE cannot predict future operating point for the system. To compensate for this, a dynamic-based estimation approach called the Forecasting-Aided State Estimation (FASE) has been introduced. This approach provides recursive (almost certain) updates of the state estimates. This tracks changes during normal system operating conditions. The FASE approach also takes care of the problem of missing data considering that the predicted states could be used pending the availability of such data. Although we will be assuming close similarities between the FASE platform and the true dynamical SE, it should be noted that they are somewhat different. In [25] a state transition model that uses Kalman filtering and an exponential smoothing algorithm for forecasting, was developed. This algorithm was developed so as to follow the changing dynamics of the system. The MASE platform on the other hand is based on the premise that for large networks, a central computational system increases the computational complexity. Thus, the power network can be divided into smaller networks and the state estimates for these smaller networks are computed independently. This reduces amount of data used by each estimator and as such reduced computational complexity and also increasing system robustness. However, it requires additional 26

34 communication overhead together with time-skewness challenges resulting from unsynchronized measurements [22]. In recent years, the need to improve SE as well as capture near dynamics of the power system network paved way for the introduction of PMUs which has been described in the previous chapter. PMUs provide synchronized measurements, thus provided more accurate and timely measurements with more time samples as compared to SCADA systems that use non-synchronized measurements. Hence SSE can be made to use measurements from both PMUs and conventional or traditional SCADA systems. In general, two ways have been described in various literatures for including PMU measurements to the SE process. One method is to use a single estimator where the PMU measurements are mixed with traditional power flow measurements and the other method makes is a two-stage scheme where an estimator which uses only PMU measurements is used to improve the state estimates obtained from the conventional SCADA measurements. In [24], SSE is done with the two stage method of adding PMU measurements. This method uses the Gauss-Newton weighted least-square approach. But it has a drawback in that it does several iterations at one time instant, increasing computational stress on the EMS/SCADA system. In [20], a method that addresses the computational challenge is discussed. It discusses the use of the twostage scheme in the FASE approach. This approach is computationally resource efficient as a single iteration is performed at each time instant. An added advantage is that it assumes the state vector contains constantly changing state variables due to constant changes in the power system network. In that work, a mixed measurement extended kalman filter (EKF) estimation algorithm is derived on a FASE platform, using a dynamic mixed measurement for the measurement model. A drawback to the mixed measurement estimation method is the fact that the measurements are of different qualities and as such combining them in a single estimator can cause the covariance matrix of the combined noise vector to be ill conditioned. Also, the dimensions of the vector and matrices used in the SE process increases the computational complexity as a result of the presence of PMU measurements. To address this challenge, a reduced order EKF (RO/EKF) estimator is proposed and used in the same work in [22]. The RO/EKF algorithm works by estimating the PMU observable state and the PMUunobservable states differently. Although the challenge of computational complexity is solved somewhat, this comes at the expense of the cost of accuracy. For this thesis work, it is proposed to use the FASE platform for incorporating the conventional and PMU measurements with the Unscented Kalman Filter (UKF) for improved filtering. 4.1 Static State Estimation (SSE) Statistical estimation is concerned with estimating the best estimate of the unknown state variables or parameters using samples of imperfect measurements. Thus the statistical criterion or 27

35 procedure needed for this becomes the problem to be solved [26]. Several state estimators differ in the sense that they have deferent objective functions for solving for the state variables. A few commonly used criteria are The weighted least-square criterion (WLS) The maximum likelihood criterion(ml) While some solution methodologies are Weighted Least square (WLS) method Least absolute value (LAV) method Weighted least absolute value (WLAV) method Least median of squares(lms) method Non-quadratic method Minimum variance method(mv) In most cases today, the WLS method is the most used because of its advantage in its statistical properties, less computational work and the fact that it uses simple model. It should be noted at this point that the use of normally distributed errors can be assumed. However, it has a draw back in its robustness in the sense that its estimates are affected by bad data and thus may not be real estimates. Other solution methods such as the Non-quadratic, WLAV and LMS, are robust but have the disadvantage of much computational work. The WLS, minimum variance and maximum criterion eventually give the same estimator [26]. 4.2 Forecasting Aided State Estimation (FASE) The state space representation, for a discrete time-variant dynamic system, is described with the state transition (prediction) model below: ( + 1) = ()() + () + () (4.1) With a measurement model at time instant, given as = h[()] + (4.2) where () is the state vector containing bus/nodal voltage magnitudes and phase angles, () R (2 1) x (2 1) is the state transition matrix and vector () relates to the trend behavior of state trajectory [21]. () represents modeling uncertainties and is given as white Gaussian noise with zero mean and covariance matrix, vector h[ ] is a non-linear load-flow function for the current network configuration, relating the state vector to the measurement vector, represent white Gaussian measurement noise, with diagonal covariance matrix expressed as 28

36 = 0 (4.3) 0 where is standard deviation of the conventional SCADA measurements and is standard deviation for PMU measurements. The error standard deviation for the PMU measurements is in general considered to be far less than that of the SCADA measurements since PMUs have more accurate measurements. Let () and ( + 1)be predictions at time instants and + 1. It should be noted that model parameters (), () and are not known a priori and are to be calculated. It is considerably safe to assume (4.1) as the memory of the system state time evolution while (4.2) serves as its refreshment for every new measurement. In other words, (4.1) represents the model of transition of states from () to ( + 1). For the dynamic state estimation process, three basic steps are considered. They are parameter identification, state forecasting and state filtering [25]. Parameter identification Considering the dynamic model described in (4.1), the parameters () and (x) are calculated using the Holt s two parameter linear exponential smoothing technique of forecasting [27]. This method gives decreasing weights to past observations. Appendix A further illustrates the Holt s initialization technique. The noise covariance matrix, is determined in other to optimize the estimation process in terms of accuracy. The elements of the matrix, which are assumed diagonal, are obtained through examining the maximum rate of change of the state variables, considering their behavior through historical data. If, for instance, the historical behavior of the state vector is given as = +, where represents the state vector, we can assume [2] that this change can be replaced by a random variable () with a Gaussian distribution. The resulting dynamic equation is then given as in (4.1). State forecasting With all parameters rightly calculated, the model described in (4.1) is now ready to make a forecast. If at time instant, the state estimated vector value is () and the true value is () with its error covariance matrix as (), the statistical characteristic of the estimation error is given as [() ()] which approximates to [(0, ()]. Using measurement information on the system behavior up to time instant and performing the conditional expectation operator on (4.1), a one step ahead predicted (forecasted ) state vector and its covariance matrix is given by (4.4) and (4.5) respectively; (+1)=()()+() (4.4) 29

37 (+1)=() () ()+(k) (4.5) where ()= (1+ ). (4.6) where (1+ ) is the th diagonal element of () ()=(1+ )(1 ) () ( 1)+(1 )( 1), (4.7) where ()= ()+(1 ) () ()= [() ( 1)]+(1 )( 1) (4.8) The forecasted state vector (+1) and its forecasted covariance matrix (+1) is then used to obtain the forecasted measurement vector (+1) and its error covariance matrix (+1) expressed as (+1)=h(+1)[ (+1)] (4.9a) (+1)=(+1)(+1) (+1), where (+1) is the Jacobian matrix expressed as State Filtering (+1)= (), at = (+1) (4.9b) When new sets of measurements ( + 1) arrive at + 1 time instant eventually, the predicted state vector ( + 1) is then filtered/ updated to obtain a new filtered estimate, ( + 1) and its corresponding error covariance (+1). The optimization/objective function for the filtering process at time instant +1 is given as ()=[ h( )] [ h( )]+ [ ] [ ] (4.10) Minimizing this objective function gives us the estimate for the state vector ( + 1). This estimate takes in to account the predicted estimate ( + 1) and the measurements ( + 1) at

38 4.3 Incorporating measurements to FASE In general, the main objectives of power system state estimation is to provide knowledge of realtime power system conditions through filtering, filling and smoothening real-time measurement acquired from the power system network. State estimation also provide representation for power system security analysis through contingency analysis, on-line power flow and load frequency control. The input to the state estimator include but are not restricted to measurements from PMUs, Network parameters such as generator parameters, line impedances and system configuration given by the topological processor etc. These known input parameter and/state measurements are not always perfect and usually noisy or contain redundant measurements. Hence, the state estimator estimates the best estimates for the unknown states. Such unknown states could be voltage magnitude and phase angle for static systems or the dynamic states of the synchronous machine(s). The state estimator also filters measurement errors and errors from model approximation. It also detects and identifies bad data and provides estimates for metered and unmetered quantities. The idea behind estimation is to formulate the network model using imperfect real-time telemetry measurements, to make raw-detection on real time measurement data so as to eliminate bad data and supplement deficient measuring points, also known as unobservable islands, with pseudo measurements. Hence, system state estimation and power flow calculations are done, thus improving complete network observability. To understand the method of using a single estimator, and subsequently process mixed measurements from PMU devices and conventional SCADA system for static state estimation, we need to relate the conventional state vector (which is in polar coordinates) to the PMU state vector (in Cartesian coordinates) through a non-linear transformation defined as = (). Hence, a single state estimator incorporating both measurements [22] is given as = h() () +, (4.11) where, represent the conventional and PMU measurements while, represent noise vectors from both measurements respectively. 31

39 5 Proposed methodology Earlier in chapter one, we established that the UKF is an efficient discrete-time recursive filter of the family of Kalman filters and that it is based on unscented transformation (UT). We also noted that unlike the EKF, it does not linearize the non-linear equation. Rather, it propagates the statistical distributions of the estimated states through non-linear equations leading to better estimates of the state and a posterior covariance matrix. The use of statistical distribution (mean) to represent the state offers the following advantages: 1) Mean and covariance of unknown distribution requires the maintenance of only a small amount of information which is sufficient for all kinds of operational activities. It gives a good compromise between computational complexity and representational flexibility 2) Mean and covariance (or covariance square root) are linear transformable quantities. i.e., the mean and covariance estimates are maintained after being subjected to linear and quasi linear transformation. 3) Sets of mean and covariance estimates can be used to characterize additional features of distribution. 5.1 Unscented Transformation (UT) An important fact in non linear estimation is the quality of the approximation of the non-linear function around the operating point. If this is not properly done, estimation errors occur and the efficiency of the estimator is questioned. To mitigate this problem, the unscented transformation (UT) assumes that It is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation [28]. The UT [29] could be illustrated with figure 5.3 below. Fig. 5.1: Unscented transformation, showing sigma points being transformed 32

40 The idea behind the UT is to obtain deterministically chosen sigma points which capture exactly the mean and covariance of the original distribution of the measurement set. Let us consider the random variables and a system in the form of : ~(, ) (5.1) = () is a an n x 1 vector of state variables described by the probability distribution with mean and covariance, () is a non linear function, is m x 1 vector of variables resulting from (). UT aims at obtaining a set of deterministically chosen vectors called sigma points. These vectors (or sigma points), containing real numbers, capture the exact mean and covariance of the original distribution of. The sigma points are then propagated through the non linear function () to obtain the mean and covariance of. The UT process can be summarized as below: 1) Select sigma points: Form an x matrix containing a set of column vector sigma points which has the state mean vector and state covariance matrix as follows = = + ( + ), = 1,, (5.2) = ( + ), = 1,, ( + is the th column of the matrix ( +, parameter is scaling parameter described as = ( + ) (5.3) is a constant, usually between and 1 i.e., It determines the spread of the sigma points away from or around the state mean. Hence, it can be used to vary the amount of higher order nonlinearities around. The closer the sigma points are specified to m, the more the higher order nonlinearities tend to be ignored. Constant is a secondary scaling value usually set to 0 or 3. It can be used to reduce the higher order errors of the mean and covariance approximations. Note, if the square root matrix of is, then =. If is positive definite as assumed in the UKF, it can be rewritten as =. The Cholesky factorization of gives, where is a lower triangular matrix. Cholesky factorization ensures sigma points are not complex numbers. 2) Nonlinear Transformation of the sigma points through the dynamic function: the sigma points set are propagated through the non linear function to obtain the propagated sigma points = (, ), = 0,1,,2 (5.4) The predicted mean vector and predicted covariance matrix for the propagated sigma points is computed as 33

41 = (5.5) = [( )( ) ] (5.6) where are the weights and can be calculated using the formulas: = =, = + (1 + ) (), = () (5.7) The variable typically has a value of 2 for Gaussian distributions. It is used to incorporate prior knowledge of the distribution of. The UT approach provides approximations that are accurate to a third order for Gaussian inputs for all nonlinearities [30]. The approximations for non-gaussian inputs are accurate to a second order. For higher order moments, their accuracy depends on the choice of and. A block diagram representation of the UT is presented in figure 5.4 below [30] Fig. 5.2: Unscented Transform (UT) block diagram 5.2 The Unscented Kalman Filter (UKF) As previously mentioned, the UKF is a recursive estimation algorithm. It uses the UT method in its recursive prediction and update structure. It is somewhat similar to the EKF however it is computationally easier to implement as the Jacobian matrix is not calculated. Let us consider the non-linear functions of state transition and observation models [1] representing a system as given by (5.8), = ( ) + 34

42 = h( ) + (5.8) where is the state vector and is the measurmrnt vector, and are system and measurement Gaussian noise with covariance matrices and, non-linear functions and h represent the system and measurements. The UKF algorithm performs estimation in three major steps 1) Selection of the sigma points 2) Kalman filter state prediction 3) Kalman filter state correction Sigma point selection: As described in (5.2), the UT procedure creates the set of sigma points, using the a posteriori estimate for the state mean and covariance, at time instant 1. The expression for the sigma points [1] is given as (5.9) = = +, = 1,2,,2 (5.9) = +, = 1,2,,2 where = + is a scaling factor that determines the spread of the sigma points around. To initialize the estimation procedure, that is at time instant = 0, the initial state and covariance ought to be defined based on priori information of the system. The estimate at = 0 is an a posteriori estimate. Prediction step: The sigma points calculated are propagated one after the other through the systems non- linear function, propagating the state estimate and covariance from one time instant to the next and thus forming the predicted a priori state vector estimate at time as; =, (5.10) We then compute the a priori covariance matrix and the state mean vector by the weighted average of the transformed points as _= (5.11) = _ _ + (5.12) (5.12) can also be written as = [ ] + (5.13) Update step: Next, we update the sigma points with the predicted state mean vector _ and covariance matrix : = _ 35

43 = _ +, = 1,2,,2 (5.14) = _ +, = 1,2,,2 We then implement the measurement update. As described in (5.8), the sigma points are propagated through the known non-linear measurement function h( ) as shown in (5.15): = h( ) (5.15) This is the predicted measurement at time instant k. The mean of the predicted measurement is computed as: = (5.16) We also compute the covariance matrix of the predicted measurements: = [( )( ) ] + (5.17) also expressed as = [ ] + (5.18) The cross covariance matrix between the measurements and state is: = _( ) (5.19) also expressed as = [ ] + (5.19b) The filter gain, the updated state mean and the covariance,are computed as = = _ + [ ] (5.20) = + where is the corrected and thus generator state. 5.3 UKF based state estimation in power systems (UKF/SE) In power systems, the UKF remains a very useful technique for state estimation. Its ability to estimate dynamic system states without calculating the Jacobian matrix makes it a unique algorithm. This enables the application of real and complex system model without fear of the errors that result from model linearization. A few examples of the application of the UKF to power systems are seen in [14] and [6], where it is used for parameter estimation of synchronous machines. In [31] the UKF is used to estimate frequency and amplitude of power signals by filtering the noise measurements. The power system can be modeled in different forms. In [1], a linear model is used to represent the smooth dynamics of 36

44 the system determined by slow load variations. However a linear model might not be sufficient to capture the true dynamics of the system introduced by the generator dynamic parameters. In general, the power system may be modeled using a set of nonlinear differential-algebraic equations with unknown initial values expressed in (5.21) below = (, ) 0 = (, ) (5.21) z= h(, ) where ( ) is the nonlinear state functions of the state transition, is the set of algebraic equations representing passive network of the power system. It is formed as a nodal equation with a nodal admittance. h( ) is contains the nonlinear measurement equations. To use these equations in solving for x given y, we need to convert them from continuous-time equations to discrete-time equations using numerical integration methods such as Euler methods, modified Euler methods, Runge-Kutta (R-K) methods, implicit integration methods, etc. The time derivative of a variable is given as: = ()() () = + ( 1) (5.22a) (5.22b) where is the time step, and 1 indicate time at = and = ( 1) respectively. Substituting (5.21) in (5.22b), we get (5.23): It can be re-written as (5.24), () = (, ) + ( 1) (5.23) = ( ) + = h ( ) +, (5.24) Let us assume that for an bus system, the initial state vector and its corresponding covariance matrix are represented as and respectively, where the subscript represent the time instant = 0. The dimension of the state vector is given by = 2 1, which corresponds to the number of unknown state variables, assuming that the reference bus angle has a value of zero. The UKF steps for the power system can be summarized as follows: Sigma point selection As described in (5.9), we use the a posteriori estimate for the state mean and covariance, at time instant 1 to calculate the set of sigma points: = = +, = 1,2,,2 (5.25) = +, = 1,2,,2 37

45 State prediction (forecasting) According to (5.22), each column of representing the sigma points at instant 1 are propagated one after the other through the state update function to form a matrix of propagated sigma points at the next time instant : = + ( ), (5.26) The predicted state mean vector and the corresponding predicted covariance matrix are then calculated using (5.11) and (5.12). State update We compute or update the existing set of sigma points as described by (5.21) which captures the distribution of the predicted state. As described in (5.22), the sigma points are then propagated through the measurement update function such as (5.27); = h( ) (5.27) Where the superscript represents the -th column of the respective matrices. We then calculate the mean and measurement covariance matrices using (5.23) and (5.24) respectively while the filter gain, state mean and covariance matrix are calculated using (5.26). This procedure is repeated for every time instant. To effectively estimating the two dynamic states we have chosen, we have done a few modifications to the standard UKF algorithm described in (5.8) to (5.20). We have randomly chosen the initial covariance matrix variable and two random numbers, each representing the initial values of each state. These variables are used for the state mean x which is in turn used for calculating the sigma points as described in equation (5.25). The model equations describing the synchronous machine as described in (2.12b) were used for the propagation of the sigma points in the prediction process as described in (5.10) and the update process as described in (5.15). It should be noted that the continuous time model equations (2.12b) were first converted to the discrete form before being implemented in (5.26). The discrete form equations will be presented in chapter 6. 38

46 6 Simulations and Results This chapter presents simulations of our proposed algorithm which has been described in chapter five. We have modeled our test system in MATLAB Simulink. Simulations of system disturbances are done to reflect the dynamic nature of our modeled system. Our proposed UKF algorithm along with the synchronous machine differential algebraic equations presented in chapter two, are used for the estimation of the dynamic states of the system. Results are then presented and discussions are done on the simulations. 6.1 Implementation of our proposed methodology The proposed methodology for this thesis is tested on a single machine infinite bus (SMIB) system described in chapter two. The choice to use this simple system configuration for our analysis and simulations is because it is very useful in understanding basic concepts and effects which can be used for further studies in large complex networks. We have assumed the generator, whose state dynamics we seek to estimate, to be a part of the whole dynamic system, though separated from the other parts of the system by the transmission network. The schematic representation for the proposed estimation of the rotor angle and rotor speed is shown below in figure 6.1. Synch. Machine Gen terminal bus t Power System PMU Vt Pt Qt Pe I UKF est. Fig. 6.1: Proposed block diagram of UKF estimator Figure 6.2 illustrates the Simulink implementation block diagram for the proposed method. It shows the connection between the synchronous machine and the embedded MATLAB function block used for implementing the UKF algorithm. 39

47 Fig. 6.2: Top level of the Implementation block in Simulink The inputs to the UKF function block are: measurable electrical output power ( ) from the machine, assumed to be measured at the terminal bus of the generator by a PMU device, the mechanical power given by the prime mover (i.e. wind turbine in our case) and the internal voltage dynamics of the exciter system represented with the variable where it is assumed to be directly measured from the generator winding with negligible measurement noise. The terminal voltage (or ) and frequency are assumed to be measured by the PMU installed on the generator terminal bus. The machine parameters (see appendix for details) are assumed to be known or measured with little or no errors. Only the dynamic states to be estimated are unknown. These input output parameters described are needed to initialize estimation using the proposed UKF algorithm. Figure 6.3 is the schematic drawing representing the power system. It shows the synchronous machine connected to the transmission line through a transformer. A 150kV voltage source connected to the transmission line represents the aggregate of the loads on the infinite system. In power systems, the load can be defined to be as the sum of the continuous power ratings of all load consuming apparatus connected to the grid. The voltage source has constant voltage and constant 40

48 frequency and as such, is referred to as an infinite bus [11]. This characteristic of an infinite bus is seen in its unlimited power capability and thus can be represented by a voltage source with zero internal impedance. The magnitude of the infinite bus voltage remains constant for any given system condition when the machine is disturbed. In real events, the loads on the power system are actually varying constantly. But the assumption that the loads are constant is a relevant approximation for this thesis work and many other power systems studies. This condition or state is generally referred to as steady state. As the system steady state conditions changes, the magnitude of the infinite bus voltage changes too and this affects the operating conditions of the entire system network. Fig. 6.3: Layout of the synchronous machine connected to the transmission line The importance of system loads, represented by the voltage source, cannot be over emphasized. During our simulations and analysis, we observed that when the voltage source was disconnected, there were swings in the frequency plot of the rotor angle. The swings observed in the rotor angle plot are as a result of the load change in the system, typified by the disconnected voltage source. If such swings of the rotor angle are extreme or excessive, the synchronous machine or generator can trip off resulting in an unstable system. The choice of using Simulink library tools to design the model for a simple single machine system was necessary so as to simulate various operating conditions of the synchronous machine. We have used the embedded MATLAB function block considering that it allows us implement dynamic state estimation simultaneously while simulating different operating conditions of the machine. Voltage source 41

49 Figure 6.4 illustrates the subsystem model representing the mechanical part of the machine. The subsystem shows the two inputs; the steady state value of the mechanical torque and the instantaneous value of the electrical torque. Figure 6.4: Mechanical part Sub-model of the synchronous machine Figure 6.5 illustrates the subsystem model representing the electrical part of the machine. Figure 6.5: Electrical part Sub-model of the synchronous machine The initial value for the dynamic states were gotten from the Machine but were not used as initial states for initiating the UKF algorithm. We ran the simulations for various states values. Our simulations showed the need to have the initializing states as close to the real states as possible to 42

50 improve the accuracy of the estimates. A list of the synchronous machine parameters and the states to be estimated can be found in appendix C. The motivation for the choice of the UKF method over the EKF, even though they both have relatively good estimations, is due to the fact that the UKF gives better approximation for the (, ) and h() functions with respect to the Extended Kalman Filter (EKF). This is because the model equations are not linearized during the estimation process as is done in the EKF. Although there is higher computational complexity when using the UKF, its advantage in providing more accurate result is compensation. As we have shown in the algorithm, only a few sigma points are evaluated to cope with the non-linear equations. The number of sigma points is limited to 2n+1 where for this work, number of states n is 2, giving a total of six sigma points to be transformed. With respect to the particle filter where considerably amount of non-deterministic random points are evaluated, the UKF s number of sigma points is small. We have fixed the UKF parameters as α = 1, υ = 3 - n and β = 2. We also varied these parameters but discovered they did not have so much effect on the estimation results. The implementation codes for the UKF algorithm are given in the appendix. Before simulation, we have converted the continuous time equations describing the power system model to their discrete form using the time derivative equations which we have described earlier. The discrete forms of the state equation are presented in (6.1) and (6.2) as ( + 1) = () + () _ (6.1) ( + 1) = () + () () + sin 2 () () _ (6.2) _ is the time step which was set to be 0.001s in the configuration parameter dropdown of Simulink file. We did set the time step in the embedded MATLAB function block to be the same with that used in the Simulink file for the purpose of simulation and to provide an acceptable base for comparison. We also set the solver to be ode3 (Bogacki-shampine). Before simulation, the Powergui tool of our model, found at the topmost layer of the model as seen in figure 6.2 is opened to reveal the machine initialization tool. The Powergui Machine Initialization Tool shows the machine information as we can see in figure 6.6. It is used to set the machine s initial steady state settings. To do this, we initiated the compute and apply button. This computes and automatically loads the Simulink file during simulation with the initial steady state values of the machine. 43

51 Fig 6.6: Powergui machine initialization tool To initialize the UKF algorithm, we set the initial covariance matrix P with the following command: % Error covariance matrix; P0=diag([10,10]); P=P0; This command initiates the algorithm by loading the initial covariance matrix variable P using P0. As observed in figure 6.2, the output power measured by the PMU is fed into the algorithm as an input. It is used in the in the UKF algorithm to correct the estimated states. With our simulations, we desire to test and analyze the robustness and effectiveness of the proposed power system state estimator. These analyses provide in-depth insights to power system transient performance as well as the nature of the machine. Through this analysis and the results presented in this thesis, it would be easy to make approximations necessary for large scale studies as well as control studies for improved system performance. We will simulate: Small signal disturbance and Three phase fault disturbance 44

52 To simulate these disturbances, we have used matlab s fault block. Keeping the input factors T m (mechanical torque supplied by prime mover) and (internal voltage) constant, the fault block, fault1, as described in figure 7.3 is used to program and implement a three phase fault. The fault clearing time is defined and varied within the fault block to illustrate different disturbance scenarios. The fault is introduced to the system closer to the generator terminals and cleared almost immediately, depending on the clearing interval. We implemented a stable fault condition with a clearing time that lasts for 0.07 seconds. With these, we have shown that the proposed estimator estimates the system states even in unpredictable and unprecedented system conditions. 6.2 Results The noise free results of our simulation using the UKF and EKF estimation methods are thus presented. While analyzing the power system, we have decided to present the results of the two filters while simulating the power system. This will enable us make proper comparison between both estimation methods and subsequently show the advantage of the UKF over the EKF. Both filters differ in the ability to track the system during system disturbances. In the EKF, linearization was done by evaluating a Jacobian in the neighborhood of the estimate. Figure 6.7 presents the machine output power measured at the terminal bus. As we illustrated in figure 6.2, it is observed from figure 6.7 that the constant input power drawn from the turbine T m is 0.8pu. When the stator resistance is neglected as we have assumed in this work, the steady state Active power (pu) x 10 4 Fig. 6.7: Output power Pt measured at the terminal bus terminal power, which also represents the air-gap power, is equal to the turbine mechanical input. In figure 6.8 the effect on the rotor angle as the generator responds to small signal disturbance is presented. As we have said, these small disturbances usually occur continually in the system due to small variations in the loads and generation. 45

53 Elec. Rad (pu) Time (seconds) Fig. 6.8: Generator rotor angle Elec. Rad/s (pu) Time (seconds) Fig 6.9: Generator rotor speed While figure 6.9 shows simulation results for the synchronous generator rotor speed, we have presented in figure 6.10a and 6.10b, the response of the rotor angle to a step input in the presence of small signal disturbance using both the UKF and EKF algorithms respectively. The true value and the estimated value of the state variables are presented and compared. The grey color represents the true system data while the blue color represents the estimated data. We can observe that both methods estimate the rotor angle dynamics properly judging by their rise time. A closer look at both estimation curves show that the UKF method gives a better estimation judging by the overshoot at 46

54 the 0.1 second of the EKF curve. Also note that during the steady state, the EKF estimate tends to deviate from the real value as opposed to the estimates when using the UKF algorithm. Fig 6.10 (a): Real rotor angle Vs estimated angle using the UKF algorithm Fig 6.10 (b): Real rotor angle Vs estimated angle using the EKF algorithm In figure 6.11, the real and estimated rotor speed is presented. A close look at the response plots show that the UKF estimator gives a better tracking performance of the machine dynamics better than the EKF estimator. 47

55 Fig 6.11 (a): Real rotor speed Vs estimated speed using the UKF algorithm Fig 6.11 (b): Real rotor speed Vs estimated speed using the EKF algorithm Note that both estimators follow the form of the output power described with figure 6.7 indicating that they use the non linear dynamic model of the system. In general, we can deduce from these results that the UKF gives a better approximation of the system dynamics than the EKF method. Next we introduce transient fault conditions to the system. The transient response with fault clearing time of 0.07 seconds is presented in figure 6.12a and 6.12b, applying both estimators to the system. 48

56 Fig 6.12 (a): Rotor angle response with simulated fault cleared in 0.07s for the UKF estimator Fig 6.12 (b): Rotor angle response with simulated fault cleared in 0.07s for the EKF estimator. We observe in both estimators that they tend to follow, relatively, the initial state conditions of the system before the fault is introduced. During the duration of 0.07 seconds for which the fault stays, the estimate tends to degrade slightly and does not accurately estimate the states. Whereas the EKF fails to adequately follow the system dynamics properly during and after the fault, the estimate in the case of the UKF is temporarily wrong during the fault but tends to follow the real value soon after the fault. This is as a result of abrupt change in the external reactance. In general, it usually takes a few milliseconds for the state estimators to track the system dynamics after such external reactance change. Observe that in figure 6.12 (a), the UKF estimator begins to track the systems dynamics just after the 7000 second. We can also observe that the swing after the fault is properly reduced soon after the fault is cleared. This is as a result of proper damping in the system. 6.3 Discussion For transient stability studies, it is generally important to consider study periods of three to five seconds after a disturbance or fault may have occurred. Studies could however be extended to ten seconds for considerably large networks. Thus we have analyzed our estimators based on this 49