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This is to certify that the thesis entitled Nbdularized Global Dynamic State Estimation for Power Systerrs presented by Mohamad Hasan Modir—Shanechi has been accepted towards fulfillment of the requirements for ’ M—degree in Wm WM Major professor Date ML. 0-7639 OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to remove charge from circulation records .\ IL ‘ ' 33' . {512'3337-455 I 3" W I '. ‘1 MODULARIZED GLOBAL DYNAMIC STATE ESTIMATION FOR POWER SYSTEMS BY Mohamad Hasan Modir Shanechi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHLIOSOPHY Department of Electrical Engineering and Systems Science 1979 ABSTRACT MODULARIZED GLOBAL DYNAMIC STATE ESTIMATION FOR POWER SYSTEMS BY M. Hasan Modir Shanechi The subject of this thesis is "Modularized global dynamic state estimation for power systems". Based on a linearized model of the electromechanical dynamics of each generator, this dynamic state estimation problent is formulated to estimate the voltage angle and frequency' at any set of load and generator buses which constitute a subarea of the power system where such a dynamic state estimate is desired. This dynamic state estimation problem is formulated such that measurement and model information from the system external to the subarea are not required. The elimination of the need for external model data and measurement is achieved by measuring the power flows on the tie lines connecting the external and internal system. Three different dynamic load models are developed which assume that the load at each load bus can be de- composed into different load types and that each load M. Hasan Modir Shanechi type can be modeled by a Markov process. Identification of the parameters of these load models and incorporating them in the dynamic state estimator eliminates the need for measurement at all load buses. The study system for the dynamic state estimator is divided into modules and the dynamics of each module is decoupled from any other module, thus eliminating the need for synchronized measurements at all points of the study area and the process of modularization permits the model to be updated on-line and with minimum on-line computation for changes in unit commitment, network con- figuration and load flow. Since these changes are local in character and thus every module will not have to be updated for each system change, and since the network reduction of all modules simultaneously requires less computation and is more accurate than a single network reduction of the network for the entire study area, the on-line computation for model updating can be kept at a reasonable level. To get all angle and frequency estimates from all modules referenced to a common reference, a global re- ferencing procedure is developed. This procedure uses the already measured tie line power flows between the modules in a least squares algorithm and not only pro- vides referencing, but also provides for bad data de- tection, identification and rejection of these tie line power flow measurements. M. Hasan Modir Shanechi A computer program for estimation of the angle and frequency at a module is developed and used against the dynamic simulation of MECS to provide; a) b) C) d) verification of the assumption that a classical stability model can be used for each generator, determination of maximum measurement error and minimum sampling rate that will result in accurate estimates of angle, frequency fluctuations and a rms coherency measure, determination of the sampling rate and the number and location of measurements necessary in order to provide accurate and fast recon- vergence of state estimates after a major dis- turbance, determination of the level of accuracy of the parameters of the linear model needed to ob- tain accurate estimates. In doing so it was also verified that neglecting voltage regulator dynamics would not affect the accuracy of the estimates unduly. ACKNOWLEDGEMENT I would like to express my gratitude to the members of my guidance committee for their invaluable contribution to my doctoral program. Specifically I would like to thank Dr. Rosenberg for his stimulating questions; Dr. Barr for the enjoyable experience of taking his classes; Dr. Park for what I learned from his guidance and leader- ship qualities in research and all the meaningful conversa- tions I had with him; not all of them restricted to academic per se; and Dr. Schlueter not so much for the help that he gave me in academic matters, although they were great and many, but most of all for his friendship and help during my stay in this country. He was much more than a research advisor to me and he will always be remembered as such. During these years I learned a lot about life and people of the United States and many other countries which I value very much and I established many friendships which I cherish dearly. I am grateful to be part of the community and most grateful to have met all those who became my friends. I must also acknowledge the financial support I received from the Division of Engineering Research, Michigan State University. Chapter 1 TABLE OF CONTENTS INTRODUCTION l-l Local Dynamic State Estimator 1-2 Global State Estimators 1-3 Research Objectives DYNAMIC STATE ESTIMATION IN POWER SYSTEM 2-1 General Constraints on the Study Area for a Dynamic State Estimator 2-2 General Constraints on Data from the External Area 2-3 Local Dynamic State Estimation Problem 2-4 Global Dynamic State Estimation MODULARIZED GLOBAL DYNAMIC STATE ESTIMATION 3-1 Modularization of the Study Area 3-2 Electrical Model of Generator 3-3 Governor-Boiler-Energy System Model 3—4 Linearized Model of the Network 3-5 Dynamic Model for a Module in the Study Area 3-6 Dynamic Load Model 3-6-1 Low Pass White Noise Model 3-6-2 Load Component Model 3-6-3 Geographical Load Model 3-7 Complete State Space Model of a Module 3-8 The Kalman Filter Equations for a Module 3-8-1 The Sampled Data Form of Equations of the System 3-8-2 The Discrete Version of the Model and its Optimal Kalman Filter Equations Page 17 17 20 22 24 27 27 29 30 32 35 41 43 44 47 48 54 S4 57 Chapter 4 GOLBAL REFERENCING FOR MODULARIZED DYNAMIC STATE ESTIMATORS 4-1 4-2 Reference by Association Reference by Line Measurement 4-2-1 Referencing Two Modules from Line Measurements 4-2-2 Least Squares Method of Referencing Two Modules 4-2-3 The Global Least Squares Referencing Procedure 4-3 Implementation 5 PERFORMANCE OF THE DYNAMIC STATE ESTIMATOR 5-1 The Test System 5-2 Determination of the Added Term to the Damping 5-3 Validation of the Linear Model 5-4 Performance of the Dynamic State 6 BIBLIOGRAPHY Estimation 5-4-1 Effect of Different Measurement Noise Level 5-4-2 Effect of Measurement Sampling Rate 5-4-3 Effect of Location and Number of Measurements Taken 5-4-4 Effect of Error in Model Parameters Page 60 64 66 69 72 78 82 88 88 90 92 95 96 99 101 109 CONCLUSION AND FUTURE INVESTIGATION 113 125 CHAPTER I INTRODUCTION Dynamic state estimation is a prerequisite for several significant improvements in the security and stability of a power system which will result by enhanc- ing the following power system operating functions: (1) - security assessment (2) - security enhancement; and (3) - control in the normal, alert, emergency and restorative operating state. A global dynamic state estimator would assist security assessment [1,2] (l) in its security monitoring task by providing an accurate estimate of electrical frequency at every bus in the system and/or a measure of coherency between any two buses as a func- tion of time. (2) in its security analysis role by providing data for possible transient security measures [3,4] and a better data base for developing dynamic equivalents for transient security analysis [5,6,7]. The global dynamic state estimator could assist in security enhancement [1,2] by providing data for new preventive and corrective controls, which would operate in the alert and emergency operating states respectively, and would (I) adjust levels of generation, using an optimal power dispatch that minimizes both the total cost of generation and a weighted sum of on- line estimates of the coherency measure [4] between pairs of buses. This power dispatch would adjust generation to relieve line over- loads and thus increase the dynamic structural integrity and security of the system; (2) perform line switching operations to improve the coherency between coherent groups based on an on-line estimate of the coherency measure between pairs of buses; (3) adjust relaying logic [4] in order to main- tain the relative level of coherency between coherent groups when loss of a line would cause a serious loss of coherency between these groups and affect system security; (4) adjust relaying logic [2] so that lines con- necting coherent groups could be opened on command to form islands when total system collapse appears imminent. Dynamic state estimation could improve control in the normal, emergency, and restorative state in several ways. Aglobal dynamic state estimator would be required for the optimal automatic generation control [8] proposed to improve the response of the power system to changes in load and compensate present power system dynamics. A local dynamic state estimator could be used to provide local state information on a particular generator and the buses it is directly connected to for discrete control strategies [9], which are used in the emergency operat- ing state to maintain synchronization after a severe con- tingency. Global dynamic state information could also be used in the emergency operating state for optimal load shedding algorithms [4] which could be used to reduce local mismatch in generation and load based on the dynamic estimate of frequency or a dynamic coherency measure de- termined from the dynamic state estimator. Finally, a global dynamic state estimator would be used in restorative operating state to reconnect sections of the system after a breakup. The reconnection would require an accurate estimate of frequency at buses which are being synchronized and then reconnected by switching in a transmission line. Global dynamic state estimation of an entire power pool or some subarea or local dynamic state estima- tion of a generator and the buses it is directly con- nected to has been very difficult for the following two reasons : dynamic dynamic dynamic tail in (1) an accurate model of power system dynamics is difficult to determine because the dynamics are nonlinear and depend on the operating condition which is constantly changing; (2) the huge size of a power system and the in- ability to decouple the dynamics of some portions of this system from other portions where a dynamic state estimator is not de- sired has made dynamic state estimation impractical. Particular difficulties that are associated with this problem are; (a) the computational burden of computing the dynamic state of the entire power system in real time; (b) the data burden of obtain- ing and constantly updating model informa- tion for this entire system; (c) the communication and instrumentation burden of making measurements at a fast sampling rate (greater than S/sec.) throughout this power system and then transmitting this data to a central computing facility. From the discussion of the applications for state estimation, there are two basic types of state estimators required, global and local state estimators which are discussed in more de- the following two subsections. 1.1 Local Dynamic State Estimators Local dynamic state estimator, would provide in- formation on the state of the generator to be controlled in relation to buses that are directly connected to it. This estimator would provide dynamic state information for use with discrete controls used to maintain synchro- nization in the emergency operating state. A very desirable feature for this estimator would be to require only measurements on the generator bus where the dis- crete control is to be applied and require model informa- tion only for the generator being controlled and the transmission lines that are directly connected to it. Zaborszky et a1. [10,11,12] have obtained a local equilibrium assumptions: (1) state estimator based on the following separate state estimators would be imple- mented to provide estimates of real and reactive power injections from the gen- erator, real and reactive load power fluctuations, and the voltage behind tran- sient reactance, but the local equilibrium state estimator would not be based on a single dynamic model of the generator voltage regulator. the governor turbine energy system, and the electrical network connected to the generator high side trans- former bus; (2) (3) The is that (l) (2) (3) all measurements on the high side transformer bus are perfect; the complex voltage at buses directly con- nected to this controlled generator are con- stant. advantages of this local dynamic estimator it provides a dynamic estimate of the accelerating power that will attempt to move the generator away from its local equilibrium reference and which must be compensated for by emergency state control procedures; it does not require any measurements of voltage or power except at the high side transformer bus at which the generator to be controlled is located; it does not require model information on transmission lines or generators except those that are connected to the high side trans- former bus where these measurements are made. The major limitations of this local equilibrium state estimator are (1) it could only be used for the discrete con- trol strategies in the emergency operating state and could not be used for security assessment, security enhancement and control in the normal, alert and restorative operat- ing state applications because it does not provide estimates of voltage angles and fre- quencies at every load and generation bus required for such applications. Each local equilibrium state estimate is referenced to equilibrium reference angle that are in no way referenced to a common bus and which are dynamic and not constant as assumed. (2) The local equilibrium state estimator is susceptible to gross errors due to bad data because this estimator does not use a composite model of the generator, turbine energy system, and electrical network, bad data detection and identification for measurement and model data is not possible. (3) A linear simple state model based on the local equilibrium state, which is required for computing on-line control,is only valid for a very short interval after the con- tingency occurs and thus the model and any resultant control based on it would be limited to a short duration after the con- tingency. A second local dynamic state estimator is that of Miller and Lewis [13,14]. The generator to be controlled is represented by a fifth order linear model (not a linearized one) with flux linkages as states. This is an accurate model of the individual generator, but the state of this model is not the local equilibrium state and thus is not useful for control strategies that attempt to modify the accelerating power experienced by the unit. However the state of this model permits accurate estimates of real and reactive power and currents Iq and Id out of the machine which would be useful in discrete controls that affect the network or the voltage regulator. Major disadvantages of this estimator are; (1) it could not be used for global dynamic state applications not only because it could not provide direct estimates of angle and fre- quency but also because the sampling rate and integration step sizes required would pose too large a computation and communica- tion burden with present technology. (2) bad data detection and identification for this estimator would be difficult because the estimator does not use a composite model of the generator, governor energy system, and electrical network and thus could not effectively check a set of redundant measure— ments and model data. 1.2 Global State Estimators: Several global dynamic state estimators reported to date have been tracking static state estimators [15, 16,17]. Measurements are routinely taken on different locations of power system and on different elements or functions of elements of the state vector. Writing the network equations using Kirchhoff's law and using these measurements to solve the resulting equations analytically, a load flow solution is obtained. Measurements are not always reliable and they usually are in error by some maximum deviation or percentage error. Therefore many more measurements are taken than are needed for the load flow solution. Assuming the system to be in steady state condition and using these redundant measurements in a least squares algorithm, an optimal static estimate of the states of the system is obtained. Therefore [15] "a static state estimator is a data processing algorithm (set of equations) for use on a digital computer to transform (process, convert) meter readings and other information available up to the present time into an estimate of the value of the static state vector at the present time". The static estimator algorithm has to be repeated in its entirety every AT seconds where AT is the sampling period of the measurements and information system. This 10 implies that the estimator has to be reinitialized every AT seconds or at best every few sampling periods. This difficulty is compounded because dynamics of the system are neglected. To overcome the problem of initialization some authors [15,16,17] have proposed using the previous estimated state as the initial value with the reasoning that [16] "if no appropriate state transition model is available one supposes a model with the state remaining the same except for an increase in uncertainty or noise". A model for the system is assumed in which state at time T + AT is regarded unchanged from its values at time T except for addition of some disturbance, where this disturbance is modelled by a stochastic process. Although [17] "these proposed techniques do pro- vide for dynamic up-dating of the system state variable estimates, they do not provide for the estimate of dynamic changes which are likely to occur. This is true due to the associated stochastic nature of load demands," and lack of a representative dynamic model of system and .load. Thus, these techniques may be classified as a dynamic tracking state estimator since they track the system's present response and do not estimate the future response. A second approach to global dynamic state estima- tion is to linearize the dynamics of the entire U.S. 11 power system for some system configuration and operating condition. The research [18] has centered on the develop- ment of separate dynamic state estimators for individual subsections of this system which either: 1. do not have complete system model information outside its own area, or 2. do not have complete set of measurements on the system outside its own area. The performance of the global state estimator, made up of these separate estimator modules, is obviously seriously degraded as the quantity of information on the model or measurement outside the area which it is to per- form state estimation is decreased. The problem of up- dating the model parameters based on system configuration or operating condition changes has not been dealt with as part of the research. 12 1.3 Research Objectives The objectives of the research in this thesis is to develop a global dynamic state estimator for angle and frequency signals at all load and generator buses in a pool or area in order to provide information for the security assessment, security enhancement and control functions described earlier in this chapter. The only application discussed for which global dynamic state estimation would be inappropriate is supplementary discrete controls, which are used in the emergency operating state and require an estimator that can provide estimates of large dynamic fluctuations in both P-f and Q-V dynamics. The global dynamic state estimator estimates angle and frequency deviations using a linearized classical stability model based on a classical generator model and a static state estimator that indicates network configura- tion, unit commitment, and load flow information. The dynamic state estimate of angle and frequency, composed of the sum of the static state estimate and the dynamic Kalman state estimate, of dynamic deviations from these static estimates, is thus more accurate and reliable than the tracking static estimate [15,16] that uses no dynamic model information. The global dynamic state estimator will not need either model information or measurements of the system external to the study area, where the dynamic state 13 estimate is desired, because the effects of the external system dynamics are represented by measurements of the power flows on lines that connect the internal and external system model which are used as inputs into the internal system model. This estimator thus overcomes the tradeoff between performance of the estimator and the amount of data (model information and measurements of the external system) which occurred in the work of Tacker [18] . Four extremely important aspects of global dynamic state estimation that have not been adequately addressed in the previous literature, and which will be investigated in this thesis, are: (1) the need to update the dynamic system model used in the estimator for changes in unit commitment, network configuration, and load flow conditions that are known from the static state estimator (2) the need to provide capability for bad-data detection and identification as in static state estimation so that the data base pro- vided by the dynamic state estimator is re- liable and can be used for the security assessment, security enhancement and control functions mentioned earlier 14 (3) the need to provide a dynamic load model as a means of eliminating the necessity to measure the dynamic load fluctuations at every bus (4) the need to keep the on-line computation and communication requirements to a minimum. The need to quickly update the system model on- line based on unit commitment, network configuration, and load flow information from the static state estimator is accomplished by modularizing the state estimation of the study area by decoupling the dynamics of subareas or modules within the study area by measuring the power flows on lines connecting each module with every other module. A local dynamic state estimator for each module is then develOped based on a linearized dynamic model unit commit- ment, network configuration and load flow conditions with the power flow measurements on lines connecting this module to other modules and the external system used as inputs. The dynamic model for each module can be easily updated because the number of load and generation buses in any module is kept small so that updating the network model and subsequent network reduction to obtain a state model of angles and frequency deviations at internal gen- erator buses is feasible. The modularization overcomes the large computational burden of repeatedly updating the entire dynamic model of the study area because 15 (l) the effects of changes in unit commitment, network configuration and load flow are local in character and thus every module will not have to be updated for each system change (2) the network reduction of all modules simultaneously requires less computation and is more accurate than a single network reduction of the network for the entire study area. The modularized dynamic state estimators are all locally referenced to a particular bus within that module and thus the global dynamic estimator could not be used for utility- or pool-wide tasks such as security assess- ment, security enhancement, or control unless the re- ferences for each modularized dynamic state estimator are referenced to a common bus in the study area. Global referencing procedures are developed that reference these modularized dynamic state estimators using the power flow measurements on the lines that connect modules. The need to detect and identify bad-data for each modularized dynamic state estimator is accompolished by using a dynamic model for each module and providing redundancy in measurements of frequency and power. Bad data detection and identification is accomplished on the global reference procedure by using a least squares 16 algorithm based on redundant measurements of power flows on all lines connecting all modules. Three different dynamic load models are proposed, discussed, and investigated in this research. The dynamic load models do not depend on voltage or frequency at a bus but are based on the same logic used for develop- ing models for load prediction [19]. The need to limit computation and communication requirements for the dynamic state estimator is accomplished by (1) modularizing the dynamic state estimation of a study area so that on-line update of the dynamic model is computationally feasible; (2) modularizing the state estimation of the study area and thus eliminating the need for synchro- nizing measurements taken in each module and eliminate the need of model and measurement from other modules in the state estimation of each module (3) development of load models that eliminate the need to communicate load measurements from every bus in the study area. CHAPTER 2 DYNAMIC STATE ESTIMATION IN POWER SYSTEM The purpose of this chapter is to discuss the objectives of, and the differences between the global and local dynamic state estimation. Specifically, the following topics are discussed: (1) The general constraints on the size of the study area for both the local and global dynamic state estimators; (2) The general constraints placed on the data available from the external area for both local and global dynamic state estimators; (3) a discussion of two local dynamic state estimator formulations that satisfy (1) and (2) and their advantages and disadvantages; (4) research objectives and a general discussion of a global dynamic state estimator that meets the constraints (1) and (2) above. 2.1 General Constraints on the Study Area for a Dynamic State Estimator The formulation of any dynamic state estimator must satisfy certain constraints on the study area, where 17 18 a dynamic state estimate is desired, and the external area, where no dynamic state estimate is needed. The local equilibrium dynamic state estimator [10,11] has been the only dynamic state estimator that has effectively dealt with these constraints. Once these constraints are clearly stated local and global dynamic state estimators can be formulated that meet these constraints. The size of a study area for either a local or global dynamic state estimator must be chosen based on the following criteria (1) The instrumentation and data acquisition system requirements to measure and collect the data sampled synchronously at a rate sufficient for the particular dynamic state estimator should be minimized; and (2) The computational requirements for performing dynamic state estimations must be minimized. The instrumentation, data acquisition, and com- putation requirements are quite different for the local and global dynamic state estimator. For a local dynamic state estimator the study area should be the external generator bus, its high side transformer bus, and all buses that are connected to this generator bus by transmission lines. The size of this study area is chosen based on the following facts 19 (l) The purpose of the local dynamic state estimator is to provide dynamic information on the state of the generator and the state of the buses directly connected to it for the discrete controls in the emergency operating state, (2) The ease in determining the state of the internal generator bus and all buses con- nected to the high side transformer bus by measurement of the voltage on this bus and complex powers flowing into this bus (high side transformer) from the generator and transmission lines. For the global dynamic state estimator the study area is generally the pool or subarea under control of some pool or coordination center. The size of the area in some cases may be constrainted by; (1) the instrumentation and data acquisition re- quirements to make synchronized measurements at several locations at a S/sec. sampling rate. This sampling rate is slower than for local dynamic state estimator because the purpose of the global estimator is to estimate dynamic fluctuations in the angle and fre- quency at all buses in the study area and need not estimate voltage magnitude or reactive power dynamics for the applications dis- cussed in Chapter 1. 20 (2) The computational requirements to update the dynamic model used for dynamic state estima- tion so that the model remains valid for changes in unit commitment, network configura- tion, or load flow conditions which occur over a time period of minutes, hours, etc. The modularization of the dynamic estimator for a study area will affect the computation requirement for updating the model for the estimator. 2.2 General Constraints on Data from the External Area The feasibility of dynamic state estimation is not only determined by the instrumentation, data acquisi- tion and computation requirement for the internal area but also by these same requirements for the external area. It is clear that for a dynamic state estimator to be feasible no model information and no measurements must be assumed to be available from the external area. The study and external area dynamics are generally tightly coupled through the power flows on the transmission lines that connect them. Since the purpose of this research is to estimate the state of the study system on-line, the need to explicitly model the external system can be eliminated if the power flows on these transmission lines are measured and used as inputs for the study system model required for the dynamic state estimation. 21 For the local dynamic state estimation, the measurement of voltage on the high side transformer bus and the complex power from the generator and all trans- mission lines permit the determination of the voltage magnitude and angle of the internal generator bus and all buses connected to the high side transformer bus by transmission lines. This determination of the state of the electrical network from measurements at one bus can be referenced to the local equilibrium state [10,11], to the voltage angle of an imaginary machine running unloaded and at the same speed as the generator, that is the internal bus angle of the machine [13,14], or the angle of the high side transformer bus. Thus there are at least three possible forms for the local dynamic state estimator. In each case the local dynamic state estimator based on the dynamic model of the generator would provide a dynamic state estimate of the generator and the state of the network directly connected to it without measure- ment or model information from the external system, which is all lines, buses, and generators not directly con- nected to the high side transformer bus for the single generator where dynamic state estimation is desired. For the global dynamic state estimation the measurements of real power injections at terminal buses of a study area connected to the external area will serve as inputs to the linear dynamic model for the study area 22 used in the global dynamic state estimator, thus eliminating the need to make measurements or model the external area. However, in this case these measurements are not sufficient to determine the global dynamic state estimate because the study area for the global dynamic state estimator is large. This difficulty can be partially alleviated, as is shown later, by modularizing the study area. 2.3 Local Dynamic State Estimation Problem The purpose of a local dynamic state estimator is to provide an accurate state estimate of the state of generator dynamics and the state of the electrical net- work directly connected to it for local control in the emergency operating state. Therefore the estimator should estimate both P-f and Q-V dynamic fluctuation for the study area and should use a model that is valid for large excursions due to contingencies. Thus both generator- exciter-voltage regulator as well as governor-turbine energy system dynamics should be included in the model used for the local dynamic state estimator. There are three possible reference frames upon which local dynamic state estimators which meet the study area and external area constraints, could be based. A local dynamic state estimator based on the local equili- brium state [10,11] is not a single dynamic state estimator based on a single dynamic model for deviations 23 from the equilibrium state but rather is a composite of (1) a dynamic estimator for the voltage behind transient reactance based on generator; exciter, voltage regulator dynamics [20]; (2) a dynamic state estimate of injected electrical power at the internal generator bus based on a model of turbine energy system dynamics [21]; (3) est- imates of load power based on a dynamic load model; and (4) measurements of complex voltage on the high side transformer bus and complex powers from the generator and on each transmission line. This local dynamic state estimator has the advantage of using the local equilibrium state so that the null state is the global equilibrium and thus the state is the essential information for any discrete controls used on the generator during the emergency operating state. The principal disadvantage is that bad data detection and identification,which would be desir- able for the assurance that the state estimate is reli- able and can be used for control, would be difficult be- cause there is no composite model of the generator, turbine energy system, and electrical network referenced to the equilibrium reference for the study area, and thus a check of the measurements and model data would be difficult due to lack of measurement redundancy in each separate estimator. A second local dynamic state estimator could be formulated based on a composite network model, generator- 24 exciter model, and turbine energy system model referenced to the angle of the high side transformer bus. Bad data detection and identification could be performed more easily to check measurement and model data because the composite model would check every measurement and model parameter against each other. However, since the state of the model is not the local equilibrium state it would not be as useful for the discrete control strategies used in emergency control. However this local dynamic state estimator could provide the data base for computing the local equilibrium state and thus would (1) provide the information needed for control (local equilibrium state) (2) perform bad data detection, identification and rejection to make the estimate reliable. This approach to obtaining a local equilibrium dynamic state estimator appears superior to the one pro- posed by Zaborszky [10,11]. This topic is not persued because the major research topic of this thesis is global dynamic state estimation which has many applica- tions and which is far less developed both theoretically and conceptually. 2.4 Global Dynamic State Estimation The purpose of the global dynamic state estimator is to provide dynamic estimates of angle and frequency at all load and generation buses in a pool or area for the 25 security assessment, security enhancement, and control functions mentioned in Chapter 1 with the exception of discrete supplementary control which requires local dynamic state estimator. The model used for the estimator only requires P-f dynamics because these applications based on global dynamic estimation do not require knowledge of Q-V dynamics and because inclusion of these dynamics would require such small integration step sizes and sampling periods that hardware requirements for global dynamic estimation would make implementation impractical. Governor turbine energy system dynamics are eliminated to reduce the order of the model and is justified because it will be shown that frequency measurements and adjustment of damping in the model can compensate for elimination of these dynamics. A linearized classical stability model will be used to develop a Kalman state estimator for the dynamic deviations in angle and frequency signals because these deviations are assumed small and because a static state estimate provides slowly time varying estimates of the nominal values of these angle and frequency signals. The dynamic estimate of the angle and frequency signals at all load and generator buses is thus the sum of the static and dynamic state estimates of the variables. The global dynamic state estimator to be developed will be based on having a classical generator model and accurate data on the unit commitment, network configura- tion. and the static estimate of the state from a static 26 state estimator. The dynamic deviations from this static state estimate will be determined using a Kalman estimator and thus should be much more accurate and reliable than a tracking static estimator due to the use of system model information. The global dynamic estimator is prcduced by: (l) modularizing the global dynamic estimator in (2) order to reduce the computation and thus permit quick on-line update of the linearized dynamic system model based on changes in unit commit- ment, network configuration, and load flow. Each of these modularized dynamic state estimators is locally referenced to a bus in the module, a global referencing procedure that references each of the modularized dynamic state estimators to a common reference bus in the study area so that the global dynamic state estimator can be used for pool or area security assessment, security enhancement, and control applications discussed in Chapter 1. Bad data detection and identification would be per- formed on each modularized dynamic state estimator and on the global referencing procedure to permit bad data rejec- tion necessary to provide the reliable data base needed for the applications discussed in chapter 1. CHAPTER 3 MODULARIZED GLOBAL DYNAMIC STATE ESTIMATION The modularized dynamic state estimation problem is now formulated. A model of generator, electrical net- work, load and measurement process will be developed and justified for this application. To make dynamic state estimation feasible, the system model is modularized and the Kalman filter equations for each module are developed. In the next chapter a referencing procedure is developed to aggregate these modularized dynamic state estimates into a global dynamic state estimate. 3.1. Modularization of the StudygArea The study area, which could be a power pool or a company's operation, is split into sections which are here called modules. These modules could be overlapping or nonoverlapping. The composite of all these modules comprise the study area. For each module the rest of the study area together with the external area con- stitutes that module's external area. It is assumed that there are M modules and N generators in the study area, and Nj generators in the jth module with M N < X Nj' as there could be some generators common 27 28 to two or more modules. Also it is assumed that there are K load buses in theNstudy area and Kj load buses in module j, with K i .E Kj' It should be noted here that the high side transfgimer bus of each generator is represented by a load bus. The modularization of the global dynamic state estimator requires that (l) the dynamic system model for each module be decoupled from the dynamics of other modules so that model information and measurements in one particular module is the only data required to produce the state estimate for that module (2) that a common, reliable reference be estab- lished for all modules so that the estimators for each module can be combined to form a global estimator for security assessment, security enhancement and control applications for large utilities or pools. The purpose of decoupling the dynamic model and state estimator for each module is (l) to eliminate the need to transmit model data and measurements to one central processor; (2) to eliminate the need to synchronize measure- ments taken in each module; which can be a very costly requirement; 29 (3) to permit rapid on-line update of model para- meters which change due to changes in unit commitment, network conditions and load flow provided by the static state estimator. The decoupling of the model for each module re- duces the computational requirements by (l) reducing the number of lines in the net- work for which synchronizing torque coeffic- ients must be updated and (2) by reducing the order of the matrix to be inverted in order to update a closed form dynamic model. 3.2. Electrical Model of Generator A classical voltage behind synchronous reactance generator model will be used for the global dynamic state estimator because (1) this estimator is only intended to provide an accurate estimate of the dynamic fluctua— tions in the deviations of voltage angle and frequency at any bus in the study area. This estimator is not intended to estimate the large transient angle and frequency deviations immediately after a contingency because; (i) the linearized power system model, which makes state estimation for a large system feasible, would be invalid for representing the effects of large excursions after a 30 contingency, (ii) a higher order nonclassical generator model would be required, which in- cludes generator and voltage regulator dynamics, if estimates of the state were re- quired immediately after a contingency, (iii) the sampling period for measurements, and the itegration step size for an estimator which models generator and voltage dynamics would be so small that the data acquisition and computational hardware requirements would make global state estimation for transient conditions impractical (2) the applications for which this global dynamic state estimator is proposed, which were dis- cussed in chapter one, do not require; (1) estimates of voltage magnitude or reactive power deviations; (ii) accurate estimate of angle and frequency deviations immediately after a contingency, but would require accurate estimates seconds after such a contingency. 3.3 Governor-Boiler-Energy Systems Model Assuming the frequency fluctuations to be so small that torque and power are proportional, the electromechanical model for the ith generator in the jth module has the form; (LID: rt (LID. fl i = 1,2,.. where S dij(t) M.. 13 1).. 13 K.. 1] PM-.(t) PGij(t) i = 1,2,...,\1. ‘3 j = 1,2,...,M s Adij(t) Awifit) 31 = Awij(t) (D.. + K..) = .1. - - 13 13 Mu.(APMift) APGi§tD M.. Awij(t) 13 13 O'Iqt ' j = 1'2’OOO'FI J the angle at the internal generator bus of the ith machine in the jth module in the synchronous frame of reference (SF) - radians the frequency at the ith machine in the jth module - radians/seconds the inertia of the generator the damping coefficient of the generator the added term for the damping of the governor-turbine-energy system and voltage regulator the mechanical input power to the generator - P.U. the electrical output power of the generator - P.U. index numbering the generators index numbering the modules. In this model, the governor-boiler turbine energy system dynamics are neglected and their effect is modeled as an added damping term. This is reasonable because the objective is to develop a model that will accurately 32 represent the synchronizing oscillations in a power system and let the measurements of real power and frequency account or correct for the slower dynamics associated with Also the governor-turbine-energy system. APMij(t) is set equal to zero since for the above reasons it has slow dynamics. The random fluctuations of APMij(t) can be represented in a compensenting noise term added to other input noise terms if needed. 3.4. Linearized Model of the Network The network equations of the jth module in polar form are linearized with the real power equations de- coupled from the reactive power equations to obtain; 7 r— m m V P H APTGj(t;7 8P1G1(t) 3P.Gi(t)-l A5§(t) s s —j . . t - a§3(t) 3§_J( ) 3PTL.(t) 8PTL.(t) s APTL (t) S 1- 573 ag.(t) 3 86.(t) ae.(t) L. 3 .. C .3 L.—3 “J S SO S 30 . t = . t gj( ) g] ( ) O . = :. t §j(t) §J( ) _ o T _ PTGj(t) — [PTGlj(t), PTG2j(t),...,PTGij(t)] T _ J 33 T s s s s <3.t=6.t 6.t,..., .t _J ( ) [ 13( l, 23( ) deJ( )1 sT s s s 6. t = 0 . t , 0 . t ,... 8 . t ._J ( l [ lj( ) 23( ) , KjJ( )] ET(t) = [E (t) E (t) E (tn -J' 13' ' 23' ""' ij VT(t) = [v .(t), v .(t),...,V .(t)] where PTGij(t) real power injection at internal generator bus i into module j in P.U. PTLij(t) real power injection at load bus i into module j in P.U. 6:j(t) as defined before with the operating point so value of dij(t) 6:j(t) the voltage phase angle at load bus i in module j in the synchronous frame of re- ference (SF), with the operating point value of 6§?(t): 1) Eij(t) the magnitude of voltage behind synchronous reactance of the ith machine in module j; Vij(t) the voltage magnitude at load bus i in module j. It should be noted that PTLij(t) is equal to - PLij(t), the power withdrawn by load at bus i if bus i is only a load bus and does not have any connec- tion with the external area for module j, that is PTLij(t) = -PLij(t). If bus i is a terminal bus for 34 module j, which implies, a transmission line connects it with the external area for that module, then PTLij(t) is the sum of the power injected to the module from external area at this bus PEij(t)' and the power withdrawn for load at this bus -PLij(t), i.e. PTLij(t) = PEij(t) - PLij(t). These equations depend on the present Operating point (0:02;) and (E3 £30) and the present electrical network configuration. This data would be obtained from the static state estimator and would be updated at a rate sufficient to maintain model validity for changes in unit commitment, network configuration or load flow con- ditions. For large changes in the network, the model may not be valid for a certain period until an update occurs based on an updated static state estimate. In this case the accuracy of the updated model, the redundancy of measurements, and the short time interval where the model is invalid, should all contribute to a very quick recon- vergence of the dynamic state estimator.3 The update rate for this network model and the level of measurement re- dundancy needed to assure reconvergence and a sufficient speed of convergence will be partially investigated in this research. It has been noted that the network model includes injections at terminal buses. Measuring these injections 35 over time and using these measurements as inputs into the linearized state model eliminates the need to model the external area because the coupling of the state of the module under study and external area are accounted for in APEij(t). It should be noted that APTLij(t) is not just the injection on the tie line that connects to terminal bus i in study area j but also the load injection at that bus. To determine APTLij(t) re- quires measurement of both the real power on the tie line and the load at terminal bus i. 3.5. Dynamic Model for a Module in the Study Area A reference bus for each modularized dynamic state estimator is necessary so that the state is not referenced to some completely independent synchronous reference, which may not be operating in synchronism with the power system. The angle and frequency fluctuations in the system can only be placed in proper perspective in terms of stability and security if these fluctuations are somehow referenced to the system module in which they occur. For module j the reference angle is chosen to be the angle of internal bus voltage of generator number one, 61j(t). When referenced to this angle the system equations are going to be in a reference frame called machine one angle frame of reference [12]. 36 The state vector for the jth module is then de- fined to be; agjm xjm = Agj(t) 6T(t) = [a (t) a (t)] —j 2: '°"' ij Tm = [ (t) (t)] 9j wlj "°"wN.j 3 Also define; T _e_j(t) [elj(t),...,eij(t)] where s.(t) _ S _ 6- (t) — dij(t) 613 13 S e..(t) = ejjm - aljm. 13 Since only the difference between angles is desired, the s superscript angles can be in any frame of reference SF or otherwise [12]. Given this notation the generator model would be d _ _ . _ ( o I + K. a) i = 1,...,Nj, j = l,2,...,M or in the vector matrix form; 37 [O H) lo . —j .X.(t) = X.(t) 4- APG.(t) ’3 -1 ’3 -1 —J 0 -M. D. -M. - '3 ‘3 “3 where F_l T -1 i. = ° I —J . — L-l _. I = diag{1, 1,...,1} .Mtj .‘= diag{Mlj' sz'ooo’Mij} D. = di D . + K . ,..., D . + K . . ‘_3 a9{( 1] 13) ( ij ijH (l) The linearized network model for the jth module would then be ”ALTE- (tp 3239. (t) 3339. (t) Ag. (t) 3 ag.(t) a_e_.(t) 3 _ J J 3PTL-(t) 3PTL.(t) LAEEP-jfib _:_J____.. :1... Afij”) _a§j(t) agjwt) ; _J gait) = gait) = §j(t) = yj(t) = Define _ agng(t) lj ‘ 3337m— and (2) 0 g]. (t) O ej (t) 0 g]. (t) O yj (t) 38 311m. (t) 1j = agj(t) then from the last of these equations _-1 _—1 A§j(t) - lj Aggpj(t) lj lj A§j(t) . (3) Substituting in the upper half of these equations re- sults in the following expression; APG. t = APTG. t = T. A6. t + S. APTL. t 4 __J( ) J( ) _j __J( ) _j J( ) ( ) where 3PTG.(t) 3PTG.(t) -l T. = [ - T. n.] -3 8g]. t) 823. (t) -J -J and 3PTG.(t) -1 S. = T. . -—j 36.(t) -j —3 Further substitution of (4) into (1) results in a closed form state model for module j; x. t = A.X. t + B. APTL. t 5 __J() —3—3() _3 J() () r A c‘ n 0 I. ‘7 0 — —J — Aj = and Ej = . -MTl T. -MTlD. -MTls. L -J -3 -J -J -J -J .1 L. .1 The invertibility of matrix lj is proved in reference [10]. 39 The ability to perform a rapid update of ‘2. and Sj each time the static state estimate is computed depends on the number of generators Nj and load buses Kj in a module since the elements of matrix raga. (t) 313;. m“ 35:]. (t) 323. (t) agwt) saga-J gagjm agju) 3 must all be updated, matrix 3PTL.(t) T. = _ e. t 3 3_J( ) must be inverted, and Ej and §j must be calculated. Increasing the number of modules M both (1) decreases the total number of synchronizing torque coefficients to be updated since these coefficients will not be cal- culated for lines that connect modules and (2) decreases the computation to form {gj}?=l and {éj}?=l since M matrix inversions each of small dimension requires less computation than one matrix inversion of large dimension due in part to the extraordinary efforts needed to invert large matrices accurately. Moreover, since the effects of unit commitment, network configuration or load flow changes are often localized, the model for each module need only be updated when the effects on model parameters is sufficient to warrantaniupdate. If there were only 40 one module for an entire power pool, the entire model could need to be updated every time a subsection of it required updating, vastly increasing the computational require- ments for model updating. Reducing the size of the modules and thus in- creasing the number of modules will however increase the total number of measurements required to perform dynamic state estimation in a study area because; (1) measurements are required in each module for (2) Reduction all the real power flows on all lines that connect this module to other modules. This condition not only imposes a requirement to measure line flows in the study area which would not necessarily be measured if one module were used but also requires measure- ment on each end of these lines to get the correct PEij value for each module. a certain level of measurement redundancy is required in each module in order to assure bad data detectability and identifiability in each module. These measurements must be in addition to line measurements because the line measurements are inputs and do not pro- vide information on the state of the module. of module size will thus reduce the computa- tion for updating the state estimate and thus reduce 41 computer hardware requirements for dynamic state estima- tion but will increase the instrumentation and data acquisition costs for retaining a reasonable level of measurement redundancy in each module. The subject of bad data detection and identification for dynamic state estimation on a power system is beyond the scope of this thesis and is a subject for further research. The level of measurement redundancy for good estimation and fast reconvergence after a disturbance however will be con- sidered in this research. 3.6. Dynamic Load Model A dynamic model for load power deviation APLj(t) in module j has not been required since it was assumed that the load power was measured and known perfectly at every bus k in the module. This assumption requires many measurements all synchronized and taken at a fast sampling rate which may not be practical. A dynamic model of the load power deviations APLj(t) is proposed in this section in order to eliminate the need to make synchronized measurements of load power at all load buses at this high (> 5/sec) sampling rate. Three different load models will be developed which assume that the load at each bus can be decomposed into different load types and that each load type can be modeled as a Markov process. These models differ from those presently being developed under EPRI and DOE support in that 42 (l) the dependence of each load type on voltage magnitude and frequency is ignored (2) the Markov models ignore all struCture informa- tion available from particular knowledge of the load type and assumes a structure that lacks any dependence on voltage and frequency at the load bus (3) the parameters and order of these Markov pro- cess models need to be identified based on measurements of load of each type and cannot be derived based on knowledge of the kind of load being modeled. Although the work done under the EPRI and DOE projects [22] may provide load models that could be used for dynamic state estimation, the Markov process models proposed here are simple and reflect the general forms of the models that are possible. A general model of the load power deviation will be proposed and the three specific load models will then be discussed and developed based on the general model form. The load power deviations in module j are assumed to satisfy P.t=. .t A__I_.._J() 513935“) Agjm = _F_'_._A_q_j(t) + g. 3 3—33' m where Wj(t) is a white process 43 Eiflj(t)} = 9 T _ E{Ej(tl)flj(t2)} - Qj5(tl - t2) with initial conditions E{qu(0)} = g E{qu(0)Aq§(0)} = yj B£qu(0) y§(t)} = 9. 3.6.1. Low Pass White Noise Model The simplest dynamic model for load power devia- tions APLj(t) is to assume that the loads at every load bus k are independent, low-passed, white noise pro- cesses with bandwidth fkj such that APij(t) = Aqkj(t) Aqkj(t) = ~fijqkj(t) + fkjij(t) E{ij(t)} = 0, Eiij(tl)ij(t2)} = 5(t1 — t2)ij such that H. = I _J — Ej = d1ag{-f1j'-f2j'...'-ijj} Ej = -§j = diag{flj,f2j,...,ijj} Qj = dlag{Qlj sz 00- Qij} T W.t=W.t,W.t,...,W.t _3( > ( 13‘ ) 23( ) ij( )> T T . t = . t = . t . t ... - - t . ggj( ) gj( ) (q13( ) q23< ) qK.3( )> J 44 A more general and realistic model might occur if the white processes Wj(t) were correlated so that Qj was not diagonal. 3.6.2. Load Component Model A second load model assumes that the load at each bus in module j can be decomposed into zj dif- ferent load types Aqij(t), i = l,2,...,9.j such as commercial, residential, agricultural, etc. and 2 MU- - j = APij(t) - h .Aqij(t)’ k 1,2,...,K. i=1 k1 3 here hi1 is the percentage of load type i (Aqij(t)) present at load bus k in module j (APij(t)) with 2. j . 2 hi. = 1. Each of these load components Aq..(t) is i=1 1 13 assumed to be independent Markov process with a dif- ferential equation of order n. that is ij’ ) (n..-n) n. ( lj - _ J 13 (t) — Z finAq.. (t) + 9 n=1 ni. Aq.. 3 13 (t) 13 ijwij where Wij(t) are scalar white noise processes with statistics E{Wij(t)} = 0 E{Wij(tl)wij(t2)} = 5(t2 - t2)Qij and E{Wi j(tl) W1 1 j(t2)} = 0 , il # iZIV illiz 6 [lin10 2 45 Now define the auxiliary variables Aqijl(t):---IAQ-- ij-l(t) by the relations Aqij2(t) = Aqij(t) (nij-l) Aqijn..-1(t) = Aqij (t) 13 T _ AElij(t) — [Aqij(t).Aqijl(t).....Aqijnij_l(t)] Then Agij(t) = gijAgij(t) + gijwij(t), 1 = 1,...,2j where F. " °. ’33 o 1 £3 . . . f3 b1n.. il 13 T — 'G‘ij "“ [0,0,009'91010 Aggregating these models for each load component, define Aa§(t) = [Ag$j(t),-...Ag§.j(t)] 3 then 46 T AP - t = H 0 A . I = p ' o o o I 3 where nlj term n2j terms nfijj term r—A—fl A A HT=[h3 0,0, 01:30 30 h3 .00 03) _kj kl! I k2, (000, ’00., kzj’ ' pace, and APL. t = H.A . t _J() _3 33() C HT H. = I _j . 3,33- L 3'3 and qu(t) satisfies the differential equation Agjm = g. qu(t) + g. wjm J J T _ flj (t) - [Ea-13(t)’ - - - .Wx .3 (tll r J a Flj 0 0 . . . 0 0 F . F. = —23 . ...] . 0 0 F . C _&jJ ’- “'1 glj 0 0 . . 0 0 E2j . G. = . —J . C9 0 -&jo Qj = diag{Qlj,...,Q£j} . In the above discussion it was assumed that Wij(t) are uncorrelated, drOpping the assumption results in a more general case and a nondiagonal Qj matrix. 47 3.6.3. Geographical Load Model The fact that each geographical area has a particular load characteristic, provides the basis for the third load model. Assuming there are tj different geographical areas, each with a load dynamic given by a Markov process Aqij(t), and that the load at each load bus is composed of a percentage (hi1) of the load Aqij(t) of each of the Rj geographical areas in module j so that I. = j APij(t) 1:1 hkiAqij(t) . Assuming a differential equation of order nij for load type Aqij(t), the structure of the geographical load model will be identical to the structure of the component load model and thus . = . A . Ag£3(t) H3 qj(t) Agjm = g]. qu(t) + g]. Wj(t) T — - Eiflj(tllflj(t2)} - 6(tl t2)gj where matrices Hj, Ej structure to those for the component load model. Here and Ej will have identical again Qj may or may not be diagonal. 48 3.7. Complete State Space Model of a Module In this section a complete state space model for the system is developed and the measurement vector is defined. It is shown that the model of a module has the following general form Em A gyt) + g gm + 9 wt) (6) gm _c_ yt) + gt) where u(t) represents the known measured signals; W(t) is a white disturbance process that is used to produce the stochastic load model or the measurement noise associated with the measurement of injected power. The process y(t) is the measurement noise on those measured signals used to produce the dynamic state estimate. The white processes W(t) and 1(t) are uncorrelated with statistics: mwtn = g (7) T -- .— E{y(t)} = Q (8) 'T' The initial conditions are assumed to be un- correlated with v(t) and W(t) with statistics 49 Etyon = g (9) E{§ 3533(0)} = 2(0). In this section the j subscript which refers to the module under consideration is dropped for the ease of notation but it is understood that the following equations are derived for an individual module. In section (3.5) the linearized model for module j was derived. It is repeated here for the ease of reference. (The j subscript is dropped.) A§_(t) 9, i A§_(t) g . = -1 ...l + _1 Aflvt). (5) mm ‘3 2 14 9. Agm 12 §. To completely specify the model APTL(t) dynamics has to be specified. As was mentioned in section (3.6) the dynamic fluctuation of APTL(t) can be followed by (1) measurements at each load bus or by (2) modeling the load dynamics and measuring the power injections into the module. The first Option is generally not feasible for a large utility or a pool due to the cost of measuring and communicating these load measurements at every load bus in the study area to a central processor. Thus in general a load model has to be developed so that (dropping subscript j) APELW) = A3§°(t) - Agguz) (10) 50 where APE9(t) are the measured line power injection into module j from other modules and the external area and the load power model is (dropping subscript j) AP_L_ * Aq(t) Ww(t) . 22-511. W(t)+§__v1(t) Q = y(t) = —G E 9 1 9 mm The measurement process statistics (8) for this set of observations are E{y(t)}=0 gytl) y_(t)}=_fg5(t -t) where Iw u 2 Q Since the AP§9(t) measurement has been used both as in- put vector and also as part of the observation vector, the resulting measurement process disturbance y(t) would be correlated with the system input disturbance T 9939 9 o 9. T _ ' _ E{_W_(tl) 31 (t2)} - 5(1:l t ) . 2 Therefore the conventional Kalman filter equations can not be used since these equations require that W(t) and y(t) should be uncorrelated. The observation vector for the modularized dynamic state estimator is thus limited to measurements of fre- quency to avoid correlation of the measurement and input disturbance processes. The observation equation then becomes z = g 5(t) + z_<_ (0)} = 2(0) E{§(0)} = Q T _ T _ B{§(0)W (t)} - E{§(0)y (nAT)} — g . The Kalman filter equation for this sampled data form of the system equations are now presented [24, 23]. Suppose the best (in the mean squares sense) estimate of the state at time tn = nAT immediately after the observation X(nAT) is taken is i(tn/n) i.e. 56 3(tn/n) = E{§(t)/X(iAT), i = 1,2,...,n} with error variance matrix E‘tn/n) = E{[§(tr)— 3(tn/n)1[§(tn) - 3(tn/n)]T} . In the time interval (nAT, (n+l)ATL no new observation is available. Therefore the best state estimate 3(t/n) is governed by the system differential equations; 3(t/n) = g 3(t/n) + g g9T + 9 9 9T 59 The algorithm starts with initial conditions; [(0/0) = )(0) and 2(0/0) = 9 (31) CHAPTER FOUR GLOBAL REFERENCING FOR MODULARIZED DYNAMIC STATE ESTIMATORS The decoupling of the dynamic model of each module from the dynamic models of the other modules by measuring the real power flows on all transmission lines that connect the modules is essential if a global dynamic state estimation is to be feasible. However, the modularized dynamic state estimators cannot provide a global dynamic state estimate for the whole area be- cause each of the modularized state estimators has a separate reference generator and these reference gen- erators have not been referenced to a global reference for the entire area. Thus, the modularized state estimators could only be used to assess security and stability margins within the subarea where it provides a dynamic state estimate and could not be used together for global security assessment, security enhancement, or control tasks in the area unless a global reference was provided. Also the dynamic state estimator for a module is vulnerable to bad data in the measurement of the in- jected power flows from other modules, because these power flows are used as input to the system and there is 60 61 no consistency check on these measurements to detect and identify bad data. To overcome these two disadvantages, two general methods of referencing the modularized dynamic state estimators are presented. It is shown that by globally referencing these modularized estimators, a global dynamic estimate of the entire study area and a procedure for bad data detection and identification on power flow measure— ments between modules are obtained. Some preliminary remarks are necessary before the referencing procedure can be discussed. In the referencing procedure the following assumptions and notations have been used; 1 - Without loss of generality it is assumed that the re- ference angle for module one 511(t) is the reference angle for the static state estimator and also it is going to be used as the reference angle for the global dynamic state estimator referencing. In case the static state estimator's reference is not the same as the angle chosen to reference the global dynamic state estimator, it is sufficient to subtract the value of this angle, as given by static state estimator, from all incoming static state angle estimates, to get them all referenced to 611(t). The value of this angle, 611(t), is then arbitrary and can conveniently be set equal to zero at all times, i.e. 611(t) = 0 V t; 62 2 — with the above convention the static state estimates of internal generator bus angles 6;j(t) and the load bus angles e£j(t) for module j are referenced to the generator bus angle 6:1(t) = 0. 3 - The dynamic angles of these same buses are 6..(t) 1] and 6kj(t) where .(t) 0 ° .— 6i] dij(t) + Adij(t) 1 — 1,2,...,N. . J j = 1'2'OOI'M 8kj(t) + Aekj(t) k 1,2,...,Kj . ekj(t) Here dij(t) and 6kj(t) are globally referenced angles and Adij(t) and Aekj(t) are the dynamic angle deviations which are globally referenced to A611(t) = 0 where 611(t) = 6:1(t) + A611(t) = 0; 4 - For module j, the angle estimates given by the modularized dynamic state estimator, Adij(t) and Aékj(t), are referenced to the deviation of the internal generator bus angle, A61j(t)' so that A6--(t) A6'-(t) - A61j(t) i = 2,3,...,Nj 1] 13 ~ j = 2,3,...,M and for angles in module one i = 2,3,...,N1 Aek1(t) = A9k1(t) - Adll(t) = A9k1(t) k = 1,2,...,K1 63 Therefore the globally referenced angles are 6--(t) 1] 6§j(t) + Adij(t) + A61j(t) 6kj(t) 0£j(t) + Aekj(t) + A61j(t) . Since the first two terms in the right of the equality sign are available from the static state estimator (the first term) and dynamic state estimator (the second term) all that is needed to reference module j to the global reference is to somehow estimate the value of Adlj(t) (the deviation of the local re- ference angle with respect to the global reference) and add it to the already available static state estimate and locally referenced modularized dynamic state estimate. A very simple method for providing a global re- ference frame for these modularized dynamic state estimators is to make one generator be common to all modules and thus all modules would be referenced to the internal gen— erator bus of this generator. Note that this procedure implies 61j(t) = 611(t) = 0, j = 2,3,...,M . If the use of a common generator for reference is impos- sible, which may often be the case for large systems, two other approaches are possible and given in the following subsections. 64 4.1. Reference by Association In this scheme (1) one generator should be common to and its internal bus angle, 6:1(t) used as reference to as many modules as possible; (2) each of the remaining modules, which do not contain this reference generator,should share a generator with a module that has the re- ference generator. The angle of the internal bus of this common generator in the module without the reference generator will be used as reference for this module and constrained to be the same as its value in the module with reference generator. For the first group of modules, those with the reference generator, the angles 61j1(t) are of course constrained to be identical, that is, 51j1(t) = 611(t) = 0 , 31.6 [1,2,...,M1] Wthh implies A61j1(t) = A611(t) = 0 Since 6 (t) = 6:1(t) + A513. (t). 316 [1,2,...,M1] 131 1 where 6:1(t) is the estimate of the internal generator bus angle for the reference generator, obtained from the static state estimator and assumed zero. 65 K. J The global state estimates {6k. }k-I and 31 ‘ N3 1 . = . {5ijl}i=l for modules 31 1,2,...,Ml are Simply the sum of the static state estimate and the local modularized dynamic state estimate a . t = 9°. t + 1% . t k = 1 2,...,K. k31( ) kjl( ) kjl( ) . 31 1.. t = 5?. t + 13.. t i = 1,2 ...,N. 131( ) lJl( ) 131( ) , 31 For the second group of modules, those sharing a common generator other than reference one for the first group 611(t): generator (1j2) in module 32 is generator (ljl) in module 31' A reference by association would require that the reference angle Glj (t) for module j2 2 = o be equal to the angle dgj (t) 62j (t) + Adfij (t) l l l where 9 e {1,2,...,Nj }, jl e {1,2,...,M }, 6 {M1+1,...,M}. l 1 32 To produce global state estimates for modules 32 E {Ml+l,...,M},A6lj (t) =A6£j (t) must be added to K3 2 “j l {ek. (t)} 2 and {61. (t)} 2 32 k=1 32 i=1 a . t = 9°. t + 15 . t +15 . t , k = 1,2,...,x. k32< ) k32( ) k32( ) 231( ) 32 .. t = 5?. t + 15.. t +15 . t i = 1,2,.. ,N. 132( ) 132‘ ) 132‘ ) 231< ). 32 66 The reference by association procedure suffers from two major difficulties which would prevent it from ever being implemented as a practical referencing pro- cedure; (l) The referencing procedure is vulnerable to bad data, because if Aégj (t) were in error 1 all the global estimates in module would j2 be in error and hence not useful. (2) (M-1) extra generator models would be needed since the reference generator models for all modules j2 E {Ml+l,...,M} would be common with generator models (£j1) in module jl € {l,...,M1} and because the reference generator models in modules j1 6 {l,...,M1} represent a single generator. 4.2. Reference by Line Measurement It may be more desirable to split the study area into completely nonoverlapping modules and for each module to treat the rest of the study area as part of external area. A procedure for global referencing is developed that utilizes the measurements of dynamic power fluctua- tions on lines that connect the modules. These measure- ments are required to decouple the dynamic models for each module and since they are already available, their use to provide a reliable accurate global reference for 67 the modularized dynamic state estimator is desirable. The develOpment of this global referencing procedure will be carried out in three stages. (1) (2) - It will be shown that module j can be referenced to module one using the following data: (I) The dynamic measurement of the real power on a single transmission line connecting them; (II) The static and the dynamic state estimates of the voltage angle at both ends of this line; (III) The static estimate of voltage magnitude at both ends of this line. Here measurement of voltage magnitude and reactive power at only one end of the line would provide this informa- tion if the static estimate of voltage magnitude at both ends of the line is not available or its use is not desirable; (IV) The line parameters. This data is used to calculate the reference angle A61j(t) that references module j to the reference angle of module 1 assuming that this angle is always zero. - It will be shown that module j can be much more accurately and reliably referenced to module 1 if a least squares procedure is used to estimate the re- ference angle A61j(t) from the data on all lines connecting module one and j. The least squares 68 estimates can be more accurate and reliable using the data on all lines rather than on just one single line because the errors in the dynamic measurement of real power, the static estimate of voltage magnitudes at two ends of the line, and the static and dynamic state estimates of voltage angles at both ends of the line can be averaged out and because bad data on a particular line can be detected, identified and eliminated from use in the referencing procedure. This least squares procedure for referencing any two modules could be used to reference (a) dynamic state estimators for different areas or pools where 1 and j would represent areas rather than modules within an area; (b) a global referencing procedure for an area where all modules j = 2,3,...,M could be referenced to module j = l. (3) - It is shown that the least squares procedures for referencing module j to module 1 by estimating A61j(t) from the data on all lines connecting these M two modules can be extended to estimating {A<51j(t)}]=2 in one step so that these estimates could be based on the data (as listed in (1) above) on all lines connecting all modules within the area rather than 69 just on data for lines connecting module 1 to a particular module j for each j. Three stages of development for the global re- ferencing procedure are presented in the next three sub- sections. 4.2.1. Referencing Two Modules from Line Measurements As was mentioned earlier, all that is needed to reference module j to module 1 is to add the devia- tion of local reference angle, A61j(t) (with reference to module 1), to the sum of locally referenced dynamic estimate of angle deviation Adij(t) and 5kj(t), and the globally referenced static estimate of the angles 5;j(t) and 9£j(t) and thus obtain global dynamic estimates dij(t) = dij(t) + Adij(t) + A61j(t) i = 1,2,...,Nj 6kj(t) = ekj(t) + Aekj(t) + A61j(t) k = 1,2,...,Kj where A6 = O and A3 . = 0 j = 1,2,...,M ll 13 A61j(t) can be obtained and module j can be referenced to module 1 if a line exists connecting load bus kl in module 1 to load bus kj in module j. Using the measurement of real power flow in this line Pk k (t) j l and the static estimates of the voltage magnitudes at both ends of the line IVE (t)| and Ivi (t)| and also ' l J the line parameters Rk k and Xk k . The angle dif- 3'1 3'1 ference 6kj(t) - 6kl(t) is obtained from the formula [19! 25]; 70 1 [R 2 2 k.k Rk k + Xk k 3 3 j 1 j 1 ' Rk.k 3 |v§ (t)!2 j l|v§j(t)||v§l(t)|cos(ekj(t) - ek1(t)) + xk.k IVE (t)I|V£ (t)lsin(6kj(t) — 9k1(t))3 3 1 3 l or the approximate equation [24] _ __li__ - - Pk.k (t) - xk k [IV§j(t)|IV£1(t)ISin(9kj(t) 9k1(t))]. 3 l J 1 Since ij (t) = 91:]. (t) + 16k]. (t) + A51j(t) 0kl(t) = 0£1(t) + A0k1(t) + A611(t) 11111 = R. l(t) = g1(t) + ilel‘t’ + EL jljmq wmwoz pcmfimusmmmz ucmuwmmwn amps: Hepmfiwumm mo wosmEHOmumm .Hlm magma 98 is 0.001 Hz. and 0.0001 Hz. since the dynamic model for the estimator does not effectively filter the noise from the measurements at these low measurement noise levels. The maximum error of angle estimates changes significantly as the maximum measurement error decreases from 0.01 Hz. to 0.001 Hz., but does not change significantly as the maximum measurement error decreases from 0.001 Hz. to 0.0001 Hz. because apparently the accuracy of the angle estimates depend more on the accuracy of the dynamic model than on frequency measurement error when this measurement error is small. A power frequency recorder [27] can presently measure frequency at a rate of five samples per second with a maximum error of 0.001 Hz. and a new Real Time Digital Data Acquisition System (RTDDAS) can measure frequency at twenty samples per second with a maximum error of some- thing less than 0.001 Hz. It is thus clear that the RTDDAS samples sufficiently fast to provide frequency estimates with maximum errors of less than 5% of the maximum fre- quency deviations due to random load fluctuation and much less than 5% Of maximum frequency deviations due to con- tingencies. The accuracy of the angle estimates using the RTDDAS are excellent and could not be appreciably improved if a more accurate frequency measurement were possible. Based on the above results and discussions, for the rest of this chapter, it is assumed that the dynamic 99 state estimator has measurements with maximum error of 30f = 0.001 Hz. available. 5-4-2 Effect of Measurement Sampling Rate As discussed in Chapter Three there are two sets of measurements required for dynamic state estimation, a) the power flows between external and internal system to be used as input to the linear model and b) the frequencies at different generator buses in the internal area to be used as the measurement vector. To investigate the effect of the rate at which these measurements are taken on the performance of the estimator the 30 steady state maximum error for fre- quency and angle at different buses of the system are calculated for different sampling rates. Table 5-2 is a comparison of these results which show that as the rate of sampling increases the error in the estimated value of frequency and angle decreases. The increase in accuracy for frequency estimates is negligible, as the frequency measurement is very accurate (30f = 0.001 Hz.). The accuracy in angle estimates improves more noticeably, since it is about twice as accurate in case of 20 per second than in case of a 5 per second sampling rate. But overall because of the excellent accuracy of 5 per second sampling rate on one hand and the higher cost of implementation for 10 and 20 per second sampling rate on the other it is con- cluded that a rate of five measurement per second is adequate for dynamic state estimation. 100 . Honum mamas ammo.o Iommo.o omma.o I oemo.o omam.o I.%mo o m mumum hummum om .. . uonnm mosmsvoum .Nm waooo.oIHmooo.o mm mmooo.OInmooo.o .Nm odooo oIHmooo.o m mumum hummum om .omm mom on .me Mom 0H .omw me m mumm mafiamfimm mmuum msflamamm ucmuommwo moons HoumEHumm on» NO wosmfiuomumm .NIm magma 101 It should be noted that as the natural modes of tie line power fluctuations are between 0.5 and 2 Hz., it is not desirable to use sampling rates less than 5 per second. 5-4-3 Effect of Location and Number of Measurements Taken To determine the effects of the number and loca- tion of measurements taken, the steady state error co- variance matrix was calculated and compared for the dif- ferent cases considered. The following results were ob- tained: a) Table 5-3 shows the maximum 30f and 30a error in the estimated values of frequency and angle at bus number three, namely the St. Clair station. The frequency of this bus was measured in every run while frequency measure— ments at other buses were dropped one by one. It can be seen that as the number of measurements decrease, the accuracy of the estimates decreases too, but this de- crease in the accuracy of the estimates is very very small. Also it is seen that the change in the accuracy of the estimates due to the location of measurements is negligible (compare lines 2 and 3 or 4 and 5). Based on these results it is concluded that as long as the frequency measurement at a bus is available to the dynamic state estimator, the accuracy of the estimated angle and frequency at that bus is almost constant regard- less of the number and location of other frequency measure- ments . 102 hmo.o mmmooo.o m .H o mmo.o ommooo.o v .m .m .H m mmo.o ommooo.o m .v .m .H v Hmo.o m~mooo.o w .m .v .m .H m omo.o nmmooo.o m .v .m .m .H N omo.o mmmcoo.o m .m .v .m .N .H H mmumwc .Nm m mom m mom um Houum mamas um uouuw woswsqoum uswEmHsmmmE mocmsvmum # wumum hommum om myopm womoum mom m>ms umnp momsm mafia .mw mom um uouum Tamed paw wosmsvmnm owumeflumm mumum mommum may sfl musmEmusmmwz hosmswmum mo coaumooq cam Hmnfisz mo uowwmm .mIm magma 103 b) Table 5-4 shows the maximum 3of and 30a error in the estimated values of angle and frequency for normal random load fluctuations for bus #6 (Monroe 3-4) when measurements at that bus are not available and measure- ments at other buses are eliminated one by one. The first line of this table gives the maximum 30f and 30a error for frequency and angle at bus #6 when all the frequency measurements are available and has been included as a benchmark against which the results for subsequent runs can be compared. In the second run frequencies at all buses except bus #6 are measured and it can be seen that the maximum errors in estimated angle and frequency increase almost three and eight times respectively. This maximum error in estimated frequency is about as big as half the maximum error in frequency due to the oscillation of frequency under random loading. Lines three and four in this table show the results for runs when an additional frequency measure- ment is eliminated. In both of these cases it is seen that the maximum error has increased from the previous case but it is seen that the maximum error is much larger when the measurement at bus 5 is eliminated because bus 5 is much more coherent with bus 6 than bus 2 and thus helps supply more information about the frequency and angle at bus 6. The additional two runs confirm that the error in angle and frequency estimates at a bus where no 104 vm.m mhmm.o ov.mH HmmHo.o m .H m mv.m ~mmm.o mm.wH mmmHo.o m .H m om.v mwhv.o om.mH wmvHo.o. v .m .m .H v no.m omom.o mm.m mmoo.o m .v .m .H m mm.~ mHm~.o eH.m mhoo.o m .v .m .N .H m H enmmo.o H hmooo.o m .m .v .m .m .H H OHDwu mummmo oHumn .Nm ow mam ow man um uouno mHMcm um nouum mocmsvmnm ucmETHSmmmE mucosvmum mumum womwum om mumum momoum mom m>mn umsu mmmom * mmmu .ww mom um uouum mHmsd can woswsvmum poumEHumm mumum hpmmum on» :H mucmfimusmmwz hocooqmum mo GOHDMOOH can Honesz mo womwmm .va THQMB 105 measurements is available continue to increase if addi- tional measurements at other buses are eliminated. From these runs it can be concluded that when the frequency at a bus is not measured the maximum error in the estimated frequency becomes comparable with the fre- quency deviations due to random load; that is, it shows that the model and the frequency measurements at other buses are unable to estimate frequency at this bus accurately enough so that the effect of random fluctuation of load on frequency would be detectable. Therefore in the normal operating state and in the absence of any dis- turbances the coherency measure, which is important in order to determine (1) weaknesses in the network and thus the dynamic structure and (2) the dynamic stability of any portion of the power system, cannot be obtained. Thus it would be imperative to measure frequency at every sig- nificant generator bus if this coherency measure were to be determined in the normal state. When a disturbance, fault or switching operation, occur then the angle fluctuations would be of a much higher magnitude than the fluctuation due to random load and thus in this case, as will be seen later, the coherency measure can be ob- tained even if some buses do not have frequency measure- ments. This run also shows that the location of the frequency measurements has a significant effect on the 106 accuracy of the estimated states. If the frequency at a bus is not measured but the frequency at a bus coherent with it is measured (line 3) the error in estimated value is far less than if a bus which is not as coherent had frequency measurement instead (line 4). This result suggests that if in implementation there is a constraint in the number of measurement devices such that not all the buses could have frequency measure- ment available, an effort has to be made to include at least one frequency measurement from every coherent group. c) By comparing the plotted error in estimated angle for different number of measurements it is concluded that when frequency measurement at any bus is available the error in estimated angle at that bus is negligible and the convergence of the estimate at that bus is immediate regardless of how many additional measurements are avail- able. Figure 5-3 shows the error in estimated angle at Karn station when the only measurement available to the estimator is the frequency measurement at this bus. It can be seen that after the insertion of the fault, the error increases momentarily but after 6 cycles the measurement corrects this error and the estimator con- verges. It should be noted that this is the worst case since Karn in the bus with the fault, so that the error in estimated angle and the time to converge is far less for other buses if they have frequency measurements avail- able and observe the effects of the same contingency. 107 .zouhmhm 2mm: hm macaw mo Zomumcmtoo .mIm museum mozouww z“ wzuh 8..» 8..» 8 ..u 8. u 8... 8. a 8.... 8. 8.8 8..u 8.» 8... 8... . OO'S-P r 00'3- I 00'!- I OO‘O U 00'! 9338030 NI 310MB UBlUHIlSE NI 80883 I OO'Z «umumzzz mam Humhzm:mm:m¢ut mo mmmzzz ONHMch ¢2H4mzcm Y 00‘8 .¢Im mezzo: ...c «exam ...—9 zcwmmmmzou .VIm mason... wozoomw 2— wt: 8..» 108 8... 8 ..u 8.... 8 ... 8... 8 ..n 8 ..u 8..~ 8 ... 8... 8.... 8. a. ‘ ...... n 8.0.. ......” N 1.9 o l ...m 8 x A o i 1 our. I"... ‘0.“ «I! II . .887- 6‘3 W0 08 N ...u. ...3 mm a 3 0 286 on :22 mam do...“ Huwhzmzwxswmwz mum—.52 OS owuwpcm zjmzcm 00's 109 'When a bus does not have frequency measurement available, the overall number of measurements are impor- tant. In general the more measurements that are avail- able the lower will be the error in estimated angle and the faster will be the convergence at a bus without fre- quency measurements. In this case the location of measure- ments are very crucial, since, with the same number of measurements, the error in estimated angle at a bus with- out frequency measurement is far less and its convergence is much faster if a bus coherent with this bus has fre- quency measurement available. Figure 5-4 is a comparison of error in estimated angle at Monroe 3-4 station (bus number 6) when only one frequency measurement (beside the frequency measurement at bus number one which is the re- ference bus and always has the frequency measurement) is available. The plot with the symbols is the error when bus number three which is very coherent with bus number six has frequency measurement and the solid plot is error when bus number two, which is not as coherent with bus number six, has measurement. It is seen that, although bus number two is the bus with the fault, since it is not very coherent with bus number six its measurement is not as effective as the measurement at bus number three. 5-4-4 Effect of Error in Model Parameters Dynamic state estimator uses a linearized model based on the operating point information given by the 110 static state estimator. When this information is erroneous due to either operating point changes as the normal slow change to follow the load or a contingency, the parameters in the linearized model are no longer exact. To find out how much the model parameters can be in error and how far the operating point can change before it becomes necessary to update the linearized model, the matrix M3 was multiplied by a constant K in the range .4 to 1.6; that is, the parameters were changed from 40% to 160% of their true value and the 30a and 30f maximum error in estimated angle and frequency at bus number six and the coherency measure between this bus and the other buses of the system were calculated. The result showed that when all buses do have frequency measurements, although the errors in estimated values increase by changing the para- meters, this error is so small that it would not pose any problem. Table 5-5 shows the values of 30a and 30f maximum error in estimated angle and frequency at bus number 6 as calculated statistically, when frequency at this bus is ngt_measured. It also shows the coherency measure between this bus and buses number three and four. Bus number three is strongly coherent with bus number six whereas bus number four is not. From this table it can be seen that as the model parameters are increased from their true value to 160% 111 Table 5-5 K 30a in 30f . C64 C63 degree in Hz. 1.6 1.0483 .0406 .243930 .031544 1.5 .9586 .0362 .243957 0.031464 1.4 .8580 .0314 .243988 0.031385 1.3 .7535 .0266 0.244029 0.031289 1.2 .6685 .0223 0.244074 0.031225 1.1 .6489 .0200 0.244123 0.031209 1.0 .7390 .0200 0.244170 0.031225 .9 .9342 .0258 0.244215 0.031432 .8 1.2075 .0333 0.244266 0.031749 .7 1.5666 .0430 0.244350 0.032311 .6 2.0959 .0556 0.244520 0.033377 .5 2.9761 .0794 0.244859 0.035637 .4 4.3670 .0904 0.245353 0.040087 112 of it the errors increase by 100% whereas when they are decreased to 40% of their true value these errors increase by more than 500%. Also from the coherency measures it is seen that when parameters increase both coherency measures decrease but this decrease is very small. In the other direction when the parameters decrease coherency measures increase and this increase is more pronounced and less proportional. It should be pointed