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Statistical model formulation for power systems Mumford, Donald Gregory 1971

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A STATISTICAL MODEL FORMULATION FOR POWER SYSTEMS by DONALD GREGORY MUMFORD B.A.Sc., University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Elec t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor ...» Members of the Committee ,•. Head of the Department . v . . . . , Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or p u b l i c a t i o n o f t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Depa rtment The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT An i n v e s t i g a t i o n has been undertaken to a s c e r t a i n how r e a d i l y a power system lends i t s e l f to s t a t i s t i c a l modelling. A nonlinear state v a r i a b l e model has been derived i n terms of measurable states. This model i s l i n e a r i n i t s c o e f f i c i e n t s which are evaluated by the l e a s t squares f i t t i n g technique of regression a n a l y s i s . The s t a t i s t i c a l model's performance i s evaluated by comparison of i t s predicted system responses with those predicted by Park's formulation, and with those produced by a laboratory power system model. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES i v LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT v i NOMENCLATURE v i i 1. INTRODUCTION 1 2. STATISTICAL MODELLING USING REGRESSION ANALYSIS 4 2.1 Features of Regression Analysis 5 2.2 Assumptions in Regression Analysis 7 2.3 Significance of Regression Equation 8 2.4 Significance of Coefficients 9 2.5 Regression Analysis in Power Systems 10 3. MATHEMATICAL MODELS - THEORETICAL AND STATISTICAL 12 3.1 Synchronous Machine State Variable Equations 13 3.2 Voltage Regulator-Exciter Equation 18 3.3 One-Machine Infinite Bus System 20 3.4 Formulation of S t a t i s t i c a l Model 22 4. LABORATORY POWER SYSTEM AND DATA ACQUISITION SYSTEM 28 4.1 Power System Laboratory Model 29 4.2 Data Acquisition Hardware . 34 4.3 Data Acquisition Software 39 5. PERFORMANCE OF THE STATISTICAL MODEL 44 5.1 System Data for S t a t i s t i c a l Model Derivation 44 5.2 S t a t i s t i c a l Investigations of the Regression Model... 47 5.3 Model Responses to Step Inputs 51 6. CONCLUSION 59 APPENDIX 3A ' 61 APPENDIX 3B 63 APPENDIX 3C . 64 APPENDIX 4A Flowchart for Data Acquisition Program 65 REFERENCES 67 i i i LIST OF TABLES / Table ^ Page 4.1 Laboratory Model Machine Specifications 29 4.2 Laboratory System Parameters 33 4.3 Noise at A/D Input 35 4.4 Multiplexer Channel Selection Decoding 38 5.1 System Operating Points Investigated 46 5.2 Steady State Values 47 5.3 Example of Variation in Coefficients 53 iv LIST OF ILLUSTRATIONS Figure . /Page 2.1 Example of Regression Model /.. 9 3.1 Rotor Angular Position 14 3.2 D.C. Shunt Motor Circuit... 17 3.3 Voltage Regulator-Exciter.... 19 3.4 Power System Schematic 20 4.1 Laboratory System Configuration 30 4.2 Active F i l t e r Schematic 34 4.3 F i l t e r Performance 35 5.1 Overall Plot of Residuals 49 5.2 Distribution of Residuals with Time 49 5.3 Example of Variance of Residuals 50 5.4 Example of Test for' Lack of F i t . . . , 50 5.5 Response to step in u-^  at operating point B 52 - S t a t i s t i c a l model from operating points A, B and C combined 5.6 Response to step in u-^  at operating point B 52 - S t a t i s t i c a l model from operating point A 5-7 Response to step in u^ at operating point B 54 - S t a t i s t i c a l model from operating point B 5.8 Response with s t a t i s t i c a l model found using 0.033 second sampling interval 57 5.9 Response with s t a t i s t i c a l model found using 0.083 second sampling interval 57 v ACKNOWLEDGEMENT I wish to thank the people who have assisted me while completing this research project. Especially, I thank Dr. B. J. Kabriel, supervisor of this project, for his interest and encouragement throughout the course of the work. Also I express a hearty thanks to Dr. Y. N. Yu for his valuable comments. The development of the data acquisition interface by Dr. A. Dunworth i s acknowledged. I appreciate the valuable discussions and proof reading offered by Mr. T. A. Curran as well as the careful proof reading of Mr. B. Prior. A special thanks to my wife Joan, not only for typing this thesis, but also for her understanding and encouragement during my graduate program. The financial support of the National Research Council of Canada is gratefully acknowledged. v i NOMENCLATURE Prime Mover L c a d.c. motor coefficient: where u L , i s the speed voltage af ej-.. m af r coefficient r armature resistance a i r series resistance in armature c i r c u i t s P 2 total resistance in armature circuit pole pairs i j f i e l d current 1 armature current; controls mechanical torque output a) mechanical speed m r T^ e l e c t r i c a l torque in d.c. motor D^ ^ d.c. motor damping coefficient Mechanical System J moment of inertia of prime mover - generator set F f r i c t i o n coefficient T £ torque loss due to f r i c t i o n ; T. = Fto i r m Regulator-Exciter T. regulator time constant A regulator gain on reference voltage input regulator gain on terminal voltage feedback u^ regulator-exciter reference voltage v i i Synchronous Machine D synchronous generator damping coefficient f i e l d resistance T' d-axis transient short c i r c u i t time constant d T^o d-axis transient open circuit time constant T" . T" d- and q-axis subtransient open circuit time constants do qo n r x , mutual reactance between stator and rotor i n d-axis ad x^, x^ d- and q-axis synchronous reactances d-axis transient reactance d x", x" d- and q-axis subtransient reactance d* q x equivalent reactance of local load and transmission system 6 < x d + x ) i , , i d- and q-axis current d q i r , f i e l d current rd P real power output of the machine Q reactive power output of the machine T £ energy conversion torque of synchronous generator mechanical torque on the rotor v^, v^ d- and q-axis voltages v • machine terminal voltage v ^ f i e l d voltage v_ a voltage proportional to f i e l d voltage r v a voltage proportional to f i e l d current r K. i j^, if; d- and q-axis flux linkages f i e l d flux linkage v i i i i^ p f l u x proportional to f i e l d flux linkage to e l e c t r i c a l angular speed / U ) q synchronous speed, 377 rad/sec ^ Au per unit speed variation ^ 6 torque angle (between q-axis and i n f i n i t e bus voltage) Transmission System r series resistance x series reactance G shunt conductance B shunt susceptance S t a t i s t i c a l Model X independent variable Y^ value of dependent Variable observed A Y^ value of dependent variable predicted by model Y mean value of observations of dependent variable e. residuals (Y, - Y.) 3 population coefficients b sample coefficients a population variance R multiple regression coefficient ix 1. INTRODUCTION Requirements f o r exact and easily-updated power system models are ever increasing with modern complex systems. One modelling problem a r i s e s when attempting to f i n d a d e t a i l e d representation of multimachine systems. For t h e o r e t i c a l models, the analysis i s very involved [1], [2] and i s l i m i t e d to only a few machines [2]. Another important problem i n modelling a system i s updating the model as the system changes. For t h e o r e t i c a l models, system parameters are measured o f f - l i n e and i f they change during system operation, f or example when a l i n e i s l o s t , the model i s no longer exact. A s t a t i s t i c a l l y derived model may f i n d a p p l i c a t i o n i n dynamic state estimation. Present s t a t i c state estimation schemes and tracking algorithms f o r system s e c u r i t y assessment [3] - [9] do not require a system model. However, i f state estimation techniques are expanded to state p r e d i c t i o n for use with l o c a l c o n t r o l l e r s then a system model w i l l no doubt be required. As data i s obtained f o r the estimation scheme, i t i s convenient to also use t h i s data f o r s t a t i s t i c a l d e r i v a t i o n of the system model. A s t a t i s t i c a l approach to modelling i s investigated i n t h i s t h e s i s . It i s introduced i n an attempt to overcome the problems of t h e o r e t i c a l models i n maintaining an up-to-date system model and pos s i b l y to f a c i l i t a t e modelling more complex multimachine systems. A s t a t i s t i c a l model has the following inherent advantages. It may be set to r e t a i n only the most s t a t i s t i c a l l y s i g n i f i c a n t system v a r i a b l e s , with, i n s i g n i f i c a n t , v a r i a b l e s r e a d i l y eliminated. Secondly, the s t a t i s t i c a l l y - d e r i v e d equations are determined using the p a r t i c u l a r system configuration operating as i t w i l l be when the model i s employed. Also s t a t i s t i c a l modelling lends i t s e l f to real-time updating, thus allowing the mathematical representation of the system to be updated as parameters change, for example, i n response to system load changes. Two ultimate aims of t h i s project are: f i r s t l y , to model systems f o r which accurate t h e o r e t i c a l representations are not obtainable; and secondly, to obtain a mathematical modelling scheme which i s f e a s i b l e f o r on-line modelling and parameter updating. The research reported i n t h i s t hesis i s concerned with the intermediate aim of i n v e s t i g a t i n g the proposed s t a t i s t i c a l scheme using a w e l l defined system i n an o f f - l i n e environment. This project involves s e t t i n g up a laboratory model power system, and d e r i v i n g two mathematical representations of t h i s system. One i s a s t a t i s t i c a l representation which i s nonlinear i n the state v a r i a b l e s , but l i n e a r i n t h e i r c o e f f i c i e n t s which are s t a t i s t i c a l l y estimated. This model constitutes the major portion of o r i g i n a l research. The other i s a t h e o r e t i c a l representation based upon Park's formulation of the synchronous machine equations. I t s purpose i s to allow a performance comparison of the newly-developed s t a t i s t i c a l model with the c l a s s i c a l t h e o r e t i c a l model. Both of these are checked against the laboratory system performance. Another facet of t h i s research i s the data a c q u i s i t i o n required to c o l l e c t observations 3 from the laboratory system for use i n der i v i n g the s t a t i s t i c a l model. The basic i n t e r f a c e and computer was a v a i l a b l e but the i n t e r f a c e required m o d i f i c a t i o n before being used. As the monitored power system sig n a l s contain considerable undesired noise, t h i s project also e n t a i l s s i g n a l conditioning and f i l t e r i n g . Data a c q u i s i t i o n required development of PDP-8 software as we l l as extensive data handling and checking programs written f o r the computing center IBM 360/Model 67. The organization of the work i s as follows. Chapter 2 contains a presentation of the ideas of regression analysis as i t i s applied to the s t a t i s t i c a l modelling i n t h i s project. In Chapter 3, the two mathematical nonlinear state v a r i a b l e representations of a power system ( t h e o r e t i c a l and s t a t i s t i c a l ) are developed. The l a b -oratory model power system and the data a c q u i s i t i o n system are discussed b r i e f l y i n Chapter 4. A comparison i s made i n Chapter 5 of the responses from the s t a t i s t i c a l model, the t h e o r e t i c a l model, and the laboratory system. Testing the data f o r v a l i d i t y of assumptions i s also discussed. Chapter 6 includes the conclusions derived from t h i s work as well as a few guide l i n e s f o r further i n v e s t i g a t i o n s . 2. STATISTICAL MODELLING USING REGRESSION ANALYSIS There are advantages to obtaining experimental models of systems using s t a t i s t i c a l techniques. I n s i g n i f i c a n t v a r i a b l e s can be detected by various s t a t i s t i c a l t e s t s , thus y i e l d i n g a model containing only s i g n i f i c a n t v a r i a b l e s . Also, since the model i s formed by data acquired from the a c t u a l system, the s t a t i s t i c a l approach lends i t s e l f to on-line modelling or on-line updating of the system model. Another advantage of using actual system data i s that the models are more r e a d i l y expressed i n terms of measurable states. Describing a system's behaviour s t a t i s t i c a l l y i s accomplished by monitoring system performance, and d e r i v i n g an equation to "best" describe t h i s observed performance. A common mathematically convenient method of determining the "best" equation i s to perform a l e a s t squares f i t to data comprised of measurements of system v a r i a b l e s . This tech-nique, which i s one method of f i t t i n g a l i n e to a set of observations or data points, simply minimizes the sum of squares of the errors. If Y i s the dependent v a r i a b l e , which the model w i l l eventually be used to p r e d i c t , then the error i s the distance measured p a r a l l e l to the Y-axis between the given data point and the f i t t e d l i n e . Regression analysis i s one technique of performing a l e a s t squares f i t . This method of s t a t i s t i c a l analysis has been chosen f o r the thesis p a r t l y because computer programs are r e a d i l y a v a i l a b l e , but mainly because regression a n a l y s i s provides many tests f or checking system data and for t e s t i n g the model produced. When choosing a modelling scheme, one must consider that systems are defined to greater 5 and l e s s e r degrees. At one extreme are completely d e t e r m i n i s t i c systems fo r which a l l theory, and therefore the model, i s completely defined. At the other extreme are the "blackbox" systems for which there e x i s t s no theory d e f i n i n g system performance from which a model may be derived. A power system i s somewhere mid-way because even though the theory i s known from which a model may be derived, the parameters of t h i s model w i l l change as the system operating point and the system configuration change. Regression analysis can t r e a t any system as a "blackbox" and use data to construct a model by t r i a l and error, but t h i s method requires extensive analysis and may y i e l d a model which allows l i t t l e or no i n s i g h t i n t o the p h y s i c a l structure and operation of the system. However, i f the form of the system model i s constrained such that i t has p h y s i c a l meaning fo r that system, then regression analysis may be used to i d e n t i f y the parameters of t h i s model with much les s analysis required than for the t r i a l - a n d - e r r o r blackbox approach. In t h i s thesis the mathematical model i s constrained to be of a form derived from theory using Park's repre-sentation of a synchronous machine (Chapter 3), and the c o e f f i c i e n t s i n these equations are estimated using regression analysis on measurements of state v a r i a b l e s made during system operation. 2.1 Features of Regression Analysis The basic concepts inherent i n regression analysis are now introduced. The t h e o r e t i c a l d e t a i l s of t h i s s t a t i s t i c a l modelling technique are not included because they are not required for the a p p l i c a t i o n of regression analysis computer programs. What i s required, though, i s an understanding of underlying assumptions and of the basic mechanisms of the analysis i n order to accurately i n t e r p r e t the r e s u l t s 6 obtained. By way of d e f i n i t i o n , when concerned with the dependence of a random v a r i a b l e Y on a quantity X which i s a v a r i a b l e but not a random v a r i a b l e , an equation that r e l a t e s Y to X i s usually c a l l e d a regression equation. Regression analysis i s applied to determine the r e l a t i o n s h i p between a dependent v a r i a b l e Y and one or more independent v a r i a b l e s X^, X 2, .... X n where X^ may be a simple system v a r i a b l e or a function of one or more v a r i a b l e s . The analysis uses many measurements of the independent v a r i a b l e s and corresponding dependent v a r i a b l e to determine the c o e f f i c i e n t s i n the r e l a t i o n s h i p . For example l e t a system be described by Y = g Q + p X x + 3 2X 2 + e , (2.1) then many observations of Y, X^, and X 2 are subjected to regression analysis to obtain estimates of the l i n e a r c o e f f i c i e n t s 3 Q» 3]_ and 32 • In equation (2.1) e represents the error the model w i l l make when used to p r e d i c t Y and, as i t i s d i f f e r e n t for each Y observed, i t i s not measurable. The population c o e f f i c i e n t s 3 Q, 3]_, and 32 can not be found exactly without examining a l l p o s s i b l e Y, , and X2 values, however they are estimated i n regression analysis by the sample c o e f f i c i e n t s b Q , b-^  and The mathematical model obtained then may be written Y = b D + b 1 X 1 + b 2 X 2 (2.2) A where Y denotes the Y values obtained when using the model f o r pr e d i c t i o n . 7 The procedure used f o r regression analysis may be explained f o r the simple two-variable case by p l o t t i n g points r e l a t i n g observed values of Y and X on a set of axes. A s t r a i g h t l i n e i s drawn through these points such that the sum of the squares of the distances ( p a r a l l e l to the Y-axis) between the points and the l i n e i s minimized. The equation of t h i s l i n e then defines the c o e f f i c i e n t s b Q and b^. M u l t i p l e r e g r e s s i o n . i n c l u d i n g many independent v a r i a b l e s consists of a s i m i l a r process except the s t r a i g h t l i n e i s replaced by hyperplanes i n m u l t i -dimensional space. The choice of independent v a r i a b l e s and therefore the form of the model chosen depends upon p r i o r knowledge of the p h y s i c a l system unless a blackbox approach i s being used i n which case the independent v a r i a b l e s are guessed. 2.2 Assumptions i n Regression Analysis In the model Y^ = 3 G + 3]_X-L + £ i , i = 1, 2, ...n describing each measured value of the dependent v a r i a b l e , i t i s assumed that: (1) e± i s a random v a r i a b l e with mean zero and variance a 2 (unknown), that i s , E ( e ± ) = 0 , V(e-j:) = a 2 . (2) and E J are uncorrelated f o r i ^ j so that cov(e^, E j ) = 0. Therefore E(Y ±) = 3 Q + 3 l X ± , V(Y ±) = a 2 . and Y^ and Y j , i ^ j are uncorrelated. (3) In addi t i o n to (1), e± i s a normally d i s t r i b u t e d random v a r i a b l e . That i s , E ± - N(o, a 2 ) . Therefore e^, Ej are not only uncorrelated, but n e c e s s a r i l y independent. Knowing the assumptions governing the errors (and therefore the data) i n regression, one i s able to t e s t the model a f t e r i t i s derived to be sure that i t adequately explains the behaviour evident i n the observed data. Also the data i t s e l f may be checked to v e r i f y / / whether or not i t does meet the assumptions inherent i n t h i s a n a l y s i s . The lack of f i t of the model may be expressed a n a l y t i c a l l y f o r the simple regression case [10]; however, i n the m u l t i p l e regression which w i l l be used, lack of f i t i s investigated by p l o t t i n g the r e s i d u a l s (or values). V e r i f i c a t i o n that the data meets required assumptions i s also accomplished by means of r e s i d u a l p l o t s . The examination of r e s i d u a l s i s explained thoroughly i n Chapter 3 of Draper and Smith [10] 2.3 S i g n i f i c a n c e of Regression Equation A f t e r the form of the equation i s established and regression analysis i s used to evaluate the c o e f f i c i e n t s , the usefulness of the regression equation as a p r e d i c t o r of system performance i s checked This i s accomplished by comparing the m u l t i p l e regression c o e f f i c i e n t , R, with tabulated values which give s i g n i f i c a n t R values for the number of v a r i a b l e s and the number of observations used. The m u l t i p l e regression c o e f f i c i e n t i s defined by 2 sum of squares due to regression S SReg (2 3) ^ sum of squares t o t a l SS •Total n - 9 where SS„ = £ (Y - Y) (2.4) R e g i = l i S S T o t a l =JA " ^ <2'5) for which the various Y values are illustrated in Figure 2.1. If R is not greater than the tabulated value for a desired level of significance, then the regression model is not useful because Y could just as well be described by i t s mean value Y. Y Y A V Y l ^~~(Yf - Y) - - - VV J \ -^-(Yf ~ Y) ^ ( Y f - Yf) = € f &. X Figure 2.1 Example of Regression Model 2.4 Significance of Coefficients When multiple regression is employed to identify coefficients in an assumed model, i t i s possible to have this analysis omit any independent variable or combination of independent variables which are found to be insignificant in the data sample from which the model i s being derived. This is done by using an F-test to check the s t a t i s t i c a l significance of the coefficients. For example in the model Y = b Q + b-jX^  + b 2X 2 (2.6) the question of whether = 0 or not may be answered by i n v e s t i g a t i n g two models, one including b^ X-^  and one omitting that term. That i s , consider equations (2.6) and (2.7) Y = b Q + b 2 X 2 ' (2.7) and measure the c o n t r i b u t i o n of b^ as though i t were added to the model l a s t . This e n t a i l s the use of a p a r t i a l F-test f o r b^. The p a r t i a l F-test which i s outlined i n d e t a i l i n Chapter 2 of Draper and Smith [10] involves f i n d i n g the d i f f e r e n c e of the sums of squares due to regression i n models (2.6) and (2.7). This type of F-test i s used during the b u i l d i n g up procedure for a regression model to omit s t a t i s t i c a l l y i n s i g n i f i c a n t v a r i a b l e s from the r e s u l t i n g model. 2.5 Regression Analysis i n Power Systems To obtain a p h y s i c a l l y meaningful system model, regression a n a l y s i s w i l l be used only to i d e n t i f y c o e f f i c i e n t s i n an assumed form of a power system model. The form of t h i s s t a t i s t i c a l model i s outlined i n Chapter 3 a f t e r development of Park's formulation on which i t i s based. In using regression analysis to evaluate the c o e f f i c i e n t s of a d i s c r e t e state v a r i a b l e model, the dependent v a r i a b l e i s chosen to be the p a r t i c u l a r state considered at time t^ +-^ a n d the independent v a r i a b l e s are the states and functions of states as defined by the form of the model at time t^, where the measurements of state v a r i a b l e s are acquired at the uniform i n t e r v a l of t ^ + ^ - t^c = At. When developing the model, the state v a r i a b l e s are measured and subjected to the 11 regression a n a l y s i s which, by l e a s t squares f i t t i n g to the acquired data, estimates the l i n e a r c o e f f i c i e n t s i n the nonlinear state v a r i a b l e equations. To obtain a system model c o n s i s t i n g of four state v a r i a b l e equations, four separate regression analyses are required. 3. MATHEMATICAL MODELS - THEORETICAL AND STATISTICAL A t h e o r e t i c a l state v a r i a b l e model f o r a one-machine i n f i n i t e bus power system i s derived. The model consists of a t h i r d order repre-sentation of the synchronous machine approximated from Park's equations, and a f i r s t order voltage regulator. Based on th i s formulation, the form of the s t a t i s t i c a l state v a r i a b l e model inc l u d i n g only measurable states i s derived. I n i t i a l development of Park's representation [11] c l o s e l y follows the development as outlined by Vongsuriya [12] and Dawson [13], except that the torque angle, 6 , i s defined as the angle between the q-axis and the i n f i n i t e bus or reference voltage at the beginning of the de r i v a t i o n . Other deviations from the references quoted include an approximation of the synchronous machine e l e c t r i c a l damping to compensate for neglecting amortisseur winding e f f e c t s i n Park's representation and the d e r i v a t i o n of a damping expression f o r the d.c. machine. A t h i r d order machine representation was chosen f o r two reasons. F i r s t l y , Dawson [13] concluded that f o r many system studies (except subtransient and switching phenonmenon) a t h i r d order repre-sentation i s s u f f i c i e n t . Secondly, to f i n d higher order models s t a t i s -t i c a l l y requires much f a s t e r system sampling and thus much more data a c q u i s i t i o n apparatus than that r e a d i l y a v a i l a b l e for t h i s p r o j e c t . / 3.1 Synchronous Machine State Variable Equations Detailed derivations of Park's equations are numerous., As a t h i r d order representation i s used, the d e r i v a t i o n s t a r t s with the t h i r d order approximation of Park's equations [12], [13]. To obtain t h i s s i m p l i f i e d form of Park's model, the following assumptions are made. (1) -Subtransient time constants are neglected. (2) The induced voltages and the voltages due to speed v a r i a t i o n s are neglected because they are small compared to the speed voltages due to cross e x c i t a t i o n s . (3) The r e l a t i v e l y small d-axis damper leakage time constant and armature resistance are neglected. These assumptions reduce Park's equations f o r a synchronous machine i n d-q coordinates to: (3.1) (3.2) Vd = -i|j qa) 0 v q = ^dwo *d = xad Vfd w 0R F (1 + T^ Qp) ^q = - x q i U) 0 + T*>> i d ( 3. 3) "o^ 1 + TdoP) (3.4) The mechanical behaviour of the machine i s expressed by the torque equation: T ± = Jp2<5 + Dp5 + T e (3.5) and the expression for the rotor angle: 6 = ' co0t + 6 (3.6) Equation (3.6) i s represented i n Figure 3.1 where the r o t a t i n g reference i s chosen to coincide with the i n f i n i t e bus voltage phasor. g-axis **- reference a-phase d-axis Figure 3.1 Rotor Angular P o s i t i o n From equations (3.1) to (3.4) and (3.5) and (3.6) the synchronous machine dynamics can be expressed i n terms of one e l e c t r i c a l and two mechanical state v a r i a b l e equations. According to the develop-ment i n Appendix 3A, the state v a r i a b l e formulation of the machine dynamics may be written as: P^F = V F " VFR (3.7) p6 = UQAW pAo) ( T ± - Dp6 - T_) (3.8) (3.9) A u x i l i a r y equations required include the energy conversion torque q^d d rq ' the terminal voltage 2 2 2 vf = v j + v z t d q (3.10) (3.11) 15 the power and r e a c t i v e power output P = v , i , + v i d d q q (3.12) (3.13) and Q = v q i d - v d i q . Also required are equations to evaluate V p R , i ^ and i . These can be solved from equation (3.14) which i s formed by combining equations (3A.7), (3A.2), and (3.4). Tdo - Tdo< xd " xd> 0 V *d = l / 0 3 o - X d / U 0 'FR 0 * i . (3.14) -x /co q o Measurable States In d e f i n i n g the form of the s t a t i s t i c a l model based on Park's formulation, i t i s required that a l l states be measurable. I t may be shown that the immeasurable state, i^-p, may be replaced by the measurable f i e l d current, i f d , as follows. From equations (3.4) and (3A.11) (3.15) i|>„ = T* x , i , , - T' (x, - x ' ) i , F do ad fd do v d d' d and ± d = r n r ^ F - w o ^ - ^ do d 1 *d (3.16) which gives IJJ^ i n terms of i ^ ^ , i ^ and i q . Equations (3.15) and (3.16) are expanded further using a u x i l i a r y equations when o u t l i n i n g the form of the s t a t i s t i c a l model. 16 Torque From d.c. machine theory [14] the torque developed by a shunt d.c. motor i s described as T, = k i (3.17) dc v a where [15] k v = | L a f i f . (3.18) The torque input to the synchronous machine, T^, i n equations (3.5) and (3.9) i s equal to the e l e c t r i c a l conversion torque of the d.c. machine i n equation (3.17) minus the mechanical torque loss i n the system. Mechanical torque loss T^=F(u m ) -a3 i s determined experimentally by evaluating the f r i c t i o n term, F, where F = ^dc = P_ L a f i a i f . (3.19) 0) 2 CD m m when using the d.c. motor prime mover to rotate the synchronous generator (with no load) at various speeds near synchronous speed [15]. Then, T. = T, - T- . (3.20) l dc f Damping The d e s c r i p t i o n of the damper winding c i r c u i t s f o r the synchronous generator i s not included i n the machine equations for the t h i r d order representation. However, the damping e f f e c t may be approximated [16], [17], [18] by D(5) = D x s i n 2 6 + D 2cos 26 (3.21) ( x d ~ X d } where D = v 2 T » (3.22) ( x e + x d) (x* - x") D = v2 _s g _ T.. # ( 3 > 2 3 ) 2 ° (x + x')2 ^ e q / The d.c. motor simulating the prime mover has an el e c t r i c a l damping which i s dependent on the change of torque with motor speed, and may be determined as follows. For a d.c. shunt motor [14] T d c = M a <3'17> e = kco (3.24) v m and in the armature cir c u i t shown in Figure 3.2, v = R i 0 + e. (3.25) a. Substituting gives R ^dc T K v V v - — T,^ + k„io_. (3.26) k v O -f V Figure 3.2 D.C. Shunt Motor Circuit 18 Assuming constant applied voltage, differentiating equation (3.26) yields - dT, = - k du> . • (3.27) k dc v m v Hence the d.c. motor damping defined as D d c = d T d c (3.28) du m may be expressed as - k 2 D d c = — . (3.29) R 3.2 Voltage Regulator-Exciter Equation A voltage regulator-exciter was modelled by an amplidyne as shown in Figure 3.3. Assuming a single time constant representation, the amplidyne equation appears as v f d K v. 1 + T.p x A^ (3.30) where K V i " K A 1 U 1 " K A 2 V f ( 3 ' 3 1 ) Therefore, PV f d = ^ (-vfd + K A l U l - K A 2v t) (3.32) describes the exciter in state variable form where the control, u-^ , i s the exciter reference voltage. This then produces a fourth order model for the synchronous generator and i t s voltage regulator. 19 amplidyne field circuits r amplidyne armature Ui • i circuit alternator field alternator armature q 0 F3 F7 F9 F4 F13 F13 © — & -F8° tM£MFJ4 J3J Figure 3.3 Voltage Regulator - Exciter Rating of F i e l d C i r c u i t s t o i l l a b e l # of turns resistance at 25° C maximum current F3 - F4 1780 980 0.12 F7 - F8 390 43 0.6 F9 - F10 85 2.6 2.2 F13 - F14 400 56 0.5 3.3 One-machine I n f i n i t e Bus System The power system modelled i s shown schematically i n Figure 3.4. V, X JB f Figure 3.4 Power System Schematic A state v a r i a b l e d e s c r i p t i o n of the generator and i t s e x c i t e r i s expressed by equations (3.7) to (3.9) and (3.32) along with t h e i r associated a u x i l i a r y equations. The transmission system may be described by the external voltage and current r e l a t i o n s h i p s [I] = [Y] [V] . (3.33) The q-axis p o s i t i o n has been described by equation (3.6) as 0 = to 0t + 6 (3.6) which i s demonstrated i n Figure 3.1. The external system and the machine must be r e f e r r e d to a common reference i n order to derive p h y s i c a l system q u a n t i t i e s from the state v a r i a b l e model outlined. Expression of external system q u a n t i t i e s in terms of d-q coordinates i s presented .in Appendix 3B using Park's transformations [12], to yield the following result. where (3.34) v d " k i k2~ v sin6 0 + " K l K 2 _ V q -k 2 k l v cos<5 o -K2 K l i q k, 1 + rG - xB K-, K, (1 + rG - xB) 2 + (xG + r B ) 2 xG + r B  (1 + rG - xB)2 + (xG + rB)2 k^r + k 2x -k^x + k 2r" (3.35) (3.36) (3.37) (3.38) An expression for and Vq is available in terms o f . i ^ and-iq and state variables (equation 3-34). Also i ^ and i may be expressed in terms of and fy^ and state variables (equation 3.14). Therefore an additional expression i s required giving ^ and ty^ in terms of state variables only, thus allowing a l l auxiliary variables to be evaluated at any time knowing the state of the system at that time. The develop-ment in Appendix 3C yields the following desired result. V ~ M1 M2" • v sin6 o + \ >. M3 M4 v Qcos6 N 2 (3.39) where M, M2 to K 1k 1 1 _2. [ -11 + (K__ A x q 2 x q uo r K l k 2 D k 2 ] [ _ J ( K 2 l)kx ] A x q (3.40) (3.41) % K l k 2 1 M 3 = — [ + (K 2 1)1^ ] (3.42) A x ' x' d d u K.k. , M 4 = — [ + (K 2 — - l ) k 2 ] (3.43) A x d x d V 2 % K l 1 N X = [ + (K 2 — - 1)K 2 ] (3.44) A T d o X d X q . X q "o K 1 K 2 1 N 2 = [ + ( K 2 — - D K , ] (3.45) A T d o x d x d x d . h \ \ 2 ( x d " V < X q - V f r x / ^ A = + oo 0 z 2 . (3.46) x q x d x q x d Therefore from equations (3.14), (3.34) and (3.39), the a u x i l i a r y system v a r i a b l e s may be found. The i n i t i a l states of a power system defined by v d , v , i ^ , i q , v Q , and 6 are determined from the steady-state operating conditions as outlined by Vongsuriya [12]. This operating point i s described by the machine terminal voltage, v t , the r e a l power, P, and r e a c t i v e power, Q, output of the machine. 3.4 Formulation of S t a t i s t i c a l Model To obtain a p h y s i c a l l y meaningful system model, regression analysis i s used to evaluate c o e f f i c i e n t s i n an assumed form of a power system model. This form i s derived using Park's formulation j u s t out-l i n e d . I t must, however, be i n terms of measurable state v a r i a b l e s so that as the system i s operating the data a c q u i s i t i o n system can record observations of these state v a r i a b l e s d i r e c t l y . If the model were i n terms of immeasurable state v a r i a b l e s , then t h e o r t i c a l a u x i l i a r y equations would be required to obtain the state v a r i a b l e value from the measurements; thus r e s u l t i n g i n a model which no longer has experimentally determined c o e f f i c i e n t s . Because the i n f i n i t e bus voltage, v Q , i s measurable and i s c o n t r o l l a b l e on the laboratory system, i t i s expressed e x p l i c i t l y i n the equations defining the form of the s t a t i s t i c a l model. This allows the model to be used to predict system response to a voltage dip from a neighbouring system. Choice of State Variables For the t h e o r e t i c a l model developed, the choice of ip-p, 6, Ato, and V f d as state v a r i a b l e s i s convenient both for the state space model d e r i v a t i o n and for evaluation of i n i t i a l conditions. Before using t h i s model to define the form of the s t a t i s t i c a l model, ^ must be replaced by a measurable state. In section 3.1 i t i s stated that i f d may be chosen to replace ipp as a state v a r i a b l e thus giving a model i n terms of measurable states only. The d e r i v a t i o n consists of s u b s t i t u t i n g equation (3.16) into (3.15) and eliminating ijjd using ^ = M j V 0 s i n 6 + M 2 v 0 c o s 6 + N-^p (3.47) from equation (3.39). The r e s u l t i n g expression i s ipp = F - [ i f d + F 2 ( M 1 v 0 s i n 6 + M 2 v D c o s 6 ) (3.48) where / co T' (x - x') F ? = ° d ° d d . (3.50) x d " % T d o < x d " x d ) N l D i f f e r e n t i a t i n g equation (3.48) y i e l d s / ( pxjj-p = F]_pif d + wQF2(M-|V0cos6Au) - I^VgSindAto) . (3.51) Substituting equations (3.48) and (3.51) into (3.7) gives a state equation for the measurable state v a r i a b l e i ^ as 1 - x a d P i f d = — t^ad^-fd + ^ d F 1 - io F 0(M 1v cos6Aco - M ?v sinSAto) ]. (3.52) O ^ -L O -^ O The torque expressions i n equation (3.9) also require expansion i n terms of measurable q u a n t i t i e s . From equation (3.17), T^ « i a . From equation (3.10) T e = V d - ( 3' 1 0> where i ^ and i ^ are expressed i n terms of and ^ by equation (3.14) so that T = -UJL-* 4\ " 1 + a 1 $J • (3.53) e °x q d TT~XT f q ° ^1 d q q do d d Sub s t i t u t i n g equations (3.39) and (3.48) into (3.53) gives T e = A l i f d 2 + ^fd^o31^ + hHc^o0056 + A,v 2sin6cos6 + A'V 2 s i n 2 6 + Aiv 2 c o s 2 6 (3.54) 4 o 5 o 6 o where the c o e f f i c i e n t s are not expanded because t h e i r values are not required when defi n i n g the form of the s t a t i s t i c a l model. From trigonometry cos 26 = 1 - s i n 2 6 (3.55) which reduces (3.54) to T e = V f d 2 + V f d ^ 5 1 1 1 6 + A 3 1 f d v 0 c o s f i + A/,v sin6cos6 + Ac-v s i n <5. (3.56) This formulation leads to a d d i t i o n a l n o n l i n e a r i t i e s due to products of state v a r i a b l e s appearing i n the expression. I t i s nevertheless an acceptable form of s o l u t i o n f or the s t a t i s t i c a l modelling scheme used here and i s r e a d i l y handled by forming the desired product at each sampling time and then obtaining the desired l i n e a r c o e f f i c i e n t s by regression a n a l y s i s . The form of the model to be exploited, then, i s defined by equations (3.8), (3.9), (3.32), (3.52) and a u x i l i a r y equations (3.17) and (3.56). The d i f f e r e n c e equation counterparts of these d i f f e r e n t i a l state equations are used because the d i s c r e t e sampling environment of the data a c q u i s i t i o n system more e a s i l y f a c i l i t a t e s a d i f f e r e n c e equation representation. In the pertinent state equations there are no int e r c e p t terms present. However for the regression a n a l y s i s , the inte r c e p t b Q i s included. To assume that s t a t i s t i c a l l y b Q = 0 requires considerable i n v e s t i g a t i o n . This int e r c e p t value i s thus retained i n a l l the regression equations for t h i s project to account for e f f e c t s of measurement o f f s e t s , even though i t i s desired that b Q i n f a c t equal zero. Maintaining the o f f s e t term, b Q, and expressing v Q e x p l i c i t l y , the d i s c r e t e state v a r i a b l e equations f o r regression analysis are as follows: i f d ( k + l ) = b 1 0 + b n i f d ( k ) + b 1 2 v f d ( k ) + b-j L3V 0(k)cos6(k)Ato(k) + b 1 4v 0(k)sin6(k)Ato(k) (3.57) 6(k+l) = b 2 Q + b 2 1 6 ( k ) + b 2 2Aco(k) (3.58) Au>(k+1) = b 3 Q + b 3 1Aco(k) + t > 3 2 v 0 2 (k) s i n 2 6 (k) ^ + b 3 3 i f d 2 ( k ) + b 3 4 v o 2 ( k ) s i n 6 ( k ) c o s 6 ( k ) / + b 3 5 i f d ( k ) v o ( k ) s i n 6 ( k ) + b 3 6± f d(k)v o(k)cos6(k) + b 3 7 i a ( k ) (3.59) v (k+1) = b 4 0 + b 4 1 v f d ( k ) + b 4 2 u x ( k ) + b 4 3 v t ( k ) . (3.60) A u x i l i a r y Equation A major consideration i n t h i s modelling involves the non-l i n e a r i t i e s i n equation (3.60) which a r i s e when s u b s t i t u t i n g f o r v^. The terminal voltage, v t , i s measurable and thus can be used to evaluate the c o e f f i c i e n t s i n equation (3.60). However, when the model i s used as a pre d i c t o r , an expression i s required to express v t as a function of state v a r i a b l e s , that i s , v f c = f(i£ d, 6, Ato, V f d ) . From the terminal condition expressed by equation (3.11) V t 2 = V + V q 2 ' ( 3 ' n ) S u b s t i t u t i o n f o r v d and v q from equations (3.1) and (3.2) gives v t 2 = * q V + * d V ( 3- 6 1> and a further s u b s t i t u t i o n of equation (3.39) gives v f c = to0 ( M^v Qsin6 + M 2 V Q c o s 5 + N-^F) + w 0 2 ( M 3 v o s i n 6 + M 4 V Q C O S 6 + N 2 ^ F ) 2 . (3.62) Expressing in equation (3.62) by equation (3.48) and substituting equation (3.55) yields an expression of the form v t = % v 0 sin6 + + ^3vo s i n ( 5 c o s ^ + B 4 i f d v 0 s i n 6 + B 5 i f d v o c o s 6 (3.63) where the values of the coefficients are not required for defining the form of the s t a t i s t i c a l model. Including an offset term, the desired auxiliary regression equation is found to be v t 2(k) = b 5 Q + b 5 1 v 0 2 ( k ) s i n 2 6 ( k ) + b 5 2 i f d 2 ( k ) + ^53vo (k)sin6(k)cos6(k) + b 5 4 i f d ( k ) v D ( k ) s i n 6 ( k ) + b 5 5 i f d ( k ) v o ( k ) c o s 6 ( k ) . (3.64) If this auxiliary equation is substituted directly into equation (3.60) then equation (3.60) is no longer linear in i t s coefficients. However, because v is a measurable quantity, equation (3.64) may be treated as an auxiliary equation and regression analysis w i l l estimate the linear coefficients. When the model is used as a predictor, vfc may be found at each time desired and i t s square root used in equation (3.60). Equations (3.57) to (3.60) and (3.64) then describe the form of the state variable model of the system with a l l equations linear in their coefficients. Regression analysis is used to estimate the unknown linear coefficients in each of the five equations, thus yielding a state variable nonlinear system model. This model is tested in Chapter 5 by investigating i t s t a t i s t i c a l l y as well by comparing i t s performance x^ith that of the theoretical model and the laboratory system. 4. LABORATORY POWER SYSTEM AND DATA ACQUISITION SYSTEM A laboratory model of a one-machine i n f i n i t e bus power / system has been assembled. This model i s used not only to check the responses predicted by both Park's model and the s t a t i s t i c a l / m o d e l but also to supply the data required i n producing the s t a t i s t i c a l model. This l a t t e r function requires considerable measurement and data a c q u i s i t i o n apparatus. The laboratory model consists of a four pole d.c. motor simulating a prime mover and d r i v i n g a small s i x pole three-phase synchronous generator. An inductive three-phase balanced transmission system connects the machine terminals to the i n f i n i t e bus. Voltage regulation i s accomplished using an amplidyne i n the e x c i t e r c i r c u i t , with a reference voltage on one input of the amplidyne and feedback from the generator terminal voltage on another input c o i l (Figure 3.3). To c o l l e c t observations for the s t a t i s t i c a l modelling, a computerized data a c q u i s i t i o n system i s employed. The computer i n t e r f a c e allows measurement of a number of analog signals through a multiplexer and an a n a l o g - t o - d i g i t a l (A/D) converter. An o p t i c a l shaft-angle encoder monitors the shaft p o s i t i o n , from which shaft speed may be derived by d i f f e r e n t i a t i o n . The c e n t r a l processor used i n the data a c q u i s i t i o n i s a D i g i t a l Equipment Corporation (DEC) PDP-8/L. The following three sections supply further d e t a i l s for the power system laboratory model, the data a c q u i s i t i o n hardware and the data a c q u i s i t i o n software. 29 4.1 Power System Laboratory Model Figure 4.1 illustrates the laboratory system configuration. Specifications for the d.c. motor, the synchronous generator and the amplidyne are displayed in Table 4.1. Table 4.1 Laboratory Model Machine Specifications Machine Specification Synchronous Generator output: 5 KVA 220 volts 13 amps 90% pf 60 Hz 1200 rpm exciter: 3.2 amp s 125 volts D.C. Motor 5.6 Kw 115 volts 56 amps 850/1200 rpm Amplidyne input: 220/440 volts 7.2/3.6 amps 3-phase - 60 Hz 1725 rpm output: 1.5 Kw 125 volts 12 amps D.C. MOTOR • SYNCHRONOUS AMPLIDYNE TRANSMISSION INFINITE (prime mover) GENERATOR (voltage regulator) LINE BUS (mains) 230 v 115 v i 1 1 S 'fd vfd u1 \AMJLT i i vt 2 vn Figure 4.1 Laboratory System Configuration Notes - broken l i n e s i n d i c a t e measurement points"". - for amplidyne d e t a i l see Figure 3.3. o Amplidyne In an attempt to minimize undesired system noise an amplidyne, rather than an a v a i l a b l e SCR e x c i t e r , was chosen to model the voltage regulator e x c i t e r c i r c u i t . However, i t appears that amplidyne-induced noise i s p l e n t i f u l and i s i n fact more d i f f i c u l t to f i l t e r than short r i s e time peaks induced by the SCR e x c i t e r . The r e g u l a t o r - e x c i t e r c i r c u i t was designed with a r e l a t i v e l y long open-loop time constant, T^, as seen i n Table 4.2. This was done because shorter time constants, of the magnitude of the generator open c i r c u i t time constant, cause the exciter-generator combination to be unstable. D.C. Motor An attempt i s made to decrease the inherent damping of the d.c. motor, thus making i t more r e a l i s t i c a l l y model a prime mover, t y p i c a l l y with low p.u. damping. From the analysis of Chapter 3 i t i s seen that d.c. motor damping i s in v e r s e l y proportional to the resistance i n the armature c i r c u i t as described by: - k v 2 D d c = — < < 3- 2 9> R Therefore a s e r i e s resistance i s placed i n the armature c i r c u i t to increase i t s t o t a l r e s i s t a n c e , R. Then 230 v o l t s d.c. i s applied to the c i r c u i t to maintain approximately 115 v o l t s across the armature and thus maintain correct armature current. Input torque to the generator i s c o n t r o l l e d by the armature current i n the d.c. motor. Torque i s then evaluated according to equation (3.17), that i s , T, = k i . The advantage of t h i s c o n t r o l over using f i e l d current dc v a & ° c o n t r o l i s that f l u c t u a t i o n s of f i e l d current i n turn cause changes i n armature current r e q u i r i n g that both be monitored to evaluate torque. System Parameters Synchronous machine e l e c t r i c a l parameters required f o r the t h e o r e t i c a l model based on Park's representation are determined using standard techniques as outlined i n Chapter 7 of IEEE Test Code [19]. Table 4.2 shows the measured values f o r the system parameters. They are displayed i n p.u. as w e l l as i n MKS units although MKS u n i t s have been used throughout t h i s t h e s i s . The base values used are: 125 v o l t s r.m.s., 8 amps r.m.s., and 377 radians/sec. The d.c. machine c o e f f i c i e n t L f l^ i s found from a p l o t of the speed voltage c o e f f i c i e n t , co L ^, which i s defined as open c i r c u i t v o l t a g e / f i e l d current [15]. The system i n e r t i a , J , i s evaluated using the re t a r d a t i o n t e s t . The f r i c t i o n damping term, F, i s evaluated using equation (3.19) [15] P L i i af a f F = (3.19) 2 co m by operating the synchronous generator at no load and d r i v i n g i t with the d.c. motor at various speeds near synchronous speed. A s i m p l i f i e d s i n g l e time constant model i s used for the amplidyne and the time constant i s found by monitoring amplidyne output for a step input. The steady state gain of each input i s also found from measured input and output voltages near the estimated operating point. Table 4.2 Laboratory System Parameters Parameter MKS units per unit DC machine parameter L * af volts-sec 2 •-^ 5 amps rad Mechanical Parameters J .62 kg - m2 F i oule-sec -.00081 com + .1637 J rad 2 Synchronous Machine Parameters x d 9.03 ohms 1.00 pu xa 5.47 ohms 0.60 pu d , Tdo < 2.00 ohms 0.22 pu 0.282 sees 125 ohms 13.85 pu 20.4 ohms 2.26 pu Exciter-Regulator Parameters KA1 KA2 TA 113 5.59 1.4 sees Transmission Line Parameters r .0313 ohms 0.0035 pu X 1.77 ohms • 0.196 pu G 0 0 B 0 0 4.2 Data Acquisition Hardware Measurements A number of the analog inputs required f i l t e r i n g before entering the A/D converter at the interface. Active f i l t e r s have been selected to perform the low pass f i l t e r i n g which entails attenu-ating noise at frequencies as low as 360 Hz without interfering with system responses or with the 60 Hz terminal waveforms. The schematic of the Philbrick f i l t e r s [20] used for these voltage inputs i s shown in Figure 4.2. F i l t e r performance curves are displayed in Figure 4.3, labelled with the input signals to which each i s applied. Table 4.3 displays a measure of ripple on the f i l t e r e d signals which are sub-mitted to the A/D converter. Ar R Ar R o v, o R bC/3 6. V, o - J + b(RCp) + (RCp)2 Figure 4.2 Active F i l t e r Schematic 10 50 100 500 1000 FREQUENCY (HZ) Figure 4.3 F i l t e r Performance Table 4.3 Noise at A/D Input Input Signal Percentage Noise ( r i p p l e ) synch, machine f i e l d v o l t s ( v£ d) 1.3% synch, machine f i e l d current ( i ^ ) 1.0% terminal voltage (v ) n e g l i g i b l e e x c i t e r reference voltage (u^) n e g l i g i b l e i n f i n i t e bus voltage (v ) n e g l i g i b l e D.C. motor armature current ( i ) a 1.1% D i r e c t current i s measured by detecting and amplifying the voltage drop across a s e r i e s r e s i s t a n c e . A l t e r n a t i n g current i s measured / by means of a current transformer. Both d.c. and a.c. voltages are fed / d i r e c t l y to the i n t e r f a c e with appropriate attenuation or a m p l i f i c a t i o n and f i l t e r i n g . A d i g i t a l shaft encoder i s f i x e d to the a c c e s s i b l e end of the synchronous machine shaft. This o p t i c a l encoder (DRC Model 29 manu-factured by Dynamics Research Corporation) outputs square pulses as the shaft rotates. Each r e v o l u t i o n of the shaft produces 1500 pulses which are fed to the i n t e r f a c e where they are used to measure the rotor p o s i t i o n with respect to some reference. Further d e t a i l s are supplied when describing the i n t e r f a c e . Interface The p r i m a r i l y TTL i n t e r f a c e , designed to be compatible with a DEC PDP-8/L computer, was constructed p r i o r to the s t a r t of t h i s research project. However, t h i s untried i n t e r f a c e had a number of bugs in c l u d i n g design d e f i c i e n c i e s which were to be r e c t i f i e d before t h i s project could proceed. F i r s t l y , a short d e s c r i p t i o n of the i n t e r f a c e i s presented, and then a look i s taken into the problems encountered. Much of t h i s i n t e r f a c e i s standard. I t consists of an A/D converter (DEC #A811 with 0.1% F.S. accuracy) following a m u l t i -channel (18 connected) FET multiplexer to measure analog s i g n a l s . For computer con t r o l functions using analog s i g n a l s , four d i g i t a l - t o -analog (D/A) converters are provided. The RTL system designed to 37 i n t e r p r e t the o p t i c a l shaft-angle encoder output i s not standard and requires some at t e n t i o n . This l o g i c allows measurement of the rotor angle i n e l e c t r i c a l units while the encoder i t s e l f i s detecting angle / / i n mechanical u n i t s . The r e s u l t for the s i x pole synchronous .machine being monitored i s that the 1/1500 r e s o l u t i o n f o r 2 IT mechanical radians provides only 1/500 r e s o l u t i o n f or 2TT e l e c t r i c a l radians. To achieve the e l e c t r i c a l angle measurement, the s h a f t -angle encoder output pulses are counted by a modulo-500 counter which i s read by t r a n s f e r r i n g i t s contents to a read buffer at a rate s p e c i f i e d by a reference pulse t r a i n . At the s t a r t of a mechancial r e v o l u t i o n , the counter i s reset to zero and i t then counts 500 pulses (1/3 revolution) when i t resets i t s e l f . In the meantime, the reference pulse has gated the counter to the read b u f f e r to determine shaft p o s i t i o n at the time of the reference pulse. This read rate may be s p e c i f i e d by one of the two a v a i l a b l e shaft encoders, by the mains frequency, or by a c r y s t a l o s c i l l a t o r i f an absolute reference i s desired. In t h i s project one state v a r i a b l e i s the angle 6 between the q-axis and the i n f i n i t e bus (mains) voltage. This can be measured d i r e c t l y by using a reference pulse t r a i n of mains frequency to gate the modulo-500 counter contents into the read buffer. In an attempt to minimize construction cost, the o r i g i n a l design of the i n t e r f a c e incorporated a number of schemes to reduce hardware expenditure. One example of t h i s which led to a dangerous design was i n the multiplexer channel s e l e c t i o n decoding l o g i c . Table 4.4(a) o u t l i n e s a portion of the decoding scheme used for m u l t i -plexer channel s e l e c t i o n . It i s noted that i f inadvertently, through 38 Table 4.4 Multiplexer Channel Selection Decoding (a) Original Channel Selection Scheme Word Addressing Channel Octal Address Channel # 0 1 2 3 4 5 6 7 8 9 10 11 1 1 1 1 4050 2 1 1 1 4044 3 1 1 1 4022 4 1 1 1 4021 5 1 1 1 2050 6 1 1 1 2044 7 1 1 1 2022 8 1 1 1 2021 (b) Modified Channel Selection Scheme Channel # Word Addressing Channel Octal Address 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 1 1 2 1 2 3 1 1 3 4 1 4 5 1 1 5 6 1 1 6 7 1 1 1 7 program error or hardware f a u l t , b i t s 0 and 1, for example, are both high, two channels may be selected at once. The r e s u l t i s a short c i r c u i t through two of the FET switches and thus the destruction of / one or more multiplexer channels. As i t i s believed that the cost of detecting and repl a c i n g shorted FET's outweighed the gain of reduced hardware for decoding, a f a i l - s a f e decoding scheme i s implemented. The r e s u l t i n g code which provides a c t i v a t i o n of only one po s s i b l e channel for every 12 b i t binary number i s outlined i n Table 4.4(b). Another minor i n t e r f a c e change which i s made for programming convenience i s the i n s t a l l a t i o n of switches on a l l the i n t e r r u p t l i n e s . Thus when programming with i n t e r r u p t on, undesired i n t e r r u p t s are e a s i l y disabled. This prevents loss of computing time i n unnecessary s e r v i c i n g of i n t e r r u p t s . 4.3 Data A c q u i s i t i o n Software Software performance i s d i c t a t e d by both the laboratory system and the s t a t i s t i c a l modelling scheme. A c e r t a i n number of observations of each v a r i a b l e i s required f or s t a t i s t i c a l a n a l y s i s . For t h i s analysis to produce an accurate mathematical model, i t i s further required that the observations be spaced close enough i n time to follow the f a s t e s t laboratory system response desired to be represented. The form of the mathematical model further defines the software by s p e c i f y i n g which v a r i a b l e s must be monitored. Data Storage When working with a basic 4-K memory computer without high speed storage devices, data storage i s a problem when a large number / of observations of many v a r i a b l e s i s required. For the laboratory system, time constants of i n t e r e s t were expected to be greater than 1/10 second and therefore i t was decided to sample a l l system v a r i a b l e s once each period of the mains voltage waveform. Furthermore, i t was decided to store eight input s i g n a l s per sampling time g i v i n g 480 samples per second. With only 3500 storage l o c a t i o n s a v a i l a b l e i t i s seen that at one observation per word storage, the system may only be monitored f o r approximately 7.5 seconds. Thus a more intense packing procedure i s desired for data storage. The packing scheme adopted uses a f u l l word to store the f i r s t value of each input, and thereafter only stores the deviation from t h i s operating point. A problem a r i s e s because for a 10 b i t A/D output the 6 b i t packing allows only 6.25% deviation when operating at f u l l s cale of - 5 v o l t s input. By working at le s s than f u l l s c a le, 10% deviations may comfortably be stored. This small allowable deviation requires software checking for overflows when packing deviations. A packing scheme which allows l a r g e r deviations but which i s more susce p t i b l e to errors i s to store the differences between successive readings. Due to the p o s s i b i l i t i e s of error ( i f one value i s wrong, a l l those following are wrong) i n the l a t t e r scheme, the deviations from an operating point were stored. The small allowable deviations are acceptable except i n d e r i v i n g speed from angle measurements where l i m i t e d r e s o l u t i o n of the shaft-angle encoder presents a problem. By s t o r i n g two observations per computer word, the system's eight measurement points may be interrogated each cycle of the mains f o r approximately 15 seconds. This provides a c q u i s i t i o n of an adequate number of observations fo r the s t a t i s t i c a l a n a l y s i s . Sampling a.c. Signals Without reconstructing the waveform sampled there are two methods of obtaining the magnitude of a.c. waveforms. They are to f i l t e r the s i g n a l and record one sample, or to continuously sample the s i g n a l and store the peak value. As f i l t e r i n g a 60 Hz waveform to an acceptable r i p p l e causes measurement response time constants longer than the system time constants, the a.c. signals are continuously sampled and t h e i r peak values stored. This too has p o t e n t i a l measurement error as the peak value may occur between samples. The data a c q u i s i t i o n program allows for three desired a.c. waveforms to be sampled with 245 ysec between samples. This r e s u l t s i n approximately 0.11% e r r o r i n detecting the peak values, which i s i n the range of the 0.1% error i n the A/D con-v e r t e r . Many more a.c. s i g n a l s could be cross-sectioned before obtaining sampling errors of the magnitude of the noise r i p p l e on the s i g n a l s . PDP-8 Program Outline The PDP-8/L program, which i s outlined by a flowchart i n Appendix 4A, begins by reading a grounded multiplexer channel and s t o r i n g the r e s u l t i n g A/D o f f s e t read. Another channel connected to an accurately known d.c. supply i s read and the A/D output stored. These o f f s e t and c a l i b r a t i o n values l a t e r allow for c o r r e c t i o n of A/D readings and for conversion of stored binary numbers back to voltage l e v e l s on the system. The program then stores ah A/D reading f o r each v a r i a b l e thus describing the system operating point. I t then begins to record observations and pack i n h a l f words t h e i r d eviation from the appropriate operating point value. A computer i n t e r r u p t at each period of the mains-voltage waveform then i n i t i a t e s the following procedure. Pack the deviations from operating point f o r the previous sampling i n t e r v a l i n the computer. Sample rotor p o s i t i o n and then a l l d.c. var i a b l e s i n succession. Sample a l l a.c. va r i a b l e s successively and continuously u n t i l the next i n t e r r u p t , s t o r i n g the peak magnitude of each a.c. waveform. As the deviations from operating point are stored, they are checked to see whether or not they may i n fac t be packed i n t o s i x binary b i t s . When an overflow i n packing the deviation from operating value i n a h a l f word occurs an overflow counter i s incremented and the program w i l l h a l t at some preset allowable number of overflows. I f the over-flow i s p o s i t i v e , the la r g e s t p o s s i b l e p o s i t i v e deviation i s stored, when negative, the most negative deviation i s stored. This minimizes the error r e s u l t i n g from using the overflowed value as a v a l i d obser-vation i n the s t a t i s t i c a l a n a l y s i s . A f t e r the computer memory i s f i l l e d the data stored i s punched on paper tape using a s i n g l e odd p a r i t y b i t for each frame. O f f - l i n e Data Handling The information on the paper tape from the PDP-8 i s con-verted onto magnetic tape for use on the computer center IBM 360/ Model 67. This data i s then supplemented with measured attenuations of am p l i f i c a t i o n s external to the i n t e r f a c e f o r each v a r i a b l e and with the accurate value of the d.c. calibration voltage. A Fortran program uses this supplementary information and the paper tape information to reconstruct a l l system voltage and current values monitored. The result is 861 observations of eight system variables for use in the s t a t i s t i c a l modelling scheme. These observations are stored on magnetic tape, readily accessible for analysis or output. 44 5. PERFORMANCE OF THE STATISTICAL MODEL The proposed s t a t i s t i c a l model i s investigated by checking i t s t a t i s t i c a l l y using the data acquired. Also i t s performance as a pr e d i c t o r i s investigated by comparison with the performance of the t h e o r e t i c a l model as w e l l as with the laboratory system. Whether the s t a t i s t i c a l model i s a good pr e d i c t o r or not i s e a s i l y determined by comparison with the laboratory system only. However, further i n v e s t i -gation of some aspects of the model may be more e a s i l y performed i n a computer using the proven t h e o r e t i c a l model. For example, a computer comparison of mathematical model responses may be used to decide whether or not the s t a t i s t i c a l model has i d e n t i f i e d a p a r t i c u l a r system time constant. Section 5.1 describes the system operating points investigated and how the data i s acquired f o r the s t a t i s t i c a l a n a l y s i s . These important points require consideration before beginning the i n v e s t i g a t i o n of the s t a t i s t i c a l model. 5.1 System Data f o r S t a t i s t i c a l Model Derivation The laboratory power system i s run at three d i f f e r e n t operating points. Its operation i s monitored using the PDP-8/L computer and associated i n t e r f a c e discussed i n Chapter 4. At each operating p o i n t responses to steps i n each of u,, i and v are monitored to be 1 a o used as comparisons f o r the responses of the mathematical models. Also system response i s recorded f or small random v a r i a t i o n s i n the three inputs u,, i and v . This i s necessary to obtain the data required to estimate the c o e f f i c i e n t s i n the equations d e f i n i n g the s t a t i s t i c a l model. Applied perturbations are required at the three inputs because these s i g n a l s are generally very w e l l regulated and thus do not provide the amount of v a r i a t i o n necessary i n the s t a t i s t i c a l a n a l y s i s . The perturbations are introduced on the c o n t r o l inputs independently and randomly. Simultaneously, the system response i s monitored by recording values of the system state v a r i a b l e s at regular time i n t e r v a l s . Separate sets of state v a r i a b l e observations are c o l l e c t e d as perturbations are applied to u^ and ± a, then to u^ and v Q , and f i n a l l y to i & and v Q i n three test runs. Hard copy of the data taken f o r each test i s produced on paper tape by the PDP-8/L computer and i s then processed using an IBM 360/Model 67 computer. Gains external to the PDP-8 i n t e r f a c e , such as r a t i o of p o t e n t i a l transformers, a m p l i f i e r s and attenuators, are measured using meters. This data i s supplied to the IBM 360 on cards to supplement the paper tape information. Thus the system voltage and current values may be reconstructed within the IBM 360 for any sampling time. These reconstructed voltage and current values are stored on magnetic tape u n t i l they are required for analysis or p l o t t i n g . The three operating points for which data i s acquired are summarized i n Table 5.1. Table 5.1 System Operating Points Investigated / / , / ' Operating Point Variable Units A • B fc P watts 1230 1635 405 Q vars 953 - 381 1660 V t volts 125 125 124 V amps 1.75 1.25 2.0 6 degrees 29.5 45.4 3.6 V f d volts 36.8 26.6 42.8 U l volts . 6.5 6.49 6.55 i a amps 15.5 18.9 9.4 V o volts 109 133 94 In Table 5.1 P, Q, and v t completely describe the i n i t i a l conditions of the system for the theoretical representation. The values of i ^ , , 6, v-, and v are calculated from the theoretical fd fd o model i n i t i a l conditions as a preliminary check on the theoretical model. Calculated and measured i n i t i a l conditions are shown in Table 5.2. It must be noted that meter accuracy is generally not better than 2 - 3 % . 47 Table 5.2 Steady State Values Steady State Values Variable Units Experimental Th e o r e t i c a l % Difference * f d amps 1.235 1.172 5.1% 6 degrees 46.0 43.4 5.9% V f d v o l t s 26.85 23.91 11% V t v o l t s 126 125 .80% V o v o l t s 133 132 .76% 5.2 S t a t i s t i c a l Investigations of the Regression Model Two important s t a t i s t i c a l i n v e s t i g a t i o n s of the model are performed using r e s i d u a l p l o t s . One i s a check of whether or not the assumptions inherent i n regression analysis (Chapter 2) have been v i o l a t e d . The other i s a t e s t of lack of f i t , i n d i c a t i n g how w e l l the p a r t i c u l a r form of equation describes the data to which i t i s f i t t e d . The philosophy used i n i n t e r p r e t i n g r e s i d u a l p l o t s i s s i m i l a r to that evident i n t e s t i n g hypotheses. That i s , i f a pl o t i n d i c a t e s an assumption i s v i o l a t e d , one concludes the assumption i s v i o l a t e d , while i f a p l o t indicates the assumption holds, one concludes only that the assumption has not been v i o l a t e d . Residual pl o t s then give evidence of lack of f i t and v i o l a t i o n of assumptions, but do not con-f i r m that the model i s p e r f e c t l y adequate or that the assumptions have been completely s a t i s f i e d . 48 The normality assumption f or the resi d u a l s (e^, i = 1, 2, ...n) i s i nvestigated using an o v e r a l l p l o t of r e s i d u a l s as shown for a t y p i c a l case i n Figure 5.1. As well as an o v e r a l l normality d i s t r i b u t i o n f o r e^, i t i s also required that the be normally d i s t r i b u t e d at any instant of time or over any i n t e r v a l of time. Rather than using a number of normality p l o t s , the d i s t r i b u t i o n of with time i s pl o t t e d as shown i n Figure 5.2. Since t h i s p l o t i n d i c a t e s uniform d i s t r i b u t i o n with time, then the r e s u l t s of the o v e r a l l p l o t may be assumed to hold during any i n t e r v a l of time. It i s noted that the d i s t r i b u t i o n of e's i n th i s t hesis depends on how the perturbations of u^, i & and V q are administered. Here the inputs are varied manually and an attempt i s made to induce random f l u c t u a t i o n s . It i s more desi r a b l e to have c o n t r o l l e d signals on the system inputs so that regulated perturbations could be administered, thus producing a better gaussian error d i s t r i b u t i o n . Also, poor r e s o l u t i o n i n speed detection requires l a r g e r than desirable perturbations from nominal values. The assumption of constant variance i s r e a d i l y checked by p l o t t i n g against the predicted value as shown i n Figure 5.3. Using a s i m i l a r p l o t , lack of f i t i s investigated by p l o t t i n g r e s i d u a l s f o r each regression equation against each of the independent v a r i a b l e s as w e l l as against the predicted value. An example i s shown i n Figure 5.4 where lack of f i t of the v,. term i n the equation f o r v. ' (Equation 3.60) i s c f d checked. It i s noted that although the normality assumption appears to be v i o l a t e d , variance and lack of f i t may be investigated q u a l i t a t i v e l y using these r e s i d u a l p l o t s . > u z LU UJ ct: X X X A RESIDUALS (ifcj — i f d ) + £j Figure 5.1 Ove r a l l Plot of Residuals Figure 5.2 D i s t r i b u t i o n of Residuals with Time 50 -o—< I "pCD cn _j c r o =J o a -• —' CD cn i UJ ct a i CD ^ 1 * * « * &X 1.2 1.3 — i 1 1 — 1.4 A 1.5 PREDICTED VRLUE ( i f d ) Figure 5.3 Example of Variance of Residuals 1.6 -a m < f ™ H 2 ° j> co i cr ZD a CD-CO L U Cr: LO CM CD I O LO X 122 124 126 128 130 INDEPENDENT VARIABLE (vt) 132 Figure 5.4 Example of Test for Lack of F i t 5.3 Model Responses to Step Inputs The performance of the t h e o r e t i c a l and s t a t i s t i c a l models were compared with each other and with the laboratory system response. For the comparisons, step inputs were applied to the regulator reference voltage, u , the d.c. motor armature current, i , and the i n f i n i t e bus i a voltage, v , as outlined i n section 5.1. E f f e c t of Operating Point From Figures 5.5, 5.6 and 5.7 i t i s evident that the s t a t i s t i c a l model y i e l d s a good steady state response, but i s not capable of p r e d i c t i n g the most rapid fluctuations i n the system's dynamic behaviour. It does, however, provide a close approximation to the system's dynamic response. By comparing Figure 5.6 and 5.7(a) i t i s observed that the s t a t i s t i c a l models give s i m i l a r p r e d i c t i o n of response whether or not they are used at the same operation point at which they are derived. This i n d i c a t e s that the model s u f f i c i e n t l y explains system n o n l i n e a r i t i e s . This i s further emphasized by comparing the c o e f f i c i e n t s f o r a t y p i c a l case (equation 3.57) given i n Table 5.3. It i s seen that the most s i g n i f i c a n t c o e f f i c i e n t s ( b ^ and b-^ i n t h i s case) vary by only small amounts of the operating point changes. Figure 5.5 displays a response of a model derived from data at a l l three operating points. This l e s s accurate p r e d i c t o r y i e l d s an acceptable response when compared with the steady state response obtained using Park's formulation. 1.5-1 i i i / / laboratory s t a t i s t i c a l theoretical 2 T 4 6 TIME (5EC5) Figure 5.5 Response to step i n u-^  at operating point B - S t a t i s t i c a l model from operating points A, B and C combined 1 .4-CL a 1.2-1.1 SyrTrw" T i — r n — r a -laboratory s t a t i s t i c a l theoretical 0 4 6 TIME (SECS) 8 10 Figure 5.6 Response to step i n u^ a t operating point B - S t a t i s t i c a l model from operating point A 53 Table 5.3 Example of Variation in Coefficients Operating Point R2 Coefficients b l l b12 b13 b14 B .9930 .932 .0031 .0014 C .9917 .904 .0043 .0028 A+B+C .9995 .929 .0033 .0034 The set of plots shown in Figure 5.7 indicate the predicted responses of a l l the measurable state variables for a step in u^ using a s t a t i s t i c a l model with coefficients estimated at the same operating point as the step i s applied. The model used is found at operating point B. The coefficients describing this model (equations (3.57) to (3.60) and (3.64)), excluding those coefficients which are non-significant to a 5% level, are as follows. b 1 Q = .00084 , b n = .932 , b±2 = .00311 , b 1 3 = .00136 , b = 0.0 b 2 0 = 0.0 , b 2 1 = 1.0 , b 2 2 = 1.0 b Qn = -.039 , b 3 1 = -.156 , b 3 2 = 0.0 , b ^ = 0.0 , b0/. = 0.0 b 3 5 = -.000478 , b, b 4 Q = -.357 , b 4 1 = .959 , b 4 2 = .971 , b ^ = -.0380 J 3 Q .v->?  u 3 1 .x_m  u 3 2 u u  u 3 3 w u  . .0000325 , b 3 7 = -.00481 J50 = 2548. , b q i = .307 , b 5 2 = -1334. , b „ = .359 , b,.^  = -14.7 , 51 b 5 5 = 96.7 . 53 54 This model was found using a 60 Hz sampling rate by combining three sets of data runs at the same operating point (B) as outlined in section 5.1. Similar s t a t i s t i c a l model performance was observed using step inputs of i and v . o 54 Figure 5 laboratory s t a t i s t i c a l t heoretical ~ i r r~ 4 6 TIME (SECS) 8 10 Responses to step i n u^ at operating point B S t a t i s t i c a l model from operating point B 55 0.005-C J LU CD a cx cn a U J LU CL CD 0 0.005--0.010 Figure 5.7(c) 2 4 6 TIME C5EC5) laboratory s t a t i s t i c a l theoretical 8 10 _32 H CO (— - J > a a 26 23 0 2 s t a t i s t i c a l theoretical "~l T™ 4 6 TIME tSECS) 8 10 Figure 5.7(d) 56 1 3 3 -T I M E (SECS) Figure 5.7(e) Effect of Sampling Rate Figures 5.8 and 5.9 indicate the effect of changes of sampling interval when obtaining data to form a model. The s t a t i s t i c a l model used to produce the response in Figure 5.7(a) used samples collected at each period of the mains. For the model producing Figure 5.8 the samples were taken every second period and in Figure 5.9 every f i f t h period. For this range of sampling frequency, change in sampling rate had negligible effect on the model produced. In forming the various s t a t i s t i c a l models from different sets of data, note was taken of changes in the value of the multiple regression coefficient, R (see section 2.3). It is noted that as more o samples were used to estimate the coefficients in the model, the R • 57 4 6 TIME (5EC5) 8 10 Figure 5.8 Response with s t a t i s t i c a l model found using 0.033 second sampling i n t e r v a l Figure 5.9 Response with s t a t i s t i c a l model found using 0.083 second sampling i n t e r v a l value tended to increase (e.g. Table 5.3). However, for equation (3.58) describing speed, R2 was only slightly significant when sampling at 7 2 each period of the mains (.016 sec), but for most sets of data this R / increased ten fold when sampling every second period (.033 sec). In the modelling scheme, speed is found by differentiating angle (i.e. to(k) = 6(k+l)-6(k) ) which i s monitored by the shaft encoder and yields poor resolution after conversion to ele c t r i c a l units. The effect is that small deviations with poor resolution produce a set of data which the particular form of equation chosen does not adequately describe. In practice, though, as long as the multiple regression coefficient i s significant, say by four times the tabulated value, an order of magnitude increase in R does not appreciably affect the model derived. Generally, then, i t i s found that the derived s t a t i s t i c a l model yields good steady state prediction, but i t responds slower than the system when predicting dynamics. Discrepancies observed in the theoretical model performance at steady state are within meter error tolerances as the theoretical and s t a t i s t i c a l models define their operating point using a different set of variables and therefore different meters. Lack of resolution in speed measurement (see Figure 5.7(c) created the major problem in the s t a t i s t i c a l modelling. 59 6. CONCLUSION An: i n v e s t i g a t i o n has been undertaken-to a s c e r t a i n how r e a d i l y a power system lends i t s e l f to s t a t i s t i c a l modelling. A nonlinear state v a r i a b l e model has been derived. This model i s l i n e a r i n i t s c o e f f i c i e n t s which are evaluated by the l e a s t squares f i t t i n g technique of regression analysis- The form of the s t a t i s t i c a l model i s based on Park's formulation of synchronous machine dynamics, with the unmeasurable state describing f i e l d f l u x , i\> , replaced by the f i e l d current, i ^ - , . As the expression F r d for v i s nonlinear i n the c o e f f i c i e n t s as well as i n the state s , an t 2 a u x i l i a r y equation was introduced to allow p r e d i c t i o n of v and thus v may be calculated at any time. An e x i s t i n g i n t e r f a c e to a PDP-8 computer was modified and used for data a c q u i s i t i o n . Signal conditioning networks were designed to eliminate undesired r i p p l e and to obtain required s i g n a l l e v e l s f o r the i n t e r f a c e . The software was designed to pack observations i n h a l f words by s t o r i n g deviations from an operating point, thus allowing an adequate number of observations to be stored i n the minimal memory a v a i l a b l e . Data handling software was also developed f o r the IBM 360. A program was w r i t t e n to i n t e r p r e t the logged data, taking i t from paper tape, reconstructing the values of system s i g n a l l e v e l s , and s t o r i n g them on magnetic tape with appropriate headings. Other programs were w r i t t e n to transform the data by combining values at each observation f o r regression analysis input; to catalogue and store intermediate s t a t i s t i c a l r e s u l t s on magnetic tape; to c a l c u l a t e and plo t r e s i d u a l s ; and to solve and p l o t the s t a t i s t i c a l model responses and the t h e o r e t i c a l model responses as w e l l as the system responses to various inputs. A l l the data acquired has been retained and, i n conjunction with the data handling routines 60 developed, t h i s data may provide a s t a r t i n g point f o r further work on s i m i l a r p r o j e c t s . The s t a t i s t i c a l models i d e n t i f i e d produce very accurate p r e d i c t i o n of the system steady state response. The models were not s e n s i t i v e to operating points, as those produced from data at one operating point pro-vided accurate p r e d i c t i o n at another. When used to p r e d i c t dynamic per-formance of the system to step inputs, the s t a t i s t i c a l model f a i l e d to pre d i c t the fa s t e s t system f l u c t u a t i o n s . I t did, however, pr e d i c t the time for the new steady state operating point to be reached and i s therefore a good dynamic model for many p r a c t i c a l a p p l i c a t i o n s . For further research, the following improvements are suggested. More rapid sampling of system v a r i a b l e s i s required to provide b e t t e r dynamic p r e d i c t i o n . More r e s o l u t i o n i s required i n the speed measurement. This may be achieved by sensing speed d i r e c t l y or by having greater accuracy i n the angle measurements. To achieve a be t t e r normal d i s t r i b u t i o n of residu a l s i t i s desirable to perturb the system using c o n t r o l l e d random inputs. However, the apparent v i o l a t i o n of the normality assumption i n th i s work did not appear to have a s i g n i f i c a n t e f f e c t on the r e s u l t s . An extension of t h i s work to more c l o s e l y track system dynamics, p r e f e r r a b l y i n an on-line environment, would be of p r a c t i c a l i n t e r e s t . The ex-tension of the modelling scheme to multimachine systems would provide a valuable c o n t r i b u t i o n as the scheme would then be of greater p r a c t i c a l i n t e r e s t . APPENDIX 3A A third order state variable model may be derived from the / simplified Park's equations (3.1) to (3.4) along with the mechanical / equations (3.5) and (3.6). Equation (3.3) may be rearranged-as [12] x , v., x , (1 + T'p) = ad fd _ _ad cT_ ± (3.3) d w R (1 + T' p) co (1 + T' p) d O F do^7 o v do ^ * _f£ _ -< — + ). (3A.1) d % R F ( 1 + TdoP> % u o < 1 + TdoP> Solving for A and B and collecting terms gives VFR ^ d 0s ^ = - (3A.2) A V F + P < x d(T' - T d) } i where v p R = (3A.3) + TdoP> A x and v = _£i v (3A.4) R F A T* It can be shown that [12] x' = _ r _ x allowing v^ to be written as: d T ' d F do V F = P { Tdo t vFR " ( xd " x d ^ d 3 } + VFR <3A-5> or v p = pijjp + v p R . (3A.6) Thus equation (3.7) is found from (3A.6) where P^p = V F " VFR ( 3 - 7 ) <JF " Tdo [ VFR " <xd " Xd> ^ ] < 3 A' 7 ) Substituting (3A.4) into(3A.6) gives R F VVT3 v M = P ( * F — ) + K C — > <3A-8> ad ad which when compared to the f i e l d voltage equation V f d = p * f d + V f d ( 3 A- 9 ) gives JL. = ijj £ d (3A.10) Xad VFR , and = i ^ d (3A.11) ad which are useful relationships when referring state equation variables to actual system quantities. The electro-mechanical relationship in equation (3.8) is derived from the expression for the rotor angle in (3.6) 9 = % t + 6. (3.6) Differentiating gives p6 = co + P6 (3A.12) o or p6 = co Aco (3A.13) r o where p9 - w co — co Aco o » ~o (3A.14) CO to o o is a per unit change in speed. APPENDIX 3B For a one-machine i n f i n i t e bus system, the transformation of transmission system quantities to the d-q coordinate system is straightforward. A simplification from Vongsuriya's derivation [12] exists because the i n f i n i t e bus voltage corresponds to the rotating reference and therefore at steady state is at the angle 6 from the q-axis. Projections onto the d and q axes are then a l l that is required to express V q in terms of Park's system. That i s , v , + iv = v sin6 + iv cos6. (3B.1) od J oq o o From the system diagram in Figure 3.4 [ 1 + (r + jx) (G + jB) ] v t = V q + (r + jx) i (3B.2) where in Park's system v = v + jv (3B.3) t d q i = i , + j i (3B.4) d J q v = v , + j v . (3B.5) o od J oq ' Substituting (3B.1) for V q in (3B.2) gives V q in Park's system, and equation (3B.3) and (3B.4) into (3B.2) puts v t and i in Park's system thus giving the system equation [ 1 + (r + jx)(G + jB) ] [v d + j v q ] = v Qsin6 + jv 0cosS + (r + j x ) ( i , + j i ) (3B.6) T. which by expanding and separating real and imaginary parts can be written as equation (3.34). APPENDIX 3C Equation (3.39), which expresses and \JJ in terms of state d q variables only, may be developed as follows. In (3.34) substitute expressions for v^, v q, i ^ and i in terms of fluxes, that i s , use equations (3.1), (3.2) for v^ and v^ and use equation (3.14) for i ^ and i q 1 1 *d = r - ^ ^ ^ o ^ d O d ) do d x d -10 X The equation resulting from these substitutions i s (3C.2) to q ° V o k l k2 -k2 k 1 v sin6 o v cos<5 o + K l K2 -K2 K± •1 1 x d T d o x. x^ (3C.3) 65 APPENDIX 4 A Flowchart for Data Acquisition Program ( START ^ Perform required flag clearing and software i n i t i a l i z a t i o n e.g. i n i t i a l i z e counters etc. Read and store calibration voltage (channel # 0) _ J Read and store A/D offset voltage (channel // 1) Turn interrupt ON Wait for interrupt INTERRUPT (at beginning of each synchronous machine el e c t r i c a l cycle) X store machine angle 66 BEFORE Pack deviations from operating point for previous cycle into half words i n storage checking for overflows i n packing. 1 Read and temporarily store DC values. Read AC quantities and temporarily store maximum values. Sample continuously u n t i l interrupt at beginning of next cycle. JL To interrupt Take nominal values from temporary storage and place at beginning of store buffer. Following nominal values store c a l i b r a t i o n voltage, offsets before and after run and it of overflows i n packing. Punch out storage buffer on paper tape with half word per tape frame plus parity b i t for odd parity. ( STOP ^ 67 REFERENCES 1. M.A. Laughton, "Matrix Analysis of Dynamic S t a b i l i t y i n Synchronous Multimachine Systems", Proc. IEE, v o l . 113, pp. 325 - 336, February 1966. 2. J.M. U n d r i l l , "Dynamic S t a b i l i t y Calculations for an A r b i t r a r y Number of Interconnected Synchronous Machines", IEEE Transactions, v o l . PAS-87, pp. 835 - 844, March 1968. 3. F.C. Schweppe and J. Wildes, "Power System S t a t i c - S t a t e Estimation, Part I: Exact Model", IEEE Transactions, v o l . PAS-89, pp. 120 - 125, January 1970. 4. F.C.. Schweppe and D.B. Rom, "Power System S t a t i c - S t a t e Estimation, Part I I : Approximate Model", IEEE Transactions, v o l . PAS-89, pp. 125 - 130, January 1970. 5. F.C. Schweppe, "Power System S t a t i c - S t a t e Estimation, Part I I I : Implementation", IEEE Transactions, v o l . PAS-89, pp. 130 - 135, January 1970. 6. R.E. Larson, W.F. Tinney, J. Peachon, "State Estimation i n Power Systems Part I: Theory and F e a s i b i l i t y " , IEEE Transactions, v o l . PAS-89, pp. 345 - 352, March 1970. 7. R.E. Larson, W.F. Tinney, L.P. Hajdu, D.S. Piercy, "State Estimation i n Power Systems Part I I : Implementation and Appl i c a t i o n s " , IEEE Transactions, v o l . PAS-89, pp. 353 - 363, March 1970. 8. O.J.M. Smith, "Power System State Estimation"- IEEE Transactions, v o l . PAS-89, pp. 363 - 379, March 1970. 9. D.S. Debs and R.E. Larson, "A Dynamic Estimator f o r Tracking the State of a Power System", IEEE Transactions, v o l . PAS-89, pp. 1670 - 1678, September/October 1970. 10. N.R. Draper and H. Smith, Applied Regression Analysis, New York; Wiley & Sons Inc., 1966. 11. R.H. Park, "Two-Reaction Theory of Synchronous Machines, Generalized Method of Analysis", AIEE Transactions, v o l . 48, pp. 716 - 730, July 1929. 12. K. Vongsuriya, "The A p p l i c a t i o n of Lyapunov Function to Power System S t a b i l i t y Analysis and Control", U.B.C. PhD. Thesis, February 1968. 68 13. G.E. Dawson, "A Dynamic Test Model for Power System Stability and Control Studies", U.B.C. PhD. Thesis, December 1969. 14. J. Hindmarsh, Ele c t r i c a l Machines, New York: Pergamon Press, 1965. 15. G.E. Dawson, "Modelling, Analogue and Tests of an Electric Machine Voltage Control System", U.B.C. M.A.Sc. Thesis, September 1966. 16. E.W. Kimbark, Power System Stability: Synchronous Machines, New York: Dover Publications Inc., 1968. 17. R.V. Shepherd, "Synchronizing and Damping Torque Coefficients of Synchronous Machines", AIEE Transactions, vol. 80, pp. 180 - 189, June 1961. 18. Y.N. Yu and K. Vongsuriya, "Nonlinear Power System Stability Study by Lyapunov Function and Zubov's Method", IEEE Transactions, vol. PAS-86, pp. 1480 - 1485, December 1967. 19. "Test Procedures for Synchronous Machines", IEEE Publication 115, 1965. 20. "Applications Manual for Operational Amplifiers", Philbrick/Nexus Research, Nimrod Press, Boston, 1968. 

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