Scholars' Mine Masters Theses Student Research & Creative Works 1972 An experimental study of power systems Nguyen Quang Duoc Follow this and additional works at: http://scholarsmine.mst.edu/masters_theses Part of the Electrical and Computer Engineering Commons Department: Recommended Citation Duoc, Nguyen Quang, "An experimental study of power systems" (1972). Masters Theses. Paper 3496. This Thesis - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Masters Theses by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. AN EXPERIMENTAL STUDY OF POWER SYSTEMS BY NGUYEN QUANG DUOC, 1939A THESIS Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI-ROLLA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN ELECTRICAL ENGINEERING 1972 T2873 74 pages c.l Approved by ii ABSTRACT For three phase AC power systems consisting of synchronous machines interconnected by transmission lines and other electrical transmission equipment, the power transmitted and the stability of the system depend upon the constants of synchronous machine and the characteristics of the transmission lines. The purpose of this work is to design and set up a model of a power system in the laboratory, measure the constants of synchronous machines, the power angles and power flows of the system. An investigation of the effects of the saturated machines is based upon the laboratory data. This work is intended primarily as a tool for studying Power Systems in a University Laboratory. It makes possible the study of electrical power systems imparting a better physical understanding to the student and improves his consequent learning. iii ACKNOWLEDGEMENTS The author would like to express his sincere gratitude to his advisor Dr. John Derald Morgan for advice, guidance and encouragement during the course of this work. The author is also grateful to the Agency for International Development for the financial support which he received. The author wished to thank Mr. Frank B. Huskey for the help he has given in the laboratory. Finally, the author wishes to thank his family and especially his wife, Dao Thi Thu Cue, for their continued encouragement and moral support during his studies. iv TABLE OF CONTENTS Page ABSTRACT...................................................... ii ACK:N'0WLEDGEJI.1ENT. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i i i LIST OF ILLUSTRATIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • v LIST OF TABLES. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • . • • • • • • • • • vii I. INTRODUCTION ••••••••••••••••••••••••••••••••••••••••••• 1 II. THEORETICAL CONSIDERATION IN .HODELING OF POWER SYSTEHS. 7 III. A. Transmission System. . • . . . . . . . . . . . . . . . . . . . . . . . . . . 7 B. Measurement of Synchronous Machine Reactances ... 16 c. calculation of Power Angle and Power Angle Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 EXPERIMENTAL RESULTS ••••••••••••••••••••• • •• • ••••• • •••• 40 A. Measurement of Machine Constants . • . . . . . . . • . . . . . . 40 B. Measurement of Power Angle .•.•.•.••..•..•......• 42 CONCLUSION ••••••••••••••••••••••••••••••••••••••••••••• 51 BIBLIOGRAPiiY ••• • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 53 VITA •••••••• • •• • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 54 APPENDICES •• • •• • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • 55 A. Laboratory Data of Machine A......•.......•...... 55 B. Sample Calculation . . . . . • . . . . . . • . . . • . . . . . . . • . . . . . . 60 rv. v LIST OF ILLUSTRATIONS Figures Page I.l. Synchronous Machine Equivalent Circuit............... 1 I.2. Two Machine Power System............................. 2 I.3. Two Machine System Equivalent Circuit................ 2 I.4. Phasor Diagram of Two Machine System................. 3 I.S. Representation of Synchronous Machine in a Transient Stability Study............................ 5 Equivalent ~ Circuit for Representing Transmission Line. • . • • . . . • . . • . . • . . . • . . . . . . . • • . . . . . . . . . . . . . . • . . . . . . 11 II. 2 . Impedance Circuit. . . • . • . . . • . . . • . . . . . . . • . . . • . • . • . . . . . . 11 II. 3. Multi -layer Coil. • • . . • . . . • . . . . . . . . . • . . • . . . • . . • • . . . • . . 13 II.4. Dimension of the Reactance Coil in CM ••...•.•.••..... 14 II.Sa. Open Circuit and Short Circuit Characteristics for Determination of xd by AIEE Definition •.......•...... 17 Open Circuit and Short Circuit Characteristics •.•.•.. for Defining xd...................................... 18 II.6. Open Circuit and Short Circuit Characteristics ....... 19 II.7. Open Circuit Characteristic Defining the Saturation Factor. . • . • . • . • . . . . . • . . • . . . . . . . . . . . • . . . • . . 19 II.B. Open Circuit and Zero Power Factor Characteristics ... 21 II.9. Symmetrical Curve of Armature Short Circuit CUrrent. • • . • . . . • . • . • . . . . . . . . . . . • . . • . . . . . . . . . . . • . . • . • • 23 II.lO. Envelope of Symmetrical Short Circuit Current ........ 23 II.ll. current Difference Plotted to Semi-logarithmic Paper................................................ 25 II.12. Locked Rotor Line to Line Test . . . . . . . . . . . . . . . . . . . . . . . 26 II.13. Oscillograms of Slip Tests .......................•..• 28 II.l4. vector Diagram of Synchronous Generator ....••..•...•. 30 II.lS. Vector Diagram of Synchronous Generator With Machine Resistance Neglected ••.•••••••••••••••.•...•. 32 II.l. II.5b. vi List of Illustrations (continued) Figures Page II.l6a. Single Line Diagram of One Machine . . . . . . . . . . . . . . . . . . . 34 II. 16b. Phasor Diagram of One Machine. . . . . . . . . . . . . . . . . . . . . . . . 34 II.l7a. Single Line Diagram of One Machine and Series Reactance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Phasor Diagram of One Machine and Series Reactance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 II.l8a. Single Line Diagram of Two Machine System . . . . . . . . . . . . 37 II.l8b. Phasor Diagram of Two Machine System . . . . . . . . . . . . . . . . . 38 III.l. Connection Circuit Diagram of Synchronous Motor ...... 42 III.2a. Single Line Diagram of Two Machine System: One Motor and One Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 III.2b. Connection Circuit Diagram of Two Machine System ..... 44 B.l. Determination of x and x From Data of Machine A . . . . . . . . . J? •••••• ~~. • • • • • • • • • • • • • • • • • • • • • • • • 59 B.2. Equivalent Circuit of Synchronous Machine . . . . . . . . . . . . 60 B.3. Symmetrical Short Circuit Transient Envelope . . . . . . . . . 64 B. 4. .6-i 65 II.l7b. X on Semi-logarithmic Scale........................ vii LIST OF TABLES Table III.l. III.2. III.3. III.4. III.5. III.6. A.l. A.2. A.3. A.4. Page Comparison Between Test and Calculated Values of Power Angle for the Unsaturated Salient Pole Synchronous l1achine. • • • • • • • . • . . • • • • • . . . • . • • • . . . • . 45 Comparison Between Test and Calculated Values of Power Angle for the Saturated Salient Pole Synchronous Machine. • • • . • • . • . • . • . • • • • • . • . . . . • . • . • . • . • . 46 Comparison Between Test and Calculated Values of Power Angle for the Unsaturated Salient Pole Synchronous Machine Connected to an Infinite Bus Through the Artificial Line Designed in Section II •••• 47 Comparison Between Test and Calculated Values of Power Angle for the Saturated Salient Pole Synchronous Machine Connected to an Infinite Bus Through the Artificial Line Designed in Section II •••• 48 Test and Calculated Values of Power Angle for the 'I'\>lo Machine System. . . • . • . . • . . • • • . • . . . • • • • • • . . • • . . • . . • . 49 Test and Calculated Values of Power Angle for the Two Machine System Through the Artificial Line Designed in Section II. . . . • • . • • . • . . . . • • . . • . . . • • • . . . . • • 50 Measurement of No-Load Voltage in Function of Field Currents. • . . • . . . • . . . • . . . . . . . . . • • . . • • . . • . . . . • . . . . 55 Measurement of Short Circuit Current in Function of Field Currents . • . • • • . • . • . • • • . • . • . • • . • • . . . . . . • . • . . • • 56 Measurement of Terminal Voltage in Function of Field current at Rated Armature Current and Zero Power Factor. • • • • • • • • • • • • . • • • . • . • • • . • • • . • • • • • • • . • . • • • • • • • • • • 57 Measurement of Short Circuit Oscillogram •....••.....•• 58 1 I. INTRODUcriON An electric power system consists of three principal components: the generating stations, the transmission system and the distribution system. By means of synchronous generators, energy in its various forms is converted into electric form and the transmission systems are the connecting links between all the generating stations and distribution systems which connect the individual loads. One problem in power system analysis is the determination of whether or not the various synchronous machines in the system will remain in synchronism with one another. The characteristics of the synchronous machines and of transmission lines obviously play an ~portant part in the problem. In power system analysis, a synchronous machine is often represented in a circuit diagram by a constant reactance in series with a constant voltage. Several synchronous machine reactances -~-~ + I E f E /0 Figure I.l. have been defined. = E Synchronous Machine Equivalent Circuit Which reactance is used depends upon which conditions are desired to be investigated for the system. 2 Consider a very simple power system, consisting of two synchronous machines A and B and series inductive reactance of the transmission line XL. The system is represented by the following circuit diagram. B Figure I.2. Two Machine Power System Combining the machine reactances and line reactance into a single reactance, the circuit diagram becomes: Figure I.3. Two Machine System Equivalent Circuit 3 Suppose that machine A is a synchronous generator and machine B is a synchronous motor, the power transmitted from the generator to the motor depend upon the phase difference 8 of the two voltages EA and EB~ these voltages are generated by the flux of the field windings of the machines, so their phase difference is approximately the same as the electrical angle between the two machine rotors. The phasor diagram of a two machine system is shown in the figure I.4 below: Figure I.4. Phasor Diagram of Two Machine System The vector equation is: (I.l) Hence, the current (I.2) since i t is assumed that there is no resistance in the line, the power transmitted from the generator to the motor is given by: 4 Real part of {E~I) p (I. 3) Substituting the value of I given by equation I.2 {I.4) P = Real part of [ (E* A E~ where is conjugate of EA Let (I. 5) (I. 6) then (I.7) E* A We have p Real part of [EA j-o x EA fj_ - EB LQ_ X J 190 T -- or E~ p /-90 E AEB :.-/_-_9_0_-8_ Real part of [ Equation I.B shows that the power P varies with the sine of the displacement angle 6 between the two rotors. The maximum power Pm that can be transmitted in the steady state with a given reactance XL and given internal voltages is: 5 p m = (I.9) This maximum occurs at a displacement angle 8 90°. In practice, this maximum is never reached. The system is stable only if the displacement angle 8 is between -90° and +90° where the s 1 ope dP · d 1s 8 · ' pos~t1ve, · t h at 1's t h e range 1n which an increase in displacement angle results in an increase in transmitted power. P m is the steady state stability limit of the system. It is the maximum power that can be transmitted and synchronism will be lost if an attempt is made to transmit more power than this limit. If the point of instability is reached by a sudden change in conditions of operation of the system, a short-circuit for example, the limiting value of power is called the transient stability limit. For transient X' d Figure I.S. Representation of Synchronous Machine in a Transient Stability Study stability studies, the synchronous machine is represented by its directaxis transient reactance xd in series with a constant voltage power 1 source which is voltage behind transient reactance . 1Bibliography No. 5, PP· 259 6 The subtransient reactances of synchronous machines are used to determine the initial current flowing on the occurrence of a short circuit. For the determination of the interrupting capacity of circuit breakers, subtransient reactance is used for synchronous generators and transient reactance is used for synchronous motors. Tests were made in the laboratory to measure the synchronous machine reactances. A transmission line was built to determine the power angle curve by tests and checked by method of analysis. 7 II. A. THEORETICAL CONSIDERATIONS IN MODELING OF POWER SYSTEHS Transmission System 1. Artificial transmission system An artificial transmission system used for making tests in the laboratory v1as constructed of reactance coils and operated normally at 220 volts. Power was obtained from the synchronous generator 11 KVA, 220 volts, 60 cycles driven hy a direct current motor. A synchronous motor de generator set was used as load: the direct current generator of this set was connected to a resistance load to provide a dead load instead of loading back on the direct current laboratory system. The field of the direct current generator was separately excited so that the load could be varied gradually by raising or lowering the voltage. The artificial transmission system was designed so that results could be obtained in per-unit comparable with those expected in an actual system using a 230 KV - 100 MVA base. 2. Calculation of transmission system constants A 230 KV transmission system of 230 mile length was selected. Base quantities were chosen to be 230 KV and 100 l~. Figure II-1 shows the equivalent pi circuit, using the resistance, inductive reactance and capacitive reactance, the pi equivalent circuit of the system was calculated. The calculation shows the per-unit impedance for the system in respect to the original base system in which it was chosen and for the artificial system base is 20 KVA and 220 volts. 8 The calculations are as follows: Base KVA· 100,000 KVA Base KV· = 230 KV 2 (Base KV) x 1000 Base KVA Base impedance 230 100 2 529 ohms . The length of the line was selected to be 230 miles, all constants of the lines are shown in Table 3, "Typical Transmission lines characteristics at 60 cycles" on page 280 of Transmission and Distribution Reference book. Resistance at S0°c r = 0.1288 ohms per mile Reactance per phase x = 0.7681 ohms per mile Shunt capacitive reactance x' = 0.1821 megohms per mile The impedances of the equivalent pi circuit are expressed in the following form: Zeq = 100 rSK r Z'eq = -j 2x' s + jlOOxSK (II .1) x (K r + jk )10 4 x where: s = length of line in hundreds of miles (II. 2) 9 K r K X k k r X = 1 - 2 2S 300x' (II.3) 1 - 2 s2 (~- r xx') 600 x' (II.4) 1 - 2 xs 1200x' (II.S) 2 rs 1200x' (II. 6) The correction factor k X is never greater than about 0.005, so it can be neglected and the shunt impedance Z' eq can be considered as a pure capacitor. From equations II.3, II.4 and II.S we get: K 1 r 2 xs - 300x' 2 = 0.7681 X 2.3 300 X 0.1821 0.9256 K X = 1 - = l 2 s2 X ( - - _E._) 600 x' xx' 2. 3 - 600 = 0.9638 2 (0.7681 0.1821 - 2 0.1288 ) 0.7681 X 0.1821 10 k r 1 l xs 2 - 1200x' _ 0.7681 2 2.3 1200 X 0.1821 X 0.9814 The equivalent impedances of the equivalent TI circuit are given by the following equations: Zeq Req + jXeq = lOOS(rK r + jxK) (II.7) x Z'eq = -jX'eq -j 2x' (k ) s r X 10 4 Numerical substitution gives: Zeq = 100 X 2.3(0.1288 X 0.9256 + j0.7681 X 0.9638) (27.42 + j170.26) ohms Z'eq - j 2 X 4 0 1821 • X 0.9814 X 10 2.3 -jl554 ohms Or in per unit system: Zeq Z'eq {0.052 + j0.322)p.u 2.94 p.u (II. 8) 11 The equivalent circuit for this line is shown in Figure II.l. Zeq 0·----r------.. ' Zeq Figure II.l. Equivalent n Circuit for Representing Transmission Line Neglecting the shunt impedance Z'eq, the equivalent circuit for representing the transmission line is shown in Figure II.2. Zeq Figure II.2. Impedance Circuit Adapted From Figure II.l by Omitting Shunt Admittances The base impedance of the system using 20 KVA 220 volts is: z Base (0.22) 0.02 2 12 zBase 2.42 ohms . The line impedance of the artificial system is: 3. z 2.42(0.052 + j0.322) z (0.125 + j0.780) ohms • Design of reactance coils a. Dimension of the coils The impedance of the artificial line per phase is: z = (0.152 + jO. 780) ohms . The inductance of the coil is XL L w X L L 21f f 0.780 27T X 60 = 2069 rnicrohenrys It is impossible to calculate the true inductance of some types of air cored coils with a fair degree of accuracy. The actual apparent inductance differs from the calculated true inductance because of distributed capacitance. For that reason, inductance calculations are generally used only for a starting point in the final design. reactance value of the reactor coil must be verified by test. The 13 The inductance of the coil is given by the following formula in which all dimensions are expressed in centimeters and inductance is in microhenrys. L = 2 2 2 2 b c 8a b 0.01257 an [(1 + ---- + ----)loge ~ -y + ---2 Y2 ] 2 2 1 32a 96a 16a (II.9) where: n: number of turns a: radius of the coil measured from the axis to the center of winding cross section b: the length of the coil c: the radial depth of winding d: the diagonal of the winding . a ••I b c Figure II.3. Multi-layer Coil 14 y 1 and y 2 are functions of~ (when b <c) or~ (when b >c). When b and c are determined, y 1 and y 2 are obtained from the table 1 shown on page 3-19 of Radio Engineering Handbook . After trying many numerical values in the formula II.9, the dimensions of the coil were determined as follows: Type of coil: Multilayer circular coil of rectangular cross section Wire: Bare diameter 2.00 mm Maximum diameter over insulation SCCE 2.2 mm (Single cotton covered enameled) b. Number of turns Number of layers: 12 Number of turns per layer: Total number of turns: c. 25 12 x 25 = 300 . Dimension of the coil Diameter of the coil: Length of the coil: D b = 15 em 6.6 em 2.6 6.6 15 Figure II.4. Dimension of the Reactance Coil in em 1 Refer to Bibliography No. 1 15 The radial depth of winding c = 2.6 em Radius of the coil measured from the axis to the center of the winding: a =D a = - c 2 15 - 2.6 2 Diagonal of the coil: I Y6.6 2 + 2.6 2 7.09 em . The ratio~ equals: c b = 2.6 6.6 0.393 c The ratio b equals 0.393, from the table we get: 0.7645 y2 = 0.242 . Substituting these values in equation II.9 we have: L 2034 microhenrys , or, the reactance: 16 XL 0.7668 ohms . This value was verified by a direct measurement. The impedance of the reactance core given by the measurement is Z (0.10 + j0.76)ohms This value is sufficiently close to the impedance of the artificial line to be designed (0.125 + j0.78 ohms) and shall be used for all further calculations. B. Measurement of Synchronous Machine Reactances 1. Direct axis reactances a. Direct axis synchronous reactance xd a-1. Unsaturated direct axis reactance The definition of unsaturated direct axis synchronous reactance given in AIEE standard is as follows: Synchronous reactance is the ratio of the field current required to circulate rated current on a sustained three phase short circuit to the field current which would produce rated voltage at no load if there were no saturation. Figure II.Sa shows the open circuit characteristic (OCC) and short circuit characteristic (SCC) and the AIEE definition gives: (II.lO) 17 Air gap line Rated voltage i Figure II.Sa. i s s Open Circuit and Short Circuit Characteristics for Determination of xd by AIEE Definition is the field current required to circulate a rated three phase short circuit current and i 0 is the field current required to produce rated voltage read from the air gap line. Another way to define unsaturated Xd is as follows: at any convenient field excitation current such as OF in Fig. II.Sb, the short circuit armature current is O'B, the excitation voltage for the same field current is OA read from the air gap line. Xd is the ratio of this voltage OA to the short circuit armature current O'B. The measurement Xd from this definition (by open circuit and short circuit tests) gives the unsaturated value of direct axis reactance. 18 Air gap line A -----------------B 0 Figure II.Sb. F 0' Open Circuit and Short Circuit Characteristics for Defining xd a-2. Saturated direct axis reactance When great accuracy is not required, the saturated value of xd at rated voltage Vt is defined by the relation: (II.ll) where, referring to Figure II.6 I sc is the armature current O'C read from the short circuit characteristic at the field current of OF corresponding to Vt on the open circuit characteristic. 19 occ Rated vt -------- sec u U) 0 Figure II.6. a-3. 0' F Open Circuit and Short Circuit Characteristics Accurate value of saturated direct axis synchronous reactance Xd The effects of saturation under load can be taken into account with good accuracy by use of a saturation factor determined from the open circuit characteristic curve. volts occ E r (a. g) E r R Figure II.7. Air gap mmf Open Circuit Characteristic Defining the Saturation Factor 20 Referring to Figure II.7, the saturation factor is defined as: E {a.g) k r E {II.l2) r where: E (a.g) is voltage corresponding to the resultant mmf R as read r from the air gap line E r is air gap voltage read from the saturation curve. The saturation factor is a function of air gap voltage E . r The saturated synchronous reactance is given by the following relation: (II.l3) where: x 1 is the leakage reactance found by application of the Potier method. x uns is the unsaturated synchronous reactance determined by relation II.ll. k is saturation factor. It is noted that the Potier reactance x leakage reactance x 1 p may be used in place of the when the open circuit characteristic is used as the saturation curve under load. The method to determine the saturated value of Xd is as follows: Find the Potier reactance x Potier triangle ABC. p by geometrical construction of a Fererring to Figure II.8, select a point C on 21 zero power factor corresponding to rated voltage Vt. Draw the horizontal line CD equal in length to the field current OC' for short circuits. Zero pf I c 0 Figure II.B. a constant Field current Open Circuit and Zero Power Factor Characteristics Through point D draw a line DA parallel to the air gap line. the vertical line AB. X X p p = The Potier reactance x p Draw is given by: Voltage drop AB per phase Zero power factor armature current per phase (AB)volts amp (I ) a (II.l4) 22 The next step is to compute the air gap voltage E ± I r a (r + jx ) (II.lS) P Knowing E , Figure II.7 gives value of E (a.g) and the saturation r r factor is determined by the relation II.l2. The saturated direct axis reactance is then: X X p + - rms X p (II.l6) k This method is based on simple cylindrical rotor theory but commonly applied as an approx~ation to salient pole machine with satisfactory results over the normal operating range of the machine. b. Transient and subtransient reactances b-1. xd, xd Three phase short circuit test The transient and subtransient reactances of a synchronous machine are found by transient short circuit test. The machine is driven as a generator, at no load and at rated voltage, and suddenly short circuited. An oscillograph is used to record the transient short circuit current. The Figure II.9 shows a symmetrical trace of a short circuit current wave. The de component is taken out by subtracting or may be eliminated by short circuiting at the instant when the d axis is 90° from the axis of phase a. The wave, whose envelope is shown in Figure II.lO may be divided into three periods: Subtransient period lasting only for the first few cycles, during which the current decrease is very rapid; 23 the transient period lasting a relatively longer t~e, the current decrement is more moderate: the steady state period, the current is constant. B / / Figure II.9. Symmetrical Curve of Armature Short Circuit current B ' .... A ..j.J s:: Q) ~""' () c ~-:::-:---___ c --~~----------~--------------------------------------------------------- 0 t Figure II.lO. Envelope of Symmetrical Short Circuit Current 24 The subtransient reactance xd determined the initial value OB of the symmetrical subtransient envelope BC. It is equal to the RMS value of the prefault open circuit voltage divided by OB 1:2 x" (II.l7) d The transient reactance xd is the ratio of voltage to current which is the initial symmetrical value neglecting the initial rapidly decaying portion; that is the current OA/1:2 in Figure II.lO. x' (II.l8) d It is noticed that the factor 12has appeared because of the peak value of current. Subtransient reactance depends on the initial (zero-t~e) current, hence the later must be determined accurately. The difference between the transient envelope and the steady state amplitude is plotted on semi log-paper in Figure II.ll. gives the initial transient difference current transient difference current ~I 0 ~1 0 Extrapolation and initial sub- . The initial transient current OA and initial subtransient current OB in relations II.l7 and II.l8 are: ~ ~I 0 + sustained short circuit current OB = ~1 0 + sustained short circuit current OA 25 ' time Figure II.ll. b-2. Current Difference Plotted to Semi-logarithmic Paper Locked rotor line-to-line test While this method is not practicable for determining the satruated value for turbine generators, i t may be applied to salient pole machines with damper windings at rated current to obtain a very good value for the saturated value of subtransient reactance. A single phase voltage is applied between any two line leads, with the field winding short circuited on itself and the rotor locked at standstill. Readings of voltage and field current are taken for various rotor positions over a pole pitch. Two rotor positions are determined, one for a maximum value of the field current and the other for minimum value. The rotor position for maximum field current is called direct axis position, the other is quadrature axis position. 26 With the rotor in the direct axis position and rated current £lowing, the subtransient reactance is equal to half the ratio of applied voltage to line current. x" d Figure II.l2. E (II.l9) 2I Locked Rotor Line to Line Tests According to Wright, this method can be used to determine xd also, under certain conditions: "Only for machines without additional rotor circuits hence only for salient pole machines without dampers does this test measure xa· Made at low currents, i t gives the unsaturated value which, when multiplied by the empirical factor 0.88 gives, approximately the saturated value." 27 2. Quadrature axis reactances a. Quadrature axis synchronous reactance x The quadrature axis synchronous reactance x q q can be measured by two methods: Slip test Maximum lagging current test • a-1. The slip test: Measurement of unsaturated quadrature axis synchronous reactance In this test, the machine is driven mechanically at a speed slightly different from synchronous speed, the field winding is open and a balanced three phase voltage of correct phase sequence is applied to its armature terminals. An oscillograph is used to record the waves of armature current, voltage applied to the armature tenninals and voltage induced in the open field winding. Figure II.l3 shows the general forms of these oscillograms. The quadrature axis synchronous reactance x q equals the minimum ratio of armature applied voltage per phase to the armature current per phase and occurs where the induced voltage in the field winding xd is maximum. x can be obtained by finding the ratio --which, in q X q the slip test, approximately equals the ratio of maximum armature current to the minimum armature current. Referring to Figure II.l3: I X q I max . m~n (II.20) 28 I i~ Terminal voltage Armature current Figure II.l3. Oscillograms of Slip Tests Using the value of xd from the open circuit and short circuit test described previously in this chapter, x q can be determined. The value thusly obtained is the unsaturated quadrature axis synchronous reactance since the test must be made at small values of armature current. 29 a-2. Maximum lagging current test The slip test introduces large errors because of the effects of the current produced in rotor circuit unless the slip is very small. In the maximum lagging current test, the machine is running as a reluctance motor - the field current is reduced to zero - then, the polarity of the field current is reversed and a small field current applied in the reversed direction causing an increase in armature current. By increasing the field current in the reversed direction, the armature current increases and reaches a maximum stable value, any further increase of the field current will cause the machine to fall out of step. The quadrature axis reactance is given by the relation: X q (II.21) = where: Vt is the armature terminal voltage per phase. I x ms q is the maximum stable armature current per phase. given by this test is the saturated value since the test can be made at normal voltage Vt. Another method for measuring the saturated value of x q is based on a vector diagram of synchronous machines derived from the machine equation. The vector equation of a synchronous generator is: (II.22) 30 where: Ef is excitation voltage vt is terminal voltage I is load current Id is direct axis component of load current I q is quadrature axis component of load current R is machine resistance a cos¢ is power factor o is power angle . Figure II.l4 is the phasor diagram drawn for a synchronous generator supplying a lagging power factor balanced load. E f X I q q Figure II.l4. Vector Diagram of Synchronous Generator 31 From Figure II.l4, we have: X I (II. 23) q q or: I Equation q = I.I. 23 I[cos(¢ +a)] becomes: I[cos(¢ + o)]X q I(cos¢coso - sin¢sino)x [Vt + R Icos¢]sino + R Isin¢coso a a = I(cos¢coso - sin¢sino)X q q Hence: Xq = (Vt + R Icoscp)sino + R Isin¢coso a a I(cos¢coso - sin¢sino) or, dividing through by sine gives: vt + R Icos¢ + R Isin¢coto a X q a (II. 24) I(cos¢coto - sin¢) The machine resistance is usually small, and we can neglect it. If we can neglect the machine resistance, the machine equation becomes: Ef = Vt + J'X I + ]'X I a q q and the vector diagram is shown in Figure II.lS. 32 I Figure II.lS. Vector Diagram of Synchronous Generator with Machine Resistance Neglected Equation II.24 giving X q X q = becomes: (II.25) I(cos¢cota -sin¢) Knowing the terminal voltage vt' load current I, power factor cos~ and power angle a, the saturated quadrature axis synchronous reactance x q is computed from equation II.24. Equation II.24 is applied not only to a generator supplying a lagging power factor load, but also to a motor operating at a leading power factor for motor case, the sign of the second term in denominator of equation II.24 changes. b. Quadrature axis transient reactance x• q Salinet pole machines have no effective quadrature axis rotor circuit hence: x'q xq For turbine generators, where xq and x~ are not equal, x; 33 may be taken equal to the saturated value of direct axis transient reactance xd. c. Quadrature axis subtransient reactance x" of quadrature q axis subtransient reactance x" may be taken equal to the direct axis q subtransient reactance x d. For a salinet pole machine with dampers x" can be detennined by q the locked rotor line-to-line test. The procedure is similiar to that for determining x". d With the rotor in the quadrature axis position and with rated current flowing, x~ is equal to 1/2 the ratio of applied voltage to line current. xq = c. E 21 for quadrature axis position Calculation of Power Angle and Power Angle Characteristics 1. One machine connected to infinite bus The power angle 8 for a machine connected to an infinite bus is the torque angle since this angle is the difference between the internal angle of the machine and the angle of the synchronously rotating reference frame which is in this case, the infinite bus. a. One machine connected to infinite bus through a line with impedance neglected In this case, the terminal voltage of the synchronous machine and the bus voltage are equal, VB = vt. II.24 to calculate power angle o We can apply the equation if quadrature axis is already 34 ·x r J q q Figure II.l6a. measured. Figure II.l6b. Single Line Diagram of one .Hachine Phasor Diagram of One Machine Equation II.24 applied to a synchronous generator is: X q = Vt + r Icos¢ + R Isin¢cot8 a a I(cos¢cot8 -sin¢) or: X I(cos¢coto - sin¢) q = vt +RIces¢ + R Isin¢cot8 a a (X Icos¢ - R Isin¢)cot8 = Vt + R Icos¢ + X Isin¢ q a a q cote = Vt + Ra Ices¢ + Xq Isin¢ X Icos¢ - R Isin¢ q a X tano V t tan ~cos¢·- R Isin¢ a + R Icos¢ + X Isin¢ q a -1 q X Icos¢ - R Isin¢ q a vt + R Ices¢ + X Isin¢ a q (II.25) 35 In the case of machine resistance neglected: o= Xs ______ Ices¢ __ tan _ 1 ____ V + X Isin¢ t q (II.26) The power delivered to the bus per phase is If the bus voltage is resolved into components Vd and Vq in phase with Id and Iq, we have (II.27) From Figure II.l6b: Ef- Vtcoso - Raicos(¢ + o) xd I q = vtsino + Raisino X q Substitution of these values in equation II.27 gives: Ef-Vtcoso - Raicos(¢ + o) Vtsin6 + Raisino p = ~-------------------------- Vtsin6 + X Vtcoso xd q E V p f t = ---xd . s~na 2 + vt X -X d q sin2a + 2xdxq v R I(cos6 _cos(¢+ 6)) t a xq xd . ~ s~no (II. 28) If the machine resistance is neglected 36 This equation is derived from synchronous generators. applied to synchronous motors with the sign changed. It can be For generator action, Ef leads Vt' for motor action Ef lags Vt. For cylindrical rotor machine where X q (II.30) b. One machine connected to infinite bus through a series impedance Consider a salient pole synchronous machine connected to an infinite bus of voltage VB through a series impedance of reactance XL per phase. small. Resistance will be neglected because it is usually The single line diagram is shown in Figure II.l2a, and \1. phasor diagram in Figure II.l2b. d axis I I I ·x I J q q Figure II.l7a. Single Line Diagram of One 1-1achine and Series Reactance Figure II.l7b. Phasor Diagram of One Machine and Series Reactance 37 The dashed phasors show the line reactance drop resolved into components due to Id and Iq. The effects of line reactance is merely to add it to the machine reactance: (II.31) (II.32) The expression of angle 8 and power P are similar to equations II.25, II.26, II.28 and II.29 except that Vt in these equations is replaced by VB and xd, xq by Xd 1 2. ,x . ql Two machine system When a system has only two synchronous machines, one acting as a motor and the other as a generator, we can use equation developed in c-1 to calculate the power angle and power angle characteristics with the terminal voltage of generator VA as the reference phasor. In this case, the angle o is angle between the internal voltage of the motor and terminal voltage of generator. The generator resistance and generator reactance are included in the determination of generator voltage. 1 Figure II.l8a. 2 Single Line Diagram of Two Machine System 38 ·x I J a _J.2 q -2 E~ -jX q2 Id 2 Figure II.l8b. Application of equations in C-1 Phasor Diagram of TWo Machine System gives only the power output of a generator, the power input of a motor is the power output of generator minus line losses. The general forms of machine power in terms of the scalar values of the internal voltages, machine impedances and impedance 1 of the line connected two machines are : For machine 1: motor (II.33) For machine 2: generator (II. 34) 1 Bibliography No. 12 39 where: z11= magnitude of impedance of machine 1 z22= magnitude of impedance of machine 2 z12= magnitude of transfer impedance i.e. magnitude of line impedance - 8 0.12: 90 8 12: impedance angle of z 0 12: 01 0 21: - - 12 12 02 01.2 Positive power is taken as the power out of machine, therefore power of generator is positive and of motor is negative. Substitution of o12 by -a 21 in equation II.9 gives: (II.35) The generator power P will be maximum when motor power will reach maximum when 8 21 o21 = = 90 - a. 90 + a. 12 and 12 • When the machine resistance and line resistance are neglected, 8 22 = 90° and a 12 = 0. The power outputs of both generator and motor will be maximum when p p o21 = E1E2 --x12 lmax ElE2 2max 90° and their values are: = xl2 40 III. A. EXPERIMENTAL RESULTS Measurement of Machine Constants 1. Machine A The following results were obtained on an 11 KVA, 220 volt, 29 ampere, 1200 rpm, 60 cycle and 3-phase machine. The equivalent armature resistance measured at ambient temperature and calculated at 75°C is R = 0.142 ohms. a The unsaturated value of direct axis reactance xd determined by open circuit tests is 3.40 ohms. The saturated value of xd determined by use of saturation factor k is 3.14. The saturation factor was determined from the open circuit characteristic and Potier reactance assuming that the terminal voltage equals rated voltage 220 volts. The current is rated current and power-factor equals 0.8. (Appendix B.3) xd The slip test determined the ratio with xd equal to the q unsaturated value, x is equal to 2.26 ohms. x- q The transient and subtransient reactances determined from the oscillogram taken by the three phase short circuit test are: xd = 0.74 ohms x" = 0.56 ohms • d 2. Machine B A 15.5 HP, 220 volt, 40 ampere, 1200 rpm, 60 cycle, 3-phase machine was used as a motor connected to machine A in the measurement of the power angle of a two machine system. The machine constants are: 41 X q 1.90 ohms xd = 2.90 ohms x' d 0.615 ohms x" d 0.548 ohms The direct axis reactance was adjusted for saturation effects, since the open circuit characteristic of a machine is essentially a direct axis magnetization curve, the quadrature axis reactance x assumed to be unaffected by saturation. q was This assumption is suitable for most practical purposes. The saturated reactance is a function of the air gap voltage, when the air gap voltage increases, the saturated reactance decreases, the air gap voltage depends upon the operating condition of the machine, i.e., the terminal voltage, the armature of measurement of saturated reactance determined only one value of reactance assuming the terminal voltage to be 220 volts, the current to be 29 ampere and power factor 0.8. Furthermore, the treatment of saturation effects as a function of a factor to be deduced from open circuit and zero power factor characteristics is not completely sufficient. Because of linearity in the unsaturated region, the unsaturated values of machine reactances are constant, therefore some analysis methods consist in heating the machine as an unsaturated one and then correcting the saturation effects by two saturation factors, one dependent, one direct axis 1 current and one on quadrature axis current • 1 Bibliography No. 12 42 B. Measur~ent of Power Angle Tables III.l and III.2 show calculated and test results when machine A is energized from an infinite bus. The machine was operated as a motor in unsaturated and saturated conditions, i.e., the bus voltage is respectively 110 and 220 volts. The connection diagram is shown in Fig. III.l; however, the model line was omitted for this test. Model line AC ~--\_.J rAN' 6 so~~ce r DC source or load Figure III.l. Connection Circuit Diagram of Synchronous Motor power source 43 Table III-3 shows the test and calculated values of power angle of the machine connected to the infinite bus through the artificial transmission line. The connection diagram is shown in Fig. III.l. Tables III-5 and III-6 give test and calculated results of two machine systems. The test circuit is shown in Fig. III.2. Excitation of the generator was varied to hold the terminal voltage constant at 220 volts, and excitation of the motor was varied so that the armature current of the motor was minimum, i.e., the power factor equal to unity. The angle a between the terminal voltage of the motor and generator is equal to zero because the two machines were connected by a line without impedance. Table III-6 shows the effects of the impedance associated with the machines. With the same power, the power angle is increased when there is an impedance associated with the machines, i.e., the power limit of the system is decreased. The calculation for the operating conditions is shown in Fig. II.l8b, the connection diagram is the same as Fig. III.2 except the addition of three reactance coils which represent the transmission line. 44 Figure III.2a. DC source Figure III.2b. Single Line Diagram of ~ro Machine System: One Motor and One Generator DC source Connection Circuit Diagram of Two Hachine System 45 Table III-1 Comparison Between Test and Calculated Values of Power Angle for the Unsaturated Salient Pole Synchronous Machine Machine A: vt (Volts) I avg (Amperes) motor, unity power factor p 8 Angle/Degrees (Watts) Test Calculated 110 5.7 1080 10.68 11.60 110 9 1715 17.80 18.10 110 15 2858 28.00 28.50 110 18 3429 33.30 33.70 110 21 4763 35.10 35.30 110 27 5144 45.50 45.60 110 31 5906 50.00 49.80 46 Table III-2 Comparison Between Test and Calculated Values of Power Angle for the Saturated Salient Pole Synchronous Machine (Refer to Fig. III.l) Machine A: vt (Volts) I avg (Amperes) motor, unity power factor l? 0 Angle/Degrees (Watts) Test Calculated 220 4.2 1600 3.80 3.85 220 6 2280 5.40 5.54 220 12 4570 11.00 11.05 220 16.5 6287 14.8 15.10 220 20.5 7810 17.20 18.62 220 22.0 8380 18.90 19.91 220 25.0 9525 20.20 22.44 220 27.0 10280 21.30 24.09 220 30.0 11430 24.50 26.49 47 Table III.3 Comparison Between Test and Calculated Values of Power Angle for the Unsaturated Salient Pole Synchronous Machine Connected to an Infinite Bus Through the Artificial Line Designed in Section II Machine A: vt (Volts) I avg (Amperes) motor, unity power factor p 8 Angle/Degrees (Watts) Test Calculated 110 5.8 1100 11.00 11.05 110 5.9 1120 11.00 11.24 110 8.25 1570 15.50 15.71 110 8.80 1675 16.50 16.76 110 9.35 1780 17.65 17.81 110 10.50 2000 19.85 20.00 110 15.00 2850 28.00 28.57 110 15.80 3000 30.00 30.10 48 Table III-4 Comparison Between Test and Calculated Values of Power Angle for the Saturated Salient Pole Synchronous Machine Connected to an Infinite Bus Through the Artificial Line Designed in Section II (Refer to Fig. III-1) Machine A: vt {Volts) I avg (Amperes) motor, unity of power p 0 Angle/Degrees (Watts) Test Calculated 220 3.75 1430 4.50 4.72 220 4.80 1830 6.00 6.05 220 8.00 3045 9.50 10.00 220 11.50 4380 13.50 14.85 220 13.00 4950 15.70 16.15 220 16.50 6285 18.86 20.26 220 18.85 7180 20.70 22.92 220 20.85 7940 23.10 25.12 220 24.50 9330 25.50 28.95 220 28.50 10800 28.60 32.88 220 31.50 12000 31.70 35.64 220 32.50 12380 34.20 36.50 49 Table III-5 Test and Calculated Values of Power Angle for the Two Machine System c1;negrees 8 Calculated 02 Calculated 2.50 2.60 3.50 6.10 2520 5.60 5.90 6.10 12.00 8.8 3360 7.50 7.70 7.80 15.50 220 11.0 4040 9.00 9.20 9.50 18.70 220 12.0 4440 10.00 10.10 10.40 20.50 220 14.5 5320 11.50 11.90 12.50 24.40 220 16.8 6080 13.00 13.30 14.40 27.70 220 19.5 6800 14.00 14.20 16.60 30.80 220 24.0 8000 15.60 15.80 20.20 36.00 220 27.50 8400 17.00 16.00 22.90 38.90 v Volts I Amp Watts 220 4.0 1120 220 7.0 220 p Test 12 Calculated Table III-6 Test and Calculated Values of Power Angle for the Two Machine System Connected Through the Artificial Line Designed in Section II (l 0 01 Calculated 82 Calculated Calculated 12 Calculated 4.00 3.70 3.50 1.60 8.80 2240 5.00 5.30 5.30 2.30 12.90 8 3000 6.80 7.10 7.00 2.90 17.00 220 10.5 3840 8.00 8.80 9.20 4.00 22.00 220 13.5 5000 11.00 11.40 11.70 4.70 27.80 220 16.0 5760 12.40 12.60 13.90 5.60 32.10 220 20.0 7000 14.50 14.70 17.20 6.90 38.80 220 22.0 7400 15.00 15.00 18.80 7.60 41.40 220 25.0 7900 15.80 15.20 21.20 8.70 45.10 v I p Volts Amp Watts 220 4 1520 220 6 220 Test U'1 0 51 IV. CONCLUSION The analysis of three phase synchronous machines operation depends upon the constants of the machines; due to saturation effects, the constants of synchronous machines have different values for different operating conditions. The laboratory work was centered on two synchronous machines in the power laboratory. All machine constants such as armature resistance, direct and quadrature axis reactance, transient and subtransient reactance were measured for both machines. The resistance was measured at ambient temperature and corrected to operating temperature. The direct axis reactance was measured at saturated and unsaturated conditions . An artificial line was designed and built for use in the test of the validity of calculated operating conditions using these measured constants. A good agreement was obtained between tests and calculations. In power system analysis, the problems to be investigated for a student who is interested in power area are: power f~ow, power limits, stability limits (transient and steady state), fault currents and critical switching time. These problems can be studied theoretically and demonstrated in the power laboratory. Accurate synchronous machine constants .which are functions of operating conditions are needed to determine these problems. saturated and unsaturated machine operating conditions were used to demonstrate the effects of saturation on the investigation of a power system. The machine data needed to demonstrate the above problems has 52 been determined for two machines in the UMR power laboratory for use in experiments by the student. and run. Sample experiments have been set up A good agreement has been obtained between calculated and experimental results. With the two machines in the laboratory and their data obtained, and with the model transmission line and its characteristics, the laboratory demonstration and experiments can be performed for or by undergraduate classes in investigation of the problems of power systems. 53 BIBLIOGRAPHY l. Henney, Radio Engineering Handbook, McGraw-Hill Book Company: New York, 1959. 2. Terman, Radio Engineers' Handbook, McGraw-Hill Book Company: New York, 1943. 3. Kimbark, Edward W., Power System Stability, Vol. I, John Wiley and Sons, Inc: New York, 1967. 4. Kimbark, Edward W., Power System Stability, Vol. 3, Dover Publication, Inc: New York, 1968. 5. Stevenson, William D., Elements of Power System Analysis, 2nd Edition, McGraw-Hill Book Company: New York, 1962. 6. Fitzgerald, A. E. and Kingsley, Charles Jr., Electrical Machinery, 2nd Edition, HcGrav.r-Hill Book Company: New York, 1961. 7. Westinghouse Transmission and Distribution Reference Book, Westinghouse. 8. Wright, Sherwin, H., "Determination of Synchronous Machine Constants by Test", AIEE Transactions, December 1931, pp. 1331-1350. 9. Robertson, B. L., Rogers, T. A. and Dalziel, C. F., "The Saturated Synchronous Machine", Electrical Engineering, July 1937, pp. 858-863. 10. Djabir, Hamdisepen, "Saturation Effects in Synchronous Hachines", AIEE Transactions, December 1954, pp. 1349-1352. 11. clarke, Edith and Lorraine, R. G., "Power Limits of Synchronous Machines", Electrical Engineering, December 1933. 12. Djabir, Hamdisepen, "Une Methode D'analyse du Fonctionnernent de la Machine Synchrone en Regime Etabli", Revue Crenerale de l'Electricite, Tome 62, September 1952, pp. 442-448. 54 VITA Nguyen-Quang-Duoc was born on March 1, 1939 in Hanam, Viet-Nam. He attended elementary school in Hanoi and High School in Saigon. He received an Electrical Engineer Diploma from National Technical Center, Phu Tho, Saigon in 1964. Afterwards he served at the National Technical Center in the Power Laboratory. He has been enrolled in the Graduate School of the University of Missouri-Rolla since June 1971 in the program of development of National Technical Center supported by the U. s. Agency for Inter- national Development and the University of Missouri-Rolla. 55 APPENDIX A Laboratory Data of Machine A 1. Data for open circuit characteristic Speed of machine: E: If: 1200 rpm line to line voltage field current Table A-1 Measurements of No-Load Voltage in Function of Field Currents E If Amperes E E Volts If Amperes Volts If Amperes 0 215 4.4 266 6.4 40 0.5 222 4.6 269 6.6 58 1.0 228 4.8 272 85 1.5 233 5.0 276 7.0 112 2.0 238 5.2 278 7.2 138 2.5 243 5.4 282 7.4 161 3.0 248 5.6 286 7.6 182 3.5 252 5.8 288 7.8 202 4.0 256 6.0 290 8.0 208 4.2 260 6.2 Volts 0 / y 6.8 56 2. Data for short circuit characteristic I : armature current of phase a Ib: armature current of phase b I armature current of phase c a c : Table A-2 Measurements of Short Circuit Current in Function of Field Currents If (Amperes) I a (Amperes) Ib (Amperes) I c (Amperes) 0 0 0 0 0.5 5.2 4.6 5.2 1.0 9.6 9.2 9.6 1.5 14.0 13.6 14.0 2.0 18.8 18.6 18.8 2.5 23.5 23.0 23.5 3.0 28.0 27.7 28.0 3.5 32.6 32.5 32.6 4.0 37.0 37.5 37.5 57 3. Data for zero power characteristic Armature current = constant = 28.5 amperes Speed: rated speed = 1200 rpm Vt = terminal voltage (line to line) Table A-3 Measurements of Terminal Voltage in Function of Field currents at Rated Armature Current and Zero Power Factor If {Amperes) vt {Volts) 4. 5.55 6.4 7.10 7.95 144 178 200 222 Data from slip test Oscillogram from slip test gives: xd = 1.5 X q 5. Three phase short circuit test steady state short circuit current I ss = 58 amp voltage before 3 phase short circuit test v 0 C Reduction from transient envelope: = 220 volts 58 Table A-4 Measurements of Short Circuit Oscillogram 6. b.i Times (sec) Average Envelope Values (amp) 0.0050 246 164 0.0125 185 103 0.0200 143 61 0.0300 115 33 0.0325 107 27 0.0425 105 24 0.0500 96 13 0.0550 95 12 0.0650 89 6 0.075 85 3 0.0825 82 0 X Average Envelope Values - I ss Measurements of power angle The power angle was measured by using a stroboscope to read the displacement angle on the axis of the machine. This angle, the mechanical angle, was multiplied by 2 to convert to electrical degrees because a 4 pole machine was used in experiments. 59 (1) (2) air gap line open circuit characteristic short circuit characteristic zero power factor characteristic (3) {4) 300 t il (!) ~ -i Rated voltage +J s:: b {/} (!) ~ ~ +J .....-! ~ 200 ::3 u +J (]) .,..; ==' u 01 m +J .....-! 0 ~ .,..; u > +J .....-! m 0 """ .c U) ~ ·r-i ~ <J) 8 100 0 1 Figure B.1. 2 c• 3 5 6 4 Field current 7 Determination of x andxuns from Data of Machine A P 8 60 APPENDIX B Sample Calculation 1. Calculation of Potier reactance Referring to Figure B.l, select a point c on the zero power factor characteristic at rated voltage (220 volts). horizontal line cd equal in length to c'O. Draw a Through point d draw the straight line da parallel to the air gap line intersecting the open circuit characteristic at point a; draw the vertical line ab. Potier reactance is: X p X p = = Voltage drop ab per phase Zero-power-factor armature current 36 /3 X (26.5) = 0.78 ohms . The equivalent circuit is shown in Figure B.2. Figure B.2. Equivalent Circuit of Synchronous Machine 61 2. Calculation of unsaturated reactance Ef(a.g) X uns I = for Ia(SC) a (SC) 29 amperes, referring to Figure B.l, Ef(a.g) equals 170 volts. X uns = 170 y'3 29 X = 3.40 ohms. 3. Calculation of saturation factor Assuming the power factor is 0.8 lagging, the rated current is: I 29{0.8 - j0.6) a = 23.20 - jl7.40 Referring to figure B.2, for generator operation, the air gap voltage is: E 220 r E r 13 /0 + j0.78(23.20 - j17.40) = 141.84 volts. Line to line air gap voltage: = 141.84 = 246 X f3 volts. 62 Saturation factor: E (a.g) k = r E r Referring to Figure B.l k = 288 246 k = 1.17 4. Calculation of saturated reactance X X - uns s X p k 3.40 + X p - 0.78 + 0.78 1.17 3.14 ohms. 5. Calculation of transient and subtransient reactance From the Table A-4, the difference ~i X between the transient envelope and the steady state short circuit current is plotted on a semilogarithm scale as a function of time (Figure B.4). of this curve gives: ~I' = 160 amperes Lli" = 240 amperes I' Lli' + I ss I' = 160 + 82 I' = 242 amperes Extrapolation 63 and I" = !J.I" + I ss I" 240 + 82 I" 322 amperes. The RMS values of transient and subtransient current are: 242 Iill.ts 12 I' 171 amperes In 322 RMS RMS I" RMS 12 = 227 .. 7 amperes. The transient and subtransient reactances are: v x• = d oc IFMS x• = 220!13 171 d x• = 0.742 ohms d v oc x" = I" d PMS XII d = 220/13 227.7 x" = 0.56 ohm d 64 400r-------------------~----------------~ 3QQ ' f.ll ' .Q) ~ (J) \ ~ ' \ c \ \ \ ·r-1 4-l \~ c<l) H H 200 u==' 100 0.05 Time in seconds Figure B.3. 0.10 Symmetrical Short Circuit Transient Envelope 65 ~I" 200 ~I' 100 Ul <D ~ Q) ~ ~ ·.-I .j.J s:: Q) ~ >-I 8 10 .OS Time in seconds Figure B.4. Ai X on Semi-logarithm Scale 66 or xd = 0.168 p.u xd 0.127 p.u on machine base. 6. Calculation of power angle a. One machine connected to infinite bus The machine was operated as a synchronous motor, and the excitation was adjusted to obtain unity power factor. The machine resistance was The power-angle in this case is: neglected. o = tan -l X I .....s_ (B.l) vt 15 amperes for I Vt The angle c 110/1:3 volts is 28° 50. The same calculation gives the results in Tables III-1 and III-2. When the machine was connected to an infinite bus through an artificial line, the x of the line. b. q is the sum of x q of the machine and reactance Tables III-3 and III-4 give the results of this case. Two machine system In a two machine syst~, one machine was operated as a generator 67 and one as a motor. power factor. line. The motor excitation was adjusted to obtain unity The two machines were connected through an artificial Refer to Figure III.l8b. = x tan- 1 cos -1 q1 rcos<f> vt + xqlisin¢ X I e vt The power-angle between the two internal voltages is: 0 = 12 81 + a. + 62 for and vt = 220 voltsj/3 I = 20 Amperes 01 = 14.70° a. = 6.9° c12 = 38.8 0 When the two machines were connected directly, with no line impedance, the angle a. equals zero. Tables III-5 and III-6 give results of these calculations.