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1、Power System Dynamics - Postgraduate Course of Tsinghua Univ. Graduate School at Shenzhen,NI Yixin Associate Professor Dept. of EEE, HKU ,Introduction,0.1 Requirements of modern power systems (P. S. ) 0.2 Recent trends of P. S. 0.3 Complexity of modern P. S. 0.4 Definitions of different types of P.
2、S. stability 0.5 Computer-aid P. S. stability analysis 0.6 Contents of our course,Introduction (1),0.1 Requirements of modern power systems (P. S. ) Satisfying load demands (as a power source) Good quality: voltage magnitude, symmetric three phase voltages, low harmonics, standard frequency etc. (as
3、 a 3-phase ac voltage source) Economic operation Secure and reliable operation with flexible controllability Loss of any one element will not cause any operation limit violations (voltage, current, power, frequency, etc. ) and all demands are still satisfied. For a set of specific large disturbances
4、, the system will keep stable after disturbances. Good energy management systems (EMS),Introduction (2),0.2 Recent trends of P. S. Systems interconnection: to obtain more benefits. It may lead to new stability issues ( e.g. low-frequency power oscillation on the tie lines; SSR caused by series-compe
5、nsated lines etc. ). Systems are often heavily loaded and very stressed. System stability under disturbances is of great concern. New technology applications in power systems. (e.g. computer/ modern control theory/ optimization theory/ IT/ AI tech. etc. ) Power electronics applications: provides fle
6、xible controller in power systems. ( e. g. HVDC transmission systems, STATCOM, UPFC, TCSC, etc.),Introduction (3),0.3 Complexity of modern P. S. Large scale, Hierarchical and distributed structure, Non-storable electric energy, Fluctuate and random loads, Highly nonlinear dynamic behavior, Unforesee
7、n emergencies, Fast transients which may lead to system collapse in seconds or minutes, Complicated control and their coordination requests. - Modern P. S. is much more complicated than ever and in the meantime it plays a significant role in modern society.,Introduction (4),Some viewpoints of Dr. Ku
8、ndur (author of the ref. book ): - The complexity of power systems is continually increasing because of the growth in interconnections and use of new technologies. At the same time, financial and regulatory constrains have forced utilities to operate the systems nearly at stability limits. - Of all
9、the complex phenomena on power systems, power system stability is the most intricate to understand and challenging to analyze. Electric power systems of the 21 century will present an even more formidable challenge as they are forced to operate closer to their stability limit.,Introduction (5),0.4 D
10、efinitions of different types of P. S. stability P. S. stability: the property of a P. S. that enable it to remain in a state of operating equilibrium under normal operating conditions and to return to an acceptable state of equilibrium after being disturbed. Classification of stability Based on siz
11、e of disturbance: large disturbance stability ( transient stability, IEEE): nonlinear system models small disturbance/signal stability ( steady-state stability, IEEE): linearized system models The time span considered: transient stability: 0 to 10 seconds mid-term stability: 10 seconds to a few minu
12、tes long-term stability(dynamics): a few minutes to 1 hour,Introduction (6),0.4 Definitions of different types of P. S. stability (cont.) Classification of stability (cont.) Based on physical nature of stability: Synchronous operation (or angle) stability: insufficient synchronizing torque - non-osc
13、illatory instability insufficient damping torque - oscillatory instability Voltage stability: insufficient reactive power and voltage controllability Subsynchronous oscillation (SSO) stability insufficient damping torque in SSO,Introduction (7) 0.5 Computer-aid P. S. stability analysis,Introduction
14、(8) 0.6 Contents of the course,Introduction Part I: Power system element models 1. Synchronous machine models 2. Excitation system models 3. Prime mover and speed governor models 4. Load models 5. Transmission line and transformer models Part II: Power system dynamics: theory and analysis 6. Transie
15、nt stability and time simulation 7. Steady-state stability and eigenvalue analysis 8. Low-frequency oscillation and control 9. *Voltage stability 10. *Subsynchronous oscillation 11. Improvement of system stability Summary,Part I Power system element models,Chapter 1 Synchronous machine models (a),Ch
16、apter 1 Synchronous machine (S. M.) models,1.1 Ideal S. M. and its model in abc coordinates 1.1.1 Ideal S. M. definition Note: * S. M. is a rotating magnetic element with complex dynamic behavior. It is the heart of P. S. It * It provides active and reactive power to loads and has strong power, freq
17、uency and voltage regulation/control capability . * To study S. M., mathematic models are developed for S. M. * Special assumptions are made to simplify the modeling.,Chapter 1 Synchronous machine (S. M.) models,1.1.1 Ideal S. M. definition (cont.): Assumptions for ideal S. M. Machine magnetic perme
18、ability (m) is a constant with magnetic saturation neglected. Eddy current, hysteresis, and skin effects are neglected, so the machine is linear. Symmetric rotor structure in direct (d) and quadratic (q) axes. Symmetric stator winding structure: the three stator windings are 120 (electric) degrees a
19、part in space with same structure. The stator and rotor have smooth surface with tooth and slot effects neglected. All windings generate sinusoidal distributed magnetic field.,Chapter 1 Synchronous machine (S. M.) models,1.1.2 Voltage equations in abc coordinates Positive direction setting: dq and a
20、bc axes, speed direction Angle definition: Y directions for abcfDQ windings i directions for abcfDQ u directions for abcfDQ (uD=uQ=0),Chapter 1 Synchronous machine (S. M.) models,1.1.2 Voltage equations in abc coordinates (cont.) Voltage equations for abc windings: where p= d / dt, t in sec. rabc: s
21、tator winding resistance, in W. iabc : stator winding current, in A. uabc: stator winding phase voltage, in V. yabc: stator winding flux linkage, in Wb. Note: * pyabc: generate emf in abc windings * uabciabc: in generator conventional direction. * iabc yabc: positive iabc generates negative yabc res
22、pectively,Chapter 1 Synchronous machine (S. M.) models,1.1.2 Voltage equations in abc coordinates (cont.) Voltage equations for fDQ windings: rfDQ: rotor winding resistance, in W. f: field winding, D: damping winding in d-axis, Q: damping winding in q-axis. ifDG, ufDG, yfDG: rotor winding currents,
23、voltages and flux linkages in A, V, Wb. Note: * uD=uQ=0 * ufDQifDQ: in load convention * ifDG yfDG: positive ifDG generates positive yfDG respectively * q-axis leads d-axis by 90 (electr.) deg.,Chapter 1 Synchronous machine (S. M.) models,1.1.2 Voltage equations in abc coordinates (cont.) Voltage eq
24、uations in matrix format: where before iabc is caused by generator convention of stator windings.,Chapter 1 Synchronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates,Chapter 1 Synchronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates (cont.) In Flux
25、linkage eqn.: Lij ( i, j = a, b, c, f, D, Q ): self and mutual inductances, L11 : stator winding self and mutual inductance , L22 : rotor winding self and mutual inductances, L12 , L21 : mutual inductances among stator and rotor windings , y, i : same definition as voltage eqn Note: * Positive iabc
26、generates negative yabc respectively. * The negative signs of iabc make Laa, Lbb, Lcc 0.,Chapter 1 Synchronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates (cont.) Stator winding self/mutual inductance (L11) Stator winding self inductance (Laa, Lbb, Lcc) Laa: reach max d-a
27、aligning (when qa=0, 180) reach min d-a perpendicular (when qa=90, 270) Laa qa: sin-curve, with period of 180 (LsLt0, for round rotor: Lt=0) (See appendix 1 of the text book for derivation),Chapter 1 Synchronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates (cont.) Stator wi
28、nding self/mutual inductance (L11) Stator winding mutual inductance Lab: reach max |.| when qa= -30, 150 reach min |.| when qa= 60, 240 Laa qa: sin-curve, with period of 180 (MsLt0, for round rotor: Lt=0) (See appendix 1 of the text book for derivation),Chapter 1 Synchronous machine (S. M.) models,1
29、.1.3 Flux linkage equations in abc coordinates (cont.) Rotor winding self/mutual inductance (L22) Rotor winding self inductance (constant: why?) Lff = Lf = const. 0 LDD = LD = const. 0 LQQ = LQ = const. 0 Rotor winding mutual inductance LfQ = LfQ = 0, LDQ = LQD = 0 LfD = LDf = MR = const. 0,Chapter
30、1 Synchronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates (cont.) Stator and rotor winding mutual inductance (L12; L21 ) abcf: (Mf=const.0, period: 360, max. when d-abc align) abcD: similar to abcf, MfMD0 abcQ:(MQ=const.0, period: 360, max. when q-abc align),Chapter 1 Sync
31、hronous machine (S. M.) models,1.1.3 Flux linkage equations in abc coordinates (summary) Time varying L-matrix : related to rotor position L11 (abcabc): 180 period; L12, L21(abcfDG): 360 period. Non-sparse L-matrix: most mutual inductances 0 L-matrix: non-user friendly, lead to abc dq0 coordinates!,
32、Chapter 1 Synchronous machine (S. M.) models,1.1.4 Generator power, torque and motion eqns. Instantaneous output power eqn. (Pe in W) Electromagnetic torque eqn. (Te in N-m, q in rad.),Chapter 1 Synchronous machine (S. M.) models,1.1.4 Generator power, torque and motion eqns. (cont.) Rotor motion eq
33、ns. According to Newtons law, we have: where Tm: input mechanical torque of generator (in N-m) Te: output electromagnetic torque (in N-m) wm/qm: rotor mechanical speed/angle (in rad/s, rad.) we/qe: rotor electrical speed/angle (in rad/s, rad.), J: rotor moment of inertia (also called rotational iner
34、tia) J= Kg-m2 In the manufacturers handbook, J is given by GD2, in ton-m2. GD2 (ton-m2) 103/4 J (Kg-m2).,Chapter 1 Synchronous machine (S. M.) models,1.1.4 Generator power, torque and motion eqns. (cont.) Rotor motion eqns. (cont.),Chapter 1 Synchronous machine (S. M.) models,1.1.5 Summary of S. M.
35、model in abc coordinates and SI units: 6 volt. DEs. (abcfDQ): 6 flux linkage AEs. (abcfDQ): 2 rotor motion eqns. (w, q): Totally 14 eqns. with 8 DEs and 6 AEs. 8th order nonlinear model. 8 state variables are: y (61) and w, q (related to 8 DEs) Totally 19 variables: u: 4 (vD=vQ=0), i: 6, y: 6, plus
36、(Tm, w, q). If 5 variables are known, remaining 14 variables can be solved. Usually uf and Tm are known (as input signals), 3 network interface eqns. (3 vabc-iabc relations from network) are known.,Chapter 1 Synchronous machine (S. M.) models,1.1.5 Summary of S. M. model in abc coordinates (cont.) R
37、equest of transformation of S. M. model: abc to dq0 coordinates: Parks transformation, Parks eqns. per unit system and S. M. pu model Reduced-order practical models: - Neglect stator abc winding transients (8th order 5th order). It can interface with network Y-matrix in Aes. - Introduce practical variables (Edq, E”dq, Ef etc.),