ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 23 Differential Algebraic Equation (DAE) model Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Review of DAE model (Polar Form) Stator currents eliminated. has dimension of power in p.u. Load flow equations are a subset of Useful in studying dynamic voltage stability, bifurcation phenomena, etc. (2) is still the set of network A.E 7m D.E + 2n A.E

Stator Equations (Rect. Form) = voltage in System Reference The Stator A.E’s in polar form now become: and Can solve (3) & (4) for

Stator Equations (contd) Solve for The Dynamic Equations are (as before): is in Rectangular co-ordinates.

Stator A.E (contd)

Stator A.E (contd) Agrees with Lec 22/9 !

Network equations (Current Balance) is Bus Admittance matrix. 1 m+1 i + - n m + -

Network Equations (contd) Apply KCL at each node. (GEN) (Load) In compact form

DAE Model In Commercial Programs the substitutions are done within the code. (5) are the network equations. Substitute (2) in (1) and (3) to get

Numerical Example 1 2 ~ ~ P-V SLACK 3 P-Q

Numerical Example (contd) is network (bus) admittance matrix. Elements of are is negative of admittance between buses i & k. is sum of all admittances connected to bus i.

Numerical Example (contd) Admittances are: 1 2 3

Numerical Example (contd) The stator algebraic equations are (Assuming ) (1) (2) (3) (4) (5)

Total Model Separate (3) into real and imaginary parts. Substitute for from (2) into (1) and (3) to get:

Reference frames We have seen Orthogonal matrix T(δ). Same holds for voltage components.

Reference Frames (contd) Machine Reference Network Reference Frame TRANSFORMATION OF VARIABLES

Reference Frames (contd) From earlier derivation Hence the block diagram.

Block Diagram Interface is the 2 x 2 Transformation matrix T.

IEEE Convention Not an industry convention. Many research papers from academia use this. To convert from industry to IEEE convention one has to make careful substitutions. We stick to industry convention.

Load Representation Power Form is a dimensionless variable, is the nominal voltage, depend on type of load. Real and Reactive powers consumed when For simplicity assume

Load Representation (contd) For computational purposes consider: α = β = 2 Constant Impedance Load (Z) α = β = 1 Constant Current Load (I) α = β = 0 Constant Power Load (P) Alternative Polynomial Model (ZIP Model) – Easily incorporated in power balance form of DAE model. Need changes if DAE model is in current balance form.

Load Representation (contd) Easily derived from:

Summary of DAE Models in Power Balance Form D.E are the same. Power Balance form have Load flow equations as a subset. Hence bifurcation phenomena can be studied easily. Current Balance form have the Network matrix along with its sparse structure. Network changes handled easily.