ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 28 Multi-machine internal–node model Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Multi machine internal-node model The Structure Preserving Model is described by DAE. Analysis of DAE systems is somewhat different. Simplification of DAE system consists in eliminating the P-Q buses by assuming a constant Impedance Model for the loads. Network Reduction (also called Kron reduction) used. The final Math Model is Differential Equations only. Forms the basis of single Machine Infinite bus (SMIB) formulation.

Augmented Y matrix Consider the network augmented by with buses n+1,…,n+m (Generator internal buses). Write the augmented bus admittance matrix as:

Augmented Y matrix (contd) is the network admittance matrix.

Converting Loads Define injected current as From Ohm’s Law since Negative sign since loads are treated as “injected” loads.

New Augmented Y Matrix (By absorbing loads) Add this to the diagonal elements of the matrix to make it . The modified augmented matrix becomes:

New Augmented Y Matrix Passive Portion of New Augmented matrix is: All elements are passive. There is “injection” only at buses n+1,…,n+m. These will be retained and others eliminated.

Network Reduction The Network equations for the new augmented network: Network buses can be eliminated, since there is no current injections at these buses.

Network Reduction (contd) From the second equation Substitute in first equation The elements of are . Since the network buses have been eliminated, we may renumber the internal nodes as 1,…,m for ease of notation.

Multi machine Model The dynamic equations are called “swing” eqns.

Multi machine Model (contd) Define Since is a function of the , the D.E’s can be integrated by any numerical algorithm. Now where

WSCC 3-machine, 9-bus system; the value of Y is half the line g

Load Flow Results Buses 2&3 are P-V Bus 1 is a Slack Bus Load-Flow Results of the WSCC 3-Machine, 9-Bus System       Buses 2&3 are P-V Bus 1 is a Slack Bus Buses 4-9 are P-a Buses

Example 3 Machine, 9 Bus System

Example (contd)

Example (contd)

Example (contd) is obtained by eliminating nodes 1 …9 Elements of are . Note: These conductances and susceptances do not correspond to any physical wires or lines but just fictional ones because of network reduction.

Summary of Multi-machine simulation Two axis machine model with IEEE Type 1 Exciter. Network with voltage dependent load. DAE formulation. Initial Conditions. DAE solution by simultaneous implicit method. Solution of non-linear algebraic equations. DAE system with flux decay model and fast exciter. Rest as in (I) (II)

Summary of Multi-machine simulation (III) DAE system with classical model and voltage dependent loads. DE model by treating loads as constant impedances. Numerical Example (IV) (V)

Numerical Example 1 2 ~ ~ 3

Numerical Example (contd)

Numerical Example (contd)

Numerical Example (contd) From power flow

Numerical Example (contd)

Numerical Example (contd) From power flow

Numerical Example (contd)

Numerical Example (contd)

Simulation Relative rotor angles with initial condition at [0;377;0;377] δ2,1 (rad/s) time (seconds)

Simulation Frequencies with initial condition at [0;377;0;377] ω (rad/s) time (seconds)