ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 31 Participation Factors Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Participation Factors While doing model reduction, we would like to know the modes which are significant. Participation of state variables in various modes. Eigen vectors do not carry that information, since the components have different dimensions. Need a dimensionless measure. Participation factors provide this. Uses entries of Right and Left eigen vectors.

Participation Factors (contd) It was shown (previous lecture) that is entry in A matrix. is the participation factor and measures the relative participation of state variable in the mode. Since measures “activity” of in mode and “weighs” the contribution of this activity to the mode, . measures the net participation. is dimensionless. Because of normalization:

Numerical Example If the eigen vectors are NOT normalized, then

Numerical Example (contd) k identifies row and state variable i identifies column and eigen value The entries are: Normalizing each column to make highest entry =1

Numerical Example (complex eigen values) Using MATLAB Since the right and left eigen vectors are not normalized, we use the formula

Numerical Example (contd) For complex eigen values, we use product of magnitude to normalize Thus

Uses of Participation Factor Study of oscillations. Which machines contribute? Where to put the power system stabilizer? How to tune the power system stabilizer. A vast literature exists on these topics (Book by Graham Rogers on P.S. oscillations). Model reduction – Selective Modal Analysis (SMA) Coherency analysis leading to dynamic equivalents.

Example 3M/C, 9 Bus System Two axis model. State Vector d- and q-axis stator currents arranged as is the angle of the slack bus; . The subscript LF stands for the load flow variables.

Example (contd) After eliminating stator algebraic variables ,and re-ordering algebraic variables, define Load Flow Jacobian Algebraic Jacobian

Example (contd) Equations are arranged so that A. The dynamic state variable equations. B. The stator algebraic equations. C F are the transmission network algebraic equations consisting of C. The real power balance equation for the slack bus. D. The reactive power balance equations at all the gen. (P-V buses) E. The reactive power balance equations for the load buses. F. The real power balance eqns at all the buses except the slack bus. This formulation is useful in the study of Bifurcation phenomena. Eliminate all algebraic equations to get Alternatively we can eliminate before getting

Example (contd) If is eliminated from the set of equations we get . if loads are constant power. Otherwise, diagonal entries of get modified to get .

Eigen Values The eigen values of for nominal operating conditions: Stable system because damping due to damper windings etc. For each eigenvalue the participation factors are computed and each column is normalized as discussed earlier. Eigenvalues of the 3-Machine System Electromechanical modes Two zero eigen values, because of zero damping.

Eigen Values and Their Participation Factors * Only P.F.’s >0.2 are included * * * * * * * Electromechanical modes * * Exciter modes

Damping in Electromechanical Modes Observations The electromechanical modes have poor damping X X

Damping of exciter Modes Damping good for exciter modes but poor for electromechanical modes ( = 0.056, 0.023). In oscillation studies, we wish to find how eigen values move in complex plane as load at a bus or system load increases.

Effect of Loading The real or reactive loads at a particular bus/buses are increased continuously. At each step, the initial conditions of the state variables are computed, after running the load flow, and linearization of the equations is done. Ideally, the increase in load is picked up by the generators through the economic load dispatch scheme. To simplify matters, the load increase is allocated among the generators (real power) in proportion to their inertia.

Effects of Loading (contd) In the case of increase of reactive power, it is picked up by the PV buses. The matrix is formed, and its eigenvalues are checked for stability. Also are computed. The step-by-step algorithm is as follows: Step-by-step algorithm Increase the load at bus/buses for a particular generating unit model. If the real load is increased, then distribute the load among the various generators in proportion to their inertias.

Step by Step Algorithm (contd) Run the load flow. Stop, if the load flow fails to converge Compute the initial conditions of the state variables, as discussed in an earlier lecture. From the linearized DAE model, compute the various matrices. Compute and the eigenvalues of If is stable, then go to step (1). If unstable, identify the states associated with the unstable eigenvalue(s) of using the participation factor method, and go to step (1).

Modes of instability As load is increased, in some cases the electromechanical mode goes unstable (oscillations, angle instability. In other instances, the exciter mode goes unstable (voltage instability). Angle and voltage instability co-exist in a large complex system. As we increase load we monitor eigenvalue movement and “track” the critical eigenvalue i.e. one which moves towards the right half plane. Also “track” Algebraic Jacobian Load Flow Jacobian

Critical Eigen Value Movement IEEE Type I Exciter, Non-zero damping * ** * System goes unstable at ** Algebraic Jacobian is singular

Critical Eigen Value Movement (contd) Static Exciter, * ** Electromechanical mode goes unstable * becomes singular **

Effects of Type of Load Voltage Dependent Loads: is nominal voltage. and are the nominal real and reactive powers. are the load indices. Three types of load are considered: Constant Power Type Constant Current Type Constant Impedance Type

Eigenvalues for Different Types of Load (nominal op. point) Comments: System is stable for all types of loads.

Eigenvalues for Increase in System is unstable for constant power type load. Stable for other types of load.

Eigenvalues for Increase in Unstable for constant power and constant current. Stable for constant impedance.