Placing Controllers in a System

Overview of Class So Far…

General Introduction Deregulation Traditional approaches to control Static devices

Type of problems

Steady State Transient Stability Inter-Area Oscillations Subsynchronous Resonance Voltage Stability

Introduction to FACTS

Detailed analysis of devices Thyristor controlled inductor SVC Statcom TCSC

System Modeling

Simplified models for use in system simulations and analysis Statcom TCSC (see Reference [1]) UPFC

Set up system equations to include FACTS devices

Get block diagrams or differential equations for device

Define device states Define device inputs Express device model in terms of existing

system states & device states Augment system equations

Use these devices to fix system problems!

OK, Say you work at an ISO & are in charge of ensuring system reliability. You've had 5 major blackouts in the last 3 years that have involved the propagation of problems from one part of the system to another. The utility members are convinced that the addition of a FACTS device or two will solve the problems & they even agree to pay!

Now what do you do? . . .

Brainstorming Activity: What things do you need to worry about? Break into groups of 2 or 3 Take 3 minutes & write down as many things as you

can No criticism allowed, Go for variety, Go for quantity not

quality

Questions:

Where do I put it? Controller Location What should it do? Controller Function

Things to consider:

More than one problem More than one system condition More than one mode More than one tool

Problems

Steady State Insure operating point is within acceptable limits

Interarea Oscillations Damp eigenvalues

Transient Stability Provide sufficient synchronizing and damping torque

Subsynchronous Resonance Avoid resonance frequencies

Voltage Stability Stabilize eigenvalues and avoid bifurcations

Interarea Oscillation Mitigation

Analysis Tools (Mostly Linear) Controllability and Observability Participations Sensitivities Power Oscillation Flows

Linearized System

x’ = Ax + Bu y = Cx + Du

Eigenvalues are i, i = 1, nstates Right Eigenvectors ri , R is matrix of ri's Left Eigenvectors (rows) i, L is matrix of i's L = R-1

Perform variable transformation to Jordan Form

x = Rz (Inverse transform z = Lx) Substituting into system equations . . .

Rz’ = AR z + B uy = CR z + Du

Multiply through by L = R-1

LRz’ = z’ = LAR z + LB u

y = CR z + Du

Controllability and Observability

Modal controllability matrix = LB tells how strongly connected the inputs (u's) are to

each of the modes Modal observability matrix = CR

tells how well we can measure or "see" each mode in the outputs (y's)

Participations

Connection strengths between modes and states

General participation pi hk = ri h lhk

link between ith (obs.) & kth (con.) states through mode h

Participation Factors pi h = ri h lh i link between mode h & state i

Eigenvalue Sensitivities

i’ = i/p

p some parameter of the system Tells how easily we can move an eigenvalue by

changing a parameter In general, i' = li A' ri

Sensitivities are also related to participations

pihk = h’ for p = aki (element of A) pih = h’ for p = akk (diagonal element)

u(t) x(t)

p

rest of system

xi

xk

Sensitivity with Controllers

The "Hybrid System"

The Power System

x’ = Ax + Bu y = Cx

assume no direct connection between y & y2

The controller transfer functionF(s,p) is the only place p shows up

Sensitivity for Hybrid System

i’ = i/p = li B {/p [F(s,p)]|s=i } C ri

related to the controllability and observability measures and to the controller transfer function (see Reference [5])

Uses of Sensitivities

Location of controllers Magnitude of ’ tells the displacement of the

eigenvalue if gain is equal to 1 Large magnitude indicates controller is a good

candidate for improving a mode Phase ’ of gives the direction of the

eigenvalue's displacement in the imaginary plane

Introduce devices likely to influence these characteristics

Simulations and Trial & Error

Tuning of a controller . . .

Adjust the phase compensation of the controller so that ’ has a phase of 180 degrees with controller in place

Adjust the gain of the controller to achieve the desired amount of damping

Power Oscillation Flows

Map where oscillations caused by a single eigenvalue appear in the system

n

x(t) ck ekt rk

k = 1

ck is the initial condition in Jordan Space

The idea is to choose ck's so that only one mode is perturbed, i.e. ci = 1 and ck= 0 for all k not equal i

then x(t) = ri eit

this solution can then be propagated through the system equations to find the power flow on key lines (or some other variable for that matter)

Placing a FACTS device using participations, sensitivities, etc.

Simple & Fast Detailed & More Accurate

Transient Stability and FACTS

Usually concerned with providing adequate damping and synchronizing torque

Often design using linear techniques and test with the nonlinear system

Nonlinear Methods

Normal forms of vector fields for extending the linear concepts to the nonlinear regions.

Second-order oscillations, participations, controllability & observability

Energy Methods

Lyapunov-based methods for determining stability indices

Tracking of energy exchanges during a disturbance

Control Strategy

Determine weak points in system Poorly damped oscillations Lack of synchronizing torque Large power swings Large energy exchanges Short critical clearing times Multi-machine instabilities

References

[1] Paserba, J. J., N. W. Miller, E. V. Larsen, and R. J. Piwko "A Thyristor Controlled Series Compensation Model for Power System Stability Analysis" IEEE Trans. on Power Delivery, Vol. 10?, (July 1994): 1471-1478.

[2] Chan, S. M. "Modal Controllability and Observability of Power-System Models" International Journal of Electric Power and Energy Systems, Vol. 6, No. 2, (April 1994): 83-89.

[3] Rouco, L., and F. L. Pagola "An Eigenvalue Sensitivity Approach to Location and Controller Design of Controllable Series Capacitors for Damping Power System Oscillations" IEEE-PES 1997 Winter Power Meeting, Paper No. PE-547-PWRS-0-01-1997.

[4] Ooi, B. T., M. Kazerani, R. Marceau, Z. Wolanski, F. D. Galiana, D. McGillis,

and G. Joos "Mid-Point SIting of FACTS Devices in Transimssion Lines" IEEE-PES 1997 Winter Power Meeting, Paper No. PE-292-PWRD-0-01-1997.

[5] Pagola, F. L., I. J. Perez-Arriaga, and G.C. Verghese "On Sensitivities, Residues, and Participations: Application to Oscillatory Stability Analysis and Control" IEEE Trans. on Power Systems, Vol. 4, No. 2, (February 1989): 278-285.

[6] Messina, A. R., J. M. Ramirez, and J. M. Canedo C. "An Investigation on the use of Power Systme Stabilizers for Damping Inter-Area Oscillations in Longitudinal Power Systems" IEEE-PES 1997 Winter Power Meeting, Paper No. PE-492-PWRS-0-01-1997.

[7] Zhou, E. Z. "Power Oscillation Flow Study of Electric Power Systems" International Journal of Electric Power and Energy Systems, Vol. 17, No. 2, (1995): 143-150.