Aulas Teóricas

Course Content -- Lesson by Lesson

COSE -- Course Content 




Quantities to control

Sensors and actuators

Control mechanisms

Computer-based modern control


Simplest control problems in EPS 


Simplest control problems in EPS

Single generator under no load

Voltage control

Frequency control

AVR Automatic Voltage Regulator

Physical system

Block diagrams


Automatic Voltage Regulator 



Root locus

Series compensation

Parallel compensation

Nonlinearities and linearizations


AVR -- continuation 



Block diagram for AVR

Issues of input/output in frequency and time

Input: step, ramp

Output: oscillatory or not, critically dampened, unstable

Rise time, overshoot, settling time, etc

Multiple frequencies

Frequency of oscillation and root location


Compensation schemes and linearization process


Compensation: series compensation and shunt compensation

Historical devices for compensation 

Introduction of zeros and derivative control

PID as a universal electronic control

Understanding derivative control as anticipated information

Use of V versus deltaV

Successive linearization


Nonlinearities and control action  


Sources of nonlinearity: magnetization curve for iron, saturation, limits and bounds

Generator loading: effects on model, KG and TECHNOLOGIES

Where to introduce disturbances or perturbations in the model

Generator as a PV bus

Rotor dynamics: voltage and power; electrical and mechanical

Imbalance of turbine output and electric demand

Spinning masses as energy storage 


Power regulation, frequency regulation 



Power imbalance for the couple turbine and demand

Storage: mechanical or kinetic of spinning masses

Amount of kinetic energy: ½Iw²

Inertia moment: dI=r²dM

Energy measured in s? Meaning of the inertia constant H

Development of the corresponding TF

Node model for the power equilibrium

Turbine TF: a simple TF with a single time constant

More of the physical model: a hydraulic regulator

Reason to include a time constant for that

Order of magnitude

The control system

Negative feedback -- review of why negative

PID and proportional control

R parameter: effects of its value

Pref, a reference entry for extra control


Frequency regulation 


Load elasticity

Regulation energy

Steady state analysis

Interpretation of frequency deviations

Secondary control or integral control

Secondary control 


Need for and role of secondary control

Integral control

Effect of gain of integral feedback

New characteristic eq

Stability issues


More than one generator in one plant 


Voltage control

Frequency control or power control

Primary control is proportional

Secondary control is integral

Tertiary control is set point control by the Dispatch

More than one generator in one plant

Presentation of the problem of interconnecting plants


The problem of interconnection 


Why interconnection has no dynamics

How to model interconnection

Minimal realization

Simplest model for two-area interconnection

Analysis of a two equal areas interconnected

Application of superposition principle

Characteristic equation

Roots and solution types

Root locus

Oscillatory responses of increasingly higher frequencies

Need for a new model


Analysis of a two equal areas interconnected 


Analysis of a two equal areas interconnected


Problem 3


Analysis in time domain 


An introduction to time-domain analysis

Standard form

Symbolic block diagram

Concepts and notation

Transfer matrix

Characteristic polynomial

Poles are eigenvalues


State space representation 


Modal matrix

Normal form

J is a diagonal of eigenvalues

Illustration for mode decoupling


Linear, constant matrix feedback

New controlled system

(A,B) controllable

Plant matrix with feedback

Where are the other controls?

Solution and exp(At)

x(t) as a convolution integral

Controllable canonical form


Controllability canonical form and optimal control 


Controllability canonical form

Its dual: observability canonical form

Importance of coefficients a’s

Role of b’s


Concepts of Optimal control

Linear quadratic regulator

Ricatti eq

The engineering problem: choosing Q’s and R’s

Limitations to the use of Optimal Control

Systems in triangle: two generators and one load

Load as a unit step at a load bus

"Complete model" for the generator

Control problem vs transient response

Linear analysis vs nonlinear analysis


Voltage regulator and frequency regulator together


"Complete model" for the generator

Control problem vs transient response

Linear analysis vs nonlinear analysis

Revision of triangle systems

Linearization formulas

AVR+AGC coupling

Possible instabilities

Stabilization by coupling: PSS

Input Output and TF for PSS

Voltage regulator and frequency regulator together

Nodal demand is an electric quantity

Role of delta and emf

Integration of the models

Simulation of the slow oscillations in interconnection power

The role PSS Power System Stabilizer

Models for PSS

The role of tertiary control


Economic Dispatch 


The role of tertiary control

Non-automatic control

Costs curves: total, marginal, efficiency, average

Key aspects of the curves

Lossless dispatch

Problem Mathematical formulation

Two ways to solve the problem:

Using a Lagrangean approach and

Using variable substitution

The results: equal incremental costs

How this can be carried out by bisection



ED with losses and bounds 


Why introduce losses into the problem

The need for penalty factors

Losses as a function of decision variables

Math formulation and results

How the problem can now be solved using the same techique

The need for ityeraions dure to PF’s

How to compute losses and PFs:

Bs methods and power flow methods

Fundamentals of those methods

Mathematical derivations


Handling inequality constraints and OPF 


Handling inequality constraints and OPF

Examples of inequality constraints in ED problems

Understanding why constraints must enter the active set and why they must leave

Kuhn Tucker conditions

The interpretation of the new added Lagrange multipliers and why the constraints must be relaxed

Introducing the OPF problem


Optimal Power Flow 


Objective of the OPF problem

Variables: what are they? (decision, dependent, auxiliary variables)

Constraints: equality and power flow equations

Inequality and transmission limitations

Simple bounds on the variables: powers, taps, ...

A first approach: reduced gradient

How to set it up and how to solve it

What is the reduced gradient an how to use it to update the decision vector


OPF by Newton and by LP 


Newton for OPF

Approach, difficulties and results

Advantages and disadvantages

LP for OPF

Successive linearization

OPF for planning and operations

The issue of system securityEvaluation of security costs

OPF leads to the UC problem


Dynamics, state, state equations and the UC problem 


Concepts of dynamics and state

State equations, state transitions, decision space

Combinatorial nature of UC problem

Examples of unit dynamics

Grid for Dynamic Programming

Nodal costs and arc costs

Bellman optimality principle

Backward DP and forward DP


Thermal units and hydro resources 


State of a thermal unit

States for operation, maintenance, ...

The importance of operation costs and operation constraints for state definition

Examples of a state transition diagram for a unit with variable startup costs, minimum up time, MUT, and minimum down time, MDT

Hydro resources

Cascade systems

Operation curves: head, flow, power

Reservoir dynamics, river dynamics, time constraints

Spill models

Minimal cost network flow problems

Nodal equations

Arc quantities: generation, pump, storage, spill

MCNF as a special LP

Basic, nonbasic, marginal true values

How to obtain these values for a given tree


Dynamic Programming applied to UC 


State transition diagram

Multiple states, multiple transitions

Problem formulation

Solution approach by Dynamic Programming

Creation of a DP grid

Node costs (or profits) and arc costs

Criterion for operations

Optimal solution and computational efficiency


Hydrothermal coordination and scheduling of a complex dynamic resource 


Hydrothermal coordination

Empiric principles for coordination

Hydro and load peak-shaving

Example of hydrothermal coordination

Marginal value of water and total value of water

Relationship between gamma and lambda


Resource coordination and revisions 


Review of primal problem and dual problem in the context of resource scheduling

Economic interpretation

Review of concepts of reserve and capacity: sources and requirements

Costs for reserve and capacity

Problem 12

Water value -- marginal costs and non-marginal

Water value as a function of reservoir location and reservoir volume

Water value for a weekly scheduling: constant value, non-constant, zero value -- interpretation in view of water constraints

Water value interaction between short-term scheduling and medium-term scheduling

Qualitative differences with respect to investment planning