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EL7047: Risk and Reliability in Electrical Energy Systems

This document introduces optimization concepts and their applications in electrical energy systems. It discusses conventional versus smart grids, including generation, transmission, distribution and control aspects. It covers basic optimization topics like objectives to minimize or maximize, examples of applications in power systems, and mathematical optimization definitions. Key optimization techniques presented include constrained optimization, Kuhn-Tucker conditions, shadow prices and dual variables. Homework is assigned on simplex, branch and bound, and using Newton's method for load flow analysis.

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Cristofer Rivera
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109 views22 pages

EL7047: Risk and Reliability in Electrical Energy Systems

This document introduces optimization concepts and their applications in electrical energy systems. It discusses conventional versus smart grids, including generation, transmission, distribution and control aspects. It covers basic optimization topics like objectives to minimize or maximize, examples of applications in power systems, and mathematical optimization definitions. Key optimization techniques presented include constrained optimization, Kuhn-Tucker conditions, shadow prices and dual variables. Homework is assigned on simplex, branch and bound, and using Newton's method for load flow analysis.

Uploaded by

Cristofer Rivera
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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EL7047: Risk and reliability in

Electrical Energy Systems


Introduction
Prof. Rodrigo Moreno
Dept. Ing. Eléctrica, Universidad de Chile

1
Contents
1. Conventional versus smart electrical energy systems
(in this ppt)
2. Basic concepts in optimisation (in this ppt)
3. Basic concepts and models in electrical energy
dispatch, planning and tariffs
4. Frequency and voltage control (AC power flow)
5. Reliability and security modelling
6. Modelling flexible electrical components
7. Advanced topics in operation and planning:
uncertainty, risk, robustness and algorighms
2
1. Conventional and “smart” electrical
energy systems

3
Electricity system (1/2)
Conventional electricity system (2/2)

Generación

Transmisión

Distribución

5
Smart grids (1/2)
Smart grids (2/2)

Control

Control

7
2. Optimisation fundamentals

8
How is optimisation useful for actual
electrical energy systems ? (1)
Minimise:

9
How is optimisation useful for actual
electrical energy systems ? (2)
Maximise:

10
Examples
• Optimisation is broadly used in power system control,
operation and planning.

Interesting applications :
• Least-cost electricity operation
• Renewables integration and new technologies
(including energy storage)
• Smart grids
• Investment under uncertainty
• Electricity network security and robustness
• Contracts, markets
• Energy management systems
• Many others….
11
Mathematical optimisation
• Mathematical optimisation or mathematical
programming refers to the selection of a best
element (with regard to some criteria) from
some set of available alternatives

• Generally speaking, optimisation includes


finding "best available" values of some
objective function given a defined domain
(defined by a set of constraints)

12
Problem:

Local optimum:

1st O. condition: ( )

f(x,y)
2nd O. condition: Hessian is
Positive-definite

f = (x-4)^2+(y-4)^2
(4,4) is optimum

Intro.mws 13
Newton’s fundamentals

14
Cauchy’s fundamentals

Scalar!

f(x,y)

15
Constrained optimization (1)

Min f = (x-4)^2+(y-4)^2
s.t.
x+y <= 5
x,y >= 0

16
Constrained optimization (2)

Min f = (x-4)^2+(y-4)^2
s.t.
(4,4) x+y <= 5
x,y >= 0

17
Constrained optimization (3)
Min f = (x-4)^2+(y-4)^2
s.t.
x+y <= 5
x,y >= 0

18
KKT conditions

Problem:

KKT:

19
KKT: equality constraints

20
Shadow prices, dual variable or
marginal costs

• This is very useful for sensitivity analysis on parameters in the


right hand side of the problem
• Indicates biding constraints at the optimum solution (in
convex problems)
• In many countries including EU, this is used to determined
prices of electricity
• Market’s marginal cost is the derivative of cost wrt volumes of
production which is usually in the right hand side (when
demand is a parameter – inelastic)
21
Homework
Research about:
• Simplex
• Branch and bound
• Usage of Newton-Raphson method for load
flow analysis (how different is this application
of the Newton’s method from that shown
previously in our class?)

22

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