EEB551 C3: Power
System Optimization
Layout
1. Introduction
2. Formulation of optimization problems;
Linear and
Non-linear programming,
3. Minimization techniques;
Optimal power flow problem
Economic dispatch problem
Introduction
Required Prior Knowledge
• Matrix algebra: scalars vectors and matrices
• Fundamentals of calculus: Constrained/Unconstrained maxima
and minima of functions
Definition
• Optimization problem is a maths model where main objective
is to minimize undesirable things (e.g. cost, energy loss,
errors, etc.) or maximize desirable things (e.g. profit, quality,
efficiency, utility, etc.), subject to some constraints.
OR
• The process of finding maxima and minima of a function
(Constrained or unconstrained)
Introduction
Power System Optimization & Its Importance
 PSE has the longest history of development among the various
areas of electrical engineering
 Value contributed by system optimization, considerable in
economical terms (hundreds of millions of dollars saved
annually in large utilities)
 Fuel cost
 Improved operational reliability
 System security
Introduction
Power System Optimization Applications
 Power System Planning
 Maintenance Scheduling (operational planning)
 Economic Dispatch/Optimal Power Flow
 Unit Commitment
 Network Reconfiguration for loss reduction
 Reactive Resources Allocation
 Optimal protection and Switching Devices Placement
 Pollution Dispatch of power plants
Formulation of an Optimization
Model
Optimization
problem
Objective
Function
Single
Multi
Constraints
Constrained
𝒎𝒂𝒙/𝒎𝒊𝒏
𝒇(𝒙);
𝒙
Unconstrained
Variables
Continuos
Discreet
OF
𝒔. 𝒕.
ℎ 𝑥 = 0;
Equality constraints
𝑔 𝑥 ≤ 0;
Inequality Constraints
𝑥𝑚𝑖𝑛 ≤ 𝑥 ≤ 𝑥𝑚𝑎𝑥 ;
Variable upper & Lower bounds
Setting Up a Mathematical
Model
Steps in Maths Modelling for optimization
• Get an overall idea of the problem
• What is the goal? What are you trying to achieve?
• Identify variables
• Identify constraints
• Identify the inputs and outputs you can control
• Specify all quantities mathematically
• Check the model for completeness and correctness
• After formulating the mathematical optimization model, you
will know the type of optimization problem it is.
• Find the solver & Syntax and convert to solver form
• Solve in Software
TYPES OF OPTIMIZATION
PROBLEMS
Power System Optimization
Problems
Linear Programming (LP)
• Linear programming (LP, also called linear optimization) is a method
to achieve the best outcome (such as maximum profit or lowest
cost) in a mathematical model whose requirements are represented
by linear relationships.
• An LP has the following forms
𝒎𝒊𝒏𝒙 𝒇𝑻 𝒙
OF
𝒔. 𝒕.
Ax ≤ 𝑏
𝐴𝑒𝑞 𝑥 = 𝑏𝑒𝑞
𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏 ;
Inequality constraints
Equality constraints
Variable upper & Lower
1) Linear Programming in Matlab
2) How to convert to solver format: https://www.mathworks.com/videos/optimizationmodeling-2-converting-to-solver-form-101560.html
3) Convexity of an OP: https://www.solver.com/convex-optimization
Linear Program in MatLab
 After formulating the mathematical optimization model, you will
know the type of optimization problem it is.
 Find the solver & Syntax and convert to solver form
 E.g. LP:
Typical Matlab Optimization
Solvers
Typical Practical example
Economic Dispatch (ED)
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