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2011 Daniel Kirschenand University of

Local and Global Optima

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Which one is the realmaximum?

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x

f(x)

A D

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Which one is the realoptimum?

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x1

x

2

B

A

C

D

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2011 Daniel Kirschen and University ofWashington

Local and Global Optima

The optimality conditions are localconditions

They do not compare separateoptima

They do not tell us which one is theglobal optimum

In general, to find the globaloptimum, we must find and compareall the optima

In large problems, this can be requireso much time that it is essentially an 2011 Daniel Kirschen 44

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Examples of ConvexFeasible Sets

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x

1

x

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x

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x

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x

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x1min x1max

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Example of Non-ConvexFeasible Sets

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1x1a x1dx1b x1c

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Example of Convex FeasibleSets

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A set is convex if, for any two points belonging to the set, all the

points on the straight line joining these two points belong to the set

x1x1mi

n

x1m

ax 2011 Daniel Kirschen 88

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xamp e o on- onvexFeasible Sets

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x1x1

a

x1

d

x1

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x1

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Example of Convex Function

x

f(x)

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Example of Convex Function

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1

x2

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Example of Non-ConvexFunction

x

f(x)

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E l f N C

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Example of Non-ConvexFunction

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C

D

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l f C

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Example of Non-ConvexFunction

x

f(x)

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Importance of Convexity If we can prove that a minimization problem is

convex: Convex feasible set

Convex objective function

Then, the problem has one and only one solution

Proving convexity is often difficult

Power system problems are usually not convex

There may be more than one solution to powersystem optimization problems

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