Spontaneous Synchronization in Power Grids Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi

Motivation

• Complex Power Grids

• Optimization

• Reliability

What is a Network?

• Generators

• Nodes

• Transmission lines

• Matrix

• What is synchronization

d·

1 =d·

2 = ... =d·

n

Building the Grid

• Newton’s Second Law

• Conservation of angular momentum

• Conservation of Energy

Id2didt2

= tmi -t ei

2Hi

wR

d2didt2

= Pmi -Pei

Ia =t

Hi =1

2Iw 2

Network Structure ● Assumptions?

● Physical network vs Effective network

● Admittance

Y0=(Y0ij) : Y0ij is the negative of the admittance between i and

Stability condition from the swing equation

Term 1: Term 2: Term 3:

Assume di =di

* +di'; Pei = Pei

* +Pei' ; Pmi = Pmi

* +Pmi'

2Hi

wR

d2di'

dt2=

¶Pmi

¶wi

wi

' -¶Pei

¶wi

wi

' -¶Pei

¶d jj=1

n

å d j'

Pei' = Diwi

' /wR + EiE j (Bij cosdij* -Gij sindij

*)dij'

j=1

n

å

¶Pmi /¶wi =Di /wR

¶Pmi /¶wi = -1/ (wRRi )

Coupled 2nd Order Equations

X =X1

X2

é

ë

êê

ù

û

úú=

d®

d·

¾®¾

é

ë

êêê

ù

û

úúú

X·

=X1

·

X2

·

é

ë

êêê

ù

û

úúú

=w®

w·

¾®¾

é

ë

êêê

ù

û

úúú

X1

·

= X2

X2

·

= -PX1 -BX2

Coupled 2nd Order Equations

B is the diagonal matrix of elementsβ

B =1

2Hi

+Di

2Hi

Simplifying coupled equations

From the different equation, this has the solution

How we guarantee stabilities? By ensuring the REAL part is negative in both terms

z·

j =0 1

-a j -b

æ

è

çç

ö

ø

÷÷z j, z j º

Z1 j

Z2 j

æ

è

çç

ö

ø

÷÷

(X®

1)el1t + (X

®

2 )el2t

Finding Stability from Matrix

• From ODE’s, we can find requirements for stability.

• Lyopanov Exponents

l j± = -b

2±

1

2b 2 - 4a j

Keeping λ Negative

• Need to keep all Re(λ) < 0

• Alter function to Λ=max (Re (λ)) ≤ 0, j≥2

• J=1 is a null eigenvalue

• Related to shift in all phases

• How can we tune variable α to keep λ negative?

Function Behavior

• For α<0

• Λ>0, not usable

• For α>0

• Λ<0, usable and decreases for increasing α

• For 0≤α≤β2/4

• Λ reaches minimum at α=β2/4

• Imaginary part is added past this value

What value for α?

• α is based on eigenvalues of the matrix P

• It is dependent on properties of the generator in the system

• We want the smallest non- zero value of α

• This will increase stability

• Choose α2

• α1 is null, we ignore this value

Parameters for changes in demand

• Droop Parameter (Off-line optimization)

• Damping Coefficient (Online optimization):

b = bopt º 2 a2

Ri =1

4Hi a2 -Dii =1,....,n

Di = 4Hi a2 -1

Ri, i =1,....,n

Testing Methods

• MATLAB script written from a simplified coupled equations.

• Assumed β equal for all nodes

• Arbitrary values for α

• 5 node system

• Solved using ODE45 function

• Integrated δ solution for varying β

• Repeated multiple times

Test Results

Test Results

• Optimal Beta

Test Results

Conclusions

• Optimal Beta differs slightly from the expected value from the literature

• Can be explained by the randomness added to alpha values

• Within 10% range

Questions?