Sums Of Squares For Power Systems Transient Stability Analysis · 2018-01-17 · Introduction Our Power System Theoretical Tools Transient Stability AnalysisConclusion Sums Of Squares

Introduction Our Power System Theoretical Tools Transient Stability Analysis Conclusion

Sums Of Squares For Power Systems TransientStability Analysis

Master project at RTE-ECN International chair in automaticcontrol and power grids

Matteo Tacchi

PhD student at LAAS-CNRS / RTE-R&D

January 17, 2018


Outline

1 Our Power System

2 Theoretical Tools

3 Transient Stability Analysis


Introduction

Acknowledgement : This research work was conducted under the supervision

of B. Marinescu and M. Anghel.

Context

Aim : asserting transient stability of power grids

Current method : large scale simulations

Issue : costly computations, transformation of the network

Tools

Lyapunov stability theory

Real algebraic geometry : Positivstellensatz

Sum-Of-Squares / Semidefinite Programming


Introduction



Context




Tools





Introduction



Context




Tools





Introduction



Context




Tools





Introduction



Context




Tools





Introduction



Context




Tools





Introduction



Context




Tools





Background

[Anghel, Milano & Papachristodoulou] Algorithmic construction ofLyapunov functions for power system stability analysis. 2013.

G1

G2

R12 + jX12 R23 + jX23

R13 + jX13G3

Our contribution

Here : no regulation =⇒ accurate but narrow stability regionWe implement the method on a regulated system with only 2 machinesWider stability region expected, computations more difficult


Background


G1

G2

R12 + jX12 R23 + jX23

R13 + jX13G3

Our contribution

Here : no regulation =⇒ accurate but narrow stability region

We implement the method on a regulated system with only 2 machinesWider stability region expected, computations more difficult


Background


G1

G2

R12 + jX12 R23 + jX23

R13 + jX13G3

Our contribution

Here : no regulation =⇒ accurate but narrow stability regionWe implement the method on a regulated system with only 2 machines

Wider stability region expected, computations more difficult


Background


G1

G2

R12 + jX12 R23 + jX23

R13 + jX13G3

Our contribution

Here : no regulation =⇒ accurate but narrow stability regionWe implement the method on a regulated system with only 2 machinesWider stability region expected, computations more difficult


Outline

1 Our Power System

2 Theoretical Tools



The Nominal System

System Description

G

SR+ jX R+ jX

2(R+ jX)

N∞

VG ∼ (vd, vq) V∞ = Vsejωst

∞


The Nominal System

Our Model

Nominal Equations

[Sauer & Pai] Power System Dynamics and Stability. Prentice Hall, 1998.

T ′d0 e

′q = −e′q − (xd − x′d)id + Efd

2Hω = Pm − (vdid + vqiq + ri2d + ri2q)

δ = ω − ωsvd = Rid −Xiq + Vs sin(δ)

vq = Riq +Xid + Vs cos(δ)

−Vs sin(δ) = (R+ r)id − (X + xq)iq

e′q − Vs cos(δ) = (R+ r)iq + (X + x′d)id

Ta ˙Efd = −Efd +Ka

(Vref −

√v2d + v2

q

)Tg ˙Pm = −Pm + Pref +Kg(ωref − ω)


The Nominal System

Our Model

Nominal Equations


T ′d0 e



δ = ω − ωs

vd = Rid −Xiq + Vs sin(δ)





(Vref −

√v2d + v2

q



The Nominal System

Our Model

Nominal Equations


T ′d0 e








(Vref −

√v2d + v2

q



The Nominal System

Our Model

Nominal Equations


T ′d0 e








(Vref −

√v2d + v2

q



The Perturbation

Introduction of a Short-circuit

Short-circuit Protocol

1) Start from nominal operating system at equilibrium

2) at t = tc, a short-circuit occurs : the state is not atequilibrium anymore

3) at t = tcl = tc + ∆t, the short-circuit is eliminated (back tonominal equations : successful auto-reclosing)

Question : What is the critical clearing time (CCT ) i.e. themaximal value of ∆t such that the system reaches anequilibrium point after the fault elimination ?


The Perturbation








The Perturbation








The Perturbation








The Perturbation








The Perturbation

The Short-circuited System

G

SR+ jX R+ jX

2(R+ jX)

N∞

VG ∼ (vd, vq) V∞ = Vsejωst

∞


The Perturbation


Short-circuit Equations

T ′d0 e



δ = ω − ωs

3vd = 2Rid − 2Xiq + Vs sin(δ)

3vq = 2Riq + 2Xid + Vs cos(δ)

−Vs sin(δ) = (2R+ 3r)id − (2X + 3xq)iq

3e′q − Vs cos(δ) = (2R+ 3r)iq + (2X + 3x′d)id


(Vref −

√v2d + v2

q



The Perturbation



T ′d0 e



δ = ω − ωs3vd = 2Rid − 2Xiq + Vs sin(δ)





(Vref −

√v2d + v2

q



The Perturbation



T ′d0 e



δ = ω − ωs3vd = 2Rid − 2Xiq + Vs sin(δ)





(Vref −

√v2d + v2

q



Preliminary Computations

Equilibrium Point

The state variables δ, ω, e′q, Efd and Pm remain constant whenthey reach the value

δeq = 1.579ωeq = 1

e′eqq = 1.055Eeqfd = 2.477

P eqm = 0.7

This equilibrium point is LAS (according to the simulations).



First Guess of CCT for a Simplified System

Simplification

We replace Vref −√v2d + v2

q by V 2ref − v2

d − v2q

Ta ˙Efd = −Efd +Ka(V2ref − v2

d − v2q )

New LAS equilibrium point :δeq = 1.539ωeq = 1

e′eqq = 1.070Eeqfd = 2.459

P eqm = 0.7




Simplification


q by V 2ref − v2

d − v2q

Ta ˙Efd = −Efd +Ka(V2ref − v2

d − v2q )

New LAS equilibrium point :δeq = 1.539ωeq = 1

e′eqq = 1.070Eeqfd = 2.459

P eqm = 0.7




Simplification


q by V 2ref − v2

d − v2q

∆t = 4.057s ∆t = 4.058s


Outline

1 Our Power System

2 Theoretical Tools



Notions of Stability

Definition : Transient Stability

Transient Stability denotes the ability of a power system’sangular characteristics to return to operating equilibrium andsychronism after a large disturbance, and within a shortamount of time (under 10s), usually going through a first-swingaperiodic drift.

Definition : A case of Lyapunov Stability

A solution x ∈ Rn to the equation F (x) = 0 is an equilibriumpoint of the system

x = F (x). (1)

It is said to be locally asymptotically stable (LAS) iff it admitsa Region Of Attraction (ROA) s.t.

x(0) ∈ ROA =⇒ x(t) −→t→+∞

x.



Definition : Transient Stability

Transient Stability denotes the ability of a power system’sangular characteristics to return to operating equilibrium andsychronism after a large disturbance, and within a shortamount of time (under 10s), usually going through a first-swingaperiodic drift.

Definition : A case of Lyapunov Stability

A solution x ∈ Rn to the equation F (x) = 0 is an equilibriumpoint of the system

x = F (x). (1)

It is said to be locally asymptotically stable (LAS) iff it admitsa Region Of Attraction (ROA) s.t.

x(0) ∈ ROA =⇒ x(t) −→t→+∞

x.



Theorem [Lyapunov - Massera]

x = 0 is a LAS equilibrium point of the previous system iff∃V : D ⊂ Rn −→ R s.t. V (0) = 0 &

x 6= 0 =⇒ V (x) > 0, V (x) := ∇V (x) · F (x) < 0

Then, any Ωc := x ∈ Rn | V (x) ≤ c s.t. Ωc ⊂ D is a positivelyinvariant subset of the ROA.


The Positivstellensatz

Some Useful Objects

Definition : Algebraic Structures

Let G = (g1, . . . , gβ),F = (f1, . . . , fα),H = (h1, . . . , hγ) ∈ Rn.

Multiplicative Monoid generated by G

M(G) :=

β∏j=1

gkjj | k1, . . . , kβ ∈ N

Quadratic Module generated by F

P(F) :=

s0 +

α∑i=1

sifi | s0, . . . , sα ∈ Σn

Ideal generated by H

I(H) :=

γ∑i=1

hkpk | p1, . . . , pγ ∈ Rn



Some Useful Objects


Let G = (g1, . . . , gβ),F = (f1, . . . , fα),H = (h1, . . . , hγ) ∈ Rn.Multiplicative Monoid generated by G

M(G) :=

β∏j=1

gkjj | k1, . . . , kβ ∈ N


P(F) :=

s0 +

α∑i=1

sifi | s0, . . . , sα ∈ Σn


I(H) :=

γ∑i=1

hkpk | p1, . . . , pγ ∈ Rn



Some Useful Objects



M(G) :=

β∏j=1

gkjj | k1, . . . , kβ ∈ N


P(F) :=

s0 +

α∑i=1

sifi | s0, . . . , sα ∈ Σn


I(H) :=

γ∑i=1

hkpk | p1, . . . , pγ ∈ Rn



Some Useful Objects



M(G) :=

β∏j=1

gkjj | k1, . . . , kβ ∈ N


P(F) :=

s0 +

α∑i=1

sifi | s0, . . . , sα ∈ Σn


I(H) :=

γ∑i=1

hkpk | p1, . . . , pγ ∈ Rn




[Bochnak, Coste & Roy] Geometrie Algebrique Reelle. Springer, 1986.

Theorem [P-satz]

The set f1(x) ≥ 0, . . . , fα(x) ≥ 0x ∈ Rn g1(x) 6= 0, . . . , gβ(x) 6= 0

h1(x) = 0, . . . , hγ(x) = 0

is empty iff ∃f ∈ P(F), g ∈M(G), h ∈ I(H) ;

f + g2 + h = 0. (2)


The Expanding Interior Algorithm

Mixing Lyapunov theory and P-satz to find the ROA

[Wloszek] Lyapunov based analysis and controller synthesis forpolynomial systems using SOS optimization. PhD, 2003.

Estimation of the ROA

x = F (x), F ∈ Rn, 0 LAS equilibrium point.Tool to estimate 0’s ROA : V ∈ Rn s.t.

V (0) = 0 & x 6= 0 =⇒ V (x) > 0

V (x) < 0 on D \ 0, D := x ∈ Rn | V (x) ≤ 1




How do we obtain a “good” estimation ?

Aim : maximize the size of D ⊂ ROA (according toLyapunov theory)

Method : maximize the size of somePβ := x | p(x) ≤ β ⊂ D with p ∈ Σn positive definite(e.g. p(x) = ‖x‖2)

Practical implementation : maximize β.

























Resulting set emptiness problem

maxV ∈Rn,V (0)=0

β s.t.

x ∈ Rn| − V (x) ≥ 0, ε‖x‖2 6= 0 = ∅x ∈ Rn|β − p(x) ≥ 0, V (x)− 1 ≥ 0, V (x)− 1 6= 0 = ∅

x ∈ Rn|1− V (x) ≥ 0, V (x) ≥ 0, ε‖x‖2 6= 0 = ∅

Equivalent SOS problem using P-satz

maxV ∈Rn,V (0)=0k1,k2,k3∈N∗

s1,...,s10∈Σn

β s.t.

s1 − V s2 + ε‖ · ‖4k1 = 0s3 + (β − p)s4 + (V − 1)s5 + (β − p)(V − 1)s6 + (V − 1)2k2 = 0s7 + (1− V )s8 + V s9 + (1− V )V s10 + ε‖ · ‖4k3 = 0




Resulting set emptiness problem

maxV ∈Rn,V (0)=0

β s.t.

x ∈ Rn| − V (x) ≥ 0, ε‖x‖2 6= 0 = ∅x ∈ Rn|β − p(x) ≥ 0, V (x)− 1 ≥ 0, V (x)− 1 6= 0 = ∅

x ∈ Rn|1− V (x) ≥ 0, V (x) ≥ 0, ε‖x‖2 6= 0 = ∅

Equivalent SOS problem using P-satz

maxV ∈Rn,V (0)=0k1,k2,k3∈N∗

s1,...,s10∈Σn

β s.t.

s1 − V s2 + ε‖ · ‖4k1 = 0s3 + (β − p)s4 + (V − 1)s5 + (β − p)(V − 1)s6 + (V − 1)2k2 = 0s7 + (1− V )s8 + V s9 + (1− V )V s10 + ε‖ · ‖4k3 = 0




Resulting simplified SOS problem

maxV ∈Rn,V (0)=0s6,s8,s9∈Σn

β s.t.

V − ε‖ · ‖2 =: s1 ∈ Σn

− ((β − p)s6 + (V − 1)) =: s5 ∈ Σn (3)

−(

(1− V )s8 + V s9 + ε‖ · ‖2)

=: s7 ∈ Σn



Expanding Interior Algorithm

A bilinear program

The issue with the SOS problem we obtain is that we want tooptimize β with V and the si as variables.=⇒ we are faced to a bilinear program (and not a linearprogram that we know how to solve)

An algorithmic solution

The EIA tackles the maximization of β by iterating two distinctsteps :

First linear search while V is fixed and the si vary

Second linear search while V and s6 vary and the other siare fixed




A bilinear program









A bilinear program









A bilinear program







Outline

1 Our Power System

2 Theoretical Tools



Reduction to a polynomial system

Recasting the System

Change of variable

z1 = sin(δ − δeq)z2 = 1− cos(δ − δeq)z3 = ω − ωrefz4 = e′q − e′,eqq

z5 = Efd − Eeqfdz6 = Pm − Pref

Constraint G(z) = z21 + z2

2 − 2z2 = 0

z = H(z), with H ∈ Rn and H(0) = 0




Change of variable




2 − 2z2 = 0





Change of variable




2 − 2z2 = 0




Resulting Expanding Interior ProgramSet emptiness problem

maxV ∈Rn,V (0)=0

β s.t.

z ∈ R6| − V (z) ≥ 0, ε‖z‖2 6= 0, G(z) = 0 = ∅z ∈ R6|β − p(z) ≥ 0, V (z)− 1 ≥ 0, V (z)− 1 6= 0, G(z) = 0 = ∅

z ∈ R6|1− V (z) ≥ 0, V (z) ≥ 0, ε‖z‖2 6= 0, G(z) = 0 = ∅

SOS problem

maxV ,q1,q2,q3∈Rn,V (0)=0

s6,s8,s9∈Σn

β s.t.

V − ε‖ · ‖2 − q1G =: s1 ∈ Σn

− ((β − p)s6 + (V − 1))− q2G =: s5 ∈ Σn (4)

−(

(1− V )s8 + V s9 + ε‖ · ‖2)− q3G =: s7 ∈ Σn



Resulting Expanding Interior ProgramSet emptiness problem

maxV ∈Rn,V (0)=0

β s.t.

z ∈ R6| − V (z) ≥ 0, ε‖z‖2 6= 0, G(z) = 0 = ∅z ∈ R6|β − p(z) ≥ 0, V (z)− 1 ≥ 0, V (z)− 1 6= 0, G(z) = 0 = ∅

z ∈ R6|1− V (z) ≥ 0, V (z) ≥ 0, ε‖z‖2 6= 0, G(z) = 0 = ∅

SOS problem

maxV ,q1,q2,q3∈Rn,V (0)=0

s6,s8,s9∈Σn

β s.t.

V − ε‖ · ‖2 − q1G =: s1 ∈ Σn

− ((β − p)s6 + (V − 1))− q2G =: s5 ∈ Σn (4)

−(

(1− V )s8 + V s9 + ε‖ · ‖2)− q3G =: s7 ∈ Σn


Our results

Plot of the estimated ROA

δmax = δeq + 0.743puωmax = ωref + 0.209pu


Our results

Plot of the estimated ROA

e′q,max = e′eqq + 0.496puEfd,max = Eeq

fd + 14.72puPm,max = Pref + 2.029pu


Our results

Plot of the resulting Lyapunov function V

CCT estimate : CCT = 3.591sactual CCT : CCT = 4.057s


Our results

Comparison to the actual ROA

ωinit = 1.2011pu ωinit = 1.2012pu

Variable δ ω e′q Efd Pm

Actual R.O.A.’s upper bound 2.465 1.211 2.966 95.42 3.888

R.O.A. estimate’s upper bound 2.282 1.209 1.566 17.18 2.729


Our results

Comparison to the actual ROA

ωinit = 1.2011pu ωinit = 1.2012pu

Variable δ ω e′q Efd Pm

Actual R.O.A.’s upper bound 2.465 1.211 2.966 95.42 3.888

R.O.A. estimate’s upper bound 2.282 1.209 1.566 17.18 2.729


Conclusion

[Tacchi, Marinescu, Anghel, Kundu, Benahmed & Cardozo]Power systems transient stability analysis using SOS programming.Submitted to Power Systems Computation Conference in october 2017.

A wider ROA

As expected the ROA and its estimation are larger with regulations

Possible further developments

Implement the same method with a better scaled polynomial p = Vend

Solve the problem with the actual voltage regulation (nosimplification)

Solve the problem using the method proposed in [Korda, Henrion &Jones] Inner approximations of the region of attraction for polynomialdynamical systems

Tackle power grids with a well-formulated sparse moment-SOShierarchy (exploiting the network structure)


Conclusion


A wider ROA








Conclusion


A wider ROA








Conclusion


A wider ROA








Conclusion


A wider ROA








Conclusion


A wider ROA








Conclusion


A wider ROA








Thank you for your attention.Questions are welcome !

Sums Of Squares For Power Systems Transient Stability Analysis · 2018-01-17 · Introduction Our Power System Theoretical Tools Transient Stability AnalysisConclusion Sums Of Squares

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