Sums Of Squares For Power Systems Transient Stability Analysis · 2018-01-17 · Introduction Our...

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

Lyapunov stability theory

Real algebraic geometry : Positivstellensatz

Sum-Of-Squares / Semidefinite Programming

Introduction

Context

Introduction

Context

Introduction

Context

Introduction

Context

Introduction

Context

Introduction

Context

Background

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

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

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

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

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

SR+ jX R+ jX

2(R+ jX)

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

)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

The Nominal System

Our Model

Nominal Equations

T ′d0 e

(Vref −

√v2d + v2

The Nominal System

Our Model

Nominal Equations

T ′d0 e

(Vref −

√v2d + v2

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 Short-circuited System

SR+ jX R+ jX

2(R+ jX)

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

The Perturbation

T ′d0 e

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

(Vref −

√v2d + v2

The Perturbation

T ′d0 e

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

(Vref −

√v2d + v2

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→+∞

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→+∞

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) :=

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) :=

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) :=

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) :=

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

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 !

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