Rubira Power Flow

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Alternative methods for solving power flow

problems

Tomas Tinoco De Rubira

Stanford University, Stanford, CA

Electric Power Research Institute, Palo Alto, CA

October 25, 2012


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Outline

1. introduction

2. available methods

3. contributions and proposed methods

4. implementation

5. experiments

6. conclusions

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1. Introduction

1.1. background

1.2. formulation

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1.1. background

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1.1. background

power flow problem

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1.1. background

power flow problem

(roughly) given gen and load powers, find [5][6]

voltage magnitudes and angles at every bus

real and reactive power flows through every branch

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1.1. background

power flow problem




essential for:

expansion, planning and daily operation[5]

real time security analysis (contingencies)[44]

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1.1. background

power flow problem




essential for:

expansion, planning and daily operation[5]

real time security analysis (contingencies)[44]

fast and reliable solution methods needed

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1.2. formulation

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1.2. formulation

network

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1.2. formulation

network

flow in flow out = 0 at each bus

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1.2. formulation

network

flow in flow out = 0 at each bus

gives power flow equations[5][6]

Pk=Pgk P

lk

m[n]

vkvm(Gkm cos(k m) +Bkm sin(k m)) = 0

Qk=Qgk Q

lk

m[n]vkvm(Gkm sin(k m) Bkm cos(k m)) = 0

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typically: [5][6]

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typically: [5][6]

partition [n] ={s} R U

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typically: [5][6]


bus i= s: slack

voltage magnitude and angle given

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typically: [5][6]


bus i= s: slack


bus i R: regulated

active power and voltage magnitude given (PV)

(target magnitude vti)

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typically: [5][6]


bus i= s: slack


bus i R: regulated

active power and voltage magnitude given (PV)

(target magnitude vti)

bus i U: unregulated

active and reactive power given (PQ)

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2. Available methods

2.1. overview

2.2. Newton-Raphson

2.3. Newton-Krylov

2.4. optimal multiplier

2.5. continuous Newton

2.6. sequential conic programming

2.7. holomorphic embedding

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2.1. overview

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2.1. overview

mid 1950s, Gauss-Seidel method[47]

low memory requirements

lacked robustness and good convergence properties

worse as network sizes

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2.1. overview

mid 1950s, Gauss-Seidel method[47]

low memory requirements

lacked robustness and good convergence properties

worse as network sizes

Newton-Raphson (NR) method[5][47]

more robust, better convergence properties

good when combined with sparse techniques

most widely used method

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improving NR

speed: fast decoupled[5], DC power flow, simplified XB and BX[47]

scalability: Newton-Krylov[13][14][15][20][26][32][35][45]

robustness: integration[34]and optimal multiplier[9][11][27][41][42]

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improving NR

speed: fast decoupled[5], DC power flow, simplified XB and BX[47]

scalability: Newton-Krylov[13][14][15][20][26][32][35][45]

robustness: integration[34]and optimal multiplier[9][11][27][41][42]

other methods genetic, neural networks and fuzzy algorithms

not competitive with NR-based method [44]

sequential conic programming[28][29]

holomorphic embedding[43]

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2.2. Newton-Raphson

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2.2. Newton-Raphson

iterative method for solving f(x) = 0, f is C1

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2.2. Newton-Raphson


linearize at xk: fk+Jk(xk+1 xk) = 0

update: xk+1= xk+pk, where Jkpk=fk

stop when fk<

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2.2. Newton-Raphson


linearize at xk: fk+Jk(xk+1 xk) = 0

update: xk+1= xk+pk, where Jkpk=fk

stop when fk<

properties: [36]

quadratic rate of convergence

only locally convergent

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forpower flow: [5][6]

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f is power mismatches, left hand side of

Pi= 0, i R U

Qi= 0, i U

x is voltage magnitudes {vi}iUand angles {i}i[n]\{s}

other variables updated separately to enforce rest of equations

heuristics to enforce Qmini Qgi Q

maxi , i R

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f is power mismatches, left hand side of

Pi= 0, i R U

Qi= 0, i U

x is voltage magnitudes {vi}iUand angles {i}i[n]\{s}

other variables updated separately to enforce rest of equations

heuristics to enforce Qmini Qgi Q

maxi , i R

If violation Qgi fixed R U vi free

more complex ones for R U [46]

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2.3. Newton-Krylov

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2.3. Newton-Krylov

size of networks , Jkpk=fk problematic for direct methods [45]

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2.3. Newton-Krylov


Krylov subspace methods[40]

GMRES[13][15][20][26], FGMRES[45]

BCGStab[20][32][35]

CGS[20]

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2.3. Newton-Krylov


Krylov subspace methods[40]

GMRES[13][15][20][26], FGMRES[45]

BCGStab[20][32][35]

CGS[20]

preconditioners

polynomial[14][32]

LU-based[20][26][35][45]

Jk changes little same preconditioner, multiple iterations[15][20][26]

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NR only locally convergent[36], can even diverge

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ensure progress with controlled update: xk+1=xk+kpk

k minimizes f(xk+kpk)22

easy in rectangular coordinates[27]

find roots of cubic polynomial

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ensure progress with controlled update: xk+1=xk+kpk

k minimizes f(xk+kpk)22

easy in rectangular coordinates[27]

find roots of cubic polynomial

extend to polar[9][11][27][41]

polar sometimes preferable[42]

can only minimize approx. off(xk+kpk)22, no guarantees

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cast power flow as ODE apply robust integration techniques [34]

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NR method forward Euler with t= 1

optimal multiplier[27] a type of variable-step forward Euler

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NR method forward Euler with t= 1

optimal multiplier[27] a type of variable-step forward Euler

Runge-Kutta[34]

more robust than NR

less iterations than the optimal multiplier method (UCTE network)

no speed comparison

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for radial networks[28]

change variables, cast power flow problem as conic program

convex, easy to solve (in theory)

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for radial networks[28]

change variables, cast power flow problem as conic program

convex, easy to solve (in theory)

for meshed networks[29](same author)

extra constraints: zero angle changes around every loop

destroy convexity

linearize and solvesequenceof conic programs (expensive)

relatively large number of iterations[29]

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embed power flow equations in system of eqs. with extra sC

[43]

s= 0 (zero net injections), s= 1 (original)

complex voltage solutionsv(s) areholomorphicfunctions ofs

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[43]



find power series ofv(s) at s= 0 (solve linear system)

applyanalytic continuation(Pade approximants [7]) to get v(s) at s= 1

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[43]



find power series ofv(s) at s= 0 (solve linear system)

applyanalytic continuation(Pade approximants [7]) to get v(s) at s= 1

very strong claims in[43]:

algorithm finds solutioniffsolution exists

few experimental results

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3. Contributions and proposed methods

3.1. line search

3.2. starting point

3.3. formulation and bus type switching

3.4. real networks

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3.1. line search

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3.1. line search

choosek forxk+1= xk+kpk (common in optimization)

can ensure progress & make NRglobally convergent[36]

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3.1. line search



most NR-based methods reviewed: no line search

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3.1. line search




suggest: should be standard practice

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3.1. line search





no need for integration[34]

or approx. rectangular optimal multiplier [9][11][41][27]

simple bracketing and bisection[38]

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3.1. line search





no need for integration[34]

or approx. rectangular optimal multiplier [9][11][41][27]

simple bracketing and bisection[38]

method: Line Search Newton-Raphson (LSNR)

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line search Newton-Raphson(LSNR)

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formulate as in NR, f(x) = 0measure progress with h= 12f

Tf (merit function)

use controlled update xk+1=xk+kpk

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formulate as in NR, f(x) = 0measure progress with h= 12f

Tf (merit function)

use controlled update xk+1=xk+kpk

k satisfiesstrong Wolfe conditions[38]

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benefits

scaling: reduce net injection by factor of 1.07 approx.

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3.2. starting point

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3.2. starting point

NR needs good x0

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3.2. starting point

NR needs good x0

oftenflat startused but can also fail, then

trial and error: tweak network & guess other x0

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3.2. starting point

NR needs good x0



propose: automate transformations (homotopy[31])

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3.2. starting point

NR needs good x0




transform network toflat network

gradually transform back

solutions for transformed networks solution for original network

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3.2. starting point

NR needs good x0




transform network toflat network

gradually transform back

solutions for transformed networks solution for original network

method: Flat Network Homotopy (FNH) (phase shifters only)

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flat network homotopy(FNH)

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formulate as f(x, t) = 0 (x and fas before)

t scales phase shifters:

t= 0 phase shifters removed (flat network)

t= 1 phase shifters in original state

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formulate as f(x, t) = 0 (x and fas before)

t scales phase shifters:

t= 0 phase shifters removed (flat network)

t= 1 phase shifters in original state

approx. solve f(x, ti) = 0 fori N, t1= 0, ti 1 asi

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how?

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how?

approx. solve sequence of problems

minimizex

Fi(x) =1

2ijU

(vj 1)2 +

1

2f(x, ti)

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indexed by i, generate approx. solutions {xi}iN

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how?


minimizex

Fi(x) =1

2ijU

(vj 1)2 +

1

2f(x, ti)

22


first term encourages trajectory ofgoodpoints

goes away: i 0 asi

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how?


minimizex

Fi(x) =1

2ijU

(vj 1)2 +

1

2f(x, ti)

22


first term encourages trajectory ofgoodpoints

goes away: i 0 asi

starting points:

subproblem i= 1 (flat network), use flat start

subproblem i >1, modify xi1

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squareformulation

pk comes from fonly: missing info (physical limits&preferences)

can lead to regions with poor or meaningless points

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squareformulation



switchingheuristics (PV - PQ)

keep formulation square

can cause jolts, cycling and prevent convergence [46]

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squareformulation






propose: add degrees of freedom, add missing info

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squareformulation






propose: add degrees of freedom, add missing info

methods: Vanishing Regulation (VR) , Penalty-Based Power Flow (PBPF)

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analogy ...

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vanishing regulation (VR)


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vanishing regulation(VR)

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s g g t o ( )

formulate as optimization, encodepreferencesin objective

minimizex

(x) =

2

jR

(vj vtj)

2 +

2

jU

(vj 1)2

subject to f(x) = 0

(specific values unclear), x also includes{vi}iR

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g g ( )


minimizex

(x) =

2

jR

(vj vtj)

2 +

2

jU

(vj 1)2

subject to f(x) = 0


apply penalty function method [23]

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g g ( )


minimizex

(x) =

2

jR

(vj vtj)

2 +

2

jU

(vj 1)2

subject to f(x) = 0


apply penalty function method [23]

approx. solve sequence

minimizex

Fi(x) =i(x) +1

2f(x)22


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solving subproblems: apply LSNR to F (x) 0


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solving subproblems: apply LSNR to Fi(x) = 0

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solving subproblems: apply LSNR to F (x) = 0


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LSNR iteration needs to solve

iGk+J

TkJk+

j

fj(xk)2fj(xk)

pk=kgk J

Tkfk,

Gk and gk are Hessian and gradient of at xk

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solving subproblems: apply LSNR to F (x) = 0


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LSNR iteration needs to solve

iGk+J

TkJk+

j

fj(xk)2fj(xk)

pk=kgk J

Tkfk,

Gk and gk are Hessian and gradient of at xk

VR ignores

jfj(xk)2fj(xk) to get descent directions easily, but then

descent directions get poorer as i 0 (JTkJk singular)

affects robustness of method

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mismatch vs regulation trade-off


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penalty-based power flow(PBPF)


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extend VR:

considerphysical limits Qmini Qgi Q

maxi , i R

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extend VR:

considerphysical limits Qmini Qgi Q

maxi , i R

improveVR:

use better optimization algorithm

keep second order terms in Newton system

use better penalties to encode preferences

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formulate as optimization, encodepreferences&physical limitsin objective


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minimizex

(x) =u(x) +q(x) +r(x)

subject to f(x) = 0

x also includes {Qi}iR

f is now mismatches at all buses except slack

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formulate as optimization, encodepreferences&physical limitsin objective


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minimizex

(x) =u(x) +q(x) +r(x)

subject to f(x) = 0

x also includes {Qi}iR

f is now mismatches at all buses except slack

penalties:

u(x) =

jUuj (vj)

q(x) =jR

qj(Qgj)

r(x) =jR

rj(vj)

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for unregulated magnitudes (quartic)


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u

j (z) =2z vmaxj vminj

vmaxj vminj

4

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u


vmaxj vminj

4

for reactive powers of regulating gens (sextic)

qj(z) =

2z Qmaxj Qminj

Qmaxj Qminj

6

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u


vmaxj vminj

4

for reactive powers of regulating gens (sextic)

qj(z) =

2z Qmaxj Qminj

Qmaxj Qminj

6

for regulated magnitudes (softmaxbetweenquartic&quadratic)

rj(z) =1

2log

e2uj (z) +e2

2j(z)

+bj,

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apply augmented Lagrangian method [8][10][22][23]


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pp y g g g [ ][ ][ ][ ]

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pp y g g g [ ][ ][ ][ ]


minimizex

Li(x, i) =i(x) iTif(x) +

1

2f(x)22,


i positive scalars, i Lagrange multiplierestimates

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[ ][ ][ ][ ]


minimizex


1

2f(x)22,



solving subproblems: applytruncatedLSNR procedure [33][37]

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minimizex


1

2f(x)22,



solving subproblems: applytruncatedLSNR procedure [33][37]

truncated LSNR iteration solves Hkpk=dk only approx.

Hk is 2xLi(xk, i), dk is xLi(xk, i)

xk is current iterate

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we use modified Conjugate Gradient [12]


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exits with pk=kdk, k0,sufficient descent directionforLi(, i)

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we use modified Conjugate Gradient [12]


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exits with pk=kdk, k0,sufficient descent directionforLi(, i)

preconditioner:

merge low-stretch spanning trees[4][21], find Hk based on resulting tree

captures important info ofHk and sparser

factorize Hk=LDLT, modify D if necessary (positive definite)

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3.4. real networks


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3.4. real networks


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most common aspect of methods from literature:

tested on smalltoy problems(IEEE networks[1])

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3.4. real networks


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in reality: networks can be large (some >45k buses)

performance on toy problems gives little info

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3.4. real networks


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in reality: networks can be large (some >45k buses)

performance on toy problems gives little info

our data: real power networks

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4. implementation


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4. implementation


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algorithms

Python[2](fast prototyping)

objects and basic numerical routines with NumPy[3]and SciPy[30]

efficient sparse LU with UMFPACK[16] [17] [18] [19]

analysis of topology and modifications with NetworkX [25]

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4. implementation


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algorithms

Python[2](fast prototyping)

objects and basic numerical routines with NumPy[3]and SciPy[30]

efficient sparse LU with UMFPACK[16] [17] [18] [19]

analysis of topology and modifications with NetworkX [25]

parsers

also in Python

handle IEEE common data format[24]and PSS RE raw format version 32

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5. experiments


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5. experiments


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power flow test cases

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method comparison (warm start)


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method comparison (flat start)


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method comparison (progress on large cases)


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method comparison (perturbing x0)


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initial angle + N(0, 0.32) (degrees)

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method comparison (perturbing x0)


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initial angle + N(0, 0.52) (degrees)

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contingency analysis (scaling net injections)


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6. conclusions


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line search:

simple to implement

not expensive

only a few function evaluations easy to parallelize

big value

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flat network homotopy:


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phase shifters are important but not enough

need more transformations to simplify problem

try to make DC power flow assumptions [39] hold:

near flat profile

near flat v profile small r and high x/r ratio

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flat network homotopy:


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phase shifters are important but not enough

need more transformations to simplify problem

try to make DC power flow assumptions [39] hold:

near flat profile

near flat v profile small r and high x/r ratio

to do

understand & v profile as net injections vary

understand & v profile as r becomes small and x/r large

use homotopy with net injections, r, x and phase shifters

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formulation and bus type switching


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moving away from square formulation is promising

more robust, no more switching, preferred solutions

VR has limitations, PBPF should solve them

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formulation and bus type switching


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moving away from square formulation is promising

more robust, no more switching, preferred solutions

VR has limitations, PBPF should solve them

to do

finish developing theory for PBPF

start implementing PBPF

parallelize function evaluations

combine with FNH

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