Alternative methods for solving power ow problems...Alternative methods for solving power ow...
Transcript of Alternative methods for solving power ow problems...Alternative methods for solving power ow...
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Alternative methods for solving power flowproblems
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
• (roughly) given gen and load powers, find [5][6]
– voltage magnitudes and angles at every bus
– real and reactive power flows through every branch
• essential for:
– expansion, planning and daily operation [5]
– real time security analysis (contingencies) [44]
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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
• 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
k
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1.2. formulation
network
k
flow in − flow out = 0 at each bus
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1.2. formulation
network
k
flow in − flow out = 0 at each bus
gives power flow equations [5][6]
∆Pk = P gk − Plk −
∑m∈[n]
vkvm(Gkm cos(θk − θm) +Bkm sin(θk − θm)) = 0
∆Qk = Qgk −Qlk −
∑m∈[n]
vkvm(Gkm sin(θk − θm)−Bkm cos(θk − θm)) = 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]
partition [n] = {s} ∪ R ∪ U
• bus i = s: slack
– voltage magnitude and angle given
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typically: [5][6]
partition [n] = {s} ∪ R ∪ U
• bus i = s: slack
– voltage magnitude and angle given
• bus i ∈ R: regulated
– active power and voltage magnitude given (PV)
(target magnitude vti)
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typically: [5][6]
partition [n] = {s} ∪ R ∪ U
• bus i = s: slack
– voltage magnitude and angle given
• 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
iterative method for solving f(x) = 0, f is C1
• linearize at xk: fk + Jk(xk+1 − xk) = 0
• update: xk+1 = xk + pk, where Jkpk = −fk
• stop when ‖fk‖∞ < ε
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2.2. Newton-Raphson
iterative method for solving f(x) = 0, f is C1
• linearize at xk: fk + Jk(xk+1 − xk) = 0
• update: xk+1 = xk + pk, where Jkpk = −fk
• stop when ‖fk‖∞ < ε
properties: [36]
• quadratic rate of convergence
• only locally convergent
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x0
update xk +1=x k+ pk
evaluate J k and f k
solve J k pk=− f k
∥ f k∥∞<ϵ
no
yes
done
k=0
k=k+1
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for power flow: [5][6]
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for power flow: [5][6]
• f is “power mismatches”, left hand side of
∆Pi = 0, i ∈ R ∪ U∆Qi = 0, i ∈ U
• x is voltage magnitudes {vi}i∈U and angles {θi}i∈[n]\{s}
• other variables updated separately to enforce rest of equations
• heuristics to enforce Qmini ≤ Qgi ≤ Qmax
i , i ∈ R
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for power flow: [5][6]
• f is “power mismatches”, left hand side of
∆Pi = 0, i ∈ R ∪ U∆Qi = 0, i ∈ U
• x is voltage magnitudes {vi}i∈U and angles {θi}i∈[n]\{s}
• other variables updated separately to enforce rest of equations
• heuristics to enforce Qmini ≤ Qgi ≤ Qmax
i , 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
size of networks ↑, Jkpk = −fk problematic for direct methods [45]
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
size of networks ↑, Jkpk = −fk problematic for direct methods [45]
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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2.4. optimal multiplier
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2.4. optimal multiplier
NR only locally convergent [36], can even diverge
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2.4. optimal multiplier
NR only locally convergent [36], can even diverge
ensure progress with controlled update: xk+1 = xk + αkpk
• αk minimizes ‖f(xk + αkpk)‖22
• easy in rectangular coordinates [27]
– find roots of cubic polynomial
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2.4. optimal multiplier
NR only locally convergent [36], can even diverge
ensure progress with controlled update: xk+1 = xk + αkpk
• αk minimizes ‖f(xk + αkpk)‖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. of ‖f(xk + αkpk)‖22, no guarantees
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2.5. continuous Newton
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2.5. continuous Newton
cast power flow as ODE → apply robust integration techniques [34]
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2.5. continuous Newton
cast power flow as ODE → apply robust integration techniques [34]
• NR method ↔ forward Euler with ∆t = 1
• optimal multiplier [27] ↔ a type of variable-step forward Euler
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2.5. continuous Newton
cast power flow as ODE → apply robust integration techniques [34]
• 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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2.6. sequential conic programming
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2.6. sequential conic programming
for radial networks [28]
• change variables, cast power flow problem as conic program
– convex, easy to solve (in theory)
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2.6. sequential conic programming
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 solve sequence of conic programs (expensive)
relatively large number of iterations [29]
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2.7. holomorphic embedding
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2.7. holomorphic embedding
embed power flow equations in system of eqs. with extra s ∈ C [43]
• s = 0 (zero net injections), s = 1 (original)
• complex voltage solutions v(s) are holomorphic functions of s
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2.7. holomorphic embedding
embed power flow equations in system of eqs. with extra s ∈ C [43]
• s = 0 (zero net injections), s = 1 (original)
• complex voltage solutions v(s) are holomorphic functions of s
find power series of v(s) at s = 0 (solve linear system)
apply analytic continuation (Pade approximants [7]) to get v(s) at s = 1
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2.7. holomorphic embedding
embed power flow equations in system of eqs. with extra s ∈ C [43]
• s = 0 (zero net injections), s = 1 (original)
• complex voltage solutions v(s) are holomorphic functions of s
find power series of v(s) at s = 0 (solve linear system)
apply analytic continuation (Pade approximants [7]) to get v(s) at s = 1
very strong claims in [43]:
• algorithm finds solution iff “solution” 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
choose αk for xk+1 = xk + αkpk (common in optimization)
• can ensure progress & make NR globally convergent [36]
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3.1. line search
choose αk for xk+1 = xk + αkpk (common in optimization)
• can ensure progress & make NR globally convergent [36]
most NR-based methods reviewed: no line search
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3.1. line search
choose αk for xk+1 = xk + αkpk (common in optimization)
• can ensure progress & make NR globally convergent [36]
most NR-based methods reviewed: no line search
suggest: should be standard practice
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3.1. line search
choose αk for xk+1 = xk + αkpk (common in optimization)
• can ensure progress & make NR globally convergent [36]
most NR-based methods reviewed: no line search
suggest: should be standard practice
• 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
choose αk for xk+1 = xk + αkpk (common in optimization)
• can ensure progress & make NR globally convergent [36]
most NR-based methods reviewed: no line search
suggest: should be standard practice
• 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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line search Newton-Raphson (LSNR)
formulate as in NR, f(x) = 0
measure progress with h = 12fTf (merit function)
use controlled update xk+1 = xk + αkpk
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line search Newton-Raphson (LSNR)
formulate as in NR, f(x) = 0
measure progress with h = 12fTf (merit function)
use controlled update xk+1 = xk + αkpk
• αk satisfies strong Wolfe conditions [38]
h( x+α p)
0 α
1) below this line
2) slope flatter than these lines
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x0
update xk +1=x k+ pk
evaluate J k and f k
solve J k pk=− f k
∥ f k∥∞<ϵ
no
yes
done
k=0
k=k+1
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x0
update xk +1=x k+αk pk
evaluate J k and f k
solve J k pk=− f k
∥ f k∥∞<ϵ
no
yes
done
k=0
k=k+1
find αk
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benefits
0 5 10 15 20 25 30 351E-8
1E-6
1E-4
1E-2
1E+0
1E+2
1E+4
1E+6
1E+8
ucteWP2002
LSNR NR
iterations
max
mi s
mat
ch (
pu)
0 5 10 15 20 251E-8
1E-6
1E-4
1E-2
1E+0
1E+2
1E+4
1E+6
1E+8
ucteWP2002 - scaled
LSNR NR
iterationsm
ax m
i sm
atch
(pu
)
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
often flat start used 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
often flat start used but can also fail, then
• trial and error: tweak network & guess other x0
propose: automate transformations (homotopy [31])
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3.2. starting point
NR needs good x0
often flat start used but can also fail, then
• trial and error: tweak network & guess other x0
propose: automate transformations (homotopy [31])
• transform network to flat network
• gradually transform back
• solutions for transformed networks → solution for original network
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3.2. starting point
NR needs good x0
often flat start used but can also fail, then
• trial and error: tweak network & guess other x0
propose: automate transformations (homotopy [31])
• transform network to flat 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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flat network homotopy (FNH)
formulate as f(x, t) = 0 (x and f as before)
t scales phase shifters:
• t = 0 phase shifters removed (flat network)
• t = 1 phase shifters in original state
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flat network homotopy (FNH)
formulate as f(x, t) = 0 (x and f as 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 for i ∈ N, t1 = 0, ti ↑ 1 as i→∞
x0
x i x*
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how?
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how?
approx. solve sequence of problems
minimizex
Fi(x) =1
2ηi∑j∈U
(vj − 1)2 +1
2‖f(x, ti)‖22
indexed by i, generate approx. solutions {xi}i∈N
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how?
approx. solve sequence of problems
minimizex
Fi(x) =1
2ηi∑j∈U
(vj − 1)2 +1
2‖f(x, ti)‖22
indexed by i, generate approx. solutions {xi}i∈N
first term encourages trajectory of good points
• goes away: ηi ↓ 0 as i→∞
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how?
approx. solve sequence of problems
minimizex
Fi(x) =1
2ηi∑j∈U
(vj − 1)2 +1
2‖f(x, ti)‖22
indexed by i, generate approx. solutions {xi}i∈N
first term encourages trajectory of good points
• goes away: ηi ↓ 0 as i→∞
starting points:
• subproblem i = 1 (flat network), use flat start
• subproblem i > 1, modify xi−1
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3.3. formulation and bus type switching
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3.3. formulation and bus type switching
square formulation
• pk comes from f only: missing info (physical limits & preferences)
• can lead to regions with poor or meaningless points
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3.3. formulation and bus type switching
square formulation
• pk comes from f only: missing info (physical limits & preferences)
• can lead to regions with poor or meaningless points
switching heuristics (PV - PQ)
• keep formulation square
• can cause “jolts”, cycling and prevent convergence [46]
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3.3. formulation and bus type switching
square formulation
• pk comes from f only: missing info (physical limits & preferences)
• can lead to regions with poor or meaningless points
switching heuristics (PV - PQ)
• keep formulation square
• can cause “jolts”, cycling and prevent convergence [46]
propose: add degrees of freedom, add missing info
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3.3. formulation and bus type switching
square formulation
• pk comes from f only: missing info (physical limits & preferences)
• can lead to regions with poor or meaningless points
switching heuristics (PV - PQ)
• keep formulation square
• can cause “jolts”, cycling and prevent convergence [46]
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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Q violations
Q violations
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power flow solutions
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good solutions
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Algorithm
name: NR
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28
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SWITCH!
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28
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SWITCH!
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28
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28
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Algorithm
name: PBPF
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vanishing regulation (VR)
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vanishing regulation (VR)
formulate as optimization, encode preferences in objective
minimizex
ϕ(x) =α
2
∑j∈R
(vj − vtj)2 +β
2
∑j∈U
(vj − 1)2
subject to f(x) = 0
β � α (specific values unclear), x also includes {vi}i∈R
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vanishing regulation (VR)
formulate as optimization, encode preferences in objective
minimizex
ϕ(x) =α
2
∑j∈R
(vj − vtj)2 +β
2
∑j∈U
(vj − 1)2
subject to f(x) = 0
β � α (specific values unclear), x also includes {vi}i∈R
apply penalty function method [23]
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vanishing regulation (VR)
formulate as optimization, encode preferences in objective
minimizex
ϕ(x) =α
2
∑j∈R
(vj − vtj)2 +β
2
∑j∈U
(vj − 1)2
subject to f(x) = 0
β � α (specific values unclear), x also includes {vi}i∈R
apply penalty function method [23]
approx. solve sequence
minimizex
Fi(x) = µiϕ(x) +1
2‖f(x)‖22
indexed by i, generate approx. solutions {xi}i∈N
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solving subproblems: apply LSNR to ∇Fi(x) = 0
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solving subproblems: apply LSNR to ∇Fi(x) = 0
LSNR iteration needs to solve(µiGk + JTk Jk +
∑j
fj(xk)∇2fj(xk))pk = −µkgk − JTk fk,
Gk and gk are Hessian and gradient of ϕ at xk
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solving subproblems: apply LSNR to ∇Fi(x) = 0
LSNR iteration needs to solve(µiGk + JTk Jk +
∑j
fj(xk)∇2fj(xk))pk = −µkgk − JTk fk,
Gk and gk are Hessian and gradient of ϕ at xk
VR ignores∑j fj(xk)∇2fj(xk) to get descent directions easily, but then
• descent directions get poorer as µi ↓ 0 (JTk Jk singular)
• affects robustness of method
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mismatch vs regulation trade-off
0 5 10 15 20 25 30 35 40 451E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
eiSP2008
mismatch dvreg
iterations
max
mi s
mat
ch (
pu)
volts
(p u
)
0 5 10 15 20 25 30 35 40 451E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
eiSP2012
mismatch dvreg
iterations
max
mi s
mat
ch (
pu)
volts
(p u
)
0 1 21E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
aesoSP2014
mismatch dvreg
iterations
max
mi s
mat
ch (
pu)
volts
(p u
)
0 10 20 30 40 50 60 701E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
ucteWP2002
mismatch dvreg
iterations
max
mi s
mat
ch (
pu)
volts
(p u
)
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penalty-based power flow (PBPF)
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penalty-based power flow (PBPF)
extend VR:
• consider physical limits Qmini ≤ Qgi ≤ Qmax
i , i ∈ R
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penalty-based power flow (PBPF)
extend VR:
• consider physical limits Qmini ≤ Qgi ≤ Qmax
i , i ∈ R
improve VR:
• use better optimization algorithm
• keep second order terms in Newton system
• use better penalties to encode preferences
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formulate as optimization, encode preferences & physical limits in objective
minimizex
ϕ(x) = ϕu(x) + ϕq(x) + ϕr(x)
subject to f(x) = 0
x also includes {Qi}i∈R
f is now mismatches at all buses except slack
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formulate as optimization, encode preferences & physical limits in objective
minimizex
ϕ(x) = ϕu(x) + ϕq(x) + ϕr(x)
subject to f(x) = 0
x also includes {Qi}i∈R
f is now mismatches at all buses except slack
penalties:
ϕu(x) =∑j∈U
ϕuj (vj)
ϕq(x) =∑j∈R
ϕqj(Qgj)
ϕr(x) =∑j∈R
ϕrj(vj)
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for unregulated magnitudes (quartic)
ϕuj (z) =
(2z − vmax
j − vminj
vmaxj − vmin
j
)4
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for unregulated magnitudes (quartic)
ϕuj (z) =
(2z − vmax
j − vminj
vmaxj − vmin
j
)4
for reactive powers of regulating gens (sextic)
ϕqj(z) =
(2z −Qmax
j −Qminj
Qmaxj −Qmin
j
)6
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for unregulated magnitudes (quartic)
ϕuj (z) =
(2z − vmax
j − vminj
vmaxj − vmin
j
)4
for reactive powers of regulating gens (sextic)
ϕqj(z) =
(2z −Qmax
j −Qminj
Qmaxj −Qmin
j
)6
for regulated magnitudes (softmax between quartic & quadratic)
ϕrj(z) =1
2log(e2ϕ
uj (z) + e2ϕ
2j(z))
+ bj,
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v t vmax ,Qmaxvmin ,Qmin
1
0
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apply augmented Lagrangian method [8][10][22][23]
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apply augmented Lagrangian method [8][10][22][23]
approx. solve sequence
minimizex
Lµi(x, λi) = µiϕ(x)− µiλTi f(x) +1
2‖f(x)‖22,
indexed by i, generate approx. solutions {xi}i∈N
µi positive scalars, λi Lagrange multiplier estimates
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apply augmented Lagrangian method [8][10][22][23]
approx. solve sequence
minimizex
Lµi(x, λi) = µiϕ(x)− µiλTi f(x) +1
2‖f(x)‖22,
indexed by i, generate approx. solutions {xi}i∈N
µi positive scalars, λi Lagrange multiplier estimates
solving subproblems: apply truncated LSNR procedure [33][37]
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apply augmented Lagrangian method [8][10][22][23]
approx. solve sequence
minimizex
Lµi(x, λi) = µiϕ(x)− µiλTi f(x) +1
2‖f(x)‖22,
indexed by i, generate approx. solutions {xi}i∈N
µi positive scalars, λi Lagrange multiplier estimates
solving subproblems: apply truncated LSNR procedure [33][37]
truncated LSNR iteration solves Hkpk = −dk only approx.
• Hk is ∇2xLµi(xk, λi), dk is ∇xLµi(xk, λi)
• xk is current iterate
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we use modified Conjugate Gradient [12]
• exits with pk = −Φkdk, Φk � 0, sufficient descent direction for Lµi(·, λi)
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we use modified Conjugate Gradient [12]
• exits with pk = −Φkdk, Φk � 0, sufficient descent direction for Lµi(·, λi)
preconditioner:
low-stretch spanning tree (resistance)
low-stretch spanning tree (reactance)
original
merge low-stretch spanning trees [4][21], find Hk based on resulting tree
• captures important info of Hk 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
most common aspect of methods from literature:
• tested on small toy problems (IEEE networks [1])
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3.4. real networks
most common aspect of methods from literature:
• tested on small toy problems (IEEE networks [1])
in reality: networks can be large (some > 45k buses)
IEEE 14 Alberta 2.5k
performance on toy problems gives little info
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3.4. real networks
most common aspect of methods from literature:
• tested on small toy problems (IEEE networks [1])
in reality: networks can be large (some > 45k buses)
IEEE 14 Alberta 2.5k
performance on toy problems gives little info
our data: real power networks
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4. implementation
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4. implementation
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
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 R©E raw format version 32
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5. experiments
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5. experiments
power flow test cases
name buses branches
ieee118 118 186
ieee300 300 411
ucteSL2002 1254 1944
ucteWL2002 1254 1944
ucteWP2002 1254 1944
aesoSL2014 2495 2823
aesoSP2014 2495 2823
aesoWP2014 2529 2869
eiSP2008 46197 60177
eiSP2012 43792 57483
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method comparison (warm start)
name LSNR PSSE PSSE Q-lim VR PBPF
ieee118
ieee300
ucteSL2002
ucteWL2002
ucteWP2002
aesoSL2014
aesoSP2014
aesoWP2014
eiSP2008
eiSP2012
converged
failed
unavailable
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method comparison (flat start)
name LSNR PSSE FNH PSSE Q-lim VR PBPF
ieee118
ieee300
ucteSL2002
ucteWL2002
ucteWP2002
aesoSL2014
aesoSP2014
aesoWP2014
eiSP2008
eiSP2012
converged
failed
unavailable
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method comparison (progress on large cases)
0 5 10 15 20 25 30 35 40 451E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
eiSP2008 - warm start
LSNR VR PSSE PSSE Q-lim
iterations
max
mi s
mat
ch (
pu)
0 5 10 15 20 25 30 35 40 451E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
eiSP2012 - warm start
LSNR VR PSSE
iterationsm
ax m
i sm
atch
(pu
)
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method comparison (perturbing x0)
aesoSL2014seed NR LSNR VR1234567891011121314151617181920
initial angle + N (0, 0.32) (degrees)
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method comparison (perturbing x0)
ucteWL2002seed NR LSNR VR1234567891011121314151617181920
initial angle + N (0, 0.52) (degrees)
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contingency analysis (scaling net injections)
1.0 1.1 1.2 1.3 1.4 1.586
88
90
92
94
96
98
100
aesoSL2014 N-1 gens (235)
VR LSNR NR
scaling factor
case
s s o
lved
(%
)
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.1686
88
90
92
94
96
98
100
ucteWL2002 N-1 gens (377)
VR LSNR NR
scaling factorca
ses
s olv
ed (
%)
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6. conclusions
line search:
• simple to implement
• not expensive
– only a few function evaluations∗ easy to parallelize
• big value
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flat network homotopy:
• phase shifters are important but not enough
• need more transformations to simplify problem
• try to make DC power flow assumptions [39] hold:
– near flat θ profile– near flat v profile– small r and high x/r ratio
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flat network homotopy:
• phase shifters are important but not enough
• need more transformations to simplify problem
• try to make DC power flow assumptions [39] hold:
– near flat θ profile– near flat v profile– small r and high x/r ratio
to do
• understand θ & v profile as net injections vary
• understand θ & 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
• 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
• 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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