main twist today: separate DC power flow : ( Y ) = 0 , v = 1 , & - - PowerPoint PPT Presentation

Fast power system analysis via implicit linearization of the power flow manifold

Allerton Conference 2015 Saverio Bolognani Florian D¨

• rfler

Power flow equations

Basic ingredients in an n-bus power system complex bus voltage uh = vhejθh ∈ C injected complex power sh = ph + jqh admittance matrix Y ∈ Cn×n Underlying nonlinear power flow equations Kirchhoff’s & Ohm’s laws nodal power balances

• =

⇒ diag(u)Yu = s Bus model specifies variables & fixed parameters PV bus: ph and vh fixed PQ bus: ph and qh fixed slack bus: vh and θh fixed ⇒ remaining ingredients are variables

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A brief history of power flow approximations

for computational tractability, analytic studies, & control/optimization design

DC power flow: ℜ(Y ) = 0, v = 1, & linearization

• B. Stott, J. Jardim, & O. Alsac, “DC Power Flow Revisited” IEEE TPS, 2009.

LinDistFlow: parameterization in flow & v2

h coordinates & linearization

M.E. Baran & F.F. Wu, Optimal sizing of capacitors placed on a radial distribution system. IEEE PES, 1988.

rectangular DC power flow: fixed-point ball for small S2/V 2

slack

• S. Bolognani & S. Zampieri, On the existence and linear approximation of the

power flow solution in power distribution networks. IEEE TPS, 2015.

many variations & extensions, sensitivity & Jacobian methods, etc.

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main twist today: separate power flow & bus model

1) derivation of linear implicit model 2) relation to other approximations 3) accuracy in the three-phase case 4) some direct applications

Linear implicit model & its advantages

today consider all of x = (v, θ, p, q) as variables implicit model for power flow manifold: F(x) = 0 linear approximant at x∗ is tangent plane: A(x − x∗) = 0 Advantages of linear implicit model:

◮ sparsity

⇒ tractable for applications with high computational burden

◮ structure-preserving

⇒ prior for distributed control,

• ptimization, estimation, etc.

◮ geometric methods

⇒ explicitly require tangent planes

q p

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• 1

theta 1 2 2 1

• 1

1 0.5

• 0.5
• 1
• 1.5
• 2

two-bus example

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The power flow manifold & linear tangent approximation

node 2 node 1

v1 = 1, θ1 = 0 y = 0.4 − 0.8j v2, θ2 p2, q2 p1, q1

1 0.5 p2

• 0.5

0.5 q2 1 1.2 1 0.8 0.6 0.4 v 2 1 power flow manifold: F(x) = 0 2 normal space spanned by ∂F(x)

∂x

• x∗=AT

3 tangent space at x∗: A(x − x∗) = 0 4 accuracy depends on curvature ∂2F(x)

∂x2

1.5 1 0.5 q2

• 0.5
• 1

1.5 1 0.5 p2

• 0.5
• 1

1.2 1 1.4 0.8 0.6 v 2

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Closer look at implicit formulae A(x − x∗) = 0

• diag Yu∗ + diag u∗NY
• ·

diag(cos θ∗) − diag(v∗) diag(sin θ∗) diag(sin θ∗) diag(v∗) diag(cos θ∗)

• shunt loads
• lossy DC flow
• rotation × scaling at operating point

× v − v∗ θ − θ∗

• =

p − p∗ q − q∗

• deviation variables

where N = I −I

• is complex conjugate in real coordinates

and A = ℜ(A) −ℑ(A) ℑ(A) ℜ(A)

• is complex rotation in real coordinates.

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. . . appears cumbersome at first glance

Special cases reveal some old friends I

flat-voltage/0-injection point: x∗ = (v∗, θ∗, p∗, q∗) = (1, 0, 0, 0) ⇒ implicit linearization: ℜ(Y ) −ℑ(Y ) −ℑ(Y ) ℜ(Y ) v θ

• =

p q

• is linear coupled power flow [D. Deka, S. Backhaus, & M. Chertkov, 2015]

⇒ ℜ(Y ) = 0 gives DC power flow: −ℑ(Y )θ = p and −ℑ(Y )v = q

2 1 !2

• 1
• 2

1.4 1.2 v 2 1 0.8 0.6 0.5

• 1
• 0.5

1 1.5 p2

power flow manifold linear coupled power flow DC power flow approximation (neglects PV coupling)

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Special cases reveal some old friends II

flat-voltage/0-injection point: x∗ = (v∗, θ∗, p∗, q∗) = (1, 0, 0, 0) ⇒ rectangular coord. ⇒ rectangular DC flow [S. Bolognani & S. Zampieri, 2015] nonlinear change to quadratic coordinates from vh to v2

h

⇒ linearization gives (non-radial) LinDistFlow [M.E. Baran & F.F. Wu, 1988]

1.5 1 0.5 q2

• 0.5
• 1

2 1 p2 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

• 1

v 2

power flow manifold linear approximation linear approximation in quadratic coordinates

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so all standard approximations are included as special cases

Extensions to more general models

Bus device models, e.g., PQ bus sh = ph + jqh = const. ⇒ implicit constraint g(x) = 0 & can be absorbed in F(x) = 0 Exponential load models sh = const. · vconst.

h

⇒ can be handled analogously Unbalanced three-phase grids with basic ingredients complex voltage uh = [ua

h ub h uc h]T ∈ C3

similar definitions for other quantities ⇒ all previous results can be analogously re-derived Matlab/Octave code available: https://github.com/saveriob/1ACPF

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Accuracy illustrated with unbalanced three-phase IEEE13

• exact solution

⋆ linear implicit model

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a glorified & highly accurate linearization . . . so what? ——– some direct applications

Fast scenario-based decision making under uncertainties

Example: feasible region for distribution network operation

• xdec
• actuation
• Prob{ xexo ∈ Xexo
• random loads

: F(x) = 0

• power flow

& Vx ≤ w constraints } ≥ 1 − ε chance

• Scenario-based approach:

sample xexo variables & build deterministic constraints ⇒ decision xdec is feasible with high probability for sufficiently many samples

Vtab P671

• S. Bolognani & F. D¨
• rfler. Fast scenario-based decision making in unbalanced

distribution networks. Power Systems Computation Conference (PSCC), June 2016.

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Distributed online optimization on power flow manifold

with Adrian Hauswirth & Gabriela Hug (ETH Z¨ urich)

power flow manifold tangent space new operating point

projected gradient step (distributed algorithm) new operating point (physical system) injections measurements

50 100 150 200 250 300 350 400 450 500 730 740 750 760 770 780 790 800 810 Objective Value [$] realized cost lower bound 50 100 150 200 250 300 350 400 450 500 1.01 1.02 1.03 1.04 1.05 1.06 Voltage Levels [p.u.]

projected gradient gradient of cost

• perating

point applied to optimal voltage control in IEEE 30 bus grid

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Cascading failures – more accuracy for similar comp. effort

with Giovanni Sansavini & Bing Li (ETH Z¨ urich)

5 10 15 20 25 30 35 0.05 0.1 0.15 0.2 Line Number Relative Error DC Linear AC ! "!! #!! $!! %!! &!!! &"!! ! &! "! '! #! (! $! )! *+,,-.-/0-1+/12345167-55+/8192+/-4.1:;1!1*;<19=>< ?00@.-/0-19A< 65 70 75 80 85 90 95 100 20 40 60 80 Percentage of reduced computation time(%) Occurence (%)

RTS24 with random loads & N-2 contingencies reduced computation time over AC (%) line number load shedding: linear AC - DC (MW)

• ccurrence (%)
• ccurrence (%)

relative error

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Cascading failures — distinct cascades under naive DC flow

with Giovanni Sansavini & Bing Li (ETH Z¨ urich)

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• 5

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• 5

DC power flow Linear AC power flow

line current (A) line current (A) voltage magnitude (p.u.) voltage angle (deg) voltage angle (deg)

N-2 contingency DC flow is optimistic compared to linear AC single line outage since voltage magnitude can (partially) compensate for other remaing line blackout

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Monitoring, state estimation, learning, & detection

Consistency equation: yk

• measurement

= H · xk system model + εk

• noise

+ dk

• attack

Attack detection: collect measurements yk over time & look for a consistent low-rank (ID × time space) input dk Results for IEEE 123 model: harder to trick an operator relying on a linearized AC model

10 20 30 40 50 10 20 30 40 50 60 Time Step ID of User

• D. Drzajic, S. Bolognani, & F. D¨
• rfler. Energy theft detection using compressive

sensing methods. ETH Z¨ urich Semester project, August 2015.

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Conclusions

Summary linear, sparse, & structure-preserving model includes all DC & LinDistFlow approximations applicable to unbalanced three-phase systems apps: monitoring, control, decision-making, . . . Ongoing & future work theory: error bounds & coord trafos applications: further develop apps Acknowledgements Sandro Zampieri, Adrian Hauswirth, Gabriela Hug, Dalibor Drzajic, Giovanni Sansavini, & Bing Li

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Saverio Bolognani

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