Solving a Class of Optimal Power Flow Problems in Tree ... · Solving a Class of Optimal Power Flow...

Solving a Class of Optimal Power Flow Problemsin Tree Networks

Amir Beck1 Yuval Beck2 Yoash Levron3 Alex Shtof3

Luba Tetruashvili3

1Tel Aviv University

2Holon Institute of Technology

3Technion - Israel Institute of Technology

July, 2018

Power networks and problems

Power network (AC)

”A mathematical model describing problems on a large electricalcircuit which conveys power from generators to consumers.”

Ingredients

I Model data - an augmented graph

I Decision variables

I Constraints

Two fundamental problems

Feasibility Power Flow

Optimization Optimal Power Flow

Power network (AC)

C1v1, s1

v2, s23

C3v3, s3

C4v4, s4

z12z 1

Model data: P = (G , z, C1, . . . , Cn).

- Topology graph G = (V ,E ).

- Edge impedances 0 6= zij ∈ C.

- Nodal constraints Ci ∈ R× C.

Decision variables:Voltage - v ∈ Cn, Power - s ∈ Cn.

Constraints:

FF(P) = {(v, s) :

si =∑

j∈N(i)

viv∗i − v∗j

z∗iji ∈ V

arg(v1) = 0

(|vi |, si ) ∈ Ci i ∈ V

Power network (AC)

C1v1, s1

v2, s23

C3v3, s3

C4v4, s4

z12z 1

Constraints:

FF(P) = {(v, s) :

si =∑

j∈N(i)

viv∗i − v∗j

z∗iji ∈ V

arg(v1) = 0

(|vi |, si ) ∈ Ci i ∈ V

Power network (AC)

C1v1, s1

v2, s23

C3v3, s3

C4v4, s4

z12z 1

Constraints:

FF(P) = {(v, s) :

si =∑

j∈N(i)

viv∗i − v∗j

z∗iji ∈ V

arg(v1) = 0

(|vi |, si ) ∈ Ci i ∈ V

Power network (AC)

C1v1, s1

v2, s23

C3v3, s3

C4v4, s4

z12z 1

Constraints:

FF(P) = {(v, s) :

si =∑

j∈N(i)

viv∗i − v∗j

z∗iji ∈ V

arg(v1) = 0

(|vi |, si ) ∈ Ci i ∈ V

Common constraints

PQ constraint - known power demand

Ci = [ui , ui ]× {si}.

Examples:Ci = [0.9, 1.1]× {−5− 1ı}.

Ci = [0.9,+∞]× {−5− 1ı}.

PV constraint - known voltage and power generation

Ci = {ui} × {pi + ı[qi, qi ]}.

Example:Ci = [1.05]× {10 + ı[−5,+∞]}.

Common constraints

Ci = [ui , ui ]× {si}.

Examples:Ci = [0.9, 1.1]× {−5− 1ı}.

Ci = [0.9,+∞]× {−5− 1ı}.

Ci = {ui} × {pi + ı[qi, qi ]}.

Example:Ci = [1.05]× {10 + ı[−5,+∞]}.

Common constraints

Ci = [ui , ui ]× {si}.

Examples:Ci = [0.9, 1.1]× {−5− 1ı}.

Ci = [0.9,+∞]× {−5− 1ı}.

Ci = {ui} × {pi + ı[qi, qi ]}.

Example:Ci = [1.05]× {10 + ı[−5,+∞]}.

The optimal power flow (OPF) problem

Input: A power network P = (G , z, C1, . . . , Cn), and a costfunction f : Cn × Cn → RObjective: Solve the optimization problem

minv,s

f (v, s)

s.t. (v, s) ∈ FF(P)

Cost function examples:

f (v, s) =∑i∈Gen

re(si ), f (v, s) =∑i∈PQ

||vi | − 0.5 · (ui + ui )|

A common setup

I Consumers have PQ constraints

I Generators have box constraints.

minv,s

f (v, s)

s.t. (v, s) ∈ FF(P)

||vi | − 0.5 · (ui + ui )|

A common setup

minv,s

f (v, s)

s.t. (v, s) ∈ FF(P)

||vi | − 0.5 · (ui + ui )|

A common setup

A non-convex and potentially non-smooth problem.

The bad news

Karsten Lehmann, Alban Grastien, Pascal Van Hentenryck.AC-Feasibility on Tree Networks is NP-HardIEEE Transactions on Power Systems, 2015

Approaches

I Heuristics1

I Interior point methodsI Convex relaxations

I Global solution methods (e.g. branch and bound)I Specialized solution2 methods for sub-classes

I Tight convex relaxationsI Our method

Stephen Frank, Ingrida Steponavice, Steffen Rebennack

Optimal power flow: a bibliographic survey I

Energy Systems, 2012

Daniel Bienstock

Progress on solving power flow problems

Optima, 2013

1Known to work well in in practice, but no theory explaining why.2A method equipped with a theory explaining why it works

Approaches

I Heuristics1

I Interior point methodsI Convex relaxations

I Global solution methods (e.g. branch and bound)I Specialized solution2 methods for sub-classes

I Tight convex relaxationsI Our method

Stephen Frank, Ingrida Steponavice, Steffen Rebennack

Optimal power flow: a bibliographic survey I

Energy Systems, 2012

Daniel Bienstock

Progress on solving power flow problems

Optima, 2013

1Known to work well in in practice, but no theory explaining why.2A method equipped with a theory explaining why it works

Our setup

Assumptions

Network assumptions

I The topology is a tree T = (V ,E ) with root(T ) = 1 and|V | > 1.

I Leaf constraints: Ci is a compact PQ or PV constraint.

I Root constraint: C1 = [ur , ur ]× C.

I Remaining constraints: Ci is a PQ constraint.

I Non-zero voltages: (u, s) ∈ Ci =⇒ u > 0.

Problem assumptions

I The objective function f is continuous.

Assumptions

Network assumptions

I The topology is a tree T = (V ,E ) with root(T ) = 1 and|V | > 1.

I Leaf constraints: Ci is a compact PQ or PV constraint.

I Root constraint: C1 = [ur , ur ]× C.

Problem assumptions

I The objective function f is continuous.

Visualization

Example

0.005ı C1 = [0.97,∞]× C Root constraint

C2 = [0.9, 1.1]× {−0.2− 0.1ı} PQ constraint

C3 = [0.9, 1.1]× {−0.4− 0.3ı} PQ constraint

C4 = {1} × (0.25 + ı[−1, 1]) PV constraint

Motivation

Only one degree of freedom. So why the effort?

Motivation

I It is challenging - to the best of our knowledge, no knownefficient solution.

I Useful as a computational step.

https://flic.kr/p/5jLLgR

The tree reduction / expansion method

The main result

I A decomposition of the feasible set into a finite union ofparameterized curves:

FF(P) =m⋃i=1

image(γi),

where γi : [0, 1]→ Cn × Cn are continuous and piecewisesmooth functions.

I An algorithm to compute a representation of {γi}mi=1.

I An efficient algorithm to compute γi(t) given i and t.

Solving OPF

I Compute a representation of the curves: {γi}mi=1.

I Choose sampling density N ∈ N.

I Grid search:

(v, s) ∈ argmin(v,s)

{f (v, s) : (v, s) = γi (j/(N − 1)),

i = 1, . . . ,m, j = 0, . . . ,N − 1}

Observation: For any (v, s) ∈ FF(P), the vector s is redundant:

si =∑

j∈N(i)

viv∗i − v∗j

z∗ij, i = 1, . . . , n.

We work with FFv (P) = {v : (v, s) ∈ FF(P)}.

Observation: Compact PQ and PV constraints are line segments inR× C.

Example: [0.9, 1.1]× {−0.4− 0.3ı} is the line segment between(0.9,−0.4− 0.3ı) and (1.1,−0.4− 0.3ı).

Observation: Line segments are curves.

Idea: Replace leaf constraints with parameterized curves -functions from [0, 1] to R× C.

Observation: Compact PQ and PV constraints are line segments inR× C.

Example: [0.9, 1.1]× {−0.4− 0.3ı} is the line segment between(0.9,−0.4− 0.3ı) and (1.1,−0.4− 0.3ı).

Observation: Line segments are curves.

Idea: Replace leaf constraints with parameterized curves -functions from [0, 1] to R× C.

Example

0.005ı

Input network

C1 = [0.97,∞]× CC2 = [0.9, 1.1]× {−0.2− 0.1ı}C3 = [0.9, 1.1]× {−0.4− 0.3ı}C4 = {1} × (0.25 + ı[−1, 1])

Curved network

C1 = [0.97,∞]× CC2 = [0.9, 1.1]× {−0.2− 0.1ı}C3 = image(ν3, σ3)

ν3(t) = 0.9 + 0.2t, σ3(t) = −0.4− 0.3ı

C4 = image(ν4, σ4)

ν4(t) = 1, σ4(t) = 0.25 + ı(−1 + 2t)

Curved network

I The topology is a tree T = (V ,E ) with root(T ) = 1.

I Leaf constraints: Ci = image(νi , σi ) given continuousνi : [0, 1]→ R and σi : [0, 1]→ C.

I Root constraint:I If |V | > 1 then C1 = [ur , ur ]× C.I If |V | = 1 then C1 = [ur , ur ]× {0}.

Next: find a relationship between FFv (P) and FFv (P ′) for asmaller network P ′.

Curved network

I The topology is a tree T = (V ,E ) with root(T ) = 1.

I Leaf constraints: Ci = image(νi , σi ) given continuousνi : [0, 1]→ R and σi : [0, 1]→ C.

I Root constraint:I If |V | > 1 then C1 = [ur , ur ]× C.I If |V | = 1 then C1 = [ur , ur ]× {0}.

Next: find a relationship between FFv (P) and FFv (P ′) for asmaller network P ′.

Tree reduction

I Reducible node - a non-leaf whose children are all leaves

I Tree reduction - the operation of removing all children ofsome reducible node.

Example

T T ′

Tree-Reduction Theorem

Let P = (T , z, C1, . . . , Cn) be a curved network.

Let j be areducible node, let T ′ be the resulting reduction, and w.l.o.g itschildren are {j + 1, . . . , n}. Define

νk(t) = |νk(t)− zkj(σk(t))∗/νk(t)|, k = j + 1, . . . , n

assume νk are invertible, and let

Uj = [uj , uj ] ∩ image(νj+1) ∩ · · · ∩ image(νn).

I If Uj = ∅ then FF(P) = ∅.I Otherwise, the there exist functions hj+1, . . . , hn, and a curveC′j , such that v ∈ FFv (P) if and only if

(v1, . . . , vj) ∈ FFv (T ′, z′, C1, . . . , Cj−1, C′j),vk = hk(vj), k = j + 1, . . . , n

j[uj , uj ]× {sj}

ν, σ

. . . n

ν, σ

Let P = (T , z, C1, . . . , Cn) be a curved network. Let j be areducible node, let T ′ be the resulting reduction, and w.l.o.g itschildren are {j + 1, . . . , n}.

Define

j[uj , uj ]× {sj}

ν, σ

. . . n

ν, σ

Let P = (T , z, C1, . . . , Cn) be a curved network. Let j be areducible node, let T ′ be the resulting reduction, and w.l.o.g itschildren are {j + 1, . . . , n}. Define

j[uj , uj ]× {sj}

ν, σ

. . . n

ν, σ

I If Uj = ∅ then FF(P) = ∅.

I Otherwise, the there exist functions hj+1, . . . , hn, and a curveC′j , such that v ∈ FFv (P) if and only if

j[uj , uj ]× {sj}

ν, σ

. . . n

ν, σ

j[uj , uj ]× {sj}

ν, σ

. . . n

ν, σ

Tree-Reduction Theorem - the definition of hk and C ′jLet

σk(t) = σk(t)− zkj|σk(t)|2

(νk(t))2,

φk = σk ◦ ν−1k ,

hk(vj) = vj − zkjφk∗(|vj |)v∗j

and C′j = image(νj , σj) with

νj(t) = (1− t) · (minUj) + t · (maxUj),

σj(t) =

sj +n∑

(φk ◦ νj)(t), j 6= 1,

0 j = 1.

Tree reduction theorem - example

The big picture (j = 2)

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒(v1, v2)T ∈ FFv (P ′),

v3 = h3(v2),

v4 = h4(v2).

z24z12

C′2z12

P ′:

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒(v1, v2)T ∈ FFv (P ′),

v3 = h3(v2),

v4 = h4(v2).

z24z12

C′2z12

P ′:Details:

I Compute ν3 and ν4. Verify invertability.

I Compute U2 = [u2, u2] ∩ image(ν3) ∩ image(ν4). Verify U2 6= ∅.I Compute the functions h3 and h4, and the curve C′2.

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒(v1, v2)T ∈ FFv (P ′),

v3 = h3(v2),

v4 = h4(v2).

z24z12

C′2z12

P ′:Details:

I Compute ν3 and ν4. Verify invertability.

I Compute U2 = [u2, u2] ∩ image(ν3) ∩ image(ν4). Verify U2 6= ∅.I Compute the functions h3 and h4, and the curve C′2.

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒v1 ∈ FFv (P ′′),

v2 = h2(v1),

v3 = h3(v2),

v4 = h4(v2).

z24z12

P ′′:

Observation

I C′1 = U1 × {0}I =⇒ FF(P ′′) = {(v1, s1) : |v1| ∈ U1, s1 = 0, arg(v1) = 0}I =⇒ FFv (P ′′) = U1.

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒v1 ∈ FFv (P ′′),

v2 = h2(v1),

v3 = h3(v2),

v4 = h4(v2).

z24z12

P ′′:

Observation

I C′1 = U1 × {0}

I =⇒ FF(P ′′) = {(v1, s1) : |v1| ∈ U1, s1 = 0, arg(v1) = 0}I =⇒ FFv (P ′′) = U1.

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒v1 ∈ FFv (P ′′),

v2 = h2(v1),

v3 = h3(v2),

v4 = h4(v2).

z24z12

P ′′:

Observation

I C′1 = U1 × {0}I =⇒ FF(P ′′) = {(v1, s1) : |v1| ∈ U1, s1 = 0, arg(v1) = 0}

I =⇒ FFv (P ′′) = U1.

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒v1 ∈ FFv (P ′′),

v2 = h2(v1),

v3 = h3(v2),

v4 = h4(v2).

z24z12

P ′′:

Observation

I C′1 = U1 × {0}I =⇒ FF(P ′′) = {(v1, s1) : |v1| ∈ U1, s1 = 0, arg(v1) = 0}I =⇒ FFv (P ′′) = U1.

Conclusion

(v1, v2, v3, v4)T ∈ FFv (P)

⇐⇒v1 ∈ U1,

v2 = h2(v1),

v3 = h3(v2),

v4 = h4(v2).

Invertability =⇒ the feasible set is a curve γ1 : [0, 1]→ R× C

The above is an algorithm to evaluate γ1(t)

Two-phase meta-algorithm

Phase 1 - tree reduction- Perform a sequence of reductions, until one-node network isobtained.- Compute hk functions (φk in practice) and Uj intervals.- If any Uj = ∅ - return ”problem infeasible”- If any νk non-invertible - return ”error”

Phase 2 - tree expansion

- Take any v1 ∈ U1.- Use expand v1 to the full v vector using hk functions.- Compute the s vector.

Inverting νk

source: Wikipedia

The remedy - spline approximation

Choose approx. density d . Let t = 1d−1(0, 1, 2, . . . , d − 1).

I νk and σk are approximated by vectors:

νk = (νk(t1), . . . , νk(td)),

σk = (σk(t1), . . . , σk(td)).

I Approximations of νk and σk are computed componentwise:

∣∣∣∣νk − zkj(σk)∗

∣∣∣∣ ,σk = . . .

I φk = σk ◦ ν−1k is approximated by a cubic spline interpolantmapping the components of νk to σk .

Multiple curves

νk(t) invertible ⇐⇒ it is strictly monotone.

νk(t)

I1 I2 I3

Clearly, νk is invertible on I1, I2, and I3.Solution: Split the network P into 3 networks.

Multiple curves

νk(t)

0 1I1 I2 I3

Clearly, νk is invertible on I1, I2, and I3.

Solution: Split the network P into 3 networks.

Multiple curves

νk(t)

0 1I1 I2 I3

Clearly, νk is invertible on I1, I2, and I3.Solution: Split the network P into 3 networks.

Multiple curves - cont.

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Eventually: multiple terminal single-node networks

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Reduction

P ′(1)

P ′(2)

P ′(3)

Reduction

infeasble

P ′′(2)

P ′′(3)

Numerical results

Experiment setup

I Test networks: IEEE radial distribution feeders.http://sites.ieee.org/pes-testfeeders/

I Modified to conform to the assumptionsI PV constraints on leaves only.I zij for each edge (i , j) ∈ E .

I Several testsI Accuracy - distance from the feasible set w.r.t the

approximation density.I Reliability - comparison with MATPOWER3.I Number of parallel networks in practice.

I Software available fromhttps://github.com/alexshtf/trem_opf_solver.

3A well-known PF and OPF solver. Stability function can be specified usingthe extension mechanism.

Accuracy

Test setup

I Check several approximation densities d ∈ [23, 212].

I For each approximation density, sample each curve atN = 1000 points.

I For each deviation measure, report the highest value amongthe samples.

Deviation measuresPQ constraints

EPQ,v (v, s) = maxj∈PQ

d(|vj |, [uj , uj ]

EPQ,s(v, s) = maxj∈PQ

|sj − sj |,

PV constraints

EPV ,v (v, s) = maxj∈PV

|uj − |vj ||

EPV ,p(v, s) = maxj∈PV

|pj − re(sj)|

EPV ,q(v, s) = maxj∈PV

im(sj), qj , qj

Accuracy

EPQ,v and EPV ,q were zero in all our experiments.

Accuracy

Reliability

Test setup

I Perturb the PQ constraints of each network:

[uj , uj ]× {re(sj) · αj + (im(sj) · βj)ı},

where αj , βj ∼ U[0, 2].

I Generate 5000 random networks from each existing network.

I Solve OPF with the “stability” objective function:

f (v, s) =∑j∈PQ

∣∣∣∣|vj | − 1

2(uj + uj)

∣∣∣∣I Solve using both MATPOWER and our solver.

I Gather statistics.

Reliability

13-node 34-node 37-node 47-node 69-node 123-node

6.46% 5.22% 2.56% 2.78% 2.34% 39.64%

The % of random networks, generated from each original network,for which our method found a solution while MATPOWER did not.

Number of networks in parallel

Observation: νk becomes more ‘wild’ when |z | increases:

νk(t) = |νk(t)− zkj(σk(t))∗/νk(t)|

Setup: Make νk non-invertible my replacing z with αz forα ∈ [1, 10].

Results - Maximum (worst) number of networks in parallel

3 5 4 1 2 2

Questions?

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