Optimal Power Flow solver based on HELM (Holomorphic ...

Optimal Power Flow solver based on HELM

(Holomorphic Embedded Load Flow)

Ian Wallace, Andreas Grothey, Ken McKinnon

School of Mathematics, University of Edinburgh

ISMP 2018, 1-6 July 2018, BordeauxT

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U N I V E R S

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OF

ED I N B U

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I Wallace, A Grothey, KIM McKinnon HELM based OPF solver

Motivation: Relaxations/Approximations of OPF

AC Optimal Power Flow + integer decisions

Interest in solving AC-OPF + integer decision(Unit Commitment, Topology Optimization, etc)

AC-OPF is nonlinear, nonconvex → difficult globaloptimization problem

⇒ Approximations (DC, PWL)Relaxations (SDP) have been proposed.

Evidence is that local solvers findglobal solution quite often

How useful are local solvers?


Motivation

Background

In 2012 Trias proposed HELM (Holomorphic EmbeddedLoad-flow Method)Claims to find the one, unique, stable, physical LF solutionout of the 22n possible solution of the algebraic LF equations.

No formal proof(?), but claim seems to hold up in practice

Questions

Can you build an AC OPF solver around HELM?

Is that competitive to other AC OPF solvers?

Does it avoid unstable OPF solutions?

Does it even avoid local solutions?(or at least increase likelyhood of finding the global one)

Supplementary perk

Interpretation of LF-based OPF solvers as reduced space SLP


Optimal Power Flow Review

Optimal Power Flow (OPF)

Optimal Power Flow is the problem of deciding on (cost)optimal electricity generator outputs to meet demand withoutoverloading the transmission network

Solved at a given time for known demand






Need to model all lines in the networkPower flows are AC: represented by complex flows (or real andreactive components)

⇒ nonlinear network constraints






Need to model all lines in the networkPower flows are AC: represented by complex flows (or real andreactive components)

⇒ nonlinear network constraintsUnlike other transportationnetworks, operator has nocontrol over routing. Routingis determined by physics!

⇒ Kirchhoffs Laws.


Power System Operation

Network

b ∈ B Buses, N = |B|

l = (bb′) ∈ L Lines

g ∈ G Generators (at bus og )

Power flows are AC. Described by real and reactive flows over lines,voltage and phase angle at buses

Variables

vb Voltage level at bus bδb Phase angle at bus bpbb′ , qbb′ Real and reactive power flow on line l = (bb′)pGg , q

Gg Real and reactive power output at generator g



Network

b ∈ B Buses, N = |B|

l = (bb′) ∈ L Lines

g ∈ G Generators (at bus og )

Parameters of the model are: line/bus characteristics, real/reactivedemands, limits on power flow and voltages

Parameters

Gbb′ ,Bbb′ conductance and susceptance of line l

(reciprocals of resistance and reactance)

Cb bus susceptancePDb ,Q

Db real and reactive power demand at bus b



Operator decides on real generation level and voltages atgenerators.

Power flows and reactive generation arrange themselves so asto satisfy power flow equations (Kirchhoff Laws)

Constraints

Kirchhoff Voltage Law (KVL)

pLbb′ = vb[Gbb′(vb − vb′ cos(δi − δj)) + Bbb′vb′ sin(δb − δb′)]

qLbb′ = vb[−Gbb′vb′ sin(δi − δj ) + Bbb′(vb′ cos(δb − δb′)− vb)]

Kirchhoff Current Law (KCL)∑

g |og=b

pGg − PDb =

∑

(b,b′)∈L

pbb′ , ∀b ∈ B,

∑

g |og=b

qGg − QDb =

∑

(b,b′)∈L

qbb′ + Cbvb2, ∀b ∈ B



Line flows and generator reactive output should satisfy

Operational Limits

Line Flow Limits at both ends of each line

(pbb′)2 + (qbb′)

2 ≤ fl2, (pb′b)

2 + (qb′b)2 ≤ fl

2

Voltage limits at buses

Vmin ≤ vb ≤ Vmax

reactive generation limits at generators

Qming ≤ qGg ≤ Qmax

g


OPF: Control formulation

The variables of the OPF model can be divided into control andstate variables

Control u (set by the system operator)pGg real power output at generators

vg voltage levels at generation busesState x (determined by Kirchhoff’s laws)vb voltage levels at non-generating busesqGg reactive power output at generators

δb phase anglespbb′ , qbb′ real and reactive power flows over lines

OPF model

minu∈U f (u)s.t. KL(u, x) = 0 (Kirchhoff’s laws)

g(x) ≤ 0 (Operational limits)







OPF model (reduced space)

minu∈U f (u)s.t. KL(u, x(u)) = 0 (Kirchhoff’s laws)

g(x(u)) ≤ 0 (Operational limits)

where x(u) is the(?) solution to the implicit functionKL(u, x(u)) = 0







OPF model (reduced space)



where x(u) is the(?) solution to the implicit functionKL(u, x(u)) = 0Nonlinear! nonconvex! May have local solutions!


The Load-Flow problem

The Load-Flow problem is the problem of finding the state x of asystem for given controls u by solving Kirchhoff’s laws

Load-Flow

Given u find x such that

KL(u, x) = 0 (Kirchhoff’s laws)

(no bounds on state variables)

Remarks:

Kirchhoffs Laws are nonlinear: There may be zero, one, ormultiple solutions.

It is often quoted that the number of solution is bounded by2N−1 (high/low voltage at every bus), however the bestknown upper bound for general systems is indeed

(2N−2N−1

)

∗

∗Guo and Salam, IEEE Trans Circ Sys, 1994


Feasible Space of Load-Flow

(from Hiskens, Davy, IEEE Trans Pow Sys 2001)


Load Flow: Issues

Load-Flow



Engineering wisdomTM:

There is only one physical solution

This solution is termed high voltage solution:it has voltages close to 1, and typically is the only solutionwithin the voltage bounds Vmin ≤ vb ≤ Vmax .

This is the only stable solution (eg dVi

dQi> 0, ∀i ∈ B)

Newton-like iteration from flat start (Vi = 1, δi = 0) willusually find this solution

Solvers: Newton or Fast Decoupled Load Flow – FDLF(simplified Newton w/o refactorizations)


OPF: Issues

OPF is nonlinear. Can and does have local solutions.

Usually solved by IPM or load-flow based solvers

Often the global solution is found by local solvers

Finding global solution important for MIP problem based onAC OPF (Unit Commitment, Line Switching, InvestmentProblems)

Received Engineering Wisdom

There is only one physical/stable/high voltage solution. (!??!)

→ There certainly are high voltage local solutions(Bukhsh, Trodden, G, McKinnon, ’13)

Question:

Can we enforced stability (in LF and thus in OPF)? Does thathelp with avoiding local solutions in OPF?


Load-flow vs OPF

Observations

LF and OPF share characteristics:Many local solutions to mathematical problem,

→ only one physical one(?)

HELM (Trias 2012) is a load-flow solver that claims to avoidnon-physical (unstable) solutions

Questions

Are the local OPF solutions “inherited” from non-physicalload-flows?

If they can be avoided in LF does that help for OPF?


HELM review

Holomorphic Embedded Load Flow Method

Proposed in 2012 by Antonio Trias(IEEE PES General meeting + Patents)

Use the rectangular coordinate (real/imaginary) version of theload-flow equations

∑

k∈B

YikVk =S∗i

V ∗i

, ∀i ∈ B

→ Given Si , solve for Vk

[Yik bus admittance matrix, Vk complex voltage at bus k ,Si = Pi + jQi complex power injection at bus i ]Introduce complex homotopy parameter z (scaling demand):

∑

k∈B

YikVk(z) =zS∗

i

V ∗i (z

∗), ∀i ∈ B

z=1: case of interest to usz=0: no-load case (easy)


HELM review

Introduce complex homotopy parameter z :

∑

k∈B

YikVk(z) =zS∗

i

V ∗i (z

∗), ∀i ∈ BPQ

Vi(z) are solution functions depending on the complexparameter z

Due to construction Vi(z) are holomorph!(⇒ know the function at one point = know it everywhere)

Vi(0) is unique (assuming that Vi(0) 6= 0,∀i) and easy toobtain.

Claim: Following this solution to z = 1 leads to the physical,stable, high voltage load flow solution.

Note:

Relation to homotopy load flow methods:(Iba, Iwamoto, Tamura ’80, Ma, Thorp ’93)


HELM review

∑

k∈B

YikVk(z) =zS∗

i

V ∗i (z

∗), ∀i ∈ BPQ

Vk(z) holomorphic ⇒ Vk(z) =∑∞

n=0 Vk [n]zn

Vk [n] =dn

dznVk(z)

∣

∣

z=0are easily obtained by successively

solving linear systems of equations (with Y as system matrix).

Matrix never changes: only one factorization neededEach solve gives coefficients Vk [n] of one order (n) for allbuses k ∈ B.

Typically radius of convergence of power series is less than 1→ cannot simply evaluate at z = 1 by summing up.


HELM review

Vk(z) =∑∞

n=0Vk [n]z

n

Typically radius of convergence of power series is less than 1.

Can use continuation of solution around singularities (poles)Pade (rational) approximants can be usedStahl’s Extremal Theorem: If there is a solution that can bereached by continuation, Pade Approximants of high enoughorder will find it

Obtaining approximations and evaluating at z = 1 is cheapand robust (given enough terms of power series)

Claim (Trias):If Pade Approximants converge, they do so to the physical,stable load flow solutionNon-convergence ⇔ no (stable) LF solution for given controlsTrias only considers pure PQ systems (+ slack bus)→ Extensions to general networks

(Subramanian ’14, Wallace, G, McKinnon ’16)I Wallace, A Grothey, KIM McKinnon HELM based OPF solver

Comparison HELM vs Newton-Load flow

HELM as load-flow solver

Only one factorization needed for entire algorithm(Newton requires one at every iteration, FDLF only needs onebut is unstable)

Number of backsolves comparable to Newton:[few Pade-terms/Newton iterations needed for well behaved system,

many needed for stressed systems close to a singularity/boundary of

feasible region]

HELM cannot take advantage of approximate solution

Guaranteed(?) to find stable/physical solution(→Seems to work in experiments)

Aim:

Build an OPF solver around HELM that inherits the desirableproperties (stability, physical solution)


Load-flow based OPF solvers


Review of Load flow OPF solvers


Review of Load flow OPF solvers

Load-flow based OPF

What is this algorithm?

Does it converge?

Under what conditions?

LF-based OPF has the reputation of not being reliable


Skeleton Algorithm: Reduced Space SLP

Reduced Space OPF



where x(u) is the (unique) solution to the implicit function

KL(u, x(u)) = 0

Step finding LP in reduced space

mind:uk+d∈U

∇f (uk)T∆u

s.t.[

∇xg(xk)dx

du(uk)

]T∆u ≤ −g(xk)

where dxdu(uk) is obtained as solution of

∇uKL(uk , xk)+∇xKL(u

k , xk)dxdu(uk) = 0


Step finding problem

We are not solving the reduced space problem directly, but replaceit with 2-step procedure:

Given uk :

1 Solve LF problem KL(uk , x(uk)) = 0 for xk (→ g(xk))

2 Linearize OPF around (uk , xk) and get step (∆u,∆x):

Full space LP

min(∆x ,∆u):uk+∆u∈U

∇f (uk)T∆u

s.t. ∇xg(xk)T∆x ≤ −g(xk)

∇uKL(uk , xk)T∆u +∇xKL(u

k , xk)∆x = 0

These 2 steps are equivalent to reduced space LP


2-Step full space Algorithm

Skeleton Algorithm

1 k = 0. Initial control u0.2 Given uk

1 Solve load flow problem for xk(uk )2 Linearize around (uk , xk). Solve full space LP for ∆u.

3 Take step u(k+1) = uk +∆u, k ← k + 1 and loop

Observations:

The above algorithm is the Load Flow based OPF of(Alsac,Bright,Paris,Stott ’90, Glavitsch & Bacher ’91)

The sequence of solving Load Flow problem + Full space LPis equivalent to solving the reduced space problem.

⇒ LF based OPF can be interpreted as reduced space SLP

⇒ Gives theoretical backup for LF-based OPF


Load-flow assumptions for reduced space method

Load-Flow



for a given u there may be multiple solutions x(u):→ x1(u), ..., xν(u).

Claim: HELM always return the unique stable solution!Not clear if that is always the same branch xi (u) (but we willassume it is)

If using a Newton based LF solver can assume we stay on thesame branch by making small enough steps in u and startingfrom previous x(u).

This assumes that we do not encounter bifurcation points(→ ∇xKL(u, x) nonsingular)


Load flow assumptions

Load-Flow



Assumption 1

The LF solutions returned by HELM are consistent (i.e. always onthe same branch of x(u)).

Assumption 2

For any point (u, x(u)) returned by HELM (or any other load flowsolver) during the algorithm we have

∇xKL(u, x(u)) is nonsingular


LF based OPF can be interpreted as reduced space SLP

Our Algorithm: Modifications

Rather than reduced space SLP do reduced space SQP(+ Trust Region + Filter + Restoration Phase)

Use HELM as load flow solver

Expected Advantages

Fast and robust convergence due to use of second orderinformation (without increase in linear algebra complexity)

Avoid unstable LF solutions due to HELM

HELM only needs single factorization for entire OPF algorithm(although more backsolves compared to Newton LF – nowarmstart)


Results

Questions

Are there unstable OPF solutions? Can HELM-OPF avoidthem?

Can HELM-OPF avoid local solutions?

Is HELM-OPF competitive?

Setup

LF algorithms implemented in Matlab with IPOPT(QP)/CPLEX (LP) as subproblem solvers.

Compared against standard LF-OPF and IPM

Random starting points (as in Bukhsh et al ’13)

General Observations

Each HELM call needs about 10-20 backsolves (to obtainpower series coefficients)

SQP converges in 5–10 iterations (SLP needed >50–100)

All OPF solutions found were stable!


Results

Compared two LF-OPF solvers: HELM and RunPF (MatPower)

Against two IPMs: RunOPF (MatPower) and IPOPT (Polarformulation)

Standard IEEE test cases. Default starting point.

RunOPF IPOPT HELM-OPF RunPF-OPFSystem Sol Time (s) Time (s) Time (s) Time (s)9-Bus 5296.69 0.15 0.13 0.12 0.1114-Bus 8081.53 0.10 0.14 0.07 0.0724-Bus 63352.21 0.12 0.19 0.16 0.1530-Bus 576.89 0.11 0.18 0.19 0.1839-Bus 41864.18 0.14 0.20 0.14 0.1357-Bus 41737.79 0.12 0.25 0.16 0.1589-Bus 5819.81 0.21 0.47 0.30 0.27118-Bus 129660.69 0.19 0.47 0.25 0.24300-Bus 719725.08 0.38 0.99 0.67 0.591354-Bus 74069.35 2.18 5.70 3.75 3.462869-Bus 133999.29 4.79 16.09 13.66 13.13

All solvers find the optimum

HELM-OPF is competitive


Results

Standard IEEE test cases. Random Starting point

V ,P ,Q uniformly between bounds. δi ∈ [ π18 ,π

18 ]

HELM-OPF ignores initial δ

Larger bounds on δ give high failures for IPOPT, RunOPF & RunPF

RunOPF IPOPT HELM-OPF RunPF-OPFSystem % Opt % Inf % Opt % Inf % Opt % Inf % Opt %Inf9-Bus 98 2 100 0 100 0 100 014-Bus 77 23 100 0 100 0 100 024-Bus 100 0 100 0 100 0 100 030-Bus 90 10 100 0 100 0 96 439-Bus 62 38 100 0 100 0 19 8157-Bus 84 16 100 0 100 0 0 10089-Bus 0 100 100 0 100 0 0 100118-Bus 95 5 100 0 100 0 96 4300-Bus 39 61 100 0 2 98 0 1001354-Bus 0 100 100 0 100 0 0 1002869-Bus 0 100 88 12 65 35 0 100

All failures for HELM/RunPF-OPF due to infeasible initial LFNo local solutions found (by any solver)


Results


V ,P ,Q uniformly between bounds. δi ∈ [π4 ,π

4 ]



RunOPF IPOPT HELM-OPF RunPF-OPFSystem % Opt % Inf % Opt % Inf % Opt % Inf % Opt %Inf9-Bus 97 3 100 0 100 0 1 9914-Bus 77 23 100 0 100 0 0 10024-Bus 100 0 100 0 100 0 0 10030-Bus 88 12 100 0 100 0 0 10039-Bus 66 34 99 1 100 0 0 10057-Bus 84 16 100 0 100 0 0 10089-Bus 0 100 62 38 100 0 0 100118-Bus 95 5 98 2 100 0 0 100300-Bus 39 61 96 4 2 98 0 100

All failures for HELM/RunPF-OPF due to infeasible initial LF

No local solutions found (by any solver)


Results


V ,P ,Q uniformly between bounds. δi ∈ [π2 ,π

2 ]



RunOPF IPOPT HELM-OPF RunPF-OPFSystem % Opt % Inf % Opt % Inf % Opt % Inf % Opt %Inf9-Bus 98 2 96 4 100 0 0 10014-Bus 78 22 100 0 100 0 0 10024-Bus 100 0 69 31 100 0 0 10030-Bus 88 12 97 3 100 0 0 10039-Bus 63 37 74 26 100 0 0 10057-Bus 85 15 88 12 100 0 0 10089-Bus 0 100 3 97 100 0 0 100118-Bus 97 3 0 100 100 0 0 100300-Bus 34 66 1 99 2 98 0 100


No local solutions found (by any solver)


Results

Modified IEEE test cases. Random starting point. δ ∈ [−π

2 ,π

2 ]

Modifications from Bukhsh, G., Trodden, McKinnon ’13

RunOPF IPOPT HELM-OPF RunPF-OPFSystem %Opt %Inf %Loc %Opt %Inf %Loc %Opt %Inf %Loc %Opt %Inf %Loc9mod 27 36 37 26 25 49 43 0 57 0 100 039mod1 14 81 5 56 44 0 100 0 0 0 100 039mod2 7 83 10 49 43 8 97 0 3 0 100 039mod3 17 54 29 18 82 0 100 0 0 0 100 039mod4 8 84 8 53 47 0 100 0 0 0 100 0118mod 79 19 2 0 100 0 100 0 0 0 100 0118mod1 53 46 1 0 100 0 100 0 0 0 100 0300mod 12 77 11 0 100 0 27 72 1 0 100 0300mod1 2 96 2 0 100 0 0 100 0 0 100 0


We now find some local solutions

HELM-OPF has fewer infeasibilities (inconclusive on localsols)

All solutions are stable (by dVdQ

criterion)


Conclusions

HELM can be used in a competitive LF-OPF solver

Somewhat more robust than other solvers for random starts

No evidence of unstable solutions(!) Instability and localsolutions do not seem to be related.

Interpretation of LF-OPF as reduced space algorithm.SQP rather than SLP makes it much more robust

References

I Wallace, A Grothey, KIM McKinnon: Solving Optimal Power

Flows using HELM, on ArXiv shortly.

I Wallace, D Roberts, A Grothey, KIM McKinnon: AlternativePV Bus Modelling with the Holomorphic Embedding Load

Flow Method, ArXiv, July 2016.


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