Table Of ContentAdjustably Robust Optimal Power Flows with
Demand Uncertainty via Path-based Flows
6
1
Jakub Marecek1, Adam Ouorou2, Guanglei Wang2∗
0
2
1 IBM Reserach – Ireland, Dublin, Ireland
n
2
a Orange Labs Research, Issy-les-Moulineaux, France
J
5 January 26, 2016
2
]
C
Abstract
O
We study the optimal power flow problem with switching (or, equiv-
.
h alently, the line expansion problem) under demand uncertainty. Specifi-
t cally,weconsidertheline-usevariablesatthefirststageandthecurrent-
a
m orpower-flowatthesecondstageoftwoaffinelyadjustablyrobustformu-
lations.
[
1
1 Introduction
v
9
3 There is a long history of research into optimisation under uncertainty in the
7
operations of power systems. Often, one considers a small discrete number of
6
realisations of uncertainty within the framework of stochastic programming.
0
At other times, one should like to consider an infinite number of realisations
.
1
of uncertainty, e.g., by replacing uncertain scalar values with uncertainty sets,
0
such as intervals. In robust optimisation, each uncertainty-immunized solution
6
1 is required to be feasible for all the realisations of the uncertainty. The ro-
: bust counterpart of an uncertain problem seeks such a solution and can often
v
be solved efficiently. This approach has a two-fold appeal: first, the robust
i
X counter-parts are often easier to solve than a scenario-expansion in stochastic
r programming,andsecond,onecanoftenderiveguaranteesastotheperformance
a
of the solutions.
Theapplicationsofrobustnessinpowersystemshave,however,beenlimited
due to the difficulties in dealing with equalities in robust optimisation. Even
very recent work considers only inequality constraints, rather than equalities
required in transmission-constrained problems. One could use two-level formu-
lations [3], also known as adjustable robustness or the decision rules approach,
but has to tackle the resulting non-convex non-linear optimisation problems.
∗GuangleiWangisthefirstauthor,despitebeinglistedthelastinthealphabeticalordering.
Email: [email protected]
1
Weproposedealingwithexpansiondecisions(line-usevariables)x forbranches
l
l∈E atthefirst,non-adjustablestage. Theadjustablevariablesarepowerflow
variables s and power reservation variables p. Demand ξ at bus k ∈ K is
k
unknown a priori, but estimates in one of two forms are available. The first
form is based on the nominal value, the dispersion range around the nominal
value, and the number of nomila values, which are allowed to deviate from the
nominal value, in the spirit of budget uncertainty [3]. The second form could be
seen either as based on the value at risk, or as a polyhedral approximation of
certain ellipsoidal uncertainty sets.
In order to solve the problem, we introduce a number of novel techniques:
• a power-flow formulation based on path-based variables for current or
power and Kirchhoff’s first law (Section 2)
• an elaborate construction of adjustably robust counter parts considering
little studied uncertainty sets (Section 3)
• methods from polynomial optimization [9] for solving the robust counter
parts exactly (Section 4)
• for comparison, an approximation of adjustable decision rules by affine
decision rules, which amounts to requiring s and p to be of the form
(cid:88)
p (ξ)=p0+ phξ , g ∈G
g g g h
h∈K
(1)
(cid:88)
s (ξ)=s0+ shξ , p∈P ,k ∈K
p p p h k
h∈K
for buses K, paths P to bus k ∈K, and generators G (Section 5).
k
Adiscussionoftheoptionsispresentedinconclusions. Thisiscomplementaryto
recentworkonadjustablerobustness[10,11],robustnessintermsofconditional
valueatrisk[18]andchance-constrainedmodels[6]. Itmaysuggestanappealing
direction for further research, e.g. making it possible to derive guarantees as to
the performance of the solutions [2].
2
1.1 Notation
Sets
N set of buses in the power network.
L set of transmission lines.
G set of generators G⊆N.
K set of demands, K⊆N.
P set of admissible paths from any generator
k
to the customer k ∈K.
P set of admissible paths from generator g ∈G to any customer
g
P set of admissible paths between demand k and generator g, where
kg
(k,g)∈K×G.
Parameters
f the investment costs for line l∈L
l
R the resistance of line l∈L
l
B the susceptance of line l∈L
l
c per-unit costs for generator g ∈G
g
d the demand of customer k ∈K
k
y an upper bound for the current flow for l∈L
l
Variables
p reserve power capacity for generator g ∈G
g
s current flow along line l∈L
l
x indicator of switching line l∈L (0 open)
l
Uncertainty sets
Ξ budget-of-uncertainty u. set (10)
d¯ nominal value of demand
k
dˆ dispersion value of demand
k
ξ a random variable in [−1,1]
k
κ parameter regulating dispersion
τ number of deviations from nominal demands
Ξ(cid:48) value-at-risk uncertainty set (11)
α quantile for the “α-tail”
3
2 A Path-Based Formulation
We propose a formulation, where for each customer, there is a scalar decision
variableforeachpathfromageneratortotheparticularcustomer,andforeach
edge on that path, a thermal limit constraint across all paths using it.
(cid:80)
Theobjectiveistominimizethetransmissionexpansioncost f x and
l∈E l l
(cid:80)
power reservation cost c p . In this paper, transmission expansion as-
g∈G g g
sumes, that the existing network is fixed and we can only expand the network
capacity by adding parallel transmission lines between existing towers.
Basedthispath-basedformulation, weproposetwovariants. Formally, vari-
able y captures the current flow along path p.
p
(cid:88) (cid:88)
min f x + c p (PB1)
l l g g
l∈E g∈G
(cid:88) (cid:112)
s.t. y ≥ d /R, k ∈K, (2)
p k
p∈Pk
(cid:88) (cid:113)
y ≤ p /R, g ∈G, (3)
p g
p∈Pg
(cid:88) (cid:88)
y ≤y¯x , l∈E, (4)
p l l
k∈Kp∈Pk:l∈p
y ≥0,p ≥0,x ∈{0,1}, k ∈K,p∈P ,g ∈G,l∈E. (5)
p g l k
Alternatively, variable s captures the apparent power provided to customer k
p
along path p:
(cid:88) (cid:88)
min f x + c p (PB2)
l l g g
l∈E g∈G
(cid:88)
s.t. s ≥d , k ∈K, (6)
p k
p∈Pk
(cid:88)
s ≤p , g ∈G, (7)
p g
p∈Pg
(cid:88) (cid:88)
s ≤y¯2R x , l∈E, (8)
p l l l
k∈Kp∈Pk:l∈p
s ≥0,p ≥0,x ∈{0,1}, k ∈K,p∈P ,g ∈G,l∈E. (9)
p g l k
Our work on this path-based formulation was partially motivated by the work
of [13], who have shown, that one may formulate the optimal power flow in the
directed current model using Kirchoff’s second law (Kirchoff’s Voltage Law),
whichrequiresthevoltagedifferencesalonganydirectedcycleC tosumtozero.
By the fact that any directed cycle C : s−t−s can be decomposed into two
paths P and P , the following proposition holds.
st ts
Proposition 1. The objective function values of path-based OPF and cycle-
based OPF are equivalent.
4
3 The Adjustable Robust Formulation under De-
mand Uncertainty
In the remaining of this paper, we derive the adjustable robust counterparts
for model (PB2). In this paper, the uncertainty comes from the prediction
of customer demands or loads. Instead of trying either to approximate one
particular multi-variate distribution of the load, as in stochastic programming,
or trying to work with any distributions on the particular support, as in early
work on robust optimisation, we consider two non-trivial uncertainty sets.
In order to construct the first uncertainty set Ξ, we express the prediction
of demands as d = d¯ +ξ dˆ , where d¯ ,dˆ are nominal value and dispersion
k k k k k k
value and ξ is a random variable in [−1,1]. Following the notion of budget
k
uncertainty [5, 4, 3], we define Ξ as:
(cid:110) (cid:111)
Ξ= ξ ∈R|K| :ξ ∈[−1,1],i∈K,(cid:107)ξ(cid:107) ≤κ,(cid:107)ξ(cid:107) ≤τ (10)
i 1 ∞
where the values of κ,τ are usually determined by the decision maker who is
responsible for the outcomes of the set Ξ. The larger the values, the larger the
set,andthehigherrobustprice. Ifκ=1,τ =|K|,thenallthepossibleoutcomes
are considered; while if both values are zero, the values of demands are just
nominal values. In this case, the problem can be reduced to the deterministic
models (PB1) or (PB2). Further, we consider an alternative uncertainty set,
(cid:110)
Ξ(cid:48) = d∈R|K| :∃λ∈RN,1λ=1, (11)
+
N (cid:41)
(cid:88) 1
d= λ d ,λ ≤ ,
i i i N(1−α)
i=1
where α is determined by the decision maker. It assumes that N observations
are available and controls the “α-tail” of the uncertainty. This seems to have
been considered only once before, see page 467 of [4]. The set Ξ(cid:48) can be seen as
apolyhedralapproximationoftheellipsoidalsetconsideredbyBen-taletal.[3].
Subsequently, we can construct the adjustable robust counterpart of the
uncertain program for (PB2) with Ξ and Ξ(cid:48). In both cases, the non-adjustable
decisionvariablesareexpansionvariablesx ,l∈E. Theadjustablevariablesare
l
power flow variables s and power reservation variables p. Hence, we formulate
the problem as
(cid:88) (cid:88)
min f x +max min c p (PB2-ARC)
l l g g
ξ∈Ξ s(ξ),p(ξ)
l∈E g∈G
(cid:92)
s.t. x∈ Proj F(ξ) (12)
x
ξ∈Ξ
where,
F(ξ)={(x,p(ξ),s(ξ)): parameterized (6),(7),(8),(9) hold}.
5
We then write out the dual of the inner minimization problem with dual asso-
ciated multipliers λ,φ,η and denote by S(x,ξ) the power generation cost.
S(x,ξ)=max (cid:88)λ d −(cid:88)y¯2xRη
k k l l l l
λ,φ,η
k∈K l∈E
s.t. c −φ ≥0,∀g∈G
g g
(cid:88)
η −λ +φ ≥0,∀(k,g)∈K×G,∀p∈P
l k g kg
l∈E:l∈p
λ,φ,η≥0.
Bystrongduality,theinnerminimizationproblemisequivalenttomaxS(x,ξ).
ξ∈U
Furthermore, (PB2-ARC) might be reduced to the following bilinear program:
(cid:88)
min γ+ fx (PB2-PP)
l l
l∈E
s.t. c −φ ≥0,g∈G, (13)
g g
(cid:88)
η −λ +φ ≥0,∀(k,g)∈K×G,p∈P , (14)
l k g kg
l∈E:l∈p
(cid:88)(λ d¯ +dˆ λ ξ )−(cid:88)y¯2Rxη ≤γ,ξ∈Ξ, (15)
k k k k k l l l l
k∈K l∈E
λ ,φ ,η ≥0,x ∈{0,1},∀k,g,l. (16)
k g l l
To deal with a infinite number of of constraints (15), the robust counterpart of
constraints (15) can be expressed as follows,
(cid:88)λ d¯ −(cid:88)y¯2R x η +κt+(cid:88)τw ≤γ, (17)
k k l l l l k
k∈K l∈E k∈K
w +t+λ dˆ ≥0,k ∈K, (18)
k k k
w +t−λ dˆ ≥0,k ∈K, (19)
k k k
w ≥0,t≥0,k ∈K. (20)
k
Alternatively, for the uncertainty set Ξ(cid:48) we can express the robust counterpart
of constraints (15) as,
N
1 (cid:88) (cid:88)
t+ w − y¯2R x η ≤γ, (21)
N(1−α) i l l l l
i=1 l∈E
(cid:88)
t+w ≥ λ di,i=1,...,N, (22)
i k k
k∈K
w ≥0,i=1,...,N. (23)
i
To conclude this section, we present the exact adjustable robust counterpart
with uncertainty set Ξ
(cid:88)
min γ+ f x [PB2-ARCΞ]
l l
l∈E
s.t. (13),(14),(16),(17),(18),(19),(20) (24)
6
and the exact adjustable robust counterpart with uncertainty set Ξ(cid:48):
(cid:88)
min γ+ f x [PB2-ARCΞ(cid:48)]
l l
l∈E
s.t. (13),(14),(16),(21),(22),(23) (25)
4 Computations
The two-stage robust counterpart is NP-Hard to solve, in general [3, 4]. Never-
theless, we derive one hierarchy of semidefinite programming relaxation due to
Lasserre [15, 1], which are asymptotically convergent, in theory, and solvable,
in practice.
Let us sketch the approach out, following the treatment of [9], on a generic
polynomial optimisation problem
min f(x)
s.t. g (x)≥0 i={1,...,m} [PP]
i
If there exist homogeneous polynomials of degree d in n-dimensional vector
x, denoted g (x),...,g (x) such that h(x) = (cid:80)k g (x)2, polynomial h(x) of
1 k i=1 i
degree 2d is sum-of-squares (SOS). We use P (S) to denote the cone of polyno-
d
mials of degree at most d that are non-negative over some S ⊆ Rn and Σ to
d
denote the cone of polynomials of degree at most d that are SOS. Choi et al.
[7] showed that each P (Rn) can be approximated as closely as desired by a
2d
sum-of-squares of polynomials. Lasserre reformulated [PP] as
max ϕ s.t. f(x)−ϕ≥0 ∀x∈S ,
G
=max ϕ s.t. f(x)−ϕ∈P (S ). [PP-D]
d G
where G = {g (x) : i = 1,...,m} and S = {x ∈ Rn : g(x) ≥ 0, ∀g ∈ G} In
i G
[17, 15, 14], the n×n matrix:
(cid:63) for i=j
(cid:63) for x ,x in the same monomial of f
R = i j
ij
(cid:63)0 ofotrhexriw,xisje,in the same constraint gk
and its associated adjacency graph G represent the so called correlative spar-
sity of [PP]. Let {I }p be the set of maximal cliques of a chordal extension
k k=1
of G following the construction in [17], i.e. I ⊂ {1,...,n}. This leads to
k
approximation Kr(I) of P (S):
G d
p
(cid:88) (cid:88)
KGr(I)= Σr(Ik)+ gjΣr−deg(gj)(Ik),
k=1 j∈Jk
7
where Σ (I ) is the set of all sum-of-squares polynomials of degree up to d
d k
supported on I and (J ,...,J ) is a partitioning of the set of polynomials
k 1 p
{g } defining S such that for every j in J , the corresponding g is supported
j j k j
on I . The support I ⊂ {1,...,n} of a polynomial contains the indices i of
k
terms x which occur in one of the monomials of the polynomial. The sparse
i
hierarchy of SDP relaxations is then given by
max ϕ [PP-SH ]∗
r
ϕ,σk(x),σr,k(x)
p
(cid:88) (cid:88)
s.t. f(x)−ϕ= σk(x)+ gj(x)σj,k(x)
k=1 j∈Jk
σ ∈Σ ((I )),σ ∈Σ (I ).
k r k j,k r−deg(gj) k
We denote the dual of [PP-SH ]∗ by [PP-SH ], in keeping with [9].
r r
Following Lasserre [15], we introduce:
Assumption 1. Let S denote the feasible set of a problem of form [PP]. Let
{I } denotethepmaximalcliquesofachordalextensionofthesparsitypattern
k k
graph of the [PP].
(i) Then, there is a M >0 such that (cid:107)x(cid:107) <M for all x∈S.
∞
(ii) Ordering conditions for index sets as in Assumption 3.2 (i) and (ii) in
[15].
(iii) Running-intersection-property, c.f. [15], holds for {I } .
k k
Notice that this assumption is easy to satisfy for our robust counter parts,
whereallvariablesareboundedfromaboveandbelow,andS ishencecompact.
(One can add redundant quadratic inequalities such that Assumption 1 (i) is
satisfied; (ii) and (iii) can be satisfied by construction of the sets {I } and the
k k
chordalextension,aspointedoutin[14].) Indeed,theresultingfull-dimensional
polynomial optimisation problems are very nicely behaved.
Following Lasserre [15] further, we have:
Proposition 2 (Asymptotic Convergence). Whenever Assumption 1 holds for
the feasible sets of [PB2-ARCΞ] and [PB2-ARCΞ(cid:48)], for the sparse hierarchy
[PB2-ARCΞ-SH ]∗ for [PB2-ARCΞ] and [PB2-ARCΞ(cid:48)-SH ]∗ for [PB2-ARCΞ(cid:48)]
r r
and their respective duals holds:
(a) inf[PB2-ARCΞ-SH ](cid:37)min([PB2-ARCΞ]) as r →∞,
r
(b) sup[PB2-ARCΞ-SH ]∗ (cid:37)min([PB2-ARCΞ])asr →∞,
r
(c) If the interior of the feasible set of [PB2-ARCΞ] is nonempty, there is no
duality gap between [PB2-ARCΞ-SH ] and [PB2-ARCΞ-SH ]∗.
r r
8
(d) If the interior of the feasible set of [PB2-ARCΞ] is nonempty, there is no
duality gap between [PB2-ARCΞ-SH ] and [PB2-ARCΞ-SH ]∗.
1 1
(e) If [PB2-ARCΞ] has a unique global minimizer, then as r tends to infinity
the components of the optimal solution of [PB2-ARCΞ-SH ] correspond-
r
ing to the linear terms converge to the unique global minimiser of [PB2-
ARCΞ].
Throughout, the analogous result holds for [PB2-ARCΞ(cid:48)].
Proof. The proof is the same as in [9]: (a,b,e) follow from Theorem 3.6 of [15],
(c) follows from Theorem 5 of [14], and (d) from [17].
5 Affine decision rule approximation
As a comparison with the exact formulations [PB2-ARCΞ] and [PB2-ARCΞ(cid:48)],
we approximate the adjustable decision rules by the affine decision rule on the
demands, which amounts to requiring s(ξ) and p(ξ) to be of the form
(cid:88)
p (ξ)=p0+ phξ , g ∈G
g g g h
h∈K
(26)
(cid:88)
s (ξ)=s0+ shξ , p∈P ,k ∈K
p p p h k
h∈K
for buses K, paths P to bus k ∈ K, and generators G. With these affine
k
functions, we rewrite the affinely adjustable robust counterpart (PB2-ARC) as
(cid:88)
min f x +γ (PB2-AAR)
l l
l∈E
(cid:88) (cid:88)
s.t. c (p0+ phξ )≤γ, ξ ∈Ξ, (27)
g g g h
g∈G h∈K
(cid:88)(s0+ (cid:88)shξ )≥d¯ +ξ dˆ , k ∈K,ξ ∈Ξ, (28)
p p h k k k
p∈Pk h∈K
(cid:88) (cid:88) (cid:88)
(s0+ shξ )−p0− phξ ≤0,g ∈G,ξ ∈Ξ, (29)
p p h g g h
p∈Pg h∈K h∈K
(cid:88) (cid:88) (cid:88)
(s0+ shξ )≤y¯2R x , l∈E,ξ ∈Ξ, (30)
p p h l l l
k∈Kp∈Pk:l∈p h∈K
(cid:88)
p0+ phξ ≥0, g ∈G, ξ ∈Ξ, (31)
g g h
h∈K
(cid:88)
s0+ shξ ≥0, p∈P , k ∈K,ξ ∈Ξ (32)
p p h k
h∈K
x ∈{0,1}, l∈E. (33)
l
9
5 5
1 1
3 3
2 2
4 4
6 6
Figure 1: The instance Garver6y with the globally optimal solution (left) and
the only other AC-feasible solution (right).
Foreachconstraint,wecanfinditsrobustcounterpart. Forexample,therobust
counterpart of constraint (28) writes
(cid:88) (cid:88)
c p0+ wkτ +w0κ≤γ, (34)
g g
g∈G k∈K
(cid:88)
wk+w0 ≥ pk,k ∈K, (35)
g
g∈G
(cid:88)
wk+w0 ≥− pk,k ∈K, (36)
g
g∈G
wk ≥0,w0 ≥0,k ∈K. (37)
See Appendix for the complete formulations w.r.t. uncertainty set Ξ and Ξ(cid:48).
Also, note that the affinely adjustable formulation has a two-fold solution:
the solution to the line expansion decisions, and a set of affine decision rules.
The performance of adjustable decision rules will be quantified by metrics on
theoptimalvaluesoftheaffinelyadjustablerobust-counterpart[PB2-AAR]and
the exact robust adjustable counterpart [PB2-ARC].
6 Computational Illustrations
Solvers: We have formulated the models twice, once in AMPL, and once in
using YALMIP. In both cases, the paths were generated by Yen’s K-shortest
paths algorithm [8] and the run-time of the path generation is not included in
the run-time of the solver. We have used IBM ILOG CPLEX for solving the
AARmodelsandSeDuMiforsolvingtheSDPrelaxationsoftheARmodels. In
solvingthesemidefiniteprograms,weemploythesparsity-exploitingtechniques
of Kim et al. [12]
10
