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1
Annealing-based Quantum Computing for
Combinatorial Optimal Power Flow
Thomas Morstyn, Senior Member, IEEE
Abstract—This paper proposes the use of annealing-based
quantum computing for solving combinatorial optimal power
flow problems. Quantum annealers provide a physical com-
puting platform which utilises quantum phase transitions to
solve specific classes of combinatorial problems. These devices
have seen rapid increases in scale and performance, and are
now approaching the point where they could be valuable for
industrial applications. This paper shows how an optimal power
flow problem incorporating linear multiphase network modelling,
discrete sources of energy flexibility, renewable generation place-
ment/sizing and network upgrade decisions can be formulated as
a quadratic unconstrained binary optimisation problem, which
can be solved by quantum annealing. Case studies with these
components integrated with the IEEE European Low Voltage
Test Feeder are implemented using D-Wave Systems’ 5,760
qubit Advantage quantum processing unit and hybrid quantum-
classical solver.
Index Terms—Distribution Network, D-Wave, Electric Vehicle,
Optimal Power Flow, Power System Planning, Quantum Anneal-
ing, Quantum Computing. Smart Charging.
I. INTRODUCTION
The optimal deployment and operation of new sources
of generation and flexibility is critical for achieving a low-
cost transition to reliable and decarbonised electrical power
systems [1]. The scope for optimised planning and oper-
ation has expanded significantly due to the emergence of
distributed energy resources (DERs), including small and
medium scale renewables and flexible loads, combined with
the new availability of substation- and customer-level sensing
and communications [2]. However, the vast potential scale of
the resulting coordination challenge has created concern over
future computational requirements [3].
Optimal power flow (OPF) problems involve finding set-
points for controllable power sources which meet demand at
minimum cost, while satisfying resource and network con-
straints [4]. In general, the nonlinear characteristics of power
networks makes this challenging, which has motivated the
development of linear approximations [5] and convex relax-
ations [6], which are accurate under specific conditions and
allow OPF problems to be solved in polynomial time. With the
rise of DERs, there has been significant work to consider the
particular features of OPF relevant at the distribution system
This work was supported by the UK Engineering and Physical Sciences
Research Council (EPSRC) (project references EP/S000887/2, EP/S031901/1
and EP/T028564/1). For the purpose of open access, the author has applied a
Creative Commons Attribution (CC BY) license to any Accepted Manuscript
version arising.
T. Morstyn is with the School of Engineering at the University of Edinburgh,
Edinburgh, EH8 9YL, United Kingdom. (email: thomas.morstyn@ed.ac.uk).
level, including unbalanced voltages, losses and reactive power
flows [7]–[9].
Combinatorial OPF is a more challenging class of problem,
which emerges when an OPF needs to be solved alongside
additional discrete decisions, such as when resources have
discrete flexibility [10], as well as where resource placement
and network investment decisions need to be made accounting
for fixed costs and limited sizing options [11]. In practice,
many DERs only offer flexibility in discrete increments, in-
cluding EV chargers [12] and heat-pumps [13] with on/off
control, and schedulable appliances with fixed operating cycles
[14]. Also, even when power converters allow continuous
control of DERs, low operating power is often associated
with low efficiency, making it desirable to impose a minimum
turn-on power [15]. Combinatorial OPF is directly relevant
for distribution system operators (DSOs) seeking to increase
the hosting capacity for clean energy technologies through
a combination of targeted network reinforcements and active
management of DERs [16].
Combinatorial optimisation problems can be solved using
exhaustive search and dynamic programming, but the curse of
dimensionality means that the computational burden increases
exponentially with the number of decision variables [17].
Mixed Integer Linear Programming (MILP) can be applied in
cases where the objective and constraints can be formulated
as linear functions of the discrete variables [18]. Significant
progress has been made towards solving large MILPs to
reasonable levels of accuracy, but in general they remain com-
putationally intensive [19]. Lagrangian Relaxation [20] and
Surrogate Lagrangian Relaxation [21] are iterative approaches
suited to problems that can be decomposed into a set of
simpler subproblems by relaxing a limited number of coupling
constraints. Combinatorial problems can also be solved using
metaheuristic methods including genetic algorithms [22], par-
ticle swarm optimisation [23], tabu search [24] and simulated
annealing [25]. However, scalability remains a challenge as
the convergence time of metaheuristic methods also increases
with the problem dimension [26].
Over the last 20 years, there has been significant progress
in the development of quantum devices which offer a funda-
mentally new computing architecture compared with classical
digital silicon-based computers. A major milestone towards
this was the recent achievement of quantum supremacy with a
54-qubit device, i.e. the practical demonstration of a quan-
tum computer solving a problem that would be infeasible
for classical computers [27]. For power systems, gate-based
quantum computing algorithms are presented for generator
unit commitment in [28], [29]. However, significant challenges
2
remain for scaling up universal gate-based quantum computers
to the point where they could be widely used for industrial
applications [30].
The challenges of scaling up gate-based quantum computers
has motivated the development of more scalable quantum
hardware architectures aimed at specific computing problems.
Quantum annealers are currently the largest quantum com-
puting devices and are capable of solving a specific class
of combinatorial optimisation, namely unconstrained quadratic
binary optimisation (QUBO) problems [31]. Quantum anneal-
ers incorporate a lattice of qubits which can be controllably
biased and coupled. Based on the quantum adiabatic theorem,
the qubit lattice is controlled so that it physically evolves
to a low-energy state which represents the solution to an
optimisation problem. This is somewhat analogous to the
process that is replicated by simulated annealing, but with
physical quantum fluctuations replacing simulated thermal
ones [32]. Also, it should be noted that before the development
of annealing-based quantum processors, quantum annealing
was used to refer to a variation of simulated annealing with
simulated quantum fluctuations [33].
Theoretically demonstrating when noisy quantum anneal-
ing has a definitive advantage over classical alternatives is
challenging [34], but a performance advantage has been
demonstrated for specific applications [35], [36]. Moreover,
quantum annealing hardware is still in its infancy, and is
rapidly improving in terms of the number of qubits and
noise level [37]. This has motivated investigations into a
range applications including protein folding [38], machine
learning [39], and wireless base station decoding [40]. The
opportunity for quantum annealing to be applied to power
system applications is noted in [41], but without a detailed
investigation. In [42], quantum annealing is demonstrated for
generator unit commitment, but power flow modelling and
network constraints are not considered. Other power system
applications of quantum annealing include grid partitioning
[43] and phasor measurement unit placement [44].
The novel contribution of this paper is to propose and
demonstrate the use of annealing-based quantum computing
for combinatorial OPF. Given the still relatively limited scale
and developing nature of quantum annealing hardware, our
focus is on its applicability to power systems rather than
the potential for speed-up with current hardware. Towards
this, a novel QUBO formulation is presented for a linear
multiphase OPF problem with controllable on/off EV charg-
ing, non-dispatchable renewable generation placement/sizing
and network upgrade decisions, which can be solved using
quantum annealing. Case studies are implemented on D-
Wave Systems’ 5,760 qubit Advantage quantum processor to
investigate how the number of required qubits scales with
the number of EVs and network constraints. D-Wave’s hybrid
quantum-classical binary quadratic model solver is then used
for a larger scale problem, which highlights the value of
co-optimising distribution network upgrades and renewable
generation investment with operational flexibility.
The rest of the paper is organised as follows: Section II
presents a brief overview of D-Wave’s implementation of
quantum annealing and its application to QUBO problems.
In Section III, the proposed QUBO formulation for the com-
binatorial OPF is developed. Case study results are presented
in Section IV. Section V concludes the paper.
II. QUA NT UM ANNEALING
This section provides an overview of quantum annealing, as
implemented by D-Wave’s quantum processors. As mentioned,
a key application is to solve QUBO problems, which can be
described by
min X
(i,j)∈E
Qij xixj+X
i∈X
cixi,(1)
where xi∈ {0,1}, i ∈ X := {1, . . . , X}are binary decision
variables, E:= {(i, j)|i, j ∈ X , i 6=j}.Qij ∈R,(i, j )∈ E
are the quadratic QUBO objective function coefficients and
ci∈R, i ∈ X are the linear QUBO objective function
coefficients.
The QUBO problem can be equivalently expressed as an
Ising model minimisation problem, through a change of vari-
ables yi= 1 −2xifor i∈ X [31], giving
min X
(i,j)∈E
Jij yiyj+X
i∈X
hiyi,(2)
Jij =−1
4Qij , hi=−1
2(ci+X
j∈X
Qij ),
where the spin values yi∈ {−1,1}, i ∈ X .
Quantum annealing is based on the natural behaviour of
coupled qubits to seek a ground state (lowest-energy state).
The quantum annealing process can be described by a time
varying Hamiltonian H(s)[34]
H(s) = A(s)HI−B(s)HP,(3)
where A(s)and B(s)are annealing path functions, which are
defined in terms of the normalised annealing time s=t/ta.
These are designed so that initially A(0) = 1 and B(0) = 0,
and after annealing A(1) = 0 and B(1) = 1.
The initial Hamiltonian HIis selected so that it has a known
ground state which is easy to prepare, for example [34]
HI=X
i
σx
i,(4)
where σx
iis the Pauli-x operator applied to qubit i. The
problem Hamiltonian HPis given by [34]
HP=X
i,j
Jij σz
i·σz
j+X
i
hiσz
i.(5)
where σz
iis the Pauli-z operator applied to qubit i. The
eigenvectors of this Hamiltonian correspond to the solutions
of the Ising model (2).
The quantum annealer first initialises the superposition state
of a qubit lattice so that H(0) = HI. The qubit couplings are
then manipulated over the annealing time so that the system
evolves towards the problem Hamiltonian. Dwave’s device
uses radio frequency superconducting quantum–interference
device (rf-SQUID) qubits [45]. The underlying physical mech-
anisms used by D-Wave’s quantum processors are described
in more detail in [46]. According to the adiabatic theorem of
3
Fig. 1: A 3 by 3 cell example (144 qubits) Pegasus topology. D-
Wave’s Advantage processor is 16 by 16 cells (5,760 qubits).
quantum computing, if the annealing time is sufficiently long
the time varying Hamiltonian will remain in the ground state
throughout. The problem Hamiltonian has classical eigenval-
ues, and thus the spin values at H(1) = Hpwill be classical
values (i.e. yi∈ {−1,1}) and these will correspond with the
optimal solution of the Ising model.
An added complexity is that the physical qubit lattices
within D-Wave’s quantum processors are sparsely connected.
In September of 2020, D-Wave released its ‘Advantage’ pro-
cessor, with 5,760 qubits connected in a Pegasus topology,
which has most qubits connected to 15 neighbours. Fig. 1
shows a 144 qubit version of the Pegasus topology [47]. The
Advantage processor allows for significantly larger problems
than its predecessor, the D-Wave 2000Q, which had 2,048
qubits arranged with most qubits connected to 6 neighbours.
The limited connectivity means that an Ising model must
be translated into an equivalent model that is compatible with
the processor’s physical qubit lattice. The translation process
is called minor embedding, and involves representing each
logical qubit of the original model with either a single physical
qubit or a chain of strongly coupled physical qubits [48]. A
example is shown in Fig. 2. Optimal minor embedding is
itself a computationally intensive problem, but heuristic tools
suitable for large problems have been developed [49]. Also,
embeddings can be reused between problems with the same
coupled decision variables, even if the linear and quadratic
weights are different.
For problems that are too large for current quantum pro-
cessors, D-Wave makes available cloud-based hybrid solvers,
which can solve QUBO models with up to a million variables
[48]. The solver code is proprietary, and is described as making
use of parallel computation on classical CPUs and GPUs,
which send queries to a quantum processor to help guide the
exploration of the solution space [50].
x1
x2
x3
x4
x5
Minor
Embedding
x1x2
x3
x4
x5
Fig. 2: A 5 qubit logical network representing a simple QUBO
problem, min (x1+x2+x3+x4+x5)2, and a 6 qubit embedding
on a subsection of the Pegasus topology.
III. PROB LE M FOR MU LATI ON
This section presents the proposed QUBO formulation for
a combinatorial OPF problem. To provide a concrete setting,
the formulation focuses on a DSO which aims to optimally
schedule EVs with controllable on/off charging, while making
decisions about the location and sizing of non-dispatchable
renewable generation and distribution network upgrades. Note
that with fairly minor modifications the formulation could be
updated to address transmission networks (e.g. by replacing
the distribution-focused multiphase linear power flow model
used here [9] with a DC linear power flow model [51]) and
a broader range of time-coupled flexible loads, such as smart
heating and schedulable appliances [52].
First, a constrained nonlinear binary formulation is pre-
sented for the combinatorial problem, which is then used to
develop the proposed QUBO formulation, which can be solved
using quantum annealing. Consider a distribution network
with a set of nodes N={0, . . . , N }, where node 0is the
point of connection with the main grid. The network has
phases Φ = {a, b, c}. The set of intervals in the optimisation
horizon is T={1, . . . , T }, where each interval has duration
τ. For interval t∈ T , the price of energy at the grid
connection point is λ0t.V={1, . . . , V }is the set of EVs,
G={1, . . . , G}is the set of potential renewable generation
sites and U={1, . . . , U }is the set of mutually exclusive
potential plans for network upgrades.
The combinatorial OPF includes long-term investment de-
cisions ahead of operation and operational scheduling over the
optimisation horizon. Computational limits generally require
that the optimisation horizon considers a shortened period rep-
resentative of longer-term operation (e.g. one or more days).
Here, only one set of investment decisions are considered,
but a potential extension is to consider multiple stages of
investment and operation (e.g. yearly). Due to the difference in
timescales, asset investment costs need to be discounted and
adjusted based on the duration of the optimisation horizon
relative to their lifetime (see e.g. [53]). The equivalent cost
cequ for the optimisation horizon of an investment with upfront
investment cost cinv is given by
cequ =cinv T
Ty r
1−(1 + r)−Ty
L,(6)
4
where Tyis the number of optimisation intervals τin one
year, Ty
Lis the asset lifetime in years and ris the discount
rate.
For EV i∈ V,Ti⊆ T is the subset of intervals when the
EV is plugged in and available for charging, xev
iv ∈ {0,1}
represents the on/off charging decision for t∈ Ti,uev
iis the
utility for energy charged, ηev
iis the charging efficiency, ρev
i
is the rated charging power, Eev
0iis the energy upon arrival,
and ¯
Eev
iis the maximum energy. DERs and loads may have
single or multi-phase connections, but for ease of presentation
it is assumed that they are wye connected. Ψv
iis the set of
node–phase pairs which EV iis connected at (e.g. if connected
at phases aand bof node nthen Ψv
i={(n, a),(n, b)}).
Each potential renewable generation site j∈ G, has a set of
sizes Sjat which generation can be installed, which determine
the rated power ρg
js and cost cg
js , s ∈ Sj(discounted and
adjusted based on the duration of the optimisation horizon
relative to the lifetime). The decision to install generation of
size sis indicated by xg
js ∈ {0,1}. It is assumed that the
renewables sources are non-dispatchable (e.g. solar or wind
not controlled by the DSO during operation). Therefore, each
source operates with a normalised generation profile over the
time horizon, ˆpg
j= (ˆpg
j1,...,ˆpg
jT ), so the total output power
vector is ρg
js ˆpg
j.
Using the linear multiphase power flow model from [9]
and a set of nominal operating points over the time horizion,
time dependent coefficients Aψωt can be obtained relating real
power injections at node–phase pair ψto the change in net
real power imports at the slack node. Similarly, a coefficient
Kψωt can be obtained which relates the impact of a power
injection at node–phase pair ψto the voltage magnitude of
another node–phase pair ω. For node–phase pair ω, let ˜vωt
be the voltage magnitude at the nominal operating point, with
upper and lower allowed limits vωand vω.
The selection of network upgrade plan k∈ U is indi-
cated by xu
k∈ {0,1}. Plans are mutually exclusive, each
being associated with a specific set of upgrades to lines and
transformers, resulting in new power flow model coefficients,
Aψkt and Kψω kt. The discounted and time horizon adjusted
cost of upgrade kis cu
k. Let ∆A
ψkt =Aψk t −Aψt and
∆K
ψωkt =Kψ ωkt −Kψωt. Also, for time t, let the impact of
upgrade kon the net import power at the nominal operating
point be given by ∆p0
kt and let the impact on the voltage
magnitude at node–phase pair ωbe given by ∆v
ωkt .
The combinatorial optimisation problem can be formulated
as
min X
i∈V
1
|Ψev
i|X
ψ∈Ψev
iX
t∈Ti
(Aψt +X
k∈U
∆A
ψkt xu
k)
·τλ0tηev
iρev
ixev
it −X
j∈G
1
|Ψg
j|X
ψ∈Ψg
jX
t∈T
(Aψt
+X
k∈U
∆A
ψkt xu
k)X
s∈Sj
xg
js τλ0tρg
js ˆpg
jt
−X
i∈V X
t∈Ti
τηev
iuev
iρev
ixev
it +X
j∈G X
s∈Sj
cg
js xg
js
+X
k∈U
cu
kxu
k+X
t∈T
τλ0t∆p0
kt xu
k,(7a)
s.t. Eev
0i+X
t∈Ti
τηev
iρev
ixev
it ≤¯
Eev
ifor i∈ V,(7b)
vω≤˜vω t +X
i∈V
1
|Ψev
i|X
ψ∈Ψev
i
ρev
i(Kψωt
+X
k∈U
∆K
ψωkt xu
k)xev
it +X
k∈U
∆v
ωkt xu
k
−X
j∈G
1
|Ψg
j|X
ψ∈Ψg
j
ρg
js (Kψωt +X
k∈U
∆K
ψωkt xu
k)
·X
s∈Sj
xg
js ˆpg
jt ≤vωfor ω∈Ω, t ∈ T ,(7c)
X
s∈S
xg
js ≤1for j∈ G,X
k∈U
xu
k≤1,(7d)
xev
it , xg
js , xu
k∈ {0,1}.(7e)
The decision variables are xev
it ,xu
k,xg
js , which are all binary
valued.
The objective (7a) is to minimise the net system cost, which
includes the cost/revenue of buying/selling energy upstream,
the utility obtained from EV charging, the cost of renewable
generation investment and the cost of network upgrades.
Constraint (7b) limits the maximum energy levels of the EVs.
Constraint (7c) limits the maximum and minimum voltage
magnitude of each node–phase pair ω∈Ω. Constraints in (7d)
specify that a single installation size can be selected for each
renewable generation site and that at most a single network
upgrade plan can be selected. Constraint (7e) specifies the
decision variables are binary.
The problem must be reformulated as a QUBO for it to
be solved using quantum annealing. The objective function
of the combinatorial OPF problem (7a) is made up of linear
and quadratic terms of the binary decision variables and can
therefore be directly incorporated into a QUBO formulation.
However, since constraints cannot be directly incorporated, the
objective must instead include equivalent penalty terms which
are high when constraints are violated and zero for feasible
solutions. A general linear inequality constraint of form Ax ≤
bcan enforced through an equivalent penalty term [54],
P(Ax −b+δ
Y−1
X
l=0
2lyl)2,with δ=s
2Y−1.(8)
Pis a penalty scalar, which will ensure the constraint is
satisfied at optimality if sufficiently large, although an overly
large penalty can increase the solution time. Trade-offs and
rules-of-thumb for penalty selection are discussed in [54].
Auxiliary slack variables yl∈ {0,1}, l ∈ {0, . . . , Y −1}are
introduced to enforce inequality, as opposed to equality. These
are arranged as a binary expansion, which efficiently enforces
the constraint with conservativeness no greater than δ, given
that the required slack quantity is no greater than s.
For specific linear constraints, simpler equivalent penalty
terms are available which make use of the properties of binary
variables. In particular, for xi∈ {0,1}, i ∈ N ={1, . . . , N },
a linear constraint Pi∈N xi≤1can be enforced with the
following penalty [54],
PX
i,j∈N ,i6=j
xixj(9)
5
Due to the network upgrades, (7) also has quadratic inequal-
ity constraints. To integrate these into a QUBO, the following
penalty term is proposed to enforce the relationship z=xy
between x, y, z ∈ {0,1},
P(xy −2zx −2zy + 3z)(10)
This penalty will be 0if z=xy, and greater than or equal to
Potherwise.
Using (8), the inequality constraints enforcing the EV
maximum energy levels (7b) can be incorporated into the
QUBO problem using equivalent penalty terms given by
Pev X
i∈V
(Eev
0i−¯
Eev
i+X
t∈Ti
τηev
iρev
ixev
it +δev
i
Yev−1
X
l=0
2lyev
il )2.
(11)
Pev is the penalty scalar and yev
il ∈ {0,1}, l ∈ {0, . . . , Y ev −1}
are binary auxiliary slack variables which enforce the con-
straint as an inequality. δev
iis the maximum conservativeness
of the constraint.
The maximum voltage magnitude constraints for each node–
phase pair in (7c) can be similarly enforced using equivalent
penalty terms given by
PvX
t∈T X
ω∈Ω˜vω t −vω+X
i∈V
1
|Ψev
i|X
ψ∈Ψev
iX
t∈Ti
ρev
t(Kψωtxev
it +X
k∈U
∆K
ψωkt xu,ev
kit ) + X
k∈U
∆v
ωkt xu
k
−X
j∈G
1
|Ψg
j|X
ψ∈Ψg
jX
s∈Sj
(Kψωtxg
js +X
k∈U
∆K
ψωkt xu,g
kjs )
·X
t∈T
ρg
js ˆpg
jt +δv
ωt
Yv−1
X
l=0
2lyv
ωtl2.(12)
Pvis the penalty scalar, yv
ωtl ∈ {0,1}, l ∈ {0, . . . , Y v−1}
are auxiliary slack variables which enforce the constraint as
an inequality, and δv
ωt is the maximum constraint conservative-
ness. xu,ev
kit and xu,g
kjs are binary auxiliary variables, which are
introduced to manage the quadratic terms in (7c). Using (10),
the required relationships xu,ev
kit =xu
kxev
it and xu,g
kjs =xu
kxg
js
can be enforced by introducing additional penalty terms given
by
Pu,ev X
k∈U X
i∈V X
t∈Ti
(xev
it xu
k−2xev
it xu,ev
kit −2xu
kxu,ev
kit −3xu,ev
kit )
+Pu,g X
k∈U X
j∈G X
s∈Sj
(xg
js xu
k−2xg
js xu,g
kjs −2xu
kxu,g
kjs −3xu,g
kjs ),
(13)
where Pu,ev and Pu,g are the penalty scalars.
Equivalent penalty terms for the minimum voltage magni-
tude constraints in (7c) are given by
PvX
t∈T X
ω∈Ω˜vω t −vω+X
i∈V
1
|Ψev
i|X
ψ∈Ψev
iX
t∈Ti
ρev
t(Kψωtxev
it +X
k∈U
∆K
ψωkt xu,ev
kit ) + X
k∈U
∆v
ωkt xu
k
−X
j∈G
1
|Ψg
j|X
ψ∈Ψg
jX
s∈Sj
(Kψωtxg
js +X
k∈U
∆K
ψωkt xu,g
kjs )
·X
t∈T
ρg
js ˆpg
jt −δv
ωt
Yv−1
X
l=0
2lyv
ωt2
(14)
yv
ωtl ∈ {0,1}, l ∈ {0, . . . , Y v−1}are the auxiliary slack
variables. Since the maximum and minimum voltage limits
have similar characteristics, the same penalty scalar, number
of auxiliary variables, and conservativeness parameter are used
for both.
Finally, using (9), equivalent penalty terms for the con-
straints in (7d) specifying that a single installation size can be
selected for each renewable generation site, and that a single
network upgrade plan can be selected, are given by
PgX
j∈G X
s,s0∈S,s6=s0
xg
js xg
js0+PuX
k,k0∈U ,k6=k0
xu
kxu
k0,(15)
where Pgand Puare the penalty scalars.
Bringing together the original objective function (7a) and
the equivalent penalty terms for constraint enforcement (11)–
(15), the proposed equivalent QUBO formulation for (7) is
given by
min (7a) + (11) + (12) + (13) + (14) + (15).
The QUBO problem has decision variables xev
it ,xu
k,xg
js ,xu,ev
kit ,
xu,g
kjs ,yev
il ,yv
ωtl,yv
ωtl, which are all binary valued.
The proposed formulation focuses on DERs with discrete
flexibility, which require combinatorial optimisation. The for-
mulation could be extended to also include continuously
controllable DERs based on a small discrete control step-
size and continuous output powers approximated by multiple
binary decision variables (arranged in binary expansions). For
example, consider a controllable generation source with output
power pc
t, limited by 0≤pc
t≤¯pc. Let the control step-size
be δcand the additional binary decision variables be xc
tm ∈
{0,1}, m ∈ {0, . . . , N c−1}, where Nc=ceil{log2(¯pc/δc)}.
The DER’s output power would be pc
t=δcPNc−1
m=0 2mxc
tm,
which could be incorporated into the objective and penalty
terms of the QUBO problem.
IV. CAS E STU DI ES
In this section, case studies are presented demonstrating the
implementation of the proposed QUBO formulation. The case
studies make use the IEEE European Low Voltage Test Feeder
[55], shown in Fig. 3, with controllable on/off EV charging,
photovoltaic (PV) generation placement/sizing and network
upgrade decisions.
The network has 55 single phase domestic loads, and up to
30 EVs with single phase 7.2 kW chargers and batteries sizes
of 75, 85 or 100 kWh. The EV arrival and departure times
are normally distributed. The mean arrival time is 6 pm and
mean departure time is 8 am, both with a standard deviation
of 2 h. The EVs’ energy upon arrival is uniformly distributed
between 10% and 30%. The network has 3 PV generation
sites which have three phase connections and can support 25
or 50 kWp. Two network upgrade plans are available, which
respectively reduce the impedance of the transformers and
6
Main Grid
Load Only
EV & Load
PV Site
Fig. 3: The distribution feeder used for the case studies, showing
the location of the main grid connection, 55 domestic loads, 30 of
which may have EV charging, and 3 sites where PV generation can
be installed.
TABLE I: Additional Case Study Parameters
uev
i0.50 £/kWh ηev
i90 %
cg
1j8.42 £cg
2j16.41 £
cu
126.32 £cu
278.96 £
Pev 0.5Yev 4
δev
i6kWh Pv2
Yv5δv
ωt 0.003 pu
Pg240 Pu300
Pu,ev 10 Pu,g 240
lines by half or three quarters. The case studies consider a
24 hour optimisation horizon from 12 pm to 12 pm the next
day, with 1 h duration intervals.
Here, operation over a single day with 1 h intervals has been
chosen since the focus is demonstrating the implementability
of the proposed QUBO formulation, but it could be refined by
considering multiple representative days and more granular
scheduling (at the cost of increased computational burden).
The upstream energy price is assumed to follow a standard UK
‘Economy 7’ tariff, with an energy price of £0.15 / kWh from 6
am to 11 pm and £0.07 / kWh from 11 pm to 6 am [56]. Smart
meter data from the UK Customer-Led Network Revolution
project is used for the domestic loads [57]. Scale-dependent
PV installation costs are calculated using data from [58],
assuming a 25-year lifetime and 5% discount rate. The network
upgrade costs are calculated using data from [59], assuming a
35-year lifetime and 5% discount rate. The voltage magnitude
limits are assumed to be 0.95 and 1.05 pu. Additional case
study parameters are provided in Table I.
A. Quantum Processor Implementation
The proposed QUBO problem is implemented on D-Wave’s
5,760 qubit Advantage quantum processor to investigate how
the number of required qubits scales with the problem size. To
do this, the problem is implemented for different numbers of
EVs and node–phase pairs where the upper and lower voltage
limits are enforced. Adding additional EVs increases the num-
ber of decision variables (EV charging decisions over the time
horizon, EV energy constraint slack variables, and auxiliary
variables associated with network upgrades), and adds new
quadratic relationships between variables, which may also add
qubits to enable minor embedding on the Pegasus topology.
Adding additional voltage constraints similarly increases the
number of decision variables and quadratic relationships.
When implemented with a single EV and voltage limits
on 3 node–phase pairs without including network upgrades,
the QUBO formulation has 746 decision variables. This in-
creases to 792 when network upgrade decisions are included.
Given voltage limits on 3 node–phase pairs, without network
upgrades each additional EV adds on average 17 decision
variables to the problem, while with network upgrades extra
EVs add an average of 44 decision variables (note that the
number of extra decision variables varies depending on the
number of intervals each EV is available for charging). Given
a single EV, enforcing voltage limits at an additional node–
phase pair increases the number of decision variables by 240,
regardless of whether or not network upgrades are considered.
Fig. 4a shows the number of assigned qubits for different
numbers of EVs, given voltage limits are enforced at 3 node–
phase pairs, and Fig. 4b which shows the number of assigned
qubits for a single EV and different numbers of node–phase
pairs with voltage limits. For each problem size, the range
of assigned qubits provided by D-Wave’s heuristic minor
embedding tool for 20 runs is shown, varying from the mean
by at most ±9.4%. Without including network constraints, the
number of assigned qubits increases approximately linearly
up to 9 EVs and 8 node–phase pairs with voltage limits,
beyond which feasible embeddings are not regularly found.
A similar relationship is seen with network upgrades, but
feasible embeddings are not regularly found above 5 EVs and 7
node–phase pairs with voltage limits. Linear scaling indicates
that the proposed formulation should meaningfully benefit as
larger annealing-based quantum processors become available.
However, the current maximum problem size is restrictive even
for relatively small-scale applications.
To investigate the performance of quantum annealing com-
pared with classical computing, the proposed QUBO prob-
lem was solved using D-Wave’s Advantage processor and
with simulated annealing, which is a state-of-the-art classical
method for combinatorial optimization problems [60]. For
quantum annealing, the problem was solved using 100 samples
and an annealing time of 100 µs. For simulated annealing,
the problem was implemented using the dwave-neal Python
package [61], and solved using a 2.3 GHz 8-core Intel Core i9
processor and 16 GB of RAM. To allow comparison between
the computation time of the two methods, simulated annealing
was implemented with one repetition and two sweeps, which
was found to result in solutions of similar energy (i.e. the
QUBO objective including penalty terms). Table II compares
the average computation time, average net utility and average
energy of the two methods, when the problem is formulated
with different numbers of EVs, voltage limits at 3 node–phase
pairs and without considering network upgrade decisions.
The averages were obtained by solving each formulation
10 times using each method. As shown, for a single EV,
quantum annealing has a lower average computation time,
higher average net utility and lower average energy. Also,
7
TABLE II: Comparison between the average computation time,
average net utility and average QUBO energy, when the QUBO
problem is solved 10 times using simulated annealing (SA) and
quantum annealing (QA). The QUBO problem was formulated with
different numbers of EVs, voltage limits at 3 node–phase pairs and
without network upgrade decisions.
No. Avg. Time (ms) Avg. Net Utility (£) Avg. Energy (×103)
EVs SA QA SA QA SA QA
156.4 49.9−78.9−70.4 2.03 1.70
384.9 50.0−47.6−40.5 2.00 2.24
594.9 51.2−14.1−4.6 2.15 2.25
7105.7 51.4−3.2 13.6 1.91 2.40
9121.5 52.2 25.5 37.4 2.10 2.47
2468
Electric Vehicles
0
2500
5000
Qubits Assigned
Upgrades
No
Yes
(a)
2468
Voltage Limits
0
2500
5000
Qubits Assigned
Upgrades
No
Yes
(b)
Fig. 4: The number of assigned qubits on D-Wave’s Pegasus topology
for (a) different numbers of EVs, given voltage limits at 3 node–phase
pairs, and (b) voltage limits enforced at different numbers of node–
phase pairs, given a single EV, with and without network upgrade
decisions.
under quantum annealing, the computation time only increases
slightly from 49.9 ms with 1 EV, to 52.2 ms with 9 EVs,
while the computation time for simulated annealing increases
from 56.4 ms to 121.5 ms. The average net utility is also
consistently higher using quantum annealing. Each method
yields similar average QUBO energies, driven mainly by slight
constraint violations. However, for simulated annealing the
average energy is more consistent for different problem sizes,
while using quantum annealing there is an upward trend with
increasing problem size, which could indicate a future scaling
challenge.
B. Hybrid Solver Implementation
D-Wave’s hybrid quantum-classical binary quadratic model
solver allows larger problems of practical interest to be solved.
To demonstrate this, case studies are presented with 30 EVs,
12:00 16:00 08:00 12:00
Time
100
0
100
175
Net Imports (kW)
Upgrades
No
Yes
Fig. 5: The net real import power for case studies with 30 EVs,
with and without network upgrade decisions, using D-Wave’s hybrid
binary quadratic solver with a time limit of 120 s.
12:00 16:00 08:00 12:00
Time (h)
25
50
75
100
Avg. EV Energy (%)
Upgrades
No
Yes
Fig. 6: The average EV energy for case studies with 30 EVs, with and
without network upgrade decisions, using D-Wave’s hybrid binary
quadratic solver with a time limit of 120 s.
which together have a significant impact on local power
demand and network voltages. Upper and lower voltage limits
are specified for 12 node–phase pairs spread throughout the
network. Case studies with and without the potential for
network upgrades are compared to show the additional value
of co-optimising operational flexibility and network investment
decisions. The hybrid solvers are heuristic, iteratively making
use of both classical and quantum computation for a specified
time limit, where upon the solution giving the lowest total
energy (QUBO objective) is returned.
First, case studies with and without network upgrades are
completed with a time limit of 120 s. Fig. 5 shows the net
power imported from the main grid for each case, and Fig.
6 shows the average energy of the 30 EVs. Fig. 7 shows the
range of the voltage magnitudes across the three phases. In
the case where network upgrades are allowed, the solution
returned by the hybrid solver specifies the moderate network
upgrade plan (reducing impedances by half) and 100 kW of
PV generation (split between the three potential sites). Without
network upgrades, only 50 kW of PV is installed, at the site
closest to the main grid.
As shown in Fig. 5, allowing network upgrades results in
greater maximum import and export of power, and from Fig. 6
it can be seen that the EVs reach a higher final average energy
level (78.6% compared with 56.8%). With network upgrades
enabled, the overall net utility over the day is £452, which
is 78% higher than the net utility without network upgrades
(£254). For the node–phase pairs where the voltage limits are
8
12:00 16:00 08:00 12:00
Time
0.95
1.00
1.05
1.10
|V| (pu)
Ph. A (No Upgrades)
Ph. B (No Upgrades)
Ph. C (No Upgrades)
(a)
12:00 16:00 08:00 12:00
Time (h)
0.95
1.00
1.05
1.10
|V| (pu)
Ph. A (Upgrades)
Ph. B (Upgrades)
Ph. C (Upgrades)
(b)
Fig. 7: The voltage magnitude range across the nodes for each phase
for case studies with 30 EVs, using D-Wave’s hybrid binary quadratic
solver with a time limit of 120 s. The voltage limits are 0.95 pu and
1.05 pu. The voltage magnitude ranges are shown for (a) the case
without network upgrade decisions and (b) the case with network
upgrade decisions.
explicitly enforced, the lowest voltage magnitude reached is
0.958 pu for the case without network upgrades, and 0.951 for
the case with network upgrades. As seen in Fig. 7a and 7b,
there are slight violations at nodes where the limits are not
enforced, with 0.947 pu the lowest voltage magnitude reached
in both cases.
Next, to show the impact of the hybrid solver time limits, the
QUBO problem is solved 30 times for a range limits between
15 s and 120 s, with and without network upgrade decisions.
Note that D-Wave’s hybrid solver imposes a minimum time
limit based on its assessment of the problem complexity, which
varied around 12 s, so 15 s was selected as a consistent
starting point. The box plots in Fig. 8a and Fig. 8b show the
distributions of the QUBO energy (returned objective function
value) and net utility. As shown in Fig. 8a, increasing the
time limit reduces the QUBO energy, with diminishing returns
starting to be seen above 60 s. Most of this reduction results
from slight improvements in constraint satisfaction, and from
Fig. 8b, it can be seen that the average net utility is fairly
stable above 30 s. The heuristic nature of the hybrid solver is
clear from the variability of the net utility, but it can be seen
that even accounting for outliers, the net utility for cases where
network upgrades are allowed is consistently higher than when
they are disallowed.
15 30 60 90 120
Time Limit (s)
200
0
200
QUBO Energy
Upgrades
No
Yes
(a)
15 30 60 90 120
Time Limit (s)
200
300
400
500
Net Utility (£)
Upgrades
No
Yes
(b)
Fig. 8: Results from solving the QUBO problem 30 times for a
range hybrid solver time limits, with and without network upgrade
decisions. Distributions are shown for (a) the QUBO energy and (b)
net utility values. The box plots show the median (centre line), in-
terquartile range (box), 1.5 times the interquartile range above/below
the box (whiskers), and outliers (circles).
V. CONCLUSION
Annealing-based quantum computing offers a new comput-
ing hardware platform with the future potential to efficiently
solve large-scale combinatorial optimisation problems. This
could be highly valuable for the power sector, particularly
for network operators aiming to integrate DER flexibility
into network planning and investment decision making. To
demonstrate this opportunity, a novel QUBO formulation
which can be solved with quantum annealing was developed
for a linear multiphase OPF problem, with controllable on/off
EV charging, renewable generation placement/sizing and net-
work upgrade decisions. Case studies based on the IEEE
European Low Voltage Test Feeder were implemented on D-
Wave’s 5,760 qubit Advantage quantum processor to show
how the problem size impacts the required number of qubits.
Although it was found that the quantum processor is too small
for distribution-scale applications, the number of qubits was
observed to grow linearly with the number of EVs and the
number of network voltage constraints, indicating that there
is a promising future opportunity given the rate of techno-
logical development of annealing-based quantum processors.
D-Wave’s hybrid quantum-classical binary quadratic model
solver was also used to solve larger case studies with 30
EVs, where EV charging flexibility has a significant impact
on distribution network power flows. In this case, combina-
torial co-optimisation of EV flexibility with generation and
network investment decisions was shown to offer substantial
9
value. The paper has focused on deterministic combinatorial
OPF as a first step, but in practice network planning and
scheduling applications may involve significant uncertainty
due to the weather-dependence of renewable generation and
the behaviour-dependence of flexible loads. An important area
for future work is to investigate how methods for robust
optimisation (see e.g. [62]) can be implemented within the
qubit limitations of quantum annealers. Another important area
for future work is the optimal selection of penalty terms, which
are necessary for constraint handling within the proposed
formulation, but may affect the computation time and solution
quality if chosen inappropriately.
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Thomas Morstyn (S’14–M’16–SM’22) received the
BEng (Hon.) degree from the University of Mel-
bourne in 2011, and the PhD degree from the
University of New South Wales in 2016, both in
electrical engineering.
He is Lecturer of Power Electronics and Smart
Grids with the School of Engineering, University
of Edinburgh. He is also the Deputy Champion
of Energy Distribution and Infrastructure for the
Scottish Energy Technology Partnership (ETP) and
an Associate Editor of IEEE Transactions on Power
Systems. His research interests include multi-agent control and market design
for integrating distributed energy resources into power system operations.
