On the Computational Viability of Quantum Optimization for PMU Placement
Eric B. Jones, Eliot Kapit, Chin-Yao Chang, David Biagioni, Deepthi Vaidhynathan, Peter Graf, Wesley Jones
OOn the Computational Viability of QuantumOptimization for PMU Placement
Eric B. Jones
Department of PhysicsColorado School of Mines andNational Renewable Energy LaboratoryGolden, Colorado 80401, USAEmail: [email protected]
Eliot Kapit
Department of PhysicsColorado School of MinesGolden, Colorado 80401, USAEmail: [email protected]
Chin-Yao Chang, David Biagioni,Deepthi Vaidhynathan, Peter Graf,and Wesley Jones
National Renewable Energy LaboratoryGolden, Colorado 80401, USAEmail: [email protected]
Abstract —Using optimal phasor measurement unit placementas a prototypical problem, we assess the computational viabilityof the current generation D-Wave Systems 2000Q quantumannealer for power systems design problems. We reformulateminimum dominating set for the annealer hardware, solve thereformulation for a standard set of IEEE test systems, andbenchmark solution quality and time to solution against theCPLEX Optimizer and simulated annealing. For some probleminstances the 2000Q outpaces CPLEX. For instances where the2000Q underperforms with respect to CPLEX and simulatedannealing, we suggest hardware improvements for the nextgeneration of quantum annealers.
I. I
NTRODUCTION
As progress continues to be made towards universal error-corrected quantum computers, many opportunities exist totest the problem solving capabilities of current and near-termquantum hardware [1]. Along with problems in chemistry, ar-tificial intelligence, and sampling, combinatorial optimizationproblems are excellent candidates to see quantum-enhancedspeedups to solution [2].Meanwhile, a new paradigm is emerging regarding howto construct next-generation energy grids that are secure,resilient, cost-effective, and which can incorporate large quan-tities of distributed renewable energy. Such systems will likelyinvolve intensive online computation, optimal control overmultiple timescales, and extensive state monitoring in orderto dynamically adapt to varying generation and demand [3].Given the complexity of this task, offline optimization andrational design of grid properties that allow more efficientonline computation and observation is crucial to the perfor-mance of future power networks. In its simplest depiction, apower grid may be modeled as an undirected graph wherebuses in the system are assigned to graph nodes and branchesare assigned to graph edges. At this level of abstraction,the first step towards designing a power system consists ofsolving a combinatorial optimization problem defined over thegraph. Many power grid-relevant combinatorial optimizationproblems are NP-complete [4] [5] [6]. It is therefore importantto identify and assess the performance of novel approximateand heuristic solution methods for combinatorial optimizationproblems particular to power systems design in instanceswhere exact solution is infeasible. Adiabatic quantum annealing (AQA) constitutes one ofthe main efforts to outperform classical solution methods onhard combinatorial optimization problems [7]. Definite run-time speedups have been demonstrated for AQA against bothsimulated annealing (SA)– in the form of a scaling advantage–and quantum Monte Carlo (QMC)– as a fixed prefactor–using the D-Wave 2X and 2000Q machines on proof-of-principle problems designed to have tall and narrow energybarriers [8] [9]. Additionally, the largest D-Wave annealer has2,048 qubits [1]. Given the large scale of available quantumannealers and their positive prospects for runtime speedupon solving combinatorial optimization problems, we thereforeassess the feasibility of speeding up the optimization of powersystems using AQA. As a prototypical example, we considerthe optimal phasor measurement unit placement (OPMUP)problem. Formulated as a graph theoretic problem, we treatthe simplest variant of OPMUP, minimum dominating set(MDS), rather than a more realistic formulation such as powerdominating set for clarity in reformulation and discussion. Wereformulate the corresponding integer linear program (ILP)into a quantum Hamiltonian operator suitable for solutionon a D-Wave quantum annealer. We analyze the scaling ofthe physical resources required to contend with the MDSformulation on standard Institute of Electrical and ElectronicsEngineers (IEEE) test power systems ranging in size from9 to 300 buses. For those problem instances that are minor-embeddable on a (16, 4)-Chimera hardware graph, we assessthe ability of the D-Wave 2000Q quantum processing unit(QPU) to accurately find ground state solutions to OPMUPusing standard annealing schedules.The OPMUP problem is applicable to next-generationpower system design due to the fact that increased incorpo-ration of renewables into the grid and demand-side manage-ment require accurate spatiotemporal state estimation of thegrid on sub-second timescales [3]. Synchrophasors, or phasormeasurement units (PMUs), are able to measure voltage andcurrent amplitude and phase angles at a rate of 30-60 Hz in aGPS synchronized manner and are therefore able to reconstructthe entire state of the grid if a measurement can be obtainedfor every bus in the power system [10]. However, placement ofa PMU at every bus is not necessary since PMUs are also able a r X i v : . [ qu a n t - ph ] J a n o deduce synchrophasor quantities at adjacent, and in someinstances further, buses [11]. Along with cost considerations,this fact implies that there is a minimum number of PMUsthat need to be placed on a given grid topology for fullobservability, which is substantially fewer than the numberof buses in the power system [12]. The MDS formulationof OPMUP, while being the simplest variant, is nonethelessNP-complete, and there exists a large body of optimizationliterature devoted to finding its solution [13]. And while alsoresiding on the “highly idealized” end of the spectrum ofdesign problems future power systems will likely need tosurmount, considering the solution of such problems by AQAshould point both towards other areas in power system designwhere quantum optimization could be helpful, and ways inwhich the attendant quantum hardware could be improved tobetter address such problems.II. Q UANTUM OPTIMIZATION
For a discrete set of possible solutions expressed as binaryvariable strings { x = ( x . . . x N ) } , a classical cost function C ( x ) to be extremized, and a set of constraints { g ( x ) = 0 } , anecessary condition for the combinatorial optimization prob-lem to be adapted to quantum hardware is that the cost functionand all constraints in the problem be represented by a quantumHamiltonian operator, ˆ H [14]. A common procedure for con-structing ˆ H begins by re-expressing the set of constraints asa (usually quadratic) penalty function P , which is then addedto C in order to write the whole problem as an extremizationproblem: H ( x ) ≡ C ( x ) + P ( x ) . Binary variables x i ∈ { , } are related to classical spin variables s i used in SA by thetransformation s i = 1 − x i . A suitable Hamiltonian operatoris then obtained by elevating the classical spin variables tosingle qubit operators, which measure qubit states in thecomputational basis: H ( s ) → ˆ H ( ˆ Z ) . Generally, a Hamiltonianobtained this way may be expanded in powers of single qubitoperators [8] ˆ H = − K (cid:88) k =1 N (cid:88) j ...j k =1 J j ...j k ˆ Z j . . . ˆ Z j k . (1)In principle, AQA is able heuristically to solve Eq. (1) toarbitrary order K . However, a particularly relevant class ofproblem Hamiltonians occurs when K = 2 . Such Hamilto-nians fall into the class of problems termed “quantum Isingmodels” and have the form ˆ H IS = − N (cid:88) j =1 J j ˆ Z j − N (cid:88) j =1 N (cid:88) j =1 J j j ˆ Z j ˆ Z j . (2) ˆ H IS is the quantum analog of the well-known classical Isingmodel and the equivalent quadratic unconstrained binary opti-mization (QUBO) problem, and the reason for its prominenceis that two-body qubit interactions have been the most straight-forward to engineer in hardware with higher-order interactionsrequiring additional overhead [15]. Both the classical Isingmodel and QUBO fall under the umbrella of binary quadratic models (BQM). Therefore, finding the quantum mechanicalground state to ˆ H IS is equivalent to solving any combinatorialoptimization problem expressed as a BQM. It will be shownin Sec. III that OPMUP can be formulated as such a BQMand so is amenable to solution on existing quantum hardware.The current generation D-Wave quantum annealer obtainsheuristic solutions to Eq. (2) by evolving the time-dependentHamiltonian ˆ H ( t ) = − A ( t ) N (cid:88) i =1 ˆ X i + B ( t ) ˆ H IS (3)adiabatically, through the parameters A ( t ) and B ( t ) , such thatat time t = 0 , A (0) >> B (0) and at time t = τ , A ( τ ) <
The minimum dominating set (MDS) representation ofOPMUP can be stated as follows. Let an electric power gridbe represented by a graph G = ( V, E ) where the nodeset V represents the buses in the system and the edge set E represents the branches. Oftentimes, an edge representsa transmission line, but this is not always the case. Wewould like to select the minimal initial number of nodes in PMU (𝑥 , = 1) 𝑥 / = 0𝐴 𝑥 , = 1 Fig. 1. MDS formulation of OPMUP. Red is a placed PMU, red and pinkare observed nodes as a result, and blue nodes are unobserved. V (or place the minimal number of PMUs) such that afterfollowing the observability rules for the graph, the full graphis observed. Denote the initial selection of nodes as a bit string x = ( x . . . x n ) ∈ { , } N where denotes an unobservedinitial node (no PMU) and denotes an observed initial node(with PMU). N = | V | is the order of the graph. The simplestset of observability rules we can take are as follows: i) a nodeis observed if it has had a PMU placed on it and ii) a nodeis observed if it shares an edge with a PMU. These rules canbe summarized using the matrix A = ˜ A + , where ˜ A is theadjacency matrix of the graph G and is the N × N identitymatrix. Then, if we define column vectors corresponding to thebit strings, x = ( x , . . . , x n ) T , and denote b = (1 , . . . , T ,the appropriate constrained optimization problem is min N (cid:88) i =1 x i (6)subject to A x ≥ b . (7)In Fig. 1, a PMU has been placed on node 1, indicated in red.As a result, itself along with nodes 2, 3, 6, and 7 are observed(pink). Nodes 4, 5, and 8, in blue, remain unobserved. Theminimum number of placed PMUs needed to observe thewhole graph is called the domination number, γ ( G ) .Eq. (6) indicates that the appropriate cost function tominimize is C ( x ) = (cid:80) i x i . Eq. (7) however is not presentlyin a form that can be enforced as a minimization problem. Inorder for this to be the case we re-write Eq. (7) as a quadraticpenalty function with N non-negative integer-valued surplusvariables { y i } P ( x, y ) = N (cid:88) i =1 α i (cid:32) N (cid:88) j =1 A ij x j − b i − y i (cid:33) . (8)We note briefly here that while derived independently, ourreformulation of minimum dominating set is similar to thetreatment by Dinnean and Hua [17]. In order to see that P appropriately accounts for Eq. (7), consider the i th term in P . If (cid:80) Nj =1 A ij x j < b i , the minimal value for the term isachieved when y i = 0 and a penalty will still be incurred because the square of the term will still be positive. If however, (cid:80) Nj =1 A ij x j ≥ b i , y i can be chosen in order to make theterm zero, thus incurring no penalty. Each y i then needs tobe represented as an expansion of binary variables so thatit too can be represented on the annealer. To understand theresources required for this, consider that the largest number y i will ever need to be is (cid:80) Nj =1 A ij (1) − b i = d i + 1 − b i . Thus, d i + 1 − b i ≥ y i ≥ . Let µ mi ≡ (cid:100) log ( d i + 1 − b i ) (cid:101) , where (cid:100)(cid:101) is the ceiling function, be the number of bits needed torepresent the upper bound to y i in a binary expansion. Then,the number of ancilla bits needed for the whole problem is (cid:80) Ni =1 µ mi , the binary expansion of y i is y i = µ mi − (cid:88) µ =0 µ y iµ , (9)and the penalty function becomes P ( x, y ) = N (cid:88) i =1 α i (cid:32) N (cid:88) j =1 A ij x j − b i − µ mi − (cid:88) µ =0 µ y iµ (cid:33) . (10)The full classical Hamiltonian to be minimized is then H ( x, y ) = C ( x ) + P ( x, y ) . (11)Since H ( x ) only contains terms constant, linear, and quadraticin the binary variables { x i } and { y iµ } , it may be suitablyprogrammed into the D-Wave as a BQM.IV. M INOR EMBEDDING OF
IEEE
TEST POWER SYSTEMS
While reformulation of an optimization problem as a BQMis a necessary condition in order to be able to use AQA forits solution, it is not sufficient. One must also be able toembed the optimization problem into the hardware graph ofthe quantum processor, where nodes of the hardware graphrepresent physical qubits and edges represent physical qubit-qubit couplings. The 2000Q D-Wave quantum processingunit (QPU) is constructed from a 2,048 qubit (16,4)-chimerahardware graph topology consisting of a × array of 8-qubit chimera unit cells, each of which organized as a K , complete bipartite graph [18]. Meanwhile, a standard set ofIEEE test power systems consists of the 9, 14, 24, 30, 39, 57,118, and 300 bus test systems shown in the first column inTable I. Column two shows the number of buses (nodes) ineach test system while column three shows the correspondingnumber of branches (edges) in the test system. The number ofancilla bits required to represent Eq. (7) is shown in columnfour. The number of nontrivial pairwise interactions inducedby the quadratic penalty function, Eq. (10), is shown in columnfive. And finally, column six shows the minimum number ofphysical qubits found by the D-Wave Ocean API minorminer tool required to embed the full problem Hamiltonian, Eq.(11), into the D-Wave 2000Q hardware graph by calling the find embedding routine 10 times. The blank entry in columnsix denotes an instance where the embedding heuristic intro-duced by Cai et al. was unable to find a suitable embedding ABLE IS
CALING OF RESOURCES FOR
MDS ON IEEE
TEST POWER SYSTEMS .System Buses Branches Ancillas Interactions QubitsIEEE 9 9 9 9 57 49IEEE 14 14 20 21 150 146IEEE 24 24 34 39 278 287IEEE 30 30 41 42 325 349IEEE 39 39 46 49 337 338IEEE 57 57 78 85 607 704IEEE 118 118 179 188 1,585 1,564IEEE 300 300 409 417 3,478 - [19]. For this formulation of MDS and for the particularconnectivity of the Chimera working graph, the number ofphysical qubits required to embed a given test system scalesroughly linearly with the number of interactions induced bythe penalty function. From this vantage point, it is clear whythe IEEE 300 bus test system cannot be embedded in the D-Wave hardware graph, since its 3,478 interactions exceed theD-Wave’s 2,048 qubits. This fact is confirmed by consideringthe instance of minor embedding a complete graph ( K n ) inthe (16, 4)-Chimera graph. It can be shown that the largestcomplete graph that can be embedded on the (16, 4)-Chimeragraph is ( K ˜ n ) where ˜ n = 1 + 4 min(16 ,
16) = 65 [18]. Acomplete graph ( K n ) has n ( n − / edges, or interactions.For ˜ n = 65 , ˜ n (˜ n − / , . Therefore, any interactiongraph with a number of edges > , cannot be embedded onthe (16, 4)-Chimera architecture, and the 300 bus test systemis ruled out.V. A SSESSMENT OF SOLUTION QUALITY
Generally speaking, an optimization heuristic holds value ifit has the prospect to reliably and quickly find optimal or near-optimal solutions to arbitrary, unstructured problem instances.With this perspective, the industry-standard CPLEX Optimizercan be regarded as an important benchmarking tool to makequantitative the terms “reliably” and “quickly”. In order tofurther characterize the viability of current and near-termquantum optimization heuristics for power system design, wecompare both the best solution found and time to best solutionfrom the D-Wave 2000Q processor with CPLEX Optimizerresults for the ILP in Eqs. (6) and (7) obtained in recent workby K. Sou [20]. We do not regard any of the numerically timedquantities as immutable fact since the results of numericalexperiments generally depend not only upon the algorithms atplay but also the particular hardware on which they are run andtheir implementation. It is nevertheless illustrative to comparethe scaling of 2000Q performance against a relatively standardoptimization package running on relatively standard hardware.We report two times-to-best metrics for the quantum an-nealer. A full quantum computation to find an optimal solutiontakes time T = T P + k ( τ + T R ) , (12)where T P is the time required to program the problem ontothe QPU and initialize the control sequence, τ (introducedin Sec. II) is the time taken for one annealing schedule to TABLE IIC
OMPARISON OF SOLUTIONS AND COMPUTATION TIMES ( IN SECONDS ) FOR
MDS ON IEEE
TEST POWER SYSTEMS FOR
SA, CPLEX,
AND
AQA.System γ SA γ AQA T CPLX ( s ) [20] T A ( s ) T ( s ) IEEE 9 3 3 0.0016 0.000070 0.0094IEEE 14 4 4 0.0066 0.000004 0.0086IEEE 24 7 7 0.0083 0.039671 0.2158IEEE 30 10 10 0.0063 0.000342 0.0130IEEE 39 13 14 0.0065 0.281507 0.4500IEEE 57 17 20 0.0140 0.078621 0.0910IEEE 118 32 49 0.0100 2.985983 4.4380 complete, T R is the time it takes to read a measurement atthe end of one annealing schedule, and k is the number oftimes the anneal-read cycle is repeated in order to ideallyobtain adequate statistics on the probabilities {| a i | } . We call T A ≡ kτ the “annealing time” and T the “QPU access time”.It is conventional to consider T A as the quantum equivalentto classical CPU time since T P has its classical equivalentin the time it takes to compile classical code and T R isfixed overhead for any quantum computation and thereforedoes not give any useful information about the scaling ofthe AQA algorithm proper [21]. However, we do report on T as well for completeness. Column one of Table II againshows the IEEE test system analyzed. Column two displaysbest-found domination numbers obtained by SA ( γ SA ) , whichcorroborate those found in Ref. [20] using CPLEX. Columnthree shows the best-found domination number obtained bythe 2000Q processor ( γ AQA ) . Column four shows the time tobest solution for the CPLEX Optimizer ( T CP LX ) running ona Mac with a 2.5 GHz CPU and 8GB of RAM. Column fiveshows the annealing time to best solution ( T A ) while columnsix shows the corresponding QPU access time ( T ) .For our AQA calculations we set the penalty strength α i = 2 ∀ i , which corresponds to a reasonably hard enforcementof Eq. (7) in the penalty function– Eq. (10)– and the chainstrength J chain = 1 . | J | M , which gauges how stronglydifferent physical qubits representing a single logical qubitinteract. | J | M is the maximum coupling strength encounteredin the Ising formulation of each problem instance. Note that | J | M = 8 for all test systems except IEEE 9 for which | J | M = 2 and IEEE 118 and 300 for which | J | M = 32 . J chain was chosen such that the fraction of logical of chains brokenupon readout of best-found solutions was bounded from aboveby . . T A was calculated in the following manner. Gridsof τ and k values were created: τ ∈ [1 µs, µs ] and k ∈ [1 , , so that the maximum annealing time consideredwas less than s – the maximum annealing time allowed bythe 2000Q. For each parameter, 20 grid points were chosen,evenly spaced on a base-12 logarithmic scale, and rounded tothe nearest integer. At each point in the { τ } × { k } grid, thelowest energy solution to Eq. (11) found by the correspondingannealing schedule was obtained, x ∗ ( τ, k ) and only solutionsthat satisfy Eq. (7) were kept. For a given IEEE test systemthen, γ AQA = min ( τ,k ) (cid:80) i x ∗ i ( τ, k ) and T A = k ∗ × τ ∗ where ( τ ∗ , k ∗ ) = arg min γ AQA .s can be seen in columns one and two of Table II, the2000Q processor is able to find the same domination numberas obtained by SA and CPLEX in the four smallest probleminstances of the seven for which a suitable minor embeddingwas found. In three of four of these instances (IEEE 9, 14,and 30), the T A is at least an order of magnitude less than T CP LX . Interestingly, these three test systems are all planargraphs. There are two non-planar problem graphs for whicha minor embedding was found (IEEE 24, and 57). While γ SA was found by the 2000Q for IEEE 24, T A was roughlyfive times larger than T CP LX on that problem instance. ForIEEE 57, the correct domination number is missed by threeadditional PMUs and T A ∼ T CP LX . Finally, while the2000Q misses γ SA by only one additional PMU for IEEE 39,the solution quality deteriorates rapidly for the larger probleminstance of IEEE 118. The failure at this problem instancepoints to an important improvement to be made in futureAQA hardware. IEEE 118 is the only embeddable probleminstance in which | J | M = 32 and | h | M = 56 . We conjecturethat when autoscaled to fit within the machine-specific analogparameter ranges h ∈ [ − , and J ∈ [ − , (or J ∈ [ − , for the VFYC solver), the resulting hardware-level energy gapsbetween different solutions become small compared to systemtemperature k B T sys . And while decreasing J chain to . | J | M for IEEE 118 resulted in an improvement in time to solution: T A ∼ . s and T ∼ . s , the best found solution wassimilarly poor γ AQA = 49 . Hence, extending parameter rangesand ensuring that energy gaps can be made large enough withrespect to system temperature to avoid thermal-noise for largeand highly-connected problem instances should be a mainfocus of next-generation hardware design.VI. C
ONCLUSION
The results presented in Sec. V corroborate that AQA holdsthe potential to outpace well-developed classical optimizationmethods on combinatorial optimization problems. Overcomingthe current limitations in hardware connectivity and thermalnoise for large and highly-connected graphs should allowquantum optimization to begin to address increasingly chal-lenging problems, and therefore become a useful tool in thedesign of future power systems.A
CKNOWLEDGMENT
This work was authored in part by the National Renew-able Energy Laboratory (NREL), operated by Alliance forSustainable Energy, LLC, for the U.S. Department of Energy(DOE) under Contract No. DE-AC36-08GO28308. This workwas supported by the Laboratory Directed Research and De-velopment (LDRD) Program at NREL. The views expressedin the article do not necessarily represent the views of theDOE or the U.S. Government. The U.S. Government retainsand the publisher, by accepting the article for publication,acknowledges that the U.S. Government retains a nonexclu-sive, paid-up, irrevocable, worldwide license to publish orreproduce the published form of this work, or allow othersto do so, for U.S. Government purposes. This research used Ising, Los Alamos National Laboratory’s D-Wave quantumannealer. Ising is supported by NNSA’s Advanced Simulationand Computing program. The authors would like to thank ScottPakin and Denny Dahl. This material is based in-part uponwork supported by the National Science Foundation underGrant No. PHY-1653820.R
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