Moment-Based Relaxation of the Optimal Power Flow Problem
aa r X i v : . [ m a t h . O C ] J a n Moment-Based Relaxation of the Optimal PowerFlow Problem
Daniel K. Molzahn, Ian A. Hiskens
University of MichiganAnn Arbor, MI 48109, [email protected], [email protected]
Abstract —The optimal power flow (OPF) problem minimizespower system operating cost subject to both engineering andnetwork constraints. With the potential to find global solutions,significant research interest has focused on convex relaxationsof the non-convex AC OPF problem. This paper investigates“moment-based” relaxations of the OPF problem developedfrom the theory of polynomial optimization problems. At thecost of increased computational requirements, moment-basedrelaxations are generally tighter than the semidefinite relaxationemployed in previous research, thus resulting in global solutionsfor a broader class of OPF problems. Exploration of the feasiblespace for test systems illustrates the effectiveness of the moment-based relaxation.
Index Terms–Optimal power flow, Global optimization,Moment relaxation, Semidefinite programming
I. I
NTRODUCTION
The optimal power flow (OPF) problem determines anoptimal operating point for an electric power system in termsof a specified objective function, subject to both networkequality constraints (i.e., the power flow equations, whichmodel the relationship between voltages and power injections)and engineering limits (e.g., inequality constraints on voltagemagnitudes, active and reactive power generations, and lineflows). Generation cost per unit time is typically chosen asthe objective function.The OPF problem is generally non-convex due to thenon-linear power flow equations [1] and may have localsolutions [2]. Non-convexity of the OPF problem has madesolution techniques an ongoing research topic. Many OPFsolution techniques have been proposed, including successivequadratic programs, Lagrangian relaxation, genetic algorithms,particle swarm optimization, and interior point methods [3].Recently, significant attention has focused on a semidefiniterelaxation of the OPF problem [4]. Using a rank relaxation,the OPF problem is reformulated as a convex semidefiniteprogram. If the relaxed problem satisfies a rank condition (i.e.,the relaxation is said to be “exact” or “tight”), the globalsolution to the original OPF problem can be determined inpolynomial time. Prior OPF solution methods do not guaranteefinding a global solution in polynomial time; semidefinite pro-gramming approaches thus have a substantial advantage overtraditional solution techniques. However, the rank conditionis not satisfied for all practical OPF problems [2], [5], [6].This paper presents alternative “moment-based” relaxationsthat globally solve a broader class of OPF problems. Currently, there is substantial research interest in determin-ing sufficient conditions for which the semidefinite relaxationof [4] is exact. Existing sufficient conditions include require-ments on power injection and voltage magnitude limits andeither radial networks (typical of distribution system mod-els) or appropriate placement of controllable phase shiftingtransformers. (See [7] and the references therein for detaileddescriptions of these conditions.)Extending this literature to mesh networks without phase-shifting transformers, [8] investigates the feasible space ofactive power injections for weakly-cyclic networks (i.e., net-works where no line belongs to more than one cycle). Al-ternative representations of the line-flow constraints (apparentpower, active power, voltage difference, and angle differencelimits), while similar in the original OPF problem, can lead tosignificantly different results for the semidefinite relaxation.Assuming no lower limits on active and reactive power in-jections at each bus and with line-flow limits represented asvoltage differences (i.e., for voltage phasors V i and V j , con-strain the flow between buses i and j as | V i − V j | ≤ ∆ V max ij ),the semidefinite relaxation of [4] is proven exact for weakly-cyclic networks with cycles of size three.While the sufficient conditions developed thus far arepromising, they only apply to a limited subset of OPF prob-lems. For more general cases, [9] proposes a method forfinding a globally optimal solution that is “hidden” in a higher-rank subspace of solutions to the semidefinite relaxation.That is, the solution to the semidefinite relaxation obtainednumerically does not satisfy the rank condition but a rankone solution exists with the same globally optimal objectivevalue. For these cases, the semidefinite relaxation is exact inthe sense that it yields the globally optimal objective valuerather than a strict lower bound, but this fact is not evident asthe semidefinite relaxation solution does not directly provideglobally optimal decision variables (i.e., the optimal voltagephasors). With a heuristic for finding such “hidden” solutions,[9] broadens the applicability of the semidefinite relaxation.However, there exist practical problems for which thesemidefinite relaxation is not exact (i.e., the semidefiniterelaxation solution has optimal objective value strictly lessthan the global minimum) [6], [9]. For such cases, [8] and [9]propose heuristics for obtaining an (only-guaranteed-locally)optimal solution from the semidefinite relaxation. Heuristicsare promising for finding local solutions, and the optimalbjective value of the semidefinite relaxation provides a metricfor the potential suboptimality of these solutions. For example,the heuristic in [9] finds a solution to a modified form of theIEEE 14-bus system that is within 0.13% of global optimality.This compares favorably to a solution from the interior-pointsolver in MATPOWER [10], which is only within 4.83% ofglobal optimality.While deserving of further study, heuristics eliminate theglobal optimality guarantee that is one of the main advantagesof the semidefinite relaxation. This paper therefore proposesan alternative moment-based convex relaxation that, whenexact, yields the global optimum. Using theory developedfor polynomial optimization problems [11], moment-basedrelaxations have the potential to globally solve a broad classof OPF problems, including many problems for which thesemidefinite relaxation of [4] is not exact. The moment-based relaxation exploits the fact that the OPF problem iscomposed of polynomials in the voltage phasor componentsand is therefore a polynomial optimization problem.The ability to globally solve a broader class of OPFproblems has a computational cost. Whereas the semidef-inite relaxation of [4] optimizes matrices composed of alldegree-two combinations of the voltage phasor components,the moment-based relaxations optimize matrices composedof higher-degree combinations. In particular, for an n -bussystem, the order- γ moment-based relaxation is solved usinga semidefinite program which has a positive semidefiniteconstraint on a k × k matrix, where k = (2 n + γ )! / ((2 n )! γ !) (i.e., this matrix is composed of all combinations of voltagecomponents up to order γ ). Thus, the computational require-ments of the moment-based relaxations can be substantiallylarger than the semidefinite relaxation of [4], especially forhigher orders of the moment-based relaxation.Fortunately, experience with small systems suggests that low(often second) order relaxations globally solve a broad class ofOPF problems, including problems for which the semidefiniterelaxation of [4] is not exact due to disconnected or otherwisenon-convex feasible spaces. Note that no fixed-order relaxationis exact for all OPF problems due to the polynomial-timecomplexity of semidefinite programs as compared to the NP-hardness of some OPF problems [4]. Indeed, as discussedin Section V, some of the NP-hard problems in [4] provideexamples where low-order moment-based relaxations are notexact. Also note that the moment-based relaxation is currentlyonly computationally tractable for small OPF problems; futurework includes exploiting sparsity to extend the moment-basedrelaxations to larger systems.After introducing the OPF problem formulation in Section IIand describing the moment-based relaxations in Section III,we explore the feasible space of the second-order moment-based relaxation for a two-bus system in Section IV. Forsome choices of parameters for this system, the semidefiniterelaxation of [4] is not exact. Since, conversely, the moment-based relaxation is exact for this problem, a comparison of thefeasible spaces of the relaxations illustrates the effectivenessof the proposed approach. Section V then presents results from the application of the moment-based relaxation to other smallOPF problems for which the semidefinite relaxation of [4] isnot exact. Section VI concludes the paper and discusses futureresearch directions.II. OPF P ROBLEM F ORMULATION
We first present the OPF problem as it is classically formu-lated. This formulation is in terms of rectangular voltage co-ordinates, active and reactive power generation, and apparent-power line-flow limits. Consider an n -bus power system,where N = { , , . . . , n } is the set of all buses, G is the setof generator buses, and L is the set of all lines. P Dk + jQ Dk represents the active and reactive load demand at each bus k ∈ N . V k = V dk + jV qk represents the voltage phasors inrectangular coordinates at each bus k ∈ N . Superscripts “max”and “min” denote specified upper and lower limits. Buses with-out generators have maximum and minimum generation set tozero (i.e., P max Gk = P min Gk = Q max Gk = Q min Gk = 0 , ∀ k ∈ N \ G ). Y = G + j B denotes the network admittance matrix.The network physics are described by the power flowequations: P Gk = f Pk ( V d , V q ) = V dk n X i =1 ( G ik V di − B ik V qi )+ V qk n X i =1 ( B ik V di + G ik V qi ) + P Dk (1a) Q Gk = f Qk ( V d , V q ) = V dk n X i =1 ( − B ik V di − G ik V qi )+ V qk n X i =1 ( G ik V di − B ik V qi ) + Q Dk (1b) Define a convex quadratic cost function for active powergeneration: f Ck ( V d , V q ) = c k ( f P k ( V d , V q )) + c k f P k ( V d , V q ) + c k (2)Define a function for squared voltage magnitude: ( V k ) = f V k ( V d , V q ) = V dk + V qk (3)Squared apparent-power line-flows ( S lm ) are polynomialfunctions of the voltage components V d and V q . We assumea π -model with series admittance g lm + jb lm and total shuntsusceptance b sh,lm for the line from bus l to bus m . (Forinductive lines and capacitive shunt susceptances, b lm is anegative quantity and b sh,lm is a positive quantity.) P lm = f Plm ( V d , V q ) = b lm ( V dl V qm − V dm V ql )+ g lm (cid:0) V dl + V ql − V ql V qm − V dl V dm (cid:1) (4a) Q lm = f Qlm ( V d , V q ) = b lm (cid:0) V dl V dm + V ql V qm − V dl − V ql (cid:1) + g lm ( V dl V qm − V dm V ql ) − b sh,lm (cid:0) V dl + V ql (cid:1) (4b) ( S lm ) = f Slm ( V d , V q ) = ( f Plm ( V d , V q )) + ( f Qlm ( V d , V q )) (4c) he classical OPF problem is then min V d ,V q X k ∈G f Ck ( V d , V q ) subject to (5a) P min Gk ≤ f Pk ( V d , V q ) ≤ P max Gk ∀ k ∈ N (5b) Q min Gk ≤ f Qk ( V d , V q ) ≤ Q max Gk ∀ k ∈ N (5c) (cid:16) V min k (cid:17) ≤ f V k ( V d , V q ) ≤ (cid:16) V max k (cid:17) ∀ k ∈ N (5d) f Slm ( V d , V q ) ≤ ( S max lm ) ∀ ( l, m ) ∈ L (5e) V q = 0 (5f) Note that this formulation limits the apparent-power flowmeasured at each end of a given line, recognizing that linelosses can cause these quantities to differ. Constraint (5f) setsthe reference bus angle to zero.III. M
OMENT -B ASED R ELAXATION O VERVIEW
The OPF problem (5) is comprised of polynomial functionsof the voltage components V d and V q and can thereforebe solved using moment-based relaxations [11]. We nextpresent the moment-based relaxation for the OPF problem (5).More detailed descriptions of moment-based relaxations areavailable in [11]Polynomial optimization problems, such as the OPF prob-lem, are a special case of a class of problems known as“generalized moment problems” [11]. Global solutions togeneralized moment problems can be approximated usingmoment-based relaxations that are formulated as semidefiniteprograms. For polynomial optimization problems that satisfya technical condition on the compactness of at least oneconstraint polynomial, the approximation converges to theglobal solution(s) as the relaxation order increases [11]. (Thistechnical condition can always be satisfied by adding largebounds on all variables and is therefore not restrictive for OPFproblems.) Note that while moment-based relaxations can findall global solutions to polynomial optimization problems, wefocus on problems with a single global optimum.Formulating the moment-based relaxation requires severaldefinitions. Define the vector ˆ x = (cid:2) V d V d . . . V qn (cid:3) ⊺ ,which contains all first-order monomials. Given a vector α = (cid:2) α α . . . α n (cid:3) ⊺ with α ∈ N n representing monomial exponents, the expression ˆ x α = V α d V α d · · · V α n qn defines themonomial associated with ˆ x and α . A polynomial g (ˆ x ) is then g (ˆ x ) = X α ∈ N n g α ˆ x α (6)where g α is the scalar coefficient corresponding to ˆ x α .Next define a linear functional L y { g } . L y { g } = X α ∈ N n g α y α (7)This functional replaces the monomials ˆ x α in a polynomialfunction g (ˆ x ) with scalar variables y α . If the argument to thefunctional L y { g } is a matrix, the functional is applied to eachelement of the matrix.Consider, for example, the vector ˆ x = (cid:2) V d V d V q (cid:3) ⊺ corresponding to the voltage components of a two-bus system.(For notational convenience, the angle reference constraint V q = 0 is used to eliminate V q .) Consider the polynomial g (ˆ x ) = − f V ( V d , V q ) = − V d + V q . (The constraint g (ˆ x ) = 0 forces the squared voltage magnitude at bus 2 toequal 1 per unit.) Then L y { g } = − y + y + y . Thus, L { g } converts a polynomial g (ˆ x ) to a linear function of y .The order- γ moment-based relaxation forms a vector x γ composed of all monomials of the voltage components up toorder γ : x γ = (cid:2) V d . . . V qn V d V d V d . . .. . . V qn V d V d V d . . . V γqn (cid:3) ⊺ (8)We now define moment and localizing matrices. The mo-ment matrix M γ ( y ) has entries y α corresponding to allmonomials x α up to order γ and is symmetric, M γ ( y ) = L y (cid:0) x γ x ⊺ γ (cid:1) . (9)Consider, for instance, a two-bus example system with x given in (10). For γ = 2 , this system has the moment matrixshown in (11). This matrix has entries y α corresponding to allmonomials x α up to degree four.Note that several terms are repeated in the moment matrixbeyond those expected for a generic symmetric matrix. In x = (cid:2) V d V d V q V d V d V d V d V q V d V d V q V q (cid:3) ⊺ (10) M ( y ) = L y ( x x ⊺ ) = y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y (11) { ( f V − . y } = y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y y + y − . y (13) M ( y ) , for instance, the terms corresponding to second-ordermonomials (e.g., y ) appear in both the second diagonalblock of M ( y ) and the first row and column. There are alsorepetitions in the off-diagonal block, whose entries correspondto third-order monomials (e.g., y ) and in the third diagonalblock of M , whose entries correspond to fourth-order mono-mials (e.g., y ). These repetitions require equality constraintsin the semidefinite program implementation.Symmetric localizing matrices are defined for each con-straint of (5). The localizing matrices consist of linear com-binations of the moment matrix entries y . Each polynomialconstraint of the form f − a ≥ in (5) (e.g., f V − V min2 ≥ )corresponds to the localizing matrix M γ − β { ( f − a ) y } = L y (cid:8) ( f − a ) x γ − β x ⊺ γ − β (cid:9) (12) where the polynomial f has degree β or β − . The bus 2lower voltage limit with V min2 = 0 . per unit for the two-busexample system, for example, has the corresponding localizingmatrix in (13).We can now form the order- γ moment-based relaxation ofthe OPF problem. min y L y (X k ∈G f Ck ) subject to (14a) M γ − n(cid:16) f Pk − P min k (cid:17) y o (cid:23) ∀ k ∈ N (14b) M γ − n(cid:16) P max k − f Pk (cid:17) y o (cid:23) ∀ k ∈ N (14c) M γ − n(cid:16) f Qk − Q min k (cid:17) y o (cid:23) ∀ k ∈ N (14d) M γ − n(cid:16) Q max k − f Qk (cid:17) y o (cid:23) ∀ k ∈ N (14e) M γ − n(cid:16) f V k − V min k (cid:17) y o (cid:23) ∀ k ∈ N (14f) M γ − n(cid:16) V max k − f V k (cid:17) y o (cid:23) ∀ k ∈ N (14g) M γ − n(cid:16) S max lm − f Slm (cid:17) y o (cid:23) ∀ ( l, m ) ∈ L (14h) M γ ( y ) (cid:23) (14i) y ... = 1 (14j) y ⋆η⋆⋆...⋆ = 0 ∀ η ≥ (14k) where (cid:23) indicates that the corresponding matrix is posi-tive semidefinite and ⋆ represents any integer in [0 , γ ] . Themoment-based relaxation is thus a semidefinite program. (Thedual form of the moment-based relaxation is a sum-of-squaresprogram; see [11] for further details on the dual formulation.)Note that the constraint (14j) enforces the fact that x = 1 .The constraint (14k) corresponds to (5f); the angle referencecan alternatively be used to eliminate all terms correspondingto V q to reduce the size of the semidefinite program. Equality constraints are modeled as two inequality con-straints. Since, for a symmetric matrix A , the constraints A (cid:23) and − A (cid:23) imply A = 0 , all entries of a localizingmatrix corresponding to an equality constrained polynomial(e.g., the power flow constraints at load buses) are zero.The order- γ moment-based relaxation yields a single globalsolution if rank ( M γ ( y )) = 1 . A solution x ∗ to the OPF prob-lem (5) can then be directly determined from the elements of y corresponding to the linear monomials (e.g., V ∗ d = y ··· ).If rank ( M γ ( y )) > , then there are either multiple globalsolutions requiring the solution extraction procedure in [11] orthe order- γ moment-based relaxation is not exact and yieldsonly a lower bound on the optimal objective value. If the order- γ moment-based relaxation is not exact, the order- ( γ + 1) moment relaxation will improve the lower bound and maygive a global solution.Note that the order γ of the moment-based relaxationmust be greater than or equal to half of the degree of anypolynomial in the OPF problem (5). All polynomials canthen be written as linear functions of the entries of M γ .For instance, the OPF problem with a linear cost functionand without apparent-power line-flow limits requires γ ≥ .Although direct implementation of (5) requires γ ≥ due tothe fourth-order polynomials in the cost function and apparent-power line-flow limits, these limits can be rewritten as second-order polynomials using a Schur complement [4]. The OPFproblem reformulated in this way only requires γ ≥ , butexperience suggests that direct implementation of (5) hasnumerical advantages.It is interesting to compare the first-order moment-basedrelaxation and the semidefinite relaxation of [4]. The first-order moment-based relaxation has a moment matrix withelements corresponding to all monomials up to second-order.The localizing matrices are the constraints multiplied by thescalar 1; the localizing matrix constraints for the first-ordermoment-based relaxation simply enforce the linear scalarequations L y { f − a } = P α ∈ N n ( f α y α ) − y a ≥ forall constraints in (5). Thus, the first-order moment-basedrelaxation is closely related to the semidefinite relaxation of [4]which has a feasible space defined by a positive semidefinitematrix constraint and linear constraints on the matrix elements.The slight difference between the formulations is that thesemidefinite relaxation of [4] uses a matrix correspondingto only the second-order monomials (e.g., the second diag-onal block in (11)), whereas the first-order moment-basedrelaxation additionally has elements corresponding to constantand linear polynomials in its moment matrix M ( y ) . Thismay yield different results for the first-order moment-basedelaxation and the semidefinite relaxation of [4] when therelaxations are not exact.IV. A PPLICATION TO A T WO -B US E XAMPLE S YSTEM
With three degrees of freedom V d , V d , and V q (the anglereference constraint (5f) forces V q = 0 ), the two-bus examplesystem in [12] allows for visualizing the entire feasible spaceof the OPF problem (5). For some choices of parameters, theOPF problem (5) for this system has a disconnected feasiblespace, and the semidefinite relaxation of [4] is not exact.(Note that this system does not satisfy the sufficient conditionsfor exactness of the semidefinite relaxation described in [7].)This section illustrates the feasible space for the second-ordermoment-based relaxation, which finds a global solution for amuch larger range of parameters for this problem.Fig. 1 gives the system’s one-line diagram assuming a100 MVA base power. The generator at bus 1 has no limitson active or reactive outputs and there is no line-flow limit.Bus 1 voltage magnitude is in the range [0 . , . per unit,while bus 2 voltage magnitude is greater than 0.95 per unitand less than the parameter V max2 . R + jX = 0.04 + j0.20V V P + jQ = 3.525 - j3.580
Fig. 1. Two-Bus System from [12]
Fig. 2 shows the feasible space for the semidefinite re-laxation of [4]. The colored conic shape is the projectionof the feasible space of the semidefinite relaxation into thespace of squared voltage components ( V d , V d , and V q ),with the colors based on a $1/MWh cost of active powergeneration at bus 1. The red line forms the (disconnected)feasible space for the OPF problem (5). With V max2 = 1 . per unit, both the semidefinite relaxation of [4] and the OPFproblem (5) have global minimum at the red square in Fig. 2,and the semidefinite relaxation is exact. The more stringentlimit of V max2 = 1 . per unit is shown by the gray plane;this constraint eliminates the feasible space to the left of theplane. The solution to the semidefinite relaxation of [4] is atthe red circle on the gray plane, while the global solution tothe classical OPF problem is at the red triangle. Thus, thesemidefinite relaxation of [4] is not exact.Conversely, the second-order moment-based relaxation isexact for both V max2 = 1 . per unit and V max2 = 1 . per unit. Fig. 3 shows a projection of the feasible space for thisproblem. The gray plane again corresponds to V max2 = 1 . per unit in the projected space. The feasible space for thesecond-order moment-based relaxation is planar with bound-aries equal to the feasible space of the OPF problem (5), whichconsists solely of the two red line segments on the left and farright of Fig. 3. (Both the colors showing the generation costand the feasible space values are recovered from the entriesof the moment matrix corresponding to the squared terms Objective Value($/hour)
Fig. 2. Feasible Space of the Semidefinite Relaxation of [4] for the Two-BusSystem Showing the Constraint V max2 = 1 . per unitFig. 3. Feasible Space of the Second-Order Moment-Based Relaxation forthe Two-Bus System Showing the Constraint V max2 = 1 . per unit in the second diagonal block of (11).) With V max2 = 1 . per unit, the second-order moment-based relaxation finds theglobal solution at the red square in Fig. 3.In the projection shown in Fig. 3, it appears that imposingthe limit V max2 = 1 . per unit will result in the second-order moment-based relaxation finding the point at the redcircle (which does not satisfy the condition rank ( M ( y )) =1 ) rather than the global optimum to the OPF problem (5),which is at the red triangle. (Points on the plane between thered line segments, such as the red circle, are not feasible forthe OPF problem, but are in the feasible space of the second-order moment-based relaxation with V max2 = 1 . per unit.)However, the second-order moment-based relaxation findsthe global solution at the red triangle and is therefore exact.While the red circle is in the feasible space when V max2 = 1 . per unit, this and nearby points are eliminated from thefeasible space when V max2 = 1 . per unit by the uppervoltage magnitude constraint (14g). That is, the localizingmatrix M (cid:8)(cid:0) V max2 − f V (cid:1) y (cid:9) is positive semidefinite forthese points when V max2 = 1 . per unit but not when V max2 = 1 . per unit. The feasible space with V max2 = 1 . per unit is the planar region between the black dashed lineand the red line on the right. (Note that there is a smallange V max2 ∈ [1 . , . per unit where the second-order relaxation does not yield a global optimum; a third-orderrelaxation finds the global solution with V max2 in this range.)With the need to have consistent higher-order terms in y thatyield positive semidefinite moment and localizing matrices,moment-based relaxations with γ > are tighter than thesemidefinite relaxation of [4]. The improved tightness hasa computational cost: the largest matrix in the semidefiniterelaxation of [4] is × in contrast to × for the second-order moment-based relaxation.V. R ESULTS FOR P REVIOUSLY P UBLISHED E XAMPLES
Section IV shows how a moment-based relaxation globallysolves a problem for which the semidefinite relaxation of [4]is not exact. The moment-based relaxation is next appliedto other small problems for which the semidefinite relax-ation of [4] is not exact. These problems were solved usingYALMIP’s moment solver [13] and SeDuMi [14].Table I lists small problems for which the semidefiniterelaxation of [4] is not exact for certain parameters. Thenumber of buses is appended to the case names. The tableshows the lowest order γ min needed for a global solution.A second-order moment-based relaxation suffices for a broadclass of problems. Third-order relaxations are occasionallyneeded for small parameter ranges. Case Parameters γ min LMBD3 [5] 50 MVA line limit 2
MLD3 [6] 100 MVA line limit 2
BGMT3 [2] 2
LH5 [1] P D = 17 . per unit 2 BGMT5 [2] Q min2 ∈ [ − , − . MVAR 2 Q min2 ∈ [ − . , − . MVAR 3 Q min2 ∈ [ − . , MVAR 2
BGMT9 [2] 2
MSL10 ex1 [8] > MSL10 ex2 [8] 2TABLE IM
OMENT -B ASED R ELAXATION R ESULTS
We have some specific comments on these examples. Thethree-bus OPF problems
LMBD3 and
MLD3 have bindingapparent-power line-flow limits. The line-flow limit in
MLD3 results in a disconnected feasible space [6]. Thus, the second-order moment-based relaxation globally solves at least someproblems for which the semidefinite relaxation of [4] is notexact due to tight line-flow limits.The semidefinite relaxation of [4] is not exact for thethree and nine-bus problems
BGMT3 and
BGMT9 due tothe presence of local optima. The second-order moment-based relaxation finds the global optimum for these problems.MATPOWER [10] with the MIPS solver initialized usinga “flat start” (unity voltage magnitudes with zero voltageangles) finds a local optimum for
BGMT9 with objective value38% greater than the globally optimal objective value. Thesemidefinite relaxation of [4] yields a lower bound that is 11%less than the global optimum. Thus, existing techniques do notperform well for this problem. Similar to the problem in Section IV, a third-order moment-based relaxation is needed to globally solve the five-busproblem
BGMT5 for a narrow parameter range.The gray region in Fig. 4 shows the non-convex feasiblespace of active power injections for the lossless five-bus OPFproblem
LH5 . The feasible space of the semidefinite relaxationof [4] shown by the black curve in Fig. 4 is not tight forsome parameters (e.g., the dashed blue line representing a load P D = 17 . per unit with generation P G more expensivethan P G ) [6]. Conversely, the feasible space for the second-order moment-based relaxation shown by the red dashed curvein Fig. 4 is tight for varying load demands P D , which resultin other lines parallel to the blue line, and generator costs. Fig. 4. Feasible Space for Five-Bus System in [1]
The ten-bus problem
MSL10 ex1 has at leasttwo global solutions (the solution in [8] and (cid:2) P G P G P G P G P G (cid:3) = (cid:2) .
99 32 .
66 0 0 38 . (cid:3) MW). A moment-based relaxation with γ > is needed tofind the multiple solutions, but only a second-order relaxationis currently computationally tractable.The second-order moment-based relaxation gives the singleglobal optimum for the related ten-bus problem MSL10 ex2 .The heuristic method proposed in [8] finds an only-locally-optimal solution to this problem with an objective valueof $153.97, which is 28.4% greater than the global min-imum of $119.95 with (cid:2) P G P G P G P G P G (cid:3) = (cid:2) .
994 0 0 40 . (cid:3) MW.The moment-based relaxation was also applied to two NP-hard OPF problems from [4]. The first problem,
LLn 1 where n represents an arbitrary number of buses, eliminates all active,reactive, and line-flow limits to minimize network losses withvoltage magnitudes constrained to 1 per unit. The second prob-lem, LLn 2 , minimizes losses for a purely resistive networkwith zero reactive power injections and voltages constrainedto the discrete set {− , } .Table II summarizes the application of the moment-basedrelaxations to these problems. Two network structures wereconsidered: a ring and a complete network (each bus connectedo every other bus). In both cases, the global solution wasobtained (or the computational capabilities were surpassedwithout yielding a solution) at the same order of the moment-based relaxation. For LLn 1 , lines have 0.1 per unit reactancesand either zero or × − per unit resistance. For LLn 2 ,lines have zero reactance and . per unit resistance. Case Parameters γ min LLn 1 , n = 2 , . . . , R = 1 × − per unit 2 LL2 1 , R = 0 per unit LL3 1 R = 0 per unit LL4 1 R = 0 per unit > LLn 1 , n = 5 , R = 0 per unit > LLn 1 , n = 7 , , R = 0 per unit > LLn 2 , n = 2 , . . . , TABLE IIM
OMENT -B ASED R ELAXATION A PPLIED TO
NP-H
ARD P ROBLEMS
Table II shows that low-degree moment-based relaxationshave some success for small problems of the forms
LLn 1 and
LLn 2 . While a second-order moment-based relaxation solves
LLn 1 with lossy networks and
LLn 2 , low-order moment-based relaxations are not well-suited for
LLn 1 with losslessnetworks. This is likely due to the fact that
LLn 1 with losslessnetworks has multiple global solutions, while
LLn 1 with lossynetworks and
LLn 2 have unique global solutions.Since there exist NP-hard OPF problems, the moment-basedrelaxation for a fixed-order γ (or any other polynomial-timerelaxation) cannot globally solve all OPF problems. However,the NP-hard problems in [4] do not represent typical powersystems. The results shown in Table I suggest that the moment-based relaxation may be exact for a broad class of OPFproblems that excludes some NP-hard problems.VI. C ONCLUSIONS AND F UTURE W ORK
Using theory developed for polynomial optimization prob-lems, this paper has proposed a moment-based relaxation forthe OPF problem. With the trade-off of increased compu-tational requirements, the moment-based relaxation globallysolves a broader class of problems than previous convexrelaxations, such as the semidefinite relaxation of [4]. Afterformulating the moment-based relaxation of the OPF problem,this paper has investigated the feasible space of the moment-based relaxation. Global solution of several small problemsfor which the semidefinite relaxation of [4] is not exactdemonstrates the moment-based relaxation’s effectiveness.There are many avenues for future work on moment-basedrelaxations, including computational improvements, develop-ing sufficient conditions for exactness of the relaxation, andextension to other power systems problems. The large size ofthe semidefinite programs used to evaluate the moment-basedrelaxation currently precludes application to problems withmore than ten buses. Future work includes the exploitation ofpower system sparsity to solve larger OPF problems. Similarto the semidefinite relaxation of [4], a matrix completiondecomposition [15] is applicable to moment-based relaxations.Further, the results of [15] indicate that only small sub-networks of typical OPF problems cause the semidefinite relaxation of [4] to not be exact. Therefore, it may be sufficientto selectively apply the moment-based relaxations to smallsubnetworks of a large OPF problem, thus further improvingcomputational tractability.Generalizing current efforts to find sufficient conditions forwhich the semidefinite relaxation of [4] is exact, future workalso includes developing sufficient conditions for which anorder- γ moment based relaxation is exact.The moment-based relaxation can also be extended toother power systems problems, including those problems forwhich the semidefinite relaxation of [4] has already shownsome success (e.g., state estimation, voltage stability, findingmultiple power flow solutions). Further, the ability of moment-based relaxations to include binary variables provides theopportunity to consider problems with discrete constraintssuch as transmission switching and unit commitment.R EFERENCES[1] B. Lesieutre and I. Hiskens, “Convexity of the Set of Feasible Injectionsand Revenue Adequacy in FTR Markets,”
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