We apply a linear relaxation procedure for polynomial optimization problems to transmission system planning. The approach recovers and improves upon existing linear models based on the DC approximation. We then consider the full AC problem, and obtain new linear models with nearly the same efficiency as the linear DC models. The new models are applied to standard test systems, and produce high-quality approximate solutions in reasonable computation time.United States. Office of Naval Research (Grant N00014-02-1-0623

Hover, Franz S.

Taylor, Joshua A.

IEEE Transactions on Power Systems

We apply a linear relaxation procedure for polynomial optimization problems to transmission system planning. The approach recovers and improves upon existing linear models based on the DC approx-imation. We then consider the full AC problem, and obtain new linear models with nearly the same efficiency as the linear DC models. The new models are applied to standard test systems, and produce high quality approximate solutions in reasonable computation time. 

Joshua A. Taylor

Franz S. Hovery

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Linear relaxations for transmission system planning

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We apply lift-and-project type relaxations for polynomial optimization problems to AC transmission system planning, and obtain new second-order cone and semidefinite models. The models are compared to linear and nonlinear approaches on a six-bus test system.United States. Office of Naval Research (ONR Grant N00014-02-1-0623

Conic relaxations for transmission system planningJoshua A. Taylor and Franz S. HoverDepartment of Mechanical EngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts 02139Email: {jatl, hover}@mit.eduAbstract—We apply lift-and-project type relaxations for poly-nomial optimization problems to AC transmission system plan-ning, and obtain new second-order cone and semidefinite models.The models are compared to linear and nonlinear approaches ona six-bus test system.Index Terms—Transmission system planning, lift-and-project,second-order cone programming, semidefinite programming, ACpower flow.I. INTRODUCTIONStatic transmission system planning entails choosing amonga set of candidate lines whose addition to a network minimizesconstruction costs while meeting physical capacity require-ments [1], [2]. The basic problem is nonlinear, and moreovernonconvex due to both the mathematical description of ACpower flow and products between line and voltage variables.Until recently, optimization-based approaches have almostexclusively been preceded by application of the DC powerflow approximation [3], [4], because the full AC version wasconsidered intractable. At present, there are few methods thatdo not first employ this simplification; of note is [5], in whichthe full AC formulation is solved using a heuristic based onnonlinear programming. An alternate recent approach is [6],where we used a lift-and-project procedure to derive efficientlinear programming relaxations for both the simplified and fullvariants of the problem.Second-order cone [7] and semidefinite [8] programmingare generalizations of linear programming, which are moreexpressive but retain its polynomial time efficiency. The recentresult of [9] has made AC power flow far more tractablevia a semidefinite programming relaxation of optimal powerflow, often capturing the true optimal solution exactly. Inthis paper we employ their power flow formulation inside atransmission system planning framework, and, using a lift-and-project procedure [6], [10], [11], construct a high-fidelitymixed-integer semidefinite programming relaxation. We thenuse the result of [12] to obtain a mixed-integer second-ordercone relaxation directly from the semidefinite model. Each isdemonstrated on the six-bus test system of [1], [5].Although mixed-integer second-order cone and semidefiniteprogramming algorithms are relatively immature today, theirsimilarities with linear programming suggest that comparablesophistication is attainable. Indeed, mixed-integer second-order cone programming is an active area of research [13]–[15], and we can anticipate that robust and efficient tools willbecome available in coming years. At the same time, the speedof semidefinite and second-order cone programming relative tononlinear programming makes our relaxations directly appli-cable within generic branch-and-bound frameworks. Our paperdoes not address this approach.II. AC POWER FLOWIn this section we provide the nonlinear AC transmissionsystem planning model and review the semidefinite powerflow relaxation of [9]. We work in rectangular coordinates,for which the voltage at bus i is represented by the complexnumber wi + jxi.Let Ω0 and Ω be the sets of existing and candidate lines.Unless otherwise specified, single subscripts are to be takenover all buses and double subscripts over members of Ω. Letcij be a line cost vector, pi, pi, qi, and qi be real and reactivebus power limits, vi and vi be voltage limits, η0ij be the numberof lines present in the existing network, and ηij be the numberof new lines which may be constructed. Let gij + jbij andpij+jqij respectively be complex admittance and power flow,and let s¯ij be capacity per line increment. Finally, let gij beconductance and define bsij = bij + bshij , where bsh is lineshunt susceptance. The AC transmission planning model inrectangular coordinates is given byNLAC minη,p,q,w,x∑i∼jcijηij (1)s.t. pij =(η0ij + ηij)(bij(wixj − wjxi)−gij(xixj + wiwj) + gij(w2i + x2i ))(2)qij =(η0ij + ηij)(gij(wixj − wjxi)+bij(xixj + wiwj)− bsij(w2i + x2i ))(3)pi≤∑jpij ≤ pi (4)qi≤∑jqij ≤ qi (5)v2i ≤ w2i + x2i ≤ v2i (6)√p2ij + q2ij ≤(η0ij + ηij)sij(i, j) ∈ Ω0 ∪ Ω (7)0 ≤ ηij ≤ ηij , ηij ∈ N (8)The basic approach of [9] is to define a column vectorof the real and imaginary bus voltages X = [wTxT ]T , andthen substitute the symmetric matrix W for XXT subject tothe constraint rank(W ) = 1 and W  0, where  denotesmatrix semidefiniteness. Dropping the rank constraint yieldsthe relaxation.A dual formulation of the power flow equations is usedin [9] to eliminate unnecessary variables; for the purpose ofintegration into a transmission planning framework, it is moreconvenient here to remain with the primal variables. The sizeof the problem can be reduced by sparsifying W and in placeof (10) using the relaxed conditionWi,i Wi,j Wi,i+n Wi,j+nWj,i Wj,j Wj,i+n Wj,j+nWi+n,i Wi+n,j Wi+n,i+n Wi+n,j+nWj+n,i Wj+n,j Wj+n,i+n Wj+n,j+n  0for each line (i, j), where n is the number of buses. We denotethe above submatrix ijW .Incorporating the substitutions into NLAC we haveNLSDAC minη,p,q,W∑i∼jcijηij (9)s.t. W  0 (10)pij =(η0ij + ηij)(bij(Wi,j+n −Wj,i+n)−gij (Wi+n,j+n +Wi,j)+gij(Wi,i +Wi+n,i+n)) (11)qij =(η0ij + ηij)(gij(Wi,j+n −Wj,i+n)+bij(Wi+n,j+n +Wi,j)−bsij(Wi,i +Wi+n,i+n))(12)v2i ≤Wi,i +Wi+n,i+n ≤ v2i (13)p2ij + q2ij ≤(η0ij + ηij)2s2ij(i, j) ∈ Ω0 ∪ Ω (14)Constraints (4), (5), (8)III. RELAXATIONSExcept for the semidefinite constraint and line capacity,NLSDAC only has bilinear nonlinearities, which are amenableto same linear relaxations as in [6]. An additional benefit ofusing second-order cone and semidefinite formulations is thatthe line capacity constraint (14) may be enforced exactly.We proceed by substituting the 4 × 4 matrix ∆ij foreach instance of the product ηij ijW . In addition to beingsymmetric, ∆ij is implicitly constrained so that ∆ij11 = ∆ji22,∆ij33 = ∆ji44, ∆ij12 = ∆ji12, ∆ij34 = ∆ji34, ∆ij13 = ∆ji24, and∆ij14 = ∆ji23. New constraints are formed by taking productsof existing constraints, for example (17) below is obtainedfrom multiplying (10) and (8), and then substituting ∆ij . Wethen have the following mixed-integer semidefinite program:SDAC minη,µ,ν,φ,ψ,W,∆∑i∼jcijηij (15)s.t. W  0 (16)0  ∆ij  ηij ijW (17)µij = bij(Wi,j+n −Wj,i+n)− gij(Wi,j +Wi+n,j+n)+ gij(Wi,i +Wi+n,i+n) (18)νij = gij(Wi,j+n −Wj,i+n)+ bij(Wi,j +Wi+n,j+n)− bsij(Wi,i +Wi+n,i+n) (19)φij = bij(∆ij14 −∆ij23)− gij(∆ij12 + ∆ij34)+ gij(∆ij11 + ∆ij33) (20)ψij = gij(∆ij14 −∆ij23)+ bij(∆ij12 + ∆ij34)− bsij(∆ij11 + ∆ij33) (21)pi≤∑jη0ijµij + φij ≤ pi (22)qi≤∑jη0ijνij + ψij ≤ qi (23)v2i ≤Wi,i +Wi+n,i+n ≤ v2i (24)v2i ηij ≤ ∆ij11 + ∆ij33 ≤ v2i ηij (25)v2jηij ≤ ∆ij22 + ∆ij44 ≤ v2jηij (26)v2i(ηij − ηij) ≤ ηij (Wi,i +Wi+n,i+n)−∆ij11 −∆ij33 ≤ v2i(ηij − ηij)(27)v2j(ηij − ηij) ≤ ηij (Wj,j +Wj+n,j+n)−∆ij22 −∆ij44 ≤ v2j(ηij − ηij)(28)µ2ij + ν2ij ≤ s2ij (i, j) ∈ Ω0 (29)φ2ij + ψ2ij ≤ η2ijs2ij (30)(ηijµij − φij)2 + (ηijνij − ψij)2 ≤(ηij − ηij)2s2ij (i, j) ∈ Ω0 (31)0 ≤ ηij ≤ ηij , ηij ∈ N (32)The new variables φ and ψ represent flows in the new,variable network, while the flows in the existing network aredenoted µ and ν, and obey AC power flow. We remark that ifthe semidefinite constraints are omitted and (29) through (31)are replaced by piecewise linear approximations, we recoverthe linear model LAC of [6].Larger, more accurate models may be constructed by takinghigher-order products of constraints and performing similarsubstitutions [10], [11]; however, the number of new con-straints and variables increases rapidly, and at some point acompromise must be struck. In this regard, SDAC is highlyappealing: it is reasonably sized as far as lift-and-projectrelaxations go, and utilizes to the full extent today’s broadestpolynomial-time conic optimization framework, semidefiniteprogramming.Using SDAC as a starting point, we can straightforwardlyconstruct a second-order cone relaxation (SOCAC) via the factthat positivity of all two-by-two principal minors is a necessarycondition for positive semidefiniteness [12]. The relaxation isthus obtained by replacing (16) and (17) with the second-ordercone constraintsWi,i ≥ 0Wi+n,i+n ≥ 0∆ijkk ≥ 0, k = 1, ..., 4W 2i,j ≤Wi,iWj,jW 2i,j+n ≤Wi,iWj+n,j+nW 2i+n,j ≤Wi+n,i+nWj,jW 2i+n,j+n ≤Wi+n,i+nWj+n,j+n∆ij122 ≤ ∆ij11∆ij22∆ij142 ≤ ∆ij11∆ij44∆ij232 ≤ ∆ij22∆ij33∆ij342 ≤ ∆ij33∆ij44(ηijWi,j −∆ij12)2≤(ηijWi,i −∆ij11)(ηijWj,j −∆ij22)(ηijWi,j+n −∆ij14)2≤(ηijWi,i −∆ij11)(ηijWj+n,j+n −∆ij44)(ηijWi+n,j −∆ij32)2≤(ηijWj,j −∆ij22)(ηijWi+n,i+n −∆ij33)(ηijWi+n,j+n −∆ij34)2≤(ηijWi+n,i+n −∆ij33)(ηijWj+n,j+n −∆ij44)for each line (i, j). Note that the line capacity constraints(29)-(31) are convex quadratic, and therefore already withinthe class of second-order cone constraints. As with analogouslinear models, it is straightforward to formulate more accuratebut less efficient binary versions of both SDAC and SOCAC[6].IV. COMPUTATIONAL RESULTSIn this section we test SDAC and SOCAC on the six-busexample of [5], which is an adaptation of Garver’s six-bussystem [1]. We specifically consider the case in which thereare preexisting lines. Also included is the linear model LACof [6]. To facilitate comparison, the following piecewise linearrelaxation for line limits, as required for LAC, has been used:|pij |+ |qij | ≤√2Sij , |pij | ≤ Sij , |qij | ≤ Sij .Mixed integer linear and second-order cone programs weresolved using the modeling language AMPL [16] and solverCPLEX [17]. Mixed integer semidefinite programs weresolved by enumerating over additions to lines {1−5, 2−3, 2−6, 3−5, 4−6}, which is the set of all lines appearing in one ormore optimal solution from LAC. CVX [18] and SeDuMi [19]were then used to identify the feasible ones; in this fashion,it is partially determined how to prune and reinforce solutionsfrom LAC.Table I shows the objective value and solution reported forthe nonlinear approach in [5] (NL) and obtained by LAC, SO-CAC, and SDAC. LAC and SOCAC each found solutions withthe same objective; SOCAC, however, found three, whereasLAC found six, indicating that SOCAC eliminated more low-quality solutions than did LAC. SDAC found three solutionswith a higher objective than the other models. The bottomportion of Table I shows a single solution obtained by eachmodel.TABLE ICONIC MODELS ON THE SIX-BUS TEST SYSTEM OF [5]Model NL [5] LAC SOCAC SDACNumber of solutions - 6 3 3Obj. 160 80 80 100Line additions2 - 6 2 2 2 13 - 5 2 1 1 24 - 6 2 0 0 1We restate that at present, only mixed-integer linear pro-gramming can be considered a mature technology, but thatmixed-integer second-order cone programming is an activefield of research [13]–[15], and we expect that mixed-integersemidefinite programming will follow. As such, the methodspresented here are not currently scalable approaches, althoughwe expect them to be in the near future.V. CONCLUSION AND FUTURE WORKWe have derived mixed-integer semidefinite and second-order cone relaxations of AC transmission system planningusing a lift-and-project procedure. On a small example, thenew models are shown to be more accurate than a similarlyconstructed linear model. The models do yet not enjoy thematurity of linear algorithms, but advancements in mixed-integer conic optimization are to be expected, in which casethese models will soon represent practical approaches to high-fidelity transmission system planning without the complica-tions of nonconvexity. At the same time, second-order coneand semidefinite programming are very efficient comparedwith nonlinear algorithms, and hence the models presentedhere should also be attractive for use with general integerprogramming techniques such as branch-and-bound.ACKNOWLEDGMENTWork is supported by the Office of Naval Research GrantN00014-02-1-0623, monitored by Dr. T. Ericsen.REFERENCES[1] L. Garver, “Transmission network estimation using linear programming,”Power Apparatus and Systems, IEEE Transactions on, vol. PAS-89,no. 7, pp. 1688–1697, Sept. 1970.[2] G. Latorre, R. Cruz, J. Areiza, and A. Villegas, “Classification ofpublications and models on transmission expansion planning,” PowerSystems, IEEE Transactions on, vol. 18, no. 2, pp. 938 – 946, May2003.[3] A. J. Wood and B. F. Wollenberg, Power generation, operation, andcontrol, 2nd ed. Knovel, 1996.[4] R. Romero, A. Monticelli, A. Garcia, and S. Haffner, “Test systemsand mathematical models for transmission network expansion planning,”IEE Proceedings - Generation, Transmission and Distribution, vol. 149,no. 1, pp. 27–36, 2002.[5] M. Rider, A. Garcia, and R. Romero, “Power system transmissionnetwork expansion planning using AC model,” Generation, TransmissionDistribution, IET, vol. 1, no. 5, pp. 731 –742, September 2007.[6] J. A. Taylor and F. S. 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