We apply relaxation procedures to polynomial optimization problems that originate in transmission system planning, and obtain new convex formulations for the AC case. The approach is novel because the optimization is efficient but also addresses the true nonlinear physics directly. We illustrate the method on a test case derived from a notional shipboard distribution system

Hover, Franz S.

Taylor, Joshua A.

DSpace@MIT

Lift-and-project relaxations of AC microgrid distribution system planningJoshua A. Taylor and Franz S. HoverMassachusetts Institute of TechnologyDepartment of Mechanical Engineeringjatl@mit.edu, hover@mit.eduKeywords: Transmission system planning, AC power flow,lift-and-project relaxation, linear programmingAbstractWe apply relaxation procedures to polynomial optimizationproblems that originate in transmission system planning, andobtain new convex formulations for the AC case. The ap-proach is novel because the optimization is efficient but alsoaddresses the true nonlinear physics directly. We illustrate themethod on a test case derived from a notional shipboard dis-tribution system.1. INTRODUCTIONTransmission system planning (TSP) is a network designproblem in which lines are selected from a candidate set tomeet physical requirements while minimizing investment andoperational costs [5, 7]. Linearized or ‘DC’ power flow is astandard simplification in the field [10], since AC flow is toocomplicated for most optimization scenarios. In network de-sign problems, however, even linearized power flow createsa nonlinear, non-convex problem because the existence of aline can be a variable. Thus TSP with linearized load flowhas traditionally been handled via further simplified forms,the so-called transportation and disjunctive models [7]. At thesame time, in applications like a ship electrical system, lineresistive characteristics are no longer negligible compared toreactances, because of the short line lengths, and the DC sim-plification is invalid for any purpose. An AC solution for TSPtherefore has significant value, but only recently has it beenapproached in full [6], using an interior point method in tan-dem with a constructive heuristic algorithm.Toward this end, our objective is to develop and applynew algorithms for power distribution design problems, thatwill improve accuracy and efficiency across several applica-tion areas. We utilize ‘lift-and-project’ relaxation proceduresin our development, specifically that of [8]. A relaxation isan approximation to an optimization problem which alwaysbounds the minimum below (or maximum above), and is typ-ically easier to solve than the original problem. The phraselift-and-project refers to lifting an optimization problem toa higher-dimensional space via the introduction of new vari-ables, and then projecting the lifted problems solution backonto the original variables. Relaxed solutions are often sub-optimal or infeasible for the original problem, but can con-tain a significant portion of the true optimal solution. Lever-aging the maturity of commercial mixed integer linear pro-gramming solvers, our method represents the most scalableoption to date for transmission system planning with full ACload flow.We emphasize that the role of relaxation in the design pro-cess is not to create a feasible solution, but rather to con-struct a near-feasible solution that captures the structure ofthe optimal one, and therefore enables a follow-on design stepthat should be easier, for example a standard nonlinear pro-gramming formulation in conjunction with branch and bound.Moreover, one often finds that the relaxed solution adds keyinsights to the original problem, which can be studied in de-tail via simulation. A relaxed solution can be used to reduceor reformulate the design variables, by relabeling the solu-tion as a fixed parameter set, and making new variables. Inmany cases, we have to identify a feasible initial solution inthe new variable set; this is a separate problem from optimiza-tion, however.Here we specialize our previous work [9] to a new bench-mark case based on a shipboard distribution system. The all-electric ship, e.g. [3], is a prototypical micro-grid design ap-plication, for which the disjunctive and transportation modelsare entirely unsuited, as noted above. Note that the modelsgiven here do not address robustness, which would be of in-terest to the all-electric ship and some other microgrids, butvia alternative cost functions such issues may be within thescope of our general approach.2. TRANSMISSION SYSTEM PLANNINGMODELSWe now derive linear models for AC transmission systemplanning which are similar in structure and size to the ex-isting linear models [7]. We are given the following problemparameters: line investment vector c, a vector of real and reac-tive generation and demand limits p, p, q, and q, normalizedflow limits s, existing network ξ0, and line construction limitsξ. Let Γ denote the set of buses, Ω0 the set of existing lines,and Ω the set of candidate lines. We follow the notationalconventions that unless otherwise specified, single subscriptsdenote members of Γ, double subscripts members of Ω, andi ∼ j summation over Ω0 ∪Ω. Let s, v, and y respectivelydenote complex powers, voltages and admittances. The basicAC power flow model is given byNLAC minξ,s,v ∑i∼ j ci jξi js.t. si j =(ξ0i j +ξi j)(viv∗i y∗i j − viv∗jy∗i j)pi ≤ Re∑j si j ≤ piqi ≤ Im∑j si j ≤ qivi ≤ |vi| ≤ vi∣∣si j∣∣≤ (ξ0i j +ξi j)si j (i, j) ∈ Ω0∪Ω0 ≤ ξi j ≤ ξi j, ξi j ∈ NNote that although line variables and parameters are non-directional, i.e. ξi j = ξ ji, si j = s ji and so on, sending and re-ceiving power flows si j and s ji are not.2.1. Linear AC modelsWe first must rewrite NLAC in terms of real, polynomialconstraints so we may begin to build a relaxation. Let y =g+ jb, v = w+ jx, s = p+ jq, and let bs = b+bsh, where bsh isthe line shunt susceptance. Applying the relaxation procedureof [8], NLACS is then is given byNLACS minξ,p,q,w,x ∑i∼ j ci jξi js.t. pi j =(ξ0i j +ξi j)(bi j(w jxi−wix j)−gi j(xix j +wiw j)+gi j(w2i + x2i ))qi j =(ξ0i j +ξi j)(gi j(w jxi−wix j)+bi j(xix j +wiw j)−bsi j(w2i + x2i ))pi ≤∑j pi j ≤ piqi ≤∑j qi j ≤ qiv2i ≤ w2i + x2i ≤ v2i√p2i j +q2i j ≤(ξ0i j +ξi j)si j(i, j) ∈ Ω0∪Ω0 ≤ ξi j ≤ ξi j, ξi j ∈ NThe line capacity constraint involving the square root rep-resents a slight obstacle: although it can be expressed polyno-mially, fourth order products of voltage variables would haveto be included, rendering the size of the resulting relaxationimpractically large. We instead approximate them so that pand q are involved linearly; a few options are apparent. Tokeep the formulation general, we introduce the constants τ1and τ2 and replace (1) withτ1i j∣∣pi j∣∣+ τ2i j∣∣qi j∣∣≤ (ξ0i j +ξi j)si jAlthough we have only used a single constraint in this ap-proximation, any set of piecewise linear constraints can alsobe used.As described in [9], define the new variables:αi = w2i + x2iδi j = ξi j (w2i + x2i)µi j = bi j(w jxi−wix j)−gi j(xix j +wiw j)+gi j(w2i + x2i )νi j = gi j(w jxi−wix j)+bi j(xix j +wiw j)−bsi j(w2i + x2i )φi j = ξi j (bi j(w jxi−wix j)−gi j(xix j +wiw j)+gi j(w2i + x2i ))ψi j = ξi j (gi j(w jxi−wix j)+bi j(xix j +wiw j)−bsi j(w2i + x2i ))These new variables have implicit constraints given bygi j(µi j −µ ji)−bi j(νi j −ν ji)=(g2i j +bi jbsi j)(αi−α j)bi j (µi j +µ ji)+gi j (νi j +ν ji)=(gi jbi j −gi jbsi j)(αi +α j)gi j (φi j −φ ji)−bi j (ψi j −ψ ji)=(g2i j +bi jbsi j)(δi j −δ ji)bi j (φi j +φ ji)+gi j (ψi j +ψ ji)=(gi jbi j −gi jbsi j)(δi j −δ ji)Let Φ denote the set on which the variables µ, ν, φ, ψ, α,and δ satisfy these equalities. Forming constraints containingup to second-order terms and substituting the new variables,we haveLAC minξ,µ,ν,φ,ψ,α,δ ∑i∼ j ci jξi js.t. {µ,ν,φ,ψ,α,δ} ∈ Φpi ≤∑j ξ0i jµi j +φi j ≤ piqi ≤∑j ξ0i jνi j +ψi j ≤ qiv2i ≤ αi ≤ v2iv2i ξi j ≤ δi j ≤ v2i ξi jv2i(ξi j −ξi j)≤ ξi jαi−δi j ≤ v2i(ξi j −ξi j)τ1i j∣∣µi j∣∣+ τ2i j∣∣νi j∣∣≤ si j (i, j) ∈ Ω0τ1i j∣∣φi j∣∣+ τ2i j∣∣ψi j∣∣≤ si jξi jτ1i j∣∣∣ξi jµi j −φi j∣∣∣+ τ2i j∣∣∣ξi jνi j −ψi j∣∣∣≤ si j(ξi j −ξi j)(i, j) ∈ Ω00 ≤ ξi j ≤ ξi j, ξi j ∈ NThrough the the relaxation procedure, the AC transmissionsystem planning problem has thus been posed as a mixed in-teger linear program.3. APPLICATIONS3.1. Computational exampleIn this section we apply LAC to a to a 19-bus, 46-candidate-line example which was abstracted from a notionalshipboard power system [2]. The outcome of this examplecalculation is not considered to be directly applicable to theelectric ship specifically, but rather as a result representativeof a microgrid design, having geographic content and highlyvariable loads. Tables 1 and 2 contain all relevant problemdata.Reactive power limits are assumed to be one tenth of realpower limits, reactance and resistance are both set to lengthover line capacity, and η to two for all lines. The base lay-out η0 was chosen to be a ring connecting the power con-verter modules. Voltage limits are not enforced in this ex-ample. Line costs are proportional to length times capacity,where length is assumed to be the ‘Manhattan distance’ (x-distance plus y-distance plus z-distance) between two buses.We use a slightly more complicated flow limit approximationthan (2.1.), which is a proper relaxation:∣∣pi j∣∣≤(ηi j +η0i j)si j,∣∣qi j∣∣≤(ηi j +η0i j)si j∣∣pi j∣∣+ ∣∣qi j∣∣≤√2(ηi j +η0i j)si j.The mixed integer linear program was solved using themodeling language AMPL [4] and solver CPLEX [1], in 1.0seconds on a laptop representative of current standards. Ta-ble 3 and Fig. 1 show which lines were selected (includingthe base design). Inspection shows that large loads are con-nected directly to large generators, as expected. Non-intuitiveconnections, or lack of connection, elsewhere are generallydue to the fact that the z-distances are not evident in the fig-ure, and that certain connections were simply not part of theallowable set in Table 1.Table 2. Bus real power limits and locationsBus p (kW) p (kW) x (m) y (m) z (m)1: fwd. mn. gen. 0 35000 18.0 2.1 5.82: fwd. aux. gen. 0 3500 23.5 -2.1 5.83: PCM 2a -500 -500 22.1 0 5.64: PCM 1a -500 -500 33.8 -2.7 35: PCM 1a -500 -500 10.4 7.0 8.26: aft mn. gen. 0 35000 -10.4 0.6 5.87: aft aux. gen. 0 3500 -43.0 -0.6 5.88: PCM 2a -500 -500 -4.7 0 5.69: PCM 1a -500 -500 8.5 -5.2 310: PCM 1a -500 -500 -18 7 8.211: PCM 2a -100 -100 -40.8 0 5.612: port rud. -50 -50 -68 4.1 3.013: stbd. rud. -50 -50 -68 -4.1 314: PCM 1a -100 -100 -30.8 -1.8 315: PCM 1a -100 -100 -50.9 6.4 8.216: port motor -37000 -37000 -14.9 4 317: stbd. motor -37000 -37000 13.4 -4 318: port rdr. -2300 -2300 30 4.2 2019: stbd. rdr. -2300 -2300 30 -4.2 20Table 3. LAC solutionLine Additions1 - 5 11 - 17 12 - 5 13 - 4 26 - 16 17 - 15 18 - 9 19 - 14 211 - 14 112 - 15 113 - 15 117 - 18 117 - 19 1Table 1. Line dataLine Length (m) s (MW) η0 Line Length (m) s (MW) η01 - 4 23.4 20.3 0 6 - 9 27.5 20.3 01 - 5 14.9 20.3 0 6 - 10 16.4 20.3 01 - 9 19.6 20.3 0 6 - 14 25.6 20.06 11 - 10 43.3 20.3 0 6 - 15 48.7 20.06 01 - 14 55.5 20.06 0 6 - 16 10.7 40 01 - 15 75.6 20.06 0 6 - 17 31.2 40 01 - 16 37.6 40 0 7 - 9 58.9 3.3 01 - 17 13.5 40 0 7 - 10 35 3.3 02 - 4 13.7 3.3 0 7 - 14 16.2 3.06 02 - 5 24.6 3.3 0 7 - 15 17.3 3.06 02 - 9 20.9 3.3 0 7 - 16 35.5 23 02 - 10 53 3.3 0 7 - 17 62.6 23 02 - 14 57.4 3.06 0 8 - 9 21 0.6 02 - 15 85.3 3.06 0 8 - 10 22.9 0.6 02 - 16 47.3 23 0 9 - 14 42.7 0.36 12 - 17 14.8 23 0 11 - 14 14.4 0.12 03 - 4 17 0.6 0 11 - 15 19.1 0.12 03 - 5 21.3 0.6 0 12 - 14 43.1 0.12 04 - 5 38.3 0.6 1 12 - 15 24.6 0.12 04 - 6 50.3 20.3 0 13 - 14 39.5 0.12 04 - 7 81.7 3.3 0 13 - 15 32.8 0.12 04 - 9 27.8 0.6 1 14 - 15 33.5 0.12 15 - 6 29.6 20.3 0 16 - 18 62.1 21.5 05 - 7 63.4 3.3 0 16 - 19 70.1 21.5 05 - 10 28.4 0.6 1 17 - 18 41.8 21.5 05 - 15 61.9 0.36 1 17 - 19 33.8 21.5 03.2. Multiple scenariosReturning to the case of a shipboard electrical system as aprototype microgrid, we note that a ship may require multiplesets of loads, for example the operating conditions of travel-ing at high speed, versus active combat. Because these aresuch different load scenarios, there is no analogue in terres-trial system design.A conservative approach would be to simply optimize witheach load bus consuming the maximum power over all sce-narios; this however, will lead to highly conservative designs.We can instead produce designs that are not overly conserva-tive by creating constraints and variables for each scenario,and optimizing the same objective as usual. Suppose we aregiven nS scenarios, and for each scenario k = 1, ...,nS we havea set of minimum and maximum bus power levels at each busi, pki , pki , qki , and qki . Using a separate set of variables for eachset of power levels, we have the following multiple scenariotransmission planning problem:MSAC minξ,s,v ∑i∼ j ci jξi js.t. ski j =(ξ0i j +ξi j)(vki vk∗i y∗i j − vki vk∗j y∗i j)∀kpki ≤ Re∑j ski j ≤ pki ∀kqki ≤ Im∑j ski j ≤ qki ∀kvi ≤∣∣∣vki∣∣∣≤ vi ∀k∣∣∣ski j∣∣∣≤ (ξ0i j +ξi j)si j (i, j) ∈ Ω0∪Ω,∀k0 ≤ ξi j ≤ ξi j, ξi j ∈ NFrom here, relaxations may be developed by identically ap-plying the procedures of the previous section.4. CONCLUSIONWe have applied a general relaxation for polynomial opti-mization problems to AC transmission system planning. Byforming the problem as a mixed integer linear program, we12 345678910111213141516171819Figure 1. Solution with buses arranged geographically ac-cording to x and y coordinates given in Table 1 (the aspectratio has been modified to aid viewing). Squares denote loadsand circles generation. Note that Buses 17 and 18 are not con-nected to Bus 3.are taking advantage of mature tools in convex optimizationto achieve efficient and reliable solutions to this difficult prob-lem.The newly obtained model has been applied here to an ACmicrogrid example based on an abstracted shipboard distri-bution system, for which the traditional simplifying assump-tions in TSP are not valid. The algorithm found a solution inmodest computation time, that forms a cogent basis for fur-ther refinement.Acknowledgements: We thank Dr. J. Chalfant for discus-sions on the electric ship example. Work is supported by theOffice of Naval Research, Grants N00014-02-1-0623, moni-tored by Dr. Terry Ericsen.REFERENCES[1] IBM ILOG CPLEX.[2] BMT Syntek. IPS electric plant one-line diagram for anotional destroyer, 2003.[3] Norbert Doerry, Henry Robey, John Amy, and ChesterPetry. Powering the future with the integrated powersystem. Naval Engineers Journal, 108(3):267–282,1996.[4] Robert Fourer, David M. Gay, and Brian W. Kernighan.AMPL: A Modeling Language for Mathematical Pro-gramming. Duxbury Press, November 2002.[5] G. Latorre, R.D. Cruz, J.M. Areiza, and A. Villegas.Classification of publications and models on transmis-sion expansion planning. Power Systems, IEEE Trans-actions on, 18(2):938 – 946, May 2003.[6] M.J. Rider, A.V. Garcia, and R. Romero. Power sys-tem transmission network expansion planning using ACmodel. Generation, Transmission Distribution, IET,1(5):731 –742, September 2007.[7] R. Romero, A. Monticelli, A. Garcia, and S. Haffner.Test systems and mathematical models for transmissionnetwork expansion planning. IEE Proceedings - Gen-eration, Transmission and Distribution, 149(1):27–36,2002.[8] H. D. Sherali and C. H. Tuncbilek. A global optimiza-tion algorithm for polynomial programming problemsusing a reformulation-linearization technique. Journalof Global Optimization, 2:101–112, 1992.[9] J. A. Taylor and F. S. Hover. Linear relaxations for trans-mission system planning. Submitted to IEEE Trans. onPower Systems, 2010.[10] A. J. Wood and B. F. Wollenberg. Power generation,operation, and control. Knovel, second edition, 1996.

Lift-and-Project Relaxations of AC Microgrid Distribution System Planning

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Lift-and-Project Relaxations of AC Microgrid Distribution System Planning

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