Approximating electrical distribution networks via mixed-integer nonlinear programming (2010).pdf

Electrical Power and Energy Systems xxx (2010) xxx–xxx

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Approximating electrical distribution networks via mixed-integernonlinear programming

Sanyogita Lakhera a,1, Uday V. Shanbhag b,⇑,1, Michael K. McInerney c

a Citibank, New York City, NYb Department of Industrial and Enterprise Systems Engineering at the University of Illinois at Urbana-Champaign, 117 Transportation Building,104 S. Mathews Ave., Urbana, IL 61801, USAc Construction Engineering Research Laboratory (CERL), USA

a r t i c l e i n f o

Article history:Received 23 July 2009Received in revised form 5 July 2010Accepted 13 August 2010Available online xxxx

Keywords:OptimizationNonlinear programmingMixed-integer nonlinear programmingDistribution system designGIS

0142-0615/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.08.020

⇑ Corresponding author.E-mail addresses: [email protected] (S. La

(U.V. Shanbhag), [email protected] Authors have been partially supported by Award N

Engineering Research Laboratory (CERL) W9132T-07-C

Please cite this article in press as: Lakhera S etPower Energ Syst (2010), doi:10.1016/j.ijepes.2

a b s t r a c t

Given urban data derived from a geographical information system (GIS), we consider the problem of con-structing an estimate of the electrical distribution system of an urban area. We employ the image data toobtain an approximate electrical load distribution over a network of a prespecificed discretization.Together with partial information about existing substations, we determine the optimal placement ofelectrical substations to sustain such a load that minimizes the cost of capital and losses. This requiressolving large-scale quadratic programs with discrete variables for which we present a novel penaliza-tion-smoothing scheme. The choice of locations allows one to determine the optimal flows in thisnetwork, as required by physical requirements which provide us with an approximation of the distribu-tion network. Furthermore, the scheme allows for approximating systems in the presence of no-go areas,such as lakes and fields. We examine the performance of our algorithm on the solution of a set of locationproblems and observe that the scheme is capable of solving large-scale instances, well beyond the realmof existing mixed-integer nonlinear programming solvers. We conclude with a case study in which astage-wise extension of this scheme is developed to reflect the temporal evolution of load.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Obtaining information pertaining to a city’s utility networks iscrucial for purposes of planning, maintenance and redesign. Yet,this information is often inaccessible to most agencies and a rel-evant question is how one may approximate such informationthrough the usage of only image data. This provides a motivationfor estimating the underlying distribution system of an urbanarea, given an image from a geographical information system(GIS). Such an image captures the parcel data as well as a subsetof existing substations in the urban area. We consider a novel ap-proach that comprises of two basic steps. First, the input image isused to construct an electrical load distribution on a grid of pre-specified discretization. Note that a grid, in this context, refers toa regular fully connected network over which a distribution net-work will be specified. Second, we solve an inverse problem thatestimates the set of lines that correspond to such a load distribu-tion. Several issues complicate such an estimation. In general, the

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khera), [email protected] (M.K. McInerney).o. US Army/Construction and

-0010.

al. Approximating electrical dis010.08.020

true distribution system is a radial network and the resulting in-verse problem falls within the realm of optimization problems infunction space and is, in general, intractable. Instead, we restrictthe set of possible networks to those that can be specified asgraphs on a grid of chosen size. Two additional complexitiesemerge from modeling distribution systems in urban areas. First,there are significant areas that cannot be covered by the distribu-tion system (such as lakes or fields). Therefore, the optimal solu-tion has to reflect these restrictions. Second, a clear evolutionpattern exists in the growth of the load and needs to be respectedin estimating the distribution system. For instance, if a particularpart of a township developed earlier than another, then the distri-bution system would have such a structure.

The resulting problem can be recast as an mathematical pro-gram in finite-dimensional space in which one seeks a set of flowsthat satisfy the substation capacity constraints, Kirchhoff’s conser-vation equations and voltage bounds. Unfortunately, this problemcan be infeasible if the substation information is inaccurate. Toavert this possibility, we consider a problem in which we deter-mine the installation of incremental substations as well as theflows that emerge from the resulting system. This optimizationproblem falls within a class of mixed-integer nonlinear problems(MINLP) and has a size that grows with the level of discretizationand the number of substations. Currently no solvers exist for

tribution networks via mixed-integer nonlinear programming. Int J Electr

http://dx.doi.org/10.1016/j.ijepes.2010.08.020

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2 S. Lakhera et al. / Electrical Power and Energy Systems xxx (2010) xxx–xxx

addressing problems arising from practical networks [17]; instead,we provide a framework that can cope with large-scale instances.

In this paper, we make the following contributions:

1. By converting the image data into electrical load data, we recastwhat was essentially an infinite-dimensional problem as afinite-dimensional discrete nonlinear optimization problem.This problem is capable of encapsulating a variety of complex-ities ranging from variable substation sizes, restricted areas andvariable feeder sizes.

2. Solving such discrete optimization problems is only possiblewhen the space of discrete variables is small. To address thischallenge, we present a penalization-smoothing scheme thatsolves a continuous form of this problem. We observe that thisallows for solving large-scale instances to local optimality withrespect to the smoothed formulation. Furthermore, the schemeis shown to scale well with the number of discrete variables.

3. The framework allows for modeling a variety of complexities.We show that restricted or no-go areas can be accommodatedwithin the framework and the obtained results do indeed reflectthe constraints.

4. Finally, we extend the framework to allow for a multi-stageapproach towards approximating the distribution system inan urban area by using a sequence of images from the GISsystem.

The remainder of this paper is organized as follows. In theremainder of this section, we review past work in this area. In Sec-tion 2, we provide an outline of our methodological framework andhas three subsections. In Section 2.1, we provide some details per-taining to the image data while in Section 2.2, we discuss the struc-tural properties of the constraints arising in such problems. InSection 2.3, we define the nonlinear discrete optimization problemand its smooth generalization. In Section 3, we compare the perfor-mance of a standard commercial MINLP solver with the smoothapproximation, particularly from the standpoint of the final valueas well as the scalability. We discuss the performance of our ap-proach when attempting to estimate the distribution system ofthe city of Champaign in Section 4. The paper concludes in Section5 with a summary and a set of possible research directions.

1.1. Background

The network flow model for distribution system design wasadequately discussed by Willis et al. [24,25]. The use of an optimalplanning approach toward the design of electrical networks wasfirst illustrated by Knight [13]. The idea here was to have a mini-mum cost network design using a network flow algorithm basedon a set of geographical locations. Other work highlighting similarapproaches include [4,15,12,26,1,23]. A succinct review of networkflow algorithms is available in [2] while power systems analysisand optimization can be found in [28].

Sun et al. [21] used the concept of a long range horizon year(target) together with a time-phased expansion process beginningwith the base year and progressing on to the target year. The idea isto determine an optimal static horizon year design using a fixedcharge transshipment problem formulation (FNCP). The branch-and-bound algorithm is then employed in the formulation whichalso includes explicit modeling of fixed charge and variable costcomponents for improved accuracy.

An efficient algorithm for the static investment planning oflarge radial distribution systems was presented by Fawzi et al.[8]. The algorithm takes into account the fixed costs, concave non-linearities in the cost functions of all elements and the operationalconstraints. It uses a concave fixed cost model to representelements with large fixed costs (substations and possibly some

Please cite this article in press as: Lakhera S et al. Approximating electrical disPower Energ Syst (2010), doi:10.1016/j.ijepes.2010.08.020

feeders) while it assumes linear cost functions for the remainingelements. The branch-and-bound algorithm was also used withthe bounding criteria dependent on the cost and operational con-straints. An iterative procedure modifying the solution of the firststep accounts for the fixed costs of the remaining elements.

Sharif et al. [22] propose an approach which makes use of boththe mixed-integer linear programming (MIP) and spanning treemethods to help estimate the future expansion paths for a radialdistribution network. The general methodology employed is to firstgenerate the spanning trees to connect the source to the demandnodes followed by a MIP approach to ascertain the spanning treewhich can be used as the optimal solution.

Modeling the planning of distribution substations as well asmethods for solving distribution planning problems are discussedin [7,16]. The solution methodology adopted is that of maximizingthe Lagrangian dual using the NETOPT program. Yahav and Oron[27] accounted for the nonlinear costs of losses and constructionthrough the solution of a nonlinear program using an off-the-shelfsolver called GINO (it uses the generalized reduced gradient solverGRG2 by Lasdon and Waren [19]).

Boulaxis et al. [3] proposes a new algorithm for solving the opti-mal feeder routing problem using the dynamic programming tech-nique and GIS facilities. Some of the factors taken intoconsideration include practical issues such as cost parameters(investments, line losses, reliability), technical constraints (voltagedrops and thermal limits) and physical routing constraints (obsta-cles, high-cost passages, existing line sections). In related work, Linet al. [14] determine substation locations and new feeders usingGIS data. This work seems closest to ours in motivation. Whilethe work by Lin et al. is far more detailed in terms of modelingthe distribution systems, our work allows far more generalityand proposes a flexible optimization framework for determiningnew locations and feeders. Related work pertaining to planningof distribution systems is seen in [9,6].

Finally, Murray and Shanbhag [17] use a local relaxation ap-proach for the problem of siting electrical substations. Throughthe solution of a sequence of appropriately defined nonlinear pro-grams, the authors are able to solve large-scale problems overthousands of grid-points. Importantly, for cases where global solu-tions are available, the algorithm compares favorably. Relatedextensions of the work to nonlinear facility location problems isfound in [18]. The nonlinear facility location problem falls withinthe realm of mixed-integer nonlinear programming problems, aclass of optimization problems that have significant applicationin distribution system planning. For instance, reconfiguration viacapacitor allocation has been modeled as a mixed-integer nonlin-ear program by Oliveira et al. [5]. Here, as well a continuous func-tion is employed for smoothing the discrete variables and a primal-dual interior point scheme is utilized for solving the resultingsmoothed problem.

Finally, its worth noting that the presence of distributed gener-ation has introduced both novel and challenging questions in thecontext of distribution system design. Popovic et al. [20] considerthe question of siting distributed generation by conducting a sen-sitivity analysis on the power flow equations. More recently, Ghoshet al. [10] employ Newton-based iterative schemes for optimal siz-ing and placement of such generation and test their schemes on amodified set of IEEE 6, 14 and 30 bus systems.

2. Methodology

Our general approach begins with data acquisition. This steptakes image data and provides a load distribution over a prescribedgrid. Integral to such a transformation is the prescription of howmuch load corresponds to a specific residential, commercial or



S. Lakhera et al. / Electrical Power and Energy Systems xxx (2010) xxx–xxx 3

industrial parcel. Using the grid-based load distribution, we con-struct an evolutionary pattern that relates how the load hasevolved historically. This would require specifying a set of vectorsthat represent the load distribution over specific time epochs.

At a particular time epoch, we solve a design subproblem thatdetermines the optimal lines and substations to meet the loadappearing in the next time epoch. Such a subproblem is in generala large-scale nonlinear integer programming problem. In the sub-sequent sections, we provide some insights of how local solutionsof such problems may be obtained. Given the location of substa-tions with prescribed capacities, we may solve a full-rank set ofequations to obtain the appropriate flows. This set of flows, if fea-sible , is taken as a proxy for the distribution system linkages. Ifthis system is infeasible with respect to substation capacities, thenwe proceed to solve a modified problem, till we attain a feasibleinteger solution. In the next three subsections, we describe variousaspects of this approach in greater detail.

2.1. Data acquisition

Our test data is the tax parcel data set of the city of Champaign.It provides information about the land usage which can be catego-rized into three major groups: residential commercial and indus-trial. We further subdivided the categories as shown in Fig. 1.

Based on the given data, we mapped the land use informationwith the energy consumption information for each type of landusage (Table 4). Fig. 2 provides the electrical load distribution pro-file. In reality, the building of a distribution system is an inherentlystage-wise activity that corresponds closely with the evolution ofthe urban load profile. Using some prior information about theevolution pattern, we determine a distribution system design. Anevolution pattern is an order over the set of subgrids (in terms ofage for instance). In reality, finding exact evolution patterns ischallenging. and may require study of historical city developmentplan. Fig. 3 gives an example of such evolution pattern and its real-ization on a grid. The schematic on the right provides an order forthe appearance of the subgrids.

To make a robust decision, one should be able to design over afamily of such evolution patterns for a city. The determination ofan appropriate evolution pattern is out of scope of this study, butone option could be the change in the historical boundary of thecity. Even if we have this information, it is difficult to estimate

Fig. 1. Parcel data for city of Champaig


the historical electrical consumption. One option to over come thiswould be to take a factor of current load distribution to representhistorical load profile. We use a similar approach for our case studyfor the city of Champaign. Once we have come is with one suchevolution pattern we can find out a evolving distribution system,we show this in Fig. 4. It can be seen that in this schematic, the griddesign evolves in accordance with the stage-wise appearance ofnew load.

2.2. Problem formulation

In this section, we formulate the optimization problem underthe assumption that the load information has been made availablefrom the GIS data set. Such an optimization problem specifies theoptimal location of substations and the related flows. In formulat-ing the problem, several concepts need clarifying. First, the rele-vant set of equations (namely the load-flow equations) arediscussed. This is followed by a brief discussion of the structureof admittance matrix of the grid. Finally, an important complexityin our setting is the requirement that we address restricted (or no-go) areas. One approach for modeling such areas is provided.

2.2.1. Load-flow equationsGiven a set of substation locations, nodal loads and a well-de-

fined branch impedance matrix, the load-flow equations providea unique solution vector of nodal voltages at nodes housing loadsand nodal currents at substation nodes. We denote the set of nodeshousing substations by SS while the set of load nodes is denoted byL. Then the optimal flow is derived from the solution of the linearequations defined by following load-flow equations:

Yv � i ¼ 0v j ¼ 1; j 2 SS

ij ¼ lj j 2 LXj

ij ¼ 0: ð1Þ

The first equation relates the current vector i, the voltage vec-tor v and the admittance matrix Y and originates from Kirchhoff’sconstraint equations. Voltage at nodes j 2 SSis assumed to be 1 pu(per-unit). Current at each node is given precisely by the load atthat node. For a given load distribution and set of substation

n (L) with superimposed grid (R).



Fig. 2. Electrical load distribution for the city of Champaign (L) with 3-D plot (R).

Fig. 3. Evolution pattern of load distribution.

Fig. 4. Evolving distribution system.


Please cite this article in press as: Lakhera S et al. Approximating electrical distribution networks via mixed-integer nonlinear programming. Int J ElectrPower Energ Syst (2010), doi:10.1016/j.ijepes.2010.08.020


Table 1Variables and parameters.

Notation Description

‘j Electrical load at node jvj Voltage at node jij Current delivered at node jY Nodal admittance matrix (relates nodal current to nodal voltage)zj Decision to install substation at node jCloss Cost of lossesCsub Capital costScap Substation capacitySS Set of substation nodel Smoothing parameterq Penalty parameter


locations, we can solve this linear system of equations. In fact,one can also show that this set of equations represents a squarenonsingular system (see [17]). Note that the notation of relevanceis summarized in Table 1.

2.2.2. The admittance matrix YAs seen in Fig. 5, for an n � n grid, every interior node is con-

nected to the four nodes while nodes on the boundaries have cor-respondingly fewer connections. The admittance matrix Y capturesthe relationship of a node with its neighbors. Each row of the firstsystem of (1) can be thought of as a current-balance relationship inwhich the current at a particular node is equal to the sum of thecurrent flows along the links connected to it. For an n � n grid,the admittance matrix Y can be represented as a symmetric blocktridiagonal matrix

Fig. 5. A 5 � 5 grid (L), nodal linkages

Fig. 6. A 5 � 5 grid with restricted area


Y ¼

A1 B1

B1 A2 B2

. .. . .

. . ..

Bn�2 An�1 Bn�1

Bn�1 An

0BBBBBBB@

1CCCCCCCA;

where the blocks Aj,j = 1, . . . ,n and Bj,j = 1, . . . ,n � 1, are tridiagonaland diagonal in structure, respectively. Fig. 5 provides the sparsitypattern for a 5 � 5 grid.

2.2.3. Restricted areaAn important complexity seen in practical problems is that cer-

tain areas are not acceptable from the standpoint of building distri-bution networks. Such areas are often referred to as restricted orno-go areas. Our framework is flexible enough to construct gridsthat do not have connectivity in such areas. As a consequence,the admittance matrix suffers some changes. In Fig. 6, we showhow a no-go area translates into a new grid. Note that any nodethat remains unconnected is essentially removed from the problemframework. Questions of infeasibility are removed by not allowingthe load to be positive at such points. Also shown in Fig. 6 is themodified admittance matrix.

2.3. Design subproblem

The location of substations over a given grid with a specifiedload distribution may be cast as a facility location problem, butwith an important difference. The cost function is nonlinear in nat-ure, while the decision variables are integers, specifically binary

(C) and admittance matrix Y (R).

(L) and admittance matrix Y (R).



Fig. 7. Smoothed step function for a variety of l’s.


variables. Such problems are inordinately hard to solve and somepreliminary results using existing solvers have not been promising[17]. In particular, for grid sizes beyond 20 � 20, the computationaleffort is enormous and renders conventional integer programmingformulations useless.

Formally, the design subproblem is given by (INLP):

minimizei;v;z

ClossesvT Yv þ CcapeT z

Yv � i ¼ 0subject to vL þ ð1� vLÞzj 6 v j 6 1; 8 j

ð1� zjÞ‘j 6 ij 6 ð1� 2zjÞ‘j þ zjScap; 8 j

zj 2 f0;1g; 8 j;

where vTYv represents the lifetime cost of losses whileCcapeT z ¼ Ccap

Pjzi gives the cost of installing capacity. Note that Ccap

provides a ratio of the capacity cost to the cost of losses. The firstconstraint is the Kirchhoff’s relationship between current and volt-age. The second constraint prescribes voltage bounds at each node.In particular, at a substation node, voltage is set to 1 while at othernode (namely load nodes), voltages are bounded between vL and 1at a load node. The third constraint specifies current at non-substa-tion nodes as being precisely the load and bounded by substationcapacity Scap at substation nodes. Finally, the decision variable per-taining to the placement of a substation at node i, given by zi, isspecified to be binary.

3. A smoothing technique for binary nonlinear programs

As mentioned in the earlier section, the problem (INLP) is alarge nonlinear integer programming problem whose solvability(to a global minimizer) is possible only for modest grid sizes. Giventhat the problems could have arbitrary sizes, we relax the require-ment of providing a global minimizer. Instead, we provide a localsolution with respect to a smoothed (and therefore continuous)nonlinear program. Such ideas were employed by Murray andShanbhag [17] to obtain starting points but not for solving the ori-ginal problem. Here, we use a different smoothing function, whichdoes have a close relationship with that in [17].

The smoothed nonlinear problem is given by NLP(l):


minimizei;v;z

ClossesvT Yv þ Ccap

Xj2N

21þ e�lzj

� 1� �

Yv � i ¼ 0subject to ð1� zjÞvL þ zj 6 v j 6 1; 8 j

ð1� zjÞ‘j 6 ij 6 ð1� 2zjÞ‘j þ zjScap; 8 j

0 6 zj 6 1; 8 j

where

Ccap 21þ e�lzj

� 1� �

represents the smoothed version of the step function representingthe installation of a substation and Ccap is the ratio of the cost ofcapital to that of losses. Finally, the relaxed decision variable per-taining to the placement of a substation at node j is zj. The smooth-ing of the step function using the exponential function is given byFig. 7:

One of the challenges in such a framework is that the relaxeddecision variables zj could be non-integral in nature. The likelihoodof such an event is reduced by adding a penalty function of theform:

Pðz;qÞ ¼Xj2N

qzjð1� zjÞ;

which essentially penalizes non-integral decision variables. Thishowever has a disadvantage of introducing a large number of localminimizers. The resulting optimization problem, nonetheless, is gi-ven by NLP(l, q)

minimizei;v;z

ClossesvT Yv þ Ccap

Xj2N

21þ e�lzj

� 1� �

þPðz; qÞ

Yv � i ¼ 0subject to ð1� zjÞvL þ zj 6 v j 6 1; 8 j

ð1� zjÞlj 6 ij 6 ð1� 2zjÞ‘j þ zjScap; 8 j

0 6 zj 6 1; 8 j

Finally, the solution to this problem may still be non-integral. Ifthat is indeed the case, then it is not clear if the load-flow equa-tions are satisfied. To avert such a possibility, we define




zi ¼zi; zi 2 f0;1g1; zi P �0; zi < �

8><>: : ð2Þ

The integer-valued z may then be used to solve the load-flowequations. If the solution to the load-flow equations is feasible withrespect to the substation capacity constraints, we have a feasibleinteger solution to (INLP). If not, we tighten the capacity con-straints by reducing Scap and repeat this procedure. The frameworkis formalized in specifying Algorithm 1. Note that in the currentframework, both l and q are kept fixed but update rules for theseparameters are part of ongoing work.

Algorithm 1. Penalization-smoothing method

initialization Scap, l, q;termin :¼ false, k :¼ 1, a :¼ 1;while termin = false do

(zk, vk, ik) solve NLP(c, l; aScap);�zk ¼ roundðzkÞ according to (2);Let ð�vk;

�ikÞ solve load-flow equations;if ½�ik�SS 6 Scap thentermin = true;

endelse

a :¼ 0.9aendk = k + 1

end

4. Numerical results

This section will examine the workings of the suggested mod-el as well as the underlying algorithms for obtaining solutions inlarge-scale instances. In estimating the underlying distributionsystem, we use the current flows from the substations as a basis.In Section 4.1, we observe how changing the load distributionleads to intuitively different flow patterns. A challenge faced insuch models is how one can deal with areas which cannot housetransmission lines. We prescribe one technique for addressingprecisely such a problem in Section 4.2 and provides someexamples to show the workings of the model In Section 4.3,we provide some examples to show the suitability of our meth-od for locating substations in the absence of such data. Finally,in Section 4.4, we examine how the penalty-smoothing algo-rithm works on a set of sample problems of steadily increasingsize. A comparison with the commercial solver CPLEX is alsogiven.

Fig. 8. (L) Uniform load distribution, (C) curren


4.1. Load-flow models and their enhancements

In this section, we provide some preliminary results pertainingto the estimation of distribution networks through the use of load-flow equations. To aid in understanding the model, we consider arelatively small area in which the substation location is known.For instance, in Fig. 8, there is exactly one substation on the lowerleft corner from which flow emerges. The positive flow specifies aninitial estimate of the distribution system. It is observed that theflow from the substation flows symmetrically in the Northernand Eastern directions. Now, consider a setting where a majorityof the load is situated in towards the top-left of the grid (Northof the substation). We model such a load by using a Gaussian loaddistribution instead of a uniform load distribution. The resultingflow pattern changes significantly with a dominant flow patterntowards the Northern direction to address the load requirements.The new pattern is shown in Fig. 9. Further complexities in the loaddistribution (as exemplified by adding two Gaussian distributions)are handled well by the existing framework (see Fig. 10). Increas-ing the scaling of the problem provides a richer characterizationof the underlying distribution system design. In Fig. 11, we usethe load distribution from our test data with a single substation.The size of the grid is increased to 25 � 25 and the resulting flowshows a concentration in the load areas and the connections be-tween the load center and the substation locations.

4.2. Modeling of restricted areas

We have modeled restricted areas as Fig. 12 shows. Specifically,in an 8 � 8 grid, we introduce a 3 � 3 region that cannot housetransmission lines. We see that flow passes around the no-go areas,precisely as required.

4.3. Optimal location of substations

This subsection provides some of the results of using such anapproach to locate substations. In most settings, we may haveeither no data (or possibly incomplete data) pertaining to the sub-stations in an existing area. The optimal placement of these substa-tions would then specify the flow along the wires.

In each figure, we display the load distribution (negative) on theleft and the current flows on the right. In Fig. 13C, we have 10 � 10grid with Ccap and l set at 1. The load centers are at locations (3, 3),(3, 8), (8, 3) and (8, 8). Since the cost of capacity is low with respectto the cost of losses, it is optimal to locate a substation at each loadcenter and our smoothing approach finds what is conceivably theglobal solution. Clearly, such a strategy is not optimal as the costof capital increases. This is observed in Fig. 13R as the number of

t flows and (R) nodal current distribution.



Fig. 9. Gaussian load distribution.

Fig. 10. Multiple Gaussians load distribution.

Fig. 11. Test area load distribution with a single substation location.

Fig. 12. Flow distribution in a 8 � 8 grid having 3 � 3 restricted area.


Please cite this article in press as: Lakhera S et al. Approximating electrical distribution networks via mixed-integer nonlinear programming. Int J ElectrPower Energ Syst (2010), doi:10.1016/j.ijepes.2010.08.020


Fig. 13. 10 � 10 grid with load distribution (L), substation locations with low capital costs (Ccap = 1) (C) and high capital costs (Ccap = 10) (R).

Fig. 14. 25 � 25 grid with symmetrically placed Gaussian loads.


substations reduces to two as the cost of capital grows by a factorof 10.

Next, we consider two scenarios (Figs. 14 and 15 in each ofwhich a 25 � 25 grid is employed with multiple Gaussian load dis-tributions placed at a variety of locations. In each instance, thesmoothing approach leads to placement of the substations at theload centers or near a clump of load centers.

4.4. Scalability of the algorithm

As noted in the earlier sections, crucial to the solution of theproblem is the ability to solve large-scale instances of the design

Fig. 15. 25 � 25 grid with asymme


subproblem given that the grid sizes can be arbitrarily high. In thissection, we provide some details on several aspect of our smooth-ing approach. First, we show how the computational effort scaleswith grid size. However such techniques lead to a local minimizerof a smoothed problem which need not be a global minimizer.Obtaining global minimizers would require the use of cutting-plane or branch-and-bound techniques which tend to scale poorlywith the size of the problem. To obtain such solutions, we use thecommercial solver CPLEX. Table 2 charts the performance of Algo-rithm 1 for obtaining a feasible integer solution to the problemNLP(l, q) where l and q are kept at values of 10 and 10,000,respectively. Note that our approach for solving this problem

trically placed Gaussian loads.



Table 2Performance of the smoothing approach.

Size Iteration Minor iteration Time Loss Substation

25 269 5338 1.03 0.0094 664 89 5314 0.72 0.0679 6

100 47 4790 0.72 0.0615 5225 4 994 0.28 0.3780 5289 44 4657 1.83 0.0531 5400 8 2642 1.21 0.2252 6625 11 5814 4.27 0.3137 5

Table 3Performance of CPLEX.

Size Iteration Minor iteration Time Loss Substation

25 8159 30,596 12.20 0.107 464 87,937 361,307 168.98 0.740 4

100 627,456 2,614,568 2053.83 0.470 4225 3,768,278 16,079,112 31742.30 1.034 4

Fig. 16. Time comparison.


requires solving NLP(l, q), given the original substation capacity.From Table 2, several observations may be made:

1. The number of major and minor iterations is relatively insensi-tive to the size of the grid. We use a sequential quadratic pro-gramming algorithm (SNOPT) [11] for solving the smoothedproblem. Such an approach solves a sequence of quadratic pro-gramming problems. The bottleneck in such an approach is thesolution of the quadratic programming subproblem. As the gridsize grows, the real challenge lies in solving such problems effi-ciently. It can be seen that while the number of grid-pointsgrows by a factor of 25, the corresponding growth in the cpu-time is far more modest.

2. An important concern that can possibly plague Algorithm 1 isthat the number of substations added can be rather large, whileensuring feasibility of the resulting integer solutions. Our preli-minary tests however show that the number of substationsadded is slightly larger than that seen in the globally optimizedsetting.

The dominant reason for developing an alternate approach forsolving the nonlinear integer programming problem is that exist-ing commercial solvers for such problems obtain global solutions


but at enormous computational cost. Table 3 details the perfor-mance of CPLEX for a set of steadily increasing grid sizes. Animmediate observation is the growth in all metrics of computa-tional effort is the rapid exponential growth. For instance, if theproblem size grows by 10 (from 25 to 225), the CPU time growsfrom 12 s to over 30,000 s. Fig. 16 compares the growth in effortof the two solvers using the logarithmic scale. A closer look atthe nature of the solutions shows that while CPLEX always findsa global solution with 4 substations, the smoothing method termi-nates with 5 or 6 substations in all the tested cases.

5. Case study: city of Champaign

In this section, we use image data of the city of Champaign toapproximate the underlying distribution system. Importantly, weemploy a stage-wise algorithm that solves a smoothed problemat each stage. Such an extension allows adherence to an underlyingtemporal evolution of the load.

5.1. Obtaining the load distribution

To approximate the test area distribution system, we began byapproximating the load distribution by using the GIS parcel dataobtained the Champaign County Regional Planning Commission.This data has information about the tax parcel polygons, parcelboundary lines and subdivision polygons. The parcel data set pro-vides information about the land, usually with some implicationfor land ownership or land usage. By aggregating the land usageinto three major categories that is, residential commercial andindustrial. We further subdivided the categories. Based on the gi-ven data, we mapped the land use information with the energyconsumption information provided in Appendix A. Using this map-ping, we created an additional mapping between, land use infor-mation and approximation of site energy consumption forresidential/commercial/industrial buildings.

5.2. Evolution pattern

We assumed an evolution of the distribution network corre-sponds with the evolution of the test area. We based the estima-tion of the city development as provided the Champaign CountyRegional Planning Commission which maps the city boundariesover the last 60 years at certain key points on the basis of old maps.Using the GIS files provides, the evolution of the city is shown inFig. 17.

Using this evolutionary data, we constructed two sets of distri-bution patterns. The first test case used the existing set of substa-tions which were sequentially introduced within the evolution.The second test case assumed that such data was not availableand prescribed an optimal placement.

5.3. Test case 1: Partial substation information

In the first test case, we assumed that information about thesubstation provided in the image data. Using the load distribution,and assuming a three stage evolution pattern we come us with fol-lowing distribution system. Here we assumed that these substa-tions have enough capacity to take care of any load requirements.

Fig. 18 shows the growth in load over the three stages. Giventhe load in stage 1, the bottom schematic in Fig. 18 shows the flowsbased on the grid placement. Using this set of lines as given andnew load emerging from the stage 2 expansion, the second sche-matic in Fig. 18 shows how the lines may be extended. Note thatwe may insert a variety of no-go areas at this instance. Here, wemerely introduce a grid of a specific size. Finally, in Fig. 19, we



Fig. 17. Evolution of the city of Champaign.

Fig. 18. Load profile (top) and distribution system (bottom) across stages 1 (top), 2 (middle) and 3 (bottom) (with substation information).


show how the designed distribution system compares with the ori-ginal area.

5.4. Test case 2: No substation information

In the second test case, we assumed that no information aboutthe location of substations is provided. Therefore, the algorithm


must find the best possible location of the substations and estimatethe distribution system. Using the load distribution, and by assum-ing a three stage evolution pattern, our algorithm provides us withthe following distribution system. Here we assumed that thesesubstations have a specified capacity. The distribution system de-sign varies significantly with the earlier one. In particular, wesequentially add substations of finite capacity and the algorithm



Fig. 19. Given location of substation.

Fig. 20. Load profile (top) and distribution system (bottom) across stages 1 (top), 2 (middle) and 3 (bottom) (without substation information).


chooses to add several substations in certain areas that representload centers. Fig. 20 provides a schematic of how the systemevolves.

6. Summary

We have presented a framework for obtaining approximationsof utility networks by employing GIS images. Our basic frameworkhas relied on obtaining an electrical load distribution over a pre-specified grid. In the presence of possibly incomplete substationdata and restricted areas, we present an optimization-based for-mulation for obtaining flows on the network, the latter usable asa proxy for the lines in the distribution system. Addressing large-scale instances of the location and flow problem requires solversthat scale well with the grid size. Unfortunately, these problemsare discrete nonlinear problems and existing solvers are capableof solving problems no more than a few hundred variables. Instead,


we present a penalization-based smoothing scheme that scaleswell with the grid size and provides solutions that are comparablewith the global solutions obtained by CPLEX. Our framework canbe seen to address a variety of load distributions and restrictionson line placements. We conclude with a case study in which astage-wise approach is adopted to the planning problem.

Acknowledgements

The authors gratefully acknowledge the data provided byChampaign County Regional Planning Commission and some preli-minary effort by Kevin Waicekauskas.

Appendix A

See Table 4.



Table 4Land usage and energy consumption.

APROP Usage Btu per square foot VA per square foot

1000 Developer held residential lots 0.00 0.001100 Single family rental dwelling 115.10 36.201150 Owner/occupied single family dwelling 115.10 36.201200 Duplex rental dwelling 123.30 36.201250 Owner/occupied duplex dwelling 123.30 36.201300 Apartment – 3–7 153.40 44.901350 Apartment – 3–7 153.40 44.901400 Apartment – 8 or more 131.50 38.601450 Apartment – 8 or more dwelling units 131.50 38.601500 Group home/fraternities/sorority 123.30 36.201700 Mobile home parks 194.50 57.001800 Condominium rental 123.30 36.201850 Owner/occupied condominium dwelling 123.30 36.202000 Industrial use 2000.00 746.672100 Developer held industrial lots 0.00 0.003000 Commercial use 128.80 37.703050 Owner/occupied commercial use 128.80 37.703100 Developer held commercial lots 0.00 0.004000 Communications or utilities use 2000.00 746.675000 Hotels and motels use 126.60 37.106000 Property exempt from taxation 123.30 36.206001 Property exempt from taxation 123.30 36.206002 Property exempt from taxation 123.30 36.206003 Property exempt from taxation 123.30 36.206005 Property exempt from taxation 123.30 36.206006 Property exempt from taxation 123.30 36.206007 Property exempt from taxation 123.30 36.206800 Property partially exempt from taxation 123.30 36.206900 Veterans and fraternal organizations 123.30 36.207000 Land used as a commons area 0.00 0.007400 Open space valuation 0.00 0.007500 Open space valuation 0.00 0.008100 Agriculture (10+ acres) 5.00 5.008150 Agriculture use with owner occupied dwelling 6.00 6.00


References

[1] Adams RN, Laughton MA. Static and time phased network synthesis. Proc IEE1974;121(2):139–47.

[2] Ahuja RK, Magnanti TL, Orlin JB. Network flows. Theory, algorithms, andapplications. Englewood Cliffs (NJ): Prentice Hall Inc.,; 1993.

[3] Boulaxis NG, Papadopoulos MP. Optimal feeder routing in distribution systemplanning using dynamic programming technique and gis facilities. IEEE TransPower Deliv 2002;17:242–7.

[4] Crawford DM, Holt SB. A mathematical optimization technique for locating andsizing distribution substations and deriving their optimal service areas. IEEETrans Power Appar Syst 1975:230–5.

[5] de Oliveira LW, Carneiro Jr S, de Oliveira EJ, Pereira JLR, Silva Jr IC, Costa JS.Optimal reconfiguration and capacitor allocation in radial distribution systemsfor energy losses minimization. Int J Electr Power Energy Syst2010;32(8):840–8.

[6] El-Fouly THM, Zeineldin HH, El-Saadany EF, Salama MMA. A new optimizationmodel for distribution substation siting, sizing, and timing. Int J Electr PowerEnergy Syst 2008;30:308–15.

[7] El-Kady MA. Computer-aided planning of distribution substation and primaryfeeders. IEEE Trans Power Appar Syst 1984;PAS-103(6):1183–92.

[8] Fawzi TH, Ali KF, El-Sobki SM. A new planning model for distribution systems.IEEE Trans Power Appar Syst 1983;PAS-102:3010–7.

[9] Farrag MA, El-Metwally MM, El-Bages MS. A new model for distributionsystem planning. Int J Electr Power Energy Syst 1997;21(7):523–31.

[10] Ghosh S, Ghoshal SP, Ghosh S. Optimal sizing and placement of distributedgeneration in a network system. Int J Electr Power Energy Syst2010;32(8):849–56.

[11] Gill PE, Murray W, Saunders MA. SNOPT: an SQP algorithm for large-scaleconstrained optimization. SIAM J Optim 2002;12(4):979–1006.

[12] Hindi KS, Brameller A. Design of low-voltage distribution networks:amathematical programming method. Proc IEE 1977;124(1):54–8.

[13] Knight UGW. The logical design of electricity networks using linearprogramming methods. Proc IEE 1970;117:2117–27.


[14] Lin MW, Tsay MT, Wu SW. Application of geographic information system forsubstation and feeder planning. Int J Electr Power Energy Syst1996;18(3):75–183.

[15] Masud E. An interactive procedure for sizing and timing distributionsubstations using optimization techniques. IEEE Trans Power Appar Syst1974;93(5):1281–6.

[16] Marshall AC, Boffy TB, Green JR, Hague H. Optimal design of electricitynetworks. Proc IEEE C 1991;138(1):69–77.

[17] Murray W, Shanbhag UV. A local relaxation approach for the siting of electricalsubstations. Comput Optim Appl 2006;33(1):7–49.

[18] Murray W, Shanbhag UV. A local relaxation method for nonlinear facilitylocation problems. In: Multiscale optimization methods and applications.Nonconvex Optim Appl, vol. 82. New York: Springer; 2006.

[19] More JJ, Wright SJ. Optimization software guide. SIAM Phila 1993:87–9.[20] Popovic DH, Greatbanks JA, Begovic M, Pregelj A. Placement of distributed

generators and reclosers for distribution network security and reliability. Int JElectr Power Energy Syst 2005;27(5–6):398–408.

[21] Sun DI, Farris DR, Cote PJ, Shoults RR, Chen MS. Optimal distribution substationand primary feeder planning via the fixed charge network formulation. IEEETrans Power Appar Syst 1982;PAS-101:602–9.

[22] Sharif SS, Salamat MA, Vanelli A. Model for future expansion of radialdistribution networks using mixed integer programming. IEEE; 1996. p. 152–5.

[23] Thompson GL, Wall DL. A branch and bound model for choosing optimalsubstation locations. IEEE Trans Power Appar Syst 1981;PAS-100(5):2683–7.

[24] Willis HL, Tram H, Engel MV, Finley L. Optimization applications to powerdistribution. IEEE Comput Appl Power 1995;10:12–7.

[25] Willis HL, Tram H, Engel MV, Finley L. Selecting and applying distributionoptimization method. IEEE Comput Appl Power 1996;1:12–7.

[26] Wall DL, Thompson GL, Northcote-Green JED. An optimization model forplanning radial distribution networks. IEEE Trans Power Appar Syst 1979;PAS-98(3):1061–6.

[27] Yahav K, Oron G. Optimal locations of electrical substations in regional energysupply systems. IEEE; 1996. p. 307–10.

[28] Zhu JZ. Optimization of power systems operation. Wiley-IEEE Press; 2009.



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