Constructive heuristic algorithm for the DC model innetwork transmission expansion planning

R. Romero, C. Rocha, J.R.S. Mantovani and I.G. S!anchez

Abstract: A novel constructive heuristic algorithm to the network expansion planning problem ispresented. The basic idea comes from Garver’s work applied to the transportation model,nevertheless the proposed algorithm is for the DCmodel. Tests results with most known systems inthe literature are carried out to show the efficiency of the method.

1 Introduction

The main objective of the transmission system planningproblem is to determine the optimal expansion plan of theelectrical system. The planning should specify the circuits(transmission lines or transformers) that have to be built toguarantee adequate operation for a specified planninghorizon. The data available is: base-year topology, candi-date circuits, forecast generation and demand of theplanning horizon, investment constraints, etc. The planningspecifies the location, quantity and when the circuits shouldbe installed. There are two types of planning: staticplanning, representing one single planning horizon, andmultistage planning in which the planning horizon isdivided into several stages. This work considers the staticplanning case only, however, the algorithm can easily beextended to the multistage planning. The solutions considera centralised planning environment, however the algorithmcan be used in an open-market environment through theaddition of adequate constrains as shown in [1]. Thetopologies found in long-term planning are further analysedwith short-term planning tools, such as AC power flow,stability analysis, reliability analysis, etc.

Usually, long-term transmission system planning ismodelled by a mathematical model called a DC model,which is a nonlinear mixed-integer problem (NLMIP) withhigh complexity, especially for large-scale and real-worldproblems. There are alternatives to the DC models, forinstance, relaxed models and hybrid models. However, thiswork employs the DC model.

The mathematical model for the transmission expansionstatic planning problem represented by DC power flowmodel is formulated as follows:

min v ¼P

ði;jÞ2Ocijnij

s:t:ð1Þ

Sf þ g ¼ d ð2Þ

fij � gijðnoij þ nijÞðyi � yjÞ ¼ 0 ð3Þ

jfijj � ðnoij þ nijÞ�ffij ð4Þ

0 � g � �gg

0 � nij � �nnij

nij integer; fij and yj unbounded

ði; jÞ 2 O

where cij, gij, nij, noij, fij and �ff ij represent, respectively, the

cost of a circuit that can be added to right-of-way i�j, thesusceptance of the circuit, the number of circuits added inright-of-way i�j, the number of circuits in the base case, thetotal power flow and the corresponding maximum powerflow for circuit in right-of-way i�j; v is the investment, S isthe branch-node transposed incidence matrix of the system,f is a vector with elements fij, yj is the phase angle in bus j, gis a vector with elements gk (generation in bus k) whosemaximum value is �gg, �nnij is the maximum number of circuitsthat can be added in right-of-way i�j and O is the set of allright-of-ways. Constraint (2) represents the conservation ofpower in each node. This constraint models Kirchhoff’scurrent law (KCL) in the equivalent DC network. Constraint(3) is an expression of Ohm’s law for the equivalent DCnetwork and so Kirchhoff’s voltage law (KVL) is implicitlytaken into account. These equations are nonlinear.

Most applied approaches in the planning problem can beclassified into three groups: heuristic algorithms (mostlyconstructive heuristics), classic optimisation methods andmeta-heuristic methods (genetic algorithms, Tabu Search,GRASP, etc). We propose a new constructive heuristicalgorithm (CHA) applied directly to the DC model. Thebasic issue when working directly with the DC model is theneed of solving a nonlinear programming (NLP) problem.As a consequence an efficient and robust algorithm isrequired and this fact has represented a limitation that maypossibly explain the insignificant number of papers in thisarea.

2 CHA in network planning

In this section we present the fundamental components of aCHA, the most-used algorithms in network expansionplanning and the fundamental characteristic of theproposed CHA.

A constructive heuristic algorithm is an iterative processthat finds a good quality solution in a step-by-step process.

The authors are with the Faculdade de Engenharia El!etrica, UniversidadeEstadual Paulista Julio de Mesquita Filho, Av. Brasil 56–Centro, CEP15385.000, Ilha Solteira, SP, Brazil and Departamento de Engenharia El!etrica,Campus de Foz de Igua-cu, Universidade Estadual do Oeste do Parana –UNIOESTE, Av. Tarquinio Joslin dos Santos 1300, CEP 85070-650, Foz deIgua-cu, PR, Brazil

r IEE, 2005

IEE Proceedings online no. 20041196

doi:10.1049/ip-gtd:20041196

Paper first received 28th January and in revised form 24th August 2004

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 277

In network transmission planning a circuit (transmissionline or transformer) is chosen by using a sensitivity indexand added to the system at each iteration. The iterativeprocess stops when a feasible high quality solution is found;consequently no more additions are needed. The maincharacteristic of a CHA is the robustness and fastconvergence.

The difference between existing algorithms is basically onthe sensitivity indicator. Usually the sensitivity indices arecreated based on local performance criteria, i.e., they lookfor the most interesting circuit and then the best circuit isadded into the system. Consequently the sensitivity indexloses the context of a global optimisation.

The general form of a CHA is the following:

(i) Assume the base topology as current topology andchoose the mathematical model.

(ii) Solve an LP or NLP (depending on the model) todetermine parameters used in the sensitivity index for achosen CHA and the operational conditions of the system.If the LP or NLP solution indicates that the system isadequately operating with the new additions, STOP. A newsolution for the model has been found. Go to step (iv).

(iii) Use a sensitivity index to identify the most attractivecircuit. Update the current topology with the chosen circuit,go to step (ii).

(iv) Sort the added circuits in decreasing order of costs.Verify by using an LP, whether the removal of a circuitkeeps the system in an adequate operational condition. Ifyes, remove the circuit, otherwise keep the circuit. Repeatcircuit removal simulation until all circuits have been tested.All added circuits that weren’t removed represent CHA’ssolution.

There are many CHAs in the literature and this workfocuses on two types: algorithms that use electrical systemperformance for building the sensitivity index, and algo-rithms that employ the relaxed version of their ownmathematical model. Algorithms like least load shedding[2] and least effort [3] belong to group 1, and Garver’s [4],Villasana–Garver–Salon (VGS) [5] and the algorithmproposed in this work belong to group 2.

The least-load-shedding algorithm employs the sensitivityindex that tries to identify the circuit that would provide themost significant reduction in load shedding. For this case,step (ii) represents an LP and operations problems arerepresented by load shedding. Moreover, the sensitivityindex is an approximate relation, owing to the fact that thechosen circuit does not guarantee the least load shedding.Although the chosen circuit provides a reduction in loadshedding, it does not mean that the circuit belongs to theoptimal topology. All these problems are partially dealtwith when the sensitivity index considers the circuit’s cost.

In the least-effort algorithm the index tries to identify thecircuit that would provide a better power-flow distributionin the system. For this case, step (ii) of the algorithm solvesa linear algebraic system, which can be replaced by an LPwhose operation problem would be the possible overloadsin the current topology. As with the previous index, it isapproximate, therefore selected and inserted circuits do notguarantee the best reduction in system overloads. Thesensitivity index, as with the previous one, represents a localobservation and this drawback is partially dealt with byconsidering the circuit’s cost. An important advantage inusing either the least-effort algorithm or the least-load-shedding algorithm is that both work directly with the DCmodel.

Garver’s algorithm works with the transportation model(TM). The TM is a relaxed version of the DCmodel createdfrom elimination of (3), which corresponds to Kirchhoff’ssecond law. The TM is a mixed-integer linear problem.Basically Garver’s algorithm consists of solving the TMafter relaxing the integrality of the investment variable, i.e.making nij � 0 and solving the problem as an LP. Observethat the LP solution, although optimal, is not feasible to theproblem, therefore Garver applied this solution as asensitivity index for a CHA. The sensitivity index is given by

IS ¼ maxfISij ¼ nij�ff ij; nij 6¼ 0g ð5Þ

where nij is the solution given by LP after relaxingintegrality of nij.

Garver’s proposal represents a broad view of the solutionprocess. At each step an LP with the current topology issolved and the operation problems are dealt by fractionalcircuit additions, which provide the least investment.However, the algorithm presents two problems. First, itemploys the transportation model and the solutions foundcan be far from solutions that satisfy the DC model, or inother words, when the topology is tested with the DCmodelthe load shedding can be very high; Secondly, when the LPsolution provides too small a fractionate nij (usually at theend of the process) the conversion of nij into a integernumber can produce serious deviations, which means thatthe information provided by relaxation is deteriorated.

One of these problems has been solved by creation of theVGS algorithm, which finds good solutions for the DCmodel [5]. In an elegant way, a bybrid formulation isemployed to identify the most important circuit at each stepof the algorithm. An LP is solved after relaxing integralityconstraints, and only circuits of current topology mustfollow both Kirchhoff’s laws. Candidate circuits followKirchhoff’s current law only. At the end of the process afeasible solution to the DC model is determined by solvingonly linear programming problems. The VGS algorithm isone of the best CHA ever proposed in the literature. Otherimportant heuristic algorithms can be found in [6, 7].

Considering the logic employed by Garver, whichconsists of solving a relaxed mathematical model to identifythe best circuit, we propose a constructive heuristicalgorithm that works directly with the DC model. Insteadof relaxed models a nonlinear programming problem has tobe solved.

3 Novel CHA for DC model

Garver’s initial proposal for the transportation model(which was extended by Villasana–Garver–Salon to workwith hybrid models) can be adapted for direct use with theDC model. The basic idea consists of solving the DC modelafter relaxing the integrality of investment variables, i.e. ifnij integer ! nij � 0, the problem becomes a NLP. There-fore the sensitivity index given by the NLP solution can beused as an indicator of potential circuits. It is expected thatthis solution is more adequate than the linear approach [4,5] since only a relaxation of integrality variables is done.The principal drawback of the method is the need forsolving a NLP at each CHA step. It is known that a largescale NLP problem is much more difficult to solve than anLP problem. There are two significant limitations in usingsuch an approach. First, there are no sufficient robustprograms for NLP, i.e. usually the majority of the algorithmand software suffers from convergence problems with largeand complex systems, and secondly, the NLP algorithmmay converge to local optimal solutions, causing deviations

278 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

in the search process. Other problems related to therelaxation of integrality variables [4–6] which affect thefinal stages of the search process, such as occurrences ofsmall fractional values of nij, remain.

The constructive algorithm presented in this work issummarised as follows:

(i). Assume the base topology as current topology andchoose the DC model.

(ii) Solve a NLP to determine parameters used in thesensitivity index (5). If the NLP solution indicates that thesystem is adequately operating with the new additionsSTOP. A new solution for the DC model has been found.Go to step (iv).

(iii) Use a sensitivity index (5) to identify the most attractivecircuit. Update the current topology with the chosen circuit,then go to step (ii).

(iv) Sort the added circuits in descending order of costs.Verify by using an LP, whether the removal of a circuitkeeps the system in adequate operation condition. If yes,remove the circuit, otherwise keep the circuit. Repeat theprocess simulating circuit removal until all circuits havebeen tested. All added circuits that weren’t removedrepresent the CHA’s solution.

The CHA for the DC model is a particular case of thealgorithm presented in Section 2. In step (ii) the followingNLP is solved:

min v ¼P

ði;jÞ2Ocijnij

s:t:Sf þ g ¼ dfij � gijðno

ij þ n1ij þ nijÞðyi � yjÞ ¼ 0

jfijj � ðnoij þ n1

ij þ nijÞ�ffij

0 � g � �gg0 � nij þ n1

ij � �nnij

nij � 0; fij and yj unboundedði; jÞ 2 O

ð6Þ

The problem is the same as system (1) with relaxedintegrality constraints, nij � 0. At each step the algorithmadds a circuit in the current topology and the circuits arestored in vector n1 ¼ fn1

ijg. The NLP changes only one

position of vector n1ij in two consecutive iterations. That

feature can be employed to initialise the NLP’s variables,i.e. the solution of a NLP can be used as initialisation tothe next NLP. The first NLP is solved considering only thecurrent topology, i.e. with the n0

ij value correspoding to the

base-case topology with all elements n1ij ¼ 0.

Step (iv) of the CHA was created with the aim to removecircuits that have been added to the system and thereafterhave become irrelevant in the final stages of the process.The occurrence of irrelevant circuits is common to all CHA,and gets worse with large and complex systems [2, 3, 6].Thus, in step (iv) a verification whether the systemadequately operates without one circuit is carried out. Theverification can be done by solving the same NLP step (ii)but nevertheless can be carried out more efficiently with thefollowing LP:

min w ¼P

rk

s:t:Sf þ gþ r ¼ dfij � gijðno

ij þ n1ijÞðyi � yjÞ ¼ 0

jfijj � ðnoij þ n1

ij�ffij

0 � g � �gg0 � r � dfij and yj unboundedði; jÞ 2 O

ð7Þ

where w represents the load shedding, n1ij is the vector with

the circuits added during the process and r is the vector withartificial generators added in each load bus. If w¼ 0, thesystem is correctly operating and the circuit that wasremoved in the simulation can be definitely removed.Otherwise, the circuit must stay in the system.

4 Tests with algorithm

The algorithm was implemented using a GAMS structurefor solving the NLP problem at each step of theconstructive algorithm. The GAMS structure employedhas three NLP solvers: SNOPT, CONOPT and MINOS.

4.1 Garver’s six-bus systemGarver’s six-bus system consists of six buses, 15 circuits and760MW of demand. The system data can be found in [4, 8]for static planning. We analysed the case with generationrescheduling. Simulations with the algorithm resulted in thefollowing configuration:

� Planning with rescheduling: v¼ 110

n3�5 ¼ 1; n4�6 ¼ 3

Table 1 presents results iteration-by-iteration; vNLP is theinvestment provided by NLP and v is the partial investmentduring iterative process. In the planning with generationrescheduling the addition sequence presented on Table 1 isthe following: n4–6, n4–6, n3–5 and n4–6.

The algorithm converges after solving five NLPs and twoLPs without removing circuits in step (iv). The solution isglobal optimal. The proposed CHA is the first to achievesuch a solution, however with high difficulty. The solverused was SNOPT. The global optimal solution was earlyfound by using Bender’s decomposition techniques andmeta-heuristics.

The CONOPT and MINOS solvers incorporated in theGAMS do not find the optimal global solution; theyidentify only the optimal local solution of v¼ 130. For thecase without generation rescheduling, both SNOPT andCONOPT find the known optimal solution of v¼ 200. The

Table 1: Garver’s system with rescheduling

Iter. Additional by NLP vNLP v

1 n2–3¼ 0.424, n3–5¼ 0.776, n4–6¼2.500 99 0

2 n2–3¼ 0.424, n3–5¼ 0.776, n4–6¼1.50 69 30

3 n2–3¼ 0.424, n3–5¼ 0.776, n4–6¼0.50 39 60

4 n2–3¼ 0.385, n4–6¼ 0.50 22.707 80

5 0 110

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 279

MINOS solver finds an initial solution of vNLP¼ 215.52,which is greater than the global optimal solution in the firstiteration, showing the difficulty of using these set of NLPsolvers.

4.2 IEEE 24-bus systemThe system consists of 24 buses, 41 circuits and 8550MWofload demand and the topology is shown in Fig. 1. The datais available in [1] and in the Appendix (Section 8), (Tables 3and 4). We present two from five possible tests representedby five generation plans (see Appendix, Section 8), i.e. thetests were carried out with generation plan G1 and G3 ofTable 3. The first plan identified by Go corresponds to thecase with generation rescheduling and the generation valuesare determined through an optimisation process. Theremaining correspond to cases without rescheduling, wherethe generation value is fixed (Table 3). The system was alsoemployed in [1]. All tests were carried out using GAMS andSNOPT solver, which gave the best performance in solvingNLP problems for the proposed algorithm.

Considering generation plan G1, the CHA found theexpansion plan of v¼ 438, 000, 000 US $ with the followingtopology:

� n01�05 ¼ 1, n03�24 ¼ 1, n06�10 ¼ 1, n07�08 ¼ 2,n14�16 ¼ 1, n15�21 ¼ 1, n15�24 ¼ 1, n16�17 ¼ 2, n16�19 ¼ 1,n17�18 ¼ 1.

The algorithm converges after solving 13 NLPs and 10 LPswithout removing circuits in step (iv). The sequence ofadditions is the following: n16�17, n14�16, n07�08, n17�18,n16�17, n16�19, n07�08, n03�24, n06�10, n01�05, n15�24, andn15�21.

Table 2 presents the iterative process. In iteration 11 thereis an investment of only 10.4 million US $ for solving theoperation problem when nij is considered nonintegervariable. However, the problem is solved with the additionof two circuits that demand an investment of 140 million US$ (almost 32 % of the total investment). Notice also that inthe present stage the value of nij given by NLP is too small.Although the solution is better than the one found in [1], itsprobability of being the global optimal solution is small.

The expansion plan found for G3 plan is v¼ 218000000US $ with the following topology:� n06�10 ¼ 1, n07�08 ¼ 2, n10�12 ¼ 1, n14�16 ¼ 1,

n16�17 ¼ 1, n20�23 ¼ 1.The algorithm converges after solving eight NLPs and six

LPs without removing circuits in step (iv). The sequence of

1 2

3

4

5

7 68

9 10

11 12

13

1415

16

1718

19 2021 22

23

24

Fig. 1

Table 2: 24 bus system without rescheduling

Iter. Addition by NLP vNLP 103

US$v 103

US$

1 n01–05¼ 0.299, n03–24¼0.312,n06–10¼ 0.535, n07–08¼2.000,n10–11¼ 0.072, n14–16¼0.745,n15–21¼ 0.083, n15–24¼0.049,n16–17¼ 1.517, n16–19¼0.523,n17–18¼ 0.637

199832.3 0

2 n01–05¼ 0.299, n03–24¼0.312,n06–10¼ 0.535, n07–08¼2.000,n10–11¼ 0.072, n14–16¼0.745,n15–21¼ 0.083, n15–24¼0.049,n16–17¼ 0.517, n16–19¼0.523,n17–18¼ 0.637

163832.3 36000

3 n01–05¼ 0.282, n03–24¼0.286,n06–10¼ 0.543, n07–08¼2.000,n10–11¼ 0.096, n15–21¼0.071,n15–24¼ 0.029, n16–17¼0.529,n16–19¼ 0.501, n17–18¼0.648

120962.3 90000

4 n01–05¼ 0.282, n03–24¼0.286,n06–10¼ 0.543, n07–08¼1.000,n10–11¼ 0.096, n15–21¼0.071,n15–24¼ 0.029, n16–17¼0.529,n16–19¼ 0.501, n17–18¼0.648

104962.32 106000

5 n01–05¼ 0.276, n03–24¼0.277,n06–10¼ 0.546, n07–08¼1.000,n10–11¼ 0.100, n15–21¼0.003,n15–24¼ 0.021, n16–17¼0.597,n16–19¼ 0.505

89006.66 126000

6 n01–05¼ 0.271, n03–24¼0.269,n06–10¼ 0.548, n07–08¼1.000,n10–11¼ 0.104, n15–24¼0.015,n16–19¼ 0.507

66670.30 162000

7 n01–05¼ 0.263, n03–24¼0.250,n06–10¼ 0.551, n07–08¼1.000,n10–11¼ 0.097, n15–24¼ 0.0001

47953.76 194000

8 n01–05¼ 0.263, n03–24¼0.250,n06–10¼ 0.551, n10–11¼0.097,n15–24¼ 0.0001

31953.76 210000

9 n01–05¼ 0.360, n06–10¼0.509,n10–11¼ 0.026, n15–24¼0.132

26837.55 260000

10 n01–05¼ 0.391, n10–11¼0.033,n15–24¼ 0.129

19536.25 276000

11 n10–11¼ 0.008, n15–24¼0.139 10441.06 298000

12 n15–21¼ 0.009 611.81 370000

13 0 438000

280 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

additions is the following: n07�08, n07�08, n20�23, n14�16,n10�12, n06�10, and n16�17.

The topologies found present smaller investments thanthe ones found in [1], thus indicating the excellentperformance of CHA in some applications. However, intwo of five tests executed in [1] the CHA presents proposalswith investment values greater than [1].

This behaviour takes place due to performance problemsof NLP solvers and also because of deviations caused bytoo small nij given by NLP in the final stages of theoptimisation process. Another important aspect of thealgorithm is that systems with low or medium complexitydo not present irrelevant circuits, meaning that the removalstep (step iv) is not executed, and that the algorithm has anexcellent performance. The same characteristic has onlybeen observed with Villasana–Garver–Salon algorithmuntil now.

5 Extensions for CHA and analysis of results

The proposed algorithm for the DC model can be easilyextended to the transmission systemmulti-stage planning byusing the same logic employed in the transportation modelof [6]. The difficulty in the implementation consists ofsolving several NLP problems, therefore an efficient solveris needed. The NLP algorithms employed in this work wereshown to be not robust and not reliable and is expected torise with multi-stage planning, since the number of variablestends to increase considerably.

Our research group is working on the development ofan interior-point algorithm for solving the non-linearproblem. We expect that the algorithm will be robust andreliable enough to extend the proposed algorithm for largescale and complex problems, and also for multi-stageplanning. In the proposed algorithm, other sensitivityindices could be employed based on results provided byNLP, in the same way as the proposal presented in [6].However, the main difficulty concerning performance is inthe final stages of the process, where too small values ofnij appear and turning this variable to integer could causeserious investment deviations. We believe that this problemcan be solved by developing a constructive heuristic in abranch-and-bound structure, whose basic module is theproposed CHA. In this approach, whenever a circuit witha low nij is added (below specified limit), the otheralternative with nij forced to zero should also be analyzed,then the process would continue searching for other newtopologies. Another alternative to deal with that problem,is to add non-linear constraints involving the nij variablesto force its values to approximate or make it equal to 0 or1. Our research group is now developing new algorithmwith both approaches.

The proposed CHA finds solutions with excellentquality to the DC model. With Garver’s six-bus systemwith generation rescheduling, global optimal solution isachieved. Notice also that for this test all other CHApresent in the literature find only local optimal solutions.The algorithm also presented an excellent performancefor the 24-bus system in two situations as shownpreviously.

The found topologies can be used as solutions proposalin planning work. However, there are many applications inthe transmission planning research area. The solutions canbe used as bounds in branch-and-bound algorithms forfinding the global optimal solution for the DC model, as

generators of high quality topologies for meta-heuristicmethods such as tabu search and genetic algorithms etc, oras elite configurations used in the implementation of pathrelinking in tabu search, to name a few. Summarising, thetopologies generated by the method can be used to improvemeta-heuristics [9, 10] performance. A more detailedexplanation about mathematical modelling can be foundin [8]. A revision of the mathematical model andoptimisation techniques employed in transmission systemsplanning can be found in [11].

6 Conclusions

This paper presented a novel constructive heuristicalgorithm that works directly with the DC model. Thealgorithm presents excellent performance for systems withlow and medium complexity. The principal advantage ofthe algorithm is that it works directly with the solutiongiven by the DC model with relaxed integer variables. Theresulting information is of excellent quality for use as asensitivity index.

It was also observed that the main difficulty in algorithmimplementation is the need for a NLP solver, which must berobust and reliable for large and complex systems and alsofor application to multistage planning.

7 References

1 Fang, R., and Hill, D.J.: ‘A new strategy for transmission expansion incompetitive electricity markets’, IEEE Trans. Power Syst., 2003, 18,(1), pp. 374–380

2 Pereira, M.V.F., and Pinto, L.M.V.G.: ‘Application of sensitivityanalysis of load supplying capability to interactive transmissionexpansion planning’, IEEE Trans. Power App. Syt., 1985, 104, (2),pp. 381–389

3 Monticelli, A., Santos, A. Jr., Pereira, M.V.F., Cunha, S.H., Parker,B.J., and Pra-ca, J.C.G.: ‘Interactive transmission network planningusing a least-effort criterion’, IEEE Trans. Power App. Syst., 1982,101, (10), pp. 3919–3925

4 Garver, L.L.: ‘Transmission network estimation using linear program-ming’, IEEE Trans.Power App. Syst., 1970, 89, pp. 1688–1697

5 Villasana, R., Garver, L.L., and Salon, S.J.: ‘Transmission networkplanning using linear programming’, IEEE Trans. Power App. Syst.,1985, 104, (2), pp. 349–356

6 Romero, R., Rocha, C., Mantovani, M., and Mantovani, J.R.S.:‘Analysis of heuristic algorithms for the transportation model in staticand multistage planning in network expansion systems’, IEE Proc.Gener. Transm. Distrib., 2003, 150, (5), pp. 521–526

7 Dechamps, C., and Jamoulle, A.: ‘Interactive computer program forplanning the expansion of meshed transmission networks’, Int. J.Electr. Power Energy Syst., 1980, 2, (2), pp. 103–108

8 Romero, R., Monticelli, A., Garcia, A., and Haffner, S.: ‘Test systemsand mathematical models for transmission network expansionplanning’, IEE Proc. Gener. Transm. Distrib., 2002, 148, (5),pp. 482–488

9 Gallego, R.A., Monticelli, A., and Romero, R.: ‘Comparative studiesof non-convex optimization methods for transmision networkexpansion planning’, IEEE Trans. Power Syst., 1998, 13, (3),pp. 822–828

10 Da Silva, E.L., Gil, H.A., and Areiza, J.M.: ‘Transmission networkexpansion planning under an improved genetic algorithm’, IEEETrans. Power Syst., 2000, 15, (4), pp. 1168–1175

11 Latorre, G., Cruz, R.D., Areiza, J.M., and Villegas, A.: ‘Classificationof publications and models on transmission expansion planning’,IEEE Trans. Power Syst., 2003, 18, (2), pp. 938–946

8 Appendix

8.1 Generation and loadTable 3.

8.2 Circuit dataTable 4.

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 281

Table 3: Generation and load in MW

Bus G0 G1 G2 G3 G4 D

1 576 576 465 576 520 324

2 576 576 576 576 520 291

3 0 0 0 0 0 540

4 0 0 0 0 0 222

5 0 0 0 0 0 213

6 0 0 0 0 0 408

7 900 900 722 900 812 375

8 0 0 0 0 0 513

9 0 0 0 0 0 525

10 0 0 0 0 0 585

11 0 0 0 0 0 0

12 0 0 0 0 0 0

13 1773 1773 1424 1457 1599 0

14 0 0 0 0 0 795

15 645 645 645 325 581 582

16 465 465 465 282 419 951

17 0 0 0 0 0 300

18 1200 1200 1200 603 718 999

19 0 0 0 0 0 543

20 0 0 0 0 0 384

21 1200 1200 1200 951 1077 0

22 900 900 900 900 900 0

23 1980 315 953 1980 1404 0

24 0 0 0 0 0 0

Table 4: Circuit data

Bus Bus nijo res. (p.u) reat. (p.u) fij MW Cij 103 US

$

1 2 1 0.0026 0.0139 175 3000

1 3 1 0.0546 0.2112 175 55000

1 5 1 0.0218 0.0845 175 22000

2 4 1 0.0328 0.1267 175 33000

2 6 1 0.0497 0.1920 175 50000

3 9 1 0.0308 0.1190 175 31000

3 24 1 0.0023 0.0839 400 50000

4 9 1 0.0268 0.1037 175 27000

5 10 1 0.0228 0.0883 175 23000

6 10 1 0.0139 0.0605 175 16000

7 8 1 0.0159 0.0614 175 16000

8 9 1 0.0427 0.1651 175 43000

8 10 1 0.0427 0.1651 175 43000

9 11 1 0.0023 0.0839 400 50000

9 12 1 0.0023 0.0839 400 50000

10 11 1 0.0023 0.0839 400 50000

10 12 1 0.0023 0.0839 400 50000

11 13 1 0.0061 0.0476 500 66000

11 14 1 0.0054 0.0418 500 58000

12 13 1 0.0061 0.0476 500 66000

12 23 1 0.0124 0.0966 500 134000

13 23 1 0.0111 0.0865 500 120000

14 16 1 0.0050 0.0389 500 54000

15 16 1 0.0022 0.0173 500 24000

15 21 2 0.0063 0.0490 500 68000

15 24 1 0.0067 0.0519 500 72000

16 17 1 0.0033 0.0259 500 36000

16 19 1 0.0030 0.0231 500 32000

17 18 1 0.0018 0.0144 500 20000

17 22 1 0.0135 0.1053 500 146000

18 21 2 0.0033 0.0259 500 36000

19 20 2 0.0051 0.0396 500 55000

20 23 2 0.0028 0.0216 500 30000

21 22 1 0.0087 0.0678 500 94000

1 8 0 0.0348 0.1344 500 35000

2 8 0 0.0328 0.1267 500 33000

6 7 0 0.0497 0.1920 500 50000

13 14 0 0.0057 0.0447 500 62000

14 23 0 0.0080 0.0620 500 86000

16 23 0 0.0105 0.0822 500 114000

19 23 0 0.0078 0.0606 500 84000

282 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005