IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013 3471

An Improved Network Model for TransmissionExpansion Planning Considering Reactive

Power and Network LossesHui Zhang, Student Member, IEEE, Gerald T. Heydt, Life Fellow, IEEE, Vijay Vittal, Fellow, IEEE, and

Jaime Quintero, Member, IEEE

Abstract—The expansion plan obtained from a DC model basedtransmission expansion planning (TEP) model could be problem-atic in the AC network because the DC model is potentially inac-curate. However, solving TEP problems using the AC model is stillextremely challenging. The motivation for this work is to developa less relaxed network model, based on which more realistic TEPsolutions are obtained. The proposed TEP model includes a linearrepresentation of reactive power, off-nominal bus voltage magni-tudes and network losses. Binary variables are added to avoid fic-titious losses. Garver’s 6-bus system is used to compare the pro-posed TEP model with the existing models. An iterative approachfor considering the criterion during the planning processis developed and demonstrated on the IEEE 118-bus system. Sim-ulation results indicate that the proposed TEP model provides abetter approximation to the AC network and is applicable to largepower system planning problems.

Index Terms—Linearized AC model, loss modeling, mixed in-teger second order cone programing, contingencymodeling,transmission expansion planning.

NOMENCLATURE

Quadratic cost coefficient of generator .

Linear cost coefficient of generator .

Series admittance of line , a negative value.

Shunt admittance of line , a positive value.

Fixed cost coefficient of generator .

Investment cost of the line .

Capacity factor of generator in year .

Hourly energy cost of generator in year .

Discount factor.

Conductance of line , a positive value.

Slope of the th piecewise linear block.

Disjunctive factor, a large positive number.

Manuscript received October 12, 2012; revised December 18, 2012; acceptedFebruary 18, 2013. Date of publication April 09, 2013; date of current ver-sion July 18, 2013. This work was supported in part by the U.S. Department ofEnergy funded project denominated “Regional Transmission Expansion Plan-ning in the Western Interconnection” under contract DOE-FOA0000068. Thisis a project under the American Recovery and Reinvestment Act. Paper no.TPWRS-01155-2012.The authors are with the School of Electrical, Computer and Energy En-

gineering, Arizona State University, Tempe, AZ 85287 USA (e-mail: hui.zhang@asu.edu; heydt@asu.edu; vijay.vittal@asu.edu; jaime.quintero.1@asu.edu).Digital Object Identifier 10.1109/TPWRS.2013.2250318

Active power flow on line .

Active power demand of load .

Active power generated by generator .

Maximum active power output of generator .

Minimum active power output of generator .

Active power loss on line .

Reactive power flow on line .

Reactive power demand of load .

Reactive power generated by generator .

Maximum reactive power output of generator .

Minimum reactive power output of generator .

Reactive power loss on line .

MVA rating of line .

Operating horizon.

Planning horizon.

Bus voltage magnitude in p.u. at bus .

Voltage magnitude deviation from 1 p.u. at bus .

Upper bound on the voltage magnitude deviation.

Lower bound on the voltage magnitude deviation.

Binary decision variable for a prospective line .

Binary variable for the th linear block of line .

Binary variable for modeling .

Phase angle difference across line .

Maximum angle difference across a line.

, Nonnegative slack variables used to replace .

th linear block of angle difference across line .

Set of generators.

Set of existing lines.

Set of prospective lines.

I. INTRODUCTION

T RANSMISSION expansion planning (TEP) is regardedas an important research area in power systems and has

been studied extensively during the past several decades. TheTEP exercise normally focused on improving the reliability

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3472 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

and security of the power system when economic impactswere not the primary concern. In contemporary power systemshowever, the increasing complexity of the network structureand the deregulated market environment have made the TEPproblem a complicated decision-making process that requirescomprehensive analysis to determine the time, location, andnumber of transmission facilities that are needed in the futurepower grid. Building the correct set of transmission lines willnot only relieve congestion in the existing network, but will alsoenhance the overall system reliability and market efficiency.Various TEP models have been developed during the past

several decades. Among these models, mathematical program-ming and heuristic methods are two major classes of solutionapproaches. Heuristic methods are usually not sensitive to themodels to be optimized and can potentially examine a largenumber of candidate solutions. The main criticism of heuristicmethods is that most of such methods do not guarantee anoptimal solution, and provide few clues regarding the qualityof the solution. Mathematical programming methods, on theother hand, can guarantee the optimality of the solution in mostcases, but tend to have stricter requirements on the model itself.In order to obtain the global optimal solution efficiently, theproblem or at least the continuous relaxation of the problemshould be convex. [1] presents a comprehensive review andclassification of the available TEP models.Due to the problem complexity, TEP using the AC network

model (ACTEP) is rarely discussed in the literature. The advan-tage of formulating TEP problems using the AC network modelis that the AC model represents the electric power network ac-curately. Nevertheless, the nonlinear and non-convex nature ofthe ACTEP model makes the problem difficult to solve and toobtain a globally optimal solution. [2] presented a mixed-in-teger nonlinear programming (MINLP) approach for solvingTEP problems using the AC network model. The interior pointmethod and a constructive heuristic algorithmwere employed tosolve the relaxed nonlinear programming problem and obtain agood solution. It is reported in [3] that by relaxing the binaryvariables, a small-scale ACTEP problem can be solved to ob-tain a local optimal solution. However, solving a MINLP-basedACTEP problem is still extremely challenging.The DC network model has been used extensively for de-

veloping TEP models [4]–[14]. One of the early works, [4]presents a linear programming (LP) approach to solve TEPproblems. A mixed integer linear programming (MILP) baseddisjunctive model in [5] eliminates the nonlinearity caused bythe binary decision variables. Based on the disjunctive model,the active power losses were included in [6] and [7]. In [8],the behavior of the demand was modeled through demand-sidebidding. A bilevel programming model appears in [9] wherethe solution to the problem is the Stackelberg equilibriumbetween two players. A transmission switching coordinatedexpansion planning model was presented in [10] where theplanning problem and the transmission switching problem aresolved alternately. The static security constraints are includedin [7], [11] and [12]. Regarding to the uncertainty modeling, atwo-stage stochastic programming model, which optimizes themathematical expectation of the weighed future scenarios wasproposed in [13] to coordinate the generation and transmissionplanning. A chance-constrained model was presented in [14] toaddress the uncertainties of loads and wind farms. This modelis by nature a risk-based game in which the planners decide theconfidence level at a specified risk.

The DC network model is essentially an approximation ofthe AC model by relaxing the reactive power and voltage con-straints. These relaxations tend to create a “gap” between thesolutions obtained from the DC model and the AC model [15].In some cases, the gap could be large and result in a TEP so-lution that is problematic in the AC network. It has also beenpointed out in [16] and [17] that the LP-based piecewise linearloss model as presented in [6] and [7] may cause the fictitiousloss problem under some circumstances. On the other hand, it isstill extremely challenging to solve a TEP problem using the ACmodel. In order to overcome the above difficulties, this paper de-velops a less relaxed networkmodel, which captures the originalAC network more accurately for TEP problems. Contributionsof this paper include:1) Develop a linearized AC model in which reactive power,off-nominal bus voltage magnitudes and network lossesare retained. Based on this linearized AC model, a novelTEP formulation (LACTEP) is proposed.

2) Present a MILP formulation for linearized network lossesmodeling to avoid fictitious losses.

3) Present an iterative approach to incorporate the con-tingency criterion effectively during the planning process.

The remainder of this paper is organized as follows: Section IIpresents the linearization of the full AC network model. By re-formulating the linearizedmodel, the LACTEPmodel is derivedin Section III. In Section IV, the proposedmodel is validated andcompared with other existing models. Concluding remarks aregiven in Section V.

II. LINEARIZATION OF THE FULL AC MODEL

The approximations made in the traditional DC model sig-nificantly simplify the full AC model, but these approximationsalso degrade the accuracy of the DC model in some cases. Inorder to improve the model accuracy, the linearized model pre-sented in this section retains a linear representation of reactivepower, off-nominal bus voltage magnitudes as well as networklosses. The linearization of the line flow equations is essentiallybased on a Taylor series and the following assumptions are as-sumed to be valid:1) The bus voltage magnitudes are always close to 1.0 per unit(p.u.).

2) The angle difference across a line is small so thatand can be applied. This assumption is

valid at the transmission level where the active power flowdominates the apparent power flow in the lines.

A. Linearization of the Power Flow Equations

If the effects of phase shifters and off-nominal transformerturns ratios are neglected, the AC power flow in branch be-tween nodes and is written as follows:

(1a)

(1b)

Based on the first assumption above, the bus voltage magnitudecan be written as

(2)

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where is expected to be small. Sub-stituting (2) into (1a) and (1b) and neglecting higher order terms

(3a)

(3b)

Notice that (3a) and (3b) still contain nonlinearities. Since ,and are expected to be small, the product andcan be treated as second order terms and therefore neg-

ligible. The linearized power flow equations for line meteredat bus are obtained as follows:

(4a)

(4b)

The power flow for the same line but metered at bus is obtainedin the same way:

(4c)

(4d)

Since and are linearized, the MVA limit for line can bewritten as a second-order cone constraint

(5)

Assuming each generator has a quadratic total cost curve

(6)

Notice that (5) and (6) are still convex and can be handled bymost commercial linear solvers such as Gurobi [21]. However,if a solver requires both the objective and the constraints to bestrictly linear, a piecewise linearized version for (5) and (6) canalso be derived.

B. Linearization of the Losses

Unlike the full AC model that inherently captures the net-work losses, the network losses for the proposed model, how-ever, need to be modeled separately. Using the second orderapproximation of and neglecting high order terms, thenetwork losses can be approximated as

(7a)

(7b)

Notice that (7a) and (7b) are non-convex constraints and needto be piecewise linearized. The following MILP formulation ispresented to achieve this objective rigorously:

(8a)

(8b)

(8c)

(8d)

(8e)

(8f)

(8g)

(8h)

(8i)

In (8b), two slack variables and are used to replace .In (8c), the sum of and is used to represent , which isexpressed as the summation of a series of linear blocks .Constraints (8d) and (8e) ensure that the right hand side of (8c)equals , while (8f)–(8i) guarantee that the linear blocks onthe left will always be filled up first as illustrated by the shadedarea in Fig. 1. This MILP formulation eliminates the fictitiouslosses using binary variables. However, addition of the binaryvariables tends to complicate the resultant model and makes itsefficient solution difficult when the problem scale is large. Al-ternatively, a relaxed model can be used by excluding (8g)–(8i)or even (8d) and (8e) to strike a balance between the computa-tion time and model accuracy.

III. LACTEP MODEL

The TEP problem is an extension of the optimal power flow(OPF) problem because it essentially solves a series of OPFproblems with different network topologies. In this section, theLACTEP model is developed based on the linearized networkmodel presented in Section II. In this model, it is assumed thatthe planners have perfect information about the existing net-work as well as the parameters of the potential lines. Notice thatthe focus of this paper is to advance network modeling. There-fore, the planning work is carried out at the peak loading hourfor a single future scenario. In real world applications, however,multiple scenarios can be developed to account for uncertaintiesand a two-stage stochastic programming planning model can bereadily formulated using the LACTEP model proposed in thispaper.

A. Objective Function

The objective function used in this paper jointly minimizesthe investment cost and the total operating cost,

(9)In (9), the first term represents the line investment cost and thesecond term corresponds to the total operating cost over a timehorizon scaled by the generator capacity factor, both in milliondollars (M$) and are discounted to the present value. Notice thatthe scaled operating cost provides only an estimate of the trueoperating cost, and can be replaced by a more accurate produc-tion cost model if the yearly load profile is available. As im-plied by the planning timeline in Fig. 2, all the selected linesare committed in the targeted planning year, and the operatingcosts are evaluated over multiple years thereafter. In reality, itis difficult to control the choice of the line to be built in a par-ticular year over the planning horizon. Issues such as project re-view process, construction and the load forecast accuracy could

3474 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

Fig. 1. Piecewise linearization of .

Fig. 2. Planning timeline.

bring too many uncertainties and make the dynamic planningprocess intractable. This paper is based on a static planningframework and focuses only on the large economic impact of theTEP project. Thus, the incremental economic benefit is lumpedinto the single targeted planning year.

B. Power Flow Constraints

In order to build the TEP model, the linearized power flowequations derived in Section III need to be reformulated.The constraints set related to the power flow equations in theLACTEP model are shown as follows:

(10a)

(10b)

(10c)

(10d)

(10e)

(10f)

(10g)

(10h)

(10i)

Constraints (10a)–(10d) represent the linearized power flowequations for existing lines and prospective lines, where

and are defined as the right hand sideof (4a) and (4b) [or (4c) and (4d)] respectively. For existinglines, the power flow is defined by and .For prospective lines, the disjunctive constraints (10c)–(10d)are used to avoid the nonlinearity that would otherwise appear.The power flow on the potential lines is forced to be zero by(10e) and (10f) if the line is not selected. The line MVA flow islimited by the second-order cone constraint (10g). Constraints(10h) and (10i) put a limit on the phase angle difference acrossexisting lines and prospective lines respectively. If the twobuses are directly connected, then is limited by and

; otherwise, (10i) is not binding.

C. Network Losses

The following constraint set extends the concept of linearizedloss modeling to the proposed TEP model:

(11a)

(11b)

(11c)

(11d)

(11e)

(11f)

(11g)

(11h)

(11i)

(11j)

(11k)

(11l)

(11m)

(11n)

(11o)

(11p)

(11q)

(11r)

Constraints (11c)–(11f) ensure that the right hand side of (11b)equals for existing lines and the selected prospectivelines respectively. Constraints (11g) and (11h) determine theupper and lower bound of a linear block for existinglines and prospective lines respectively. For existing linesand the selected prospective lines, is bounded by zeroand , otherwise, (11h) is not binding. The active andreactive power losses for existing lines are given by (11i) and(11j), respectively. For prospective lines, the active and reactivepower losses are determined by (11k)–(11l) and (11m)–(11n),respectively. Constraints (11o)–(11r) guarantee that the linearblocks on the left will be filled up first. Constraints (11a)–(11r)

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present a full MILP formulation that linearizes the networklosses rigorously without generating fictitious losses. Re-laxed models can be formed by removing (11o)–(11r) or even(11c)–(11f). The linearized line losses are then split in half andattached to the two terminal buses as “virtual demands”. Theterms corresponding to the network losses are added to thenodal balance equations as follows:

(11s)

(11t)

D. Generator Capacity Limits

In the planning study, all the generators in the system areassumed to be online. The generator outputs are limited by theirminimum and maximum generating capacities as shown in(12a) and (12b). Unit commitment is regarded as an operationalproblem and is therefore not considered in this model. Thegenerator limits are

(12a)

(12b)

The complete LACTEP model is described by (9)–(12).

E. Computational Burden and Modeling

The computational burden is a major concern in MIP prob-lems. Typically, increase the number of binary variables couldpotentially slow the solution process. Therefore, the candidateline set should be carefully selected and only the applicabletransmission corridors should be included. With a large-scaleMIP problem, the solver may have trouble finding an initial fea-sible solution. In this case, providing a feasible starting pointwill help reduce the overall simulation time.The contingency modeling is another major source of

the computational burden. In fact, a complete analysisin the TEP model for a well-designed power system is gener-ally unnecessary because the number of contingencies that willcause serious overloads is generally limited. The mod-eling approach used in [7] was to explicitly invoke the set of net-work constraints for all possible operating conditions and sat-isfy all the constraints when solving the optimization problem.However, the model presented in this paper is more compli-cated. If the approach in [7] were used, the size of the problemcould easily become too large to be solvable. Moreover, the TEPproblem uses only a relaxed network model, which means thatthe solution that satisfies the criterion in the TEP modelmay not represent the actual case in the AC network. In orderto make the planned system comply with the criterionwithout imposing too much computational burden, an iterativeapproach is proposed in Fig. 3.Using this approach in Fig. 3, the original problem is de-

composed into a master problem, which solves the optimizationmodel and a sub-problem, which verifies the network security.The master problem passes the TEP solution and the generatordispatch to the sub-problem, while the sub-problem passes thenetwork violations back to the master problem. The approachsolves the two problems iteratively until there is no violation orall the violations identified in the sub-problem are within presetlimits.

Fig. 3. Iterative approach for the contingency modeling.

TABLE ICANDIDATE LINE DATA FOR GARVER’S 6-BUS SYSTEM

IV. ILLUSTRATIVE RESULTS

In this section, Garver’s 6-bus system and the IEEE 118-bussystem are studied and the simulation results are demonstrated.The work presented in this paper is programmed using AMPL[20]. The DC lossless, DC lossy and the LACTEP models aresolved by Gurobi 5.0.2 [21]. The ACTEP models are solved byKnitro 8.0 [22]. PowerWorld [23] is used for AC power flowand the contingency analysis. All simulations are doneon a Linux workstation with an Intel , 4-core CPU at3.40 GHz with 16 GB of RAM.

A. Garver’s 6-Bus System

Garver’s 6-bus system has 6 existing lines, 5 loads and 3 gen-erators [2]. Initially, the generator connected at bus 6 is iso-lated from the main system. The system parameters are listed inTables I and II. It is assumed that at most 3 lines are allowed ineach transmission corridor. The total number of candidate linesis 39. The objective function is to minimize the line investmentcost only. The bus voltage magnitude range is 1.00 –1.05 p.u.The following two cases are analyzed:1) Case 1: Compare the TEP solutions given by the LACTEPmodel and other existing models.

2) Case 2: Network losses sensitivity analysis.Case 1: In this case, the TEP solution obtained from the

LACTEP model is compared with the solutions obtained fromother available TEP models. The full MILP approach is usedfor modeling the network losses. The number of linear blocksis 7. The comparison results are shown in Table III.The two DC-based TEP models in Table III seem to be su-

perior in the sense that the investment costs are less. However,the reactive power needed for these two models in the AC net-work actually exceeds the amount that the three generators can

3476 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

TABLE IIGENERATOR AND LOAD DATA FOR GARVER’S 6-BUS SYSTEM

TABLE IIITEP RESULTS COMPARISON OF GARVER’S SYSTEM

Fig. 4. TEP results of Garver’s 6-bus system.

supply. In order to make the AC power flow converge, an ad-ditional 189 MVArs and 129 MVArs are needed for the DClossless and the DC lossy model respectively. Meanwhile, over-loads and under voltage issues are observed in the system, whichrequire additional investment for network reinforcement. Thesolution obtained from the LACTEP model requires buildingmore lines than the DC-based models do, but needs no addi-tional reactive power and there are no overloads and under-voltage problems in the AC power flow. The expanded Garver’s6-bus system with all indices within the preset limits is plottedin Fig. 4.As a non-convex global optimization problem, multiple

starting points are tried to obtain a good solution for theACTEP model. As shown in Table III, the best objectivevalue for the ACTEP model after five thousand restarts isstill much higher than the objective function given by theLACTEP model. It will also be computationally too expensive

TABLE IVEFFECTS OF NUMBER OF LINEAR BLOCKS

to apply the ACTEP model to larger power system planningproblems. This comparison reveals that the solutions given bythe DC-based TEP models may not represent the actual case inthe AC network and additional network reinforcement is likelyto be needed. The LACTEP model better approximates the ACnetwork and therefore provides a more realistic TEP solution.For small systems such as the 6-bus example, reactive power

can be a critical issue to make the AC power flow converge.As indicated by Table III, the LACTEP model chooses to buildmore lines to provide reactive power support. In reality, in-creasing generator reactive power capacity and installing VArsupport devices can certainly be considered as alternative so-lutions if a DC-based TEP solution is adopted, but one shouldbe aware that it may not be easy to increase reactive power ca-pacity of existing generators, and can be costly to install VArsupport devices at high voltage buses, too. For real world appli-cations, different solution options can be compared to find themost cost effective TEP plan. For large systems with meshedtopology, using LACTEP model is more appropriate because itdispatches generators more accurately, gives a better estimationof line flows, and therefore provides a realistic TEP solution,which DC-based models usually fail to do.Case 2: As discussed in Section II, the linearized network

losses can be rigorously modeled using the MILP formulation.However, addition of the binary variables also increases thecomplexity of the TEP model. The number of linear blocks cansignificantly affect the solution time as well as the model accu-racy. Table IV shows how the number of linear blocks changesthe size of the problem and the TEP solution. The full MILPformulation is used for the results shown in Table IV.The variable types in Table IV show that the size of the

problem increases as the number of linear blocks increases.This behavior coincides with the intuition that more variablesare needed to model the additional linear blocks. It shouldbe noted that the linearization intrinsically overestimates thelosses in the system. If too few linear blocks are used, e.g.,1, then the overestimation can be significant and the problemwill be infeasible with the given set of candidate line set. Thisis reflected from both the trends of losses and the objectivevalues listed in Table IV. It is worth noticing that due to themixed-integer nature of the problem, the change in solutiontime does not follow a linear pattern. When too few linearblocks are used, the TEP results may contain unnecessary linesdue to the significant overestimation of the network losses. Itmay also take the solver a long time to branch out an initialfeasible solution. On the other hand, too many linear blockswill impose unnecessary computational burden and slow the

ZHANG et al.: IMPROVED NETWORK MODEL FOR TRANSMISSION EXPANSION PLANNING 3477

TABLE VCOMPARISON OF DIFFERENT NETWORK LOSSES MODELS

TABLE VIZONAL DATA OF THE IEEE 118-BUS SYSTEM

TABLE VIITEP PLANNING CRITERION FOR THE IEEE 118-BUS SYSTEM

solution time. The key idea of the study is to find the numberof linear blocks that gives the best balance between the modelaccuracy and the solution time. In this case, 7 is an appropriatenumber.The results contained in Table V compare the accuracy of

the relaxed losses models and the solution time. The numberof linear blocks used for this study is 7.Among all the loss modeling approaches listed in Table V,

the full MILP formulation is the most accurate and serves as abasis of the study. The approach relaxes the constraints forprioritizing the lower linear blocks. This approach reduces thesolution time by approximately 41%, but the drawback is thatit creates 2.4 MW fictitious active power losses. The ap-proach relaxes the constraints for modeling the absolute value.It reduces the solution time by approximately 35%, and createsonly 0.2MW fictitious losses. The approach relaxes both theconstraints that were relaxed in and . It reduces the solu-tion time by approximately 38%, but creates 2.5 MW fictitiouslosses. Additionally, if losses are ignored, the solution time willbe significantly reduced by 91%, but the TEP solution no longersatisfies preset the voltage requirement. Except for the no losscase, the TEP solutions remain the same for all other loss mod-eling approaches. One explanation is that the impact of fictitiouslosses is not significant enough to change the TEP results in thiscase. The study results show that the approach is consideredas the best trade-off between model accuracy and solution time.

B. IEEE 118-Bus System

The IEEE 118-bus system [24] is used to demonstrate the po-tential of applying the proposed LACTEP model to large powersystems. The system has 186 existing branches, 54 generatorsand 91 loads. The line ratings are reduced to create congestions.The system is divided into three zones as shown in Fig. 5 with

the zonal data listed in Table VI. The load assumed is the peakloading level. The discount rate is assumed to be 10%, and thenumber of linear blocks used for loss modeling is 10. The plan-ning horizon is ten years. The objective function jointly mini-mizes the line investment cost and the scaled ten-year total op-erating cost. The average capacity factors published in [25] areused in this paper. The capital costs of transmission lines are as-sumed to be proportional to the length of the lines. Due to theabsence of real data, all prospective lines are assumed to sharethe same corridor and have the same parameters as the existinglines. The planning criteria are given in Table VII. The detailedplanning procedure is described in the following steps.Step 1) Run a regular AC power flow on the system to be

planned, and identify the lines that are overloadedor heavily loaded. These lines will form the initialcandidate line set.

Step 2) Use the candidate line set and run the LACTEPmodel. Obtain the TEP solution and update thesystem.

Step 3) Rerun a regular AC power flow on the expandedsystem and identify any overloaded lines/trans-formers. Notice that it is still possible to observesome violations in this step because the networkmodel used in the TEP problem is essentially a re-laxation of the AC network model. If this happens,one should slightly reduce the line ratings used inthe TEP problem and redo Step 2 to Step 3. If noviolation is identified in this step, then proceed toStep 4.

Step 4) Perform a complete analysis on the expandedsystem. Identify the worst contingency and take theline out of service. Form a new candidate line setand return to Step 2. Do this iteratively until all vi-olations are within the preset threshold (as specifiedin Table VII). It is assumed that the generator dis-patch do not change during this process.

The flowchart of the iterative approach is plotted in Fig. 6.Table VIII shows the 15 initial candidate lines and their cost

data. The candidate lines for the contingency analysis arenot included in the table.The cost of building a transmission line can be roughly esti-

mated by its length, cost per mile and the cost multipliers [26].Assuming all lines are 230-kV double circuit lines, then the cap-ital cost of a transmission line is calculated as

(13)

where 1.5 is cost per mile of 230-kV double circuit lines andis the transmission length cost multipliers. For lines longer

than 10 miles, 3–10 miles and shorter than 3 miles, the valuesare 1.0, 1.2 and 1.5, respectively. Notice that (13) only givesa rough estimate of the line capital cost, more factors need beincluded in order to obtain a better estimate. The TEP resultsare demonstrated in Tables IX and X for and thecontingency case, respectively.It is observed from Table IX that four lines need to be added

in order to relieve the overloads in the original system with alllines in service . The investment cost is 43 M$, and theestimated 10-year total operating cost is 1567.4 M$, which isapproximately 156.7 M$ per year. The original system is thenexpanded using the TEP solution in Table IX and solved usingthe AC power flow with all indices within the limits. Therefore,

3478 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

Fig. 5. Single line diagram of IEEE 118-bus system [24].

Fig. 6. Flowchart of the iterative approach for considering contingency.

TABLE VIIIINITIAL CANDIDATE LINES FOR THE IEEE 118-BUS SYSTEM

with the four lines being added, the system is secure.Meanwhile, it is worth mentioning that the TEP solution given

TABLE IXTEP RESULTS FOR

TABLE XITERATIVE PLANNING PROCESS FOR

by the DC lossless model requires building no line for this case.However, significant overloads and undervoltage issues are ob-served in the AC power flow.In order for the system to comply with the criterion, the

planning process needs to proceed to Step 4. In this case, onlyline (do not include transformers) contingencies are considered.During the contingency, the monitored violations monitored areoverloads, loss of loads as well as undervoltages. The iterativeplanning process is elaborated in Table X.In Table X, the second column lists the lines that are man-

ually outaged in each iteration. The contingencies in the table

ZHANG et al.: IMPROVED NETWORK MODEL FOR TRANSMISSION EXPANSION PLANNING 3479

are ranked in the order of the severity of overload caused in thesystem. The line that causes severe overloads and results in alarge number of associated overloaded lines will be addressedfirst. The third column shows the type of the violations and thelast column provides the solution to mitigate the potential over-loads or loss of loads. After eleven iterations, all indices arewithin the limits set in Table VII for the contingency case. Thesystem complies with the contingency criterion. Mathe-matically, this iterative approach does not guarantee an optimalsolution, but in terms of the computational burden, this approachattains the same goal more efficiently.

V. CONCLUSIONS

This paper presents a new approach to linearize the full ACnetwork model, based on which a TEP model is developed. Theproposed LACTEP model retains a linear representation of re-active power, off-nominal bus voltage magnitudes and networklosses. A MILP formulation for network losses modeling is de-veloped to eliminate fictitious losses. An iterative approach isalso presented to incorporate the contingency criterion inTEP problems.The simulation results of Garver’s 6-bus system show that

additional network reinforcements may be needed if DC-basedTEP models are adopted. The proposed LACTEP model ap-proximates the AC network more accurately, and therefore pro-vides more realistic planning results. The loss modeling sen-sitivity study shows that the approach tends to give thebest trade-off between accuracy and solution time. The ficti-tious losses are not significant enough to alter the planning re-sult for the Garver’s 6-bus system. However, this conclusion canbe case dependent. The simulation results of the IEEE 118-bussystem show that the proposed LACTEP model can be appliedto solve large power system planning problems and the itera-tive approach is a computationally effective way to include the

criterion in the TEP study.

ACKNOWLEDGMENT

The authors would like to thank Dr. H. D. Mittelmann forproviding the ORION computing platform. The authors wouldalso like to thankMr.M. Bailey andMr. K.Moyer fromWesternElectricity Coordinating Council (WECC) for their inputs to thispaper.

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Hui Zhang (S’09) received the B.E. degree from Hohai University, Nanjing,China, in 2008 and the M.S. degree from Arizona State University, Tempe, AZ,USA, in 2010, both in electrical engineering. He is currently pursuing the Ph.D.degree at Arizona State University.

Gerald T. Heydt (S’62–M’64–SM’80–F’91–LF’08) received the Ph.D. degreein electrical engineering from Purdue University, West Lafayette, IN, USA, in1970.He is a Regents’ Professor at Arizona State University, Tempe, AZ, USA.

Vijay Vittal (S’78–F’97) received the Ph.D. degree from Iowa State University,Ames, IA, USA, in 1982.He is currently the Director of the Power Systems Engineering Research

Center (PSERC).

Jaime Quintero (M’06) received the Ph.D. degree in electrical engineeringfrom Washington State University, Pullman, WA, USA, in 2005.Currently, he is a postdoctoral researcher at Arizona State University, Tempe,

AZ, USA.