1

Market-Driven Security-Constrained TransmissionNetwork Expansion Planning

Lina P. Garces, Student Member, IEEE, Ruben Romero, Senior Member, IEEE, and

Jesus Marıa Lopez-Lezama Student Member, IEEE.

Abstract—This paper presents a Bi-level Programming (BP)approach to solve the Transmission Network Expansion Planning(TNEP) problem. The proposed model is envisaged under amarket environment and considers security constraints. Theupper-level of the BP problem corresponds to the transmissionplanner which procures the minimization of the total investmentand load shedding cost. This upper-level problem is constrainedby a single lower-level optimization problem which models amarket clearing mechanism that includes security constraints.Results on the Garver‘s 6-bus and IEEE 24-bus RTS test systemsare presented and discussed. Finally, some conclusions are drawn.

Index Terms—Bi-level programming, electricity market, trans-mission network expansion planning, security constraints.

NOMENCLATURE

The following notation is used throughout this paper:

A. Indices and sets

s(i) Bus index where the i-th generating unit is located.

s(j) Bus index where the j-th demand is located.

ΩL+Set of all prospective transmission lines.

ΩD Set of indices of the demands.

ΩG Set of indices of the generating units.

ΩN Set of all networks buses.

ΨDs Set of indices of the demands located at bus s.

ΨGs Set of indices of the generating units located at bus

s.

Ωi Set of indices of the blocks of the i-th generating

unit.

Ωj Set of indices of the blocks of the j-th demand.

ΩL Set of indices of all right-of-way of the system.

ΩLc Set of transmission lines considered as contingencies

of the system.

B. Parameters

bk Susceptance of one circuit in the right-of-way k.

cUj load shedding cost for consumer j (e/MWh).

ck Investment cost of building one circuit in the right-

of-way k (e).

L. P. Garces is supported by FAPESP, Project 2009/17428-8, Sao Paulo,Brazil.

L. P. Garces and R. Romero are with the Paulista StateUniversity, Ilha Solteira, Brazil (e-mails: linaneg@aluno.feis.unesp.br,ruben@dee.feis.unesp.br).

J. M. Lopez-Lezama is with Univerisity of Antioquia (UdeA), EfficientEnergy Management Research Group (GIMEL), Medellın, Colombia. (e-mail:lezama@udea.edu.co).

cmax Budget for investment in transmission expansion

(e).

dj Maximum power consumed by the j-th demand

(MW).

dj Minimum power consumed by the j-th demand

(MW).

djh Size of the h-th block of the j-th demand (MW).

fk Capacity of one circuit in the right-of-way k (MW).

gib Size of the b-th block of the i-th generating unit

(MW).

nk Maximum number of lines that can be added in the

right-of-way k .

o(k) Sending-end bus of the right-of-way k.

r(k) Receiving-end bus of the right-of-way k.

λDjh Price bid by the h-th block of the j-th demand

(e/MWh).

λGib Price offered by the b-th block of the i-th generating

unit (e/MWh).

C. Variables

d0jh Power consumed by the h-th block of the j-th demand

in normal operating condition (MW).

dcjh Power consumed by the h-th block of the j-th demand

under contingency (MW).

f0k Power flow through right-of-way k in normal operat-

ing condition (MW).

fck Power flow through right-of-way k under contingency

(MW).

g0ib Power produced by the b-th block of the i-th generat-

ing unit in normal operating condition (MW).

gcib Power produced by the b-th block of the i-th generat-

ing unit under contingency (MW).

nk Integer variable that corresponds to the number of

lines added to the right-of-way k.

r0j Load shed by the j-th demand in normal operating

condition (MW).

rcj Load shed by the j-th demand under contingency

(MW).

θ0s Voltage angle at bus s in normal operating condition

(radians).

θcs Voltage angle at bus s under contingency (radians).

I. INTRODUCTION

The Transmission Network Expansion Planning (TNEP)

problem consists in finding the number of circuits (lines and

transformers) that must be added to a current network in

2010 IEEE/PES Transmission and Distribution Conference and Exposition: Latin America 427

978-1-4577-0487-1/10/$26.00 ©2010 IEEE

2

order to make its operation viable for a pre-defined horizon

planning. This procedure is carried out by the Transmission

System Operator (TSO), which in most cases, procures the

minimization of the investment cost while attending a set of

constraints.

In the specialized literature there are a bunch of models and

methodologies that have been used to solve the TNEP prob-

lem. Pioneering works [1] and [2] are based on linear program-

ming models. Additionally, in [3] and [4] linear programming

is also used to solve the TNEP considering transmission losses

and electricity markets, respectively. Recently, in [5], it was

presented a model based on BP for the TNEP within a market

environment, where the target of the transmission planner is

to minimize network investment cost while facilitating energy

trading. The approach presented in this paper is based on the

previous reference, being the main difference the inclusion of

security constraints. Furthermore, the mathematical structures

of both models are also distinct. In the model presented in [5],

the upper-level problem is constrained by a collection of lower-

level market clearing problems considering different scenarios.

Nevertheless, in this work the upper-level is constrained by a

single lower-level problem which models a market clearing

mechanism that includes security constraints. In addition, it is

worth to mention that the proposed model considers not only

contingencies upon the base case, but also, on the new circuits

proposed in the expansion plan.

Within the framework described above, this paper presents

a Bi-level Programming (BP) approach to solve the TNEP

problem. The proposed model is envisaged under a market

environment and considers security constraints. The upper-

level of the BP problem corresponds to the transmission

planner which procures the minimization of the sum of total

investment and load shedding costs. This upper-level problem

is constrained by a lower-level optimization problem which

models a market clearing mechanism. This market clearing

mechanism takes into account a security criterion, which

establishes that the new expanded system must continue to

operate adequately after a single outage of a circuit takes place.

Such outage is selected from a set of pre-defined contingencies

established by the planner.

The main contributions of this paper are as follows:

1) To provide a network expansion model involving contin-

gency constraints under a market environment.

2) To use duality theory and linearization schemes in order

to turn the BP problem into a single level programming

problem that can be solved by using commercially avail-

able software.

3) To analise the results of the proposed approach in order

to show its applicability in real power systems.

The remaining of this paper is organized as follows. Section

II presents the proposed bi-level model and the corresponding

transformation into an equivalent single-level problem. Section

III presents the methodology to solve the TNEP considering

electricity markets and security constraints. Section IV pro-

vides results for two test systems (Garver’s 6-bus and 24-

bus RTS); these results are discussed and analyzed in detail.

Finally, Section V provides some relevant conclusions.

II. MATHEMATICAL MODEL FORMULATION

A. Bi-level Model

The decision making problem pertaining to a transmission

planner that jointly minimizes network investment and the total

load shedding cost (load shedding cost in normal operation

plus load shedding cost in contingency condition) can be

formulated as a bi-level programming model, [6] and [7]. The

upper-level problem represents the decisions to be made by the

planner with the target of deciding transmission investments

while minimizing the total load shedding cost (considering the

base case and each selected single circuit contingency of the

system) as well as the total investment cost. This upper-level

problem is constrained by a single lower-level problem. Such

lower-level problem represents a market clearing mechanism

that involves security constraints and considers investment de-

cisions as known. Thus, the proposed model has the following

mathematical structure:

1) Upper level problem: The upper-level problem, repre-

senting the target of the transmission planner, is as follows:

Min∑

k∈ΩL+

cknk +∑

j∈ΩD

cUj

[r0j +

∑c∈ΩLc

rcj

](1)

subject to: ∑k∈ΩL+

cknk ≤ cmax (2)

0 ≤ nk ≤ nk ∀k (3)

r0j and rc

j ∈ Υlower problem (4)

In this level, the investment decisions (nk) are taken. The

objective function described by (1) is composed by two

terms. These terms correspond to the minimization of the

total investment cost, as well as the total load shedding cost,

respectively. The upper-level constraints are composed by:

(a) Constraint (2) enforces a limit in the maximum investment

cost,

(b) Constraints (3) enforce the maximum number of circuits

that can be added in each right-of-way, and

(c) Constraints (4) state that the decision variables r0j and rc

j

belong to a lower-level problem.

2) Lower level problem: The lower level problem repre-

sents a market clearing procedure involving a set of security

constraints. Such constraints are modeled as proposed in [8].

Objective function:

Max SW 0 + SW c =∑j∈ΩD

∑h∈Ωj

λDjhd0jh −

∑i∈ΩG

∑b∈Ωi

λGibg0ib −

∑j∈ΩD

cUj r0

j

+∑

c∈ΩLc

[ ∑j∈ΩD

∑h∈Ωj

λDjhdcjh −

∑i∈ΩG

∑b∈Ωi

λGibgcib

−∑

j∈ΩD

cUj rc

j

](5)

In this case, the objective function (5), to be maximized,

is the sum of the declared social welfare for both, under

428 2010 IEEE/PES Transmission and Distribution Conference and Exposition: Latin America

3

contingency and normal operating conditions. This lower-

level problem is constrained by (6) - (36). Note that dual

variables are provided after the corresponding equalities

or inequalities separated by a colon.

Constraints:1) Constraints in normal operation:

∑i∈ΨG

s

∑b∈Ωi

g0ib −

∑k|o(k)=s

f0k +

∑k|r(k)=s

f0k

+∑

j∈ΨDs

r0j =

∑j∈ΨD

s

∑h∈Ωj

d0jh : λ0

s ∀s (6)

f0k = bk(nk + nbase

k )(θ0o(k) − θ0

r(k)) : φ0k ∀k (7)

f0k ≥ −(nk + nbase

k )fk : φ0

k∀k (8)

f0k ≤ (nk + nbase

k )fk : φ0k ∀k (9)

g0ib ≤ gib : ϕ0

ib ∀i,∀b (10)

d0jh ≤ ¯djh : β0

jh ∀j, ∀h (11)∑h∈Ωj

d0jh ≥ dj : ρ0

j ∀j (12)

r0j ≤ dj : α0

j ∀j (13)

θ0s ≥ −π : ξ0

s∀s ∈ ΩN\s : slack (14)

θ0s ≤ π : ξ0

s ∀s ∈ ΩN\s : slack (15)

θ0s = 0 : χ0

s s : slack (16)

g0ib ≥ 0 ∀i,∀b (17)

d0jh ≥ 0 ∀j,∀h (18)

r0j ≥ 0 ∀j (19)

Constraints (6) enforce the power balance in each bus.

Constraints (7) represent the power flow in each right-

of-way. Constraints (8) and (9) enforce limits in the

power flow. Constraint (10) and (11) determine the size

of the blocks of the generating units and demands,

respectively. The minimum demand consumption is

determined by (12) while (13) imposes the maximum

limit on load shedding. Constraints (14) and (15) impose

limits on the voltage angles at every bus. Finally,

Constraints (17), (18) and (19) state that the blocks

of generation and demand, and the load shedding are

positive, respectively.

In the same way, the set of security constraints are

imposed as follows:

2) Security constraints:{ ∑i∈ΨG

s

∑b∈Ωi

gcib −

∑k|o(k)=s

f ck +

∑k|r(k)=s

f ck

+∑

j∈ΨDs

rcj =

∑j∈ΨD

s

∑h∈Ωj

dcjh : λc

s ∀s (20)

f ck = bk(nk + nbase

k )(θco(k) − θc

r(k))

: φckc ∀k �= c (21)

f ck = bk(nk + nbase

k − 1)(θco(k) − θc

r(k))

: μckc for k = c and (nk + nbase

k − 1) ≥ 0 (22)

f ck ≥ −(nk + nbase

k )fk : φc

kc∀k �= c (23)

f ck ≤ (nk + nbase

k )fk : φckc ∀k �= c (24)

f ck ≥ −(nk + nbase

k − 1)fk

: μckc

for k = c and (nk + nbasek − 1) ≥ 0 (25)

f ck ≤ (nk + nbase

k − 1)fk

: μckc for k = c and (nk + nbase

k − 1) ≥ 0 (26)

gcib ≤ gib : ϕc

ib ∀i,∀b (27)

dcjh ≤ djh : βc

jh ∀j,∀h (28)∑h∈Ωj

dcjh ≥ dj : ρc

j ∀j (29)

rcj ≤ dj : αc

j ∀j (30)

θcs ≥ −π : ξc

s∀s ∈ ΩN\s : slack (31)

θcs ≤ π : ξc

s ∀s ∈ ΩN\s : slack (32)

θcs = 0 : χc

s s : slack (33)

gcib ≥ 0 ∀i,∀b (34)

dcjh ≥ 0 ∀j, ∀h (35)

rcj ≥ 0 ∀j

}∀c ∈ ΩLc (36)

The above set of constraints is similar to (6)-(19).

However, note that some constraints are added with the

objective of representing the outage of a circuit in the

system. Also note that, the size of this problem depends

on the number of single contingencies selected.

B. Transforming the bi-level model into a single-level opti-mization problem

Observe that for a given set of decision variables (nk) from

the upper-level problem, the problem given by (5) - (36) is

continuous and convex. Therefore, it can be transformed into

a set constraints which corresponds to the primal constraints,

the constraints of its dual problem and the strong duality

condition, [9]. In this way, the bi-level problem (1) - (36)

can be transformed into a single-level optimization problem,

substituting the lower-level problem by the aforementioned

set of constraints and incorporating it into the upper-level

problem.

1) Dual problem corresponding to the lower level problem:The dual problem associated to the lower-level problem is:

Max∑

k∈ΩL

(φ0k − φ0

k)(nk + nbase

k )fk +∑

i∈ΩG

∑b∈Ωi

ϕ0ibgib

+∑

j∈ΩD

∑h∈Ωj

β0jhdjh +

∑j∈ΩD

(α0j dj + ρ0

jdj) +∑

s∈ΩN

(ξ0s − ξ0

s)π

+∑

c∈ΩLc

{ ∑k∈ΩL

[∑k �=c

(φckc − φc

kc)(nk + nbase

k )fk

+∑k=c

(μckc − μc

kc)(nk + nbase

k − 1)fk

]+

∑i∈ΩG

∑b∈Ωi

ϕcibgib

+∑

j∈ΩD

[( ∑h∈Ωj

βcjhdjh

)+ αc

j dj + ρcjdj

]+

∑s∈ΩN

(ξcs − ξc

s)π

}

(37)

GARCES et al.: MARKET-DRIVEN SECURITY-CONSTRAINED 429

4

subject to:

λ0s(i) + ϕ0

ib ≥ −λGib ∀i,∀b (38)

− λ0s(j) + β0

jh + ρ0j ≥ λDjh ∀j,∀h (39)

λ0s(j) + α0

j ≥ −cUj ∀j (40)

− λ0o(k) + λ0

r(k) + φ0k + φ0

k + φ0

k= 0 ∀k (41)

−∑

k|o(k)=s

bk(nk + nbasek )φ0

k +∑

k|r(k)=s

bk(nk + nbasek )φ0

k

+ ξ0s + ξ0

s= 0 ∀s ∈ ΩN\s : slack (42)

−∑

k|o(k)=s

bk(nk + nbasek )φ0

k +∑

k|r(k)=s

bk(nk + nbasek )φ0

k

+ χ0s = 0 s : slack (43)

λcs(i) + ϕc

ib ≥ −λGib ∀i,∀b, ∀c (44)

− λcs(j) + βc

jh + ρcj ≥ λDjh ∀j,∀h, ∀c (45)

λcs(j) + αc

j ≥ −cUj ∀j, ∀c (46)

− λco(k) + λc

r(k) + φckc + φc

kc + φc

kc= 0 ∀k,∀c |k �=c (47)

− λco(k) + λc

r(k) + μckc + μc

kc + μckc

= 0 ∀k, ∀c |k=c (48)

−∑

k|o(k)=s and k �=c

bk(nk + nbasek )φc

kc

+∑

k|r(k)=s and k �=c

bk(nk + nbasek )φc

kc

−∑

k|o(k)=s and k=c

bk(nk + nbasek − 1)μc

kc

+∑

k|r(k)=s and k=c

bk(nk + nbasek − 1)μc

kc

+ ξcs + ξc

s= 0 ∀s ∈ ΩN\s : slack,∀c (49)

−∑

k|o(k)=s and k �=c

bk(nk + nbasek )φc

kc

+∑

k|r(k)=s and k �=c

bk(nk + nbasek )φc

kc

−∑

k|o(k)=s and k=c

bk(nk + nbasek − 1)μc

kc

+∑

k|r(k)=s and k=c

bk(nk + nbasek − 1)μc

kc

+ χcs = 0 s : slack,∀c (50)

ϕ0ib ≥ 0 ∀i,∀b (51)

β0jh ≥ 0 ∀j,∀h (52)

α0j ≥ 0, ρ0

j ≤ 0 ∀j (53)

φ0k free, φ0

k ≥ 0, φ0

k≤ 0 ∀k (54)

λ0s free, ξ0

s ≥ 0, ξ0

s≤ 0, χ0

s free ∀s (55)

ϕcib ≥ 0 ∀i,∀b,∀c (56)

βcjh ≥ 0 ∀j,∀h, ∀c (57)

αcj ≥ 0, ρc

j ≤ 0 ∀j,∀c (58)

φckc free, φc

kc ≥ 0, φc

kc≤ 0 ∀k,∀c (59)

μckc free, μc

kc ≥ 0, μckc

≤ 0 ∀k,∀c (60)

λcs free, ξc

s ≥ 0, ξc

s≤ 0, χc

s free ∀s,∀c (61)

2) Strong duality condition: The strong duality condition

states that a primal feasible solution and a dual feasible

solution are optimal solutions of the primal and dual problem,

respectively, if and only if the values of the objective functions

of both problems are equal. That is, (5) = (37). Mathematically,

∑j∈ΩD

∑h∈Ωj

λDjhd0jh −

∑i∈ΩG

∑b∈Ωi

λGibg0ib −

∑j∈ΩD

cUj r0

j

+∑

c∈ΩLc

[ ∑j∈ΩD

∑h∈Ωj

λDjhdcjh −

∑i∈ΩG

∑b∈Ωi

λGibgcib

−∑

j∈ΩD

cUj rc

j

]=

∑k∈ΩL

(φ0k − φ0

k)(nk + nbase

k )fk

+∑

i∈ΩG

∑b∈Ωi

ϕ0ibgib +

∑j∈ΩD

∑h∈Ωj

β0jhdjh

+∑

j∈ΩD

(α0j dj + ρ0

jdj) +∑

s∈ΩN

(ξ0s − ξ0

s)π

+∑

c∈ΩLc

{ ∑k∈ΩL

[∑k �=c

(φckc − φc

kc)(nk + nbase

k )fk

+∑k=c

(μckc − μc

kc)(nk + nbase

k − 1)fk

]+

∑i∈ΩG

∑b∈Ωi

ϕcibgib

+∑

j∈ΩD

[( ∑h∈Ωj

βcjhdjh

)+ αc

j dj + ρcjdj

]+

∑s∈ΩN

(ξcs − ξc

s)π

}

(62)

Finally, the equivalent single-level problem is given by:

Max (1) : Upper-level objective function (63)

subject to:

(2) − (3), : Upper-level constraints (64)

(6) − (19), : Lower-level primal constraints (65)

(20) − (36), : Lower-level dual constraints (66)

(62), : Strong duality condition (67)

The above formulation corresponds to a mixed-integer nonlin-

ear programming problem which can be linearized using some

concepts for the transformation of integer variables into binary

variables and linearization schemes, [5], [9], [10], [11].

III. METHODOLOGY

To address the TNEP within a market environment, con-

sidering security constraints, it is necessary to first create a

list of contingencies to be evaluated in the above BP model.

Therefore, the proposed methodology considers the following

two steps:

1) First Step: This step consists in solving the BP model

without security constraints, as proposed in [5]. In this

case the scenario of maximum demand is considered.

From the optimal solution, the power flows in all circuits

(prospectives and existing) are obtained. The list of

contingencies is created with the circuits that present the

higher values of loading (close to its maximum capacity).

Note that, in order to avoid unnecessary duplication of

430 2010 IEEE/PES Transmission and Distribution Conference and Exposition: Latin America

5

constraints, there should be no more than one circuit per

right-of-way in the contingency list.

2) Second Step: Fixing the investment decision nk found

in the first step, the next step consists in solving the

proposed BP model (1) - (36) considering the pre-defined

contingency list.

The optimal solution is found adding the investment plans

proposed in the first and second steps. The resulting investment

plan is such that: a) it maximizes social welfare, facilitating

energy trading, and b) it allows the system to operate ade-

quately not only under normal conditions, but also, after any

of the pre-defined contingencies takes place.

IV. TEST AND RESULTS

A. Garver’s 6-bus Test System

The proposed methodology is analyzed using the Garver’s

6-bus system depicted in Figure 1. The data used in the

simulations can be found in [1] and [5].

bus 4

bus 2

bus 5

bus 3

bus 6

80MW

160MW

360MW

40MW

240MW

600MW

240MW

bus 1

150MW

Figure 1. Garver’s 6-bus test system.

1) First step: Solution of the bi-level TNEP without securityconstraints and selection of single contingencies: Initially, the

model proposed in [5] is solved considering only the scenario

of maximum demand of the system. The optimal solution

obtained for this system is presented in Table I.

TABLE ISOLUTION OF THE BI-LEVEL TNEP WITHOUT SECURITY CONSTRAINTS

Investment plan 2-3 (1), 3-5 (1), 4-6 (2)

Investment cost 19.308 Me

Average Social Welfare 276.56 Me

CPU time 0.83 seconds

The power flows through the right-of-ways (considering the

investment plan obtained in this step) are presented in Table

II.

For illustrative purposes, the list of pre-defined contin-

gencies will be created with those circuits pertaining to the

right-of-ways with a loading percentage greater than 50%.

TABLE IIPOWER FLOWS THROUGH THE RIGHT-OF-WAYS IN THE INVESTMENT PLAN

OBTAINED (FIRST STEP)

Right-of-wayPower flow Capacity of the Loading

(MW) Right-of-way (MW) Percentage

1-2 24 100 24%

1-4 -10 80 12.5%

1-5 56 100 56%

2-3 154 200 77%

2-4 -38 100 38%

3-5 170 200 85%

4-6 -200 200 100%

Therefore, the security constraints of the proposed model

consider the single outage of a circuit in the right-of-ways

1 − 5, 2 − 3, 3 − 5, and 4 − 6.

2) Second step: Solution of the bi-level TNEP with securityconstraints: Considering the contingency list defined in the

previous step, the proposed BP model with security constraints

is simulated. The number of circuits in each right-of-way is

limited to 3. The optimal solution yields the building of one

circuit in the right-of-way 1 − 5 and 4 − 6 (investment cost

of 9.654 Me). With this solution the system has no load

shedding and the average social welfare under both, normal

and contingency operation, are the maximum possible. The

computing time used in the simulation of this system was

0.813 seconds.

The final solution corresponds to the sum of the expansion

plans found in the first and second step. In this case, the

proposed methodology suggests the building of lines in the

right-of-way 1 − 5 (one circuit), 2 − 3 (one circuit), 3 − 5(one circuit), 4 − 6 (three circuits) corresponding to a total

investment cost of 28.962 Me. This solution proposes the

building of new circuits and reinforcements in the network,

interconnecting load and generation areas. It was found that a

higher reinforcement is built in the interconnection of node 6

(node with the highest generation) with the rest of the system.

All simulations have been solved using CPLEX 8.1 under

GAMS [12] on a Dell Optiplex computer with a processor

clocking at 3 GHZ and 3 GB of RAM.

B. IEEE 24-bus RTS test system

The IEEE 24-bus RTS test system depicted in Figure 2

is also simulated. Line lengths, resistances, reactances and

capacities of lines as well as the generation and demand data

can be found in [13], [14] and [5].

1) First step: Solution of the bi-level TNEP without securityconstraints and selection of single contingencies: The model

proposed in [5] considering the maximum demand scenario is

solved. The optimal solution corresponds to that presented in

Table III.

Power flows through the right-of-ways are obtained with

the aim of identifying the contingencies to be considered in

the second step. In this case, the circuits that have a loading

percentage greater than 80% are considered. Thus, the selected

single contingencies in the system correspond to one circuit

in the right-of-ways: 1 − 5, 3 − 24, 6 − 10, 7 − 8, 9 − 12,

GARCES et al.: MARKET-DRIVEN SECURITY-CONSTRAINED 431

6

10

1821

22

17

23

19 2016

14

24

13

11

3 9

6

85

4

721

12

15

138 kV

230 kV

Synch.Cond.

Figure 2. IEEE 24-bus RTS test system.

TABLE IIISOLUTION OF THE BI-LEVEL TNEP WITHOUT SECURITY CONSTRAINTS

Investment plan 7-8 (1), 6-7 (1), 13-14 (1)

Investment cost 9.022 Me

Average Social Welfare 669.510 Me

CPU time 8.25 seconds

10− 11, 10− 12, 11− 13, 14− 16, 15− 21, 15− 24, 16− 17,

and 17 − 18.2) Second step: Solution to the bi-level TNEP with security

constraints: In this step, the proposed model is simulated

considering the investment plan and contingency list obtained

in the previous step. The number of circuits in each right-of-

way is limited to 5. The optimal solution found is presented

in Table IV.

TABLE IVSOLUTION OF THE BI-LEVEL TNEP WITH SECURITY CONSTRAINTS

Investment plan

1-2 (2), 1-5 (2), 3-9 (1),

6-10 (1), 7-8 (2)

10-12 (1), 2-8 (2)

Investment cost 28.803 Me

CPU time 14.1720 seconds

As with the previous system, the final solution consists in

the addition of the investment plans obtained in the first and

second step. That is: to build two circuits in the right-of-way

1−2, two circuits in 1−5, two circuits in 2−8, one circuit in

3− 9, one circuit in 6− 7, one circuit in 6− 10, three circuits

in 7−8, one circuit in 10−12, and one circuit in 13−14. This

investment plan has associated a total cost of 37.825 Me. Note

that most of the reinforcements proposed in the network are

located in the south of the system, which corresponds to the

higher consumption area. Such reinforcements are associated

with nodes 1, 2 and 7, avoiding a high penalization cost

due to load shedding. This expansion plan presents no load

shedding when considered both, normal operation and the

selected contingencies. Also, the average social welfare of

the market obtained with this investment plan is as high as

possible.

All simulations have been solved using CPLEX 8.1 under

GAMS [12] on a Dell Optiplex computer with a processor

clocking at 3 GHZ and 3 GB of RAM.

V. CONCLUSIONS

This paper provides a market-driven bi-level programming

approach to solve the TNEP problem considering security

constraints. In this approach the planner is represented as

the upper-level problem, which procures the minimization of

the total investment cost and load shedding cost in both,

normal and contingency conditions. The lower-level problem

corresponds to a market clearing procedure adapted to in-

clude a set of selected contingencies. It was found that the

investment plans obtained using the proposed methodology

mainly reinforce the areas of the system involving high energy

consumption. These solutions present a high social welfare in

both, normal operation and under contingency. An advantage

of the proposed methodology, when compared to others that

also deal with security constraints, relays in a reduced com-

putational burden, since not all of the possible contingencies

are considered. This makes sense, because not all outages

might lead to load shedding. Consequently, the planner can

take advantage of his experience and knowledge of the system

to select a set of possible contingencies. On the other hand,

there is a trade-off between the number of contingencies

selected and the computational time. As expected, the more

contingencies are selected, the more computational time is

spent to find the solution, however, this relationship is linear.

VI. ACKNOWLEDGEMENTS

The authors gratefully acknowledge the advice of pro-

fessors Antonio Conejo and Raquel Garcıa-Bertrand of the

Universidad de Castilla La Mancha, Spain, in the development

of the first stage of this work. We also acknowledge the

support of Fundacao de Ensino, Pesquisa e Extensao de

Ilha Solteira (FEPISA), Sao Paulo, Brazil; and Antioquia

University, Medellın, Colombia.

REFERENCES

[1] L. L. Garver, “Transmission network estimation using linear program-ming,” IEEE Trans. Power Apparat. Syst., vol. 89, no. 7, pp. 1688-1697,Sep. 1970.

[2] R. Villasana, L. L. Garver and S. J. Salon, “Transmission networkplanning using linear programming,” IEEE Trans. Power Apparat. Syst.,vol. 104, no. 2, pp. 349-356, Feb. 1985.

[3] N. Alguacil, A. L. Motto and A. J. Conejo, “Transmission expansionplanning: a mixed-integer LP approach,” IEEE Trans. Power Syst.,vol. 18, no. 3, pp. 1070-1077, Aug. 2003.

[4] S. de la Torre, A. J. Conejo and J. Contreras, “Transmission expansionplanning in electricity markets,” IEEE Trans. Power Syst., vol. 23, no. 1,pp. 238-248, Feb. 2008.

432 2010 IEEE/PES Transmission and Distribution Conference and Exposition: Latin America

7

[5] L. P. Garces, A. J. Conejo, R. Garcıa-Bertrand and R. Romero, “Abilevel approach to transmission expansion planning within a marketenvironment” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1513-1522,Aug. 2009.

[6] B. Colson, P. Marcotte and G. Savard, “An overview of bilevel opti-mization,” Ann. Oper. Res., vol. 153, no. 1, pp. 235-256, Sep. 2007.

[7] J. Fortuny-Amat and B. McCarl, “A representation and economic in-terpretation of a two-level programming problem,” J. Oper. Res. Soc.,vol. 32, no. 9, pp. 783-792, Sep. 1981.

[8] I. de J. Silva, M. J. Rider, R. Romero, A. V. Garcia and C. A. Murari,“Transmission network expansion planning with security constraints,”IEE Proc. Gener. Transm. Distrib., vol. 152, no. 6, pp. 828-836,Nov. 2005.

[9] A. L. Motto, J. M. Arroyo and F. D. Galiana, “A mixed-integer LP pro-cedure for the analysis of electric grid security under disruptive threat,”IEEE Trans. Power Syst., vol. 20, no. 3, pp. 1357-1365, Aug. 2005.

[10] G. C. Oliveira, S. Binato and M. V. F. Pereira, “Value-based transmissionexpansion planning of hydrothermal systems under uncertainty,” IEEETrans. Power Syst., vol. 22, no. 4, pp. 1429-1435, Nov. 2007.

[11] L. A. Barroso, R. D. Carneiro, S. Granville, M. V. Pereira, andM. H. C. Fampa, “Nash equilibrium in strategic bidding: a binaryexpansion approach,” IEEE Trans. Power Syst., vol. 21, no. 1, pp. 629-638, May. 2006.

[12] R. E. Rosenthal, GAMS, a user’s guide. Washington: GAMS Develop-ment Corporation, 2008.

[13] Reliability Test System Task Force, “The IEEE Reliability Test System- 1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010-1020,Aug. 1999.

[14] R. Fang and D. J. Hill, “A new strategy for transmission expansion incompetitive electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 1,pp. 374-380, Feb. 2003.

Lina P. Garces (S’07) received the BScEE andMScEE degree from Universidad Tecnologica dePereira, Pereira, Colombia, in 2003 and 2005, re-spectively, and the Ph.D degree from the Universi-dade Estadual Paulista Julio de Mesquita Filho, IlhaSolteira, Brazil, in 2010. Currently, she is a Postdoc-toral Research Assistant at department of ElectricalEngineering at Universidade Estadual Paulista Juliode Mesquita Filho, Ilha Solteira, Brazil.

Her research interests include transmission expan-sion planning, operation and control of electrical

power systems, as well as optimization techniques.

Ruben Romero (SM’08) received the B.Sc. and P.E.degrees in 1978 and 1984, respectively, from theNational University of Engineering, Lima, Peru, andthe M.Sc. and Ph.D. degrees from the UniversidadeEstadual de Campinas, Campinas, Brazil, in 1990and 1993, respectively. Currently, he is a Professorof Electrical Engineering at Universidade Estad-ual Paulista Julio de Mesquita Filho, Ilha Solteira,Brazil.

His research interests include methodologies forthe optimization, planning and control of electrical

power systems, applications of artificial intelligence in power system, as wellas operations research.

Jesus Marıa Lopez-Lezama (S’07) received hisB.Sc. and M.Sc. degrees from the National Univer-sity of Colombia in 2001 and 2006 respectively. Heis a Professor of the Department of Electrical Engi-neering at the University of Antioquia in MedellınColombia. Currently he is pursuing his Ph.D. degreeat the Universidade Estadual Paulista (UNESP), SP,Brazil.

His major research interests are planning andoperation of electrical power systems and distributedgeneration.

GARCES et al.: MARKET-DRIVEN SECURITY-CONSTRAINED 433