Efficient method for AC transmission network expansion planning

Electric Power Systems Research 80 (2010) 1056–1064

Contents lists available at ScienceDirect

Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Efficient method for AC transmission network expansion planning M. Rahmani a,b,∗ , M. Rashidinejad a , E.M. Carreno c , R. Romero b a

Electrical Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran Faculdade de Engenharia de Ilha Solteira, UNESP - Univ Estadual Paulista, Departamento de Engenharia Elétrica, Ilha Solteira, SP, Brazil c Centro de Engenharia, Universidade Estadual do Oeste de Parana, UNIOESTE, Foz do Iguac¸u - PR – Brazil b

a r t i c l e

i n f o

Article history: Received 18 January 2009 Received in revised form 7 December 2009 Accepted 18 January 2010 Available online 10 February 2010 Keywords: Transmission expansion Real genetic algorithm Reactive power planning Power system planning VAr-plant

a b s t r a c t A combinatorial mathematical model in tandem with a metaheuristic technique for solving transmission network expansion planning (TNEP) using an AC model associated with reactive power planning (RPP) is presented in this paper. AC-TNEP is handled through a prior DC model while additional lines as well as VAr-plants are used as reinforcements to cope with real network requirements. The solution of the reinforcement stage can be obtained by assuming all reactive demands are supplied locally to achieve a solution for AC-TNEP and by neglecting the local reactive sources, a reactive power planning (RPP) will be managed to find the minimum required reactive power sources. Binary GA as well as a real genetic algorithm (RGA) are employed as metaheuristic optimization techniques for solving this combinatorial TNEP as well as the RPP problem. High quality results related with lower investment costs through case studies on test systems show the usefulness of the proposal when working directly with the AC model in transmission network expansion planning, instead of relaxed models. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Transmission network expansion planning (TNEP) is a crucial issue especially in restructured power systems. In the new environment of the electricity industry, open access to transmission networks introduces some new challenges to all market participants. As electricity consumption grows rapidly, additional transmission lines are required to facilitate alternative paths for power transfer from power plants to load centers. TNEP via simplified models such as the transportation model, hybrid model, linear disjunctive model, and DC model [1], among others, will usually fail to support a solution that can handle real network requirements. A complete and comprehensive model including major aspects of real networks, an AC model, is employed in this research. In fact, transmission planning using an AC model should be associated with reactive power planning (RPP). Without considering reactive sources or VAr-plants, the AC-TNEP problem may have an optimum solution in which only the generators satisfy reactive load demands. Although in case of generator capability of supporting reactive load demands, transferring such an amount of reactive power may reduce the available transfer capability (ATC) that lead to more new transmission lines. While by allocating VAr-plants close to the

∗ Corresponding author at: Faculdade de Engenharia de Ilha Solteira, UNESP - Univ Estadual Paulista, Departamento de Engenharia Elétrica, Av Brasil 56, CEP 15385-000 Ilha Solteira, SP, Brazil. E-mail address: [email protected] (M. Rahmani). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.01.012

load centers, this may be possible to prevent the necessity of additional transmission lines. On the other hand, without considering the VAr-plants, load bus voltages may differ from their specified magnitudes, which may not only cause unacceptable power quality but also increase real power losses. Increasing power losses may require more transmission line additions. That is why we propose a concurrent planning approach for the TNEP and RPP problem as a combined methodology, the so-called TNERPP. Besides increasing the capacity of transmission lines and power loss reduction due to inclusion of RPP in TNEP, voltage profile enhancement and voltage stability margin improvement can be attained. Another important research topic is the integrated planning of generation sources and network expansion, but this topic is outside the scope of this work. The objective function of TNERPP is to determine where, how many and when new equipment such as transmission lines and reactive power sources must be added to the network in order to make its operation viable for a pre-defined planning horizon at minimum cost. In fact, such a combinatorial problem seems so complicated that it needs to be solved via mixed integer nonlinear programming. In the existing TNEP approaches, the transmission network planning problem is first handled using both classical optimization techniques [2–6] and evolutionary algorithms [7–14]. Subsequently, the expanded network is reinforced considering the reactive power source allocation [15–19]. It should be noted that there are a few reports in existing planning methods discussing both stages in an integrated way. In [20] the authors use a constructive heuristics algorithm to solve AC-TNEP; in their study, when transmission lines are constructed, reactive power sources

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

are allocated to weak buses. Even with the fast convergence of their approach, the obtained solution may not be optimal. Solving the TNERPP problem considering the AC model in one stage is too difficult; particularly in transmission network expansion where the initial topology has isolated areas, the solution might not be easily achieved. When there are a large number of isolated areas, convergence problems appear in the solution of the nonlinear subproblems, as do convergence problems for local optimum, and an increase in processing time. It seems that a better way to tackle such a problem is to start with a DC model (DC-TNEP) which presents an initial system that is totally connected and then include candidate lines to try to find a new solution for transmission lines as well as reactive sources via the AC model. In this research the TNERPP problem is managed in three phases: in the first phase, a DC-TNEP is solved to introduce initial solution; in the second phase, by assuming all reactive demands are procured locally, new lines will be added through an AC-TNEP to support power losses. Finally, in the third phase, by removing local reactive sources in the second phase, reactive power planning is handled to find the location and minimum capacities of the VAr-plants needed for feasible system operation. The real genetic algorithm (RGA) is employed to solve the TNERPP problem in the first phase as well as in the second phase. Binary GA is applied for VAr-plant placement in the third phase, while the capacities of VAr-plants are obtained through an OPF. In general, the genetic algorithm (GA) technique seems suitable for multi-objective optimization problems, resulting in good solutions while maintaining low computational effort [21,22]. This paper is organized as follows: in Section 2 mathematical modeling of TNERPP is presented, in Section 3 a solution algorithm is introduced, while in Section 4 the implementation of the proposed methodology is described. In Section 5 a modified Garver system is employed for illustrative examples associated with further analysis through the implementation of the proposed approach. Section 6 considers a modified IEEE 24-bus system and a south Brazilian system that is used as a complementary case study, and concluding remarks are presented in Section 7.

min v1 = f (q, u)

P(V , , n) − P G + P D = 0

(3)

Q (V , , n) − Q G + Q D − q = 0

(4)

¯ P - G ≤ PG ≤ PG Q G ≤ Q G ≤ Q¯ G q ≤ q ≤ q¯ ¯ V - ≤V ≤V

(5)

0

from

0

to

(N + N )S (N + N )S

(6) (7) (8) 0

(9)

≤ (N + N )S¯

(10)

≤ (N + N )S¯ 0

0 ≤ n ≤ n¯

(11)

The first objective function (1) considers only the expansion costs of transmission lines, while the second objective function (2) considers the minimum costs of the VAr-plants that will be installed. The limits for real and reactive power for generators are expressed by Eqs. (5) and (6), respectively; for VAr-plants and voltage magnitudes by Eqs. (7) and (8), respectively. Capacity limits (MVA) of the line flows are presented by Eqs. (9) and (10), while new added circuit constraints are shown in Eq. (11). The cost of VAr-plants can be defined as follows: f (q, u) =



(c0k + c1k qk )uk

(12)

k ∈ ˝l

where k ∈ ˝l represents the kth load bus, ˝l is the set of all load buses, and c0k and c1k are the installation costs and unit costs for a VAr-plant at bus k, respectively. qk is the MVAr size of a VAr-plant installed at bus k, and uk is a binary variable that indicates whether or not to install a reactive power source at bus k. Eqs. (3) and (4) represent the conventional equations of AC power flow considering n, the number of circuits (lines and transformers), and q, the size of VAr-plants treated as variables. The elements of vectors P(V, , n) and Q(V, , n) are calculated by Eqs. (13) and (14), respectively:



Vj [Gij (n) cos ij + Bij (n) sin ij ]

(13)

j ∈ NB

A mathematical model for the TNERPP problem can be formulated via Eqs. (1)–(11), where c and n represent the circuit costs vector and the added lines vector, respectively. N and N0 are diagonal matrices containing vector n and the existing circuits in the base configuration, respectively. f(q,u) is the cost function of VAr-plants; q is the MVAr size of the VAr-plants vector. u is a binary vector that indicates whether or not to install reactive power sources at load buses. v0 is the investment due to the addition of new circuits to the network, and v1 is the cost of VAr-plants. n¯ is a vector containing the maximum number of circuits that can be added.  is the phase angles vector, while PG and QG are the existing real and reactive power generation vectors. PD and QD are real and reactive power demand vectors; V is the voltage magnitudes vector; P¯ G , Q¯ G and V¯ are the vectors of maximum real and reactive power generation , and V limits and voltage magnitudes, respectively; P - are the - G, Q -G vectors of minimum real and reactive power generation limits and voltage magnitudes, respectively. In this paper, 105% and 95% of the nominal value are used for the maximum and minimum voltage magnitude limits, respectively. Sfrom , Sto , and S are the apparent power flow vectors (MVA) through the branches in both terminals and their limits: min v0 = c T n

s.t.

Pi (V, , n) = Vi

2. TNERPP mathematical modeling

1057

Qi (V, , n) = Vi



Vj [Gij (n) sin ij − Bij (n) cos ij ]

(14)

j ∈ NB

where i, j ∈ NB represent ith and jth buses and NB is the set of all buses, and ij represents the circuit between buses i and j.  ij =  i −  j represents the phase angle difference between buses i and j. The elements of bus admittance matrix (G and B) are

⎧ ⎫ 0 0 ⎪ ⎨ Gij (n) = −(nij gij + nij gij ) ⎪ ⎬  G= 0 0 (nij gij + nij gij ) ⎪ G (n) = ⎪ ⎩ ii ⎭

(15)

j ∈ ˝l

⎧ ⎫ 0 0 ⎪ ⎪ ⎨ Bij (n) = −(nij bij + nij bij ) ⎬  B= sh 0 0 sh 0 sh B (n) = bi + [nij (bij + bij ) + nij (bij + (bij ) )] ⎪ ⎪ ⎩ ii ⎭

(16)

j ∈ ˝l

(1)

where gij , bij and bsh are the conductance, susceptance, and shunt ij susceptance of the transmission line or transformer ij (if ij is a trans= 0), respectively, and bsh is the shunt susceptance at bus former bsh ij i i, while the proposed model does not consider the phase shifters. Elements (ij) of vectors Sfrom and Sto of (9) and (10) are given by the following relationship:

(2)

Sijfrom =



2

(Pijfrom ) + (Qijfrom )

2

1058

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

and decoding, and it is faster and more accurate than binary GA. The proposed RGA approach includes problem codification, selection, recombination, mutation, and fitness evaluation, which are explained briefly in the following sections. Obviously, other metaheuristics can be used to solve the mathematical model and search strategy of the proposed solution.

Fig. 1. Codification proposal.

Pijfrom = Vi2 gij − Vi Vj (gij cos ij + bij sin ij )

3.1. Problem codification

+ bij ) − Vi Vj (gij sin ij − bij cos ij ) Qijfrom = −Vi2 (bsh ij

Sijto =

2

(Pijto ) + (Qijto )

2

Pijto = Vj2 gij − Vi Vj (gij cos ij − bij sin ij ) + bij ) + Vi Vj (gij sin ij + bij cos ij ) Qijto = −Vj2 (bsh ij Integer variable n, the number of circuits added in branch ij, and the binary variable u, the connection or disconnection of VAr-plants to a load bus, are the most important decision variables where any feasible operation solution of power systems depends on their values. The remaining variables only represent the operating state of a feasible solution in which a feasible investment proposal, defined through specified values of n and u, can include several feasible operation states.

One of the most important factors in representing a candidate solution in the proposed approach is codification, since proper codification may prevent complexity in the implementation of RGA. An individual is defined as a proposed solution to the planning problem, or, better put, the topology made up of all transmission lines added to the system corresponding to an investment proposal. In TNEP, the individual in the RGA is represented by a vector. Each vector consists of the number of new lines that are proposed to be added to respective branches, where each member of this vector can vary from zero to the maximum number of lines. Thus, in the codification shown in Fig. 1, branch 1–4 has three new lines; branch 2–3 has one new line, etc. The proposed approach in this paper does not demand similar characteristics of transmission lines between two buses, and it can work with various types of circuits. In this case, the lines selected are similar to the previous form of existing line. The number of individuals in the RGA population, depends on the dimension of the system.

3. Solution algorithm 3.2. Selection Mathematically, GA can be considered as a technique for optimizing the combination of large scale and complex problems with a high probability of finding the global optimum among many local optimal solutions. In this paper, two types of GA are used for different planning stages. An ordinary (binary) GA is used for VAr-plant allocation: by employing a fitness function, it seeks the most attractive load buses for minimum reactive allocation, while a real genetic algorithm (RGA) is used for AC-TNEP as well as DC-TNEP. Since binary GA is a well-known algorithm, readers are referred to related papers [23,24]; here only fitness evaluation will be described. In the following section, RGA with particular characteristics is described. RGA does not require any binary coding

Chromosome selection is a random process in which individual chromosomes are mapped to the adjacent segments of a line and the length of each segment on the line corresponds to the levels of fitness (i.e. fitness values) of each individual. As part of the trial process, a random number is generated and the individual chromosome position on the line that corresponds to a random number will be selected. This technique is analogous to a roulette wheel where each slice is proportional in size to the fitness value. For chromosome selection, a ranking process is applied where the population is sorted in accordance to its corresponding fitness values. The fitness level assigned to each individual chromosome depends only on its

Fig. 2. Different schemes of recombination.

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

1059

ranking position and not on the actual fitness value. It is assumed that the number of individuals in the population is N, while P is the position of each individual in the population. Eq. (17) can be used to calculate the rank of each individual: Rank(P) = 1 − SP +

2(SP − 1)(P − 1) N−1

(17)

Minimum and maximum values of P are 1 and N, respectively, while SP is a random number between 1 and 2 [25]. Fig. 3. A typical membership function.

3.3. Recombination In recombination, a next generation is created, which in RGA is different from ordinary GA. There are three kinds of recombination [25] adopted in RGA, designated by Eqs. (18)–(20): O1 = P1 + (1 − )P2 O2 = P2 + (1 − )P1

 ∈ {0, 1}

O1 = 1 P1 + (1 − 1 )P2 O2 = 2 P2 + (1 − 2 )P1

1 , 2 ∈ [−0.25, 1.25]

O1 = P1 + (1 − )P2 O2 = P2 + (1 − )P1

 ∈ [−0.25, 1.25]

(18)

(19)

(20)

where P1 , P2 are the two parents, Q1 , Q2 are their two offspring, and , 1 and 2 are randomly generated numbers. A typical individual chromosome with two genes, in which two parents can merge based on three forms, is shown in Fig. 2a–c. The RGA recombination shown in Fig. 2a generates the offspring that are located on the corners of the hypercube defined by the parents. The line recombination shown in Fig. 2b can generate any offspring from the parents on the specified line, and Fig. 2c shows the intermediate recombination capable of producing any point within the hypercube that is slightly larger than the one defined by the parents. The line recombination is similar to intermediate recombination, except that only one  value is used for all variables [25,26].

To calculate fitness values, fuzzy numbers are adopted due to the linguistic and qualitative behavior of the aforementioned objectives. The fitness of a solution is assessed against the degree of membership of a specific solution to a given membership function [26,27], where higher membership values imply a better solution, as described in Fig. 3. Membership values, which lie between 0 and ¯ is a strictly mono1, can be represented by Eq. (22). Where hi (f (X)) tonically decreasing function that can be either linear or nonlinear,

⎧ ⎨

1 fi (X) = hi (f (X)) ⎩ 0

if if if

⎫

fi (X) < fimin ⎬ min fi ≤ fi (X) ≤ fimax ⎭ fi (X) > fimax

(22)

Depending on the objective function, fitness evaluation can be based upon the following equations: • When using RGA with the investment cost of transmission line as the objective, fitness = w1 1 (v0 ) + w2 (dV ) + 2 (dG ) + 3 (dS )

(23)

• When using binary GA with the investment cost of reactive power sources as the objective,

3.4. Mutation

fitness = w1 1 (v1 ) + w2 (dV ) + 2 (dG ) + 3 (dS )

The reason for the mutation stage is to introduce artificial diversification in the population and to avoid premature convergence to a local optimum. The mutation operator that is used is known as dynamic or non-uniform mutation and has been successfully used in a number of studies [23]. In order to achieve a high degree of precision in the proposed method, dynamic mutation is designated for fine-tuning to provide a degree of control. For example, in mutation if gene Pk is selected from parent P, then there is an equal chance for the resulting gene to be either of the following choices:

where all terms of the objective function are fuzzy numbers, with the appropriate membership function. w1 and w2 are weights that can be selected by the decision maker, such that the following equation can be satisfied:

⎧  ⎫ t c⎪ ⎪ ⎨ Ok = Pk − r(Pk − ak ) 1 − ⎬ T  c ⎪ ⎩ Ok = Pk + r(bk − Pk ) 1 − t ⎪ ⎭

(21)

T

where ak and bk are, respectively, the lower and upper band of Pk , r is a uniform random number between (0,1), t is the number of the current generation, T is maximum number of generations, and c is a parameter determining the degree of non-uniformity. 3.5. Fitness evaluation The fitness function defined in this paper aims to minimize the following objectives: 1. Deviation from constraint limits; 2. Costs of installation of new transmission lines; 3. Costs of installation of VAr-plants.

w1 + w2 = 1

(24)

(25)

v0 is the total cost of new lines, v1 is the cost of VAr-plants, and dV , dG , and dS are the sum of deviations from voltage limits, power generation limits, and branch flow limits, respectively. The range of fitness function is from 0 to 1; when fitness is equal to one, the cost of installation is zero (v0 = 0 or v1 = 0) and the deviation from problem constraints is zero. In other words, all variables are inside their limits (dV = 0; dG = 0; dS = 0). 4. Implementation of the proposed approach The overall implementation procedure for solving a TNERPP problem using the proposed algorithm is shown in Fig. 4, in which the various steps to find the optimum solution are described as follows: I. Read network data and candidate lines and VAr-plants that might be installed on the network. II. Solve the following DC-TNEP, which is a simplified form of the AC model, to find the minimum cost of initial topology for the

1060

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

TNERPP problem:

min

v = cT n

s.t.

Sf + P G − P D = 0 f − b(n0 + n) = 0 |f | ≤ (n0 + n)f¯ ¯ P - G ≤ PG ≤ PG 0 ≤ n ≤ n¯ n is an interger and  is unbounded

where f and f¯ are the vectors of total power flow and the corresponding maximum branch flow, respectively. S is the branch-node incidence transposed matrix of the power system, while the other parameters are described above in Section 2. Although the search area is very large, this kind of problem has been well developed in many studies [1–14], in which it is solved effectively by introducing some heuristics methods and the linearization of the DC model. In the proposed method, n will be replaced by a new topology suggested by RGA in order to convert DC-TNEP to a linear programming problem. Therefore each proposed topology will be treated as an LP problem in order to calculate the fitness function. In fact, after some RGA generation the optimal solution (topology) will be obtained and inserted in the system. After this phase, a completely connected system feasible for the DC model is available, but it could be unfeasible for the AC model. III. Solve the following mathematical problem, a constrained AC power flow, to impose the feasibility operation requirements:

P(V , ) − P G + P D = 0 Q (V , ) − Q G + Q D = 0 ¯ P - G ≤ PG ≤ PG Q G ≤ Q G ≤ Q¯ G ¯ V - ≤V ≤V from ¯ S ≤S S to ≤ S¯  is unbounded

This problem considers active and reactive power balances as well as voltage, generators, and apparent power flow constraints. When this problem is satisfied, it seems there is no need to reinforce the transmission network. Otherwise, a power system that may need extra transmission lines or VAr-plants should move to the next step. This problem is solved only once (see Fig. 4). IV. Solve the following AC-TNEP considering all reactive power demand supplied locally to search for new lines that are necessary for real power loss support:

min

v = cT n

s.t.

P(V , , n) − P G + P D = 0 Q (V , , n) − Q G = 0 ¯ P - G ≤ PG ≤ PG Q G ≤ Q G ≤ Q¯ G ¯ V - ≤V ≤V (N + N 0 )S from ≤ (N + N 0 )S¯ (N + N 0 )S to ≤ (N + N 0 )S¯ 0 ≤ n ≤ n¯ n is an integer and  is unbounded

This problem is a mixed integer nonlinear programming (NLP), but since the search area is quite limited, the solution in this step can be derived after a few repetitions of NLP. As in step II, RGA is employed to handle this NLP problem. In this phase, we identify only the reinforcements needed in transmission lines. Independent of the size of the system, the number of reinforcements in this phase is low, so the processing time is low. V. Considering the new transmission lines added in steps II and IV, and the neglected local reactive sources, which were added in the previous section, complete a simple reactive power planning to find the minimum number of VAr-plants that are needed for feasible operation of the power system, and then add the proposed VAr-plants to the network. The problem that is solved in this stage is as follows: min

s.t.

v1 = f (q, u) P(V , ) − P G + P D = 0 Q (V , ) − Q G + Q D − q = 0 ¯ P - G ≤ PG ≤ PG Q G ≤ Q G ≤ Q¯ G q ≤ q ≤ q¯ ¯ V - ≤V ≤V S from ≤ S¯ S to ≤ S¯ u is an integer and  is unbounded

This problem is also well addressed in many papers [15–19], while in our proposed methodology this step is solved through a binary GA. Each topology of VAr-plants allocated by the GA embedded in the above problem and converted to an NLP problem can be solved with an NLP solver. VI. A solution to the problem becomes feasible by including the transmission lines derived in steps II and IV as well as the VArplants in step V, and further reinforcement is not needed. The algorithm will therefore stop at this stage. The previous proposal can be easily extended by selecting a reduced set of quality topologies for the DC model at the end of the phase 1, instead of choosing only the best one. Phases 2 and 3 can thus be implemented for each of those topologies, thereby increasing the possibility of finding the optimal solution in complex systems. 5. Illustrative examples The proposed approach is implemented in the MATLAB platform, while for illustrative examples, a modified Garver 6-bus system is employed. This system has 6 buses and 15 branch candidates, its total demands are 760 MW and 152 MVAr, and it is assumed that a maximum of five lines can be added to each branch. The modified Garver system data is given in [20], and two different base cases have been investigated: Base Case 1: This uses the original topology proposed by Garver. Base Case 2: This does not consider the original topology proposed by Garver. In these examples the VAr-plant fixed costs and VAr-plant variable costs are c0 = 100$ and c1 = 0.3$/k var, respectively. To implement RGA in this study, a random initial population of 50 individuals is used for DC-TNEP; 20 for AC-TNEP. Crossover and mutation rates are 76% and 3%, respectively, and the selection process is terminated when there is no new individual in the population. Table 1 shows the results for the following examples.

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

1061

Fig. 4. Implementation flow diagram.

Example 1. The modified Garver system with Base Case 1. This example is solved in three phases as follows: • DC-TNEP—the solution in this step is obtained using RGA with the following topology: n2–6 = 2; n3–5 = 1 with an investment cost of US$ 80,000. Since the obtained solution may not be feasible for the AC model, the transmission network must be reinforced with additional lines or VAr-plants. The number of LP solved in this step to generate the solution is about 185 through 5 RGA generations. • AC-TNEP—assuming all reactive demands are supplied locally, AC-TNEP is solved using RGA, and n4–6 with an investment cost of US$ 30,000 is added to the network. The number of NLP solved in this step to generate the solution is about 45 through 5 RGA generations. • RPP—a binary GA is used to allocate minimum reactive power sources for the feasible operation of the transmission network. Two reactive sources are added to the system with an investment Table 1 Results for the modified Garver system. Example 1: TNERPP modified Garver—Base Case 1 v0 = US$ 110, 000 v1 = US$ 15, 150 Lines addition in DC-TNEP phase: n3–5 = 1; n2–6 = 2 LPs solved; 185 Lines addition in AC-TNEP phase: n4–6 = 1 NLPs solved; 45 VAR source allocation phase: 30.38 MVAr at bus 4 and 19.44 MVar at bus 5 NLPs solved; 5 Example 2: TNERPP Modified Garver—Base Case 2 v0 = US$ 190, 000 v1 = US$ 9290 Lines addition in DC-TNEP phase: n1–5 = 1; n2–3 = 1; n2–6 = 1; n3–5 = 2; n4–6 = 2 LPS solved; 195 Lines addition in AC-TNEP phase: n2–3 = 1 NLPs solved; 40 VAR source allocation phase: 3.50 MVAr at bus 4 and 26.79 MVAr at bus 5 NLPs solved; 7

of about US$ 15,150. The sizes of the VAr-plants are 30.38 MVAr and 19.44 MVAr at buses 4 and 5, respectively. This solution is obtained after solving 5 NLP by binary GA. Fig. 5 shows the solution results for this example. Example 2. Modified Garver system with Base Case 2: this example is handled assuming that no line is constructed and no VAr-plant is installed, while only generators and loads are specified. As in Example 1, the solution is found in three phases as follows: • DC-TNEP—using RGA the following topology with investment cost of US$ 170,000 is obtained: n1–5 = 1; n2–3 = 1; n2–6 = 1; n3–5 = 2; n4–6 = 2. This solution is not feasible for the AC model since other lines as well as a VAr-plant must be added for feasible operation. This topology is obtained after 195 solutions of LP through 8 generations of RGA. • AC-TNEP—assuming all reactive power demand is supplied locally, RGA is used to find additional lines for feasible operation. At this stage one new line between buses 2 and 3 is added to the network. This solution is obtained after 5 RGA generations solving 40 NLP. • RPP—after executing a binary GA, two VAr-plants are added to the network at buses 4 and 5 with sizes 3.50 MVAr and 26.79 MVAr, respectively. The total cost of the VAr-plants is about US$ 9290. This solution is obtained by solving 7 NLP. Fig. 6 shows the complete results for this example. Comparing these results with the ones reported in [20], the line investment costs in Example 2, Fig. 6, and in Example 1, with a different topology, are equal. In these examples, our aim is to present a simple algorithm to find a high quality solution for the TNERPP problem. It should be considered that CHA algorithms are unable to find high quality solutions for this problem especially for largescale systems, this fact is shown in the case studies section. In these examples and in the next case studies, one may note that how it is possible to solve, for example, only 195 LP after 8 generations of RGA, while starting with 50 initial solutions requires solving 400 LP problems. The reason is that in GA a new generation may consist of previous individuals. Therefore to speed up the process, a pool of all individuals is constructed. To calculate the fitness function for repeated individuals we do not carry out all the calculations but simply pull out the fitness values from the pool. All calculations are carried out for each new individual, and finally their data and fitness values will be inserted into the pool.

1062

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

Fig. 5. Solution for modified Garver system with Base Case 1.

6. Case studies 6.1. IEEE 24-bus An IEEE 24-bus system is chosen with 41 transmission lines. In this case VAr-plant fixed costs and VAr-plant variable costs are c0 = 1000$ and c1 = 3 $/k var, respectively. The IEEE 24-bus system is investigated in the following three steps:

• DC-TNEP—the initial random population size for this system is 100. Three lines, n7–8 = 2; n6–10 = 1, are added through 280 LP by 9 RGA generations. This solution is not feasible in the AC model and may require additional transmission lines or VAr-plants. • In AC-TNEP—assuming all reactive demand is supplied locally, the system has a feasible operation and there is no need for additional lines.

Fig. 6. Solution for modified Garver system with Base Case 2.

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

• In the next step, the minimum size of VAr-plants is obtained via RPP, while binary GA with 20 random initial solutions is used to find the minimum costs of VAr-plants for feasible operation of the network. The results that are generated with about 112 solutions of NLP show that 5 reactive sources should be installed at load buses. The size of the VAr-plants in buses 3, 4 11, 12, and 24 are 306.71 MVAr, 76.04 MVAr, 172.36 MVAr, 835.42 MVAr, and 296.48 MVAr, respectively. Therefore line investment cost is US$ 48 million and the investment cost of VAr-plants is about US$ 5.06 million. By comparing these results with those reported in [20], in which the line investment is US$ 86 million and VAr installation cost is US$ 2.55 million, the average investment cost of transmission lines and VAr-plants decreases by 40%. It is important to highlight that the main difference between the solution presented in [20] and the solution presented in this work is in the lines added. In [20] one line was added in paths 7–8, 6–10, and 14–16, totaling an investment of US$ 86 million. In contrast, in this work two lines are added in 7–8 and one in 6–10, totaling an investment of US$ 48 million. 6.2. Tests with the 46-bus south Brazilian system This system has 46 buses and 79 circuits. The original data of the southern Brazilian system (for the DC model) is in [1]. In fact, some modifications in the data of the system had to be introduced in order to use the AC model. The reactive power (in each bus) is 15% of the real power, the upper limits of the transmission lines are increased by 20%, and the resistance and susceptance are 10% and 1% of the impedance of the transmission line, respectively. This system has a total demand of 6880 MW and 1032 MVAr. Carrying out line and reactive power source planning, the obtained results are as follows: • DC-TNEP—while the initial solution is 100, after 15 generations of RGA and solving 2370 LP the following lines are added: n5–11 = 2; n20–21 = 1; n42–43 = 1; n46–11 = 1. • AC-TNEP—with 50 initial solutions and after 6 generation of RGA, n24–34 = 1 is added. In this step the reactive demand is supplied locally and 190 NLP are solved. • In the next step, reactive power planning is carried out with 50 initial solutions. The solution is achieved with the solution of 760 NLP, and after 11 generations of binary GA, a total of 47.12 MVAr reactive sources are installed at buses 4, 9, 13, 20, 21, 24, 26, 43, and 45. Therefore the installation line and VAR costs are about US$ 47.48 million and US$ 14.136 million, respectively, adding a total investment of US$ 61.62 million. However, in [20] the line and VAR costs are about US$ 102.58 and 10.765 million, respectively, adding a total investment of US$ 113.34 million. So, this proposal found a solution that is US$ 51.7 million cheaper, showing the superiority of a metaheuristic when compared with a CHA to find better quality (lower costs) solutions. In CPU time, this proposal is slower than the proposal presented in [20]. The metaheuristic solves 3320 subproblems (950 NLP problems and 2370 LP problems) meanwhile, in [20] only 59 subproblems are solved (NLP) using a heuristic. The CPU time of [20] is not published but according to the author it took about 4 s of CPU time, while the proposal in this work takes 225 s of CPU time, both programs were implemented in MATLAB. So, the tests have shown that a metaheuristic takes more time than a CHA. Unfortunately, it is not possible to compare the quality of the solution found in this work with other proposals because the models used are different. In this work, we use the AC model, a more exact representation

1063

model; meanwhile, other works available in the literature generally use the DC model. However, when comparing CPU time, this proposal solves less subproblems than the metaheuristics presented in [8,10,13], where the DC model is used. Another important issue that should be considered is that we could eliminate the first stage, DC-TNEP, through the optimization process while starting with only the AC model from the beginning. However starting with DC-TNEP reduces computational efforts considerably. For example, in the IEEE 24-bus system, by solving just 280 LP problems to satisfy AC network requirements, we do not need any NLP in the second stage. 7. Concluding remarks In this paper a combinatorial model associated with a metaheuristic technique for solving the TNERPP problem using DC-TNEP prior to AC-TNEP model for a transmission expansion system is presented. Despite the fact that the proposed metaheuristic can find better solution than CHA but the main contribution of this paper is to propose an efficient approach with less effort for transmission network planning when using AC model. Although it is possible to solve the TNERPP in one stage, the easiest and by far the fastest way is to start with the solution of a DC model and then reinforce the expanded transmission network using new transmission lines as well as reactive power sources. A new methodology is also proposed to solve the reinforcement stage in two steps: the transmission network is reinforced with new lines and then with reactive power sources. The ordinary genetic algorithm and the real genetic algorithm are used to solve different steps of the TNERPP problem. A set of examples using a modified Garver system and a general analysis of the results are presented. The IEEE 24-bus system and a real south Brazilian system are employed for case studies where the results show a successful performance of the proposed methodology. Results from the case studies in this paper are compared with the results presented in [20], which is the only paper available at the time working with the AC model in the transmission network expansion problem. The effectiveness of the proposal is demonstrated when it found solution of better quality, meaning lower expansion costs for the TNERPP problem. Additionally, this work demonstrates that it is possible to work directly with the AC model in transmission network expansion, integrating the tools used for expansion and planning into one single framework like it was proposed in [20] instead of using relaxed models in the planning stage and later the AC model when the operation is analyzed, doubling the analysis efforts. References [1] R. Romero, A. Monticelli, A. Garcia, S. Haffner, Test systems and mathematical models for transmission network expansion planning, IEE Proc., Gen. Trans. Distrib. 149 (1) (2002) 27–36. [2] R. Romero, A. Monticelli, A hierarchical decomposition approach for transmission network expansion planning, IEEE Trans. Power Syst. 9 (1) (1994) 373–380. [3] M.J. Rider, A.V. Garcia, R. Romero, Transmission system expansion planning by a branch-and-bound algorithm, IET Gener. Transm. Distrib. 2 (1) (2008) 90–99. [4] S. Binato, M.V.F. Pereira, S. Granville, A new Benders decomposition approach to solve power transmission network design problems, IEEE Trans. Power Syst. 16 (2) (2001) 235–240. [5] N. Alguacil, A.L. Motto, A.J. Conejo, Transmission expansion planning: a mixedinteger LP approach, IEEE Trans. Power Syst. 18 (3) (2003) 1070–1077. [6] I.G. Sanchez, R. Romero, J.R.S. Mantovani, A. Garcia, Interior point algorithm for linear programming used in transmission network synthesis, Electric Power Syst. Res. 76 (2005) 9–16. [7] R.A. Gallego, A.B. Alves, A. Monticelli, R. Romero, Parallel simulated annealing applied to long term transmission network expansion planning, IEEE Trans. Power Syst. 12 (1) (1997) 181–188. [8] R.A. Gallego, A. Monteicelli, R. Romero, Transmission systems expansion planning by an extended genetic algorithms, IEE Proc., Gen. Trans. Distrib. 145 (3) (1998) 329–335.

1064

M. Rahmani et al. / Electric Power Systems Research 80 (2010) 1056–1064

[9] R. Romero, M.J. Rider, I. de, J. Silva, A metaheuristic to solve the transmission expansion planning, IEEE Trans. Power Syst. 22 (4) (2007) 2289–2291. [10] E.L. Da Silva, J.M.A. Orteiz, G.C. De Oliveira, S. Binato, Transmission network expansion planning under a Tabu search approach, IEEE Trans. Power Syst. 16 (1) (2001) 62–68. [11] S. Binato, G.C. De oliveira, J.L. De araujo, A greedy randomized adaptive search procedure for transmission expansion planning, IEEE Trans. Power Syst. 16 (2) (2001) 247–253. [12] T. Al-Saba, I. El-Amin, The application of artificial intelligent tools to the transmission expansion problem, Electric Power Syst. Res. 62 (2002) 117–126. [13] H.A. Gil, E.L. da Silva, A reliable approach for solving the transmission network expansion planning problem using genetic algorithms, Electric Power Syst. Res. 58 (2001) 45–51. [14] J.R.S. Mantovani, A.V. Garcia, A heuristic method for reactive power planning, IEEE Trans. Power Syst. 11 (1) (1996) 68–74. [15] M. Delfanti, G. Granelli, P. Marannino, M. Montagna, Optimal capacitor placement using deterministic and genetic algorithms, IEEE Trans. Power Syst. 15 (3) (2000) 1041–1046. [16] C.T. Hsu, Y.H. Yan, C.S. Chen, S.L. Her, Optimal reactive power planning for distribution systems with nonlinear loads, in: Proc. IEEE Region 10 Int. Conf. Computer, Communication, Control and Power Engineering, vol. 5 (1), Beijing, China, October 19–21, 1993, pp. 330–333. [17] W. Zhang, Y. Liu, Y. Liu, Optimal VAr planning in area power system, in: Proc. Int. Conf. Power System Technology, October 13–17, 2002, pp. 2072–2075.

[18] J. Urdaneta, J.F. Gomez, E. Sorrentino, L. Flores, R. Diaz, A hybrid genetic algorithm for optimal reactive power planning based upon successive linear programming, IEEE Trans. Power Syst. 14 (4) (1999) 1292–1298. [19] R.A. Gallego, A.J. Monticelli, R. Romero, Optimal capacitor placement in radial distribution networks, IEEE Trans. Power Syst. 16 (4) (2001) 630–637. [20] M.J. Rider, A.V. Garcia, R. Romero, Power system transmission network expansion planning using AC model, IET Gener. Transm. Distrib. 1 (5) (2007) 731– 742. [21] Y. Wang, H. Cheng, C. Wang, Z. Hu, L. Yao, Z. Ma, Z. Zhu, Pareto optimality-based multi-objective transmission planning considering transmission congestion, Electric Power Syst. Res. 78 (2008) 1619–1626. [22] J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, Univ. Michigan Press, Ann Arbor, MI, 1975. [23] D.E. Goldberg, Genetics Algorithms in Search, in: Optimization and Machine Learning, Addison-Wesley, 1989. [24] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer, Berlin, Germany, 1996. [25] H. Pohlheim, Geatbx: Genetic and Evolutionary Toolbox for Use with Matlab, 1994–1999, www.Geatbx.com. [26] L.-X. Wang, A Course in Fuzzy Systems and Control, Prentice Hall, 1997. [27] J. Choi, A.R.A. El-Keib, T. Tran, A fuzzy branch and bound-based transmission system expansion planning for the highest satisfaction level of the decision maker, IEEE Trans. Power Syst. 20 (1) (2005) 475–484.