Genetic Algorithm-Based Investigation of Load Growth Factor Effect on the Network Loss in TNEP

Ahad Kazemi a, Saeed Jalilzadeh b, Meisam Mahdavi b and Hosein Haddadian b a Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran

b Electrical Engineering Department, Zanjan University, Zanjan, Iran sa_jalilzadeh@yahoo.com , kazemi@iust.ac.ir , meysam@znu.ac.ir , haddadian@znu.ac.ir

Abstract- Transmission network expansion planning (TNEP) is a basic part of power network Expansion planning. Up to now, various papers have been presented on the field of static transmission network expansion planning (STNEP). But, in all of them, the effect of load growth on the network loss in transmission networks with different voltage levels has not been investigated. Therefore, in this paper, STNEP has been studied considering the effect of load growth factor on the network loss in a transmission network with two different voltage levels using genetic algorithm (GA). Finally the proposed idea is examined on Garver's six-bus network. The results show that the load growth factor has important effect on the network loss rate and subsequent type of the network arrangement.

I. INTRODUCTION

Transmission network expansion planning (TNEP) is an important component of power system planning. It determines the characteristic and performance of the future electric power network and influences the operation of power system directly. Its task is to minimize the network construction and operational cost, while meeting imposed technical, economic and reliability constraints [1], [2], [3]. Determination of investment cost for power system expansion is very difficult work because this cost should be determined from grid owners with agreement of customer and considering the various reliability criteria [4]. After Garver’s paper that was published in 1970 [5], the different methods such as GRASP [3], bender decomposition [6], genetic algorithm [1], [7], [8], [9], Tabu search [10], HIPER [11], branch and bound algorithm [12], sensitivity analysis [13], simulated annealing [14], neural network [15] and Kernel-oriented algorithm [16] were proposed to solve the STNEP problem. But, in all of these methods, STNEP problem has been solved regardless to effect of inflation rate on the network loss.

The network loss has important role in determining the configuration and arrangement of the network. Also, considering the network loss in transmission expansion planning decreases the operational costs considerably and the network satisfies the requirement of delivering electric power more safely and reliably to load centers. The load growth factor can affect on the network loss because this parameter has important role in rate of the loss growth for the years after

expansion. Accordingly, determining the effect quantity of this factor has a particular importance. Therefore, in this paper, the goal is to evaluate the effect of load growth factor on the network loss in static expansion planning of a transmission network with different voltage levels. Thus, the loss cost and also the expansion cost of related substations from the voltage level point of view have been included in the objective function. The studied voltage levels in this paper are 230 and 400 kV. This paper is organized as follows: the mathematical model of the problem is represented in Sec. 2. Section 3 describes completely chromosome structure and the proposed GA based method for solution of the STNEP problem. The method of choosing selection, crossover and mutation operators for solving the problem is described in Sec. 4. The characteristics of case study system and applying of the proposed method are given in Sec. 5 and 6, respectively. Finally, in Sec. 7 conclusion is represented.

II. MATHEMATICAL MODEL OF THE PROBLEM Due to evaluating the load growth effect on the network loss

in a transmission network with various voltage levels and subsequent adding expansion cost of substations to expansion costs, the proposed objective function is defined as follows:

∑∑∑=Ψ∈Ω∈

++=NY

iloss

kk

jiijijT i

CCSnCLC1,

(1)

and: 8760×××= lossMWhloss kClossC (2)

∑Ω∈

=ji

ijij IRloss,

2

(3)

where:

TC , CLij, kCS , lossC , Loss, CMWh, kloss, Rij, Iij, Ω , Ψ , ijn , NY are total expansion cost of network, construction cost of each line in branch i-j. (is different for 230 and 400 kV lines), expansion cost of kth substation, annual loss cost of network, total loss of network, cost of one MWh ($US/MWh), loss coefficient, resistance of branch i-j, flow of branch i-j, set of all corridors, set of all substations, number of all new circuits in corridor i-j, expanded network adequacy (in year) respectively.

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The calculation method of kloss and kCS has been

represented in Appendices A and B, respectively. According to [15], [16] the problem constraints are:

0))(( 0 =−+− jiijijijij nnf θθγ (4)

0=−+ dgSf (5) ijij

0ijij f)nn(f +≤ ijij nn ≤≤0

gg ≤≤0 Line_Loading≤ LLmax N-1 Safe Criterion Where, Ω∈),( ji and: S, f, g, d, θ, ijγ , 0

ijn , ijn , ijf , Line_Loading, LLmax, g , N are

branch-node incidence matrix, active power matrix in each corridor, generation vector, demand vector, phase angle of each bus, total susceptance in corridor i-j, number of initial circuits in corridor i-j, maximum number of constructible circuits in corridor i-j, maximum of transmissible active power through corridor i-j which will have two different rates according to voltage level of candidate line, loading of lines at planning horizon year and start of operation time, maximum loading of lines at planning horizon year, generated power limit in generator buses, number of network buses respectively. In this work, the objective function is different from those which are mentioned in other papers and in part of the problem constraints,

ijf has been considered as an addition constraint. In addition to above-mentioned changes, also Line_Loading constraint has been considered as a new constraint in order to ensure adequacy of the network after expansion. It should be noted that LLmax is an experimental parameter that is determined according to load growth factor and its rate is between 0 and 1. Unknowns parameters of the problem are discrete time type and consequently the optimization problem is an integer programming problem. In this study, the decimal codification genetic algorithm (DCGA) is being used for solution of the STNEP problem due to flexibility, simple implementation and the advantages which were mentioned in [7].

III. DCGA AND CHROMOSOME STRUCTURE OF THE PROBLEM

The standard genetic algorithm manipulates the binary strings which may be the solutions of the problem. It doesn’t need a good initial estimation for sake of problem solution, In other words, the solution of a complex problem can be started with weak initial estimations and then be corrected in evolutionary process of fitness. This algorithm can be used to solve many practical problems such as transmission network expansion planning. The genetic algorithm generally includes the three fundamental genetic operators of reproduction,

crossover and mutation. These operators conduct the chromosomes toward better fitness. There are three methods for coding the transmission lines based on the genetic algorithm method [7]: 1) Binary codification for each corridor 2) Binary codification with independent bits for each line 3) Decimal codification for each corridor. Although binary codification is conventional in genetic algorithm but in here, the third method has been used due to preventing the production of completely different offspring from their parents and subsequent occurrence of divergence in mentioned algorithm. In this method crossover can take place only at the boundary of two integer numbers. Mutation operator selects one of existed integer numbers in chromosome and then changes its value randomly. Reproduction operator, similar to standard form, reproduces each chromosome proportional to value of its objective function. Selected chromosome is as shown in Fig. 1. In part 1, each gene includes number of existed circuits (both of constructed and new circuits) in each corridor. Genes of part 2 describe voltage levels of existed genes in part 1. It should be noted that the binary digits of 0 and 1 have been used for representing voltage levels of 230 and 400 kV, respectively. A typical chromosome for a network with 6 corridors has been shown in Fig.1. In the first corridor one 400 kV transmission circuit, in the second corridor two 230 kV transmission circuits, in the third corridor three 230 kV transmission circuits and finally in the sixth corridor two 230 kV transmission circuits have been predicted.

Fig. 1. A typical chromosome

IV. SELECTION-CROSSOVER-MUTATION PROCESS

The most commonly used strategy to select pairs of individuals that has applied in this paper is the method of roulette-wheel selection, in which every string is assigned a slot in a simulated wheel sized in proportion to the string’s relative fitness. This ensures that highly fit strings have a greater probability to be selected to form the next generation through crossover and mutation. After selection of the pairs of parent strings, the crossover operator is applied to each of these pairs. The crossover operator involves the swapping of genetic material (bit-values) between the two parent strings. Based on predefined probability, known as crossover probability, an even number of chromosomes are chosen randomly. A random position is then chosen for each pair of the chosen chromosomes. The two chromosomes of each pair swap their genes after that random position. Crossover can be classified to three types of single, multiple and uniform. In this work, uniform crossover is used with probability of 1. Each individuals (children) resulting from each crossover operation

1 2 3 1 1 2 1 0 0 1 1 0

Part 1 Part 2

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will now be subjected to the mutation operator in the final step to forming the new generation. The mutation operator enhances the ability of the GA to find a near optimal solution to a given problem by maintaining a sufficient level of genetic variety in the population, which is needed to make sure that the entire solution space is used in the search for the best solution. In a sense, it serves as an insurance policy; it helps prevent the loss of genetic material. This operator randomly flips or alters one or more bit values usually with very small probability known as a mutation probability (typically between 0.001 and 0.01). In a binary coded GA, it is simply done by changing the gene from 1 to 0 or vice versa. In DCGA, as in this study, the gene value is randomly increased or decreased by 1 providing not to cross its limits. Practical experience has shown that in the transmission expansion planning application the rate of mutation has to be larger than ones reported in the literature for other application of the GA. In this work mutation is used with probability of 0.1 per bit. After mutation, the production of new generation is completed and it is ready to start the process all over again with fitness evaluation of each chromosome. The process continues and it is terminated by either setting a target value for the fitness function to be achieved, or by setting a definite number of generations to be produced. In this study, a more suitable criteria termination has accomplished that is production of predefined generations after obtaining the best fitness and finding no better solution. In this work a maximum number of 1000 generations has chosen.

V. CASE STUDY

To prove the validity of the proposed planning technique, it was applied to the Garver's 6-bus system. The configuration of the test system before expansion is given in Fig. 2 and Tables 1 and 2.

Fig. 2. Garver's 6-bus network

TABLE 1

CONFIGURATION OF THE SUBSTATIONS, GENERATION AND LOAD Generation

(MW) Load (MW)

Voltage Level (kV) Substation

100 80 230/63 1 0 240 400/230 2

250 40 400/63 3 0 160 230/63 4 0 240 400/63 5

450 0 400/63 6

TABLE 2 CONFIGURATION OF THE LINES

From bus To bus Length (Km) 1 2 200 1 3 190 1 4 300 1 5 100 1 6 340 2 3 100 2 4 200 2 5 150 2 6 150 3 4 300 3 5 100 3 6 240 4 5 315 4 6 150 5 6 300

VI. SIMULATION RESULTS

In order to evaluate the effect of load growth factor on the network loss and subsequent transmission expansion planning, the proposed idea is tested on the case study system, considering and neglecting the network loss, based on three scenarios. The load growth factor (LGF) in each scenario is different. In scenarios 1, 2 and 3, load growth factor is 4%, 7% and 10% respectively. Also, the inflation rate (IFR) and horizon planning are 10% and 10 years (year 2017) respectively for the both scenarios.

A. Scenario 1

In this scenario, the load growth factor is as follows: LGF= 4% The proposed method is applied to the case study system

and the results (lines which must be added to the network up to planning horizon year) are given in Tables 3 and 5. Also, Tables 4 and 6 show the expansion costs. The first and second configurations are obtained neglecting and considering the network loss respectively.

TABLE 3 FIRST CONFIGURATION: NEGLECTING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

3 230 2-6 2 230 4-6

TABLE 4 EXPANSION COST OF NETWORK WITH THE FIRST CONFIGURATION

0 Expansion Cost of Substations 37.16 M$ Expansion Cost of Lines 37.16 M$ Total Expansion Cost of Network

TABLE 5 SECOND CONFIGURATION: CONSIDERING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

2 400 2-6 1 400 4-6

2

3

4

5 1

6

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TABLE 6 EXPANSION COST OF NETWORK WITH THE SECOND CONFIGURATION

6.99 M$ Expansion Cost of Substations 47.79 M$ Expansion Cost of Lines 54.78 M$ Total Expansion Cost of Network

According to Table 4, expansion cost of substations has

obtained zero. The reason is that the voltage level of proposed lines for network expansion has been existed in their both first and end substations and therefore substations have not required expansion from voltage level point of view. Total expansion cost (sum of expansion costs and losses cost) of expanded network with the two proposed configurations has been shown in Fig. 3. As shown in Fig. 3, the total expansion cost of network with second configuration is more than that of the first one until about 9 years after planning horizon (year 2026), but afterward, the total expansion cost of network with first configuration becomes more than another one. The reason is that the loss cost of second configuration (all its lines are 400 kV) becomes less than that of the first one (all its lines are 230 kV), about 9 years after planning horizon.

The Investment return curve of this configuration in comparison with the first configuration is shown in Fig. 4. In fact this curve is equal to subtraction of cost curve of two mentioned configurations in Fig. 3.

0 1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90

Year after network expansion time (2017)

Net

wor

k ex

pans

ion

with

loss

cos

t (M

$)

1st configuration (without losses consideration: LGF=4%)2nd configuration (with losses consideration: IFR=10%)

Fig. 3. Sum of expansion costs and annual loss cost of the network with two

proposed configurations in scenario 1

0 1 2 3 4 5 6 7 8 9 10-20

-15

-10

-5

0

5

Year after network expansion time (2017)

Cap

ital s

ave

(M$)

Fig. 4. Investment return curve by choosing of the second configuration in

comparison with the first one

B. Scenario 2 In here, the load growth factor is as follows: LGF=7% Similar to the previous scenario, the proposed method is

applied to the case study system and the results (lines which must be added to the network up to planning horizon year) are given in Tables 7 and 9. Also, Tables 8 and 10 show the expansion costs. The first and second configurations are obtained neglecting and considering the network loss, respectively.

TABLE 7 FIRST CONFIGURATION: NEGLECTING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

3 230 2-6 3 230 4-6 1 400 3-5

TABLE 8 EXPANSION COST OF NETWORK WITH THE FIRST CONFIGURATION

0 Expansion Cost of Substations 55.79 M$ Expansion Cost of Lines 55.79 M$ Total Expansion Cost of Network

TABLE 9 SECOND CONFIGURATION: CONSIDERING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

3 400 2-6 3 230 4-6 1 400 3-5

TABLE 10 EXPANSION COST OF NETWORK WITH THE SECOND CONFIGURATION

0 Expansion Cost of Substations 81.28 M$ Expansion Cost of Lines 81.28 M$ Total Expansion Cost of Network

Similar to scenario 1, As shown in Fig. 5, the total

expansion cost of network with second configuration is more than that of the first one until about 8 years after planning horizon (year 2025), but afterward, the total expansion cost of network with first configuration becomes more than another one. The reason is that the loss cost of second configuration (its 400 kV lines is more than its 230 kV) becomes less than that of the first one (most of its lines are 230 kV), about 8 years after planning horizon. The Investment return curve of this configuration in comparison with the first configuration is shown in Fig. 6.

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0 1 2 3 4 5 6 7 8 9 1050

70

90

110

130

150

Year after network expansion time (2017)

Net

wor

k ex

pans

ion

with

loss

cos

t (M

$)1st configuration (without losses consideration: LGF=7%)2nd configuration (with losses consideration: IFR=10%)

Fig. 5. Sum of expansion costs and annual loss cost of the network with two

proposed configurations in scenario 2

0 1 2 3 4 5 6 7 8 9 10-30

-25

-20

-15

-10

-5

0

5

10

15

Year after network expansion time (2017)

Cap

ital s

ave

(M$)

Fig. 6. Investment return curve by choosing of the second configuration in

comparison with the first one

C. Scenario 3 In here, the load growth factor is as follows: LGF=10% After testing of the proposed method on the Garver's

network the following results were obtained:

TABLE 11 FIRST CONFIGURATION: NEGLECTING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

4 230 2-6 3 230 4-6 2 400 3-5 1 230 3-6

TABLE 12 EXPANSION COST OF NETWORK WITH THE FIRST CONFIGURATION

0 Expansion Cost of Substations 85.99 M$ Expansion Cost of Lines 85.99 M$ Total Expansion Cost of Network

TABLE 13 SECOND CONFIGURATION: CONSIDERING THE NETWORK LOSS

Number of Circuits

Voltage Level (kV) Corridor

3 400 2-6 4 230 4-6 2 400 3-5

TABLE 14 EXPANSION COST OF NETWORK WITH THE SECOND CONFIGURATION

0 Expansion Cost of Substations 99.92 M$ Expansion Cost of Lines 99.92 M$ Total Expansion Cost of Network

In here too, According to Fig. 7, the total expansion cost of

network with second configuration (its 400 kV lines is more than its 230 kV) is more than that of the first one (most of its lines are 230 kV) until about 5 years after planning horizon (year 2022), but afterward, the total expansion cost of network with first configuration becomes more than another one. The Investment return curve of this configuration in comparison with the first configuration is shown in Fig. 8.

0 1 2 3 4 5 6 7 8 9 1080

100

120

140

160

180

200

220

240

Year after network expansion time (2017)

Net

wor

k ex

pans

ion

with

loss

cos

t (M

$)

1st configuration (without losses consideration): LGF=10%2nd configuration (with losses consideration: IFR=10%)

Fig. 7. Sum of expansion costs and annual loss cost of the network with two

proposed configurations in scenario 3

0 1 2 3 4 5 6 7 8 9 10-20

-10

0

10

20

30

40

Year after network expansion time (2022)

Cap

ital s

ave

(M$)

Fig. 8. Investment return curve by choosing of the second configuration in

comparison with the first one

As expected, regardless to the network loss, increasing of the load growth factor is cased more 230 and 400 kV lines are added to the network. The reason is that, increasing of this factor is cased increasing of the transmitted power trough the lines. Also, can say, the load growth factor has important effect on the network loss. Because increasing of it is caused the cost curve of the second configuration cuts the curve of the first one earlier and subsequent more 400 kV lines are appended to the network. Thus, the load growth factor plays important role in determining of the network configuration.

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VII. CONCLUSION In this research, the effect of load growth factor on the

network loss in the static transmission network expansion planning has been studied using decimal codification genetic algorithm. According to simulation results in three proposed scenarios, it is concluded that, the parameter of load growth factor has important effect on the network loss rate. Also, regardless to the network loss, increasing of this parameter is cased more 230 and 400 kV lines are added to the network Therefore, this parameter plays important role in determining of the network configuration and arrangement. Increasing the rate of this parameter is caused increasing of the network loss effect on transmission expansion planning i.e. is cased the more 400 kV lines are appended to transmission network.

APPENDIX A. Calculation Method for Losses Coefficient (kloss)

This coefficient that simulates ratio of load changes to peak load is equal to area square of under the load duration curve (LDC). Load duration curve for a typical network is shown in Fig. 9. Coordinate axis are normalized therefore this coefficient will be between 0 and 1.

Fig. 9. Load duration curve for a typical network

B. Calculation Method for Expansion Cost of Substations (CSk) In the transmission network expansion planning it is

assumed that power plants and substations have enough adequacy for providing required power of loads and only the lines should be expanded. Thus, in here, substations have been expanded only from voltage level point of view. Therefore the goal of evaluation of expansion cost of the substations is calculating the expansion cost of substations that their voltage levels are not match to the voltage levels of their related candidate lines. For example, construction of a 400 kV line in corridor which it’s first and end substations are 230/63 kV causes their expansion to 400/230 kV one. For calculation of this cost, DC Load Flow (DCLF) program is run with presence of candidate lines. Then according to transmitted power through the lines and using Kirchhoff current low (KCL) the power of transmission substations is calculated. In accordance with this obtained powers and the standard capacities of transformers, number of required transformers is determined. Therefore, total expansion cost of substations can be achieved.

C. Other Required Data Losses cost in now = 36.1( MWh$ ) Number of initial population = 5 N=1000 LLmax =50% Inflation rate = 10%

REFERENCES [1] A. R. Abdelaziz, “Genetic agorithm bsed Power tansmission epansion

panning,” the 7th IEEE Int. Conf. Electronics, Circuits and Systems, vol. 2, pp. 642-645, December 2000.

[2] V. A. Levi, and M. S. Ćalović, “Linear-programming-based decomposition method for optimal planning of transmission network investments,” IEE Proc. Generation., Transmission and Distribution, vol. 140, No. 6, pp. 516-522, November 1993.

[3] S. Binato, G. C. Oliveira, and J. L. Araújo, “A Greedy Randomized Adaptive Search Procedure for Transmission Expansion Planning,” IEEE Trans. Power Systems, vol. 16, No. 2, pp. 247-253, May 2001.

[4] J. Choi, T. Mount, and R. Thomas, “Transmission system expansion plans in view point of deterministic, probabilistic and security reliability criteria,” 39th Hawaii International Conf. on System Sciences, pp. 1-10, y 2006.

[5] L. L. Garver, “Transmission net estimation using linear programming,” IEEE Trans. Power Apparatus and Systems, vol. PAS-89, No. 7, pp. 1688-1696, September/October 1970.

[6] S. Binato, M. V. F. Periera, and S. Granville, “A new Benders decomposition approach to solve power transmission network design problems,” IEEE Trans. Power System, vol. 16, pp. 235-240, May 2001.

[7] R. A. Gallego, A. Monticelli, and R. Romero, “Transmission system expansion planning by an extended genetic algorithm,” IEE Proc. Generation, Transmission and Distribution, vol. 145, No. 3, pp. 329-335, May 1998.

[8] P. Zhiqi, Z. Yao, and Z. Fenglie, “Application of an improved genetic algorithm in transmission network expansion planning,” the 6th Int. Conf. Advances in Power System Control, Operation and Management, vol. 1, pp. 318-326, November 2003.

[9] E. L. da Silva, H. A. Gil, and J. M. Areiza, “Transmission Network Expansion Planning Under an Improved Genetic Algorithm,” IEEE Trans. Power Systems, vol. 15, No. 3, pp. 1168-1175, August 2000.

[10] R. A. Gallego, R. Romero, and A. J. Monticelli, “Tabu Search Algorithm for Network Synthesis,” IEEE Trans. Power Systems, vol. 15, No. 2, pp. 490-495, May 2000.

[11] R. Romero, and A. Monticelli, “A hierarchical decomposition approach for transmission network expansion planning,” IEEE Trans. Power Systems, vol. 9, No. 1, pp. 373-380, February 1994.

[12] STY. Lee, K. L. Hocks, and H. Hnyilicza, “Transmission expansion of branch and bound integer programming with optimal cost capacity curves,” IEEE Trans. Power Apparatus and Systems, vol. PAS- 93, pp. 1390-1400, August 1970.

[13] M. V. F. Periera, and L. M. V. G. Pinto, “Application of sensitivity analysis of load supplying capacity to interactive transmission expansion planning,” IEEE Trans. Power Apparatus and Systems, vol. PAS-104, pp. 381-389, February 1985.

[14] R. Romero, R. A. Gallego, and A. Monticelli, “Transmission system expansion planning by simulated annealing,” IEEE Trans. Power Systems, vol. 11, No. 1, pp. 364-369, February 1996.

[15] T. Al-Saba and I. El-Amin, “The application of artificial intelligent tools to the transmission expansion problem,” Electrical Power Systems Research., No. 62, pp. 117-126, 2002.

[16] J. Contreras, and F. F. Wu, “A Kernel-Oriented Algorithm for Transmission Expansion Planning,” IEEE Trans. Power Systems, vol. 15, No. 4, pp. 1434-1440, November 2000.

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