new formulation and analysis of the system planning expansion model

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2009; 19:240–257Published online 18 October 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/etep.210
*CyE-

Co

New formulation and analysis of the system planningexpansion model

A. Sadegheih*,y

Department of Industrial Engineering, University of Yazd, P.O. Box 89195-741, Yazd, Iran

SUMMARY

In this paper, an electrical power transmission network planning expansion is formulated for mixed-integerprogramming, a genetic algorithm and tabu search (TS). Compared with other optimisation methods, GA’s aresuitable for traversing large search spaces since they can do this relatively rapidly and because the use of mutationdiverts the method away from local minima, which will tend to become more common as the search space increasesin size. GA’s give an excellent trade-off between solution quality and computing time and flexibility for taking intoaccount specific constraints in real situations. TS has emerged as a new, highly efficient, search paradigm forfinding quality solutions to combinatorial problems. It is characterised by gathering knowledge during the search,and subsequently profiting from this knowledge. The attractiveness of the technique comes from its ability toescape local optimality. The cost function of this problem consists of the capital investment cost in discrete form,the cost of transmission losses and the power generation costs. The DC load flow equations for the network areembedded in the constraints of the mathematical model to avoid sub-optimal solutions that can arise if theenforcement of such constraints is done in an indirect way. The solution of the model gives the best line additions,and also provides information regarding the optimal generation at each generation point. This method of solution isdemonstrated on the expansion of a 10 bus-bar system to 18 bus-bars. Finally, an empirical analysis of the effects ofparameter values on genetic algorithm performance is tested. Copyright # 2007 John Wiley & Sons, Ltd.

key words: system planning; tabu search; genetic algorithm; mathematical programming; artificialintelligence; iterative improvement methods

1. INTRODUCTION

Networks arise in numerous settings and in a variety of forms. Transportation, electrical and

communication networks pervade our daily lives. Network representations also are widely used for

problems in such diverse areas as manufacturing systems analysis, logistics, project planning, facilities

location and resource management. In fact, a network representation provides such a powerful visual

and conceptual aid for portraying the relationships between the components of systems that it is used in

virtually every field of scientific, social and economic endeavour.

orrespondence to: A. Sadegheih, Department of Industrial Engineering, University of Yazd, P.O. Box 89195-741, Yazd, Iran.mail: [email protected]

pyright # 2007 John Wiley & Sons, Ltd.

THE SYSTEM PLANNING EXPANSION MODEL 241

One of the most significant developments in operational research in recent years has been the rapid

advance in both the methodology and application of network optimisation models. Many of the

network flow problems (i.e. transportation, minimum-cost flow and transmission network planning,

etc.) can be formulated as different forms of mathematical programming, for example linear, non-linear

and integer.

The structure of a typical electrical power or energy system is very large and complex. Nevertheless,

it can be divided into five fundamental components as follows: (i) energy source; (ii) energy converter;

(iii) transmission system; (iv) distribution system; (v) load.

The objective of system planning is to optimise the facilities necessary to provide an adequate

electrical energy supply at the lowest reasonable cost. Transmission planning is closely related to

generation planning. The objectives of transmission planning are based on existing systems, future

load, generation scenarios, right-of-way constraints, costs, line capabilities, etc. Transmission planning

is an important part of power system planning. Its task is to determine an optimal network configuration

according to load growth and a generation planning scheme for the planning period so as to meet the

requirements of delivering electricity safely and economically.

In general, transmission planning should answer the following questions: (i) where to build a new

transmission line? (ii) When to build it? (iii) What type of transmission line to build?

Transmission network planning is generally divided into two stages: (i) scheme formation; (ii)

scheme evaluation.

At the scheme formation stage, the topology and capacity of the transmission lines are determined.

Whereas, at the scheme evaluation stage an analysis is carried out of the network’s characteristics

such as load flow, short circuit current capacity, stability analysis, reliability, etc. There are many

good and well-established methods of analysis that can be applied at the scheme evaluation stage.

However, satisfactory methods for scheme formation are still evolving and they are the subject of

much current research [1–28]. Scheme formation can be a complex task subject to many constraints

and a non-linear object function. ‘Optimal’ network designs are important because they can result in

large cost savings. There is clearly a need and strong justification for the development of methods for

the design of networks that are as near to ‘optimal’ as is possible. This paper aims to provide such

methods.

System network planning expansion is a complex mathematical optimisation problem because it

involves, typically, a large number of problem variables. The commonly used methods reported in the

literature can be categorised into mathematical programming, heuristic based, artificial intelligence

(AI) and iterative improvement methods.

As long ago as 1960, Knight [8] used such a method in which starting from the geographical

positions of the substations required to interconnect, a set of equations is obtained and solved by linear

programming to obtain a minimum-cost power transmission network design. The drawback of this

method is that the load flow constraints are not taken into consideration. Garver [9] proposed a method

which starts by converting the electrical network expansion problem into a linear programming

problem. The mathematical programming technique used in solving the linear network model

minimises a loss function defined as power times a guide number summed over all network links. The

overload path with the largest overload is selected for circuit addition. The drawback of this method is

that the model has no user interaction and is fixed by program formulation. Villasana et al. [7] and Serna

et al. [10] also proposed methods used a DC linear power flow model and a transportation model,

respectively. In both methods, the model is intractable.

Berg and Sharaf [11] proposed a method, using the admittance approach and linear programming, for

planning transmission capacity additions. The method consists of two phases. In the first phase

Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

242 A. SADEGHEIH

admittance addition is made, while in the second phase VAR allocation is specified. In this method,

losses have been excluded. Kaltenbach et al. [12] proposed a model which uses a combination of linear

and dynamic programming techniques to find the minimum-cost capacity addition to accommodate a

given change in demand and generation. The drawback of this method is that a very large number of

decision variables is required.

Farrag and El-Metwally [13] proposed a method, using mixed-integer programming, in which the

objective function contains both capital cost represented in its discrete form and the transmission loss

cost in a linear form. Kirchhoff’s first and second laws are included in the constraints, in addition to the

line security constraints. In this method, the loss term is linearised and a large number of decision

variables is required. Sharifnia and Ashtiani [14] proposed a method for the synthesis of a

minimum-cost secure network. In this method, the loss terms are linearised in the constraints and a

large number of decision variables is required. Adam et al. [15] proposed a method which is based on an

interpretation of fixed cost transportation type models, and includes both network security (in the

transmission network) and cost of loss (in the distribution network). The drawback of this method is

that the loss term is in a linearised form and it requires a large number of decision variables, due to the

use of the mixed-integer linear programming technique as the solution tool.

Lee et al. [16] proposed a method which is based on static expansion of networks using the zero-one

integer programming technique and Romero and Monticelli [17] proposed a zero-one implicit

enumeration method for optimising investments in transmission expansion planning. These methods

require a large number of decision variables and are computationally very expensive. Padiyar et al. [18]

made a comparison of the computation times required by four different optimisation techniques: the

transportation model; linear; zero-one and non-linear programming. The use of zero-one and

non-linear programming requires high CPU times compared to other methods which makes them

ineffective for large-scale systems [19] and all of the methods reviewed are fixed by program

formulation.

Yousef and Hackam [20] proposed a model capable of dealing with both static and dynamic modes of

transmission planning, using non-linear programming. The cost function includes the investment and

transmission loss cost. Again, this method requires long computation times and a large number of

decision variables [21].

El-Sobki et al. [22] proposed a heuristic method which is a systematic procedure to cancel the

ineffective lines from the network. The process is directed in a good manner such that the

minimum-cost network will be obtained containing the most effective routes with the best number of

circuits. The DC-load flow model is used. The drawback of this method is that power losses are not

taken into account.

Albuyeh and Skiles [23] presented a planning method involving three integral parts. The first is a

network model using a fast decoupled load flow relating the changes in active and reactive powers to

changes in bus angles and voltages, respectively. In the second part, a selection contingency analysis is

employed to determine the maximum overload on each branch and the maximum voltage deviation for

each bus. Finally, the line cost, maximum overload and a sensitivity matrix are combined into two

formulae to determine the branch to be added and the susceptance of that branch. The procedure is

repeated until the contingency analysis shows no overload. In this method, losses have been included as

a linear term.

Ekwue [24] proposed a method derived on the basis of a DC-load flow approach. The method

determines the number of lines of each specification to be added to a network to eliminate system

overloads at minimum cost. A static optimisation procedure, based on the steepest-descent algorithm,

is then used to determine the new admittances to be implemented along these rights of way. In this


DOI: 10.1002/etep


method, the model is only applicable to already connected systems and not expansion as considered

here.

In general, a characteristic of heuristic techniques is that strictly speaking an optimal solution is not

sought, instead the goal is a ‘good’ solution. Whilst this may be seen as an advantage from the practical

point of view, it is a distinct disadvantage if there are good alternative techniques that target the optimal

solution.

With the development of AI theory and techniques, some AI-based approaches to transmission

network planning have been proposed in recent years. These include the use of expert systems [25] and

artificial neural network (ANN)-based [26] methods. The main advantage of the expert system-based

method lies in its ability to simulate the experience of planning experts in a formal way. However,

knowledge acquisition is always a very difficult task in applying this method. Moreover, maintenance

of the large knowledge base is very difficult. Research into the application of the ANN to the planning

of transmission networks is in the preliminary stages, and much work remains to be done. The potential

advantage of the ANN is its inherent parallel processing nature.

In recent years, there has been a lot of interest in the application of simulated annealing (SA) and

tabu search (TS) to solving some difficult or poorly characterised optimisation problems of a

multi-modal or combinatorial nature. SA is powerful in obtaining good solutions to large-scale

optimisation problems and has been applied to the planning of transmission networks [27]. In this

reference, the transmission network planning is first formulated as a mixed integer non-linear

programming and then solved using SA. The strength of GA’s is that they are free from limitations

about the search space, for example continuity, differentiability and unimodality and they are very

flexible in the choice of an objective function. Furthermore, GA’s can work on very large and

complex spaces. These properties give GA the ability to solve many complex real-world problems.

TS has emerged as a highly efficient, search paradigm for finding quickly high quality solutions to

combinatorial problems [28–31]. It is characterised by gathering knowledge during the search, and

subsequently profiting from this knowledge. TS has been applied successfully to many complicated

combinatorial optimisation problems in many areas including power systems [32,33], The drawback

of this method is that its effectiveness depends very much on the strategy for tabu list manipulation.

Obviously, how to specify the size of the tabu list in the searching process plays an important role in

the search for good solutions. In general, the tabu list size should grow with the size of a given

problem.

From the above review, in this paper, the application of mixed integer programming, a genetic

algorithm and TS are proposed to solve the system network planning problem.

GA’s are based in concept natural genetic and evolutionary mechanisms working on populations of

solutions in contrast to other search techniques that work on a single solution. Searching not on the real

parameter solution space but on a bit string encoding of it, they mimic natural chromosome genetics by

applying genetics-like operators in search of the global optimum. An important aspect of GA’s is that

although they do not require any prior knowledge or any space limitations such as smoothness,

convexity or unimodality of the function to be optimised, they exhibit very good performance in the

majority of applications [34]. They only require an evaluation function to assign a quality value (fitness

value) to every solution produced. Another interesting feature is that they are inherently parallel

(solutions are individuals unrelated with each other), therefore their implementation on parallel

machines reduces significantly the CPU time required.

Compared with other optimisation methods, GA’s are suitable for traversing large search spaces

since they can do this relatively rapidly and because the use of mutation diverts the method away from

local minima, which will tend to become more common as the search space increases in size. GA’s give


DOI: 10.1002/etep

244 A. SADEGHEIH

an excellent trade-off between solution quality and computing time and flexibility for taking into

account specific constraints in real situations.

2. FORMULATION

In this paper, this is a novel model of the problem of minimum-cost expansion of power transmission

networks that is solved by mixed-integer programming. The model explicitly takes into consideration

the capital investment cost in its discrete form and the cost of transmission losses. The model is also

formulated to be applied with or without the cost of power generation. The DC load flow equations for

the network are embedded in the constraints of this mathematical model to avoid sub-optimal solutions

that can arise if the enforcement of such constraints is done in an indirect way. The solution of the

model gives the best line additions, and also provides information regarding the optimal generation

(MW) at each generation point. This new formulation is demonstrated on an example expansion

problem.

Minimise:

Z ¼XNP

i¼1

XNSðiÞ

j¼1

ðCijðZþij þ Z�

ij Þ þ LijðPþij þ P�

ij ÞÞ þXNE

i¼1

SiðPþEi þ P�

EiÞ þXk2NG

CkzPGk (1)

Subject to:

Co

(i) th

Xi2SPðsðiÞ¼

pyright

e power balance constraint at bus-bar k¼ 1,2,. . .NB� 1 or the power flow conservation

equation at each bus-bar upholding Kirchhoff’s First Law:

kÞk

XNSðiÞ

j¼1

ðP�ij � Pþ

ij ÞþX

i2SPðkÞeðiÞ¼k

XNSðiÞ

j¼1

ðPþij�P�

ij ÞþX

j2SEðkÞsðjÞ¼k

ðP�Ej � Pþ

EjÞþX

j2SEðkÞeðjÞ¼k

ðPþEj � P�

EjÞ ¼ PLk � PGk

(2Þ
(ii) th e loop equation l¼ 1, 2, . . ., LBE, containing only existing lines, this constraint upholds
Kirchhoff’s Second Law for existing lines:Xi2LEð‘Þ

XEiðPþEi � P�

EiÞ ¼ 0 (3Þ

(iii) th
e loop equations for loop l containing one proposed line i:
Xk2LPð‘Þ

XEkðPþEk � P�

EkÞ þXNSðiÞ

k¼1

XPikðPþ

ik � P�ikÞ � Kð1 �

XNSðiÞ

k¼1

ðZþik þ Z�

ik ÞÞ (4Þ

Xk2LPð‘Þ

XEkðPþEk � P�

EkÞ þXNSðiÞ

k¼1

XPikðPþ

ik � P�ikÞ � K

XNSðiÞ

k¼1

ðZþik þ Z�

ik Þ � 1

!

l ¼ 1; 2; . . . ;LBP

(5Þ

(iv) th
e exclusivity constraint for each proposed line i. This constraint forces the program to select
one state only for each proposed line, or delete all its states. The exclusivity constraints result

# 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

Copyright


from the fact that the capacity of any line can take on only one value. That value, however, may

be any of the discrete capacities in the cost-capacity curve. The exclusivity constraints prevent

the capacity from assuming more than one value.

XNSðiÞ

j¼1

ðZþij þ Z�

ij Þ � 1 (6Þ

(v) th
e overload constraint for each existing line i:
PþEi þ P�

Ei � EMi i ¼ 1; 2; . . . ;NE (7Þ
(vi) th e overload constraint for the state j of each proposed line i:
P0MijðZþ

ij þ Z�ij Þ � Pþ

ij þ P�ij � PMijðZþ

ij þ Z�ij Þ i ¼ 1; 2; . . . ;NP

j¼ 1; 2; . . . ;NSðiÞ (8Þ

(vii) th
e generator capacity limit at each bus-bar k:
PGk � PMk (9Þ
(viii) th e availability constraint at each bus-bar k—this controls the number of lines connected to
each bus-bar according to parameter MPk:

Xi2SPðkÞ

XNSðiÞ

j¼1

ðZþij þ Z�

ij Þ � MPk k ¼ 1; 2; . . . ;NB (10Þ

and

PþEi;P

�Ei;P

þij ;P

�ij ;PGk � 0 8i 2 NE; 8i; j 2 NP;NSðiÞ; 8k 2 NG (11Þ

Zþij ; Z

�ij ¼ 0; 1 8i; j 2 NP; NSðiÞ (12Þ

The objective function Z consists of the capital investment cost in its discrete form, the cost of

transmission losses and cost of generation. The cost function for any proposed line i is given as:

ZPi ¼XNSðiÞ

j¼1

CijZþij þ AijðPþ

ij Þ2

(13)

CijZþij is the capital cost obtained from cost-capacity curves. If the different types of transmission lines

are explicitly considered as the variables in the optimisation problem, the computation will be

extremely complex. This complexity is greatly reduced by considering the capacity to be added as the

only variable in each line. The determination of the optimal cost-capacity curve for each line is made as

follows:

(i) a
vailable line types are determined;
(ii) f
or each combination (state), the equivalent admittance, resistance and cost are determined.
The above procedure results in the determination of a number of states for each proposed route. Each

state has a definite capacity, resistance, admittance and cost (Figure 1).

# 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

1

2

3

State

Capacity/MW

Capital cost

4

P1 P2 P3 P40

Figure 1. The cost-capacity curve for a line.

246 A. SADEGHEIH

The second part of the cost function is AijðPþij Þ[2] which is non-linear. For existing lines, the cost

function consists only of an equivalent non-linear component:

ZEi ¼ aiðPþEiÞ

2(14)

To get rid of the non-linearity in these cost functions, the following methods are applied:

Co

(i) a

pyrig

s the power flow on any state of a proposed line is known within two limits, linearisation is

made between these two limits (see Figure 2). This gives:

ZPi ¼XNSðiÞ

j¼1

CijZþij þ LijP

þij (15Þ

(ii) s
ince transmission losses are only a small fraction of the total cost, a curve fitting technique is
used for existing lines (see Figure 3). This gives:

ZEi ¼ SiPþEi (16Þ

Transmission loss cost

Capacity/MW

Lij

P2P3

0

Figure 2. Linearisation of transmission loss cost for state j of proposed line i.

ht # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

Transmission loss cost

Capacity/MW

Si

0

Figure 3. Linearisation of transmission loss cost for an existing line i.

Copyrig


Combining Equations (15), (16) and cost of generation, the final cost function to be minimised

becomes:

Z¼XNP

i¼1

XNSðiÞ

j¼1

ðCijZþij þLijP

þij Þþ

XNE

i¼1

SiPþEiþ

Xk2NG

CkPGk (17)

3. ILLUSTRATIVE EXAMPLE

This example is an actual system in the western part of China [34], see Figure 4. The original network

has 10 bus-bars and nine lines. The system consists of seven existing load buses and three existing

generator buses.

The system is to be expanded to 18 bus-bars with four new load buses added and four new generator

buses. Table I gives the specifications for the existing and proposed lines in the network. The net

generation for each of the bus-bars is given in Table II.

In this example, the cost of a circuit is defined as being directly proportional to the line length.

The application of the developed method has been made in the light of the following factors:

(i) o

h

nly one line type is assumed;

(ii) th
e maximum number of states¼ 4;
(iii) th
e cost of a circuit is proportional to the line length, therefore, the line length can be used to
replace the cost in comparison analysis.

The total cost of this network plan with the mixed integer programming is 13 558.98.

4. APPLICATION OF GENETIC ALGORITHM

The chromosome structure used to represent a particular set of possible transmission line power

capacities, for the mixed-integer transmission network planning using GA has 27 state variables

t # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

3

4

7

6

5

8

9

10

656

259

88MW

213199

38

154

276

121

55

1

2

G

G

GBus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

MW

MW

MW

MW

MW

MW

MW

MW

MW

Key:

Bus-bar

Power generation at bus-bar

Load at bus-bar

Existing transmission line

G

Figure 4. The original 10 bus-bar network.

248 A. SADEGHEIH

(genes) Pij as follows:

P1;2;P1;11;P2;3; . . . ;P17;18 (18)

Each individual line capacity is encoded by sufficient bits to cover its allowable range of values. The

bit strings for each Pij are concatenated to form a chromosome. The initial population is generated

randomly, that is each bit in each chromosome is set randomly to either 1 or 0. Whenever a new

chromosome is generated, it is checked to see that in decoded form it produces valid values for the Pij’s.

When an invalid value is produced, the chromosome is discarded and another one is generated.

Genetic algorithm which works by:

Co

1. A

pyr

starting population is formed from 50 randomly generated individuals.

2. A
simulation model is used as a fitness function and validates an individuals’ quality—gives a
fitness value to individuals.

3. T
he selected individuals are used for generating new individuals with genetic operators.
4. N
on-selected individuals are replaced with new individuals made with genetic operators.
5. P
oint 2 repeats if the stopping criteria is not satisfied.
ight # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

Table I. Transmission line specifications for the 18 bus-bar system.

Linenumber

Frombus-bar

To bus-bar Reactanceper unit

Circuitcapacity (MW)

Existingline

New circuitsallowed

Length (km)

1 1 2 0.0176 230 Yes 0 702 1 11 0.0102 230 No 1 403 2 3 0.0348 230 Yes 0 1384 3 4 0.0404 230 Yes 0 1555 3 7 0.0325 230 Yes 0 1296 4 7 0.0501 230 No 1 2007 4 16 0.0501 230 No 3 2008 5 6 0.0267 230 Yes 0 1069 5 11 0.0153 230 No 2 60

10 5 12 0.0102 230 No 1 4011 6 7 0.0126 230 Yes 0 5012 6 13 0.0126 230 No 1 5013 6 14 0.0554 230 No 2 22014 7 8 0.0151 230 Yes 1 6015 7 9 0.0318 230 No 1 12616 7 13 0.0126 230 No 2 5017 7 15 0.0448 230 No 2 17818 8 9 0.0102 230 Yes 1 4019 9 10 0.0501 230 Yes 2 20020 9 16 0.0501 230 No 2 20021 10 18 0.0255 230 No 1 10022 11 12 0.0126 230 No 2 5023 11 13 0.0255 230 No 1 10024 12 13 0.0153 230 No 1 6025 14 15 0.0428 230 No 2 17026 16 17 0.0153 230 No 2 6027 17 18 0.0140 230 No 1 55


During the selection stage:

Co

1. I

pyr

t selected a fixed number of chromosomes from the current population.

2. U
sing genetic operators, evolution occurs in the selected population in order to obtain a new
population.

The spreadsheet model is developed for solving this problem. In the next step for solving the system

planning using a GA, Equation (2) as Kirchhoff’s First Law and Equations (7–11) must be satisfied.

Equations (3–5) as Kirchhoff’s Second Law are used to penalise solutions in the cost function.

The final step in the implementation of the system planning using a GA is the fitness function. The

fitness value of a chromosome is a measure of how well it meets the desired objective. In this case, the

objective is the minimisation of the network’s cost function. Choosing and formulating an appropriate

objective function is crucial to the efficient solution of any given genetic algorithm problem. When

designing an objective function for an optimisation problem with constraints, penalty functions can be

introduced and applied to individuals that violate the imposed constraints. The fitness function in

Equation (1) with penalty functions is used to calculate the fitness value of each individual.

In the GA approach, the parameters that influence its performance include population size, crossover

rate and mutation rate. A population size of 50, crossover rate 0.5 and mutation rate 0.006 for the

system network planning are used. Figure 5 shows the results obtained with the genetic algorithm. The

ight # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

Table II. Net generation at each bus-bar for the expanded 18 bus-bar network.

Nodes (bus-bars) Generator output (MW) Load (MW) Net output (MW)

1 0 55 �552 360 84 2763 0 154 �1544 0 38 �385 760 639 1216 0 199 �1997 0 213 �2138 0 88 �889 0 259 �259

10 750 94 65611 540 700 �16012 0 190 �19013 0 110 �11014 540 32 50815 0 200 �20016 495 132 36317 0 400 �40018 142 0 142

250 A. SADEGHEIH

total cost of this network plan is 13 558.98. This result is the same as that obtained with mixed-integer

programming.

5. EMPIRICAL ANALYSIS OF THE EFFECTS OF PARAMETER VALUES

ON GA PERFORMANCE

In the experiments performed so far in this paper, the GA’s have used a population size of 50, crossover

rate of 0.5 and a mutation rate of 0.006, that is a general purpose or utilitarian set of values.

1 2 3 4

16

11 13 7 8 9 10

12 5 6 14 15 17

18

Figure 5. The topological configuration of the network using genetic algorithm.


DOI: 10.1002/etep


For the 18 bus-bar network planning, the following GA parameter values were used in analysis of

variance experiments to analyse the effects of the parameters on the number of iterations required to

reach the optimal fitness. (Population Size: 20, 50, 80, Crossover Rate: 0.2, 0.4, 0.5, 0.6, 0.8, Mutation

Rate: 0.001, 0.006, 0.011, 0.016).

For each combination of the parameters, the GA is run for eight different random initial

populations—these eight populations being different for each combination. Thus, in total, the GA is

run 3� 5� 4� 8¼ 480 times. The values chosen for the population size are representative of the range

of values typically seen in the literature. If the mutation rate is too low, then many genes that

would have been useful are never tried out. If it is too high, there will be too much random

configuration, offspring will start losing their similarity to their parents, and the algorithm will lose the

ability to learn from previous searches. It is normally the case that the mutation rate needs to be small to

be effective—as in the range of values used here. The crossover rates chosen are representative of the

entire range.

The data provide sufficient evidence to conclude that the means for the five different crossover rates

are not significantly different and that the crossover rate is therefore an insignificant factor across the

range [0.2:0.8] in the performance of GA.

The analysis of variance of mutation rate shows that mutation is a significant factor. This shows that

there is clearly little difference in performance in the two mid-values of mutation rate [0.006 and

0.011]. The analysis of variance of population size indicates insignificance of this factor.

These tests show that the number of iterations required to reach the minimum cost is a very

good fit to a log-normal distribution. In particular, the chi-square (x2) value is not significant at the 5%

level. The Anderson–Darling normality test produces a p-value of 0.004 when testing the

untransformed (not logged) data for normality. This indicates that the data observed would be

extremely unlikely if the distribution was normal. However, the logged data (natural logarithm)

produces a fairly high p-value of 0.107 which indicates that the data could well have come from a log

normal distribution.

This result means that methods of statistical analysis based on assumptions of normality may be

applied to the log of the number of iterations required by the GA to reach the optimal solution. Hence,

experimental x2 is insignificant at the 5% level, so the number of iterations appear to form a log-normal

distribution.

Observation (i), which tells us that the value of the crossover rate is not significant in the broad range

[0.2:0.8], implies that the selection of a ‘good’ value for the crossover rate is not an issue—provided it

is in this range. This result does of course raise the question, ‘Is crossover necessary’? To address this

question, experiments were performed for the 18 bus-bar transmission network planning with crossover

rate¼ 0, that is no crossover, with a population size of 50 and mutation rate of 0.006. This omission of

crossover resulted in a mean number of iterations of 10 758 which is substantially more than the mean

of 6255 produced before across the crossover range [0.2:0.8]. The analysis of variance shows that

crossover rate becomes a significant factor when a value of zero is included. Clearly, crossover makes a

significant contribution to the GA and should not be omitted.

Observation (ii) tells us that the value of the mutation rate is important. As the mutation rate increases

so the GA is replaced by a more random search. As mutation tends towards 0 so movement around the

search space, and subsequent diversity in the population, is lost due to the comparative rarity in actually

applying mutation.

Whilst there may be concern that in practical applications a ‘good’ value must be identified for the

mutation rate, this concern is reduced by knowing from Observation (iii) that the good values appear to

occupy a ‘good range’, rather than being some difficult to isolate single value.


DOI: 10.1002/etep

252 A. SADEGHEIH

Observation (iv) tells us that the effect of the population size does not vary across the range [20,50,

80] when using a ‘good’ mutation rate. This changes with the ‘poorer’ mutation rates. This shows that if

the mutation rate is too low, then a larger population is required to get the necessary diversity.

Observation (v) tells us that changing crossover rate does not alter the significance of population size

for the ‘poorer’ mutation values, that is changing crossover rate does not compensate for the ‘poorer’

mutation rate.

6. IMPLEMENTATION OF THE TABU SEARCH ALGORITHM

Tabu search was developed by Glover [29–31]. TS has emerged as a new, highly efficient, search

paradigm for finding quality solutions to combinatorial problems. It is characterised by gathering

knowledge during the search, and subsequently profiting from this knowledge. The attractiveness of the

technique comes from its ability to escape local optimality. TS has now become an established

optimisation approach that is rapidly spreading to many new fields. For example, successful

applications of TS have been reported recently in solving some power system problems, such as

hydro-thermal scheduling [32], alarm processing [33], flexible manufacturing systems, neural network

training, network optimal [26], etc. The drawback of this method is that its effectiveness depends very

much on the strategy for tabu list manipulation. At each iteration, the neighbourhood of the current

solution is explored and the best solution in the neighbourhood is selected as the new current solution.

TS is different from other local search techniques, in that the procedure does not stop when no

improvement is possible. The best solution in the neighbourhood is selected as the current solution,

even if it is not better than the current solution. This strategy allows escape from local optima and,

consequently, exploration of a larger proportion of the solution space. TS is a restricted neighbourhood

search technique, and its computational flowchart is an iterative process.

The notations used in the developed algorithm are introduced below:

P0 in

Copyright #

itial solution

P c
urrent solution
P0 n
eighbor solution
P0best b
est neighbor solution
Pbest b
est solution
M(P) a
move that yields solution P
The steps of the TS algorithm are follows:

Step 1. C
hoose the initial solution P0. Current solution P¼P0 and best solution Pbest¼P0. i¼ 0 and
start with empty tabu lists.

Step 2. R
epeat.
Step 2.1. Generate the neighbour, P0, for current solution y and call the simulation model

to calculate the cost function Z(P0).Step 2.2. Select the best neighbour y0

best. If Z(P0best)�Z(Pbest) then Pbest¼P0

best and go to

Step 2.4.

Step 2.3. If (M(P0best) is tabu) and Z(P0

best)�Z(Pbest) then Z(P0best)¼1 and go to

Step 2.2. else current solution P¼P0best.

20
07 John Wil ey & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257
DOI: 10.1002/etep

Table III. Different number of iterations and total unit cost.

Iterations 4000 5000 4500 6700 3500 4600Total unit cost 13 608.47 13 608.47 13 608.47 13 608.47 13 608.47 13 608.47

Lines 1–(MW) �2Number 1

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Step 2.4. i¼ iþ 1. Keep M(P0best) on associated tabu list for associated tabu tenure.

Step 2.5. Until i� imax.

215

20

1–11 2–3160 61 �1 1

07 John Wil

Among all the trial solutions thus produced, TS seeks the one that most improves the objective

function. In certain situations, if there are no improving moves, a local optimum does exist. In this case,

TS chooses the one that least degrades the objective function. A move is a transition from one solution

to another. Moves are generally expected to produce an improvement to the solutions. Many kinds of

moves are currently available [30–31].

Another important aspect of TS is the tabu list. The tabu list stores characteristics of previous moves

so that these characteristics can be used to classify certain moves as tabu (to be avoided) for a number of

iterations. By accepting moves that do not produce improvements, it is possible to return to solutions

which have already been found, which is known as cycling. Obviously, how to specify the size of the

tabu list in the searching process plays an important role in the search for good solutions. In general, the

tabu list size should grow with the size of a given problem. Up to now, the tabu list size has been

determined experimentally [28]. In addition, for how many iterations a move should be retained in the

tabu list is also an important issue. Many methods to implement and manage the tabu list have been

developed. For strategies for the short term memory and long memory in TS, see Glover [30] and

Glover et al. [29].

For this problem, a 27 state variables string was used to represent all possible topological

configurations. Therefore, there were 40 neighbours of P and 40 different possible moves. The

neighbourhood was constructed with adjacent solutions created from a current solution by changing a

bit between 0 and 1.

The objective function which in this case is total unit cost, Z(P) was applied to the neighbourhood of

P. Then the last improving move that had been classified tabu would be selected as the next solution.

There are three common tabu list types: fixed, and dynamic and hashing function or a combination of

three. In this paper, a fixed tabu list size was used. A tabu list size of 15 was empirically determined for

this problem. The result was obtained after experimenting with different numbers of iterations (see

Table III). Therefore the tabu search algorithm obtained a value of 13 608.47. It can be seen that an

improvement of 0.36% is achieved by the genetic algorithm and mixed integer programming.

Table IV and Figures 5 and 6 show that there are some substantial differences in the individual line

power flows although the fundamental structure is similar with only one difference in network

topology, two differences in number of lines, and one difference in direction of flow.

Table IV. Tabu search solution.

3–4 3–7 4–16 5–6 5–12 6–7 6–14 7–8 7–13 8–9 9–10 10–18 14–15 16–17 17–18116 23 �154 �69 190 40 �308 �260 110 �348 �607 49 200 209 �1911 1 1 1 1 1 2 2 1 2 3 1 1 1 1

ey & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep

1 2 3 4

16

11 13 7 8 9 10

12 5 6 14 15 17

18

Figure 6. The topological configuration of the network using tabu search.

254 A. SADEGHEIH

7. CONCLUSION

If GA’s are to be recommended for general use in the planning of transmission networks, then a greater

understanding of the effects of these potentially critical parameters on the convergence of the GA to the

optimal solution should be provided. This is the suggestion of this paper. It could be argued that the results

demonstrate the validity of the tabu search method since it achieve a ‘very near optimal’ solution. However,

the tabu search approach does not provide us with a generic engineering tool. The results support the

argument that genetic algorithms reduce the need for tabu search method since ultimately it targets an

optimal solution whilst being easily adapted to different applications and, consequently, a generic

engineering tool. The results presented here support the extension of this argument into the field of

topological configuration of network system in particular. The power transmission network planning

problem has been modelled by the mixed integer programming model. The GA has three user-defined

parameters—population size, mutation rate and crossover rate. The results were obtained with a

population size of 50, crossover rate of 0.5 and mutation rate of 0.006—a utilitarian set of parameters.

Further experiments showed that the value of the crossover rate does not appear to be highly critical, with a

value between 0.2 and 0.8 giving the best results in terms of rate of convergence. However, initial

experiments showed that the rate of convergence can be very sensitive to the mutation rate. Overall tabu

search needed longer computation times compared to the genetic algorithm. The drawback of tabu search

is that its effectiveness depends very much on the strategy for tabu list manipulation. Finally, if the

mutation rate is too high, then a smaller population is required. Put the other way round, if the population

size is small a higher mutation rate is required to compensate for the reduced potential for diversity across a

smaller population, whilst as population size grows, the need for mutation is reduced.

8. LIST OF SYMBOLS AND ABBREVIATIONS

8.1. Symbols

ai c

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ost of transmission losses of unit power transmitted on existing line i

Aij c
ost of transmission on losses for unit power transmitted on state j of proposed line i
2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:240–257

DOI: 10.1002/etep


Ck c

Copyright #

ost of generating a unit of power at bus-bar k

Cij c
apital cost of state j of proposed line i
e(i)¼ k s
et of lines that end at bus-bar k
EMi m
aximum power flow of existing line i
K a
large positive integer number
Lij l
inearised cost coefficient representing transmission losses cost of state j of proposed line i
MPk m
inimum number of proposed lines connected to bus-bar k
PþEi o
riented power flow on existing line i from its ‘start’ to its ‘end’
P�Ei o
riented power flow on existing line i from its ‘end’ to its ‘start’
PGk p
ower generation at bus-bar k
Pþij o
riented power flow on state j of proposed line i from its ‘start’ to its ‘end’
P�ij o
riented power flow on state j of proposed line i from its ‘end’ to its ‘start’
PLk l
oad at bus-bar k
PMk m
aximum power output of generator k
PMij m
aximum power flow of state j of proposed line i
P0Mij m
inimum power flow of state j of proposed line i
Si l
inearised cost coefficient representing transmission losses cost of existing line i
s(i)¼ k s
et of lines that start from bus-bar k
XEi r
eactance of existing line i
XPij r
eactance of state j of proposed line i
Z t
otal system cost (capital, transmission losses and generation)
ZEi c
ost function for existing line i
Zþij z
ero-one integer variable assigned to state j of proposed line i from its ‘start’ to its ‘end’
Z�ij z
ero-one integer variable assigned to state j of proposed line i from its ‘end’ to its ‘start’
ZPi c
ost function for proposed line i
8.2. Abbreviations

LBE n
umber of basic loops containing existing lines only
LBP n
umber of basic loops containing existing lines plus one proposed line
LE(l) s
et of existing lines forming basic loop l which contains existing lines only
LP(l) s
et of existing lines forming basic loop l which contains one proposed line
NB n
umber of bus-bars
NE to
tal number of existing lines
NG s
et of generation bus-bars
NP to
tal number of proposed lines
NS(i) n
umber of states of proposed line i
SE(k) s
et of existing lines connected to bus-bar k
SP(k) s
et of proposed lines connected to bus-bar k
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DOI: 10.1002/etep


AUTHOR’S BIOGRAPHY

A. Sadegheih was born on May 10, 1960, in Yazd. He received his B.S., M.S. degrees andPh.D. in Power Engineering, System Engineering and Industrial Engineering from IsfahanUniversity of Technology, Iran, and university of Cardiff, Wales, UK, in 1985, 1987 and 1999,respectively. In May of 1988, he joined University of Yazd as faculty member where he ispresently an Associate Professor in the Industrial Engineering Department. His researchinterests include the application of artificial intelligence to power systems operation andsystem planning.


DOI: 10.1002/etep

new formulation and analysis of the system planning expansion model

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