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Multi-objective transmission network planning by a hybrid GA approach with fuzzy decision analysis
Electrical Power and Energy Systems 25 (2003) 187–192 www.elsevier.com/locate/ijepes
Multi-objective transmission network planning by a hybrid GA approach with fuzzy decision analysis T.S. Chunga,*, K.K. Lia, G.J. Chenb, J.D. Xieb, G.Q. Tangb a
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China b Department of Electrical Engineering, Southeast University, Nanjing, People’s Republic of China Received 28 March 2001; revised 8 May 2002; accepted 29 May 2002
Abstract In this paper, power system transmission network planning is formulated as a multi-objective mathematical optimization problem. In this context, three objectives: investment cost, reliability and environmental impact are considered in the optimization. To overcome the drawback of conventional mathematical optimization method in arriving at local optimum solution, a genetic algorithm approach is developed, from which several possible planning schemes are generated. This is followed by a fuzzy decision analysis method to select the optimum solution scheme. Case study results on a large 220 kV transmission network planning problem are presented to show the methodology’s feasibility and efficiency. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Transmission network planning; Genetic algorithms; Multi-objective fuzzy decision
1. Introduction Multiple objectives are often considered simultaneously in practical transmission network expansion planning. These objectives may be conflicting each other. It is difficult to find a single solution, which is optimal for all objectives. In this context, a suitable compromise solution should be determined. The multi-objective optimization method is an ideal solution to this kind of problem [1]. In conventional algorithms, multi-objective optimization problems are transformed into single objective optimization problems. These conventional algorithms are often faced with drawbacks, such as local optimum, dimension disasters, etc., which make the algorithm inefficient. Genetic algorithms (GA), as a powerful search and optimization method based on natural evolution, may overcome the drawbacks of conventional mathematical optimization methods [2]. In recent years, GAs have been used widely in power systems [3 –6]. Moreover, GAs are suitable for multi-criteria methods and can generate a set of possible optimal schemes instead of only one single optimal scheme. In this paper, a multi-objective GA is proposed, in which three selected objectives: investment cost, reliability and environmental impact, are considered. * Corresponding author. Tel.: þ 852-27666160; fax: þ 852-23301544. E-mail address: [email protected] (T.S. Chung).
In general, several possible optimal planning schemes are generated in a practical transmission network planning. Then the best scheme is selected from them considering different criteria. Because objectives are usually conflicting, it is difficult to choose the best scheme even for an experienced planning engineer especially when the objective number is large. So, a decision model for transmission network planning should be set up. Much work has been done on the decision analysis in power systems [7 –10]. In Ref. [7], possible decision analysis applications are reviewed. A systematic approach for decision making in a fuzzy environment is presented in Ref. [8]. In Ref. [9], types of conflicts among planning objectives are illustrated and categorized. In Ref. [10], a flexible decision method was proposed, which made a compromise between economical objective and applicable objective. Because the factors considered in transmission network planning are usually fuzzy, a multi-objective fuzzy decision method for transmission network planning is applied in this paper. Finally, the proposed method is illustrated using a practical example, and tested for its feasibility and efficiency. The paper is organized as follows. In Section 2, a multiobjective optimization model for transmission network planning is formulated. In Section 3, a GA is developed to solve the model. In Section 4, a fuzzy decision method is used to make the decision analysis. In Section 5, a practical system case study is presented.
0142-0615/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 0 6 1 5 ( 0 2 ) 0 0 0 7 9 - 0
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T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
Nomenclature decision vector negative deviation positive deviation objective function of optimization total investment cost of all additions expected energy not supplied per year incremental land area of the new additions expected energy not supplied per year of new network scheme
X di2 diþ F F1 F2 F3 eens
The first objective function minimizes the total investment cost of all additions. The second objective function minimizes the expected energy not supplied per year in power systems. The third objective function minimizes the incremental land area of the new additions. So, the mathematical optimal model for multi-objective transmission network planning is formulated as min F ¼
In general, the mathematical model of a multi-objective programming is formulated as min F ¼
m X
ðpi diþ þ qi di2 Þ
ð1Þ
i¼1
gi ðXÞ þ di2 ; diþ
2
$ 0;
diþ
subject to
¼ gi
di2 diþ
¼0
ð1 # i # mÞ
i¼1
eensðXÞ þ d22 2 d2þ ¼ eens n X
f ðxi Þli þ d32 2 d3þ ¼ f
i¼1
di2 ; diþ $ 0; di2 diþ ¼ 0;
i ¼ 1; 2; 3
ð1 # i # mÞ di2
where X is the decision vector; is the negative deviation, which indicates the unfinished quantity of objective i; diþ is the positive deviation, which indicates the excessive quantity of objective i; gi ðxÞ is the objective function of objective i; gi is the corresponding objective value; m is the number of objectives; pi ; qi are the priorities of objective i for the positive deviation and the negative deviation, respectively. di2 diþ ¼ 0 is the requirement of the objective programming so that one of di2 ; diþ must be 0. In this paper, three objectives ðF1 ; F2 ; F3 Þ that consider investment cost, reliability index and environmental impact index are considered. They can be expressed as follows n X
ci xi li
ð2Þ
min F2 ðXÞ ¼ eensðXÞ
ð3Þ
i¼1
min F3 ðXÞ ¼
ci xi li þ d12 2 d1þ ¼ c
where c ; eens and f are expected objective values corresponding to three objectives, respectively.
X$0
min F1 ðXÞ ¼
ð5Þ
X$0
subject to di2
ðpi diþ þ qi di2 Þ
i¼1
n X
2. Multi-objective optimization model for transmission network planning
3 X
n X
f ðxi Þli
ð4Þ
i¼1
where X ¼ ðx1 ; x2 ; …; xi ; …; xn Þ; xi is the number of possible circuit additions in path i; n is the number of right-of-way, in which circuits can be added; ci is the present investment cost of new additions per kilometer; li is the length of addition i; eensðXÞ is the expected energy not supplied per year of new network scheme. f ðxi Þ is the incremental land area after xi additions in path i is built up.
3. Application of genetic algorithms to multi-objective transmission network planning 3.1. Genetic algorithms GAs based on natural evolution, have been successfully applied to mathematical optimization problems [3 –6]. It is demonstrated that GAs are especially suitable for solving complicated optimization problems. GAs work with a set of individuals (called chromosomes), which represent possible solutions to the optimization problem. These individuals evolve through an iterative procedure, which is called reproduction. In the procedure of evolvement, three operators: selection, crossover and mutation, are used. The procedure of GAs can be depicted as 1. Chromosome coding. Each chromosome represents one possible solution to the problem. And each variable is coded using one bit (called as gene) or several bits of chromosomes. Generally, a binary coding strategy is used. 2. Initialization. The initial population of individuals is generated in this step. Each bit of individuals is generated through a random generator. 3. Calculating fitness values. The fitness function is made up of objective function and penalty items. Fitness values of initial individuals are calculated, respectively. 4. Reproduction. A new set of individuals is produced through the effects of three operators. The operator selection, selects the preferable individuals from the
T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
prior individual set through a preliminary strategy and keeps the individual number to be constant. The purpose of using the operator crossover is to make the offspring get better combinations of genes. First, two individuals are taken out randomly from the prior individual set. Then, a random number decides whether the two individuals will make a crossover operation. If it does, another random number will decide the position of the crossover operation. The operator mutation is used to seek the optimal solution in a global space for fear that a local optimum will come up. For each individual, a random number decides that if they will make a mutation operation and which bit the operation will act on. 5. Replace the old individual set with the new one. 6. Repeat steps (3) – (5), until a satisfactory solution is reached. 3.2. Chromosome coding A binary coding strategy is adopted. Three bits (a gene group) are used to represent one possible addition path for the circuit number of one possible addition path is no more than seven. That is, the length of each individual is three times the total possible addition paths. The possible addition paths are ordered according to their connected bus numbers so that relations between each gene group and the possible addition paths can be set up by order. 3.3. Fitness function In general, GAs only consider the maximization problem. While formulation (5) is a minimization problem, a transformation should be done. The fitness function is expressed as U ¼ U0 2 F
ð6Þ
where U0 is a huge number.
189
Pgi # Pmax gi where Pgi is the output active power of generator i; Pmax is gi the output capacity of generator i; Pij is the active power flow of circuit ij; Pmax is the transmission capacity of circuit ij ij: In this paper, only single faults are considered.
4. Decision analysis using multi-objective fuzzy decision method In the procedure of planning a practical power network, several possible planning schemes could be generated. Then the best one is selected among them considering different factors. The factors concerned are so many that sometimes it is difficult for a planner to have all things considered merely by his past experiences. In this context, a scientific decision model for transmission network planning should be set up. In this paper, a multi-objective fuzzy decision method is proposed, which can take into account the fuzzy characteristics of different factors. The method is divided into three steps: boundary decision, normalization and decision analysis, which will be depicted in detail in the following paragraphs. 4.1. Boundary decision Suppose the scheme set is A ¼ {A1 ; A2 ; …; An }; the objective set is F ¼ {F1 ; F2 ; …; Fm }: A decision matrix can be formed 2 Xnm
x11
6 6x 6 21 6 ¼6 . 6 . 6 . 4 xn1
x12 x22 .. . xn2
· · · x1m
3
7 · · · x2m 7 7 7 .. 7 7 . 7 5 · · · xnm
ð9Þ
3.4. Operation simulation To calculate the expected energy not supplied per year, an operation simulation procedure should be conducted. X eens ¼ Zi Pi ð7Þ where Pi is the fault probability of circuit i: Zi is the dissatisfied energy of the network supply when the circuit i does not work for a fault, which can be calculated by the difference between load demands and the load supply capacity (LSC). The LSC can be formulated as LSC ¼ max
n X i¼1
subject to Pij # Pmax ij
Pgi
ð8Þ
xij is the value of objective j for the scheme i; i ¼ 1; 2; …; n; j ¼ 1; 2; …; m: In all, objectives can be divided into three objective sets. The first set is D1 ; which represents the objective: the larger, the better. The second set is D2 ; which represents the objective: the smaller, the better. The third set is D3 ; which represents the objective: the closer to unity, the better. In this paper, three objectives of transmission network planning all belong to D2 : In this step, the ideal optimal scheme and the possible worst scheme are decided according to system characteristics, possible schemes and historical information, etc. Actually, it is often difficult to compare two schemes in transmission network planning. A practical method is to obtain them from n possible schemes. Let the ideal optimal scheme is E ¼ ðe1 ; e2 ; …; em Þ; the possible worst scheme is L ¼ ðl1 ; l2 ; …; lm Þ:
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Then, ej ¼ max xij ;
lj ¼ min xij j [ D1
ej ¼ min xij ;
lj ¼ max xij j [ D2
ej ¼ minlxij 2 aj l;
p is the distance parameter. It is called as Haimin distance when p ¼ 1 and Euler distance when p ¼ 2: Similarly, define Dðyi ; LÞ as the weight distance between scheme i and the worst scheme L: 2 31=p m X p Dðyi ; LÞ ¼ ui 4 ðwj lrij 2 0lÞ 5 ð15Þ
lj ¼ lxij 2 aj l j [ D3
j¼1
To calculate the optimal value of the ui ; an objective is given
4.2. Normalization The decision matrix should be normalized to eliminate the impact of different units among different objectives before the decision analysis is being done. Suppose the decision matrix Xnm \ is translated into the matrix Ynm after normalization. 3 2 y11 y12 · · · y1m 7 6 7 6y 6 21 y22 · · · y2m 7 7 6 ð10Þ Ynm ¼ 6 . .. 7 .. 7 6 . 7 6 . . . 5 4 yn1 yn2 · · · ynm For j [ D1 or D2 ; yij ¼
xij 2 lj aj 2 l j
i ¼ 1; 2; …; n
ð11Þ
For j [ D3 ; yij ¼ 1 2
lxij 2 aj l llj 2 aj l
i ¼ 1; 2; …; n
That is minimizing the sum of the weight distance to the optimum and the weight distance to the worst scheme. It can be solved by Lagrange method. The equation is derived and solved dFðUi Þ ¼0 dUi Then, ui ¼ 1þ
1 8 m 92=p X > p> > > > > ðw lr 2 1lÞ > > < j¼1 j ij = > > > > :
m X
ðwj rij Þp
j¼1
i ¼ 1; 2; …; n
ð17Þ
> > > > ;
5. Application example
4.3. Decision analysis For scheme i; let ui be the membership value to the optimum and vj be the membership value to the worst. The following equation can be deduced: ð13Þ
So, the decision problem can be translated into the problem of calculating ui for it represents the membership value to the optimum. And then, the scheme with the largest ui will be selected as the best scheme. Let the weightP vector of the objective set be W ¼ ðw1 ; w2 ; …; wm ÞT ð m i¼1 wi ¼ 1; wi is the weight of objective i:Þ and yi ¼ ðyi1 ; yi2 ; …; yim ÞT be the membership vector to the optimum for the scheme i Define Dðyi ; EÞ as the weight distance between scheme i and the optimal scheme E: 2 31=p m X ð14Þ Dðyi ; EÞ ¼ ui 4 ðwj lrij 2 1lÞp 5 j¼1
ð16Þ
ð12Þ
yij is the membership value to the optimum of objective i for the scheme j: And the matrix Y is called as the membership matrix to the optimum.
ui ¼ 1 2 v i
Min Fðui Þ ¼ D2 ðyi ; EÞ þ D2 ðyi ; LÞ
Fig. 1 shows a modified area network of Jiang Su Province in China. The voltage level is 220 kV. The solid lines in Fig. 1 are existing circuits and the dotted lines are possible additions. The corridor widths for single circuit, double circuits, triple circuits and quadruple circuits are 48.27, 68.27, 88.27 and 108.27 m, respectively. The network data are listed in Appendix A. The expected objective values are decided by both analyzing results of single objective programming and based on previous planning experiences. In this case, the expected objective value of present investment cost is chosen as Yuan 60 million (Yuan one is approximately US$1/8.6); the expected energy not supplied is chosen as 0.144 p.u.; the expected value of incremental land area is chosen as 7 km2. In the procedure of using the GA, the population of chromosomes is taken as 40. The crossover probability is taken as 0.7. The mutation probability is taken as 0.05 at the first iteration and then it decreases gradually with the increase of iteration time until at last it reaches 0.01. The number of iterations is taken as 100. After running the GA, three good schemes are generated. Table 1 is the description of three schemes. The objective values of different schemes are shown in Table 2.
T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
191
Fig. 1. Network of a 220 kV transmission system under study.
From Table 2, an ideal optimal scheme and a possible worst scheme are generated. The ideal optimal scheme is E ¼ ð56:0; 0:1421; 6:76Þ The possible worst scheme is L ¼ ð68:6; 0:1467; 8:00Þ The decision matrix is 2 3 64:6 0:1435 7:80 6 7 7 X33 ¼ 6 4 68:6 0:1421 8:00 5 56:0 0:1467 6:76 After normalization, Y33 is gained 2 3 0:3175 0:6957 0:1613 6 7 Y33 ¼ 6 1 0 7 4 0 5 1
0
1
where p ¼ 1 is used and w ¼ ð0:35; 0:35; 0:3Þ: According to Eq. (17), the membership vector to the optimum can be Table 1 Description of three schemes New transmission lines Scheme 1 Scheme 2 Scheme 3
5– 6, 6–9, 6 –13, 7–9, 8–9, 10 –11 5– 6, 6–9, 6 –13, 7–9, 8–9, 8 –9, 10–11 5– 6, 6–9, 6 –13, 8–9, 10–11
calculated U ¼ ð0:3131; 0:2248; 0:7752Þ It is observed that u3 . u1 . u2 : According to the criteria that the larger the membership value to the optimum the more optimal the scheme is, Scheme 3 is the most optimal solution. In this example, only three objectives are studied. If more objectives are considered, the efficiency of using multiobjective fuzzy decision method will be more remarkable.
6. Conclusions Transmission network planning is a multi-objective mathematical optimal problem, which should be solved by a multi-objective mathematical model. In this paper, a multi-objective mathematical optimization model for transmission network planning is introduced. Three objectives investment cost, reliability and environmental impact, are considered. A GA method is employed to solve the model, which is shown to be suitable for the multi-objective, Table 2 Objective values of different schemes Objective
F1 (Yuan £ 106)
F2 (p.u.)
F3 (km2)
Scheme 1 Scheme 2 Scheme 3
64.6 68.6 56.0
0.1435 0.1421 0.1467
7.80 8.00 6.76
F1 : investment cost; F2 : reliability index of the power network; F3 : incremental land area.
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T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
Table A1 Load data Bus no.
Pl (MW)
Bus no.
Pl (MW)
1 2 3 4 5 6 7
70 95 99 109 130 0 200
8 9 10 11 12 13
0 40 137 0 141 179
Table A2 Power supply data Bus no.
Pg (MW)
1 6 8 11
200 500 250 250
generated from the GA. Then a fuzzy decision method is adopted to select the best scheme. Results from a practical case study show that the methodology is feasible and efficient.
Acknowledgments The authors would like to acknowledge Hong Kong Polytechnic University for research grant and the cooperation of Yancheng Power Supply Bureau in this study.
Appendix A See Tables A1 – A3.
References
Table A3 Length of transmission lines Start Bus no.
End Bus no.
Length (km)
1 2 2 3 3 3 4 4 5 6 6 6 7 7 7 8 9 10 11
2 3 11 5 6 7 5 6 6 7 9 13 8 9 11 9 12 11 12
24.4 25.3 70 90 90 70 30 38 15 37 15 70 10.1 21.5 64.5 10 31.5 30 82.2
non-linear, discrete optimal problem and can avoid shortcomings of conventional mathematical optimization method. To consider the conflicts among different objectives, several possible optimal planning schemes are first
[1] Chen YL, Liu CC. Multi-objective var planning using the goalattainment methods. IEE Proc Gen, Trans Distrib 1994;141(3): 227 –32. [2] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading, MA: Addison-Wesley; 1985. [3] Miranda V, Ranito JV, Procenca LM. Genetic algorithms in optimal multistage distribution network planning. IEEE Trans Power Syst 1994;9(4):1927– 33. [4] Mori H. A genetic approach to power system topological observability. Proceedings of the IEEE International Symposium on Circuits and Systems, New York, NY, USA, IEEE; 1991. p. 1141–4. [5] Yeh E-C, Venkata SS, Sumic Z. Improved distribution system planning using computational evolution. IEEE Trans Power Syst 1996;11(2):668– 74. [6] Ignacio JR-R, Jose LB-A. Genetic algorithms applied to the design of large power distribution systems. IEEE Trans Power Syst 1998;13(2): 696 –703. [7] Esser WF, Ghose P, Chen K. Decision analysis for electric power systems engineering and management. IEEE Trans Power Apparatus Syst 1977;PAS-96(2):447–56. [8] Dhar SB. Power system long-range decision analysis under fuzzy environment. IEEE Trans Power Apparatus Syst 1979;PAS-98(2): 585 –96. [9] Enrique OC, Peter D, Pablo A, Hyde MM. Conflicting objectives and risk in power system planning. IEEE Trans Power Syst 1993;8(3): 887 –93. [10] Zhang HM, Fang YL, Liao PH. Flexible strategy approach for transmission system expansion planning. J Chin Electric Mach 1998; 18(1):48–50 in Chinese.
Multi-objective transmission network planning by a hybrid GA approach with fuzzy decision analysis T.S. Chunga,*, K.K. Lia, G.J. Chenb, J.D. Xieb, G.Q. Tangb a
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China b Department of Electrical Engineering, Southeast University, Nanjing, People’s Republic of China Received 28 March 2001; revised 8 May 2002; accepted 29 May 2002
Abstract In this paper, power system transmission network planning is formulated as a multi-objective mathematical optimization problem. In this context, three objectives: investment cost, reliability and environmental impact are considered in the optimization. To overcome the drawback of conventional mathematical optimization method in arriving at local optimum solution, a genetic algorithm approach is developed, from which several possible planning schemes are generated. This is followed by a fuzzy decision analysis method to select the optimum solution scheme. Case study results on a large 220 kV transmission network planning problem are presented to show the methodology’s feasibility and efficiency. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Transmission network planning; Genetic algorithms; Multi-objective fuzzy decision
1. Introduction Multiple objectives are often considered simultaneously in practical transmission network expansion planning. These objectives may be conflicting each other. It is difficult to find a single solution, which is optimal for all objectives. In this context, a suitable compromise solution should be determined. The multi-objective optimization method is an ideal solution to this kind of problem [1]. In conventional algorithms, multi-objective optimization problems are transformed into single objective optimization problems. These conventional algorithms are often faced with drawbacks, such as local optimum, dimension disasters, etc., which make the algorithm inefficient. Genetic algorithms (GA), as a powerful search and optimization method based on natural evolution, may overcome the drawbacks of conventional mathematical optimization methods [2]. In recent years, GAs have been used widely in power systems [3 –6]. Moreover, GAs are suitable for multi-criteria methods and can generate a set of possible optimal schemes instead of only one single optimal scheme. In this paper, a multi-objective GA is proposed, in which three selected objectives: investment cost, reliability and environmental impact, are considered. * Corresponding author. Tel.: þ 852-27666160; fax: þ 852-23301544. E-mail address: [email protected] (T.S. Chung).
In general, several possible optimal planning schemes are generated in a practical transmission network planning. Then the best scheme is selected from them considering different criteria. Because objectives are usually conflicting, it is difficult to choose the best scheme even for an experienced planning engineer especially when the objective number is large. So, a decision model for transmission network planning should be set up. Much work has been done on the decision analysis in power systems [7 –10]. In Ref. [7], possible decision analysis applications are reviewed. A systematic approach for decision making in a fuzzy environment is presented in Ref. [8]. In Ref. [9], types of conflicts among planning objectives are illustrated and categorized. In Ref. [10], a flexible decision method was proposed, which made a compromise between economical objective and applicable objective. Because the factors considered in transmission network planning are usually fuzzy, a multi-objective fuzzy decision method for transmission network planning is applied in this paper. Finally, the proposed method is illustrated using a practical example, and tested for its feasibility and efficiency. The paper is organized as follows. In Section 2, a multiobjective optimization model for transmission network planning is formulated. In Section 3, a GA is developed to solve the model. In Section 4, a fuzzy decision method is used to make the decision analysis. In Section 5, a practical system case study is presented.
0142-0615/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 0 6 1 5 ( 0 2 ) 0 0 0 7 9 - 0
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T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
Nomenclature decision vector negative deviation positive deviation objective function of optimization total investment cost of all additions expected energy not supplied per year incremental land area of the new additions expected energy not supplied per year of new network scheme
X di2 diþ F F1 F2 F3 eens
The first objective function minimizes the total investment cost of all additions. The second objective function minimizes the expected energy not supplied per year in power systems. The third objective function minimizes the incremental land area of the new additions. So, the mathematical optimal model for multi-objective transmission network planning is formulated as min F ¼
In general, the mathematical model of a multi-objective programming is formulated as min F ¼
m X
ðpi diþ þ qi di2 Þ
ð1Þ
i¼1
gi ðXÞ þ di2 ; diþ
2
$ 0;
diþ
subject to
¼ gi
di2 diþ
¼0
ð1 # i # mÞ
i¼1
eensðXÞ þ d22 2 d2þ ¼ eens n X
f ðxi Þli þ d32 2 d3þ ¼ f
i¼1
di2 ; diþ $ 0; di2 diþ ¼ 0;
i ¼ 1; 2; 3
ð1 # i # mÞ di2
where X is the decision vector; is the negative deviation, which indicates the unfinished quantity of objective i; diþ is the positive deviation, which indicates the excessive quantity of objective i; gi ðxÞ is the objective function of objective i; gi is the corresponding objective value; m is the number of objectives; pi ; qi are the priorities of objective i for the positive deviation and the negative deviation, respectively. di2 diþ ¼ 0 is the requirement of the objective programming so that one of di2 ; diþ must be 0. In this paper, three objectives ðF1 ; F2 ; F3 Þ that consider investment cost, reliability index and environmental impact index are considered. They can be expressed as follows n X
ci xi li
ð2Þ
min F2 ðXÞ ¼ eensðXÞ
ð3Þ
i¼1
min F3 ðXÞ ¼
ci xi li þ d12 2 d1þ ¼ c
where c ; eens and f are expected objective values corresponding to three objectives, respectively.
X$0
min F1 ðXÞ ¼
ð5Þ
X$0
subject to di2
ðpi diþ þ qi di2 Þ
i¼1
n X
2. Multi-objective optimization model for transmission network planning
3 X
n X
f ðxi Þli
ð4Þ
i¼1
where X ¼ ðx1 ; x2 ; …; xi ; …; xn Þ; xi is the number of possible circuit additions in path i; n is the number of right-of-way, in which circuits can be added; ci is the present investment cost of new additions per kilometer; li is the length of addition i; eensðXÞ is the expected energy not supplied per year of new network scheme. f ðxi Þ is the incremental land area after xi additions in path i is built up.
3. Application of genetic algorithms to multi-objective transmission network planning 3.1. Genetic algorithms GAs based on natural evolution, have been successfully applied to mathematical optimization problems [3 –6]. It is demonstrated that GAs are especially suitable for solving complicated optimization problems. GAs work with a set of individuals (called chromosomes), which represent possible solutions to the optimization problem. These individuals evolve through an iterative procedure, which is called reproduction. In the procedure of evolvement, three operators: selection, crossover and mutation, are used. The procedure of GAs can be depicted as 1. Chromosome coding. Each chromosome represents one possible solution to the problem. And each variable is coded using one bit (called as gene) or several bits of chromosomes. Generally, a binary coding strategy is used. 2. Initialization. The initial population of individuals is generated in this step. Each bit of individuals is generated through a random generator. 3. Calculating fitness values. The fitness function is made up of objective function and penalty items. Fitness values of initial individuals are calculated, respectively. 4. Reproduction. A new set of individuals is produced through the effects of three operators. The operator selection, selects the preferable individuals from the
T.S. Chung et al. / Electrical Power and Energy Systems 25 (2003) 187–192
prior individual set through a preliminary strategy and keeps the individual number to be constant. The purpose of using the operator crossover is to make the offspring get better combinations of genes. First, two individuals are taken out randomly from the prior individual set. Then, a random number decides whether the two individuals will make a crossover operation. If it does, another random number will decide the position of the crossover operation. The operator mutation is used to seek the optimal solution in a global space for fear that a local optimum will come up. For each individual, a random number decides that if they will make a mutation operation and which bit the operation will act on. 5. Replace the old individual set with the new one. 6. Repeat steps (3) – (5), until a satisfactory solution is reached. 3.2. Chromosome coding A binary coding strategy is adopted. Three bits (a gene group) are used to represent one possible addition path for the circuit number of one possible addition path is no more than seven. That is, the length of each individual is three times the total possible addition paths. The possible addition paths are ordered according to their connected bus numbers so that relations between each gene group and the possible addition paths can be set up by order. 3.3. Fitness function In general, GAs only consider the maximization problem. While formulation (5) is a minimization problem, a transformation should be done. The fitness function is expressed as U ¼ U0 2 F
ð6Þ
where U0 is a huge number.
189
Pgi # Pmax gi where Pgi is the output active power of generator i; Pmax is gi the output capacity of generator i; Pij is the active power flow of circuit ij; Pmax is the transmission capacity of circuit ij ij: In this paper, only single faults are considered.
4. Decision analysis using multi-objective fuzzy decision method In the procedure of planning a practical power network, several possible planning schemes could be generated. Then the best one is selected among them considering different factors. The factors concerned are so many that sometimes it is difficult for a planner to have all things considered merely by his past experiences. In this context, a scientific decision model for transmission network planning should be set up. In this paper, a multi-objective fuzzy decision method is proposed, which can take into account the fuzzy characteristics of different factors. The method is divided into three steps: boundary decision, normalization and decision analysis, which will be depicted in detail in the following paragraphs. 4.1. Boundary decision Suppose the scheme set is A ¼ {A1 ; A2 ; …; An }; the objective set is F ¼ {F1 ; F2 ; …; Fm }: A decision matrix can be formed 2 Xnm
x11
6 6x 6 21 6 ¼6 . 6 . 6 . 4 xn1
x12 x22 .. . xn2
· · · x1m
3
7 · · · x2m 7 7 7 .. 7 7 . 7 5 · · · xnm
ð9Þ
3.4. Operation simulation To calculate the expected energy not supplied per year, an operation simulation procedure should be conducted. X eens ¼ Zi Pi ð7Þ where Pi is the fault probability of circuit i: Zi is the dissatisfied energy of the network supply when the circuit i does not work for a fault, which can be calculated by the difference between load demands and the load supply capacity (LSC). The LSC can be formulated as LSC ¼ max
n X i¼1
subject to Pij # Pmax ij
Pgi
ð8Þ
xij is the value of objective j for the scheme i; i ¼ 1; 2; …; n; j ¼ 1; 2; …; m: In all, objectives can be divided into three objective sets. The first set is D1 ; which represents the objective: the larger, the better. The second set is D2 ; which represents the objective: the smaller, the better. The third set is D3 ; which represents the objective: the closer to unity, the better. In this paper, three objectives of transmission network planning all belong to D2 : In this step, the ideal optimal scheme and the possible worst scheme are decided according to system characteristics, possible schemes and historical information, etc. Actually, it is often difficult to compare two schemes in transmission network planning. A practical method is to obtain them from n possible schemes. Let the ideal optimal scheme is E ¼ ðe1 ; e2 ; …; em Þ; the possible worst scheme is L ¼ ðl1 ; l2 ; …; lm Þ:
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Then, ej ¼ max xij ;
lj ¼ min xij j [ D1
ej ¼ min xij ;
lj ¼ max xij j [ D2
ej ¼ minlxij 2 aj l;
p is the distance parameter. It is called as Haimin distance when p ¼ 1 and Euler distance when p ¼ 2: Similarly, define Dðyi ; LÞ as the weight distance between scheme i and the worst scheme L: 2 31=p m X p Dðyi ; LÞ ¼ ui 4 ðwj lrij 2 0lÞ 5 ð15Þ
lj ¼ lxij 2 aj l j [ D3
j¼1
To calculate the optimal value of the ui ; an objective is given
4.2. Normalization The decision matrix should be normalized to eliminate the impact of different units among different objectives before the decision analysis is being done. Suppose the decision matrix Xnm \ is translated into the matrix Ynm after normalization. 3 2 y11 y12 · · · y1m 7 6 7 6y 6 21 y22 · · · y2m 7 7 6 ð10Þ Ynm ¼ 6 . .. 7 .. 7 6 . 7 6 . . . 5 4 yn1 yn2 · · · ynm For j [ D1 or D2 ; yij ¼
xij 2 lj aj 2 l j
i ¼ 1; 2; …; n
ð11Þ
For j [ D3 ; yij ¼ 1 2
lxij 2 aj l llj 2 aj l
i ¼ 1; 2; …; n
That is minimizing the sum of the weight distance to the optimum and the weight distance to the worst scheme. It can be solved by Lagrange method. The equation is derived and solved dFðUi Þ ¼0 dUi Then, ui ¼ 1þ
1 8 m 92=p X > p> > > > > ðw lr 2 1lÞ > > < j¼1 j ij = > > > > :
m X
ðwj rij Þp
j¼1
i ¼ 1; 2; …; n
ð17Þ
> > > > ;
5. Application example
4.3. Decision analysis For scheme i; let ui be the membership value to the optimum and vj be the membership value to the worst. The following equation can be deduced: ð13Þ
So, the decision problem can be translated into the problem of calculating ui for it represents the membership value to the optimum. And then, the scheme with the largest ui will be selected as the best scheme. Let the weightP vector of the objective set be W ¼ ðw1 ; w2 ; …; wm ÞT ð m i¼1 wi ¼ 1; wi is the weight of objective i:Þ and yi ¼ ðyi1 ; yi2 ; …; yim ÞT be the membership vector to the optimum for the scheme i Define Dðyi ; EÞ as the weight distance between scheme i and the optimal scheme E: 2 31=p m X ð14Þ Dðyi ; EÞ ¼ ui 4 ðwj lrij 2 1lÞp 5 j¼1
ð16Þ
ð12Þ
yij is the membership value to the optimum of objective i for the scheme j: And the matrix Y is called as the membership matrix to the optimum.
ui ¼ 1 2 v i
Min Fðui Þ ¼ D2 ðyi ; EÞ þ D2 ðyi ; LÞ
Fig. 1 shows a modified area network of Jiang Su Province in China. The voltage level is 220 kV. The solid lines in Fig. 1 are existing circuits and the dotted lines are possible additions. The corridor widths for single circuit, double circuits, triple circuits and quadruple circuits are 48.27, 68.27, 88.27 and 108.27 m, respectively. The network data are listed in Appendix A. The expected objective values are decided by both analyzing results of single objective programming and based on previous planning experiences. In this case, the expected objective value of present investment cost is chosen as Yuan 60 million (Yuan one is approximately US$1/8.6); the expected energy not supplied is chosen as 0.144 p.u.; the expected value of incremental land area is chosen as 7 km2. In the procedure of using the GA, the population of chromosomes is taken as 40. The crossover probability is taken as 0.7. The mutation probability is taken as 0.05 at the first iteration and then it decreases gradually with the increase of iteration time until at last it reaches 0.01. The number of iterations is taken as 100. After running the GA, three good schemes are generated. Table 1 is the description of three schemes. The objective values of different schemes are shown in Table 2.
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191
Fig. 1. Network of a 220 kV transmission system under study.
From Table 2, an ideal optimal scheme and a possible worst scheme are generated. The ideal optimal scheme is E ¼ ð56:0; 0:1421; 6:76Þ The possible worst scheme is L ¼ ð68:6; 0:1467; 8:00Þ The decision matrix is 2 3 64:6 0:1435 7:80 6 7 7 X33 ¼ 6 4 68:6 0:1421 8:00 5 56:0 0:1467 6:76 After normalization, Y33 is gained 2 3 0:3175 0:6957 0:1613 6 7 Y33 ¼ 6 1 0 7 4 0 5 1
0
1
where p ¼ 1 is used and w ¼ ð0:35; 0:35; 0:3Þ: According to Eq. (17), the membership vector to the optimum can be Table 1 Description of three schemes New transmission lines Scheme 1 Scheme 2 Scheme 3
5– 6, 6–9, 6 –13, 7–9, 8–9, 10 –11 5– 6, 6–9, 6 –13, 7–9, 8–9, 8 –9, 10–11 5– 6, 6–9, 6 –13, 8–9, 10–11
calculated U ¼ ð0:3131; 0:2248; 0:7752Þ It is observed that u3 . u1 . u2 : According to the criteria that the larger the membership value to the optimum the more optimal the scheme is, Scheme 3 is the most optimal solution. In this example, only three objectives are studied. If more objectives are considered, the efficiency of using multiobjective fuzzy decision method will be more remarkable.
6. Conclusions Transmission network planning is a multi-objective mathematical optimal problem, which should be solved by a multi-objective mathematical model. In this paper, a multi-objective mathematical optimization model for transmission network planning is introduced. Three objectives investment cost, reliability and environmental impact, are considered. A GA method is employed to solve the model, which is shown to be suitable for the multi-objective, Table 2 Objective values of different schemes Objective
F1 (Yuan £ 106)
F2 (p.u.)
F3 (km2)
Scheme 1 Scheme 2 Scheme 3
64.6 68.6 56.0
0.1435 0.1421 0.1467
7.80 8.00 6.76
F1 : investment cost; F2 : reliability index of the power network; F3 : incremental land area.
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Table A1 Load data Bus no.
Pl (MW)
Bus no.
Pl (MW)
1 2 3 4 5 6 7
70 95 99 109 130 0 200
8 9 10 11 12 13
0 40 137 0 141 179
Table A2 Power supply data Bus no.
Pg (MW)
1 6 8 11
200 500 250 250
generated from the GA. Then a fuzzy decision method is adopted to select the best scheme. Results from a practical case study show that the methodology is feasible and efficient.
Acknowledgments The authors would like to acknowledge Hong Kong Polytechnic University for research grant and the cooperation of Yancheng Power Supply Bureau in this study.
Appendix A See Tables A1 – A3.
References
Table A3 Length of transmission lines Start Bus no.
End Bus no.
Length (km)
1 2 2 3 3 3 4 4 5 6 6 6 7 7 7 8 9 10 11
2 3 11 5 6 7 5 6 6 7 9 13 8 9 11 9 12 11 12
24.4 25.3 70 90 90 70 30 38 15 37 15 70 10.1 21.5 64.5 10 31.5 30 82.2
non-linear, discrete optimal problem and can avoid shortcomings of conventional mathematical optimization method. To consider the conflicts among different objectives, several possible optimal planning schemes are first
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