transmission network optimal planning based on plant growth simulation algorithm

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2009; 19:291–301Published online 17 October 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/etep.214
*CoandChiyE-

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Transmission network optimal planning based on plant growthsimulation algorithm

Chun Wang*,y and Haozhong Cheng*

Department of Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

SUMMARY

A new method, which employs a plant growth simulation algorithm (PGSA) as optimal means, is presented in thispaper for solving the optimal planning problem of a transmission network in a single stage (horizon year). Thetransmission network planning is formulated as an integer programming problem subjected to the constraints suchas overload, N� 1 security, and right-of-way, and an efficient solving approach based upon PGSA is developed.PGSA is a new and highly efficient random search algorithm, which characterizes the growth process of plantphototropism and can quickly find high quality solutions of the combinatorial optimization problems. The testresults for two systems have verified the feasibility and efficiency of the developed PGSA based transmissionnetwork optimal planning method. Copyright # 2007 John Wiley & Sons, Ltd.

key words: transmission network planning; plant growth simulation algorithm (PGSA); random search

1. INTRODUCTION

The transmission network planning problem is a nonlinear, large-scale combinatorial optimization

problem. It draws more and more attention of the researchers. The optimal planning methods of

transmission network can be broadly divided into three categories: the heuristic methods [1]; the

traditional mathematical optimization methods such as linear programming [2], integer programming

[3], mixed integer programming [4], and Benders decomposition [5] etc; and the random search

methods, which are also called modern heuristic method and include genetic algorithms (GAs) [6,7],

simulated annealing (SA) [8], tabu search (TS) [9], and particle swarm (PS) [10], etc. The heuristic

methods are based on intuitive analysis and have the advantages of straightforwardness, flexibility, high

speed of computation, and the easy involvement of personnel in decision making. The main

disadvantage of the heuristic methods is that they are not rigorous from the mathematical viewpoint and

only can give a good design scheme based on experience and analysis [1]. The traditional mathematical

optimization methods can take into account the interaction between variables, and are more rigorous in

theory. However, because the number of network planning variables is very large and constraints are

rrespondence to: ChunWang and Haozhong Cheng, Department of Electrical Engineering, School of Electronic InformationElectrical Engineering, Shanghai Jiao Tong University, No. 800, Dongchuan Road, Shanghai 200240, People’s Republic ofna.mail: [email protected]

pyright # 2007 John Wiley & Sons, Ltd.

292 C. WANG AND H. CHENG

very complex, the traditional mathematical optimization methods are very difficult in solving

large-scale planning problems. Generally, many simplifications must be conducted in using these

approaches for solving practical planning problems. The advantages of the random search methods are

that they can solve some difficult or poorly characterized practical problems, and that they probably

find the global optimal solution, not just a good solution as the heuristic methods do. Disadvantages are

that they have to spend much of computation time in solving real large-scale systems, and that they

include some external parameters such as barrier factor, crossover rate, mutation rate, etc. It is difficult

to determine the external parameters in many cases and they affect the calculation efficiency and

convergence, which probably cause occasional computational difficulty.

In this paper, a new random search method is proposed for solving the transmission network optimal

planning problem. First, the transmission network optimal planning problem is formulated as an integer

programming problem. Secondly, a new plant growth simulation algorithm (PGSA) based method is

developed for solving this problem. Finally, the feasibility and efficiency of the developed PGSA based

transmission network optimal planning method are tested by applying it to the planning of two systems.

2. MATHEMATICAL MODEL

The transmission network optimal planning problem can be broadly categorized into static and

dynamic planning problems. The static planning problem, which is dealt with in this paper, is also

known as the single stage or horizon year planning problem. It involves determining the network

connection scheme for a particular generation and load pattern in a future horizon year. The objective of

the single stage planning problem is to determine the most economical planning scheme to meet the

load demand in the horizon year subject to the security or reliability constraints. Generally, the security

or reliability constraints are the overload constraints in the planning network. In this paper, not only the

overload constraints, but also the N� 1 security constraints are considered. The single stage (horizon

year) transmission network optimal planning problem can be expressed in Equations (1)–(3):

min f Xð Þ ¼Xni¼1

Aixi (1)

s: t:

Eu ¼ g� d

Pij j � Pmaxi

0 � xi � xmaxi

8<:

(2)

Eu ¼ g� d

Pi

�� Pmaxi

�(3)

where X is a n dimensional solution vector of the optimal planning problem; n is the number of

candidate right-of-way; xi is the ith element of X, and the value of xi denotes the number of candidate

lines added to the right-of-way i; Ai is the construction investment cost of a line that is a candidate for

addition to the right-of-way i; g is the node generator output vector; d is the node load vector; E is the

matrix whose elements are the imaginary parts of the nodal admittance matrix; Pi is the active power

flow in a circuit of the right-of-way i; Pmaxi is maximum active power that can be transmitted in a circuit

of the right-of-way i; xmaxi is upper bound constraint of circuit number that can be added in right-of-way

Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:291–301

DOI: 10.1002/etep

TRANSMISSION NETWORK OPTIMAL PLANNING BASED ON PGSA 293

i; the character above which there is a sign ‘^’ expresses the corresponding parameter while opening

any circuit in the planning network. Equation (1) is the objective function; Equation (2) is the overload

and right-of-way constraints; Equation (3) is N� 1 security constraints, which means that the planning

network should be safe and the state of all the equipments should be in desired range after the opening

of any circuit.

Moreover, the optimal scheme must be a connected network, that is, no isolated islands exist.

3. PLANT GROWTH SIMULATION ALGORITHM

The PGSA is based on the plant growth process, where a plant grows a trunk from its root; and some

branches will grow from the nodes on the trunk; and then some new branches will grow from the nodes

on the branches. Such process repeats and repeats, and stops until a plant is formed. Based on an

analogy with the plant growth process, an algorithm can be specified where the system to be optimized

first ‘grows’ beginning at the root of a plant and then continually ‘grows’ branches until the optimal

solution is found.

3.1. The growth laws of a plant

About the growth laws of a plant, the following facts have been proved by the biological experiments:

Co

(1) I

pyrig

n the growth process of a plant, the higher is the morphactin concentration of a node, the greater

is the probability to grow a new branch on the node.

(2) T
he morphactin concentration of any node on a plant is not given beforehand and is not fixed; it
is determined by the environmental information of the node, and the environmental information

of a node depends on its relative position on the plant. The morphactin concentrations of all

nodes of the plant are allotted again according to the new environment information after it grows

a new branch.

3.2. Probability model of plant growth

By simulating the growth process of plant phototropism, a probability model is established [11]. In this

model, a function gðYÞ is introduced for describing the environment of the node Y on a plant. The

smaller is the value of gðYÞ; the better is the environment of the node Y for growing a new branch. The

main outline of the model is as follows: A plant grows a trunkM from its root B0. Assuming there are k

nodes BM1, BM2, . . ., BMk that have better environment than the root B0, on the trunk M, which means

the function gðYÞ of the nodes BM1, BM2, . . ., BMk, and B0 satisfy gðBMiÞ < gðB0Þ ði ¼ 1; 2; � � � ; kÞ,then the morphactin concentrations CM1, CM2, . . ., CMk of the nodes BM1, BM2, . . ., BMk can be

calculated using Equation (4).

CMi ¼ gðB0Þ�gðBMiÞD1

ði ¼ 1; 2; � � � ; kÞ

D1 ¼Pki¼1

ðgðB0Þ � gðBMiÞÞ

8><>: (4)

The significance of Equation (4) is that the morphactin concentration of any node depends on the

relative magnitude of the gap of the environmental functions between the root and the corresponding

node in overall nodes, which really describes the relationship between the morphactin concentration

and the environment.

ht # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2009; 19:291–301

DOI: 10.1002/etep

Figure 1. Morphactin concentration state space.


From Equation (4), we can derivePk

i¼1 CMi ¼ 1, which means that the morphactin concentrations

CM1, CM2, . . ., CMk of the nodes BM1, BM2, . . ., BMk form a state space shown in Figure 1. Selecting a

random number b in the interval [0,1], b is like a ball thrown to the interval [0,1] and will drop into one

of CM1, CM2,. . ., CMk in Figure 1, then the corresponding node that is called the preferential growth

nodewill take priority of growing a new branch in the next step. In other words,BMTwill take priority of

growing a new branch if the selected b satisfies 0 � b �PT

i¼1 CMiðT ¼ 1Þ orPT�1

i¼1 CMi < b �PT

i¼1

CMiðT ¼ 2; 3; � � � kÞ. For example, if random number b drops into CM2, which meansP1i¼1 CMi < b �

P2i¼1 CMi, then the node BM2 will grow a new branch m. Assuming there are q

nodes Bm1;Bm2; � � � ;Bmq, which have better environment than the root B0, on the branch m, their

corresponding morphactin concentrations are Cm1, Cm2, . . ., Cmq. Now, not only the morphactin

concentrations of the nodes on branch m need to be calculated, but also the morphactin concentrations

of the nodes except BM2 (the morphactin concentration of the node BM2 becomes zero after it growing

the branch m) on trunk M need to be recalculated after growing the branch m. The calculation can be

done using Equation (5), which is gained from Equation (4) by adding the related terms of the nodes on

branch m and abandoning the related terms of the node BM2.

CMi ¼gðB0Þ � gðBMiÞ

D1 þ D2

ði ¼ 1; 3; � � � ; kÞ

Cmj ¼gðB0Þ � gðBmjÞ

D1 þ D2

ðj ¼ 1; 2; � � � ; qÞ

D1 ¼Pk

i¼1; i 6¼2

ðgðB0Þ � gðBMiÞÞ

D2 ¼Pqj¼1

ðgðB0Þ � gðBmjÞÞ

8>>>>>>>>>>>><>>>>>>>>>>>>:

(5)

We can also derivePk

i¼1; i 6¼2 CMi þPq

j¼1 Cmj ¼ 1 from Equation (5). Now, the morphactin

concentrations of the nodes (except BM2) on trunk M and branch m will form a new state space. (The

shape is the same as Figure 1, only the nodes are more than that in Figure 1.) A new preferential growth

node, which will grow a new branch in the next step, can be gained in a similar way as BM2.

Such process is repeated until there is no new branch to grow, and then a plant is formed.

From the viewpoint of optimal mathematics, the nodes in a plant can express the possible solutions;

gðYÞ can express the objective function; the length of the trunk and the branch can express the search

domain of possible solutions; the root of a plant can express the initial solution; the preferential growth

node corresponds to the basic point of the next searching iteration. In this way, the growth process of

plant phototropism can be applied to solve the problem of integer programming.


DOI: 10.1002/etep


4. TRANSMISSION NETWORK PLANNING BASED ON PGSA

A general procedure of the PGSA based transmission network optimal planning is as follows:

Step 1. Input data: Input all data needed in the computation, including initial network, candidate

right-of-way, etc.

Step 2. Produce an initial solution: Produce an initial solutionX0 (X0 is a vector), which corresponds

to the root of a plant, by assigning all elements of X0 to be their upper limits. Calculate the initial

objective function f ðX0Þ. Set the initial value of the basic point XB, which corresponds to the initial

preferential growth node of a plant, and the initial value of the best solution Xbest equal to X0, and set

Fbest that is used to save the objective function value of the best solution Xbest equal to f ðX0Þ, namely,

XB¼Xbest¼X0 and Fbest ¼ f ðX0Þ.

Step 3. Search for the new feasible solutions: Search for the new possible solutions by starting from

the basic point XB ¼ ½xB1 ; xB2 ; � � � ; xBi ; � � � xBn �, and carry out the check of the constraints for the found

possible solutions, then save the feasible solutions in the set of the feasible solution. In PGSA, it does

not need to construct an optimization goal by incorporating the constraints into the objective function in

barrier terms. The constraints are dealt with in the following way: for the found any possible solution,

the network connectivity constraint, and the overload constraints and N� 1 security constraints are

checked in turn. A detailed procedure about this step is shown in Figure 2.

Step 4. Save the best solution: Find the local optimal oneXminL from the feasible solutions just gained

in Step 3. Compare f ðXminL Þ with Fbest, if f ðXmin

L Þ < Fbest, then set Xbest ¼ XminL and Fbest ¼ f ðXmin

L Þ.

Step 5. Check stop criterion: Output the optimal results if satisfying the stop criterion L � Lmax,

otherwise continue Step 6. Where L is the consecutive iterative number within which the best solution

remains unchanged; Lmax is a given allowable consecutive iterative number, the choice of Lmax depends

on the size and difficulty of the solved problem.

Step 6. Calculate the probabilities of feasible solutions: Calculate the probabilities C1, C2, . . ., CN of

all feasible solutions in the set of feasible solution using the method introduced in Section 3, which

corresponds to determining the morphactin concentrations of the nodes on a plant.

Step 7. Determine a new basic point for the next iteration: Calculate the accumulating probabi-

litiesP

C1;P

C2; . . . ;P

CN of the solutions X1, X2, . . ., XN according toP

Cj ¼Pj

i¼1 Ci. Select

a random number b from the interval [0 1], b must belong to one of the intervals

0;P

C1�½ ;P

C1;P

C2�ð ; . . . ;P

CN�1;P

CN �ð , then the feasible solution, the accumulating prob-

ability of which is equal to the upper limit of the corresponding interval, will be the new basic pointXB

for the next iteration, which corresponds to the new preferential growth node of a plant for the next step.

Return to Step 3.

5. CASE STUDY

5.1. The 18-bus system

The first test system is an 18-bus system from Reference [1], which is a simplified actual system in the

western part of China. The network path schematic diagram of the test system is shown in Figure 3, in

which the solid lines express existing lines, and the dashed lines express expandable lines. Ten nodes


DOI: 10.1002/etep

Figure 2. Flow chart for searching the new feasible solutions by starting from a basic point.


DOI: 10.1002/etep


Figure 3. Network path schematic diagram.

Table I. Results of transmission network planning.

Branch Nodes of two terminals Optimal schemethat just satisfyoverload con-

straints

Optimal scheme that satisfy both overloadconstraints and N� 1 security constraints

1 2 1 2 3 4

1 1–2 0 0 1 1 1 12 1–11 1 1 2 2 1 17 4–16 1 1 1 1 1 19 5–11 0 0 0 0 0 010 5–12 1 1 1 1 2 212 6–13 1 0 1 0 1 013 6–14 2 2 2 2 2 214 7–8 1 1 2 2 2 216 7–13 0 1 1 2 1 217 7–15 0 0 1 1 1 118 8–9 1 1 2 2 2 219 9–10 2 2 3 3 3 321 10–18 0 0 1 1 1 122 11–12 0 0 1 1 1 125 14–15 1 1 2 2 2 226 16–17 2 2 2 2 2 227 17–18 1 1 1 1 1 1

Construction investment cost 1615 1615 2573 2573 2573 2573

Table II. Comparisons among three methods.

Items GA TS Proposed method

Optimal schemes Scheme 1 Schemes 1 and 2 Schemes 1 and 2Computation time (seconds) 73 26 15

Schemes 1 and 2 denote the optimal schemes 1 and 2 listed in the Columns 3 and 4 of Table I.


DOI: 10.1002/etep


Figure 4. Optimal planning scheme of 77-bus practical system (only consider overload constraints).


DOI: 10.1002/etep



and nine lines are included in the initial network. In a future horizon year, the system will be expanded

to 18 nodes. All nodal and branch data can be found in Reference [1]. For simplicity, it is supposed that

the construction investment cost of a candidate line is proportional to its length, and therefore the line

length can be used to replace the cost in comparison analysis.

Adopting the same calculation condition as that in Reference [1] and the developed PGSA based

approach, two optimal planning schemes, which satisfy the overload constraints (exclusion of N� 1

security constraints check in Figure 2), can be attained in a single run only by setting Lmax � 83. The

two optimal planning schemes have the same objective function and are listed in Table I (Columns 3

and 4). The method presented in Reference [1] only obtained optimal planning scheme 1. The computing

time for planning this system is about 15 seconds on a personal computer with a 2.2GHz processor and

256MB random access memory in a Microsoft Windows XP environment while Lmax ¼ 90.

To meet N� 1 security constraints, suppose the number of expandable line on each right-of way is 3.

Using the developed PGSA based approach, four optimal planning schemes, which satisfy both the

overload constraints and the N� 1 security constraints, are found in a single run while Lmax ¼ 200. The

four optimal schemes also have the same objective function value and are also listed in Table I

(Columns 5–8). To the best of our knowledge, there are not the other papers that have ever reported the

four optimal schemes.

For comparison, the GA and TS are also applied to solve this problem. Table II gives the results that

satisfy the overload constraints. For GA application, parameters were selected as population size to be

50, the crossover ratio to be 0.5, the mutation ratio to be 0.03. For TS application, the parameters were

selected as the neighborhood sampling number to be 36, the tabu tenure to be 30. From Table II, it is

observed that the proposed method and TS have the same best solution, while GA attains one of two

optimal schemes. The computation time of the proposed method is less than those of the GA and TS.

So, it could be concluded that the performance of the proposed PGSA based method is better than both

the GA and TS methods.

5.2. The 77-bus system

To demonstrate the applicability of the proposed methodology in large-scale transmission network

planning, it was applied to a practical transmission network planning. The network has 77 buses and 93

right-of-ways for the addition of new circuits, and the total number of circuit (including existing circuit

and addition circuit) on each right-of way is 4. The total demand for this system is 47 258.7MW.

Figure 4 gives the optimal planning scheme that only considers overload constraints. This test case

verifies that the proposed method can be well applied to practical large-scale transmission network

planning.

6. CONCLUSIONS

In this paper, a new method is proposed for the optimal planning of a transmission network. At first,

based upon the DC power-flow model, the problem is formulated as an integer programming problem,

in which both the overload and the N� 1 security constraints are considered. Then, a PGSA based

approach is developed for solving the problem.

The test results on two systems have shown that the developed PGSA based approach is feasible and

efficient, and has potential for practical applications.


DOI: 10.1002/etep


The main advantages of the proposed method over previously published random approaches are that

it is not necessary to construct an optimization goal by incorporating the constraints into the objective

function in barrier terms, which averts the trouble to determine the barrier factors and makes the

increase/decrease of constraints convenient, and that it does not need any external parameters such as

crossover rate and mutation rate in GA, and that it has a guiding search direction that continuously

changes as the change of the objective function.

7. LIST OF SYMBOLS AND ABBREVIATIONS

7.1. Symbols

b r

Copyright

andom number in the interval [0,1]

d n
ode load vector
E m
atrix whose elements are the imaginary parts of the nodal admittance matrix
f(x) o
bjective function
g n
ode generator output vector
L c
onsecutive iterative number within which the best solution remains unchanged
Lmax g
iven allowable consecutive iterative number
n n
umber of candidate right-of-way
Pmaxi m
aximum active power that can be transmitted in a circuit of the right-of-way i
X s
olution vector of the optimal planning problem
X0 i
nitial solution
XB b
asic point
Xbest b
est solution
xmaxi u
pper bound constraint of circuit number that can be added in right-of-way i
7.2. Abbreviations

PGSA p

#

lant growth simulation algorithm

ACKNOWLEDGEMENTS

This work is jointly supported by National Natural Science Foundation of China (50177017), Shanghai KeyScience and Technology Research Program (041612012), and Post-doctoral Science Foundation of China(20060400648).

REFERENCES

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7. Gil HA, Silva ELD. A reliable approach for solving the transmission network expansion planning problem using geneticalgorithms. Electric Power Systems Research 2001; 58(1):45–51.

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AUTHORS’ BIOGRAPHIES

ChunWang received his M.S. and Ph.D. from Kanazawa University, Japan, in 2000 and 2003,respectively, in Electrical Engineering. Since 2003, he has been with the Electrical Engineeringand Automation Department, Nanchang University, Nanchang, People’s Republic of China.Currently he is Post-Doctoral Researcher of Electrical Engineering at Shanghai Jiao TongUniversity. His main research interests are Power System Planning and Power SystemOptimization.

Haozhong Cheng received his Ph.D. in Electrical Engineering from Shanghai Jiao TongUniversity, Shanghai, People’s Republic of China in 1998. Now he is Professor of ElectricalEngineering at Shanghai Jiao Tong University. His research interests include Power SystemPlanning and Power Quality.


DOI: 10.1002/etep

transmission network optimal planning based on plant growth simulation algorithm

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