A DIFFERENTIAL EVOLUTION ALGORITHM FOR MULTISTAGE TRANSMISSION EXPANSION PLANNING T. Sum-Im(1), G. A. Taylor(1), M. R. Irving(1) and Y. H. Song(2) (1) Brunel Institute of Power Systems, School of Engineering and Design, Brunel University, UK (2) Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, UK ABSTRACT In previous research by the authors of this paper [6] a novel Differential Evolution Algorithm (DEA) was applied directly to the DC power flow based model in order to solve the static Transmission Expansion Planning (TEP) problem. The DEA performed well with regard to both low and medium complexity transmission systems as demonstrated on the Garver six-bus and IEEE 25-bus test systems, respectively. As a consequence of the successful results obtained with regard to the static TEP problem, the DEA is selected again to solve the multistage TEP problem with DC model, which is classed as a mixed integer nonlinear optimisation problem. The problem is more complex and difficult to solve than the static TEP problem because it considers not only the optimal number of lines and location that should be added to an existing network but also the most appropriate times to carry out the investment. In this paper, the effectiveness of the proposed enhancement is initially demonstrated via the analysis of the medium complexity transmission test systems as described in figures 2 and 3. The analysis is performed within the mathematical programming environment of MATLAB using both a DEA and a Conventional Genetic Algorithm (CGA) and a detailed comparison of accuracy and performance is presented. Keywords: Multistage Planning, Transmission Expansion Planning, Differential Evolution Algorithm is not interested in determining when the circuits should
1. INTRODUCTION
be installed or what investment is appropriate at the Cost-effective transmission expansion planning is a
beginning of the planning horizon. On the other hand, a
major
system
time-phased or multiple years approach considers an
optimisation problems. The main purpose of TEP
optimal expansion strategy across the whole planning
problem is to determine the optimal expansion plan of
period. Such a planning approach can be typically
the electrical transmission system. Furthermore, TEP
referred to as multistage planning. Multistage planning
should specify the new circuits that have to be added to
models have been developed by many researchers over
an existing network to guarantee adequate operation for a
the last few years [3-6]. The multistage TEP problem is a
specified planning horizon. Usually, TEP can be
very complex and large-scale problem, because it has to
categorised as static or dynamic (multistage) planning
consider not only the number of added circuits and
according to the treatment of the study period. Static
placement but also expansion timing considerations. It
planning involves a single planning horizon, whereas
requires the consideration of many variables and
dynamic planning is a derived generalisation that
constraints, and requires enormous computational effort
considers the separation of the planning horizon into
in order to achieve an optimal solution, especially for a
multiple stages.
large-scale system or real world transmission system [3].
In solving the static planning problem, the planner
Detailed examples of multistage planning models have
considers only a single planning horizon and determines
been presented by Romero et al. [4] and Escobar et al.
the number of circuits that should be added to each
[5]. An
branch of the electrical transmission system. The planner
transportation model in static and multistage TEP was
challenge
with
regard
to
power
analysis of
heuristic algorithms
for a
presented by Romero et al. [4] in 2003. In this paper, a
the most economical scheduling. The transmission
Constructive Heuristic Algorithm (CHA) was applied to
expansion investment plan is obtained with reference to
the transportation model of Garver, who was one of the
the base year. Considering an annual interest rate I, the
first researchers to apply a CHA in order to solve the
present values of the investment costs for the base year t0
transportation model. The approach was extensively
with a horizon of T stages are the following:
analysed and the excellent results obtained demonstrated
c ( x ) = (1 + I ) − ( t1 − t0 ) c1 ( x ) + (1 + I ) − (t2 − t0 ) c2 ( x )
that the CHA was an efficient and effective technique to minimise both static and multistage TEP problems. In
+ ... + (1 + I ) − ( tT − t0 ) cT ( x )
2004, Escobar et al. [5] proposed an efficient Genetic Algorithm (GA) to solve the multistage and coordinated TEP problem, which is a mixed integer nonlinear
(1)
= δ c ( x ) + δ c ( x ) + ... + δ c ( x ) 1 inv 1
2 inv 2
T inv T
where
programming problem. The proposed GA has a set of
t δ inv = (1 + I )
specialised genetic operators and utilises an efficient
− ( tt − t0 )
form of generation for the initial population that finds
Using the above relations, the multistage planning for the
high quality suboptimal topologies for large size and
DC model assumes the as following:
high complexity systems.
T ⎡ ⎤ t min v = ∑ ⎢δ inv cijt nijt ⎥ ∑ t =1 ⎣ ( i , j )∈Ω ⎦
In the last few years, DEA has been applied to solve a wide range of power system problems such as static TEP [6], short-term scheduling of hydrothermal power
where
systems [7] and power system planning [8]. In 2006,
v: ctij:
Cost
of
a
candidate
circuit
added
to
the
right-of-way i-j at stage t
problems. The authors considered only static planning, but with power system reliability, social welfare and
The present value of the expansion investment cost of the transmission system,
Dong et al. [8] presented a Differential Evolution (DE) based method for solving transmission system planning
(2)
ntij:
The number of circuits added to the right-of-way i-j of stage t
system expansion flexibility as the primary objectives in the planning problem. The results obtained were
Ω:
Set of all candidate right-of-ways for expansion
compared with a GA and an Evolution Strategy (ES). In
δ
The discount factor used to find the present value
t inv:
of an investment at stage t.
previous research by the same authors [6], a novel DEA was applied in order to solve the static TEP problem using DC model. The results obtained also demonstrated
2.2 Transmission Expansion Planning Constraints
that DEA performed well with regard to low and medium
The above objective function represents the capital costs
complexity transmission systems. In a number of
of the new transmission lines constraints must be
previous cases, DEA has proven to be reliable and
included in the mathematical formulation in order to
provides
acceptable
ensure that the solutions satisfy transmission planning
computational effort. Therefore in this paper, DEA is
requirements. The constraints are formulated in the
proposed to solve the multistage TEP problem using a
following equations (3)-(9).
optimal
solutions
with
DC model. DC Nodal Power Flow Balance 2. DC MODEL FOR THE MULTISTAGE TRANSMISSION EXPANSION PLANNING
This
linear
equality
constraint
represents
the
conservation of real power at each node.
d t + B tθ t = g t
2.1 The Objective Function
(3)
A DC model can be applied to multistage planning in
where
order to determine the financial investment according to
d t: Real power demand vector in all nodes at stage t
B t: The susceptance matrix whose elements are the
Right-of-Way
imaginary parts of the nodal admittance of existing
It is significant in TEP that planners need to know the
ones and the added lines to the existing network at
exact location and capacity of the new transmission lines.
stage t
Therefore a right-of-way constraint has to be included in
θ : Bus voltage phase angle vector at stage t
the planning problem.
g:
Real power generation vector in the existing power
defines the line location and the maximum number of
plants at stage t
lines that can be installed in a specified location. It is
t
t
Mathematically, this constraint
represented as follows: Power Flow Limit on Transmission Lines
fijt ≤ (nij0 +
t
∑n ) f s ij
max ij
0 ≤ nijt ≤ nijt ,max (4)
T
∑n
s =1
t =1
t ij
≤ nijmax
(7) (8)
In the DC power flow model, each branch element power
where
flow can be included in constraint (4) can be described as
nijt: Total number of circuits added to the right-of-way i-j at stage t
follows: (nij0 fijt
nijt,max:
t
+
=
∑ s =1
xij
nijs )
Maximum number of circuits that can be added in
the right-of-way i-j at stage t × (θit
− θ tj )
(5)
nijmax:
Maximum total number of circuits that can be
added in the right-of-way i-j
where fijt: fij
Total branch power flow in the right-of-way i-j at
Bus Voltage Phase Angle Limit
stage t
The voltage bus magnitude is not a factor in this analysis
max
nsij:
: Maximum branch power flow in the right-of-way i-j
planning. Therefore, only the voltage phase angle will be
Number of circuits added to the right-of-way i-j at
considered and included as a transmission planning
stage s (from stage 1 to stage t)
constraint. The calculated phase angle should be less
0
n ij: Number of circuits in the original base system in the right-of-way i-j xij:
as the DC power flow model is used for the transmission
than the predefined maximum phase angles as follows: θical ≤ θimax
Reactance of a circuit in the right-of-way i-j
θit: Voltage phase angle of the terminal bus i at stage t θjt: Voltage phase angle of the terminal bus j at stage t
(9)
3. OVERVIEW OF DIFFERENTIAL EVOLUTION
Power Generation Limit
A DEA is an evolutionary computation algorithm that
In this research, the power generation limits must be
was originally introduced by Storn and Price in 1995 [9].
included in the TEP constraints. They are represented as
It is a novel evolution algorithm as it employs real-coded
follows:
variables and typically relies on mutation as the search
git ,min ≤ git ≤ git ,max
(6)
where
operator. More recently DEA has evolved to share many features with CGA [2]. The major similarity between these two types of algorithm is that they both maintain
git: Real power generation at node i at stage t
populations of potential solutions and use a selection
git,min and
mechanism for choosing the best individuals from the
git,max:
Lower and upper real power generation
limits at node i at stage t respectively
population. The main differences are as follows [10]:
z z
z
DEA operates directly on floating point vectors
3.2 Mutation
while CGA relies mainly on binary strings;
After the population is initialised, the operators of
CGA relies mainly on recombination to explore the
mutation, recombination (also known as crossover), and
search space, while DEA uses mutation as the
selection create the population for the next generation
dominant operator;
P(G+1) by using the current population P(G). Once every
DEA is an abstraction of evolution at individual
generation, each parameter vector of the current
behavioural level, stressing the behavioural link
population becomes a target vector, which is compared
between an individual and its offspring, while CGA
with a mutant vector. The mutation operator generates
maintains the genetic link.
mutant vectors (Xi/) by perturbing a randomly selected vector (Xa) with the difference of two other randomly
DEA uses a population P of size NP that consists of floating point encoded individuals (10) that evolve over G generations to reach an optimal solution. At every
selected vectors (Xb and Xc). X i/(G ) = X a(G ) + F ( X b(G ) − X c(G ) ), i = 1,..., N P (13)
generation G, DEA maintains a population P(G) of NP
where a, b and c are randomly chosen indices, which a, b
vectors of candidate solutions with regard to the problem
and c ∈ {1,…,NP} and a ≠ b ≠ c ≠ i. Xa, Xb and Xc are
at hand.
selected anew for each parent vector. F is a user-defined
P ( G ) = [ X 1( G ) , ..., X
(G ) i
, ..., X
(G ) NP
constant that is known as the scaling mutation factor and
(10)
]
is typically chosen from within the range [0,2] .
The size of the population NP is held constant throughout the optimisation process. Each individual or candidate
3.3 Crossover
solution Xi is a vector, containing as many parameters
In this step a crossover process is used in the proposed
(11) as the problem decision parameters D.
DEA because it helps to increase the diversity among the
X i( G ) = [ X 1,(Gi ) ,..., X D( G,i) ]T , i = 1,..., N p
mutant parameter vectors. At the generation G, the
(11)
crossover operation generates trial vectors Xi//(G) by
3.1 Initialisation
mixing the parameters of the mutant vectors Xi/(G) with
DEA optimisation process is performed via sharing three
the target vectors Xi(G) according to a selected probability
basic genetic operations; mutation, crossover and
distribution.
selection. The first step in DEA optimisation process is the initialisation of the population of candidate solutions. Typically, each decision parameter in every vector of the
⎧ /(G ) / ⎪ X j ,i if η j ≤ CR or j = q G) X //( = ⎨ (G ) j ,i ⎪ X j ,i otherwise ⎩
(14)
initial population is assigned a randomly chosen value
where i = 1,…,NP and j = 1,…,D; ηj/ is a uniformly
from within corresponding feasible bounds.
distributed random number within the range [0,1)
min X (0) + η j ( X max − X min j ,i = X j j j )
generated anew for each value of j. q is a randomly
(12) Xj,i(0)
chosen index ∈ {1,…,D} that ensures that the trial vector th
gets at least one parameter from the mutant vector. The
parameter of the i individual of the initial population.
crossover constant CR is a user-supplied parameter,
Xjmin
th
which is usually selected from within the range [0,1].
decision parameter, respectively. ηj denotes a uniformly
This constant controls the diversity of the population and
distributed random number within the range [0,1]
aids the algorithm to escape from local optima.
where i = 1,…,NP and j = 1,…,D;
is the j
th
and
Xjmax
are the lower and upper bounds of the j
generated anew for each value of j (for each decision parameter). Once every vector of the population has been
3.4 Selection
initialised,
Finally, the selection operator is applied in the last stage
the
corresponding
fitness
calculated and stored for future reference.
values
are
of the DEA. The selection operator chooses the vectors that will compose the population in the next generation.
This operator compares the fitness of the trial vector and
predetermined convergence criterion is satisfied. The
the corresponding target vector and selects the one that
computation
provides a better solution. The fitter of the two vectors is
transmission expansion planning program is illustrated
then allowed to advance into the next generation
by the flowchart in figure 1.
processes
of
DEA
for
multistage
according to the following conditional operator: //( G ) if f ( X i//(G ) ) f ( X i(G ) ) X X i(G 1) i X i(G ) otherwise
4.3 Control Parameters Setting (15)
A suitable selection of control parameters is very significant for algorithm performance and success in
The DEA optimisation process is repeated across
finding an optimal solution. The optimal control
generations to improve the fitness of individuals. The
parameters are problem-specific [11]. So the set of
overall optimisation process is stopped whenever the
control parameters should be selected carefully for each
maximum number of generations is reached or any other
problem. In this research, parameter tuning adjusts the
predetermined convergence criterion is satisfied.
control parameters through testing until the best settings are found. In this paper, the DEA parameter settings are
4. DEA FOR THE MULTISTAGE TRANSMISSION
selected from within the following ranges of values: F = [0.5,0.9], CR = [0.55,0.95] and NP = [3*D,10*D].
EXPANSION PLANNING In this paper, DEA can be applied to solve the multistage TEP problem. The purpose of DEA is to search for an individual vector Xk that optimises the fitness function, where the fitness function is the objective function from equation (2). 4.1 Initialisation Step The first step of DEA is to create an initial population. DEA operates on a population P of individuals that represent possible solutions to the optimisation problem at hand. The population P comprises NP individuals of D decision parameters and the size of the population is chosen by the user. The population is generated according to equation (12), where each parameter of each individual is assigned a feasible value.
P [ X1 , X 2 ,..., X k ,..., X N P ]
(16)
X k [n1t , n2t ,..., nlt ,..., nDt ]
(17)
where nlt is the number of the newly added lines in the possible right-of-way l at time stage t. 4.2 Optimisation Step New individuals are generated using the mutation operator (13), crossover operator (14) and selection operator (15). The process is repeated until the maximum number of generations is reached or any other
Figure 1 DEA Applied to Multistage TEP
5. TEST SYSTEMS AND NUMERICAL RESULTS
Investment cost of stage P2 with reference to base year 2007: 2 v2 inv c2 ( x) (1 I ) (2010 2007) c2
The effectiveness of the proposed method is tested by solving the multistage TEP problem on the IEEE 25-bus
(1 0.1) 3 27.81 106
test system and 46-bus test system within the
0.7513 27.81 106 20.89 106
mathematical programming environment of MATLAB. The multistage TEP problem is analysed without the resizing of power generation for both test systems in this
Using the above investment costs v1 and v2, the total
paper.
investment cost of the IEEE 25-bus test system is v = v1 + v2 = 114.38 million US$
5.1 IEEE 25-Bus Test System The first test system used in this paper is the IEEE 25-bus system as shown in figure 2. The system
The number of additional transmission lines determined
comprises 25 buses, 36 possible rights-of-way or
by DEA is as follows:
branches and 2750 MW of load demand for the entire planning horizon [12]. The new bus is bus 25, connected
Stage P1
to bus 5 and bus 24. In figure 2, the solid lines represent
n1-2 = 2, n5-25 = 1, n7-13 = 1, n8-22 = 2, n12-14 = 2,
existing circuits in the base case topology and the dotted
n13-18 = 2, n13-20 = 2, n17-19 = 1, n24-25 = 1
lines represent new paths for circuit addition. The addition of parallel transmission lines to existing lines is permitted in this analysis and a maximum of 4 lines are
Stage P2 n1-2 = 1, n8-22 = 1, n12-23 = 3, n13-18 = 1, n13-20 = 1
permitted in each branch. Two planning stages P1 and P2 are considered in this case. The P1 stage is the duration
5.2 46-Bus Test System
from year 2007 until 2010 and year 2007 is the base year
The second test system is the 46-bus system illustrated in
for this stage. The P2 stage is the duration from year
figure 3. The system consists of 46 buses, 79 circuits and
2010 until 2012 and year 2010 is the base year for this
6880 MW of load demand for the entire planning horizon
stage. Furthermore, the total transmission expansion
[13]. In figure 3, the solid lines represent existing circuits
investment plan is obtained with reference to the base
in the base case topology and the dotted lines represent
year 2007 and an annual interest rate value I = 10 % is
the possible addition of new transmission lines. The
used to calculate the investment cost. Hence, the total
addition of parallel transmission lines to existing lines is
investment cost can be calculated by using equation (1).
again allowed in this case with a limit of 4 lines in each
The proposed DEA method can determine the best
branch. Three planning stages P1, P2 and P3 are
solution of the TEP problem for the first test system. The
considered in this case. The P1 stage is the first stage that
best solution with the present value of investment
is the duration from year 2007 until 2010 and year 2007
projected to the base year 2007 is v = 114.38 million US$
is the base year for this stage. The P2 stage is the
which can be calculated as follows:
duration from year 2010 until 2012 and year 2010 is the base year for the second stage. The P3 stage is the
Investment cost of stage P1 with reference to base
duration from year 2012 until 2015 and year 2012 is the
year 2007:
base year for the third stage. Furthermore, the total
v1 c ( x) (1 I ) 1 inv 1
(2007 2007)
(1 0.1)0 93.49 106 93.49 10
6
c1
transmission expansion investment plan is obtained with reference to the base year 2007 and the annual interest rate value I = 10 %. Hence, the total investment cost can again be calculated by using equation (1). The proposed DEA method can determine the best solution of the TEP
problem for the second test system. The solution with the
Table 1 Results of IEEE 25-Bus Test System
present value of investment projected to the base year 2007 is v = 204.16 million US$ which can be calculated
Methods
as follows:
Best cost
CPU Times
million US$
Investment cost of stage P1 with reference to base
DEA
114.38
12 min 59 sec
CGA
114.53
13 min 48 sec
year 2007: 1 v1 inv c1 ( x) (1 I ) (2007 2007) c1
Table 2 Results of 46-Bus Test System
(1 0.1)0 111.10 106 Methods
111.10 106
Best cost
CPU Times
million US$
Investment cost of stage P2 with reference to base
DEA
204.16
20 min 52 sec
CGA
212.34
22 min 25 sec
year 2007: 2 v2 inv c2 ( x) (1 I ) (2010 2007) c2 3
(1 0.1) 76.09 10
6. CONCLUSION
6
0.7513 76.09 106 57.17 106
In this paper a novel DEA optimisation method is proposed to solve the multistage TEP problem, without considering the resizing of power generation. The results
Investment cost of stage P3 with reference to base
obtained for the two transmission system test cases
year 2007:
illustrate that the DEA is efficient and effectively
3 v3 inv c3 ( x ) (1 I ) (2012 2007) c3
(1 0.1) 5 57.81 106 0.6209 57.81 106 35.89 106
minimises the total investment cost in a multistage TEP problem. As the numerical results of the test cases indicate, the total investment costs of the DEA are less expensive than the CGA for both the 25-bus and 46-bus systems. Furthermore, the DEA requires less computing
Using the above investment costs v1, v2 and v3, the total
time than CGA for calculation in both cases. As a
investment cost of 46-bus test system is
consequence of these successful results, the effectiveness of DEA will be applied in order to solve multistage TEP
v = v1 + v2 + v3 = 204.16 million US$
problems in large-scale, high complexity, real world TEP problems in the future research work.
The number of additional transmission lines determined by DEA is as follows:
Stage P1 n2-3 = 1, n5-11 = 1, n14-15 = 1, n19-25 = 1, n20-21 = 1, n24-25 = 2, n31-32 = 1, n40-41 = 1, n42-43 = 2
Stage P2 n5-6 = 2, n9-10 = 1, n31-41 = 1
Stage P3 n6-46 = 1, n26-29 = 1, n28-31 = 1, n29-30 = 1
Figure 2 IEEE 25-Bus Test System
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Figure 3 46-Bus Test System
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AUTHOR’S ADDRESS
Power Systems, vol.19, no.2, pp. 735-744, May 6.
2004.
Mr. Thanathip Sum-Im
Sum-Im, T., Taylor, G.A., Irving, M.R. and Song,
Brunel Institute of Power Systems
Y.H., “A Comparative Study of State-of-the-art
School of Engineering and Design, Brunel University
Transmission Expansion Planning Tools,” Proc. the
Uxbridge, Middlesex UB8 3PH, United Kingdom
st
41 International Universities Power Engineering
Email:
[email protected]
