K. Karthikeyan1,*,
S. Kannan,1
S. Baskar, 2
C. Thangaraj3
J. Electrical Systems 9-2 (2013): 203-211
Regular paper
Application of Self-adaptive
DifferentialEvolution Algorithm to
Generation Expansion Planning Problem
JES
Journal of
Electrical
Systems
Generation Expansion Planning (GEP) is one of the most important decision-making activities in
electric utilities. Least-cost GEP is to determine the minimum-cost capacity addition plan (i.e., the
type and number of candidate plants) that meets forecasted demand within a pre specified
reliability criterion over a planning horizon. In this paper, Differential Evolution (DE), and Selfadaptive Differential Evolution (SaDE) algorithms have been applied to the GEP problem. The
original GEP problem has been modified by incorporating Virtual Mapping Procedure (VMP). The
GEP problem of a synthetic test systems for 6-year, 14-year and 24-year planning horizons having
five types of candidate units have been considered. The results have been compared with Dynamic
Programming (DP) method. The SaDE performs well than DE algorithm.
Keywords: Generation Expansion Planning, Self-adaptive Differential Evolution, Differential Evolution,
Dynamic Programming, Virtual Mapping Procedure.
I. Introduction
The Generation Expansion Planning (GEP) problem for an electric utility is the problem
of determining WHAT generation plants should be constructed, WHERE and WHEN they
should be committed over a “long-range” planning horizon [1], [2]. GEP is a challenging
problem due to its non-linear, large-scale, highly constrained, and discrete nature of the
decision variables (unit size and allocation) [2], [3].
The basic objective of the GEP is to determine the best investment plan that minimizes
present value of the investment and operating costs such that it will meet the load demand.
Earlier for finding the solution of the GEP problem, the methods like DP [3], tunnel
constrained DP [4], Branch and bound method [5], and Benders-decomposition [6] were
applied. Some of the emerging techniques for GEP problem are reviewed in [7].
Genetic Algorithm (GA) and its variants are applied to the GEP in [8], [9], [10]. Hybrid
approaches like GA with Immune algorithm [11] and DP [12] are also applied. Eight metaheuristic techniques were applied and the results are compared with DP in [13]. The authors
concluded that Differential Evolution (DE) [14] outperformed other meta-heuristic
techniques. The Opposition based DE (ODE) has been applied in [15]. The Elitist Nondominated Sorting Genetic Algorithm-II (NSGA-II) has been applied to multiobjective GEP problem [16]-[17]. The environmental constraints and impact of various
incentive mechanisms are considered in [18]-[20]. The effectiveness and efficiency of capand-trade and carbon tax policies are compared in a GEP framework [21]. The game theory
and modified game theory along with Meta-heuristic techniques has been applied to GEP in
partially deregulated and Competitive environment in [22]-[26].
In addition to GEP, DE has been widely applied in variety of fields [27], [28]. Even
though it produces better results, it has certain weaknesses. One of these weaknesses is that
for a specific problem choosing the best among different mutation strategies and its
associated control parameters, scaling factor F and crossover rate (CR) are very difficult
[29]. It needs time consuming trial and error procedure. In order to avoid this extensive
*
Corresponding author: K. Karthikeyan, Department of Electrical Engineering, Kalasalingam University,
Krishnankoil – 626126, Tamil Nadu, India. Email: karthikeyanju@gmail.com
1
Department of Electrical Engineering, Kalasalingam University, Krishnankoil – 626126, Tamil Nadu, India
2
Department of Electrical Engineering, Thiagarajar College of Engineering, Madurai, Tamil Nadu, India
3
Former Vice-chancellor of Anna University of Technology Chennai, Chennai, Tamil Nadu, India
Copyright © JES 2013 on-line : journal/esrgroups.org/jes
J. Electrical Systems 9-2 (2013): 203-211
computational time, P. N. Suganthan et.al, proposed Self-adaptive Differential Evolution
(SaDE) [30].
In this paper, SaDE has been applied to the GEP problem and the results are compared
with DP and DE. The rest of the paper is organized as follows: Section II describes the GEP
problem formulation and Section III describes implementation of the GEP problem. Section
IV provides test results, and section V concludes.
II. Generation Expansion Planning (GEP) Problem Formulation
The GEP problem is equivalent to finding a set of best decision vectors over a planning
horizon that minimizes the investment and operating costs under relevant constraints.
A. Cost Objective
The cost objective is represented by the following expression
T
min Cost = ∑ [I (U t ) + M ( X t ) + O( X t ) - S (U t )]
(1)
t =1
where,
X t = X t -1 + U t
I (U t ) = (1 + d )
(t = 1,2,...T )
-2t
(2)
N
∑ (CI i × U t ,i )
(3)
i =1
S (U t ) = (1 + d )
-T '
N
∑ (CI i × δi × U t ,i )
(4)
i =1
M (X t ) =
∑ ((1 + d )
1
1.5 + t ' + s '
s '= 0
O ( X t ) = EENS × OC ×
(∑ ( X
∑ ((1 + d )
1
s '= 0
t
× FC) + MC
1.5 + t ' + s '
)
))
(5)
(6)
The outage cost calculation of (6), used in (1), depends on Expected Energy Not Served
(EENS). The equivalent energy function method [1] is used to calculate EENS (and also to
calculate loss of load probability, LOLP, used in the constraint objective).
t ' = 2(t - 1) and T ' = 2 × T - t '
(7)
and
total cost, $;
Cost
N-dimensional vector of introduced units in the stage t (1 stage = 2 years);
Ut
the number of introduced units of type i in stage t;
Ut, i
cumulative capacity vector of existing units in stage t, (MW);
Xt
is the investment cost of the introduced unit at the t-thstage, $;
I(Ut)
M(Xt) total operation and maintenance cost of existing and the newly introduced
units, $;
variable used to indicate that the maintenance cost is calculated at the middle
s'
of each year;
O(Xt) outage cost of the existing and the introduced units, $;
S(Ut) salvage value of the introduced unit at t-thinterval, $;
discount rate;
d
capital investment cost of i-thunit, $;
CIi
salvage factor of i-thunit;
δi
length of the planning horizon (in stages);
T
total number of different types of units;
N
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K. Karthikeyanet al: Application of SaDE Alg to Gen. Expansion Planning Problem
FC
MC
EENS
OC
fixed operation and maintenance cost of the units, $/MW;
variable operation and maintenance cost of the units, $;
Expected energy not served, MWhrs;
value of outage cost constant, $/ MWhrs
B. Constraints
1) Construction limit:Let Utrepresent the units to be committed in the expansion plan at
stage t that must satisfy
0 ≤ U t ≤ U max, t
(8)
whereUmax,t is the maximum construction capacity of the units at stage t.
2) Reserve Margin:The selected units must satisfy the minimum and maximum reserve
margin.
(1 + R min ) × Dt ≤
N
∑ X t ,i ≤ (1 + Rmax ) × Dt
(9)
i =1
where
Rmin minimum reserve margin;
Rmax maximum reserve margin;
demand at the t-thstage in megawatts (MW);
Dt
cumulative capacity of i-thunit at stage t.
Xt,i
3) Fuel Mix Ratio:The GEP has different types of generating units such as coal, liquefied
natural gas (LNG), oil, and nuclear. The selected units along with the existing units of each
type must satisfy the fuel mix ratio
j
FM min
≤ X t, j
N
∑ X t, i ≤
i
where
j
FM min
=1
j
FM max
j = 1, 2, ..., N
(10)
minimum fuel mix ratio of j-thtype;
j
maximum fuel mix ratio of j-thtype;
FM max
type of the unit (e.g., oil, LNG, coal, nuclear).
j
4) Reliability Criterion:The introduced units along with the existing units must satisfy a
reliability criterion on loss of load probability (LOLP)
LOLP( X t ) ≤ ε
(11)
where ε is the reliability criterion, a fraction, for maximum allowable LOLP. Minimum
reserve margin constraint avoids the need for a separate demand constraint.
III. Implementation of the GEP problem
A. Virtual Mapping Procedure (VMP)
To improve the convergence characteristics of the algorithms VMP is introduced in the
GEP problem. VMP is concerned with the solution representation; it transforms each
combination of candidate units for every year into a dummy decision variable, referred to as
the VMP variable [13]. The value of the VMP variable for a specific candidate solution is
the rank of that solution when all solutions are sorted in ascending order of capacity.
To illustrate, consider the range of the decision vector lies between [0] and Umax,t as
given in [9]. The capacity vector is [200 450 500 1000 700]. Let a candidate solution be
U1= [5 0 1 1 0]; then its corresponding capacity will be 2500 MW (5×200 + 0×450 +
1×500 + 1×1000 + 0×700). If U1 changes to [5 0 1 2 0] (via mutation operators), then the
capacity is increased from 2500 MW to 3500 MW, a difference of 1000 MW, which is a
large deviation.
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J. Electrical Systems 9-2 (2013): 203-211
The solution [5 0 1 1 0] with capacity of 2500 has VMP value of 190. A change (by
mutation) to VMP value of 198 would correspond to a solution of [2 1 2 0 1] with capacity
of 2550, a relatively small change in capacity. Without VMP, to get a solution with
capacity of 2550 MW, all 5 variable values must be changed. This may take a very large
number of iterations, and in general, moving from sub-optimal to optimal may take much
iteration.
Since five different types of units are assumed to be available for each stage, the size of
the decision vector increases by multiples of 5 as the number of stages increases. However,
when using VMP, the number of decision variables obtained is a multiple of its number of
stages; the size of the decision vector becomes 3 for a 3-stage problem. Thus, a size
reduction of 80%, [(15-3)/15, for 3 stages] is realized.
Hence, the dimensionality of the problem, in terms of the number of decision variables,
is reduced, and computational time and memory requirements reduce accordingly.
B. Differential Evolution (DE)
Price and Storn [14] proposed DE. It is a population-based stochastic direct search
algorithm like GA but differs from GA with respect to the mechanics of mutation,
crossover and selection.
At each iteration J, DE employs the mutation and crossover operations to produce a trail
vector Vi,J for each individual vector Xi,J, also called target vector, in the current population.
Based on the mutation rule, the DE has been classified into 6 different strategies [14].
They are:
i) DE/best/1
U i,J +1 = X best + F × ( X r1, J − X r 2 , J )
(13)
ii) DE/rand/1
U i,J +1 = X r 3 ,J + F × ( X r1, J − X r 2 , J )
(14)
iii) DE/rand-to-best/1
U i,J +1 = X i, J + F × ( X best − X i, J ) + F × ( X r1, J − X r 2 , J )
(15)
iv) DE/best/2
U i,J +1 = X best + F × ( X r1, J − X r 2 , J + X r 3 , J − X r 4 , J )
(16)
v) DE/rand/2
U i,J +1 = X r 5 , J + F × ( X r1, J − X r 2 , J + X r 3 , J − X r 4 , J )
(17)
vi) DE/Current-to-rand/1
U i,J +1 = X r1, J + F × ( X r1, J − X r 2 , J ) + K × ( X r 3 , J − X i, J )
(18)
where,
X
is the set of population
Ui, J+1 is the mutated i-th individual for the next iteration
Xi,J
is i-th individual of current iteration
is the best individuals among the population X
Xbest
F
is a constant [0, 2]
K
is a constant taken equal to F
J
is the current iteration
Xr1,J, Xr2,J, Xr3,J, Xr4,J, and Xr5,J are the randomly selected populations in
the current iteration.
After mutation, the exponential crossover is applied on the individuals by the rule
⎧⎪U k ,i,J +1 if (rand j [0,1] ≤ CR) or (k = k rand )
, k = 1,2,..., n
(19)
Vi,J +1 = ⎨
⎪⎩X k,i,J otherwise
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K. Karthikeyanet al: Application of SaDE Alg to Gen. Expansion Planning Problem
where, CR is the crossover rate [0, 1]
The parents for the next iteration are selected based on one-to-one greedy selection
scheme for minimization problems such as GEP as follows:
if f (Vi,J +1 ) > f ( X i,J +1 )
⎧⎪Vi,J +1
X i,J + 1 = ⎨
(20)
if f ( X i ,J +1 ) > f (Vi,J +1 )
⎪⎩ X i,J +1
where,
f(Vi,J+1) is the fitness function value of i-th individual of the population in which
the mutation and crossover operators are applied
f(Xi,J+1) is the fitness function value of i-thindividual in the original population
The loss of best individuals in the subsequent iteration is avoided by this selection
mechanism, as the best individuals replace the worst individuals.
C. Self-adaptive Differential Evolution (SaDE)
SaDE [29-31] probabilistically selects one out of several available learning strategies
(“DE/rand/1”, “DE/rand-to-best/2”, “DE/rand/2”, and “DE/current to rand/1”) for each
individual in the current population based on their previous experiences of generating
promising solutions.
The initial probabilities pi, i=1, 2…numst are taken to be equal, so that each strategy
will have equal probability to be applied to every individual in the initial population. Then
the probability of pi is updated as: [33]
௡௦೔
‫݌‬௜ ൌ
(21)
where,
nsi
௡௦೔ ା௡௙೔
the number of trial vectors successfully entering the next iteration while
generated by each strategy
the number of trial vectors discarded while generated by each strategy.
nfi
The number of population NP is taken as user specified value so as to deal with
different type of problem. The value of F is taken with normal distributions of mean 0.5
and standard deviation 0.3 for different individuals in the current population. The value of
CR is taken with normal distributions of mean CRm and standard deviation 0.1 and CRm is
dynamically adapted based on the previous learning experience. [30-31].
IV. Test Results
All the algorithms DE, SaDE and DP were implemented using MATLAB version 7.2,
on a desktop PC with Intel Core i5 processor having 3.10GHz speed and 4 GB RAM.
A. Test System Description
The forecasted peak demand and other technical data are taken from[9]. The test system
with 15 existing power plants and 5 types of candidate options is considered for a 6-years,
14-years and 24-years planning horizon. The planning horizon comprises of stages with 2year intervals.
B. Parameters for GEP
The lower and upper bounds for reserve margin are set at 20% and 40% respectively.
The salvage factor (δ) for oil, LNG, coal, PWR, and PHWR are taken as 0.1, 0.1, 0.15, 0.2
and 0.2, respectively. Cost of EENS is set at 0.05 $/kWh. The discount rate is 8.5%. It is
assumed that the date of availability of new generation is two years beyond the current date.
The year k investment cost is assumed to incur in the beginning of year k; the year k
maintenance cost is assumed to incur in the middle of year k and is calculated by using the
equivalent energy function method [1]. The year k salvage cost is estimated at the end of
the planning horizon.
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J. Electrical Systems 9-2 (2013): 203-211
C. Parameters for DE and SaDE
The best parameters for the algorithms DE and SaDE are chosen through 50
independent test runs. Their values for the entire planning horizon are tabulated in Table 1.
The number of initial populations is chosen based on the number of VMP decision variable,
n. In SaDE, scaling factor F and crossover rate CR are initialized to 0.5 and 0.5
respectively.
Table 1. Best Parameters for DE and SaDE Algorithm
Sl. No
Algorithm
DE
20×n
20000×n
DE/rand/1/bin
0.5
0.5
Parameter
1
2
3
4
5
Number of Populations NP
Maximum function evaluations
Mutation Strategy
Scaling Factor F
Crossover rate CR
SaDE
10×n
20000×n
Adaptive
N(0.5,0.3)
Adaptive
D. Results and Discussion
The results obtained for the 6-year planning horizon with VMP is given in Table 2. The
solution obtained by the DP is the optimal solution and it is taken as the reference for
calculating the error percentage.
Table 2. Results for 6-year Planning Horizon
Cost x 1010 $
Best
Worst
1.2009
1.2009
1.2009
1.2009
1.2009
Algorithm
DE
SaDE
DP
Error (%)
SR (%)
0
0
0
100
100
100
Mean Execution Time
(in Seconds)
171.241
116.626
6.1
Since the planning horizon is shorter, in other words the dimensionality of the problem
is less; both DE and SaDE are performing better. The Success Rate (SR) is the ratio of the
number of times the optimal solution found to the number of test runs.
The DE and SaDE results for the 6-year planning horizon with different population size
are tabulated in Table 3. In this case all other parameters for DE and SaDE are kept same.
Even for smaller population size say 10×n (30 candidates), the SaDE finds the optimal
solution with the SR of 90%.
Table 3. Results for 6- year Planning Horizon with different population size
Cost x 1010 $
Population size
20×n*
10×n
3×n
Algorithm
SaDE
DE
SaDE
DE
SaDE
DE
Best
Worst
1.2009
1.2009
1.2009
1.2009
1.2009
1.2009
1.2009
1.2009
1.2012
1.2014
1.2014
1.2024
Error (%)
SR (%)
0
0
0-0.03
0-0.04
0-0.04
0-0.12
100
100
90
70
50
30
* ‘n’ is the number of VMP variable, for 6-year planning horizon n=3
Since the error percentage is less and SR is more than 90%, to reduce the computational
time the population size are changed to 10×n for the 14-year and 24-year planning
horizons. SaDE needs time to find the best mutation strategy and its associated parameters
during its run, hence increasing the iterations and function evaluations are best rather than
increasing the population size [31].
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K. Karthikeyanet al: Application of SaDE Alg to Gen. Expansion Planning Problem
The results for the 14-year and 24-year planning horizons with VMP are summarized in
Table 4 and Table 5 respectively. The SaDE performs better with less error rate, since its
learning strategies and the associated parameters are automatically adapted during the
evolving process.
Table 4. Results for 14- year Planning Horizon
Algorithm
DE
SaDE
DP
Cost x 1010 $
Best
Worst
2.182
2.1943
2.1808
2.1920
2.1797
Error (%)
SR (%)
0.11-0.67
0.05-0.56
0
0
0
100
Mean Execution Time
(in Seconds)
1080.7
779.562
1725.2
Since the execution time required for running the Dynamic Programming is
exponentially increases with increase in dimensionality of the program, it is very difficult to
get the results for the 24-year planning horizon. In [13], it has been given that for 24-year
planning horizon the best cost value is 2.9206×1010. SaDE provides better result than this
result. From the Table 5, it can be clearly said that even the worst cost value of the SaDE is
better than the best cost value of the DE.
Table 5. Results for 24- year Planning Horizon
Algorithm
DE
SaDE
DP
Mean Execution Time
Cost x 1010 $
Error (%)*
SR (%)
(in Seconds)
Best
Worst
2.925
2.9527
0.64-1.59
3941.61
2.9065
2.9186
0-0.42
3010.7
Unknown
*2.9065 have been considered as the best and error has been calculated.
V. Conclusions
The Differential Evolution (DE) and Self-adaptive Differential Evolution (SaDE) have
been applied to solve the GEP problem. The optimal solution is obtained by incorporating
Virtual Mapping Procedure. The results were validated with DP. The performance of the
DE and SaDE are compared in terms of their success rate and error percentage. For the
shorter planning horizon (6-year), all the algorithms are able to find the optimal solution.
For the longer planning horizon (14-year), and very long planning horizon (24- year), SaDE
performs very well and produces better results than DE. Since the mutation strategy and its
control parameters are adapted in the SaDE according to the number of successful
candidates entering the next iteration, it is able to perform better. SaDE is able to perform
better than DE even with lesser population size.
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Acknowledgment
The authors gratefully acknowledge the management of Kalasalingam University, Krishnankoil and
Thiagarajar College of Engineering, Madurai, Tamil Nadu, India for their constant support and encouragement
extended during this research. They are also thankful to the Department of Science and Technology, New Delhi,
for its support through the project: SR/S4/MS/364/06. The MATLAB code for the basic DE was taken from R.
Storn website (http://www.icsi.berkeley.edu/~storn/code.html#matl). Thanks to Professor. P. N. Suganthan,
Associate Professor, School of Electrical and Electronic Engineering, Nanyang Technological University,
Singapore for providing the MATLAB code for SaDE through mail.
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