A Hybrid Method for MultiArea Generation Expansion using Tabusearch and Dynamic Programming
ABSTRACT This paper combines Tabu search with an optimization technique using dynamic programming for the solution of generation expansion and placement considering reliability in multiarea power systems. Instead of random selection, initial solution for Tabu search is obtained from optimizing a simplified problem utilizing dynamic programming and reliability assessment technique called global decomposition. The comparison between random initial solutions and the proposed method is made. The method is implemented for an actual 12area power system.
 [Show abstract] [Hide abstract]
ABSTRACT: This dissertation aims to address two optimization problems involving power system reliabilty analysis, namely multiarea power system adequacy planning and transformer maintenance optimization. A new simulation method for power system reliability evaluation is proposed. The proposed method provides reliability indexes and distributions which can be used for risk assessment. Several solution methods for the planning problem are also proposed. The first method employs sensitivity analysis with Monte Carlo simulation. The procedure is simple yet effective and can be used as a guideline to quantify effectiveness of additional capacity. The second method applies scenario analysis with a statespace decomposition approach called global decomposition. The algorithm requires less memory usage and converges with fewer stages of decomposition. A system reliability equation is derived that leads to the development of the third method using dynamic programming. The main contribution of the third method is the approximation of reliability equation. The fourth method is the stochastic programming framework. This method offers modeling flexibility. The implementation of the solution techniques is presented and discussed. Finally, a probabilistic maintenance model of the transformer is proposed where mathematical equations relating maintenance practice and equipment lifetime and cost are derived. The closedform expressions insightfully explain how the transformer parameters relate to reliability. This mathematical model facilitates an optimum, costeffective maintenance scheme for the transformer.01/2008;
Page 1
1
TO BE PRESENTED AT 2006 POWERCON, CHONGQING, CHINA, OCT 2006
A Hybrid Method for MultiArea Generation
Expansion using Tabusearch and Dynamic
Programming
Panida Jirutitijaroen, Student Member, IEEE, and Chanan Singh, Fellow, IEEE
Abstract— This paper combines Tabu search with an
optimization technique using dynamic programming for the
solution of generation expansion and placement considering
reliability in multiarea power systems. Instead of random
selection, initial solution for Tabu search is obtained from
optimizing a simplified problem utilizing dynamic programming
and reliability assessment technique called global decomposition.
The comparison between random initial solutions and the
proposed method is made. The method is implemented for an
actual 12area power system.
Index Terms— Multiarea Power System, Reliability, Tabu
Search, Power System Optimization, Global Decomposition,
Generation Adequacy, Dynamic Programming.
I. NOMENCLATURE
A. Indices
s
t
i,j
I
Source node
Sink node
Network nodes
{1,2,…,n} Set of network nodes
B. Parameters
Capacity of existing generation arc i (MW)
Cost of an additional unit at node i ($/MW)
C
Capacity of an additional generator (MW)
G
N
Total number of additional generators
iL
Capacity of load arc i (MW)
ij T
Capacity of existing tie line arc ij (MW)
ij a
Cost of a tie line between nodes i and j ($/MW)
C
Capacity of an additional tie line (MW)
T
N
Total number of additional tie lines
R Total available budget ($)
n
Number of areas in the network
i G
ia
G
G
T
T
This work was supported by Power System Engineering Research Center
and NSF Grant No. ECS0406794.
P. Jirutitijaroen and C. Singh are with the Department of Electrical and
Computer Engineering, Texas A&M University, College Station, TX 77843
USA (email: pjirut@ece.tamu.edu; singh@ece.tamu.edu).
C. Decision variables
X
Flow from node i to j
X
Number of additional generators at node i, integer
X
Number of additional tie lines between nodes i and j
where (i,j) = (j,i), integer
ij
G
i
T
ij
II. INTRODUCTION
Multiarea reliability analysis has two major approaches,
MonteCarlo simulation and statespace decomposition. In
MonteCarlo simulation, failure and repair history of
components are created using their probability distributions
and reliability indices are estimated by statistical inferences.
The idea of statespace decomposition [8] [10] [11] [12] [13],
is to efficiently classify the system state space into three sets;
acceptable sets (A sets), unacceptable sets (L sets), and
unclassified sets (U sets) while the reliability indices are
calculated concurrently. The
Decomposition is based on the fact that decomposition
depends on state capacities and not state probabilities. With
this approach, the additional generators in prospective areas
can be included in the system for one time decomposition.
This global state space is valid for all generation
combinations. The unavailability (forced outage rate) of
additional generators is also considered in the formulation.
The major advantage of this technique is that decomposition
is performed only once. Reliability indices of each
combination can be evaluated by allocating zero probability to
the omitted states.
Both classical and heuristic optimization techniques have
been applied to solve generation expansion problem [1] [2] [3]
[4] [5] [6] [7]. In particular, [1] proposes dynamic
programming to optimally locate the prospective generators in
multiarea power systems
decomposition as a reliability evaluation tool. One of the
contributions of [1] is to explicitly derive loss of load
probability equation in terms of decision variables that is the
number of additional generators in the system. However, in
this approach the optimization is applied to a subset of state
space and may not guarantee global optimality since the
reliability index is simplified and approximated.
Tabu search [14] is one of many heuristic techniques
applied to generation expansion problem. It has been
concept of Global
while utilizing global
Page 2
2
recognized as an efficient method for combinatorial
optimization problems. The algorithm is powerful due to the
flexible forms of memory in the search space. The search
performance, however, depends on a good starting solution.
This paper combines Tabu search with the solution from [1] to
obtain optimal solution. The comparison between using
randomly generated starting solutions and solution from [1] is
made. In the following, problem formulation is first
introduced. Next, concept of global decomposition is
described. Tabu search algorithm and parameters are
presented. The method is then implemented for an actual 12
area power system. Concluding remarks are given in the last
section.
III. PROBLEM FORMULATION
For multiarea reliability evaluation, power system is
modeled as a network flow problem where each node in the
network represents an area in the system and each arc
represents tie line connection between areas. Source and sink
nodes are introduced to represent generation capacity and load
as shown in Fig. 1. The capacity of every arc in the network is
a random variable because generation, tie line and load
capacity are random with discrete probability distributions.
For computational efficiency, all arc capacities are rounded
off to a fixed increment.
icv
Fig. 1. Power System Network Capacity Flow Model
The decision variables of network flow problem are integer
as the number of additional generators is an integer value. The
standard formulation is derived in the following with the
objective of minimizing loss of load probability subject to cost
and network capacity constraints. The additional generators
have capacity of
MW. The objective function to
minimize loss of load probability is given below.
Min
( )
{
t nttt
XXXXXf
<=
:,...,,Pr
121
The problem has the following constraints;
Capacity constraints
−
Flow in generation arc
GX
≤
−
Flow in tie line
TXX
+≤−
−
Flow in load arc
i it
LX ≤
Conservation of flow at node i in the network
jisi
XX
=+
∑∑
∈
≠
Maximum number of additional generators
G
C
n ntt
LXLXL
<<
UU
...
221
}
(1)
G
i
G
isi
XC
+
Ii∈∀
(2)
T
ij
T
ij ijji
XC
jiIji
≠∈∀
,,
(3)
Ii∈∀
(4)
it
ij
Ij
ij
ij
Ij
XX
+
≠
∈
Ii∈∀
(5)
G
Ii
G
i
NX
=
∑
∈
(6)
Maximum number of additional transmission line
X =
∑∑
∈∈
T
ji
IiJj
T
ij
N
≠
(7)
Budget constraint
Xa
Ii
∈
RXa
ji
IiJj
T
ij
T
ij
G
i
G
i
≤+∑∑
∈
∑
≠
∈
(8)
Non negativity
0,,
≥
T
ij
G
iij
XXX
Iji
∈∀ ,
(9)
The expression within the parenthesis in equation 1
represents the system loss of load event. The problem is thus
formulated to minimize the loss of load probability index
subject to cost constraint. If the optimal system reliability
obtained through this process does not satisfy the
requirements, the cost constraint can be relaxed to allow more
additional generators in the system and the LOLP can be re
optimized
All possible additional generation units are included in each
prospective area of the system before performing global
decomposition. In the global decomposition process,
constraints (2) to (7) and (9) have already been included. The
problem has only one additional constraint which is the
budget constraint (8).
IV. CONCEPT OF GLOBAL DECOMPOSITION
The system state space consists of generation states in each
area and interarea tie line states. It is defined as (10).
⎡
=Ω
mm
K
21
where
M
Maximum state of arc k
m
Minimum state of arc k
N Number of arcs in the network
A system state, y, can assume any value between its
minimum and maximum state as shown in (11).
[
yyy
K
21
where
kkk
Mym
≤≤
ky
State of arc k
Global decomposition approach analytically partitions the
state space into the following three different sets of states.
1. Sets of acceptable states (A sets): The success states
that all area loads are satisfied.
2. Sets of unacceptable states (L sets): The failure
states or Loss of load states that some area loads are
not satisfied.
3. Set of unclassified states (U sets): The states that
have not been classified into A or L sets.
The process of partitioning the state space into A and L sets
involves determining maximum flow in the network. Ford
Fulkerson algorithm is implemented with breadthfirst search
for finding existing flow in the system. At the beginning of the
decomposition, state space (Ω, as in (10)) is the first U set,
unclassified set. At every step of decomposition, one A set, N
L sets and N U sets are generated from one U set. The A sets
⎥⎦
⎤
⎢⎣
N
N
m
MMM
K
21
(10)
k
k
]
N
y
=
(11)
……. …….
ijtv
Power system network
Tie line between areas
st
L
v
Page 3
3
will be deleted to minimize memory usage since the goal of
this evaluation is to extract all L sets for system loss of load
probability calculation. The U sets have to be kept and
partitioned further to A, U and L sets.
The concept of global decomposition is based on the fact
that decomposition depends only on state capacities and not
the state probabilities. This allows us to include the maximum
possible number of generators in each area in the
decomposition. The sets obtained from this state space are
valid for all scenarios of additional generators. Probability of
each scenario can then be evaluated by allowing zero
probability for the excluded states because of the omission of
corresponding additional generators included in the original
decomposition.
V. TABU SEARCH ALGORITHM
Tabu search is an intelligent search procedure that has been
widely applied to combinatorial optimization problems. The
procedure starts with an initial solution. Neighborhood
solutions are then created by some prespecified neighborhood
function. Objective function value of these neighborhood
solutions is evaluated. The decision on moving from current
solution to the next solution is made based on adaptive
memory in Tabu list and current aspiration level. This list is
vital since it prevents cycling in the search procedure.
In this application, neighborhood function is simply a
random sampling of location to add and drop one generator.
Objective function is calculated from global decomposition
technique. From computational experiment, it is efficient to
sample 8 neighborhood solutions and keep 3 moves in Tabu
list. The algorithm is presented in the following.
Initialization,
0
=
k
o Initial feasible solution,
o Initialize Tabu list,
T , and aspiration level,
o Initialize best solution,
x
where
k
ix
Number of additional units in area i at iteration k
k
xv
Current solution vector at iteration k
k
T
Tabu list
k
A
Aspiration level at iteration k which is the reliability
Index in this application
() ⋅
f
Objective function value of solution vector
While iteration
maximum iterations do the following,
<
k
Step 1. Generate neighborhood solutions, {
where
{
Set of neighborhood solutions
() ⋅
N
Neighborhood function generation
Step 2. Compute objective function values,
the best neighborhood solution,
where
(
xfx
minarg
=
Step 3. Check move with
T
o If not in
T
:
−
xx
←
,
fA
←
[]
A =
00
2
0
1
0
nxxxx
K
v
=
0
( )
x
00
f
0*
x
vv←
}()
1
−
⊂
knbhd
xNx
vv
}
nbhd
xv
(){
}
nbhd
xfv
and find
bestnbhd
x
,
v
){}
nbhd bestnbhd
vv
,
1
−
k
1
−
k
bestnbhdk
,
vv
()
bestnbhdk
x
,
v
, and update
k
T
−
If
x
Advance
k
T
Check aspiration criteria, if (
bestnbhdk
xx
←
,
x
update
T
Otherwise, advance
Note that in this application, a move is stored as number of
the area that a unit is added in the solution vector. A move is
checked by comparing the area that a unit is dropped in the
solution vector with number of areas in the Tabu list. This
criterion prevents cycling since it checks whether the area that
a generator is dropped has a generator added in the previous
iteration or not. If a generator has been added to this area in
previous iteration, we rather not drop it out in current
iteration.
()
( )
x
*,
fx
x
f
bestnbhd
vv
v
<
then update best solution,
bestnbhd
,*
v←
−
1
+← kk
and go to step 1.
o If in
−
1
−
:
)
1,
−
<
kbestnbhd
Axfv
A
, then
)
, and
,
vv
best nbhd
x
,*
vv←
,
(
bestnbhdk
xf
,
v
←
k
−
and go to step 1.
1
+← kk
VI. A TWELVEAREA TEST SYSTEM
A 12area power system is shown in Fig. 2. The test system
is a multiarea representation of an actual power system [9]
that has 137 generation units and 169 tie line connections
between areas. Transfer capabilities between areas are shown
in Appendix A. All the tie lines in the system is assumed to
have a mean repair time of 8 hours and a failure rate of 10 per
year.
Fig. 2. Twelve Area Test System
TABLE I shows area generation and loads as well as
availability and cost per generator in prospective areas which
are areas 1 to 5, and 9 to 12. It is assumed that the additional
generators have capacity of 200 MW each. The expansion
budget is assumed to be $1 billion. Maximum number of
additional units allowed in each area is four units, which gives
495 possible generator combinations. System loss of load
probability before unit additions is 0.010169.
TABLE I
GENERATION AND LOAD PARAMETERS OF TWELVE AREA TEST SYSTEM
Area
j (MW) (MW)
Load Generation FOR
of additional
Cost
($m)
1 2 8
63 9
5 11 7
104 12
Page 4
4
units
0.025
0.025
0.025
0.025
0.025



0.025
0.025
0.025
0.025
1
2
3
4
5
6
7
8
9
10
11
12
1900
18300
10250
2200
600
0
0
0
1200
2400
2850
850
2550
23600
15100
3100
900
550
3500
400
2100
3100
4150
900
250
250
250
250
250



250
250
250
250
TABLE II shows the solution obtained from [1] and solutions
from random sampling for comparison in the next section.
These solutions are used as starting solutions in Tabu search
procedure in the next section.
TABLE II
SOLUTION FROM AN OPTIMIZATION METHOD AND FROM RANDOM SAMPLING
Area 1 2 3 4 5 9 1
0
0
1
1
0
1
1
0
0
0
1
1
2
0
0
2
0
Solution from optimization
Random Sampling 1
Random Sampling 2
Random Sampling 3
0
1
0
0
0
1
0
0
1
0
0
1
3
0
1
1
0
0
0
1
0
1
0
0
VII. COMPARISON RESULTS
Each initial solution in TABLE II is used in Tabu search
procedure. The algorithm iterates for 10 times. The
comparison between results using initial solution obtained
from dynamic programming and those from random sampling
are made. Fig. 3. shows objective function values at each
iteration resulting from different starting solutions.
012 345 6789 10
4.5
5
5.5
6
6.5
7
7.5
8
8.5x 10
3
Iteration
LOLP
Solution from DP
Random Sampling 1
Random Sampling 2
Random Sampling 3
Fig. 3. Comparison of algorithm efficiency produced from different initial
solutions
The optimal solution found from enumeration is to locate 2
generators in area 2 and 2 generators in area 4. Random
sampling 1, 2, and 3 reach optimal solution at the 6th, 5th, and
6th iteration. Initial solution from optimization reaches optimal
solution at the 3rd iteration.
Even though the difference in number of iterations is small,
initial solution from optimization provide better assurance of
getting good solution than those from random sampling. In
actual application, optimal solution is not known; therefore,
there will be no guarantee at which iteration it will be reached.
Initial solution from optimization at least offers a solution that
is close to optimal and likely to achieve it in timely manner.
VIII. CONCLUSION
A combination of heuristic technique and classical
optimization is proposed to find an optimal generation
location in multi area power systems. The problem has
reliability objective function that complicates the optimization
process. In [1], dynamic programming is applied to the
simplified problem. Global decomposition is used as
reliability evaluation tool. The produced solution is near
optimal and can be used as a good starting solution for
heuristic techniques. Tabu search algorithm is implemented in
this paper to search for the optimal solution.
The proposed approach is efficient and ensures a better
optimal solution when initial solution from optimization is
used. The comparison between randomly selected initial
solutions and initial solution from [1] is made. The algorithm
reaches optimal solution faster with initial solution from
optimization procedure than with random initial solutions.
Other metaheuristic techniques such as Particle Swamp
Optimization (PSO) or Simulated Annealing (SA) can also be
applied along with classical optimization procedure.
APPENDIX
A. Transfer Capability of a 12area Power System
Transfer capability of a 12area power system [9] is shown
in TABLE A.I.
Page 5
5
TABLE A.I
TRANSFER CAPABILITY
From Area To Area Transfer Capability
(MW)
4550
300
100
150
1050
150
900
450
400
200
50
50
300
200
150
400
50
650
350
950
150
150
150
100
1
1
1
1
2
2
2
2
3
3
3
4
4
4
4
5
5
5
7
7
9
9
10
10
2
3
6
10
3
8
9
10
7
10
11
5
7
10
11
6
10
11
11
12
10
11
11
12
REFERENCES
[1] P. Jirutitijaroen and C. Singh, “Reliability and Cost TradeOff in Multi
Area Power System Generation
Programming and Global Decomposition”, to appear in IEEE Trans.
Power Syst.
[2] P. Jirutitijaroen and C. Singh, “A Method for Generation Adequacy
Planning in MultiArea Power Systems Using Dynamic Programming
and Global Decomposition”, to appear in Proc. Power Engineering
Society General Meeting, Montreal, Canada, Jun. 2006.
[3] S. Kannan, S. M. R. Slochanal and N. P. Padhy, “Application and
Comparison of Metaheuristic Techniques to Generation Expansion
Planning Problem”, IEEE Trans. Power Syst., vol. 20, no. 1, pp. 466
475, Feb. 2005.
[4] N. S. Rau and F. Zeng, “Adequacy and Responsibility of Locational
Generation and Transmission – Optimization Procedure”, IEEE Trans.
Power Syst., vol. 19, no. 4, pp. 20932101, Nov. 2004.
[5] H. M. Khodr et al., “A Linear Programming Methodology for the
Optimization of Electric Power–Generation Schemes”, IEEE Trans.
Power Syst., vol. 17, no. 3, pp. 864869, Aug. 2002.
[6] H. T. Firmo and L. F. L. Legey, “Generation Expansion Planning: An
Iterative Genetic Algorithm Approach”, IEEE Trans. Power Syst., vol.
17, no. 3, pp. 901906, Aug. 2002.
[7] J. Zhu and M. Chow, “A Review of Emerging Techniques on Generation
Expansion Planning”, IEEE Trans. Power Syst., vol. 12, no. 4, pp. 1722
1728, Nov. 1997.
[8] Z. Deng and C. Singh, “A New Approach to Reliability Evaluation of
Interconnected Power Systems Including Planned Outages and
Frequency Calculations”, IEEE Trans. Power Syst., vol. 7, no. 2, pp.
734743, May 1992.
[9] S. Sung, “MultiArea Power System Reliability Modeling”, Ph. D.
Dissertation, Dept. Elec. Comp. Eng., Texas A&M Univ., College
Station, Texas, 1992.
[10] C. Singh and Z. Deng, “A New Algorithm for MultiArea Reliability
Evaluation–Simultaneous DecompositionSimulation
Electric Power System Research, vol. 21, pp. 129136, 1991.
[11] A. LagoGonzalez and C. Singh, “The Extended Decomposition
Simulation Approach for MultiArea Reliability Calculations”, IEEE
Trans. Power Syst., vol. 5, no. 3, pp. 10241031, Aug. 1990.
Expansion Using Dynamic
Approach”,
[12] C. Singh and A. LagoGonzalez, “Improved Algorithm for MultiArea
Reliability Evaluation Using the DecompositionSimulation Approach”,
IEEE Trans. Power Syst., vol. 4, no. 1, pp. 321328, Feb. 1989.
[13] P. Doulliez and E. Jamoulle, “Transportation Networks with Random
Arc Capacities”, Reveu Francaise d’Automatique, Informatique et
Recherche Operationelle, vol. 3, pp 4566, 1972.
[14] C. Rego and F. Glover, www.tabusearch.net.