Energy and Power En gineering, 2010, 2, 223-229
doi:10.4236/epe.2010.24033 Published Online November 2010 (
Copyright © 2010 SciRes. EPE
A Novel Particle Swarm Optimization for Optimal
Scheduling of Hydrothermal System
Wenping Chang1,2
1School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, China
2Henan Mechanical and Electrical Engineering College, Xinxiang, China
Received April 18, 2010; revised July 5, 2010; accepted August 14, 2010
A fuzzy adaptive particle swarm optimization (FAPSO) is presented to determine the optimal operation of
hydrothermal power system. In order to solve the shortcoming premature and easily local optimum of the
standard particle swarm optimization (PSO), the fuzzy adaptive criterion is applied for inertia weight based
on the evolution speed factor and square deviation of fitness for the swarm, in each iteration process, the in-
ertia weight is dynamically changed using the fuzzy rules to adapt to nonlinear optimization process. The
performance of FAPSO is demonstrated on hydrothermal system comprising 1 thermal unit and 4 hydro
plants, the comparison is drawn in PSO, FAPSO and genetic algorithms (GA) in terms of the solution quality
and computational efficiency. The experiment showed that the proposed approach has higher quality solu-
tions and strong ability in global search.
Keywords: Hydrothermal Generation Scheduling, Particle Swarm Optimization, Fuzzy, Adaptability
1. Introduction
Optimal operation is one of the important optimization
issues in a hydrothermal power system. Its objective is to
minimize the operation cost of thermal units in a given
period of time while all constraints are satisfied. Since
the cost of power generation is huge, the economic con-
sequence of operation scheduling is significant. A num-
ber of methods have been proposed for solving the
hydrothermal scheduling problem. Some examples of
these methods are nonlinear programming (NLP) [1-3],
Dynamic Programming (DP) [4-6], Lagrangian Relaxa-
tion (LR) [7-9], Tabu Search [10], Expert Systems [11],
Artificial Neural Networks (ANN) [12], Genetic Algo-
rithms (GA) [13-15]. Among these methods, DP would
provide a good framework for optimizing the decisions.
The main difficulty with DP is the curse of dimensional-
ity that is the exponential increase in computation, it
suffers from the “curse of dimensionality”. The LR has
shown great potential for the optimal problem, but the
disadvantage of LR is its inherent sub-optimality.
Particle swarm optimization (PSO) is an evolutionary
computation technique [16], it has lots of advantage:
simple concept, easy implementation, robustness to con-
trol parameters and computational efficiency. Recently,
PSO have been successfully used in many areas [17-21].
Although PSO has many strong points, it has some
shortcoming such as premature convergence. To over-
come these problems, many methods have been devel-
oped. Shi and Eberhart introduced an inertia weight in
1998 [22], the inertia weight method could improve the
performance of algorithm, but a fixed inertia weight does
not satisfy the balance between global and local search,
so many attempts have been tried by using various inertia
weight strategies to make it performance better. Shi and
Eberhart proposed a linear decreasing strategy in 1999
[23], which does not truly reflect the actual search proc-
ess to find the optimum.
In this paper, a fuzzy adaptive particle swarm optimi-
zation (FAPSO) is proposed to the operation of hydro-
thermal power system, which is designed to dynamically
adjust the inertia weight as the environment changing.
This can improve the global and local search ability of
the PSO and overcome the disadvantages of the PSO.
The correlative examination indicates that the FAPSO
has fast convergent velocity and powerful search capa-
bilities to generate satisfactory results.
2. Problem Formulation
The objective of optimal operation to hydrothermal power
Copyright © 2010 SciRes. EPE
system is usually to minimum the thermal cost function,
while satisfying physical and operational constraints.
2.1. Objective Function
Hydrothermal scheduling is the optimization of a prob-
lem with non-linear objective function, the objective
function to be minimized can be written as:
min(( ,))
[(,) (,)]
ithith i
where T is number of operating periods, N is number of
thermal plants, F is composite cost function, P
th(i,t) is
loading of ith thermal unit at time t, ai,bi,ci are thermal
generation cost coefficients.
2.2. Constraints
The constraints must be satisfied the optimization proc-
ess, which are as follows:
1) Total generation meets the system demand
(, )(, )()
thh D
iN jM
PitPjt Pt
 (2)
where M is number of hydropower stations, Ph(j,t) is
loading of jth hydro unit at time t, PD(t) is load demand
at time t.
2) System spinning reserve limits
(, )(, )()()
iN jM
RitR jtPtRt
 (3)
where Rreq(t) is required system spinning capacity re-
3) Thermal plant loading limits
(, )
ith i
PPitP (4)
where ,mini
P is minimum power output of thermal unit i,
P is maximum power output of thermal unit i.
4) Ramp rate constraints
(,1)(, )(,1)
thi ththi
PitPit Pit  (5)
where i
is ramp rate maximum permitted for the
thermal unit i.
5) Hydro plant loading limits
,min ,max
jh j
where ,minj
P is minimum power output of hydro unit j,
P is maximum power output of hydro unit j.
6) Reservoir level limits
where (,)
Vjt is storages of reservoir j during time t,
V is minimum storage of reservoir j, ,maxj
V is
maximum storage of reservoir j.
7) Reservoir discharge limits
,min ,max
where Q(j,t) is water discharge of reservoir j at time t,
Q is minimum release of reservoir j, ,maxj
Q is
maximum release of reservoir j.
8) Water balance equation
(, 1)(,)(,)(,)(,)
 
where (,)
jt is inflow of reservoir j during time t,
is water delay time between reservoir j and i.
9) Initial and terminal reservoir levels
() T
where 0
V is initial storages of reservoir j, T
V is ter-
minal storages of reservoir j.
10) Power generation of hydro plant
The hydro generator power output is written in terms
of reservoir volume instead of the reservoir net head and
the rate of water discharge, Q, a frequently used func-
tional is
,1 23
(,)(,)(,)(,) (,)
(,) (,)
hj jj
PjtcVjtcQ jtcVjtQjt
cVjt cQjt c
 (11)
where 16
CC are hydro power generation coeffi-
3. Implementation of Fapso for Optimization
3.1. Overview of the PSO
PSO is a population-based optimization algorithm, which
was firstly proposed by Kennedy and Eberhant in 1995
[24]. It simulates the food searching activities of a swarm
of birds (particles), and every particle has its own loca-
tion and velocity, individuals are evolved by cooperation
and competition among themselves through generations,
particles move around the search space until they find the
optimal solution.
In n-dimensional search space, the location of the par-
ticle is represented as the vector 1, 2
mmm mn
xx x
and its velocity is represented as the vector
1, 2
mmm mn
Vvv v
 . Pbest and Gbest, respectively, are
the best location of individual m and all particles’ best
location so far. Using the information, the velocity of
Copyright © 2010 SciRes. EPE
individual is modified as shown below:
() ()
() ()
kkkk k
mm mbestm
mbest m
arand GX
 
 (12)
where 12
,aa are learning factors.
Using (12), a certain velocity that gradually gets close
to Pbest and Gbest can be calculated. Each individual
moves from the current location to the next one using the
following equation:
 (13)
The search mechanism of the PSO using the modified
velocity and location of individual m is showed in Fig-
ure 1.
In PSO, it is the inertia weight to balance between
global and local search ability. A large inertia weight
facilitates a global search while a small inertia weight
facilitates a local search, concept of linearly decreasing
inertial weight for particle swarm optimization (LDWPSO)
is introduced in [23], which ω is given by
max min
 (14)
where ω is inertia weight, iter is the number of iterations,
itermax is the maximum number of iterations.
In the LDWPSO approach, particle’s fitness is best
until the preceding iteration, its velocity is kept un-
changed in the next iteration, otherwise, the particle’s
velocity and position are changed according to (12) and
(13), which does not truly reflect the search process to
find the optimum.
In order to obtain a better search process, the inertia
weight should be nonlinearly, dynamically changed to
balance between global and local search ability. There-
fore, fuzzy adaptive PSO is presented to design a fuzzy
system to dynamically adjust the inertia weight.
Figure 1. The search mechanism of PSO.
3.2. Fitness Function
Several techniques for handling constraints have been
proposed in the specialized literature [25]. One of them
considers the objective function penalization. Using this
technique, the fitness function is formed by the objective
function (1) plus penalty terms for particles that have
violated some inequality constraints and equation con-
straints. Such fitness function can be expressed as:
[] []
ii jP
JR gRh
 
where R1, R2 are coefficients of punishment, gj is value of
violate inequality constraint, hp is value of violate equa-
tion constraint.
3.3. Fuzzy Adaptive PSO
In this section, a fuzzy adaptive particle swarm optimiza-
tion will be described to fit a wider range of optimal
problems. To design a fuzzy system to dynamically adapt
the inertia weight, two variables are selected as inputs to
the fuzzy system: the evolution speed factor and the
square deviation of fitness for swarm; the output variable
of system is the change of the inertia weight.
1) Evolution speed factor
The evolution speed factor (ESF) measures the per-
formance of the particle evolution process so far by the
PSO. In case of a minimization problem, 1kk
best best
ESF is used as an input to bound the limit between 0 and
1 as:
2) The square deviation of fitness
The square deviation of fitness the distribution of parti-
cles, it is by equation:
max{ ,}
kk kk
mgmean meanw
where k
is fitness of individual m at iteration k,
F is mean fitness of individuals at iteration k,k
the best fitness of individual at iteration k, k
is the
worst fitness of individual at iteration k.
3) Current inertia weight correction
The change of the inertia weight need both positive
and negative corrections, this paper presents the range
(0.04, +0.04) for the inertia weight correction (ω).
Two input variables and one output variable are de-
fined to have three fuzzy sets: LOW (L), MIDDLE (M)
and HIGH (H) with associated membership function as
left-Triangle, Triangle and right-Triangle, respectively in
Figure 2.
The three membership functions are:
Left-triangle membership function:
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Figure 2. Triangular membership function.
Triangle membership function:
Right-triangle membership function:
The whole fuzzy system for dynamically adapting the
inertia weight can be described as Table 1, where there
are 9 IF/THEN rules for 2 input variables and 3 linguistic
values of each input values. Degrees of membership of
ESF (μESF) and σ(μσ) are calculated from (18)-(20), re-
spectively. Lasen product is used as fuzzy implication
operator for the individual rules, using arithmetic product,
the degree of fulfillment for rule r is DOFr = μESF•μσ. For
each rule, fuzzy inertia weight correction will be calcu-
lated by DOF. Finally, the center-of-area method is the
most well known and simple defuzzification method
which is implemented to determine the output crispy
value [26]. The function of modify inertia weigh de-
scribed as:
 (21)
The proposed algorithm flowchart is depicted in Fig-
ure 3. The implementation steps are as follows:
Step 1. Initialization of the swarm
For a population size n, the particles are randomly
Figure 3. The framework flow chat of FAPSO.
Table 1. Fuzzy rules for inertia weight correction.
Input Output
Rule No.
ESF σ ∆ω
1 S S S
2 S M M
3 S L L
4 M S L
5 M M M
6 M L S
7 L S S
8 L M S
9 L L M
generated in the range 0-1 and located between the
maximum and the minimum extreme.
1) Hydro plant
If there are M reservoirs, the particles is initialized
randomly within the effective real storage of reservoir,
the mth particle is represented as a matrix as follows:
()((1, ),(2, ),,(, ))
Vtv tvtvMt
The jth reservoir is allocated a value of Vhj as given
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below to satisfy the constraint given by (6).
,min,max ,min
(,)() ()
hj jj
vjt VrandVV  (23)
The hydroelectric power outputs can easily be com-
puted using (11).
2) Thermal plant
If there are N thermal plants, the particles is initialized
randomly within the effective real power generation lim-
its, the ith particle is represented as a matrix as follows:
()((1,),(2, ),,(, ))
thith thth
PtP tPtPNt (24)
Step 2. Defining the fitness function
FAPSO algorithm conventionally searches for the op-
timal solution by minimum a given fitness function. The
hydrothermal co-ordination problem, the evaluation
function is combinations of the thermal cost function and
penalty function terms that take into account the various
systems, unit and hydraulic network constraint violations.
The evaluation function should be different in the feasi-
ble and infeasible search domains. The evaluation func-
tion is given by (15).
Step 3. Initialization of pbest and gbest
The fitness values obtained above for the initial parti-
cles, the best position of a particle is taken as pbest, and
the best position of all pbests is taken as gbest.
Step 4. Calculate ESF and σ
When the best, the worst and mean of fitness values
obtained, the evolution speed factor and square deviation
of fitness can be used respectively Equations (16) and
(17) calculated.
Step 5. Modify inertia weight using fuzzy rules
The inertia weight correction (
) can be obtained
using fuzzy rules, modify the inertia weight according to
Equation (21).
Step6. Update the swarm
Modify the velocity vector of each particle using
Equation (12), the particle position vector updated using
Step 7. Stopping criteria
There are different criteria available to stop optimiza-
tion algorithm. In this paper, maximum number of itera-
tions is adopted as the stopping criterion.
4. Simulation Result
4.1. Testing Strategies
In this study, the maximum number of generations is set
as 100, the population size is set as 30, and the general
parameters of PSO are set as: c1 = c2 = 2 for all the PSO
runs, the same ωmax = 0.9, ωmin = 0.3 are employed for
both FAPSO and PSO in order to examine their per-
The program is implemented in MATLAB 7.0.1 and
the simulations are carried on a Pentium personal
computer with 256 M RAM.
4.2. Optimal Scheduling of Hydrothermal Power
The test hydrothermal power system consists of a multi-
chain cascade of 4 hydro units and a number of thermal
units represented by an equivalent thermal plant. The
scheduling period is 24 hours, with an hour intervals.
The data of load demanded are shown as Tabl e 2. The
system and unit parameters including costing functions
and operating constraints are presented in [27].
FAPSO algorithm and PSO algorithm are used to
simulate the operation course 20 times. The inflow
course, water level-capacity curve, and the downstream
stage discharge curve are all foregone. Tab le 3 gives the
comparison results of FAPSO PSO and GA about two
indexes: cost output and average execution time.
From Table 3, it is obvious that the results of FAPSO
are best than that of PSO and GA with the object of the
cost output and its convergence speed is very faster than
PSO. The corresponding hourly hydro plant discharge is
shown in Figure 4.
The results show that, among the three algorithms, the
FAPSO offers the best solution quality, the proposed
approach is superiority among its competitors.
Table 2. Data of load demanded (MW).
hour load hour load hour load
1 1370 9 2240 17 2130
2 1390 10 2320 18 2140
3 1360 11 2230 19 2240
4 1290 12 2310 20 2280
5 1290 13 2230 21 2240
6 1410 14 2200 22 2120
7 1650 15 2130 23 1850
8 2000 16 2070 24 1590
Table 3. Comparison of results.
Algorithm Cost ($) Average time used (S)
PSO 921920 10.67
FAPSO 914660 4.73
GA [27] 926707 -
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Figure 4. Hourly hydr o plant discharge trajector i es.
5. Conclusions
This paper presents an application of FAPSO for solving
the hydrothermal optimal scheduling problem. FAPSO
generates better solutions than other methods, mainly
because it is implemented to dynamically adjust the iner-
tia weight by using “IF-THEN” rules, this can improve
the global and local search ability of the PSO and over-
come the disadvantages of the PSO.
The proposed FAPSO is applied to the optimal opera-
tion of hydrothermal power plant. For comparison,
simulations are conducted for FAPSO PSO and GA. The
experimental results indicate that the proposed algorithm
can improve the computational efficiency and produce
more satisfactory output, which offers a new way to
solve hydrothermal optimal operation problem.
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