330 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, NO. 3, JULY 2012
Hybrid Simulated Annealing–Tabu Search Method
for Optimal Sizing of Autonomous Power Systems
With Renewables
Yiannis A. Katsigiannis, Pavlos S. Georgilakis, Senior Member, IEEE, and Emmanuel S. Karapidakis, Member, IEEE
Abstract—Small autonomous power systems (SAPS) that in-
clude renewable energy sources are a promising option for isolated
power generation at remote locations. The optimal sizing problem
of SAPS is a challenging combinatorial optimization problem, and
its solution may prove a very time-consuming process. This paper
initially investigates the performance of two popular metaheuristic
methods, namely, simulated annealing (SA) and tabu search (TS),
for the solution of SAPS optimal sizing problem. Moreover, this
paper proposes a hybrid SA-TS method that combines the ad van-
tages of each one of the above-mentioned metaheuristic m ethods.
The proposed m ethod has been successfully applied to design
an SAPS in Chania region, Greece. In the study, the objective
function is the minimization of SAPS cost of energy (€/kWh), and
the d esign variables are: 1) wind turbines size, 2) photovoltaics
size, 3) diesel generator size, 4) biodiesel generator size, 5) fuel cells
size, 6) batteries size, 7) converter size, and 8) dispatch strategy.
The performance of the proposed hybrid optimization method-
ology is studied for a large number of alternative scenarios via
sensitivity analysis, and the conclusion is that the proposed hybrid
SA-TS improves the obtained solutions, in terms of quality and
convergence, compared to the solutions provided by individual SA
or individual TS methods.
Index Terms—Hybrid power s ystems , optimal sizing, opti-
mization methods, power generation dispatch, renewable energy
sources, simulated annealing (SA), small autonomous power
systems (SAPS), solar energy, tabu search (TS), wind e nergy.
I. INTRODU CTION
T
ODAY more diverse challenges have emerged: climate
change, econom ic recession, and security of energy
supply. Moreover, the rapid depletion of fossil fuels and their
high and volatile prices have necessitated an urgent need for
alternative energy sources to meet the energy demands [1].
Renewable energy sources (RES), such as wind an d solar, are
clean, inexhaustible, and environmentally friendly alternative
energy sources with negligible fuel cost. However, RES tech-
nologies, such as wind turbines (WTs) and solar photovoltaics
(PVs), are dependen t on a resou rce that is u npredictable and
depends on weather an d climatic changes, and the produ ctio n
Manuscript received July 14, 2011; revised November 21, 2011; accepted
January 12, 2012. Date of publication April 06, 2012; date of current version
June 15, 2012.
Y. A. Katsigiannis and E . S. Karapidakis are with the Department of Nat-
ural Resources and Environment, Technological Educational Institute of Crete,
Chania 73133, Greece (e-mail: karapidakis@chania.teicrete.gr).
P. S. Georgilakis is with the School of Electrical and Computer Engineering,
National Technical University of Athens (NTUA), Athens 15780, Greece
(e-mail: pgeorg@power.ece.ntua.gr).
Di
gital Object Identifier 10.1109/TSTE.2012.2184840
of WTs and PVs may not match w ith the load demand, so there
is an impact on the reliability of the electri c energy system.
This reliability problem can be solved by a pr oper com bination
of the two resources (WTs and PVs) together w ith the use
of an energy storage system, such as batteries, as a type of
energy-balancing medium [2]. Such a system, which is called
a sm all autonomous power system (SAPS), is a promising
option for isolated power gener ation at rem ote locations [3].
To be more precise, an SAPS is an isolated hybrid system
with renewable energy sources, conventional power sources
(usually diesel generators), and energy storage. Proper sizing
of the overall SAPS is challen ging, d ue to the large n umber of
design options and the uncertainty in key parameters, such as
load size and future fuel price. Renewable energy sources add
further complexity because their power output is unpredictable
and intermittent.
The problem of SA PS optimal sizing belongs to the cate-
gory of combinatorial optimization problem s, since the sizes of
system’s components, which constitute the design variables, can
take only discrete values. For the solution of this problem, var-
ious deterministic optimization techniques have been proposed
[4]; h owev e r, these methods may provide suboptimal solutions,
which are u sua lly comb ined with increased computational com-
plexity. The most direct method for solving the SAPS sizing
problem is the complete enumeration method that is used by
HOMER software [5]; however,itcanprovetobeextremely
time consuming. Moreover, a recent review of computer tools
has shown that there is no energy tool that addresses all issues
related to SAP S optimal sizing, b ut instead the “ideal ” en e rgy
tool highly depends on the specific objectives that must be ful-
filled [6].
In recent years, new methods have been developed, in order
to solve many types of com plex optim ization p roblem s, partic-
ularly tho se of combinatorial nature. These methods are called
metaheuristics and include genetic algorithms (GAs), simulated
annealing (SA), tabu search (TS), and particle swarm o ptimiza-
tion (PSO) amo ng others. Metaheuristics orchestrate an inter-
action between local improvement procedures and higher-level
strategies to create a process capable of escaping from local op-
tima and performing a robust search of a solution space. From
the area of m etaheuristics, GAs [7]–[9], SA [10], TS [11], as
well as PSO [12], have been proposed for the solution of SAPS
optimal sizing.
This paper initially investigates the applicatio n of two met a -
heuristic methods, n amely SA a nd TS, for solving the SAPS
sizing problem. SA transposes the technique of annealing to the
1949-3029/$31.00 © 2012 IEEE
KATSIGIANNIS et al.: HYBRID SA-TS METHOD FOR OPTIMAL SIZING OF AUTONOMOUS POWER SYSTEMS WITH RENEWA
BLES 331
solution of an optimization problem. TS is a powerful iterative
optimization procedure that is characterized by its ability to
escape from l ocal optima (which usually cause conventional
algorithms to terminate) by using a flexible memory system.
Moreover, this paper proposes a h ybrid optimization method-
ology that combines the above-men tio ned methods (SA and
TS) for solving the SAPS sizing problem. Hybrid methods
that contain SA and TS have been applied in various areas,
including unit commitment [13], optimal capacitor placement
[14], and nonperm utatio n flowsho p scheduling [15]. The
proposed method has been successfully applied to design an
SAPS in the Chania region, Greece. In the study, the objective
function is the minimization of SAPS cost of energy (€/kWh),
and the design variables are: 1) WTs size, 2) PVs size, 3) diesel
generator size, 4) biodiesel generator size, 5) fuel cells size,
6) batteries s ize, 7) converter size, and 8) dispatch strategy.
The performance of the proposed hybrid SA-TS o ptimiza-
tion m etho dology is studied for a large number of alternative
scenarios, and it is concluded that it im pr oves the obtained
solutions, in terms of quality and convergence, com pared to the
solutions provided by individual SA or indiv idual T S method.
II. P
ROBLEM FORMULATION
The SAPS optim al sizing problem has to fulfil the objective
defined by (1) subject to the constraints (3)–(9). The computa-
tions of the objectiv e function and the constraints of the problem
are related with the results obtained by simulating t he operation
of SAPS for a given time step
, taking into account compo-
nents’ type, cost, and technical characteristics.
A. Objective Function
Minimization of t he system’s cost of energy (COE)
(1)
The COE (€/kWh) of SAPS is calculated as follows:
(2)
where
(€) is the total annualized cost and
(kWh) is the to tal annual useful electric energy prod uction.
takes into account the annualized capital costs, the annu-
alized replacement costs, the annual operation and maintenance
(O&M) costs, an d the annual fuel costs (if a p plicable) of the
system’s components. COE is adopted since it is a very good
measure of system cost for SAPS sizing [16].
B. Constraints
1) Initial cost constraint
(3)
where IC (€) is the initial installation cost of the syst em,
and
(€) is the m aximum acceptable initial cost of
the system.
2) Unmet load constraint
(4)
where
is the annual unmet load fraction, (kW)
is the unmet l oad during the sim ul a tion time step
(h),
(kWh) is the total annual electric e nergy demand,
and
is the maximum allowable annual unmet load
fraction. In this paper,
so 52 560 summa-
tions are needed for the entire year to compute
,as(4)
implies.
3) Capacity shortage constraint
(5)
where
is the annual capacity shortage fraction,
(kW) is the capacity shortage during ,and is the
maximum allowable annual capacity shortage fraction. Ca-
pacity sh ort a ge is d e fined as a shortfall that occurs betw een
the required am oun t of operating capacity (load plus re-
quired operating reserve) and the actual operating capacity
the system can provide. Operating reserve in an SAPS with
RES technologies is the surplus electrical generation ca-
pacity (above that required to meet the current electric
load) t hat is operating and is able to respond instantly to
a sudden increase in the electric load or a sudden decrease
in the renewable power output.
4) Fuel consumption constraint
(6)
where
is the fuel consumption of a generator
during
,and is the maximum allowable
annual fuel con sumption of th e generator.
5) Minimum renewable fraction constraint
(7)
where
is the RES fraction of the system,
(kWh) is the total annual renewable energy production,
(kWh) is the total annual energy production of th e
system, and
is the m inimum allowable RES frac-
tion.
6) Components’ size constraints
(8)
(9)
where
is the size of system’s component ,
and
is the maximum allowable size of .
III. SA PS C
OMPONENTS AND MODELING
The considered SAPS has to serve electrical load, and it can
contain seven component t ypes:
1) WTs.
2) Amorphous silicon (a-Si) PVs.
332 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, N O. 3, JULY 2012
3) Generator with diesel fuel.
4) Generator with biodiesel fuel.
5) Fuel cells (FCs) combined with reformer, using methanol
as a fuel.
6) Lead-acid batteries.
7) Converter.
The modeling of SAPS componen ts is im plemen ted as fol-
lows. The WT m odeling is imp lem ented u sing a power curve
profile that is b a sed on Fuhrländer’s FL 30 model. The selected
WT has the following characteristics: rated power 30-kW AC,
cut-in speed
3 m/s, and cut-out speed 25 m /s. For
the WT power curve fitting, a n inth-order polynomial expres-
sion has been selected, as it provides accurate correlation with
real data, while it presents exclusively positive values for the
generated power in the interval
.
In the PV modeling, the output of the PV array
(in kW)
is calculated from [17]
(10)
where
is the PV derating factor, is the nomin al PV
array power in kW
under standard test conditions (STCs),
is the global solar radiation inciden t on t he PV array in
kW/m
, is the solar radiation under STC (1 kW/m ),
is the temperature of the PV cells, is the STC tem-
perature (25
C), and is the PV temperature coefficient
(
C for a-Si). The PV derating factor is a scaling
factor applied to the PV array output to account for losses, such
as dust cover, aging and unreliability of the PV array, and is
considered to be equal to 0.80.
can be estimated f rom the
ambient temperature
(in C) and the global solar radiation
on a horizontal p lane
(in kW/m ) using (11) [18]
(11)
where NOCT is the normal operating cell temperature, which is
usually obtaining the value of 4 8
C.
The d iesel generator fuel consumption
(L/kWh) is a ssumed
to be a linear function of its electrical pow er output [19]
(12)
where
is generator’s rated power and is generator’s
output power. When biodiesel is used instead of diesel, fuel con-
sumption is increased [20]. In t his paper, a 10% increase in fuel
consumption has b een considered. Moreover, a 30% minimum
allowable load ratio of
has been assumed for each type
of generator. Methanol has been selected as FC fuel because
it presents economic, environmental, and reliability advantages
for autonom ous power system s [21], while the overall FC effi-
ciency has been considered 50%. In the three types of control-
lable generators (diesel, biodiesel, and FCs), lifetime before re-
placement d epends on the total num ber of their operating hours,
as calculated by the simulation process.
Lead-acid batteries have been modeled as devices capable of
storing a certain amount of dc electricity at fixed roun d-trip en-
ergy efficiency, with limits on how quickly they can be charged
or discharged, how deep ly they can be discharged without
causing damage, and how much energy they can cycle before
needing r eplacem ent. Batteries have been modeled according to
the following technical characteristics per com po nent: capacity
1000 Ah, voltage 6 V, round-trip efficiency 80%, maximum
charge and discharge current equal to
,minimumstate
of charge 20%, and lifetime throughp ut 9600 kWh. Finally,
converter efficiency has been taken equal to 90%.
The sim ulation process examines a particular system config-
uration, in which comp onen ts sizes satisfy constraints (8) and
(9). The necessary inputs for th e simu lati on are: 1) annual time
series data for wind speed, solar radiation, ambient temperature,
and l oad, 2) com ponent characteristics, 3) constraint bounds,
and 4) general parameters (project li feti me, discount ra te) . T he
specific values for these data are described in Section V-A. In
the simulation, for every time step
, the available renewable
power (from WTs and PVs) is calculated and then is compared
with the load. In case of excess, the surplus ren e wable en ergy
is charging the batteries, if they are not fully charged. If renew-
able power sources are not capable to fully serve the load, the
remaining electric load has to be supplied by contro llable gen -
erators and/or batteries. From all possible combinations, it is se-
lected the one that supplies the load at the least cost. When the
whole year’s simulation has been completed, it is d etermined
whether the system is feasible, i.e., it is checked if it satisfies the
constraints (3 )–(7). After the end of simulation , the life-cy cle
cost of the system is calculated b y taking into account: 1) the an-
nual results o f the simulation, 2) the capital, replacement, O&M,
and fuel cost (if applicable) of each component, 3) the compo-
nents’ lifetim e , 4) the project lifetime, and 5) the discount rate.
Having computed the life-cycle cost, the total annualized cost
can be calculated, which represents the hypothetical an-
nual cost value that if it occurred each year of the SAPS life-
time, it would yield a net present cost equivalent to the actual
life-cycle cost [16]. The total annualusefulelectricenergypro-
duction
that is also needed for the calculation of
COE (2) is provided by simulation ’s results.
An additional aspect of system operation arises, which is
whether (and how) the controllable generators should charge
the battery bank. Two common control strategies that can be
used are the load following (LF) strategy and the cycle charging
(CC) strategy. It has been found [22] that over a wide range
of conditions, the better of these two strategies is virtu a lly as
cost-effective as an ideal predictive strategy, which assumes
the existence of perfect knowledge in future load an d wind
conditions. In the LF strategy, batteries are charged exclusively
by nondispatchable RES t echno logies (WTs and PVs), whereas
controllable generators (diesel, biodiesel, or FCs) produce only
enough power to meet the lo a d, if needed. An exception may
occur if the required load is less than the minimum power of the
operating controllable generators (due to minimum allowable
load ratio constraint); then the excess generated power is stored
in the batteries. LF strategy tends to be optimal in systems with
a lot of renewable power, when the renewable po wer outpu t
sometimes exceeds the load. In the CC strategy, whenever the
controllable generators need to oper a te to serve the load, they
operate at full output power in order to achieve their maximum
efficiency at their rated (maximum) power [see (12)]. A set-
point state of charge,
, has also to be set in this strategy.
The charging of the battery by the controllable generators
KATSIGIANNIS et al.: HYBRID SA-TS METHOD FOR OPTIMAL SIZING OF AUTONOMOUS POWER SYSTEMS WITH RENEWA
BLES 333
will not stop until it reaches the specified . The battery,
however, can be charged further by the nondispatch able R ES
technologies (WTs and PVs), if
. In this paper,
three alternative values of
have been considered: 80%,
90%, and 100%, so the total number of examined dispatch
strategies is 4. The CC strategy tends to be optimal in systems
with little or no renewable power. In such systems, controllable
generators produce the largest portion of power output. In order
to increase the overall efficiency of the system, it is better for
these generators to operate at their maximum efficiency (rated
power), even if the battery losses are taken into account, r ather
than operating with significantly less efficiency (power) in
order to meet the load at each time step.
IV. P
ROPOSED METHODOLOGY
A. Overview of Simulated Annealing
In physics, annealing refers to the process of h eating up a
solidtoahightemperaturefollowedbyslowcoolingachieved
by decreasing t he tem perature of the environm ent in steps. In the
SA algorithm, the Metropolis algorithm [23] is utilized for sim-
ulating the evolution of a physical system at a given temperature
. By repeatedly observing this Metropolis rule of acceptance,
a sequence of configur ations i s gene rated , which const itutes a
Markov chain.
A finite time implementation of the SA algorithm can be real-
ized by generating homogeneous Markov chains of finite length
for a finite sequence of descending values of
. To achieve
this, a set of parameters that governs the convergence of the al-
gorithm has to be specified. These parameters form a cooling
schedule. The parameters of the cooling schedule are:
1) Th e initial tem perature
.
2) Th e length of the hom ogen eous Markov chains
.
3) The law of decrease of
.
4) Th e criterion for program termination.
From the above parameters, the law of decrease is the one that
draws the most attention. The geometrical law of decrease is a
widely accepted one
(13)
where
and are the tem peratures at the and itera-
tion o f the algorithm, respectively, while
is constant .
Typical values for
lie between 0.5 and 0.99 [23]. An alterna-
tive soluti on consists of resorting to an adaptive law of the form
[23]
(14)
where
is the s tandard deviation of the values generated at
the
th Markov chain, and is a constant called distance pa-
rameter. Small
-values lead to small decrements in . Typical
values of
are between 0.1 and 1.
B. Proposed Simula ted An neal ing for SA PS Op timal Sizing
The proposed SA methodology for the optimal sizing of
SAPS is composed of the following steps:
1) Setting of the parameters of the coo lin g sched ule.
2) Random generation of an initial solution and calculation of
its energy.
3) If equilibrium is achieved, go to Step 6; otherwise, repeat
Steps 4 and 5.
4) Fin din g of a trial solution that is a neighbor to the curren t
solution of the a lgorithm , and calcu lation of its energy.
The trial solution is generated by changing randomly the
value of an SAPS parameter (component size o r dispatch
strategy) in the current solution.
5) Perform ing of the acceptance test according to Metropolis
algorithm.
6) If the stopping criterion is satisfied, the algorithm stops;
otherwise,
is decreased.
The calculation of the energy of a solution depends on its fea-
sibility. If the solution is feasible, its en ergy is consider ed equal
to the value o f the objectiv e function (COE). On the other hand,
if the solution d oes not satisfy the constraints of the problem,
the corresponding value of COE is considered equal to a mar-
ginal value COE
, which has to be greater than any obtained
value of C O E, while at the same tim e it has to remain at the
same order of magnitude. In this paper, COE
is set equ al
to 1 €/kWh. The total energy of a nonfeasible so lut ion is then
obtained by sum ming the value of COE
with the overall
normalized penalty function .
C. Overview of Tabu Search
TS is a powerful optimization procedure that has been suc-
cessfully applied to a number of combinatorial problems. It uses
an operation called move to define the neighborhood of any
given solution. TS can be viewed as an iterative technique that
explores a s et of problem solutions by repeatedly making moves
from one solution to another, in the manner of a greatest-descent
algorithm [23]. TS is characterized by the ability to escape from
local optim a and the occurrence of cycles, which usually cause
simple descent algorithms to termin ate. T his go al is obtained b y
using a finite-size list of forbidden moves, called tabu moves,
derived from the recent history of the search.
The t wo m a in com ponents of the TS are the tabu list restric-
tions and the aspiration criteria of the solution associated with
these restrictions. Tabu lists are managed by recording moves in
the order in which they are made. I f a new attribute enters into
the tabu list, the oldest one is released from the tabu list. The
proper choice of the tabu list size is critical to the success of the
algorithm and it depends on the specificproblem.
Aspiration criteria can override tabu restrictions. That is, if
a certain move is forbidden, the aspirat ion criteria, when satis-
fied, can reactivate this move. T he most widely used aspiration
criterion removes a tabu classification from a trial move when a
move yields a solution better than the best obtained so far. How-
ever, other aspiration criteria have been also proposed [23].
D. Proposed Tabu Search for SAPS Op tim al Sizing
In the proposed TS methodology for SAPS optimal sizing, the
neighborhood of a current solution contains all configurations
of similar co mp onen t sizes, as well as the alternative dispatch
strategies optio ns. More specifically, a move is definedbyse-
lecting each time the next l arger size (if permitted) and the pre-
vious sm aller size (if permitted ) of a comp onent size, while for
334 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 3, N O. 3, JULY 2012
the dispatch s tr ategy a move i s defined by examining the three
remaining o ptions of the current strategy. Since the SAPS con-
tains seven com ponents (Section III), at maximum 14 configu-
rations with different component sizes are considered that are
added to the three remaining dispatch strategies; consequently
the maximum number of configurations that belong to the n eigh -
borhood of current solution is 17.
The TS algorithm is composed of the following steps:
1) S e tt ing of the tabu list size.
2) Generation of an initial feasible solution, and calculation
of its COE.
3) S e tt ing of the global best solution equal to the initial solu-
tion (current solution).
4) F inding o f a set of feasible t rial solutions that are neighbors
to the current solution and sorting of them in ascendin g
order of COE.
5) C hecking if t he selected move of the first trial solution
belongs to the tabu list. If it belongs and the aspiration
criterion (Step 6) is not satisfied, a selection of the next
solution of the sorted set of trial solutions has to be done.
Otherwise, the solution is accepted (current solution) and
the update o f the tabu list is p e rform e d by adding in it the
chosen move, a nd by removing from it the oldest m ove,
with respect to tabu list size.
6) Ex amination o f the aspiration criterion. In the prop osed
algorithm, a move aspiration is satisfied if the move yields
a solution better than the best obtained so far.
7) U pdate of the global b est solution if the best acceptable
solution found from the t rial set has a lower COE value.
8) Repeat Steps 4–7 . Stop the procedure if the termination
criterion is satisfied. In this paper, the search is terminated
if a maximum predefined allowable number of iterations is
reached.
E. Proposed Hybrid S imu lated Annealing–Tabu Search for
SAPS Optimal Sizing
Hybrid optimization method s combine the advantages of in-
dividual optimization methods in order to find the optimal so-
lution in a f a st and effective m anner . SA is a stochastic m e tho d
that excels at gravitating towards the global optimum. However,
SA is not especially fast at finding the optimum in a given so-
lution region. For this reason, SA is often combined with local
search. More specifically, SA is utilized to find the region of the
optimum, and then the local optimizer takes over to find the op-
timum. During th e local search procedure, the qual ity of the ini -
tial soluti on is essential for its successful implementation. Then
the local search method is proceeding iteratively from one s olu-
tion to another u ntil a chosen termination criterion is satisfied.
This pap er proposes a hybrid SA-TS optimization method-
ology for the solution of the SAPS sizing problem . In this
methodology, SA provides the initial solution. Moreover, i n
order to improve the quality of results, in this paper, the con-
ventional local search method has been replaced by TS. TS can
be seen as an extension of local search, as its inherent adaptive
memory ensures that the search will not return periodically and
stack to the same solutions.
TABLE I
C
ONSTRAINT VALUES FOR CASE STUDY SYSTEM
V. R ESULTS AND DISCUSSION
A. Case Study System
In th e considered SAPS, the project lifetime and t he discount
rate are assum ed to be 25 years and 5%, respectively. The simu-
lation time step
is taken equal to 10 min (1/6 h). The annual
wind, solar, and am bient temperature data needed for the esti-
mation of WT and PV performa nce refer to measurements of
the Technological Educational Institute of Crete for the moun-
tainous region of Keramia (alti tude 500 m), in Chania, C rete,
Greece. The annual SAPS peak load has been considered equal
to 120 kW, whereas the n ecessary SA PS load profile was com-
puted by downscaling the actual annual load profile of Crete is-
land, which is the l argest autonomous power system of Greece,
with 600-MW peak load and 17% min/max annual load. A n ad-
ditional noise has been added in t he load profile, in order to
reduce the min/max annual load ratio fro m 17% (Crete pow er
system) to 12% (SAPS).
The WT hub height has been considered 25 m, and the PVs
do n ot include the tracking system. The o peratin g reserve inputs,
needed for the calculation of system’s capacity shortage, have
been considered as 5% of the average 10-min load, 30% of the
average 10-min WT output, and 15% of the average 10-min PV
output. The values of parameters involv ed in co nstraints (3)– (7 )
are shown in Table I.
The cost, lifetime, and size characteristics for each compo-
nent are presented in Table II. For each component, the min-
imum size is equal to zero. Moreover, with the exception of
diesel and biodiesel generators, all components have constant
increment of their size, as Table II shows. The considered sizes
for the generators are 0, 5, 10, 20 , 30, 50, 80, 100, and 120 kW.
For the SAPS sizing problem of Table II, the complete enumer-
ation method r equires
(15)
i.e., approximately
evaluations in order to find the
optimal COE (in (15), Disp. denotes the number of dispatch
strategies). The computational time for each COE evaluation is
3 s. Consequently, the evaluatio ns of the complete enumeration
method require approxim ately 362 years. That is why it is es-
sential to develop alternative optimization methods to solve the
SAPS sizing problem in a fast and effective way.
B. Simulated Annealing
In the proposed SA algorithm,
has been set equal to 2.
The termination criterion is satisfied either if
reaches ,
