PSO-Based Multi-Criteria Economic Dispatch
Considering Wind Power Penetration Subject to
Lingfeng Wang and Chanan Singh
Department of Electrical and Computer Engineering
Texas A&M University
College Station, TX 77843
Emails: firstname.lastname@example.org, email@example.com
Abstract— Signiﬁcant attention has been paid to the renewable desired. Particle swarm optimization (PSO) is a salient meta-
energy resources such as wind power in recent years. It has heuristics which has drawn much attention recently due to
potential beneﬁts in curbing emissions and reducing the con- its ability to solve highly nonlinear and complex engineering
sumption of irreplaceable fuel reserves. However, the penetration
of wind power into traditional fuel-based generation systems optimization problems as well as its outstanding convergence
will also have some implications such as security concerns due performance. In this study, a multi-objective particle swarm
to its unpredictable nature. Thus, in economic power dispatch optimization (MOPSO) algorithm is developed to derive the
with wind power penetration, a reasonable tradeoff between tradeoff solutions for economic dispatch accounting for wind
system risk and operational cost is desired. In this paper, a bi- penetration. Furthermore, considering the different attitudes of
objective economic dispatch problem considering wind penetra-
tion is formulated, which treats economic and security impacts dispatchers towards the wind power integration, we used the
as conﬂicting objectives. Different fuzzy membership functions fuzzy constraints to indicate the security level in terms of wind
are used to reﬂect the dispatcher’s attitude toward the wind penetration and wind power cost. Different fuzzy constraints
power penetration. A modiﬁed multi-objective particle swarm including linear and quadratic functions can be used to reﬂect
optimization (MOPSO) algorithm is adopted to develop a power the dispatcher’s optimistic, neutral, or pessimistic attitude
dispatch scheme which is able to achieve compromise between
economic and security requirements. Numerical simulations in- toward wind penetration.
cluding sensitivity analysis are reported based on a typical IEEE The remainder of the paper is organized as follows. Section
test power system to show the validity and applicability of the 2 presents the wind penetration model. Section 3 formulates
proposed approach. the target economic dispatch problem including its multiple
objectives as well as design constraints. Section 4 introduces
the inner working of particle swarm optimization algorithms.
I. I NTRODUCTION The proposed MOPSO algorithm is discussed in Section 4.
Economic dispatch (ED) is a crucial function in the mod- Simulation results and analysis are presented in Section 5. Fi-
ern Energy Management System (EMS). Its objective is to nally, conclusions are drawn and future research is suggested.
schedule the power generation properly in order to minimize
the total operational cost [7-10, 14, 15]. More recently, wind II. W IND P OWER P ENETRATION M ODEL
power has drawn much attention as a promising renewable Because of its unpredictable and variable characteristics,
energy resource, which has shown some prospects in cutting the integration of wind power into the traditional thermal
fuel consumption and reducing the pollutants emission [1-6, generation systems will inevitably bring about the operator’s
11, 17, 18, 20]. However, accurate forecasting of the expected concern on system security. Fuzzy deﬁnition regarding wind
generation from a wind park is nearly impossible primarily penetration is a viable way to represent the penetration level
due to the stochastic nature of the wind as well as the highly of the wind power, since it is usually difﬁcult to determine the
nonlinear transform from wind speed to electrical energy. This optimal amount of wind power that should be integrated into
inevitably brings about the security concerns about incorpo- the conventional power grids .
rating wind power in the traditional power system, and risk As shown in Figure 1, a membership function µ regarding
mitigation issues should be properly addressed. For instance, the wind penetration is deﬁned to indicate the system security
the dynamic stability may be violated due to wind ﬂuctuations. level. It can be mathematically expressed in the following :
Therefore, in order to achieve the compromise between system
1, W ≤ W (PD )min
risk and total running cost, it is highly desirable to investigate
W (PD )max −W
how to dispatch the power in an appropriate manner for the µ= , Wmin ≤ W ≤ Wmax
W (PD )max −W (PD )min
power system with wind penetration. In this paper, the problem 0, W ≥ W (PD )max
is formulated as a bi-objective optimization problem through (II.1)
simultaneous minimization of both risk level and operational where W is the wind power incorporated in economic dis-
cost. For this purpose, an effective optimization algorithm is patch; W (PD )min is the lower bound of wind power penetra-
W ( PD ) min W ( PD ) max W W ( PD ) min W1 W2 W3 W ( PD ) max W
WC ( PD ) min WC ( PD ) max WC
Fig. 1. Fuzzy linear representation of the security level in terms of wind Fig. 2. Fuzzy quadratic representation of the security level in terms of wind
penetration and wind power cost. power penetration.
tion, below which the system is deemed secure; W (PD )max is shape is shown in Figure 3.
the upper bound of wind power penetration, above which the
system is considered as insecure due to the wind perturbations. 1, W C ≤ W C(PD )min
µ= ac W C 2 + bc W C + cc , W Cmin ≤ W C ≤ W Cmax
Both W (PD )min and W (PD )max are dependent on the total
0, W C ≥ W C(PD )max
load demand in the power dispatch.
The above deﬁned membership function can also be repre- where ac , bc , and cc determine the curve shape of the quadratic
sented in terms of the operational cost for incorporating wind function deﬁned in terms of the running cost of wind power.
1, W C ≤ W C(PD )min
W Cmax −W C 1
µ= W Cmax −W Cmin , W Cmin ≤ W C ≤ W Cmax
0, W C ≥ W C(PD )max
where W C is the running cost of wind power in the power
dispatch; W C(PD )min is the lower bound cost for producing
WC ( PD ) min WC1 WC2 WC3 WC ( PD ) max WC
wind power, below which the system is seen as secure;
W (PD )max is the upper bound cost for including wind power,
above which the system is considered as insecure due to the
wind intermittency. In a similar fashion, both W C(PD )min Fig. 3. Fuzzy quadratic representation of the security level in terms of wind
and W C(PD )max are dependent on the total load demand in
the power dispatch. In this study, sensitivity studies are also
carried out to illustrate the impact of different allowable ranges
of wind power penetration as well as different running costs III. P ROBLEM F ORMULATION
of wind power on the ﬁnal solutions obtained. The problem of economic power dispatch with wind pen-
To reﬂect dispatcher’s different attitudes toward wind power etration consideration can be formulated as a bi-criteria op-
penetration, a quadratic membership function can be deﬁned timization model. The two conﬂicting objectives, i.e., total
as follows : operational cost and system risk level, should be minimized
simultaneously while fulﬁlling certain system constraints. This
1, W ≤ W (PD )min bi-objective optimization problem is formulated mathemati-
µ= aw W 2 + bw W + cw , Wmin ≤ W ≤ Wmax cally in this section.
0, W ≥ W (PD )max
A. Problem objectives
where aw , bw , and cw are the coefﬁcients of the quadratic
function, which determine its curve shape reﬂecting the dis- There are two objectives that should be minimized simulta-
patcher’s attitude toward wind power. As shown in Figure 2, neously, that is, system security level and the total operational
by selecting different coefﬁcients aw , bw , and cw , different cost.
curve shapes of the quadratic function can be deﬁned. For • Objective 1: Minimization of system risk level
the identical risk level µ0 , the wind power costs for different From the security level function deﬁned in (II.1) and (II.
deﬁned functions w1 < w2 < w3 . The curves corresponding 2), we know that the larger the value of membership
to these three values reﬂect the pessimistic, neutral, and function µ is, the more secure the system will become.
optimistic attitudes of the dispatcher toward the wind power If the wind penetration is restricted under a certain level,
integration, respectively. the system can be considered as secure. On the contrary,
In a similar fashion, the security level can also be deﬁned if excessive wind penetration is introduced into the power
in terms of the operational cost of wind power. Its function dispatch, the system may become insecure. Thus, here we
deﬁne an objective function which should be minimized For normal system operations, real power output of each
in order to ensure system security: generator is restricted by lower and upper bounds as
1 min max
R(µ) = (III.1) PGi ≤ PGi ≤ PGi (III.8)
where PGi and PGi are the minimum and maximum
• Objective 2: Minimization of operational cost
power from generator i, respectively.
The cost curves of different generators are represented
• Constraint 2: Power balance constraint
by quadratic functions with sine components. The super-
The total power generation and the wind power must
imposed sine components represent the rippling effects
cover the total demand PD and the real power loss
produced by the steam admission valve openings. The
in transmission lines PL . For the linear membership
total $/h fuel cost F C(PG ) can be represented as follows:
function, this relation can be represented by
F C(PG ) = ai + bi PGi + 2
ci PGi PGi + Wmax − µ ∗ ∆W = PD + PL (III.9)
min For the quadratic membership function, the relation can
+ |di sin[ei (PGi − PGi )]| (III.2)
be expressed by
where M is the number of generators committed to the
operating system, ai , bi , ci , di , ei are the cost coefﬁcients bw µ − (cw − 4aw )
of the i-th generator, and PGi is the real power output of PGi − ± = PD +PL (III.10)
the ith generator. PG is the vector of real power outputs
of generators and deﬁned as The sign of the last term in (III.10) is determined by
the curve shape of the deﬁned quadratic function. The
PG = [PG1 , PG2 , . . . , PGM ] (III.3) transmission losses can be calculated based on the Kron’s
The running cost of wind power can be represented in loss formula as follows:
terms of the value of membership function µ which indi- M M M
cates the system security level. For the linear membership PL = PGi Bij PGj + B0i PGi + B00 (III.11)
function case, i=1 j=1 i=1
M where Bij , B0i , B00 are the transmission network power
W C(PG , µ) = Cw (Wav − (PD + PL − PGi )) loss B-coefﬁcients. It should be noted that the transfer
i loss of the wind power is not considered in this study.
− µ ∗ ∆W C + W Cmax (III.4) • Constraint 3: Available wind power constraint
The wind power used for dispatch should not exceed the
where Wav is the available wind power from the wind
available wind power from the wind park:
farm, Cw is the coefﬁcient of penalty cost for not using
all the available wind power, PD is the load demand, PL M
is the transmission loss, and 0 ≤ PD + PL − PGi ≤ Wav (III.12)
∆W C = W Cmax − W Cmin . (III.5) • Constraint 4: Security level constraint
For the quadratic membership function case, From the deﬁnition of membership function shown from
(II.1) to (II.4), the values of µ should be within the
interval of [0, 1]:
W C(PG , µ) = Cw (Wav − (PD + PL − PGi ))
0 ≤ µ ≤ 1. (III.13)
bc µ − (cc − 4ac )
C. Problem statement
− ± (III.6)
In summary, the objective of economic power dispatch op-
The sign of the last term in (III.6) is determined by the timization considering wind penetration is to minimize R(µ)
curve shape of the deﬁned quadratic function. Thus, the and T OC(PG , µ) simultaneously subject to the constraints
total operational cost T OC can be calculated as (III.7)–(III.11).
T OC(PG , µ) = F C(PG ) + W C(PG , µ) (III.7)
IV. M ECHANISM OF PARTICLE S WARM O PTIMIZATION
Particle swarm optimization (PSO) is a population-based
B. Problem constraints
stochastic optimization technique, which was inspired by the
Due to the physical or operational limits in practical sys- movement pattern in a bird ﬂock or ﬁsh school [12, 13]. In
tems, there is a set of constraints that should be satisﬁed PSO, individuals (i.e., particles) move around in a multidi-
throughout the system operations for a feasible solution. mensional search space to approach the optima, where each
• Constraint 1: Generation capacity constraint point represents a solution to the target problem. Initially a
bunch of particles are randomly created and set into motion V. T HE P ROPOSED A PPROACH
through this space. In their movement, each particle adjusts The standard PSO algorithm is not suited to resolve multi-
its position based on its own experience as well as the expe- objective optimization problems in that no absolute global op-
rience of a neighboring particle by utilizing the best position timum exists there, but rather a set of non-dominated solutions.
encountered by itself and its neighbors. At each generation, Thus, to render the PSO algorithm capable of dealing with MO
they observe the “ﬁtness” of themselves and their neighbors problems, some modiﬁcations become necessary. When using
and move toward those with a better position. In this way, PSO stochastic search based algorithms to optimize multi-objective
combines both local and global search methods together in problems, two key issues usually arise in the algorithm design.
order to improve its search effectiveness and efﬁciency. Unlike First, the ﬁtness evaluation should be suitably designed to
other evolutionary computation algorithms including Genetic guide the search toward the set of Pareto-optimal solutions.
Algorithms (GA), PSO has no evolution operators such as Second, the diversity of the population should be maintained
crossover and mutation. The optima is obtained via following by refraining the search from premature convergence. In this
the current optimum particles by the potential particles. This study, the classic PSO algorithm is revised accordingly to
simple algorithm turns out to be highly effective in a diverse facilitate a multi-objective optimization approach, i.e., multi-
set of optimization problems. objective particle swarm optimization (MOPSO). The Pareto-
Let x and v denote a particle coordinates (posi- dominance concept is used to evaluate the ﬁtness of each
tion) and its corresponding ﬂight speed (velocity) in the particle and thus determine which particles should be selected
search space. Therefore, the i-th particle is represented as to store in the archive of non-dominated solutions. Somehow
xi = [xi1 , xi1 , . . . , xid , . . . , xiM , xi,M +1 ] in the (M + 1)- similar to the elitism used in evolutionary algorithms and the
dimensional space. Each particle keeps track of its coor- tabu list used in tabu searches, the best historical solutions
dinates in the solution space which are associated with found by the population are recorded continuously in the
the best solution it has achieved so far. This ﬁtness archive in order to serve as the non-dominated solutions
value is called pbest. The best previous position of the generated in the past. Furthermore, due to the global attraction
i-th particle is recorded and represented as pbesti = mechanism in PSO, the historical archive of previously found
[pbesti1 , pbesti2 , . . . , pbestid , . . . , pbestiM , pbesti,M +1 ]. An- non-dominated solutions would make the search converge
other “best” value that is tracked by the particle swarm toward globally non-dominated solutions highly possible.
optimizer is the best value obtained so far by any particle
in the neighbors of the particle. When a particle takes all the
A. Constrained PSO
population as its topological neighbors, the best value is a
global best and is called gbest. The index of the best particle Because the standard PSO does not take into account how
among all the particles in the group is represented by the to deal with the constraints, the constraints handling mech-
gbestd . The rate of the velocity for particle i is represented anism should be added to ensure the solution feasibility in
as vi = (vi1 , vi2 , . . . , vid , . . . , viM , vi,M +1 ). The modiﬁed constrained optimization problems such as power dispatch. In
velocity and position of each particle can be calculated using the proposed MOPSO, a simple constraint checking procedure
the current velocity and the distance from pbestid to gbestd called rejecting strategy is incorporated. When an individual is
as shown in the following formulas: evaluated, the constraints are ﬁrst checked to determine if it is
a feasible candidate solution. If it satisﬁes all of the constraints,
it is then compared with the non-dominated solutions in the
(t+1) (t) (t)
vid = w ∗ vid + c1 ∗ rand() ∗ (pbestid − xid ) archive. Here the concept of Pareto dominance is applied
+ c2 ∗ Rand() ∗ (gbestd − xid ),
(IV.14) to determine if it is eligible to be chosen to store in the
archive of non-dominated solutions. The constraint satisfaction
checking scheme used in the proposed algorithm proves to be
quite effective in ensuring the feasibility of the non-dominated
(t+1) (t) (t+1)
xid = xid +χ∗vid , i = 1, 2, . . . , N, d = 1, 2, . . . , M +1. solutions.
where N is the number of particles in a group, M + 1 B. Individual (particle) representation
is the number of members in a particle, t is the pointer
It is crucial to appropriately encode the individuals of
of generations, χ ∈ [0, 1] is the constriction factor which
the population in PSO for handling the economic dispatch
controls the velocity magnitude, w is the inertia weight factor,
application. The power output of each generating unit and
c1 and c2 are acceleration constants, rand() and Rand() are
(t) the value of membership function are chosen to represent
uniform random values in a range [0, 1], vi is the velocity
(t) the particle position in each dimension, and positions in
of particle i at generation t, and xi is the current position
different dimensions constitute an individual (particle), which
of particle i at generation t. The particle swarm optimization
is a candidate solution for the target problem. The position in
concept consists of, at each time step, changing the velocity
each dimension is real-coded. The i-th individual PGi can be
of (accelerating) each particle toward its pbest and gbest
represented as follows:
locations. Acceleration is weighted by a random term, with
separate random numbers being generated for acceleration PGi = [PGi1 , PGi2 , . . . , PGid , . . . , PGiM , µi ], i = 1, 2, . . . , N
toward pbest and gbest locations. (V.16)
where M is the number of generators and N is the population For the value of membership function µ,
size; PGid is the power generated by the d-th unit in i-th (t+1) (t+1) (t+1)
individual; and µi is the value of the membership function in µi = µi + RT µi (V.22)
i-th individual. Thus, the dimension of a population is N × where RT is the turbulence factor. The turbulence term
(M + 1). is used here to enhance the diversity of solutions.
• Step 11: Update the archive which stores non-dominated
C. Algorithm steps solutions according to Pareto-optimality based selection
The proposed MOPSO is applied to the constrained power criteria .
dispatch problem in order to derive out the optimal or near- • Step 12: If the current individual is dominated by the
optimal solutions. Its computational steps include: pbest in the memory, then keep the pbest in the memory;
Otherwise, replace the pbest in the memory with the
• Step 1: Specify the lower and upper bounds of generation
power of each unit as well as the range of security level.
• Step 13: If the maximum iterations are reached, then go
• Step 2: Randomly initialize the individuals of the popu-
to Step 14. Otherwise, go to Step 7.
lation. Note that the speed and position of each particle
• Step 14: Output a set of Pareto-optimal solutions from
should be initialized such that each candidate solution
the archive as the ﬁnal solutions.
(particle) locates within the feasible search space.
• Step 3: For each individual PGi of the population, the
transmission loss PLi is calculated based on B-coefﬁcient VI. S IMULATION AND E VALUATION OF THE P ROPOSED
loss formula. A PPROACH
• Step 4: Evaluate each individual PGi in the population In this study, a typical IEEE 30-bus test system with 6-
based on the concept of Pareto-dominance. generators  is used to investigate the effectiveness of
• Step 5: Store the non-dominated members found thus far the proposed MOPSO approach. The system conﬁguration is
in the archive. shown in Figure 4. The system parameters including fuel cost
• Step 6: Initialize the memory of each particle in which a coefﬁcients and generator capacities are listed in Table I. The
single local best pbest is stored. The memory is contained sinusoidal term in (III. 2) is not considered in this study due
in another archive. to its relatively minor impact on the total fuel costs. The B-
• Step 7: Increment iteration counter. coefﬁcients are shown in (VI.23). The load demand PD used
• Step 8: Choose the personal best position pbest for in the simulations is 2.834 p.u., the available wind power Wav
each particle based on the memory record; Choose the is 0.5668 p.u, and the coefﬁcient of penalty cost Cw is set 20
global best gbest from the fuzziﬁed region using binary $/p.u.
tournament selection . The niching and ﬁtness sharing
mechanism is also applied throughout this process for
enhancing the diversity of solutions.
• Step 9: Update the member velocity v of each individual
PGi based on (IV.14). For the output of each generator,
(t+1) (t) (t)
vid = w ∗ vi + c1 ∗ rand() ∗ (pbestid − PGid )
+ c2 ∗ Rand() ∗ (gbestd − PGid ),
i = 1, . . . , N ; d = 1, . . . , M (V.17)
where N is the population size, and M is the number of
generating units. For the value of membership function
(t+1) (t) (t)
vi,M +1 = w ∗ vi + c1 ∗ rand() ∗ (pbesti,M +1 − µi )
+ c2 ∗ Rand() ∗ (gbestM +1 − µi ), (V.18)
• Step 10: Modify the member position of each individual
PGi based on (IV.15). For the output of each generator, Fig. 4. IEEE 30-bus test power system
(t+1) (t) (t+1)
PGid = PGid + χvid (V.19)
For the value of membership function µ,
(t+1) (t) (t+1)
µi = µi + χvi,M +1 (V.20) Since PSO algorithms are sometimes quite sensitive to
Following this, add the turbulence factor into the current certain parameters, the simulation parameters should be appro-
position. For the output of each generator, priately chosen. In the simulations, both the population size
and archive size are set to 100, and the number of generations
(t+1) (t+1) (t+1)
PGid = PGid + RT PGid (V.21) is set to 500. The acceleration constants c1 and c2 are chosen
0.1382 −0.0299 0.0044 −0.0022 −0.0010 −0.0008
−0.0299 0.0487 −0.0025 0.0004 0.0016 0.0041
0.0044 −0.0025 0.0182 −0.0070 −0.0066 −0.0066
Bij = (VI.23)
−0.0022 0.0004 −0.0070 0.0137 0.0050 0.0033
−0.0010 0.0016 −0.0066 0.0050 0.0109 0.0005
−0.0008 0.0041 −0.0066 0.0033 0.0005 0.0244
F UEL COST COEFFICIENTS AND GENERATOR CAPACITIES 1
Quadratic function (Pessimistic)
0.9 Linear function (Neutral)
Generator i ai bi ci min
PGi Quadratic function (Optimistic)
G1 10 200 100 0.05 0.50 0.8
G2 10 150 120 0.05 0.60 0.7
G3 20 180 40 0.05 1.00
G4 10 100 60 0.05 1.20
G5 20 180 40 0.05 1.00
G6 10 150 100 0.05 0.60
as 1. Both turbulence factor and niche radius are set to 0.02. 0.2
The inertia weight factor w decreases when the number of
wmax − wmin 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65
w = wmax − × iter (VI.24) Wind power (p.u.)
where itermax is the number of generations and iter is the
Fig. 5. Different curve shapes of membership functions.
current number of iterations. From the above equation, we
can appreciate that the value of w will decrease as the iteration TABLE II
number increases. In the search process, the most efﬁcient way E XAMPLE SOLUTIONS FOR DIFFERENT DESIGN SCENARIOS .
to locate the optimal or near-optimal solutions in a complex
large search space is ﬁrst to move to the smaller solution Generators/objectives Pessimistic Linear Optimistic
PG1 0.0678 0.0500 0.0605
space as promptly as possible, and then seek out the desired PG2 0.2463 0.2430 0.2425
solution in this space via thorough search. The parameter w is PG3 0.3863 0.4029 0.4221
deﬁned to regulate the size of search step of each particle. PG4 0.9164 0.9300 0.9354
PG5 0.4034 0.3990 0.3490
At ﬁrst, the value of w is set relatively large in order to PG6 0.3166 0.2929 0.2897
drive the particle to the solution area quickly. Then, when W 0.5043 0.5232 0.5408
the particle approaches the desired solution, the size of each Cost ($/hour) 518.893 515.436 512.147
Risk level 6.5864 6.49894 6.31094
search step becomes smaller in order to prevent the particle
from ﬂying past the target position during the ﬂight. In this
way, the desired solutions can be sought out through gradual
reﬁnement. The simulation program is coded using C++ and fronts evolved using the proposed MOPSO are shown in
executed in a 2.20 GHz Pentium-4 processor with 512 MB of Figure 6.
RAM. In simulations, the minimum and maximum allowable As shown in the ﬁgure, the Pareto-optimal solutions are
wind power penetrations are set as 10% and 20% of the total widely distributed on the tradeoff surface due to the diver-
load demand, respectively. The running cost of wind power sity preserving mechanisms used in the proposed MOPSO
is calculated based on its linear relationship with the amount algorithm. Unlike the single-objective optimization, in multi-
of wind power integrated, i.e., W C = σW . The coefﬁcient objective optimization the decision maker can choose a suit-
σ indicating the running cost of wind power is set 50 $/p.u. able solution based on his/her preference from a pool of non-
in the simulation. The parameters used in the simulations are dominated solutions. We can also appreciate that for the same
listed below and different function curves are shown in Figure risk level calculated from different membership functions,
5. the optimistic design has the lowest operational cost since it
includes the largest amount of wind power among all of the
• Quadratic representation (optimistic design): aw =
−9.9607, bw = 4.94, cw = 0.4;
• Linear representation (neutral design): Wmin =0.2834,
Wmax =0.5668; B. Sensitivity analysis
• Quadratic representation (pessimistic design): aw = A study on sensitivity analysis is carried out in order to
4.9803, bw = −7.7629, cw = 2.8. illustrate the impacts of different allowable ranges of wind
The illustrative non-dominated solutions derived in different power penetration as well as different running costs of wind
design scenarios are listed in Table II and the Pareto-optimal power on the ﬁnal non-dominated solutions derived. Here the
Quadratic function (Pessimistic) [6%,16%]
18 Linear function (Neutral) 18 [8%,18%]
Quadratic function (Optimistic) [10%,20%]
500 510 520 530 540 550 560 500 510 520 530 540 550 560 570 580 590
Operational cost ($/hour) Operational cost ($/hour)
Fig. 6. Pareto fronts obtained based on different membership functions. Fig. 7. Pareto fronts obtained for different wind penetration ranges.
linear membership function is used. As indicated in (II.1), for 20
Running cost = 40 $/p.u.
the same range of wind penetration, we can see that the more 18 Running cost = 50 $/p.u.
wind power is integrated, the more insecure the system will Running cost = 60 $/p.u.
become. We herein quantify the impact of wind penetration
through numerical simulations by changing the permissible 14
ranges of wind power penetration [W (PD )min , W (PD )max ]. 12
In the simulations for determining the impact of different
allowable wind penetration ranges, the running cost of wind
power keeps unchanged, i.e., σ = 50$/p.u. In a similar 8
fashion, in the simulations for examining the impact of running 6
costs of wind power, the penetration range of wind power is
ﬁxed, i.e., [W (PD )min , W (PD )max ] = [10%∗PD , 20%∗PD ]. 4
The derived Pareto fronts are shown in Figure 7 and Figure 8, 2
respectively. From the ﬁgures, we can appreciate that the non-
dominated solutions vary with the different ranges of allowable 500 510 520 530 540 550 560 570
Operational cost ($/hour)
wind penetration as well as the running costs of wind power.
In Figure 7, at the same risk level, the design scenario with the
largest value of maximum allowable wind penetration Wmax Fig. 8. Pareto fronts obtained for different running costs of wind power.
has the lowest cost since the most portion of wind power is
integrated. In Figure 8, at the identical risk level, the scenario
with the lowest running cost of wind power results in the wind power penetration level and operational costs are used
lowest overall cost since the same amount of wind power is in the construction of economic dispatch models. A multi-
integrated with the lowest cost. objective particle swarm optimization (MOPSO) algorithm
is developed to derive the optimal tradeoff solutions with
VII. C ONCLUDING R EMARKS respect to the two speciﬁed design objectives. Different design
The paper is primarily intended to investigate the integration scenarios can be formulated according to dispatcher’s attitudes
of wind power into conventional power networks and its toward wind power integration in terms of risk and cost. A nu-
impact on the generation resource management due to its merical application example is used to demonstrate the validity
nondispatchable characteristic. Wind power is environmentally and applicability of the proposed optimization procedure. In
attractive since it can help to spur the reductions in fossil fuel the further investigations, probabilistic methods may also be
and natural gas consumption. Wind power needs less opera- adopted to handle various operational and planning problems
tional cost since it does not consume fossil fuels and natural in a more effective fashion due to the intermittency of wind
gases. However, due to the intermittent and variable nature power.
of the wind power, it is usually quite difﬁcult to determine
how much wind power should be used in order to guarantee ACKNOWLEDGMENT
both power system security and operational cost reduction. In The authors would like to thank the ﬁnancial support from
this paper, fuzzy representations of system security in terms of NSF for this research (Grant No. ECS0406794: Exploring the
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