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An Economic Dispatch Model Incorporating Wind Power
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Abstract
In solving the electrical power systems economic dispatch
(ED) problem, the goal is to find the optimal allocation of
output power among the various generators available to serve the
system load. With the continuing search for alternatives to conventional
energy sources, it is necessary to include wind energy
conversion system (WECS) generators in the ED problem. This
paper develops a model to include the WECS in the ED problem,
and in addition to the classic economic dispatch factors, factors to
account for both overestimation and underestimation of available
wind power are included. With the stochastic wind speed characterization
based on the Weibull probability density function, the
optimization problem is numerically solved for a scenario involving
two conventional and two wind-powered generators. Optimal solutions
are presented for various values of the input parameters, and
these solutions demonstrate that the allocation of system generation
capacity may be influenced by multipliers related to the risk
of overestimation and to the cost of underestimation of available
wind power.
Index Terms—Economic dispatch, penalty cost, reserve cost,
Weibull probability density function, wind energy.
I. INTRODUCTION
ECONOMIC dispatch (ED) deals with the minimum cost
of power production in electrical power system analysis
[1], [2]. More specifically, in solving the ED problem, one
seeks to find the optimal allocation of the electrical power output
from various available generators. Prior to the widespread
use of alternate sources of energy, the ED problem involved
only conventional thermal energy power generators, which use
depletable resources such as fossil fuels. It has become apparent
that there is a need for alternatives to thermal energy power
generation, and one of the sources that is now seeing more
widespread use, particularly outside of the United States, is the
wind energy. One of the major benefits of wind energy is that,
after the initial land and capital costs, there is essentially no cost
involved in the production of power from wind energy conversion
systems (WECS). In addition, the impacts of WECS are
generally considered to be environmentally friendlier than the
impacts of thermal energy sources.
The primary problem associated with the incorporation of
wind power into the ED model is the fact that the future wind
speed, which is the power source for the WECS, is an unknown
at any given time. A similar comment might be made about
Manuscript received July 12, 2006; revised September 29, 2007. Paper no.
TEC-00234-2006.
The authors are with the Department of Electrical Engineering and Computer
Science, University of Wisconsin–Milwaukee, Milwaukee, WI 53211 USA
(e-mail: jhetzer[at]uwm.edu; yu[at]uwm.edu; kalu[at]uwm.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2007.914171
the volatility of the prices of conventional energy sources, such
as coal or oil, or about the future system load; however, even
though these inputs, too, are unknowns, their variability is still
much lower than that of the future wind speed. Several investigations
have looked at the prediction of wind speed for use
in determining the available wind power. These investigations
have been based on such foundations as fuzzy logic [3], neural
networks [4], and time series [5]. Because the focus of this paper
is on the ED problem and not on wind power forecasting, fuzzy
logic or similar theories to develop the wind speed profile will
not be used, but a known probability distribution function (PDF)
for the wind speed will be assumed, and then, transformed to the
corresponding wind power distribution for use in the ED model.
With themorewidespread use of theWECS, the power system
operator now faces the problem of not only allocating system
power among conventional generators, but also among various
available wind-powered generators. The primary characteristic
that differentiates wind-powered from conventional generators
in the ED problem is the stochastic nature of wind speed. After
an ED model that incorporates both thermal and wind energy
sources is developed, it is still necessary to characterize the
stochastic nature of the wind speed in order to analyze the
problem with numerical results.
The objective of this paper is to incorporate wind-powered
generators into the classical economic dispatch problem and to
investigate the problem via numerical solutions. In Section II,
the economic dispatch model, which is fundamentally a classic
optimization problem,will be developed to include both conventional
generators and wind-powered generators. Section III will
discuss the characterization of wind speed as a random variable
and will introduce theWeibull probability density function (pdf)
as the basis for numerical solutions of the EDmodel. TheWECS
power input–output equation and the transformation from the
wind speed random variable to the wind power random variable
is presented in Section IV. In Section V, the numerical solution
to the ED problem using the MATLAB is discussed. Section VI
presents a discussion of the numerical results achieved when
several different wind and cost scenarios are applied to the ED
model. Finally, in Section VII, conclusions are drawn, based on
the results found from the numerical analyses in Section VI.
II. ECONOMIC DISPATCH MODEL INCLUDING THE WECS
As applied to electrical power systems, the economic dispatch
problem is a classic mathematical optimization problem.
The goal is to obtain an optimum allocation of power output
among the available generators with given constraints. The sum
of the outputs from the available generators must equal the system
load plus any system losses. In addition, certain constraints
0885-8969/$25.00 © 2008 IEEE
604 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 2, JUNE 2008
may be placed on the generators in the model. These constraints
typically take the form of minimum and maximum generator
outputs. The ED equations are valid for a given time period
within which the generator outputs, loads, and losses are considered
constant. From the point of view of the system operator,
the economic dispatch problem may take different forms, depending
on the extent of ownership by the system operator of
the conventional and wind-powered generators.
If the wind-powered generators are owned by the system operator
that is performing the economic dispatch, there is little
or no incremental cost associated with the wind-powered generators.
This incremental cost forms the basis for an economic
dispatch; so effectively, the system operator will want to use
all available wind energy. On the other hand, because of the
uncertainty in the availability of wind energy at any time in
the future, even if the system operator owns the wind-powered
generators, the ED model must still provide some check on the
overscheduling wind power, and this is the reason why some
factor in the model must account for the reserve necessary in
case the scheduled wind power is not available.
In [6], an economic dispatch model to include wind-powered
generators is developed by using concepts from the fuzzy set theory.
To the classic ED equations, the authors of [6] add a penalty
cost factor for not using the available wind power capacity. A
fuzzy wind border is defined, where wind values below a certain
minimum are considered fully acceptable based on security
concerns and wind values above a certain maximum are considered
unacceptable for system security reasons. Values within
these two limits form the fuzzy border, where the membership
value goes from 1 for the minimum value to 0 for the maximum
value.
In a manner somewhat similar to the fuzzy set theory approach,
this paper will use probability functions to characterize
the wind speed profiles, and an additional factor for overestimation
of the available wind power will be added. In general,
losses are ignored in the model; however, they could be added
in the system load and losses term L, if necessary.
In this paper, the ED model will be developed in the most
general case, so that it is adaptable to all situations, regardless
of who owns the generation facilities. In the most general form,
the system operator will have certain conventional generators
and certain wind generators available. Because of the uncertainty
of the wind energy available at any given time, factors for
overestimation and underestimation of available wind energy
must be included in the model. The factor for overestimation is
easily explainable in that, if a certain amount of wind power is
assumed and that power is not available at the assumed time,
power must be purchased from an alternate source or loads must
be shed. In the case of the underestimation penalty, if the available
wind power is actually more than what was assumed, that
power will be wasted, and it is reasonable for the system operator
to pay a cost to the wind power producer for the waste
of available capacity. The surplus wind power is usually sold
to adjacent utilities, or by fast redispatch and automatic gain
control (AGC), the output of nonwind generators is correspondingly
reduced. Only if this cannot be achieved, then dummy
load resistors need to be connected to “waste” the surplus energy.
Obviously, these practicalities can be modeled by a simple
underestimation penalty cost function.
Putting the aforementioned discussion in the format of an
optimization problem, the mathematical model directly follows.
This model is valid in any given ED time period; however,
to lessen confusion at this point, the time dependence of the
equations is suppressed.
Minimize
M
i
Ci(pi) +
N
i
Cwj(wi) +
N
i
Cp,wj(Wi,av − wi)
+
N
i
Cr,w,i(wi −Wi,av ) (1)
subject to
pi,min ≤ pi ≤ pi,max (2)
0 ≤ wi ≤ wr,i (3)
M
i
pi +
N
i
wi = L (4)
where
M number of conventional power generators;
N number of wind-powered generators;
pi power from the ith conventional generator;
wi scheduled wind power from the ith wind-powered generator;
Wi,av available wind power from the ith wind-powered generator.
This is a random variable, with a value range
of 0 ≤ Wi,av ≤ wr and probabilities varying with the
given pdf. We considered Weibull pdf for wind variation;
wr,i rated wind power from the ith wind-powered
generator;
Ci cost function for the ith conventional generator;
Cw,i cost function for the ith wind-powered generator. This
factor will typically take the form of a payment to the
wind farm operator for the wind-generated power actually
used;
Cp,w,i penalty cost function for not using all available power
from the ith wind-powered generator;
Cr,w,i required reserve cost function, relating to uncertainty
of wind power. This is effectively a penalty associated
with the overestimation of the available wind power;
L system load and losses.
Taking a closer look at the objective function (1), the first
term is the traditional sum of the fuel costs of the conventional
generators. The second term is the direct cost for the power
derived from the wind-powered generators. The existence and
size of this term will depend on the ownership of the windpowered
generators. If the generators are owned by the system
operator, this term may not even exist if it accounts only for
the incremental fuel cost, which is zero for the wind; however,
if the system operator is paying for the wind power from the
owner of the wind farm, a direct cost will be involved. The third
term, which will be explained in more detail next, accounts for
HETZER et al.: ECONOMIC DISPATCH MODEL INCORPORATING WIND POWER 605
not using all the available wind power. As with the previous
term, the costs associated with this term will depend on who
owns the wind-powered generators. Finally, the fourth term in
the objective function relates to the price that must be paid
for overestimation of the available wind power. Without regard
to ownership of the wind-powered generators, the ED model
must account for the possibility that a reserve would need to be
drawn on if all the available wind power is inadequate to cover
the amount of the wind power scheduled in a given time period.
For the conventional generators, a quadratic cost functionwill
be assumed, which is practical formost of the cases, and is given
by
Ci(pi) = ai
2 p2i
+ bipi + ci (5)
where ai , bi , and ci are cost coefficients for the ith conventional
energy source, which are found from the input–output curves of
the generators and are dependent on the particular type of fuel
used [1], [2].
In the case where the WECS is owned by the system operator,
this function may not exist as the power requires no fuel, unless
the operator wants to assign some payback cost to the initial outlay
for theWECS or unless the system operator wants to assign
this as a maintenance and renewal cost. But in a nonutilityowned
WECS, the wind generation will have a cost that must
be based on the special contractual agreements. The output of
the wind generator is constrained by an upper and lower limit,
decided by the system operator based on the agreements for the
optimal operation of the system [7]. For simplicity, this can be
considered to be proportional to the scheduled wind power or
totally neglected. We neglected this in all our studies for the
system-operator-owned WECS, and considered to be proportional
to the scheduled wind power for the nonutility-owned
WECS. The ED model is thus developed in the most general
sense to make it adaptable to all situations, regardless of who
owns the generation facilities.
A linear cost function will be assumed for the wind-generated
power actually used as
Cw,i(wi) = diwi (6)
where di is the direct cost coefficient for the ith wind generator.
It will be assumed that the penalty cost for not using all the
available wind power will be linearly related to the difference
between the available wind power and the actual wind power
used. The penalty cost function will then take the following
form
Cp,w,i(Wi,av − wi) = kp,i(Wi,av − wi)
= kp,i
wr , i
wi
(w − wi)fW (w)dw (7)
where
kp,i penalty cost (underestimation) coefficient for the ith
wind generator;
fW (w) WECS wind power pdf.
As with the direct cost, if the system operator owns the windpowered
generators, the penalty cost may not exist.
The reserve requirement cost will be similar to the penalty
cost (7), in that it is an integral over the pdf of the wind power
random variable, except that, in this case, it is a cost due to the
available wind power being less than the scheduled wind power.
Both (7) and (8) can be modeled in the MATLAB using the
built-in “quad” function
Cr,w,i(wi −Wi,av) = kr,i(wi −Wi,av )
= kr,i
wi
o
(wi − w)fW (w)dw (8)
where kp,i is the reserve cost (overestimation) coefficient for the
ith wind-powered generator.
To avoid unnecessary complexity in the model, it is assumed
that the difference between the available wind power and the
scheduled wind power, multiplied by the wind power output
probability function is linearly related to the reserve cost.
To obtain a numerical value for the reserve and penalty costs,
it is necessary to find or assume the pdf for the wind power
output. In general, of course, the wind speed is an unknown at
any future time; however, in order to obtain some quantitative
results, some known probability function for the wind speed
will be assumed. This leads to the next section.
III. WIND SPEED CHARACTERIZATION
In order to be able to rationally approach the economic dispatch
with WECS problem, some characterization of the uncertain
nature of the wind speed is needed. Prior research [8]
has shown that the wind speed profile at a given location most
closely follow a Weibull distribution over time. The pdf for a
Weibull distribution is given by
fv (v) =
k
c
v
c
(k−1)
(e)−(vc)k
, 0 < v < ∞ (9)
where
V wind speed random variable;
v wind speed;
c scale factor at a given location (units of wind speed);
k shape factor at a given location (dimensionless).
For later use in conjunction with the wind power probability
function, the Weibull PDF is given by
FV (v) =
v
0
fV (τ ) dτ = 1− e
−(v/c)k
. (10)
The Weibull distribution function with a shape factor of 2 is
also known as the Rayleigh distribution. In [9], the advantages
of the Weibull distribution are noted as follows: 1) it is a twoparameter
distribution, which is more general than the singleparameter
Rayleigh distribution, but less complicated than the
five-parameter bivariate normal distribution; 2) it has been previously
shown to provide a good fit to observed wind speed
data; and 3) if the k and c parameters are known at one height, a
methodology exists to find the corresponding parameters at another
height. The characteristics of the wind depend on various
factors like geography, topography, etc., and can be estimated
by the observed frequency of wind speed in the target region.
606 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 2, JUNE 2008
Fig. 1. Weibull probability density functions for k factors of 1 (top) and 3
(bottom), each with c factors of 5, 10, 15, and 20.
Methods of estimating the Weibull shape and scale factors using
the available wind speed data are given in [9] and [10]. The
shape parameter varied from 1.0 to 3.0 and the scale parameter
ranged from 5 to 20 in [8], which are used in this paper for
analysis.
It will be shown later that, due to the characteristics ofWECS
generators, the continuous wind speed distributions will become
mixed discrete and continuous distributions in the transformation
to wind power distributions.
Fig. 1 shows the Weibull pdf with shape factors of 1and 3.
Within each of these plots, curves of scale factor 5, 10, 15, and
20 are indicated.
Before moving on to look at another potential wind speed
probability distribution to be used with the ED model, some
comments on Fig. 1 may be made. The mean of the Weibull
function is
μ = cΓ(1 + k
−1 )
and the variance is
σ2
v = c2Γ(1 + 2k
−1 ) − μ2
and where the gamma function is
Γ(x) =
∞
0
yx−1e
−y dy, y > 0.
For the Rayleigh distribution, k = 2 and
μ =
√
π
2
and σ2
v = C2(1 − π
4
).
From the aforementioned, it is seen that, as the c factor of
theWeibull function increases, the mean and standard deviation
also increase in a linear relationship. The topic of wind speed
prediction is outside the scope of this paper; however, in the
analysis section of this paper, some assumptions about possible
outcomes of prediction algorithms will be made, and it will be
seen how the potential outcomes will affect the allocation of
system generation capacity among available generators.
IV. WECS INPUT/OUTPUT AND PROBABILITY FUNCTIONS
Once the uncertain nature of the wind is characterized as
a random variable, the output power of the WECS may also
be characterized as a random variable through a transformation
from wind speed to output power. Ignoringminor nonlinearities,
the output of the WECS with a given wind speed input may be
stated as [11]
w = 0, for v < vi and v > vo (11)
w = wr
(v − vi)
(vr − vi) , for vi ≤ v ≤ vr (12)
w = wr , for vr ≤ v ≤ vo (13)
where
w WECS output power (typical units of kilowatt or
megawatt);
wr WECS rated power;
vi cut-in wind speed (typical units of miles/hour or miles/
second);
vr rated wind speed;
vo cut-out wind speed.
Thus, it is seen that the WECS has: 1) no power output up
to cut-in wind speed (11); 2) a linear power output relationship
between cut-in and rated wind speed (12); 3) a constant rated
power output between the rated wind speed and cut-out wind
speed (13); and 4) once again has no power output with wind
speeds greater than the cut-out wind speed (11).
Due to the fact that the WECS power output has a constant
zero value below the cut-in wind speed and also above the cutout
wind speed, and due to the fact that the power output is
constant between rated wind speed and cut-out wind speed, the
power output random variable will be discrete in these ranges
of wind speed. The WECS power output is a mixed random
variable, which is continuous between values of zero and rated
power, and is discrete at values of zero and rated power output.
If it is assumed that the wind speed has a given distribution,
such as theWeibull, it is then necessary to convert that distribution
to a wind power distribution. This transformation may be
accomplished in the following manner, with V as the wind speed
random variable and W as the wind power random variable. For
a linear transformation, in general [12], such as that described
in (12)
W = T(V ) = aV + b (14)
and
fW (w) = fV [T
−1 (w)]
dT −1 (w)
dw
= fV
w − b
a
1
a
(15)
where
T a transformation, in general;
W wind power random variable;
V wind speed random variable;
w wind power (a realization of the wind power random
variable);
HETZER et al.: ECONOMIC DISPATCH MODEL INCORPORATING WIND POWER 607
Fig. 2. Wind power output mixed probability function for the Weibull wind
speed distribution. Discrete probabilities at 0 and 1; continuous probability
function between 0 and 1.
v wind speed (a realization of the wind speed random
variable).
For the Weibull function, the discrete portions of the WECS
power output random variable will have the following values,
found directly from the Weibull PDF
Pr{W = 0} = FV (vi) + (1 − FV (vo ))
= 1− exp
−
vi
c
k
+ exp
−
vo
c
k
(16)
and
Pr{W = wr } = FV (vo ) − FV (vr )
= exp
−
vr
c
k
− exp
−
vo
c
k
. (17)
To make the transformation from the wind speed random
variable to the WECS power output random variable in the
linear portion of the curve a bit less cumbersome, the following
ratios are defined:
ρ = w
wr
ratio of wind power output to rated wind
power; and
l = (vr −vi )
vi
ratio of linear range of wind speed to cut-in
wind speed.
Using (14), the Weibull PDF of the WECS power output
random variable in the continuous range then takes the form
fW (w) = klvi
c
(1 + ρl)vi
c
k−1
exp
−
(1 + ρl)vi
c
k
.
(18)
Before proceeding to the numerical solutions of the ED problem,
it may be instructive to look at the relationship between the
critical wind speed values related to the WECS generator power
output—cut-in, rated, and cut-out—and the critical values that
define the wind speed probability profiles – the c and k values
in the case of theWeibull distribution function. For Fig. 2, vi =
5, vr = 15, and vo = 45 will be used. These numbers are not
for any particular wind turbine; however, they are reflective of
general numbers on a mile per hour unit basis (see [13]). Although
the plot between 0 and 1 is a continuous pdf, for clarity,
individual markers are shown along the continuous line so that
these continuous portions of the probability function may be
associated with the corresponding discrete probability markers
shown at both ends of the probability function.
In Fig. 2, the discrete and continuous portions of the wind
power output probability function based on the Weibull wind
speed pdf with k = 2 and c factors of 10, 15, and 20 are plotted.
As the c factor in the Weibull distribution function is increased,
a greater proportion of the wind speed profile will be located at
higher values of wind speed. This translates to a lower discrete
probability of zero power, a higher discrete probability of rated
power, and less power in the continuous portion of the plot.
As with any other mixed discrete and continuous probability
function, the sum of the discrete probabilities at zero and rated
power, plus the integral from 0 to 1 of the continuous function
will sum to 1.
V. NUMERICAL SOLUTION
The classic solution of the ED problem without the inclusion
of the WECS is to take the partial derivatives of the objective
(cost) function with respect to each generator output. Except
for the generators operating at a fixed minimum, the solution is
found where the partial derivatives, also known as incremental
costs, are equal for all generators. In addition, other constraints
of the ED problem, most importantly, the load balance equation
must be satisfied. This method could potentially be used with
the inclusion of the WECS generators; however, the difficulty
arises in that the derivatives with respect to the generator outputs
for the objective cost components (7), (8) are not as easily found
as those for the objective cost components (5), (6), due to the
fact that the solutions to the integrals cannot be derived in the
closed form.
Given this background, the optimization problem will be
solved numerically for the case of two conventional and two
WECS generators. Using the model stated in (1)–(4), this numerical
method demonstrates solutions for the case of two conventional
generators and two wind-powered generators. The
input for the conventional generator and direct WECS costs are
rather straightforward applications of (5) and (6). As for (7)
and (8), the wind power probability functions (16)–(18) for the
cases of the Weibull distribution are set up. These functions are
then used as inputs for (7) and (8). The constraints (2)–(4) are
set up, and then, the optimum minimal solution of an objective
function subject to linear and/or nonlinear constraints is found
by using the MATLAB optimization toolbox.
As the optimization itself is a challenging topic to be explored,
this paper focuses only on the results of ED problem with
the WECS but not on the optimization technique itself. However,
several optimization algorithms applicable to the ED problem
based on classical calculus or modern stochastic searching
optimization techniques, including the Lagrangian relaxation
(LA) [2], direct search method (DSM) [7], evolution programming
(EP) [14], particle swarm optimization (PSO) [15], genetic
608 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 2, JUNE 2008
algorithm (GA) [16], and simulated annealing (SA) [17] arewell
documented in references to name a few.
VI. ANALYSIS
In the preceding discussion, the model that incorporates both
conventional and wind-powered generators into the ED problem
has been developed. The use of theWeibull probability distribution
to model the wind speed has been explained, and the wind
speed distributions have been transformed to wind power distributions
using the linear wind power equations. A MATLAB
program based on the ED model with two conventional and two
WECS generators was developed to provide a numerical tool to
investigate how variations in the wind speed profiles and variations
in the many different cost coefficients in the model will
affect the optimum solution of the ED problem. Because of the
number of variables in the model and the need to provide an
analyzable graphic output, in general, all factors must be held
constant except the one under investigation. In both the text and
figures that follow, the abbreviations CG1 and CG2 for the conventional
generators, and WG1 and WG2 for the wind-powered
generators will be used.
Regarding the values of the parameters that are used later, because
the primary focus is on the effects that changes to various
parameters have on the optimum scheduled outputs for the generators
and on the relationships among the various factors in the
ED model, specific dimensional unit values, such as miles/hour,
meters/sec, or dollars/kilowatthour will not be assigned to the
values. The relationships among the factors are the important aspects
and these relationships may bemore easily studied without
the use of a specific system of units.