Comparison of Lagrangian and ANN methods for load scheduling in Electrical Power
typically possess number of units and they
need to commit units as electricity cannot
be stored and demand is a random variable
process fluctuating with the time of the
day and the day of the week. A problem
that must be frequently resolved by an
electricity utility is to economically
determine a schedule of which units to be
used to meet the demand forecasted and
satisfy operating constraints over a short
time horizon. The accuracy affects the
economic operation and reliability of the
system greatly. Under prediction of load
forecast leads to insufficient reserve
capacity preparation and in turn, increases
the operating cost by using expensive
peaking units. On the other hand, over
prediction of load forecast leads to the
unnecessarily large reserve capacity,
which is also related to high operating
cost. This paper presents an approach for
Lagrangian algorithm regression method
and artificial neural network
Index Terms--- load forecast, shortterm load scheduling, STLF, Lagrangian
algorithm, and artificial neural network
Load forecasting is an important
component for power system energy
forecasting helps the electric utility to
make unit commitment decisions, reduce
spinning reserve capacity and schedule
device maintenance plan properly.
Besides playing a key role in reducing the
generation cost, it is also essential to the
reliability of power systems. All these
considerations may make for conflicting
requirement and usually a compromise has
to be made for optimal system operation
Here, economy of operation also
called the economic dispatch problem is
to minimize the total cost of generating
real power (production cost) at various
stations while satisfying the loads and the
losses in the transmission links.
Normally, hydro plants operate in
conjunction with thermal plants. While
there is negligible operating cost at a
hydro plant, there is a limitation of
availability of water over the period of
time which must be used to save
maximum fuel at the thermal plants. The
system operators use the load forecasting
result as a basis of off-line network
analysis to determine if the system might
be vulnerable. If so, corrective actions
should be prepared such as load
shedding, power purchases and bringing
peaking unit on line
In load flow problems, two variables are
specified at each bus and the solution is
then obtained for the remaining variables.
The specified variables are real and
reactive powers at PQ buses, real powers
and voltage magnitudes at PV buses and
voltage magnitude and angle at the slack
bus. The additional variables to be
specified for load flow solution are t he tap
settings of regulating transformers
If the specified variables are allowed to
vary in a region constrained by practical
consideration(upper and lower limit on
active and reactive
generations bus
,voltage limits, and range of transformer
tap settings)there results an infinite
number of load flow solutions, each
pertaining to one set of values of
specified variables. The best choice in
some sense of the values of specified
variables leads to the best load flow
solution. The first problem in power
system is called the 'Unit Commitment
(UC ) problem and the second is called
the 'Load Scheduling' (LS) problem. One
must first solve the UC problem before
proceeding with the LS problem. LM has
been successfully applied to the complex
UC problem including various hard
constraints (e.g. Ramp rate constraints,
m i n i m u m u p a n d d o w n time, e t c .).
Lagrangian relaxation with evolutionary
programming and quadratic programming
(LREQP) for ramp rate constrained unit
commitment (RUC) problem is one of
the most successful approaches.
Most of the developed methods can be
broadly categorized into three groups,
namely parametric, nonparametric, and
artificial intelligence based methods. In
the parametric methods, a mathematical
or statistical relationship is developed
between the load and the factors affecting
it. Some examples of these models are
time functions, polynomial functions,
linear regressions, Fourier series. In
time-series methods, the load is treated
as a time series signal, with known
periodicity such as seasonal, weekly, or
daily. Such repetitive cycle gives a rough
prediction of the load at the given season,
day of the week, and time of the day. The
difference between the estimated and
actual load can be considered as a
stochastic process, which can be then
analyzed using Kalman- Filter methods.
Nonparametric methods forecast the load
directly from historical data.
II Optimal Operation of Generating
Mathematical formulation
Generator Operating Cost
The major component of generator
operating cost is the fuel input/hour,
while maintenance contributes only to a
small extent. The fuel cost is meaningful
in the case of thermal and nuclear
stations, but for hydro stations where the
energy storage is 'apparently free’; the
operating cost as such is not meaningful.
We concentrate on fuel fired stations.
Fig 1: Input-output curve of a generating unit
The input-output curve of a unit can be
expressed in a million kilocalories per hour
or directly in terms of rupees per hour
versus output in megawatts. The cost curve
can be determined experimentally. A
typical curve is shown in Fig. 1 where
(MW)min is the minimum loading limit
below which it is uneconomical (or may
be technically infeasible) to operate the
unit and (MW)max is the maximum
outpu t limit. The input-output curve has
discontinuities at steam valve openings
which have not been indicated in the
figure. By fitting a suitable degree
polynomial, an analytical expression for
operating cost can be written as
Ci(PGi )Rs/Hour at output PGi
where the suffix i stands for the unit
number. It generally suffices to fit a second
degree polynomial
degree to represent IC curve in the
inverse form
PGi= Ai+Bi(ICi)+Ci(ICi)2
Let us assume that it is known a priori
which generators are to run to meet a
particular load demand on the station
Ci(PGi) =1/2ai PGi2+ bPGi +di Rs/Hr----(1)
∑ PGi,max≥Pd
The slope of the cost curve is dCi/d
PGi, called the incremental fuel cost(IC)
and is expressed in units of rupees per
megawatt hour (Rs/MWh).A typical plot
of incremental fuel cost versus power
output is shown in Fig.2.If the cost curve
is approximated as a quadratic as in Eq.
(1), we have
where PGi,max,is the rated real power
capacity of the ith generator and Pd is the
total power demand on the station.
Further, the load on each generator is to
be constrained within lower and upper
limits, i.e
(IC)i=ai Pgi + bi
Considerations of spinning reserve
require that
PGi,min≤PGi,≤PGi,max where
∑PGi,max>Pd by proper margin ----(6)
Fig.2: Incremental fuel cost versus power output
for the unit whose input-output curve is shown in
i.e. a linear relationship. For better
accuracy incremental fuel cost may be
expressed by a number of short line
Alternatively, we can fit a polynomial of
Since the operating cost is insensitive to
reactive loading of a generator, the
manner in which the reactive load of the
station is shared among various online
generators does not affect the operating
economy. The question that has now to
be answered is: 'Wh at is the optimal
manner in which the load demand Pd
must be shared by the generators on the
bus keeping SC constraint optimized?'
This is answered by minimizing the
operating cost
C= ∑Ci(PGi) +SC
under the equality constraint of meeting
the load demand i,e
∑PGi – Pd=0
-------- (7)
Where k = the number of generators on the
Further, the loading of each generator is
constrained by the inequality constraint
of Eq. (3). Since Ci(PGi) is non-linear
and C, is independent of PGj, this is a
separable non-linear programming
Lagrangian Algorithm
1)µ=∑Ci(PGi) - £(∑PGi – Pd)
1) Choose trial value of £ i.e,
2) solve for PGi from eq.(4)
3) If │∑PGi - Pd│< €(a specified
value)the optimal solution is
reached Otherwise,
4) 4)Increment IC by
∆(IC) │∑PGi - Pd│< € or
decrement IC by
∆(IC), If │∑PGi - Pd│>0
And repeat from step 2
This step is possible because
PGi is monotonically increasing function
of (IC)
Effect of equality constraint
Where £ is langrangian multiplier
2)Minimization is achieved by
d µ /dPGi=0; or
dCi/dPGi=£; i=1,2…..k
where dCi/dPGi is the incremental cost of
ith generator(Units=Rs/MWh), a function
of generator loading P Gi
dC1/dPG1= dC2/dPG2... dCi/dPGi=£--(8)
i,e. the optimal loading of generators
corresponds to the equal incremental cost
point of all the generators. Eq.(8) is called
the co-ordinate equations numbering k are
solved simultaneously with the load
demand equation(4) to yield a solution for
the lagrangian multiplier £ and the optimal
loading of k generators
Flow Chart for Computer
Simulation using Lagrangian
As IC is increased or decreased in
iterative process, if a particular generator
loading reaches its P Gimax or PGimin, its
loading from now on is held fixed at this
value and the balance load is shared
between remaining units based on equal IC
Most of the conventional ANN-based
load forecasting methods deals with 24hour-ahead load forecasting or next day
peak load forecasting
by using
forecasted temperature. The drawback of
this method is that when rapid changes in
temperature of the forecasted day occur,
load power changes significantly, which
leads to high forecast error. In addition,
conventional neural networks use all
similar Dayís data throughout the training
process. However, training of the neural
networks using all similar Dayís data is a
complex task and it does not suit
learning of neural network.
In the above fig. input signal are
represented by X, Y is hidden layer V, W
is the interconnection between input layer
and hidden layer and Z is the output
layer. In depends largely on the proper
selection case of STLF, the performance
of the forecast of the load affecting
variables. In STLF, the key variables are
time, forecasted weather variables, and
historical load. Therefore, it is vital to
identify the input variables, which have
significant impacts on the system load.
This is particularly important since
inclusion of irrelevant inputs or inputs
with no significant impact on the target
outputs can distort the forecast
performance, increase the training time,
increase network complexity and reduce
the network execution time. One
approach to identify the most affecting
input variables are by evaluating the
statistical correlation between such input
variables and the target output. In this
paper, the linear correlation coefficient
index is calculated between each input
variable and the
output target.
Correlation coefficient with absolute
values near 1 implies a high influence of
an input variable to the target output.
It is vital to identify the input variables,
which have significant impacts on the
system load. This is particularly
important since inclusion of irrelevant
inputs or inputs with no significant
impact on the target outputs can distort
the forecast performance, increase the
complexity and reduce the work
execution time. In case of STLF, the
performance of the forecast depends
largely on the proper selection of the
load affecting variables. In STLF, the
key variables are time, forecasted weather
variables, and historical load. Therefore,
in this approach, the forecast time interval
is taken to be one hour i.e. forecast is
done for each hour. Therefore, it is
assumed that the hourly values of the
weather parameters can capture the most
conservative conditions that may happen
during this hour.
Training is the process by which ANN
parameters such as weights and biases. In
general, the training data set should cover
a wide range of input patterns sufficient
enough to train the network to recognize
and predict the relationship between input
variables and target output. Typically,
are trained
supervised pattern, i.e. the desired output
is given for each input and the training
process then adjusts the weights and
biases to match the desired output. A
new method for selecting the training
vector is presented in this paper. In this
method, the minimum distance between
the forecasted input variable and its
desired outcome is calculated for the
entire historical database.
In theory, a two-layer feed forward
network can be used for predictions. For
the number of neurons in each layer are
10 and 20, the number of units in the
input and output layers are fixed by the
number of inputs and the number of
outputs, respectively. Since the target
output is the forecasted hourly load, the
model has one output representing the
forecasted load of the target hour. Due
to seasonal load variations, four case
studies, related to the different four
seasons are performed. The number of
inputs for each case is dependent on the
number of the effective parameters as
determined by the correlation analysis.
One of the challenges in the design of
ANN is the proper selection of the
number of neurons in the hidden layer,
which affects the learning capability and
varies by the complexity of the problem.
In general, a tradeoff between accuracy
and generalization ability can be achieved
by selecting the proper number of hidden
units. While, there is no rigorous set of
rules to determine the optimal number of
hidden units, the fundamental rule is to
select the minimum number of hidden
neurons just enough to ensure the
complexity of the problem, but not too
many to cause over fitting of the training
set and losing generalization ability.
Self-Organization: An ANN can
create its own organization
representation of the information
it receives during learning time.
c) Real Time Operation: ANN
c o m p u t a t i o n s ma y b e carried out
in parallel, and special hardware
devices are being designed and
manufactured which take advantage
of this capability.
The method starts by setting the
estimated optimal number of hidden
neurons as the square root of the
product of the number of inputs times
the number of the outputs. Then the
number of hidden neurons is gradually
incremented by one. The parameters of
three layer artificial neural network for
24 hour ahead load forecasting are as
given below:
No. of layers: 3 (Input layer,
Hidden layer, Output layer)
No of neurons in hidden layer:
10 to 20
No of neurons in output layer: 1
Activation function of hidden
layer: logsig
Activation function of output
layer: Linear
Training algorithm: BackPropagation
Learning rate (·): 0.1
a) Adaptive learning: An ability to
learn how to do tasks based on the
data given for training or initial
No of data sets in each epoch: 63
No. of epochs for training: 100
numHiddenNeurons = 50;
% adjust as desired net =
net.divideParam.trainRatio =
net.divideParam.valRatio =
net.divideParam.testRatio =
% Train and Apply Network
[net,tr] =
outputs = sim(net,inputs);
% Plot plotperf(tr) plotfit(net,inputs,targets)
Lagrangian with all coupling
results as it can detect inbetween-hours shutdown of
some units, earlier shutdown,
and/or later startup of some
other units
Main advantages of using NNs
• its capability of dealing with stochastic
variations of the scheduled operating point
with increasing data
• Very fast and on-line processing and
• Implicit nonlinear modeling and filtering
of system data However, NNs for power
system should be viewed as an additional
tool instead of a replacement for
conventional or other AI based power
system techniques. Currently NNs rely on
conventional simulations in order to
produce training vectors and analysis the
training vectors, especially with noisy
data. There are some remain major
challenge to be tackled using NNs for
power system: training time, selection of
training vector, upgrading of trained
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