A Hybrid Neural Network and Simulated Annealing Approach to the Unit Commitment Problem
ABSTRACT In this paper, the authors present an approach combining the feedforward neural network and the simulated annealing method to solve unit commitment, a mixed integer combinatorial optimisation problem in power system. The artificial neural network (ANN) is used to determine the discrete variables corresponding to the state of each unit at each time interval. The simulated annealing method is used to generate the continuous variables corresponding to the power output of each unit and the production cost. The type of neural network used in this method is a multilayer perceptron trained by the backpropagation algorithm. A set of load profiles as inputs and the corresponding unitcommitment schedules as outputs (satisfying the minimum updown, spinning reserve and crew constraints) are utilized to train the network. A method to generate the training patterns is also presented. The experimental result demonstrates that the proposed approach can solve unit commitment in a reduced computational time with an optimum generation schedule.
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ABSTRACT: During the past decade, the methodology of parameter design for signalresponse systems has received increasing attention in engineering applications. Several approaches, based on statistical techniques, are presented to resolve the problems. These approaches intend to make the signalresponse relationship insensitive to the noise variation by choosing a combination of control factors. In this work, we propose an alternative methodology consisting of three phases based on soft computing to optimize signalresponse systems. The first phase involves training a backpropagation neural network to construct the response model of a signalresponse system. The response model is then used to predict the responses of a specific experimental run. The second phase develops performance measures for three types of signalresponse systems to evaluate the predicted responses. The third phase involves transforming performance measures into energy values and presenting the optimization process, which minimizes the energy value by performing a simulated annealing algorithm. The proposed methodology is illustrated with a numerical example.International Journal of Advanced Manufacturing Technology 07/2007; 33(11):10771086. · 1.78 Impact Factor  SourceAvailable from: 144.118.25.24[Show abstract] [Hide abstract]
ABSTRACT: The primary objective of this work is to propose a general and computationally efficient methodology for dynamic scheduling and optimal control of primary HVAC&R systems that would be of practical benefit to plant operators and can be implemented online. This objective is achieved in several discrete steps. First, a generalized static optimization scheme is developed using Sequential Quadratic Programming along with physical models for each component. Next, in order to enhance computational efficiency, an experimental design technique, namely response surface methodology, is adopted to predict expected minimal cost (or energy consumption) for different plant operating modes. Then the optimal dynamic scheduling strategy over several hours of the planning horizon is determined using the computationally efficient Dijkstra’s algorithm which results in minimal operating cost. The fourth phase involves analyzing the effect of various stochastic factors that impact the optimal operation, such as the uncertainty in load prediction, and the uncertainties associated with various component models and response surface models using Monte Carlo methods. Finally, a decision analysis framework is adopted to study the robustness of the determined optimal operating strategy in terms of its benefit as against a riskaverse operating strategy. This methodology is illustrated for a hybrid cooling plant operated under two electric rate structures, namely, realtime pricing (RTP) and timeofuse (TOU) with electric demand. We find that under the RTP rate, leastcost paths are equally leastenergy paths, while this is not so under TOU with electricity demand rate. Also under RTP rate, the relative benefit of optimal scheduling and control during a hot day is not as significant as during a mild day; while under TOU with electricity demand, it can yield very significant benefit. The decision analysis involves formulating an additive multiattribute utility function which includes expected value and variability of the predicted operating cost as well as the penalty of possible loss of cooling capability. Though several riskaverse operating strategies can be defined, we select a simple one involving operating an additional chiller than the optimal deterministic strategy. How the assumption of different weights results in different scheduling decisions to be made is studied.
Page 1
A Hybrid Neural Network and Simulated Annealing Approach to the Unit
Commitment Problem
R. Nayak1 and J. D. Sharma2
Machine Learning Research Centre1
School of Comp. Science, QUT
Brisbane, QLD 4001, Australia
nayak@fit.qut.edu.au
Professor, Dept of Electrical Engineering2
University of Roorkee
Roorkee, UP 247667, India
Abstract – In this paper, the authors present an approach combining the feedforward neural network and
the simulated annealing method to solve unit commitment, a mixed integer combinatorial optimisation
problem in power system. The artificial neural network (ANN) is used to determine the discrete variables
corresponding to the state of each unit at each time interval. The simulated annealing method is used to
generate the continuous variables corresponding to the power output of each unit and the production
cost. The type of neural network used in this method is a multilayer perceptron trained by the back
propagation algorithm. A set of load profiles as inputs and the corresponding unitcommitment schedules
as outputs (satisfying the minimum updown, spinning reserve and crew constraints) are utilized to train
the network. A method to generate the training patterns is also presented. The experimental result
demonstrates that the proposed approach can solve unit commitment in a reduced computational time
with an optimum generation schedule.
Key words: Power system planning, Unit Commitment, Artificial Neural Networks, Simulated
Annealing.
1. INTRODUCTION
The formulation of a generation allocation plan for power system operations suffers from
various problems such as increase in the number, type and size of generating facilities, and
variations in load demands. Due to these complexities, power system operators have to face a
wide range of decisionmaking problems. One of the problems is the scheduling of generators in
a power system at any given time. It is not economical for a power system to run all the units that
are required to satisfy the peak load during low load periods. The unit commitment problem is a
plan of units to be selected from the generating facilities to meet the predicted demand in a
reliable and an economical manner. The minimum updown time, spinning reserve and crew
constraints should be considered for the reliable operation of a power system. Similarly, the
production cost should be at the global minimum for the commitment of units in a scheduling
period for the economic operation of a power system.
The unit commitment problem is formulated as a combinatorial optimisation problem
involving a large number of calculations. A detailed literature synopsis of solution methods for
unit commitment is summarized by Sheble’ and Fahd [13]. Some of the solution methods for unit
commitment are Priority Ordering methods [15], Mixed Integer Programming methods [16],
Linear and Dynamic Programming methods [14], Risk Analysis methods [3], Branch and Bound
methods [4] and Lagrangian Relaxation methods [1].
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2
memory) is required to solve unit commitment with the first four methods. The later three
methods also take a large amount of computational time to solve unit commitment and may not
give the global solution. In recent years, unit commitment has been compared to the annealing of
a metal and solved by the simulated annealing method. At present, the speed of a unit
commitment solution by the simulated annealing method is a limiting factor [18]. Researchers
have also investigated the effectiveness of artificial neural networks in solving unit commitment
[11,12,17]. The difference between our approach and the other approaches lies in the manner,
neural networks are used to solve unitcommitment. Ouyang et.al utilizes neural networks to
generate a preschedule according to the input load profile and then refines the schedule,
where the commitment states of some of the units are uncertain, using a dynamic search [11].
The approach taken by Sasaki et.al utilizes the Hopfield neural network. In their approach, a
large number of inequality constraints included in unit commitment are handled by the
dedicated neural network instead of including them in the energy function. Once the states of
generators are determined by the network, their outputs are adjusted according to the priority
order in fuel cost per unit output [12].
An alternative hybrid neural networksimulated annealing approach is presented in this
paper to solve unit commitment. Considering the characteristics of the daily unit commitment, a
threelayer feedforward neural network is constructed [10]. The network is trained by a set of
load profiles as the input data and the corresponding unit commitment (satisfying the minimum
updown, spinning reserve and crew constraints) as the desired output. The commitment states
(on/off) of the units are determined for any load demand by the trained neural network.
Further the simulated annealing method generates a global production cost, and an
optimized production level of each committed unit determined by the trained neural network.
The global generation is required to reduce the production cost at its minimum because of the
variations in cost factors and nonlinearity in unitload curves (operating costs Vs unit power
outputs). The simulated annealing method works iteratively in two phases, firstly the production
cost is evaluated at a number of randomly generated feasible states, and in the second phase, the
generated states are replaced by local searches to obtain a global value. The annealing concept is
used to jump out of local minima to a global minimum.
The proposed method is tested for the daily scheduling of a power system consisting of
10 units. For comparison, the same load profiles are also scheduled by the simulated annealing
method. The experimental result demonstrates that the proposed approach can solve the unit
commitment problem in a reduced computational time with an optimum generation schedule.
The case studies demonstrate that the twostage hybrid method benefits from advantages of both
approaches.
2. MATHEMATICAL MODELLING
Unit commitment problem is a type of scheduling operation that fits between economic
dispatch, and maintenance and production scheduling in the management of generation
resources.
2.1 Nomenclature
pit
: the real power generation of unit i in hour t (continuous variable)
uit : the commitment of unit i in hour t (discrete variable)
1 is unit on line, 0 is unit off line
As the size of a power system increases, more computational effort (CPU time and
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2.2 Mathematical Formulation
Unit commitment problem is formulated as the production cost of units considering the
fuel cost and the transition (startup and startdown) cost. The various factors that affect the
performance of unit commitment are formulated as constraints in the mathematical model.
Fuel cost model
The fuel cost is the production cost of operating generators to meet the load demand of a
system during a specified time period. This cost depends on the heat rate, fuel price (constants)
and unitload curves. A unitload curve represents the incremental or total operating cost of a
generating unit as the function of megawatt power level. The incorporation of minimum and
maximum limits is important to model the curve (Figure 1). The cost curve is assumed to be non
linear. The curves are approximated by the quadratic function.
FCit (pit )
SCit (uit )
a1i, a2i , a3i
b1i, b2i , b3i
Dt
ptmax(i)
ptmin(i)
Rtov
Rtun
mut(i)
mdt(i)
:
:
:
:
:
:
:
:
:
:
:
the fuel cost function of unit i in hour t
the start up cost function of unit i in hour t
the fuel cost function coefficients of unit i
the start up cost function coefficients of unit i
the system power demand in hour t
the upper generation limit of unit i in hour t
the lower generation limit of unit i in hour t
the upper system capacity reserve requirement in hour t
the lower system capacity reserve requirement in hour t
the minimum up time of unit i
the minimum down time of unit i
it
FC = 1ia
+ 2i
a
(itp ) + 3i
a
(itp
2) +. . . . .
( ) 1
Start up cost model
Some units are required to start or shut down in a power system with the change in load
demand for the economic operation. The addition of a unit in the system involves labour and
money. It is necessary to consider this type of cost for the economic scheduling. The start up cost
is expressed as:
it
SC (it
u ) = itu (1it 1
u
) 1i
b
(12i
b
3i
b
t
)
exp
( ) 2
Consider a power system consisting of I thermal units to be scheduled over an H hour planning
horizon at time interval t, unit commitment can be expressed as:
The Object:
( ) 3
)]
uti
(
SCti
+ )
pti
(
FCti uti
[(
I
∑
1=i
H
∑
1=t
Minimize
uti
pti
Subject to the following Constraints:
1.
Equality Constraints
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1.1
The power balance constraints
This constraint signifies that generation of units in a power system should always match
the load demand.
) 4 (1
1
.....H = t ,
Dt
=
pti
uti
I
∑
=i
2.
2.1
Inequality Constrains
The spinning reserve constrains
These constraints signify the availability of spinning reserves as a basic requirement for
the reliable operation of a system.
(a) ReserveUp Margin
) 5 (
max
1....H = t ,
Rtov
Dt

pt
(i)
uti
I
∑
1=i
≥
(b) ReserveDown Margin
) 6 (
min
1....H = t ,
Rtun
Dt
+
pt
(i)
uti

I
∑
1=i
≥
2.2
(a)
Unit wise constrains
The MinimumUp Time
This constraint signifies the minimum time for a committed unit to be turned off and to
remove from online.
) 7 (
1
≤
uid
Did
t
mut(i) t=d
∑
Where
id
D
= (d t+mut(i))
2
= mut(i)
, (d = (t mut(i)),....(t 2))
id
D
D2
1, (d = (t 1))
id
= mut(i)
2
1, (d = t)
(b)
The MinimumDown Time
This constraint signifies the minimum time for a decommitted unit to be turned on and
to bring online.
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5
) 8 (
2
1) (mdt(i)
uid
Did
t
mdt(i)  t=d
≤
∑
Where
id
D
= (d t+mdt(i)), (d = (t mdt(i)),...(t  ))
= mdt(i) , (d =(t  ))
= mdt(i) , (d = t)
2
id
D
D
id
2
2
2
11
1
(c)
The Power Generation Limits
These limits define the region within which a unit must be dispatched.
) 9 (
min max
pt
(i)
pti
pt
(i)
≥≥
(d)
Crew Constrains
The unit i should not be started more than once a day to take the minimum labour cost.
)10(12
2
1
)
1
....I = i ,
u ti

uti
(
H
∑
=t
≤
optimum solutions with one being the global optimum solution.
3. NEURAL NETWORK COMPUTING
An artificial neural network is a computational paradigm based on a biological metaphor
that mimics the computation of human brain. A neuron is a fundamental element to access and
transmit activities in the neural system of a human. The major components of a neuron include a
central cell body, small processes called dendrites and one large process called axon. A thick,
fatty myclin sheath surrounds many large axons. The thousands of fibers are typically grouped
together to form nerves and transmit information in the form of electrochemical pulses from one
place to another within one organism. Along the axon, the information is coded and transmitted
in the form of “all/none” or “on/off” electrical pulses (called action/spike potential). There is
nothing inherent in a neuron that governs the direction of information flow, i.e. same neuron
works as a sensor and a motor neuron. Neurons have the capability to receive, process and
transmit electrochemical signals over neural pathways [7]. An artificial neural network is
basically a large number of highly connected but relatively simple processing elements
communicating via messages, each element (neuron) replicating the property of human neuron.
Artificial neural networks have made progress by employing models based on the ability of
human brain. Artificial neural networks are characterized as.
1. The architecture
A neural network is formed by interconnected slabs arranged in a particular manner. A
feedforward neural network receives inputs through the input slab and yields outputs
through the output slab.
The objective function of unitcommitment is a multimodal i.e. a number of local
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2. Highly parallel distributed control
Control is the transfer function that describes the output of a neuron at given inputs.
3. Learning paradigm
Learning is a training technique that adjusts the underlying network’s parameters
appropriately to respond the problem to be learned, until the problem is being learned or
some termination criteria are met.
4. PROPOSED APPROACH
Unit commitment is a mixed integer nonlinear programming problem consisting of
discrete and continuous variables. In the present method both variables are calculated separately,
discrete variables are solved as a sequencing problem by an artificial neural network and
continuous variables are solved as a scheduling problem by the simulated annealing method
(Figure 2).
4.1 Sequencing problem solution
In a sequencing problem, a multilayer feedforward network using backpropagation error
oflearning algorithm determines the discrete variables (corresponding to the state of each unit at
each time interval), while the continuous variables are treated as constants. There are several
types of neural networks having three common characteristics: individual neurons, connections
between neurons and the learning algorithm. The capability of closely approximating the
underlying function makes backpropagation networks an obvious choice for this type of problem
[6].
Network Architecture
The threelayer feedforward network is configured in a regular and fully connected
manner. The number of neurons in the input layer is dependent on the number of hours (over
that I thermal units are to be scheduled). A load demand profile is input to neurons in the input
layer and is normalized using the following formula:
j
DN =
j
D 
j
D
j
D

j
D
min
maxmin
( ) 11
Where
j
D
DN
Dmax
Dmin =
power system times the number of hours that a generator will be scheduled over (I * H), i.e.
commitment schedule of all units.
Training Details
The network is trained by the back propagation learning algorithm using gradient descent
[7]. A set of load profiles and their corresponding commitment schedules satisfying all the
=
=
=
=
index for hour
Load demand
Normalized load demand
Maximum demand among all patterns
Minimum demand among all patterns
The number of neurons in the output layer corresponds to the number of generators in a
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constraints (described above in section 2) are used to train the neural network. The commitments
of units, used as target outputs to train the network, are generated offline by the rigorous
algorithm as shown in Figure 3.
A major concern in the application of a neural network to a problem domain is to
converge the network to the global minimum and prevent it from being trapped in local minima.
A common approach to overcoming this problem is to make the slope of the sigmoid function
sharper with time. In other words, a careful selection of control parameters such as learning rate
and momentum helps to solve the problem.
Another concern is to prevent the network from overfitting or memorization of data
instead of generalization. Too many parameters (a large network) and a few training patterns will
allow the network to fit the training data very closely but will not necessarily lead to an optimal
generalization. The gradient learning process employed by the back propagation algorithm works
as follows: initially all hidden nodes in the network do the same work, i.e. they all attempt to fit
the major features of the data. The nodes then start to differentiate, with some of the nodes
beginning to fit the second most important aspect of data. This process of differentiation
continues as long as the error remains in the network and training continues. It is only at later
times of training that the network tries to overfit the sampling data.
The problem is to determine when the network has extracted all the useful information
from data and begins to overfit. The method we employ is to divide the training patterns into two
sets: a training set which is used to determine values of weights and biases, and a cross
validation set which is used to decide when to stop training. The performance on the cross
validation set is monitored. As long as this performance on the crossvalidation set improves,
training continues. When it ceases to improve, training is stopped.
Network performance
The trained backpropagation network is able to produce the corresponding schedule
pattern for the unseen or seen input load profiles. A commitment schedule contains the on/off
state of each unit for every time interval within the time span, where one (1) shows the on state
and zero (0) shows the off state.
The quality of predictions on unseen data obtained from a neural network depends on the
number and quality of patterns used for training. Although there is no general method to find out
the required number of patterns to obtain an optimal generalized network. There are statistical
arguments that suggest that the number of training patterns required to fully determine the
weights in a network is approximately proportional to the number of weights in a network [2].
Considering that the daily load demand for a power system usually follows a specific pattern, if
we effectively utilize previous scheduling information, a large training set may not be quite
necessary [11]. We have also experienced that a training set consisting of 20 to 30 patterns
covering different types of situations was sufficient to generalize this size of network.
4.2 Scheduling problem solution
In a scheduling problem, continuous variables (corresponding to the operating level of
generators and production cost) are determined by simulated annealing method, while
considering discrete variables as constants. The algorithm for the simulated annealing method is
based on the stochastic method that depends only on function evaluation [9]. The method works
iteratively in two phases: in the first phase the objective function is evaluated over a number of
randomly sampled feasible states, while in the second phase these states are manipulated by local
searches to yield a possible candidate for the global optimal. The implementation is based on the
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random searches and the quadratic interpolation for nonlinear optimization (Figure 4). The
annealing concept is used to jump out of local minima to the global minimum.
The concept used in the simulated annealing method is based on the analogy that exists
between the process to solidify fluids up to their equilibrium and combinatorial optimisation
methods. When a fluid/metal is cooled slowly (annealed), its energy tends to assume the globally
minimal value out of local minima. The various states of a metal during annealing correspond to
various feasible solutions of the underlying optimization problem, and the energy of a state is
analogous to the cost function of a feasible optimized solution. The probability of a metal at
temperature T to reach in equilibrium follows the Boltzmann distribution [5]:
P(
i
E ) =
(
i
E
B
k
T )
E
j = 1
N
∑
(
j
T)
B
k
exp
exp
()12
Where
large number of trials to evaluate the cost function in an optimization problem, which
resembles the first phase of our method (Figure 4). Metropolis proposed an algorithm to
simulate a system of atoms in equilibrium [8]. An atom is randomly displaced to generate a
trial state at each iteration, and the induced change in the energy (∆e) of the system is
computed. If ∆e<0, the trial state is accepted as new state. Otherwise the trial state is
accepted with the probability:
kB
N
Ei
Ej
The denominator in the above equation is computationally equivalent to generate a
=
=
=
=
Boltzmann's constants
Number of all possible states
Energy of the particle at the ith state
Energy of the particle at the jth state
p( ) = (
B
k
T)
∆
∆
exp()13
the simulated system of atoms will evolve into the Boltzmann distribution after sufficient
iteration [8]. On comparing with unitcommitment, ∆ is computationally equivalent to induced
change in the objective cost function (equation 3) from current value to the value formed by a
small random displacement i.e. [F(M)  F(P)]. As the concept of temperature in physical
annealing has no equivalence in unitcommitment, the control parameter kBT is replaced by [F(P)
 F(L)]. This term is decremental and tends to zero, as P state is tending towards L by making the
neighborhood small.
F(M) = The greatest value of function evaluated
F(L) = The lowest value of function evaluated
F(P) = local minimum function value of function evaluated
5. EXPERIMENTAL RESULTS AND DISCUSSIONS
Each trial state in our method is generated using quadratic interpolation. It is shown that
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generating units. The cost coefficients and other characteristics of each unit are summarized in
Table 1. The unitload curve is modeled as a nonlinear function (Figure 1). In the dailyload
profiles, which are considered for training and generalization of the neural network, generation
changes in every three hours. A sample load profile used for training is shown in Figure 5 and
the commitment schedule (generated by the algorithm shown in Figure 3) is given in Table 2.
The threelayer feedforward network, input layer consisting of eight (8) neurons, hidden
layer consisting of eighteen (18) neurons and output layer consisting of eighty (80) neurons, was
used for this particular sample. A total of twentyfive (25) patterns were used to train the neural
network and ten (10) patterns were used for crossvalidation to monitor the training performance.
The computation time to train the network was 33 minutes 45.7 sec on an IBM PC486 EISA (32
MB RAM and 50 MHz CPU). The rootmeansquareerror (RMSE) on the crossvalidation set
was continued to reduce until 5000 epochs. The RMSE of 0.00768 was recorded on the training
set at 5000 epoch. The learning rate and momentum term during training were set to 0.3 and 0.4
respectively.
The trained network was able to produce the target commitment schedules for both load
profiles: (1) that had been used for training (2) and those that had not previously been presented
to the network. The simulation result is reported in Table 3 for two load profiles. As a
comparison, the same load patterns were scheduled by using the simulated annealing method
only i.e. both sequencing and scheduling problems are solved by simulated annealing method.
Case 1 represents the scheduling of a load pattern that has previously been utilized in the
training process (Figure 5).
Case 2 represents the scheduling of a load pattern that has been not utilized by the network
during training (Figure 6).
The computing time reported in Table 3 for the proposed method does not include the
time spent on training of the network. The computation time only indicates time to predict the
commitment schedule by the trained network and time to calculate the production cost and
operating level.
There were no discrepancies in the commitment schedule generated by the trained
network for case 1 and the target schedule for the load profile. The corresponding commitment
generated by the trained neural network for case 2 is summarized in Table 4. When the results
are compared with the verification set given in Table 5 (generated by Figure 3), they are found to
be very similar except for a minor difference, as neurons are allowed to generate output in
continuous values rather than discrete (as the verification set Table 5) in the simulation. So it is
assumed that the output value close to zero is off state of the unit, and close to one is on state.
The evaluated expected production level of each unit for a day obtained by simulated
annealing for case 2 is given in Table 6. It can be observed from Table 6 that the unit 8 has not
been committed for the production at any time. From Table 1 it is clear that the unit 8 has high
fuel and startup cost coefficient and lesser capacity. It is more economical if the unit 8 is not
run in the power system when the rest of units are capable to satisfy the power demand.
The case studies demonstrate that if the existing knowledge is used as the starting point
in predicting the new commitment states, a large portion of overall processing time is reduced for
the simulated annealing method to solve unit commitment.
The proposed approach benefits with following advantages by using neural networks as
compared to the conventional algorithms:
• A multilayer feedforward network is a universal approximater i.e. for any given function
there is a neural network capable of approximating the function with arbitrary precision. A
neural network supported solution is able to find the optimum commitment schedule for any
The proposed approach has been applied to a sample power system consisting of ten
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load profile in a small amount of time, assuming the network is well trained. The learning
and generalization capability of a neural network makes it a better alternative to other
existing methods.
• The blind search associated with the conventional methods is eliminated and available
information regarding the unit commitment scheduling in a specified power system is
utilized extensively. A substantial amount of time is saved in predicting the commitment
schedule by use of neural networks. Most of the computation time is spent on training the
neural network which can be done offline well before the actual scheduling. Once the neural
network is trained, the actual processing to predict the unit commitment schedule for a load
profile and production cost for the power system takes little time.
• The processing time will not be very much affected by the size of a power system unlike
other heuristic guided methods. As proven by the neuralcomputing community, neural
networks have the ability to represent extremely large data sets [2], thus the applicability of
the proposed method to the unit commitment problem of large power system seems to be
feasible.
• Neural networks are composed of many interconnected weak processing elements that work
simultaneously to achieve an outcome. An approach to gain speed, as well as quality of
schedule, is through massively parallel realization of the neural network. The various
constraints affecting the unit commitment problem can be mapped onto the different
processors and can be calculated simultaneously. With this realization, the ANN supported
methods can achieve faster and more accurate results with linear scalability for unit
commitment of a large power system.
6. CONCLUSIONS
In the present method a hybrid approach, feedforward neural network and simulated
annealing method is presented to solve the unit commitment problem. The case studies
demonstrate that the twostage hybrid method can benefit from the advantages of both
approaches. This method assumes no specific problem structures and is flexible in handling unit
commitment constraints. The simulation result shows that if the existing knowledge for
scheduling of a power system is used as the starting point to predict the new commitment states,
a large portion of the overall processing time of the simulated annealing method is greatly
reduced. The blind search to find the optimized group of working units is partially reduced in
simulated annealing method. This method gives highly near optimal solution even in the case of
a multimodel objective function, where other methods can be trapped at local searches.
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Unit
Pmin
Pmax
Fuel Cost
Coefficients
1.4
1.5
1.35
1.4
1.45
1.35
105 1.39
100 1.32
49 1.26
82 1.21
Startup Cost
Coefficients
85 .588
101 .594
114 .57
94 .65
113 .639
176 .568
267 .749
282 .749
187 .617
227 .642
Min
Up/Down
Time
3
3
3
1
2
3
4
5
6
7
8
9
10
Hour
3
6
9
12
15
18
21
24
15
20
30
25
50
75
250 520
50
120 320
75
60
80
100
120
150
280
15
25
40
32
29
72
.0051
.00396
.00393
.00382
.00212
.00261
.00127
.00135
.00289
.00148
.2
.2
.2
.18 3
.18 3
.15 3
.09 3
.09 3
.13 3
.11 3
3
3
3
3
3
3
3
3
3
3
150
200
U1
1
1
1
1
0
0
0
1
t
U2
0
0
0
0
0
0
0
0
t
U3
1
1
1
1
1
0
0
0
t
U4
1
1
1
1
1
1
1
1
t
U5
1
1
1
1
1
1
1
1
t
U6
1
0
0
1
1
1
0
0
t
U7
1
1
1
1
1
1
1
1
t
U8
1
1
1
0
0
0
0
0
t
U9
1
1
1
1
1
1
1
1
t
U10
1
1
1
1
1
1
1
1
t
The proposed method
(ANN generalization + SA)
Cost of Solution
(money unit)
Simulated Annealing Method
(SA)
Case
Computation
time
(sec)
47.3
46.1
Cost of Solution
(money unit)
Computational
time
(min)
48
41
1
2
15.029.12
14,014.75
15,831.58
14,039.13
Page 13
13
Hour
3
6
9
12
15
18
21
24
Hour
3
6
9
12
15
18
21
24
h
3
6
9
12
15
18
21
24
U1
0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.399 0.999 0.999
0.999 0.499 0.999 0.999 0.999 0.999 0.999 0.391 0.999 0.999
0.999 0.248 0.999 0.999 0.999 0.999 0.999 0.379 0.999 0.999
0.145 0.247 0.999 0.999 0.999 0.999 0.999 0.323 0.999 0.999
0.361 0.247 0.019 0.999 0.999 0.999 0.999 0.186 0.999 0.999
0.354 0.248 0.089 0.999 0.999 0.914 0.999 0.186 0.999 0.262
0.962 0.249 0.935 0.999 0.999 0.999 0.999 0.186 0.999 0.999
0.999 0.999 0.908 0.999 0.999 0.999 0.999 0.235 0.999 0.999
t
U2
t
U3
t
U4
t
U5
t
U6
t
U7
t
U8
t
U9
t
U10
t
U1
1
1
1
0
0
0
1
1
t
U2
1
0
0
0
0
0
0
1
t
U3
1
1
1
1
0
0
1
1
t
U4
1
1
1
1
1
1
1
1
t
U5
1
1
1
1
1
1
1
1
t
U6
1
1
1
1
1
1
1
1
t
U7
1
1
1
1
1
1
1
1
t
U8
0
0
0
0
0
0
0
0
t
U9
1
1
1
1
1
1
1
1
t
U10
1
1
1
1
1
0
1
1
t
Dt
1480
1400
1340
1100
800
750
1150
1420
P1
37.31
45.83
45.24
0
0
0
36.69
51.32
t
P2
68.5
0
0
0
0
0
0
59.7
t
P3
57.87
75.08
73.76
54.10
0
0
55.96
82.88
t
P4
65.71
120
65.08
76.79
49.2
70.23
69.38
94.86
t
P5
120.9
117.2
121.0
121.8
76.52
87.27
125.7
113.8
t
P6
201.5
199.3
242.0
143.2
124.1
143.8
189.9
202.7
t
P7
493.8
437.9
410.9
363.9
289.0
292.6
321.1
476.3
t
P8
0
0
0
0
0
0
0
0
t
P9
255.3
228.7
199.6
179.3
184.2
156.0
198.6
167.5
t
P10
178.9
175.8
182.1
160.7
76.8
0
152.6
170.8
t
LIST OF TABLES
Table 1
Table 2
Characteristics of generators in the sample power system
One of the unit commitment schedules used in training of the
neural network (case 1)
Page 14
14
Table 3
Table 4
Simulated result
The unit commitment schedule determined by the trained network
(case 2)
Table 5 The verificationset of the unitcommitment schedule to test the
network performance (case 2)
Table 6 Demand and the corresponding production levels for the power
system
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