978-1-4244-5961-2/10/$26.00 ©2010 IEEE 4395
2010 Sixth International Conference on Natural Computation (ICNC 2010)
Use of Differential Evolution in Low NOx
Combustion Optimization of a Coal-Fired Boiler
Ligang Zheng, Yugui Zhang, Shuijun Yu, Minggao Yu
School of Safety Science and Engineering and Key Lab of
Gas Geology and Gas Control, Henan Polytechnic University
Jiaozuo 454000, China
Abstract—The present work focuses on low NOx emissions
combustion modification of a 300MW dual-furnaces coal-fired
utility boiler through a combination of support vector regression
(SVR) and a novel and modern differential evolution
optimization technique (DE). SVR, used as a more versatile type
of regression tool, was employed to build a complex model
between NOx emissions and operating conditions by using
available experimental results in a case boiler. The trained SVR
model performed well in predicting the NOx emissions with an
average relative error of less than 1.14% compared with the
experimental results in the case boiler. The optimal ten inputs
(namely operating conditions to be optimized by operators of the
boiler) of NOx emissions characteristics model were regulated by
DE so that low NOx emissions were achieved, given that the boiler
load is determined. Two cases were optimized in this work to
check the possibility of reducing NOx emissions by DE under high
and low boiler load. The time response of DE was typical of 20
sec, at the same time with the better quality of optimized results.
Remarkable good results were obtained when DE was used to
optimize NOx emissions of this boiler, supporting its applicability
for the development of an advanced on-line and real-time low
NOx emissions combustion optimization software package in
modern power plants.
China United Engineering Corporation
Hangzhou 310022, China
Keywords- support vector regression; differential evolution;
boiler; combustion optimization
Combustion optimization in coal-fired boiler (i.e.
equipment for generating steam to drive gas turbine) is
important for coal utilization with high efficiency and low
emissions. Of low emissions, NOx is one of principal pollutants.
According to combustion principle, low NOx emission can be
achieved by low oxygen combustion and avoidance of high
temperature. This can be realized by regulating the air/fuel
distribution (i.e. distribution of the primary air for pulverized
coal transportation and that of the secondary air for oxidizer
required for complete coal combustion).
Currently, Low NOx combustion optimization is under
study by using artificial intelligence . First, an objective
function is established. Then, the optimization algorithms
(mainly the evolutionary algorithms-based global optimization)
are applied on the objective function. The objective function
give rise to the relationship between the NOx emission emitted
from the coal-fired boiler and the primary air the secondary air
distribution. The objective function is non-linear because of the
complexity of combustion process. Most models for predicting
NOx emission are based on artificial neural network such as the
back-propagation neural network (BPNN), radial basis function
neural network (RBFNN), general regression neural network
(GRNN) . The support vector regression (SVR)  was
recently also employed to predict NOx emission and gave rise
to better accuracy compared to the BPNN and GRNN.
Once the model is established, the global optimization
algorithm was then employed to choose the optimal inputs of
the model so as to achieve the lowest NOx emission. The
currently used global optimization algorithms consist of genetic
algorithm (GA), particle swarm optimization (PSO), ant colony
optimization (ACO) and estimation of distribution algorithm
The aim of the present study is for the first time to check
the applicability of differential evolution algorithm to optimize
the NOx emission model. The optimal inputs (i.e. design
variables of the differential evolution algorithm) for the NOx
emission model given by the differential evolution algorithm
will provide suggestion for the combustion engineers to
regulate the combustion parameters in order to achieve low
OVERVIEW OF DIFFERENTIAL EVOLUTION
Differential evolution  is one of the most successful EAs
for the global continuous optimization problem. DE extracts
the differential information (i.e. distance and direction
information) from the current population of solutions to guide
its further search.
DE was mostly applied to scientific applications involving
heat transfer, chemical engineering , electricity load
economic dispatch , and in areas of environmental science.
Considering xi as the target point a trial point yi is found
from two points (parents), the point xi, i.e., the target point and
the point ui determined by the mutation operation. In its
mutation phase DE randomly selects three distinct points xr1, xr2
and xr3 from the population S. The integers r1, r2, r3 are chosen
randomly from the interval (0, NP-1) and different from the
target point index i. None of these points should coincide with
the current target point xi.
Mutation: The weighted difference of any two points is then
added to the third point which can be mathematically described
where F > 0 is a model parameter, conceptualized as scaling
factor, and xr1 is known as the base vector. If the point ui
beyond the search space then the mutation operation is
repeated. It can be seen that the point ui is generated by
combining the base vector and a randomly sampled vector
differentials (xr2 - xr3).
Crossover: The trial point yi is found from its parents xi and
ui using the following crossover rule:
CR is another model constant of DE, also known as crossover
rate; rnbr∈ [1, …, D] is a randomly chosen index which
ensures the trial vector gets at least one parameter from the
mutant vector even if CR=0. Thus, it does not become an exact
replica of the original parent vector.
Selection: the selection operator compares the fitness of
the trial vector and the fitness of corresponding target vector,
and selects the one which performs better (global minimum),
the selection process is repeated for each pair of target/trial
vector until the next population is completed. Fig. 1 presented
the principle of the differential evolution algorithm.
The algorithm implemented in this study starts with
specifying the parameters, namely, scaling factor (F), crossover
rate (CR), population size (NP), maximum number of
successive iterations (Scmax) without improvement in the best
function value and maximum number of generations (Genmax).
The initial population is randomly generated within the entire
solution space and the fitness function value (here refers to the
x NP Parameter vectors form generation G
o Newly generated parameter vector v
Figure 1. Pricinple of defferential evolution.
calculated NOx emissions concentration from the SVR model)
is assigned to each individual. New individual is created using
the differential operators discussed above. Each individual in
population has to be compared with its possible replacement
(new individual) and if an improvement occurs, it is replaced.
This process is repeated until stopping criteria are reached.
The main difference between GA and DE is that for a GA
the crossover operator is the main search step while a DE uses
it as a secondary operator and mutation is the main search step.
The main steps are given in the following.
Step 0. Set scaling factor F, crossover parameter CR and
population scale NP. Give the maximum number of
iterations, SSmax. Set counter k=1.
Step 1. Randomly generate individuals (points)
from solution space.
) 0 (
) 0 (
Step 2. While (stopping criterion is not met, namely k < SSmax)
Generate NP new individuals (points) as follows: For
each preset vector
.,...,2 , 1,NPi =
Step 2.1. (First selection). Chose three vectors
randomly from the current population,
where r1, r2, r3 ∈(1, 2,…, NP).
Step 2.2. (Mutation). Generate a vector
according to eq. (1).
,2 ,i1 ,inii
Step 2.3. (Crossover). Generate a new vector
,2 ,1 ,niiii
according to eq. (2).
from the preset vector
,2 ,i1 ,inii
Step 2.4. (Second selection).
to eq. (3).
) 1( +
is generated according
Step 3. Output the best results.
Prior to the optimization, the objective function which
accurately predict the NOx emission from a coal combusted
boiler was needed. According to our experience, the NOx
emissions were correlated with the process variables such as air
velocity, coal feed rate, coal quality etc. Therefore, the NOx
concentration emitted from a boiler which will be modeled by
support vector regression (SVR) has nineteen inputs and one
output, as shown in Table I. Four levels of Primary air
velocities (PA), six levels of Secondary air velocities (SA)
entering the air ports, the boiler load (electricity power), four
speeds of pulverizers, and four coal quality parameters
constitutes the input variables of the model . The NOx
concentration is the output variable of the SVR model. Table I
presented the data structure of the training samples for
establishing the support vector regression model. The design
variables among the inputs were shown in Table 2. Due to
pages limit, the details of building SVR model were omitted
here. Interested readers can refer to literature .
Because of the complexity of combustion process, the NOx
emission predictive model must be given by the support vector
regression model. Hence, the accuracy of the objective function
must be justified. After well-chosen control parameters of the
predictive model, the predicted NOx emission reproduced the
actual value well, as shown in Fig. 2. The model presented the
max relative error (the difference of the predicted value and the
actual value subdivided by the actual value) of 16.15%,
averaged relative error of 1.14% and the coefficient factor R of
0.9624, as shown in Table 3. On the whole, the model is very
good in terms of the predictive accuracy. In , comparison
between the SVR model with BPNN and RBFNN models was
THE DATA STRUCTURE OF THE TRAINING SAMPLES 
Variables NOx con. PA SA
% Unit mg/m3 m/s m/s
DESIGN VARIABLES USED BY DIFFERENTIAL EVOLUTION
Figure 2. Performance assessment of the objective function.
THE SUMMARY OF THE SVR MODEL
IV. RESULTS AND DISCUSSION
Once the objective function is determined, the differential
evolution can be employed to this objective function. Then the
optimal inputs for this objective function will be found by the
DE. This section will report the results given by DE. DE has
several control parameters: scaling factor F, the crossover rate
CR, population size NP. We used the trial-and-error method to
select the control parameters.
Figure 3. Dependence of avearaged solution on (F, CR) for case A.
0.2 0.40.6 0.81
Figure 4. Dependence of Solution on Parameters Pair (F,CR) for case B.
The software was written using MATLAB 7.7 (2008b).
Simulations were executed on a Pentium-IV 2.0 GHz dual
cores and 2G DDR palmtop computer.
The scaling factor F and the crossover rate CR are the main
control parameters of DE. The detailed and extensive
parametric study is useful to determine the effect of the control
parameters of DE on the performance of DE. Generally, as for
F, it lies in the range 0.4 to 1.0. Initially F=0.5 can be tried the
F is increased if the population converges prematurely. As for
CR, it lies in the range 0.1 to 1.0. In this work, two dimensional
grid of F and CR was tried, i.e. F in range of [0.05, 1] with the
step of 0.05, CR in range of [0.025, 0.975] with the step of 0.05.
This process is very time-demanding. However, the extensive
computation will give a clear observation of the sensitivity of
the differential evolution to the control parameters F and CR.
The combinational effect of F and CR on the solutions is
given in Fig. 3 and Fig. 4. Herein after, in each case study, the
results represent the average of 10 runs of the proposed method.
We have tried two cases in this work. For case A, the boiler
emitted very high NOx concentration. For case B, the boiler
emitted a lower NOx concentration. As shown in Fig. 3 and Fig.
4, the dependence of DE on F and CR is not a monotonic trend.
However, F and CR located in a large zone of the grid will
present good results (i.e. lower NOx concentration for the
global minimum in this study).Therefore, it is convenient for us
to choose the scaling factor F and the crossover rate CR. From
both Fig. 3 and Fig. 4, the median-size scaling factor F is
preferable. Moreover, the larger the crossover rate CR, the
wider the range of the scaling factor F. The optimized solution
changed with F and CR in the range of [549, 644.6] mg/m3 for
case A and [541.6, 630.3] mg/m3 for case B. Hence, the
solution was significantly influenced by F and CR, and it is
important to perform parametric study in order to choose a best
pair of control parameters.
On the other hand, the stability of DE algorithm was also
studied in this work. As we know, a large number of random
numbers were used in computation. Therefore, the solution
probably changed from one run to another run. In the present
study, the standard deviation of the 10 runs represented by std
was employed to demonstrate the stability of DE algorithm, as
shown in Fig. 5 and Fig. 6. The standard deviation of solution
for 10 runs changed with F and CR in the range of [0, 37.2]
mg/m3 for case A and [0, 31.7] mg/m3 for case B. Obviously,
the smaller the standard deviation, the better the DE algorithm.
The DE algorithm is robust when an appropriate pair of (F, CR)
Note that the parameter F can be seen as the step length used
to produce the mutants described by eq.(1). On the other hand,
the parameter CR determines the probability with which a
mutant is accepted. Thus, for example, if CR ≈ 1, then
(essentially) DE should change the population at each iteration,
independently of the choice of the value of F. In this work,
after intensive computational tests, we consider CR= 0.4 and
The performance of DE is always sensitive to the selected
population size. This is easily conceivable because DE employs
a one-to-one reproduction strategy. Therefore, if a very large
population size is selected, then DE exhausts the fitness
evaluations very quickly without being able to locate the
optimum. Storn and Price  suggested a larger population size
(between 5N to 10N). To investigate the sensitivity of the
proposed algorithm to variations of population size, we
experimented with different population sizes. Results, reported
in Table 4 and Table 5, show how the performance of DE
changes with the population size.
Averaged NOx emission and its standard deviation, and
CPU-time were the average of 50 different runs. From Table 4
and Table 5, it is clear that for all population size the CPU-time
required to obtain global optimum using DE is less than 30
seconds. When the population size increased from 20 to 100,
the optimized result changed from 563.4 mg/m3 to 550.9
mg/m3 for case A and from 560.7 mg/m3 to 541.6 mg/m3 for
case B. When F and CR were preset to 0.3 and 0.4 respectively,
the standard deviation of optimized result changed with
population size NP from 25.2 mg/m3 to 9.4 mg/m3 for case A
and from 28.2 mg/m3 to 0.0 mg/m3 for case B. The averaged
CPU-time was slightly increased with the population size NP,
as shown in Table 4 and Table 5.
Figure 5. Dependence of std of solution on (F,CR) for case A.
0.20.4 0.6 0.81
Figure 6. Dependence of std of solution on (F,CR) for case B.
DEPENDENCE OF SOLUTIONS ON NP FOR CASE A
Averaged CPU-time [sec]
DEPENDENCE OF SOLUTIONS ON NP FOR CASE B
Averaged CPU-time [sec]
4399 Download full-text
Time history (sec)
15 20 2530
Optimized NOx emission (mg/m3)
Figure 7. Convergence curve of various NPs for case A.
Time history (sec)
Optimized NOx emission (mg/m3)
Figure 8. Convergence curve of various NPs for case B.
Fig. 7 and Fig. 8 presented convergence histories for both
cases with various population sizes. Five groups of population
size were tried. On the whole, DE algorithm converged to a
steady solution very quickly. With the exception of NP=40 for
case A, the larger the population size NP, the smaller the steady
solution. On the other hand, the steady solutions for both cases
changed no longer when the population size NP is large enough
(e.g. NP=80 for case A and NP=60 for case B).The suggested
population size between 5N to 10N by Storn and Price  was
applicable for our study.
The performance of differential evolution on combustion
optimization in a coal-fired boiler was checked. The influence
of control parameters, i.e. the scaling factor F, the crossover
rate CR and the population size NP was extensively
investigated. The results indicated that the optimized solution
changed significantly with F and CR in the range of [549,
644.6] mg/m3 for case A and [541.6, 630.3] mg/m3 for case B.
The stability of algorithm represented by the standard
deviation of solution for 10 runs was also examined, changing
remarkably with F and CR in the range of [0, 37.2] mg/m3 for
case A and [0, 31.7] mg/m3 for case B. When the population
size increased from 20 to 100, the optimized result changed
from 563.4 mg/m3 to 550.9 mg/m3 for case A and from 560.7
mg/m3 to 541.6 mg/m3 for case B. With the exception of
NP=40 for case A, the larger the population size NP, the
smaller the steady solution. On the other hand, the steady
solutions for both cases changed no longer when the
population size NP is large enough. The suggested population
size between 5N to 10N by Storn and Price was applicable for
The authors wish to thank NSF (No. 646102) and Doctoral
Foundation of HPU for the financial support. This work is an
extension of the previous study under supervision by
Academician KF Cen and Dr. Zhou of Zhejiang University,
China. The first author’s thanks are extended to Prof. KF Cen
and Dr. H Zhou and Dr. CL Wang of Hangzhou Dianzi
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