A Novel Real-coded Quantum-inspired Genetic Algorithm and Its Application in Data Reconciliation *

Traditional quantum-inspired genetic algorithm (QGA) has drawbacks such as premature convergence, heavy computational cost, complicated coding and decoding process etc. In this paper, a novel real-coded quantum-inspired genetic algorithm is proposed based on interval division thinking. Detailed comparisons with some similar approaches for some standard benchmark functions test validity of the proposed algorithm. Besides, the proposed algorithm is used in two typical nonlinear data reconciliation problems (distilling process and extraction process) and simulation results show its efficiency in nonlinear data reconciliation problems


Introduction
Due to the influence of many factors in the petrochemical manufacture process, the data obtained through chemical manufacture devices or measuring meters may be corrupted by kinds of noises, which impose great impacts on the decision-makings of control process.Therefore, the collected data should be preprocessed before been applied to manufacture process analysis and data reconciliation technique receives widespread applications in the petrochemical industry.The nonlinear data reconciliation has always been the research focus of data reconciliation techniques.Normally, there are two major directions, i.e. two-steps matrix projection method 1 and independent logistics based Simpson method 2 .Zhou et al. 3 proposed a modified outlier detection method to efficiently decrease the effect of outliers on the reconciled results through distinguishing the outliers of each variable individually and modifying the weight accordingly.LI and Rong 4 improved the efficiency and accuracy of mixed integer linear programming (MILP) approach by reducing the number of binary variables and giving accurate weights for suspected gross errors candidates.
Gao et al. 5 proposed a new nonlinear dynamic data reconciliation method to get high robustness and simple calculation by introducing a penalty function matrix in a conventional least-square objective function and assigning small weights for outliers and large weights for normal measurements.Mei et al. 6 overcame the defects of the nodal test (NT) and the measurement test (MT).Their method avoided some artificial manipulation and more than one gross error problems by combining NT and MT.Zhou et al. 7 solved efficiently gross error effect on data reconciliation by using several technologies including linearization method, penalty function, virtual observation equation, and equivalent weights method.Jiang et al. 8 proposed a new bias detection strategy which reduced greatly the number of parameters to be estimated and avoided sequential detections and iterations by detecting the presence of measurement bias and its occurrence time.As discussed above, although there are many nonlinear data reconciliation techniques, most of the existing approaches require complicated matrix calculation and transformation.Moreover, when the matrix cannot or is difficult to be solved, the data reconciliation process will become extremely complex.Fortunately, the evolutionary algorithms (EAs) provide new solutions to this problem, which can improve the optimization efficiency and avoid the complex matrix computation process.For instance, genetic algorithms (GAs) were applied into the data reconciliation and achieved satisfactory performance [9][10][11] .Although, GA ensures colony evolves and solutions change continually, it lacks a strong capacity of producing better offspring and causes slow convergence near global optimum, sometimes may be trapped into local optimum.In this paper, the concepts of quantum computing are adopted to improve the performance of GA.
The quantum-inspired evolutionary algorithms (QEAs) 12 are based on the principles of quantum computing, which can strike right balance between exploration and exploitation more easily when compared with conventional EAs.Meanwhile, the QEAs can explore search space with a smaller number of individuals and exploit global solution within a short span of time [13][14][15] .Due to the distinguished characteristics, several sub-branches appear in the recent years, i.e.Quantum-inspired Genetic Algorithm (QGA) 16 , Quantum-inspired Immune Clone Algorithm (QICA) 17 , and Quantum-inspired Particle Swarm Optimization (QPSO) 18 .QGA combines the advantages of quantum computing and GA, which is designed to address some intrinsic problems of genetic algorithms, and has been widely used in many fields [19][20][21][22][23][24] .Generally, QGA has the characteristics of small population size, fast convergence speed, and robust searching ability.However, in QGA, the chromosome is usually represented by binary code, which has the disadvantage of low computation efficiency due to the repeated encoding and decoding process.
As a sequence, in this paper, following the research of QEAs and GAs, a novel real-coded QGA is proposed.This method adopts real numbers instead of binary code in order to improve algorithm searching ability and population diversity.Therefore, the complex coding and decoding processes can be avoided.Furthermore, the interval division approach is used, which can improve the searching capabilities and reduce computation cost by interval parallel computing.
This paper is organized as follows.Section 1 is the introduction; the QGA will be reviewed in detail.In section 2, the principles and procedures of the proposed algorithm will be discussed, followed by the numerical case studies before being applied into two nonlinear data reconciliation cases.The results and future work will be summarized in the last section.

2
Real-coded Quantum-inspired Genetic Algorithm

Quantum-inspired Genetic Algorithm (QGA)
It will be very instructive to review the classical QGA first before introducing the proposed algorithm.QGA has stronger search ability and quicker convergence speed since it introduces the concepts of quantum bit and quantum rotation gate.In the QGA, the state of a unit is depicted by quantum bit and angle, which are defined as below.
Quantum bit, the smallest unit in the QGA, is defined as a pair of numbers as shown in Eq. (1), The modulus 2 α and 2 β give the probabilities that the quantum bit exists in states "0" and "1", respectively, which satisfies Eq. (2), Published by Atlantis Press Copyright: the authors A string of quantum bits consists of a quantum bit individual, which can be defined by Eq. (3), 1  2 Therefore, a chromosome can be represented as a string of quantum bit individuals as shown in Eq. ( 4), A quantum bit individual is able to represent a linear superposition of all possible solutions due to its probabilistic representation.This quantum bit representation has better characteristic of generating diversity in population than other representations.
Because of the normalization condition, the quantum angle can be represented by Eq. ( 5), The fundamental update mechanism of QGA is evolving quantum bits and angles, by which the updated quantum bits should still satisfy the normalization condition.The quantum rotation gate update equation could be calculated by Eq. ( 6), Although the quantum bit and rotation gate representation has better characteristics of population diversity, the premature convergence problem could still appear because of the poor performance of binary representations.Therefore, in this paper, a novel quantum-inspired genetic algorithm is proposed, which will be discussed extensively in the next section.

The Proposed Algorithm
The proposed algorithm in this section adopts real numbers instead of chromosome in order to improve algorithm searching ability and population diversity.The detailed encoding rules and procedures are presented as follows.

Suppose the boundaries of variable
, and divide this interval into L consecutive subintervals such as, ( ) ( then algorithm will works on these L subintervals simultaneously. For subintervals 1 2 , , L R R R L , the chromosomes can be coded by Eq. ( 7), As a sequence, an individual can be expressed by Eq. ( 8), , , , Compared with Eqs. ( 3)-(4), Eqs. ( 7)-( 8) not only greatly reduces the length of individual coding, but also uses overall interval division covering the entire searching space.The searching efficiency could be improved when subinterval number increases, but algorithm will become more complicated and the associated computational cost (CPU time) will also increase and this result will be seen in the posterior case studies.. So, the number of subintervals which can be adjusted manually or automatically by the program is an important factor for algorithm performance.

Procedures
There are six steps in the proposed algorithm.
Step-1(initialization). Set n as the population size, g as iteration variable ( 1 g = when algorithm starts), ma G as the maximum iteration number,

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Copyright: the authors Initially, all the values are set as the median of subintervals, that is, where Step-2(Fitness Evaluation).Suppose the subinterval evaluation of one individual is the best, fitness evaluation strategy is shown in Eq. ( 11) 25 , where is the real value of the corresponding chromosome derived, and are the corresponding upper and lower boundaries, θ is the corresponding angle of the best individual.
Step-3(Judgement).If the satisfactory solution is acquired or the iteration number reaches , then the algorithm stops.Otherwise, go to Step-4.
In the next section, the proposed algorithm will be tested with several benchmark functions before being applied into real industrial process data reconciliation.

Benchmark Functions Test
In this section, the proposed algorithm will be compared with two QGAs using several benchmark functions listed in Tab.1.

sin
For these QGAs, Parameters are selected samely as follows, maximum iteration number max 1000 G = , population dimension 50 n = , population size 10 N = , limit probability 0.01 δ = . The stop criterion of algorithm is the evolvement reaches the maximum iteration number.If the absolute error between the results and global optimal value is less than 0.001, the optimization process will be considered as success.Besides, for testing subinterval number effect on the final performance of the proposed algorithm, three intervals 2, 3, and 5 are selected respectively.The tested results are shown in Tab.2.
Published by Atlantis Press Copyright: the authors From Tab.2 it can be seen that the proposed algorithm has better performance that QGA, and if the number of initial subintervals chosen appropriately (such as L=3) it also has better general performance (less MIN and CT) than RCQGA 25 .These results not only illuminate validity of the proposed algorithm, but also indicate the number of initial subintervals effect on it.

Distilling Process Data Reconciliation
Conventional distilling process is shown in Fig. 1.There are 10 constraint equations in this typical bilinear process, which are listed in Tab.3.
There are 21 variables in this process; the actual experiential values of each variable are as follows, The objective function is defined by Eq. ( 15), ) ) Where , , X Y Z are the measured value vector, ˆˆ, , X Y Z are the reconciled value vector, are the covariance matrixes of measurement errors of , , X Y Z which are assumed to be given or estimated.
The number of subintervals is 4, and maximum iteration number is .The results are shown in Tab. 4  The simulation results denote that the reconciled value of each amount is no longer change positively after 786 generations.According to the criteria of data reconciliation, reconciled value of every variable satisfies balance equations, the objective function decreased and stabilized after 786 generations (4.3 seconds), reconciliation goal is achieved.Besides, from the results, it is possible to use a great number of generations to obtain good results.

Extraction Process of Composing Juice Data Reconciliation
Extraction process of composing juice is shown in Fig. 3.

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Copyright: the authors  From the above simulation results, it can be seen that in the three methods, NLP has the best reconciliation precision, but its run time is so longer.Crowe has the shortest run time, but unideal precision.The proposed algorithm obtains better reconciliation results with shorter run time, and shows that it has the best global performance in nonlinear data reconciliation.

Conclusions
A novel quantum-inspired genetic algorithm with realcoded, interval division thinking was proposed in this paper, and some benchmark functions simulation illustrated that it could avoid some drawbacks of traditional QGA.Furthermore, the proposed algorithm was applied in industrial manufacture process data reconciliation, and simulation results showed that it had global performance in nonlinear data reconciliation and could be used in the relevant researches and applications.
population.denotes the j-th individual of the s-th generation as shown in Eq. (9), , 1 p j= ,2,L g j

θ
are the component of subinterval and corresponding angle respectively.Crossover operation is implemented only in the subintervals where this subinterval located.Then the subinterval component of each individual of next generation will be obtained by Eqs.(12) and (13),

Fig. 3 .
Fig. 3. Extraction progress of composing juice The process has 4 equipments, 7 components and 88 variables which are 11 matter flow variables ( 1,2, ,11 j F j L ) = and 77 component variables , ( 1, 2, ,11.1j k x j k L , 2, , 7).L = =Parameters of the proposed algorithm are as follows, the number of subintervals is 3, and maximum iteration number is max 1000 G = .The reconciliation results compared with Crowe 1 and NLP26 are shown in Tab.5 (a part of variables are listed, * represents unmeasured variable) and Tab.6.

x
-reconciliation value of i x , i σ -standard deviation of i x ,n -number of measurable variables.run time of each loop.

Table 2 .
Benchmark functions testing results OV-Optimum Value, MIN-Mean Iteration Number, CT-CPU Time

Table 3 .
Constraint equations of rectification process

Table 4 .
and Fig.2.Data reconciliation results with the proposed algorithm

Table 5 .
Reconciliation results using Crowe, NLP and the proposed algorithm

Table 6 .
Compare results of Crowe, NLP and the proposed algorithm