Hydraulic turbine governing system identification using TâS fuzzy ...
Engineering Applications of Artificial Intelligence 26 (2013) 2073–2082
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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Hydraulic turbine governing system identification using T–S fuzzy model optimized by chaotic gravitational search algorithm Chaoshun Li n, Jianzhong Zhou, Jian Xiao, Han Xiao School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 November 2012 Received in revised form 28 February 2013 Accepted 11 April 2013 Available online 27 May 2013
Hydraulic turbine governing system (HTGS) is a complicated nonlinear system that controls the frequency and power output of hydroelectric generating unit (HGU). The modeling of HTGS is an important and difficult task, because some components, like hydraulic turbine and governor actuator, are with strong nonlinearity. In this paper, a novel Takagi–Sugeno (T–S) fuzzy model identification method based on chaotic gravitational search algorithm (CGSA) is proposed and applied in the modeling of HTGS. In the proposed method, fuzzy c-regression model clustering algorithm is used to partition the input space and identify the coarse antecedent membership function (MF) parameters at first. And then, a novel CGSA is proposed to search better MF parameters around the coarse results, in which chaotic search has been embedded in the iteration of basic GSA to search and replace the current best solution of GSA. The performance of the proposed fuzzy model identification method is validated by benchmark problems, and the results show that the accuracies of identified models have been improved significantly compared with the other existing models. Finally, the proposed approach has been applied to approximate the dynamic behaviors of HTGS of a HGU in a hydropower station of Jiangxi Province of China. The experimental results show that our approach can identify the HTGS satisfactorily with acceptable accuracy. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Takagi–Sugeno model System identification Heuristic algorithms Hydraulic turbine governing system Chaotic gravitational search algorithm Fuzzy c-regression model
1. Introduction Hydraulic turbine governing system (HTGS) is a complex nonlinear system which is influenced by water, mechanical and electrical factors. In this system, turbine governor is the controller and hydroelectric generating unit is the controlled object. Thus the closeloop control system's performance will be directly related to the safety of the power plant and the power quality sending to the grid. The precise modeling of HTGS is difficult for that the nonlinear analytical mathematic model of hydraulic turbine has not been established, and there are plenty of nonlinear factors existing in this system. The characteristic of hydraulic turbine can be expressed by nonlinear flow function and moment function, and neural network was applied in approximating these functions for modeling of hydraulic turbine (Chen et al., 2010). However, this approximation is based on the experimental data of model turbine, and the modeling of turbine is only the premise of modeling HTGS. Giving the fact that the structure and parameter of hydraulic turbine model remain unknown and strong nonlinearity is existing, fuzzy modeling can be a proper modeling approach for HTGS.
n
Corresponding author. Tel.: +86 27 87543992. E-mail address: [email protected] (C. Li).
0952-1976/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2013.04.002
The most important task to accomplish a fuzzy model is to perform structure identification, which is concerned with the determination of rules and parameter estimation (Li et al., 2009). In order to perform structure identification and parameter estimation, clustering algorithm has been widely used in literature. Clustering algorithm, i.e. fuzzy c-means (FCM) (Sugeno and Yasukawa, 1993), subtractive clustering algorithm (Yager and Filev, 1994), provides an effective way to partition input space of fuzzy model according to the number of rules, and then the membership functions (MF) parameters can be calculated for different rules. However, most traditional clustering methods are hyper-spherical-shaped cluster methods, in which the distance from data to point-shaped cluster center is based on the Euclidean norm and thus forms hyper-spherical-shaped clusters in geometric sense. Since it is more reasonable for the data space of T– S fuzzy model to be partitioned into hyper-plane-shaped clusters than into hyper-spherical-shaped clusters a series of fuzzy cregression models (FCRM) clustering algorithms have been developed with hyper-plane-shaped clusters for fuzzy modeling (Kim et al., 1997, 1998; Kung and Su, 2007), results have proved the effectiveness of this kind of clustering. In Li et al. (2009), a novel fuzzy c-regression model clustering algorithm (NFCRMA) was proposed, while an analysis solution for the FCRM clustering model had been developed, and the results indicated an improvement in fuzzy modeling application. From another point of view,
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structure identification can be formulated as a search problem in multidimensional space where each solution represents a possible structure with different membership functions and related parameters. Genetic algorithms (GA) (Du and Zhang, 2008), tabu search algorithm (TSA) (Bagis, 2008), cooperative random learning particle swarm optimization (CRPSO) (Zhao et al., 2010), quantuminquired differential evolution (QDE) (Su and Yang, 2011) and Artificial bee algorithm (Su et al., 2012) have been utilized in structure and parameter identification for T–S fuzzy model. Gravitational search algorithm (GSA) is a heuristic optimization method based on the law of gravity and mass interactions, proposed by Rashedi et al. (2009). As a global optimization method, GSA has shown superiority over traditional heuristic optimization methods like particle swarm optimization (PSO) and GA in function optimization (Rashedi et al., 2009), system identification (Li and Zhou, 2011), economic dispatch of power systems (Shaw et al., 2012), power flow optimization (Duman et al., 2012), and so on. GSA has also been used in fuzzy modeling as reported in Li et al. (2012), in which GSA is used to optimize a hyper-plane clustering model and the clustering algorithm based on GSA is used for structure identification for fuzzy modeling. Although the basic GSA is a relatively excellent optimization method, some researchers still try to improve GSA by avoiding local minima and premature convergence. Chaos and chaotic search have been proved to be a useful tool for improvement of heuristic optimization methods (Liu et al., 2005; Alatas et al., 2009). For the improvement of searching ability of GSA, chaotic GSAs were developed recently. A two-phase chaotic GSA was proposed for parameter identification problem, while chaotic search was applied in the end of gravitational search to enhance the search ability of GSA (Li et al., 2012). A modified GSA was proposed by Han and Chang (2012), in which chaotic numbers were used to modify the random numbers in position updating equation to enhance the ability of escaping from premature convergence of GSA. Besides chaotic GSAs, other variances of GSA have been developed, including new GSA proposed in Sarafrazi et al. (2011), while an operator called “Disruption”, originating from astrophysics was proposed to improve the exploration and exploitation abilities of GSA. Based on the discussion above, this paper is motivated by the following questions: excellent clustering algorithms, like NFCRMA, and global optimization method are all powerful tool for identification of T–S fuzzy model, will the combination of NFCRMA and global optimization produces better fuzzy modeling results; given the fact that GAs and TSA are relatively fundamental global optimization methods, will the implement of more effective global optimization methods, like chaotic GSA, improves the fuzzy modeling performance? For the discussed issues, clustering algorithm and global search algorithm for T–S fuzzy model identification, we have done some relative works. From our previous works on T–S fuzzy model identification methods based on clustering techniques and works on advanced heuristic search methods, the motivation of this paper may be clearer. In the researches of T–S fuzzy model identification, we proposed the NFCRMA for T–S fuzzy modeling (Li et al., 2009), and discovered that clustering models with hyperplane prototypes could improve the performance of T–S fuzzy model identification. In Li et al. (2012), we proposed another kind of clustering model with hyper-plane prototypes and solved this model by GSA, for the model could not been solved by Lagrange method of multipliers. In the researches of GSA, we noticed GSA was an excellent global search technique, and tried to improve GSA by combining the moving strategy of agents of PSO and GSA (Li and Zhou, 2011), and by implementing chaotic search to refine the solution obtained by GSA (Li et al., 2012). Based on our previous works, it is believed that the combination of suitable
clustering model and an excellent global search algorithm could improve the performance of T–S fuzzy model identification. In this paper, an evolving T–S fuzzy model based on a novel CGSA and NFCRMA is developed. In this approach, NFCRMA is used to partition fuzzy space and obtain the initial MF parameters, and then, MF parameters of all rules are encoded in one agent position and optimized by the proposed CGSA. In this CGSA, chaotic search is embedded in the iterations of GSA to refine the current best solution of GSA. The evaluation method used in parameter optimization considers the most important evaluation criterion accuracy, in terms of the mean square error (MSE) between the predicted output and the target output. The CGSA proposed in this paper is inspired by the existing works of CGSAs (Li et al., 2012; Han and Chang, 2012), however the differences between them are obvious. In Han and Chang (2012), chaotic number, not chaotic search, was used to improve GSA. In Li et al. (2012), we used chaotic search to refine the best solution after the end of GSA optimization, while in this paper, chaotic search is embedded in the searching process of GSA for seeking and replacing the current best solution in each iteration of GSA. The rest of this paper is organized as follows: in Section 2, T–S fuzzy model, the NFCRMA for fuzzy space partition and the related parameter identification methods are introduced; in Section 3, the CGSA is proposed and the optimization scheme for fuzzy model is designed; in Section 4, the proposed approach is validated by benchmark problem of fuzzy modeling; in Section 5, the proposed approach is applied in fuzzy model identification for the HTGS of a hydroelectric generating unit of Zhelin hydropower station in Jiangxi Province of China; finally, the conclusion is drawn in Section 6.
2. T–S fuzzy model identification with NFCRMA 2.1. T–S fuzzy model Takagi and Sugeno proposed the well-known T–S fuzzy model in Takagi and Sugeno (1985) to describe complicated nonlinear system. The T–S fuzzy model of this system can be described by the following IF-THEN fuzzy rules: I Rule i: IF x1 is A1i and…and xNI is AN i THEN I yi ¼ p0i þ p1i x1 þ ⋯ þ pN i xN I
ð1Þ
where i¼ 1,2,…,NR, NR is the number of fuzzy rule, x ¼ ½x1 ; x2 ; …; xNI is the system input, NI is the dimension of input vector, yi is the ith output, and pji ðj ¼ 1; …; N I Þ is the consequent parameter of the ith output. The final output of T–S fuzzy model can be expressed by a weighted mean defuzzification as follows: ∑N¼R 1 wi yi _ y ¼ iN ∑i ¼R 1 wi
ð2Þ
where the weight wi represents the overall truth value of the premise of the jth implication for the input which can be calculated as: NI
wi ¼ ∏ μðAji Þ
ð3Þ
j¼1
where μðAji Þ is the grade of membership function, which can be described by a bell-shaped Gaussian function as: 0 !2 1 xj −cji A j @ ; i ¼ 1; …; NR ; j ¼ 1; …; N I μðAi Þ ¼ exp − ð4Þ sji where cji and sji represent the center and width of the membership function respectively.
C. Li et al. / Engineering Applications of Artificial Intelligence 26 (2013) 2073–2082
2.2. Novel fuzzy c-regressive model clustering algorithm There have been many researches applying FCM to fuzzy modeling, but the clusters developed by FCM algorithm are hyper-sphere-shaped, not hyper-plane-shaped. Since hyperplane-shaped clusters are more suitable to be used in input– output fuzzy modeling, fuzzy c-regressive model clustering algorithm has been adopted to partition the input–output data pairs into hyper-plane-shaped clusters in fuzzy modeling (Kim et al., 1997; Kung and Su, 2007). In T–S fuzzy modeling, we assume that the N data pairs ðxk ; yk Þ ðk ¼ 1; …; NÞ are sampled, while xk is the input vector and yk is the output. In order to get NR rules, those data pairs are grouped in NR clusters, while the data samples in ith cluster are accorded with a linear regression model, which is actually a hyperplane function, that is i
i
yk ¼ f ðxk ; θi Þ ¼ ai1 xk1 þ ai2 xk2 þ ⋯ þ aiNI xkNI þ b0 ¼ ½xk 1⋅θTi
ð5Þ
where i¼ 1,2,…NR, xk ¼ ½xk1 ; …; xkNI is the kth input vector and i θi ¼ ½ai1 ; …; aiM ; b0 is the parameter vector of linear model. The distance (measure of error) of every sampled ðxk ; yk Þ to the ith regression model with parameter θi is defined as follows: dik ðθi Þ ¼ jyk −½xk 1⋅θTi j
ð6Þ
The objective function of FCRM is defined as: NR
N
J m ðU; θÞ ¼ ∑ ∑ ðμik Þmf ðdik ðθi ÞÞ2
ð7Þ
Once the antecedent fuzzy set parameters have been identified, the consequent parameters can be obtained from the following matrix equation, while all NS input–output data for fuzzy identification have been calculated according to Eq. (2): y ¼ Ap
NR
∑ μik ¼ 1;
i¼1
k ¼ 1; 2; …; N
ð8Þ
As deduced in Li et al. (2009), the analytical solution of Eq. (7) could be expressed by the following equations: μik ¼
θij ¼
1 2=ðmf −1Þ R ∑N j ¼ 1 ½ðdik ðθ i ÞÞ=ðdjk ðθ j ÞÞ
Σ nk ¼ 1 ðμik Þmf ðyk −Σ t≠j θit ⋅x^ kt Þ⋅x^ kj 2
Σ nk ¼ 1 ðμik Þmf ⋅x^ kj
ð9Þ
ð10Þ
By iterating Eqs. (9) and (10), the fuzzy membership degree μik and affine line model parameter θij will vary towards the direction that minimizes J m ðU; θÞ gradually. The specific steps of NFCRMA are shown in Li et al. (2009). 2.3. Model identification We use the NFCRMA to separate the input–output space, and obtain the membership functions of the fuzzy sets in the antecedent part of each fuzzy rule, as described by Eq. (4). As mentioned in Kim et al. (1997), in view of this equation, the fuzzy set centers cji ð1 ≤i ≤N R ; 1 ≤j ≤N I Þ and the respective standard deviations sji can be easily obtained by using μik, the membership of ðxk ; yk Þ belonging to ith clustering representing hyper-plane, as follows: cji
¼
ΣN k ¼ 1 μik ⋅xkj ΣN k ¼ 1 μik
; i ¼ 1; …; N R ; j ¼ 1; …; NI
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u2Σ μik ⋅ðxkj −cji Þ2 j ; i ¼ 1; …; N R ; j ¼ 1; …; NI si ¼ t k ¼ 1N Σ k ¼ 1 μik
ð11Þ
ð13Þ
NI 0 I where p ¼ ½p01 ; …; pN 1 ; …; pN R ; …; pN R , x k ¼ ½xk1 ; xk2 ; …; xkN I is the kth input vector of fuzzy model, y ¼ ½y1 ; y2 ; …; yNS T is the target system output, A is the coefficient matrix, which can be calculated according to Eqs. (2)–(4). In this paper, the least square method (LSM) is used to solve the matrix equation in Eq. (13) to identify the consequent parameter vector p.
3. T–S fuzzy model optimization based on CGSA 3.1. Chaotic gravitational search algorithm GSA is a newly developed stochastic search algorithm based on the law of gravity and mass interactions. In GSA, the search agents are a collection of masses which interact with each other based on the Newtonian gravity and the laws of motion (Rashedi et al., 2009). Assumed there are N agents (masses), position of the ith agent 1 d n is Li ¼ ðli ; …; li ; …; li Þ; i ¼ 1; …; N. Inertial mass is calculated according to objective function value of the agent as:
k¼1i¼1
where mf ∈ð1; ∞Þ is the fuzzy weighting exponent; the value of mf is usually set as 2, μik ∈½0; 1 is the fuzzy membership degree of the kth data pair belonging to the ith cluster. It is assumed that μik is constrained with the following equation:
mi ¼
obji −worst best−worst
ð14Þ
where obji is the objective function value of the ith agent. For optimization problem seeking minimal value, best ¼ minobjj , worst ¼ maxobjj . According to Newtonian gravitation theory, the force acting on the ith mass from the jth mass is defined: F dij ðtÞ ¼ GðtÞ
M pi ðtÞ M aj ðtÞ d d ðl ðtÞ−li ðtÞÞ jjX i ðtÞ; X j ðtÞjj2 j
ð15Þ
where Mi and Mj are masses of agents, M i ¼ mi =Σ N j ¼ 1 mj , G(t) is the gravitational constant at time t. It must be pointed out that the gravitational constant G(t) is important in determining the performance of GSA, G(t) is defined a function of time t: t GðtÞ ¼ G0 exp −β⋅ ð16Þ maxt where G0 is the initial value, β is the constant, t is the current iterations, and max_t is the maximum iterations. For the ith agent, the randomly weighted sum of the forces exerted from other agents: F di ðtÞ ¼ ∑randj F dij ðtÞ
ð17Þ
j≠i
Based on law of motion, the acceleration of the ith agent is calculated by: adi ðtÞ ¼
F di ðtÞ M ii ðtÞ
ð18Þ
where, Mii is the inertial mass of the ith agent. Then, the searching strategy on this concept can be described by following equations: vdi ðt þ 1Þ ¼ randi vdi ðtÞ þ adi ðtÞ
ð12Þ
2075
d
d
li ðt þ 1Þ ¼ li ðtÞ þ vdi ðt þ 1Þ
ð19Þ ð20Þ
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Step 4: calculate the mass of ith agent mi by Eq. (14), i¼ 1,…,N. Step 5: calculate the gravitational constant G by Eq. (16). Step 6: calculate the gravitation of ith agent that other agent act on by Eqs. (15) and (17), i ¼1,…,N. Step 7: calculate the acceleration ai for ith agent by Eq. (18). Step 8: update the velocity and position for all agents according to Eqs. (19) and (20). Step 9: in order to improve the computing efficiency, employ chaotic search with little probability in the former phase of evolvement due to quick convergence of GSA, and use chaotic search in all probability in the latter phase of evolvement. The probability to progress chaotic search may denote as follows: p ¼ 1−1=ð1−lnðtÞÞ. Generate a random number r between (0, 1) interval, if r op, then go to Step 11 to start the chaotic search, else go to Step 18. Step 10: start the chaotic search around the current best solution Lbest, and ICMIC map is used to illuminate the process of chaotic search. randomly initialize n chaotic variables cxðkÞ ∈ d ½−1; 0Þ∪ð0; 1; d ¼ 1; 2; …; n; Xbest ¼Lbest ¼ (pg,1,pg,d,pg,n); Fbest ¼fg； set iteration counter k¼ 0 for chaotic search, and the maximum number of chaotic search is Cmax.
In above equations, li represents the position of ith agent in dth dimension, vdi is the velocity, adi is the acceleration, randi is a random number among [0, 1]. To improve the performance of GSA, the chaotic system is incorporated into GSA by means of utilizing the chaotic search as a local search procedure of GSA. Based on the definition of chaotic map and illumination of chaotic search, the CGSA is proposed. A chaotic map is a discrete-time dynamical system that could be expressed as: cxðnþ1Þ ¼ f ðcxðnÞ Þ
ð21Þ (0)
By giving initial state cx , the chaotic sequence could be gotten through running the chaotic map. It has been reported in He et al. (2009, 2000) that chaotic map with infinite collapses (ICMIC) map with a symmetrical region ½−1; 0Þ∪ð0; 1 shows advantage in chaotic search. The ICMIC map is described by following equation: a cxðkþ1Þ ¼ sin for a 4 0; cxðkÞ∈½−1; 0Þ∪ð0; 1 ð22Þ cxðkÞ This paper proposes a new chaotic gravitational search algorithm (CGSA) by introducing chaotic maps with ergodicity, irregularity and the stochastic property in GSA to improve the global convergence by escaping from the local solutions. The chaotic search is based on chaotic sequences that generated by ICMIC map. Chaotic search has been proved to be effective of improving the searching ability of heuristic optimization by numerous experiments. The procedure of this novel approach can be summarized as follows: Step 1: initialize the positions for a population with N agents d L ¼ fli ; i ¼ 1; …; N; d ¼ 1; …; ngrandomly; initialize velocity of agents V¼ {vij, i¼ 1,…,N, d ¼1,…,n}, set vij ¼0; set global optimal solution Lbest; the global optimal objective function value fg; set the iteration counter t¼ 0, the maximum number of iteration Tmax. Step 2: calculate the objective function value for N agents, fi ¼ obj(Li), i ¼1,…,N. Step 3: refresh fg and Lbest, for a minimization problem, if fi ofg, then fg ¼fi and Lbest ¼Li.
Table 1 Comparison of different models for the Box and Jenkins examplea. Model
No. of rules
MSE
Box and Jenkins (1970) Sugeno and Yasukawa (1993) Evsukoff et al. (2002) Kim et al. (1997) Kim et al. (1998) Kung and Su (2007) Bagis (2008) Du and Zhang (2008) Zhao et al. (2010) Su and Yang (2011) Su et al. (2012) Li et al. (2012) Li et al. (2009) Our model
6 3 2 6 6 2 4 4 3.96 4 3 3 4 4
0.202 0.19 0.090 0.062 0.055 0.0518 0.148 0.06 0.1428 0.112 0.0548 0.0534 0.0498 0.035
a
In literature Zhao et al. (2010), number of rules is a mean value.
70 Target Prediction
y (t)
65 60 55 50 45 0
50
100
150 t
200
250
300
0
50
100
150 t
200
250
300
1.5 1 Error
0.5 0 -0.5 -1 -1.5
Fig. 1. Comparison of prediction of our model and the target output for Box and Jenkins example.
C. Li et al. / Engineering Applications of Artificial Intelligence 26 (2013) 2073–2082
2077
10 Training
Target Prediction
Testing
y (t)
5
0
0
100
200
300
400
500 t
600
700
800
900
1000
0
100
200
300
400
500 t
600
700
800
900
1000
0.1
Error
0.05 0 -0.05 -0.1
Fig. 2. Comparison of prediction of our model and the target output for the nonlinear differential equation example references.
Step 11: adjust the search radius δðkÞ , and optimization varid ables interval (xmin d xmax d), and xmaxd ¼ pg;d þ δðkÞ =2, xmini ¼ d pg;d −δðkÞ =2; d Step 12: generate the chaotic variables cxdðkþ1Þ for the next iteration according to Eq. (22). Step 13: project chaotic variables to optimization intervals (xmin i, xmax i) and get optimization variables: zdðkþ1Þ ¼
xmaxd þ xmind xmaxd −xmind ðkÞ þ cxd 2 2
Step 14: evaluate the new solution Z kþ1 ¼ ½z1ðkþ1Þ ; …; zdðkþ1Þ ; ðkþ1Þ …; zm by calculating objective function value f(Zk+1). Step 15: if f(Zk+1)oFbest, then Xbest ¼Zk+1 and Fbest ¼f(Zk+1). Step 16: decrease the searching radius δdðkþ1Þ ¼ w δðkÞ ; d 0 o w o1. Step 17: k¼ k+1; if k4 Cmax, go out of chaotic search and go to Step 19, otherwise go back to step 12. Step 18: renew the current best solution, Lbest ¼Xbest, fg ¼Fbest. Step 19: t ¼t+1; if toTmax, go to Step 2, or else go out of the algorithm, and return the best solution Lbest, together with the best objective function value fg.
3.2. Encoding scheme for T–S fuzzy model In order to optimize the fuzzy model by CGSA, the optimization variables should be selected and encoded in the position vector of agents. Because the consequent parameters can be calculated by least square method, it is only necessary to encode antecedent MF parameters of centers and widths, which are necessary to represent a Gaussian MF, into the position vector. Assume the number of rules is NR, the number of input is NI, and there are total NR NI optimization variables that have to be encoded. The position of kth agent dimension is a NR NI dimension vector, which can be described as: NI NI NI 1 1 1 I Lk ¼ ½c11 ; …; cN 1 ; s1 ; …; s1 ; …; cNR ; …; cNR ; sN R ; …; sN R I where cN 1 1 first rule; s1
c11
ð23Þ
represent the MF parameters of Gaussian center of the I sN 1 represent the MF parameters of Gaussian width of NI 1 I the first rule; c1NR cN N R and sNR sN R are corresponding MF parameters for the NR rule.
3.3. Optimization of T–S fuzzy model The CGSA is used to optimize and adjust the MF parameters around the parameters of the fuzzy model. The initialization of positions of agents in CGSA is depended on the MF parameters obtained by NFCRMA. Assume that the MF parameters identified by NFCRMA are expressed as vector Lopt, and then boundaries Lmin and Lmax could be set to make sure the center is Lopt. The positions of agents are randomly generated in the boundaries of Lmin and Lmax. The object function for optimization of T–S fuzzy model is defined based on the mean square error (MSE) between the predicted output of fuzzy model and target output: J MSE ¼
1 NS ∑ ðy −y^ Þ2 NS k ¼ 1 k k
ð24Þ
where NS is the data length, y^ k the predicted output and yk the target output. The proposed approach for fuzzy modeling in this paper could be summarized as: (1) Prepare input and output data of the target system, select NI inputs for fuzzy model, set number of rules NR. NS data pairs ðxk ; yk Þ are prepared for fuzzy modeling, while xk ¼ ðxk1 ; …; xkNI Þ is the input of fuzzy model. (2) NFCRMA is used to partition the dataset xk (k ¼1,…,NS) into NR clusters, while fuzzy membership μik (i¼1,…,NR, k¼ 1,…,NS) is obtained, then the MF parameters cji and sji is calculated by Eqs. (11) and (12), get Lopt according to Eq. (23). (3) Set the population size N, maximum iteration number Nitg, gravitational constant function parameter G0 and β, and chaotic search number Nitc for CGSA; initialize population of agents randomly between Lmin and Lmax. (4) Start the iteration of CGSA, the specific step is shown above. For agents the objective function is calculated as following steps: decode the location vector to obtain cji and sji ; calculate μðAji Þ and wi for xk; calculate coefficient matrix A, and then solve Eq. (13) by LSM to get consequent parameters vector p; calculate model output y^ ¼ Ap, calculate objective function by Eq. (24).
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4.2. A nonlinear differential equation
(5) Get the best solution Lbest and the corresponding MSE; decoding Lbest to get MF parameters.
4. Applications to benchmark problem
This example is taken from Evsukoff et al. (2002) in which the plant to be identified is given by the second-order highly nonlinear difference equation
4.1. Box–Jenkins system
yðkÞ ¼
In this subsection, we use the Box–Jenkins data set (Box and Jenkins, 1970), which consists of 296 input–output measurements of a gas-furnace process. At each sampling time k, the input xðkÞ of this process is the gas flow rate and the output yðkÞ is the output CO2 concentration. For comparison with conventional fuzzy models, we choose uðkÞ, uðk−1Þ, uðk−2Þ, yðk−1Þ, yðk−2Þ, yðk−3Þ as the input of the fuzzy model (Kim et al., 1997; Kung and Su, 2007). The number of fuzzy rules NR ¼4. Parameters of CGSA are set as: population size N ¼ 30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼100. The performance of our fuzzy model is shown in Fig. 1, where the prediction of our model and the target output are compared, and also the errors between prediction and target output are shown. It is noted that our model has a MSE of 0.035, better than the result identified by NFCRMA in Li et al. (2009) and other results that occur in literature. The detailed comparison of performances of different methods is given in Table 1. The fuzzy model identification approach proposed in this paper is based on NFCRMA and CGSA, the fuzzy model identified by NFCRMA is optimized by CGSA, so we expect our model in this paper performs better than that obtained by NFCRMA. The results listed in Table 1 verify that the CGSA can search better MF parameters to obtain more accurate fuzzy model.
Frequency (Hz)
54
a
52 50
46
MSE
12 8 100 4 5.6 5 4 4 4
Training data
Testing data
0.5072 0.6184 0.1577 0.0341 0.002 1.27e−4 0.0102 0.0149 8.45e−5
0.2447 0.2037 0.0185 0.0378 0.0035 1.27e−4 0.0128 0.0115 6.5e−5
0
10
20
30 Time (s)
40
50
60
0
10
20
30 Time (s)
40
50
60
0.05
0
-0.05
In literature Zhao et al. (2010), number of rules is a mean value.
Generation setpoint
Target Prediction
48
Error
Sugeno and Yasukawa (1993) Wang and Lee (1999) Evsukoff et al. (2002) Bagis (2008) Zhao et al. (2010) Su and Yang (2011) Li et al. (2012) Li et al. (2009) Our model
No. of rules
Fig. 4. Fuzzy model training of 48–52 Hz up frequency disturbance process.
Speed regulation
+
-
Unit generation
Rs Speed setpoint
Guide vane position +
+
Governor controller -
ð25Þ
Much effort has been paid to check the robustness of our fuzzy model. We use training data to build the model, and testing data to check the model. We get 500 points training data from Eq. (25), assuming a random input signal u(k) uniformly distributed in the interval [−2, 2], and 500 points testing data, applying a sinusoidal input signal uðkÞ ¼ sin ð2kπ=25Þ. The model has three inputs u(k), y(k−1), y(k−2), and a single output y(k). After the learning process is finished, the model is tested by applying testing data to drive it. The rule number of the fuzzy model is four. Parameters of CGSA are set as: population size N ¼30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼ 100. The prediction of our model and the target system are shown in Fig. 2, while outputs comparison and the respective errors are exhibited. Fig. 2 reflects both the training process and testing process of our fuzzy model. It is easy to find out that our model match well with the target system and the errors of both training
Table 2 Comparison of different models for the nonlinear differential equation examplea. Model
yðk−1Þ⋅yðk−2Þ⋅ðyðk−1Þ þ 2:5Þ þ uðkÞ 1 þ y2 ðk−1Þ þ y2 ðk−2Þ
Turbine control actuator
Turbine & water column
Generator
Unit speed Fig. 3. Hydraulic turbine governing system with speed regulation.
Power output to grid
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Frequency (Hz)
54 52 Target Prediction
50 48 46 0
5
10
15
20
25 Time (s)
30
35
40
45
50
0
5
10
15
20
25 Time (s)
30
35
40
45
50
1
Error
0.5 0 -0.5 -1
Fig. 5. Fuzzy model testing by 48–52 Hz up frequency disturbance process.
Frequency (Hz)
54 Target Prediction
52 50 48 46 0
5
10
15
20 25 Time (s)
30
35
40
45
0
5
10
15
20 25 Time (s)
30
35
40
45
0.1
Error
0.05 0 -0.05 -0.1
Fig. 6. Fuzzy model training of 52–48 Hz down frequency disturbance process.
and testing are puny. The MSE of training process is as small as 8.45e−5, and MSE for testing data is 6.5e−5. In order to show the effectiveness of the proposed approach, the MSE of our model is compared with different models in Table 2. The results listed in Table 2 show that the modeling accuracy has been improved significantly through the efforts we make in this paper. 4.3. Comparison with related works As the development of optimization techniques, some excellent global optimization algorithms have been applied in T–S fuzzy model identification (Du and Zhang, 2008; Bagis, 2008; Zhao et al., 2010; Su and Yang, 2011; Su et al., 2012). There are two ways to
apply optimization algorithms in T–S fuzzy model identification, namely direct way and indirect way. For direct way, the structure or parameters of T–S fuzzy model are optimized directly. For example, in Du and Zhang (2008), the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the MF parameters of the T–S fuzzy model were all represented in one chromosome and evolved together by genetic algorithm. And TSA (Bagis, 2008), CRPSO (Zhao et al., 2010) and QDE (Su and Yang, 2011) were all used to identify the T–S fuzzy model directly. For indirect way, optimization algorithm can be used to solve clustering model and thus enhancing the fuzzy space partition performance for T–S fuzzy model identification. For example, the artificial bee colony (ABC) algorithm (Su et al., 2012) was used to optimize clustering model of
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FCM, and then the new version of fuzzy c-means clustering technique (VABC-FCM) was applied in T–S fuzzy identification. In Li et al. (2012), we propose a hyper-plane clustering algorithm based on GSA, and apply this technique in T–S fuzzy model identification. In this paper, the CGSA are used to optimize MF parameters of T–S fuzzy model. However, the usage of optimization technique is different with those in literature. We use NFCRMA to identify T–S fuzzy model at first, then apply CGSA to optimize the identified model. We can analyze the identification results listed in Tables 1 and 2. For identification of Box and Jenkins system, fuzzy models identified by indirect optimization techniques (Su et al., 2012; Li et al., 2012) achieve better performance than those identified by direct optimization techniques (Du and Zhang, 2008; Bagis, 2008; Zhao et al., 2010; Su and Yang, 2011). T–S fuzzy model is complicated and there are many parameters that need optimization. This makes the direct optimization of T–S fuzzy model a difficult task. If we can get a coarse result at first and set a search range for optimization technique, the optimization will be very effective. By comparing identification results listed in Tables 1 and 2, the outstanding performances of our approach in this paper can be confirmed.
5. Application to hydraulic turbine governing system Hydraulic turbine governing system (HTGS) is a complicated nonlinear system, mainly containing four parts, namely governor controller, turbine control actuator, turbine and water column, and generator. Fig. 3 shows the schematic of the overall system. HTGS is a control system of hydroelectric generating unit (HGU) that governs the speed of turbine according to the setpoint of output power and setpoint of speed. The main task of HTGS is to adjust the power output to grid and to track the frequency of grid. In general, the HTGS should be given a comprehensive test and performance evaluation after installation and overhaul of HGU. The modeling and simulation of HTGS is important for it provide a way to test and evaluate the governing system of HGU. Hydraulic turbine is the key component in HTGS, and it is a very complicated nonlinear system, which do not has any analytic
expression until now. However, it is usually described as moment function and flow function of guide vane position y, generator unit speed x and water head h, shown as following: (
mt ¼ f 1 ðy; x; hÞ
ð26Þ
q ¼ f 2 ðy; x; hÞ
Although the analytic expression of nonlinear functions f 1 ð⋅Þ and f 2 ð⋅Þ are unknown, moment and flow characteristic curves are provide by manufacturer through experiments. Based on the characteristic curve of turbine, we can calculate moment and flow of the hydraulic turbine at certain time by means of interpolation or nonlinear fitting. Neural network has been applied in modeling of hydraulic turbine by fitting the characteristic curves of moment and flow (Chen et al., 2010). In this paper, we try to build the fuzzy model of hydraulic turbine for accurate modeling of HTGS of the fourth generating unit (4F) of Zhelin hydropower station, in Jiangxi province of China. In order to build the model for turbine and generator, output of governor actuator and generator should be collected. After the overhaul of HGU, the HTGS should be tested in a series experiments, including frequency disturbance experiment under no-load condition. In this experiment, the setpoint frequency is changed suddenly, and then the governor of HTGS controls the frequency (speed) of the unit by adjusting the guide vane position. We could sample signal of guide vane position y and frequency of generator x for fuzzy modeling. Frequency disturbance experiments of 4F unit were recorded and unit frequency x, guide vane position y were collected. There are two types of frequency disturbance, up frequency disturbance and down frequency disturbance. In experiment of fuzzy modeling, we designed three groups of experiments. In the first group, a 48–52 Hz up frequency disturbance process was used for training of fuzzy model, while another 48–52 Hz up frequency disturbance process was used for model testing. In the second group, a 52–48 Hz down frequency disturbance process was used for training, with a 50–48 Hz down frequency disturbance process for testing. In the third group, a 48–52 Hz up frequency disturbance process was used for training, with a 52–48 Hz down frequency disturbance process for testing.
Frequency (Hz)
51 Target Prediction
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30 35 Time (s)
40
45
50
55
15
20
25
30 35 Time (s)
40
45
50
55
Error
0.5
0
-0.5 10
Fig. 7. Fuzzy model training of 50–48 Hz down frequency disturbance process.
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50 48 46 0
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60
70
0
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30 40 Time (s)
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60
70
0.1
Error
0.05 0 -0.05 -0.1
Fig. 8. Fuzzy model training of 48–52 Hz up frequency disturbance process.
54 Frequency (Hz)
Target 52
Prediction
50 48 46 0
10
20
30 Time (s)
40
50
60
0
10
20
30 Time (s)
40
50
60
Error
0.5
0
-0.5
Fig. 9. Fuzzy model training of 52–48 Hz down frequency disturbance process.
The sampling rate is 50 Hz. y(k−1), x(k−1), x(k−6) Are selected as input of fuzzy model. The number of fuzzy rules NR ¼3, parameters of CGSA is set as: population size N ¼30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼ 100. In the first group, output of both the fuzzy model and the target system are shown in Fig. 4 while outputs comparison and the respective errors are exhibited. The trained fuzzy model possesses high accuracy with a MSE of 3.38e−5. The fuzzy model is verified by testing data, the output of the trained fuzzy model almost coincide with the target system, as shown in Fig. 5. The MSE between prediction of fuzzy model and target is as small as 0.00534. In the second group, output of both the fuzzy model and the target system are shown in Fig. 6 while outputs comparison and the respective errors are exhibited. The trained fuzzy model
possesses high accuracy with a MSE of 3.83e−5. Fig. 7 shows the testing result of the trained fuzzy model, while another down frequency disturbance process is used for target. When checked by testing data, the fuzzy model is found out to match well with the actual system output with a MSE of 0.0017. In the third group, output of both the fuzzy model and the target system are shown in Fig. 8 while outputs comparison and the respective errors are exhibited. The trained fuzzy model possesses high accuracy with a MSE of 6.41e−5. Fig. 9 shows the validation of the trained fuzzy model. It is found that our fuzzy model could predict the system output accurately, with a MSE of 0.0008. In this example, the up frequency disturbance process is used to train the model, while down frequency distance process used for testing. The prediction result proves the generalization ability of the trained fuzzy model.
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6. Conclusions In this paper, a new fuzzy model identification approach based on a novel fuzzy c-regression clustering algorithm (NFCRMA) and chaotic gravitational search algorithm (CGSA) is presented for the accurate modeling of hydraulic turbine governing system. In this approach, NFCRMA is used to partition the input–output space of fuzzy model into clusters with hyper-plane prototype and extract membership function (MF) parameters of different rule, and then a CGSA which embeds chaotic search in gravitational search is proposed to optimize the MF parameters of fuzzy model. Benchmark problems of nonlinear system modeling have shown that the proposed modeling approach improves the modeling accuracy significantly by comparison with previous results existing in literature. Finally this approach is applied to fuzzy modeling of HTGS of a HGU of Zhelin Hydropower station in Jiangxi Province of China, and the results demonstrate the effectiveness and accuracy of the proposed approach.
Acknowledgments This paper is supported by the National Natural Science Foundation of China (No. 51109088), the Research Fund for the Doctoral Program of Higher Education of China (No. 20110142120020) and the Fundamental Research Funds for the Central Universities, HUST (No. 2013QN114). References Alatas, B., Akin, E., Ozer, A.B., 2009. Chaos embedded particle swarm optimization algorithms. Chaos Solitons Fract. 40 (4), 1715–1734. Bagis, A., 2008. Fuzzy rule base design using tabu search algorithm for nonlinear system modeling. ISA Trans. 47 (1), 32–44. Box, G.E.P., Jenkins, G.M., 1970. Time Series Analysis, Forecasting and Control. Holden Day, San Francisco, CA. Chen, G., Liu, Y., Meng, Z. Development of a nonlinear real-time simulation system for hydroelectric generating unit. In: Proceedings of 2010 Power and Energy Engineering Conference, pp. 1–4. Du, H., Zhang, N., 2008. Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification. Appl. Soft Comput. 8 (1), 676–686. Duman, Serhat, Güvenç, Uğur, Sönmez, Yusuf, Yörükeren, Nuran, 2012. Optimal power flow using gravitational search algorithm. Energy Convers. Manage. 59, 86–95.
Evsukoff, A., Branco, A.C.S., Galichet, S., 2002. Structure identification and parameter optimization for non-linear fuzzy modeling. Fuzzy Sets Syst. 132 (2), 173–188. Han, X., Chang, X., 2012. A chaotic digital secure communication based on a modified gravitational search algorithm filter. Inf. Sci. 208, 14–27. He, D., He, C., Jiang, L.G., Zhu, H., Hu, G. A chaotic map with infinite collapses. In: Proceedings of 2000 IEEE TENCON, pp. 95–99. He, Y., Zhou, J., Xiang, X., Chen, H., Qin, H., 2009. Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling. Chaos Solitons Fract. 42 (5), 3169–3176. Kim, E., Park, M., Ji, S., Park, M., 1997. A new approach to fuzzy modeling. IEEE Trans. Fuzzy Syst. 5 (3), 328–337. Kim, E., Park, M., Kim, S., Park, M., 1998. A transformed input-domain approach to fuzzy modeling. IEEE Trans. Fuzzy Syst. 6 (4), 596–604. Kung, C.C., Su, J.Y., 2007. Affine Takagi–Sugeno fuzzy modelling algorithm by fuzzy c-regression models clustering with a novel cluster validity criterion. IET Control Theory Appl. 1 (5), 1255–1265. Li, C., Zhou, J., 2011. Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm. Energy Convers. Manage 52, 374–381. Li, C., Zhou, J., Xiang, X., Li, Q., An, X., 2009. T–S fuzzy model identification based on a novel fuzzy c-regression model clustering algorithm. Eng. Appl. Artif. Intell. 22 (4–5), 646–653. Li, C., Zhou, J., Xiao, J., Xiao, H., 2012. Parameters identification of chaotic system by chaotic gravitational search algorithm. Chaos, Solitons Fract. 45 (4), 539–547. Li, C., Zhou, J., Fu, B., Kou, P., Xiao, J., 2012. T–S fuzzy model identification with gravitational search based hyper-plane clustering algorithm. IEEE Trans. Fuzzy Syst. 20 (2), 305–317. Liu, B., Wang, L., Jin, Y.H., Tang, F., Huang, D.X, 2005. Improved particle swarm optimization combined with chaos. Chaos Soliton Fract. 25 (5), 1261–1271. Rashedi, E., Nezamabadi-pour, H., Saryazdi, S., 2009. GSA: a gravitational search algorithm. Inf. Sci. 179 (13), 2232–2248. Sarafrazi, S., Nezamabadi-Pour, H., Saryazdi, S., 2011. Disruption: a new operator in gravitational search algorithm. Sci. Iran. 18, 539–548 3 D. Shaw, B., Mukherjee, V., Ghoshal, S.P., 2012. A novel opposition-based gravitational search algorithm for combined economic and emission dispatch problems of power systems. Int. J. Electr. Power 35 (1), 21–33. Su, H., Yang, Y., 2011. Differential evolution and quantum-inquired differential evolution for evolving Takagi–Sugeno fuzzy models. Expert Syst. Appl. 38, 6447–6451. Su, Z., Wang, P., Shen, J., Zhang, Y., Chen, L., 2012. Convenient T–S fuzzy model with enhanced performance using a novel swarm intelligent fuzzy clustering technique. J. Process Control 22, 108–124. Sugeno, M., Yasukawa, T., 1993. A fuzzy-logic-based approach to qualitative modeling. IEEE Trans. Fuzzy Syst. 1 (1), 7–31. Takagi, T., Sugeno, M., 1985. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 15 (1), 116–132. Wang, S., Lee, C., 1999. Fuzzy system modeling using linear distance rules. Fuzzy Set. Syst 108 (2), 179–191. Yager, R., Filev, D., 1994. Generation of fuzzy rules by mountain method. J. Intell. Fuzzy Syst. 2, 209–219. Zhao, L., Qian, F., Yang, Y., Zeng, Y., Su, H., 2010. Automatically extracting T–S fuzzy models using cooperative random learning particle swarm optimization. Appl. Soft Comput. 10, 938–944.
Contents lists available at ScienceDirect
Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Hydraulic turbine governing system identification using T–S fuzzy model optimized by chaotic gravitational search algorithm Chaoshun Li n, Jianzhong Zhou, Jian Xiao, Han Xiao School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 November 2012 Received in revised form 28 February 2013 Accepted 11 April 2013 Available online 27 May 2013
Hydraulic turbine governing system (HTGS) is a complicated nonlinear system that controls the frequency and power output of hydroelectric generating unit (HGU). The modeling of HTGS is an important and difficult task, because some components, like hydraulic turbine and governor actuator, are with strong nonlinearity. In this paper, a novel Takagi–Sugeno (T–S) fuzzy model identification method based on chaotic gravitational search algorithm (CGSA) is proposed and applied in the modeling of HTGS. In the proposed method, fuzzy c-regression model clustering algorithm is used to partition the input space and identify the coarse antecedent membership function (MF) parameters at first. And then, a novel CGSA is proposed to search better MF parameters around the coarse results, in which chaotic search has been embedded in the iteration of basic GSA to search and replace the current best solution of GSA. The performance of the proposed fuzzy model identification method is validated by benchmark problems, and the results show that the accuracies of identified models have been improved significantly compared with the other existing models. Finally, the proposed approach has been applied to approximate the dynamic behaviors of HTGS of a HGU in a hydropower station of Jiangxi Province of China. The experimental results show that our approach can identify the HTGS satisfactorily with acceptable accuracy. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Takagi–Sugeno model System identification Heuristic algorithms Hydraulic turbine governing system Chaotic gravitational search algorithm Fuzzy c-regression model
1. Introduction Hydraulic turbine governing system (HTGS) is a complex nonlinear system which is influenced by water, mechanical and electrical factors. In this system, turbine governor is the controller and hydroelectric generating unit is the controlled object. Thus the closeloop control system's performance will be directly related to the safety of the power plant and the power quality sending to the grid. The precise modeling of HTGS is difficult for that the nonlinear analytical mathematic model of hydraulic turbine has not been established, and there are plenty of nonlinear factors existing in this system. The characteristic of hydraulic turbine can be expressed by nonlinear flow function and moment function, and neural network was applied in approximating these functions for modeling of hydraulic turbine (Chen et al., 2010). However, this approximation is based on the experimental data of model turbine, and the modeling of turbine is only the premise of modeling HTGS. Giving the fact that the structure and parameter of hydraulic turbine model remain unknown and strong nonlinearity is existing, fuzzy modeling can be a proper modeling approach for HTGS.
n
Corresponding author. Tel.: +86 27 87543992. E-mail address: [email protected] (C. Li).
0952-1976/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2013.04.002
The most important task to accomplish a fuzzy model is to perform structure identification, which is concerned with the determination of rules and parameter estimation (Li et al., 2009). In order to perform structure identification and parameter estimation, clustering algorithm has been widely used in literature. Clustering algorithm, i.e. fuzzy c-means (FCM) (Sugeno and Yasukawa, 1993), subtractive clustering algorithm (Yager and Filev, 1994), provides an effective way to partition input space of fuzzy model according to the number of rules, and then the membership functions (MF) parameters can be calculated for different rules. However, most traditional clustering methods are hyper-spherical-shaped cluster methods, in which the distance from data to point-shaped cluster center is based on the Euclidean norm and thus forms hyper-spherical-shaped clusters in geometric sense. Since it is more reasonable for the data space of T– S fuzzy model to be partitioned into hyper-plane-shaped clusters than into hyper-spherical-shaped clusters a series of fuzzy cregression models (FCRM) clustering algorithms have been developed with hyper-plane-shaped clusters for fuzzy modeling (Kim et al., 1997, 1998; Kung and Su, 2007), results have proved the effectiveness of this kind of clustering. In Li et al. (2009), a novel fuzzy c-regression model clustering algorithm (NFCRMA) was proposed, while an analysis solution for the FCRM clustering model had been developed, and the results indicated an improvement in fuzzy modeling application. From another point of view,
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structure identification can be formulated as a search problem in multidimensional space where each solution represents a possible structure with different membership functions and related parameters. Genetic algorithms (GA) (Du and Zhang, 2008), tabu search algorithm (TSA) (Bagis, 2008), cooperative random learning particle swarm optimization (CRPSO) (Zhao et al., 2010), quantuminquired differential evolution (QDE) (Su and Yang, 2011) and Artificial bee algorithm (Su et al., 2012) have been utilized in structure and parameter identification for T–S fuzzy model. Gravitational search algorithm (GSA) is a heuristic optimization method based on the law of gravity and mass interactions, proposed by Rashedi et al. (2009). As a global optimization method, GSA has shown superiority over traditional heuristic optimization methods like particle swarm optimization (PSO) and GA in function optimization (Rashedi et al., 2009), system identification (Li and Zhou, 2011), economic dispatch of power systems (Shaw et al., 2012), power flow optimization (Duman et al., 2012), and so on. GSA has also been used in fuzzy modeling as reported in Li et al. (2012), in which GSA is used to optimize a hyper-plane clustering model and the clustering algorithm based on GSA is used for structure identification for fuzzy modeling. Although the basic GSA is a relatively excellent optimization method, some researchers still try to improve GSA by avoiding local minima and premature convergence. Chaos and chaotic search have been proved to be a useful tool for improvement of heuristic optimization methods (Liu et al., 2005; Alatas et al., 2009). For the improvement of searching ability of GSA, chaotic GSAs were developed recently. A two-phase chaotic GSA was proposed for parameter identification problem, while chaotic search was applied in the end of gravitational search to enhance the search ability of GSA (Li et al., 2012). A modified GSA was proposed by Han and Chang (2012), in which chaotic numbers were used to modify the random numbers in position updating equation to enhance the ability of escaping from premature convergence of GSA. Besides chaotic GSAs, other variances of GSA have been developed, including new GSA proposed in Sarafrazi et al. (2011), while an operator called “Disruption”, originating from astrophysics was proposed to improve the exploration and exploitation abilities of GSA. Based on the discussion above, this paper is motivated by the following questions: excellent clustering algorithms, like NFCRMA, and global optimization method are all powerful tool for identification of T–S fuzzy model, will the combination of NFCRMA and global optimization produces better fuzzy modeling results; given the fact that GAs and TSA are relatively fundamental global optimization methods, will the implement of more effective global optimization methods, like chaotic GSA, improves the fuzzy modeling performance? For the discussed issues, clustering algorithm and global search algorithm for T–S fuzzy model identification, we have done some relative works. From our previous works on T–S fuzzy model identification methods based on clustering techniques and works on advanced heuristic search methods, the motivation of this paper may be clearer. In the researches of T–S fuzzy model identification, we proposed the NFCRMA for T–S fuzzy modeling (Li et al., 2009), and discovered that clustering models with hyperplane prototypes could improve the performance of T–S fuzzy model identification. In Li et al. (2012), we proposed another kind of clustering model with hyper-plane prototypes and solved this model by GSA, for the model could not been solved by Lagrange method of multipliers. In the researches of GSA, we noticed GSA was an excellent global search technique, and tried to improve GSA by combining the moving strategy of agents of PSO and GSA (Li and Zhou, 2011), and by implementing chaotic search to refine the solution obtained by GSA (Li et al., 2012). Based on our previous works, it is believed that the combination of suitable
clustering model and an excellent global search algorithm could improve the performance of T–S fuzzy model identification. In this paper, an evolving T–S fuzzy model based on a novel CGSA and NFCRMA is developed. In this approach, NFCRMA is used to partition fuzzy space and obtain the initial MF parameters, and then, MF parameters of all rules are encoded in one agent position and optimized by the proposed CGSA. In this CGSA, chaotic search is embedded in the iterations of GSA to refine the current best solution of GSA. The evaluation method used in parameter optimization considers the most important evaluation criterion accuracy, in terms of the mean square error (MSE) between the predicted output and the target output. The CGSA proposed in this paper is inspired by the existing works of CGSAs (Li et al., 2012; Han and Chang, 2012), however the differences between them are obvious. In Han and Chang (2012), chaotic number, not chaotic search, was used to improve GSA. In Li et al. (2012), we used chaotic search to refine the best solution after the end of GSA optimization, while in this paper, chaotic search is embedded in the searching process of GSA for seeking and replacing the current best solution in each iteration of GSA. The rest of this paper is organized as follows: in Section 2, T–S fuzzy model, the NFCRMA for fuzzy space partition and the related parameter identification methods are introduced; in Section 3, the CGSA is proposed and the optimization scheme for fuzzy model is designed; in Section 4, the proposed approach is validated by benchmark problem of fuzzy modeling; in Section 5, the proposed approach is applied in fuzzy model identification for the HTGS of a hydroelectric generating unit of Zhelin hydropower station in Jiangxi Province of China; finally, the conclusion is drawn in Section 6.
2. T–S fuzzy model identification with NFCRMA 2.1. T–S fuzzy model Takagi and Sugeno proposed the well-known T–S fuzzy model in Takagi and Sugeno (1985) to describe complicated nonlinear system. The T–S fuzzy model of this system can be described by the following IF-THEN fuzzy rules: I Rule i: IF x1 is A1i and…and xNI is AN i THEN I yi ¼ p0i þ p1i x1 þ ⋯ þ pN i xN I
ð1Þ
where i¼ 1,2,…,NR, NR is the number of fuzzy rule, x ¼ ½x1 ; x2 ; …; xNI is the system input, NI is the dimension of input vector, yi is the ith output, and pji ðj ¼ 1; …; N I Þ is the consequent parameter of the ith output. The final output of T–S fuzzy model can be expressed by a weighted mean defuzzification as follows: ∑N¼R 1 wi yi _ y ¼ iN ∑i ¼R 1 wi
ð2Þ
where the weight wi represents the overall truth value of the premise of the jth implication for the input which can be calculated as: NI
wi ¼ ∏ μðAji Þ
ð3Þ
j¼1
where μðAji Þ is the grade of membership function, which can be described by a bell-shaped Gaussian function as: 0 !2 1 xj −cji A j @ ; i ¼ 1; …; NR ; j ¼ 1; …; N I μðAi Þ ¼ exp − ð4Þ sji where cji and sji represent the center and width of the membership function respectively.
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2.2. Novel fuzzy c-regressive model clustering algorithm There have been many researches applying FCM to fuzzy modeling, but the clusters developed by FCM algorithm are hyper-sphere-shaped, not hyper-plane-shaped. Since hyperplane-shaped clusters are more suitable to be used in input– output fuzzy modeling, fuzzy c-regressive model clustering algorithm has been adopted to partition the input–output data pairs into hyper-plane-shaped clusters in fuzzy modeling (Kim et al., 1997; Kung and Su, 2007). In T–S fuzzy modeling, we assume that the N data pairs ðxk ; yk Þ ðk ¼ 1; …; NÞ are sampled, while xk is the input vector and yk is the output. In order to get NR rules, those data pairs are grouped in NR clusters, while the data samples in ith cluster are accorded with a linear regression model, which is actually a hyperplane function, that is i
i
yk ¼ f ðxk ; θi Þ ¼ ai1 xk1 þ ai2 xk2 þ ⋯ þ aiNI xkNI þ b0 ¼ ½xk 1⋅θTi
ð5Þ
where i¼ 1,2,…NR, xk ¼ ½xk1 ; …; xkNI is the kth input vector and i θi ¼ ½ai1 ; …; aiM ; b0 is the parameter vector of linear model. The distance (measure of error) of every sampled ðxk ; yk Þ to the ith regression model with parameter θi is defined as follows: dik ðθi Þ ¼ jyk −½xk 1⋅θTi j
ð6Þ
The objective function of FCRM is defined as: NR
N
J m ðU; θÞ ¼ ∑ ∑ ðμik Þmf ðdik ðθi ÞÞ2
ð7Þ
Once the antecedent fuzzy set parameters have been identified, the consequent parameters can be obtained from the following matrix equation, while all NS input–output data for fuzzy identification have been calculated according to Eq. (2): y ¼ Ap
NR
∑ μik ¼ 1;
i¼1
k ¼ 1; 2; …; N
ð8Þ
As deduced in Li et al. (2009), the analytical solution of Eq. (7) could be expressed by the following equations: μik ¼
θij ¼
1 2=ðmf −1Þ R ∑N j ¼ 1 ½ðdik ðθ i ÞÞ=ðdjk ðθ j ÞÞ
Σ nk ¼ 1 ðμik Þmf ðyk −Σ t≠j θit ⋅x^ kt Þ⋅x^ kj 2
Σ nk ¼ 1 ðμik Þmf ⋅x^ kj
ð9Þ
ð10Þ
By iterating Eqs. (9) and (10), the fuzzy membership degree μik and affine line model parameter θij will vary towards the direction that minimizes J m ðU; θÞ gradually. The specific steps of NFCRMA are shown in Li et al. (2009). 2.3. Model identification We use the NFCRMA to separate the input–output space, and obtain the membership functions of the fuzzy sets in the antecedent part of each fuzzy rule, as described by Eq. (4). As mentioned in Kim et al. (1997), in view of this equation, the fuzzy set centers cji ð1 ≤i ≤N R ; 1 ≤j ≤N I Þ and the respective standard deviations sji can be easily obtained by using μik, the membership of ðxk ; yk Þ belonging to ith clustering representing hyper-plane, as follows: cji
¼
ΣN k ¼ 1 μik ⋅xkj ΣN k ¼ 1 μik
; i ¼ 1; …; N R ; j ¼ 1; …; NI
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u2Σ μik ⋅ðxkj −cji Þ2 j ; i ¼ 1; …; N R ; j ¼ 1; …; NI si ¼ t k ¼ 1N Σ k ¼ 1 μik
ð11Þ
ð13Þ
NI 0 I where p ¼ ½p01 ; …; pN 1 ; …; pN R ; …; pN R , x k ¼ ½xk1 ; xk2 ; …; xkN I is the kth input vector of fuzzy model, y ¼ ½y1 ; y2 ; …; yNS T is the target system output, A is the coefficient matrix, which can be calculated according to Eqs. (2)–(4). In this paper, the least square method (LSM) is used to solve the matrix equation in Eq. (13) to identify the consequent parameter vector p.
3. T–S fuzzy model optimization based on CGSA 3.1. Chaotic gravitational search algorithm GSA is a newly developed stochastic search algorithm based on the law of gravity and mass interactions. In GSA, the search agents are a collection of masses which interact with each other based on the Newtonian gravity and the laws of motion (Rashedi et al., 2009). Assumed there are N agents (masses), position of the ith agent 1 d n is Li ¼ ðli ; …; li ; …; li Þ; i ¼ 1; …; N. Inertial mass is calculated according to objective function value of the agent as:
k¼1i¼1
where mf ∈ð1; ∞Þ is the fuzzy weighting exponent; the value of mf is usually set as 2, μik ∈½0; 1 is the fuzzy membership degree of the kth data pair belonging to the ith cluster. It is assumed that μik is constrained with the following equation:
mi ¼
obji −worst best−worst
ð14Þ
where obji is the objective function value of the ith agent. For optimization problem seeking minimal value, best ¼ minobjj , worst ¼ maxobjj . According to Newtonian gravitation theory, the force acting on the ith mass from the jth mass is defined: F dij ðtÞ ¼ GðtÞ
M pi ðtÞ M aj ðtÞ d d ðl ðtÞ−li ðtÞÞ jjX i ðtÞ; X j ðtÞjj2 j
ð15Þ
where Mi and Mj are masses of agents, M i ¼ mi =Σ N j ¼ 1 mj , G(t) is the gravitational constant at time t. It must be pointed out that the gravitational constant G(t) is important in determining the performance of GSA, G(t) is defined a function of time t: t GðtÞ ¼ G0 exp −β⋅ ð16Þ maxt where G0 is the initial value, β is the constant, t is the current iterations, and max_t is the maximum iterations. For the ith agent, the randomly weighted sum of the forces exerted from other agents: F di ðtÞ ¼ ∑randj F dij ðtÞ
ð17Þ
j≠i
Based on law of motion, the acceleration of the ith agent is calculated by: adi ðtÞ ¼
F di ðtÞ M ii ðtÞ
ð18Þ
where, Mii is the inertial mass of the ith agent. Then, the searching strategy on this concept can be described by following equations: vdi ðt þ 1Þ ¼ randi vdi ðtÞ þ adi ðtÞ
ð12Þ
2075
d
d
li ðt þ 1Þ ¼ li ðtÞ þ vdi ðt þ 1Þ
ð19Þ ð20Þ
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Step 4: calculate the mass of ith agent mi by Eq. (14), i¼ 1,…,N. Step 5: calculate the gravitational constant G by Eq. (16). Step 6: calculate the gravitation of ith agent that other agent act on by Eqs. (15) and (17), i ¼1,…,N. Step 7: calculate the acceleration ai for ith agent by Eq. (18). Step 8: update the velocity and position for all agents according to Eqs. (19) and (20). Step 9: in order to improve the computing efficiency, employ chaotic search with little probability in the former phase of evolvement due to quick convergence of GSA, and use chaotic search in all probability in the latter phase of evolvement. The probability to progress chaotic search may denote as follows: p ¼ 1−1=ð1−lnðtÞÞ. Generate a random number r between (0, 1) interval, if r op, then go to Step 11 to start the chaotic search, else go to Step 18. Step 10: start the chaotic search around the current best solution Lbest, and ICMIC map is used to illuminate the process of chaotic search. randomly initialize n chaotic variables cxðkÞ ∈ d ½−1; 0Þ∪ð0; 1; d ¼ 1; 2; …; n; Xbest ¼Lbest ¼ (pg,1,pg,d,pg,n); Fbest ¼fg； set iteration counter k¼ 0 for chaotic search, and the maximum number of chaotic search is Cmax.
In above equations, li represents the position of ith agent in dth dimension, vdi is the velocity, adi is the acceleration, randi is a random number among [0, 1]. To improve the performance of GSA, the chaotic system is incorporated into GSA by means of utilizing the chaotic search as a local search procedure of GSA. Based on the definition of chaotic map and illumination of chaotic search, the CGSA is proposed. A chaotic map is a discrete-time dynamical system that could be expressed as: cxðnþ1Þ ¼ f ðcxðnÞ Þ
ð21Þ (0)
By giving initial state cx , the chaotic sequence could be gotten through running the chaotic map. It has been reported in He et al. (2009, 2000) that chaotic map with infinite collapses (ICMIC) map with a symmetrical region ½−1; 0Þ∪ð0; 1 shows advantage in chaotic search. The ICMIC map is described by following equation: a cxðkþ1Þ ¼ sin for a 4 0; cxðkÞ∈½−1; 0Þ∪ð0; 1 ð22Þ cxðkÞ This paper proposes a new chaotic gravitational search algorithm (CGSA) by introducing chaotic maps with ergodicity, irregularity and the stochastic property in GSA to improve the global convergence by escaping from the local solutions. The chaotic search is based on chaotic sequences that generated by ICMIC map. Chaotic search has been proved to be effective of improving the searching ability of heuristic optimization by numerous experiments. The procedure of this novel approach can be summarized as follows: Step 1: initialize the positions for a population with N agents d L ¼ fli ; i ¼ 1; …; N; d ¼ 1; …; ngrandomly; initialize velocity of agents V¼ {vij, i¼ 1,…,N, d ¼1,…,n}, set vij ¼0; set global optimal solution Lbest; the global optimal objective function value fg; set the iteration counter t¼ 0, the maximum number of iteration Tmax. Step 2: calculate the objective function value for N agents, fi ¼ obj(Li), i ¼1,…,N. Step 3: refresh fg and Lbest, for a minimization problem, if fi ofg, then fg ¼fi and Lbest ¼Li.
Table 1 Comparison of different models for the Box and Jenkins examplea. Model
No. of rules
MSE
Box and Jenkins (1970) Sugeno and Yasukawa (1993) Evsukoff et al. (2002) Kim et al. (1997) Kim et al. (1998) Kung and Su (2007) Bagis (2008) Du and Zhang (2008) Zhao et al. (2010) Su and Yang (2011) Su et al. (2012) Li et al. (2012) Li et al. (2009) Our model
6 3 2 6 6 2 4 4 3.96 4 3 3 4 4
0.202 0.19 0.090 0.062 0.055 0.0518 0.148 0.06 0.1428 0.112 0.0548 0.0534 0.0498 0.035
a
In literature Zhao et al. (2010), number of rules is a mean value.
70 Target Prediction
y (t)
65 60 55 50 45 0
50
100
150 t
200
250
300
0
50
100
150 t
200
250
300
1.5 1 Error
0.5 0 -0.5 -1 -1.5
Fig. 1. Comparison of prediction of our model and the target output for Box and Jenkins example.
C. Li et al. / Engineering Applications of Artificial Intelligence 26 (2013) 2073–2082
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10 Training
Target Prediction
Testing
y (t)
5
0
0
100
200
300
400
500 t
600
700
800
900
1000
0
100
200
300
400
500 t
600
700
800
900
1000
0.1
Error
0.05 0 -0.05 -0.1
Fig. 2. Comparison of prediction of our model and the target output for the nonlinear differential equation example references.
Step 11: adjust the search radius δðkÞ , and optimization varid ables interval (xmin d xmax d), and xmaxd ¼ pg;d þ δðkÞ =2, xmini ¼ d pg;d −δðkÞ =2; d Step 12: generate the chaotic variables cxdðkþ1Þ for the next iteration according to Eq. (22). Step 13: project chaotic variables to optimization intervals (xmin i, xmax i) and get optimization variables: zdðkþ1Þ ¼
xmaxd þ xmind xmaxd −xmind ðkÞ þ cxd 2 2
Step 14: evaluate the new solution Z kþ1 ¼ ½z1ðkþ1Þ ; …; zdðkþ1Þ ; ðkþ1Þ …; zm by calculating objective function value f(Zk+1). Step 15: if f(Zk+1)oFbest, then Xbest ¼Zk+1 and Fbest ¼f(Zk+1). Step 16: decrease the searching radius δdðkþ1Þ ¼ w δðkÞ ; d 0 o w o1. Step 17: k¼ k+1; if k4 Cmax, go out of chaotic search and go to Step 19, otherwise go back to step 12. Step 18: renew the current best solution, Lbest ¼Xbest, fg ¼Fbest. Step 19: t ¼t+1; if toTmax, go to Step 2, or else go out of the algorithm, and return the best solution Lbest, together with the best objective function value fg.
3.2. Encoding scheme for T–S fuzzy model In order to optimize the fuzzy model by CGSA, the optimization variables should be selected and encoded in the position vector of agents. Because the consequent parameters can be calculated by least square method, it is only necessary to encode antecedent MF parameters of centers and widths, which are necessary to represent a Gaussian MF, into the position vector. Assume the number of rules is NR, the number of input is NI, and there are total NR NI optimization variables that have to be encoded. The position of kth agent dimension is a NR NI dimension vector, which can be described as: NI NI NI 1 1 1 I Lk ¼ ½c11 ; …; cN 1 ; s1 ; …; s1 ; …; cNR ; …; cNR ; sN R ; …; sN R I where cN 1 1 first rule; s1
c11
ð23Þ
represent the MF parameters of Gaussian center of the I sN 1 represent the MF parameters of Gaussian width of NI 1 I the first rule; c1NR cN N R and sNR sN R are corresponding MF parameters for the NR rule.
3.3. Optimization of T–S fuzzy model The CGSA is used to optimize and adjust the MF parameters around the parameters of the fuzzy model. The initialization of positions of agents in CGSA is depended on the MF parameters obtained by NFCRMA. Assume that the MF parameters identified by NFCRMA are expressed as vector Lopt, and then boundaries Lmin and Lmax could be set to make sure the center is Lopt. The positions of agents are randomly generated in the boundaries of Lmin and Lmax. The object function for optimization of T–S fuzzy model is defined based on the mean square error (MSE) between the predicted output of fuzzy model and target output: J MSE ¼
1 NS ∑ ðy −y^ Þ2 NS k ¼ 1 k k
ð24Þ
where NS is the data length, y^ k the predicted output and yk the target output. The proposed approach for fuzzy modeling in this paper could be summarized as: (1) Prepare input and output data of the target system, select NI inputs for fuzzy model, set number of rules NR. NS data pairs ðxk ; yk Þ are prepared for fuzzy modeling, while xk ¼ ðxk1 ; …; xkNI Þ is the input of fuzzy model. (2) NFCRMA is used to partition the dataset xk (k ¼1,…,NS) into NR clusters, while fuzzy membership μik (i¼1,…,NR, k¼ 1,…,NS) is obtained, then the MF parameters cji and sji is calculated by Eqs. (11) and (12), get Lopt according to Eq. (23). (3) Set the population size N, maximum iteration number Nitg, gravitational constant function parameter G0 and β, and chaotic search number Nitc for CGSA; initialize population of agents randomly between Lmin and Lmax. (4) Start the iteration of CGSA, the specific step is shown above. For agents the objective function is calculated as following steps: decode the location vector to obtain cji and sji ; calculate μðAji Þ and wi for xk; calculate coefficient matrix A, and then solve Eq. (13) by LSM to get consequent parameters vector p; calculate model output y^ ¼ Ap, calculate objective function by Eq. (24).
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4.2. A nonlinear differential equation
(5) Get the best solution Lbest and the corresponding MSE; decoding Lbest to get MF parameters.
4. Applications to benchmark problem
This example is taken from Evsukoff et al. (2002) in which the plant to be identified is given by the second-order highly nonlinear difference equation
4.1. Box–Jenkins system
yðkÞ ¼
In this subsection, we use the Box–Jenkins data set (Box and Jenkins, 1970), which consists of 296 input–output measurements of a gas-furnace process. At each sampling time k, the input xðkÞ of this process is the gas flow rate and the output yðkÞ is the output CO2 concentration. For comparison with conventional fuzzy models, we choose uðkÞ, uðk−1Þ, uðk−2Þ, yðk−1Þ, yðk−2Þ, yðk−3Þ as the input of the fuzzy model (Kim et al., 1997; Kung and Su, 2007). The number of fuzzy rules NR ¼4. Parameters of CGSA are set as: population size N ¼ 30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼100. The performance of our fuzzy model is shown in Fig. 1, where the prediction of our model and the target output are compared, and also the errors between prediction and target output are shown. It is noted that our model has a MSE of 0.035, better than the result identified by NFCRMA in Li et al. (2009) and other results that occur in literature. The detailed comparison of performances of different methods is given in Table 1. The fuzzy model identification approach proposed in this paper is based on NFCRMA and CGSA, the fuzzy model identified by NFCRMA is optimized by CGSA, so we expect our model in this paper performs better than that obtained by NFCRMA. The results listed in Table 1 verify that the CGSA can search better MF parameters to obtain more accurate fuzzy model.
Frequency (Hz)
54
a
52 50
46
MSE
12 8 100 4 5.6 5 4 4 4
Training data
Testing data
0.5072 0.6184 0.1577 0.0341 0.002 1.27e−4 0.0102 0.0149 8.45e−5
0.2447 0.2037 0.0185 0.0378 0.0035 1.27e−4 0.0128 0.0115 6.5e−5
0
10
20
30 Time (s)
40
50
60
0
10
20
30 Time (s)
40
50
60
0.05
0
-0.05
In literature Zhao et al. (2010), number of rules is a mean value.
Generation setpoint
Target Prediction
48
Error
Sugeno and Yasukawa (1993) Wang and Lee (1999) Evsukoff et al. (2002) Bagis (2008) Zhao et al. (2010) Su and Yang (2011) Li et al. (2012) Li et al. (2009) Our model
No. of rules
Fig. 4. Fuzzy model training of 48–52 Hz up frequency disturbance process.
Speed regulation
+
-
Unit generation
Rs Speed setpoint
Guide vane position +
+
Governor controller -
ð25Þ
Much effort has been paid to check the robustness of our fuzzy model. We use training data to build the model, and testing data to check the model. We get 500 points training data from Eq. (25), assuming a random input signal u(k) uniformly distributed in the interval [−2, 2], and 500 points testing data, applying a sinusoidal input signal uðkÞ ¼ sin ð2kπ=25Þ. The model has three inputs u(k), y(k−1), y(k−2), and a single output y(k). After the learning process is finished, the model is tested by applying testing data to drive it. The rule number of the fuzzy model is four. Parameters of CGSA are set as: population size N ¼30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼ 100. The prediction of our model and the target system are shown in Fig. 2, while outputs comparison and the respective errors are exhibited. Fig. 2 reflects both the training process and testing process of our fuzzy model. It is easy to find out that our model match well with the target system and the errors of both training
Table 2 Comparison of different models for the nonlinear differential equation examplea. Model
yðk−1Þ⋅yðk−2Þ⋅ðyðk−1Þ þ 2:5Þ þ uðkÞ 1 þ y2 ðk−1Þ þ y2 ðk−2Þ
Turbine control actuator
Turbine & water column
Generator
Unit speed Fig. 3. Hydraulic turbine governing system with speed regulation.
Power output to grid
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Frequency (Hz)
54 52 Target Prediction
50 48 46 0
5
10
15
20
25 Time (s)
30
35
40
45
50
0
5
10
15
20
25 Time (s)
30
35
40
45
50
1
Error
0.5 0 -0.5 -1
Fig. 5. Fuzzy model testing by 48–52 Hz up frequency disturbance process.
Frequency (Hz)
54 Target Prediction
52 50 48 46 0
5
10
15
20 25 Time (s)
30
35
40
45
0
5
10
15
20 25 Time (s)
30
35
40
45
0.1
Error
0.05 0 -0.05 -0.1
Fig. 6. Fuzzy model training of 52–48 Hz down frequency disturbance process.
and testing are puny. The MSE of training process is as small as 8.45e−5, and MSE for testing data is 6.5e−5. In order to show the effectiveness of the proposed approach, the MSE of our model is compared with different models in Table 2. The results listed in Table 2 show that the modeling accuracy has been improved significantly through the efforts we make in this paper. 4.3. Comparison with related works As the development of optimization techniques, some excellent global optimization algorithms have been applied in T–S fuzzy model identification (Du and Zhang, 2008; Bagis, 2008; Zhao et al., 2010; Su and Yang, 2011; Su et al., 2012). There are two ways to
apply optimization algorithms in T–S fuzzy model identification, namely direct way and indirect way. For direct way, the structure or parameters of T–S fuzzy model are optimized directly. For example, in Du and Zhang (2008), the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the MF parameters of the T–S fuzzy model were all represented in one chromosome and evolved together by genetic algorithm. And TSA (Bagis, 2008), CRPSO (Zhao et al., 2010) and QDE (Su and Yang, 2011) were all used to identify the T–S fuzzy model directly. For indirect way, optimization algorithm can be used to solve clustering model and thus enhancing the fuzzy space partition performance for T–S fuzzy model identification. For example, the artificial bee colony (ABC) algorithm (Su et al., 2012) was used to optimize clustering model of
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FCM, and then the new version of fuzzy c-means clustering technique (VABC-FCM) was applied in T–S fuzzy identification. In Li et al. (2012), we propose a hyper-plane clustering algorithm based on GSA, and apply this technique in T–S fuzzy model identification. In this paper, the CGSA are used to optimize MF parameters of T–S fuzzy model. However, the usage of optimization technique is different with those in literature. We use NFCRMA to identify T–S fuzzy model at first, then apply CGSA to optimize the identified model. We can analyze the identification results listed in Tables 1 and 2. For identification of Box and Jenkins system, fuzzy models identified by indirect optimization techniques (Su et al., 2012; Li et al., 2012) achieve better performance than those identified by direct optimization techniques (Du and Zhang, 2008; Bagis, 2008; Zhao et al., 2010; Su and Yang, 2011). T–S fuzzy model is complicated and there are many parameters that need optimization. This makes the direct optimization of T–S fuzzy model a difficult task. If we can get a coarse result at first and set a search range for optimization technique, the optimization will be very effective. By comparing identification results listed in Tables 1 and 2, the outstanding performances of our approach in this paper can be confirmed.
5. Application to hydraulic turbine governing system Hydraulic turbine governing system (HTGS) is a complicated nonlinear system, mainly containing four parts, namely governor controller, turbine control actuator, turbine and water column, and generator. Fig. 3 shows the schematic of the overall system. HTGS is a control system of hydroelectric generating unit (HGU) that governs the speed of turbine according to the setpoint of output power and setpoint of speed. The main task of HTGS is to adjust the power output to grid and to track the frequency of grid. In general, the HTGS should be given a comprehensive test and performance evaluation after installation and overhaul of HGU. The modeling and simulation of HTGS is important for it provide a way to test and evaluate the governing system of HGU. Hydraulic turbine is the key component in HTGS, and it is a very complicated nonlinear system, which do not has any analytic
expression until now. However, it is usually described as moment function and flow function of guide vane position y, generator unit speed x and water head h, shown as following: (
mt ¼ f 1 ðy; x; hÞ
ð26Þ
q ¼ f 2 ðy; x; hÞ
Although the analytic expression of nonlinear functions f 1 ð⋅Þ and f 2 ð⋅Þ are unknown, moment and flow characteristic curves are provide by manufacturer through experiments. Based on the characteristic curve of turbine, we can calculate moment and flow of the hydraulic turbine at certain time by means of interpolation or nonlinear fitting. Neural network has been applied in modeling of hydraulic turbine by fitting the characteristic curves of moment and flow (Chen et al., 2010). In this paper, we try to build the fuzzy model of hydraulic turbine for accurate modeling of HTGS of the fourth generating unit (4F) of Zhelin hydropower station, in Jiangxi province of China. In order to build the model for turbine and generator, output of governor actuator and generator should be collected. After the overhaul of HGU, the HTGS should be tested in a series experiments, including frequency disturbance experiment under no-load condition. In this experiment, the setpoint frequency is changed suddenly, and then the governor of HTGS controls the frequency (speed) of the unit by adjusting the guide vane position. We could sample signal of guide vane position y and frequency of generator x for fuzzy modeling. Frequency disturbance experiments of 4F unit were recorded and unit frequency x, guide vane position y were collected. There are two types of frequency disturbance, up frequency disturbance and down frequency disturbance. In experiment of fuzzy modeling, we designed three groups of experiments. In the first group, a 48–52 Hz up frequency disturbance process was used for training of fuzzy model, while another 48–52 Hz up frequency disturbance process was used for model testing. In the second group, a 52–48 Hz down frequency disturbance process was used for training, with a 50–48 Hz down frequency disturbance process for testing. In the third group, a 48–52 Hz up frequency disturbance process was used for training, with a 52–48 Hz down frequency disturbance process for testing.
Frequency (Hz)
51 Target Prediction
50 49 48 47 10
15
20
25
30 35 Time (s)
40
45
50
55
15
20
25
30 35 Time (s)
40
45
50
55
Error
0.5
0
-0.5 10
Fig. 7. Fuzzy model training of 50–48 Hz down frequency disturbance process.
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Frequency (Hz)
54 52 Target Prediction
50 48 46 0
10
20
30
40 Time (s)
50
60
70
0
10
20
30 40 Time (s)
50
60
70
0.1
Error
0.05 0 -0.05 -0.1
Fig. 8. Fuzzy model training of 48–52 Hz up frequency disturbance process.
54 Frequency (Hz)
Target 52
Prediction
50 48 46 0
10
20
30 Time (s)
40
50
60
0
10
20
30 Time (s)
40
50
60
Error
0.5
0
-0.5
Fig. 9. Fuzzy model training of 52–48 Hz down frequency disturbance process.
The sampling rate is 50 Hz. y(k−1), x(k−1), x(k−6) Are selected as input of fuzzy model. The number of fuzzy rules NR ¼3, parameters of CGSA is set as: population size N ¼30, maximum iteration number Nitg ¼200, gravitational constant function parameter G0 ¼20 and β ¼8, chaotic search number Nitc ¼ 100. In the first group, output of both the fuzzy model and the target system are shown in Fig. 4 while outputs comparison and the respective errors are exhibited. The trained fuzzy model possesses high accuracy with a MSE of 3.38e−5. The fuzzy model is verified by testing data, the output of the trained fuzzy model almost coincide with the target system, as shown in Fig. 5. The MSE between prediction of fuzzy model and target is as small as 0.00534. In the second group, output of both the fuzzy model and the target system are shown in Fig. 6 while outputs comparison and the respective errors are exhibited. The trained fuzzy model
possesses high accuracy with a MSE of 3.83e−5. Fig. 7 shows the testing result of the trained fuzzy model, while another down frequency disturbance process is used for target. When checked by testing data, the fuzzy model is found out to match well with the actual system output with a MSE of 0.0017. In the third group, output of both the fuzzy model and the target system are shown in Fig. 8 while outputs comparison and the respective errors are exhibited. The trained fuzzy model possesses high accuracy with a MSE of 6.41e−5. Fig. 9 shows the validation of the trained fuzzy model. It is found that our fuzzy model could predict the system output accurately, with a MSE of 0.0008. In this example, the up frequency disturbance process is used to train the model, while down frequency distance process used for testing. The prediction result proves the generalization ability of the trained fuzzy model.
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6. Conclusions In this paper, a new fuzzy model identification approach based on a novel fuzzy c-regression clustering algorithm (NFCRMA) and chaotic gravitational search algorithm (CGSA) is presented for the accurate modeling of hydraulic turbine governing system. In this approach, NFCRMA is used to partition the input–output space of fuzzy model into clusters with hyper-plane prototype and extract membership function (MF) parameters of different rule, and then a CGSA which embeds chaotic search in gravitational search is proposed to optimize the MF parameters of fuzzy model. Benchmark problems of nonlinear system modeling have shown that the proposed modeling approach improves the modeling accuracy significantly by comparison with previous results existing in literature. Finally this approach is applied to fuzzy modeling of HTGS of a HGU of Zhelin Hydropower station in Jiangxi Province of China, and the results demonstrate the effectiveness and accuracy of the proposed approach.
Acknowledgments This paper is supported by the National Natural Science Foundation of China (No. 51109088), the Research Fund for the Doctoral Program of Higher Education of China (No. 20110142120020) and the Fundamental Research Funds for the Central Universities, HUST (No. 2013QN114). References Alatas, B., Akin, E., Ozer, A.B., 2009. Chaos embedded particle swarm optimization algorithms. Chaos Solitons Fract. 40 (4), 1715–1734. Bagis, A., 2008. Fuzzy rule base design using tabu search algorithm for nonlinear system modeling. ISA Trans. 47 (1), 32–44. Box, G.E.P., Jenkins, G.M., 1970. Time Series Analysis, Forecasting and Control. Holden Day, San Francisco, CA. Chen, G., Liu, Y., Meng, Z. Development of a nonlinear real-time simulation system for hydroelectric generating unit. In: Proceedings of 2010 Power and Energy Engineering Conference, pp. 1–4. Du, H., Zhang, N., 2008. Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification. Appl. Soft Comput. 8 (1), 676–686. Duman, Serhat, Güvenç, Uğur, Sönmez, Yusuf, Yörükeren, Nuran, 2012. Optimal power flow using gravitational search algorithm. Energy Convers. Manage. 59, 86–95.
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