Direct modeling method of generalized Hamiltonian system and simulation simplified

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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 31 (2012) 901 – 908

www.elsevier.com/locate/procedia

2011 2nd International Conference on Advances in Energy Engineering

Direct modeling method of generalized Hamiltonian system and simulation simplified Tianmao Xua*, Yun Zengb, Lixiang Zhangb, Jing Qianc a Computing Center, Kunming University of Science and Technology, Kunming 650500, China Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming 650500, China c College of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China

b

Abstract Generalized Hamiltonian model can give dynamics mechanism of inner parameters connected, however, it will meet many difficult while an actual physical system is converted to Generalized Hamiltonian model. And due to complexity of the structure matrix and damping matrix of high order Hamiltonian system, analysis and application to inner parameters connected mechanism is not ease and convenience. In this paper one kind of Hamiltonian model, four order and single input affine nonlinear system, is given basic form, it is a formulation formulize form, that is generalize Hamiltonian model can be directly derived by formulate calculation. The hydro turbine Hamiltonian model is taken as case to introduce this modeling method. At last, a simulation simplified method based-physical background is proposed, and used to simplify the hydro turbine Hamiltonian model.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: geneeralized Hamiltonian model; direct modeling; hydro turbine; simulation simplified

1. Introductions The generalized Hamiltonian model, in contrast traditional differential equations model, their differential part is equivalent, but Hamiltonian function, energy flow and inner relative mechanism reflected by system matrix provide more dynamic information about object system. In recently, the generalized Hamiltonian theory could be used to describe practical system with energy dissipation in

*

Corresponding author. Tel: +86-0871-5916105; fax: +86-0871-5916108. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.1119

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inner and energy exchanging with outside environment in [1-2]. However, the complexity of inner connected mechanism result in structure and damping of high order Hamiltonian model more complex in form, and rigorous mathematical analysis means for its mechanism is lack or undeveloped in currently. High order Hamiltonian model can only simple analyze by means of connected factors form of structure and damping matrix in [3]. In modelling theory for generalized Hamiltonian, the direct method is that Hamiltonian model is derived in mathematic deduce from basic dynamic characteristic of practical system. But, it is difficult and even impossible for most actual system, so the generalized Hamiltonian realization theory is developed, which transforms the differential equation into Hamiltonian model. Hamiltonian realization methods for many types physical system are developed, such as linear system [4], nonlinear system [5-6], time varying and time invariant [7-8], discrete system [9-10] and differential algebraic system [11-12], and so on. However, applied these theories and methods need good well mathematical theoretical basis, it may be a main wide gap while engineering discipline expert and scholar apply the generalized Hamiltonian theory. Based on above reasons, this paper gives a direct modelling method of generalized Hamiltonian system to build the bridge between theory and engineering application. The main works include, give a type of generalized Hamiltonian model standard form from our prior works and modelling steps by direct calculation. An actual example is used to introduce the modelling procedure. A simplified method basedsimulation is proposed, which is suit for high order and complex Hamiltonian model, and is used to simplify built Hamiltonian model. 2. Four-order generalized Hamiltonian model Simple input, four-order affine nonlinear system is follow. 

x  f ( x)  g ( x)u

(1)

Where u is the input control, g(x) is the input matrix, and its form is equation (2). g  0 0 0 g 4 

T

(2)

Hamiltonian function is H(x). According to Ref. [6], system (1) can be converted into generalized Hamiltonian model by orthogonal decomposition realization method, and its general form is follow: 

x  J ( x )  R( x )

J ( x) 

1 H

2

New control is up:

 0  J  12   J 13    J 14

J 12 0  J 23  J 24

J 13 J 23 0  J 34

H  g ( x )u p x

J 14  J 24  ; J 34   0 

(3)

0 0 0  r ( x )  0 r( x ) 0 0  R( x )   0 0 r( x ) 0   0 0 0 r ( x ) 

up  u  s H

Where J 12   x2 Hf1   x1 Hf 2 , J 13   x3 Hf 1   x1 Hf 3 , J 23   x3 Hf 2   x2 Hf 3 , sH  J 14   x4 Hf 1   x1 Hf 4   x1 Hg 4 s H , J 24   x4 Hf 2   x2 Hf 4   x2 Hg 4 s H ,

(4)

1 2 s( x ) H , Lg H

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J 34   x4 Hf 3   x3 Hf 4   x 3 Hg 4 s H ,

H

2



 

 



2 2 2 2  x1 H  x2 H  x3 H  x 4 H  , r(x) and s(x) 2

can be derived by decomposed Z  f ( x ), H / H , which determine system damping matrix form, Lg H  g (x ), H is the Lie derivative, up is new equivalent input control.

Original differential equation model (1) could be obtained if system (3) is expanded, which verifies the derivation process whether is correct. Although their differential equation are equivalent, but structure matrix J(x) reflects inner relative mechanism of parameters, damping matrix R(x) reflects damping characteristic on port, and input matrix g(x) reflects the relationship mechanism between system and outside environment. Thus, generalized Hamiltonian system gives the dynamic mechanism of system motion, and includes more information. Now, this kind of affine nonlinear system can be transformed into generalized Hamiltonian model following below steps: Step 1: According to physical system feature select Hamiltonian function H(x). Step 2: Decomposed Z  f ( x ), H / H

2

to obtain r(x) and s(x), and energy flow of dissipative

system are consistent with that of actual system. Step 3: Calculation Lie derivative by equation Lg H  g (x ), H . Step 4: Calculation new control up. Step 5: All above calculation results are substituted into (3)-(4) to obtain Hamiltonian model. 3. Hydro turbine Hamiltonian model 3.1. Differential equation of hydro turbine Single penstock single machine, elastic water column, hydraulic system differential equation is follow: 

x  f ( x)  g ( x)u x2   2  2  1 g  2 x1  [h0  ( f p  r2 ) x32 ] 3 x4 Z n Te  Te  2 ； f ( x)   4 gr 2 2 [h0  ( f p  2 ) x3 ]    3 x1  Z nTe x4   1    ( x4  g 0 )   T y  

(5)

0 0   g( x)   0  1 T   y

Where x1 and x2 are intermediate variable, x3=q, x4=g, q is the hydro turbine flow in per unit, g is the guide vane opening in per unit, g0 and gr is the initial and rated of guide vane opening in per unit, correspondingly, h0 is static head of hydraulic power plant in per unit, fp is the head loss coefficient in penstock, Zn is the surge impedance of penstock in per unit, Te is the elastic time constant in second, Ty is the main servomotor time constant in second. Hydraulic system parameters can be calculated from equation (5) at first, then calculates the turbine power by power model of hydro turbine. In this paper, power model of hydro turbine with inner energy loss will be used, it is given in [13]:

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2 2 g r2 x32 (6) ( x3  qnl )  At ' k h ' ( x33  qnl3 )  pz 0 [e n1 ( qz  x3 )  e n1 ( qz qnl ) ] 2 x4 Where pt is power of hydro turbine in per unit, γ=9.81, qnl is no-load flow of hydro turbine in per unit, qz is flow at maximum efficiency in some head in per unit, At´ is gain coefficient, kh´ is head loss coefficient of hydro turbine channel, Δpz0 is impact loss in per unit, n1 is coefficient of impact loss. The second term in above (6) is hydraulic loss power of hydro turbine channel, the third term is impact loss power in runner district.

pt  At ' 

3.2. Hamiltonian modeling Hydro turbine system (5) satisfies the form of system (1), so its Hamiltonian model can be built by using method introduced in section 2. Step 1: selected Hamiltonian function To easy connect with generator, the nature output of Hamiltonian system should be near to that turbine’s. Based above this reasonable, we select Hamiltonian function of hydro turbine as follow: 2 2 g r2 x32 1 ( x3  qnl )  Ty At ' k h ' ( x33  qnl3 ) x4  Ty pz 0 [e n1 ( qz  x3 )  e n1 ( qz qnl ) ]x4  ( x12  x22 ) x4 2 (7) Last item denote energy produced by intermediate variable in transient. Variable x3 and x4 are flow and guide vane opening, after generating units start and connect with power system, x3>0, x4>0, and x3>qnl , thus selected Hamiltonian function is positive function. The nature output of Hamiltonian system is: H yH  g T   pt (8) x The nature output between Hamiltonian system and hydro turbine is only negative sign, and easy to connect with generator. Step 2: Dissipative decomposition 2 is that separates items with dissipative The purpose of decomposed Z  f ( x ), H / H characteristics from Z, and is denoted as r(x), another items in Z relate to inner generated energy, it is denoted as s(x). Therefore, this decomposition relies on physical background and characteristics of actual system. Follow basic expressions are given directly:

H ( x )  Ty At ' 

H H H dq H H ； ( x )  x1 ； ( x )  x2 ； f 3 ( x )  Ty pt ( x)  x1 x3 q dt x2 x4 Z can be decomposed as follow: 1 Z [ f 1 x1 H  f 2  x2 H  f 3  x3 H  f 4  x4 H ] 2 H

1



r( x ) 

H

1 H

2

2

[

d 1 2 1 2 H dq ( x1  x 2 )   x 4 p t  g 0 p t ]   r ( x )  s( x ) dt 2 2 q dt

(9)

2 2   g r2 x32 qnl  At ' k h ' ( x33  qnl3 ) x 4  x 4 pz 0 [e n1 ( qz  x3 )  e n1 ( qz qnl ) ]  g 0 p t  (10)  At '  x4  

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s( x ) 

1 H

2

  H dq g 2 x3 d 1 1  At '  r 3  ( x12  x22 )   x t q t  2 2 d d 4  

(11)

In above decomposition, the item x4pt in (9) is replaced by pt expression Eq.(6), and its dissipative items are separated. Step 3: Calculating Lie derivative Lie derivative is follow: Lg H  g ( x ), H  g 4 ( x ) x4 H   pt

(12)

Step 4: Calculating new control up  u 

1 s( x ) H Lg H

2

u

 g 2 x3 d 1 1 1  H dq  At '  r 3  ( x12  x22 )   pt  q dt x4 2 dt 2 

(13)

Step 5: All above calculation results are substituted into (3)-(4) to obtain Hamiltonian model. 4. Hamiltonian model simplified Simplifying to complex system is difficult in theory analysis according to characteristic and feature of physical system sometimes. Because many complex systems have determined practical background, they may be simulated with simulation tools. Therefore, we propose a simplification method for high order complex system, that is that obtain change characteristic and feature of parameters and sub expressions in transient with simulation method, according to their numerical feature simulate system. In order to obtain more and close to actual performances, the simulation object is composed by hydro turbine, generator, governor and excitement, which is a complete hydro turbine generating units. Taking the YEG power plant as case, initial pt=0.5(pu). After power regulation, the output power is pt=0.8(pu), changes of relative factors in structure matrix J(x) in transient are shown as Fig. 1.

Fig.1 Changes of relative factors

The relative factor J34 is more than other obviously, it is not given in Fig.1 in order to identify changes of other factors in same figure. Fig.1 is a local amplification figure to given some detail.

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Simplification steps: Step1: If relative factor value in steady and transient is far less than other’s, then it may be omitted. From Fig.1, steady value of factor J12 and J13 is zero approximately, and their transient change is also far less than other’s. So these two factors may be omitted. J12=0,

J13=0

Setp2：If sub expressions value in steady and transient is far less than other’s, then it may be omitted. Denoted : C1   x3 Hf 2 , C2   x2 Hf 3 ,

C3   x4 Hf 1 , C4   x1 Hf 4 ,

C5   x4 Hf 2 ,

C6   x2 Hf 4

C7   x4 Hf 3 , C8   x3 Hf 4 ,

C0  s( x ) H

2

  x3 Hf 3  At ' 

x H g r2 x33 , C9   1 C0 , pmTy x4

C10  

 x2 H pmTy

C0 , C11  

 x3 H pmTy

C0

In same regulation, responses of sub expressions are shown in Fig. 2.

Fig.2 Changes of sub expressions

Some sub expressions are not identified in Fig. 2, which can be omitted. They are C1, C2, C3, C4, and C6. Now, relative factors of structure matrix are simplified as follow: J12=0,

J13=0,

J 14   x1 Hg 4 s H ,

J23=0 , J 24   x4 Hf 2   x2 Hg 4 s H ,

J 34   x4 Hf 3   x3 Hf 4   x3 Hg 4 s H

Simplified hydro turbine Hamiltonian model can be written as follow: 

x  J ( x )  R( x )

H  g ( x )u p x

(14)

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J ( x) 

1 H

2

0   0   0    x1 Hg 4 s H

0

0

0 0   x4 Hf 2   x2 Hg 4 s H

0 0  J 34

 x1 Hg 4 s H

  x4 Hf 2   x2 Hg 4 s H   J 34  0 

(15)

0 0 0  r ( x )  0 r( x ) 0 0  R( x )   0 0 r( x ) 0   0 0 0 r( x ) 

(16)

 g 2 x3 d 1 1  H dq 1  At '  r 3  ( x12  x 22 )   dt 2 2 pt  q dt x4 

(17)

New control is up: up  u 

Where r(x) expression is Eq.(10). Above simplified steps give actually simplification method in various angles. In practical application, structure matrix can be simplified in different degree according to study require. 5. Conclusions This paper gives a direct modeling method of generalized Hamiltonian system based on differential equation of affine nonlinear system, which avoids from complex mathematical transforms in modelling procedure, and is easily applied by engineering scholar and expert. The Hamiltonian modelling of hydro turbine is taken as case to illustrate some key steps in this method. The simplification method with simulation proposed in this paper is universality to high order complex system. Modern mathematical theory and method emerge in endlessly, which usually exceed the knowledge field of engineering experts and scholars. Thus these new mathematical theory and method applied to actual engineering need long time and step by step. For shorting time the theory applied to engineering, the research idea in this paper is a new explore. Acknowledgements The research reported here is financially supported by the National Natural Science Foundation of China under Grant No. 50839003, 51179079. And part works is financially supported by the Natural Science Foundation of Yunnan Province No. 2008GA027. References [1] Daizhan Cheng, Zairong Xi, Qiang Lu, et al. Geometrical structure and application of generalized controlled Hamiltonian systems. Science China： Series E, 2000,30(4):341-355. [2] Van der Schaft, A J. L2-gain and passivity techniques in nonlinear control. Berlin: Springer, 1996. [3] Yun Zeng, Lixiang Zhang, Fengrong Yu, Jing Qian. The Hamiltonian Model of Generator Described with Practical Parameters, Proceedings of The CSEE, 2009, 29(27):54-58. [4] J. Basto Goncalves. Realization Theory for Hamiltonian System. SIAM J. Control and Optimization, 1987,25(1):63-73． [5] P. E. Crouch, M. Irving. Dynamical realizations of homogeneous Hamiltonian systems. SIAM J. Control and Optimization,

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