Part III. Speed control solutions for hydrogenerators
4. Cascade control of the hydrogenerator
Mannesmann-Rexroth solution [III-33] is based on a pole-allocation technique, resulting a second order with lag mathematical model, which can be approximated with a first order system.
The external part corresponds to the hydraulic subsystem and the synchronous generator coupled to the power system (chapter 2). It is accepted that the servo-system can be stabilized in the form of P1(s):
2 1 2
2 ) 1
( Ts T s
s k P
s s
s
+
= +
ς (3.4.2)
The load-type disturbance d1(t) acts at this level: the whole water column (having the height of the dam) [III-17], [III-22] is weighing on the system and modelled as a deterministic disturbance. The worst case is at 10% open blades of the turbine, the best case is when the blades are completely open. Suppose that the modifications of d1(t)=x3(t) are given by:
) ( ) ( )
( 1 10
1 t d t d t
d
Te& + = (3.4.3)
d10(t) - constant. So, the state-space mathematical model of P1 can be expressed as:
) ( )
(
) ( )
( ) ( )
(
1
0 1
t x c t y
t d L t u b t x A t x
= T
+ +
& =
(3.4.4) in the case of an additive to x1(t) - type disturbance d1(t):
[ ]
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
=
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
−
=
e T
i
e i i
T L
c T g b T T T
A
/ 1
0 0 ,
0 1 0 ,
0 0 / ,
/ 1 0 0
/ 1 0 / 1
0 0
0 0 1
2
2 (3.4.5)
The P2 part of the plant corresponds to the hydro-turbine HT (T in figure), the penstock system (PsS) and the SG connected to the PS. For normal operating regimes the simplified linearized models were synthesized in chapter 2 table 3.2.1. It is assumed that the turbine is ideal, without losses and at full load, and P2 is given by [III-17] – [III-24]:
2 / 1
) 1
( *2
*
2
w w
P sT
k sT s
P +
= − (3.4.6)
where =kw is the plant gain, Tw represents the water starting time. The HG (including the turbine) connected to the PS can be approximated with the transfer function :
* 2
kP
)
*(
*
2 s
P )
) ( (
*
*
* 2
* 2
m m
P
sT s k
P = +
α (3.4.7)
where Tm stands for the mechanical time constant of the HG-SG depending on the rotating part’s inertia. The parameter αm has the values between 0.3 ≤ αm ≤ 1.3 with its extreme values αm = 0 (before synchronization of the SG with the PS); the largest value for αm is for SGs operating connected to the PS with infinite load. In the design phase of the controller, values of αm=1 are considered. In many situations the extreme case of αm = 0 needs an alternative controller.
Connecting these subsystems, the resulting transfer function for the plant results in P2(s):
) sT )(
2 / sT 1 (
) sT 1 ( ) k
s ( P
m m w
w 2
2 + P +
= −
α (3.4.8)
Load disturbances d2(t) are induced by the power sysem and characterise the active power request. The transfer function of the plant controlled by the GPC is
) )(
2 / 1 (
) 1 ( )
2 1 ) (
( 2 2 2
m m w
w P
s s
s
p sT sT
sT k
s T s T s k
H + +
− +
= +
ς α (3.4.9)
In accordance with IEEE Committee reports [III-23], [III-24], the resulted model (3.4.9) is widely used in studies regarded to modelling speed control systems of hydrogenerators, and current approaches with this respect include suboptimal control problems solved by various control techniques. The simultaneous rejection of the independently acting disturbances d1(t) and d2(t) by a single controller can become a difficult task. This is the reason why in the presented control structure their rejection becomes an independent task for two cascade controllers.
• Disturbance rejection in the cascade control structure
Under these conditions the combined cascade control structure can be an attractive solution to ensure the control system performance enhancement. The proposed cascade control structure is presented in Figure 3.3.2. The structure contains two controllers:
- the inner, a state-feedback controller that realizes the stabilized electro-hydraulic system (EHS) and insures adequate transients and behaviour regarding disturbance d1(t),
- the main GPC controller, which deals with reference tracking and rejection of disturbance d2(t).
The stabilized EHS must ensure an aperiodic response or a slightly oscillating one, (0<< ζ <1). The dynamics is imposed by the hydraulic part of the system. In steady-state regime d1(t) can be considered almost constant or slowly variable.
In case of disadvantageous behaviour of the system (see for example [III-25], [III-35], [III-36], [III-37]) the random disturbance on the loop, d2(t), induced by the electrical PS can be characterized as an oscillating disturbance with a relatively small value for ζ, which can be considered as the output of a second order lag filter with the transfer function [III-38]:
0 2 02
02
dist(s) 2 s s
F = + +
ςω ω
ω and d2(s)=Fdist(s)v(s) (3.4.10) v(s) - step or impulse. ω0 characterizes the frequency, specific for the power system in different functioning regime. Generally the value of ω0 is not constant, but in a given point of the power system its variation range is not too large. In discrete form, they can be approximated also as random variations, described by discrete relation given in [III-38]:
) ) ( (
) ) (
( 1
1
2 t
q D
q t C
d = −− ξ (3.4.11)
where ξ(t) is a zero mean white noise.
Using spectral analysis techniques it is possible to detect the frequency around which the main disturbance is located. In the literature it is mentioned that such a disturbance filter can be tuned according to [III-38]:
) 1
)(
1 )(
1 (
1 )
( ) (
1 1
1 1
1
−
−
−
−
−
+
−
= −
q e a q
e a q
q D
q C
j
jα α (3.4.12)
where α=2πf0 h (h the sampling period, f0 – a characteristic frequency of power system [III-35]) and 0 << a <1 .
o The validation of the proposed control solution. Simulation results
The case study is dedicated to the speed control of a real-like application. The parameters of the plant in (3.4.8) and (3.4.9) according to [III-19], [III-33] are: g0 = 0.0625, Ti1 = 0.001872 , Ti2 = 0.0756 , kω = 1, Tw = 2.2 sec, αm = 1 (the SG is accepted to operate connected to the power system with infinite load), Tm = 6.8 sec.
The gain components for minimax control are calculated as presented in section 3.2.
For this calculus a Matlab program was written. As a first step, the value of the parameter γ was established using interval halving (the parameter ρ is chosen to be ρ=0.1) applied to the CARE, and the final minimal value is γ=γmin=0.028.
Solution to the CARE leads finally to the min-max gain:
[
43.706 99.311 40.879]
, =[
149.58 345.35 140.65]
= d
u K
K (3.4.13)
Frequency domain analysis can be performed in order to check the performance of the control loop. Here only the closed-loop transfer function P1(s) is presented which is further needed for the cascade control structure:
4 CI 2
10 86 . 4 s 1459 s
6 . ) 441
s (
P = + + ⋅ (3.4.14)
In its simplified form the stabilized EHS can be modelled according to (4.1.1).
Accepting the first-order Pade approximation for the dead-time element, the sub-plant P2 can be characterized by a transfer function:
s 4 .
2 e 4
) s 8 . 6 1 )(
s 1 . 1 1 (
s 2 . 2 ) 1
s (
P −
+ +
= + (3.4.15)
Since the stabilized EHS has a very small (neglectable) time constant compared to the rest of the process, and it disappears at sampling, at the design phase of the GPC outer loop only the steady-state value of P1(s) is taken into consideration, so the composed process transfer function is
PT(s)= 0.0101·P 2 (s) .
The second step is to apply GPC to the process, having the following parameters:
1 . 0 ,
1 ,
1 ,
70 ,
1 2
1 = N = Nu = = u =
N δ λ (3.4.16)
Filter T(q-1) was chosen for simplicity to be T=1. A sampling time of h=1.45 sec was chosen (so the discrete dead time results as TD=3). The R and S polynomials are computed also with a Matlab program, and the results in this case are:
2 1
1
4 3
2 1
1
z 2004 . 7 z 3389 . 23 1385 . 11 ) z ( S
z 0041 . 0 z 0095 . 0 z 0094 . 0 z 0093 . 0 183 . 0 ) z ( R
−
−
−
−
−
−
−
−
+
−
=
+ +
+ +
= (3.4.17)
The IMC structure parameters result:
4 3
2 1
3 2
1 1
1 1
W 1
1 r
z 1896 . 0 z 214 . 1 z 815 . 2 z 788 . 2 1
z 09 . 1 z 84 . 5 z 05 . 10 465 . ) 5
z ( C
) z ( T ) z ( F , ) z ( T ) z ( F
−
−
−
−
−
−
− −
−
−
−
−
+
− +
−
− +
= −
=
=
(3.4.18)
The testing through simulation of the proposed cascade control structure was performed in two different ways.
• First the inner loop was simulated as a continuous subsystem (this simulation is sustained by the real implementation solutions – analogue electronics, local microcontroller). A comparative test was made:
- the control structure using minimax controller,
- an LQ controller having tuning parameters Q=1 and R=0.01.
Zero reference signal was supposed, and the initial conditions for the state variables were chosen for:
, 1 ) 0 ( , 2 ) 0 ( , 1 ) 0
( 2 3
1 = x = x =
x (3.4.19)
The figure 3.4.2 presents the output signal’s and the states’ evolution without disturbance. An exponentially decreasing disturbance of type d1 is applied to both systems, acting as in (3.4.10). Its time evolution is depicted in figure 3.4.3. Applying this disturbance, the states and outputs are presented for both minimax and LQ control (Figure 3.4.4). It can be noticed that rejection of the disturbance is better in the case of the minimax control.
(a)
(b)
Figure 3.4.2. Outputs and states of minimax and LQ control
• Secondly, the GPC structure was tested through simulation. The following scenario was chosen: a step reference followed by two non-simultaneous disturbances, d1 – type step disturbance (acting at time moment 100 sec) having the amplitude of -1, then d2 – type disturbance (amplitude -0.05), according to the specific application and modelled with a second-order filter having the transfer function:
Figure 3.4.3. Generated disturbance d1
dist 2
s 4 . 0 s 5 . 0 1 ) 1 s (
F = + + (3.4.20)
The amplitude of the step is -0.05, acting at time moment of 170 sec. (this is similar to the one presented for example in [III-34], [III-35], Figure 3.4.5 emphasizes the simulation results.
Figure 3.4.4. Disturbance rejection in minimax and LQ loops
From the analysis of the simulation it results a good behavior of the system both
Figure 3.4.5. Simulation of cascade control system
5. Part conclusions and contributions
Part III of the thesis presents a cascade control solution dedicated to the speed control of hydro-generators (HGs). Preceded by a detailed presentation of mathematical modelling aspects of HGs, a cascade control structure is presented for speed control and internal stabilization of the servo part. A two-stage cascade control structure with an internal minimax state controller dedicated for rejecting internally located deterministic disturbances and a main GPC loop is introduced. In case of many applications the maximal value of the inner disturbance can be estimated or calculated. Since the design effort in case of the minimax controller is increased, a computer aided design is used both in this case and for the GPC controller as well. An IMC form for the RST polymial representation was deduced. The use of the GPC controller under IMC representation based on the RST structure has the advantage of easy implementation.
Simulations give good results and sustain the efficiency of both the inner loop and of the GPC controller, regarding disturbances that are specific for the aimed applications. Then, the control solution is applied to the speed control of hydro turbine generators. The solution involves a cascade control structure with an internal minimax state-feedback controller to reject internally located deterministic disturbances and a main GPC loop. The proposed approach is justified because in case of many applications (e.g. the presented one) the maximal value of the inner disturbance can be estimated or calculated.
Since the design effort in case of the minimax controller is increased, a computer-aided design is used both in this case and for GPC in IMC structure representation. Digital simulation results of the case study show that the control system ensures good performances.
The results validate this control structure and its design method.
Part IV. Contributions
“Things should be made as simple as possible, but not any simpler”
(Alber Einstein)
1. Contributions of the PhD thesis
The las part of the thesis, synthesizes the contributions of the thesis, the main conclusions that can be drawn, and enumerates some possible further research directions. A short overview of the contributions was introduced in the first part of the thesis, in chapter 3.
The central point of the thesis is energy generation and energy management. Starting from this idea, two applications are presented, and the according problems which should be solved. The first application, dominating the thesis, is a series architecture hybrid solar vehicle. The main task is to solve the optimal energy management (and so achieve minimal fuel consumption). Part II is divided into three main chapters, each dealing with a different task: mathematical modelling of the hybrid vehicle, control strategies for fuel consumption optimization, mathematical modelling and control of the electric drive. The second application is oriented towards the power generation sector, and deals with the mathematical modelling and control of a hydro-generator. In what follows, the contributions of each part are presented in more detail.
A. Contributions in Part II
Part II of the thesis is oriented towards energy management optimization aspects in a series structure hybrid solar vehicle. Part II is structured on three main parts: mathematical modelling of the vehicle, fuel consumption optimization strategies and control of the electric drive.
• Chapter 2: is dedicated to the mathematical modelling of a series hybrid solar vehicle (HSV). The work is divided into three main parts: first a simplified non-linear and linearized mathematical model is built, based on the modelling of the vehicle components relevant from the point of view of fuel consumption optimization. The second part concentrates on a more complex mathematical model, namely to model the linear system as PieceWise Affine (PWA) systems. One way to treat non-linearitites is to apply feedback linearization, presented in this chapter for the HSV.
Regarding PWA systems, chapter 2 presents a complex Piecewise Bilinear model of the vehicle, both in continuous and in discrete time, where also controllability and
stability of the system were studied. Finally, the third part of the chapter presents numerical data and simulation results of both the simplified and the complex mathematical model.
• Chapter 3: focuses on optimal control solutions for fuel consumption minimization of the HSV. After a brief literature review, which presents the latest trend in this topic, two solutions are approached. First a Dynamic Programming (DP) solution is given, which delivers the global optimum for the optimization problem, but it is not feasible in real-time. Another solution applicable also in practice is Model Predictive Control (MPC), which also takes into account the physical constraints of the plant. After defining the cost function, simulations were run for different values of the tuning factors Q and R. Three simulation results were presented, considered as representative since they refer to significantly different cases, conclusions are drawn based on these results. The important remark must be made that for energy optimization MPC solutions are a viable alternative, so further research and improvement is of high actuality.
• Chapter 4: introduces an efficient cascade control structure for electrical drives used for electrical traction vehicle in two variants – without and with a forcing feed forward term for the current reference. The numerical data regarded to the application is based on a real application of a HSV. In order to ensure superior performances, for controller design different variants of the modulus optimum tuning method were used: the MO-method for the inner loop, and for the outer loop the Extended Symmetrical Optimum method (ESO-m) and a correction of it based on the tuning method named a Double Parameterization of the ESO method (2p-ESO-m) was used. Simulations were performed using the Matlab/Simulink environment, for a reference drive cycle, derived from the NEDC Cycle. The simulated cases reflect a very good behaviour of the system both regarding reference tracking and also sensitivity to parameter changes.
• Appendix 2 presents a Youla parameterization approach of the MO-m, ESO-m and 2p-SO-methods.
B. Contributions in Part III
Part III of the thesis deals with a control application from the power generation sector, namely cascade control of a hydro-generator.
• Chapter 2: Presents a detailed description of mathematical modelling aspects of HGs.
• Chapter 3: Introduces a cascade control structure for speed control and internal stabilization of the servo part. A two-stage cascade control structure with an internal minimax state controller dedicated for rejecting internally located deterministic disturbances and a main GPC loop is introduced. In case of many applications the maximal value of the inner disturbance can be estimated or calculated. Since the design effort in case of the minimax controller is increased, a computer aided design is used both in this case and for the GPC controller as well. An IMC form for the RST polymial representation was deduced. The use of the GPC controller under IMC representation based on the RST structure has the advantage of easy implementation.
• Chapter 4: Presents the application of the Minimax-GPC cascade control structure to the hydro-generation application. The solution involves a cascade control structure with an internal minimax state-feedback controller to reject internally located deterministic disturbances and a main GPC loop. The proposed approach is justified because in case of many applications (e.g. the presented one) the maximal value of the inner disturbance can be estimated or calculated. Simulations give good results and sustain the efficiency of both the inner loop and of the GPC controller, regarding disturbances that are specific for the aimed applications. The results validate this control structure and its design method.
2. Possible further research directions
The approach control topics and given solutions can support further research topics. Some of them would be:
- Develop piecewise affine and LPV mathematical models for other structures of hybrid electric vehicles as well;
- Implement and perform real experiments of control systems and development methods on the hybrid electric vechicle (see Part II);
- Develop new “auto-calibration” methods for PI and PID controllers based on ESO-m and 2p-SO-m (see Part II);
- Developing Combined Control strategies in speed control of HG and try to implement them on real applications (see part III);
- Handle real applications in constraint cases.
The relevance of the presented design methods and according results is sustained by different papers published during the past years.
Appendix I. Youla parameterization approach of the MO, ESO and 2p-SO methods
1. General aspects
The Youla parameterization (called as Q-parameterization) is a design method applied for both stable and unstable plants [A1-1]-[A1-6]. The Youla parameterization requires polynomials relative to the system’s properties. The disadvantage of the method consists in fact that in case of high order, non-minimum phase or unstable plants the controller results as a non-conventional one.
In this chapter the controller design method based on MO-m, ESO-m and 2p-SO-m [A1-6]
are transposed in a Youla-parameterization form. The study is justified by the possibility of imposing favorable forms on the expressions of S(s) and T(s) that ensure the desired system performances.
2. Co-prime factorization
If G(s) is a bounded rational form G(jω) <∞, with real coefficients there exists a co-prime factorization over the set of all bounded rational forms with real coefficients
) (
) ) (
( M s
s s N
G = with G(s)∈ϕ (A1-1)
1 ) ( ) ( ) ( )
(s X s +M s Y s =
N (Bezout’s Identity) (A1-2)
ϕ
∈ ) ( ), ( ), ( ),
(s X s M s Y s
N , φ – the set of all bounded rational forms with real coefficients.
The set of all stabilising controllers for a plant given byG(s)in form of (A1-1) is defined as:
) ( ) ( ) (
) ( ) ( ) ) (
( Y s N s Q s s Q s M s s X
C −
= + where Q(s)∈ϕ (A1-3)
(s)
Q - represents the so-called e Youla parameterization polynomial.
If P(s) is stable, the co-prime factorization (A1-1) and (A1-2) can be particularised as:
1 ) ( , 0 ) ( , 1 ) ( , ) ( )
(s =P s M s = X s = Y s =
N (A1-4)
Accordingly, the controller (A1-3) is determined as:
) ( ) ( 1
) ) (
( P s Q s
s s Q
C = − , (A1-5)
Figures A1-1 present the basic block diagram (a) and the restructured block diagram (b) regarding the Youla parameterization of controller design.
F u y
r0 r
_
d2
e C C
d1 r u y
_
d1
P(s) Q(s)
P(s)
d2
e
C(s)
(a) (b)
Figure A1-1. Basic control structures
The plants are characterized by bounded P(s) transfer functions, P(jω) <∞. For stable plants the Youla parameterization controller design consists in establishing Q(s) so that well stated requirements are fulfilled for S(s):
) ( ) ( 1 )
(s P s Q s
S = − . (A1-6)
It results that S(s) and C(s) depend only on Q(s) and P(s), and in controller design the following steps are taken [A1-6]:
Step (1): For the given plant P(s), calculation of C(s) is performed with Q(s) as parameter.
Calculations of S(s) and T(s) follow.
Step (2): Establishing of a Q(s) through which the imposed performances for S(s) or T(s) are ensured.
Step (3): Establishing the controller C(s) that fulfills the imposed requirements.
Step (4): Verification of desired performances.
3. Youla parameterization of the MO-m
The plant transfer function is considered to be of one of the following two forms:
(
1)(
1 1)
)
( sT sT
s k
P p
+
= +
Σ
(1) or
(
1)(
1)
(1 )) (
2
1 sT
sT sT
s k
P p
+ +
= +
Σ
(2) (A1-7) The design is exemplified only for (A1-7) (1), the case (2) being solved similarly.
(1) Calculation of C(s) with Q(s) parameter. Replacing (A1-7) into (A1-5), results:
( )( )
(
1 sT)(
1 sT)
k Q(s) sT 1 sT ) 1s ( Q ) s ( C
p 1
1
− + +
+
= +
Σ
Σ (A1-8)
with the realizable expression for S(s) and T(s):
( )( )
(
1 sT)(
11 sT1p)
) s ( Q k sT 1 sT ) 1
s (
S + +
− +
= +
Σ
Σ (A1-9)
( )(
1)
p
sT 1 sT 1
) s ( Q ) k
s (
T = + +
Σ
(A1-10) (2) Establish the stable Q(s). Using (A1-9) or (A1-10) results:
(
1)(
1 1 )1 ( )
( T s sT sT
s k Q
p
+ +
= Σ
)
or 1 [1 ( )](
1)(
1 1)
( S s sT sT
s k Q
p
+ +
−
= Σ
)
(A1-11)Using T(s) with its specific MO-m form and replacing into (A1-10) results Q(s):
2
2 2
2 1 ) 1
(s T s T s
T
Σ Σ +
= + (A1-12)
( )( )
2 2
1
2 2 1
1 1
) 1
( T s T s
sT sT
s k Q
p Σ Σ
Σ
+ +
+
= + (A1-13)
(1) Establish a controller C(s). Finally, the controller results as a PI controller:
) s T 1 s( T k 2 ) 1 s (
C 1
p
+
=
Σ
,
Σ
= k T k
p
c 2
1 , Tc =T1 (A1-14) and Tc' =T2 (A1-15) Remarks. 1. If the design requirement is imposed in form of zero steady-state error then:
( )( )
(
1 sT)(
1 sT)
0) s ( Q k sT 1 sT 0 1
) s ( S
0 1 s
p 1 0
s =
+ +
− +
⇔ +
=
Σ = Σ
= , and results:
kp
Q 1
) 0 ( =
and a PID-controller is obtained [A1-6]. The controller structure becomes more complicated if the design requirement refers to a very restrictive transfer function Hd2(s).
4. Youla parameterization of the ESO-m The transfer function of the plant is given as:
(
+ Σ)
= s sT s k
P p
) 1
( (a) or
(
1)(
1 1)
)
( s sT sT
s k
P p
+
= +
Σ
(b) (A1-16) The design is exemplified only for transfer function (A1-16) (a).
(1) Calculation of C(s) with Q(s) parameter. Replacing, results C(s), S(s) and T(s)
( )
(
1 sT)
k Q(s) ssT 1 ) s
s ( Q ) s ( C
− p
+
= +
Σ
Σ (A1-17)
( )
(
Σ)
Σ
+
−
= +
sT 1 s
) s ( Q k sT 1 ) s s (
S p (A1-18)
(
+ Σ)
= s1 sT ) s ( Q ) k
s (
T p (A1-19)
(2) Establish a stable Q(s):
(
+ Σ)
= T s s sT s k
Q
p
1 ) 1 ( )
( or = −S s s
(
+sTΣ)
s k Q
p
1 )) ( 1 1 ( )
( (A1-20)
The system performances are imposed through T(s), which is specific for ESO-m:
3 3 2 / 3 2 2 2 /
3 T s T s
s T 1
T ) 1
s ( T
Σ Σ
Σ
Σ
β + β
+ β +
β
= + (A1-21)
(3) Establish the controller C(s). The resulting controller is a PI type controller )
s T 1 s( T k ) 1
s (
C 2
p 2 /
3 Σ
Σ
β β +
= , 2
2 / 3
1
Σ
= k T k
p
c β , Tc =βTΣ and 2
' T
Tc =
(A1-22), (A1-23), (A1-24) 5. Youla parameterization of the 2p-SO-m
The plant transfer function is (A1-7) (a) or (b), the design is exemplified only for (A1-7) (a).
(1) Calculation of C(s) with Q(s) parameter. Replacing, results C(s), S(s) and T(s)
( )( )
(
1 sT1)(
1sT sT1)
sTk Q(s)) s ( Q ) s ( C
p 1
1
− + +
+
= +
Σ
Σ (A1-25)
( )
(
Σ+ Σ)
+
− +
= +
sT 1 ) sT 1 (
) s ( Q k sT 1 ) sT 1 ) ( s ( S
1
p
1 (A1-26) ,
(
+ Σ)
= +
sT 1 ) sT 1 (
) s ( Q ) k
s ( T
1
p (A1-27)
(2) Establish a stable Q(s):
(
+ Σ)
+
= T s sT sT
s k Q
p
1 ) 1 )(
1 ( )
( 1 or = −S s +sT
(
+sTΣs k Q
p
1 ) 1 ))(
( 1 1 ( )
( 1
)
(A1-28)The system performances are imposed through the choice of T(s) as a proportional-derivative-with 3rd order lag model. Accepting the use of a PI controller:
1 s a s a s a
sT ) 1
s ( T
1 2 2 3 3
c
+ + +
= + (A1-29)
where 1 , 1
,
, 2 1 1 0
1
3 + =
+ =
=
= Σ Σ a
k k
T k a k
k k
T a T
k k
T a T
c p
c c p c
p c
p
(A1-30) (3) Establish the controller C(s).
( )( )
] 1 T s a s a T a [ a s ) T a (
sT 1 sT 1 ) sT 1 ( k
) 1 s ( C
c 1 2 2 c 1 c 3 1
c 1
p +
+ −
− −
+ +
= + Σ (A1-31)
Using successive replacements, one gets
1 ,
) , 1 , (
) 1
( 0
' 1 2 / 2 3 ' 2 / 3 2 2 2
2 / 3 3 ' 2 / 3 3 3
3 = = =
= + + =
= Σ Σ Σ TΣ a TΣ a
m a T
m T
a T β β β β β (A1-32)
( )
1 ) 1
( 1
) 1 1 ( )
( 3/2 '3 3 3/2 '2 2 ' 1
+ +
+
+ +
= +
Σ Σ
Σ
Σ Σ
s T s T s
T
s T sT
sT s k
Q m
p β β β
β (A2-33)
For the second case the supplementary time constant is Tc' =T2.
In this case, for Youla parameterization design the following choice can be made:
- Place the zero
T m
z
Σ
−
= β 1
3 as a function of the values of {β, TΣ , m= TΣ/ T1 };
- The poles of the system are function of {β, TΣ , m}:
1 )
( = 3/2 '3 3+ 3/2 '2 2+ ' +
Δ s β TΣ s β TΣ s βTΣs (A1-34)
Σ
−
= T
p1 1/12
β ,
[ ]
'2
2 / 2 1 2 '
' 2 2 / ' 1
2 / 1 3
,
2 2
4 )
( ) (
Σ
Σ Σ
Σ ± − −
−
= −
T
T T
p T
β
β β
β β
β (A1-35)
The poles’ placement given by (A1-35) leads to a Q(s) of form (A1-33), and finally in step (3), the controller is expressed as a PI controller.
6. Conclusions
Appendix I. presents in detail the way of transposing the positive results gained from classical design methods based on modulus conditions (MO-m, SO-m) or conditions derived from these (ESO-m and 2E-SO-m) into a Youla-parameterization formulation. If the imposed conditions are adequately chosen, the controller is easy to implement. If the conditions are inadequate then the controller structure results as more difficult to comprehend and the solutions are less accepted in the practice.
Appendix II. References
Gen.
No.
References Chapter No.
[1] Arsie I., Martino R. Di, Rizzo G., Sorrentino M. (2006): A model for a hybrid solar vehicle prototype, Workshop on Hybrid Solar Vehicles, November 6, University of Salerno, Italy, 2006, pp. 35-42.
[I-1], [II-11]
[2] Koot M., Kessels J., de Jager B. (2005): Energy management in a vehicle with dueal storage power net, Preprints of the 16th IFAC World Congress July 4-8, 2005, Prague, Czech Republic, Electronic format, Paper ID-04079.
[I-2], [II-27]
[3] G. Maggetto, J. van Mierlo (2001). Electric vehicles, hybrid electric vehicles and fuel cell electric vehicles: state of the art and perspectives, Ann. Chim. Sci. Mat, Vol. 26(4), pp. 9-26.
[I-3], [II-89]
[4] P.Bauer, Zs. Preitl, T. Peter, P. Gaspar, Z. Szabo, J. Bokor (2006). Control oriented modelling of a series hybrid solar vehicle, Workshop on Hybrid Solar Vehicles, November 6, 2006, University of Salerno, Italy, pp. 19-26.
[I-4], [II-30]
[5] P. Bauer, Zs. Preitl, P. Gáspár, Z. Szabó, J. Bokor (2007). Improved model of a series hybrid solar vehicle, Second International Workshop on Hybrid and Solar Vehicles, September 14, University of Salerno, Italy, pp. 1054-1059.
[I-5], [II-31]
[6] Bauer P., Preitl Zs., Gaspar P., Szabo Z., Bokor J. (2007): Modelling of a Series Hybrid Electric Vehicle. Proceedings of the Third AUTOCOM Workshop on Hybrid Electric Vehicle Modeling and Control. Istanbul, Turkey, 12. June 2007. (CD)
[I-6], [II-32]
[7] M.W.T. Koot, J.T.B.A. Kessels, A.G. de Jager, W.P.M.H. Heemels, P.P.J.
van den Bosch, M. Steinbuch (2005). Energy Management Strategies for Vehicular Electric Power Systems, IEEE Trans. on Vehicular Technology, 54(3), 771-782.
[I-7], [II-28]
[8] G. Rousseau, D. Sinoquet, P. Rouchon (2006): Hybrid electrical vehicles:
from optimisation toward real-time control strategies, Workshop on Hybrid Solar Vehicles, November 6, University of Salerno, Italy, 2006.
[I-8], [II-68]
[9] I. Arsie, M. Graziosi, C. Pianese, G. Rizzo, M. Sorrentino (2004):
Optimisation of supervisory control strategy for parallel hybrid vehicle with provisional load estimate, Proc. 7th Int. Symp. Adv. Vehicle Control (AVEC), Arnhem, The Netherlands, Aug. 2004.
[I-9], [II-8]
[10] J. Jonsson, Z. Jansson (2004): Fuel optimized predictive following in low speed condition, Proceedings of IFAC Symposium “Advances in Automotive Control”, University of Salerno, Italy, April 19-23, 2004, pp.134-139.
[I-10], [II-72]
[11] E. Münz, M. Schwaiger, V. Krebs (2004): Controlling Hybrid Electric Vehicles using differential game theory, Proceedings of IFAC Symposium “Advances in Automotive Control”, University of Salerno, Italy, April 19-23, 2004, pp.440-445.
[I-11], [II-73]
[12] Preitl, Zs., P. Bauer, J. Bokor (2006). Fuel Consumption Optimization for Hybrid Solar Vehicle, Workshop on Hybrid Solar Vehicles, November 6, 2006, University of Salerno, Italy, pp. 11-18.
[II-12], [II-78]
[13] Preitl Zs. Bauer P., Kulcsár B. Rizzo G., Bokor J. (2007): Control Solutions for Hybrid Solar Vehicle Fuel Consumption Minimization. IEEE Intelligent Vehicles Symposium, Istanbul, Turkey. 2007. pp. 767-772.
IEEE Catalog Number: 07TH8947. ISBN: 1-4244-1068-1
[I-13], [II-42]
[14] Preitl Zs., P. Bauer, B. Kulcsár, J. Bokor (2006) Predictive Control for Fuel Consumption of Hybrid Solar vehicles, HUCI 2006, 7th International Symposium of Hungarian Researchers on Computational Intelligence, November 25-26, 2006, Budapest.
[I-14], [II-33]
[15] O.F. Bay, G. Bal, S. Demirbas (1996). Fuzzy Logic Based Control of a Brushless DC Servo Motor Drive, Proc. Of the 7th International Power Electronics & Motion Control Conference Exhibition, Budapest, Hungary, vol. 3, pp.448-452.
[I-15], [II-39]
[16] R.M. Crowder (1998). Electric Drives and their Controls, Oxford University Press Inc., New York.
[I-16], [II-35]
[17] M. Ehsani, K.M. Rahman, M.D. Bellar, A.J. Severinsky (2001). Evaluation of Soft Switching for EV and HEV Motor Drives, IEEE Transactions on Industrial Electronics, Vol. 48, No.1, February 2001, pp.82-90.
[I-37], [II-36]
[18] C. Mi, M. Filippa, J. Shen, N. Natarajan (2004). Modeling and Control of a Variable-Speed Constant Frequency Synchronous Generator With Brushless Exciter, IEEE Transactions on Industry Applications, Vol.40, No.2, March/April 2004.
[I-18], [II-38]
[19] Preitl Zs., Bauer P., Bokor J.(2007): Cascade control solution for traction motor for hybrid electric vehicles. Acta Polytechnica Hungarica. 4 (3) : 75-88. 2007. ISSN 1785-8860
[I-19], [II-83]
[20] Preitl, Zs.: Improving Disturbance Rejection by Means of a Double Parameterization of the Symmetrical Optimum Method, Scientific Bulletin of the “Politehnica” University of Timişoara, Series Automation and Computers, Politehnica Publishing House, Timişoara, ISSN 1224-600X, vol. 50(64), 2005, pp. 25-34
[I-20], [II-84]
[21] Zs. Preitl, Bauer P., Bokor J. (2007): A Simple Control Solution for Traction Motor Used in Hybrid Vehicles, SACI 2007 - 4th International Symposium on Applied Computational Intelligence and Informatics, 2007 Mai 17-18, Timisoara, Romania, pp. 157-162.
[I-21], [II-34]
[22] Zs. Preitl, R. Bars, R. Haber: An Applied GPC Cascade Control Solution for Hydro-Turbines, International Journal of Tomography and Statistics, Vol.6, no. S07, Summer 2007, ISSN 0972-9976, pp. 27-32.
[I-28], [III-1]
[23] Zs. Preitl, R. Bars, R. Haber, (2006) An applied GPC Cascade Control Solution for Hydro-Turbines, IIFAFACC WoWorrkksshhoopp onon CoConnttrrooll AApppplliiccaattiioonnss ofof OOppttiimmiissaattiioonn,, AApprriill 2266--2828,, 22000066,, CCaacchhaann,, FFrraannccee
[I-29], [III-2]
[24] Zs. Preitl, B. Kulcsár, J. Bokor (2008): Piecewise Linear Parameter Varying Mathematical Model of a Hybrid Solar Vehicle, IV 2008 – IEEE Intelligent Vehicles Symposium, Eindhoven, The Netherlands, June 4-6, 2008.
[I-30], [II-122]
[25] Preitl, Zs., Bars, R., (2006) A Youla-parameterization Approach for Controller Design based on ESO and 2E-SO Methods for Low Order Benchmarks, Studies In Informatics and Control, September 2006, Volume 15, number 3, ISSN 1220-1766.
[I-31], [II-85]
[26] Preitl, Zs., Bars, R., (2005) A Youla-parameterization Approach for Controller Design based on ESO and 2E-SO Methods for Electrical Drives (Low Order Benchmarks), The 2nd Romanian-Hungarian Joint Symposium on Applied Computational Intelligence, SACI-2005, Timisoara, Romania, 12-14 May, 2005,
[I-32], [A1-6]
[27] Zs. Preitl (2005) 2p-SO módszer alapján hangolt PI(PID) szabályozós struktúrák érzékenységi vizsgálata, ENELKO 2005 Konferencia, Rendező és kiadó az Erdélyi Magyar Műszaki Tudományos Tarsaság) Kolozsvár, 2005 okt. 7-9, ISBN 973-7840-06-2.
[I-33], [II-86]
[28] Zsuzsa Preitl, Ruth Bars, István Vajk, Robert Haber (2005) Cascade GPC with Minimax Optimal Inner Control Loop, Proceedings of the 6-th International Carpathian Control Conference, Miskolc-Lilafured, Hungary, may 24-27, 2005, pp.323-328.
[I-34]
[29] Zs. Preitl, R. Bars, I. Vajk, R. Haber (2004) Internal Model Representation for Generalized Predictive Control with Constraint Handling, Proceedings of the 4-th International Conference on Systems Design and Application ISDA 2004, Budapest, Hungary, 2004 Aug. 26-28, pp. 681-685.
[I-35], [III-32]
[30] Zs. Preitl, R. Bars (2005): Influence of tuning parameter variations upon GPC control loop, IFAC 1st Int. Workshop on Advanced Control Circuits and Systems ACCS’05 Cairo, Egypt, 2005 March 6-10, electronic format.
[I-36]
[31] A. Barta, R. Bars, I. Vajk, Zs. Preitl: Practical controller design considering input saturation. ICCC’05, International Carpatian Control Conference, Lillafüred, May 24-27, 2005. Proc. ISBN 963-661-644-2, Vol.1, pp. 51-56.
[I-37]
[32] Precup, R.-E., Preitl, Zsuzsa, Kilyeni St. (2005) Fuzzy Control Solution for Hydro Turbine Generators, ICCA’05 The 5th International Conference on Control & Automation, Hungarian Academy of Science, Budapest, Hungary, June 26-29, 2005.
[I-38], [III-33]
[33] Wellington R.P. (1996), Model Solar Vehicles Provide Motivation for School Students, Solar Energy Vol.58, N.1-3.
[II-1]
[34] Sasaki K., Yokota M., Nagayoshi H., Kamisako K. (1997), Evaluation of an Electric Motor and Gasoline Engine Hybrid Car Using Solar Cells, Solar Energy Material and Solar Cells (47), 1997.
[II-2]
[35] Seal M.R. (1995), Viking 23 - zero emissions in the city, range and performance on the freeway. Northcon - Conference Record 1995. IEEE, RC-108.p 264-268.
[II-3]
[36] S.Letendre, R.Perez, Christy Herig, Vehicle Integrated PV: a Clean and Secure Fuel for Hybrid Electric Vehicles, Proc. of Annual Meeting of the American Solar Energy Society, June 21-26, 2003, Austin, TX.
[II-4]
[37] S. Letendre: Ushering in an area of solar powered mobility, Workshop on Hybrid Solar Vehicles, November 6, University of Salerno, Italy, 2006.
[II-5]
[38] Hammad M., Khatib T. (1996), Energy Parameters of a Solar Car for Jordan, Energy Conversion Management, V.37, No.12.
[II-6]
[39] Saitoh, T.; Hisada, T.; Gomi, C.; Maeda, C. (1992), Improvement of urban air pollution via solar-assisted super energy efficient vehicle. 92 ASME JSES KSES Int Sol Energy Conf. Publ by ASME, New York, NY, USA.p 571-577.
[II-7]
[40] G.Gutmann (1999). Hybrid electric vehicles and electrochemical storage systems – a technology push – pull couple, Journal of Power Sources, Vol. 84, pp. 275-279, 1999.
[II-9]
[41] K. DAVID HUANG, SHENG-CHUNG TZENG: A new parallel – type hybrid electric – vehicle, Applied Energy, Vol. 79, pp. 51-64, 2004.
[II-10]
[42] Arsie I., Marotta M., Pianese C., Rizzo G., Sorrentino M., Optimal Design of a Hybrid Electric Car with Solar Cells, Proc. of 1st AUTOCOM Workshop on Preventive and Active Safety Systems for Road Vehicles, Istanbul, Sept.19-21, 2005.
[II-12]