Part II. Modelling and control of a hybrid vehicle
3. Control solutions for fuel consumption optimization for a hybrid solar vehicle
3.2 Model Predictive Control applied for the hybrid solar vehicle model
modified with this value, namely:
dSOC dm dt
dSOC dt dm
dt dSOC
m fb
fb
fb = =
&
(2.3.4)
The time mean value of (2.3.4) is calculated during the simulation, which represents the average amount of consumed fuel per SOC unit:
∫
= T dt
dSOC dm T
dSOC dm
0
1 (2.3.5)
The fuel amount needed for charging the battery (or fuel excess, respectively) is finally defined by:
) 7
. 0
( final
needed
fb SOC
dSOC
m − = dm − (2.3.6)
where SOCfinal represents the SOC level at the end of the simulation. Using this fuel equivalent, some performance indices of a control solution can be evaluated.
Plant
ControllerMPC
Figure 2.3.6. Bloc diagram of a SISO MPC Toolbox Application
The first element to be defined is the plant model that is used in the predictive controller. The model used for MPC control system design is the linearized mathematical model presented in section 2.1. of Part II, with the numerical values given by equation (2.2.8).
The system is both controllable and observable.
To recall, the inputs, outputs and states of the plant are enumerated once again, and the discrete state-space equation as well [II-78], [II-42]:
Inputs: - u1: ICE power,
- u2: Battery nominal power;
State variables: - x1: state of dynamics of EM, - x2: SOC,
- x3: state of dynamics of ICE;
Measured disturbance input: - dm: PV panel power.
Outputs: - o1: Drive power, - o2: SOC,
- o3: Fuel rate.
And the resulting state-space equation is (2.2.8):
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⎤
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⋅
⋅
−
⋅
⋅ +
+
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=
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⎡ + + +
−
−
−
−
−
) (
) (
) (
100 0 0
0 1 0
0 0 800
) (
) (
) (
) ( 0
0 10 321 . 6 ) (
) ( 0
10 638 . 2
10 517 . 1 0
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9048 . 0 0 0
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3 2 1
3 2 1
4
2 1 7
11 4 6
3 2 1
3 2 1
k x
k x
k x
k y
k y
k y
k k d
u k u k x
k x
k x
k x
k x
k x
m
(2.2.8)
Constraints act upon the system inputs and outputs, enumerated in equations (2.3.7).
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
≤
≤
≤
≤
≤
≤
−
≤
≤
−
≤
≤
3 . 7 y 0
8 . 0 y 6 . 0
58000 y
40000
14000 u
26000 93000 u
0
3 2
1 2 1
(2.3.7)
The PV power is treated as a measured disturbance, depending on the actual irradiation value.
Further, a quadratic cost function is defined in the form of:
∑
∑
= =+ Δ + +
− +
= Nu
i N
i
R k i k u Q
k i k r k i k y k
J
0
2 1
2 ˆ( | )
)
| ( )
| ( ˆ )
( (2.3.8)
Where yˆ(k+ik)are the predictions at time k of the output y, r(k+ik)is the reference trajectory vector, Δuˆ(k+ik)are the changes of the future input vector, N is the prediction horizon, Nu is the control horizon, Q and R are weighting matrices.
The choice of the weighting factors, prediction and control horizon is crucial; the aim is to get a balance between good tracking and acceptable control signals [II-82]. As a starting point, it is advisable to normalize all signals in the cost function, and then start systematically tuning each element of the diagonal matrices Q and R, so that a desired trade-off is achieved.
In what follows, simulation results are presented concerning different tuning parameter values. The reference signals in all cases are: r1 – drive power demand calculated from the drive cycle, r2 = 0.7 SOC value, r3=0 for fuel rate. The Pd drive power demand is calculated from the time-velocity characteristics from figure 2.3.7 (representing the urban part of the NEDC [II-9]), based on the basic dynamical relations of vehicle motion described by equations (2.2.1)-(2.2.3).
0 100 200 300 400 500 600 700 800
0 10 20 30 40 50 60
T i m e [ s e c ]
S p e e d [ k m / h ]
U r b a n p a r t o f N E D C
Figure 2.3.7. The urban part of the NEDC (first 800 seconds)
The prediction horizon was N=10 and control horizon Nu=4 for all simulation cases.
For getting conclusive results, the Q and R matrices were fixed for three simulation examples.
In what follows, simulation results and calculations are presented for the three cases treated also in [II-42].
• The first experiment (figures 2.3.7 to 2.3.10):
⎥⎦
⎢ ⎤
⎣
=⎡
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
= − −
−
4 4 2
10 0
0 R 10
, 0.001 0
0
0 100 0
0 0 10
Q (2.3.9)
0 100 200 300 400 500 600 700 800
-4 -3 -2 -1 0 1 2 3 4x 104
T i m e [ s e c ]
D r i v e p o w e r [ W ]
D r i v e p o w e r r e f e r e n c e t r a c k i n g
Output Pd Reference Pd
Figure 2.3.7. Drive power reference tracking
0 100 200 300 400 500 600 700 800
0.675 0.68 0.685 0.69 0.695 0.7 0.705 0.71
T i m e [ s e c ]
S t a t e o f c h a r g e [ r. u. ]
S t a t e o f c h a r g e
Figure 2.3.8. State of Charge
0 100 200 300 400 500 600 700 800
-20 0 20 40 60 80 100 120 140
T i m e [ s e c ]
F u e l [ g ]
F u e l c o n s u m p t i o n
Figure 2.3.9 Fuel consumption
0 100 200 300 400 500 600 700 800 -2
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
3x 104
T i m e [ s e c ] C o n t r o l s i g n a l s P I C E [ W ] , P b n [ W ]
C o n t r o l s i g n a l s P
I C E a n d P b n
PICE Pbn
Figure 2.3.10. Control signals PICE and Pbn
Synthesising the simulation results, one gets:
- The total fuel consumption is mf=136.4328g;
- the final SOC is SOCfinal=0.6773;
- the fuel needed for bringing the SOC to 0.7 following the drive tendency:
mfb-needed=79.8137g;
mtotal=mf+mfb-needed=216.2465g.
• The second experiment (figures 2.3.11 to 2.3.14):
⎥⎦
⎢ ⎤
⎣
=⎡
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
= − −
−
5 5 3
10 0
0 R 10
, 0.001 0
0
0 100 0
0 0 10
Q (2.3.10)
0 100 200 300 400 500 600 700 800
-4 -3 -2 -1 0 1 2
3x 104 D r i v e p o w e r r e f e r e n c e t r a c k i n g
T i m e [ s e c ]
D r i v e p o w e r [ W ]
Output Pd Reference P
d
Figure 2.3.11. Drive power reference tracking
0 100 200 300 400 500 600 700 800 0.67
0.675 0.68 0.685 0.69 0.695 0.7 0.705 0.71
S t a t e o f c h a r g e
T i m e [ s e c ]
S t a t e o f c h a r g e [ r. u. ]
Figure 2.3.12. State of Charge
0 100 200 300 400 500 600 700 800
-20 0 20 40 60 80 100 120
F u e l c o n s u m p t i o n
T i m e [ s e c ]
F u e l [ g ]
Figure 2.3.13. Fuel consumption
0 100 200 300 400 500 600 700 800
-1.5 -1 -0.5 0 0.5 1 1.5
2x 104 C o n t r o l s i g n a l s P
I C E a n d P b n
T i m e [ s e c ]
P I C E a n d P b n [ W ]
PICE Pbn
Figure 2.3.14. Control signals PICE and Pbn
Synthesising the simulation results, one gets:
- The total fuel consumption is mf=112.2535g;
- the final SOC is SOCfinal=0.6731;
- the fuel needed for bringing the SOC to 0.7 following the drive tendency:
mfb-needed= 150.8056g;
mtotal=mf+mfb-needed=263.0591g.
• The third experiment (figures 2.3.15 to 2.3.18)
⎥⎦
⎢ ⎤
⎣
=⎡
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
= − −
−
5 5
4
10 0
0 R 10
, 0.001 0
0
0 1000 0
0 0
10
Q (2.3.11)
0 100 200 300 400 500 600 700 800
-4 -3 -2 -1 0 1
2x 104 D r i v e p o w e r r e f e r e n c e t r a c k i n g
T i m e [ s e c ]
D r i v e p o w e r [ W ]
Output P d Reference P
d
Figure 2.3.15. Drive power reference tracking
0 100 200 300 400 500 600 700 800
0.692 0.694 0.696 0.698 0.7 0.702 0.704 0.706 0.708
S t a t e o f c h a r g e
T i m e [ s e c ]
S t a t e o f c h a r g e [ r. u. ]
Figure 2.3.16. State of Charge
0 100 200 300 400 500 600 700 800 -50
0 50 100 150 200 250 300 350
F u e l c o n s u m p t i o n
T i m e [ s e c ]
F u e l [ g ]
Figure 2.3.17. Fuel consumption
0 100 200 300 400 500 600 700 800
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
2.5x 104 C o n t r o l s i g n a l s P I C E a n d P b n
T i m e [ s e c ] P I C E a n d Pb n [ W ]
PICE Pbn
Figure 2.3.18. Control signals PICE and Pbn
Synthesising the simulation results, one gets:
- The total fuel consumption is mf=341.3378g;
- the final SOC is SOCfinal=0.6946;
- the fuel needed for bringing the SOC to 0.7 following the drive tendency:
mfb-needed=77.7170g;
mtotal=mf+mfb-needed=419.0548g.
One simulation was performed for HSV model without any controller at all, when the ICE delivers all the energy needed for the EM. In this case the total fuel consumption was mf=274.7437g.
Based on the simulation results it can be concluded that for all tuning parameters the total fuel consumed, including the fuel equivalent, gives a value larger than the global optimum. The smallest total fuel is in the case of the first simulation, namely mtotal=216.2465g. This value is smaller than the value without controller. Also in the second
case, the total fuel is smaller than this value: mtotal =263.0591g, which reflects also an acceptable result.
In this case, the tracking performances regarding Pd are good; the overshoots have values between the two other simulations. For the third simulation, the tracking is best of all three, with an exception at the start. However, the fuel consumption is extremely large, compared to the other cases, it is bigger than the value without controller. This case is not acceptable, it is a good counterexample.
The final values of the SOC largely differ in the first two cases from the third one, reflecting the influence of the tuning weights.
The control signals differ slightly in aspect in the first two cases from the third one. It can also be noticed a difference in the fuel consumption, in the first two cases there is a step-wise evolution (meaning there is no fuel consumption at those moments), whereas in the third one there is an almost continuous and gradual increase, ICE functioning at full load. All these differences reflect the importance of proper balancing of the tuning parameters, which defines the switching between the energy sources.
To conclude the chapter, it can be stated that based on the tendencies from the literature regarding hybrid vehicles’ energy management, for the existing linearised model presented in the previous chapter, two control strategies were applied. The first strategy is Dynamic Programming, which delivers the global optimum solution for a constrained optimization problem, but it cannot be implemented in real time. Still it represents a very good comparison for all other sub-optimal solutions. The second control strategy was Model Predictive Control, which delivers a sub-optimal solution to the constrained optimization problem, but it can be applied in real-time successfully. Different simulations were performed and the results are encouraging. Further development of the mathematical model and of the control algorithms used should give even better results, which could be implemented and tested on prototypes.